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* lib/msun/Makefile b/lib/msun/Makefile:

Browse Source 
. Disconnect imprecise.c from the build.  This file can be deleted.
  . Add b_tgammal.c to the build for ld80 and ld128 targets.  The ld128
    is a 'git mv' of imprecise.c to ld128/b_tgammal.c.

* lib/msun/ld80/b_expl.c:
  . New file.  Implement __exp__D for ld80 targets.  This is based on
    bsdsrc/b_exp.c.

* lib/msun/ld80/b_logl.c:
  . New file.  Implement __log__D for ld80 targets.  This is based on
    bsdsrc/b_log.c.

* lib/msun/ld80/b_tgammal.c b/lib/msun/ld80/b_tgammal.c
  . New file.  Implement tgammal(x) for ld80 targets.

Submitted by:           Steve Kargl
Differential Revision:  https://reviews.freebsd.org/D33444
Reviewed by:            pfg
This commit is contained in:
Mark Murray 2021-12-14 09:08:57 +00:00
parent 455b2ccda3
commit 03a88e3de9
5 changed files with 908 additions and 2 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

3
lib/msun/Makefile
View File

@ -69,7 +69,6 @@ COMMON_SRCS= b_tgamma.c \
e_pow.c e_powf.c e_rem_pio2.c \
e_rem_pio2f.c e_remainder.c e_remainderf.c e_scalb.c e_scalbf.c \
e_sinh.c e_sinhf.c e_sqrt.c e_sqrtf.c fenv.c \
imprecise.c \
k_cos.c k_cosf.c k_exp.c k_expf.c k_rem_pio2.c k_sin.c k_sinf.c \
k_tan.c k_tanf.c \
s_asinh.c s_asinhf.c s_atan.c s_atanf.c s_carg.c s_cargf.c s_cargl.c \
@ -116,7 +115,7 @@ SYMBOL_MAPS= ${SYM_MAPS}
COMMON_SRCS+= s_copysignl.c s_fabsl.c s_llrintl.c s_lrintl.c s_modfl.c
.if ${LDBL_PREC} != 53
# If long double != double use these; otherwise, we alias the double versions.
COMMON_SRCS+= catrigl.c \
COMMON_SRCS+= b_tgammal.c catrigl.c \
e_acoshl.c e_acosl.c e_asinl.c e_atan2l.c e_atanhl.c \
e_coshl.c e_fmodl.c e_hypotl.c \
e_lgammal.c e_lgammal_r.c e_powl.c \

0
lib/msun/src/imprecise.c → lib/msun/ld128/b_tgammal.c
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113
lib/msun/ld80/b_expl.c Normal file
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@ -0,0 +1,113 @@
/*-
* SPDX-License-Identifier: BSD-3-Clause
*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/*
* See bsdsrc/b_exp.c for implementation details.
*
* bsdrc/b_exp.c converted to long double by Steven G. Kargl.
*/
#include "fpmath.h"
#include "math_private.h"
static const union IEEEl2bits
p0u = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-01L),
p1u = LD80C(0xb60b60b60b60b59a, -9, -2.77777777777777775377e-03L),
p2u = LD80C(0x8ab355e008a3cfce, -14, 6.61375661375629297465e-05L),
p3u = LD80C(0xddebbc994b0c1376, -20, -1.65343915327882529784e-06L),
p4u = LD80C(0xb354784cb4ef4c41, -25, 4.17535101591534118469e-08L),
p5u = LD80C(0x913e8a718382ce75, -30, -1.05679137034774806475e-09L),
p6u = LD80C(0xe8f0042aa134502e, -36, 2.64819349895429516863e-11L);
#define p1 (p0u.e)
#define p2 (p1u.e)
#define p3 (p2u.e)
#define p4 (p3u.e)
#define p5 (p4u.e)
#define p6 (p5u.e)
#define p7 (p6u.e)
/*
* lnhuge = (LDBL_MAX_EXP + 9) * log(2.)
* lntiny = (LDBL_MIN_EXP - 64 - 10) * log(2.)
* invln2 = 1 / log(2.)
