Physics · Article

Biot–Savart and Ampère’s laws

Compute magnetic fields from complete steady currents, derive original segment, loop and coaxial examples, and use a two-wire counterexample to distinguish circulation from local field. Connect the integral laws to boundary conditions, topology and the vector potential.

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After reading, you can

State the current-conservation and time-independence assumptions of magnetostatics and distinguish a complete circuit from a selected current portion.
Apply the oriented Biot–Savart integral with a source-to-observer displacement, correct units and a defined source domain.
Derive the selected straight-segment and circular-loop axial fields, preserving their normalizations, limits and excluded singular points.
Use Ampère’s law with consistent contour and surface orientation, extracting a local field only when symmetry and boundary conditions justify it.
Resolve enclosed current and magnetic field through an original coaxial return-current geometry, including continuity and branch limits.
Explain why equal circulation does not mean equal local field, and connect magnetic-source laws to vector potentials, topology and the limits of a steady approximation.

Start with a complete steady current

The magnetic-force lesson treated E and B as prescribed fields. We now calculate B from specified currents. In classical magnetostatics, the charge and current distributions and fields are time independent. Charge continuity then requires ∇·J = 0: current cannot simply stop at an endpoint while a stationary charge distribution is assumed.

For localized currents, the source description must include the return paths that make the current continuous. A short straight segment can be one contribution to the field of a complete circuit, but it is not by itself an isolated physical steady current. A terminal model with charge accumulation would instead be a time-dependent problem.

We work in vacuum or a prescribed nonmagnetic medium with constant μ₀ and compatible macroscopic current density. Magnetized matter introduces additional source bookkeeping and constitutive relations, developed in the following lesson. No universal substitution B = μH is assumed for arbitrary nonlinear or hysteretic materials.

The original examples use positive symbolic scales I₀ and L₀ and retain μ₀ symbolically. In the revised SI, μ₀ is experimentally determined rather than an exactly fixed numerical 4π × 10⁻⁷. No numerical material constant or practical magnet performance is needed for the dimensionless calculations.

Orient the Biot–Savart integral

For a complete steady volume current in the stated uniform medium and with suitable boundary behavior, B(r) = (μ₀/4π)∫ J(r′) × (r − r′)/|r − r′|³ d³r′. The primed position labels the source, and the unprimed position labels the observer. Reversing that displacement without also changing the formula reverses the field incorrectly.

For a thin wire sufficiently far from its cross-section, replace J d³r′ by I dℓ′, with dℓ′ oriented along positive conventional current. Then dB = (μ₀I/4π)dℓ′ × R/R³, where R = r − r′. Sum vector components over the actual source; adding positive magnitudes alone generally misses cancellation or reinforcement.

The units check is μ₀I dℓ/R²: (T·m/A)(A)(m)/(m²) = T. Superposition is linear in the specified current in this fixed medium. Reversing a complete current reverses its contribution to B, while changing an observation position also changes the displacement and geometric factors.

An ideal filament has a singular self-field on its line. The thin-wire expression should not be evaluated there as an ordinary field or used directly as its own force field. A finite cross-section, external-field separation or other justified regularization is needed for a source-region or self-force question. The original plots exclude filament source points explicitly.

Integrate a finite straight portion

Consider the portion of a circuit along the z axis from z′ = −4L₀ to +4L₀ carrying +I₀. Observe at z = 0 and cylindrical radius r > 0. At an observer on +x, dℓ′ = dz′ẑ and R = rx̂ − z′ẑ, so dℓ′ × R = r dz′ŷ. The portion’s field is azimuthal in the positive circulation direction.

Its magnitude is B_portion = (μ₀I₀/4π)∫ from −4L₀ to 4L₀ r dz′/(r² + z′²)^(3/2). An antiderivative is z′/[r√(r² + z′²)]. Evaluating both endpoints gives B_portion = (μ₀I₀/2πr)[4L₀/√(r² + 16L₀²)]. The two ends contribute symmetrically; neither is silently sent to infinity.

Define R = r/L₀ and B₀ = μ₀I₀/(2πL₀). The dimensionless result is B_portion/B₀ = 4/[R√(R² + 16)]. For comparison, the ideal infinite-wire field in the same scales is B_infinite/B₀ = 1/R. Their ratio is 4/√(R² + 16).

This is exactly the selected straight portion’s Biot–Savart contribution. The unspecified return portion of the real circuit produces an additional field and can break the cylindrical symmetry of the complete source. We do not use the open portion alone as a complete magnetostatic source satisfying Ampère’s law everywhere.

Interpret a segment limit with its source

At R = 3, √(R² + 16) = 5, so B_portion/B₀ = 4/15. The infinite-wire value there is 1/3, and the portion-to-infinite ratio is 4/5. This exact control checks the integral, the length scale and the factor of two from the two endpoints.

When the observer is much closer to the middle of a very long straight portion than to its ends, its ratio approaches one. Whether that contribution also approximates the complete circuit field requires the return path to be sufficiently remote or otherwise controlled. Long relative to one distance is not a universal statement about every part of a circuit.

Far from this finite straight portion, its contribution falls as its length divided by r². That is not the complete far field of a localized closed loop. The other portions then matter, and cancellations give the appropriate multipole behavior of the complete current distribution.

The singular R = 0 is outside the ideal-filament domain. The native segment curves begin at R = 1/2 and stop at R = 6, while the exact R = 3 control is separately retained. A plotted cutoff defines a displayed domain; it does not remove the physical need to describe source geometry or a finite wire radius.

A finite portion is not a complete current source

Compare the selected finite straight portion’s contribution with an ideal infinite-wire field, then show their ratio. The portion belongs to an unspecified complete circuit; its return field is not included, and the ideal source line is excluded.

Portion contribution versus a different infinite-wire model

Selected finite straight portion only: Bφ/B₀
Separate ideal infinite wire: Bφ/B₀=1/R
Separate exact R=3 control(s)

The comparison ratio, with its source scope visible

Finite-portion/infinite-wire field ratio
Separate exact R=3 control(s)

Straight portion z∈[−4L₀,+4L₀],current+I₀; observations at z=0,R>0. Only this portion contribution is computed; complete return field is not included. B₀=μ₀I₀/(2πL₀). Source lineR0 excluded.
Positive symbolic μ₀,I₀,L₀. Positiveφ is counterclockwise viewed from +z for the selected+z current. Both plots cover only1/2≤R≤6; no ideal source-line value is assigned atR0.
The finite contribution is4/[R√(R²+16)] in B₀ units; the ideal infinite wire is1/R. Their ratio4/√(R²+16) is a comparison of different source selections, not the total field of an unspecified closed circuit. AtR3 the separate exact controls are4/15,1/3 and4/5.
All marks read exact admitted records, including rational controls, original binary64 function-specific qualifications,12-significant-digit displays and retained values. No fitted conservation, near-zero snap, field solver or sampled-quadrature proof is used.

Complete labelled data equivalents preserve every admitted scalar, record, branch, mask and source constraint. Original manuscript tables remain separate.

All 129 finite-portion and infinite-wire comparison records (129 rows)

Straight portion z∈[−4L₀,+4L₀],current+I₀; observations at z=0,R>0. Only this portion contribution is computed; complete return field is not included. B₀=μ₀I₀/(2πL₀). Source lineR0 excluded.

finiteSegment: exact controls, scales and source limits (7 rows)

finiteSegment: exact controls, scales and source limits
Quantity/source condition	Admitted value
R=r/L₀	3Exact fraction: `3`Retained binary64: `3.0`
Finite-portion Bφ/B₀	4/15Exact fraction: `4/15`Retained binary64: `0.26666666666666666`
Ideal infinite-wire Bφ/B₀	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`
Finite-portion/infinite-wire ratio	4/5Exact fraction: `4/5`Retained binary64: `0.8`
source Portion Only	true
complete Return Field Computed	false
source Line R0 Excluded	true

Focus a labelled diagram or table region and use arrow keys, or scroll sideways. Fixed-size readable graphics remain within their own regions. Full data disclosures retain all records, exact values, one-sided flags and original precision qualifications.

Derive the field on a circular loop axis

Take a circular loop of radius a in the xy plane with conventional current I counterclockwise as viewed from +z. On the z axis, contributions from opposite elements cancel their transverse components. Their z components add in +z, both above and below the loop. Axis symmetry establishes the direction before the magnitude integral is evaluated.

For an element of arc length dℓ, the source-to-observer distance is √(a² + z²). The z component of dℓ × R is a dℓ for the stated orientation. Therefore B_z = (μ₀I/4π)∮ a dℓ/(a² + z²)^(3/2) = μ₀Ia²/[2(a² + z²)^(3/2)]. The loop integral supplies ∮dℓ = 2πa.

Choose the original loop a = 3L₀ and I = 2I₀. Its center field is B_c = μ₀I₀/(3L₀). With Z = z/L₀, B_z/B_c = 27/(9 + Z²)^(3/2). This B_c normalization is different from B₀ = μ₀I₀/(2πL₀) used for the straight and coaxial examples; the axis label must say which one is used.

At Z = 0 the ratio is one. At Z = ±4, the denominator is 25^(3/2) = 125, so the ratio is 27/125 on either side. There is no source on the axis, even at the origin: the wire is at radius 3L₀. The center is regular, in contrast to the excluded line of an ideal straight filament.

A regular loop-axis field in its own normalization

Show the positive, even axial field of the specified current loop in its own center-field normalization. Retain the regular center and the exact controls on both sides; the plot does not assign the field away from the axis.

Positive, even B_z on both sides of the loop

Positive, even loop-axis B_z/B_c; B_c distinct from B₀
All three exact controls: Z=−4,0,+4

Separate loop radius3L₀,current2I₀,counterclockwise viewed from +z. B_c=μ₀I₀/(3L₀),not B₀. Field is positive+z on both axis halves, regular at the center; no off-axis field is assigned.
B_z/B_c=27/(9+Z²)^(3/2),Z=z/L₀. The regular center hasB_z/B_c=1; exact controls atZ=±4 are27/125. The positive even scalar is the signed+z component, not a field pointing away from the loop on both sides.
The current loop is a complete steady source. This plot covers only its axis; off-axis fields and finite-device error estimates are not supplied. Its separate B_c normalization must not be silently replaced by the straight-wire B₀.
All marks read exact admitted records, including rational controls, original binary64 function-specific qualifications,12-significant-digit displays and retained values. No fitted conservation, near-zero snap, field solver or sampled-quadrature proof is used.

Complete labelled data equivalents preserve every admitted scalar, record, branch, mask and source constraint. Original manuscript tables remain separate.

All 129 loop-axis records in the B_c normalization (129 rows)

Separate loop radius3L₀,current2I₀,counterclockwise viewed from +z. B_c=μ₀I₀/(3L₀),not B₀. Field is positive+z on both axis halves, regular at the center; no off-axis field is assigned.

All 129 loop-axis records in the B_c normalization
Row	Z=z/L₀	Loop-axis B_z/B_c (separate normalization)
0	-6Exact fraction: `-6`Retained binary64: `-6.0`	0.0894427191Retained binary64: `0.08944271909999159`binary64;12 significant-digit display; exact zero sign canonicalized only
1	-189/32Exact fraction: `-189/32`Retained binary64: `-5.90625`	0.0928768731769Retained binary64: `0.09287687317691615`binary64;12 significant-digit display; exact zero sign canonicalized only
2	-93/16Exact fraction: `-93/16`Retained binary64: `-5.8125`	0.096477064471Retained binary64: `0.09647706447098157`binary64;12 significant-digit display; exact zero sign canonicalized only
3	-183/32Exact fraction: `-183/32`Retained binary64: `-5.71875`	0.10025259196Retained binary64: `0.10025259196002366`binary64;12 significant-digit display; exact zero sign canonicalized only
4	-45/8Exact fraction: `-45/8`Retained binary64: `-5.625`	0.104213311622Retained binary64: `0.10421331162222675`binary64;12 significant-digit display; exact zero sign canonicalized only
5	-177/32Exact fraction: `-177/32`Retained binary64: `-5.53125`	0.10836966778Retained binary64: `0.10836966777956844`binary64;12 significant-digit display; exact zero sign canonicalized only
6	-87/16Exact fraction: `-87/16`Retained binary64: `-5.4375`	0.11273272542Retained binary64: `0.11273272541960044`binary64;12 significant-digit display; exact zero sign canonicalized only
7	-171/32Exact fraction: `-171/32`Retained binary64: `-5.34375`	0.117314203339Retained binary64: `0.11731420333925156`binary64;12 significant-digit display; exact zero sign canonicalized only
8	-21/4Exact fraction: `-21/4`Retained binary64: `-5.25`	0.122126507903Retained binary64: `0.12212650790322065`binary64;12 significant-digit display; exact zero sign canonicalized only
9	-165/32Exact fraction: `-165/32`Retained binary64: `-5.15625`	0.127182767148Retained binary64: `0.12718276714816618`binary64;12 significant-digit display; exact zero sign canonicalized only
10	-81/16Exact fraction: `-81/16`Retained binary64: `-5.0625`	0.132496864891Retained binary64: `0.13249686489060267`binary64;12 significant-digit display; exact zero sign canonicalized only
11	-159/32Exact fraction: `-159/32`Retained binary64: `-4.96875`	0.138083474409Retained binary64: `0.1380834744092699`binary64;12 significant-digit display; exact zero sign canonicalized only
12	-39/8Exact fraction: `-39/8`Retained binary64: `-4.875`	0.14395809117Retained binary64: `0.14395809116960265`binary64;12 significant-digit display; exact zero sign canonicalized only
13	-153/32Exact fraction: `-153/32`Retained binary64: `-4.78125`	0.150137063936Retained binary64: `0.1501370639364056`binary64;12 significant-digit display; exact zero sign canonicalized only
14	-75/16Exact fraction: `-75/16`Retained binary64: `-4.6875`	0.156637623478Retained binary64: `0.15663762347829968`binary64;12 significant-digit display; exact zero sign canonicalized only
15	-147/32Exact fraction: `-147/32`Retained binary64: `-4.59375`	0.163477907901Retained binary64: `0.1634779079010968`binary64;12 significant-digit display; exact zero sign canonicalized only
16	-9/2Exact fraction: `-9/2`Retained binary64: `-4.5`	0.170676983454Retained binary64: `0.17067698345391663`binary64;12 significant-digit display; exact zero sign canonicalized only
17	-141/32Exact fraction: `-141/32`Retained binary64: `-4.40625`	0.178254859428Retained binary64: `0.17825485942839164`binary64;12 significant-digit display; exact zero sign canonicalized only
18	-69/16Exact fraction: `-69/16`Retained binary64: `-4.3125`	0.186232495514Retained binary64: `0.18623249551444782`binary64;12 significant-digit display; exact zero sign canonicalized only
19	-135/32Exact fraction: `-135/32`Retained binary64: `-4.21875`	0.194631799683Retained binary64: `0.19463179968272995`binary64;12 significant-digit display; exact zero sign canonicalized only
20	-33/8Exact fraction: `-33/8`Retained binary64: `-4.125`	0.203475614331Retained binary64: `0.20347561433082678`binary64;12 significant-digit display; exact zero sign canonicalized only
21	-129/32Exact fraction: `-129/32`Retained binary64: `-4.03125`	0.212787688056Retained binary64: `0.21278768805565235`binary64;12 significant-digit display; exact zero sign canonicalized only
22	-63/16Exact fraction: `-63/16`Retained binary64: `-3.9375`	0.222592629996Retained binary64: `0.2225926299961136`binary64;12 significant-digit display; exact zero sign canonicalized only
23	-123/32Exact fraction: `-123/32`Retained binary64: `-3.84375`	0.232915843228Retained binary64: `0.23291584322833797`binary64;12 significant-digit display; exact zero sign canonicalized only
24	-15/4Exact fraction: `-15/4`Retained binary64: `-3.75`	0.243783433192Retained binary64: `0.24378343319197046`binary64;12 significant-digit display; exact zero sign canonicalized only
25	-117/32Exact fraction: `-117/32`Retained binary64: `-3.65625`	0.255222086585Retained binary64: `0.25522208658479195`binary64;12 significant-digit display; exact zero sign canonicalized only
26	-57/16Exact fraction: `-57/16`Retained binary64: `-3.5625`	0.267258915592Retained binary64: `0.26725891559220677`binary64;12 significant-digit display; exact zero sign canonicalized only
27	-111/32Exact fraction: `-111/32`Retained binary64: `-3.46875`	0.279921261731Retained binary64: `0.27992126173085763`binary64;12 significant-digit display; exact zero sign canonicalized only
28	-27/8Exact fraction: `-27/8`Retained binary64: `-3.375`	0.293236453001Retained binary64: `0.2932364530007922`binary64;12 significant-digit display; exact zero sign canonicalized only
29	-105/32Exact fraction: `-105/32`Retained binary64: `-3.28125`	0.307231507485Retained binary64: `0.30723150748511074`binary64;12 significant-digit display; exact zero sign canonicalized only
30	-51/16Exact fraction: `-51/16`Retained binary64: `-3.1875`	0.321932776046Retained binary64: `0.32193277604638104`binary64;12 significant-digit display; exact zero sign canonicalized only
31	-99/32Exact fraction: `-99/32`Retained binary64: `-3.09375`	0.337365516394Retained binary64: `0.3373655163935259`binary64;12 significant-digit display; exact zero sign canonicalized only
32	-3Exact fraction: `-3`Retained binary64: `-3.0`	0.353553390593Retained binary64: `0.3535533905932738`binary64;12 significant-digit display; exact zero sign canonicalized only
33	-93/32Exact fraction: `-93/32`Retained binary64: `-2.90625`	0.370517878154Retained binary64: `0.3705178781542135`binary64;12 significant-digit display; exact zero sign canonicalized only
34	-45/16Exact fraction: `-45/16`Retained binary64: `-2.8125`	0.388277597214Retained binary64: `0.3882775972138499`binary64;12 significant-digit display; exact zero sign canonicalized only
35	-87/32Exact fraction: `-87/32`Retained binary64: `-2.71875`	0.406847527223Retained binary64: `0.40684752722275946`binary64;12 significant-digit display; exact zero sign canonicalized only
36	-21/8Exact fraction: `-21/8`Retained binary64: `-2.625`	0.426238127976Retained binary64: `0.4262381279755768`binary64;12 significant-digit display; exact zero sign canonicalized only
37	-81/32Exact fraction: `-81/32`Retained binary64: `-2.53125`	0.446454352032Retained binary64: `0.44645435203199096`binary64;12 significant-digit display; exact zero sign canonicalized only
38	-39/16Exact fraction: `-39/16`Retained binary64: `-2.4375`	0.467494550658Retained binary64: `0.46749455065826817`binary64;12 significant-digit display; exact zero sign canonicalized only
39	-75/32Exact fraction: `-75/32`Retained binary64: `-2.34375`	0.489349277558Retained binary64: `0.4893492775584918`binary64;12 significant-digit display; exact zero sign canonicalized only
40	-9/4Exact fraction: `-9/4`Retained binary64: `-2.25`	0.512Retained binary64: `0.512`binary64;12 significant-digit display; exact zero sign canonicalized only
41	-69/32Exact fraction: `-69/32`Retained binary64: `-2.15625`	0.535417733583Retained binary64: `0.5354177335833474`binary64;12 significant-digit display; exact zero sign canonicalized only
42	-33/16Exact fraction: `-33/16`Retained binary64: `-2.0625`	0.559561624921Retained binary64: `0.5595616249205255`binary64;12 significant-digit display; exact zero sign canonicalized only
43	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	0.584377515834Retained binary64: `0.5843775158342336`binary64;12 significant-digit display; exact zero sign canonicalized only
44	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	0.609796533217Retained binary64: `0.609796533217142`binary64;12 significant-digit display; exact zero sign canonicalized only
45	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	0.635733760067Retained binary64: `0.6357337600670846`binary64;12 significant-digit display; exact zero sign canonicalized only
46	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	0.662087054908Retained binary64: `0.6620870549076695`binary64;12 significant-digit display; exact zero sign canonicalized only
47	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	0.688736098038Retained binary64: `0.6887360980383765`binary64;12 significant-digit display; exact zero sign canonicalized only
48	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	0.7155417528Retained binary64: `0.7155417527999327`binary64;12 significant-digit display; exact zero sign canonicalized only
49	-45/32Exact fraction: `-45/32`Retained binary64: `-1.40625`	0.742345837007Retained binary64: `0.7423458370071108`binary64;12 significant-digit display; exact zero sign canonicalized only
50	-21/16Exact fraction: `-21/16`Retained binary64: `-1.3125`	0.76897140241Retained binary64: `0.7689714024097063`binary64;12 significant-digit display; exact zero sign canonicalized only
51	-39/32Exact fraction: `-39/32`Retained binary64: `-1.21875`	0.79522361691Retained binary64: `0.7952236169096769`binary64;12 significant-digit display; exact zero sign canonicalized only
52	-9/8Exact fraction: `-9/8`Retained binary64: `-1.125`	0.820891333759Retained binary64: `0.8208913337591622`binary64;12 significant-digit display; exact zero sign canonicalized only
53	-33/32Exact fraction: `-33/32`Retained binary64: `-1.03125`	0.845749412828Retained binary64: `0.8457494128281079`binary64;12 significant-digit display; exact zero sign canonicalized only
54	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`	0.869561830523Retained binary64: `0.8695618305230164`binary64;12 significant-digit display; exact zero sign canonicalized only
55	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`	0.89208557712Retained binary64: `0.8920855771199931`binary64;12 significant-digit display; exact zero sign canonicalized only
56	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	0.913075294254Retained binary64: `0.9130752942544301`binary64;12 significant-digit display; exact zero sign canonicalized only
57	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`	0.932288553415Retained binary64: `0.9322885534145126`binary64;12 significant-digit display; exact zero sign canonicalized only
58	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	0.949491622095Retained binary64: `0.949491622095043`binary64;12 significant-digit display; exact zero sign canonicalized only
59	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	0.964465512451Retained binary64: `0.9644655124510275`binary64;12 significant-digit display; exact zero sign canonicalized only
60	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	0.977012063226Retained binary64: `0.9770120632257652`binary64;12 significant-digit display; exact zero sign canonicalized only
61	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	0.986959774913Retained binary64: `0.9869597749125953`binary64;12 significant-digit display; exact zero sign canonicalized only
62	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	0.994169105415Retained binary64: `0.9941691054148616`binary64;12 significant-digit display; exact zero sign canonicalized only
63	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	0.998536942354Retained binary64: `0.9985369423543109`binary64;12 significant-digit display; exact zero sign canonicalized only
64	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`
65	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	0.998536942354Retained binary64: `0.9985369423543109`binary64;12 significant-digit display; exact zero sign canonicalized only
66	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	0.994169105415Retained binary64: `0.9941691054148616`binary64;12 significant-digit display; exact zero sign canonicalized only
67	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	0.986959774913Retained binary64: `0.9869597749125953`binary64;12 significant-digit display; exact zero sign canonicalized only
68	3/8Exact fraction: `3/8`Retained binary64: `0.375`	0.977012063226Retained binary64: `0.9770120632257652`binary64;12 significant-digit display; exact zero sign canonicalized only
69	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	0.964465512451Retained binary64: `0.9644655124510275`binary64;12 significant-digit display; exact zero sign canonicalized only
70	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	0.949491622095Retained binary64: `0.949491622095043`binary64;12 significant-digit display; exact zero sign canonicalized only
71	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	0.932288553415Retained binary64: `0.9322885534145126`binary64;12 significant-digit display; exact zero sign canonicalized only
72	3/4Exact fraction: `3/4`Retained binary64: `0.75`	0.913075294254Retained binary64: `0.9130752942544301`binary64;12 significant-digit display; exact zero sign canonicalized only
73	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	0.89208557712Retained binary64: `0.8920855771199931`binary64;12 significant-digit display; exact zero sign canonicalized only
74	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	0.869561830523Retained binary64: `0.8695618305230164`binary64;12 significant-digit display; exact zero sign canonicalized only
75	33/32Exact fraction: `33/32`Retained binary64: `1.03125`	0.845749412828Retained binary64: `0.8457494128281079`binary64;12 significant-digit display; exact zero sign canonicalized only
76	9/8Exact fraction: `9/8`Retained binary64: `1.125`	0.820891333759Retained binary64: `0.8208913337591622`binary64;12 significant-digit display; exact zero sign canonicalized only
77	39/32Exact fraction: `39/32`Retained binary64: `1.21875`	0.79522361691Retained binary64: `0.7952236169096769`binary64;12 significant-digit display; exact zero sign canonicalized only
78	21/16Exact fraction: `21/16`Retained binary64: `1.3125`	0.76897140241Retained binary64: `0.7689714024097063`binary64;12 significant-digit display; exact zero sign canonicalized only
79	45/32Exact fraction: `45/32`Retained binary64: `1.40625`	0.742345837007Retained binary64: `0.7423458370071108`binary64;12 significant-digit display; exact zero sign canonicalized only
80	3/2Exact fraction: `3/2`Retained binary64: `1.5`	0.7155417528Retained binary64: `0.7155417527999327`binary64;12 significant-digit display; exact zero sign canonicalized only
81	51/32Exact fraction: `51/32`Retained binary64: `1.59375`	0.688736098038Retained binary64: `0.6887360980383765`binary64;12 significant-digit display; exact zero sign canonicalized only
82	27/16Exact fraction: `27/16`Retained binary64: `1.6875`	0.662087054908Retained binary64: `0.6620870549076695`binary64;12 significant-digit display; exact zero sign canonicalized only
83	57/32Exact fraction: `57/32`Retained binary64: `1.78125`	0.635733760067Retained binary64: `0.6357337600670846`binary64;12 significant-digit display; exact zero sign canonicalized only
84	15/8Exact fraction: `15/8`Retained binary64: `1.875`	0.609796533217Retained binary64: `0.609796533217142`binary64;12 significant-digit display; exact zero sign canonicalized only
85	63/32Exact fraction: `63/32`Retained binary64: `1.96875`	0.584377515834Retained binary64: `0.5843775158342336`binary64;12 significant-digit display; exact zero sign canonicalized only
86	33/16Exact fraction: `33/16`Retained binary64: `2.0625`	0.559561624921Retained binary64: `0.5595616249205255`binary64;12 significant-digit display; exact zero sign canonicalized only
87	69/32Exact fraction: `69/32`Retained binary64: `2.15625`	0.535417733583Retained binary64: `0.5354177335833474`binary64;12 significant-digit display; exact zero sign canonicalized only
88	9/4Exact fraction: `9/4`Retained binary64: `2.25`	0.512Retained binary64: `0.512`binary64;12 significant-digit display; exact zero sign canonicalized only
89	75/32Exact fraction: `75/32`Retained binary64: `2.34375`	0.489349277558Retained binary64: `0.4893492775584918`binary64;12 significant-digit display; exact zero sign canonicalized only
90	39/16Exact fraction: `39/16`Retained binary64: `2.4375`	0.467494550658Retained binary64: `0.46749455065826817`binary64;12 significant-digit display; exact zero sign canonicalized only
91	81/32Exact fraction: `81/32`Retained binary64: `2.53125`	0.446454352032Retained binary64: `0.44645435203199096`binary64;12 significant-digit display; exact zero sign canonicalized only
92	21/8Exact fraction: `21/8`Retained binary64: `2.625`	0.426238127976Retained binary64: `0.4262381279755768`binary64;12 significant-digit display; exact zero sign canonicalized only
93	87/32Exact fraction: `87/32`Retained binary64: `2.71875`	0.406847527223Retained binary64: `0.40684752722275946`binary64;12 significant-digit display; exact zero sign canonicalized only
94	45/16Exact fraction: `45/16`Retained binary64: `2.8125`	0.388277597214Retained binary64: `0.3882775972138499`binary64;12 significant-digit display; exact zero sign canonicalized only
95	93/32Exact fraction: `93/32`Retained binary64: `2.90625`	0.370517878154Retained binary64: `0.3705178781542135`binary64;12 significant-digit display; exact zero sign canonicalized only
96	3Exact fraction: `3`Retained binary64: `3.0`	0.353553390593Retained binary64: `0.3535533905932738`binary64;12 significant-digit display; exact zero sign canonicalized only
97	99/32Exact fraction: `99/32`Retained binary64: `3.09375`	0.337365516394Retained binary64: `0.3373655163935259`binary64;12 significant-digit display; exact zero sign canonicalized only
98	51/16Exact fraction: `51/16`Retained binary64: `3.1875`	0.321932776046Retained binary64: `0.32193277604638104`binary64;12 significant-digit display; exact zero sign canonicalized only
99	105/32Exact fraction: `105/32`Retained binary64: `3.28125`	0.307231507485Retained binary64: `0.30723150748511074`binary64;12 significant-digit display; exact zero sign canonicalized only
100	27/8Exact fraction: `27/8`Retained binary64: `3.375`	0.293236453001Retained binary64: `0.2932364530007922`binary64;12 significant-digit display; exact zero sign canonicalized only
101	111/32Exact fraction: `111/32`Retained binary64: `3.46875`	0.279921261731Retained binary64: `0.27992126173085763`binary64;12 significant-digit display; exact zero sign canonicalized only
102	57/16Exact fraction: `57/16`Retained binary64: `3.5625`	0.267258915592Retained binary64: `0.26725891559220677`binary64;12 significant-digit display; exact zero sign canonicalized only
103	117/32Exact fraction: `117/32`Retained binary64: `3.65625`	0.255222086585Retained binary64: `0.25522208658479195`binary64;12 significant-digit display; exact zero sign canonicalized only
104	15/4Exact fraction: `15/4`Retained binary64: `3.75`	0.243783433192Retained binary64: `0.24378343319197046`binary64;12 significant-digit display; exact zero sign canonicalized only
105	123/32Exact fraction: `123/32`Retained binary64: `3.84375`	0.232915843228Retained binary64: `0.23291584322833797`binary64;12 significant-digit display; exact zero sign canonicalized only
106	63/16Exact fraction: `63/16`Retained binary64: `3.9375`	0.222592629996Retained binary64: `0.2225926299961136`binary64;12 significant-digit display; exact zero sign canonicalized only
107	129/32Exact fraction: `129/32`Retained binary64: `4.03125`	0.212787688056Retained binary64: `0.21278768805565235`binary64;12 significant-digit display; exact zero sign canonicalized only
108	33/8Exact fraction: `33/8`Retained binary64: `4.125`	0.203475614331Retained binary64: `0.20347561433082678`binary64;12 significant-digit display; exact zero sign canonicalized only
109	135/32Exact fraction: `135/32`Retained binary64: `4.21875`	0.194631799683Retained binary64: `0.19463179968272995`binary64;12 significant-digit display; exact zero sign canonicalized only
110	69/16Exact fraction: `69/16`Retained binary64: `4.3125`	0.186232495514Retained binary64: `0.18623249551444782`binary64;12 significant-digit display; exact zero sign canonicalized only
111	141/32Exact fraction: `141/32`Retained binary64: `4.40625`	0.178254859428Retained binary64: `0.17825485942839164`binary64;12 significant-digit display; exact zero sign canonicalized only
112	9/2Exact fraction: `9/2`Retained binary64: `4.5`	0.170676983454Retained binary64: `0.17067698345391663`binary64;12 significant-digit display; exact zero sign canonicalized only
113	147/32Exact fraction: `147/32`Retained binary64: `4.59375`	0.163477907901Retained binary64: `0.1634779079010968`binary64;12 significant-digit display; exact zero sign canonicalized only
114	75/16Exact fraction: `75/16`Retained binary64: `4.6875`	0.156637623478Retained binary64: `0.15663762347829968`binary64;12 significant-digit display; exact zero sign canonicalized only
115	153/32Exact fraction: `153/32`Retained binary64: `4.78125`	0.150137063936Retained binary64: `0.1501370639364056`binary64;12 significant-digit display; exact zero sign canonicalized only
116	39/8Exact fraction: `39/8`Retained binary64: `4.875`	0.14395809117Retained binary64: `0.14395809116960265`binary64;12 significant-digit display; exact zero sign canonicalized only
117	159/32Exact fraction: `159/32`Retained binary64: `4.96875`	0.138083474409Retained binary64: `0.1380834744092699`binary64;12 significant-digit display; exact zero sign canonicalized only
118	81/16Exact fraction: `81/16`Retained binary64: `5.0625`	0.132496864891Retained binary64: `0.13249686489060267`binary64;12 significant-digit display; exact zero sign canonicalized only
119	165/32Exact fraction: `165/32`Retained binary64: `5.15625`	0.127182767148Retained binary64: `0.12718276714816618`binary64;12 significant-digit display; exact zero sign canonicalized only
120	21/4Exact fraction: `21/4`Retained binary64: `5.25`	0.122126507903Retained binary64: `0.12212650790322065`binary64;12 significant-digit display; exact zero sign canonicalized only
121	171/32Exact fraction: `171/32`Retained binary64: `5.34375`	0.117314203339Retained binary64: `0.11731420333925156`binary64;12 significant-digit display; exact zero sign canonicalized only
122	87/16Exact fraction: `87/16`Retained binary64: `5.4375`	0.11273272542Retained binary64: `0.11273272541960044`binary64;12 significant-digit display; exact zero sign canonicalized only
123	177/32Exact fraction: `177/32`Retained binary64: `5.53125`	0.10836966778Retained binary64: `0.10836966777956844`binary64;12 significant-digit display; exact zero sign canonicalized only
124	45/8Exact fraction: `45/8`Retained binary64: `5.625`	0.104213311622Retained binary64: `0.10421331162222675`binary64;12 significant-digit display; exact zero sign canonicalized only
125	183/32Exact fraction: `183/32`Retained binary64: `5.71875`	0.10025259196Retained binary64: `0.10025259196002366`binary64;12 significant-digit display; exact zero sign canonicalized only
126	93/16Exact fraction: `93/16`Retained binary64: `5.8125`	0.096477064471Retained binary64: `0.09647706447098157`binary64;12 significant-digit display; exact zero sign canonicalized only
127	189/32Exact fraction: `189/32`Retained binary64: `5.90625`	0.0928768731769Retained binary64: `0.09287687317691615`binary64;12 significant-digit display; exact zero sign canonicalized only
128	6Exact fraction: `6`Retained binary64: `6.0`	0.0894427191Retained binary64: `0.08944271909999159`binary64;12 significant-digit display; exact zero sign canonicalized only

All three exact loop-axis controls (3 rows)

Separate loop radius3L₀,current2I₀,counterclockwise viewed from +z. B_c=μ₀I₀/(3L₀),not B₀. Field is positive+z on both axis halves, regular at the center; no off-axis field is assigned.

