Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
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/* lgammal expanding around zeros.
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2021-01-02 19:32:25 +00:00
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Copyright (C) 2015-2021 Free Software Foundation, Inc.
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Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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Prefer https to http for gnu.org and fsf.org URLs
Also, change sources.redhat.com to sourceware.org.
This patch was automatically generated by running the following shell
script, which uses GNU sed, and which avoids modifying files imported
from upstream:
sed -ri '
s,(http|ftp)(://(.*\.)?(gnu|fsf|sourceware)\.org($|[^.]|\.[^a-z])),https\2,g
s,(http|ftp)(://(.*\.)?)sources\.redhat\.com($|[^.]|\.[^a-z]),https\2sourceware.org\4,g
' \
$(find $(git ls-files) -prune -type f \
! -name '*.po' \
! -name 'ChangeLog*' \
! -path COPYING ! -path COPYING.LIB \
! -path manual/fdl-1.3.texi ! -path manual/lgpl-2.1.texi \
! -path manual/texinfo.tex ! -path scripts/config.guess \
! -path scripts/config.sub ! -path scripts/install-sh \
! -path scripts/mkinstalldirs ! -path scripts/move-if-change \
! -path INSTALL ! -path locale/programs/charmap-kw.h \
! -path po/libc.pot ! -path sysdeps/gnu/errlist.c \
! '(' -name configure \
-execdir test -f configure.ac -o -f configure.in ';' ')' \
! '(' -name preconfigure \
-execdir test -f preconfigure.ac ';' ')' \
-print)
and then by running 'make dist-prepare' to regenerate files built
from the altered files, and then executing the following to cleanup:
chmod a+x sysdeps/unix/sysv/linux/riscv/configure
# Omit irrelevant whitespace and comment-only changes,
# perhaps from a slightly-different Autoconf version.
git checkout -f \
sysdeps/csky/configure \
sysdeps/hppa/configure \
sysdeps/riscv/configure \
sysdeps/unix/sysv/linux/csky/configure
# Omit changes that caused a pre-commit check to fail like this:
# remote: *** error: sysdeps/powerpc/powerpc64/ppc-mcount.S: trailing lines
git checkout -f \
sysdeps/powerpc/powerpc64/ppc-mcount.S \
sysdeps/unix/sysv/linux/s390/s390-64/syscall.S
# Omit change that caused a pre-commit check to fail like this:
# remote: *** error: sysdeps/sparc/sparc64/multiarch/memcpy-ultra3.S: last line does not end in newline
git checkout -f sysdeps/sparc/sparc64/multiarch/memcpy-ultra3.S
2019-09-07 05:40:42 +00:00
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<https://www.gnu.org/licenses/>. */
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Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
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#include <float.h>
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#include <math.h>
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#include <math_private.h>
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Do not include fenv_private.h in math_private.h.
Continuing the clean-up related to the catch-all math_private.h
header, this patch stops math_private.h from including fenv_private.h.
Instead, fenv_private.h is included directly from those users of
math_private.h that also used interfaces from fenv_private.h. No
attempt is made to remove unused includes of math_private.h, but that
is a natural followup.
(However, since math_private.h sometimes defines optimized versions of
math.h interfaces or __* variants thereof, as well as defining its own
interfaces, I think it might make sense to get all those optimized
versions included from include/math.h, not requiring a separate header
at all, before eliminating unused math_private.h includes - that
avoids a file quietly becoming less-optimized if someone adds a call
to one of those interfaces without restoring a math_private.h include
to that file.)
There is still a pitfall that if code uses plain fe* and __fe*
interfaces, but only includes fenv.h and not fenv_private.h or (before
this patch) math_private.h, it will compile on platforms with
exceptions and rounding modes but not get the optimized versions (and
possibly not compile) on platforms without exception and rounding mode
support, so making it easy to break the build for such platforms
accidentally.
I think it would be most natural to move the inlines / macros for fe*
and __fe* in the case of no exceptions and rounding modes into
include/fenv.h, so that all code including fenv.h with _ISOMAC not
defined automatically gets them. Then fenv_private.h would be purely
the header for the libc_fe*, SET_RESTORE_ROUND etc. internal
interfaces and the risk of breaking the build on other platforms than
the one you tested on because of a missing fenv_private.h include
would be much reduced (and there would be some unused fenv_private.h
includes to remove along with unused math_private.h includes).
Tested for x86_64 and x86, and tested with build-many-glibcs.py that
installed stripped shared libraries are unchanged by this patch.
* sysdeps/generic/math_private.h: Do not include <fenv_private.h>.
* math/fromfp.h: Include <fenv_private.h>.
* math/math-narrow.h: Likewise.
* math/s_cexp_template.c: Likewise.
* math/s_csin_template.c: Likewise.
* math/s_csinh_template.c: Likewise.
* math/s_ctan_template.c: Likewise.
* math/s_ctanh_template.c: Likewise.
* math/s_iseqsig_template.c: Likewise.
* math/w_acos_compat.c: Likewise.
* math/w_acosf_compat.c: Likewise.
* math/w_acosl_compat.c: Likewise.
* math/w_asin_compat.c: Likewise.
* math/w_asinf_compat.c: Likewise.
* math/w_asinl_compat.c: Likewise.
* math/w_ilogb_template.c: Likewise.
* math/w_j0_compat.c: Likewise.
* math/w_j0f_compat.c: Likewise.
* math/w_j0l_compat.c: Likewise.
* math/w_j1_compat.c: Likewise.
* math/w_j1f_compat.c: Likewise.
* math/w_j1l_compat.c: Likewise.
* math/w_jn_compat.c: Likewise.
* math/w_jnf_compat.c: Likewise.
* math/w_llogb_template.c: Likewise.
* math/w_log10_compat.c: Likewise.
* math/w_log10f_compat.c: Likewise.
* math/w_log10l_compat.c: Likewise.
* math/w_log2_compat.c: Likewise.
* math/w_log2f_compat.c: Likewise.
* math/w_log2l_compat.c: Likewise.
* math/w_log_compat.c: Likewise.
* math/w_logf_compat.c: Likewise.
* math/w_logl_compat.c: Likewise.
* sysdeps/aarch64/fpu/feholdexcpt.c: Likewise.
* sysdeps/aarch64/fpu/fesetround.c: Likewise.
* sysdeps/aarch64/fpu/fgetexcptflg.c: Likewise.
* sysdeps/aarch64/fpu/ftestexcept.c: Likewise.
* sysdeps/ieee754/dbl-64/e_atan2.c: Likewise.
* sysdeps/ieee754/dbl-64/e_exp.c: Likewise.
* sysdeps/ieee754/dbl-64/e_exp2.c: Likewise.
* sysdeps/ieee754/dbl-64/e_gamma_r.c: Likewise.
* sysdeps/ieee754/dbl-64/e_jn.c: Likewise.
* sysdeps/ieee754/dbl-64/e_pow.c: Likewise.
* sysdeps/ieee754/dbl-64/e_remainder.c: Likewise.
* sysdeps/ieee754/dbl-64/e_sqrt.c: Likewise.
* sysdeps/ieee754/dbl-64/gamma_product.c: Likewise.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: Likewise.
* sysdeps/ieee754/dbl-64/s_atan.c: Likewise.
* sysdeps/ieee754/dbl-64/s_fma.c: Likewise.
* sysdeps/ieee754/dbl-64/s_fmaf.c: Likewise.
* sysdeps/ieee754/dbl-64/s_llrint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_llround.c: Likewise.
* sysdeps/ieee754/dbl-64/s_lrint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_lround.c: Likewise.
* sysdeps/ieee754/dbl-64/s_nearbyint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_sin.c: Likewise.
* sysdeps/ieee754/dbl-64/s_sincos.c: Likewise.
* sysdeps/ieee754/dbl-64/s_tan.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_lround.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_nearbyint.c: Likewise.
* sysdeps/ieee754/dbl-64/x2y2m1.c: Likewise.
* sysdeps/ieee754/float128/float128_private.h: Likewise.
* sysdeps/ieee754/flt-32/e_gammaf_r.c: Likewise.
* sysdeps/ieee754/flt-32/e_j1f.c: Likewise.
* sysdeps/ieee754/flt-32/e_jnf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/s_llrintf.c: Likewise.
* sysdeps/ieee754/flt-32/s_llroundf.c: Likewise.
* sysdeps/ieee754/flt-32/s_lrintf.c: Likewise.
* sysdeps/ieee754/flt-32/s_lroundf.c: Likewise.
* sysdeps/ieee754/flt-32/s_nearbyintf.c: Likewise.
* sysdeps/ieee754/k_standardl.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_expl.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_j1l.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-128/gamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_nearbyintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/x2y2m1l.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_expl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_j1l.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_rintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c: Likewise.
* sysdeps/ieee754/ldbl-96/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-96/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-96/gamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_fma.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-96/x2y2m1l.c: Likewise.
* sysdeps/powerpc/fpu/e_sqrt.c: Likewise.
* sysdeps/powerpc/fpu/e_sqrtf.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_ceil.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_floor.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_nearbyint.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_round.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_roundeven.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_trunc.c: Likewise.
