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gcc/libquadmath/math/lgammaq.c
Joseph Myers 4239f144ce Update most of libquadmath/math/ from glibc, automate update (PR libquadmath/68686). 
libquadmath sources are mostly based on glibc sources at present, but
derived from them by a manual editing / substitution process and with
subsequent manual merges.  The manual effort involved in merges means
they are sometimes incomplete and long-delayed.

Since libquadmath was first created, glibc's support for this format
has undergone significant changes so that it can also be used in glibc
to provide *f128 functions for the _Float128 type from TS 18661-3.
This makes it significantly easier to use it for libquadmath in a more
automated fashion, since glibc has a float128_private.h header that
redefines many identifiers as macros as needed for building *f128
functions.

Simply using float128_private.h directly in libquadmath, with
unmodified glibc sources except for changing function names in that
one header to be *q instead of *f128, would be tricky, given its
dependence on lots of other glibc-internal headers (whereas
libquadmath supports non-glibc systems), and also given how some libm
functions in glibc are built from type-generic templates using a
further set of macros rather than from separate function
implementations for each type.

So instead this patch adds a script update-quadmath.py to convert
glibc sources into libquadmath ones, and the script reads
float128_private.h to identify many of the substitutions it should
make.  quadmath-imp.h is updated with various new internal
definitions, taken from glibc as needed; this is the main place
expected to need updating manually when subsequent merges from glibc
are done using the script.  No attempt is made to make the script
output match the details of existing formatting, although the
differences are of a size that makes a rough comparison (ignoring
whitespace) possible.

Two new public interfaces are added to libquadmath, exp2q and
issignalingq, at a new QUADMATH_1.2 symbol version, since those
interfaces are used internally by some of the glibc sources being
merged into libquadmath; although there is a new symbol version, no
change however is made to the libtool version in the libtool-version
file.  Although there are various other interfaces now in glibc libm
but not in libquadmath, this patch does nothing to add such interfaces
(although adding many of them would in fact be easy to do, given the
script).

One internal file (not providing any public interfaces),
math/isinf_nsq.c, is removed, as no longer used by anything in
libquadmath after the merge.

Conditionals in individual source files on <fenv.h> availability or
features are moved into quadmath-imp.h (providing trivial macro
versions of the functions if real implementations aren't available),
to simplify the substitutions in individual source files.  Note
however that I haven't tested for any configurations lacking <fenv.h>,
so further changes could well be needed there.

Two files in libquadmath/math/ are based on glibc sources but not
updated in this patch: fmaq.c and rem_pio2q.c.  Both could be updated
after further changes to the script (and quadmath-imp.h as needed); in
the case of rem_pio2q.c, based on two separate glibc source files,
those separate files would naturally be split out into separate
libquadmath source files in the process (as done in this patch with
expq_table.h and tanq_kernel.c, where previously two glibc source
files had been merged into one libquadmath source file).  complex.c,
nanq.c and sqrtq.c are not based on glibc sources (though four of the
(trivial) functions in complex.c could readily be replaced by instead
using the four corresponding files from glibc, if desired).

libquadmath also has printf/ and strtod/ sources based on glibc, also
mostly not updated for a long time.  Again the script could no doubt
be made to generate those automatically, although that would be a
larger change (effectively some completely separate logic in the
script, not sharing much if anything with the existing code).

Bootstrapped with no regressions on x86_64-pc-linux-gnu.

	PR libquadmath/68686
	* Makefile.am: (libquadmath_la_SOURCES): Remove math/isinf_nsq.c.
	Add math/exp2q.c math/issignalingq.c math/lgammaq_neg.c
	math/lgammaq_product.c math/tanq_kernel.c math/tgammaq_product.c
	math/casinhq_kernel.c.
	* Makefile.in: Regenerate.
	* libquadmath.texi (exp2q, issignalingq): Document.
	* quadmath-imp.h: Include <errno.h>, <limits.h>, <stdbool.h> and
	<fenv.h>.
	(HIGH_ORDER_BIT_IS_SET_FOR_SNAN, FIX_FLT128_LONG_CONVERT_OVERFLOW)
	(FIX_FLT128_LLONG_CONVERT_OVERFLOW, __quadmath_kernel_tanq)
	(__quadmath_gamma_productq, __quadmath_gammaq_r)
	(__quadmath_lgamma_negq, __quadmath_lgamma_productq)
	(__quadmath_lgammaq_r, __quadmath_kernel_casinhq, mul_splitq)
	(math_check_force_underflow_complex, __glibc_likely)
	(__glibc_unlikely, struct rm_ctx, SET_RESTORE_ROUNDF128)
	(libc_feholdsetround_ctx, libc_feresetround_ctx): New.
	(feraiseexcept, fenv_t, feholdexcept, fesetround, feupdateenv)
	(fesetenv, fetestexcept, feclearexcept): Define if not supported
	through <fenv.h>.
	(__quadmath_isinf_nsq): Remove.
	* quadmath.h (exp2q, issignalingq): New.
	* quadmath.map (QUADMATH_1.2): New.
	* quadmath_weak.h (exp2q, issignalingq): New.
	* update-quadmath.py: New file.
	* math/isinf_nsq.c: Remove file.
