The compiler may substitute calls to sin or cos with calls to sincos, thus
we should have the same optimized implementations for sincos. The
optimized implementations may produce results that differ, that also makes
sure that the sincos call aggrees with the sin and cos calls.
Both float, double, and _Float128 are assumed to be supported
(float and double already only uses builtins). Only long double
is parametrized due GCC bug 29253 which prevents its usage on
powerpc.
It allows to remove i686, ia64, x86_64, powerpc, and sparc arch
specific implementation.
On ia64 it also fixes the sNAN handling:
math/test-float64x-fabs
math/test-ldouble-fabs
Checked on x86_64-linux-gnu, i686-linux-gnu, powerpc-linux-gnu,
powerpc64-linux-gnu, sparc64-linux-gnu, and ia64-linux-gnu.
Converting double precision constants to float is now affected by the
runtime dynamic rounding mode instead of being evaluated at compile
time with default rounding mode (except static object initializers).
This can change the computed result and cause performance regression.
The known correctness issues (increased ulp errors) are already fixed,
this patch fixes remaining cases of unnecessary runtime conversions.
Add float M_* macros to math.h as new GNU extension API. To avoid
conversions the new M_* macros are used and instead of casting double
literals to float, use float literals (only required if the conversion
is inexact).
The patch was tested on aarch64 where the following symbols had new
spurious conversion instructions that got fixed:
__clog10f
__gammaf_r_finite@GLIBC_2.17
__j0f_finite@GLIBC_2.17
__j1f_finite@GLIBC_2.17
__jnf_finite@GLIBC_2.17
__kernel_casinhf
__lgamma_negf
__log1pf
__y0f_finite@GLIBC_2.17
__y1f_finite@GLIBC_2.17
cacosf
cacoshf
casinhf
catanf
catanhf
clogf
gammaf_positive
Fixes bug 28713.
Reviewed-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
I used these shell commands:
../glibc/scripts/update-copyrights $PWD/../gnulib/build-aux/update-copyright
(cd ../glibc && git commit -am"[this commit message]")
and then ignored the output, which consisted lines saying "FOO: warning:
copyright statement not found" for each of 7061 files FOO.
I then removed trailing white space from math/tgmath.h,
support/tst-support-open-dev-null-range.c, and
sysdeps/x86_64/multiarch/strlen-vec.S, to work around the following
obscure pre-commit check failure diagnostics from Savannah. I don't
know why I run into these diagnostics whereas others evidently do not.
remote: *** 912-#endif
remote: *** 913:
remote: *** 914-
remote: *** error: lines with trailing whitespace found
...
remote: *** error: sysdeps/unix/sysv/linux/statx_cp.c: trailing lines
s_cosf.c and s_sinf.c have
if (abstop12 (y) < abstop12 (pio4))
where abstop12 takes a float argument, but pio4 is static const double.
pio4 is used only in calls to abstop12 and never in arithmetic. Apply
-static const double pio4 = 0x1.921FB54442D18p-1;
+static const float pio4 = 0x1.921FB6p-1f;
to fix:
FAIL: math/test-float-cos
FAIL: math/test-float-sin
FAIL: math/test-float-sincos
FAIL: math/test-float32-cos
FAIL: math/test-float32-sin
FAIL: math/test-float32-sincos
when compiling with GCC 12.
Reviewed-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
The macro TAYLOR_SIN adds the term `-0.5*da*a^2 + da` in hopes
of regaining some precision as a function of da. However the
comment says we add the term `-0.5*da*a^2 + 0.5*da` which is
different. This fix updates the comment to reflect the
code and also simplifies the calculation by replacing `a` with `x`
because they always have the same value.
Signed-off-by: Akila Welihinda <akilawelihinda@ucla.edu>
Reviewed-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
The error handling is moved to sysdeps/ieee754 version with no SVID
support. The compatibility symbol versions still use the wrapper with
SVID error handling around the new code. There is no new symbol version
nor compatibility code on !LIBM_SVID_COMPAT targets (e.g. riscv).
Only ia64 is unchanged, since it still uses the arch specific
__libm_error_region on its implementation.
Checked on x86_64-linux-gnu, i686-linux-gnu, and aarch64-linux-gnu.
This implementation is based on 'An Improved Algorithm for hypot(a,b)'
by Carlos F. Borges [1] using the MyHypot3 with the following changes:
- Handle qNaN and sNaN.
