fix: avoid cancellation near sin(x) = 1 in math/base/special/coversin and hacoversin

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• evaluates math/base/special/coversin and math/base/special/hacoversin as cos(x)^2 / (1 + sin(x)) (halved for hacoversin) where sin(x) > 0 (JS and C), instead of 1 - sin(x).

1 - sin(x) cancels wherever sin(x) is close to one:

var coversin = require( '@stdlib/math/base/special/coversin' );

coversin( 26.706218176635318 );
// develop: 0.000003592862648615913
// this PR: 0.0000035928626486120986 (256-bit BigFloat: 0.0000035928626486120986)

1 - sin(x) = (1 - sin(x))(1 + sin(x)) / (1 + sin(x)) = cos(x)^2 / (1 + sin(x)). Near the cancellation, cos(x) is near a zero, which cos computes with full relative accuracy after its own argument reduction, and the denominator is near 2. Where sin(x) <= 0, 1 - sin(x) doesn't cancel and is unchanged.

• regenerates the fixtures in 256-bit BigFloat. They came from Julia's Float64 1 .- sin.(x) (the expression being replaced) and were checked at 0 ULPs, so they pinned the cancellation. Against the exact values, develop is off by up to about 2^22 ULPs on medium_positive.json. This PR passes every set at 2 to 3 ULPs, and those are the new tolerances.
• adds near_extrema.json, 4200 values within 10^-8 to 10^-1 of ±π/2 + 2πk for k = -10..10. develop is off by up to about 2^40 ULPs there; this PR passes at 2.
• changes the coversin( 3.141592653589793/2.0 ) examples from 0.0 to ~0.0. The double nearest π/2 is not π/2, and 1 - sin of it is 1.87e-33, which the doctest now sees.

Related Issues

Does this pull request have any related issues?

None. The same change for covercos and hacovercos, where 1 + sin(x) cancels near sin(x) = -1, follows separately.

Questions

Any questions for reviewers of this pull request?

No.

Other

Any other information relevant to this pull request? This may include screenshots, references, and/or implementation notes.

test.js and test.native.js pass.

Checklist

Please ensure the following tasks are completed before submitting this pull request.

• Read, understood, and followed the contributing guidelines.

AI Assistance

When authoring the changes proposed in this PR, did you use any kind of AI assistance?

• Yes
• No

If you answered "yes" above, how did you use AI assistance?

• Code generation (e.g., when writing an implementation or fixing a bug)
• Test/benchmark generation
• Documentation (including examples)
• Research and understanding

[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/coversin	$\\color{green}174/174$ $\\color{green}+100.00\\%$	$\\color{green}7/7$ $\\color{green}+100.00\\%$	$\\color{green}2/2$ $\\color{green}+100.00\\%$	$\\color{green}174/174$ $\\color{green}+100.00\\%$
math/base/special/hacoversin	$\\color{green}174/174$ $\\color{green}+100.00\\%$	$\\color{green}7/7$ $\\color{green}+100.00\\%$	$\\color{green}2/2$ $\\color{green}+100.00\\%$	$\\color{green}174/174$ $\\color{green}+100.00\\%$

The above coverage report was generated for the changes in this PR.