	* math/casinhq_kernel.c, math/exp2q.c, math/expq_table.h,
	math/issignalingq.c, math/lgammaq_neg.c, math/lgammaq_product.c,
	math/tanq_kernel.c, math/tgammaq_product.c: New files.  Generated
	from glibc sources with update-quadmath.py.
	* math/acoshq.c, math/acosq.c, math/asinhq.c, math/asinq.c,
	math/atan2q.c, math/atanhq.c, math/atanq.c, math/cacoshq.c,
	math/cacosq.c, math/casinhq.c, math/casinq.c, math/catanhq.c,
	math/catanq.c, math/cbrtq.c, math/ccoshq.c, math/ceilq.c,
	math/cexpq.c, math/cimagq.c, math/clog10q.c, math/clogq.c,
	math/conjq.c, math/copysignq.c, math/coshq.c, math/cosq.c,
	math/cosq_kernel.c, math/cprojq.c, math/crealq.c, math/csinhq.c,
	math/csinq.c, math/csqrtq.c, math/ctanhq.c, math/ctanq.c,
	math/erfq.c, math/expm1q.c, math/expq.c, math/fabsq.c,
	math/fdimq.c, math/finiteq.c, math/floorq.c, math/fmaxq.c,
	math/fminq.c, math/fmodq.c, math/frexpq.c, math/hypotq.c,
	math/ilogbq.c, math/isinfq.c, math/isnanq.c, math/j0q.c,
	math/j1q.c, math/jnq.c, math/ldexpq.c, math/lgammaq.c,
	math/llrintq.c, math/llroundq.c, math/log10q.c, math/log1pq.c,
	math/log2q.c, math/logbq.c, math/logq.c, math/lrintq.c,
	math/lroundq.c, math/modfq.c, math/nearbyintq.c,
	math/nextafterq.c, math/powq.c, math/remainderq.c, math/remquoq.c,
	math/rintq.c, math/roundq.c, math/scalblnq.c, math/scalbnq.c,
	math/signbitq.c, math/sincos_table.c, math/sincosq.c,
	math/sincosq_kernel.c, math/sinhq.c, math/sinq.c,
	math/sinq_kernel.c, math/tanhq.c, math/tanq.c, math/tgammaq.c,
	math/truncq.c, math/x2y2m1q.c: Regenerate from glibc sources with
	update-quadmath.py.

From-SVN: r265822
2018-11-05 23:03:55 +00:00

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/* lgammal
*
* Natural logarithm of gamma function
*
*
*
* SYNOPSIS:
*
* long double x, y, lgammal();
* extern int sgngam;
*
* y = lgammal(x);
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of the absolute
* value of the gamma function of the argument.
* The sign (+1 or -1) of the gamma function is returned in a
* global (extern) variable named sgngam.
*
* The positive domain is partitioned into numerous segments for approximation.
* For x > 10,
* log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2)
* Near the minimum at x = x0 = 1.46... the approximation is
* log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z)
* for small z.
* Elsewhere between 0 and 10,
* log gamma(n + z) = log gamma(n) + z P(z)/Q(z)
* for various selected n and small z.
*
* The cosecant reflection formula is employed for negative arguments.
*
*
*
* ACCURACY:
*
*
* arithmetic domain # trials peak rms
* Relative error:
* IEEE 10, 30 100000 3.9e-34 9.8e-35
* IEEE 0, 10 100000 3.8e-34 5.3e-35
* Absolute error:
* IEEE -10, 0 100000 8.0e-34 8.0e-35
* IEEE -30, -10 100000 4.4e-34 1.0e-34
* IEEE -100, 100 100000 1.0e-34
*
* The absolute error criterion is the same as relative error
* when the function magnitude is greater than one but it is absolute
* when the magnitude is less than one.
*
*/
/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
#include "quadmath-imp.h"
#ifdef HAVE_MATH_H_SIGNGAM
# include <math.h>
#endif
__float128
lgammaq (__float128 x)
{
#ifndef HAVE_MATH_H_SIGNGAM
int signgam;
#endif
return __quadmath_lgammaq_r (x, &signgam);
}
static const __float128 PIL = 3.1415926535897932384626433832795028841972E0Q;
static const __float128 MAXLGM = 1.0485738685148938358098967157129705071571E4928Q;
static const __float128 one = 1;
static const __float128 huge = FLT128_MAX;
/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
1/x <= 0.0741 (x >= 13.495...)
