Fix ACOS precision issue for boundary values (x=±1.0)

[Dayuxiaoshui]

The ACOS operator was producing incorrect results for boundary values due to poor precision of ASIN's Taylor series expansion near x=±1.0.

Root cause:

• ASIN used a 6-term Taylor series that converges slowly near boundaries
• ACOS was implemented as acos(x) = π/2 - asin(x), inheriting ASIN errors
• At x=1.0, ASIN error of 0.354874 (22.6%) caused ACOS to output 0.354874 instead of 0.0

Solution:

• Modified ASIN to use system library function (asinf) for |x| >= 0.9
• Modified ACOS to use system library function (acosf) for |x| >= 0.9
• For |x| < 0.9, continue using Taylor series (accurate in this range)

This ensures high precision for boundary values while maintaining the existing behavior for values in the middle range.

Fixes #18580

[gemini-code-assist]

Summary of Changes

Hello @Dayuxiaoshui, I'm Gemini Code Assist¹! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request significantly enhances the accuracy of the ACOS and ASIN trigonometric functions, particularly at their domain boundaries. By strategically switching to system library functions for extreme input values while retaining the efficient Taylor series for central values, the change ensures robust and precise mathematical computations without sacrificing performance where the series remains effective.

Highlights

• Precision Improvement for ACOS and ASIN: This pull request resolves a precision issue in the ACOS operator, which previously produced incorrect results for boundary values like x=±1.0. The root cause was identified as the poor convergence of ASIN's 6-term Taylor series expansion near these boundaries, affecting ACOS due to its implementation as "π/2 - asin(x)".
• Hybrid Implementation for ASIN and ACOS: A hybrid approach has been implemented for both ASIN and ACOS. For input values where "|x| >= 0.9", the system library functions ("asinf" and "acosf") are now utilized to ensure high precision. For values where "|x| < 0.9", the existing Taylor series expansion, which is accurate in this range, continues to be used.

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[gemini-code-assist]

Code Review

This pull request addresses a precision issue with ACOS and ASIN operators for boundary values (x=±1.0) by switching to system library functions when |x| >= 0.9. The changes for tir.asin look good, with an explicit domain check and conditional logic to use either the system library or a Taylor series expansion. For tir.acos, I've suggested adding a similar explicit domain check to ensure consistent and robust behavior for out-of-domain inputs, mirroring the implementation of tir.asin.

[tlopex]

Thanks for the fix! Would you mind adding a test to cover the input? The test should be at test_tir_intric

[Dayuxiaoshui]

@tlopex Ok, no problem, I'll go add some tests first.

[Dayuxiaoshui]

@tlopex

[tlopex]

LGTM Thanks!