All three exact loop-axis controls
Row	Z=z/L₀	Loop-axis B_z/B_c (separate normalization)
0	-4Exact fraction: `-4`Retained binary64: `-4.0`	27/125Exact fraction: `27/125`Retained binary64: `0.216`
1	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`
2	4Exact fraction: `4`Retained binary64: `4.0`	27/125Exact fraction: `27/125`Retained binary64: `0.216`

circularLoop: exact controls, scales and source limits (5 rows)

Separate loop radius3L₀,current2I₀,counterclockwise viewed from +z. B_c=μ₀I₀/(3L₀),not B₀. Field is positive+z on both axis halves, regular at the center; no off-axis field is assigned.

circularLoop: exact controls, scales and source limits
Quantity/source condition	Admitted value
radius Over L0	3Exact fraction: `3`Retained binary64: `3.0`
Signed current/I₀	2Exact fraction: `2`Retained binary64: `2.0`
normalization	Bc=mu0 I0/(3L0), distinct from B0
field On Both Axis Halves	positive-z
regular Center	true

Separate axis symmetry from a dipole limit

The loop’s axial field is an even function of z and points in the same +z direction at positive and negative z for the chosen current. This does not imply that B points in +z everywhere outside the axis or that its field lines fail to return. The axial formula is a statement on a one-dimensional set of observation points.

For |z| much larger than a, B_z approaches μ₀Ia²/(2|z|³). With the loop moment m_loop = Iπa², this is μ₀m_loop/(2π|z|³) on the axis. The far-field approximation follows from expanding the exact axial denominator; it does not replace the exact expression near the loop.

The magnetic-dipole interpretation connects this source calculation to the following lesson on dipoles and magnetic materials. Here the moment is fixed by a specified current loop and orientation. Intrinsic spin, material magnetization and nonlinear magnetic response are separate physical models.

At a generic point away from the axis, the geometry does not reduce to the same scalar formula. One must retain the full vector integral or use another justified method. A convenient axis result and a visually symmetric sketch are insufficient grounds for assigning the field throughout space.

State Ampère’s law with both orientations

For the complete magnetostatic problem in the stated medium, ∮_C B·dℓ = μ₀I_S, where I_S = ∫_S J·n̂ dA is the signed current through an oriented surface bounded by C. The contour and normal follow the right-hand convention: looking from the positive normal side, the positive contour runs counterclockwise.

Stokes’ theorem gives the equivalent smooth local equation ∇ × B = μ₀J. Reversing the surface normal requires reversing the contour orientation; both sides then change sign. The law constrains a signed circulation rather than the integral of |B| along an arbitrary curve.

For steady complete currents, different admissible spanning surfaces with the same oriented boundary give the same net current. Join two such surfaces with opposite orientations to form a closed surface and use ∇·J = 0. This surface independence would fail for an incompatible open current ending in an accumulating charge if the displacement contribution were omitted.

The magnetic flux condition ∇·B = 0 is also required. Ampère’s law alone does not determine every component of B or establish a unique boundary-value solution. Source symmetry, regularity, behavior at infinity and boundary conditions complete the magnetic problem.

Use symmetry before taking B outside the integral

For an isolated ideal infinite straight current along +z in a uniform medium, translational and cylindrical source symmetry, with the appropriate boundary conditions and no added external field, give B = B_phi(r)φ̂. A centered circle of radius r then has constant tangential field and ∮B·dℓ = 2πrB_phi(r).

A circle outside a thin wire encloses I, so B_phi = μ₀I/(2πr). Inside a finite-radius uniform-current cylinder, the enclosed current instead grows with area and must be computed at that radius. The total wire current is not enclosed by every interior contour.

The infinite-wire construction is a defined idealization. It may approximate a region of a long real circuit when ends and returns have controlled influence. A uniform background field can be added mathematically unless the boundary conditions exclude it; the current distribution alone does not remove every homogeneous solution.

Ampère’s integral law remains valid for complete steady sources without useful symmetry. What fails is the step of replacing its integrand by one unknown constant times the contour length. The two-wire example below makes that failure quantitative while keeping the total circulation fixed.

Specify the coaxial current ledger

Choose an infinite coaxial geometry. The inner solid cylinder has radius 2L₀ and carries +I₀ uniformly in +z. A vacuum gap extends from radius 2L₀ to 3L₀. The return conductor is an annulus from 3L₀ to 5L₀ carrying −I₀ uniformly. Outside that annulus the net axial current enclosed is zero.

Use current-density unit J_c = I₀/(πL₀²). The inner density is +(1/4)J_c because its area is 4πL₀². The return density is −(1/16)J_c because its area is (25 − 9)πL₀² = 16πL₀². These are bounded bulk currents with abrupt idealized material edges, not delta-function current sheets.

With R = r/L₀, the enclosed-current fraction is R²/4 for 0 ≤ R ≤ 2, one in the gap, 1 − (R² − 9)/16 = (25 − R²)/16 in the return, and zero outside. The fraction counts the return current actually enclosed so far, rather than subtracting the whole return as soon as its inner boundary is crossed.

The continuity of the enclosed-current integral at R = 2, 3 and 5 is an immediate source check. Its derivative changes where the bulk current density changes. Current is translationally invariant along z, so these specified axial bulk currents satisfy ∇·J = 0 in the distributional steady description without introducing a current endpoint.

Current support and enclosed fraction in the selected infinite coaxial model
Region	Radial domain	J_z/J_c in the open region	I_enclosed/I₀
Inner cylinder	0 ≤ R < 2	1/4	R²/4
Vacuum gap	2 < R < 3	0	1
Return annulus	3 < R < 5	−1/16	(25 − R²)/16
Exterior	R > 5	0	0

Resolve the coaxial field and its limits

Cylindrical symmetry and a centered contour give B_phi/B₀ = (I_enclosed/I₀)/R for R > 0. Substituting the source fractions yields R/4 in the inner cylinder, 1/R in the gap, (25 − R²)/(16R) in the return, and zero outside. At the axis, the Cartesian field is zero by the regular limiting solution.

The field magnitude is continuous: it is B₀/2 at R = 2, B₀/3 at R = 3 and zero at R = 5 from either adjacent side. In the inner cylinder it grows linearly from the axis instead of diverging like an ideal filament. At R = 0, an azimuthal unit direction is undefined, so no direction arrow is assigned to the zero field.

The derivative and current density have one-sided limits at the idealized edges. For example, d(B_phi/B₀)/dR changes from +1/4 to −1/4 at R = 2; at R = 5 it changes from −1/8 to zero. These are branch limits, not a claim of a single classical derivative or uniquely assigned stepped current density at an interface.

For every radius, circulation divided by μ₀I₀ equals the enclosed-current fraction. The complete return makes the ideal exterior field vanish here because of the specified symmetry and source balance. Zero total current alone would not prove a zero local exterior field for a noncoaxial finite circuit or arbitrary arrangement.

Enclosed forward and return current determine B

Track the progressively enclosed inner and return currents and the corresponding azimuthal magnetic field. Preserve continuous field limits, the regular zero at the axis, and the distinction between one-sided current-density or slope limits and an ideal current sheet.

Progressively enclosed inner and return current

Inner bulk: 0≤R≤2
Current-free gap: 2≤R≤3
Return bulk: 3≤R≤5
Outside: R≥5; plotted to7
All nine controls; paired rows at2,3,5 have the same continuous plotted value

Continuous azimuthal field, regular zero at the axis

Inner bulk: 0≤R≤2
Current-free gap: 2≤R≤3
Return bulk: 3≤R≤5
Outside: R≥5; plotted to7
All nine controls; paired rows at2,3,5 have the same continuous plotted value

Infinite uniform bulk coax: radius2 inner+I₀,gap2<R<3,return annulus3<R<5 with−I₀,outsidezero net current. No ideal sheet. B is continuous; J and slope endpoint entries are one-sided limits only. AtR0 Cartesian B=0 and azimuth is undefined.
Current-density unit is I₀/(πL₀²). Inner density+1/4,return density−1/16; total inner/return currents are+I₀ and−I₀. The source records retain all 324 branch samples and nine controls, including paired metadata atR2,3,5.
Both plotted quantities are continuous atR2,3,5. Source current-density and field-slope entries at these radii are separately labelled one-sided branch limits in every relevant table row; no unique ordinary interface density/slope is assigned. This is different from an ideal surface-current sheet.
AtR0 the Cartesian field is zero, whileφ direction is undefined. The scalar graph reaches the regular limiting coefficient0 without assigning an azimuthal unit vector. Circulation/(μ₀I₀) equals the retained enclosed-current fraction; it is not a second independent local field determination.
All marks read exact admitted records, including rational controls, original binary64 function-specific qualifications,12-significant-digit displays and retained values. No fitted conservation, near-zero snap, field solver or sampled-quadrature proof is used.

Complete labelled data equivalents preserve every admitted scalar, record, branch, mask and source constraint. Original manuscript tables remain separate.

Current support and enclosed fraction in the selected infinite coaxial model

Exact field and circulation controls in the common coaxial scales

Four source branch identities (4 rows)

Infinite uniform bulk coax: radius2 inner+I₀,gap2<R<3,return annulus3<R<5 with−I₀,outsidezero net current. No ideal sheet. B is continuous; J and slope endpoint entries are one-sided limits only. AtR0 Cartesian B=0 and azimuth is undefined.

Four source branch identities
Branch row	Source key
0	inner
1	gap
2	return
3	outside

Complete inner coax branch (65 rows) (65 rows)

Complete inner coax branch (65 rows)
Row	R=r/L₀	Signed enclosed I/I₀	Bφ/B₀	Circulation/(μ₀I₀)	Branch J_z/[I₀/(πL₀²)]; one-sided at interfaces	Branch db/dR; one-sided at interfaces	J/slope are interface one-sided limits only?	Azimuthal direction defined?	Cartesian field zero at center?
0AtR=0 the Cartesian field is exactlyzero; azimuthal direction is undefined. No azimuthal unit vector is assigned there.	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	false	true
1	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	1/4096Exact fraction: `1/4096`Retained binary64: `0.000244140625`	1/128Exact fraction: `1/128`Retained binary64: `0.0078125`	1/4096Exact fraction: `1/4096`Retained binary64: `0.000244140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
2	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	1/1024Exact fraction: `1/1024`Retained binary64: `0.0009765625`	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	1/1024Exact fraction: `1/1024`Retained binary64: `0.0009765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
3	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	9/4096Exact fraction: `9/4096`Retained binary64: `0.002197265625`	3/128Exact fraction: `3/128`Retained binary64: `0.0234375`	9/4096Exact fraction: `9/4096`Retained binary64: `0.002197265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
4	1/8Exact fraction: `1/8`Retained binary64: `0.125`	1/256Exact fraction: `1/256`Retained binary64: `0.00390625`	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	1/256Exact fraction: `1/256`Retained binary64: `0.00390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
5	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	25/4096Exact fraction: `25/4096`Retained binary64: `0.006103515625`	5/128Exact fraction: `5/128`Retained binary64: `0.0390625`	25/4096Exact fraction: `25/4096`Retained binary64: `0.006103515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
6	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	9/1024Exact fraction: `9/1024`Retained binary64: `0.0087890625`	3/64Exact fraction: `3/64`Retained binary64: `0.046875`	9/1024Exact fraction: `9/1024`Retained binary64: `0.0087890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
7	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	49/4096Exact fraction: `49/4096`Retained binary64: `0.011962890625`	7/128Exact fraction: `7/128`Retained binary64: `0.0546875`	49/4096Exact fraction: `49/4096`Retained binary64: `0.011962890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
8	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
9	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	81/4096Exact fraction: `81/4096`Retained binary64: `0.019775390625`	9/128Exact fraction: `9/128`Retained binary64: `0.0703125`	81/4096Exact fraction: `81/4096`Retained binary64: `0.019775390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
10	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	25/1024Exact fraction: `25/1024`Retained binary64: `0.0244140625`	5/64Exact fraction: `5/64`Retained binary64: `0.078125`	25/1024Exact fraction: `25/1024`Retained binary64: `0.0244140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
11	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	121/4096Exact fraction: `121/4096`Retained binary64: `0.029541015625`	11/128Exact fraction: `11/128`Retained binary64: `0.0859375`	121/4096Exact fraction: `121/4096`Retained binary64: `0.029541015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
12	3/8Exact fraction: `3/8`Retained binary64: `0.375`	9/256Exact fraction: `9/256`Retained binary64: `0.03515625`	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	9/256Exact fraction: `9/256`Retained binary64: `0.03515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
13	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	169/4096Exact fraction: `169/4096`Retained binary64: `0.041259765625`	13/128Exact fraction: `13/128`Retained binary64: `0.1015625`	169/4096Exact fraction: `169/4096`Retained binary64: `0.041259765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
14	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	49/1024Exact fraction: `49/1024`Retained binary64: `0.0478515625`	7/64Exact fraction: `7/64`Retained binary64: `0.109375`	49/1024Exact fraction: `49/1024`Retained binary64: `0.0478515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
15	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	225/4096Exact fraction: `225/4096`Retained binary64: `0.054931640625`	15/128Exact fraction: `15/128`Retained binary64: `0.1171875`	225/4096Exact fraction: `225/4096`Retained binary64: `0.054931640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
16	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	1/8Exact fraction: `1/8`Retained binary64: `0.125`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
17	17/32Exact fraction: `17/32`Retained binary64: `0.53125`	289/4096Exact fraction: `289/4096`Retained binary64: `0.070556640625`	17/128Exact fraction: `17/128`Retained binary64: `0.1328125`	289/4096Exact fraction: `289/4096`Retained binary64: `0.070556640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
18	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	81/1024Exact fraction: `81/1024`Retained binary64: `0.0791015625`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	81/1024Exact fraction: `81/1024`Retained binary64: `0.0791015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
19	19/32Exact fraction: `19/32`Retained binary64: `0.59375`	361/4096Exact fraction: `361/4096`Retained binary64: `0.088134765625`	19/128Exact fraction: `19/128`Retained binary64: `0.1484375`	361/4096Exact fraction: `361/4096`Retained binary64: `0.088134765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
20	5/8Exact fraction: `5/8`Retained binary64: `0.625`	25/256Exact fraction: `25/256`Retained binary64: `0.09765625`	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	25/256Exact fraction: `25/256`Retained binary64: `0.09765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
21	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	441/4096Exact fraction: `441/4096`Retained binary64: `0.107666015625`	21/128Exact fraction: `21/128`Retained binary64: `0.1640625`	441/4096Exact fraction: `441/4096`Retained binary64: `0.107666015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
22	11/16Exact fraction: `11/16`Retained binary64: `0.6875`	121/1024Exact fraction: `121/1024`Retained binary64: `0.1181640625`	11/64Exact fraction: `11/64`Retained binary64: `0.171875`	121/1024Exact fraction: `121/1024`Retained binary64: `0.1181640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
23	23/32Exact fraction: `23/32`Retained binary64: `0.71875`	529/4096Exact fraction: `529/4096`Retained binary64: `0.129150390625`	23/128Exact fraction: `23/128`Retained binary64: `0.1796875`	529/4096Exact fraction: `529/4096`Retained binary64: `0.129150390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
24	3/4Exact fraction: `3/4`Retained binary64: `0.75`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
25	25/32Exact fraction: `25/32`Retained binary64: `0.78125`	625/4096Exact fraction: `625/4096`Retained binary64: `0.152587890625`	25/128Exact fraction: `25/128`Retained binary64: `0.1953125`	625/4096Exact fraction: `625/4096`Retained binary64: `0.152587890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
26	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	169/1024Exact fraction: `169/1024`Retained binary64: `0.1650390625`	13/64Exact fraction: `13/64`Retained binary64: `0.203125`	169/1024Exact fraction: `169/1024`Retained binary64: `0.1650390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
27	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	729/4096Exact fraction: `729/4096`Retained binary64: `0.177978515625`	27/128Exact fraction: `27/128`Retained binary64: `0.2109375`	729/4096Exact fraction: `729/4096`Retained binary64: `0.177978515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
28	7/8Exact fraction: `7/8`Retained binary64: `0.875`	49/256Exact fraction: `49/256`Retained binary64: `0.19140625`	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	49/256Exact fraction: `49/256`Retained binary64: `0.19140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
29	29/32Exact fraction: `29/32`Retained binary64: `0.90625`	841/4096Exact fraction: `841/4096`Retained binary64: `0.205322265625`	29/128Exact fraction: `29/128`Retained binary64: `0.2265625`	841/4096Exact fraction: `841/4096`Retained binary64: `0.205322265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
30	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	225/1024Exact fraction: `225/1024`Retained binary64: `0.2197265625`	15/64Exact fraction: `15/64`Retained binary64: `0.234375`	225/1024Exact fraction: `225/1024`Retained binary64: `0.2197265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
31	31/32Exact fraction: `31/32`Retained binary64: `0.96875`	961/4096Exact fraction: `961/4096`Retained binary64: `0.234619140625`	31/128Exact fraction: `31/128`Retained binary64: `0.2421875`	961/4096Exact fraction: `961/4096`Retained binary64: `0.234619140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
32	1Exact fraction: `1`Retained binary64: `1.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
33	33/32Exact fraction: `33/32`Retained binary64: `1.03125`	1089/4096Exact fraction: `1089/4096`Retained binary64: `0.265869140625`	33/128Exact fraction: `33/128`Retained binary64: `0.2578125`	1089/4096Exact fraction: `1089/4096`Retained binary64: `0.265869140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
34	17/16Exact fraction: `17/16`Retained binary64: `1.0625`	289/1024Exact fraction: `289/1024`Retained binary64: `0.2822265625`	17/64Exact fraction: `17/64`Retained binary64: `0.265625`	289/1024Exact fraction: `289/1024`Retained binary64: `0.2822265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
35	35/32Exact fraction: `35/32`Retained binary64: `1.09375`	1225/4096Exact fraction: `1225/4096`Retained binary64: `0.299072265625`	35/128Exact fraction: `35/128`Retained binary64: `0.2734375`	1225/4096Exact fraction: `1225/4096`Retained binary64: `0.299072265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
36	9/8Exact fraction: `9/8`Retained binary64: `1.125`	81/256Exact fraction: `81/256`Retained binary64: `0.31640625`	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	81/256Exact fraction: `81/256`Retained binary64: `0.31640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
37	37/32Exact fraction: `37/32`Retained binary64: `1.15625`	1369/4096Exact fraction: `1369/4096`Retained binary64: `0.334228515625`	37/128Exact fraction: `37/128`Retained binary64: `0.2890625`	1369/4096Exact fraction: `1369/4096`Retained binary64: `0.334228515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
38	19/16Exact fraction: `19/16`Retained binary64: `1.1875`	361/1024Exact fraction: `361/1024`Retained binary64: `0.3525390625`	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	361/1024Exact fraction: `361/1024`Retained binary64: `0.3525390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
39	39/32Exact fraction: `39/32`Retained binary64: `1.21875`	1521/4096Exact fraction: `1521/4096`Retained binary64: `0.371337890625`	39/128Exact fraction: `39/128`Retained binary64: `0.3046875`	1521/4096Exact fraction: `1521/4096`Retained binary64: `0.371337890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
40	5/4Exact fraction: `5/4`Retained binary64: `1.25`	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
41	41/32Exact fraction: `41/32`Retained binary64: `1.28125`	1681/4096Exact fraction: `1681/4096`Retained binary64: `0.410400390625`	41/128Exact fraction: `41/128`Retained binary64: `0.3203125`	1681/4096Exact fraction: `1681/4096`Retained binary64: `0.410400390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
42	21/16Exact fraction: `21/16`Retained binary64: `1.3125`	441/1024Exact fraction: `441/1024`Retained binary64: `0.4306640625`	21/64Exact fraction: `21/64`Retained binary64: `0.328125`	441/1024Exact fraction: `441/1024`Retained binary64: `0.4306640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
43	43/32Exact fraction: `43/32`Retained binary64: `1.34375`	1849/4096Exact fraction: `1849/4096`Retained binary64: `0.451416015625`	43/128Exact fraction: `43/128`Retained binary64: `0.3359375`	1849/4096Exact fraction: `1849/4096`Retained binary64: `0.451416015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
44	11/8Exact fraction: `11/8`Retained binary64: `1.375`	121/256Exact fraction: `121/256`Retained binary64: `0.47265625`	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	121/256Exact fraction: `121/256`Retained binary64: `0.47265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
45	45/32Exact fraction: `45/32`Retained binary64: `1.40625`	2025/4096Exact fraction: `2025/4096`Retained binary64: `0.494384765625`	45/128Exact fraction: `45/128`Retained binary64: `0.3515625`	2025/4096Exact fraction: `2025/4096`Retained binary64: `0.494384765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
46	23/16Exact fraction: `23/16`Retained binary64: `1.4375`	529/1024Exact fraction: `529/1024`Retained binary64: `0.5166015625`	23/64Exact fraction: `23/64`Retained binary64: `0.359375`	529/1024Exact fraction: `529/1024`Retained binary64: `0.5166015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
47	47/32Exact fraction: `47/32`Retained binary64: `1.46875`	2209/4096Exact fraction: `2209/4096`Retained binary64: `0.539306640625`	47/128Exact fraction: `47/128`Retained binary64: `0.3671875`	2209/4096Exact fraction: `2209/4096`Retained binary64: `0.539306640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
48	3/2Exact fraction: `3/2`Retained binary64: `1.5`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	3/8Exact fraction: `3/8`Retained binary64: `0.375`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
49	49/32Exact fraction: `49/32`Retained binary64: `1.53125`	2401/4096Exact fraction: `2401/4096`Retained binary64: `0.586181640625`	49/128Exact fraction: `49/128`Retained binary64: `0.3828125`	2401/4096Exact fraction: `2401/4096`Retained binary64: `0.586181640625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
50	25/16Exact fraction: `25/16`Retained binary64: `1.5625`	625/1024Exact fraction: `625/1024`Retained binary64: `0.6103515625`	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	625/1024Exact fraction: `625/1024`Retained binary64: `0.6103515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
51	51/32Exact fraction: `51/32`Retained binary64: `1.59375`	2601/4096Exact fraction: `2601/4096`Retained binary64: `0.635009765625`	51/128Exact fraction: `51/128`Retained binary64: `0.3984375`	2601/4096Exact fraction: `2601/4096`Retained binary64: `0.635009765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
52	13/8Exact fraction: `13/8`Retained binary64: `1.625`	169/256Exact fraction: `169/256`Retained binary64: `0.66015625`	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	169/256Exact fraction: `169/256`Retained binary64: `0.66015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
53	53/32Exact fraction: `53/32`Retained binary64: `1.65625`	2809/4096Exact fraction: `2809/4096`Retained binary64: `0.685791015625`	53/128Exact fraction: `53/128`Retained binary64: `0.4140625`	2809/4096Exact fraction: `2809/4096`Retained binary64: `0.685791015625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
54	27/16Exact fraction: `27/16`Retained binary64: `1.6875`	729/1024Exact fraction: `729/1024`Retained binary64: `0.7119140625`	27/64Exact fraction: `27/64`Retained binary64: `0.421875`	729/1024Exact fraction: `729/1024`Retained binary64: `0.7119140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
55	55/32Exact fraction: `55/32`Retained binary64: `1.71875`	3025/4096Exact fraction: `3025/4096`Retained binary64: `0.738525390625`	55/128Exact fraction: `55/128`Retained binary64: `0.4296875`	3025/4096Exact fraction: `3025/4096`Retained binary64: `0.738525390625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
56	7/4Exact fraction: `7/4`Retained binary64: `1.75`	49/64Exact fraction: `49/64`Retained binary64: `0.765625`	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	49/64Exact fraction: `49/64`Retained binary64: `0.765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
57	57/32Exact fraction: `57/32`Retained binary64: `1.78125`	3249/4096Exact fraction: `3249/4096`Retained binary64: `0.793212890625`	57/128Exact fraction: `57/128`Retained binary64: `0.4453125`	3249/4096Exact fraction: `3249/4096`Retained binary64: `0.793212890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
58	29/16Exact fraction: `29/16`Retained binary64: `1.8125`	841/1024Exact fraction: `841/1024`Retained binary64: `0.8212890625`	29/64Exact fraction: `29/64`Retained binary64: `0.453125`	841/1024Exact fraction: `841/1024`Retained binary64: `0.8212890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
59	59/32Exact fraction: `59/32`Retained binary64: `1.84375`	3481/4096Exact fraction: `3481/4096`Retained binary64: `0.849853515625`	59/128Exact fraction: `59/128`Retained binary64: `0.4609375`	3481/4096Exact fraction: `3481/4096`Retained binary64: `0.849853515625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
60	15/8Exact fraction: `15/8`Retained binary64: `1.875`	225/256Exact fraction: `225/256`Retained binary64: `0.87890625`	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	225/256Exact fraction: `225/256`Retained binary64: `0.87890625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
61	61/32Exact fraction: `61/32`Retained binary64: `1.90625`	3721/4096Exact fraction: `3721/4096`Retained binary64: `0.908447265625`	61/128Exact fraction: `61/128`Retained binary64: `0.4765625`	3721/4096Exact fraction: `3721/4096`Retained binary64: `0.908447265625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
62	31/16Exact fraction: `31/16`Retained binary64: `1.9375`	961/1024Exact fraction: `961/1024`Retained binary64: `0.9384765625`	31/64Exact fraction: `31/64`Retained binary64: `0.484375`	961/1024Exact fraction: `961/1024`Retained binary64: `0.9384765625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
63	63/32Exact fraction: `63/32`Retained binary64: `1.96875`	3969/4096Exact fraction: `3969/4096`Retained binary64: `0.968994140625`	63/128Exact fraction: `63/128`Retained binary64: `0.4921875`	3969/4096Exact fraction: `3969/4096`Retained binary64: `0.968994140625`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
64Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1Exact fraction: `1`Retained binary64: `1.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	true	true	false

Complete gap coax branch (65 rows) (65 rows)