- Tune the 'widely varying operands' to avoid spurious underflow
due the multiplication and fix the return value for upwards
rounding mode.
- Handle required underflow exception for subnormal results.
The main advantage of the new algorithm is its precision. With a
random 1e9 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc
current implementation shows around 0.05% results with an error of
1 ulp (453266 results) while the new implementation only shows
0.0001% of total (1280).
Checked on aarch64-linux-gnu and x86_64-linux-gnu.
[1] https://arxiv.org/pdf/1904.09481.pdf
This implementation is based on 'An Improved Algorithm for hypot(a,b)'
by Carlos F. Borges [1] using the MyHypot3 with the following changes:
- Handle qNaN and sNaN.
- Tune the 'widely varying operands' to avoid spurious underflow
due the multiplication and fix the return value for upwards
rounding mode.
- Handle required underflow exception for subnormal results.
The main advantage of the new algorithm is its precision. With a
random 1e8 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc
current implementation shows around 0.02% results with an error of
1 ulp (23158 results) while the new implementation only shows
0.0001% of total (111).
[1] https://arxiv.org/pdf/1904.09481.pdf
Improve hypot performance significantly by using fma when available. The
fma version has twice the throughput of the previous version and 70% of
the latency. The non-fma version has 30% higher throughput and 10%
higher latency.
Max ULP error is 0.949 with fma and 0.792 without fma.
Passes GLIBC testsuite.
This implementation is based on the 'An Improved Algorithm for
hypot(a,b)' by Carlos F. Borges [1] using the MyHypot3 with the
following changes:
- Handle qNaN and sNaN.
- Tune the 'widely varying operands' to avoid spurious underflow
due the multiplication and fix the return value for upwards
rounding mode.
- Handle required underflow exception for denormal results.
The main advantage of the new algorithm is its precision: with a
random 1e9 input pairs in the range of [DBL_MIN, DBL_MAX], glibc
current implementation shows around 0.34% results with an error of
1 ulp (3424869 results) while the new implementation only shows
0.002% of total (18851).
The performance result are also only slight worse than current
implementation. On x86_64 (Ryzen 5900X) with gcc 12:
Before:
"hypot": {
"workload-random": {
"duration": 3.73319e+09,
"iterations": 1.12e+08,
"reciprocal-throughput": 22.8737,
"latency": 43.7904,
"max-throughput": 4.37184e+07,
"min-throughput": 2.28361e+07
}
}
After:
"hypot": {
"workload-random": {
"duration": 3.7597e+09,
"iterations": 9.8e+07,
"reciprocal-throughput": 23.7547,
"latency": 52.9739,
"max-throughput": 4.2097e+07,
"min-throughput": 1.88772e+07
}
}
Co-Authored-By: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Checked on x86_64-linux-gnu and aarch64-linux-gnu.
[1] https://arxiv.org/pdf/1904.09481.pdf
Use a more optimized comparison for check for NaN and infinite and
add an inlined issignaling implementation for float. With gcc it
results in 2 FP comparisons.
The file Copyright is also changed to use GPL, the implementation was
completely changed by 7c10fd3515 to use double precision instead of
scaling and this change removes all the GET_FLOAT_WORD usage.
Checked on x86_64-linux-gnu.
The largest errors over the full binary32 range are after this
patch (on x86_64):
RNDN: libm wrong by up to 9.00e+00 ulp(s) [9] for x=0x1.04c39cp+6
RNDZ: libm wrong by up to 9.00e+00 ulp(s) [9] for x=0x1.04c39cp+6
RNDU: libm wrong by up to 9.00e+00 ulp(s) [9] for x=0x1.04c39cp+6
RNDD: libm wrong by up to 8.98e+00 ulp(s) [9] for x=0x1.4b7066p+7
Inputs that were yielding huge errors have been added to "make check".