*/
static const union IEEEl2bits
ln2hiu = LD80C(0xb17217f700000000, -1, 6.93147180369123816490e-01L),
ln2lou = LD80C(0xd1cf79abc9e3b398, -33, 1.90821492927058781614e-10L),
lnhugeu = LD80C(0xb18b0c0330a8fad9, 13, 1.13627617309191834574e+04L),
lntinyu = LD80C(0xb236f28a68bc3bd7, 13, -1.14057368561139000667e+04L),
invln2u = LD80C(0xb8aa3b295c17f0bc, 0, 1.44269504088896340739e+00L);
#define ln2hi (ln2hiu.e)
#define ln2lo (ln2lou.e)
#define lnhuge (lnhugeu.e)
#define lntiny (lntinyu.e)
#define invln2 (invln2u.e)
/* returns exp(r = x + c) for |c| < |x| with no overlap. */
static long double
__exp__D(long double x, long double c)
{
long double hi, lo, z;
int k;
if (x != x) /* x is NaN. */
return(x);
if (x <= lnhuge) {
if (x >= lntiny) {
/* argument reduction: x --> x - k*ln2 */
z = invln2 * x;
k = z + copysignl(0.5L, x);
/*
* Express (x + c) - k * ln2 as hi - lo.
* Let x = hi - lo rounded.
*/
hi = x - k * ln2hi; /* Exact. */
lo = k * ln2lo - c;
x = hi - lo;
/* Return 2^k*[1+x+x*c/(2+c)] */
z = x * x;
c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 +
z * (p5 + z * (p6 + z * p7))))));
c = (x * c) / (2 - c);
return (ldexpl(1 + (hi - (lo - c)), k));
} else {
/* exp(-INF) is 0. exp(-big) underflows to 0. */
return (isfinite(x) ? ldexpl(1., -5000) : 0);
}
} else
/* exp(INF) is INF, exp(+big#) overflows to INF */
return (isfinite(x) ? ldexpl(1., 5000) : x);
}

375
lib/msun/ld80/b_logl.c Normal file
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@ -0,0 +1,375 @@
/*-
* SPDX-License-Identifier: BSD-3-Clause
*
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/*
* See bsdsrc/b_log.c for implementation details.
*
* bsdrc/b_log.c converted to long double by Steven G. Kargl.
*/
#define N 128
/*
* Coefficients in the polynomial approximation of log(1+f/F).
* Domain of x is [0,1./256] with 2**(-84.48) precision.
*/
static const union IEEEl2bits
a1u = LD80C(0xaaaaaaaaaaaaaaab, -4, 8.33333333333333333356e-02L),
a2u = LD80C(0xcccccccccccccd29, -7, 1.25000000000000000781e-02L),
a3u = LD80C(0x9249249241ed3764, -9, 2.23214285711721994134e-03L),
a4u = LD80C(0xe38e959e1e7e01cf, -12, 4.34030476540000360640e-04L);
#define A1 (a1u.e)
#define A2 (a2u.e)
#define A3 (a3u.e)
#define A4 (a4u.e)
/*
* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
* Used for generation of extend precision logarithms.
* The constant 35184372088832 is 2^45, so the divide is exact.
* It ensures correct reading of logF_head, even for inaccurate
* decimal-to-binary conversion routines. (Everybody gets the
* right answer for integers less than 2^53.)
* Values for log(F) were generated using error < 10^-57 absolute
* with the bc -l package.