Complete gap coax branch (65 rows)
Row	R=r/L₀	Signed enclosed I/I₀	Bφ/B₀	Circulation/(μ₀I₀)	Branch J_z/[I₀/(πL₀²)]; one-sided at interfaces	Branch db/dR; one-sided at interfaces	J/slope are interface one-sided limits only?	Azimuthal direction defined?	Cartesian field zero at center?
0Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	true	true	false
1	129/64Exact fraction: `129/64`Retained binary64: `2.015625`	1Exact fraction: `1`Retained binary64: `1.0`	64/129Exact fraction: `64/129`Retained binary64: `0.49612403100775193`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/16641Exact fraction: `-4096/16641`Retained binary64: `-0.2461390541433808`	false	true	false
2	65/32Exact fraction: `65/32`Retained binary64: `2.03125`	1Exact fraction: `1`Retained binary64: `1.0`	32/65Exact fraction: `32/65`Retained binary64: `0.49230769230769234`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/4225Exact fraction: `-1024/4225`Retained binary64: `-0.24236686390532544`	false	true	false
3	131/64Exact fraction: `131/64`Retained binary64: `2.046875`	1Exact fraction: `1`Retained binary64: `1.0`	64/131Exact fraction: `64/131`Retained binary64: `0.48854961832061067`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/17161Exact fraction: `-4096/17161`Retained binary64: `-0.2386807295612144`	false	true	false
4	33/16Exact fraction: `33/16`Retained binary64: `2.0625`	1Exact fraction: `1`Retained binary64: `1.0`	16/33Exact fraction: `16/33`Retained binary64: `0.48484848484848486`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1089Exact fraction: `-256/1089`Retained binary64: `-0.23507805325987144`	false	true	false
5	133/64Exact fraction: `133/64`Retained binary64: `2.078125`	1Exact fraction: `1`Retained binary64: `1.0`	64/133Exact fraction: `64/133`Retained binary64: `0.48120300751879697`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/17689Exact fraction: `-4096/17689`Retained binary64: `-0.2315563344451354`	false	true	false
6	67/32Exact fraction: `67/32`Retained binary64: `2.09375`	1Exact fraction: `1`Retained binary64: `1.0`	32/67Exact fraction: `32/67`Retained binary64: `0.47761194029850745`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/4489Exact fraction: `-1024/4489`Retained binary64: `-0.22811316551570507`	false	true	false
7	135/64Exact fraction: `135/64`Retained binary64: `2.109375`	1Exact fraction: `1`Retained binary64: `1.0`	64/135Exact fraction: `64/135`Retained binary64: `0.4740740740740741`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/18225Exact fraction: `-4096/18225`Retained binary64: `-0.22474622770919067`	false	true	false
8	17/8Exact fraction: `17/8`Retained binary64: `2.125`	1Exact fraction: `1`Retained binary64: `1.0`	8/17Exact fraction: `8/17`Retained binary64: `0.47058823529411764`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-64/289Exact fraction: `-64/289`Retained binary64: `-0.22145328719723184`	false	true	false
9	137/64Exact fraction: `137/64`Retained binary64: `2.140625`	1Exact fraction: `1`Retained binary64: `1.0`	64/137Exact fraction: `64/137`Retained binary64: `0.46715328467153283`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/18769Exact fraction: `-4096/18769`Retained binary64: `-0.2182321913794022`	false	true	false
10	69/32Exact fraction: `69/32`Retained binary64: `2.15625`	1Exact fraction: `1`Retained binary64: `1.0`	32/69Exact fraction: `32/69`Retained binary64: `0.463768115942029`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/4761Exact fraction: `-1024/4761`Retained binary64: `-0.21508086536441923`	false	true	false
11	139/64Exact fraction: `139/64`Retained binary64: `2.171875`	1Exact fraction: `1`Retained binary64: `1.0`	64/139Exact fraction: `64/139`Retained binary64: `0.460431654676259`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/19321Exact fraction: `-4096/19321`Retained binary64: `-0.2119973086279178`	false	true	false
12	35/16Exact fraction: `35/16`Retained binary64: `2.1875`	1Exact fraction: `1`Retained binary64: `1.0`	16/35Exact fraction: `16/35`Retained binary64: `0.45714285714285713`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1225Exact fraction: `-256/1225`Retained binary64: `-0.2089795918367347`	false	true	false
13	141/64Exact fraction: `141/64`Retained binary64: `2.203125`	1Exact fraction: `1`Retained binary64: `1.0`	64/141Exact fraction: `64/141`Retained binary64: `0.45390070921985815`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/19881Exact fraction: `-4096/19881`Retained binary64: `-0.20602585383029023`	false	true	false
14	71/32Exact fraction: `71/32`Retained binary64: `2.21875`	1Exact fraction: `1`Retained binary64: `1.0`	32/71Exact fraction: `32/71`Retained binary64: `0.4507042253521127`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/5041Exact fraction: `-1024/5041`Retained binary64: `-0.20313429875024797`	false	true	false
15	143/64Exact fraction: `143/64`Retained binary64: `2.234375`	1Exact fraction: `1`Retained binary64: `1.0`	64/143Exact fraction: `64/143`Retained binary64: `0.44755244755244755`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/20449Exact fraction: `-4096/20449`Retained binary64: `-0.2003031933101863`	false	true	false
16	9/4Exact fraction: `9/4`Retained binary64: `2.25`	1Exact fraction: `1`Retained binary64: `1.0`	4/9Exact fraction: `4/9`Retained binary64: `0.4444444444444444`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-16/81Exact fraction: `-16/81`Retained binary64: `-0.19753086419753085`	false	true	false
17	145/64Exact fraction: `145/64`Retained binary64: `2.265625`	1Exact fraction: `1`Retained binary64: `1.0`	64/145Exact fraction: `64/145`Retained binary64: `0.4413793103448276`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/21025Exact fraction: `-4096/21025`Retained binary64: `-0.19481569560047562`	false	true	false
18	73/32Exact fraction: `73/32`Retained binary64: `2.28125`	1Exact fraction: `1`Retained binary64: `1.0`	32/73Exact fraction: `32/73`Retained binary64: `0.4383561643835616`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/5329Exact fraction: `-1024/5329`Retained binary64: `-0.19215612685306813`	false	true	false
19	147/64Exact fraction: `147/64`Retained binary64: `2.296875`	1Exact fraction: `1`Retained binary64: `1.0`	64/147Exact fraction: `64/147`Retained binary64: `0.43537414965986393`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/21609Exact fraction: `-4096/21609`Retained binary64: `-0.1895506501920496`	false	true	false
20	37/16Exact fraction: `37/16`Retained binary64: `2.3125`	1Exact fraction: `1`Retained binary64: `1.0`	16/37Exact fraction: `16/37`Retained binary64: `0.43243243243243246`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1369Exact fraction: `-256/1369`Retained binary64: `-0.18699780861943024`	false	true	false
21	149/64Exact fraction: `149/64`Retained binary64: `2.328125`	1Exact fraction: `1`Retained binary64: `1.0`	64/149Exact fraction: `64/149`Retained binary64: `0.42953020134228187`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/22201Exact fraction: `-4096/22201`Retained binary64: `-0.1844961938651412`	false	true	false
22	75/32Exact fraction: `75/32`Retained binary64: `2.34375`	1Exact fraction: `1`Retained binary64: `1.0`	32/75Exact fraction: `32/75`Retained binary64: `0.4266666666666667`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/5625Exact fraction: `-1024/5625`Retained binary64: `-0.18204444444444445`	false	true	false
23	151/64Exact fraction: `151/64`Retained binary64: `2.359375`	1Exact fraction: `1`Retained binary64: `1.0`	64/151Exact fraction: `64/151`Retained binary64: `0.423841059602649`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/22801Exact fraction: `-4096/22801`Retained binary64: `-0.17964124380509627`	false	true	false
24	19/8Exact fraction: `19/8`Retained binary64: `2.375`	1Exact fraction: `1`Retained binary64: `1.0`	8/19Exact fraction: `8/19`Retained binary64: `0.42105263157894735`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-64/361Exact fraction: `-64/361`Retained binary64: `-0.1772853185595568`	false	true	false
25	153/64Exact fraction: `153/64`Retained binary64: `2.390625`	1Exact fraction: `1`Retained binary64: `1.0`	64/153Exact fraction: `64/153`Retained binary64: `0.41830065359477125`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/23409Exact fraction: `-4096/23409`Retained binary64: `-0.1749754367978128`	false	true	false
26	77/32Exact fraction: `77/32`Retained binary64: `2.40625`	1Exact fraction: `1`Retained binary64: `1.0`	32/77Exact fraction: `32/77`Retained binary64: `0.4155844155844156`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/5929Exact fraction: `-1024/5929`Retained binary64: `-0.17271040647664024`	false	true	false
27	155/64Exact fraction: `155/64`Retained binary64: `2.421875`	1Exact fraction: `1`Retained binary64: `1.0`	64/155Exact fraction: `64/155`Retained binary64: `0.4129032258064516`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/24025Exact fraction: `-4096/24025`Retained binary64: `-0.17048907388137358`	false	true	false
28	39/16Exact fraction: `39/16`Retained binary64: `2.4375`	1Exact fraction: `1`Retained binary64: `1.0`	16/39Exact fraction: `16/39`Retained binary64: `0.41025641025641024`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1521Exact fraction: `-256/1521`Retained binary64: `-0.168310322156476`	false	true	false
29	157/64Exact fraction: `157/64`Retained binary64: `2.453125`	1Exact fraction: `1`Retained binary64: `1.0`	64/157Exact fraction: `64/157`Retained binary64: `0.40764331210191085`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/24649Exact fraction: `-4096/24649`Retained binary64: `-0.16617306990141587`	false	true	false
30	79/32Exact fraction: `79/32`Retained binary64: `2.46875`	1Exact fraction: `1`Retained binary64: `1.0`	32/79Exact fraction: `32/79`Retained binary64: `0.4050632911392405`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/6241Exact fraction: `-1024/6241`Retained binary64: `-0.1640762698285531`	false	true	false
31	159/64Exact fraction: `159/64`Retained binary64: `2.484375`	1Exact fraction: `1`Retained binary64: `1.0`	64/159Exact fraction: `64/159`Retained binary64: `0.4025157232704403`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/25281Exact fraction: `-4096/25281`Retained binary64: `-0.16201890747992564`	false	true	false
32	5/2Exact fraction: `5/2`Retained binary64: `2.5`	1Exact fraction: `1`Retained binary64: `1.0`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4/25Exact fraction: `-4/25`Retained binary64: `-0.16`	false	true	false
33	161/64Exact fraction: `161/64`Retained binary64: `2.515625`	1Exact fraction: `1`Retained binary64: `1.0`	64/161Exact fraction: `64/161`Retained binary64: `0.39751552795031053`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/25921Exact fraction: `-4096/25921`Retained binary64: `-0.15801859496161413`	false	true	false
34	81/32Exact fraction: `81/32`Retained binary64: `2.53125`	1Exact fraction: `1`Retained binary64: `1.0`	32/81Exact fraction: `32/81`Retained binary64: `0.3950617283950617`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/6561Exact fraction: `-1024/6561`Retained binary64: `-0.15607376924249353`	false	true	false
35	163/64Exact fraction: `163/64`Retained binary64: `2.546875`	1Exact fraction: `1`Retained binary64: `1.0`	64/163Exact fraction: `64/163`Retained binary64: `0.39263803680981596`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/26569Exact fraction: `-4096/26569`Retained binary64: `-0.15416462794986638`	false	true	false
36	41/16Exact fraction: `41/16`Retained binary64: `2.5625`	1Exact fraction: `1`Retained binary64: `1.0`	16/41Exact fraction: `16/41`Retained binary64: `0.3902439024390244`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1681Exact fraction: `-256/1681`Retained binary64: `-0.1522903033908388`	false	true	false
37	165/64Exact fraction: `165/64`Retained binary64: `2.578125`	1Exact fraction: `1`Retained binary64: `1.0`	64/165Exact fraction: `64/165`Retained binary64: `0.3878787878787879`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/27225Exact fraction: `-4096/27225`Retained binary64: `-0.15044995408631773`	false	true	false
38	83/32Exact fraction: `83/32`Retained binary64: `2.59375`	1Exact fraction: `1`Retained binary64: `1.0`	32/83Exact fraction: `32/83`Retained binary64: `0.3855421686746988`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/6889Exact fraction: `-1024/6889`Retained binary64: `-0.14864276382638988`	false	true	false
39	167/64Exact fraction: `167/64`Retained binary64: `2.609375`	1Exact fraction: `1`Retained binary64: `1.0`	64/167Exact fraction: `64/167`Retained binary64: `0.38323353293413176`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/27889Exact fraction: `-4096/27889`Retained binary64: `-0.14686794076517623`	false	true	false
40	21/8Exact fraction: `21/8`Retained binary64: `2.625`	1Exact fraction: `1`Retained binary64: `1.0`	8/21Exact fraction: `8/21`Retained binary64: `0.38095238095238093`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-64/441Exact fraction: `-64/441`Retained binary64: `-0.14512471655328799`	false	true	false
41	169/64Exact fraction: `169/64`Retained binary64: `2.640625`	1Exact fraction: `1`Retained binary64: `1.0`	64/169Exact fraction: `64/169`Retained binary64: `0.378698224852071`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/28561Exact fraction: `-4096/28561`Retained binary64: `-0.14341234550610973`	false	true	false
42	85/32Exact fraction: `85/32`Retained binary64: `2.65625`	1Exact fraction: `1`Retained binary64: `1.0`	32/85Exact fraction: `32/85`Retained binary64: `0.3764705882352941`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/7225Exact fraction: `-1024/7225`Retained binary64: `-0.14173010380622839`	false	true	false
43	171/64Exact fraction: `171/64`Retained binary64: `2.671875`	1Exact fraction: `1`Retained binary64: `1.0`	64/171Exact fraction: `64/171`Retained binary64: `0.3742690058479532`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/29241Exact fraction: `-4096/29241`Retained binary64: `-0.14007728873841524`	false	true	false
44	43/16Exact fraction: `43/16`Retained binary64: `2.6875`	1Exact fraction: `1`Retained binary64: `1.0`	16/43Exact fraction: `16/43`Retained binary64: `0.37209302325581395`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/1849Exact fraction: `-256/1849`Retained binary64: `-0.13845321795565171`	false	true	false
45	173/64Exact fraction: `173/64`Retained binary64: `2.703125`	1Exact fraction: `1`Retained binary64: `1.0`	64/173Exact fraction: `64/173`Retained binary64: `0.3699421965317919`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/29929Exact fraction: `-4096/29929`Retained binary64: `-0.13685722877476694`	false	true	false
46	87/32Exact fraction: `87/32`Retained binary64: `2.71875`	1Exact fraction: `1`Retained binary64: `1.0`	32/87Exact fraction: `32/87`Retained binary64: `0.367816091954023`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/7569Exact fraction: `-1024/7569`Retained binary64: `-0.1352886775003303`	false	true	false
47	175/64Exact fraction: `175/64`Retained binary64: `2.734375`	1Exact fraction: `1`Retained binary64: `1.0`	64/175Exact fraction: `64/175`Retained binary64: `0.3657142857142857`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/30625Exact fraction: `-4096/30625`Retained binary64: `-0.1337469387755102`	false	true	false
48	11/4Exact fraction: `11/4`Retained binary64: `2.75`	1Exact fraction: `1`Retained binary64: `1.0`	4/11Exact fraction: `4/11`Retained binary64: `0.36363636363636365`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-16/121Exact fraction: `-16/121`Retained binary64: `-0.1322314049586777`	false	true	false
49	177/64Exact fraction: `177/64`Retained binary64: `2.765625`	1Exact fraction: `1`Retained binary64: `1.0`	64/177Exact fraction: `64/177`Retained binary64: `0.3615819209039548`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/31329Exact fraction: `-4096/31329`Retained binary64: `-0.13074148552459383`	false	true	false
50	89/32Exact fraction: `89/32`Retained binary64: `2.78125`	1Exact fraction: `1`Retained binary64: `1.0`	32/89Exact fraction: `32/89`Retained binary64: `0.3595505617977528`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/7921Exact fraction: `-1024/7921`Retained binary64: `-0.12927660648907965`	false	true	false
51	179/64Exact fraction: `179/64`Retained binary64: `2.796875`	1Exact fraction: `1`Retained binary64: `1.0`	64/179Exact fraction: `64/179`Retained binary64: `0.3575418994413408`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/32041Exact fraction: `-4096/32041`Retained binary64: `-0.12783620985612185`	false	true	false
52	45/16Exact fraction: `45/16`Retained binary64: `2.8125`	1Exact fraction: `1`Retained binary64: `1.0`	16/45Exact fraction: `16/45`Retained binary64: `0.35555555555555557`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/2025Exact fraction: `-256/2025`Retained binary64: `-0.12641975308641976`	false	true	false
53	181/64Exact fraction: `181/64`Retained binary64: `2.828125`	1Exact fraction: `1`Retained binary64: `1.0`	64/181Exact fraction: `64/181`Retained binary64: `0.35359116022099446`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/32761Exact fraction: `-4096/32761`Retained binary64: `-0.12502670858642898`	false	true	false
54	91/32Exact fraction: `91/32`Retained binary64: `2.84375`	1Exact fraction: `1`Retained binary64: `1.0`	32/91Exact fraction: `32/91`Retained binary64: `0.3516483516483517`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/8281Exact fraction: `-1024/8281`Retained binary64: `-0.12365656321700277`	false	true	false
55	183/64Exact fraction: `183/64`Retained binary64: `2.859375`	1Exact fraction: `1`Retained binary64: `1.0`	64/183Exact fraction: `64/183`Retained binary64: `0.34972677595628415`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/33489Exact fraction: `-4096/33489`Retained binary64: `-0.12230881782077697`	false	true	false
56	23/8Exact fraction: `23/8`Retained binary64: `2.875`	1Exact fraction: `1`Retained binary64: `1.0`	8/23Exact fraction: `8/23`Retained binary64: `0.34782608695652173`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-64/529Exact fraction: `-64/529`Retained binary64: `-0.12098298676748583`	false	true	false
57	185/64Exact fraction: `185/64`Retained binary64: `2.890625`	1Exact fraction: `1`Retained binary64: `1.0`	64/185Exact fraction: `64/185`Retained binary64: `0.34594594594594597`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/34225Exact fraction: `-4096/34225`Retained binary64: `-0.11967859751643535`	false	true	false
58	93/32Exact fraction: `93/32`Retained binary64: `2.90625`	1Exact fraction: `1`Retained binary64: `1.0`	32/93Exact fraction: `32/93`Retained binary64: `0.34408602150537637`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/8649Exact fraction: `-1024/8649`Retained binary64: `-0.1183951901953983`	false	true	false
59	187/64Exact fraction: `187/64`Retained binary64: `2.921875`	1Exact fraction: `1`Retained binary64: `1.0`	64/187Exact fraction: `64/187`Retained binary64: `0.3422459893048128`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/34969Exact fraction: `-4096/34969`Retained binary64: `-0.11713231719523005`	false	true	false
60	47/16Exact fraction: `47/16`Retained binary64: `2.9375`	1Exact fraction: `1`Retained binary64: `1.0`	16/47Exact fraction: `16/47`Retained binary64: `0.3404255319148936`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-256/2209Exact fraction: `-256/2209`Retained binary64: `-0.11588954277953825`	false	true	false
61	189/64Exact fraction: `189/64`Retained binary64: `2.953125`	1Exact fraction: `1`Retained binary64: `1.0`	64/189Exact fraction: `64/189`Retained binary64: `0.3386243386243386`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/35721Exact fraction: `-4096/35721`Retained binary64: `-0.11466644270877076`	false	true	false
62	95/32Exact fraction: `95/32`Retained binary64: `2.96875`	1Exact fraction: `1`Retained binary64: `1.0`	32/95Exact fraction: `32/95`Retained binary64: `0.3368421052631579`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1024/9025Exact fraction: `-1024/9025`Retained binary64: `-0.11346260387811634`	false	true	false
63	191/64Exact fraction: `191/64`Retained binary64: `2.984375`	1Exact fraction: `1`Retained binary64: `1.0`	64/191Exact fraction: `64/191`Retained binary64: `0.33507853403141363`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-4096/36481Exact fraction: `-4096/36481`Retained binary64: `-0.11227762396864122`	false	true	false
64Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/9Exact fraction: `-1/9`Retained binary64: `-0.1111111111111111`	true	true	false

Complete return coax branch (129 rows) (129 rows)

