<complex>: Improve numerical accuracy of sqrt and log

[statementreply]

Goals:

• Fix sqrt overflow when input is huge.
• Fix sqrt inaccuracy when input is tiny, due to internal underflow.
• Fix log inaccuracy when input is tiny, due to internal underflow.
• Improve accuracy of log when |z| ≈ 1, and fix log(complex<float>{1, 0}) != 0 under certain combinations of compile time and runtime settings. (DevCom-1093507 is a symptom of the inaccuracy. It is itself non-bug.)
  • Use hardware FMA on x86/x64 when available.
  • Use hardware FMA on arm/arm64 when available.
• Add test coverage.

The changes contain algorithm described in W. Kahan (1987), Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit.

[StephanTLavavej]

This looks good to me! 😺 I'll validate and push changes for a number of superficial issues that I noticed, nothing affecting the core logic.

[StephanTLavavej]

I've pushed a change for compatibility with our internal test harness - it doesn't contain the same option-parsing machinery that the Python/lit test harness does, so it attempts to use ARM64-only options on x86 and x64 (and vice versa). clang-cl emits an "unknown command line argument" warning that upgrades to an error, so we need to disable that. (MSVC will emit a warning from the compiler driver, but that doesn't get upgraded to an error by /WX, so we don't need to make a change for it.)

Unfortunately, significant changes are needed to be compatible with /clr:pure (yet again we pay a price for not having ported its builds and tests to GitHub yet). I think I know how to fix it, but not in time for merging tomorrow, so I'm moving this back to WIP.

[StephanTLavavej]

I've pushed a merge, resolving conflicts with the test harness rewrite. The tests.py changes now have comments referring to features.py. The previous changes to config.py have been replaced by different changes to features.py (with the same effect).

This doesn't do anything about /clr:pure yet, but should get the tests here passing again.

[StephanTLavavej]

Changelog

• Fused src/float_multi_prec.hpp and src/math_algorithms.cpp into <complex>, because
  /clr:pure refused to work with the separately compiled code.
• Introduced macros _FMP_USING_X86_X64_INTRINSICS and _FMP_USING_ARM64_INTRINSICS,
  defined within <complex> only.
• /clr:pure doesn't have access to any intrinsics (beyond a few special ones like _InterlockedIncrement).
• Clang makes it difficult to use the FMA intrinsics needed here. This may be solvable in the future, but for now, I'm activating the fallback for Clang. @statementreply's profiling indicates that the intrinsics are worth ~5% performance (~3 ns out of ~60 ns for complex<double>::log) so they're "nice to have" but not critical.
• To reduce the throughput impact, I'm including <emmintrin.h> (which is relatively small)
  and then manually declaring _mm_fmsub_sd (which is declared by <immintrin.h>, which is large). We usually like to centralize intrinsic declarations for the STL in <intrin0.h>, but the definition of __m128d can't simply be repeated - we may be able to solve this in the future (with more significant changes to vcruntime).
• For ARM64, I'm including <arm64_neon.h>, again because of type definitions.
• Instead of blocking /fp:fast, I'm using #pragma float_control(precise, on, push), #pragma float_control(pop) around the entire namespace _Float_multi_prec. As @statementreply explained to me, the correctness of this logic is massively damaged by /fp:fast, which is why this is necessary.
• To support C++14 mode, we can't use terse static_assert anymore.
• Because _High_half (and _Sqr_error_fallback, indirectly) use _Bit_cast, they need to be marked as _CONSTEXPR_BIT_CAST for CUDA.
• I'm changing _Sqr_x2 from constexpr to inline and removing its usage of is_constant_evaluated - nothing requires this to be constexpr (yet).
• I've reworked how _Sqr_x2 chooses the intrinsic versus fallback implementations (for clarity and to deal with the new system that detects /clr:pure and Clang).
• I've additionally added code, like we used in <bit>, to detect __AVX2__ and avoid the runtime ISA test.
• The contents of src/math_algorithms.cpp are now within std::_Math_algorithms (we can't use unnamed namespaces in headers). This makes some _STD qualification unnecessary (on non-functions, and _Ugly functions).
• To support C++14 mode, we can't have explicit specializations of non-inline variable templates. I've converted _Hypot_leg_huge and _Hypot_leg_tiny to use helper structs.
• This header-only code no longer needs __std_math_log_hypot and __std_math_log_hypotf; we can call _Math_algorithms::_Log_hypot directly.
• <xutility> needs to leave _CONSTEXPR_BIT_CAST defined for <complex> to use.

[StephanTLavavej]

Thanks again for these accuracy improvements and bug fixes! 🎯 😸 This will ship in VS 2019 16.9 Preview 3.