hundreds of similar aliasing bugs, but all except this one seem to have
been fixed by Cygnus and/or NetBSD before the modified version of fdlibm
was imported into FreeBSD in 1994.
PR: standards/113147
Submitted by: Steve Kargl <sgk@troutmask.apl.washington.edu>
involving moving the check for x == 0). The savings in cycles are
smaller for cbrtf() than for cbrt(), and positive in all measured cases
with gcc-3.4.4, but still very machine/compiler-dependent.
Warning, after symbol versioning is enabled, going back is not easy
(use WITHOUT_SYMVER at your own risk).
Change the default thread library to libthr.
There most likely still needs to be a version bump for at least the
thread libraries. If necessary, this will happen later.
that ints have >= 31 value bits elsewhere. s/int/int32_t/ seems to
have been done too globally for all other files in msun/src before
msun/ was imported into FreeBSD.
Minor fixes in comments.
e_lgamma_r.c:
Describe special cases in more detail:
- exception for lgamma(0) and lgamma(neg.integer)
- lgamma(-Inf) = Inf. This is wrong but is required by C99 Annex F. I
hope to change this.
(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.
Details: As a side-effect of restoring a saved FP environment,
fesetenv() overwrites the tag word, which indicates which i387
registers are in use. Normally this isn't a problem because
the calling convention requires the register stack to be empty
on function entry and exit. However, fesetenv() is inlined, so we
need to tell gcc explicitly that the i387 registers get clobbered.
PR: 85101
In e_log.c, there was just a off-by-1 (1 ulp) error in the comment
about the threshold. The precision of the threshold is unimportant,
but the magic numbers in the code are easier to understand when the
threshold is described precisely.
In e_logf.c, mistranslation of the magic numbers gave an off-by-1
(1 * 16 ulps) error in the intended negative bound for the threshold
and an off-by-7 (7 * 16 ulps) error in the intended positive bound for
the threshold, and the intended bounds were not translated from the
double precision bounds so they were unnecessarily small by a factor
of about 2048.
The optimization of using the simple Taylor approximation for args
near a power of 2 is dubious since it only applies to a relatively
small proportion of args, but if it is done then doing it 2048 times
as often _may_ be more efficient. (My benchmarks show unexplained
dependencies on the data that increase with further optimizations
in this area.)
2**-28 as a side effect, by merging with the float precision version
of tanh() and the double precision version of sinh().
For tiny x, tanh(x) ~= x, and we used the expression x*(one+x) to
return this value (x) and set the inexact flag iff x != 0. This
doesn't work on ia64 since gcc -O does the dubious optimization
x*(one+x) = x+x*x so as to use fma, so the sign of -0.0 was lost.
Instead, handle tiny x in the same as sinh(), although this is imperfect:
- return x directly and set the inexact flag in a less efficient way.
- increased the threshold for non-tinyness from 2**-55 to 2**-28 so that
many more cases are optimized than are pessimized.
Updated some comments and fixed bugs in others (ranges for half-open
intervals mostly had the open end backwards, and there were nearby style
bugs).
functions are only for compatibility with obsolete standards. They
shouldn't be used, so they shouldn't be optimized. Use the generic
versions instead.
This fixes scalbf() as a side effect. The optimized asm version left
garbage on the FP stack. I fixed the corresponding bug in the optimized
asm scalb() and scalbn() in 1996. NetBSD fixed it in scalb(), scalbn()
and scalbnf() in 1999 but missed fixing it in scalbf(). Then in 2005
the bug was reimplemented in FreeBSD by importing NetBSD's scalbf().
The generic versions have slightly different error handling:
- the asm versions blindly round the second parameter to a (floating
point) integer and proceed, while the generic versions return NaN
if this rounding changes the value. POSIX permits both behaviours
(these functions are XSI extensions and the behaviour for a bogus
non-integral second parameter is unspecified). Apart from this
and the bug in scalbf(), the behaviour of the generic versions seems
to be identical. (I only exhusatively tested
generic_scalbf(1.0F, anyfloat) == asm_scalb(1.0F, anyfloat). This
covers many representative corner cases involving NaNs and Infs but
doesn't test exception flags. The brokenness of scalbf() showed up
as weird behaviour after testing just 7 integer cases sequentially.)
to scalbf(), but ldexpf() cannot be implemented in that way since the
types of the second parameter differ. ldexpf() can be implemented as
a weak or strong reference to scalbnf() (*) but that was already done
long before rev.1.10 was committed. The old implementation uses a
reference, so rev.1.10 had no effect on applications. The C files for
the scalb() family are not used for amd64 or i386, so rev.1.10 had even
less effect for these arches.
