/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __tgammal = tgammal
#include "libm.h"
#include <sys/isa_defs.h>
#if defined(_BIG_ENDIAN)
#define H0_WORD(x) ((unsigned *) &x)[0]
#define H3_WORD(x) ((unsigned *) &x)[3]
#define CHOPPED(x) (long double) ((double) (x))
#else
#define H0_WORD(x) ((((int *) &x)[2] << 16) | \
(0x0000ffff & (((unsigned *) &x)[1] >> 15)))
#define H3_WORD(x) ((unsigned *) &x)[0]
#define CHOPPED(x) (long double) ((float) (x))
#endif
struct LDouble {
long double h, l;
};
/* INDENT OFF */
/* Primary interval GTi() */
static const long double P1[] = {
+0.709086836199777919037185741507610124611513720557L,
+4.45754781206489035827915969367354835667391606951e-0001L,
+3.21049298735832382311662273882632210062918153852e-0002L,
-5.71296796342106617651765245858289197369688864350e-0003L,
+6.04666892891998977081619174969855831606965352773e-0003L,
+8.99106186996888711939627812174765258822658645168e-0004L,
-6.96496846144407741431207008527018441810175568949e-0005L,
+1.52597046118984020814225409300131445070213882429e-0005L,
+5.68521076168495673844711465407432189190681541547e-0007L,
+3.30749673519634895220582062520286565610418952979e-0008L,
};
static const long double Q1[] = {
+1.0+0000L,
+1.35806511721671070408570853537257079579490650668e+0000L,
+2.97567810153429553405327140096063086994072952961e-0001L,
-1.52956835982588571502954372821681851681118097870e-0001L,
-2.88248519561420109768781615289082053597954521218e-0002L,
+1.03475311719937405219789948456313936302378395955e-0002L,
+4.12310203243891222368965360124391297374822742313e-0004L,
-3.12653708152290867248931925120380729518332507388e-0004L,
+2.36672170850409745237358105667757760527014332458e-0005L,
};
static const long double P2[] = {
+0.428486815855585429730209907810650135255270600668084114L,
+2.62768479103809762805691743305424077975230551176e-0001L,
+3.81187532685392297608310837995193946591425896150e-0002L,
+3.00063075891811043820666846129131255948527925381e-0003L,
+2.47315407812279164228398470797498649142513408654e-0003L,
+3.62838199917848372586173483147214880464782938664e-0004L,
+3.43991105975492623982725644046473030098172692423e-0006L,
+4.56902151569603272237014240794257659159045432895e-0006L,
+2.13734755837595695602045100675540011352948958453e-0007L,
+9.74123440547918230781670266967882492234877125358e-0009L,
};
static const long double Q2[] = {
+1.0L,
+9.18284118632506842664645516830761489700556179701e-0001L,
-6.41430858837830766045202076965923776189154874947e-0003L,
-1.24400885809771073213345747437964149775410921376e-0001L,
+4.69803798146251757538856567522481979624746875964e-0003L,
+7.18309447069495315914284705109868696262662082731e-0003L,
-8.75812626987894695112722600697653425786166399105e-0004L,
-1.23539972377769277995959339188431498626674835169e-0004L,
+3.10019017590151598732360097849672925448587547746e-0005L,
-1.77260223349332617658921874288026777465782364070e-0006L,
};
static const long double P3[] = {
+0.3824094797345675048502747661075355640070439388902L,
+3.42198093076618495415854906335908427159833377774e-0001L,
+9.63828189500585568303961406863153237440702754858e-0002L,
+8.76069421042696384852462044188520252156846768667e-0003L,
+1.86477890389161491224872014149309015261897537488e-0003L,
+8.16871354540309895879974742853701311541286944191e-0004L,
+6.83783483674600322518695090864659381650125625216e-0005L,
-1.10168269719261574708565935172719209272190828456e-0006L,
+9.66243228508380420159234853278906717065629721016e-0007L,
+2.31858885579177250541163820671121664974334728142e-0008L,
};
static const long double Q3[] = {
+1.0L,
+8.25479821168813634632437430090376252512793067339e-0001L,
-1.62251363073937769739639623669295110346015576320e-0002L,
-1.10621286905916732758745130629426559691187579852e-0001L,
+3.48309693970985612644446415789230015515365291459e-0003L,
+6.73553737487488333032431261131289672347043401328e-0003L,
-7.63222008393372630162743587811004613050245128051e-0004L,
