fix: evaluate large arguments accurately in math/base/special/fresnel, fresnelc and fresnels

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• fixes the large-argument branch of math/base/special/fresnel, fresnelc and fresnels (JS and C; the three packages carry the same code).

Two problems, both inherited from Cephes:

1. For x > 36974, the functions returned exactly 0.5. The oscillating part of C(x) and S(x) is f(x) sin(πx²/2) - g(x) cos(πx²/2) with f(x) ≈ 1/(πx), which is 3.2e-6 at x = 1e5:

   var fresnelc = require( '@stdlib/math/base/special/fresnelc' );
   
   fresnelc( 100000.37 );
   // develop: 0.5
   // this PR: 0.5000006792295584 (mpmath: 0.5000006792295584)

2. Below that, sincos( HALF_PI * x2 ) used the rounded x*x. An error of ε x² there becomes πεx²/2 radians in the argument, which at x = 1e4 is about 1.7e-8. On the medium fixtures ([100, 36968]), develop is off by up to 81920 ULPs.

The fix reduces x² modulo 4 (the period of sin/cos( πu/2 )) exactly. A Veltkamp split x = hi + lo makes hi*hi, 2*hi*lo and lo*lo exact, and each term is reduced with fmod. The asymptotic branch then runs up to x = 1e16, beyond which 1/(πx) is below half an ulp of 0.5.

• replaces the C/Cephes fixtures, which were generated by the code being fixed and checked at 0 ULPs, with a Julia runner that evaluates both integrals in 2048-bit BigFloat: the power series below x = 10, and the asymptotic expansions of f and g (Abramowitz and Stegun 7.3.27–28) above it. I spot-checked it against mpmath. Same ranges as before, except that huge.json now runs log-spaced up to 1e15:

set	develop	this PR
small [1e-4, 9.99]	20 ULPs	5 ULPs
medium [100, 36968]	81920	0
large [4e4, 6e4]	~1.7e11 (returns 0.5)	0
huge [6e4, 1e15]	~1.7e11 (returns 0.5)	0

Related Issues

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None.

Questions

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No.

Other

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test.js and test.native.js pass.

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Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/fresnel	$\\color{red}604/616$ $\\color{green}+98.05\\%$	$\\color{red}28/36$ $\\color{green}+77.78\\%$	$\\color{green}7/7$ $\\color{green}+100.00\\%$	$\\color{red}604/616$ $\\color{green}+98.05\\%$
math/base/special/fresnelc	$\\color{red}446/455$ $\\color{green}+98.02\\%$	$\\color{red}22/28$ $\\color{green}+78.57\\%$	$\\color{green}5/5$ $\\color{green}+100.00\\%$	$\\color{red}446/455$ $\\color{green}+98.02\\%$
math/base/special/fresnels	$\\color{red}446/455$ $\\color{green}+98.02\\%$	$\\color{red}22/28$ $\\color{green}+78.57\\%$	$\\color{green}5/5$ $\\color{green}+100.00\\%$	$\\color{red}446/455$ $\\color{green}+98.02\\%$

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

[kgryte]

This is where it would benefit you to attend office hours and seek alignment before opening many PRs. You should not be deleting existing fixtures, as you need to understand their intent.

In this case (and others), we need the reference fixtures in order to access how well our implementation aligns with the reference implementation. If we notice large divergences, that tells us that something may be off with our implementation. Our special function tests are not just about total accuracy, they are also about helping us assess drift from a reference implementation.

So, in general, your approach in all these PRs is off. There is room for both reference implementation fixtures and arbitrary precision fixtures. One does not preclude the existence of the other. What it means is that we may need to duplicate tests so that we have separate test files exercising the different fixtures.

But again, this is where you need to seek alignment, rather than opening now dozens of similar PRs, none of which have been merged.

[Abhist17]

@kgryte thank you, that is clear, and I am sorry for the volume. I misread what the reference fixtures are for. I've moved this PR and the others like it (#16109–#16117, #16119) to draft, and I won't open more until we have agreed on the approach.

Here is how I'd rework them. Please correct me if any of it is off:

1. Restore every reference fixture I deleted or regenerated (Cephes, Boost/C++, SciPy, Julia Float64) and the runners that produce them, unchanged.
2. Keep the existing tests against those fixtures.
3. Put the arbitrary-precision (Julia BigFloat) fixtures in their own directory, with a separate test file per implementation exercising them, so the two kinds of check stay apart.

One thing I'd like your view on before I touch the tests. In some packages the change deliberately departs from the reference implementation where the reference itself is inaccurate. Here, Cephes returns exactly 0.5 for x > 36974, so this PR differs from the large/huge Cephes fixtures by up to ~1e11 ULPs. Similarly, coversin near sin(x) = 1 differs from Julia's Float64 1 .- sin.(x) by up to ~2^22 ULPs. For the reference tests there, would you rather I:

• (a) keep them over the full range with a tolerance widened to the measured divergence and a comment explaining why, or
• (b) restrict them to the region where the implementation still follows the reference, and leave the rest to the arbitrary-precision tests?