Reviewed-by: Adhemeral Zanella <adhemerval.zanella@linaro.org>
C2X adds new <math.h> functions for floating-point maximum and
minimum, corresponding to the new operations that were added in IEEE
754-2019 because of concerns about the old operations not being
associative in the presence of signaling NaNs. fmaximum and fminimum
handle NaNs like most <math.h> functions (any NaN argument means the
result is a quiet NaN). fmaximum_num and fminimum_num handle both
quiet and signaling NaNs the way fmax and fmin handle quiet NaNs (if
one argument is a number and the other is a NaN, return the number),
but still raise "invalid" for a signaling NaN argument, making them
exceptions to the normal rule that a function with a floating-point
result raising "invalid" also returns a quiet NaN. fmaximum_mag,
fminimum_mag, fmaximum_mag_num and fminimum_mag_num are corresponding
functions returning the argument with greatest or least absolute
value. All these functions also treat +0 as greater than -0. There
are also corresponding <tgmath.h> type-generic macros.
Add these functions to glibc. The implementations use type-generic
templates based on those for fmax, fmin, fmaxmag and fminmag, and test
inputs are based on those for those functions with appropriate
adjustments to the expected results. The RISC-V maintainers might
wish to add optimized versions of fmaximum_num and fminimum_num (for
float and double), since RISC-V (F extension version 2.2 and later)
provides instructions corresponding to those functions - though it
might be at least as useful to add architecture-independent built-in
functions to GCC and teach the RISC-V back end to expand those
functions inline, which is what you generally want for functions that
can be implemented with a single instruction.
Tested for x86_64 and x86, and with build-many-glibcs.py.
Avoid defining f64xfmaf128 twice when building s_fmaf128.c.
This can be reproduced on powerpc64le whenever f128 functions do not
have IFUNC enabled, e.g. using "--with-cpu=power8 --disable-multi-arch", or
when using "-with-cpu=power9".
Fixes: b3f27d8150 ("Add narrowing fma functions")
This patch adds the narrowing fused multiply-add functions from TS
18661-1 / TS 18661-3 / C2X to glibc's libm: ffma, ffmal, dfmal,
f32fmaf64, f32fmaf32x, f32xfmaf64 for all configurations; f32fmaf64x,
f32fmaf128, f64fmaf64x, f64fmaf128, f32xfmaf64x, f32xfmaf128,
f64xfmaf128 for configurations with _Float64x and _Float128;
__f32fmaieee128 and __f64fmaieee128 aliases in the powerpc64le case
(for calls to ffmal and dfmal when long double is IEEE binary128).
Corresponding tgmath.h macro support is also added.
The changes are mostly similar to those for the other narrowing
functions previously added, especially that for sqrt, so the
description of those generally applies to this patch as well. As with
sqrt, I reused the same test inputs in auto-libm-test-in as for
non-narrowing fma rather than adding extra or separate inputs for
narrowing fma. The tests in libm-test-narrow-fma.inc also follow
those for non-narrowing fma.
The non-narrowing fma has a known bug (bug 6801) that it does not set
errno on errors (overflow, underflow, Inf * 0, Inf - Inf). Rather
than fixing this or having narrowing fma check for errors when
non-narrowing does not (complicating the cases when narrowing fma can
otherwise be an alias for a non-narrowing function), this patch does
not attempt to check for errors from narrowing fma and set errno; the
CHECK_NARROW_FMA macro is still present, but as a placeholder that
does nothing, and this missing errno setting is considered to be
covered by the existing bug rather than needing a separate open bug.
missing-errno annotations are duly added to many of the
auto-libm-test-in test inputs for fma.
This completes adding all the new functions from TS 18661-1 to glibc,
so will be followed by corresponding stdc-predef.h changes to define
__STDC_IEC_60559_BFP__ and __STDC_IEC_60559_COMPLEX__, as the support
for TS 18661-1 will be at a similar level to that for C standard
floating-point facilities up to C11 (pragmas not implemented, but
library functions done). (There are still further changes to be done
to implement changes to the types of fromfp functions from N2548.)
Tested as followed: natively with the full glibc testsuite for x86_64
(GCC 11, 7, 6) and x86 (GCC 11); with build-many-glibcs.py with GCC
11, 7 and 6; cross testing of math/ tests for powerpc64le, powerpc32
hard float, mips64 (all three ABIs, both hard and soft float). The
different GCC versions are to cover the different cases in tgmath.h
and tgmath.h tests properly (GCC 6 has _Float* only as typedefs in
glibc headers, GCC 7 has proper _Float* support, GCC 8 adds
__builtin_tgmath).
As described in bug 28358, the round-to-odd computations used in the
libm functions that round their results to a narrower format can yield
spurious underflow exceptions in the following circumstances: the
narrowing only narrows the precision of the type and not the exponent
range (i.e., it's narrowing _Float128 to _Float64x on x86_64, x86 or
ia64), the architecture does after-rounding tininess detection (which
applies to all those architectures), the result is inexact, tiny
before rounding but not tiny after rounding (with the chosen rounding
mode) for _Float64x (which is possible for narrowing mul, div and fma,
not for narrowing add, sub or sqrt), so the underflow exception
resulting from the toward-zero computation in _Float128 is spurious
for _Float64x.