*/
static double logF_head[N+1] = {
0.,
.007782140442060381246,
.015504186535963526694,
.023167059281547608406,
.030771658666765233647,
.038318864302141264488,
.045809536031242714670,
.053244514518837604555,
.060624621816486978786,
.067950661908525944454,
.075223421237524235039,
.082443669210988446138,
.089612158689760690322,
.096729626458454731618,
.103796793681567578460,
.110814366340264314203,
.117783035656430001836,
.124703478501032805070,
.131576357788617315236,
.138402322859292326029,
.145182009844575077295,
.151916042025732167530,
.158605030176659056451,
.165249572895390883786,
.171850256926518341060,
.178407657472689606947,
.184922338493834104156,
.191394852999565046047,
.197825743329758552135,
.204215541428766300668,
.210564769107350002741,
.216873938300523150246,
.223143551314024080056,
.229374101064877322642,
.235566071312860003672,
.241719936886966024758,
.247836163904594286577,
.253915209980732470285,
.259957524436686071567,
.265963548496984003577,
.271933715484010463114,
.277868451003087102435,
.283768173130738432519,
.289633292582948342896,
.295464212893421063199,
.301261330578199704177,
.307025035294827830512,
.312755710004239517729,
.318453731118097493890,
.324119468654316733591,
.329753286372579168528,
.335355541920762334484,
.340926586970454081892,
.346466767346100823488,
.351976423156884266063,
.357455888922231679316,
.362905493689140712376,
.368325561158599157352,
.373716409793814818840,
.379078352934811846353,
.384411698910298582632,
.389716751140440464951,
.394993808240542421117,
.400243164127459749579,
.405465108107819105498,
.410659924985338875558,
.415827895143593195825,
.420969294644237379543,
.426084395310681429691,
.431173464818130014464,
.436236766774527495726,
.441274560805140936281,
.446287102628048160113,
.451274644139630254358,
.456237433481874177232,
.461175715122408291790,
.466089729924533457960,
.470979715219073113985,
.475845904869856894947,
.480688529345570714212,
.485507815781602403149,
.490303988045525329653,
.495077266798034543171,
.499827869556611403822,
.504556010751912253908,
.509261901790523552335,
.513945751101346104405,
.518607764208354637958,
.523248143765158602036,
.527867089620485785417,
.532464798869114019908,
.537041465897345915436,
.541597282432121573947,
.546132437597407260909,
.550647117952394182793,
.555141507540611200965,
.559615787935399566777,
.564070138285387656651,
.568504735352689749561,
.572919753562018740922,
.577315365035246941260,
.581691739635061821900,
.586049045003164792433,
.590387446602107957005,
.594707107746216934174,
.599008189645246602594,
.603290851438941899687,
.607555250224322662688,
.611801541106615331955,
.616029877215623855590,
.620240409751204424537,
.624433288012369303032,
.628608659422752680256,
.632766669570628437213,
.636907462236194987781,
.641031179420679109171,
.645137961373620782978,
.649227946625615004450,
.653301272011958644725,
.657358072709030238911,
.661398482245203922502,
.665422632544505177065,
.669430653942981734871,
.673422675212350441142,
.677398823590920073911,
.681359224807238206267,
.685304003098281100392,
.689233281238557538017,
.693147180560117703862
};
static double logF_tail[N+1] = {
0.,
-.00000000000000543229938420049,
.00000000000000172745674997061,
-.00000000000001323017818229233,
-.00000000000001154527628289872,
-.00000000000000466529469958300,
.00000000000005148849572685810,
-.00000000000002532168943117445,
-.00000000000005213620639136504,
-.00000000000001819506003016881,
.00000000000006329065958724544,
.00000000000008614512936087814,
-.00000000000007355770219435028,
.00000000000009638067658552277,
.00000000000007598636597194141,
.00000000000002579999128306990,
-.00000000000004654729747598444,
-.00000000000007556920687451336,
.00000000000010195735223708472,
-.00000000000017319034406422306,
-.00000000000007718001336828098,
.00000000000010980754099855238,
-.00000000000002047235780046195,
-.00000000000008372091099235912,
.00000000000014088127937111135,
.00000000000012869017157588257,
.00000000000017788850778198106,
.00000000000006440856150696891,
.00000000000016132822667240822,
-.00000000000007540916511956188,
-.00000000000000036507188831790,
.00000000000009120937249914984,
.00000000000018567570959796010,
-.00000000000003149265065191483,
-.00000000000009309459495196889,
.00000000000017914338601329117,
-.00000000000001302979717330866,