* sysdeps/riscv/rvd/s_finite.c: Likewise.
* sysdeps/riscv/rvd/s_fmax.c: Likewise.
* sysdeps/riscv/rvd/s_fmin.c: Likewise.
* sysdeps/riscv/rvd/s_fpclassify.c: Likewise.
* sysdeps/riscv/rvd/s_isinf.c: Likewise.
* sysdeps/riscv/rvd/s_isnan.c: Likewise.
* sysdeps/riscv/rvd/s_issignaling.c: Likewise.
* sysdeps/riscv/rvf/fegetround.c: Likewise.
* sysdeps/riscv/rvf/feholdexcpt.c: Likewise.
* sysdeps/riscv/rvf/fesetenv.c: Likewise.
* sysdeps/riscv/rvf/fesetround.c: Likewise.
* sysdeps/riscv/rvf/feupdateenv.c: Likewise.
* sysdeps/riscv/rvf/fgetexcptflg.c: Likewise.
* sysdeps/riscv/rvf/ftestexcept.c: Likewise.
* sysdeps/riscv/rvf/s_ceilf.c: Likewise.
* sysdeps/riscv/rvf/s_finitef.c: Likewise.
* sysdeps/riscv/rvf/s_floorf.c: Likewise.
* sysdeps/riscv/rvf/s_fmaxf.c: Likewise.
* sysdeps/riscv/rvf/s_fminf.c: Likewise.
* sysdeps/riscv/rvf/s_fpclassifyf.c: Likewise.
* sysdeps/riscv/rvf/s_isinff.c: Likewise.
* sysdeps/riscv/rvf/s_isnanf.c: Likewise.
* sysdeps/riscv/rvf/s_issignalingf.c: Likewise.
* sysdeps/riscv/rvf/s_nearbyintf.c: Likewise.
* sysdeps/riscv/rvf/s_roundevenf.c: Likewise.
* sysdeps/riscv/rvf/s_roundf.c: Likewise.
* sysdeps/riscv/rvf/s_truncf.c: Likewise.
2018-09-03 21:09:04 +00:00
|
|
|
#include <fenv_private.h>
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static const _Float128 lgamma_zeros[][2] =
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
2016-09-02 16:01:07 +00:00
|
|
|
{ L(-0x2.74ff92c01f0d82abec9f315f1a08p+0), L(0xe.d3ccb7fb2658634a2b9f6b2ba81p-116) },
|
|
|
|
|
{ L(-0x2.bf6821437b20197995a4b4641eaep+0), L(-0xb.f4b00b4829f961e428533e6ad048p-116) },
|
|
|
|
|
{ L(-0x3.24c1b793cb35efb8be699ad3d9bap+0), L(-0x6.5454cb7fac60e3f16d9d7840c2ep-116) },
|
|
|
|
|
{ L(-0x3.f48e2a8f85fca170d4561291236cp+0), L(-0xc.320a4887d1cb4c711828a75d5758p-116) },
|
|
|
|
|
{ L(-0x4.0a139e16656030c39f0b0de18114p+0), L(0x1.53e84029416e1242006b2b3d1cfp-112) },
|
|
|
|
|
{ L(-0x4.fdd5de9bbabf3510d0aa40769884p+0), L(-0x1.01d7d78125286f78d1e501f14966p-112) },
|
|
|
|
|
{ L(-0x5.021a95fc2db6432a4c56e595394cp+0), L(-0x1.ecc6af0430d4fe5746fa7233356fp-112) },
|
|
|
|
|
{ L(-0x5.ffa4bd647d0357dd4ed62cbd31ecp+0), L(-0x1.f8e3f8e5deba2d67dbd70dd96ce1p-112) },
|
|
|
|
|
{ L(-0x6.005ac9625f233b607c2d96d16384p+0), L(-0x1.cb86ac569340cf1e5f24df7aab7bp-112) },
|
|
|
|
|
{ L(-0x6.fff2fddae1bbff3d626b65c23fd4p+0), L(0x1.e0bfcff5c457ebcf4d3ad9674167p-112) },
|
|
|
|
|
{ L(-0x7.000cff7b7f87adf4482dcdb98784p+0), L(0x1.54d99e35a74d6407b80292df199fp-112) },
|
|
|
|
|
{ L(-0x7.fffe5fe05673c3ca9e82b522b0ccp+0), L(0x1.62d177c832e0eb42c2faffd1b145p-112) },
|
|
|
|
|
{ L(-0x8.0001a01459fc9f60cb3cec1cec88p+0), L(0x2.8998835ac7277f7bcef67c47f188p-112) },
|
|
|
|
|
{ L(-0x8.ffffd1c425e80ffc864e95749258p+0), L(-0x1.e7e20210e7f81cf781b44e9d2b02p-112) },
|
|
|
|
|
{ L(-0x9.00002e3bb47d86d6d843fedc352p+0), L(0x2.14852f613a16291751d2ab751f7ep-112) },
|
|
|
|
|
{ L(-0x9.fffffb606bdfdcd062ae77a50548p+0), L(0x3.962d1490cc2e8f031c7007eaa1ap-116) },
|
|
|
|
|
{ L(-0xa.0000049f93bb9927b45d95e1544p+0), L(-0x1.e03086db9146a9287bd4f2172d5ap-112) },
|
|
|
|
|
{ L(-0xa.ffffff9466e9f1b36dacd2adbd18p+0), L(-0xd.05a4e458062f3f95345a4d9c9b6p-116) },
|
|
|
|
|
{ L(-0xb.0000006b9915315d965a6ffea41p+0), L(0x1.b415c6fff233e7b7fdc3a094246fp-112) },
|
|
|
|
|
{ L(-0xb.fffffff7089387387de41acc3d4p+0), L(0x3.687427c6373bd74a10306e10a28ep-112) },
|
|
|
|
|
{ L(-0xc.00000008f76c7731567c0f0250fp+0), L(-0x3.87920df5675833859190eb128ef6p-112) },
|
|
|
|
|
{ L(-0xc.ffffffff4f6dcf617f97a5ffc758p+0), L(0x2.ab72d76f32eaee2d1a42ed515d3ap-116) },
|
|
|
|
|
{ L(-0xd.00000000b092309c06683dd1b9p+0), L(-0x3.e3700857a15c19ac5a611de9688ap-112) },
|
|
|
|
|
{ L(-0xd.fffffffff36345ab9e184a3e09dp+0), L(-0x1.176dc48e47f62d917973dd44e553p-112) },
|
|
|
|
|
{ L(-0xe.000000000c9cba545e94e75ec57p+0), L(-0x1.8f753e2501e757a17cf2ecbeeb89p-112) },
|
|
|
|
|
{ L(-0xe.ffffffffff28c060c6604ef3037p+0), L(-0x1.f89d37357c9e3dc17c6c6e63becap-112) },
|
|
|
|
|
{ L(-0xf.0000000000d73f9f399bd0e420f8p+0), L(-0x5.e9ee31b0b890744fc0e3fbc01048p-116) },
|
|
|
|
|
{ L(-0xf.fffffffffff28c060c6621f512e8p+0), L(0xd.1b2eec9d960bd9adc5be5f5fa5p-116) },
|
|
|
|
|
{ L(-0x1.000000000000d73f9f399da1424cp+4), L(0x6.c46e0e88305d2800f0e414c506a8p-116) },
|
|
|
|
|
{ L(-0x1.0ffffffffffff3569c47e7a93e1cp+4), L(-0x4.6a08a2e008a998ebabb8087efa2cp-112) },
|
|
|
|
|
{ L(-0x1.1000000000000ca963b818568887p+4), L(-0x6.ca5a3a64ec15db0a95caf2c9ffb4p-112) },
|
|
|
|
|
{ L(-0x1.1fffffffffffff4bec3ce234132dp+4), L(-0x8.b2b726187c841cb92cd5221e444p-116) },
|
|
|
|
|
{ L(-0x1.20000000000000b413c31dcbeca5p+4), L(0x3.c4d005344b6cd0e7231120294abcp-112) },
|
|
|
|
|
{ L(-0x1.2ffffffffffffff685b25cbf5f54p+4), L(-0x5.ced932e38485f7dd296b8fa41448p-112) },
|
|
|
|
|
{ L(-0x1.30000000000000097a4da340a0acp+4), L(0x7.e484e0e0ffe38d406ebebe112f88p-112) },
|
|
|
|
|
{ L(-0x1.3fffffffffffffff86af516ff7f7p+4), L(-0x6.bd67e720d57854502b7db75e1718p-112) },
|
|
|
|
|
{ L(-0x1.40000000000000007950ae900809p+4), L(0x6.bec33375cac025d9c073168c5d9p-112) },
|
|
|
|
|
{ L(-0x1.4ffffffffffffffffa391c4248c3p+4), L(0x5.c63022b62b5484ba346524db607p-112) },
|
|
|
|
|
{ L(-0x1.500000000000000005c6e3bdb73dp+4), L(-0x5.c62f55ed5322b2685c5e9a51e6a8p-112) },
|
|
|
|
|
{ L(-0x1.5fffffffffffffffffbcc71a492p+4), L(-0x1.eb5aeb96c74d7ad25e060528fb5p-112) },
|
|
|
|
|
{ L(-0x1.6000000000000000004338e5b6ep+4), L(0x1.eb5aec04b2f2eb663e4e3d8a018cp-112) },
|
|
|
|
|
{ L(-0x1.6ffffffffffffffffffd13c97d9dp+4), L(-0x3.8fcc4d08d6fe5aa56ab04307ce7ep-112) },
|
|
|
|
|
{ L(-0x1.70000000000000000002ec368263p+4), L(0x3.8fcc4d090cee2f5d0b69a99c353cp-112) },