Fixed by making ROUND_TO_ODD call feclearexcept (FE_UNDERFLOW) in the
problem cases (as indicated by an extra argument to the macro); there
is never any need to preserve underflow exceptions from this part of
the computation, because the conversion of the round-to-odd value to
the narrower type will underflow in exactly the cases in which the
function should raise that exception, but it may be more efficient to
avoid the extra manipulation of the floating-point environment when
not needed.
Tested for x86_64 and x86, and with build-many-glibcs.py.
include/math.h has a mechanism to redirect internal calls to various
libm functions, that can often be inlined by the compiler, to call
non-exported __* names for those functions in the case when the calls
aren't inlined, with the redirection being disabled when
NO_MATH_REDIRECT. Add fma to the functions to which this mechanism is
applied.
At present, libm-internal fma calls (generally to __builtin_fma*
functions) are only done when it's known the call will be inlined,
with alternative code not relying on an fma operation being used in
the caller otherwise. This patch is in preparation for adding the TS
18661 / C2X narrowing fma functions to glibc; it will be natural for
the narrowing function implementations to call the underlying fma
functions unconditionally, with this either being inlined or resulting
in an __fma* call. (Using two levels of round-to-odd computation like
that, in the case where there isn't an fma hardware instruction, isn't
optimal but is certainly a lot simpler for the initial implementation
than writing different narrowing fma implementations for all the
various pairs of formats.)
Tested with build-many-glibcs.py that installed stripped shared
libraries are unchanged by the patch (using
<https://sourceware.org/pipermail/libc-alpha/2021-September/130991.html>
to fix installed library stripping in build-many-glibcs.py). Also
tested for x86_64.
This patch adds the narrowing square root functions from TS 18661-1 /
TS 18661-3 / C2X to glibc's libm: fsqrt, fsqrtl, dsqrtl, f32sqrtf64,
f32sqrtf32x, f32xsqrtf64 for all configurations; f32sqrtf64x,
f32sqrtf128, f64sqrtf64x, f64sqrtf128, f32xsqrtf64x, f32xsqrtf128,
f64xsqrtf128 for configurations with _Float64x and _Float128;
__f32sqrtieee128 and __f64sqrtieee128 aliases in the powerpc64le case
(for calls to fsqrtl and dsqrtl when long double is IEEE binary128).
Corresponding tgmath.h macro support is also added.
The changes are mostly similar to those for the other narrowing
functions previously added, so the description of those generally
applies to this patch as well. However, the not-actually-narrowing
cases (where the two types involved in the function have the same
floating-point format) are aliased to sqrt, sqrtl or sqrtf128 rather
than needing a separately built not-actually-narrowing function such
as was needed for add / sub / mul / div. Thus, there is no
__nldbl_dsqrtl name for ldbl-opt because no such name was needed
(whereas the other functions needed such a name since the only other
name for that entry point was e.g. f32xaddf64, not reserved by TS
18661-1); the headers are made to arrange for sqrt to be called in
that case instead.
The DIAG_* calls in sysdeps/ieee754/soft-fp/s_dsqrtl.c are because
they were observed to be needed in GCC 7 testing of
riscv32-linux-gnu-rv32imac-ilp32. The other sysdeps/ieee754/soft-fp/
files added didn't need such DIAG_* in any configuration I tested with
build-many-glibcs.py, but if they do turn out to be needed in more
files with some other configuration / GCC version, they can always be
added there.
I reused the same test inputs in auto-libm-test-in as for
non-narrowing sqrt rather than adding extra or separate inputs for
narrowing sqrt. The tests in libm-test-narrow-sqrt.inc also follow
those for non-narrowing sqrt.