-1.35792670669190631476784768961953711773073251336e-0004L,
+3.19610150954223587006220730065608156460205690618e-0005L,
-1.82096553862822346610109522015129585693354348322e-0006L,
};
static const long double
#if defined(__x86)
GZ1_h = 0.938204627909682449364570100414084663498215377L,
GZ1_l = 4.518346116624229420055327632718530617227944106e-20L,
GZ2_h = 0.885603194410888700264725126309883762587560340L,
GZ2_l = 1.409077427270497062039119290776508217077297169e-20L,
GZ3_h = 0.936781411463652321613537060640553022494714241L,
GZ3_l = 5.309836440284827247897772963887219035221996813e-21L,
#else
GZ1_h = 0.938204627909682449409753561580326910854647031L,
GZ1_l = 4.684412162199460089642452580902345976446297037e-35L,
GZ2_h = 0.885603194410888700278815900582588658192658794L,
GZ2_l = 7.501529273890253789219935569758713534641074860e-35L,
GZ3_h = 0.936781411463652321618846897080837818855399840L,
GZ3_l = 3.088721217404784363585591914529361687403776917e-35L,
#endif
TZ1 = -0.3517214357852935791015625L,
TZ3 = 0.280530631542205810546875L;
/* INDENT ON */
/* INDENT OFF */
/*
* compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845]
* ...assume yh got 53 or 24(i386) significant bits
*/
/* INDENT ON */
static struct LDouble
GT1(long double yh, long double yl) {
long double t3, t4, y;
int i;
struct LDouble r;
y = yh + yl;
for (t4 = Q1[8], t3 = P1[8] + y * P1[9], i = 7; i >= 0; i--) {
t4 = t4 * y + Q1[i];
t3 = t3 * y + P1[i];
}
t3 = (y * y) * t3 / t4;
t3 += (TZ1 * yl + GZ1_l);
t4 = TZ1 * yh;
r.h = CHOPPED((t4 + GZ1_h + t3));
t3 += (t4 - (r.h - GZ1_h));
r.l = t3;
return (r);
}
/* INDENT OFF */
/*
* compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374]
* ...assume yh got 53 significant bits
*/
/* INDENT ON */
static struct LDouble
GT2(long double yh, long double yl) {
long double t3, t4, y;
int i;
struct LDouble r;
y = yh + yl;
for (t4 = Q2[9], t3 = P2[9], i = 8; i >= 0; i--) {
t4 = t4 * y + Q2[i];
t3 = t3 * y + P2[i];
}
t3 = GZ2_l + (y * y) * t3 / t4;
r.h = CHOPPED((GZ2_h + t3));
r.l = t3 - (r.h - GZ2_h);
return (r);
}
/* INDENT OFF */
/*
* compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000]
* ...assume yh got 53 significant bits
*/
/* INDENT ON */
static struct LDouble
GT3(long double yh, long double yl) {
long double t3, t4, y;
int i;
struct LDouble r;
y = yh + yl;
for (t4 = Q3[9], t3 = P3[9], i = 8; i >= 0; i--) {
t4 = t4 * y + Q3[i];
t3 = t3 * y + P3[i];
}
t3 = (y * y) * t3 / t4;
t3 += (TZ3 * yl + GZ3_l);
t4 = TZ3 * yh;
r.h = CHOPPED((t4 + GZ3_h + t3));
t3 += (t4 - (r.h - GZ3_h));
r.l = t3;
return (r);
}
/* INDENT OFF */
/* Hex value of GP[0] shoule be 3FB55555 55555555 */
static const long double GP[] = {
+0.083333333333333333333333333333333172839171301L,
-2.77777777777777777777777777492501211999399424104e-0003L,
+7.93650793650793650793635650541638236350020883243e-0004L,
-5.95238095238095238057299772679324503339241961704e-0004L,
+8.41750841750841696138422987977683524926142600321e-0004L,
-1.91752691752686682825032547823699662178842123308e-0003L,
+6.41025641022403480921891559356473451161279359322e-0003L,
-2.95506535798414019189819587455577003732808185071e-0002L,
+1.79644367229970031486079180060923073476568732136e-0001L,
-1.39243086487274662174562872567057200255649290646e+0000L,
+1.34025874044417962188677816477842265259608269775e+0001L,
-1.56803713480127469414495545399982508700748274318e+0002L,
+2.18739841656201561694927630335099313968924493891e+0003L,
-3.55249848644100338419187038090925410976237921269e+0004L,
+6.43464880437835286216768959439484376449179576452e+0005L,
-1.20459154385577014992600342782821389605893904624e+0007L,
+2.09263249637351298563934942349749718491071093210e+0008L,
-2.96247483183169219343745316433899599834685703457e+0009L,
+2.88984933605896033154727626086506756972327292981e+0010L,
-1.40960434146030007732838382416230610302678063984e+0011L, /* 19 */
};
static const long double T3[] = {
+0.666666666666666666666666666666666634567834260213L, /* T3[0] */
