#pragma STDC CX_LIMITED_RANGE ON
the "ON" needs to be in caps. gcc doesn't understand this pragma
anyway and assumes it is always on in any case, but icc supports
it and cares about the case.
conj() instead of using expressions like z * I. The latter is bad for
several reasons:
1. It is implemented using arithmetic, which is unnecessary, and can
generate floating point exceptions, contrary to the requirements on
these functions.
2. gcc implements complex multiplication using a formula that breaks
down for infinities, e.g., it gives INFINITY * I == nan + inf I.
- When y/x is huge, it's faster and more accurate to return pi/2
instead of pi - pi/2.
- There's no need for 3 lines of bit fiddling to compute -z.
- Fix a comment.
at compile time regardless of the dynamic precision, and there's
no way to disable this misfeature at compile time. Hence, it's
impossible to generate the appropriate tables of constants for the
long double inverse trig functions in a straightforward way on i386;
this change hacks around the problem by encoding the underlying bits
in the table.
Note that these functions won't pass the regression test on i386,
even with the FPU set to extended precision, because the regression
test is similarly damaged by gcc. However, the tests all pass when
compiled with a modified version of gcc.
Reported by: bde
- Adjust several constants for float precision. Some thresholds
that were appropriate for double precision were never changed
when these routines were converted to float precision. This
has an impact on performance but not accuracy. (Submitted by bde.)
- Reduce the degrees of the polynomials used. A smaller degree
suffices for float precision.
- In asinf(), use double arithmetic in part of the calculation to
avoid a corner case and some complicated arithmetic involving a
division and some buggy constants. This improves performance and
accuracy.
Max error (ulps):
asinf acosf atanf
before 0.925 0.782 0.852
after 0.743 0.804 0.852
As bde points out, it's cheaper for asin*() and acos*() to use
polynomials instead of rational functions, but that's a task for
another day.
spurious optimizations. gcc doesn't support FENV_ACCESS, so when it
folds constants, it assumes that the rounding mode is always the
default and floating point exceptions never matter.
1. architecture-specific files
2. long double format-specific files
3. bsdsrc
4. src
5. man
The original order was virtually the opposite of this.
This should not cause any functional changes at this time. The
difference is only significant when one wants to override, say, a
generic foo.c with a more specialized foo.c (as opposed to foo.S).
- fma(x, y, z) returns z, not NaN, if z is infinite, x and y are finite,
x*y overflows, and x*y and z have opposite signs.
- fma(x, y, z) doesn't generate an overflow, underflow, or inexact exception
if z is NaN or infinite, as per IEEE 754R.
- If the rounding mode is set to FE_DOWNWARD, fma(1.0, 0.0, -0.0) is -0.0,
not +0.0.
This makes little difference in float precision, but in double
precision gives a speedup of about 30% on amd64 (A64 CPU) and i386
(A64). This depends on fabs[f]() being inline and efficient. The
bit fiddling (or any use of SET_HIGH_WORD(), which libm does too
much because it was best on old 32-bit machines) always causes
packing overheads and sometimes causes stalls in the packing, since
it operates on only part of a variable in the double precision case.
It apparently did cause stalls in a critical path here.
fabs(x+0.0)+fabs(y+0.0) when mixing NaNs. This improves
consistency of the result by making it harder for the compiler to reorder
the operands. (FP addition is not necessarily commutative because the
order of operands makes a difference on some machines iff the operands are
both NaNs.)
e_rem_pio2.c:
This case goes up to about 2**20pi/2, but the comment about it said that
it goes up to about 2**19pi/2.
It went too far above 2**pi/2, giving a multiplier fn with 21 significant
bits in some cases. This would be harmful except for a numerical
accident. It happens that the terms of the approximation to pi/2,
when rounded to 33 bits so that multiplications by 20-bit fn's are
exact, happen to be rounded to 32 bits so multiplications by 21-bit
fn's are exact too, so the bug only complicates the error analysis (we
might lose a bit of accuracy but have bits to spare).
e_rem_pio2f.c:
The bogus comment in e_rem_pio2.c was copied and the code was changed
to be bug-for-bug compatible with it, except the limit was made 90
ulps smaller than necessary. The approximation to pi/2 was not
modified except for discarding some of it.
