Underflow exception is only raised when the result is
invalid, but fmod is always exact. x87 has a denormalization
exception, but that's nonstandard. And the superflous *1.0
will be optimized away by any compiler that does not honor
signaling nans.
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.
Some long double consts were stored in two doubles as a workaround
for x86_64 and i386 with the following comment:
/* Long double constants are slow on these arches, and broken on i386. */
This is most likely old gcc bug related to the default x87 fpu
precision setting (it's double instead of double extended on BSD).
up to 30% faster exp2 by avoiding slow frndint and fscale functions.
expm1 also takes a much more direct path for small arguments (the
expected usage case).
unlike some implementations, these functions perform the equivalent of
gcc's -ffloat-store on the result before returning. this is necessary
to raise underflow/overflow/inexact exceptions, perform the correct
rounding with denormals, etc.
unlike trig functions, these are easy to do in asm because they do not
involve (arbitrary-precision) argument reduction. fpatan automatically
takes care of domain issues, and in asin and acos, fsqrt takes care of
them for us.
infinities were getting converted into nans. the new code simply tests
for infinity and replaces it with a large magnitude value of the same
sign.
also, the fcomi instruction is apparently not part of the i387
instruction set, so avoid using it.
these are functions that have direct fpu approaches to implementation
without problematic exception or rounding issues. x86_64 lacks
float/double versions because i'm unfamiliar with the necessary sse
code for performing these operations.
A faster workaround for spurious inexact exceptions
when the result cannot be represented. The old code
actually could be wrong, because gcc reordered the
integer conversion and the exception check.
Note that the new fesetround has slightly different semantics:
Storing the floating-point environment with fnstenv makes the
next fldenv (or fldcw) "non-signaling", so unmasked and pending
exceptions does not invoke the exception handler.
(These are rare since exceptions are handled immediately and by
default all exceptions are masked anyway. But if one manually
unmasks an exception in the control word then either sets the
corresponding exception flag in the status word or the execution
of an exception raising floating-point operation gets interrupted
then it may happen).
So the old implementation did not trap in some rare cases
where the new implementation traps.
However POSIX does not specify anything like the x87 exception
handling traps and the fnstenv/fldenv pair is significantly slower
than the fnstcw/fldcw pair (new code is about 5x faster here and
it's dominated by the function call overhead).
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.
