fix: avoid cancellation near the edges of the support in stats/base/dists/cosine/logpdf

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• evaluates stats/base/dists/cosine/logpdf as 2*ln( cospi(z/2) ) - ln(s) instead of ln( 1 + cospi(z) ) - ln(2s) (JS, factory and C),
• fixes the test fixtures, one of which tested nothing.

1 + cos(pi*z) cancels as z = (x-mu)/s approaches -1 or 1. Close to the edges of the support the result is -Infinity, although the log-density is finite there:

var logpdf = require( '@stdlib/stats/base/dists/cosine/logpdf' );

logpdf( 1.0-1.0e-9, 0.0, 1.0 );
// develop: -Infinity
// this PR: -40.54336631987778

logpdf( 1.0-1.0e-5, 0.0, 1.0 );
// develop: -22.122685445187134
// this PR: -22.122685519452897

1 + cos(pi*z) = 2*cos(pi*z/2)^2 holds exactly. On [-1, 1], cospi(z/2) is non-negative, so the log needs no subtraction.

Fixtures and tests:

• runner.jl now evaluates the textbook formula ln((1 + cos(pi*z)) / (2s)) in 2048-bit BigFloat instead of Distributions.logpdf.

• Its x values were drawn from ranges that mostly missed the support. positive_mean.json had 1,000 of 1,000 entries null (x in [-20, -10] with mu in [0, 10]), so its test checked nothing, and negative_mean.json had 964 of 1,000. All three now draw x inside the support.

• There's a new edges.json with 1 - |z| log-spaced over [1e-15, 0.1] on both sides. mu = 0 and s is a power of two, so z is exact.

• Max error in JS and native, against the exact values:

  fixture	develop	this PR
  positive_mean	22,613 ULP	1,062 ULP
  negative_mean	11,063 ULP	1,530 ULP
  large_variance	23,776 ULP	261 ULP
  edges	-Infinity	1 ULP

  The tolerances are set to those values. The remaining error is at results close to 0 (e.g. -0.0013). There, a 1-ULP change in s moves the exact result by about 1/|result| ULP, so that's conditioning. On develop, 986 of the 4,022 assertions in test.logpdf.js fail. With this change they all pass.

Related Issues

Does this pull request have any related issues?

None.

Questions

Any questions for reviewers of this pull request?

cosine/pdf has the same 1 + cospi(z) (it returns 0 near the edges); that one is #15875. cosine/cdf and logcdf cancel too: 1 + z + sinpi(z)/pi near z = -1 is about pi^2 (1+z)^3 / 6, but fixing that needs a series for u - sin(u), so I'd do it separately.

Other

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

No.

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

Disclosure

If you answered "yes" to using AI assistance, please provide a short disclosure indicating how you used AI assistance. This helps reviewers determine how much scrutiny to apply when reviewing your contribution. Example disclosures: "This PR was written primarily by Claude Code." or "I consulted ChatGPT to understand the codebase, but the proposed changes were fully authored manually by myself.".

This PR was written primarily by Claude Code. I found the bug by scanning stats/base/dists for ln( 1.0 + ... ).

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/cosine/logpdf	$\\color{green}318/318$ $\\color{green}+100.00\\%$	$\\color{green}31/31$ $\\color{green}+100.00\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{green}318/318$ $\\color{green}+100.00\\%$

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