.00000000000023097385217586939,
.00000000000023999540484211737,
.00000000000015393776174455408,
-.00000000000036870428315837678,
.00000000000036920375082080089,
-.00000000000009383417223663699,
.00000000000009433398189512690,
.00000000000041481318704258568,
-.00000000000003792316480209314,
.00000000000008403156304792424,
-.00000000000034262934348285429,
.00000000000043712191957429145,
-.00000000000010475750058776541,
-.00000000000011118671389559323,
.00000000000037549577257259853,
.00000000000013912841212197565,
.00000000000010775743037572640,
.00000000000029391859187648000,
-.00000000000042790509060060774,
.00000000000022774076114039555,
.00000000000010849569622967912,
-.00000000000023073801945705758,
.00000000000015761203773969435,
.00000000000003345710269544082,
-.00000000000041525158063436123,
.00000000000032655698896907146,
-.00000000000044704265010452446,
.00000000000034527647952039772,
-.00000000000007048962392109746,
.00000000000011776978751369214,
-.00000000000010774341461609578,
.00000000000021863343293215910,
.00000000000024132639491333131,
.00000000000039057462209830700,
-.00000000000026570679203560751,
.00000000000037135141919592021,
-.00000000000017166921336082431,
-.00000000000028658285157914353,
-.00000000000023812542263446809,
.00000000000006576659768580062,
-.00000000000028210143846181267,
.00000000000010701931762114254,
.00000000000018119346366441110,
.00000000000009840465278232627,
-.00000000000033149150282752542,
-.00000000000018302857356041668,
-.00000000000016207400156744949,
.00000000000048303314949553201,
-.00000000000071560553172382115,
.00000000000088821239518571855,
-.00000000000030900580513238244,
-.00000000000061076551972851496,
.00000000000035659969663347830,
.00000000000035782396591276383,
-.00000000000046226087001544578,
.00000000000062279762917225156,
.00000000000072838947272065741,
.00000000000026809646615211673,
-.00000000000010960825046059278,
.00000000000002311949383800537,
-.00000000000058469058005299247,
-.00000000000002103748251144494,
-.00000000000023323182945587408,
-.00000000000042333694288141916,
-.00000000000043933937969737844,
.00000000000041341647073835565,
.00000000000006841763641591466,
.00000000000047585534004430641,
.00000000000083679678674757695,
-.00000000000085763734646658640,
.00000000000021913281229340092,
-.00000000000062242842536431148,
-.00000000000010983594325438430,
.00000000000065310431377633651,
-.00000000000047580199021710769,
-.00000000000037854251265457040,
.00000000000040939233218678664,
.00000000000087424383914858291,
.00000000000025218188456842882,
-.00000000000003608131360422557,
-.00000000000050518555924280902,
.00000000000078699403323355317,
-.00000000000067020876961949060,
.00000000000016108575753932458,
.00000000000058527188436251509,
-.00000000000035246757297904791,
-.00000000000018372084495629058,
.00000000000088606689813494916,
.00000000000066486268071468700,
.00000000000063831615170646519,
.00000000000025144230728376072,
-.00000000000017239444525614834
};
/*
* Extra precision variant, returning struct {double a, b;};
* log(x) = a + b to 63 bits, with 'a' rounded to 24 bits.
*/
static struct Double
__log__D(long double x)
{
int m, j;
long double F, f, g, q, u, v, u1, u2;
struct Double r;
/*
* Argument reduction: 1 <= g < 2; x/2^m = g;
* y = F*(1 + f/F) for |f| <= 2^-8
*/
g = frexpl(x, &m);
g *= 2;
m--;
if (m == DBL_MIN_EXP - 1) {
j = ilogbl(g);
m += j;
g = ldexpl(g, -j);
}
j = N * (g - 1) + 0.5L;
F = (1.L / N) * j + 1;
f = g - F;
g = 1 / (2 * F + f);
u = 2 * f * g;
v = u * u;
q = u * v * (A1 + v * (A2 + v * (A3 + v * A4)));
if (m | j) {
u1 = u + 513;
u1 -= 513;
} else {
u1 = (float)u;
}
u2 = (2 * (f - F * u1) - u1 * f) * g;
u1 += m * (long double)logF_head[N] + logF_head[j];
u2 += logF_tail[j];
u2 += q;
u2 += logF_tail[N] * m;
r.a = (float)(u1 + u2); /* Only difference is here. */
r.b = (u1 - r.a) + u2;
return (r);
}

419
lib/msun/ld80/b_tgammal.c Normal file
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@ -0,0 +1,419 @@
/*-
* SPDX-License-Identifier: BSD-3-Clause
*
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/*
* The original code, FreeBSD's old svn r93211, contain the following
* attribution:
*
* This code by P. McIlroy, Oct 1992;
*
* The financial support of UUNET Communications Services is greatfully
* acknowledged.
