fix: avoid cancellation in math/base/special/vercos and havercos

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• fixes the loss of precision in math/base/special/vercos and math/base/special/havercos (JS and C) for arguments near odd multiples of pi. It is the counterpart of fix: avoid cancellation in math/base/special/versin and haversin #15609 (versin/haversin).

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

vercos( Math.PI + 1.0e-6 ); // 5.000444502911705e-13 (correct to 2 ULP: 5.000000000172716e-13)

1 + cos(x) cancels when cos(x) is close to -1, so the relative error grows like 2e-16/(x - (2k+1)pi)^2, and right next to an odd multiple of pi no digits are correct. The fix uses the exact half-angle identities:

• vercos(x) = 2*cos(x/2)^2
• havercos(x) = cos(x/2)^2

x/2 is exact and cos handles its own argument reduction, so both functions are within 2 ULP of a 256-bit reference on every fixture, including huge |x|.

Related Issues

Does this pull request have any related issues?

This pull request has the following related issues:

• fix: avoid cancellation in math/base/special/versin and haversin #15609

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.

• Fixtures: the runners used to evaluate 1.0 .+ cos.( x ) in double precision, which recorded the cancellation. They now evaluate the same textbook formula at 4096 bits and round to Float64, independent of the identity the implementation uses. There is a new near_odd_pi.json (4,000 points at (2k+1)*pi +/- d for k in -10:9 and d log-spaced over [1e-12, 0.1]). On develop, 3,990 of those 4,000 points are more than 2 ULP off. With this change, all of them are within 2 ULP. The JS fixture tests now allow 2 ULP instead of exact equality. The native tests already allowed 2 ULP, and 8 for huge values.
• All tests pass in test.js and test.native.js for both packages, and the EditorConfig lint is clean.

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• Read, understood, and followed the contributing guidelines.

AI Assistance

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• Yes
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• Test/benchmark generation
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This PR was written primarily by Claude Code. The bug was found by comparing every stats/base/dists function against SciPy on random parameters.

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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/havercos	$\\color{green}181/181$ $\\color{green}+0.00\\%$	$\\color{green}5/5$ $\\color{green}+0.00\\%$	$\\color{green}2/2$ $\\color{green}+0.00\\%$	$\\color{green}181/181$ $\\color{green}+0.00\\%$
math/base/special/vercos	$\\color{green}181/181$ $\\color{green}+0.00\\%$	$\\color{green}5/5$ $\\color{green}+0.00\\%$	$\\color{green}2/2$ $\\color{green}+0.00\\%$	$\\color{green}181/181$ $\\color{green}+0.00\\%$

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

[kgryte]

@anandkaranubc Mind taking a look at this? Quick glance suggests some clean-up is needed:

1. Implementation method moved from source and added to JSDoc.
2. Use of abs2.
3. Empty lines separating comments and expressions in Julia.
4. ...etc.

Furthermore, I am not sure about the test fixture generation approach. Simply hardcoding the half-angle implementation when generating test fixtures seems incorrect. Instead, we should compute the quantities at higher-precision using the textbook formula, truncate to double, and then compare that we, in fact, retain and improve accuracy by moving to using the identity. Stated otherwise, use of the identity is an implementation detail. We should NOT be generating test fixtures which simply regurgitate that identity. That degrades the value of the tests.

[Abhist17]

Thanks for the review. Addressed in 9441496:

1. The method is now documented under ## Method in the JSDoc and in the C doc comment, and the inline comment is gone.
2. The functions now return 2.0 * abs2( cos( x / 2.0 ) ) and abs2( cos( x / 2.0 ) ), and the C versions use stdlib_base_abs2.
3. Agreed on the fixtures. The runners now evaluate the textbook formula, 1 + cos(x), at 4096 bits (setprecision( BigFloat, 4096 )) and round to double. The identity no longer appears in the fixture generation. That is enough precision that the cancellation doesn't affect the reference even at the points closest to odd multiples of pi, and the regenerated values are the same doubles as before. Against them, the implementation is within 2 ULP on every fixture, while develop misses 3,990 of the 4,000 near_odd_pi points. I also added the empty lines between comments and expressions in the Julia runners.

I made the same changes to #15609 (814681b).