(*) scalbnf() raises the radix to the given exponent, while ldexpf()
raises 2 to the given exponent. Thus the functions are equivalent
except possibly for their error handling iff the radix is 2. Standards
more or less require identical error handling. Under FreeBSD, the
functions are equivalent except for more details being missing in
scalbnf()'s man page.
commit broke the 2**24 cases where |x| > DBL_MAX/2. There are exponent
range problems not just for denormals (underflow) but for large values
(overflow). Doubles have more than enough exponent range to avoid the
problems, but I forgot to convert enough terms to double, so there was
an x+x term which was sometimes evaluated in float precision.
Unfortunately, this is a pessimization with some combinations of systems
and compilers (it makes no difference on Athlon XP's, but on Athlon64's
it gives a 5% pessimization with gcc-3.4 but not with gcc-3.3).
Exlain the problem better in comments.
algorithm for the second step significantly to also get a perfectly
rounded result in round-to-nearest mode. The resulting optimization
is about 25% on Athlon64's and 30% on Athlon XP's (about 25 cycles
out of 100 on the former).
Using extra precision, we don't need to do anything special to avoid
large rounding errors in the third step (Newton's method), so we can
regroup terms to avoid a division, increase clarity, and increase
opportunities for parallelism. Rearrangement for parallelism loses
the increase in clarity. We end up with the same number of operations
but with a division reduced to a multiplication.
Using specifically double precision, there is enough extra precision
for the third step to give enough precision for perfect rounding to
float precision provided the previous steps are accurate to 16 bits.
(They were accurate to 12 bits, which was almost minimal for imperfect
rounding in the old version but would be more than enough for imperfect
rounding in this version (9 bits would be enough now).) I couldn't
find any significant time optimizations from optimizing the previous
steps, so I decided to optimize for accuracy instead. The second step
needed a division although a previous commit optimized it to use a
polynomial approximation for its main detail, and this division dominated
the time for the second step. Use the same Newton's method for the
second step as for the third step since this is insignificantly slower
than the division plus the polynomial (now that Newton's method only
needs 1 division), significantly more accurate, and simpler. Single
precision would be precise enough for the second step, but doesn't
have enough exponent range to handle denormals without the special
grouping of terms (as in previous versions) that requires another
division, so we use double precision for both the second and third
steps.
distributed non-large args, this saves about 14 of 134 cycles for
Athlon64s and about 5 of 199 cycles for AthlonXPs.
Moved the check for x == 0 inside the check for subnormals. With
gcc-3.4 on uniformly distributed non-large args, this saves another
5 cycles on Athlon64s and loses 1 cycle on AthlonXPs.
Use INSERT_WORDS() and not SET_HIGH_WORD() when converting the first
approximation from bits to double. With gcc-3.4 on uniformly distributed
non-large args, this saves another 4 cycles on both Athlon64s and and
AthlonXPs.
Accessing doubles as 2 words may be an optimization on old CPUs, but on
current CPUs it tends to cause extra operations and pipeline stalls,
especially for writes, even when only 1 of the words needs to be accessed.
Removed an unused variable.
function approximation for the second step. The polynomial has degree
2 for cbrtf() and 4 for cbrt(). These degrees are minimal for the final
accuracy to be essentially the same as before (slightly smaller).
Adjust the rounding between steps 2 and 3 to match. Unfortunately,
for cbrt(), this breaks the claimed accuracy slightly although incorrect
rounding doesn't. Claim less accuracy since its not worth pessimizing
the polynomial or relying on exhaustive testing to get insignificantly
more accuracy.
This saves about 30 cycles on Athlons (mainly by avoiding 2 divisions)
so it gives an overall optimization in the 10-25% range (a larger
percentage for float precision, especially in 32-bit mode, since other
overheads are more dominant for double precision, surprisingly more
in 32-bit mode).
- in preparing for the third approximation, actually make t larger in
magnitude than cbrt(x). After chopping, t must be incremented by 2
ulps to make it larger, not 1 ulp since chopping can reduce it by
almost 1 ulp and it might already be up to half a different-sized-ulp
smaller than cbrt(x). I have not found any cases where this is
essential, but the think-time error bound depends on it. The relative
smallness of the different-sized-ulp limited the bug. If there are
cases where this is essential, then the final error bound would be
5/6+epsilon instead of of 4/6+epsilon ulps (still < 1).