Peak relative error 1.5e-36 */
static const __float128 ls2pi = 9.1893853320467274178032973640561763986140E-1Q;
#define NRASY 12
static const __float128 RASY[NRASY + 1] =
{
8.333333333333333333333333333310437112111E-2Q,
-2.777777777777777777777774789556228296902E-3Q,
7.936507936507936507795933938448586499183E-4Q,
-5.952380952380952041799269756378148574045E-4Q,
8.417508417507928904209891117498524452523E-4Q,
-1.917526917481263997778542329739806086290E-3Q,
6.410256381217852504446848671499409919280E-3Q,
-2.955064066900961649768101034477363301626E-2Q,
1.796402955865634243663453415388336954675E-1Q,
-1.391522089007758553455753477688592767741E0Q,
1.326130089598399157988112385013829305510E1Q,
-1.420412699593782497803472576479997819149E2Q,
1.218058922427762808938869872528846787020E3Q
};
/* log gamma(x+13) = log gamma(13) + x P(x)/Q(x)
-0.5 <= x <= 0.5
12.5 <= x+13 <= 13.5
Peak relative error 1.1e-36 */
static const __float128 lgam13a = 1.9987213134765625E1Q;
static const __float128 lgam13b = 1.3608962611495173623870550785125024484248E-6Q;
#define NRN13 7
static const __float128 RN13[NRN13 + 1] =
{
8.591478354823578150238226576156275285700E11Q,
2.347931159756482741018258864137297157668E11Q,
2.555408396679352028680662433943000804616E10Q,
1.408581709264464345480765758902967123937E9Q,
4.126759849752613822953004114044451046321E7Q,
6.133298899622688505854211579222889943778E5Q,
3.929248056293651597987893340755876578072E3Q,
6.850783280018706668924952057996075215223E0Q
};
#define NRD13 6
static const __float128 RD13[NRD13 + 1] =
{
3.401225382297342302296607039352935541669E11Q,
8.756765276918037910363513243563234551784E10Q,
8.873913342866613213078554180987647243903E9Q,
4.483797255342763263361893016049310017973E8Q,
1.178186288833066430952276702931512870676E7Q,
1.519928623743264797939103740132278337476E5Q,
7.989298844938119228411117593338850892311E2Q
/* 1.0E0L */
};
/* log gamma(x+12) = log gamma(12) + x P(x)/Q(x)
-0.5 <= x <= 0.5
11.5 <= x+12 <= 12.5
Peak relative error 4.1e-36 */
static const __float128 lgam12a = 1.75023040771484375E1Q;
static const __float128 lgam12b = 3.7687254483392876529072161996717039575982E-6Q;
#define NRN12 7
static const __float128 RN12[NRN12 + 1] =
{
4.709859662695606986110997348630997559137E11Q,
1.398713878079497115037857470168777995230E11Q,
1.654654931821564315970930093932954900867E10Q,
9.916279414876676861193649489207282144036E8Q,
3.159604070526036074112008954113411389879E7Q,
5.109099197547205212294747623977502492861E5Q,
3.563054878276102790183396740969279826988E3Q,
6.769610657004672719224614163196946862747E0Q
};
#define NRD12 6
static const __float128 RD12[NRD12 + 1] =
{
1.928167007860968063912467318985802726613E11Q,
5.383198282277806237247492369072266389233E10Q,
5.915693215338294477444809323037871058363E9Q,
3.241438287570196713148310560147925781342E8Q,
9.236680081763754597872713592701048455890E6Q,
1.292246897881650919242713651166596478850E5Q,
7.366532445427159272584194816076600211171E2Q
/* 1.0E0L */
};
/* log gamma(x+11) = log gamma(11) + x P(x)/Q(x)
-0.5 <= x <= 0.5
10.5 <= x+11 <= 11.5
Peak relative error 1.8e-35 */
static const __float128 lgam11a = 1.5104400634765625E1Q;
static const __float128 lgam11b = 1.1938309890295225709329251070371882250744E-5Q;
#define NRN11 7
static const __float128 RN11[NRN11 + 1] =
{
2.446960438029415837384622675816736622795E11Q,
7.955444974446413315803799763901729640350E10Q,
1.030555327949159293591618473447420338444E10Q,
6.765022131195302709153994345470493334946E8Q,
2.361892792609204855279723576041468347494E7Q,
4.186623629779479136428005806072176490125E5Q,
3.202506022088912768601325534149383594049E3Q,
6.681356101133728289358838690666225691363E0Q
};
#define NRD11 6
static const __float128 RD11[NRD11 + 1] =
{
1.040483786179428590683912396379079477432E11Q,
3.172251138489229497223696648369823779729E10Q,
3.806961885984850433709295832245848084614E9Q,
2.278070344022934913730015420611609620171E8Q,
7.089478198662651683977290023829391596481E6Q,
1.083246385105903533237139380509590158658E5Q,
6.744420991491385145885727942219463243597E2Q
/* 1.0E0L */
};
/* log gamma(x+10) = log gamma(10) + x P(x)/Q(x)
-0.5 <= x <= 0.5
9.5 <= x+10 <= 10.5
Peak relative error 5.4e-37 */
static const __float128 lgam10a = 1.280181884765625E1Q;
static const __float128 lgam10b = 8.6324252196112077178745667061642811492557E-6Q;
#define NRN10 7
static const __float128 RN10[NRN10 + 1] =
{
-1.239059737177249934158597996648808363783E14Q,