|
|
|
|
|
{ L(-0x1.7fffffffffffffffffffe0d30fe7p+4), L(0x7.2f577cca4b4c8cb1dc14001ac5ecp-112) },
|
|
|
|
|
{ L(-0x1.800000000000000000001f2cf019p+4), L(-0x7.2f577cca4b3442e35f0040b3b9e8p-112) },
|
|
|
|
|
{ L(-0x1.8ffffffffffffffffffffec0c332p+4), L(-0x2.e9a0572b1bb5b95f346a92d67a6p-112) },
|
|
|
|
|
{ L(-0x1.90000000000000000000013f3ccep+4), L(0x2.e9a0572b1bb5c371ddb3561705ap-112) },
|
|
|
|
|
{ L(-0x1.9ffffffffffffffffffffff3b8bdp+4), L(-0x1.cad8d32e386fd783e97296d63dcbp-116) },
|
|
|
|
|
{ L(-0x1.a0000000000000000000000c4743p+4), L(0x1.cad8d32e386fd7c1ab8c1fe34c0ep-116) },
|
|
|
|
|
{ L(-0x1.afffffffffffffffffffffff8b95p+4), L(-0x3.8f48cc5737d5979c39db806c5406p-112) },
|
|
|
|
|
{ L(-0x1.b00000000000000000000000746bp+4), L(0x3.8f48cc5737d5979c3b3a6bda06f6p-112) },
|
|
|
|
|
{ L(-0x1.bffffffffffffffffffffffffbd8p+4), L(0x6.2898d42174dcf171470d8c8c6028p-112) },
|
|
|
|
|
{ L(-0x1.c000000000000000000000000428p+4), L(-0x6.2898d42174dcf171470d18ba412cp-112) },
|
|
|
|
|
{ L(-0x1.cfffffffffffffffffffffffffdbp+4), L(-0x4.c0ce9794ea50a839e311320bde94p-112) },
|
|
|
|
|
{ L(-0x1.d000000000000000000000000025p+4), L(0x4.c0ce9794ea50a839e311322f7cf8p-112) },
|
|
|
|
|
{ L(-0x1.dfffffffffffffffffffffffffffp+4), L(0x3.932c5047d60e60caded4c298a174p-112) },
|
|
|
|
|
{ L(-0x1.e000000000000000000000000001p+4), L(-0x3.932c5047d60e60caded4c298973ap-112) },
|
|
|
|
|
{ L(-0x1.fp+4), L(0xa.1a6973c1fade2170f7237d35fe3p-116) },
|
|
|
|
|
{ L(-0x1.fp+4), L(-0xa.1a6973c1fade2170f7237d35fe08p-116) },
|
|
|
|
|
{ L(-0x2p+4), L(0x5.0d34b9e0fd6f10b87b91be9aff1p-120) },
|
|
|
|
|
{ L(-0x2p+4), L(-0x5.0d34b9e0fd6f10b87b91be9aff0cp-120) },
|
|
|
|
|
{ L(-0x2.1p+4), L(0x2.73024a9ba1aa36a7059bff52e844p-124) },
|
|
|
|
|
{ L(-0x2.1p+4), L(-0x2.73024a9ba1aa36a7059bff52e844p-124) },
|
|
|
|
|
{ L(-0x2.2p+4), L(0x1.2710231c0fd7a13f8a2b4af9d6b7p-128) },
|
|
|
|
|
{ L(-0x2.2p+4), L(-0x1.2710231c0fd7a13f8a2b4af9d6b7p-128) },
|
|
|
|
|
{ L(-0x2.3p+4), L(0x8.6e2ce38b6c8f9419e3fad3f0312p-136) },
|
|
|
|
|
{ L(-0x2.3p+4), L(-0x8.6e2ce38b6c8f9419e3fad3f0312p-136) },
|
|
|
|
|
{ L(-0x2.4p+4), L(0x3.bf30652185952560d71a254e4eb8p-140) },
|
|
|
|
|
{ L(-0x2.4p+4), L(-0x3.bf30652185952560d71a254e4eb8p-140) },
|
|
|
|
|
{ L(-0x2.5p+4), L(0x1.9ec8d1c94e85af4c78b15c3d89d3p-144) },
|
|
|
|
|
{ L(-0x2.5p+4), L(-0x1.9ec8d1c94e85af4c78b15c3d89d3p-144) },
|
|
|
|
|
{ L(-0x2.6p+4), L(0xa.ea565ce061d57489e9b85276274p-152) },
|
|
|
|
|
{ L(-0x2.6p+4), L(-0xa.ea565ce061d57489e9b85276274p-152) },
|
|
|
|
|
{ L(-0x2.7p+4), L(0x4.7a6512692eb37804111dabad30ecp-156) },
|
|
|
|
|
{ L(-0x2.7p+4), L(-0x4.7a6512692eb37804111dabad30ecp-156) },
|
|
|
|
|
{ L(-0x2.8p+4), L(0x1.ca8ed42a12ae3001a07244abad2bp-160) },
|
|
|
|
|
{ L(-0x2.8p+4), L(-0x1.ca8ed42a12ae3001a07244abad2bp-160) },
|
|
|
|
|
{ L(-0x2.9p+4), L(0xb.2f30e1ce812063f12e7e8d8d96e8p-168) },
|
|
|
|
|
{ L(-0x2.9p+4), L(-0xb.2f30e1ce812063f12e7e8d8d96e8p-168) },
|
|
|
|
|
{ L(-0x2.ap+4), L(0x4.42bd49d4c37a0db136489772e428p-172) },
|
|
|
|
|
{ L(-0x2.ap+4), L(-0x4.42bd49d4c37a0db136489772e428p-172) },
|
|
|
|
|
{ L(-0x2.bp+4), L(0x1.95db45257e5122dcbae56def372p-176) },
|
|
|
|
|
{ L(-0x2.bp+4), L(-0x1.95db45257e5122dcbae56def372p-176) },
|
|
|
|
|
{ L(-0x2.cp+4), L(0x9.3958d81ff63527ecf993f3fb6f48p-184) },
|
|
|
|
|
{ L(-0x2.cp+4), L(-0x9.3958d81ff63527ecf993f3fb6f48p-184) },
|
|
|
|
|
{ L(-0x2.dp+4), L(0x3.47970e4440c8f1c058bd238c9958p-188) },
|
|
|
|
|
{ L(-0x2.dp+4), L(-0x3.47970e4440c8f1c058bd238c9958p-188) },
|
|
|
|
|
{ L(-0x2.ep+4), L(0x1.240804f65951062ca46e4f25c608p-192) },
|
|
|
|
|
{ L(-0x2.ep+4), L(-0x1.240804f65951062ca46e4f25c608p-192) },
|
|
|
|
|
{ L(-0x2.fp+4), L(0x6.36a382849fae6de2d15362d8a394p-200) },
|
|
|
|
|
{ L(-0x2.fp+4), L(-0x6.36a382849fae6de2d15362d8a394p-200) },
|
|
|
|
|
{ L(-0x3p+4), L(0x2.123680d6dfe4cf4b9b1bcb9d8bdcp-204) },
|
|
|
|
|
{ L(-0x3p+4), L(-0x2.123680d6dfe4cf4b9b1bcb9d8bdcp-204) },
|
|
|
|
|
{ L(-0x3.1p+4), L(0xa.d21786ff5842eca51fea0870919p-212) },
|
|
|
|
|
{ L(-0x3.1p+4), L(-0xa.d21786ff5842eca51fea0870919p-212) },
|
|
|
|
|
{ L(-0x3.2p+4), L(0x3.766dedc259af040be140a68a6c04p-216) },
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
};
|
|
|
|
|
|
2016-09-02 16:01:07 +00:00
|
|
|
static const _Float128 e_hi = L(0x2.b7e151628aed2a6abf7158809cf4p+0);
|
|
|
|
|
static const _Float128 e_lo = L(0xf.3c762e7160f38b4da56a784d9048p-116);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
|
|
|
|
|
approximation to lgamma function. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static const _Float128 lgamma_coeff[] =
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
2016-09-02 16:01:07 +00:00
|
|
|
L(0x1.5555555555555555555555555555p-4),
|
|
|
|
|
L(-0xb.60b60b60b60b60b60b60b60b60b8p-12),
|
|
|
|
|
L(0x3.4034034034034034034034034034p-12),
|
|
|
|
|
L(-0x2.7027027027027027027027027028p-12),
|
|
|
|
|
L(0x3.72a3c5631fe46ae1d4e700dca8f2p-12),
|
|
|
|
|
L(-0x7.daac36664f1f207daac36664f1f4p-12),
|
|
|
|
|
L(0x1.a41a41a41a41a41a41a41a41a41ap-8),
|
|
|
|
|
L(-0x7.90a1b2c3d4e5f708192a3b4c5d7p-8),
|
|
|
|
|
L(0x2.dfd2c703c0cfff430edfd2c703cp-4),
|
|
|
|
|
L(-0x1.6476701181f39edbdb9ce625987dp+0),
|
|
|
|
|
L(0xd.672219167002d3a7a9c886459cp+0),
|
|
|
|
|
L(-0x9.cd9292e6660d55b3f712eb9e07c8p+4),
|
|
|
|
|
L(0x8.911a740da740da740da740da741p+8),
|
|
|
|
|
L(-0x8.d0cc570e255bf59ff6eec24b49p+12),
|
|
|
|
|
L(0xa.8d1044d3708d1c219ee4fdc446ap+16),
|
|
|
|
|
L(-0xe.8844d8a169abbc406169abbc406p+20),
|
|
|
|
|
L(0x1.6d29a0f6433b79890cede62433b8p+28),
|
|
|
|
|
L(-0x2.88a233b3c8cddaba9809357125d8p+32),
