feat: add c implementation of math/base/special/kernel-betainc

[nirmaljb]

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Progresses #649 .

Description

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This pull request:

• Progresses a part of : [RFC]: Add C implementations to base special math functions (tracking issue) #649

Related Issues

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This pull request has the following related issues:

• [RFC]: Add C implementations to base special math functions (tracking issue) #649

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I used codex to find references, but the proposed changes were fully authored manually by myself.

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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
math/base/special/kernel-betainc	$\color{red}1644/1787$ $\color{green}+0.00%$	$\color{red}195/226$ $\color{green}+0.00%$	$\color{red}14/16$ $\color{green}+0.00%$	$\color{red}1644/1787$ $\color{green}+0.00%$

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

[nirmaljb]

/stdlib update-copyright-years

[nirmaljb]

Thank you for the suggestions! @gunjjoshi

[nirmaljb]

Fixed the issues
Thank you for pointing out! @gunjjoshi

[nirmaljb]

Note: The following three Rocco Romeo cases are not added out because they require long double (80-bit extended) precision to match the Wolfram reference values. Boost passes these by internally promoting double to long double (its default promote_double policy). JavaScript and C double-only computation loses too much precision in the intermediate x^a * y^b / Beta(a,b) power term for these extreme x values.

1. beta( 2, 4, ldexp(1+1/1024, -351) ): rel error ~9.3e-10 (~4.2M eps)
2. beta( 2, 4, ldexp(1+1/2048, -351) ): rel error ~2.4e-7 (~1B eps)
3. ibeta( 3, 5, ldexp(1+15/16, -268) ): rel error ~6.5e-3 (~29B eps)

[nirmaljb]

@kgryte @Planeshifter Could you please review this PR?

[Neerajpathak07]

@nirmaljb Given that the C implementation for this package is expected to be pretty involved we'll need a bit-of time to think this through for the correct fit. Overall, thank you for following the templating as per referenced from the boost implementation.

[nirmaljb]

Thank you, @Neerajpathak07, for taking the time to review this function. It's a huge function and would require line-by-line review, and I really appreciate you doing that.
I would be working on reviewing it myself, and I look forward to more review changes. Hope to get this merged as soon as possible.

[anandkaranubc]

They seem high ULP diff numbers here. Is there a reason why that is the case?

[rautelaKamal]

Hey @nirmaljb, pulled your branch locally since kernel-betainc is a dependency for some of my downstream work, wanted to make sure things look good end to end.

Overall looks solid. Two small things I noticed:

In examples/c/example.c around line 39, the printf statement seems to be missing a \n at the end, output lines will run together when executed.

The ULP tolerances in test.native.js get fairly wide in some spots (e.g., 24k+ ULP for the large_range upper regularized tests around line 574). Just wanted to check, is that expected behavior matching the JS tests, or is there room to tighten those?

Great work on this overall, its a complex port.