- in preparing for the third approximation, round more carefully (but
still sloppily to avoid branches) so that the claimed error bound of
0.667 ulps is satisfied in all cases tested for cbrt() and remains
satisfied in all cases for cbrtf(). There isn't enough spare precision
for very sloppy rounding to work:
- in cbrt(), even with the inadequate increment, the actual error was
0.6685 in some cases, and correcting the increment increased this
a little. The fix uses sloppy rounding to 25 bits instead of very
sloppy rounding to 21 bits, and starts using uint64_t instead of 2
words for bit manipulation so that rounding more bits is not much
costly.
- in cbrtf(), the 0.667 bound was already satisfied even with the
inadequate increment, but change the code to almost match cbrt()
anyway. There is not enough spare precision in the Newton
approximation to double the inadequate increment without exceeding
the 0.667 bound, and no spare precision to avoid this problem as
in cbrt(). The fix is to round using an increment of 2 smaller-ulps
before chopping so that an increment of 1 ulp is enough. In cbrt(),
we essentially do the same, but move the chop point so that the
increment of 1 is not needed.
Fixed comments to match code:
- in cbrt(), the second approximation is good to 25 bits, not quite 26 bits.
- in cbrt(), don't claim that the second approximation may be implemented
in single precision. Single precision cannot handle the full exponent
range without minor but pessimal changes to renormalize, and although
single precision is enough, 25 bit precision is now claimed and used.
Added comments about some of the magic for the error bound 4/6+epsilon.
I still don't understand why it is 4/6+ and not 6/6+ ulps.
Indent comments at the right of code more consistently.
the second step of approximating cbrt(x). It turns out to be neither
very magic not nor very good. It is just the (2,2) Pade approximation
to 1/cbrt(r) at r = 1, arranged in a strange way to use fewer operations
at a cost of replacing 4 multiplications by 1 division, which is an
especially bad tradeoff on machines where some of the multiplications
can be done in parallel. A Remez rational approximation would give
at least 2 more bits of accuracy, but the (2,2) Pade approximation
already gives 6 more bits than needed. (Changed the comment which
essentially says that it gives 3 more bits.)
Lower order Pade approximations are not quite accurate enough for
double precision but are plenty for float precision. A lower order
Remez rational approximation might be enough for double precision too.
However, rational approximations inherently require an extra division,
and polynomial approximations work well for 1/cbrt(r) at r = 1, so I
plan to switch to using the latter. There are some technical
complications that tend to cost a division in another way.
This gives an optimization of between 9 and 22% on Athlons (largest
for cbrt() on amd64 -- from 205 to 159 cycles).
We extracted the sign bit and worked with |x|, and restored the sign
bit as the last step. We avoided branches to a fault by using accesses
to FP values as bits to clear and restore the sign bit. Avoiding
branches is usually good, but the bit access macros are not so good
(especially for setting FP values), and here they always caused pipeline
stalls on Athlons. Even using branches would be faster except on args
that give perfect branch misprediction, since only mispredicted branches
cause stalls, but it possible to avoid touching the sign bit in FP
values at all (except to preserve it in conversions from bits to FP
not related to the sign bit). Do this. The results are identical
except in 2 of the 3 unsupported rounding modes, since all the
approximations use odd rational functions so they work right on strictly
negative values, and the special case of -0 doesn't use an approximation.
<cbrt(x) in bits> ~= <x in bits>/3 + BIAS.
Keep the large comments only in the double version as usual.
Fixed some style bugs (mainly grammar and spelling errors in comments).
It was because I forgot to translate the part of the double precision
algorithm that chops t so that t*t is exact. Now the maximum error
is the same as for double precision (almost exactly 2.0/3 ulps).
The maximum error was 3.56 ulps.
The bug was another translation error. The double precision version
has a comment saying "new cbrt to 23 bits, may be implemented in
precision". This means exactly what it says -- that the 23 bit second
approximation for the double precision cbrt() may be implemented in
single (i.e., float) precision. It doesn't mean what the translation
assumed -- that this approximation, when implemented in float precision,
is good enough for the the final approximation in float precision.
First, float precision needs a 24 bit approximation. The "23 bit"
approximation is actually good to 24 bits on float precision args, but
only if it is evaluated in double precision. Second, the algorithm
requires a cleanup step to ensure its error bound.