|
|
|
|
|
L(0x5.0dde6f27500939a85c40939a85c4p+36),
|
|
|
|
|
L(-0xb.4005bde03d4642a243581714af68p+40),
|
|
|
|
|
L(0x1.bc8cd6f8f1f755c78753cdb5d5c9p+48),
|
|
|
|
|
L(-0x4.bbebb143bb94de5a0284fa7ec424p+52),
|
|
|
|
|
L(0xe.2e1337f5af0bed90b6b0a352d4fp+56),
|
|
|
|
|
L(-0x2.e78250162b62405ad3e4bfe61b38p+64),
|
|
|
|
|
L(0xa.5f7eef9e71ac7c80326ab4cc8bfp+68),
|
|
|
|
|
L(-0x2.83be0395e550213369924971b21ap+76),
|
|
|
|
|
L(0xa.8ebfe48da17dd999790760b0cep+80),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
};
|
|
|
|
|
|
|
|
|
|
#define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
|
|
|
|
|
|
|
|
|
|
/* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
|
|
|
|
|
the integer end-point of the half-integer interval containing x and
|
|
|
|
|
x0 is the zero of lgamma in that half-integer interval. Each
|
|
|
|
|
polynomial is expressed in terms of x-xm, where xm is the midpoint
|
|
|
|
|
of the interval for which the polynomial applies. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static const _Float128 poly_coeff[] =
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
|
|
|
|
/* Interval [-2.125, -2] (polynomial degree 23). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0x1.0b71c5c54d42eb6c17f30b7aa8f5p+0),
|
|
|
|
|
L(-0xc.73a1dc05f34951602554c6d7506p-4),
|
|
|
|
|
L(-0x1.ec841408528b51473e6c425ee5ffp-4),
|
|
|
|
|
L(-0xe.37c9da26fc3c9a3c1844c8c7f1cp-4),
|
|
|
|
|
L(-0x1.03cd87c519305703b021fa33f827p-4),
|
|
|
|
|
L(-0xe.ae9ada65e09aa7f1c75216128f58p-4),
|
|
|
|
|
L(0x9.b11855a4864b5731cf85736015a8p-8),
|
|
|
|
|
L(-0xe.f28c133e697a95c28607c9701dep-4),
|
|
|
|
|
L(0x2.6ec14a1c586a72a7cc33ee569d6ap-4),
|
|
|
|
|
L(-0xf.57cab973e14464a262fc24723c38p-4),
|
|
|
|
|
L(0x4.5b0fc25f16e52997b2886bbae808p-4),
|
|
|
|
|
L(-0xf.f50e59f1a9b56e76e988dac9ccf8p-4),
|
|
|
|
|
L(0x6.5f5eae15e9a93369e1d85146c6fcp-4),
|
|
|
|
|
L(-0x1.0d2422daac459e33e0994325ed23p+0),
|
|
|
|
|
L(0x8.82000a0e7401fb1117a0e6606928p-4),
|
|
|
|
|
L(-0x1.1f492f178a3f1b19f58a2ca68e55p+0),
|
|
|
|
|
L(0xa.cb545f949899a04c160b19389abp-4),
|
|
|
|
|
L(-0x1.36165a1b155ba3db3d1b77caf498p+0),
|
|
|
|
|
L(0xd.44c5d5576f74302e5cf79e183eep-4),
|
|
|
|
|
L(-0x1.51f22e0cdd33d3d481e326c02f3ep+0),
|
|
|
|
|
L(0xf.f73a349c08244ac389c007779bfp-4),
|
|
|
|
|
L(-0x1.73317bf626156ba716747c4ca866p+0),
|
|
|
|
|
L(0x1.379c3c97b9bc71e1c1c4802dd657p+0),
|
|
|
|
|
L(-0x1.a72a351c54f902d483052000f5dfp+0),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.25, -2.125] (polynomial degree 24). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0xf.2930890d7d675a80c36afb0fd5e8p-4),
|
|
|
|
|
L(-0xc.a5cfde054eab5c6770daeca577f8p-4),
|
|
|
|
|
L(0x3.9c9e0fdebb07cdf89c61d41c9238p-4),
|
|
|
|
|
L(-0x1.02a5ad35605fcf4af65a6dbacb84p+0),
|
|
|
|
|
L(0x9.6e9b1185bb48be9de1918e00a2e8p-4),
|
|
|
|
|
L(-0x1.4d8332f3cfbfa116fd611e9ce90dp+0),
|
|
|
|
|
L(0x1.1c0c8cb4d9f4b1d490e1a41fae4dp+0),
|
|
|
|
|
L(-0x1.c9a6f5ae9130cd0299e293a42714p+0),
|
|
|
|
|
L(0x1.d7e9307fd58a2ea997f29573a112p+0),
|
|
|
|
|
L(-0x2.921cb3473d96178ca2a11d2a8d46p+0),
|
|
|
|
|
L(0x2.e8d59113b6f3409ff8db226e9988p+0),
|
|
|
|
|
L(-0x3.cbab931625a1ae2b26756817f264p+0),
|
|
|
|
|
L(0x4.7d9f0f05d5296d18663ca003912p+0),
|
|
|
|
|
L(-0x5.ade9cba12a14ea485667b7135bbp+0),
|
|
|
|
|
L(0x6.dc983a5da74fb48e767b7fec0a3p+0),
|
|
|
|
|
L(-0x8.8d9ed454ae31d9e138dd8ee0d1a8p+0),
|
|
|
|
|
L(0xa.6fa099d4e7c202e0c0fd6ed8492p+0),
|
|
|
|
|
L(-0xc.ebc552a8090a0f0115e92d4ebbc8p+0),
|
|
|
|
|
L(0xf.d695e4772c0d829b53fba9ca5568p+0),
|
|
|
|
|
L(-0x1.38c32ae38e5e9eb79b2a4c5570a9p+4),
|
|
|
|
|
L(0x1.8035145646cfab49306d0999a51bp+4),
|
|
|
|
|
L(-0x1.d930adbb03dd342a4c2a8c4e1af6p+4),
|
|
|
|
|
L(0x2.45c2edb1b4943ddb3686cd9c6524p+4),
|
|
|
|
|
L(-0x2.e818ebbfafe2f916fa21abf7756p+4),
|
|
|
|
|
L(0x3.9804ce51d0fb9a430a711fd7307p+4),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.375, -2.25] (polynomial degree 25). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0xd.7d28d505d6181218a25f31d5e45p-4),
|
|
|
|
|
L(-0xe.69649a3040985140cdf946829fap-4),
|
|
|
|
|
L(0xb.0d74a2827d053a8d44595012484p-4),
|
|
|
|
|
L(-0x1.924b0922853617cac181afbc08ddp+0),
|
|
|
|
|
L(0x1.d49b12bccf0a568582e2d3c410f3p+0),
|
|
|
|
|
L(-0x3.0898bb7d8c4093e636279c791244p+0),
|
|
|
|
|
L(0x4.207a6cac711cb53868e8a5057eep+0),
|
|
|
|
|
L(-0x6.39ee63ea4fb1dcab0c9144bf3ddcp+0),
|
|
|
|
|
L(0x8.e2e2556a797b649bf3f53bd26718p+0),
|
|
|
|
|
L(-0xd.0e83ac82552ef12af508589e7a8p+0),
|
|
|
|
|
L(0x1.2e4525e0ce6670563c6484a82b05p+4),
|
|
|
|
|
L(-0x1.b8e350d6a8f2b222fa390a57c23dp+4),
|
|
|
|
|
L(0x2.805cd69b919087d8a80295892c2cp+4),
|
|
|
|
|
L(-0x3.a42585424a1b7e64c71743ab014p+4),
|
|
|
|
|
L(0x5.4b4f409f98de49f7bfb03c05f984p+4),
|
|
|
|
|
L(-0x7.b3c5827fbe934bc820d6832fb9fcp+4),
|
|
|
|
|
L(0xb.33b7b90cc96c425526e0d0866e7p+4),
|
|
|
|
|
L(-0x1.04b77047ac4f59ee3775ca10df0dp+8),
|
|
|
|
|
L(0x1.7b366f5e94a34f41386eac086313p+8),
|
|
|
|
|
L(-0x2.2797338429385c9849ca6355bfc2p+8),
|
|
|
|
|
L(0x3.225273cf92a27c9aac1b35511256p+8),
|
|
|
|
|
L(-0x4.8f078aa48afe6cb3a4e89690f898p+8),
|
|
|
|
|
L(0x6.9f311d7b6654fc1d0b5195141d04p+8),
|
|
|
|
|
L(-0x9.a0c297b6b4621619ca9bacc48ed8p+8),
|
|
|
|
|
L(0xe.ce1f06b6f90d92138232a76e4cap+8),
|
|
|
|
|
L(-0x1.5b0e6806fa064daf011613e43b17p+12),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.5, -2.375] (polynomial degree 27). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0xb.74ea1bcfff94b2c01afba9daa7d8p-4),
|
|
|
|
|
L(-0x1.2a82bd590c37538cab143308de4dp+0),
|
|
|
|
|
L(0x1.88020f828b966fec66b8649fd6fcp+0),
|
|
|
|
|
