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a conforming compiler for an arch with excess precision floating point
(FLT_EVAL_METHOD!=0; presently i386 is the only such arch supported)
computes all intermediate results in the types float_t and double_t
rather than the nominal type of the expression. some incorrect
compilers, however, only keep excess precision in registers, and
convert down to the nominal type when spilling intermediate results to
memory, yielding unpredictable results that depend on the compiler's
choices of what/when to spill. in particular, this happens on old gcc
versions with -ffloat-store, which we need in order to work around
bugs where the compiler wrongly keeps explicitly-dropped excess
by explicitly converting to double_t where expressions are expected be
be evaluated in double_t precision, we can avoid depending on the
compiler to get types correct when spilling; the nominal and
intermediate precision now match. this commit should not change the
code generated by correct compilers, or by old ones on non-i386 archs
where double_t is defined as double.
this fixes a serious bug in argument reduction observed on i386 with
gcc 4.2: for values of x outside the unit circle, sin(x) was producing
results outside the interval [-1,1]. changes made in commit
0ce946cf808274c2d6e5419b139e130c8ad4bd30 were likely responsible for
breaking compatibility with this and other old gcc versions.
patch by Szabolcs Nagy.
method: if there is a large difference between the scale of x and y
then the larger magnitude dominates, otherwise reduce x,y so the
argument of sqrt (x*x+y*y) does not overflow or underflow and calculate
the argument precisely using exact multiplication. If the argument
has less error than 1/sqrt(2) ~ 0.7 ulp, then the result has less error
than 1 ulp in nearest rounding mode.
the original fdlibm method was the same, except it used bit hacks
instead of dekker-veltkamp algorithm, which is problematic for long
double where different representations are supported. (the new hypot
and hypotl code should be smaller and faster on 32bit cpu archs with
fast fpu), the new code behaves differently in non-nearest rounding,
but the error should be still less than 2ulps.
ld80 and ld128 are supported
this also fixes overflow/underflow raising and excess
precision issues (as those are handled well in scalbn)
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.