New complex acos function.

[s-oboyle]

Unlike asin and atan, acos needs more than just a call to the equivalent inverse-hyperbolic function.
Doing it this way fixes all the under/overflow issues.

Perf

We have a slightly suspicious result here. Despite this being basically a wrapper around (the fairly large) acosh with an extra fma and some sign flips, it has much less perf than acosh, as seen here, where I would expect them to be quite similar.
We could have either hit a register usage boundary, or maybe acosh hasn't been inlined.
Also possible is the values that math_bench test (which try to guess real-life usage) now hit a slowpath in acosh more often.
To be investigated, as this is nearly ~1.5x slower than anticipated.

Operations/SM/cycle:
cacos():

H100	old	new	new/old
fp64	0.1518	0.1430	0.94
fp32	0.4461	0.4072	0.91

Correctness

GPU fp64:
Max ulp real error (4.772,0.3392) @ (4174.277773,1.009244847)	(0x40b04e471c22e769,0x3ff025ddec967ac6)
	Ours = (0.0002417771195,-9.029843829)    Ref = (0.0002417771195,-9.029843829)
	Ours = (0x3f2fb0b1a45a9274,0xc0220f47b0bd75df)               Ref = (0x3f2fb0b1a45a926f,0xc0220f47b0bd75df)

Max ulp imag error (0.7242,4.233) @ (-4.243991582e-314,0.0009757882705)	(0x8000000200000000,0x3f4ff9815ad08602)
	Ours = (1.570796327,-0.0009757881156)    Ref = (1.570796327,-0.0009757881156)
	Ours = (0x3ff921fb54442d19,0xbf4ff98105af1dab)               Ref = (0x3ff921fb54442d18,0xbf4ff98105af1daf)

GPU fp32:
Max ulp real error (4.768,0.2087) @ (1.67785461e+35,4.062130506e+31)	(0x7a0141d9,0x74002da1)
	Ours = (0.0002421027166,-81.80113983)    Ref = (0.0002421026438,-81.80113983)
	Ours = (0x397ddcf4,0xc2a39a2f)               Ref = (0x397ddcef,0xc2a39a2f)

Max ulp imag error (0.6333,4.976) @ (-2.802596929e-45,0.007756583858)	(0x80000002,0x3bfe2af1)
	Ours = (1.570796251,-0.007756503765)    Ref = (1.570796371,-0.007756506093)
	Ours = (0x3fc90fda,0xbbfe2a45)               Ref = (0x3fc90fdb,0xbbfe2a4a)

CPU fp64:
Max ulp real error (4.855,0.004554) @ (8476.368757,1.467965354e-247)    (0x40c08e2f336b8800,0xcb06c1784f32580)
        Ours = (1.731832824e-251,-9.738184602)    Ref = (1.731832824e-251,-9.738184602)
        Ours = (0xbdfbe1a4050c7e0,0xc02379f35503c149)               Ref = (0xbdfbe1a4050c7db,0xc02379f35503c149)

Max ulp imag error (0.7242,4.184) @ (-0,0.003891582592) (0x8000000000000000,0x3f6fe13d7ebcb800)
        Ours = (1.570796327,-0.003891572769)    Ref = (1.570796327,-0.003891572769)
        Ours = (0x3ff921fb54442d19,0xbf6fe13838bc3a57)               Ref = (0x3ff921fb54442d18,0xbf6fe13838bc3a53)

CPU fp32:
Max ulp real error (5.348,0.09948) @ (66712.41406,-8356.245117) (0x47824c35,0xc60290fb)
        Ours = (0.1246087849,11.80907726)    Ref = (0.1246087477,11.80907726)
        Ours = (0x3dff32e4,0x413cf1fb)               Ref = (0x3dff32df,0x413cf1fb)

Max ulp imag error (0.1838,3.693) @ (-0.4162324369,-0.1135988012)       (0xbed51c6b,0xbde8a67d)
        Ours = (1.996576786,0.1244143695)    Ref = (1.996576786,0.1244143993)
        Ours = (0x3fff8fd4,0x3dfeccf6)               Ref = (0x3fff8fd4,0x3dfeccfa)

[copy-pr-bot]

This pull request requires additional validation before any workflows can run on NVIDIA's runners.

Pull request vetters can view their responsibilities here.

Contributors can view more details about this message here.

[coderabbitai]

No actionable comments were generated in the recent review. 🎉

ℹ️ Recent review info

⚙️ Run configuration

Configuration used: Path: .coderabbit.yaml

Review profile: CHILL

Plan: Enterprise

Run ID: 868d7bcb-fb59-43e1-ba9b-e9325a35558b

📥 Commits

Reviewing files that changed from the base of the PR and between 3f2087c and 01ee399.

📒 Files selected for processing (1)

• libcudacxx/include/cuda/std/__complex/inverse_trigonometric_functions.h

---

📝 Walkthrough

Summary by CodeRabbit

• Refactor
  • Streamlined the internal implementation of the inverse cosine function for complex numbers in the CUDA standard library, improving computational efficiency and reducing code dependencies while maintaining equivalent behavior.

important:

Walkthrough

The cuda::std::acos implementation for complex numbers is refactored to replace explicit special-case branches with sign-normalized arithmetic: extract signbits, compute acosh on absolute components, apply conditional pi correction via fma for negative real inputs, and restore original signs.

Changes

Complex acos arithmetic optimization

Layer / File(s)	Summary
Header includes updated libcudacxx/include/cuda/std/__complex/inverse_trigonometric_functions.h	Added __cmath/fma.h, retained predicate headers (isinf, isnan, signbit), removed headers from the previous log/sqrt code path.
acos(complex<_Tp>) sign-normalized acosh computation libcudacxx/include/cuda/std/__complex/inverse_trigonometric_functions.h	Rewrote acos(const complex<_Tp>&) to use signbit/fabs normalization, compute acosh on magnitudes, reconstruct real/imag per quadrant, conditionally apply a pi high/low fma correction when original real<0, and flip imag sign for original imag<0. Removed prior explicit isinf/isnan/zero checks and log + sqrt path.

Suggested reviewers

• davebayer
• fbusato

---

_{Comment @coderabbitai help to get the list of available commands and usage tips.}

[coderabbitai]

Actionable comments posted: 1

---

ℹ️ Review info

⚙️ Run configuration

Configuration used: Path: .coderabbit.yaml

Review profile: CHILL

Plan: Enterprise

Run ID: fa3d28f3-03af-4c48-9c2d-954c73bc2ca3

📥 Commits

Reviewing files that changed from the base of the PR and between 52f2794 and 3f2087c.

📒 Files selected for processing (1)

• libcudacxx/include/cuda/std/__complex/inverse_trigonometric_functions.h

[s-oboyle]

/ok to test 01ee399

[github-actions]

🥳 CI Workflow Results

🟩 Finished in 1h 53m: Pass: 100%/116 | Total: 2d 18h | Max: 1h 15m | Hits: 70%/516144

See results here.