-4.725899566371458992365624673357356908719E13Q,
-7.283906268647083312042059082837754850808E12Q,
-5.802855515464011422171165179767478794637E11Q,
-2.532349691157548788382820303182745897298E10Q,
-5.884260178023777312587193693477072061820E8Q,
-6.437774864512125749845840472131829114906E6Q,
-2.350975266781548931856017239843273049384E4Q
};
#define NRD10 7
static const __float128 RD10[NRD10 + 1] =
{
-5.502645997581822567468347817182347679552E13Q,
-1.970266640239849804162284805400136473801E13Q,
-2.819677689615038489384974042561531409392E12Q,
-2.056105863694742752589691183194061265094E11Q,
-8.053670086493258693186307810815819662078E9Q,
-1.632090155573373286153427982504851867131E8Q,
-1.483575879240631280658077826889223634921E6Q,
-4.002806669713232271615885826373550502510E3Q
/* 1.0E0L */
};
/* log gamma(x+9) = log gamma(9) + x P(x)/Q(x)
-0.5 <= x <= 0.5
8.5 <= x+9 <= 9.5
Peak relative error 3.6e-36 */
static const __float128 lgam9a = 1.06045989990234375E1Q;
static const __float128 lgam9b = 3.9037218127284172274007216547549861681400E-6Q;
#define NRN9 7
static const __float128 RN9[NRN9 + 1] =
{
-4.936332264202687973364500998984608306189E13Q,
-2.101372682623700967335206138517766274855E13Q,
-3.615893404644823888655732817505129444195E12Q,
-3.217104993800878891194322691860075472926E11Q,
-1.568465330337375725685439173603032921399E10Q,
-4.073317518162025744377629219101510217761E8Q,
-4.983232096406156139324846656819246974500E6Q,
-2.036280038903695980912289722995505277253E4Q
};
#define NRD9 7
static const __float128 RD9[NRD9 + 1] =
{
-2.306006080437656357167128541231915480393E13Q,
-9.183606842453274924895648863832233799950E12Q,
-1.461857965935942962087907301194381010380E12Q,
-1.185728254682789754150068652663124298303E11Q,
-5.166285094703468567389566085480783070037E9Q,
-1.164573656694603024184768200787835094317E8Q,
-1.177343939483908678474886454113163527909E6Q,
-3.529391059783109732159524500029157638736E3Q
/* 1.0E0L */
};
/* log gamma(x+8) = log gamma(8) + x P(x)/Q(x)
-0.5 <= x <= 0.5
7.5 <= x+8 <= 8.5
Peak relative error 2.4e-37 */
static const __float128 lgam8a = 8.525146484375E0Q;
static const __float128 lgam8b = 1.4876690414300165531036347125050759667737E-5Q;
#define NRN8 8
static const __float128 RN8[NRN8 + 1] =
{
6.600775438203423546565361176829139703289E11Q,
3.406361267593790705240802723914281025800E11Q,
7.222460928505293914746983300555538432830E10Q,
8.102984106025088123058747466840656458342E9Q,
5.157620015986282905232150979772409345927E8Q,
1.851445288272645829028129389609068641517E7Q,
3.489261702223124354745894067468953756656E5Q,
2.892095396706665774434217489775617756014E3Q,
6.596977510622195827183948478627058738034E0Q
};
#define NRD8 7
static const __float128 RD8[NRD8 + 1] =
{
3.274776546520735414638114828622673016920E11Q,
1.581811207929065544043963828487733970107E11Q,
3.108725655667825188135393076860104546416E10Q,
3.193055010502912617128480163681842165730E9Q,
1.830871482669835106357529710116211541839E8Q,
5.790862854275238129848491555068073485086E6Q,
9.305213264307921522842678835618803553589E4Q,
6.216974105861848386918949336819572333622E2Q
/* 1.0E0L */
};
/* log gamma(x+7) = log gamma(7) + x P(x)/Q(x)
-0.5 <= x <= 0.5
6.5 <= x+7 <= 7.5
Peak relative error 3.2e-36 */
static const __float128 lgam7a = 6.5792388916015625E0Q;
static const __float128 lgam7b = 1.2320408538495060178292903945321122583007E-5Q;
#define NRN7 8
static const __float128 RN7[NRN7 + 1] =
{
2.065019306969459407636744543358209942213E11Q,
1.226919919023736909889724951708796532847E11Q,
2.996157990374348596472241776917953749106E10Q,
3.873001919306801037344727168434909521030E9Q,
2.841575255593761593270885753992732145094E8Q,
1.176342515359431913664715324652399565551E7Q,
2.558097039684188723597519300356028511547E5Q,
2.448525238332609439023786244782810774702E3Q,
6.460280377802030953041566617300902020435E0Q
};
#define NRD7 7
static const __float128 RD7[NRD7 + 1] =
{
1.102646614598516998880874785339049304483E11Q,
6.099297512712715445879759589407189290040E10Q,
1.372898136289611312713283201112060238351E10Q,
1.615306270420293159907951633566635172343E9Q,
1.061114435798489135996614242842561967459E8Q,
3.845638971184305248268608902030718674691E6Q,
7.081730675423444975703917836972720495507E4Q,
5.423122582741398226693137276201344096370E2Q
/* 1.0E0L */
};
/* log gamma(x+6) = log gamma(6) + x P(x)/Q(x)
-0.5 <= x <= 0.5
5.5 <= x+6 <= 6.5
Peak relative error 6.2e-37 */