*
* bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
*/
/*
* See bsdsrc/t_tgamma.c for implementation details.
*/
#include <float.h>
#if LDBL_MAX_EXP != 0x4000
#error "Unsupported long double format"
#endif
#ifdef __i386__
#include <ieeefp.h>
#endif
#include "fpmath.h"
#include "math.h"
#include "math_private.h"
/* Used in b_log.c and below. */
struct Double {
long double a;
long double b;
};
#include "b_logl.c"
#include "b_expl.c"
static const double zero = 0.;
static const volatile double tiny = 1e-300;
/*
* x >= 6
*
* Use the asymptotic approximation (Stirling's formula) adjusted for
* equal-ripples:
*
* log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
*
* Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
* premature round-off.
*
* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
*/
/*
* The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
* first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
* variables are clearly misnamed.
*/
static const union IEEEl2bits
ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
#define ln2pi_hi (ln2pi_hiu.e)
#define ln2pi_lo (ln2pi_lou.e)
static const union IEEEl2bits
Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
#define Pa0 (Pa0u.e)
#define Pa1 (Pa1u.e)
#define Pa2 (Pa2u.e)
#define Pa3 (Pa3u.e)
#define Pa4 (Pa4u.e)
#define Pa5 (Pa5u.e)
#define Pa6 (Pa6u.e)
#define Pa7 (Pa7u.e)
#define Pa8 (Pa8u.e)
#define Pa9 (Pa9u.e)
static struct Double
large_gam(long double x)
{
long double p, z, thi, tlo, xhi, xlo;
long double logx;
struct Double u;
z = 1 / (x * x);
p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
p = p / x;
u = __log__D(x);
u.a -= 1;
/* Split (x - 0.5) in high and low parts. */
x -= 0.5L;
xhi = (float)x;
xlo = x - xhi;
/* Compute t = (x-.5)*(log(x)-1) in extra precision. */
thi = xhi * u.a;
tlo = xlo * u.a + x * u.b;
/* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
tlo += ln2pi_lo;
tlo += p;
u.a = ln2pi_hi + tlo;
u.a += thi;
u.b = thi - u.a;
u.b += ln2pi_hi;
u.b += tlo;
return (u);
}
/*
* Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
* [1.066.., 2.066..] accurate to 4.25e-19.
*
* Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
*/
static const union IEEEl2bits
a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
#define a0_hi (a0_hiu.e)
#define a0_lo (a0_lou.e)
static const union IEEEl2bits
P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
#define P0 (P0u.e)
#define P1 (P1u.e)
#define P2 (P2u.e)
#define P3 (P3u.e)
#define P4 (P4u.e)
#define P5 (P5u.e)
#define P6 (P6u.e)
#define P7 (P7u.e)
#define P8 (P8u.e)
#define Q1 (Q1u.e)
#define Q2 (Q2u.e)
#define Q3 (Q3u.e)
#define Q4 (Q4u.e)
#define Q5 (Q5u.e)
#define Q6 (Q6u.e)
#define Q7 (Q7u.e)
#define Q8 (Q8u.e)
static struct Double
ratfun_gam(long double z, long double c)
{
long double p, q, thi, tlo;
struct Double r;
q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
z * (Q6 + z * (Q7 + z * Q8)))))));
p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
z * (P6 + z * (P7 + z * P8)))))));
p = p / q;
/* Split z into high and low parts. */
thi = (float)z;
tlo = (z - thi) + c;
tlo *= (thi + z);
/* Split (z+c)^2 into high and low parts. */
thi *= thi;
q = thi;
thi = (float)thi;
tlo += (q - thi);
/* Split p/q into high and low parts. */
r.a = (float)p;
r.b = p - r.a;
tlo = tlo * p + thi * r.b + a0_lo;
thi *= r.a; /* t = (z+c)^2*(P/Q) */
r.a = (float)(thi + a0_hi);
r.b = ((a0_hi - r.a) + thi) + tlo;
return (r); /* r = a0 + t */
}
/*
* x < 6
*
* Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
* 2.066124]. Use a rational approximation centered at the minimum
* (x0+1) to ensure monotonicity.