Complete return coax branch (129 rows)
Row	R=r/L₀	Signed enclosed I/I₀	Bφ/B₀	Circulation/(μ₀I₀)	Branch J_z/[I₀/(πL₀²)]; one-sided at interfaces	Branch db/dR; one-sided at interfaces	J/slope are interface one-sided limits only?	Azimuthal direction defined?	Cartesian field zero at center?
0Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1Exact fraction: `1`Retained binary64: `1.0`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-17/72Exact fraction: `-17/72`Retained binary64: `-0.2361111111111111`	true	true	false
1	193/64Exact fraction: `193/64`Retained binary64: `3.015625`	65151/65536Exact fraction: `65151/65536`Retained binary64: `0.9941253662109375`	65151/197632Exact fraction: `65151/197632`Retained binary64: `0.32965815252590674`	65151/65536Exact fraction: `65151/65536`Retained binary64: `0.9941253662109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-139649/595984Exact fraction: `-139649/595984`Retained binary64: `-0.23431669306558564`	false	true	false
2	97/32Exact fraction: `97/32`Retained binary64: `3.03125`	16191/16384Exact fraction: `16191/16384`Retained binary64: `0.98822021484375`	16191/49664Exact fraction: `16191/49664`Retained binary64: `0.3260107925257732`	16191/16384Exact fraction: `16191/16384`Retained binary64: `0.98822021484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-35009/150544Exact fraction: `-35009/150544`Retained binary64: `-0.23254995217345095`	false	true	false
3	195/64Exact fraction: `195/64`Retained binary64: `3.046875`	64375/65536Exact fraction: `64375/65536`Retained binary64: `0.9822845458984375`	12875/39936Exact fraction: `12875/39936`Retained binary64: `0.32239082532051283`	64375/65536Exact fraction: `64375/65536`Retained binary64: `0.9822845458984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-5617/24336Exact fraction: `-5617/24336`Retained binary64: `-0.230810322156476`	false	true	false
4	49/16Exact fraction: `49/16`Retained binary64: `3.0625`	3999/4096Exact fraction: `3999/4096`Retained binary64: `0.976318359375`	3999/12544Exact fraction: `3999/12544`Retained binary64: `0.3187978316326531`	3999/4096Exact fraction: `3999/4096`Retained binary64: `0.976318359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-8801/38416Exact fraction: `-8801/38416`Retained binary64: `-0.2290972511453561`	false	true	false
5	197/64Exact fraction: `197/64`Retained binary64: `3.078125`	63591/65536Exact fraction: `63591/65536`Retained binary64: `0.9703216552734375`	63591/201728Exact fraction: `63591/201728`Retained binary64: `0.31523140069796957`	63591/65536Exact fraction: `63591/65536`Retained binary64: `0.9703216552734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-141209/620944Exact fraction: `-141209/620944`Retained binary64: `-0.22741020124197994`	false	true	false
6	99/32Exact fraction: `99/32`Retained binary64: `3.09375`	15799/16384Exact fraction: `15799/16384`Retained binary64: `0.96429443359375`	15799/50688Exact fraction: `15799/50688`Retained binary64: `0.311691130050505`	15799/16384Exact fraction: `15799/16384`Retained binary64: `0.96429443359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-35401/156816Exact fraction: `-35401/156816`Retained binary64: `-0.22574864809713294`	false	true	false
7	199/64Exact fraction: `199/64`Retained binary64: `3.109375`	62799/65536Exact fraction: `62799/65536`Retained binary64: `0.9582366943359375`	62799/203776Exact fraction: `62799/203776`Retained binary64: `0.30817662531407036`	62799/65536Exact fraction: `62799/65536`Retained binary64: `0.9582366943359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-142001/633616Exact fraction: `-142001/633616`Retained binary64: `-0.2241120805030176`	false	true	false
8	25/8Exact fraction: `25/8`Retained binary64: `3.125`	975/1024Exact fraction: `975/1024`Retained binary64: `0.9521484375`	39/128Exact fraction: `39/128`Retained binary64: `0.3046875`	975/1024Exact fraction: `975/1024`Retained binary64: `0.9521484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-89/400Exact fraction: `-89/400`Retained binary64: `-0.2225`	false	true	false
9	201/64Exact fraction: `201/64`Retained binary64: `3.140625`	61999/65536Exact fraction: `61999/65536`Retained binary64: `0.9460296630859375`	61999/205824Exact fraction: `61999/205824`Retained binary64: `0.3012233753109453`	61999/65536Exact fraction: `61999/65536`Retained binary64: `0.9460296630859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-142801/646416Exact fraction: `-142801/646416`Retained binary64: `-0.2209119204970174`	false	true	false
10	101/32Exact fraction: `101/32`Retained binary64: `3.15625`	15399/16384Exact fraction: `15399/16384`Retained binary64: `0.93988037109375`	15399/51712Exact fraction: `15399/51712`Retained binary64: `0.29778387995049505`	15399/16384Exact fraction: `15399/16384`Retained binary64: `0.93988037109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-35801/163216Exact fraction: `-35801/163216`Retained binary64: `-0.21934736790510734`	false	true	false
11	203/64Exact fraction: `203/64`Retained binary64: `3.171875`	61191/65536Exact fraction: `61191/65536`Retained binary64: `0.9337005615234375`	61191/207872Exact fraction: `61191/207872`Retained binary64: `0.29436864993842365`	61191/65536Exact fraction: `61191/65536`Retained binary64: `0.9337005615234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-143609/659344Exact fraction: `-143609/659344`Retained binary64: `-0.21780587978354243`	false	true	false
12	51/16Exact fraction: `51/16`Retained binary64: `3.1875`	3799/4096Exact fraction: `3799/4096`Retained binary64: `0.927490234375`	3799/13056Exact fraction: `3799/13056`Retained binary64: `0.29097732843137253`	3799/4096Exact fraction: `3799/4096`Retained binary64: `0.927490234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-9001/41616Exact fraction: `-9001/41616`Retained binary64: `-0.21628700499807765`	false	true	false
13	205/64Exact fraction: `205/64`Retained binary64: `3.203125`	60375/65536Exact fraction: `60375/65536`Retained binary64: `0.9212493896484375`	12075/41984Exact fraction: `12075/41984`Retained binary64: `0.2876095655487805`	60375/65536Exact fraction: `60375/65536`Retained binary64: `0.9212493896484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-5777/26896Exact fraction: `-5777/26896`Retained binary64: `-0.2147903033908388`	false	true	false
14	103/32Exact fraction: `103/32`Retained binary64: `3.21875`	14991/16384Exact fraction: `14991/16384`Retained binary64: `0.91497802734375`	14991/52736Exact fraction: `14991/52736`Retained binary64: `0.2842650182038835`	14991/16384Exact fraction: `14991/16384`Retained binary64: `0.91497802734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-36209/169744Exact fraction: `-36209/169744`Retained binary64: `-0.21331534546140068`	false	true	false
15	207/64Exact fraction: `207/64`Retained binary64: `3.234375`	59551/65536Exact fraction: `59551/65536`Retained binary64: `0.9086761474609375`	59551/211968Exact fraction: `59551/211968`Retained binary64: `0.2809433499396135`	59551/65536Exact fraction: `59551/65536`Retained binary64: `0.9086761474609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-145249/685584Exact fraction: `-145249/685584`Retained binary64: `-0.21186171205862447`	false	true	false
16	13/4Exact fraction: `13/4`Retained binary64: `3.25`	231/256Exact fraction: `231/256`Retained binary64: `0.90234375`	231/832Exact fraction: `231/832`Retained binary64: `0.2776442307692308`	231/256Exact fraction: `231/256`Retained binary64: `0.90234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-569/2704Exact fraction: `-569/2704`Retained binary64: `-0.21042899408284024`	false	true	false
17	209/64Exact fraction: `209/64`Retained binary64: `3.265625`	58719/65536Exact fraction: `58719/65536`Retained binary64: `0.8959808349609375`	58719/214016Exact fraction: `58719/214016`Retained binary64: `0.2743673370215311`	58719/65536Exact fraction: `58719/65536`Retained binary64: `0.8959808349609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-146081/698896Exact fraction: `-146081/698896`Retained binary64: `-0.2090167921979808`	false	true	false
18	105/32Exact fraction: `105/32`Retained binary64: `3.28125`	14575/16384Exact fraction: `14575/16384`Retained binary64: `0.88958740234375`	2915/10752Exact fraction: `2915/10752`Retained binary64: `0.27111235119047616`	14575/16384Exact fraction: `14575/16384`Retained binary64: `0.88958740234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1465/7056Exact fraction: `-1465/7056`Retained binary64: `-0.20762471655328799`	false	true	false
19	211/64Exact fraction: `211/64`Retained binary64: `3.296875`	57879/65536Exact fraction: `57879/65536`Retained binary64: `0.8831634521484375`	57879/216064Exact fraction: `57879/216064`Retained binary64: `0.26787896178909953`	57879/65536Exact fraction: `57879/65536`Retained binary64: `0.8831634521484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-146921/712336Exact fraction: `-146921/712336`Retained binary64: `-0.20625238651422925`	false	true	false
20	53/16Exact fraction: `53/16`Retained binary64: `3.3125`	3591/4096Exact fraction: `3591/4096`Retained binary64: `0.876708984375`	3591/13568Exact fraction: `3591/13568`Retained binary64: `0.2646668632075472`	3591/4096Exact fraction: `3591/4096`Retained binary64: `0.876708984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-9209/44944Exact fraction: `-9209/44944`Retained binary64: `-0.2048994304022784`	false	true	false
21	213/64Exact fraction: `213/64`Retained binary64: `3.328125`	57031/65536Exact fraction: `57031/65536`Retained binary64: `0.8702239990234375`	57031/218112Exact fraction: `57031/218112`Retained binary64: `0.2614757555751174`	57031/65536Exact fraction: `57031/65536`Retained binary64: `0.8702239990234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-147769/725904Exact fraction: `-147769/725904`Retained binary64: `-0.20356548524322776`	false	true	false
22	107/32Exact fraction: `107/32`Retained binary64: `3.34375`	14151/16384Exact fraction: `14151/16384`Retained binary64: `0.86370849609375`	14151/54784Exact fraction: `14151/54784`Retained binary64: `0.2583053446261682`	14151/16384Exact fraction: `14151/16384`Retained binary64: `0.86370849609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-37049/183184Exact fraction: `-37049/183184`Retained binary64: `-0.20225019652371387`	false	true	false
23	215/64Exact fraction: `215/64`Retained binary64: `3.359375`	56175/65536Exact fraction: `56175/65536`Retained binary64: `0.8571624755859375`	11235/44032Exact fraction: `11235/44032`Retained binary64: `0.25515534156976744`	56175/65536Exact fraction: `56175/65536`Retained binary64: `0.8571624755859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-5945/29584Exact fraction: `-5945/29584`Retained binary64: `-0.20095321795565171`	false	true	false
24	27/8Exact fraction: `27/8`Retained binary64: `3.375`	871/1024Exact fraction: `871/1024`Retained binary64: `0.8505859375`	871/3456Exact fraction: `871/3456`Retained binary64: `0.25202546296296297`	871/1024Exact fraction: `871/1024`Retained binary64: `0.8505859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-2329/11664Exact fraction: `-2329/11664`Retained binary64: `-0.19967421124828533`	false	true	false
25	217/64Exact fraction: `217/64`Retained binary64: `3.390625`	55311/65536Exact fraction: `55311/65536`Retained binary64: `0.8439788818359375`	55311/222208Exact fraction: `55311/222208`Retained binary64: `0.2489154305875576`	55311/65536Exact fraction: `55311/65536`Retained binary64: `0.8439788818359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-149489/753424Exact fraction: `-149489/753424`Retained binary64: `-0.1984128458875746`	false	true	false
26	109/32Exact fraction: `109/32`Retained binary64: `3.40625`	13719/16384Exact fraction: `13719/16384`Retained binary64: `0.83734130859375`	13719/55808Exact fraction: `13719/55808`Retained binary64: `0.24582497133027523`	13719/16384Exact fraction: `13719/16384`Retained binary64: `0.83734130859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-37481/190096Exact fraction: `-37481/190096`Retained binary64: `-0.1971687989226496`	false	true	false
27	219/64Exact fraction: `219/64`Retained binary64: `3.421875`	54439/65536Exact fraction: `54439/65536`Retained binary64: `0.8306732177734375`	54439/224256Exact fraction: `54439/224256`Retained binary64: `0.24275381706621005`	54439/65536Exact fraction: `54439/65536`Retained binary64: `0.8306732177734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-150361/767376Exact fraction: `-150361/767376`Retained binary64: `-0.19594175475907508`	false	true	false
28	55/16Exact fraction: `55/16`Retained binary64: `3.4375`	3375/4096Exact fraction: `3375/4096`Retained binary64: `0.823974609375`	675/2816Exact fraction: `675/2816`Retained binary64: `0.23970170454545456`	3375/4096Exact fraction: `3375/4096`Retained binary64: `0.823974609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-377/1936Exact fraction: `-377/1936`Retained binary64: `-0.1947314049586777`	false	true	false
29	221/64Exact fraction: `221/64`Retained binary64: `3.453125`	53559/65536Exact fraction: `53559/65536`Retained binary64: `0.8172454833984375`	53559/226304Exact fraction: `53559/226304`Retained binary64: `0.23666837528280543`	53559/65536Exact fraction: `53559/65536`Retained binary64: `0.8172454833984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-151241/781456Exact fraction: `-151241/781456`Retained binary64: `-0.1935374480456993`	false	true	false
30	111/32Exact fraction: `111/32`Retained binary64: `3.46875`	13279/16384Exact fraction: `13279/16384`Retained binary64: `0.81048583984375`	13279/56832Exact fraction: `13279/56832`Retained binary64: `0.23365357545045046`	13279/16384Exact fraction: `13279/16384`Retained binary64: `0.81048583984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-37921/197136Exact fraction: `-37921/197136`Retained binary64: `-0.19235958931904878`	false	true	false
31	223/64Exact fraction: `223/64`Retained binary64: `3.484375`	52671/65536Exact fraction: `52671/65536`Retained binary64: `0.8036956787109375`	52671/228352Exact fraction: `52671/228352`Retained binary64: `0.2306570557735426`	52671/65536Exact fraction: `52671/65536`Retained binary64: `0.8036956787109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-152129/795664Exact fraction: `-152129/795664`Retained binary64: `-0.19119754067043376`	false	true	false
32	7/2Exact fraction: `7/2`Retained binary64: `3.5`	51/64Exact fraction: `51/64`Retained binary64: `0.796875`	51/224Exact fraction: `51/224`Retained binary64: `0.22767857142857142`	51/64Exact fraction: `51/64`Retained binary64: `0.796875`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-149/784Exact fraction: `-149/784`Retained binary64: `-0.19005102040816327`	false	true	false
33	225/64Exact fraction: `225/64`Retained binary64: `3.515625`	51775/65536Exact fraction: `51775/65536`Retained binary64: `0.7900238037109375`	2071/9216Exact fraction: `2071/9216`Retained binary64: `0.22471788194444445`	51775/65536Exact fraction: `51775/65536`Retained binary64: `0.7900238037109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-6121/32400Exact fraction: `-6121/32400`Retained binary64: `-0.18891975308641976`	false	true	false
34	113/32Exact fraction: `113/32`Retained binary64: `3.53125`	12831/16384Exact fraction: `12831/16384`Retained binary64: `0.78314208984375`	12831/57856Exact fraction: `12831/57856`Retained binary64: `0.22177475110619468`	12831/16384Exact fraction: `12831/16384`Retained binary64: `0.78314208984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-38369/204304Exact fraction: `-38369/204304`Retained binary64: `-0.18780346933980735`	false	true	false
35	227/64Exact fraction: `227/64`Retained binary64: `3.546875`	50871/65536Exact fraction: `50871/65536`Retained binary64: `0.7762298583984375`	50871/232448Exact fraction: `50871/232448`Retained binary64: `0.2188489468612335`	50871/65536Exact fraction: `50871/65536`Retained binary64: `0.7762298583984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-153929/824464Exact fraction: `-153929/824464`Retained binary64: `-0.18670190572299095`	false	true	false
36	57/16Exact fraction: `57/16`Retained binary64: `3.5625`	3151/4096Exact fraction: `3151/4096`Retained binary64: `0.769287109375`	3151/14592Exact fraction: `3151/14592`Retained binary64: `0.21594024122807018`	3151/4096Exact fraction: `3151/4096`Retained binary64: `0.769287109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-9649/51984Exact fraction: `-9649/51984`Retained binary64: `-0.18561480455524776`	false	true	false
37	229/64Exact fraction: `229/64`Retained binary64: `3.578125`	49959/65536Exact fraction: `49959/65536`Retained binary64: `0.7623138427734375`	49959/234496Exact fraction: `49959/234496`Retained binary64: `0.21304841020742357`	49959/65536Exact fraction: `49959/65536`Retained binary64: `0.7623138427734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-154841/839056Exact fraction: `-154841/839056`Retained binary64: `-0.1845419137697603`	false	true	false
38	115/32Exact fraction: `115/32`Retained binary64: `3.59375`	12375/16384Exact fraction: `12375/16384`Retained binary64: `0.75531005859375`	2475/11776Exact fraction: `2475/11776`Retained binary64: `0.21017323369565216`	12375/16384Exact fraction: `12375/16384`Retained binary64: `0.75531005859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1553/8464Exact fraction: `-1553/8464`Retained binary64: `-0.1834829867674858`	false	true	false
39	231/64Exact fraction: `231/64`Retained binary64: `3.609375`	49039/65536Exact fraction: `49039/65536`Retained binary64: `0.7482757568359375`	49039/236544Exact fraction: `49039/236544`Retained binary64: `0.2073144954004329`	49039/65536Exact fraction: `49039/65536`Retained binary64: `0.7482757568359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-155761/853776Exact fraction: `-155761/853776`Retained binary64: `-0.1824377822754446`	false	true	false
40	29/8Exact fraction: `29/8`Retained binary64: `3.625`	759/1024Exact fraction: `759/1024`Retained binary64: `0.7412109375`	759/3712Exact fraction: `759/3712`Retained binary64: `0.2044719827586207`	759/1024Exact fraction: `759/1024`Retained binary64: `0.7412109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-2441/13456Exact fraction: `-2441/13456`Retained binary64: `-0.1814060642092747`	false	true	false
41	233/64Exact fraction: `233/64`Retained binary64: `3.640625`	48111/65536Exact fraction: `48111/65536`Retained binary64: `0.7341156005859375`	48111/238592Exact fraction: `48111/238592`Retained binary64: `0.2016454868562232`	48111/65536Exact fraction: `48111/65536`Retained binary64: `0.7341156005859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-156689/868624Exact fraction: `-156689/868624`Retained binary64: `-0.1803876015399068`	false	true	false
42	117/32Exact fraction: `117/32`Retained binary64: `3.65625`	11911/16384Exact fraction: `11911/16384`Retained binary64: `0.72698974609375`	11911/59904Exact fraction: `11911/59904`Retained binary64: `0.19883480235042736`	11911/16384Exact fraction: `11911/16384`Retained binary64: `0.72698974609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-39289/219024Exact fraction: `-39289/219024`Retained binary64: `-0.17938216816421945`	false	true	false
43	235/64Exact fraction: `235/64`Retained binary64: `3.671875`	47175/65536Exact fraction: `47175/65536`Retained binary64: `0.7198333740234375`	9435/48128Exact fraction: `9435/48128`Retained binary64: `0.19603972739361702`	47175/65536Exact fraction: `47175/65536`Retained binary64: `0.7198333740234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-6305/35344Exact fraction: `-6305/35344`Retained binary64: `-0.17838954277953825`	false	true	false
44	59/16Exact fraction: `59/16`Retained binary64: `3.6875`	2919/4096Exact fraction: `2919/4096`Retained binary64: `0.712646484375`	2919/15104Exact fraction: `2919/15104`Retained binary64: `0.19326006355932204`	2919/4096Exact fraction: `2919/4096`Retained binary64: `0.712646484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-9881/55696Exact fraction: `-9881/55696`Retained binary64: `-0.17740950876185005`	false	true	false
45	237/64Exact fraction: `237/64`Retained binary64: `3.703125`	46231/65536Exact fraction: `46231/65536`Retained binary64: `0.7054290771484375`	46231/242688Exact fraction: `46231/242688`Retained binary64: `0.1904956157700422`	46231/65536Exact fraction: `46231/65536`Retained binary64: `0.7054290771484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-158569/898704Exact fraction: `-158569/898704`Retained binary64: `-0.17644185404760634`	false	true	false
46	119/32Exact fraction: `119/32`Retained binary64: `3.71875`	11439/16384Exact fraction: `11439/16384`Retained binary64: `0.69818115234375`	11439/60928Exact fraction: `11439/60928`Retained binary64: `0.18774619222689076`	11439/16384Exact fraction: `11439/16384`Retained binary64: `0.69818115234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-39761/226576Exact fraction: `-39761/226576`Retained binary64: `-0.17548637101899583`	false	true	false
47	239/64Exact fraction: `239/64`Retained binary64: `3.734375`	45279/65536Exact fraction: `45279/65536`Retained binary64: `0.6909027099609375`	45279/244736Exact fraction: `45279/244736`Retained binary64: `0.1850116043410042`	45279/65536Exact fraction: `45279/65536`Retained binary64: `0.6909027099609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-159521/913936Exact fraction: `-159521/913936`Retained binary64: `-0.17454285639257017`	false	true	false
48	15/4Exact fraction: `15/4`Retained binary64: `3.75`	175/256Exact fraction: `175/256`Retained binary64: `0.68359375`	35/192Exact fraction: `35/192`Retained binary64: `0.18229166666666666`	175/256Exact fraction: `175/256`Retained binary64: `0.68359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-25/144Exact fraction: `-25/144`Retained binary64: `-0.1736111111111111`	false	true	false
49	241/64Exact fraction: `241/64`Retained binary64: `3.765625`	44319/65536Exact fraction: `44319/65536`Retained binary64: `0.6762542724609375`	44319/246784Exact fraction: `44319/246784`Retained binary64: `0.17958619683609958`	44319/65536Exact fraction: `44319/65536`Retained binary64: `0.6762542724609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-160481/929296Exact fraction: `-160481/929296`Retained binary64: `-0.17269094023863227`	false	true	false
50	121/32Exact fraction: `121/32`Retained binary64: `3.78125`	10959/16384Exact fraction: `10959/16384`Retained binary64: `0.66888427734375`	10959/61952Exact fraction: `10959/61952`Retained binary64: `0.17689501549586778`	10959/16384Exact fraction: `10959/16384`Retained binary64: `0.66888427734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-40241/234256Exact fraction: `-40241/234256`Retained binary64: `-0.1717821528584113`	false	true	false
51	243/64Exact fraction: `243/64`Retained binary64: `3.796875`	43351/65536Exact fraction: `43351/65536`Retained binary64: `0.6614837646484375`	43351/248832Exact fraction: `43351/248832`Retained binary64: `0.17421794624485595`	43351/65536Exact fraction: `43351/65536`Retained binary64: `0.6614837646484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-161449/944784Exact fraction: `-161449/944784`Retained binary64: `-0.17088456197395382`	false	true	false
52	61/16Exact fraction: `61/16`Retained binary64: `3.8125`	2679/4096Exact fraction: `2679/4096`Retained binary64: `0.654052734375`	2679/15616Exact fraction: `2679/15616`Retained binary64: `0.1715548155737705`	2679/4096Exact fraction: `2679/4096`Retained binary64: `0.654052734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-10121/59536Exact fraction: `-10121/59536`Retained binary64: `-0.16999798441279226`	false	true	false
53	245/64Exact fraction: `245/64`Retained binary64: `3.828125`	42375/65536Exact fraction: `42375/65536`Retained binary64: `0.6465911865234375`	8475/50176Exact fraction: `8475/50176`Retained binary64: `0.16890545280612246`	42375/65536Exact fraction: `42375/65536`Retained binary64: `0.6465911865234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-6497/38416Exact fraction: `-6497/38416`Retained binary64: `-0.1691222407330279`	false	true	false
54	123/32Exact fraction: `123/32`Retained binary64: `3.84375`	10471/16384Exact fraction: `10471/16384`Retained binary64: `0.63909912109375`	10471/62976Exact fraction: `10471/62976`Retained binary64: `0.1662696900406504`	10471/16384Exact fraction: `10471/16384`Retained binary64: `0.63909912109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-40729/242064Exact fraction: `-40729/242064`Retained binary64: `-0.16825715513252693`	false	true	false
55	247/64Exact fraction: `247/64`Retained binary64: `3.859375`	41391/65536Exact fraction: `41391/65536`Retained binary64: `0.6315765380859375`	41391/252928Exact fraction: `41391/252928`Retained binary64: `0.1636473620951417`	41391/65536Exact fraction: `41391/65536`Retained binary64: `0.6315765380859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-163409/976144Exact fraction: `-163409/976144`Retained binary64: `-0.1674025553606845`	false	true	false
56	31/8Exact fraction: `31/8`Retained binary64: `3.875`	639/1024Exact fraction: `639/1024`Retained binary64: `0.6240234375`	639/3968Exact fraction: `639/3968`Retained binary64: `0.1610383064516129`	639/1024Exact fraction: `639/1024`Retained binary64: `0.6240234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-2561/15376Exact fraction: `-2561/15376`Retained binary64: `-0.1665582726326743`	false	true	false
57	249/64Exact fraction: `249/64`Retained binary64: `3.890625`	40399/65536Exact fraction: `40399/65536`Retained binary64: `0.6164398193359375`	40399/254976Exact fraction: `40399/254976`Retained binary64: `0.15844236320281124`	40399/65536Exact fraction: `40399/65536`Retained binary64: `0.6164398193359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-164401/992016Exact fraction: `-164401/992016`Retained binary64: `-0.1657241415461041`	false	true	false
58	125/32Exact fraction: `125/32`Retained binary64: `3.90625`	9975/16384Exact fraction: `9975/16384`Retained binary64: `0.60882568359375`	399/2560Exact fraction: `399/2560`Retained binary64: `0.155859375`	9975/16384Exact fraction: `9975/16384`Retained binary64: `0.60882568359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1649/10000Exact fraction: `-1649/10000`Retained binary64: `-0.1649`	false	true	false
59	251/64Exact fraction: `251/64`Retained binary64: `3.921875`	39399/65536Exact fraction: `39399/65536`Retained binary64: `0.6011810302734375`	39399/257024Exact fraction: `39399/257024`Retained binary64: `0.15328918700199204`	39399/65536Exact fraction: `39399/65536`Retained binary64: `0.6011810302734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-165401/1008016Exact fraction: `-165401/1008016`Retained binary64: `-0.1640856891160458`	false	true	false
60	63/16Exact fraction: `63/16`Retained binary64: `3.9375`	2431/4096Exact fraction: `2431/4096`Retained binary64: `0.593505859375`	2431/16128Exact fraction: `2431/16128`Retained binary64: `0.15073164682539683`	2431/4096Exact fraction: `2431/4096`Retained binary64: `0.593505859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-10369/63504Exact fraction: `-10369/63504`Retained binary64: `-0.16328105316200553`	false	true	false
61	253/64Exact fraction: `253/64`Retained binary64: `3.953125`	38391/65536Exact fraction: `38391/65536`Retained binary64: `0.5858001708984375`	38391/259072Exact fraction: `38391/259072`Retained binary64: `0.14818660449604742`	38391/65536Exact fraction: `38391/65536`Retained binary64: `0.5858001708984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-166409/1024144Exact fraction: `-166409/1024144`Retained binary64: `-0.162485939477261`	false	true	false
62	127/32Exact fraction: `127/32`Retained binary64: `3.96875`	9471/16384Exact fraction: `9471/16384`Retained binary64: `0.57806396484375`	9471/65024Exact fraction: `9471/65024`Retained binary64: `0.1456539124015748`	9471/16384Exact fraction: `9471/16384`Retained binary64: `0.57806396484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-41729/258064Exact fraction: `-41729/258064`Retained binary64: `-0.1617001984003968`	false	true	false
63	255/64Exact fraction: `255/64`Retained binary64: `3.984375`	37375/65536Exact fraction: `37375/65536`Retained binary64: `0.5702972412109375`	7475/52224Exact fraction: `7475/52224`Retained binary64: `0.14313342524509803`	37375/65536Exact fraction: `37375/65536`Retained binary64: `0.5702972412109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-6697/41616Exact fraction: `-6697/41616`Retained binary64: `-0.1609236831987697`	false	true	false
64	4Exact fraction: `4`Retained binary64: `4.0`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-41/256Exact fraction: `-41/256`Retained binary64: `-0.16015625`	false	true	false
65	257/64Exact fraction: `257/64`Retained binary64: `4.015625`	36351/65536Exact fraction: `36351/65536`Retained binary64: `0.5546722412109375`	36351/263168Exact fraction: `36351/263168`Retained binary64: `0.13812849586575876`	36351/65536Exact fraction: `36351/65536`Retained binary64: `0.5546722412109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-168449/1056784Exact fraction: `-168449/1056784`Retained binary64: `-0.15939775772532513`	false	true	false
66	129/32Exact fraction: `129/32`Retained binary64: `4.03125`	8959/16384Exact fraction: `8959/16384`Retained binary64: `0.54681396484375`	8959/66048Exact fraction: `8959/66048`Retained binary64: `0.1356437742248062`	8959/16384Exact fraction: `8959/16384`Retained binary64: `0.54681396484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-42241/266256Exact fraction: `-42241/266256`Retained binary64: `-0.15864806802475812`	false	true	false
67	259/64Exact fraction: `259/64`Retained binary64: `4.046875`	35319/65536Exact fraction: `35319/65536`Retained binary64: `0.5389251708984375`	35319/265216Exact fraction: `35319/265216`Retained binary64: `0.1331706986003861`	35319/65536Exact fraction: `35319/65536`Retained binary64: `0.5389251708984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-169481/1073296Exact fraction: `-169481/1073296`Retained binary64: `-0.157907045213995`	false	true	false
68	65/16Exact fraction: `65/16`Retained binary64: `4.0625`	2175/4096Exact fraction: `2175/4096`Retained binary64: `0.531005859375`	435/3328Exact fraction: `435/3328`Retained binary64: `0.1307091346153846`	2175/4096Exact fraction: `2175/4096`Retained binary64: `0.531005859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-425/2704Exact fraction: `-425/2704`Retained binary64: `-0.15717455621301776`	false	true	false
69	261/64Exact fraction: `261/64`Retained binary64: `4.078125`	34279/65536Exact fraction: `34279/65536`Retained binary64: `0.5230560302734375`	34279/267264Exact fraction: `34279/267264`Retained binary64: `0.1282589499521073`	34279/65536Exact fraction: `34279/65536`Retained binary64: `0.5230560302734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-170521/1089936Exact fraction: `-170521/1089936`Retained binary64: `-0.15645047048634048`	false	true	false
70	131/32Exact fraction: `131/32`Retained binary64: `4.09375`	8439/16384Exact fraction: `8439/16384`Retained binary64: `0.51507568359375`	8439/67072Exact fraction: `8439/67072`Retained binary64: `0.1258200143129771`	8439/16384Exact fraction: `8439/16384`Retained binary64: `0.51507568359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-42761/274576Exact fraction: `-42761/274576`Retained binary64: `-0.15573465998484937`	false	true	false
71	263/64Exact fraction: `263/64`Retained binary64: `4.109375`	33231/65536Exact fraction: `33231/65536`Retained binary64: `0.5070648193359375`	33231/269312Exact fraction: `33231/269312`Retained binary64: `0.12339219938212928`	33231/65536Exact fraction: `33231/65536`Retained binary64: `0.5070648193359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-171569/1106704Exact fraction: `-171569/1106704`Retained binary64: `-0.15502699908918735`	false	true	false
72	33/8Exact fraction: `33/8`Retained binary64: `4.125`	511/1024Exact fraction: `511/1024`Retained binary64: `0.4990234375`	511/4224Exact fraction: `511/4224`Retained binary64: `0.12097537878787878`	511/1024Exact fraction: `511/1024`Retained binary64: `0.4990234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-2689/17424Exact fraction: `-2689/17424`Retained binary64: `-0.15432736455463728`	false	true	false
73	265/64Exact fraction: `265/64`Retained binary64: `4.140625`	32175/65536Exact fraction: `32175/65536`Retained binary64: `0.4909515380859375`	6435/54272Exact fraction: `6435/54272`Retained binary64: `0.11856942806603774`	32175/65536Exact fraction: `32175/65536`Retained binary64: `0.4909515380859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-6905/44944Exact fraction: `-6905/44944`Retained binary64: `-0.15363563545745818`	false	true	false
74	133/32Exact fraction: `133/32`Retained binary64: `4.15625`	7911/16384Exact fraction: `7911/16384`Retained binary64: `0.48284912109375`	7911/68096Exact fraction: `7911/68096`Retained binary64: `0.11617422462406016`	7911/16384Exact fraction: `7911/16384`Retained binary64: `0.48284912109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-43289/283024Exact fraction: `-43289/283024`Retained binary64: `-0.15295169314263102`	false	true	false
75	267/64Exact fraction: `267/64`Retained binary64: `4.171875`	31111/65536Exact fraction: `31111/65536`Retained binary64: `0.4747161865234375`	31111/273408Exact fraction: `31111/273408`Retained binary64: `0.11378964770599251`	31111/65536Exact fraction: `31111/65536`Retained binary64: `0.4747161865234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-173689/1140624Exact fraction: `-173689/1140624`Retained binary64: `-0.152275421172972`	false	true	false
76	67/16Exact fraction: `67/16`Retained binary64: `4.1875`	1911/4096Exact fraction: `1911/4096`Retained binary64: `0.466552734375`	1911/17152Exact fraction: `1911/17152`Retained binary64: `0.11141557835820895`	1911/4096Exact fraction: `1911/4096`Retained binary64: `0.466552734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-10889/71824Exact fraction: `-10889/71824`Retained binary64: `-0.1516067052795723`	false	true	false
77	269/64Exact fraction: `269/64`Retained binary64: `4.203125`	30039/65536Exact fraction: `30039/65536`Retained binary64: `0.4583587646484375`	30039/275456Exact fraction: `30039/275456`Retained binary64: `0.10905189939591078`	30039/65536Exact fraction: `30039/65536`Retained binary64: `0.4583587646484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-174761/1157776Exact fraction: `-174761/1157776`Retained binary64: `-0.15094543331352525`	false	true	false
78	135/32Exact fraction: `135/32`Retained binary64: `4.21875`	7375/16384Exact fraction: `7375/16384`Retained binary64: `0.45013427734375`	1475/13824Exact fraction: `1475/13824`Retained binary64: `0.10669849537037036`	7375/16384Exact fraction: `7375/16384`Retained binary64: `0.45013427734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1753/11664Exact fraction: `-1753/11664`Retained binary64: `-0.1502914951989026`	false	true	false
79	271/64Exact fraction: `271/64`Retained binary64: `4.234375`	28959/65536Exact fraction: `28959/65536`Retained binary64: `0.4418792724609375`	28959/277504Exact fraction: `28959/277504`Retained binary64: `0.10435525253690037`	28959/65536Exact fraction: `28959/65536`Retained binary64: `0.4418792724609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-175841/1175056Exact fraction: `-175841/1175056`Retained binary64: `-0.14964478288694327`	false	true	false
80	17/4Exact fraction: `17/4`Retained binary64: `4.25`	111/256Exact fraction: `111/256`Retained binary64: `0.43359375`	111/1088Exact fraction: `111/1088`Retained binary64: `0.10202205882352941`	111/256Exact fraction: `111/256`Retained binary64: `0.43359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-689/4624Exact fraction: `-689/4624`Retained binary64: `-0.14900519031141868`	false	true	false
81	273/64Exact fraction: `273/64`Retained binary64: `4.265625`	27871/65536Exact fraction: `27871/65536`Retained binary64: `0.4252777099609375`	27871/279552Exact fraction: `27871/279552`Retained binary64: `0.0996988038003663`	27871/65536Exact fraction: `27871/65536`Retained binary64: `0.4252777099609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-176929/1192464Exact fraction: `-176929/1192464`Retained binary64: `-0.14837261334514082`	false	true	false
82	137/32Exact fraction: `137/32`Retained binary64: `4.28125`	6831/16384Exact fraction: `6831/16384`Retained binary64: `0.41693115234375`	6831/70144Exact fraction: `6831/70144`Retained binary64: `0.09738537864963503`	6831/16384Exact fraction: `6831/16384`Retained binary64: `0.41693115234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-44369/300304Exact fraction: `-44369/300304`Retained binary64: `-0.147746949757579`	false	true	false
83	275/64Exact fraction: `275/64`Retained binary64: `4.296875`	26775/65536Exact fraction: `26775/65536`Retained binary64: `0.4085540771484375`	1071/11264Exact fraction: `1071/11264`Retained binary64: `0.09508167613636363`	26775/65536Exact fraction: `26775/65536`Retained binary64: `0.4085540771484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-7121/48400Exact fraction: `-7121/48400`Retained binary64: `-0.1471280991735537`	false	true	false
84	69/16Exact fraction: `69/16`Retained binary64: `4.3125`	1639/4096Exact fraction: `1639/4096`Retained binary64: `0.400146484375`	1639/17664Exact fraction: `1639/17664`Retained binary64: `0.09278759057971014`	1639/4096Exact fraction: `1639/4096`Retained binary64: `0.400146484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-11161/76176Exact fraction: `-11161/76176`Retained binary64: `-0.14651596303297626`	false	true	false
85	277/64Exact fraction: `277/64`Retained binary64: `4.328125`	25671/65536Exact fraction: `25671/65536`Retained binary64: `0.3917083740234375`	25671/283648Exact fraction: `25671/283648`Retained binary64: `0.09050301782490974`	25671/65536Exact fraction: `25671/65536`Retained binary64: `0.3917083740234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-179129/1227664Exact fraction: `-179129/1227664`Retained binary64: `-0.1459104445516037`	false	true	false
86	139/32Exact fraction: `139/32`Retained binary64: `4.34375`	6279/16384Exact fraction: `6279/16384`Retained binary64: `0.38323974609375`	6279/71168Exact fraction: `6279/71168`Retained binary64: `0.08822785521582734`	6279/16384Exact fraction: `6279/16384`Retained binary64: `0.38323974609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-44921/309136Exact fraction: `-44921/309136`Retained binary64: `-0.1453114486827804`	false	true	false
87	279/64Exact fraction: `279/64`Retained binary64: `4.359375`	24559/65536Exact fraction: `24559/65536`Retained binary64: `0.3747406005859375`	24559/285696Exact fraction: `24559/285696`Retained binary64: `0.08596200156810035`	24559/65536Exact fraction: `24559/65536`Retained binary64: `0.3747406005859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-180241/1245456Exact fraction: `-180241/1245456`Retained binary64: `-0.14471888208013772`	false	true	false
88	35/8Exact fraction: `35/8`Retained binary64: `4.375`	375/1024Exact fraction: `375/1024`Retained binary64: `0.3662109375`	75/896Exact fraction: `75/896`Retained binary64: `0.08370535714285714`	375/1024Exact fraction: `375/1024`Retained binary64: `0.3662109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-113/784Exact fraction: `-113/784`Retained binary64: `-0.1441326530612245`	false	true	false
89	281/64Exact fraction: `281/64`Retained binary64: `4.390625`	23439/65536Exact fraction: `23439/65536`Retained binary64: `0.3576507568359375`	23439/287744Exact fraction: `23439/287744`Retained binary64: `0.08145782362099645`	23439/65536Exact fraction: `23439/65536`Retained binary64: `0.3576507568359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-181361/1263376Exact fraction: `-181361/1263376`Retained binary64: `-0.1435526715720419`	false	true	false
90	141/32Exact fraction: `141/32`Retained binary64: `4.40625`	5719/16384Exact fraction: `5719/16384`Retained binary64: `0.34906005859375`	5719/72192Exact fraction: `5719/72192`Retained binary64: `0.07921930407801418`	5719/16384Exact fraction: `5719/16384`Retained binary64: `0.34906005859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-45481/318096Exact fraction: `-45481/318096`Retained binary64: `-0.14297884915245712`	false	true	false
91	283/64Exact fraction: `283/64`Retained binary64: `4.421875`	22311/65536Exact fraction: `22311/65536`Retained binary64: `0.3404388427734375`	22311/289792Exact fraction: `22311/289792`Retained binary64: `0.07698970295936396`	22311/65536Exact fraction: `22311/65536`Retained binary64: `0.3404388427734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-182489/1281424Exact fraction: `-182489/1281424`Retained binary64: `-0.142411098902471`	false	true	false
92	71/16Exact fraction: `71/16`Retained binary64: `4.4375`	1359/4096Exact fraction: `1359/4096`Retained binary64: `0.331787109375`	1359/18176Exact fraction: `1359/18176`Retained binary64: `0.07476892605633803`	1359/4096Exact fraction: `1359/4096`Retained binary64: `0.331787109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-11441/80656Exact fraction: `-11441/80656`Retained binary64: `-0.1418493354493156`	false	true	false
93	285/64Exact fraction: `285/64`Retained binary64: `4.453125`	21175/65536Exact fraction: `21175/65536`Retained binary64: `0.3231048583984375`	4235/58368Exact fraction: `4235/58368`Retained binary64: `0.07255688048245613`	21175/65536Exact fraction: `21175/65536`Retained binary64: `0.3231048583984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-7345/51984Exact fraction: `-7345/51984`Retained binary64: `-0.14129347491535857`	false	true	false
94	143/32Exact fraction: `143/32`Retained binary64: `4.46875`	5151/16384Exact fraction: `5151/16384`Retained binary64: `0.31439208984375`	5151/73216Exact fraction: `5151/73216`Retained binary64: `0.07035347465034965`	5151/16384Exact fraction: `5151/16384`Retained binary64: `0.31439208984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-46049/327184Exact fraction: `-46049/327184`Retained binary64: `-0.14074343488679153`	false	true	false
95	287/64Exact fraction: `287/64`Retained binary64: `4.484375`	20031/65536Exact fraction: `20031/65536`Retained binary64: `0.3056488037109375`	20031/293888Exact fraction: `20031/293888`Retained binary64: `0.06815861824912892`	20031/65536Exact fraction: `20031/65536`Retained binary64: `0.3056488037109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-184769/1317904Exact fraction: `-184769/1317904`Retained binary64: `-0.14019913438308101`	false	true	false
96	9/2Exact fraction: `9/2`Retained binary64: `4.5`	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	19/288Exact fraction: `19/288`Retained binary64: `0.06597222222222222`	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-181/1296Exact fraction: `-181/1296`Retained binary64: `-0.1396604938271605`	false	true	false
97	289/64Exact fraction: `289/64`Retained binary64: `4.515625`	18879/65536Exact fraction: `18879/65536`Retained binary64: `0.2880706787109375`	18879/295936Exact fraction: `18879/295936`Retained binary64: `0.06379419874567474`	18879/65536Exact fraction: `18879/65536`Retained binary64: `0.2880706787109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-185921/1336336Exact fraction: `-185921/1336336`Retained binary64: `-0.1391274350163432`	false	true	false
98	145/32Exact fraction: `145/32`Retained binary64: `4.53125`	4575/16384Exact fraction: `4575/16384`Retained binary64: `0.27923583984375`	915/14848Exact fraction: `915/14848`Retained binary64: `0.061624461206896554`	4575/16384Exact fraction: `4575/16384`Retained binary64: `0.27923583984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1865/13456Exact fraction: `-1865/13456`Retained binary64: `-0.13859988109393578`	false	true	false
99	291/64Exact fraction: `291/64`Retained binary64: `4.546875`	17719/65536Exact fraction: `17719/65536`Retained binary64: `0.2703704833984375`	17719/297984Exact fraction: `17719/297984`Retained binary64: `0.0594629241838488`	17719/65536Exact fraction: `17719/65536`Retained binary64: `0.2703704833984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-187081/1354896Exact fraction: `-187081/1354896`Retained binary64: `-0.13807775652153376`	false	true	false
100	73/16Exact fraction: `73/16`Retained binary64: `4.5625`	1071/4096Exact fraction: `1071/4096`Retained binary64: `0.261474609375`	1071/18688Exact fraction: `1071/18688`Retained binary64: `0.057309503424657536`	1071/4096Exact fraction: `1071/4096`Retained binary64: `0.261474609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-11729/85264Exact fraction: `-11729/85264`Retained binary64: `-0.13756098705197972`	false	true	false
101	293/64Exact fraction: `293/64`Retained binary64: `4.578125`	16551/65536Exact fraction: `16551/65536`Retained binary64: `0.2525482177734375`	16551/300032Exact fraction: `16551/300032`Retained binary64: `0.05516411582764505`	16551/65536Exact fraction: `16551/65536`Retained binary64: `0.2525482177734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-188249/1373584Exact fraction: `-188249/1373584`Retained binary64: `-0.13704949970296684`	false	true	false
102	147/32Exact fraction: `147/32`Retained binary64: `4.59375`	3991/16384Exact fraction: `3991/16384`Retained binary64: `0.24359130859375`	3991/75264Exact fraction: `3991/75264`Retained binary64: `0.05302667942176871`	3991/16384Exact fraction: `3991/16384`Retained binary64: `0.24359130859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-47209/345744Exact fraction: `-47209/345744`Retained binary64: `-0.13654322273126937`	false	true	false
103	295/64Exact fraction: `295/64`Retained binary64: `4.609375`	15375/65536Exact fraction: `15375/65536`Retained binary64: `0.2346038818359375`	3075/60416Exact fraction: `3075/60416`Retained binary64: `0.050897113347457626`	15375/65536Exact fraction: `15375/65536`Retained binary64: `0.2346038818359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-7577/55696Exact fraction: `-7577/55696`Retained binary64: `-0.13604208560758402`	false	true	false
104	37/8Exact fraction: `37/8`Retained binary64: `4.625`	231/1024Exact fraction: `231/1024`Retained binary64: `0.2255859375`	231/4736Exact fraction: `231/4736`Retained binary64: `0.048775337837837836`	231/1024Exact fraction: `231/1024`Retained binary64: `0.2255859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-2969/21904Exact fraction: `-2969/21904`Retained binary64: `-0.13554601899196494`	false	true	false
105	297/64Exact fraction: `297/64`Retained binary64: `4.640625`	14191/65536Exact fraction: `14191/65536`Retained binary64: `0.2165374755859375`	14191/304128Exact fraction: `14191/304128`Retained binary64: `0.0466612742003367`	14191/65536Exact fraction: `14191/65536`Retained binary64: `0.2165374755859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-190609/1411344Exact fraction: `-190609/1411344`Retained binary64: `-0.13505495470983686`	false	true	false
106	149/32Exact fraction: `149/32`Retained binary64: `4.65625`	3399/16384Exact fraction: `3399/16384`Retained binary64: `0.20745849609375`	3399/76288Exact fraction: `3399/76288`Retained binary64: `0.04455484479865772`	3399/16384Exact fraction: `3399/16384`Retained binary64: `0.20745849609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-47801/355216Exact fraction: `-47801/355216`Retained binary64: `-0.1345688257285708`	false	true	false
107	299/64Exact fraction: `299/64`Retained binary64: `4.671875`	12999/65536Exact fraction: `12999/65536`Retained binary64: `0.1983489990234375`	12999/306176Exact fraction: `12999/306176`Retained binary64: `0.042455973035117056`	12999/65536Exact fraction: `12999/65536`Retained binary64: `0.1983489990234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-191801/1430416Exact fraction: `-191801/1430416`Retained binary64: `-0.13408756613460698`	false	true	false
108	75/16Exact fraction: `75/16`Retained binary64: `4.6875`	775/4096Exact fraction: `775/4096`Retained binary64: `0.189208984375`	31/768Exact fraction: `31/768`Retained binary64: `0.040364583333333336`	775/4096Exact fraction: `775/4096`Retained binary64: `0.189208984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-481/3600Exact fraction: `-481/3600`Retained binary64: `-0.13361111111111112`	false	true	false
109	301/64Exact fraction: `301/64`Retained binary64: `4.703125`	11799/65536Exact fraction: `11799/65536`Retained binary64: `0.1800384521484375`	11799/308224Exact fraction: `11799/308224`Retained binary64: `0.038280601121262456`	11799/65536Exact fraction: `11799/65536`Retained binary64: `0.1800384521484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-193001/1449616Exact fraction: `-193001/1449616`Retained binary64: `-0.13313939691614882`	false	true	false
110	151/32Exact fraction: `151/32`Retained binary64: `4.71875`	2799/16384Exact fraction: `2799/16384`Retained binary64: `0.17083740234375`	2799/77312Exact fraction: `2799/77312`Retained binary64: `0.036203952814569534`	2799/16384Exact fraction: `2799/16384`Retained binary64: `0.17083740234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-48401/364816Exact fraction: `-48401/364816`Retained binary64: `-0.13267236086136572`	false	true	false
111	303/64Exact fraction: `303/64`Retained binary64: `4.734375`	10591/65536Exact fraction: `10591/65536`Retained binary64: `0.1616058349609375`	10591/310272Exact fraction: `10591/310272`Retained binary64: `0.03413456580033003`	10591/65536Exact fraction: `10591/65536`Retained binary64: `0.1616058349609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-194209/1468944Exact fraction: `-194209/1468944`Retained binary64: `-0.13220994129115882`	false	true	false
112	19/4Exact fraction: `19/4`Retained binary64: `4.75`	39/256Exact fraction: `39/256`Retained binary64: `0.15234375`	39/1216Exact fraction: `39/1216`Retained binary64: `0.032072368421052634`	39/256Exact fraction: `39/256`Retained binary64: `0.15234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-761/5776Exact fraction: `-761/5776`Retained binary64: `-0.13175207756232687`	false	true	false
113	305/64Exact fraction: `305/64`Retained binary64: `4.765625`	9375/65536Exact fraction: `9375/65536`Retained binary64: `0.1430511474609375`	1875/62464Exact fraction: `1875/62464`Retained binary64: `0.030017289959016393`	9375/65536Exact fraction: `9375/65536`Retained binary64: `0.1430511474609375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-7817/59536Exact fraction: `-7817/59536`Retained binary64: `-0.13129871002418705`	false	true	false
114	153/32Exact fraction: `153/32`Retained binary64: `4.78125`	2191/16384Exact fraction: `2191/16384`Retained binary64: `0.13372802734375`	2191/78336Exact fraction: `2191/78336`Retained binary64: `0.027969260620915034`	2191/16384Exact fraction: `2191/16384`Retained binary64: `0.13372802734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-49009/374544Exact fraction: `-49009/374544`Retained binary64: `-0.13084977999914563`	false	true	false
115	307/64Exact fraction: `307/64`Retained binary64: `4.796875`	8151/65536Exact fraction: `8151/65536`Retained binary64: `0.1243743896484375`	8151/314368Exact fraction: `8151/314368`Retained binary64: `0.025928211522801302`	8151/65536Exact fraction: `8151/65536`Retained binary64: `0.1243743896484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-196649/1507984Exact fraction: `-196649/1507984`Retained binary64: `-0.13040522976371102`	false	true	false
116	77/16Exact fraction: `77/16`Retained binary64: `4.8125`	471/4096Exact fraction: `471/4096`Retained binary64: `0.114990234375`	471/19712Exact fraction: `471/19712`Retained binary64: `0.023894074675324676`	471/4096Exact fraction: `471/4096`Retained binary64: `0.114990234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-12329/94864Exact fraction: `-12329/94864`Retained binary64: `-0.1299650025299376`	false	true	false
117	309/64Exact fraction: `309/64`Retained binary64: `4.828125`	6919/65536Exact fraction: `6919/65536`Retained binary64: `0.1055755615234375`	6919/316416Exact fraction: `6919/316416`Retained binary64: `0.021866782969255663`	6919/65536Exact fraction: `6919/65536`Retained binary64: `0.1055755615234375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-197881/1527696Exact fraction: `-197881/1527696`Retained binary64: `-0.1295290424272892`	false	true	false
118	155/32Exact fraction: `155/32`Retained binary64: `4.84375`	1575/16384Exact fraction: `1575/16384`Retained binary64: `0.09613037109375`	315/15872Exact fraction: `315/15872`Retained binary64: `0.019846270161290324`	1575/16384Exact fraction: `1575/16384`Retained binary64: `0.09613037109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1985/15376Exact fraction: `-1985/15376`Retained binary64: `-0.12909729448491156`	false	true	false
119	311/64Exact fraction: `311/64`Retained binary64: `4.859375`	5679/65536Exact fraction: `5679/65536`Retained binary64: `0.0866546630859375`	5679/318464Exact fraction: `5679/318464`Retained binary64: `0.017832470860128617`	5679/65536Exact fraction: `5679/65536`Retained binary64: `0.0866546630859375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-199121/1547536Exact fraction: `-199121/1547536`Retained binary64: `-0.128669704614303`	false	true	false
120	39/8Exact fraction: `39/8`Retained binary64: `4.875`	79/1024Exact fraction: `79/1024`Retained binary64: `0.0771484375`	79/4992Exact fraction: `79/4992`Retained binary64: `0.015825320512820512`	79/1024Exact fraction: `79/1024`Retained binary64: `0.0771484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-3121/24336Exact fraction: `-3121/24336`Retained binary64: `-0.12824621959237345`	false	true	false
121	313/64Exact fraction: `313/64`Retained binary64: `4.890625`	4431/65536Exact fraction: `4431/65536`Retained binary64: `0.0676116943359375`	4431/320512Exact fraction: `4431/320512`Retained binary64: `0.013824755391373802`	4431/65536Exact fraction: `4431/65536`Retained binary64: `0.0676116943359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-200369/1567504Exact fraction: `-200369/1567504`Retained binary64: `-0.12782678704488154`	false	true	false
122	157/32Exact fraction: `157/32`Retained binary64: `4.90625`	951/16384Exact fraction: `951/16384`Retained binary64: `0.05804443359375`	951/80384Exact fraction: `951/80384`Retained binary64: `0.011830712579617835`	951/16384Exact fraction: `951/16384`Retained binary64: `0.05804443359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-50249/394384Exact fraction: `-50249/394384`Retained binary64: `-0.12741135543024057`	false	true	false
123	315/64Exact fraction: `315/64`Retained binary64: `4.921875`	3175/65536Exact fraction: `3175/65536`Retained binary64: `0.0484466552734375`	635/64512Exact fraction: `635/64512`Retained binary64: `0.00984312996031746`	3175/65536Exact fraction: `3175/65536`Retained binary64: `0.0484466552734375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-8065/63504Exact fraction: `-8065/63504`Retained binary64: `-0.12699987402368354`	false	true	false
124	79/16Exact fraction: `79/16`Retained binary64: `4.9375`	159/4096Exact fraction: `159/4096`Retained binary64: `0.038818359375`	159/20224Exact fraction: `159/20224`Retained binary64: `0.007861946202531646`	159/4096Exact fraction: `159/4096`Retained binary64: `0.038818359375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-12641/99856Exact fraction: `-12641/99856`Retained binary64: `-0.12659229290177856`	false	true	false
125	317/64Exact fraction: `317/64`Retained binary64: `4.953125`	1911/65536Exact fraction: `1911/65536`Retained binary64: `0.0291595458984375`	1911/324608Exact fraction: `1911/324608`Retained binary64: `0.005887100749211356`	1911/65536Exact fraction: `1911/65536`Retained binary64: `0.0291595458984375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-202889/1607824Exact fraction: `-202889/1607824`Retained binary64: `-0.12618856292728559`	false	true	false
126	159/32Exact fraction: `159/32`Retained binary64: `4.96875`	319/16384Exact fraction: `319/16384`Retained binary64: `0.01947021484375`	319/81408Exact fraction: `319/81408`Retained binary64: `0.003918533805031447`	319/16384Exact fraction: `319/16384`Retained binary64: `0.01947021484375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-50881/404496Exact fraction: `-50881/404496`Retained binary64: `-0.12578863573434595`	false	true	false
127	319/64Exact fraction: `319/64`Retained binary64: `4.984375`	639/65536Exact fraction: `639/65536`Retained binary64: `0.0097503662109375`	639/326656Exact fraction: `639/326656`Retained binary64: `0.0019561863244514106`	639/65536Exact fraction: `639/65536`Retained binary64: `0.0097503662109375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-204161/1628176Exact fraction: `-204161/1628176`Retained binary64: `-0.1253924637139965`	false	true	false
128Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	5Exact fraction: `5`Retained binary64: `5.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	true	true	false

Complete outside coax branch (65 rows) (65 rows)

