stillwater-sc / universal / 35523692073

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Commit Message

fix(internal): stop the error-free transformations depending on a build flag (#1580)

* fix(internal): stop the error-free transformations depending on a build flag

mingw-w64's software std::fma is not correctly rounded. Measured against
Dekker's product over 200000 random pairs in [0.5, 2), 15763 of them -- 7.9% --
come back with a wrong roundoff:

    dekker = -5.24664541231088145084e-17
    fma    = -5.24664574318312647205e-17     wrong from the 9th digit

y == fma(a, b, -x) is an error-free transformation only if fma rounds once, so
on that build two_prod is not error-free and every algorithm above it silently
loses its exactness guarantee.

This was already known: cmake/toolchains/x86_64-w64-mingw32.cmake passes -mfma
so gcc emits the hardware FMA3 instruction instead of calling the routine, and
the comment there names floatcascade as what it broke. With -mfma the
disagreements go to zero, and every configuration this project builds and tests
is correct.

A build that does not use that toolchain file is not correct, and it fails
silently -- no error, no warning, just wrong low-order bits. Correctness should
not rest on a flag the caller has to remember. UNIVERSAL_TRUSTED_FMA in
utility/directives.hpp answers "is std::fma correctly rounded here?" the same
way LONG_DOUBLE_SUPPORT answers its question (#1534), and the affected sites
fall back to Dekker's product, which needs nothing from libm.

Affected, measured on MinGW without -mfma, before -> after:

    expansion_ops two_prod (ereal)   5333/50000 wrong  ->  0
    floatcascade two_prod (dd/qd)    7942/100000 wrong ->  0

The ereal suites fail outright there; the cascade suites happen to pass, so that
half is a latent inexactness rather than an observed failure -- but it is the
same identity on the same broken fma, and it is what the toolchain comment was
written about.

tracked_bounded uses fma for three exact residuals, two of which are consulted
only for their SIGN, which an incorrectly round... (continued)

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