Tested as followed: natively with the full glibc testsuite for x86_64
(GCC 11, 7, 6) and x86 (GCC 11); with build-many-glibcs.py with GCC
11, 7 and 6; cross testing of math/ tests for powerpc64le, powerpc32
hard float, mips64 (all three ABIs, both hard and soft float). The
different GCC versions are to cover the different cases in tgmath.h
and tgmath.h tests properly (GCC 6 has _Float* only as typedefs in
glibc headers, GCC 7 has proper _Float* support, GCC 8 adds
__builtin_tgmath).
We stopped adding "Contributed by" or similar lines in sources in 2012
in favour of git logs and keeping the Contributors section of the
glibc manual up to date. Removing these lines makes the license
header a bit more consistent across files and also removes the
possibility of error in attribution when license blocks or files are
copied across since the contributed-by lines don't actually reflect
reality in those cases.
Move all "Contributed by" and similar lines (Written by, Test by,
etc.) into a new file CONTRIBUTED-BY to retain record of these
contributions. These contributors are also mentioned in
manual/contrib.texi, so we just maintain this additional record as a
courtesy to the earlier developers.
The following scripts were used to filter a list of files to edit in
place and to clean up the CONTRIBUTED-BY file respectively. These
were not added to the glibc sources because they're not expected to be
of any use in future given that this is a one time task:
https://gist.github.com/siddhesh/b5ecac94eabfd72ed2916d6d8157e7dchttps://gist.github.com/siddhesh/15ea1f5e435ace9774f485030695ee02
Reviewed-by: Carlos O'Donell <carlos@redhat.com>
This patch is using the corresponding GCC builtin for roundevenf,
roundeven and roundevenl if the USE_FUNCTION_BUILTIN macros are defined
to one in math-use-builtins.h.
These builtin functions is supported since GCC 10.
The code of the generic implementation is not changed.
Signed-off-by: Shen-Ta Hsieh <ibmibmibm.tw@gmail.com>
Reviewed-by: H.J. Lu <hjl.tools@gmail.com>
This patch redirect roundeven function for futhermore changes.
Signed-off-by: Shen-Ta Hsieh <ibmibmibm.tw@gmail.com>
Reviewed-by: H.J. Lu <hjl.tools@gmail.com>
This patch replaced obsolete AC_TRY_COMPILE to AC_COMPILE_IFELSE or
AC_PREPROC_IFELSE.
It has been confirmed that GNU 'autoconf' 2.69 suppressed obsolete
warnings, updated the following files:
- configure
- sysdeps/mach/configure
- sysdeps/mach/hurd/configure
- sysdeps/s390/configure
- sysdeps/unix/sysv/linux/configure
and didn't change the following files:
- sysdeps/ieee754/ldbl-opt/configure
- sysdeps/unix/sysv/linux/powerpc/configure
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
The symbol has never been exported, so no compatibility symbol is
needed. Removing this file prevents ld from creation an exported
symbol in case GLIBC_2_0 expands to a symbol version which
does not have a local: *; directive in the symbol version map file.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
These workload traces cover the whole "long double" range.
This patch was prepared with the help of Adhemerval Zanella.
Reviewed-by: Carlos O'Donell <carlos@redhat.com>
With this patch, the maximal known error for tgamma is now reduced to 9 ulps
for dbl-64, for all rounding modes. Since exhaustive testing is not possible
for dbl-64, it might be that there are still cases with an error larger than
9 ulps, but all known cases are fixed (intensive tests were done to find cases
with large errors).
Tested on x86_64 and powerpc (and by Adhemerval Zanella on aarch64, arm,
s390x, sparc, and i686).
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
All of the isnan functions are in libc.so due to printf_fp, so move
__isnanf128 there too for consistency.
Reviewed-by: Tulio Magno Quites Machado Filho <tuliom@ascii.art.br>
Reviewed-by: Florian Weimer <fweimer@redhat.com>
Previous commit was missing deleted files in sysdeps/ieee754/dbl-64.
Finally remove all mpa related files, headers, declarations, probes, unused
tables and update makefiles.
Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Finally remove all mpa related files, headers, declarations, probes, unused
tables and update makefiles.
Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Remove slow paths in tan. Add ULP annotations. Merge 'number' into 'mynumber'.
Remove unused entries from tan constants.
Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
This patch series removes all remaining slow paths and related code.
