fix: avoid cancellation near sin(x) = -1 in math/base/special/covercos and hacovercos

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• evaluates math/base/special/covercos and math/base/special/hacovercos as cos(x)^2 / (1 - sin(x)) (halved for hacovercos) where sin(x) < 0 (JS and C), instead of 1 + sin(x). This is the mirror image of fix: avoid cancellation near sin(x) = 1 in math/base/special/coversin and hacoversin #16116.

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

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

covercos( -1.5918310322213998 );
// develop: 0.0002212212592344054
// this PR: 0.00022122125923442057 (256-bit BigFloat: 0.00022122125923442057)

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, 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)). This PR passes every set at 2 to 3 ULPs, and those are the new tolerances.
• adds near_extrema.json (values within 10^-8 to 10^-1 of ±π/2 + 2πk), which passes at 2 ULPs.

Related Issues

Does this pull request have any related issues?

None. Companion to #16116.

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/covercos	$\\color{green}174/174$ $\\color{green}+100.00\\%$	$\\color{green}7/7$ $\\color{green}+100.00\\%$	$\\color{green}2/2$ $\\color{green}+100.00\\%$	$\\color{green}174/174$ $\\color{green}+100.00\\%$
math/base/special/hacovercos	$\\color{green}174/174$ $\\color{green}+100.00\\%$	$\\color{green}7/7$ $\\color{green}+100.00\\%$	$\\color{green}2/2$ $\\color{green}+100.00\\%$	$\\color{green}174/174$ $\\color{green}+100.00\\%$

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