*
* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
* It also has correct monotonicity.
*/
static const union IEEEl2bits
xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
#define x0 (xm1u.e)
static const double
left = -0.3955078125; /* left boundary for rat. approx */
static long double
small_gam(long double x)
{
long double t, y, ym1;
struct Double yy, r;
y = x - 1;
if (y <= 1 + (left + x0)) {
yy = ratfun_gam(y - x0, 0);
return (yy.a + yy.b);
}
r.a = (float)y;
yy.a = r.a - 1;
y = y - 1 ;
r.b = yy.b = y - yy.a;
/* Argument reduction: G(x+1) = x*G(x) */
for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
t = r.a * yy.a;
r.b = r.a * yy.b + y * r.b;
r.a = (float)t;
r.b += (t - r.a);
}
/* Return r*tgamma(y). */
yy = ratfun_gam(y - x0, 0);
y = r.b * (yy.a + yy.b) + r.a * yy.b;
y += yy.a * r.a;
return (y);
}
/*
* Good on (0, 1+x0+left]. Accurate to 1 ulp.
*/
static long double
smaller_gam(long double x)
{
long double d, rhi, rlo, t, xhi, xlo;
struct Double r;
if (x < x0 + left) {
t = (float)x;
d = (t + x) * (x - t);
t *= t;
xhi = (float)(t + x);
xlo = x - xhi;
xlo += t;
xlo += d;
t = 1 - x0;
t += x;
d = 1 - x0;
d -= t;
d += x;
x = xhi + xlo;
} else {
xhi = (float)x;
xlo = x - xhi;
t = x - x0;
d = - x0 - t;
d += x;
}
r = ratfun_gam(t, d);
d = (float)(r.a / x);
r.a -= d * xhi;
r.a -= d * xlo;
r.a += r.b;
return (d + r.a / x);
}
/*
* x < 0
*
* Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
* At negative integers, return NaN and raise invalid.
*/
static const union IEEEl2bits
piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
#define pi (piu.e)
static long double
neg_gam(long double x)
{
int sgn = 1;
struct Double lg, lsine;
long double y, z;
y = ceill(x);
if (y == x) /* Negative integer. */
return ((x - x) / zero);
z = y - x;
if (z > 0.5)
z = 1 - z;
y = y / 2;
if (y == ceill(y))
sgn = -1;
if (z < 0.25)
z = sinpil(z);
else
z = cospil(0.5 - z);
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
if (x < -1753) {
if (x < -1760)
return (sgn * tiny * tiny);
y = expl(lgammal(x) / 2);
y *= y;
return (sgn < 0 ? -y : y);
}
y = 1 - x;
if (1 - y == x)
y = tgammal(y);
else /* 1-x is inexact */
y = - x * tgammal(-x);
if (sgn < 0) y = -y;
return (pi / (y * z));
}
/*
* xmax comes from lgamma(xmax) - emax * log(2) = 0.
* static const float xmax = 35.040095f
* static const double xmax = 171.624376956302725;
* ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
* ld128: 1.75554834290446291700388921607020320e+03L,
*
* iota is a sloppy threshold to isolate x = 0.
*/
static const double xmax = 1755.54834290446291689;
static const double iota = 0x1p-116;
long double
tgammal(long double x)
{
struct Double u;
ENTERI();
if (x >= 6) {
if (x > xmax)
RETURNI(x / zero);
u = large_gam(x);
RETURNI(__exp__D(u.a, u.b));
}
if (x >= 1 + left + x0)
RETURNI(small_gam(x));
if (x > iota)
RETURNI(smaller_gam(x));
if (x > -iota) {
if (x != 0)
u.a = 1 - tiny; /* raise inexact */
RETURNI(1 / x);
}
if (!isfinite(x))
RETURNI(x - x); /* x is NaN or -Inf */
RETURNI(neg_gam(x));
}