Complete outside coax branch (65 rows)
Row	R=r/L₀	Signed enclosed I/I₀	Bφ/B₀	Circulation/(μ₀I₀)	Branch J_z/[I₀/(πL₀²)]; one-sided at interfaces	Branch db/dR; one-sided at interfaces	J/slope are interface one-sided limits only?	Azimuthal direction defined?	Cartesian field zero at center?
0Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	5Exact fraction: `5`Retained binary64: `5.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	true	true	false
1	161/32Exact fraction: `161/32`Retained binary64: `5.03125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
2	81/16Exact fraction: `81/16`Retained binary64: `5.0625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
3	163/32Exact fraction: `163/32`Retained binary64: `5.09375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
4	41/8Exact fraction: `41/8`Retained binary64: `5.125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
5	165/32Exact fraction: `165/32`Retained binary64: `5.15625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
6	83/16Exact fraction: `83/16`Retained binary64: `5.1875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
7	167/32Exact fraction: `167/32`Retained binary64: `5.21875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
8	21/4Exact fraction: `21/4`Retained binary64: `5.25`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
9	169/32Exact fraction: `169/32`Retained binary64: `5.28125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
10	85/16Exact fraction: `85/16`Retained binary64: `5.3125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
11	171/32Exact fraction: `171/32`Retained binary64: `5.34375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
12	43/8Exact fraction: `43/8`Retained binary64: `5.375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
13	173/32Exact fraction: `173/32`Retained binary64: `5.40625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
14	87/16Exact fraction: `87/16`Retained binary64: `5.4375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
15	175/32Exact fraction: `175/32`Retained binary64: `5.46875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
16	11/2Exact fraction: `11/2`Retained binary64: `5.5`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
17	177/32Exact fraction: `177/32`Retained binary64: `5.53125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
18	89/16Exact fraction: `89/16`Retained binary64: `5.5625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
19	179/32Exact fraction: `179/32`Retained binary64: `5.59375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
20	45/8Exact fraction: `45/8`Retained binary64: `5.625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
21	181/32Exact fraction: `181/32`Retained binary64: `5.65625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
22	91/16Exact fraction: `91/16`Retained binary64: `5.6875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
23	183/32Exact fraction: `183/32`Retained binary64: `5.71875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
24	23/4Exact fraction: `23/4`Retained binary64: `5.75`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
25	185/32Exact fraction: `185/32`Retained binary64: `5.78125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
26	93/16Exact fraction: `93/16`Retained binary64: `5.8125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
27	187/32Exact fraction: `187/32`Retained binary64: `5.84375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
28	47/8Exact fraction: `47/8`Retained binary64: `5.875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
29	189/32Exact fraction: `189/32`Retained binary64: `5.90625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
30	95/16Exact fraction: `95/16`Retained binary64: `5.9375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
31	191/32Exact fraction: `191/32`Retained binary64: `5.96875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
32	6Exact fraction: `6`Retained binary64: `6.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
33	193/32Exact fraction: `193/32`Retained binary64: `6.03125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
34	97/16Exact fraction: `97/16`Retained binary64: `6.0625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
35	195/32Exact fraction: `195/32`Retained binary64: `6.09375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
36	49/8Exact fraction: `49/8`Retained binary64: `6.125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
37	197/32Exact fraction: `197/32`Retained binary64: `6.15625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
38	99/16Exact fraction: `99/16`Retained binary64: `6.1875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
39	199/32Exact fraction: `199/32`Retained binary64: `6.21875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
40	25/4Exact fraction: `25/4`Retained binary64: `6.25`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
41	201/32Exact fraction: `201/32`Retained binary64: `6.28125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
42	101/16Exact fraction: `101/16`Retained binary64: `6.3125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
43	203/32Exact fraction: `203/32`Retained binary64: `6.34375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
44	51/8Exact fraction: `51/8`Retained binary64: `6.375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
45	205/32Exact fraction: `205/32`Retained binary64: `6.40625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
46	103/16Exact fraction: `103/16`Retained binary64: `6.4375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
47	207/32Exact fraction: `207/32`Retained binary64: `6.46875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
48	13/2Exact fraction: `13/2`Retained binary64: `6.5`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
49	209/32Exact fraction: `209/32`Retained binary64: `6.53125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
50	105/16Exact fraction: `105/16`Retained binary64: `6.5625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
51	211/32Exact fraction: `211/32`Retained binary64: `6.59375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
52	53/8Exact fraction: `53/8`Retained binary64: `6.625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
53	213/32Exact fraction: `213/32`Retained binary64: `6.65625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
54	107/16Exact fraction: `107/16`Retained binary64: `6.6875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
55	215/32Exact fraction: `215/32`Retained binary64: `6.71875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
56	27/4Exact fraction: `27/4`Retained binary64: `6.75`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
57	217/32Exact fraction: `217/32`Retained binary64: `6.78125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
58	109/16Exact fraction: `109/16`Retained binary64: `6.8125`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
59	219/32Exact fraction: `219/32`Retained binary64: `6.84375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
60	55/8Exact fraction: `55/8`Retained binary64: `6.875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
61	221/32Exact fraction: `221/32`Retained binary64: `6.90625`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
62	111/16Exact fraction: `111/16`Retained binary64: `6.9375`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
63	223/32Exact fraction: `223/32`Retained binary64: `6.96875`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false
64	7Exact fraction: `7`Retained binary64: `7.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false

All nine coax controls, including paired one-sided metadata rows (9 rows)

All nine coax controls, including paired one-sided metadata rows
Row	R=r/L₀	Signed enclosed I/I₀	Bφ/B₀	Circulation/(μ₀I₀)	Branch J_z/[I₀/(πL₀²)]; one-sided at interfaces	Branch db/dR; one-sided at interfaces	J/slope are interface one-sided limits only?	Azimuthal direction defined?	Cartesian field zero at center?
0	1Exact fraction: `1`Retained binary64: `1.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	false	true	false
1Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1Exact fraction: `1`Retained binary64: `1.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	true	true	false
2Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	true	true	false
3Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/9Exact fraction: `-1/9`Retained binary64: `-0.1111111111111111`	true	true	false
4Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1Exact fraction: `1`Retained binary64: `1.0`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-17/72Exact fraction: `-17/72`Retained binary64: `-0.2361111111111111`	true	true	false
5	4Exact fraction: `4`Retained binary64: `4.0`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-41/256Exact fraction: `-41/256`Retained binary64: `-0.16015625`	false	true	false
6Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	5Exact fraction: `5`Retained binary64: `5.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	true	true	false
7Current density and slope in this row are branch one-sided limits only. B and enclosed current are continuous; this is not an ideal current sheet.	5Exact fraction: `5`Retained binary64: `5.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	true	true	false
8	6Exact fraction: `6`Retained binary64: `6.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	true	false

The three exact coax interface radii (3 rows)

The three exact coax interface radii
Row	value
0	2Exact fraction: `2`Retained binary64: `2.0`
1	3Exact fraction: `3`Retained binary64: `3.0`
2	5Exact fraction: `5`Retained binary64: `5.0`

coax: exact controls, scales and source limits (5 rows)

coax: exact controls, scales and source limits
Quantity/source condition	Admitted value
inner Current Over I0	1Exact fraction: `1`Retained binary64: `1.0`
return Current Over I0	-1Exact fraction: `-1`Retained binary64: `-1.0`
total Current Over I0	0Exact fraction: `0`Retained binary64: `0.0`
ideal Surface Current Sheet	false
endpoint Rule	B continuous; plotted current density and field slopes at R2,3,5 are branch one-sided limits only. R0 Cartesian field zero; azimuth undefined.

Exact field and circulation controls in the common coaxial scales
R	I_enclosed/I₀	B_phi/B₀	Circulation/(μ₀I₀)
1	1/4	1/4	1/4
2	1	1/2	1
3	1	1/3	1
4	9/16	9/64	9/16
5	0	0	0
6	0	0	0

Two wires test the symmetry shortcut

Now take two ideal infinite +I₀ currents parallel to z at (x,y) = (−2L₀,0) and (+2L₀,0). In coordinates X = x/L₀ and Y = y/L₀, their superposed field in units B₀ is b = Σ over a = −2,+2 of (−Y, X − a)/[(X − a)² + Y²]. Each term is the azimuthal field of its own wire expressed in common Cartesian axes.

A circle of radius 3L₀ centered at the origin encloses both currents. Its positive circulation is therefore 2μ₀I₀. But at the point (X,Y) = (3,0), the tangential field is (1 + 1/5)B₀ = (6/5)B₀. At (0,3), the two y components cancel and the tangential field is (6/13)B₀. Those values cannot both equal the constant-field guess (2/3)B₀.

Write X = 3 cos θ and Y = 3 sin θ on that circle. Resolving the exact Cartesian sum gives b_t = 6(13 − 8 cos²θ)/(169 − 144 cos²θ) and b_r = −48 cos θ sin θ/(169 − 144 cos²θ). The radial component is generally nonzero, and the tangential component varies. At θ = π/4 they are 54/97 and −24/97, respectively.

Ampère’s law fixes the integral, so the mean tangential component around this circle is (2/3)B₀. It does not assign that mean as the local field. The native plots retain both circle components and separate exact controls. The analytic circulation follows from the complete source law; a numerical sum over plotted samples is not used as its proof.

Same enclosing contour, different local magnetic field components
θ	Point on the radius-3 circle	B_t/B₀	B_r/B₀
0	(3,0)	6/5	0
π/4	(3/√2,3/√2)	54/97	−24/97
π/2	(0,3)	6/13	0
Full-contour mean	Integral divided by 6πL₀	2/3	Not determined by Ampère circulation

A null and a source exclusion are different

At the midpoint (X,Y) = (0,0), the two wire fields cancel exactly. The total field there is the zero vector, even though both individual contributions are nonzero. This is a field null in the valid observation domain. It must not be removed as if it were a source singularity or normalized into an arbitrary arrow.

The two points (−2,0) and (+2,0) are instead on ideal filaments, where the corresponding model field is singular and excluded. A zero produced by cancellation and a value absent because the model is singular carry different physical meanings. A mask or an accessible table must retain that distinction.

The displayed integer grid −3 ≤ X,Y ≤ 3 has 49 candidate positions. Excluding the two source points leaves 47 records: one exact midpoint null and 46 nonzero field vectors. Their arrows use a common physical vector scale of one-quarter of the dimensionless field, so arrow lengths communicate those sampled field strengths rather than unit directions.

Equal physical x and y scales preserve the Cartesian directions. The grid arrows are local field samples, not integrated field lines or charged-particle orbits. A line of constant density of plotted arrows does not establish constant |B|, and a field-line picture would require its own trajectory integration and labeling.

Fixed circulation does not make the local field constant

Compare the complete two-wire field on an equal-scale Cartesian grid with its varying tangential and radial components around one enclosing circle. Keep the exact midpoint null and excluded source points distinct. The contour mean follows from circulation and is not the local field.

Physical field vectors, one null and two excluded sources

Physical vectors: stored head−tail, common1/4 scale; not unit directions
R=3 integration contour; positive orientation counterclockwise from+z; not a field line
Excluded ideal source: open crossed circle
Exact field null: filled point, no direction

Two varying local components and one analytic mean

Local counterclockwise tangential B_t/B₀
Local outward radial B_r/B₀
Analytic mean of B_t/B₀=2/3, from circulation; not local B or sampled quadrature
Separate exact π/4 controls:54/97 and−24/97

Two complete infinite+I₀z-directed filament idealizations at(−2,0),(2,0),return at infinity. Source points excluded,midpoint null retained. Grid arrows use actual stored head/tail and common physical-vector scale1/4,not normalized directions. CircleR3 encloses both currents counterclockwise viewed from +z; analytic circulation/mean is not a sampled quadrature result.
View from +z,+x right,+y up. All46 nonzero arrows use stored physical tails/heads with common scale1/4: displacement/L₀=(1/4)(B/B₀). They retain differing lengths, not unit directions. Arrowhead paint scales to short shafts; it does not replace their physical endpoints.
The two red open crossed circles are excluded ideal sources, not zeros. The midpoint violet filled dot is the exact retained null, with no direction arrow. The violet dashed R3 contour is an integration path, not a magnetic field line. Positive contour orientation is counterclockwise.
The circle plot retains both B_t/B₀ and outward B_r/B₀. Their variation disproves taking one constantB outside the integral. Exact cardinal controls and the separateπ/4 controls54/97,−24/97 are retained; no generic near-zero snap is applied.
Ampère’s law with each+I₀ filament enclosed once gives circulation4πB₀L₀ and analytic mean B_t/B₀=2/3. The horizontal mean is not the local tangential field, and is not inferred by quadrature of the129 samples.
All marks read exact admitted records, including rational controls, original binary64 function-specific qualifications,12-significant-digit displays and retained values. No fitted conservation, near-zero snap, field solver or sampled-quadrature proof is used.

Complete labelled data equivalents preserve every admitted scalar, record, branch, mask and source constraint. Original manuscript tables remain separate.

Same enclosing contour, different local magnetic field components

Both excluded ideal-filament source locations (2 rows)

Two complete infinite+I₀z-directed filament idealizations at(−2,0),(2,0),return at infinity. Source points excluded,midpoint null retained. Grid arrows use actual stored head/tail and common physical-vector scale1/4,not normalized directions. CircleR3 encloses both currents counterclockwise viewed from +z; analytic circulation/mean is not a sampled quadrature result.

Both excluded ideal-filament source locations
Row	X=x/L₀	Y=y/L₀	Signed current/I₀
0	-2Exact fraction: `-2`Retained binary64: `-2.0`	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`
1	2Exact fraction: `2`Retained binary64: `2.0`	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`

All 47 retained grid records:46 physical vectors and one null (47 rows)

All 47 retained grid records:46 physical vectors and one null
Row	X=x/L₀	Y=y/L₀	Signed Bₓ/B₀	Signed Bᵧ/B₀	\|B\|²/B₀²	Nonzero field direction defined?	Physical arrow tail X	Physical arrow tail Y	Physical arrow head X	Physical arrow head Y	Common-scale physical vector, not unit direction?
0	-3Exact fraction: `-3`Retained binary64: `-3.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	33/85Exact fraction: `33/85`Retained binary64: `0.38823529411764707`	-21/85Exact fraction: `-21/85`Retained binary64: `-0.24705882352941178`	18/85Exact fraction: `18/85`Retained binary64: `0.21176470588235294`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-987/340Exact fraction: `-987/340`Retained binary64: `-2.902941176470588`	-1041/340Exact fraction: `-1041/340`Retained binary64: `-3.0617647058823527`	true
1	-3Exact fraction: `-3`Retained binary64: `-3.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	68/145Exact fraction: `68/145`Retained binary64: `0.4689655172413793`	-54/145Exact fraction: `-54/145`Retained binary64: `-0.3724137931034483`	52/145Exact fraction: `52/145`Retained binary64: `0.3586206896551724`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	-418/145Exact fraction: `-418/145`Retained binary64: `-2.882758620689655`	-607/290Exact fraction: `-607/290`Retained binary64: `-2.093103448275862`	true
2	-3Exact fraction: `-3`Retained binary64: `-3.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	7/13Exact fraction: `7/13`Retained binary64: `0.5384615384615384`	-9/13Exact fraction: `-9/13`Retained binary64: `-0.6923076923076923`	10/13Exact fraction: `10/13`Retained binary64: `0.7692307692307693`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	-149/52Exact fraction: `-149/52`Retained binary64: `-2.8653846153846154`	-61/52Exact fraction: `-61/52`Retained binary64: `-1.1730769230769231`	true
3	-3Exact fraction: `-3`Retained binary64: `-3.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	-6/5Exact fraction: `-6/5`Retained binary64: `-1.2`	36/25Exact fraction: `36/25`Retained binary64: `1.44`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	0Exact fraction: `0`Retained binary64: `0.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-3/10Exact fraction: `-3/10`Retained binary64: `-0.3`	true
4	-3Exact fraction: `-3`Retained binary64: `-3.0`	1Exact fraction: `1`Retained binary64: `1.0`	-7/13Exact fraction: `-7/13`Retained binary64: `-0.5384615384615384`	-9/13Exact fraction: `-9/13`Retained binary64: `-0.6923076923076923`	10/13Exact fraction: `10/13`Retained binary64: `0.7692307692307693`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	1Exact fraction: `1`Retained binary64: `1.0`	-163/52Exact fraction: `-163/52`Retained binary64: `-3.1346153846153846`	43/52Exact fraction: `43/52`Retained binary64: `0.8269230769230769`	true
5	-3Exact fraction: `-3`Retained binary64: `-3.0`	2Exact fraction: `2`Retained binary64: `2.0`	-68/145Exact fraction: `-68/145`Retained binary64: `-0.4689655172413793`	-54/145Exact fraction: `-54/145`Retained binary64: `-0.3724137931034483`	52/145Exact fraction: `52/145`Retained binary64: `0.3586206896551724`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	2Exact fraction: `2`Retained binary64: `2.0`	-452/145Exact fraction: `-452/145`Retained binary64: `-3.117241379310345`	553/290Exact fraction: `553/290`Retained binary64: `1.9068965517241379`	true
6	-3Exact fraction: `-3`Retained binary64: `-3.0`	3Exact fraction: `3`Retained binary64: `3.0`	-33/85Exact fraction: `-33/85`Retained binary64: `-0.38823529411764707`	-21/85Exact fraction: `-21/85`Retained binary64: `-0.24705882352941178`	18/85Exact fraction: `18/85`Retained binary64: `0.21176470588235294`	true	-3Exact fraction: `-3`Retained binary64: `-3.0`	3Exact fraction: `3`Retained binary64: `3.0`	-1053/340Exact fraction: `-1053/340`Retained binary64: `-3.097058823529412`	999/340Exact fraction: `999/340`Retained binary64: `2.9382352941176473`	true
7	-2Exact fraction: `-2`Retained binary64: `-2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	34/75Exact fraction: `34/75`Retained binary64: `0.4533333333333333`	-4/25Exact fraction: `-4/25`Retained binary64: `-0.16`	52/225Exact fraction: `52/225`Retained binary64: `0.2311111111111111`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-283/150Exact fraction: `-283/150`Retained binary64: `-1.8866666666666667`	-76/25Exact fraction: `-76/25`Retained binary64: `-3.04`	true
8	-2Exact fraction: `-2`Retained binary64: `-2.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	3/5Exact fraction: `3/5`Retained binary64: `0.6`	-1/5Exact fraction: `-1/5`Retained binary64: `-0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	-37/20Exact fraction: `-37/20`Retained binary64: `-1.85`	-41/20Exact fraction: `-41/20`Retained binary64: `-2.05`	true
9	-2Exact fraction: `-2`Retained binary64: `-2.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	18/17Exact fraction: `18/17`Retained binary64: `1.0588235294117647`	-4/17Exact fraction: `-4/17`Retained binary64: `-0.23529411764705882`	20/17Exact fraction: `20/17`Retained binary64: `1.1764705882352942`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	-59/34Exact fraction: `-59/34`Retained binary64: `-1.7352941176470589`	-18/17Exact fraction: `-18/17`Retained binary64: `-1.0588235294117647`	true
10	-2Exact fraction: `-2`Retained binary64: `-2.0`	1Exact fraction: `1`Retained binary64: `1.0`	-18/17Exact fraction: `-18/17`Retained binary64: `-1.0588235294117647`	-4/17Exact fraction: `-4/17`Retained binary64: `-0.23529411764705882`	20/17Exact fraction: `20/17`Retained binary64: `1.1764705882352942`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	1Exact fraction: `1`Retained binary64: `1.0`	-77/34Exact fraction: `-77/34`Retained binary64: `-2.264705882352941`	16/17Exact fraction: `16/17`Retained binary64: `0.9411764705882353`	true
11	-2Exact fraction: `-2`Retained binary64: `-2.0`	2Exact fraction: `2`Retained binary64: `2.0`	-3/5Exact fraction: `-3/5`Retained binary64: `-0.6`	-1/5Exact fraction: `-1/5`Retained binary64: `-0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	2Exact fraction: `2`Retained binary64: `2.0`	-43/20Exact fraction: `-43/20`Retained binary64: `-2.15`	39/20Exact fraction: `39/20`Retained binary64: `1.95`	true
12	-2Exact fraction: `-2`Retained binary64: `-2.0`	3Exact fraction: `3`Retained binary64: `3.0`	-34/75Exact fraction: `-34/75`Retained binary64: `-0.4533333333333333`	-4/25Exact fraction: `-4/25`Retained binary64: `-0.16`	52/225Exact fraction: `52/225`Retained binary64: `0.2311111111111111`	true	-2Exact fraction: `-2`Retained binary64: `-2.0`	3Exact fraction: `3`Retained binary64: `3.0`	-317/150Exact fraction: `-317/150`Retained binary64: `-2.1133333333333333`	74/25Exact fraction: `74/25`Retained binary64: `2.96`	true
13	-1Exact fraction: `-1`Retained binary64: `-1.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	7/15Exact fraction: `7/15`Retained binary64: `0.4666666666666667`	-1/15Exact fraction: `-1/15`Retained binary64: `-0.06666666666666667`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-53/60Exact fraction: `-53/60`Retained binary64: `-0.8833333333333333`	-181/60Exact fraction: `-181/60`Retained binary64: `-3.0166666666666666`	true
14	-1Exact fraction: `-1`Retained binary64: `-1.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	36/65Exact fraction: `36/65`Retained binary64: `0.5538461538461539`	-2/65Exact fraction: `-2/65`Retained binary64: `-0.03076923076923077`	4/13Exact fraction: `4/13`Retained binary64: `0.3076923076923077`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	-56/65Exact fraction: `-56/65`Retained binary64: `-0.8615384615384616`	-261/130Exact fraction: `-261/130`Retained binary64: `-2.0076923076923077`	true
15	-1Exact fraction: `-1`Retained binary64: `-1.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	3/5Exact fraction: `3/5`Retained binary64: `0.6`	1/5Exact fraction: `1/5`Retained binary64: `0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	-17/20Exact fraction: `-17/20`Retained binary64: `-0.85`	-19/20Exact fraction: `-19/20`Retained binary64: `-0.95`	true
16	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	4/9Exact fraction: `4/9`Retained binary64: `0.4444444444444444`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	1/6Exact fraction: `1/6`Retained binary64: `0.16666666666666666`	true
17	-1Exact fraction: `-1`Retained binary64: `-1.0`	1Exact fraction: `1`Retained binary64: `1.0`	-3/5Exact fraction: `-3/5`Retained binary64: `-0.6`	1/5Exact fraction: `1/5`Retained binary64: `0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	1Exact fraction: `1`Retained binary64: `1.0`	-23/20Exact fraction: `-23/20`Retained binary64: `-1.15`	21/20Exact fraction: `21/20`Retained binary64: `1.05`	true
18	-1Exact fraction: `-1`Retained binary64: `-1.0`	2Exact fraction: `2`Retained binary64: `2.0`	-36/65Exact fraction: `-36/65`Retained binary64: `-0.5538461538461539`	-2/65Exact fraction: `-2/65`Retained binary64: `-0.03076923076923077`	4/13Exact fraction: `4/13`Retained binary64: `0.3076923076923077`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	2Exact fraction: `2`Retained binary64: `2.0`	-74/65Exact fraction: `-74/65`Retained binary64: `-1.1384615384615384`	259/130Exact fraction: `259/130`Retained binary64: `1.9923076923076923`	true
19	-1Exact fraction: `-1`Retained binary64: `-1.0`	3Exact fraction: `3`Retained binary64: `3.0`	-7/15Exact fraction: `-7/15`Retained binary64: `-0.4666666666666667`	-1/15Exact fraction: `-1/15`Retained binary64: `-0.06666666666666667`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	true	-1Exact fraction: `-1`Retained binary64: `-1.0`	3Exact fraction: `3`Retained binary64: `3.0`	-67/60Exact fraction: `-67/60`Retained binary64: `-1.1166666666666667`	179/60Exact fraction: `179/60`Retained binary64: `2.9833333333333334`	true
20	0Exact fraction: `0`Retained binary64: `0.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	6/13Exact fraction: `6/13`Retained binary64: `0.46153846153846156`	0Exact fraction: `0`Retained binary64: `0.0`	36/169Exact fraction: `36/169`Retained binary64: `0.21301775147928995`	true	0Exact fraction: `0`Retained binary64: `0.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	3/26Exact fraction: `3/26`Retained binary64: `0.11538461538461539`	-3Exact fraction: `-3`Retained binary64: `-3.0`	true
21	0Exact fraction: `0`Retained binary64: `0.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0Exact fraction: `0`Retained binary64: `0.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	true	0Exact fraction: `0`Retained binary64: `0.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	1/8Exact fraction: `1/8`Retained binary64: `0.125`	-2Exact fraction: `-2`Retained binary64: `-2.0`	true
22	0Exact fraction: `0`Retained binary64: `0.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	0Exact fraction: `0`Retained binary64: `0.0`	4/25Exact fraction: `4/25`Retained binary64: `0.16`	true	0Exact fraction: `0`Retained binary64: `0.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	1/10Exact fraction: `1/10`Retained binary64: `0.1`	-1Exact fraction: `-1`Retained binary64: `-1.0`	true
23Exact field null: retain the point and values, with no normalized direction or arrow.	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	false	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	true
24	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`	-2/5Exact fraction: `-2/5`Retained binary64: `-0.4`	0Exact fraction: `0`Retained binary64: `0.0`	4/25Exact fraction: `4/25`Retained binary64: `0.16`	true	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`	-1/10Exact fraction: `-1/10`Retained binary64: `-0.1`	1Exact fraction: `1`Retained binary64: `1.0`	true
25	0Exact fraction: `0`Retained binary64: `0.0`	2Exact fraction: `2`Retained binary64: `2.0`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0Exact fraction: `0`Retained binary64: `0.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	true	0Exact fraction: `0`Retained binary64: `0.0`	2Exact fraction: `2`Retained binary64: `2.0`	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	2Exact fraction: `2`Retained binary64: `2.0`	true
26	0Exact fraction: `0`Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`	-6/13Exact fraction: `-6/13`Retained binary64: `-0.46153846153846156`	0Exact fraction: `0`Retained binary64: `0.0`	36/169Exact fraction: `36/169`Retained binary64: `0.21301775147928995`	true	0Exact fraction: `0`Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`	-3/26Exact fraction: `-3/26`Retained binary64: `-0.11538461538461539`	3Exact fraction: `3`Retained binary64: `3.0`	true
27	1Exact fraction: `1`Retained binary64: `1.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	7/15Exact fraction: `7/15`Retained binary64: `0.4666666666666667`	1/15Exact fraction: `1/15`Retained binary64: `0.06666666666666667`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	true	1Exact fraction: `1`Retained binary64: `1.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	67/60Exact fraction: `67/60`Retained binary64: `1.1166666666666667`	-179/60Exact fraction: `-179/60`Retained binary64: `-2.9833333333333334`	true
28	1Exact fraction: `1`Retained binary64: `1.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	36/65Exact fraction: `36/65`Retained binary64: `0.5538461538461539`	2/65Exact fraction: `2/65`Retained binary64: `0.03076923076923077`	4/13Exact fraction: `4/13`Retained binary64: `0.3076923076923077`	true	1Exact fraction: `1`Retained binary64: `1.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	74/65Exact fraction: `74/65`Retained binary64: `1.1384615384615384`	-259/130Exact fraction: `-259/130`Retained binary64: `-1.9923076923076923`	true
29	1Exact fraction: `1`Retained binary64: `1.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	3/5Exact fraction: `3/5`Retained binary64: `0.6`	-1/5Exact fraction: `-1/5`Retained binary64: `-0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	1Exact fraction: `1`Retained binary64: `1.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	23/20Exact fraction: `23/20`Retained binary64: `1.15`	-21/20Exact fraction: `-21/20`Retained binary64: `-1.05`	true
30	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	-2/3Exact fraction: `-2/3`Retained binary64: `-0.6666666666666666`	4/9Exact fraction: `4/9`Retained binary64: `0.4444444444444444`	true	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`	-1/6Exact fraction: `-1/6`Retained binary64: `-0.16666666666666666`	true
31	1Exact fraction: `1`Retained binary64: `1.0`	1Exact fraction: `1`Retained binary64: `1.0`	-3/5Exact fraction: `-3/5`Retained binary64: `-0.6`	-1/5Exact fraction: `-1/5`Retained binary64: `-0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	1Exact fraction: `1`Retained binary64: `1.0`	1Exact fraction: `1`Retained binary64: `1.0`	17/20Exact fraction: `17/20`Retained binary64: `0.85`	19/20Exact fraction: `19/20`Retained binary64: `0.95`	true
32	1Exact fraction: `1`Retained binary64: `1.0`	2Exact fraction: `2`Retained binary64: `2.0`	-36/65Exact fraction: `-36/65`Retained binary64: `-0.5538461538461539`	2/65Exact fraction: `2/65`Retained binary64: `0.03076923076923077`	4/13Exact fraction: `4/13`Retained binary64: `0.3076923076923077`	true	1Exact fraction: `1`Retained binary64: `1.0`	2Exact fraction: `2`Retained binary64: `2.0`	56/65Exact fraction: `56/65`Retained binary64: `0.8615384615384616`	261/130Exact fraction: `261/130`Retained binary64: `2.0076923076923077`	true
33	1Exact fraction: `1`Retained binary64: `1.0`	3Exact fraction: `3`Retained binary64: `3.0`	-7/15Exact fraction: `-7/15`Retained binary64: `-0.4666666666666667`	1/15Exact fraction: `1/15`Retained binary64: `0.06666666666666667`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	true	1Exact fraction: `1`Retained binary64: `1.0`	3Exact fraction: `3`Retained binary64: `3.0`	53/60Exact fraction: `53/60`Retained binary64: `0.8833333333333333`	181/60Exact fraction: `181/60`Retained binary64: `3.0166666666666666`	true
34	2Exact fraction: `2`Retained binary64: `2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	34/75Exact fraction: `34/75`Retained binary64: `0.4533333333333333`	4/25Exact fraction: `4/25`Retained binary64: `0.16`	52/225Exact fraction: `52/225`Retained binary64: `0.2311111111111111`	true	2Exact fraction: `2`Retained binary64: `2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	317/150Exact fraction: `317/150`Retained binary64: `2.1133333333333333`	-74/25Exact fraction: `-74/25`Retained binary64: `-2.96`	true
35	2Exact fraction: `2`Retained binary64: `2.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	3/5Exact fraction: `3/5`Retained binary64: `0.6`	1/5Exact fraction: `1/5`Retained binary64: `0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	2Exact fraction: `2`Retained binary64: `2.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	43/20Exact fraction: `43/20`Retained binary64: `2.15`	-39/20Exact fraction: `-39/20`Retained binary64: `-1.95`	true
36	2Exact fraction: `2`Retained binary64: `2.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	18/17Exact fraction: `18/17`Retained binary64: `1.0588235294117647`	4/17Exact fraction: `4/17`Retained binary64: `0.23529411764705882`	20/17Exact fraction: `20/17`Retained binary64: `1.1764705882352942`	true	2Exact fraction: `2`Retained binary64: `2.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	77/34Exact fraction: `77/34`Retained binary64: `2.264705882352941`	-16/17Exact fraction: `-16/17`Retained binary64: `-0.9411764705882353`	true
37	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	-18/17Exact fraction: `-18/17`Retained binary64: `-1.0588235294117647`	4/17Exact fraction: `4/17`Retained binary64: `0.23529411764705882`	20/17Exact fraction: `20/17`Retained binary64: `1.1764705882352942`	true	2Exact fraction: `2`Retained binary64: `2.0`	1Exact fraction: `1`Retained binary64: `1.0`	59/34Exact fraction: `59/34`Retained binary64: `1.7352941176470589`	18/17Exact fraction: `18/17`Retained binary64: `1.0588235294117647`	true
38	2Exact fraction: `2`Retained binary64: `2.0`	2Exact fraction: `2`Retained binary64: `2.0`	-3/5Exact fraction: `-3/5`Retained binary64: `-0.6`	1/5Exact fraction: `1/5`Retained binary64: `0.2`	2/5Exact fraction: `2/5`Retained binary64: `0.4`	true	2Exact fraction: `2`Retained binary64: `2.0`	2Exact fraction: `2`Retained binary64: `2.0`	37/20Exact fraction: `37/20`Retained binary64: `1.85`	41/20Exact fraction: `41/20`Retained binary64: `2.05`	true
39	2Exact fraction: `2`Retained binary64: `2.0`	3Exact fraction: `3`Retained binary64: `3.0`	-34/75Exact fraction: `-34/75`Retained binary64: `-0.4533333333333333`	4/25Exact fraction: `4/25`Retained binary64: `0.16`	52/225Exact fraction: `52/225`Retained binary64: `0.2311111111111111`	true	2Exact fraction: `2`Retained binary64: `2.0`	3Exact fraction: `3`Retained binary64: `3.0`	283/150Exact fraction: `283/150`Retained binary64: `1.8866666666666667`	76/25Exact fraction: `76/25`Retained binary64: `3.04`	true
40	3Exact fraction: `3`Retained binary64: `3.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	33/85Exact fraction: `33/85`Retained binary64: `0.38823529411764707`	21/85Exact fraction: `21/85`Retained binary64: `0.24705882352941178`	18/85Exact fraction: `18/85`Retained binary64: `0.21176470588235294`	true	3Exact fraction: `3`Retained binary64: `3.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	1053/340Exact fraction: `1053/340`Retained binary64: `3.097058823529412`	-999/340Exact fraction: `-999/340`Retained binary64: `-2.9382352941176473`	true
41	3Exact fraction: `3`Retained binary64: `3.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	68/145Exact fraction: `68/145`Retained binary64: `0.4689655172413793`	54/145Exact fraction: `54/145`Retained binary64: `0.3724137931034483`	52/145Exact fraction: `52/145`Retained binary64: `0.3586206896551724`	true	3Exact fraction: `3`Retained binary64: `3.0`	-2Exact fraction: `-2`Retained binary64: `-2.0`	452/145Exact fraction: `452/145`Retained binary64: `3.117241379310345`	-553/290Exact fraction: `-553/290`Retained binary64: `-1.9068965517241379`	true
42	3Exact fraction: `3`Retained binary64: `3.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	7/13Exact fraction: `7/13`Retained binary64: `0.5384615384615384`	9/13Exact fraction: `9/13`Retained binary64: `0.6923076923076923`	10/13Exact fraction: `10/13`Retained binary64: `0.7692307692307693`	true	3Exact fraction: `3`Retained binary64: `3.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	163/52Exact fraction: `163/52`Retained binary64: `3.1346153846153846`	-43/52Exact fraction: `-43/52`Retained binary64: `-0.8269230769230769`	true
43	3Exact fraction: `3`Retained binary64: `3.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	6/5Exact fraction: `6/5`Retained binary64: `1.2`	36/25Exact fraction: `36/25`Retained binary64: `1.44`	true	3Exact fraction: `3`Retained binary64: `3.0`	0Exact fraction: `0`Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`	3/10Exact fraction: `3/10`Retained binary64: `0.3`	true
44	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	-7/13Exact fraction: `-7/13`Retained binary64: `-0.5384615384615384`	9/13Exact fraction: `9/13`Retained binary64: `0.6923076923076923`	10/13Exact fraction: `10/13`Retained binary64: `0.7692307692307693`	true	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	149/52Exact fraction: `149/52`Retained binary64: `2.8653846153846154`	61/52Exact fraction: `61/52`Retained binary64: `1.1730769230769231`	true
45	3Exact fraction: `3`Retained binary64: `3.0`	2Exact fraction: `2`Retained binary64: `2.0`	-68/145Exact fraction: `-68/145`Retained binary64: `-0.4689655172413793`	54/145Exact fraction: `54/145`Retained binary64: `0.3724137931034483`	52/145Exact fraction: `52/145`Retained binary64: `0.3586206896551724`	true	3Exact fraction: `3`Retained binary64: `3.0`	2Exact fraction: `2`Retained binary64: `2.0`	418/145Exact fraction: `418/145`Retained binary64: `2.882758620689655`	607/290Exact fraction: `607/290`Retained binary64: `2.093103448275862`	true
46	3Exact fraction: `3`Retained binary64: `3.0`	3Exact fraction: `3`Retained binary64: `3.0`	-33/85Exact fraction: `-33/85`Retained binary64: `-0.38823529411764707`	21/85Exact fraction: `21/85`Retained binary64: `0.24705882352941178`	18/85Exact fraction: `18/85`Retained binary64: `0.21176470588235294`	true	3Exact fraction: `3`Retained binary64: `3.0`	3Exact fraction: `3`Retained binary64: `3.0`	987/340Exact fraction: `987/340`Retained binary64: `2.902941176470588`	1041/340Exact fraction: `1041/340`Retained binary64: `3.0617647058823527`	true

All 129 contour records, both components and exact cardinal masks (129 rows)