+0.400000000000000000000000000040853636176634934140L, /* T3[1] */
+0.285714285714285714285696975252753987869020263448L, /* T3[2] */
+0.222222222222222225593221101192317258554772129875L, /* T3[3] */
+0.181818181817850192105847183461778186703779262916L, /* T3[4] */
+0.153846169861348633757101285952333369222567014596L, /* T3[5] */
+0.133033462889260193922261296772841229985047571265L, /* T3[6] */
};
static const long double c[] = {
0.0L,
1.0L,
2.0L,
0.5L,
1.0e-4930L, /* tiny */
4.18937683105468750000e-01L, /* hln2pim1_h */
8.50099203991780329736405617639861397473637783412817152e-07L, /* hln2pim1_l */
0.418938533204672741780329736405617639861397473637783412817152L, /* hln2pim1 */
2.16608493865351192653179168701171875e-02L, /* ln2_32hi */
5.96317165397058692545083025235937919875797669127130e-12L, /* ln2_32lo */
46.16624130844682903551758979206054839765267053289554989233L, /* invln2_32 */
#if defined(__x86)
1.7555483429044629170023839037639845628291e+03L, /* overflow */
#else
1.7555483429044629170038892160702032034177e+03L, /* overflow */
#endif
};
#define zero c[0]
#define one c[1]
#define two c[2]
#define half c[3]
#define tiny c[4]
#define hln2pim1_h c[5]
#define hln2pim1_l c[6]
#define hln2pim1 c[7]
#define ln2_32hi c[8]
#define ln2_32lo c[9]
#define invln2_32 c[10]
#define overflow c[11]
/*
* |exp(r) - (1+r+Et0*r^2+...+Et10*r^12)| <= 2^(-128.88) for |r|<=ln2/64
*/
static const long double Et[] = {
+5.0000000000000000000e-1L,
+1.66666666666666666666666666666828835166292152466e-0001L,
+4.16666666666666666666666666666693398646592712189e-0002L,
+8.33333333333333333333331748774512601775591115951e-0003L,
+1.38888888888888888888888845356011511394764753997e-0003L,
+1.98412698412698413237140350092993252684198882102e-0004L,
+2.48015873015873016080222025357442659895814371694e-0005L,
+2.75573192239028921114572986441972140933432317798e-0006L,
+2.75573192239448470555548102895526369739856219317e-0007L,
+2.50521677867683935940853997995937600214167232477e-0008L,
+2.08767928899010367374984448513685566514152147362e-0009L,
};
/*
* long double precision coefficients for computing log(x)-1 in tgamma.
* See "algorithm" for details
*
* log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
* j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
* T1(n) = T1[2n,2n+1] = n*log(2)-1,
* T2(j) = T2[2j,2j+1] = log(z[j]),
* T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7 + ... + T3[6]s^15
* Note
* (1) the leading entries are truncated to 24 binary point.
* (2) Remez error for T3(s) is bounded by 2**(-136.54)
*/
static const long double T1[] = {
-1.000000000000000000000000000000000000000000e+00L,
+0.000000000000000000000000000000000000000000e+00L,
-3.068528175354003906250000000000000000000000e-01L,
-1.904654299957767878541823431924500011926579e-09L,
+3.862943053245544433593750000000000000000000e-01L,
+5.579533617547508924291635313615100141107647e-08L,
+1.079441487789154052734375000000000000000000e+00L,
+5.389068187551732136437452970422650211661470e-08L,
+1.772588670253753662109375000000000000000000e+00L,
+5.198602757555955348583270627230200282215294e-08L,
+2.465735852718353271484375000000000000000000e+00L,
+5.008137327560178560729088284037750352769117e-08L,
+3.158883035182952880859375000000000000000000e+00L,
+4.817671897564401772874905940845299849351090e-08L,
+3.852030217647552490234375000000000000000000e+00L,
+4.627206467568624985020723597652849919904913e-08L,
+4.545177400112152099609375000000000000000000e+00L,
+4.436741037572848197166541254460399990458737e-08L,
+5.238324582576751708984375000000000000000000e+00L,
+4.246275607577071409312358911267950061012560e-08L,
+5.931471765041351318359375000000000000000000e+00L,
+4.055810177581294621458176568075500131566384e-08L,
};
/*
* T2[2i,2i+1] = log(1+i/64+1/128)
*/
static const long double T2[] = {
+7.7821016311645507812500000000000000000000e-03L,
+3.8810890398166212900061136763678127453570e-08L,