First asin/acos, tan, atan, atan2 implementations are updated, and the final
patch removes the unused mpa files, headers and probes. Passes buildmanyglibc.
Remove slow paths from asin/acos. Add ULP annotations based on previous slow
path checks (which are approximate). Update AArch64 and x86_64 libm-test-ulps.
Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Remove the wordsize-64 implementations by merging them into the main dbl-64
directory. The second patch just moves all wordsize-64 files and removes a
few wordsize-64 uses in comments and Implies files.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Remove the wordsize-64 implementations by merging them into the main dbl-64
directory. The first patch adds special cases needed for 32-bit targets
(FIX_INT_FP_CONVERT_ZERO and FIX_DBL_LONG_CONVERT_OVERFLOW) to the
wordsize-64 versions. This has no effect on 64-bit targets since they don't
define these macros.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Make the tests use TEST_COND_intel96 to decide on whether to build the
unnormal tests instead of the macro in nan-pseudo-number.h and then
drop the header inclusion. This unbreaks test runs on all
architectures that do not have ldbl-96.
Also drop the HANDLE_PSEUDO_NUMBERS macro since it is not used
anywhere.
I used these shell commands:
../glibc/scripts/update-copyrights $PWD/../gnulib/build-aux/update-copyright
(cd ../glibc && git commit -am"[this commit message]")
and then ignored the output, which consisted lines saying "FOO: warning:
copyright statement not found" for each of 6694 files FOO.
I then removed trailing white space from benchtests/bench-pthread-locks.c
and iconvdata/tst-iconv-big5-hkscs-to-2ucs4.c, to work around this
diagnostic from Savannah:
remote: *** pre-commit check failed ...
remote: *** error: lines with trailing whitespace found
remote: error: hook declined to update refs/heads/master
Add support to treat pseudo-numbers specially and implement x86
version to consider all of them as signaling.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
asin and acos have slow paths for rounding the last bit that cause some
calls to be 500-1500x slower than average calls.
These slow paths are rare, a test of a trillion (1.000.000.000.000)
random inputs between -1 and 1 showed 32870 slow calls for acos and 4473
for asin, with most occurrences between -1.0 .. -0.9 and 0.9 .. 1.0.
The slow paths claim correct rounding and use __sin32() and __cos32()
(which compare two result candidates and return the closest one) as the
final step, with the second result candidate (res1) having a small offset
applied from res. This suggests that res and res1 are intended to be 1
ULP apart (which makes sense for rounding), barring bugs, allowing us to
pick either one and still remain within 1 ULP of the exact result.
Remove the slow paths as the accuracy is better than 1 ULP even without
them, which is enough for glibc.
Also remove code comments claiming correctly rounded results.
After slow path removal, checking the accuracy of 14.400.000.000 random
asin() and acos() inputs showed only three incorrectly rounded
(error > 0.5 ULP) results:
- asin(-0x1.ee2b43286db75p-1) (0.500002 ULP, same as before)
- asin(-0x1.f692ba202abcp-4) (0.500003 ULP, same as before)
- asin(-0x1.9915e876fc062p-1) (0.50000000001 ULP, previously exact)
The first two had the same error even before this commit, and they did
not use the slow path at all.
Checking 4934 known randomly found previously-slow-path asin inputs
shows 25 calls with incorrectly rounded results, with a maximum error of
0.500000002 ULP (for 0x1.fcd5742999ab8p-1). The previous slow-path code
rounded all these inputs correctly (error < 0.5 ULP).
The observed average speed increase was 130x.
Checking 36240 known randomly found previously-slow-path acos inputs
shows 42 calls with incorrectly rounded results, with a maximum error of
0.500000008 ULP (for 0x1.f63845056f35ep-1). The previous "exact"
slow-path code showed 34 calls with incorrectly rounded results, with the
same maximum error of 0.500000008 ULP (for 0x1.f63845056f35ep-1).
The observed average speed increase was 130x.
The functions could likely be trimmed more while keeping acceptable
accuracy, but this at least gets rid of the egregiously slow cases.
Tested on x86_64.
The tls.h inclusion is not really required and limits possible
definition on more arch specific headers.
This is a cleanup to allow inline functions on sysdep.h, more
specifically on i386 and ia64 which requires to access some tls
definitions its own.
No semantic changes expected, checked with a build against all
affected ABIs.