All 129 contour records, both components and exact cardinal masks
Row	Circle sample index	Exact intended θ/π sampling label	Retained binary64 θ	X=x/L₀	Y=y/L₀	Sine with declared cardinal mask	Cosine with declared cardinal mask	Signed B_t/B₀ (positive counterclockwise)	Signed outward B_r/B₀	Exact analytic cardinal mask?
0	0	0Exact fraction: `0`Retained binary64: `0.0`	0Retained binary64: `0.0`binary64;12 significant-digit display; exact zero sign canonicalized only	3Exact fraction: `3`Retained binary64: `3.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`	6/5Exact fraction: `6/5`Retained binary64: `1.2`	0Exact fraction: `0`Retained binary64: `0.0`	true
1	1	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	0.0490873852123Retained binary64: `0.04908738521234052`binary64;12 significant-digit display; exact zero sign canonicalized only	2.99638636862Retained binary64: `2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	0.147203022982Retained binary64: `0.14720302298225404`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0490676743274Retained binary64: `0.049067674327418015`binary64;12 significant-digit display; exact zero sign canonicalized only	0.998795456205Retained binary64: `0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	1.18814547616Retained binary64: `1.188145476158511`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0928093754866Retained binary64: `-0.09280937548656867`binary64;12 significant-digit display; exact zero sign canonicalized only	false
2	2	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	0.0981747704247Retained binary64: `0.09817477042468103`binary64;12 significant-digit display; exact zero sign canonicalized only	2.98555418002Retained binary64: `2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	0.294051420989Retained binary64: `0.2940514209886818`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0980171403296Retained binary64: `0.0980171403295606`binary64;12 significant-digit display; exact zero sign canonicalized only	0.995184726672Retained binary64: `0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	1.15455491769Retained binary64: `1.1545549176859664`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.17746602462Retained binary64: `-0.17746602462049874`binary64;12 significant-digit display; exact zero sign canonicalized only	false
3	3	3/64Exact fraction: `3/64`Retained binary64: `0.046875`	0.147262155637Retained binary64: `0.14726215563702155`binary64;12 significant-digit display; exact zero sign canonicalized only	2.96752952989Retained binary64: `2.967529529894343`binary64;12 significant-digit display; exact zero sign canonicalized only	0.440191423366Retained binary64: `0.44019142336608524`binary64;12 significant-digit display; exact zero sign canonicalized only	0.146730474455Retained binary64: `0.14673047445536175`binary64;12 significant-digit display; exact zero sign canonicalized only	0.989176509965Retained binary64: `0.989176509964781`binary64;12 significant-digit display; exact zero sign canonicalized only	1.10438096927Retained binary64: `1.1043809692671784`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.24792736336Retained binary64: `-0.24792736336047408`binary64;12 significant-digit display; exact zero sign canonicalized only	false
4	4	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	0.196349540849Retained binary64: `0.19634954084936207`binary64;12 significant-digit display; exact zero sign canonicalized only	2.94235584121Retained binary64: `2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	0.585270966048Retained binary64: `0.5852709660483848`binary64;12 significant-digit display; exact zero sign canonicalized only	0.195090322016Retained binary64: `0.19509032201612825`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980785280403Retained binary64: `0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	1.04416626665Retained binary64: `1.0441662666512623`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.301318877642Retained binary64: `-0.30131887764212295`binary64;12 significant-digit display; exact zero sign canonicalized only	false
5	5	5/64Exact fraction: `5/64`Retained binary64: `0.078125`	0.245436926062Retained binary64: `0.2454369260617026`binary64;12 significant-digit display; exact zero sign canonicalized only	2.91009375958Retained binary64: `2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	0.72894053971Retained binary64: `0.7289405397097917`binary64;12 significant-digit display; exact zero sign canonicalized only	0.242980179903Retained binary64: `0.24298017990326387`binary64;12 significant-digit display; exact zero sign canonicalized only	0.970031253195Retained binary64: `0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980067282262Retained binary64: `0.9800672822617179`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.337700240994Retained binary64: `-0.3377002409938321`binary64;12 significant-digit display; exact zero sign canonicalized only	false
6	6	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	0.294524311274Retained binary64: `0.2945243112740431`binary64;12 significant-digit display; exact zero sign canonicalized only	2.8708210072Retained binary64: `2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	0.870854031763Retained binary64: `0.870854031763387`binary64;12 significant-digit display; exact zero sign canonicalized only	0.290284677254Retained binary64: `0.29028467725446233`binary64;12 significant-digit display; exact zero sign canonicalized only	0.956940335732Retained binary64: `0.9569403357322088`binary64;12 significant-digit display; exact zero sign canonicalized only	0.916802849799Retained binary64: `0.9168028497993616`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.359067650093Retained binary64: `-0.35906765009302066`binary64;12 significant-digit display; exact zero sign canonicalized only	false
7	7	7/64Exact fraction: `7/64`Retained binary64: `0.109375`	0.343611696486Retained binary64: `0.3436116964863836`binary64;12 significant-digit display; exact zero sign canonicalized only	2.82463219555Retained binary64: `2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	1.01066956018Retained binary64: `1.0106695601766602`binary64;12 significant-digit display; exact zero sign canonicalized only	0.336889853392Retained binary64: `0.33688985339222005`binary64;12 significant-digit display; exact zero sign canonicalized only	0.941544065183Retained binary64: `0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	0.8574011812Retained binary64: `0.8574011812000413`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.36826905947Retained binary64: `-0.368269059470106`binary64;12 significant-digit display; exact zero sign canonicalized only	false
8	8	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0.392699081699Retained binary64: `0.39269908169872414`binary64;12 significant-digit display; exact zero sign canonicalized only	2.77163859753Retained binary64: `2.77163859753386`binary64;12 significant-digit display; exact zero sign canonicalized only	1.1480502971Retained binary64: `1.1480502970952693`binary64;12 significant-digit display; exact zero sign canonicalized only	0.382683432365Retained binary64: `0.3826834323650898`binary64;12 significant-digit display; exact zero sign canonicalized only	0.923879532511Retained binary64: `0.9238795325112867`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803445295387Retained binary64: `0.8034452953866823`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.368218363885Retained binary64: `-0.36821836388474133`binary64;12 significant-digit display; exact zero sign canonicalized only	false
9	9	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	0.441786466911Retained binary64: `0.44178646691106466`binary64;12 significant-digit display; exact zero sign canonicalized only	2.71196787937Retained binary64: `2.71196787937033`binary64;12 significant-digit display; exact zero sign canonicalized only	1.28266528029Retained binary64: `1.2826652802908463`binary64;12 significant-digit display; exact zero sign canonicalized only	0.42755509343Retained binary64: `0.4275550934302821`binary64;12 significant-digit display; exact zero sign canonicalized only	0.903989293123Retained binary64: `0.9039892931234433`binary64;12 significant-digit display; exact zero sign canonicalized only	0.755490613796Retained binary64: `0.7554906137960373`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.36147543592Retained binary64: `-0.3614754359197161`binary64;12 significant-digit display; exact zero sign canonicalized only	false
10	10	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	0.490873852123Retained binary64: `0.4908738521234052`binary64;12 significant-digit display; exact zero sign canonicalized only	2.64576379305Retained binary64: `2.6457637930450653`binary64;12 significant-digit display; exact zero sign canonicalized only	1.41419021048Retained binary64: `1.4141902104779929`binary64;12 significant-digit display; exact zero sign canonicalized only	0.471396736826Retained binary64: `0.47139673682599764`binary64;12 significant-digit display; exact zero sign canonicalized only	0.881921264348Retained binary64: `0.881921264348355`binary64;12 significant-digit display; exact zero sign canonicalized only	0.713457339879Retained binary64: `0.7134573398794789`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.350098959157Retained binary64: `-0.35009895915671696`binary64;12 significant-digit display; exact zero sign canonicalized only	false
11	11	11/64Exact fraction: `11/64`Retained binary64: `0.171875`	0.539961237336Retained binary64: `0.5399612373357456`binary64;12 significant-digit display; exact zero sign canonicalized only	2.57318583Retained binary64: `2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	1.54230823258Retained binary64: `1.542308232579665`binary64;12 significant-digit display; exact zero sign canonicalized only	0.514102744193Retained binary64: `0.5141027441932217`binary64;12 significant-digit display; exact zero sign canonicalized only	0.85772861Retained binary64: `0.8577286100002721`binary64;12 significant-digit display; exact zero sign canonicalized only	0.676924529232Retained binary64: `0.6769245292321222`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.335653346111Retained binary64: `-0.33565334611128717`binary64;12 significant-digit display; exact zero sign canonicalized only	false
12	12	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	0.589048622548Retained binary64: `0.5890486225480862`binary64;12 significant-digit display; exact zero sign canonicalized only	2.49440883691Retained binary64: `2.4944088369076356`binary64;12 significant-digit display; exact zero sign canonicalized only	1.66671069906Retained binary64: `1.6667106990588065`binary64;12 significant-digit display; exact zero sign canonicalized only	0.55557023302Retained binary64: `0.5555702330196022`binary64;12 significant-digit display; exact zero sign canonicalized only	0.831469612303Retained binary64: `0.8314696123025452`binary64;12 significant-digit display; exact zero sign canonicalized only	0.645322782685Retained binary64: `0.6453227826850789`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.319281969174Retained binary64: `-0.31928196917414164`binary64;12 significant-digit display; exact zero sign canonicalized only	false
13	13	13/64Exact fraction: `13/64`Retained binary64: `0.203125`	0.63813600776Retained binary64: `0.6381360077604268`binary64;12 significant-digit display; exact zero sign canonicalized only	2.40962259444Retained binary64: `2.409622594441935`binary64;12 significant-digit display; exact zero sign canonicalized only	1.78709791348Retained binary64: `1.7870979134773002`binary64;12 significant-digit display; exact zero sign canonicalized only	0.595699304492Retained binary64: `0.5956993044924334`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803207531481Retained binary64: `0.8032075314806449`binary64;12 significant-digit display; exact zero sign canonicalized only	0.618048288686Retained binary64: `0.6180482886858799`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.301796556882Retained binary64: `-0.30179655688214413`binary64;12 significant-digit display; exact zero sign canonicalized only	false
14	14	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	0.687223392973Retained binary64: `0.6872233929727672`binary64;12 significant-digit display; exact zero sign canonicalized only	2.31903136009Retained binary64: `2.319031360088211`binary64;12 significant-digit display; exact zero sign canonicalized only	1.90317985249Retained binary64: `1.9031798524909365`binary64;12 significant-digit display; exact zero sign canonicalized only	0.634393284164Retained binary64: `0.6343932841636455`binary64;12 significant-digit display; exact zero sign canonicalized only	0.773010453363Retained binary64: `0.773010453362737`binary64;12 significant-digit display; exact zero sign canonicalized only	0.59452384969Retained binary64: `0.5945238496900943`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.28375954762Retained binary64: `-0.2837595476200214`binary64;12 significant-digit display; exact zero sign canonicalized only	false
15	15	15/64Exact fraction: `15/64`Retained binary64: `0.234375`	0.736310778185Retained binary64: `0.7363107781851077`binary64;12 significant-digit display; exact zero sign canonicalized only	2.22285337606Retained binary64: `2.2228533760648777`binary64;12 significant-digit display; exact zero sign canonicalized only	2.01467686454Retained binary64: `2.014676864541055`binary64;12 significant-digit display; exact zero sign canonicalized only	0.671558954847Retained binary64: `0.6715589548470183`binary64;12 significant-digit display; exact zero sign canonicalized only	0.740951125355Retained binary64: `0.7409511253549592`binary64;12 significant-digit display; exact zero sign canonicalized only	0.574227266834Retained binary64: `0.5742272668338242`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.265551467115Retained binary64: `-0.2655514671149646`binary64;12 significant-digit display; exact zero sign canonicalized only	false
16	16	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0.785398163397Retained binary64: `0.7853981633974483`binary64;12 significant-digit display; exact zero sign canonicalized only	2.12132034356Retained binary64: `2.121320343559643`binary64;12 significant-digit display; exact zero sign canonicalized only	2.12132034356Retained binary64: `2.121320343559643`binary64;12 significant-digit display; exact zero sign canonicalized only	0.707106781187Retained binary64: `0.7071067811865476`binary64;12 significant-digit display; exact zero sign canonicalized only	0.707106781187Retained binary64: `0.7071067811865476`binary64;12 significant-digit display; exact zero sign canonicalized only	0.556701030928Retained binary64: `0.5567010309278352`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.247422680412Retained binary64: `-0.2474226804123712`binary64;12 significant-digit display; exact zero sign canonicalized only	false
17	17	17/64Exact fraction: `17/64`Retained binary64: `0.265625`	0.83448554861Retained binary64: `0.8344855486097889`binary64;12 significant-digit display; exact zero sign canonicalized only	2.01467686454Retained binary64: `2.014676864541055`binary64;12 significant-digit display; exact zero sign canonicalized only	2.22285337606Retained binary64: `2.222853376064877`binary64;12 significant-digit display; exact zero sign canonicalized only	0.740951125355Retained binary64: `0.7409511253549591`binary64;12 significant-digit display; exact zero sign canonicalized only	0.671558954847Retained binary64: `0.6715589548470183`binary64;12 significant-digit display; exact zero sign canonicalized only	0.541552078078Retained binary64: `0.5415520780778695`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.229531696146Retained binary64: `-0.22953169614640892`binary64;12 significant-digit display; exact zero sign canonicalized only	false
18	18	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	0.883572933822Retained binary64: `0.8835729338221293`binary64;12 significant-digit display; exact zero sign canonicalized only	1.90317985249Retained binary64: `1.9031798524909365`binary64;12 significant-digit display; exact zero sign canonicalized only	2.31903136009Retained binary64: `2.3190313600882106`binary64;12 significant-digit display; exact zero sign canonicalized only	0.773010453363Retained binary64: `0.7730104533627369`binary64;12 significant-digit display; exact zero sign canonicalized only	0.634393284164Retained binary64: `0.6343932841636455`binary64;12 significant-digit display; exact zero sign canonicalized only	0.528446786213Retained binary64: `0.5284467862129726`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.211972876718Retained binary64: `-0.21197287671839937`binary64;12 significant-digit display; exact zero sign canonicalized only	false
19	19	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	0.932660319034Retained binary64: `0.9326603190344698`binary64;12 significant-digit display; exact zero sign canonicalized only	1.78709791348Retained binary64: `1.7870979134773004`binary64;12 significant-digit display; exact zero sign canonicalized only	2.40962259444Retained binary64: `2.4096225944419345`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803207531481Retained binary64: `0.8032075314806448`binary64;12 significant-digit display; exact zero sign canonicalized only	0.595699304492Retained binary64: `0.5956993044924335`binary64;12 significant-digit display; exact zero sign canonicalized only	0.517104116847Retained binary64: `0.517104116847037`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.194796194149Retained binary64: `-0.1947961941489689`binary64;12 significant-digit display; exact zero sign canonicalized only	false
20	20	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	0.981747704247Retained binary64: `0.9817477042468103`binary64;12 significant-digit display; exact zero sign canonicalized only	1.66671069906Retained binary64: `1.666710699058807`binary64;12 significant-digit display; exact zero sign canonicalized only	2.49440883691Retained binary64: `2.4944088369076356`binary64;12 significant-digit display; exact zero sign canonicalized only	0.831469612303Retained binary64: `0.8314696123025452`binary64;12 significant-digit display; exact zero sign canonicalized only	0.55557023302Retained binary64: `0.5555702330196023`binary64;12 significant-digit display; exact zero sign canonicalized only	0.507288441884Retained binary64: `0.5072884418838717`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.17802117899Retained binary64: `-0.17802117898961156`binary64;12 significant-digit display; exact zero sign canonicalized only	false
21	21	21/64Exact fraction: `21/64`Retained binary64: `0.328125`	1.03083508946Retained binary64: `1.030835089459151`binary64;12 significant-digit display; exact zero sign canonicalized only	1.54230823258Retained binary64: `1.542308232579665`binary64;12 significant-digit display; exact zero sign canonicalized only	2.57318583Retained binary64: `2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	0.85772861Retained binary64: `0.8577286100002721`binary64;12 significant-digit display; exact zero sign canonicalized only	0.514102744193Retained binary64: `0.5141027441932217`binary64;12 significant-digit display; exact zero sign canonicalized only	0.498802809184Retained binary64: `0.49880280918407266`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.161646700822Retained binary64: `-0.16164670082216517`binary64;12 significant-digit display; exact zero sign canonicalized only	false
22	22	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	1.07992247467Retained binary64: `1.0799224746714913`binary64;12 significant-digit display; exact zero sign canonicalized only	1.41419021048Retained binary64: `1.4141902104779933`binary64;12 significant-digit display; exact zero sign canonicalized only	2.64576379305Retained binary64: `2.645763793045065`binary64;12 significant-digit display; exact zero sign canonicalized only	0.881921264348Retained binary64: `0.8819212643483549`binary64;12 significant-digit display; exact zero sign canonicalized only	0.471396736826Retained binary64: `0.4713967368259978`binary64;12 significant-digit display; exact zero sign canonicalized only	0.491482964995Retained binary64: `0.49148296499543775`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.145657786623Retained binary64: `-0.14565778662337517`binary64;12 significant-digit display; exact zero sign canonicalized only	false
23	23	23/64Exact fraction: `23/64`Retained binary64: `0.359375`	1.12900985988Retained binary64: `1.1290098598838318`binary64;12 significant-digit display; exact zero sign canonicalized only	1.28266528029Retained binary64: `1.2826652802908467`binary64;12 significant-digit display; exact zero sign canonicalized only	2.71196787937Retained binary64: `2.71196787937033`binary64;12 significant-digit display; exact zero sign canonicalized only	0.903989293123Retained binary64: `0.9039892931234433`binary64;12 significant-digit display; exact zero sign canonicalized only	0.42755509343Retained binary64: `0.4275550934302822`binary64;12 significant-digit display; exact zero sign canonicalized only	0.485192220669Retained binary64: `0.48519222066852785`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.130030346599Retained binary64: `-0.13003034659880072`binary64;12 significant-digit display; exact zero sign canonicalized only	false
24	24	3/8Exact fraction: `3/8`Retained binary64: `0.375`	1.1780972451Retained binary64: `1.1780972450961724`binary64;12 significant-digit display; exact zero sign canonicalized only	1.1480502971Retained binary64: `1.1480502970952695`binary64;12 significant-digit display; exact zero sign canonicalized only	2.77163859753Retained binary64: `2.77163859753386`binary64;12 significant-digit display; exact zero sign canonicalized only	0.923879532511Retained binary64: `0.9238795325112867`binary64;12 significant-digit display; exact zero sign canonicalized only	0.382683432365Retained binary64: `0.38268343236508984`binary64;12 significant-digit display; exact zero sign canonicalized only	0.479817136768Retained binary64: `0.4798171367682244`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.114734426669Retained binary64: `-0.11473442666895743`binary64;12 significant-digit display; exact zero sign canonicalized only	false
25	25	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	1.22718463031Retained binary64: `1.227184630308513`binary64;12 significant-digit display; exact zero sign canonicalized only	1.01066956018Retained binary64: `1.0106695601766602`binary64;12 significant-digit display; exact zero sign canonicalized only	2.82463219555Retained binary64: `2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	0.941544065183Retained binary64: `0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	0.336889853392Retained binary64: `0.33688985339222005`binary64;12 significant-digit display; exact zero sign canonicalized only	0.475263947549Retained binary64: `0.4752639475490124`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0997364253884Retained binary64: `-0.09973642538841015`binary64;12 significant-digit display; exact zero sign canonicalized only	false
26	26	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	1.27627201552Retained binary64: `1.2762720155208536`binary64;12 significant-digit display; exact zero sign canonicalized only	0.870854031763Retained binary64: `0.870854031763387`binary64;12 significant-digit display; exact zero sign canonicalized only	2.8708210072Retained binary64: `2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	0.956940335732Retained binary64: `0.9569403357322089`binary64;12 significant-digit display; exact zero sign canonicalized only	0.290284677254Retained binary64: `0.29028467725446233`binary64;12 significant-digit display; exact zero sign canonicalized only	0.47145563276Retained binary64: `0.4714556327596614`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0850005837166Retained binary64: `-0.08500058371660249`binary64;12 significant-digit display; exact zero sign canonicalized only	false
27	27	27/64Exact fraction: `27/64`Retained binary64: `0.421875`	1.32535940073Retained binary64: `1.325359400733194`binary64;12 significant-digit display; exact zero sign canonicalized only	0.72894053971Retained binary64: `0.7289405397097919`binary64;12 significant-digit display; exact zero sign canonicalized only	2.91009375958Retained binary64: `2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	0.970031253195Retained binary64: `0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.242980179903Retained binary64: `0.24298017990326398`binary64;12 significant-digit display; exact zero sign canonicalized only	0.46832954499Retained binary64: `0.46832954498989426`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0704899646682Retained binary64: `-0.07048996466818105`binary64;12 significant-digit display; exact zero sign canonicalized only	false
28	28	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	1.37444678595Retained binary64: `1.3744467859455345`binary64;12 significant-digit display; exact zero sign canonicalized only	0.585270966048Retained binary64: `0.585270966048385`binary64;12 significant-digit display; exact zero sign canonicalized only	2.94235584121Retained binary64: `2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980785280403Retained binary64: `0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	0.195090322016Retained binary64: `0.19509032201612833`binary64;12 significant-digit display; exact zero sign canonicalized only	0.465835509997Retained binary64: `0.46583550999658746`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0561670756741Retained binary64: `-0.056167075674099266`binary64;12 significant-digit display; exact zero sign canonicalized only	false
29	29	29/64Exact fraction: `29/64`Retained binary64: `0.453125`	1.42353417116Retained binary64: `1.423534171157875`binary64;12 significant-digit display; exact zero sign canonicalized only	0.440191423366Retained binary64: `0.44019142336608524`binary64;12 significant-digit display; exact zero sign canonicalized only	2.96752952989Retained binary64: `2.967529529894343`binary64;12 significant-digit display; exact zero sign canonicalized only	0.989176509965Retained binary64: `0.989176509964781`binary64;12 significant-digit display; exact zero sign canonicalized only	0.146730474455Retained binary64: `0.14673047445536175`binary64;12 significant-digit display; exact zero sign canonicalized only	0.463934329729Retained binary64: `0.46393432972877835`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0419942415741Retained binary64: `-0.04199424157413743`binary64;12 significant-digit display; exact zero sign canonicalized only	false
30	30	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	1.47262155637Retained binary64: `1.4726215563702154`binary64;12 significant-digit display; exact zero sign canonicalized only	0.294051420989Retained binary64: `0.2940514209886823`binary64;12 significant-digit display; exact zero sign canonicalized only	2.98555418002Retained binary64: `2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	0.995184726672Retained binary64: `0.9951847266721968`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0980171403296Retained binary64: `0.09801714032956077`binary64;12 significant-digit display; exact zero sign canonicalized only	0.462596630632Retained binary64: `0.4625966306322258`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0279338048805Retained binary64: `-0.02793380488051291`binary64;12 significant-digit display; exact zero sign canonicalized only	false
31	31	31/64Exact fraction: `31/64`Retained binary64: `0.484375`	1.52170894158Retained binary64: `1.521708941582556`binary64;12 significant-digit display; exact zero sign canonicalized only	0.147203022982Retained binary64: `0.14720302298225438`binary64;12 significant-digit display; exact zero sign canonicalized only	2.99638636862Retained binary64: `2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	0.998795456205Retained binary64: `0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0490676743274Retained binary64: `0.049067674327418126`binary64;12 significant-digit display; exact zero sign canonicalized only	0.461802012129Retained binary64: `0.4618020121288596`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.013948208327Retained binary64: `-0.013948208327026385`binary64;12 significant-digit display; exact zero sign canonicalized only	false
32	32	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1.57079632679Retained binary64: `1.5707963267948966`binary64;12 significant-digit display; exact zero sign canonicalized only	0Exact fraction: `0`Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	6/13Exact fraction: `6/13`Retained binary64: `0.46153846153846156`	0Exact fraction: `0`Retained binary64: `0.0`	true
33	33	33/64Exact fraction: `33/64`Retained binary64: `0.515625`	1.61988371201Retained binary64: `1.6198837120072371`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.147203022982Retained binary64: `-0.14720302298225402`binary64;12 significant-digit display; exact zero sign canonicalized only	2.99638636862Retained binary64: `2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	0.998795456205Retained binary64: `0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0490676743274Retained binary64: `-0.04906767432741801`binary64;12 significant-digit display; exact zero sign canonicalized only	0.461802012129Retained binary64: `0.4618020121288596`binary64;12 significant-digit display; exact zero sign canonicalized only	0.013948208327Retained binary64: `0.013948208327026352`binary64;12 significant-digit display; exact zero sign canonicalized only	false
34	34	17/32Exact fraction: `17/32`Retained binary64: `0.53125`	1.66897109722Retained binary64: `1.6689710972195777`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.294051420989Retained binary64: `-0.2940514209886819`binary64;12 significant-digit display; exact zero sign canonicalized only	2.98555418002Retained binary64: `2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	0.995184726672Retained binary64: `0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0980171403296Retained binary64: `-0.09801714032956065`binary64;12 significant-digit display; exact zero sign canonicalized only	0.462596630632Retained binary64: `0.4625966306322257`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0279338048805Retained binary64: `0.027933804880512867`binary64;12 significant-digit display; exact zero sign canonicalized only	false
35	35	35/64Exact fraction: `35/64`Retained binary64: `0.546875`	1.71805848243Retained binary64: `1.718058482431918`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.440191423366Retained binary64: `-0.4401914233660849`binary64;12 significant-digit display; exact zero sign canonicalized only	2.96752952989Retained binary64: `2.967529529894343`binary64;12 significant-digit display; exact zero sign canonicalized only	0.989176509965Retained binary64: `0.989176509964781`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.146730474455Retained binary64: `-0.14673047445536164`binary64;12 significant-digit display; exact zero sign canonicalized only	0.463934329729Retained binary64: `0.46393432972877824`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0419942415741Retained binary64: `0.041994241574137385`binary64;12 significant-digit display; exact zero sign canonicalized only	false
36	36	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	1.76714586764Retained binary64: `1.7671458676442586`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.585270966048Retained binary64: `-0.5852709660483846`binary64;12 significant-digit display; exact zero sign canonicalized only	2.94235584121Retained binary64: `2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980785280403Retained binary64: `0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.195090322016Retained binary64: `-0.1950903220161282`binary64;12 significant-digit display; exact zero sign canonicalized only	0.465835509997Retained binary64: `0.46583550999658757`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0561670756741Retained binary64: `0.056167075674099225`binary64;12 significant-digit display; exact zero sign canonicalized only	false
37	37	37/64Exact fraction: `37/64`Retained binary64: `0.578125`	1.81623325286Retained binary64: `1.8162332528565992`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.72894053971Retained binary64: `-0.7289405397097917`binary64;12 significant-digit display; exact zero sign canonicalized only	2.91009375958Retained binary64: `2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	0.970031253195Retained binary64: `0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.242980179903Retained binary64: `-0.24298017990326387`binary64;12 significant-digit display; exact zero sign canonicalized only	0.46832954499Retained binary64: `0.46832954498989426`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0704899646682Retained binary64: `0.07048996466818103`binary64;12 significant-digit display; exact zero sign canonicalized only	false
38	38	19/32Exact fraction: `19/32`Retained binary64: `0.59375`	1.86532063807Retained binary64: `1.8653206380689396`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.870854031763Retained binary64: `-0.8708540317633865`binary64;12 significant-digit display; exact zero sign canonicalized only	2.8708210072Retained binary64: `2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	0.956940335732Retained binary64: `0.9569403357322089`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.290284677254Retained binary64: `-0.29028467725446216`binary64;12 significant-digit display; exact zero sign canonicalized only	0.47145563276Retained binary64: `0.4714556327596614`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0850005837166Retained binary64: `0.08500058371660243`binary64;12 significant-digit display; exact zero sign canonicalized only	false
39	39	39/64Exact fraction: `39/64`Retained binary64: `0.609375`	1.91440802328Retained binary64: `1.9144080232812801`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.01066956018Retained binary64: `-1.0106695601766598`binary64;12 significant-digit display; exact zero sign canonicalized only	2.82463219555Retained binary64: `2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	0.941544065183Retained binary64: `0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.336889853392Retained binary64: `-0.33688985339221994`binary64;12 significant-digit display; exact zero sign canonicalized only	0.475263947549Retained binary64: `0.47526394754901236`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0997364253884Retained binary64: `0.0997364253884101`binary64;12 significant-digit display; exact zero sign canonicalized only	false
40	40	5/8Exact fraction: `5/8`Retained binary64: `0.625`	1.96349540849Retained binary64: `1.9634954084936207`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.1480502971Retained binary64: `-1.1480502970952693`binary64;12 significant-digit display; exact zero sign canonicalized only	2.77163859753Retained binary64: `2.77163859753386`binary64;12 significant-digit display; exact zero sign canonicalized only	0.923879532511Retained binary64: `0.9238795325112867`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.382683432365Retained binary64: `-0.3826834323650897`binary64;12 significant-digit display; exact zero sign canonicalized only	0.479817136768Retained binary64: `0.47981713676822435`binary64;12 significant-digit display; exact zero sign canonicalized only	0.114734426669Retained binary64: `0.11473442666895739`binary64;12 significant-digit display; exact zero sign canonicalized only	false
41	41	41/64Exact fraction: `41/64`Retained binary64: `0.640625`	2.01258279371Retained binary64: `2.012582793705961`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.28266528029Retained binary64: `-1.2826652802908456`binary64;12 significant-digit display; exact zero sign canonicalized only	2.71196787937Retained binary64: `2.7119678793703303`binary64;12 significant-digit display; exact zero sign canonicalized only	0.903989293123Retained binary64: `0.9039892931234434`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.42755509343Retained binary64: `-0.42755509343028186`binary64;12 significant-digit display; exact zero sign canonicalized only	0.485192220669Retained binary64: `0.48519222066852774`binary64;12 significant-digit display; exact zero sign canonicalized only	0.130030346599Retained binary64: `0.1300303465988006`binary64;12 significant-digit display; exact zero sign canonicalized only	false
42	42	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	2.06167017892Retained binary64: `2.061670178918302`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.41419021048Retained binary64: `-1.414190210477993`binary64;12 significant-digit display; exact zero sign canonicalized only	2.64576379305Retained binary64: `2.6457637930450653`binary64;12 significant-digit display; exact zero sign canonicalized only	0.881921264348Retained binary64: `0.881921264348355`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.471396736826Retained binary64: `-0.4713967368259977`binary64;12 significant-digit display; exact zero sign canonicalized only	0.491482964995Retained binary64: `0.49148296499543787`binary64;12 significant-digit display; exact zero sign canonicalized only	0.145657786623Retained binary64: `0.14565778662337517`binary64;12 significant-digit display; exact zero sign canonicalized only	false
43	43	43/64Exact fraction: `43/64`Retained binary64: `0.671875`	2.11075756413Retained binary64: `2.110757564130642`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.54230823258Retained binary64: `-1.5423082325796647`binary64;12 significant-digit display; exact zero sign canonicalized only	2.57318583Retained binary64: `2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	0.85772861Retained binary64: `0.8577286100002721`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.514102744193Retained binary64: `-0.5141027441932216`binary64;12 significant-digit display; exact zero sign canonicalized only	0.498802809184Retained binary64: `0.49880280918407255`binary64;12 significant-digit display; exact zero sign canonicalized only	0.161646700822Retained binary64: `0.1616467008221651`binary64;12 significant-digit display; exact zero sign canonicalized only	false
44	44	11/16Exact fraction: `11/16`Retained binary64: `0.6875`	2.15984494934Retained binary64: `2.1598449493429825`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.66671069906Retained binary64: `-1.6667106990588059`binary64;12 significant-digit display; exact zero sign canonicalized only	2.49440883691Retained binary64: `2.494408836907636`binary64;12 significant-digit display; exact zero sign canonicalized only	0.831469612303Retained binary64: `0.8314696123025453`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.55557023302Retained binary64: `-0.555570233019602`binary64;12 significant-digit display; exact zero sign canonicalized only	0.507288441884Retained binary64: `0.5072884418838717`binary64;12 significant-digit display; exact zero sign canonicalized only	0.17802117899Retained binary64: `0.1780211789896114`binary64;12 significant-digit display; exact zero sign canonicalized only	false
45	45	45/64Exact fraction: `45/64`Retained binary64: `0.703125`	2.20893233456Retained binary64: `2.2089323345553233`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.78709791348Retained binary64: `-1.7870979134773002`binary64;12 significant-digit display; exact zero sign canonicalized only	2.40962259444Retained binary64: `2.409622594441935`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803207531481Retained binary64: `0.8032075314806449`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.595699304492Retained binary64: `-0.5956993044924334`binary64;12 significant-digit display; exact zero sign canonicalized only	0.517104116847Retained binary64: `0.517104116847037`binary64;12 significant-digit display; exact zero sign canonicalized only	0.194796194149Retained binary64: `0.1947961941489689`binary64;12 significant-digit display; exact zero sign canonicalized only	false
46	46	23/32Exact fraction: `23/32`Retained binary64: `0.71875`	2.25801971977Retained binary64: `2.2580197197676637`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.90317985249Retained binary64: `-1.9031798524909362`binary64;12 significant-digit display; exact zero sign canonicalized only	2.31903136009Retained binary64: `2.3190313600882115`binary64;12 significant-digit display; exact zero sign canonicalized only	0.773010453363Retained binary64: `0.7730104533627371`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.634393284164Retained binary64: `-0.6343932841636454`binary64;12 significant-digit display; exact zero sign canonicalized only	0.528446786213Retained binary64: `0.5284467862129726`binary64;12 significant-digit display; exact zero sign canonicalized only	0.211972876718Retained binary64: `0.21197287671839937`binary64;12 significant-digit display; exact zero sign canonicalized only	false
47	47	47/64Exact fraction: `47/64`Retained binary64: `0.734375`	2.30710710498Retained binary64: `2.3071071049800045`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.01467686454Retained binary64: `-2.0146768645410553`binary64;12 significant-digit display; exact zero sign canonicalized only	2.22285337606Retained binary64: `2.2228533760648768`binary64;12 significant-digit display; exact zero sign canonicalized only	0.740951125355Retained binary64: `0.740951125354959`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.671558954847Retained binary64: `-0.6715589548470184`binary64;12 significant-digit display; exact zero sign canonicalized only	0.541552078078Retained binary64: `0.5415520780778695`binary64;12 significant-digit display; exact zero sign canonicalized only	0.229531696146Retained binary64: `0.22953169614640898`binary64;12 significant-digit display; exact zero sign canonicalized only	false
48	48	3/4Exact fraction: `3/4`Retained binary64: `0.75`	2.35619449019Retained binary64: `2.356194490192345`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.12132034356Retained binary64: `-2.1213203435596424`binary64;12 significant-digit display; exact zero sign canonicalized only	2.12132034356Retained binary64: `2.121320343559643`binary64;12 significant-digit display; exact zero sign canonicalized only	0.707106781187Retained binary64: `0.7071067811865476`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.707106781187Retained binary64: `-0.7071067811865475`binary64;12 significant-digit display; exact zero sign canonicalized only	0.556701030928Retained binary64: `0.556701030927835`binary64;12 significant-digit display; exact zero sign canonicalized only	0.247422680412Retained binary64: `0.2474226804123711`binary64;12 significant-digit display; exact zero sign canonicalized only	false
49	49	49/64Exact fraction: `49/64`Retained binary64: `0.765625`	2.4052818754Retained binary64: `2.405281875404685`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.22285337606Retained binary64: `-2.2228533760648768`binary64;12 significant-digit display; exact zero sign canonicalized only	2.01467686454Retained binary64: `2.0146768645410558`binary64;12 significant-digit display; exact zero sign canonicalized only	0.671558954847Retained binary64: `0.6715589548470186`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.740951125355Retained binary64: `-0.7409511253549589`binary64;12 significant-digit display; exact zero sign canonicalized only	0.574227266834Retained binary64: `0.574227266833824`binary64;12 significant-digit display; exact zero sign canonicalized only	0.265551467115Retained binary64: `0.26555146711496436`binary64;12 significant-digit display; exact zero sign canonicalized only	false
50	50	25/32Exact fraction: `25/32`Retained binary64: `0.78125`	2.45436926062Retained binary64: `2.454369260617026`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.31903136009Retained binary64: `-2.319031360088211`binary64;12 significant-digit display; exact zero sign canonicalized only	1.90317985249Retained binary64: `1.9031798524909365`binary64;12 significant-digit display; exact zero sign canonicalized only	0.634393284164Retained binary64: `0.6343932841636455`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.773010453363Retained binary64: `-0.773010453362737`binary64;12 significant-digit display; exact zero sign canonicalized only	0.59452384969Retained binary64: `0.5945238496900943`binary64;12 significant-digit display; exact zero sign canonicalized only	0.28375954762Retained binary64: `0.2837595476200214`binary64;12 significant-digit display; exact zero sign canonicalized only	false
51	51	51/64Exact fraction: `51/64`Retained binary64: `0.796875`	2.50345664583Retained binary64: `2.5034566458293663`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.40962259444Retained binary64: `-2.4096225944419345`binary64;12 significant-digit display; exact zero sign canonicalized only	1.78709791348Retained binary64: `1.7870979134773004`binary64;12 significant-digit display; exact zero sign canonicalized only	0.595699304492Retained binary64: `0.5956993044924335`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.803207531481Retained binary64: `-0.8032075314806448`binary64;12 significant-digit display; exact zero sign canonicalized only	0.618048288686Retained binary64: `0.6180482886858798`binary64;12 significant-digit display; exact zero sign canonicalized only	0.301796556882Retained binary64: `0.301796556882144`binary64;12 significant-digit display; exact zero sign canonicalized only	false
52	52	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	2.55254403104Retained binary64: `2.552544031041707`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.49440883691Retained binary64: `-2.494408836907636`binary64;12 significant-digit display; exact zero sign canonicalized only	1.66671069906Retained binary64: `1.6667106990588065`binary64;12 significant-digit display; exact zero sign canonicalized only	0.55557023302Retained binary64: `0.5555702330196022`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.831469612303Retained binary64: `-0.8314696123025453`binary64;12 significant-digit display; exact zero sign canonicalized only	0.645322782685Retained binary64: `0.645322782685079`binary64;12 significant-digit display; exact zero sign canonicalized only	0.319281969174Retained binary64: `0.3192819691741418`binary64;12 significant-digit display; exact zero sign canonicalized only	false
53	53	53/64Exact fraction: `53/64`Retained binary64: `0.828125`	2.60163141625Retained binary64: `2.6016314162540475`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.57318583Retained binary64: `-2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	1.54230823258Retained binary64: `1.5423082325796653`binary64;12 significant-digit display; exact zero sign canonicalized only	0.514102744193Retained binary64: `0.5141027441932218`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.85772861Retained binary64: `-0.857728610000272`binary64;12 significant-digit display; exact zero sign canonicalized only	0.676924529232Retained binary64: `0.6769245292321221`binary64;12 significant-digit display; exact zero sign canonicalized only	0.335653346111Retained binary64: `0.3356533461112871`binary64;12 significant-digit display; exact zero sign canonicalized only	false
54	54	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	2.65071880147Retained binary64: `2.650718801466388`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.64576379305Retained binary64: `-2.645763793045065`binary64;12 significant-digit display; exact zero sign canonicalized only	1.41419021048Retained binary64: `1.4141902104779933`binary64;12 significant-digit display; exact zero sign canonicalized only	0.471396736826Retained binary64: `0.4713967368259978`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.881921264348Retained binary64: `-0.8819212643483549`binary64;12 significant-digit display; exact zero sign canonicalized only	0.713457339879Retained binary64: `0.7134573398794788`binary64;12 significant-digit display; exact zero sign canonicalized only	0.350098959157Retained binary64: `0.35009895915671685`binary64;12 significant-digit display; exact zero sign canonicalized only	false
55	55	55/64Exact fraction: `55/64`Retained binary64: `0.859375`	2.69980618668Retained binary64: `2.6998061866787286`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.71196787937Retained binary64: `-2.71196787937033`binary64;12 significant-digit display; exact zero sign canonicalized only	1.28266528029Retained binary64: `1.282665280290846`binary64;12 significant-digit display; exact zero sign canonicalized only	0.42755509343Retained binary64: `0.42755509343028203`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.903989293123Retained binary64: `-0.9039892931234433`binary64;12 significant-digit display; exact zero sign canonicalized only	0.755490613796Retained binary64: `0.7554906137960373`binary64;12 significant-digit display; exact zero sign canonicalized only	0.36147543592Retained binary64: `0.3614754359197161`binary64;12 significant-digit display; exact zero sign canonicalized only	false
56	56	7/8Exact fraction: `7/8`Retained binary64: `0.875`	2.74889357189Retained binary64: `2.748893571891069`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.77163859753Retained binary64: `-2.77163859753386`binary64;12 significant-digit display; exact zero sign canonicalized only	1.1480502971Retained binary64: `1.1480502970952697`binary64;12 significant-digit display; exact zero sign canonicalized only	0.382683432365Retained binary64: `0.3826834323650899`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.923879532511Retained binary64: `-0.9238795325112867`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803445295387Retained binary64: `0.8034452953866823`binary64;12 significant-digit display; exact zero sign canonicalized only	0.368218363885Retained binary64: `0.3682183638847414`binary64;12 significant-digit display; exact zero sign canonicalized only	false
57	57	57/64Exact fraction: `57/64`Retained binary64: `0.890625`	2.7979809571Retained binary64: `2.7979809571034093`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.82463219555Retained binary64: `-2.824632195549062`binary64;12 significant-digit display; exact zero sign canonicalized only	1.01066956018Retained binary64: `1.010669560176661`binary64;12 significant-digit display; exact zero sign canonicalized only	0.336889853392Retained binary64: `0.33688985339222033`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.941544065183Retained binary64: `-0.9415440651830207`binary64;12 significant-digit display; exact zero sign canonicalized only	0.8574011812Retained binary64: `0.8574011812000407`binary64;12 significant-digit display; exact zero sign canonicalized only	0.36826905947Retained binary64: `0.36826905947010585`binary64;12 significant-digit display; exact zero sign canonicalized only	false
58	58	29/32Exact fraction: `29/32`Retained binary64: `0.90625`	2.84706834232Retained binary64: `2.84706834231575`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.8708210072Retained binary64: `-2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	0.870854031763Retained binary64: `0.8708540317633872`binary64;12 significant-digit display; exact zero sign canonicalized only	0.290284677254Retained binary64: `0.2902846772544624`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.956940335732Retained binary64: `-0.9569403357322088`binary64;12 significant-digit display; exact zero sign canonicalized only	0.916802849799Retained binary64: `0.9168028497993616`binary64;12 significant-digit display; exact zero sign canonicalized only	0.359067650093Retained binary64: `0.3590676500930207`binary64;12 significant-digit display; exact zero sign canonicalized only	false
59	59	59/64Exact fraction: `59/64`Retained binary64: `0.921875`	2.89615572753Retained binary64: `2.8961557275280905`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.91009375958Retained binary64: `-2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	0.72894053971Retained binary64: `0.7289405397097922`binary64;12 significant-digit display; exact zero sign canonicalized only	0.242980179903Retained binary64: `0.24298017990326407`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.970031253195Retained binary64: `-0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980067282262Retained binary64: `0.9800672822617179`binary64;12 significant-digit display; exact zero sign canonicalized only	0.337700240994Retained binary64: `0.33770024099383245`binary64;12 significant-digit display; exact zero sign canonicalized only	false
60	60	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	2.94524311274Retained binary64: `2.945243112740431`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.94235584121Retained binary64: `-2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	0.585270966048Retained binary64: `0.5852709660483858`binary64;12 significant-digit display; exact zero sign canonicalized only	0.195090322016Retained binary64: `0.1950903220161286`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.980785280403Retained binary64: `-0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	1.04416626665Retained binary64: `1.0441662666512623`binary64;12 significant-digit display; exact zero sign canonicalized only	0.301318877642Retained binary64: `0.30131887764212356`binary64;12 significant-digit display; exact zero sign canonicalized only	false
61	61	61/64Exact fraction: `61/64`Retained binary64: `0.953125`	2.99433049795Retained binary64: `2.9943304979527716`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.96752952989Retained binary64: `-2.967529529894343`binary64;12 significant-digit display; exact zero sign canonicalized only	0.440191423366Retained binary64: `0.4401914233660854`binary64;12 significant-digit display; exact zero sign canonicalized only	0.146730474455Retained binary64: `0.1467304744553618`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.989176509965Retained binary64: `-0.989176509964781`binary64;12 significant-digit display; exact zero sign canonicalized only	1.10438096927Retained binary64: `1.1043809692671784`binary64;12 significant-digit display; exact zero sign canonicalized only	0.24792736336Retained binary64: `0.24792736336047416`binary64;12 significant-digit display; exact zero sign canonicalized only	false
62	62	31/32Exact fraction: `31/32`Retained binary64: `0.96875`	3.04341788317Retained binary64: `3.043417883165112`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.98555418002Retained binary64: `-2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	0.294051420989Retained binary64: `0.2940514209886825`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0980171403296Retained binary64: `0.09801714032956083`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.995184726672Retained binary64: `-0.9951847266721968`binary64;12 significant-digit display; exact zero sign canonicalized only	1.15455491769Retained binary64: `1.1545549176859657`binary64;12 significant-digit display; exact zero sign canonicalized only	0.17746602462Retained binary64: `0.17746602462049893`binary64;12 significant-digit display; exact zero sign canonicalized only	false