+2.3167014122009277343750000000000000000000e-02L,
+4.5159525100885049160962289916579411752759e-08L,
+3.8318812847137451171875000000000000000000e-02L,
+5.1454999148021880325123797290345960518164e-08L,
+5.3244471549987792968750000000000000000000e-02L,
+4.2968824489897120193786528776939573415076e-08L,
+6.7950606346130371093750000000000000000000e-02L,
+5.5562377378300815277772629414034632394030e-08L,
+8.2443654537200927734375000000000000000000e-02L,
+1.4673873663533785068668307805914095366600e-08L,
+9.6729576587677001953125000000000000000000e-02L,
+4.9870874110342446056487463437015041543346e-08L,
+1.1081433296203613281250000000000000000000e-01L,
+3.3378253981382306169323211928098474801099e-08L,
+1.2470346689224243164062500000000000000000e-01L,
+1.1608714804222781515380863268491613205318e-08L,
+1.3840228319168090820312500000000000000000e-01L,
+3.9667438227482200873601649187393160823607e-08L,
+1.5191602706909179687500000000000000000000e-01L,
+1.4956750178196803424896884511327584958252e-08L,
+1.6524952650070190429687500000000000000000e-01L,
+4.6394605258578736449277240313729237989366e-08L,
+1.7840760946273803710937500000000000000000e-01L,
+4.8010080260010025241510941968354682199540e-08L,
+1.9139480590820312500000000000000000000000e-01L,
+4.7091426329609298807561308873447039132856e-08L,
+2.0421552658081054687500000000000000000000e-01L,
+1.4847880344628820386196239272213742113867e-08L,
+2.1687388420104980468750000000000000000000e-01L,
+5.4099564554931589525744347498478964801484e-08L,
+2.2937405109405517578125000000000000000000e-01L,
+4.9970790654210230725046139871550961365282e-08L,
+2.4171990156173706054687500000000000000000e-01L,
+3.5325408107597432515913513900103385655073e-08L,
+2.5391519069671630859375000000000000000000e-01L,
+1.9284247135543573297906606667466299224747e-08L,
+2.6596349477767944335937500000000000000000e-01L,
+5.3719458497979750926537543389268821141517e-08L,
+2.7786844968795776367187500000000000000000e-01L,
+1.3154985425144750329234012330820349974537e-09L,
+2.8963327407836914062500000000000000000000e-01L,
+1.8504673536253893055525668970003860369760e-08L,
+3.0126130580902099609375000000000000000000e-01L,
+2.4769140784919125538233755492657352680723e-08L,
+3.1275570392608642578125000000000000000000e-01L,
+6.0778104626049965596883190321597861455475e-09L,
+3.2411944866180419921875000000000000000000e-01L,
+1.9992407776871920760434987352182336158873e-08L,
+3.3535552024841308593750000000000000000000e-01L,
+2.1672724744319679579814166199074433006807e-08L,
+3.4646672010421752929687500000000000000000e-01L,
+4.7241991051621587188425772950711830538414e-08L,
+3.5745584964752197265625000000000000000000e-01L,
+3.9274281801569759490140904474434669956562e-08L,
+3.6832553148269653320312500000000000000000e-01L,
+2.9676011119845105154050398826897178765758e-08L,
+3.7907832860946655273437500000000000000000e-01L,
+2.4325502905656478345631019858881408009210e-08L,
+3.8971674442291259765625000000000000000000e-01L,
+6.7171126157142136040035208670510556529487e-09L,
+4.0024316310882568359375000000000000000000e-01L,
+1.0181870233355751019951311700799406124957e-09L,
+4.1065990924835205078125000000000000000000e-01L,
+1.5736916335153056203175822787661567534220e-08L,
+4.2096924781799316406250000000000000000000e-01L,
+4.6826136472066367161506795972449857268707e-08L,
+4.3117344379425048828125000000000000000000e-01L,
+2.1024120852577922478955594998480144051225e-08L,
+4.4127452373504638671875000000000000000000e-01L,
+3.7069828842770746441661301225362605528786e-08L,
+4.5127463340759277343750000000000000000000e-01L,
+1.0731865811707192383079012478685922879010e-08L,
+4.6117568016052246093750000000000000000000e-01L,
+3.4961647705430499925597855358603099030515e-08L,
+4.7097969055175781250000000000000000000000e-01L,
+2.4667033200046897856056359251373510964634e-08L,
+4.8068851232528686523437500000000000000000e-01L,
+1.7020465042442243455448011551208861216878e-08L,
+4.9030393362045288085937500000000000000000e-01L,
