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the freebsd fma code failed to raise underflow exception in some
cases in nearest rounding mode (affects fmal too) e.g.
fma(-0x1p-1000, 0x1.000001p-74, 0x1p-1022)
and the inexact exception may be raised spuriously since the fenv
is not saved/restored around the exact multiplication algorithm
(affects x86 fma too).
another issue is that the underflow behaviour when the rounded result
is the minimal normal number is target dependent, ieee754 allows two
ways to raise underflow for inexact results: raise if the result before
rounding is in the subnormal range (e.g. aarch64, arm, powerpc) or if
the result after rounding with infinite exponent range is in the
subnormal range (e.g. x86, mips, sh).
to avoid all these issues the algorithm was rewritten with mostly int
arithmetics and float arithmetics is only used to get correct rounding
and raise exceptions according to the behaviour of the target without
any fenv.h dependency. it also unifies x86 and non-x86 fma.
fmaf is not affected, fmal need to be fixed too.
this algorithm depends on a_clz_64 and it required a few spurious
instructions to make sure underflow exception is raised in a particular
corner case. (normally FORCE_EVAL(tiny*tiny) would be used for this,
but on i386 gcc is broken if the expression is constant
https://gcc.gnu.org/bugzilla/show_bug.cgi?id=57245
and there is no easy portable fix for the macro.)
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the issue is described in commits 1e5eb73545ca6cfe8b918798835aaf6e07af5beb
and ffd8ac2dd50f99c3c83d7d9d845df9874ec3e7d5
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only fma used these macros and the explicit union is clearer
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in lgammal don't handle 1 and 2 specially, in fma use the new ldshape
union instead of ld80 one.
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The underflow exception is not raised correctly in some
cornercases (see previous fma commit), added comments
with examples for fmaf, fmal and non-x86 fma.
In fmaf store the result before returning so it has the
correct precision when FLT_EVAL_METHOD!=0
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1) in downward rounding fma(1,1,-1) should be -0 but it was 0 with
gcc, the code was correct but gcc does not support FENV_ACCESS ON
so it used common subexpression elimination where it shouldn't have.
now volatile memory access is used as a barrier after fesetround.
2) in directed rounding modes there is no double rounding issue
so the complicated adjustments done for nearest rounding mode are
not needed. the only exception to this rule is raising the underflow
flag: assume "small" is an exactly representible subnormal value in
double precision and "verysmall" is a much smaller value so that
(long double)(small plus verysmall) == small
then
(double)(small plus verysmall)
raises underflow because the result is an inexact subnormal, but
(double)(long double)(small plus verysmall)
does not because small is not a subnormal in long double precision
and it is exact in double precision.
now this problem is fixed by checking inexact using fenv when the
result is subnormal
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The long double adjustment was wrong:
The usual check is
mant_bits & 0x7ff == 0x400
before doing a mant_bits++ or mant_bits-- adjustment since
this is the only case when rounding an inexact ld80 into
double can go wrong. (only in nearest rounding mode)
After such a check the ++ and -- is ok (the mantissa will end
in 0x401 or 0x3ff).
fma is a bit different (we need to add 3 numbers with correct
rounding: hi_xy + lo_xy + z so we should survive two roundings
at different places without precision loss)
The adjustment in fma only checks for zero low bits
mant_bits & 0x3ff == 0
this way the adjusted value is correct when rounded to
double or *less* precision.
(this is an important piece in the fma puzzle)
Unfortunately in this case the -- is not a correct adjustment
because mant_bits might underflow so further checks are needed
and this was the source of the bug.
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Some code assumed ldexp(x, 1) is faster than 2.0*x,
but ldexp is a wrapper around scalbn which uses
multiplications inside, so this optimization is
wrong.
This commit also fixes fmal which accidentally
used ldexp instead of ldexpl loosing precision.
There are various additional changes from the
work-in-progress const cleanups.
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correctly rounded double precision fma using extended
precision arithmetics for ld80 systems (x87)
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this is necessary to support archs where fenv is incomplete or
unavailable (presently arm). fma, fmal, and the lrint family should
work perfectly fine with this change; fmaf is slightly broken with
respect to rounding as it depends on non-default rounding modes to do
its work.
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thanks to the hard work of Szabolcs Nagy (nsz), identifying the best
(from correctness and license standpoint) implementations from freebsd
and openbsd and cleaning them up! musl should now fully support c99
float and long double math functions, and has near-complete complex
math support. tgmath should also work (fully on gcc-compatible
compilers, and mostly on any c99 compiler).
based largely on commit 0376d44a890fea261506f1fc63833e7a686dca19 from
nsz's libm git repo, with some additions (dummy versions of a few
missing long double complex functions, etc.) by me.
various cleanups still need to be made, including re-adding (if
they're correct) some asm functions that were dropped.
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