63	63	63/64Exact fraction: `63/64`Retained binary64: `0.984375`	3.09250526838Retained binary64: `3.0925052683774528`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.99638636862Retained binary64: `-2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	0.147203022982Retained binary64: `0.1472030229822539`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0490676743274Retained binary64: `0.049067674327417966`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.998795456205Retained binary64: `-0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	1.18814547616Retained binary64: `1.188145476158511`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0928093754866Retained binary64: `0.09280937548656856`binary64;12 significant-digit display; exact zero sign canonicalized only	false
64	64	1Exact fraction: `1`Retained binary64: `1.0`	3.14159265359Retained binary64: `3.141592653589793`binary64;12 significant-digit display; exact zero sign canonicalized only	-3Exact fraction: `-3`Retained binary64: `-3.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	6/5Exact fraction: `6/5`Retained binary64: `1.2`	0Exact fraction: `0`Retained binary64: `0.0`	true
65	65	65/64Exact fraction: `65/64`Retained binary64: `1.015625`	3.1906800388Retained binary64: `3.1906800388021335`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.99638636862Retained binary64: `-2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.147203022982Retained binary64: `-0.14720302298225318`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0490676743274Retained binary64: `-0.049067674327417724`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.998795456205Retained binary64: `-0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	1.18814547616Retained binary64: `1.188145476158511`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0928093754866Retained binary64: `-0.09280937548656812`binary64;12 significant-digit display; exact zero sign canonicalized only	false
66	66	33/32Exact fraction: `33/32`Retained binary64: `1.03125`	3.23976742401Retained binary64: `3.2397674240144743`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.98555418002Retained binary64: `-2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.294051420989Retained binary64: `-0.29405142098868176`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0980171403296Retained binary64: `-0.09801714032956059`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.995184726672Retained binary64: `-0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	1.15455491769Retained binary64: `1.1545549176859664`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.17746602462Retained binary64: `-0.1774660246204987`binary64;12 significant-digit display; exact zero sign canonicalized only	false
67	67	67/64Exact fraction: `67/64`Retained binary64: `1.046875`	3.28885480923Retained binary64: `3.2888548092268146`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.96752952989Retained binary64: `-2.967529529894343`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.440191423366Retained binary64: `-0.44019142336608474`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.146730474455Retained binary64: `-0.14673047445536158`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.989176509965Retained binary64: `-0.989176509964781`binary64;12 significant-digit display; exact zero sign canonicalized only	1.10438096927Retained binary64: `1.1043809692671784`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.24792736336Retained binary64: `-0.2479273633604738`binary64;12 significant-digit display; exact zero sign canonicalized only	false
68	68	17/16Exact fraction: `17/16`Retained binary64: `1.0625`	3.33794219444Retained binary64: `3.3379421944391554`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.94235584121Retained binary64: `-2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.585270966048Retained binary64: `-0.585270966048385`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.195090322016Retained binary64: `-0.19509032201612836`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.980785280403Retained binary64: `-0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	1.04416626665Retained binary64: `1.0441662666512623`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.301318877642Retained binary64: `-0.3013188776421232`binary64;12 significant-digit display; exact zero sign canonicalized only	false
69	69	69/64Exact fraction: `69/64`Retained binary64: `1.078125`	3.38702957965Retained binary64: `3.3870295796514958`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.91009375958Retained binary64: `-2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.72894053971Retained binary64: `-0.7289405397097914`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.242980179903Retained binary64: `-0.24298017990326382`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.970031253195Retained binary64: `-0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980067282262Retained binary64: `0.9800672822617179`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.337700240994Retained binary64: `-0.33770024099383206`binary64;12 significant-digit display; exact zero sign canonicalized only	false
70	70	35/32Exact fraction: `35/32`Retained binary64: `1.09375`	3.43611696486Retained binary64: `3.436116964863836`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.8708210072Retained binary64: `-2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.870854031763Retained binary64: `-0.8708540317633864`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.290284677254Retained binary64: `-0.2902846772544621`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.956940335732Retained binary64: `-0.9569403357322089`binary64;12 significant-digit display; exact zero sign canonicalized only	0.916802849799Retained binary64: `0.9168028497993613`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.359067650093Retained binary64: `-0.3590676500930204`binary64;12 significant-digit display; exact zero sign canonicalized only	false
71	71	71/64Exact fraction: `71/64`Retained binary64: `1.109375`	3.48520435008Retained binary64: `3.485204350076177`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.82463219555Retained binary64: `-2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.01066956018Retained binary64: `-1.0106695601766602`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.336889853392Retained binary64: `-0.3368898533922201`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.941544065183Retained binary64: `-0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	0.8574011812Retained binary64: `0.8574011812000413`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.36826905947Retained binary64: `-0.3682690594701061`binary64;12 significant-digit display; exact zero sign canonicalized only	false
72	72	9/8Exact fraction: `9/8`Retained binary64: `1.125`	3.53429173529Retained binary64: `3.5342917352885173`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.77163859753Retained binary64: `-2.7716385975338604`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.1480502971Retained binary64: `-1.148050297095269`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.382683432365Retained binary64: `-0.38268343236508967`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.923879532511Retained binary64: `-0.9238795325112868`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803445295387Retained binary64: `0.8034452953866827`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.368218363885Retained binary64: `-0.3682183638847416`binary64;12 significant-digit display; exact zero sign canonicalized only	false
73	73	73/64Exact fraction: `73/64`Retained binary64: `1.140625`	3.5833791205Retained binary64: `3.5833791205008576`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.71196787937Retained binary64: `-2.7119678793703303`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.28266528029Retained binary64: `-1.2826652802908454`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.42755509343Retained binary64: `-0.4275550934302818`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.903989293123Retained binary64: `-0.9039892931234434`binary64;12 significant-digit display; exact zero sign canonicalized only	0.755490613796Retained binary64: `0.7554906137960373`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.36147543592Retained binary64: `-0.3614754359197161`binary64;12 significant-digit display; exact zero sign canonicalized only	false
74	74	37/32Exact fraction: `37/32`Retained binary64: `1.15625`	3.63246650571Retained binary64: `3.6324665057131984`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.64576379305Retained binary64: `-2.6457637930450653`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.41419021048Retained binary64: `-1.4141902104779929`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.471396736826Retained binary64: `-0.47139673682599764`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.881921264348Retained binary64: `-0.881921264348355`binary64;12 significant-digit display; exact zero sign canonicalized only	0.713457339879Retained binary64: `0.7134573398794789`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.350098959157Retained binary64: `-0.35009895915671696`binary64;12 significant-digit display; exact zero sign canonicalized only	false
75	75	75/64Exact fraction: `75/64`Retained binary64: `1.171875`	3.68155389093Retained binary64: `3.6815538909255388`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.57318583Retained binary64: `-2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.54230823258Retained binary64: `-1.5423082325796647`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.514102744193Retained binary64: `-0.5141027441932216`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.85772861Retained binary64: `-0.8577286100002721`binary64;12 significant-digit display; exact zero sign canonicalized only	0.676924529232Retained binary64: `0.6769245292321222`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.335653346111Retained binary64: `-0.33565334611128705`binary64;12 significant-digit display; exact zero sign canonicalized only	false
76	76	19/16Exact fraction: `19/16`Retained binary64: `1.1875`	3.73064127614Retained binary64: `3.730641276137879`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.49440883691Retained binary64: `-2.4944088369076365`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.66671069906Retained binary64: `-1.6667106990588059`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.55557023302Retained binary64: `-0.555570233019602`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.831469612303Retained binary64: `-0.8314696123025455`binary64;12 significant-digit display; exact zero sign canonicalized only	0.645322782685Retained binary64: `0.6453227826850794`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.319281969174Retained binary64: `-0.319281969174142`binary64;12 significant-digit display; exact zero sign canonicalized only	false
77	77	77/64Exact fraction: `77/64`Retained binary64: `1.203125`	3.77972866135Retained binary64: `3.77972866135022`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.40962259444Retained binary64: `-2.409622594441935`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.78709791348Retained binary64: `-1.7870979134772997`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.595699304492Retained binary64: `-0.5956993044924332`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.803207531481Retained binary64: `-0.8032075314806449`binary64;12 significant-digit display; exact zero sign canonicalized only	0.618048288686Retained binary64: `0.6180482886858799`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.301796556882Retained binary64: `-0.3017965568821441`binary64;12 significant-digit display; exact zero sign canonicalized only	false
78	78	39/32Exact fraction: `39/32`Retained binary64: `1.21875`	3.82881604656Retained binary64: `3.8288160465625602`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.31903136009Retained binary64: `-2.3190313600882115`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.90317985249Retained binary64: `-1.9031798524909358`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.634393284164Retained binary64: `-0.6343932841636453`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.773010453363Retained binary64: `-0.7730104533627371`binary64;12 significant-digit display; exact zero sign canonicalized only	0.59452384969Retained binary64: `0.5945238496900944`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.28375954762Retained binary64: `-0.2837595476200215`binary64;12 significant-digit display; exact zero sign canonicalized only	false
79	79	79/64Exact fraction: `79/64`Retained binary64: `1.234375`	3.87790343177Retained binary64: `3.877903431774901`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.22285337606Retained binary64: `-2.222853376064877`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.01467686454Retained binary64: `-2.0146768645410553`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.671558954847Retained binary64: `-0.6715589548470184`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.740951125355Retained binary64: `-0.7409511253549591`binary64;12 significant-digit display; exact zero sign canonicalized only	0.574227266834Retained binary64: `0.574227266833824`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.265551467115Retained binary64: `-0.26555146711496447`binary64;12 significant-digit display; exact zero sign canonicalized only	false
80	80	5/4Exact fraction: `5/4`Retained binary64: `1.25`	3.92699081699Retained binary64: `3.9269908169872414`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.12132034356Retained binary64: `-2.121320343559643`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.12132034356Retained binary64: `-2.1213203435596424`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.707106781187Retained binary64: `-0.7071067811865475`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.707106781187Retained binary64: `-0.7071067811865477`binary64;12 significant-digit display; exact zero sign canonicalized only	0.556701030928Retained binary64: `0.5567010309278351`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.247422680412Retained binary64: `-0.2474226804123712`binary64;12 significant-digit display; exact zero sign canonicalized only	false
81	81	81/64Exact fraction: `81/64`Retained binary64: `1.265625`	3.9760782022Retained binary64: `3.9760782021995817`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.01467686454Retained binary64: `-2.014676864541056`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.22285337606Retained binary64: `-2.2228533760648768`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.740951125355Retained binary64: `-0.7409511253549589`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.671558954847Retained binary64: `-0.6715589548470187`binary64;12 significant-digit display; exact zero sign canonicalized only	0.541552078078Retained binary64: `0.5415520780778696`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.229531696146Retained binary64: `-0.22953169614640914`binary64;12 significant-digit display; exact zero sign canonicalized only	false
82	82	41/32Exact fraction: `41/32`Retained binary64: `1.28125`	4.02516558741Retained binary64: `4.025165587411922`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.90317985249Retained binary64: `-1.9031798524909378`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.31903136009Retained binary64: `-2.3190313600882098`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.773010453363Retained binary64: `-0.7730104533627367`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.634393284164Retained binary64: `-0.6343932841636459`binary64;12 significant-digit display; exact zero sign canonicalized only	0.528446786213Retained binary64: `0.5284467862129727`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.211972876718Retained binary64: `-0.21197287671839962`binary64;12 significant-digit display; exact zero sign canonicalized only	false
83	83	83/64Exact fraction: `83/64`Retained binary64: `1.296875`	4.07425297262Retained binary64: `4.074252972624263`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.78709791348Retained binary64: `-1.7870979134772993`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.40962259444Retained binary64: `-2.4096225944419354`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.803207531481Retained binary64: `-0.803207531480645`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.595699304492Retained binary64: `-0.5956993044924331`binary64;12 significant-digit display; exact zero sign canonicalized only	0.517104116847Retained binary64: `0.5171041168470369`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.194796194149Retained binary64: `-0.19479619414896876`binary64;12 significant-digit display; exact zero sign canonicalized only	false
84	84	21/16Exact fraction: `21/16`Retained binary64: `1.3125`	4.12334035784Retained binary64: `4.123340357836604`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.66671069906Retained binary64: `-1.6667106990588065`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.49440883691Retained binary64: `-2.4944088369076356`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.831469612303Retained binary64: `-0.8314696123025452`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.55557023302Retained binary64: `-0.5555702330196022`binary64;12 significant-digit display; exact zero sign canonicalized only	0.507288441884Retained binary64: `0.5072884418838717`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.17802117899Retained binary64: `-0.17802117898961148`binary64;12 significant-digit display; exact zero sign canonicalized only	false
85	85	85/64Exact fraction: `85/64`Retained binary64: `1.328125`	4.17242774305Retained binary64: `4.172427743048944`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.54230823258Retained binary64: `-1.5423082325796653`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.57318583Retained binary64: `-2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.85772861Retained binary64: `-0.857728610000272`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.514102744193Retained binary64: `-0.5141027441932218`binary64;12 significant-digit display; exact zero sign canonicalized only	0.498802809184Retained binary64: `0.49880280918407255`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.161646700822Retained binary64: `-0.16164670082216517`binary64;12 significant-digit display; exact zero sign canonicalized only	false
86	86	43/32Exact fraction: `43/32`Retained binary64: `1.34375`	4.22151512826Retained binary64: `4.221515128261284`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.41419021048Retained binary64: `-1.4141902104779935`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.64576379305Retained binary64: `-2.645763793045065`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.881921264348Retained binary64: `-0.8819212643483549`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.471396736826Retained binary64: `-0.47139673682599786`binary64;12 significant-digit display; exact zero sign canonicalized only	0.491482964995Retained binary64: `0.49148296499543775`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.145657786623Retained binary64: `-0.1456577866233752`binary64;12 significant-digit display; exact zero sign canonicalized only	false
87	87	87/64Exact fraction: `87/64`Retained binary64: `1.359375`	4.27060251347Retained binary64: `4.270602513473625`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.28266528029Retained binary64: `-1.2826652802908474`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.71196787937Retained binary64: `-2.7119678793703295`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.903989293123Retained binary64: `-0.9039892931234431`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.42755509343Retained binary64: `-0.4275550934302825`binary64;12 significant-digit display; exact zero sign canonicalized only	0.485192220669Retained binary64: `0.48519222066852796`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.130030346599Retained binary64: `-0.13003034659880083`binary64;12 significant-digit display; exact zero sign canonicalized only	false
88	88	11/8Exact fraction: `11/8`Retained binary64: `1.375`	4.31968989869Retained binary64: `4.319689898685965`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.1480502971Retained binary64: `-1.148050297095271`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.77163859753Retained binary64: `-2.7716385975338595`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.923879532511Retained binary64: `-0.9238795325112865`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.382683432365Retained binary64: `-0.38268343236509034`binary64;12 significant-digit display; exact zero sign canonicalized only	0.479817136768Retained binary64: `0.4798171367682245`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.114734426669Retained binary64: `-0.1147344266689576`binary64;12 significant-digit display; exact zero sign canonicalized only	false
89	89	89/64Exact fraction: `89/64`Retained binary64: `1.390625`	4.3687772839Retained binary64: `4.368777283898306`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.01066956018Retained binary64: `-1.0106695601766598`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.82463219555Retained binary64: `-2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.941544065183Retained binary64: `-0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.336889853392Retained binary64: `-0.33688985339221994`binary64;12 significant-digit display; exact zero sign canonicalized only	0.475263947549Retained binary64: `0.47526394754901236`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0997364253884Retained binary64: `-0.0997364253884101`binary64;12 significant-digit display; exact zero sign canonicalized only	false
90	90	45/32Exact fraction: `45/32`Retained binary64: `1.40625`	4.41786466911Retained binary64: `4.417864669110647`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.870854031763Retained binary64: `-0.8708540317633873`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.8708210072Retained binary64: `-2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.956940335732Retained binary64: `-0.9569403357322088`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.290284677254Retained binary64: `-0.29028467725446244`binary64;12 significant-digit display; exact zero sign canonicalized only	0.47145563276Retained binary64: `0.4714556327596614`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0850005837166Retained binary64: `-0.08500058371660249`binary64;12 significant-digit display; exact zero sign canonicalized only	false
91	91	91/64Exact fraction: `91/64`Retained binary64: `1.421875`	4.46695205432Retained binary64: `4.466952054322987`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.72894053971Retained binary64: `-0.7289405397097923`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.91009375958Retained binary64: `-2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.970031253195Retained binary64: `-0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.242980179903Retained binary64: `-0.24298017990326412`binary64;12 significant-digit display; exact zero sign canonicalized only	0.46832954499Retained binary64: `0.4683295449898944`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0704899646682Retained binary64: `-0.07048996466818111`binary64;12 significant-digit display; exact zero sign canonicalized only	false
92	92	23/16Exact fraction: `23/16`Retained binary64: `1.4375`	4.51603943954Retained binary64: `4.516039439535327`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.585270966048Retained binary64: `-0.585270966048386`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.94235584121Retained binary64: `-2.942355841209691`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.980785280403Retained binary64: `-0.9807852804032303`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.195090322016Retained binary64: `-0.19509032201612866`binary64;12 significant-digit display; exact zero sign canonicalized only	0.465835509997Retained binary64: `0.4658355099965875`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0561670756741Retained binary64: `-0.05616707567409936`binary64;12 significant-digit display; exact zero sign canonicalized only	false
93	93	93/64Exact fraction: `93/64`Retained binary64: `1.453125`	4.56512682475Retained binary64: `4.565126824747668`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.440191423366Retained binary64: `-0.4401914233660869`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.96752952989Retained binary64: `-2.9675295298943425`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.989176509965Retained binary64: `-0.9891765099647809`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.146730474455Retained binary64: `-0.1467304744553623`binary64;12 significant-digit display; exact zero sign canonicalized only	0.463934329729Retained binary64: `0.46393432972877824`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0419942415741Retained binary64: `-0.04199424157413757`binary64;12 significant-digit display; exact zero sign canonicalized only	false
94	94	47/32Exact fraction: `47/32`Retained binary64: `1.46875`	4.61421420996Retained binary64: `4.614214209960009`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.294051420989Retained binary64: `-0.29405142098868137`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.98555418002Retained binary64: `-2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.995184726672Retained binary64: `-0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0980171403296Retained binary64: `-0.09801714032956045`binary64;12 significant-digit display; exact zero sign canonicalized only	0.462596630632Retained binary64: `0.4625966306322257`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0279338048805Retained binary64: `-0.027933804880512815`binary64;12 significant-digit display; exact zero sign canonicalized only	false
95	95	95/64Exact fraction: `95/64`Retained binary64: `1.484375`	4.66330159517Retained binary64: `4.663301595172349`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.147203022982Retained binary64: `-0.14720302298225407`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.99638636862Retained binary64: `-2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.998795456205Retained binary64: `-0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0490676743274Retained binary64: `-0.04906767432741803`binary64;12 significant-digit display; exact zero sign canonicalized only	0.461802012129Retained binary64: `0.4618020121288596`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.013948208327Retained binary64: `-0.013948208327026358`binary64;12 significant-digit display; exact zero sign canonicalized only	false
96	96	3/2Exact fraction: `3/2`Retained binary64: `1.5`	4.71238898038Retained binary64: `4.71238898038469`binary64;12 significant-digit display; exact zero sign canonicalized only	0Exact fraction: `0`Retained binary64: `0.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`	6/13Exact fraction: `6/13`Retained binary64: `0.46153846153846156`	0Exact fraction: `0`Retained binary64: `0.0`	true
97	97	97/64Exact fraction: `97/64`Retained binary64: `1.515625`	4.7614763656Retained binary64: `4.76147636559703`binary64;12 significant-digit display; exact zero sign canonicalized only	0.147203022982Retained binary64: `0.147203022982253`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.99638636862Retained binary64: `-2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.998795456205Retained binary64: `-0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0490676743274Retained binary64: `0.04906767432741766`binary64;12 significant-digit display; exact zero sign canonicalized only	0.461802012129Retained binary64: `0.4618020121288597`binary64;12 significant-digit display; exact zero sign canonicalized only	0.013948208327Retained binary64: `0.013948208327026253`binary64;12 significant-digit display; exact zero sign canonicalized only	false
98	98	49/32Exact fraction: `49/32`Retained binary64: `1.53125`	4.81056375081Retained binary64: `4.81056375080937`binary64;12 significant-digit display; exact zero sign canonicalized only	0.294051420989Retained binary64: `0.29405142098868026`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.98555418002Retained binary64: `-2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.995184726672Retained binary64: `-0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0980171403296Retained binary64: `0.09801714032956009`binary64;12 significant-digit display; exact zero sign canonicalized only	0.462596630632Retained binary64: `0.4625966306322257`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0279338048805Retained binary64: `0.027933804880512707`binary64;12 significant-digit display; exact zero sign canonicalized only	false
99	99	99/64Exact fraction: `99/64`Retained binary64: `1.546875`	4.85965113602Retained binary64: `4.859651136021712`binary64;12 significant-digit display; exact zero sign canonicalized only	0.440191423366Retained binary64: `0.44019142336608585`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.96752952989Retained binary64: `-2.9675295298943425`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.989176509965Retained binary64: `-0.9891765099647809`binary64;12 significant-digit display; exact zero sign canonicalized only	0.146730474455Retained binary64: `0.14673047445536194`binary64;12 significant-digit display; exact zero sign canonicalized only	0.463934329729Retained binary64: `0.46393432972877824`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0419942415741Retained binary64: `0.041994241574137475`binary64;12 significant-digit display; exact zero sign canonicalized only	false
100	100	25/16Exact fraction: `25/16`Retained binary64: `1.5625`	4.90873852123Retained binary64: `4.908738521234052`binary64;12 significant-digit display; exact zero sign canonicalized only	0.585270966048Retained binary64: `0.5852709660483849`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.94235584121Retained binary64: `-2.9423558412096913`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.980785280403Retained binary64: `-0.9807852804032304`binary64;12 significant-digit display; exact zero sign canonicalized only	0.195090322016Retained binary64: `0.1950903220161283`binary64;12 significant-digit display; exact zero sign canonicalized only	0.465835509997Retained binary64: `0.46583550999658746`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0561670756741Retained binary64: `0.05616707567409925`binary64;12 significant-digit display; exact zero sign canonicalized only	false
101	101	101/64Exact fraction: `101/64`Retained binary64: `1.578125`	4.95782590645Retained binary64: `4.957825906446392`binary64;12 significant-digit display; exact zero sign canonicalized only	0.72894053971Retained binary64: `0.7289405397097912`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.91009375958Retained binary64: `-2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.970031253195Retained binary64: `-0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.242980179903Retained binary64: `0.24298017990326376`binary64;12 significant-digit display; exact zero sign canonicalized only	0.46832954499Retained binary64: `0.46832954498989426`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0704899646682Retained binary64: `0.07048996466818098`binary64;12 significant-digit display; exact zero sign canonicalized only	false
102	102	51/32Exact fraction: `51/32`Retained binary64: `1.59375`	5.00691329166Retained binary64: `5.006913291658733`binary64;12 significant-digit display; exact zero sign canonicalized only	0.870854031763Retained binary64: `0.8708540317633862`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.8708210072Retained binary64: `-2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.956940335732Retained binary64: `-0.9569403357322089`binary64;12 significant-digit display; exact zero sign canonicalized only	0.290284677254Retained binary64: `0.29028467725446205`binary64;12 significant-digit display; exact zero sign canonicalized only	0.47145563276Retained binary64: `0.4714556327596614`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0850005837166Retained binary64: `0.08500058371660238`binary64;12 significant-digit display; exact zero sign canonicalized only	false
103	103	103/64Exact fraction: `103/64`Retained binary64: `1.609375`	5.05600067687Retained binary64: `5.056000676871073`binary64;12 significant-digit display; exact zero sign canonicalized only	1.01066956018Retained binary64: `1.0106695601766589`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.82463219555Retained binary64: `-2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.941544065183Retained binary64: `-0.9415440651830209`binary64;12 significant-digit display; exact zero sign canonicalized only	0.336889853392Retained binary64: `0.3368898533922196`binary64;12 significant-digit display; exact zero sign canonicalized only	0.475263947549Retained binary64: `0.47526394754901247`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0997364253884Retained binary64: `0.09973642538841`binary64;12 significant-digit display; exact zero sign canonicalized only	false
104	104	13/8Exact fraction: `13/8`Retained binary64: `1.625`	5.10508806208Retained binary64: `5.105088062083414`binary64;12 significant-digit display; exact zero sign canonicalized only	1.1480502971Retained binary64: `1.14805029709527`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.77163859753Retained binary64: `-2.77163859753386`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.923879532511Retained binary64: `-0.9238795325112866`binary64;12 significant-digit display; exact zero sign canonicalized only	0.382683432365Retained binary64: `0.38268343236509`binary64;12 significant-digit display; exact zero sign canonicalized only	0.479817136768Retained binary64: `0.4798171367682244`binary64;12 significant-digit display; exact zero sign canonicalized only	0.114734426669Retained binary64: `0.11473442666895746`binary64;12 significant-digit display; exact zero sign canonicalized only	false
105	105	105/64Exact fraction: `105/64`Retained binary64: `1.640625`	5.1541754473Retained binary64: `5.154175447295755`binary64;12 significant-digit display; exact zero sign canonicalized only	1.28266528029Retained binary64: `1.2826652802908465`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.71196787937Retained binary64: `-2.71196787937033`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.903989293123Retained binary64: `-0.9039892931234433`binary64;12 significant-digit display; exact zero sign canonicalized only	0.42755509343Retained binary64: `0.42755509343028214`binary64;12 significant-digit display; exact zero sign canonicalized only	0.485192220669Retained binary64: `0.48519222066852785`binary64;12 significant-digit display; exact zero sign canonicalized only	0.130030346599Retained binary64: `0.1300303465988007`binary64;12 significant-digit display; exact zero sign canonicalized only	false
106	106	53/32Exact fraction: `53/32`Retained binary64: `1.65625`	5.20326283251Retained binary64: `5.203262832508095`binary64;12 significant-digit display; exact zero sign canonicalized only	1.41419021048Retained binary64: `1.4141902104779929`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.64576379305Retained binary64: `-2.6457637930450653`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.881921264348Retained binary64: `-0.881921264348355`binary64;12 significant-digit display; exact zero sign canonicalized only	0.471396736826Retained binary64: `0.4713967368259976`binary64;12 significant-digit display; exact zero sign canonicalized only	0.491482964995Retained binary64: `0.49148296499543775`binary64;12 significant-digit display; exact zero sign canonicalized only	0.145657786623Retained binary64: `0.1456577866233751`binary64;12 significant-digit display; exact zero sign canonicalized only	false
107	107	107/64Exact fraction: `107/64`Retained binary64: `1.671875`	5.25235021772Retained binary64: `5.252350217720435`binary64;12 significant-digit display; exact zero sign canonicalized only	1.54230823258Retained binary64: `1.5423082325796647`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.57318583Retained binary64: `-2.5731858300008166`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.85772861Retained binary64: `-0.8577286100002722`binary64;12 significant-digit display; exact zero sign canonicalized only	0.514102744193Retained binary64: `0.5141027441932216`binary64;12 significant-digit display; exact zero sign canonicalized only	0.498802809184Retained binary64: `0.49880280918407255`binary64;12 significant-digit display; exact zero sign canonicalized only	0.161646700822Retained binary64: `0.16164670082216512`binary64;12 significant-digit display; exact zero sign canonicalized only	false
108	108	27/16Exact fraction: `27/16`Retained binary64: `1.6875`	5.30143760293Retained binary64: `5.301437602932776`binary64;12 significant-digit display; exact zero sign canonicalized only	1.66671069906Retained binary64: `1.6667106990588056`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.49440883691Retained binary64: `-2.4944088369076365`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.831469612303Retained binary64: `-0.8314696123025455`binary64;12 significant-digit display; exact zero sign canonicalized only	0.55557023302Retained binary64: `0.5555702330196018`binary64;12 significant-digit display; exact zero sign canonicalized only	0.507288441884Retained binary64: `0.5072884418838716`binary64;12 significant-digit display; exact zero sign canonicalized only	0.17802117899Retained binary64: `0.17802117898961134`binary64;12 significant-digit display; exact zero sign canonicalized only	false
109	109	109/64Exact fraction: `109/64`Retained binary64: `1.703125`	5.35052498815Retained binary64: `5.350524988145116`binary64;12 significant-digit display; exact zero sign canonicalized only	1.78709791348Retained binary64: `1.7870979134772984`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.40962259444Retained binary64: `-2.409622594441936`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.803207531481Retained binary64: `-0.8032075314806453`binary64;12 significant-digit display; exact zero sign canonicalized only	0.595699304492Retained binary64: `0.5956993044924328`binary64;12 significant-digit display; exact zero sign canonicalized only	0.517104116847Retained binary64: `0.5171041168470369`binary64;12 significant-digit display; exact zero sign canonicalized only	0.194796194149Retained binary64: `0.19479619414896862`binary64;12 significant-digit display; exact zero sign canonicalized only	false
110	110	55/32Exact fraction: `55/32`Retained binary64: `1.71875`	5.39961237336Retained binary64: `5.399612373357457`binary64;12 significant-digit display; exact zero sign canonicalized only	1.90317985249Retained binary64: `1.9031798524909367`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.31903136009Retained binary64: `-2.3190313600882106`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.773010453363Retained binary64: `-0.7730104533627369`binary64;12 significant-digit display; exact zero sign canonicalized only	0.634393284164Retained binary64: `0.6343932841636456`binary64;12 significant-digit display; exact zero sign canonicalized only	0.528446786213Retained binary64: `0.5284467862129726`binary64;12 significant-digit display; exact zero sign canonicalized only	0.211972876718Retained binary64: `0.21197287671839946`binary64;12 significant-digit display; exact zero sign canonicalized only	false
111	111	111/64Exact fraction: `111/64`Retained binary64: `1.734375`	5.44869975857Retained binary64: `5.448699758569798`binary64;12 significant-digit display; exact zero sign canonicalized only	2.01467686454Retained binary64: `2.014676864541055`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.22285337606Retained binary64: `-2.222853376064877`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.740951125355Retained binary64: `-0.7409511253549591`binary64;12 significant-digit display; exact zero sign canonicalized only	0.671558954847Retained binary64: `0.6715589548470183`binary64;12 significant-digit display; exact zero sign canonicalized only	0.541552078078Retained binary64: `0.5415520780778695`binary64;12 significant-digit display; exact zero sign canonicalized only	0.229531696146Retained binary64: `0.22953169614640892`binary64;12 significant-digit display; exact zero sign canonicalized only	false
112	112	7/4Exact fraction: `7/4`Retained binary64: `1.75`	5.49778714378Retained binary64: `5.497787143782138`binary64;12 significant-digit display; exact zero sign canonicalized only	2.12132034356Retained binary64: `2.121320343559642`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.12132034356Retained binary64: `-2.121320343559643`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.707106781187Retained binary64: `-0.7071067811865477`binary64;12 significant-digit display; exact zero sign canonicalized only	0.707106781187Retained binary64: `0.7071067811865474`binary64;12 significant-digit display; exact zero sign canonicalized only	0.556701030928Retained binary64: `0.556701030927835`binary64;12 significant-digit display; exact zero sign canonicalized only	0.247422680412Retained binary64: `0.24742268041237103`binary64;12 significant-digit display; exact zero sign canonicalized only	false
113	113	113/64Exact fraction: `113/64`Retained binary64: `1.765625`	5.54687452899Retained binary64: `5.546874528994478`binary64;12 significant-digit display; exact zero sign canonicalized only	2.22285337606Retained binary64: `2.2228533760648768`binary64;12 significant-digit display; exact zero sign canonicalized only	-2.01467686454Retained binary64: `-2.014676864541056`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.671558954847Retained binary64: `-0.6715589548470187`binary64;12 significant-digit display; exact zero sign canonicalized only	0.740951125355Retained binary64: `0.7409511253549589`binary64;12 significant-digit display; exact zero sign canonicalized only	0.574227266834Retained binary64: `0.574227266833824`binary64;12 significant-digit display; exact zero sign canonicalized only	0.265551467115Retained binary64: `0.26555146711496436`binary64;12 significant-digit display; exact zero sign canonicalized only	false
114	114	57/32Exact fraction: `57/32`Retained binary64: `1.78125`	5.59596191421Retained binary64: `5.595961914206819`binary64;12 significant-digit display; exact zero sign canonicalized only	2.31903136009Retained binary64: `2.3190313600882098`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.90317985249Retained binary64: `-1.9031798524909378`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.634393284164Retained binary64: `-0.6343932841636459`binary64;12 significant-digit display; exact zero sign canonicalized only	0.773010453363Retained binary64: `0.7730104533627365`binary64;12 significant-digit display; exact zero sign canonicalized only	0.59452384969Retained binary64: `0.594523849690094`binary64;12 significant-digit display; exact zero sign canonicalized only	0.28375954762Retained binary64: `0.2837595476200211`binary64;12 significant-digit display; exact zero sign canonicalized only	false
115	115	115/64Exact fraction: `115/64`Retained binary64: `1.796875`	5.64504929942Retained binary64: `5.64504929941916`binary64;12 significant-digit display; exact zero sign canonicalized only	2.40962259444Retained binary64: `2.4096225944419354`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.78709791348Retained binary64: `-1.7870979134772997`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.595699304492Retained binary64: `-0.5956993044924332`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803207531481Retained binary64: `0.803207531480645`binary64;12 significant-digit display; exact zero sign canonicalized only	0.618048288686Retained binary64: `0.6180482886858801`binary64;12 significant-digit display; exact zero sign canonicalized only	0.301796556882Retained binary64: `0.30179655688214424`binary64;12 significant-digit display; exact zero sign canonicalized only	false
116	116	29/16Exact fraction: `29/16`Retained binary64: `1.8125`	5.69413668463Retained binary64: `5.6941366846315`binary64;12 significant-digit display; exact zero sign canonicalized only	2.49440883691Retained binary64: `2.4944088369076356`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.66671069906Retained binary64: `-1.6667106990588065`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.55557023302Retained binary64: `-0.5555702330196022`binary64;12 significant-digit display; exact zero sign canonicalized only	0.831469612303Retained binary64: `0.8314696123025452`binary64;12 significant-digit display; exact zero sign canonicalized only	0.645322782685Retained binary64: `0.6453227826850789`binary64;12 significant-digit display; exact zero sign canonicalized only	0.319281969174Retained binary64: `0.31928196917414164`binary64;12 significant-digit display; exact zero sign canonicalized only	false
117	117	117/64Exact fraction: `117/64`Retained binary64: `1.828125`	5.74322406984Retained binary64: `5.743224069843841`binary64;12 significant-digit display; exact zero sign canonicalized only	2.57318583Retained binary64: `2.573185830000816`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.54230823258Retained binary64: `-1.5423082325796655`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.514102744193Retained binary64: `-0.5141027441932219`binary64;12 significant-digit display; exact zero sign canonicalized only	0.85772861Retained binary64: `0.857728610000272`binary64;12 significant-digit display; exact zero sign canonicalized only	0.676924529232Retained binary64: `0.6769245292321221`binary64;12 significant-digit display; exact zero sign canonicalized only	0.335653346111Retained binary64: `0.33565334611128717`binary64;12 significant-digit display; exact zero sign canonicalized only	false
118	118	59/32Exact fraction: `59/32`Retained binary64: `1.84375`	5.79231145506Retained binary64: `5.792311455056181`binary64;12 significant-digit display; exact zero sign canonicalized only	2.64576379305Retained binary64: `2.6457637930450644`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.41419021048Retained binary64: `-1.4141902104779938`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.471396736826Retained binary64: `-0.4713967368259979`binary64;12 significant-digit display; exact zero sign canonicalized only	0.881921264348Retained binary64: `0.8819212643483548`binary64;12 significant-digit display; exact zero sign canonicalized only	0.713457339879Retained binary64: `0.7134573398794786`binary64;12 significant-digit display; exact zero sign canonicalized only	0.350098959157Retained binary64: `0.3500989591567167`binary64;12 significant-digit display; exact zero sign canonicalized only	false
119	119	119/64Exact fraction: `119/64`Retained binary64: `1.859375`	5.84139884027Retained binary64: `5.841398840268521`binary64;12 significant-digit display; exact zero sign canonicalized only	2.71196787937Retained binary64: `2.7119678793703295`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.28266528029Retained binary64: `-1.2826652802908476`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.42755509343Retained binary64: `-0.42755509343028253`binary64;12 significant-digit display; exact zero sign canonicalized only	0.903989293123Retained binary64: `0.9039892931234431`binary64;12 significant-digit display; exact zero sign canonicalized only	0.755490613796Retained binary64: `0.7554906137960368`binary64;12 significant-digit display; exact zero sign canonicalized only	0.36147543592Retained binary64: `0.36147543591971604`binary64;12 significant-digit display; exact zero sign canonicalized only	false
120	120	15/8Exact fraction: `15/8`Retained binary64: `1.875`	5.89048622548Retained binary64: `5.890486225480862`binary64;12 significant-digit display; exact zero sign canonicalized only	2.77163859753Retained binary64: `2.7716385975338595`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.1480502971Retained binary64: `-1.148050297095271`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.382683432365Retained binary64: `-0.3826834323650904`binary64;12 significant-digit display; exact zero sign canonicalized only	0.923879532511Retained binary64: `0.9238795325112865`binary64;12 significant-digit display; exact zero sign canonicalized only	0.803445295387Retained binary64: `0.8034452953866816`binary64;12 significant-digit display; exact zero sign canonicalized only	0.368218363885Retained binary64: `0.3682183638847414`binary64;12 significant-digit display; exact zero sign canonicalized only	false
121	121	121/64Exact fraction: `121/64`Retained binary64: `1.890625`	5.93957361069Retained binary64: `5.939573610693203`binary64;12 significant-digit display; exact zero sign canonicalized only	2.82463219555Retained binary64: `2.8246321955490625`binary64;12 significant-digit display; exact zero sign canonicalized only	-1.01066956018Retained binary64: `-1.01066956017666`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.336889853392Retained binary64: `-0.33688985339222`binary64;12 significant-digit display; exact zero sign canonicalized only	0.941544065183Retained binary64: `0.9415440651830208`binary64;12 significant-digit display; exact zero sign canonicalized only	0.8574011812Retained binary64: `0.8574011812000413`binary64;12 significant-digit display; exact zero sign canonicalized only	0.36826905947Retained binary64: `0.36826905947010596`binary64;12 significant-digit display; exact zero sign canonicalized only	false
122	122	61/32Exact fraction: `61/32`Retained binary64: `1.90625`	5.98866099591Retained binary64: `5.988660995905543`binary64;12 significant-digit display; exact zero sign canonicalized only	2.8708210072Retained binary64: `2.8708210071966267`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.870854031763Retained binary64: `-0.8708540317633875`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.290284677254Retained binary64: `-0.2902846772544625`binary64;12 significant-digit display; exact zero sign canonicalized only	0.956940335732Retained binary64: `0.9569403357322088`binary64;12 significant-digit display; exact zero sign canonicalized only	0.916802849799Retained binary64: `0.9168028497993616`binary64;12 significant-digit display; exact zero sign canonicalized only	0.359067650093Retained binary64: `0.35906765009302083`binary64;12 significant-digit display; exact zero sign canonicalized only	false
123	123	123/64Exact fraction: `123/64`Retained binary64: `1.921875`	6.03774838112Retained binary64: `6.037748381117884`binary64;12 significant-digit display; exact zero sign canonicalized only	2.91009375958Retained binary64: `2.910093759583632`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.72894053971Retained binary64: `-0.7289405397097926`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.242980179903Retained binary64: `-0.24298017990326418`binary64;12 significant-digit display; exact zero sign canonicalized only	0.970031253195Retained binary64: `0.970031253194544`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980067282262Retained binary64: `0.9800672822617179`binary64;12 significant-digit display; exact zero sign canonicalized only	0.337700240994Retained binary64: `0.33770024099383256`binary64;12 significant-digit display; exact zero sign canonicalized only	false
124	124	31/16Exact fraction: `31/16`Retained binary64: `1.9375`	6.08683576633Retained binary64: `6.086835766330224`binary64;12 significant-digit display; exact zero sign canonicalized only	2.94235584121Retained binary64: `2.942355841209691`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.585270966048Retained binary64: `-0.5852709660483861`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.195090322016Retained binary64: `-0.19509032201612872`binary64;12 significant-digit display; exact zero sign canonicalized only	0.980785280403Retained binary64: `0.9807852804032303`binary64;12 significant-digit display; exact zero sign canonicalized only	1.04416626665Retained binary64: `1.0441662666512606`binary64;12 significant-digit display; exact zero sign canonicalized only	0.301318877642Retained binary64: `0.3013188776421231`binary64;12 significant-digit display; exact zero sign canonicalized only	false
125	125	125/64Exact fraction: `125/64`Retained binary64: `1.953125`	6.13592315154Retained binary64: `6.135923151542564`binary64;12 significant-digit display; exact zero sign canonicalized only	2.96752952989Retained binary64: `2.9675295298943425`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.440191423366Retained binary64: `-0.4401914233660872`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.146730474455Retained binary64: `-0.1467304744553624`binary64;12 significant-digit display; exact zero sign canonicalized only	0.989176509965Retained binary64: `0.9891765099647809`binary64;12 significant-digit display; exact zero sign canonicalized only	1.10438096927Retained binary64: `1.1043809692671767`binary64;12 significant-digit display; exact zero sign canonicalized only	0.24792736336Retained binary64: `0.2479273633604746`binary64;12 significant-digit display; exact zero sign canonicalized only	false
126	126	63/32Exact fraction: `63/32`Retained binary64: `1.96875`	6.18501053675Retained binary64: `6.1850105367549055`binary64;12 significant-digit display; exact zero sign canonicalized only	2.98555418002Retained binary64: `2.9855541800165906`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.294051420989Retained binary64: `-0.29405142098868153`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0980171403296Retained binary64: `-0.0980171403295605`binary64;12 significant-digit display; exact zero sign canonicalized only	0.995184726672Retained binary64: `0.9951847266721969`binary64;12 significant-digit display; exact zero sign canonicalized only	1.15455491769Retained binary64: `1.1545549176859664`binary64;12 significant-digit display; exact zero sign canonicalized only	0.17746602462Retained binary64: `0.17746602462049857`binary64;12 significant-digit display; exact zero sign canonicalized only	false
127	127	127/64Exact fraction: `127/64`Retained binary64: `1.984375`	6.23409792197Retained binary64: `6.234097921967246`binary64;12 significant-digit display; exact zero sign canonicalized only	2.99638636862Retained binary64: `2.996386368615517`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.147203022982Retained binary64: `-0.14720302298225427`binary64;12 significant-digit display; exact zero sign canonicalized only	-0.0490676743274Retained binary64: `-0.04906767432741809`binary64;12 significant-digit display; exact zero sign canonicalized only	0.998795456205Retained binary64: `0.9987954562051724`binary64;12 significant-digit display; exact zero sign canonicalized only	1.18814547616Retained binary64: `1.188145476158511`binary64;12 significant-digit display; exact zero sign canonicalized only	0.0928093754866Retained binary64: `0.09280937548656881`binary64;12 significant-digit display; exact zero sign canonicalized only	false
128	128	2Exact fraction: `2`Retained binary64: `2.0`	6.28318530718Retained binary64: `6.283185307179586`binary64;12 significant-digit display; exact zero sign canonicalized only	3Exact fraction: `3`Retained binary64: `3.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`	6/5Exact fraction: `6/5`Retained binary64: `1.2`	0Exact fraction: `0`Retained binary64: `0.0`	true