In float precision, any reasonable algorithm works for the cleanup
step. Use the same algorithm as for double precision, although this
is much more than enough and is a significant pessimization, and don't
optimize or simplify anything using double precision to implement the
float case, so that the whole double precision algorithm can be verified
in float precision. A maximum error of 0.667 ulps is claimed for cbrt()
and the max for cbrtf() using the same algorithm shouldn't be different,
but the actual max for cbrtf() on amd64 is now 0.9834 ulps. (On i386
-O1 the max is 0.5006 (down from < 0.7) due to extra precision.)
The threshold for not being tiny was too small. Use the usual 2**-12
threshold. As for sinhf, use a different method (now the same as for
sinhf) to set the inexact flag for tiny nonzero x so that the larger
threshold works, although this method is imperfect. As for sinhf,
this change is not just an optimization, since the general code that
we fell into has accuracy problems even for tiny x. On amd64, avoiding
it fixes tanhf on 2*13495596 args with errors of between 1 and 1.3
ulps and thus reduces the total number of args with errors of >= 1 ulp
from 37533748 to 5271278; the maximum error is unchanged at 2.2 ulps.
The magic number 22 is log(DBL_MAX)/2 plus slop. This is bogus for
float precision. Use 9 (log(FLT_MAX)/2 plus less slop than for
double precision). Unlike for coshf and tanhf, this is just an
optimization, and MAX isn't misspelled EPSILON in the commit log.
I started testing with nonstandard rounding modes, and verified that
the chosen thresholds work for all modes modulo problems not related
to thresholds. The best thresholds are not very dependent on the mode,
at least for tanhf.
shares its low half with pio2_hi. pio2_hi is rounded down although
rounding to nearest would be a tiny bit better, so pio4_hi must be
rounded down too. It was rounded to nearest, which happens to be
different in float precision but the same in double precision.
This fixes about 13.5 million errors of more than 1 ulp in asinf().
The largest error was 2.81 ulps on amd64 and 2.57 ulps on i386 -O1.
Now the largest error is 0.93 ulps on amd65 and 0.67 ulps on i386 -O1.
sqrt(2)/2-1. For log1p(), fixed the approximation to sqrt(2)/2-1.
The end result is to fix an error of 1.293 ulps in
log1pf(0.41421395540 (hex 0x3ed413da))
and an error of 1.783 ulps in
log1p(-0.292893409729003961761) (hex 0x12bec4 00000001)).
The former was the only error of > 1 ulp for log1pf() and the latter
is the only such error that I know of for log1p().
The approximations don't need to be very accurate, but the last 2 need
to be related to the first and be rounded up a little (even more than
1 ulp for sqrt(2)/2-1) for the following implementation-detail reason:
when the arg (x) is not between (the approximations to) sqrt(2)/2-1
and sqrt(2)-1, we commit to using a correction term, but we only
actually use it if 1+x is between sqrt(2)/2 and sqrt(2) according to
the first approximation. Thus we must ensure that
!(sqrt(2)/2-1 < x < sqrt(2)-1) implies !(sqrt(2)/2 < x+1 < sqrt(2)),
where all the sqrt(2)'s are really slightly different approximations
to sqrt(2) and some of the "<"'s are really "<="'s. This was not done.
In log1pf(), the last 2 approximations were rounded up by about 6 ulps
more than needed relative to a good approximation to sqrt(2), but the
actual approximation to sqrt(2) was off by 3 ulps. The approximation
to sqrt(2)-1 ended up being 4 ulps too small, so the algoritm was
broken in 4 cases. The result happened to be broken in 1 case. This
is fixed by using a natural approximation to sqrt(2) and derived
approximations for the others.
In logf(), all the approximations made sense, but the approximation
to sqrt(2)/2-1 was 2 ulps too small (a tiny amount, since we compare
with a granularity of 2**32 ulps), so the algorithm was broken in 2
cases. The result was broken in 1 case. This is fixed by rounding
up the approximation to sqrt(2)/2-1 by 2**32 ulps, so 2**32 cases are
now handled a little differently (still correctly according to my
assertion that the approximations don't need to be very accurate, but
this has not been checked).
on assignment.
Extra precision on i386's broke hi+lo decomposition in the usual way.
It caused all except 1 of the 62343 errors of more than 1 ulp for
log1pf() on i386's with gcc -O [-fno-float-store].
according to the highest nonzero bit in a denormal was missing.
fdlibm ilogbf() and ilogb() have always had the adjustment, but only
use a small part of their method for handling denormals; use the
normalization method in log[f]() for the main part.
