(1) tgamma(-Inf) returned +Inf and failed to raise any exception, but
should always have raised an exception, and should behave like
tgamma(negative integer).
(2) tgamma(negative integer) returned +Inf and raised divide-by-zero,
but should return NaN and raise "invalid" on any IEEEish system.
(3) About half of the 2**52 negative intgers between -2**53 and -2**52
were misclassified as non-integers by using floor(x + 0.5) to round
to nearest, so tgamma(x) was wrong (+-0 instead of +Inf and now NaN)
on these args. The floor() expression is hard to use since rounding
of (x + 0.5) may give x or x + 1, depending on |x| and the current
rounding mode. The fixed version uses ceil(x) to classify x before
operating on x and ends up being more efficient since ceil(x) is
needed anyway.
(4) On at least the problematic args in (3), tgamma() raised a spurious
inexact.
(5) tgamma(large positive) raised divide-by-zero but should raise overflow.
(6) tgamma(+Inf) raised divide-by-zero but should not raise any exception.
(7) Raise inexact for tiny |x| in a way that has some chance of not being
optimized away.
The fix for (5) and (6), and probably for (2), also prevents -O optimizing
away the exception.
PR: 112180 (2)
Standards: Annex F in C99 (IEC 60559 binding) requires (1), (2) and (6).
values in more detail, and change the style of this comment to be closer
to fdlibm and C99:
- tgamma(-Inf) was undocumented and is wrong (+Inf, should be NaN)
- tgamma(negative integer) is as intended (+Inf) but not best for IEEE-754
(NaN)
- tgamma(-0) was documented as being wrong (+Inf) but was correct (-Inf)
- documentation of setting of exceptions (overflow, etc.) was more
complete here than in most of libm, but was further from matching
the actual setting than in most of libm, due to various bugs here
(primarily, always evaluating +Inf one/zero and getting unwanted
divide-by-zero exceptions from this). Now the actual behaviour with
gcc -O0 is documented. Optimization still breaks setting of exceptions
all over libm, so nothing can depend on this working.
- tgamma(NaN)'s exception was documented as being wrong (invalid) but was
correct (no exception with IEEEish NaNs).
Finish (?) rev.1.5. gamma was not renamed to tgamma in one place.
Finish (?) rev.1.6. errno.h was not completely removed.
to doubles as bits. fdlibm-1.1 had similar aliasing bugs, but these
were fixed by NetBSD or Cygnus before a modified version of fdlibm was
imported in 1994. TRUNC() is only used by tgamma() and some
implementation-detail functions. The aliasing bugs were detected by
compiling with gcc -O2 but don't seem to have broken tgamma() on i386's
or amd64's. They broke my modified version of tgamma().
Moved the definition of TRUNC() to mathimpl.h so that it can be fixed
in one place, although the general version is even slower than necessary
because it has to operate on pointers to volatiles to handle its arg
sometimes being volatile. Inefficiency of the fdlibm macros slows
down libm generally, and tgamma() is a relatively unimportant part of
libm. The macros act as if on 32-bit words in memory, so they are
hard to optimize to direct actions on 64-bit double registers for
(non-i386) machines where this is possible. The optimization is too
hard for gcc on amd64's, and declaring variables as volatile makes it
impossible.
