this function never existed historically; since the float/double
functions it's based on are nonstandard and deprecated, there's really
no justification for its existence except that glibc has it. it can be
added back if there's ever really a need...
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.
apparently initializing a variable is not "using" it but assigning to
it is "using" it. i don't really like this fix, but it's better than
trying to make a bigger cleanup just before a release, and it should
work fine (tested against nsz's math tests).
old: 2*atan2(sqrt(1-x),sqrt(1+x))
new: atan2(fabs(sqrt((1-x)*(1+x))),x)
improvements:
* all edge cases are fixed (sign of zero in downward rounding)
* a bit faster (here a single call is about 131ns vs 162ns)
* a bit more precise (at most 1ulp error on 1M uniform random
samples in [0,1), the old formula gave some 2ulp errors as well)
this is a nonstandard function so it's not clear what conditions it
should satisfy. my intent is that it be fast and exact for positive
integral exponents when the result fits in the destination type, and
fast and correctly rounded for small negative integral exponents.
otherwise we aim for at most 1ulp error; it seems to differ from pow
by at most 1ulp and it's often 2-5 times faster than pow.
special care is made to avoid any inexact computations when either arg
is zero (in which case the exact absolute value of the other arg
should be returned) and to support the special condition that
hypot(±inf,nan) yields inf.
hypotl is not yet implemented since avoiding overflow is nontrivial.
the old formula atan2(1,sqrt((1+x)/(1-x))) was faster but
could give nan result at x=1 when the rounding mode is
FE_DOWNWARD (so 1-1 == -0 and 2/-0 == -inf), the new formula
gives -0 at x=+-1 with downward rounding.
this has not been tested heavily, but it's known to at least assemble
and run in basic usage cases. it's nearly identical to the
corresponding i386 code, and thus expected to be just as correct or
just as incorrect.
old code saved/restored the fenv (the new code is only as slow
as that when inexact is not set before the call, but some other
flag is set and the rounding is inexact, which is rare)
before:
bench_nearbyint_exact 5000000 N 261 ns/op
bench_nearbyint_inexact_set 5000000 N 262 ns/op
bench_nearbyint_inexact_unset 5000000 N 261 ns/op
after:
bench_nearbyint_exact 10000000 N 94.99 ns/op
bench_nearbyint_inexact_set 25000000 N 65.81 ns/op
bench_nearbyint_inexact_unset 10000000 N 94.97 ns/op
the fscale instruction is slow everywhere, probably because it
involves a costly and unnecessary integer truncation operation that
ends up being a no-op in common usages. instead, construct a floating
point scale value with integer arithmetic and simply multiply by it,
when possible.
for float and double, this is always possible by going to the
next-larger type. we use some cheap but effective saturating
arithmetic tricks to make sure even very large-magnitude exponents
fit. for long double, if the scaling exponent is too large to fit in
the exponent of a long double value, we simply fallback to the
expensive fscale method.
on atom cpu, these changes speed up scalbn by over 30%. (min rdtsc
timing dropped from 110 cycles to 70 cycles.)
exponents (base 2) near 16383 were broken due to (1) wrong cutoff, and
(2) inability to fit the necessary range of scalings into a long
double value.
as a solution, we fall back to using frndint/fscale for insanely large
exponents, and also have to special-case infinities here to avoid
inf-inf generating nan.
thankfully the costly code never runs in normal usage cases.
zero, one, two, half are replaced by const literals
The policy was to use the f suffix for float consts (1.0f),
but don't use suffix for long double consts (these consts
can be exactly represented as double).
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.