static const __float128 lgam6a = 4.7874908447265625E0Q;
static const __float128 lgam6b = 8.9805548349424770093452324304839959231517E-7Q;
#define NRN6 8
static const __float128 RN6[NRN6 + 1] =
{
-3.538412754670746879119162116819571823643E13Q,
-2.613432593406849155765698121483394257148E13Q,
-8.020670732770461579558867891923784753062E12Q,
-1.322227822931250045347591780332435433420E12Q,
-1.262809382777272476572558806855377129513E11Q,
-7.015006277027660872284922325741197022467E9Q,
-2.149320689089020841076532186783055727299E8Q,
-3.167210585700002703820077565539658995316E6Q,
-1.576834867378554185210279285358586385266E4Q
};
#define NRD6 8
static const __float128 RD6[NRD6 + 1] =
{
-2.073955870771283609792355579558899389085E13Q,
-1.421592856111673959642750863283919318175E13Q,
-4.012134994918353924219048850264207074949E12Q,
-6.013361045800992316498238470888523722431E11Q,
-5.145382510136622274784240527039643430628E10Q,
-2.510575820013409711678540476918249524123E9Q,
-6.564058379709759600836745035871373240904E7Q,
-7.861511116647120540275354855221373571536E5Q,
-2.821943442729620524365661338459579270561E3Q
/* 1.0E0L */
};
/* log gamma(x+5) = log gamma(5) + x P(x)/Q(x)
-0.5 <= x <= 0.5
4.5 <= x+5 <= 5.5
Peak relative error 3.4e-37 */
static const __float128 lgam5a = 3.17803955078125E0Q;
static const __float128 lgam5b = 1.4279566695619646941601297055408873990961E-5Q;
#define NRN5 9
static const __float128 RN5[NRN5 + 1] =
{
2.010952885441805899580403215533972172098E11Q,
1.916132681242540921354921906708215338584E11Q,
7.679102403710581712903937970163206882492E10Q,
1.680514903671382470108010973615268125169E10Q,
2.181011222911537259440775283277711588410E9Q,
1.705361119398837808244780667539728356096E8Q,
7.792391565652481864976147945997033946360E6Q,
1.910741381027985291688667214472560023819E5Q,
2.088138241893612679762260077783794329559E3Q,
6.330318119566998299106803922739066556550E0Q
};
#define NRD5 8
static const __float128 RD5[NRD5 + 1] =
{
1.335189758138651840605141370223112376176E11Q,
1.174130445739492885895466097516530211283E11Q,
4.308006619274572338118732154886328519910E10Q,
8.547402888692578655814445003283720677468E9Q,
9.934628078575618309542580800421370730906E8Q,
6.847107420092173812998096295422311820672E7Q,
2.698552646016599923609773122139463150403E6Q,
5.526516251532464176412113632726150253215E4Q,
4.772343321713697385780533022595450486932E2Q
/* 1.0E0L */
};
/* log gamma(x+4) = log gamma(4) + x P(x)/Q(x)
-0.5 <= x <= 0.5
3.5 <= x+4 <= 4.5
Peak relative error 6.7e-37 */
static const __float128 lgam4a = 1.791748046875E0Q;
static const __float128 lgam4b = 1.1422353055000812477358380702272722990692E-5Q;
#define NRN4 9
static const __float128 RN4[NRN4 + 1] =
{
-1.026583408246155508572442242188887829208E13Q,
-1.306476685384622809290193031208776258809E13Q,
-7.051088602207062164232806511992978915508E12Q,
-2.100849457735620004967624442027793656108E12Q,
-3.767473790774546963588549871673843260569E11Q,
-4.156387497364909963498394522336575984206E10Q,
-2.764021460668011732047778992419118757746E9Q,
-1.036617204107109779944986471142938641399E8Q,
-1.895730886640349026257780896972598305443E6Q,
-1.180509051468390914200720003907727988201E4Q
};
#define NRD4 9
static const __float128 RD4[NRD4 + 1] =
{
-8.172669122056002077809119378047536240889E12Q,
-9.477592426087986751343695251801814226960E12Q,
-4.629448850139318158743900253637212801682E12Q,
-1.237965465892012573255370078308035272942E12Q,
-1.971624313506929845158062177061297598956E11Q,
-1.905434843346570533229942397763361493610E10Q,
-1.089409357680461419743730978512856675984E9Q,
-3.416703082301143192939774401370222822430E7Q,
-4.981791914177103793218433195857635265295E5Q,
-2.192507743896742751483055798411231453733E3Q
/* 1.0E0L */
};
/* log gamma(x+3) = log gamma(3) + x P(x)/Q(x)
-0.25 <= x <= 0.5
2.75 <= x+3 <= 3.5
Peak relative error 6.0e-37 */
static const __float128 lgam3a = 6.93145751953125E-1Q;
static const __float128 lgam3b = 1.4286068203094172321214581765680755001344E-6Q;
#define NRN3 9
static const __float128 RN3[NRN3 + 1] =
{
-4.813901815114776281494823863935820876670E11Q,
-8.425592975288250400493910291066881992620E11Q,
-6.228685507402467503655405482985516909157E11Q,
-2.531972054436786351403749276956707260499E11Q,
-6.170200796658926701311867484296426831687E10Q,
-9.211477458528156048231908798456365081135E9Q,
-8.251806236175037114064561038908691305583E8Q,
-4.147886355917831049939930101151160447495E7Q,