L(-0x3.32279f040eb694970e9db24863dcp+0),
|
|
|
|
|
L(0x5.57ac82517767e68a721005853864p+0),
|
|
|
|
|
L(-0x9.c2aedcfe22833de43834a0a6cc4p+0),
|
|
|
|
|
L(0x1.12c132f1f5577f99e1a0ed3538e1p+4),
|
|
|
|
|
L(-0x1.ea94e26628a3de3597f7bb55a948p+4),
|
|
|
|
|
L(0x3.66b4ac4fa582f58b59f96b2f7c7p+4),
|
|
|
|
|
L(-0x6.0cf746a9cf4cba8c39afcc73fc84p+4),
|
|
|
|
|
L(0xa.c102ef2c20d75a342197df7fedf8p+4),
|
|
|
|
|
L(-0x1.31ebff06e8f14626782df58db3b6p+8),
|
|
|
|
|
L(0x2.1fd6f0c0e710994e059b9dbdb1fep+8),
|
|
|
|
|
L(-0x3.c6d76040407f447f8b5074f07706p+8),
|
|
|
|
|
L(0x6.b6d18e0d8feb4c2ef5af6a40ed18p+8),
|
|
|
|
|
L(-0xb.efaf542c529f91e34217f24ae6a8p+8),
|
|
|
|
|
L(0x1.53852d873210e7070f5d9eb2296p+12),
|
|
|
|
|
L(-0x2.5b977c0ddc6d540717173ac29fc8p+12),
|
|
|
|
|
L(0x4.310d452ae05100eff1e02343a724p+12),
|
|
|
|
|
L(-0x7.73a5d8f20c4f986a7dd1912b2968p+12),
|
|
|
|
|
L(0xd.3f5ea2484f3fca15eab1f4d1a218p+12),
|
|
|
|
|
L(-0x1.78d18aac156d1d93a2ffe7e08d3fp+16),
|
|
|
|
|
L(0x2.9df49ca75e5b567f5ea3e47106cp+16),
|
|
|
|
|
L(-0x4.a7149af8961a08aa7c3233b5bb94p+16),
|
|
|
|
|
L(0x8.3db10ffa742c707c25197d989798p+16),
|
|
|
|
|
L(-0xe.a26d6dd023cadd02041a049ec368p+16),
|
|
|
|
|
L(0x1.c825d90514e7c57c7fa5316f947cp+20),
|
|
|
|
|
L(-0x3.34bb81e5a0952df8ca1abdc6684cp+20),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.625, -2.5] (polynomial degree 28). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0x3.d10108c27ebafad533c20eac32bp-4),
|
|
|
|
|
L(0x1.cd557caff7d2b2085f41dbec5106p+0),
|
|
|
|
|
L(0x3.819b4856d399520dad9776ea2cacp+0),
|
|
|
|
|
L(0x6.8505cbad03dc34c5e42e8b12eb78p+0),
|
|
|
|
|
L(0xb.c1b2e653a9e38f82b399c94e7f08p+0),
|
|
|
|
|
L(0x1.50a53a38f148138105124df65419p+4),
|
|
|
|
|
L(0x2.57ae00cbe5232cbeeed34d89727ap+4),
|
|
|
|
|
L(0x4.2b156301b8604db85a601544bfp+4),
|
|
|
|
|
L(0x7.6989ed23ca3ca7579b3462592b5cp+4),
|
|
|
|
|
L(0xd.2dd2976557939517f831f5552cc8p+4),
|
|
|
|
|
L(0x1.76e1c3430eb860969bce40cd494p+8),
|
|
|
|
|
L(0x2.9a77bf5488742466db3a2c7c1ec6p+8),
|
|
|
|
|
L(0x4.a0d62ed7266e8eb36f725a8ebcep+8),
|
|
|
|
|
L(0x8.3a6184dd3021067df2f8b91e99c8p+8),
|
|
|
|
|
L(0xe.a0ade1538245bf55d39d7e436b1p+8),
|
|
|
|
|
L(0x1.a01359fae8617b5826dd74428e9p+12),
|
|
|
|
|
L(0x2.e3b0a32caae77251169acaca1ad4p+12),
|
|
|
|
|
L(0x5.2301257c81589f62b38fb5993ee8p+12),
|
|
|
|
|
L(0x9.21c9275db253d4e719b73b18cb9p+12),
|
|
|
|
|
L(0x1.03c104bc96141cda3f3fa4b112bcp+16),
|
|
|
|
|
L(0x1.cdc8ed65119196a08b0c78f1445p+16),
|
|
|
|
|
L(0x3.34f31d2eaacf34382cdb0073572ap+16),
|
|
|
|
|
L(0x5.b37628cadf12bf0000907d0ef294p+16),
|
|
|
|
|
L(0xa.22d8b332c0b1e6a616f425dfe5ap+16),
|
|
|
|
|
L(0x1.205b01444804c3ff922cd78b4c42p+20),
|
|
|
|
|
L(0x1.fe8f0cea9d1e0ff25be2470b4318p+20),
|
|
|
|
|
L(0x3.8872aebeb368399aee02b39340aep+20),
|
|
|
|
|
L(0x6.ebd560d351e84e26a4381f5b293cp+20),
|
|
|
|
|
L(0xc.c3644d094b0dae2fbcbf682cd428p+20),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.75, -2.625] (polynomial degree 26). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0x6.b5d252a56e8a75458a27ed1c2dd4p-4),
|
|
|
|
|
L(0x1.28d60383da3ac721aed3c5794da9p+0),
|
|
|
|
|
L(0x1.db6513ada8a66ea77d87d9a8827bp+0),
|
|
|
|
|
L(0x2.e217118f9d348a27f7506a707e6ep+0),
|
|
|
|
|
L(0x4.450112c5cbf725a0fb9802396c9p+0),
|
|
|
|
|
L(0x6.4af99151eae7810a75df2a0303c4p+0),
|
|
|
|
|
L(0x9.2db598b4a97a7f69aeef32aec758p+0),
|
|
|
|
|
L(0xd.62bef9c22471f5ee47ea1b9c0b5p+0),
|
|
|
|
|
L(0x1.379f294e412bd62328326d4222f9p+4),
|
|
|
|
|
L(0x1.c5827349d8865f1e8825c37c31c6p+4),
|
|
|
|
|
L(0x2.93a7e7a75b7568cc8cbe8c016c12p+4),
|
|
|
|
|
L(0x3.bf9bb882afe57edb383d41879d3ap+4),
|
|
|
|
|
L(0x5.73c737828cee095c43a5566731c8p+4),
|
|
|
|
|
L(0x7.ee4653493a7f81e0442062b3823cp+4),
|
|
|
|
|
L(0xb.891c6b83fc8b55bd973b5d962d6p+4),
|
|
|
|
|
L(0x1.0c775d7de3bf9b246c0208e0207ep+8),
|
|
|
|
|
L(0x1.867ee43ec4bd4f4fd56abc05110ap+8),
|
|
|
|
|
L(0x2.37fe9ba6695821e9822d8c8af0a6p+8),
|
|
|
|
|
L(0x3.3a2c667e37c942f182cd3223a936p+8),
|
|
|
|
|
L(0x4.b1b500eb59f3f782c7ccec88754p+8),
|
|
|
|
|
L(0x6.d3efd3b65b3d0d8488d30b79fa4cp+8),
|
|
|
|
|
L(0x9.ee8224e65bed5ced8b75eaec609p+8),
|
|
|
|
|
L(0xe.72416e510cca77d53fc615c1f3dp+8),
|
|
|
|
|
L(0x1.4fb538b0a2dfe567a8904b7e0445p+12),
|
|
|
|
|
L(0x1.e7f56a9266cf525a5b8cf4cb76cep+12),
|
|
|
|
|
L(0x2.f0365c983f68c597ee49d099cce8p+12),
|
|
|
|
|
L(0x4.53aa229e1b9f5b5e59625265951p+12),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-2.875, -2.75] (polynomial degree 24). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0x8.a41b1e4f36ff88dc820815607d68p-4),
|
|
|
|
|
L(0xc.da87d3b69dc0f2f9c6f368b8ca1p-4),
|
|
|
|
|
L(0x1.1474ad5c36158a7bea04fd2f98c6p+0),
|
|
|
|
|
L(0x1.761ecb90c555df6555b7dba955b6p+0),
|
|
|
|
|
L(0x1.d279bff9ae291caf6c4b4bcb3202p+0),
|
|
|
|
|
L(0x2.4e5d00559a6e2b9b5d7fe1f6689cp+0),
|
|
|
|
|
L(0x2.d57545a75cee8743ae2b17bc8d24p+0),
|
|
|
|
|
L(0x3.8514eee3aac88b89bec2307021bap+0),
|
|
|
|
|
L(0x4.5235e3b6e1891ffeb87fed9f8a24p+0),
|
|
|
|
|
L(0x5.562acdb10eef3c9a773b3e27a864p+0),
|
|
|
|
|
L(0x6.8ec8965c76efe03c26bff60b1194p+0),
|
|
|
|
|
L(0x8.15251aca144877af32658399f9b8p+0),
|
|
|
|
|
L(0x9.f08d56aba174d844138af782c0f8p+0),
|
|
|
|
|
L(0xc.3dbbeda2679e8a1346ccc3f6da88p+0),
|
|
|
|
|
L(0xf.0f5bfd5eacc26db308ffa0556fa8p+0),
|
|
|
|
|
L(0x1.28a6ccd84476fbc713d6bab49ac9p+4),
|
|
|
|
|
L(0x1.6d0a3ae2a3b1c8ff400641a3a21fp+4),
|
|
|
|
|
L(0x1.c15701b28637f87acfb6a91d33b5p+4),
|
|
|
|
|
L(0x2.28fbe0eccf472089b017651ca55ep+4),
|
|
|
|
|
L(0x2.a8a453004f6e8ffaacd1603bc3dp+4),
|
|
|
|
|
L(0x3.45ae4d9e1e7cd1a5dba0e4ec7f6cp+4),
|
|
|
|
|
L(0x4.065fbfacb7fad3e473cb577a61e8p+4),