The same rough error analysis that justifies the limit of 2**20pi/2
for double precision only justifies a limit of 2**18pi/2 for float
precision. We depended on exhaustive testing to check the magic numbers
for float precision. More exaustive testing shows that we can go up
to 2**28pi/2 using a 53+25 bit approximation to pi/2 for float precision,
with a the maximum error for cosf() and sinf() unchanged at 0.5009
ulps despite the maximum error in rem_pio2f being ~0.25 ulps. Implement
this.
gives an average speedup of about 12 cycles or 17% for
9pi/4 < |x| <= 2**19pi/2 and a smaller speedup for larger x, and a
small speeddown for |x| <= 9pi/4 (only 1-2 cycles average, but that
is 4%).
Inlining this is less likely to bust caches than inlining the float
version since it is much smaller (about 220 bytes text and rodata) and
has many fewer branches. However, the float version was already large
due to its manual inlining of the branches and also the polynomial
evaluations.
__kernel_rem_pio2(). This simplifies analysis of aliasing and thus
results in better code for the usual case where __kernel_rem_pio2()
is not called. In particular, when __ieee854_rem_pio2[f]() is inlined,
it normally results in y[] being returned in registers. I couldn't
get this to work using the restrict qualifier.
In float precision, this saves 2-3% in most cases on amd64 and i386
(A64) despite it not being inlined in float precision yet. In double
precision, this has high variance, with an average gain of 2% for
amd64 and 0.7% for i386 (but a much larger gain for usual cases) and
some losses.
this function and its callers cosf(), sinf() and tanf() don't waste time
converting values from doubles to floats and back for |x| > 9pi/4.
All these functions were optimized a few years ago to mostly use doubles
internally and across the __kernel*() interfaces but not across the
__ieee754_rem_pio2f() interface.
This saves about 40 cycles in cosf(), sinf() and tanf() for |x| > 9pi/4
on amd64 (A64), and about 20 cycles on i386 (A64) (except for cosf()
and sinf() in the upper range). 40 cycles is about 35% for |x| < 9pi/4
<= 2**19pi/2 and about 5% for |x| > 2**19pi/2. The saving is much
larger on amd64 than on i386 since the conversions are not easy to
optimize except on i386 where some of them are automatic and others
are optimized invalidly. amd64 is still about 10% slower in cosf()
and tanf() in the lower range due to conversion overhead.
This also gives a tiny speedup for |x| <= 9pi/4 on amd64 (by simplifying
the code). It also avoids compiler bugs and/or additional slowness
in the conversions on (not yet supported) machines where double_t !=
double.
e_rem_pio2.c:
Float and double precision didn't work because init_jk[] was 1 too small.
It needs to be 2 larger than you might expect, and 1 larger than it was
for these precisions, since its test for recomputing needs a margin of
47 bits (almost 2 24-bit units).
init_jk[] seems to be barely enough for extended and quad precisions.
This hasn't been completely verified. Callers now get about 24 bits
of extra precision for float, and about 19 for double, but only about
8 for extended and quad. 8 is not enough for callers that want to
produce extra-precision results, but current callers have rounding
errors of at least 0.8 ulps, so another 1/2**8 ulps of error from the
reduction won't affect them much.
Add a comment about some of the magic for init_jk[].
e_rem_pio2.c:
Double precision worked in practice because of a compensating off-by-1
error here. Extended precision was asked for, and it executed exactly
the same code as the unbroken double precision.
e_rem_pio2f.c:
Float precision worked in practice because of a compensating off-by-1
error here. Double precision was asked for, and was almost needed,
since the cosf() and sinf() callers want to produce extra-precision
results, at least internally so that their error is only 0.5009 ulps.
However, the extra precision provided by unbroken float precision is
enough, and the double-precision code has extra overheads, so the
off-by-1 error cost about 5% in efficiency on amd64 and i386.
variations (e500 currently), this provides a gcc-level FPU emulation and is an
alternative approach to the recently introduced kernel-level emulation
(FPU_EMU).