-1.010851868928346082547075956946476932162E6Q,
-8.333374463411801009783402800801201603736E3Q
};
#define NRD3 9
static const __float128 RD3[NRD3 + 1] =
{
-5.216713843111675050627304523368029262450E11Q,
-8.014292925418308759369583419234079164391E11Q,
-5.180106858220030014546267824392678611990E11Q,
-1.830406975497439003897734969120997840011E11Q,
-3.845274631904879621945745960119924118925E10Q,
-4.891033385370523863288908070309417710903E9Q,
-3.670172254411328640353855768698287474282E8Q,
-1.505316381525727713026364396635522516989E7Q,
-2.856327162923716881454613540575964890347E5Q,
-1.622140448015769906847567212766206894547E3Q
/* 1.0E0L */
};
/* log gamma(x+2.5) = log gamma(2.5) + x P(x)/Q(x)
-0.125 <= x <= 0.25
2.375 <= x+2.5 <= 2.75 */
static const __float128 lgam2r5a = 2.8466796875E-1Q;
static const __float128 lgam2r5b = 1.4901722919159632494669682701924320137696E-5Q;
#define NRN2r5 8
static const __float128 RN2r5[NRN2r5 + 1] =
{
-4.676454313888335499356699817678862233205E9Q,
-9.361888347911187924389905984624216340639E9Q,
-7.695353600835685037920815799526540237703E9Q,
-3.364370100981509060441853085968900734521E9Q,
-8.449902011848163568670361316804900559863E8Q,
-1.225249050950801905108001246436783022179E8Q,
-9.732972931077110161639900388121650470926E6Q,
-3.695711763932153505623248207576425983573E5Q,
-4.717341584067827676530426007495274711306E3Q
};
#define NRD2r5 8
static const __float128 RD2r5[NRD2r5 + 1] =
{
-6.650657966618993679456019224416926875619E9Q,
-1.099511409330635807899718829033488771623E10Q,
-7.482546968307837168164311101447116903148E9Q,
-2.702967190056506495988922973755870557217E9Q,
-5.570008176482922704972943389590409280950E8Q,
-6.536934032192792470926310043166993233231E7Q,
-4.101991193844953082400035444146067511725E6Q,
-1.174082735875715802334430481065526664020E5Q,
-9.932840389994157592102947657277692978511E2Q
/* 1.0E0L */
};
/* log gamma(x+2) = x P(x)/Q(x)
-0.125 <= x <= +0.375
1.875 <= x+2 <= 2.375
Peak relative error 4.6e-36 */
#define NRN2 9
static const __float128 RN2[NRN2 + 1] =
{
-3.716661929737318153526921358113793421524E9Q,
-1.138816715030710406922819131397532331321E10Q,
-1.421017419363526524544402598734013569950E10Q,
-9.510432842542519665483662502132010331451E9Q,
-3.747528562099410197957514973274474767329E9Q,
-8.923565763363912474488712255317033616626E8Q,
-1.261396653700237624185350402781338231697E8Q,
-9.918402520255661797735331317081425749014E6Q,
-3.753996255897143855113273724233104768831E5Q,
-4.778761333044147141559311805999540765612E3Q
};
#define NRD2 9
static const __float128 RD2[NRD2 + 1] =
{
-8.790916836764308497770359421351673950111E9Q,
-2.023108608053212516399197678553737477486E10Q,
-1.958067901852022239294231785363504458367E10Q,
-1.035515043621003101254252481625188704529E10Q,
-3.253884432621336737640841276619272224476E9Q,
-6.186383531162456814954947669274235815544E8Q,
-6.932557847749518463038934953605969951466E7Q,
-4.240731768287359608773351626528479703758E6Q,
-1.197343995089189188078944689846348116630E5Q,
-1.004622911670588064824904487064114090920E3Q
/* 1.0E0 */
};
/* log gamma(x+1.75) = log gamma(1.75) + x P(x)/Q(x)
-0.125 <= x <= +0.125
1.625 <= x+1.75 <= 1.875
Peak relative error 9.2e-37 */
static const __float128 lgam1r75a = -8.441162109375E-2Q;
static const __float128 lgam1r75b = 1.0500073264444042213965868602268256157604E-5Q;
#define NRN1r75 8
static const __float128 RN1r75[NRN1r75 + 1] =
{
-5.221061693929833937710891646275798251513E7Q,
-2.052466337474314812817883030472496436993E8Q,
-2.952718275974940270675670705084125640069E8Q,
-2.132294039648116684922965964126389017840E8Q,
-8.554103077186505960591321962207519908489E7Q,
-1.940250901348870867323943119132071960050E7Q,
-2.379394147112756860769336400290402208435E6Q,
-1.384060879999526222029386539622255797389E5Q,
-2.698453601378319296159355612094598695530E3Q
};
#define NRD1r75 8
static const __float128 RD1r75[NRD1r75 + 1] =
{
-2.109754689501705828789976311354395393605E8Q,
-5.036651829232895725959911504899241062286E8Q,
-4.954234699418689764943486770327295098084E8Q,
-2.589558042412676610775157783898195339410E8Q,
-7.731476117252958268044969614034776883031E7Q,
-1.316721702252481296030801191240867486965E7Q,
-1.201296501404876774861190604303728810836E6Q,
-5.007966406976106636109459072523610273928E4Q,
-6.155817990560743422008969155276229018209E2Q
/* 1.0E0L */
};
/* log gamma(x+x0) = y0 + x^2 P(x)/Q(x)
-0.0867 <= x <= +0.1634
1.374932... <= x+x0 <= 1.625032...