|
|
|
|
|
L(0x4.f3d1473020927acac1944734a39p+4),
|
|
|
|
|
L(0x6.54bb091245815a36fb74e314dd18p+4),
|
|
|
|
|
L(0x7.d7f445129f7fb6c055e582d3f6ep+4),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Interval [-3, -2.875] (polynomial degree 23). */
|
2016-09-02 16:01:07 +00:00
|
|
|
L(-0xa.046d667e468f3e44dcae1afcc648p-4),
|
|
|
|
|
L(0x9.70b88dcc006c214d8d996fdf5ccp-4),
|
|
|
|
|
L(0xa.a8a39421c86d3ff24931a0929fp-4),
|
|
|
|
|
L(0xd.2f4d1363f324da2b357c8b6ec94p-4),
|
|
|
|
|
L(0xd.ca9aa1a3a5c00de11bf60499a97p-4),
|
|
|
|
|
L(0xf.cf09c31eeb52a45dfa7ebe3778dp-4),
|
|
|
|
|
L(0x1.04b133a39ed8a09691205660468bp+0),
|
|
|
|
|
L(0x1.22b547a06edda944fcb12fd9b5ecp+0),
|
|
|
|
|
L(0x1.2c57fce7db86a91df09602d344b3p+0),
|
|
|
|
|
L(0x1.4aade4894708f84795212fe257eep+0),
|
|
|
|
|
L(0x1.579c8b7b67ec4afed5b28c8bf787p+0),
|
|
|
|
|
L(0x1.776820e7fc80ae5284239733078ap+0),
|
|
|
|
|
L(0x1.883ab28c7301fde4ca6b8ec26ec8p+0),
|
|
|
|
|
L(0x1.aa2ef6e1ae52eb42c9ee83b206e3p+0),
|
|
|
|
|
L(0x1.bf4ad50f0a9a9311300cf0c51ee7p+0),
|
|
|
|
|
L(0x1.e40206e0e96b1da463814dde0d09p+0),
|
|
|
|
|
L(0x1.fdcbcffef3a21b29719c2bd9feb1p+0),
|
|
|
|
|
L(0x2.25e2e8948939c4d42cf108fae4bep+0),
|
|
|
|
|
L(0x2.44ce14d2b59c1c0e6bf2cfa81018p+0),
|
|
|
|
|
L(0x2.70ee80bbd0387162be4861c43622p+0),
|
|
|
|
|
L(0x2.954b64d2c2ebf3489b949c74476p+0),
|
|
|
|
|
L(0x2.c616e133a811c1c9446105208656p+0),
|
|
|
|
|
L(0x3.05a69dfe1a9ba1079f90fcf26bd4p+0),
|
|
|
|
|
L(0x3.410d2ad16a0506de29736e6aafdap+0),
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const size_t poly_deg[] =
|
|
|
|
|
{
|
|
|
|
|
23,
|
|
|
|
|
24,
|
|
|
|
|
25,
|
|
|
|
|
27,
|
|
|
|
|
28,
|
|
|
|
|
26,
|
|
|
|
|
24,
|
|
|
|
|
23,
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const size_t poly_end[] =
|
|
|
|
|
{
|
|
|
|
|
23,
|
|
|
|
|
48,
|
|
|
|
|
74,
|
|
|
|
|
102,
|
|
|
|
|
131,
|
|
|
|
|
158,
|
|
|
|
|
183,
|
|
|
|
|
207,
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
/* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static _Float128
|
|
|
|
|
lg_sinpi (_Float128 x)
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
2016-09-02 16:01:07 +00:00
|
|
|
if (x <= L(0.25))
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
return __sinl (M_PIl * x);
|
|
|
|
|
else
|
2016-09-02 16:01:07 +00:00
|
|
|
return __cosl (M_PIl * (L(0.5) - x));
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static _Float128
|
|
|
|
|
lg_cospi (_Float128 x)
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
2016-09-02 16:01:07 +00:00
|
|
|
if (x <= L(0.25))
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
return __cosl (M_PIl * x);
|
|
|
|
|
else
|
2016-09-02 16:01:07 +00:00
|
|
|
return __sinl (M_PIl * (L(0.5) - x));
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
static _Float128
|
|
|
|
|
lg_cotpi (_Float128 x)
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
|
|
|
|
return lg_cospi (x) / lg_sinpi (x);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Compute lgamma of a negative argument -50 < X < -2, setting
|
|
|
|
|
*SIGNGAMP accordingly. */
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128
|
|
|
|
|
__lgamma_negl (_Float128 x, int *signgamp)
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
{
|
|
|
|
|
/* Determine the half-integer region X lies in, handle exact
|
|
|
|
|
integers and determine the sign of the result. */
|
Use floor functions not __floor functions in glibc libm.
Similar to the changes that were made to call sqrt functions directly
in glibc, instead of __ieee754_sqrt variants, so that the compiler
could inline them automatically without needing special inline
definitions in lots of math_private.h headers, this patch makes libm
code call floor functions directly instead of __floor variants,
removing the inlines / macros for x86_64 (SSE4.1) and powerpc
(POWER5).
The redirection used to ensure that __ieee754_sqrt does still get
called when the compiler doesn't inline a built-in function expansion
is refactored so it can be applied to other functions; the refactoring
is arranged so it's not limited to unary functions either (it would be
reasonable to use this mechanism for copysign - removing the inline in
math_private_calls.h but also eliminating unnecessary local PLT entry
use in the cases (powerpc soft-float and e500v1, for IBM long double)
where copysign calls don't get inlined).
The point of this change is that more architectures can get floor
calls inlined where they weren't previously (AArch64, for example),
without needing special inline definitions in their math_private.h,
and existing such definitions in math_private.h headers can be
removed.
Note that it's possible that in some cases an inline may be used where
an IFUNC call was previously used - this is the case on x86_64, for
example. I think the direct calls to floor are still appropriate; if
there's any significant performance cost from inline SSE2 floor
instead of an IFUNC call ending up with SSE4.1 floor, that indicates
that either the function should be doing something else that's faster
than using floor at all, or it should itself have IFUNC variants, or
that the compiler choice of inlining for generic tuning should change
to allow for the possibility that, by not inlining, an SSE4.1 IFUNC
might be called at runtime - but not that glibc should avoid calling
floor internally. (After all, all the same considerations would apply
to any user program calling floor, where it might either be inlined or
left as an out-of-line call allowing for a possible IFUNC.)
Tested for x86_64, and with build-many-glibcs.py.