twoWires: exact controls, scales and source limits (12 rows)

twoWires: exact controls, scales and source limits
Quantity/source condition	Admitted value
physical Vector Scale	1/4Exact fraction: `1/4`Retained binary64: `0.25`
grid Retained Records	47
nonzero Grid Vectors	46
circle Radius Over L0	3Exact fraction: `3`Retained binary64: `3.0`
Exact intended θ/π sampling label	1/4Exact fraction: `1/4`Retained binary64: `0.25`
Signed B_t/B₀ (positive counterclockwise)	54/97Exact fraction: `54/97`Retained binary64: `0.5567010309278351`
Signed outward B_r/B₀	-24/97Exact fraction: `-24/97`Retained binary64: `-0.24742268041237114`
analytic Circulation Over B0 L0	12.5663706144Retained binary64: `12.566370614359172`Symbolic: `4*pi`
mean Tangential B Over B0	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`
circulation Proof	Ampere/stationary-source winding: each +I0 line enclosed once; total2I0, circulation=2mu0I0=4piB0L0. Not inferred from sampled quadrature.
equal Physical Axes Required	true
complete Two Infinite Wire Field	true

Return path: curl-free does not set circulation

Outside a straight ideal wire, J = 0 and ∇ × B = 0 locally. Yet a closed path winding once around the excluded wire has circulation μ₀I. There is no contradiction with Stokes’ theorem: a spanning surface crossing the wire passes through the source, while a surface avoiding the excluded source cannot be freely substituted as if the entire enclosed disk were source free.

A curl-free field has a scalar-potential description locally, and a globally single-valued scalar potential on a suitably simply connected source-free domain. In a multiply connected domain, additional circulation data can remain. Treating every local curl-free region as globally conservative would erase the current-linked topology.

The magnetic flux law ∇·B = 0 supplies a different condition from circulation. Field lines need not be closed circles in every configuration; they have no ordinary magnetic monopole source in the classical theory used here. A zero net flux through a closed surface also does not imply a zero field at every point on it.

These distinctions mirror the earlier electrostatic boundary-value discussion: differential equations, domain topology and boundary data must be considered together. A source-free subregion can still contain a field produced by currents elsewhere, and two locally valid fields can differ by an allowed homogeneous field unless the remaining conditions select one.

Distinguish a current sheet from a bulk step

For an interface with normal n̂ from region 1 to region 2 in vacuum, magnetic flux continuity gives n̂·(B₂ − B₁) = 0. A true ideal tangential surface current K gives n̂ × (B₂ − B₁) = μ₀K. These are oriented one-sided conditions, obtained from a small pillbox and a narrow Ampèrian rectangle.

K has units A/m, distinct from the bulk current density J with units A/m². In the same uniform medium, the tangential jump can also be written (B₂ − B₁)_t = μ₀K × n̂. The cross-product order follows from the chosen normal and must not be reversed by copying an unsigned magnitude formula.

Our coaxial example has no such delta-function K at its material edges. Its bounded J_z changes abruptly, but the integrated enclosed current and B remain continuous. The derivative can jump, and the adjacent J values have distinct one-sided limits. Modeling the finite return annulus as an infinitely thin current sheet would be a different source problem.

The conditions for magnetic media require their own free and bound current and H bookkeeping. Likewise the field at an ideal filament is not obtained by assigning one of these finite sheet limits to a line singularity. Preserve the dimension and support of each source before choosing an interface or self-field formula.

Know the idealizations behind coil formulas

For an ideal infinitely long solenoid represented by a continuous uniform azimuthal current sheet, symmetry and an external-field condition give B = μ₀nI inside and zero outside, where n is turns per unit length. The Ampèrian rectangle compares the axial interior and exterior fields; the source-sheet jump supplies the difference.

A long finite real coil can approximate that central interior field, but it has end and return fields. Declaring its exterior field exactly zero would generally be wrong. The distance to the ends, winding geometry and required accuracy determine whether the idealization is adequate; no numerical error bound follows merely from the word long.

For an ideal continuously wound toroid with appropriate rotational symmetry, a circular contour in the toroidal winding region encloses NI and gives B_phi(r) = μ₀NI/(2πr). The field is not constant across a thick radial cross-section. The ideal interior hole and exterior can have zero field when their linked current and the imposed symmetry and boundary conditions justify it.

Finite turn spacing, leads, gaps and magnetic core response alter a real toroid. A constant-permeability replacement is an additional material assumption, not part of Ampère’s law itself. The current-defined vacuum examples connect naturally to inductors and magnetic materials while keeping those later constitutive and time-dependent topics separate.

Connect source fields to parallel-wire forces

The field produced by one ideal long straight wire at transverse distance d has magnitude μ₀|I₁|/(2πd). Applying the previous lesson’s external-field force to a second parallel wire gives force magnitude per unit length μ₀|I₁I₂|/(2πd) in the corresponding ideal geometry. The vector products determine attraction for currents in the same direction and repulsion for opposite directions.

Conventional current already includes the signed charge and drift of the carriers. One should not reverse this force again merely because ordinary metallic carriers are negative. In a macroscopic wire the force is transmitted through electromagnetic and material interactions, with supports and source circuitry setting the actual motion.

The result uses the other wire’s field, not a singular ideal filament self-field. Real finite wires require their full return paths and may have electrostatic forces if their charge or potentials are relevant. Neutral-current assumptions used for a simple magnetic illustration are additional constraints; steady current alone does not prove every part of a real circuit is electrically neutral.

Mechanical motion of circuits can exchange energy with sources and induced electric fields. The field-source calculation and J × B force must be embedded in that complete energy and momentum balance. A force between ideal current elements taken in isolation is not a complete argument about a closed electromagnetic system.

Return path: derive the vector Poisson equation

A convenient way to satisfy magnetic flux continuity is to introduce B = ∇ × A. Under the Coulomb gauge ∇·A = 0, the vector identity ∇ × (∇ × A) = ∇(∇·A) − ∇²A turns magnetostatic Ampère’s law into ∇²A = −μ₀J. In Cartesian components, this is a Poisson equation for each component of A.

For a localized stationary divergence-free J in all space with suitable decay and no additional boundary field, the Green-function solution is A(r) = (μ₀/4π)∫ J(r′)/|r − r′| d³r′. The sign follows from ∇²[1/(4π|r − r′|)] = −δ(r − r′). This is the magnetic counterpart of a source integral, with vector rather than scalar source data.

To check the gauge, use ∇_r(1/R) = −∇_{r′}(1/R), integrate by parts and use ∇′·J = 0. The volume term and the boundary term vanish under the stated source and decay conditions, giving ∇·A = 0. An incompatible truncated current or omitted boundary term would invalidate that inference.

Taking the observer curl gives ∇ × [J(r′)/R] = ∇(1/R) × J(r′) = J(r′) × (r − r′)/R³. The Biot–Savart formula follows with its original cross-product order. This derivation connects the integral solution to both local magnetic equations rather than treating one memorized source formula as an unrelated rule.

Return path: boundary data and gauge remain

The free-space source integral selects a particular boundary behavior. In a bounded region, imposed boundaries or currents outside the integration domain can add a homogeneous solution. Specifying J inside that region does not by itself eliminate a uniform external field or every topological circulation allowed by the domain.

The vector potential is also gauge dependent: A + ∇χ produces the same B wherever the derivatives are defined appropriately. Within Coulomb gauge, residual gauge functions satisfy ∇²χ = 0 together with the relevant boundary conditions. Gauge freedom changes a representation, while missing physical boundary data can change the field itself.

An infinite ideal wire needs separate care. Its physical B falls as 1/r, but a corresponding A can contain a logarithm and need not vanish at radial infinity. The localized-source decay assumptions used for the all-space uniqueness argument cannot be copied unchanged into that infinite-source problem.

A solution audit therefore distinguishes physical source completion, domain and boundary conditions, regularity, topological data and gauge choice. The following Maxwell and gauge-potential lesson develops the time-dependent framework. Here the static Poisson construction is sufficient to show why local equations and source integrals must agree under matching assumptions.

Audit the steady approximation before extending it

For a time-dependent configuration, the vacuum Ampère–Maxwell law includes μ₀ε₀∂E/∂t alongside μ₀J. Its divergence is consistent with charge continuity through Gauss’s law. A charging-capacitor example cannot omit this term and still expect current through every spanning surface to agree.

Magnetoquasistatic approximations can neglect some displacement effects under a controlled scale ordering, but that is not identical to declaring every field and current exactly time independent. Induction and finite propagation require their own checks. A slowly varying source does not automatically make every static result exact at every distance.

For a present magnetostatic calculation, record the complete current support, direction, medium and observation domain. Select Biot–Savart for a source integral or Ampère’s law when the geometry actually permits the desired symmetry reduction. Test limits, units, source exclusions, interface conditions and the current enclosed at each contour.

Finally separate exact mathematical controls from approximation claims. The segment contribution, loop axis, ideal coax and two-wire counterexample answer different questions. Their agreement with source and circulation laws supports their stated models; it does not establish a finite apparatus’s performance or replace the material and time-dependent physics left to subsequent lessons.

Worked examples

A finite portion supplies four-fifths of a comparison field

For the selected length-8L₀ straight portion, evaluate its midplane field at r = 3L₀ and compare it with the infinite wire.

Set R = 3 and evaluate √(R² + 16) = 5.
The selected contribution is B_portion/B₀ = 4/(3·5) = 4/15.
The infinite-wire comparison is B_infinite/B₀ = 1/3.
Their ratio is (4/15)/(1/3) = 4/5; an unspecified real return path still has its own field.

The straight portion gives (4/15)B₀, four-fifths of the infinite-wire value at that point. It is a source contribution, not a claimed complete isolated steady circuit.

A loop-axis control on either side

For the radius-3L₀ loop carrying 2I₀, find its field at z = ±4L₀.

The center normalization is B_c = μ₀I₀/(3L₀).
At Z = ±4, 9 + Z² = 25 and (9 + Z²)^(3/2) = 125.
Therefore B_z/B_c = 27/125 on both sides.
The current is counterclockwise viewed from +z, so both axial values point in +z.

B = (27/125)B_c ẑ at either specified axis point. The symmetry is an axial statement, not a uniform-field claim away from the axis.

Enclose only part of the return current

At R = 4 in the selected coaxial return annulus, calculate enclosed current, field and circulation.

The return area enclosed from radius3 to4 is (16 − 9)πL₀² = 7πL₀².
At return density −I₀/(16πL₀²), this contributes −(7/16)I₀.
The inner +I₀ leaves enclosed current (9/16)I₀.
Divide its fraction by R = 4 to get B_phi/B₀ = 9/64; Ampère circulation is (9/16)μ₀I₀.

I_enclosed = (9/16)I₀, B_phi = (9/64)B₀ and circulation = (9/16)μ₀I₀. The un-enclosed outer return current is not subtracted early.

Fixed circulation, variable local field

Compare the two-wire field tangent to the radius-3L₀ contour at (3L₀,0) and (0,3L₀).

At (3,0), distances from the two dimensionless wire positions are5 and1; their +y fields add to 1/5 + 1 = 6/5.
At (0,3), each squared source distance is13. Their x components add to −6/13, while their y components cancel.
The positive contour tangent at (0,3) is −x, so B_t/B₀ = 6/13 there.
Both points belong to the same contour enclosing2I₀, whose mean tangential field is (2/3)B₀.

The local tangential values differ, (6/5)B₀ and (6/13)B₀, even though one circulation integral fixes their contour mean. Taking a single B outside the integral would be unjustified.

A bulk edge is not a current sheet

Check B and its radial derivative on the two sides of the coaxial radius R = 5.

The return branch is b(R) = (25 − R²)/(16R), which gives b(5) = 0.
The outside branch is b = 0, so the field is continuous.
The inner derivative there is −(25/25 + 1)/16 = −1/8; the outside derivative is zero.
The bounded current density ends there, but no delta-function surface current has been added.

B is continuous and its derivative has distinct one-sided limits. Assigning a magnetic-field jump would incorrectly insert a source sheet absent from the model.

Try it yourself

Can a finite current segment ending in empty space be the complete source of an exactly stationary problem without a return or charge accumulation?
Show the answer and reasoning
No.
Stationary charge continuity requires a compatible divergence-free current. An isolated endpoint would accumulate or remove charge.
Which displacement appears in J(r′) × R/R³ for the stated Biot–Savart law?
Show the answer and reasoning
R = r − r′, from source to observer.
Reversing it alone reverses the field sign. The source integration variable and observation point have different roles.
At R = 3, is the selected portion’s field (1/3)B₀ or (4/15)B₀?
Show the answer and reasoning
(4/15)B₀.
The finite-end factor is4/5. The value1/3 belongs to the separate infinite-wire comparison.
Does the chosen loop’s B_z reverse sign when z changes from +4L₀ to −4L₀?
Show the answer and reasoning
No.
The exact axial expression is positive and even in z for the specified counterclockwise current viewed from +z.
What fraction of the inner current is enclosed at R = 1 in the uniform radius-2 cylinder?
Show the answer and reasoning
1/4.
Uniform current density gives the area fraction R²/4. The fullI₀ is not enclosed by every interior contour.
Is the coaxial field singular at its axis because the exterior wire formula contains1/r?
Show the answer and reasoning
No. Its regular Cartesian field is zero there.
Inside the finite-radius uniform-current cylinder, B_phi/B₀ = R/4. The azimuthal direction itself is undefined at the exact axis.
Does zero enclosed current prove that B vanishes at every point of an arbitrary contour?
Show the answer and reasoning
No.
It fixes one signed circulation. Fields produced elsewhere may be nonzero, with components whose integral cancels.
In the two-wire grid, why is the midpoint retained while the two wire locations are excluded?
Show the answer and reasoning
The midpoint is an exact cancellation null in the valid domain; the filament locations are source singularities.
A zero vector and an undefined ideal-source value are different physical and mathematical cases.
Can a contour around an excluded current filament have nonzero circulation even where curl B = 0 at each ordinary point along the contour?
Show the answer and reasoning
Yes.
The source-linked domain is multiply connected. Stokes’ theorem cannot treat a spanning disk through the source as an everywhere source-free surface.
May the magnetostatic Ampère law omit the displacement term in a charging-capacitor problem without further justification?
Show the answer and reasoning
No.
Time-dependent electric flux and charge continuity require the Ampère–Maxwell term or a controlled approximation that retains the needed consistency.

What to carry forward

Magnetostatics requires compatible complete steady currents. A finite source portion is a contribution, while returns and boundary fields remain part of the full problem.
Biot–Savart is a vector source integral with an oriented source-to-observer displacement. Thin-filament source points are excluded from its ordinary observation domain.
Finite straight portions, infinite wires and loop axes have distinct geometries, limits and normalizations. A plotted or asymptotic result must keep its domain.
Ampère’s law fixes circulation with a linked contour and surface orientation. Extracting a local field requires justified symmetry and boundary conditions.
A coaxial return must be counted progressively. B can be continuous through bounded bulk-current steps even while its derivative has different one-sided limits.
The two-wire example has fixed total circulation and varying tangential and radial fields. A contour mean is not a pointwise value.
Magnetic flux continuity, topology, boundary behavior and current support complete the local equations. A vector potential adds gauge freedom without replacing physical boundary data.
Changing currents, magnetic materials, finite coil details and induced electric fields need their own models. Conservation laws and the exact-versus-approximate distinction remain the organizing checks.

References

Sources for the concepts and teaching approaches in this article.