Peak relative error 4.0e-36 */
static const __float128 x0a = 1.4616241455078125Q;
static const __float128 x0b = 7.9994605498412626595423257213002588621246E-6Q;
static const __float128 y0a = -1.21490478515625E-1Q;
static const __float128 y0b = 4.1879797753919044854428223084178486438269E-6Q;
#define NRN1r5 8
static const __float128 RN1r5[NRN1r5 + 1] =
{
6.827103657233705798067415468881313128066E5Q,
1.910041815932269464714909706705242148108E6Q,
2.194344176925978377083808566251427771951E6Q,
1.332921400100891472195055269688876427962E6Q,
4.589080973377307211815655093824787123508E5Q,
8.900334161263456942727083580232613796141E4Q,
9.053840838306019753209127312097612455236E3Q,
4.053367147553353374151852319743594873771E2Q,
5.040631576303952022968949605613514584950E0Q
};
#define NRD1r5 8
static const __float128 RD1r5[NRD1r5 + 1] =
{
1.411036368843183477558773688484699813355E6Q,
4.378121767236251950226362443134306184849E6Q,
5.682322855631723455425929877581697918168E6Q,
3.999065731556977782435009349967042222375E6Q,
1.653651390456781293163585493620758410333E6Q,
4.067774359067489605179546964969435858311E5Q,
5.741463295366557346748361781768833633256E4Q,
4.226404539738182992856094681115746692030E3Q,
1.316980975410327975566999780608618774469E2Q,
/* 1.0E0L */
};
/* log gamma(x+1.25) = log gamma(1.25) + x P(x)/Q(x)
-.125 <= x <= +.125
1.125 <= x+1.25 <= 1.375
Peak relative error = 4.9e-36 */
static const __float128 lgam1r25a = -9.82818603515625E-2Q;
static const __float128 lgam1r25b = 1.0023929749338536146197303364159774377296E-5Q;
#define NRN1r25 9
static const __float128 RN1r25[NRN1r25 + 1] =
{
-9.054787275312026472896002240379580536760E4Q,
-8.685076892989927640126560802094680794471E4Q,
2.797898965448019916967849727279076547109E5Q,
6.175520827134342734546868356396008898299E5Q,
5.179626599589134831538516906517372619641E5Q,
2.253076616239043944538380039205558242161E5Q,
5.312653119599957228630544772499197307195E4Q,
6.434329437514083776052669599834938898255E3Q,
3.385414416983114598582554037612347549220E2Q,
4.907821957946273805080625052510832015792E0Q
};
#define NRD1r25 8
static const __float128 RD1r25[NRD1r25 + 1] =
{
3.980939377333448005389084785896660309000E5Q,
1.429634893085231519692365775184490465542E6Q,
2.145438946455476062850151428438668234336E6Q,
1.743786661358280837020848127465970357893E6Q,
8.316364251289743923178092656080441655273E5Q,
2.355732939106812496699621491135458324294E5Q,
3.822267399625696880571810137601310855419E4Q,
3.228463206479133236028576845538387620856E3Q,
1.152133170470059555646301189220117965514E2Q
/* 1.0E0L */
};
/* log gamma(x + 1) = x P(x)/Q(x)
0.0 <= x <= +0.125
1.0 <= x+1 <= 1.125
Peak relative error 1.1e-35 */
#define NRN1 8
static const __float128 RN1[NRN1 + 1] =
{
-9.987560186094800756471055681088744738818E3Q,
-2.506039379419574361949680225279376329742E4Q,
-1.386770737662176516403363873617457652991E4Q,
1.439445846078103202928677244188837130744E4Q,
2.159612048879650471489449668295139990693E4Q,
1.047439813638144485276023138173676047079E4Q,
2.250316398054332592560412486630769139961E3Q,
1.958510425467720733041971651126443864041E2Q,
4.516830313569454663374271993200291219855E0Q
};
#define NRD1 7
static const __float128 RD1[NRD1 + 1] =
{
1.730299573175751778863269333703788214547E4Q,
6.807080914851328611903744668028014678148E4Q,
1.090071629101496938655806063184092302439E5Q,
9.124354356415154289343303999616003884080E4Q,
4.262071638655772404431164427024003253954E4Q,
1.096981664067373953673982635805821283581E4Q,
1.431229503796575892151252708527595787588E3Q,
7.734110684303689320830401788262295992921E1Q
/* 1.0E0 */
};
/* log gamma(x + 1) = x P(x)/Q(x)
-0.125 <= x <= 0
0.875 <= x+1 <= 1.0
Peak relative error 7.0e-37 */
#define NRNr9 8
static const __float128 RNr9[NRNr9 + 1] =
{
4.441379198241760069548832023257571176884E5Q,
1.273072988367176540909122090089580368732E6Q,
9.732422305818501557502584486510048387724E5Q,
-5.040539994443998275271644292272870348684E5Q,
-1.208719055525609446357448132109723786736E6Q,
-7.434275365370936547146540554419058907156E5Q,
-2.075642969983377738209203358199008185741E5Q,
-2.565534860781128618589288075109372218042E4Q,
-1.032901669542994124131223797515913955938E3Q,
};
#define NRDr9 8
static const __float128 RDr9[NRDr9 + 1] =
{
-7.694488331323118759486182246005193998007E5Q,
-3.301918855321234414232308938454112213751E6Q,
-5.856830900232338906742924836032279404702E6Q,
-5.540672519616151584486240871424021377540E6Q,
-3.006530901041386626148342989181721176919E6Q,
-9.350378280513062139466966374330795935163E5Q,
-1.566179100031063346901755685375732739511E5Q,
-1.205016539620260779274902967231510804992E4Q,
