fix: use expm1 in stats/base/dists/lognormal/variance

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• evaluates stats/base/dists/lognormal/variance as expm1( sigma^2 ) * exp( 2*mu + sigma^2 ) instead of ( exp( sigma^2 ) - 1 ) * exp( 2*mu + sigma^2 ) (JS and C).

exp(sigma^2) - 1 cancels when sigma is small, and is 0 for sigma^2 < 2^-53:

var variance = require( '@stdlib/stats/base/dists/lognormal/variance' );

variance( 0.0, 1.0e-9 );
// develop: 0.0
// this PR: 1.0e-18

variance( 0.0, 1.0e-5 );
// develop: 1.000000082840371e-10
// this PR: 1.0000000001500002e-10

stdev calls variance, so it's fixed by this change too. Its fixtures had the same Distributions.jl cancellation, so they're regenerated the same way (with a new small_sigma.json). The tolerances are 18 ULP on data and 1 on small_sigma.

Fixtures and tests:

• runner.jl now evaluates the textbook formula in 2048-bit BigFloat instead of Distributions.var, which uses the same double-precision expression. Distributions is dropped from REQUIRE, and data.json is regenerated with the same input ranges.
• There's a new small_sigma.json with sigma log-spaced over [1e-150, 0.1]. It passes at 3 ULP in JS and native.
• On data.json, the max error is 24 ULP (develop: 45). It comes at sigma ≈ 4.2, where the rounding of 2*mu + sigma^2 is multiplied by the size of the exponent, so the tolerance goes from 0 to 24. The old 0 only held because the fixture used the same double-precision formula.
• On develop, 1,002 of the 1,110 assertions in test.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?

lognormal/skewness has the same exp(sigma^2) - 1 cancellation, and lognormal/kurtosis has a worse one: e^{4s} + 2e^{3s} + 3e^{2s} - 6 with s = sigma^2. With u = expm1(s) the latter is exactly u*(16 + u*(15 + u*(6 + u))), with no cancellation. I'll send both 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)
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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 exp(...) - 1.

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@stdlib-js/reviewers

[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/lognormal/stdev	$\\color{green}169/169$ $\\color{green}+0.00\\%$	$\\color{green}5/5$ $\\color{green}+0.00\\%$	$\\color{green}2/2$ $\\color{green}+0.00\\%$	$\\color{green}169/169$ $\\color{green}+0.00\\%$
stats/base/dists/lognormal/variance	$\\color{green}179/179$ $\\color{green}+0.00\\%$	$\\color{green}9/9$ $\\color{green}+0.00\\%$	$\\color{green}2/2$ $\\color{green}+0.00\\%$	$\\color{green}179/179$ $\\color{green}+0.00\\%$

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

[kgryte]

These changes seem reasonable. Thanks for flagging, @Abhist17.