Approved by: cognet (mentor)
MFp4: e500
fabs(), a conditional branch, and sign adjustments of 3 variables for
x < 0 when the branch is taken. In double precision, even when the
branch is perfectly predicted, this saves about 10 cycles or 10% on
amd64 (A64) and i386 (A64) for the negative half of the range, but
makes little difference for the positive half of the range. In float
precision, it also saves about 4 cycles for the positive half of the
range on i386, and many more cycles in both halves on amd64 (28 in the
negative half and 11 in the positive half for tanf), but the amd64
times for float precision are anomalously slow so the larger
improvement is only a side effect.
Previous commits arranged for the x < 0 case to be handled simply:
- one part of the rounding method uses the magic number 0x1.8p52
instead of the usual 0x1.0p52. The latter is required for large |x|,
but it doesn't work for negative x and we don't need it for large |x|.
- another part of the rounding method no longer needs to add `half'.
It would have needed to add -half for negative x.
- removing the "quick check no cancellation" in the double precision
case removed the need to take the absolute value of the quadrant
number.
Add my noncopyright in e_rem_pio2.c
FP-to-FP method to round to an integer on all arches, and convert this
to an int using FP-to-integer conversion iff irint() is not available.
This is cleaner and works well on at least ia64, where it saves 20-30
cycles or about 10% on average for 9Pi/4 < |x| <= 32pi/2 (should be
similar up to 2**19pi/2, but I only tested the smaller range).
After the previous commit to e_rem_pio2.c removed the "quick check no
cancellation" non-optimization, the result of the FP-to-integer
conversion is not needed so early, so using irint() became a much
smaller optimization than when it was committed.
An earlier commit message said that cos, cosf, sin and sinf were equally
fast on amd64 and i386 except for cos and sin on i386. Actually, cos
and sin on amd64 are equally fast to cosf and sinf on i386 (~88 cycles),
while cosf and sinf on amd64 are not quite equally slow to cos and sin
on i386 (average 115 cycles with more variance).
9pi/2 < |x| < 32pi/2 since it is only a small or negative optimation
and it gets in the way of further optimizations. It did one more
branch to avoid some integer operations and to use a different
dependency on previous results. The branches are fairly predictable
so they are usually not a problem, so whether this is a good
optimization depends mainly on the timing for the previous results,
which is very machine-dependent. On amd64 (A64), this "optimization"
is a pessimization of about 1 cycle or 1%; on ia64, it is an
optimization of about 2 cycles or 1%; on i386 (A64), it is an
optimization of about 5 cycles or 4%; on i386 (Celeron P2) it is an
optimization of about 4 cycles or 3% for cos but a pessimization of
about 5 cycles for sin and 1 cycle for tan. I think the new i386
(A64) slowness is due to an pipeline stall due to an avoidable
load-store mismatch (so the old timing was better), and the i386
(Celeron) variance is due to its branch predictor not being too good.
the the double to int conversion operation which is very slow on these
arches. Assume that the current rounding mode is the default of
round-to-nearest and use rounding operations in this mode instead of
faking this mode using the round-towards-zero mode for conversion to
int. Round the double to an integer as a double first and as an int
second since the double result is needed much earler.
Double rounding isn't a problem since we only need a rough approximation.
We didn't support other current rounding modes and produce much larger
errors than before if called in a non-default mode.
This saves an average about 10 cycles on amd64 (A64) and about 25 on
i386 (A64) for x in the above range. In some cases the saving is over
25%. Most cases with |x| < 1000pi now take about 88 cycles for cos
and sin (with certain CFLAGS, etc.), except on i386 where cos and sin
(but not cosf and sinf) are much slower at 111 and 121 cycles respectivly
due to the compiler only optimizing well for float precision. A64
hardware cos and sin are slower at 105 cycles on i386 and 110 cycles
on amd64.
the same as lrint() except it returns int instead of long. Though the
extern lrint() is fairly fast on these arches, it still takes about
12 cycles longer than the inline version, and 12 cycles is a lot in
applications where [li]rint() is used to avoid slow conversions that
are only a couple of times slower.
This is only for internal use. The libm versions of *rint*() should
also be inline, but that would take would take more header engineering.
Implementing irint() instead of lrint() also avoids a conflict with
the extern declaration of the latter.