-2.724583156305709733221564484006088794284E2Q
/* 1.0E0 */
};
/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
static __float128
neval (__float128 x, const __float128 *p, int n)
{
__float128 y;
p += n;
y = *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}
/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
static __float128
deval (__float128 x, const __float128 *p, int n)
{
__float128 y;
p += n;
y = x + *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}
__float128
__quadmath_lgammaq_r (__float128 x, int *signgamp)
{
__float128 p, q, w, z, nx;
int i, nn;
*signgamp = 1;
if (! finiteq (x))
return x * x;
if (x == 0)
{
if (signbitq (x))
*signgamp = -1;
}
if (x < 0)
{
if (x < -2 && x > -50)
return __quadmath_lgamma_negq (x, signgamp);
q = -x;
p = floorq (q);
if (p == q)
return (one / fabsq (p - p));
__float128 halfp = p * 0.5Q;
if (halfp == floorq (halfp))
*signgamp = -1;
else
*signgamp = 1;
if (q < 0x1p-120Q)
return -logq (q);
z = q - p;
if (z > 0.5Q)
{
p += 1;
z = p - q;
}
z = q * sinq (PIL * z);
w = __quadmath_lgammaq_r (q, &i);
z = logq (PIL / z) - w;
return (z);
}
if (x < 13.5Q)
{
p = 0;
nx = floorq (x + 0.5Q);
nn = nx;
switch (nn)
{
case 0:
/* log gamma (x + 1) = log(x) + log gamma(x) */
if (x < 0x1p-120Q)
return -logq (x);
else if (x <= 0.125)
{
p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1);
}
else if (x <= 0.375)
{
z = x - 0.25Q;
p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
p += lgam1r25b;
p += lgam1r25a;
}
else if (x <= 0.625)
{
z = x + (1 - x0a);
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x <= 0.875)
{
z = x - 0.75Q;
p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else
{
z = x - 1;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
p = p - logq (x);
break;
case 1:
if (x < 0.875Q)
{
if (x <= 0.625)
{
z = x + (1 - x0a);
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x <= 0.875)
{
z = x - 0.75Q;
p = z * neval (z, RN1r75, NRN1r75)
/ deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else
{
z = x - 1;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
p = p - logq (x);
}
else if (x < 1)
{
z = x - 1;
p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9);
}
else if (x == 1)
p = 0;
else if (x <= 1.125Q)
{
z = x - 1;
p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1);
}
else if (x <= 1.375)
{
z = x - 1.25Q;
p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
p += lgam1r25b;
p += lgam1r25a;
}
else
{
/* 1.375 <= x+x0 <= 1.625 */
z = x - x0a;
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
break;
case 2:
if (x < 1.625Q)
{
z = x - x0a;
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x < 1.875Q)
{
z = x - 1.75Q;
p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else if (x == 2)
p = 0;
else if (x < 2.375Q)
{
z = x - 2;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
else
{
z = x - 2.5Q;
p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
p += lgam2r5b;
p += lgam2r5a;
}
break;
case 3:
if (x < 2.75)
{
z = x - 2.5Q;
p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
p += lgam2r5b;
p += lgam2r5a;
}
else
{
z = x - 3;
p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3);
p += lgam3b;
p += lgam3a;
}
break;
case 4:
z = x - 4;
p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4);
p += lgam4b;
p += lgam4a;
break;
case 5:
z = x - 5;
p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5);
p += lgam5b;
p += lgam5a;
break;
case 6:
z = x - 6;
p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6);
p += lgam6b;
p += lgam6a;
break;
case 7:
z = x - 7;
p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7);
p += lgam7b;
p += lgam7a;
break;
case 8:
z = x - 8;
p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8);
p += lgam8b;
p += lgam8a;
break;
case 9:
z = x - 9;
p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9);
p += lgam9b;
p += lgam9a;
break;
case 10:
z = x - 10;
p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10);
p += lgam10b;
p += lgam10a;
break;
case 11:
z = x - 11;
p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11);
p += lgam11b;
p += lgam11a;
break;
case 12:
z = x - 12;
p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12);
p += lgam12b;
p += lgam12a;
break;
case 13:
z = x - 13;
p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13);
p += lgam13b;
p += lgam13a;
break;
}
return p;
}
if (x > MAXLGM)
return (*signgamp * huge * huge);
if (x > 0x1p120Q)
return x * (logq (x) - 1);
q = ls2pi - x;
q = (x - 0.5Q) * logq (x) + q;
if (x > 1.0e18Q)
return (q);
p = 1 / (x * x);
q += neval (p, RASY, NRASY) / x;
return (q);
}
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