* include/math.h [!_ISOMAC && !(__FINITE_MATH_ONLY__ &&
__FINITE_MATH_ONLY__ > 0) && !NO_MATH_REDIRECT] (MATH_REDIRECT):
New macro.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_LDBL): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_F128): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_UNARY_ARGS): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (sqrt): Redirect using MATH_REDIRECT.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (floor): Likewise.
* sysdeps/aarch64/fpu/s_floor.c: Define NO_MATH_REDIRECT before
header inclusion.
* sysdeps/aarch64/fpu/s_floorf.c: Likewise.
* sysdeps/ieee754/dbl-64/s_floor.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_floor.c: Likewise.
* sysdeps/ieee754/float128/s_floorf128.c: Likewise.
* sysdeps/ieee754/flt-32/s_floorf.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_floorl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_floorl.c: Likewise.
* sysdeps/m68k/m680x0/fpu/s_floor_template.c: Likewise.
* sysdeps/powerpc/powerpc32/power4/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/powerpc/powerpc32/power4/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/powerpc/powerpc64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/powerpc/powerpc64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_floor.c: Likewise.
* sysdeps/riscv/rvf/s_floorf.c: Likewise.
* sysdeps/sparc/sparc64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/sparc/sparc64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/powerpc/fpu/math_private.h [_ARCH_PWR5X] (__floor):
Remove macro.
[_ARCH_PWR5X] (__floorf): Likewise.
* sysdeps/x86_64/fpu/math_private.h [__SSE4_1__] (__floor): Remove
inline function.
[__SSE4_1__] (__floorf): Likewise.
* math/w_lgamma_main.c (LGFUNC (__lgamma)): Use floor functions
instead of __floor variants.
* math/w_lgamma_r_compat.c (__lgamma_r): Likewise.
* math/w_lgammaf_main.c (LGFUNC (__lgammaf)): Likewise.
* math/w_lgammaf_r_compat.c (__lgammaf_r): Likewise.
* math/w_lgammal_main.c (LGFUNC (__lgammal)): Likewise.
* math/w_lgammal_r_compat.c (__lgammal_r): Likewise.
* math/w_tgamma_compat.c (__tgamma): Likewise.
* math/w_tgamma_template.c (M_DECL_FUNC (__tgamma)): Likewise.
* math/w_tgammaf_compat.c (__tgammaf): Likewise.
* math/w_tgammal_compat.c (__tgammal): Likewise.
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (sin_pi): Likewise.
* sysdeps/ieee754/dbl-64/k_rem_pio2.c (__kernel_rem_pio2):
Likewise.
* sysdeps/ieee754/dbl-64/lgamma_neg.c (__lgamma_neg): Likewise.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (sin_pif): Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c (__lgamma_negf): Likewise.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Likewise.
* sysdeps/ieee754/ldbl-128/e_powl.c (__ieee754_powl): Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c (__lgamma_negl):
Likewise.
* sysdeps/ieee754/ldbl-128/s_expm1l.c (__expm1l): Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_lgammal_r.c (__ieee754_lgammal_r):
Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_powl.c (__ieee754_powl): Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c (__lgamma_negl):
Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_expm1l.c (__expm1l): Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_truncl.c (__truncl): Likewise.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (sin_pi): Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c (__lgamma_negl): Likewise.
* sysdeps/powerpc/power5+/fpu/s_modf.c (__modf): Likewise.
* sysdeps/powerpc/power5+/fpu/s_modff.c (__modff): Likewise.
2018-09-14 13:09:01 +00:00
|
|
|
int i = floorl (-2 * x);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
if ((i & 1) == 0 && i == -2 * x)
|
2016-09-02 16:01:07 +00:00
|
|
|
return L(1.0) / L(0.0);
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
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i -= 4;
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*signgamp = ((i & 2) == 0 ? -1 : 1);
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SET_RESTORE_ROUNDL (FE_TONEAREST);
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/* Expand around the zero X0 = X0_HI + X0_LO. */
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
|
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_Float128 xdiff = x - x0_hi - x0_lo;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
|
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|
|
/* For arguments in the range -3 to -2, use polynomial
|
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|
|
approximations to an adjusted version of the gamma function. */
|
|
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|
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if (i < 2)
|
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{
|
Use floor functions not __floor functions in glibc libm.
Similar to the changes that were made to call sqrt functions directly
in glibc, instead of __ieee754_sqrt variants, so that the compiler
could inline them automatically without needing special inline
definitions in lots of math_private.h headers, this patch makes libm
code call floor functions directly instead of __floor variants,
removing the inlines / macros for x86_64 (SSE4.1) and powerpc
(POWER5).
The redirection used to ensure that __ieee754_sqrt does still get
called when the compiler doesn't inline a built-in function expansion
is refactored so it can be applied to other functions; the refactoring
is arranged so it's not limited to unary functions either (it would be
reasonable to use this mechanism for copysign - removing the inline in
math_private_calls.h but also eliminating unnecessary local PLT entry
use in the cases (powerpc soft-float and e500v1, for IBM long double)
where copysign calls don't get inlined).
The point of this change is that more architectures can get floor
calls inlined where they weren't previously (AArch64, for example),
without needing special inline definitions in their math_private.h,
and existing such definitions in math_private.h headers can be
removed.
Note that it's possible that in some cases an inline may be used where
an IFUNC call was previously used - this is the case on x86_64, for
example. I think the direct calls to floor are still appropriate; if
there's any significant performance cost from inline SSE2 floor
instead of an IFUNC call ending up with SSE4.1 floor, that indicates
that either the function should be doing something else that's faster
than using floor at all, or it should itself have IFUNC variants, or
that the compiler choice of inlining for generic tuning should change
to allow for the possibility that, by not inlining, an SSE4.1 IFUNC
might be called at runtime - but not that glibc should avoid calling
floor internally. (After all, all the same considerations would apply
to any user program calling floor, where it might either be inlined or
left as an out-of-line call allowing for a possible IFUNC.)
Tested for x86_64, and with build-many-glibcs.py.
* include/math.h [!_ISOMAC && !(__FINITE_MATH_ONLY__ &&
__FINITE_MATH_ONLY__ > 0) && !NO_MATH_REDIRECT] (MATH_REDIRECT):
New macro.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_LDBL): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_F128): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (MATH_REDIRECT_UNARY_ARGS): Likewise.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (sqrt): Redirect using MATH_REDIRECT.
[!_ISOMAC && !(__FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0)
&& !NO_MATH_REDIRECT] (floor): Likewise.
* sysdeps/aarch64/fpu/s_floor.c: Define NO_MATH_REDIRECT before
header inclusion.
* sysdeps/aarch64/fpu/s_floorf.c: Likewise.
* sysdeps/ieee754/dbl-64/s_floor.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_floor.c: Likewise.
* sysdeps/ieee754/float128/s_floorf128.c: Likewise.
* sysdeps/ieee754/flt-32/s_floorf.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_floorl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_floorl.c: Likewise.
* sysdeps/m68k/m680x0/fpu/s_floor_template.c: Likewise.
* sysdeps/powerpc/powerpc32/power4/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/powerpc/powerpc32/power4/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/powerpc/powerpc64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/powerpc/powerpc64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_floor.c: Likewise.
* sysdeps/riscv/rvf/s_floorf.c: Likewise.
* sysdeps/sparc/sparc64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/sparc/sparc64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/s_floor.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/s_floorf.c: Likewise.
* sysdeps/powerpc/fpu/math_private.h [_ARCH_PWR5X] (__floor):
Remove macro.
[_ARCH_PWR5X] (__floorf): Likewise.
* sysdeps/x86_64/fpu/math_private.h [__SSE4_1__] (__floor): Remove
inline function.
[__SSE4_1__] (__floorf): Likewise.
* math/w_lgamma_main.c (LGFUNC (__lgamma)): Use floor functions
instead of __floor variants.
* math/w_lgamma_r_compat.c (__lgamma_r): Likewise.
* math/w_lgammaf_main.c (LGFUNC (__lgammaf)): Likewise.
* math/w_lgammaf_r_compat.c (__lgammaf_r): Likewise.
* math/w_lgammal_main.c (LGFUNC (__lgammal)): Likewise.
* math/w_lgammal_r_compat.c (__lgammal_r): Likewise.
* math/w_tgamma_compat.c (__tgamma): Likewise.
* math/w_tgamma_template.c (M_DECL_FUNC (__tgamma)): Likewise.
* math/w_tgammaf_compat.c (__tgammaf): Likewise.
* math/w_tgammal_compat.c (__tgammal): Likewise.
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (sin_pi): Likewise.
* sysdeps/ieee754/dbl-64/k_rem_pio2.c (__kernel_rem_pio2):
Likewise.
* sysdeps/ieee754/dbl-64/lgamma_neg.c (__lgamma_neg): Likewise.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (sin_pif): Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c (__lgamma_negf): Likewise.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Likewise.
* sysdeps/ieee754/ldbl-128/e_powl.c (__ieee754_powl): Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c (__lgamma_negl):
Likewise.
* sysdeps/ieee754/ldbl-128/s_expm1l.c (__expm1l): Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_lgammal_r.c (__ieee754_lgammal_r):
Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_powl.c (__ieee754_powl): Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c (__lgamma_negl):
Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_expm1l.c (__expm1l): Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_truncl.c (__truncl): Likewise.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (sin_pi): Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c (__lgamma_negl): Likewise.
