[SYCL][libdevice] improve complex trig and pi ulp

[cperkinsintel]

Some of the inverse trig functions (acos, asin, acosh asinh) over complex numbers in SYCL have two problems:

• the PI constant is calculated by a rounding transcendental, and drops 1 ULP before ever being used
• when there is a large negative real part, the intermediate sum z + sqrt(z^2 -+ 1) cancels catastrophically, its true
  value is smaler than the rounding error of its two terms.

[Copilot]

Note

Copilot was unable to run its full agentic suite in this review.

Copilot review overview

Review effort: Lite
Findings: 1 · 6

Open (7)

• id<1> does not support arithmetic/indexing the way this kernel uses it (4 * i and d_in[i]).… · New
• Same issue as in complex_math.hpp: __has_include(<version>) should be guarded (e.g.,… · New
• __has_include is not guaranteed to be available on all preprocessors, and using it unguarded can… · New
• These explicit specializations define non-inline variables in a header. If this header is… · New
• These explicit specializations define non-inline variables in a header. If this header is… · New
• These explicit specializations define non-inline variables in a header. If this header is… · New
• These explicit specializations define non-inline variables in a header. If this header is… · New

What changed in this PR

This PR improves correctness of complex inverse trig/hyperbolic functions for large-magnitude inputs on device by avoiding catastrophic cancellation and using a more reliable value of π, and adds/extends end-to-end coverage for the affected cases.

Changes:

• Add a new DeviceLib E2E test to compare std::acos/asin/acosh/asinh on device vs host for large arguments.
• Update complex math implementations (acos/acosh/asinh and π constant selection) to avoid cancellation for negative real parts and fix π rounding issues.
• Extend existing complex math test cases with additional “large argument” inputs for inverse functions.

File	Description
sycl/​test-e2e/​DeviceLib/​std_complex_inverse_trig_large_args.cpp	Adds a new E2E test exercising large-argument inverse complex functions on device.
sycl/​test-e2e/​Complex/​sycl_complex_math_test_cases.hpp	Refactors default test values and appends large-argument cases for inverse functions.
sycl/​include/​sycl/​ext/​oneapi/​experimental/​complex/​detail/​complex_math.hpp	Switches π computation to a constant and updates inverse functions to avoid cancellation on negative real inputs.
libdevice/​fallback-complex.hpp	Mirrors cancellation-avoidance logic and π constant usage for float complex fallbacks.
libdevice/​fallback-complex-fp64.hpp	Mirrors cancellation-avoidance logic and π constant usage for double complex fallbacks.
libdevice/​device_complex.h	Introduces a shared π definition used by libdevice complex implementations.

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[cperkinsintel]

The failure in llvm/sycl/test-e2e/USM/memops2d/memcpy2d_device_to_host.cpp (or device_to_device.cpp) on Arc is being seen by other PR as well.
#23385

[KseniyaTikhomirova]

@cperkinsintel could you please add libdevice owner to review this first?

[jinge90]

Hi, @cperkinsintel
I borrowed the algorithm from LLVM libcxx long time ago, so the same floating-point cancle issue may still exist there, maybe the fix applied in this PR can also help libcxx.
Thanks very much.

[cperkinsintel]

@jinge90 - yeah, I think something like this could be applied to libcxx upstream. I've never opened a PR against that, have you?

@KseniyaTikhomirova - can you review as the representative from llvm-reviewers-runtime ?

[jinge90]

@jinge90 - yeah, I think something like this could be applied to libcxx upstream. I've never opened a PR against that, have you?

@KseniyaTikhomirova - can you review as the representative from llvm-reviewers-runtime ?

Hi, @cperkinsintel
I haven't opened any PR about this.
Thanks very much.

[KseniyaTikhomirova]

It is hard for me to estimate correctness of applied math changes so I reviewed this from language perspective only.

[KseniyaTikhomirova]

SYCL RT changes LGTM

[sys-ce-bb]

@intel/llvm-gatekeepers please consider merging