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this could perhaps use some additional testing for corner cases, but
it seems to be correct.
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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).
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The old scalbn.c was wrong and slow, the new one is just slow.
(scalbn(0x1p+1023,-2097) should give 0x1p-1074, but the old code gave 0)
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correctly rounded double precision fma using extended
precision arithmetics for ld80 systems (x87)
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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.
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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.
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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.
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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.
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Simple wrappers around round is enough because
spurious inexact exception is allowed.
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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.
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untested; may need followup-fixes.
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if all exception flags will be cleared, we can avoid the expensive
store/reload of the environment and just use the fnclex instruction.
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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).
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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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otherwise, the standard C lgamma function will clobber a symbol in the
namespace reserved for the application.
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standard functions cannot depend on nonstandard symbols
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long double and float bessel functions are no longer xsi extensions
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a double precision nan, when converted to extended (80-bit) precision,
will never end in 0x400, since the corresponding bits do not exist in
the original double precision value. thus there's no need to waste
time and code size on this check.
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presumably broken gcc may generate calls to these, and it's said that
ffmpeg makes use of sincosf.
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the fsqrt opcode is correctly rounded, but only in the fpu's selected
precision mode, which is 80-bit extended precision. to get a correctly
rounded double precision output, we check for the only corner cases
where two-step rounding could give different results than one-step
(extended-precision mantissa ending in 0x400) and adjust the mantissa
slightly in the opposite direction of the rounding which the fpu
already did (reported in the c1 flag of the fpu status word).
this should have near-zero cost in the non-corner cases and at worst
very low cost.
note that in order for sqrt() to get used when compiling with gcc, the
broken, non-conformant builtin sqrt must be disabled.
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