* sysdeps/powerpc/power5+/fpu/s_modf.c (__modf): Likewise.
* sysdeps/powerpc/power5+/fpu/s_modff.c (__modff): Likewise.
2018-09-14 13:09:01 +00:00
|
|
|
int j = floorl (-8 * x) - 16;
|
2016-09-02 16:01:07 +00:00
|
|
|
_Float128 xm = (-33 - 2 * j) * L(0.0625);
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 x_adj = x - xm;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
size_t deg = poly_deg[j];
|
|
|
|
|
size_t end = poly_end[j];
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 g = poly_coeff[end];
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
for (size_t j = 1; j <= deg; j++)
|
|
|
|
|
g = g * x_adj + poly_coeff[end - j];
|
|
|
|
|
return __log1pl (g * xdiff / (x - xn));
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* The result we want is log (sinpi (X0) / sinpi (X))
|
|
|
|
|
+ log (gamma (1 - X0) / gamma (1 - X)). */
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 x_idiff = fabsl (xn - x), x0_idiff = fabsl (xn - x0_hi - x0_lo);
|
|
|
|
|
_Float128 log_sinpi_ratio;
|
2016-09-02 16:01:07 +00:00
|
|
|
if (x0_idiff < x_idiff * L(0.5))
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Use log not log1p to avoid inaccuracy from log1p of arguments
|
|
|
|
|
close to -1. */
|
|
|
|
|
log_sinpi_ratio = __ieee754_logl (lg_sinpi (x0_idiff)
|
|
|
|
|
/ lg_sinpi (x_idiff));
|
|
|
|
|
else
|
|
|
|
|
{
|
|
|
|
|
/* Use log1p not log to avoid inaccuracy from log of arguments
|
|
|
|
|
close to 1. X0DIFF2 has positive sign if X0 is further from
|
|
|
|
|
XN than X is from XN, negative sign otherwise. */
|
2016-09-02 16:01:07 +00:00
|
|
|
_Float128 x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * L(0.5);
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 sx0d2 = lg_sinpi (x0diff2);
|
|
|
|
|
_Float128 cx0d2 = lg_cospi (x0diff2);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
log_sinpi_ratio = __log1pl (2 * sx0d2
|
|
|
|
|
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
|
|
|
|
|
}
|
|
|
|
|
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 log_gamma_ratio;
|
|
|
|
|
_Float128 y0 = 1 - x0_hi;
|
|
|
|
|
_Float128 y0_eps = -x0_hi + (1 - y0) - x0_lo;
|
|
|
|
|
_Float128 y = 1 - x;
|
|
|
|
|
_Float128 y_eps = -x + (1 - y);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* We now wish to compute LOG_GAMMA_RATIO
|
|
|
|
|
= log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
|
|
|
|
|
accurately approximates the difference Y0 + Y0_EPS - Y -
|
|
|
|
|
Y_EPS. Use Stirling's approximation. First, we may need to
|
|
|
|
|
adjust into the range where Stirling's approximation is
|
|
|
|
|
sufficiently accurate. */
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 log_gamma_adj = 0;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
if (i < 20)
|
|
|
|
|
{
|
|
|
|
|
int n_up = (21 - i) / 2;
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 ny0, ny0_eps, ny, ny_eps;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
ny0 = y0 + n_up;
|
|
|
|
|
ny0_eps = y0 - (ny0 - n_up) + y0_eps;
|
|
|
|
|
y0 = ny0;
|
|
|
|
|
y0_eps = ny0_eps;
|
|
|
|
|
ny = y + n_up;
|
|
|
|
|
ny_eps = y - (ny - n_up) + y_eps;
|
|
|
|
|
y = ny;
|
|
|
|
|
y_eps = ny_eps;
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 prodm1 = __lgamma_productl (xdiff, y - n_up, y_eps, n_up);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
log_gamma_adj = -__log1pl (prodm1);
|
|
|
|
|
}
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 log_gamma_high
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
= (xdiff * __log1pl ((y0 - e_hi - e_lo + y0_eps) / e_hi)
|
2016-09-02 16:01:07 +00:00
|
|
|
+ (y - L(0.5) + y_eps) * __log1pl (xdiff / y) + log_gamma_adj);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 y0r = 1 / y0, yr = 1 / y;
|
|
|
|
|
_Float128 y0r2 = y0r * y0r, yr2 = yr * yr;
|
|
|
|
|
_Float128 rdiff = -xdiff / (y * y0);
|
|
|
|
|
_Float128 bterm[NCOEFF];
|
|
|
|
|
_Float128 dlast = rdiff, elast = rdiff * yr * (yr + y0r);
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
bterm[0] = dlast * lgamma_coeff[0];
|
|
|
|
|
for (size_t j = 1; j < NCOEFF; j++)
|
|
|
|
|
{
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 dnext = dlast * y0r2 + elast;
|
|
|
|
|
_Float128 enext = elast * yr2;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
|
|
|
bterm[j] = dnext * lgamma_coeff[j];
|
|
|
|
|
dlast = dnext;
|
|
|
|
|
elast = enext;
|
|
|
|
|
}
|
2016-07-20 20:20:51 +00:00
|
|
|
_Float128 log_gamma_low = 0;
|
Fix lgamma (negative) inaccuracy (bug 2542, bug 2543, bug 2558).
The existing implementations of lgamma functions (except for the ia64
versions) use the reflection formula for negative arguments. This
suffers large inaccuracy from cancellation near zeros of lgamma (near
where the gamma function is +/- 1).
This patch fixes this inaccuracy. For arguments above -2, there are
no zeros and no large cancellation, while for sufficiently large
negative arguments the zeros are so close to integers that even for
integers +/- 1ulp the log(gamma(1-x)) term dominates and cancellation
is not significant. Thus, it is only necessary to take special care
about cancellation for arguments around a limited number of zeros.
Accordingly, this patch uses precomputed tables of relevant zeros,
expressed as the sum of two floating-point values. The log of the
ratio of two sines can be computed accurately using log1p in cases
where log would lose accuracy. The log of the ratio of two gamma(1-x)
values can be computed using Stirling's approximation (the difference
between two values of that approximation to lgamma being computable
without computing the two values and then subtracting), with
appropriate adjustments (which don't reduce accuracy too much) in
cases where 1-x is too small to use Stirling's approximation directly.
In the interval from -3 to -2, using the ratios of sines and of
gamma(1-x) can still produce too much cancellation between those two
parts of the computation (and that interval is also the worst interval
for computing the ratio between gamma(1-x) values, which computation
becomes more accurate, while being less critical for the final result,
for larger 1-x). Because this can result in errors slightly above
those accepted in glibc, this interval is instead dealt with by
polynomial approximations. Separate polynomial approximations to
(|gamma(x)|-1)(x-n)/(x-x0) are used for each interval of length 1/8
from -3 to -2, where n (-3 or -2) is the nearest integer to the
1/8-interval and x0 is the zero of lgamma in the relevant half-integer
interval (-3 to -2.5 or -2.5 to -2).
Together, the two approaches are intended to give sufficient accuracy
for all negative arguments in the problem range. Outside that range,
the previous implementation continues to be used.
Tested for x86_64, x86, mips64 and powerpc. The mips64 and powerpc
testing shows up pre-existing problems for ldbl-128 and ldbl-128ibm
with large negative arguments giving spurious "invalid" exceptions
(exposed by newly added tests for cases this patch doesn't affect the
logic for); I'll address those problems separately.
[BZ #2542]
[BZ #2543]
[BZ #2558]
* sysdeps/ieee754/dbl-64/e_lgamma_r.c (__ieee754_lgamma_r): Call
__lgamma_neg for arguments from -28.0 to -2.0.
* sysdeps/ieee754/flt-32/e_lgammaf_r.c (__ieee754_lgammaf_r): Call
__lgamma_negf for arguments from -15.0 to -2.0.
* sysdeps/ieee754/ldbl-128/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -48.0 or -50.0 to -2.0.
* sysdeps/ieee754/ldbl-96/e_lgammal_r.c (__ieee754_lgammal_r):
Call __lgamma_negl for arguments from -33.0 to -2.0.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: New file.
* sysdeps/ieee754/dbl-64/lgamma_product.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_productf.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_product.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_productl.c: Likewise.
* sysdeps/generic/math_private.h (__lgamma_negf): New prototype.
(__lgamma_neg): Likewise.
(__lgamma_negl): Likewise.
(__lgamma_product): Likewise.
(__lgamma_productl): Likewise.
* math/Makefile (libm-calls): Add lgamma_neg and lgamma_product.
* math/auto-libm-test-in: Add more tests of lgamma.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
2015-09-10 22:27:58 +00:00
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for (size_t j = 0; j < NCOEFF; j++)
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log_gamma_low += bterm[NCOEFF - 1 - j];
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log_gamma_ratio = log_gamma_high + log_gamma_low;
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return log_sinpi_ratio + log_gamma_ratio;
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}
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