+5.4424740957290971159645746860530583309571e-08L,
+4.9982786178588867187500000000000000000000e-01L,
+7.7705606579463314152470441415126573566105e-09L,
+5.0926184654235839843750000000000000000000e-01L,
+5.5247449548366574919228323824878565745713e-08L,
+5.1860773563385009765625000000000000000000e-01L,
+2.8574195534496726996364798698556235730848e-08L,
+5.2786707878112792968750000000000000000000e-01L,
+1.0839714455426392217778300963558522088193e-08L,
+5.3704142570495605468750000000000000000000e-01L,
+4.0191927599879229244153832299023744345999e-08L,
+5.4613238573074340820312500000000000000000e-01L,
+5.1867392242179272209231209163864971792889e-08L,
+5.5514144897460937500000000000000000000000e-01L,
+5.8565892217715480359515904050170125743178e-08L,
+5.6407010555267333984375000000000000000000e-01L,
+3.2732129626227634290090190711817681692354e-08L,
+5.7291972637176513671875000000000000000000e-01L,
+2.7190020372374006726626261068626400393936e-08L,
+5.8169168233871459960937500000000000000000e-01L,
+5.7295907882911235753725372340709967597394e-08L,
+5.9038740396499633789062500000000000000000e-01L,
+4.2637180036751291708123598757577783615014e-08L,
+5.9900814294815063476562500000000000000000e-01L,
+4.6697932764615975024461651502060474048774e-08L,
+6.0755521059036254882812500000000000000000e-01L,
+3.9634179246672960152791125371893149820625e-08L,
+6.1602985858917236328125000000000000000000e-01L,
+1.8626341656366315928196700650292529688219e-08L,
+6.2443327903747558593750000000000000000000e-01L,
+8.9744179151050387440546731199093039879228e-09L,
+6.3276666402816772460937500000000000000000e-01L,
+5.5428701049364114685035797584887586099726e-09L,
+6.4103114604949951171875000000000000000000e-01L,
+3.3371431779336851334405392546708949047361e-08L,
+6.4922791719436645507812500000000000000000e-01L,
+2.9430743363812714969905311122271269100885e-08L,
+6.5735805034637451171875000000000000000000e-01L,
+2.2361985518423140023245936165514147093250e-08L,
+6.6542261838912963867187500000000000000000e-01L,
+1.4155960810278217610006660181148303091649e-08L,
+6.7342263460159301757812500000000000000000e-01L,
+4.0610573702719835388801017264750843477878e-08L,
+6.8135917186737060546875000000000000000000e-01L,
+5.2940532463479321559568089441735584156689e-08L,
+6.8923324346542358398437500000000000000000e-01L,
+3.7773385396340539337814603903232796216537e-08L,
};
/*
* S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w)
*/
static const long double S[] = {
#if defined(__x86)
+1.0000000000000000000000000e+00L,
+1.0218971486541166782081522e+00L,
+1.0442737824274138402382006e+00L,
+1.0671404006768236181297224e+00L,
+1.0905077326652576591003302e+00L,
+1.1143867425958925362894369e+00L,
+1.1387886347566916536971221e+00L,
+1.1637248587775775137938619e+00L,
+1.1892071150027210666875674e+00L,
+1.2152473599804688780476325e+00L,
+1.2418578120734840485256747e+00L,
+1.2690509571917332224885722e+00L,
+1.2968395546510096659215822e+00L,
+1.3252366431597412945939118e+00L,
+1.3542555469368927282668852e+00L,
+1.3839098819638319548151403e+00L,
+1.4142135623730950487637881e+00L,
+1.4451808069770466200253470e+00L,
+1.4768261459394993113155431e+00L,
+1.5091644275934227397133885e+00L,
+1.5422108254079408235859630e+00L,
+1.5759808451078864864006862e+00L,
+1.6104903319492543080837174e+00L,
+1.6457554781539648445110730e+00L,
+1.6817928305074290860378350e+00L,
+1.7186192981224779156032914e+00L,
+1.7562521603732994831094730e+00L,
+1.7947090750031071864148413e+00L,
+1.8340080864093424633989166e+00L,
+1.8741676341102999013002103e+00L,
+1.9152065613971472938202589e+00L,
+1.9571441241754002689657438e+00L,
#else
+1.00000000000000000000000000000000000e+00L,
+1.02189714865411667823448013478329942e+00L,
+1.04427378242741384032196647873992910e+00L,
+1.06714040067682361816952112099280918e+00L,
+1.09050773266525765920701065576070789e+00L,
+1.11438674259589253630881295691960313e+00L,
+1.13878863475669165370383028384151134e+00L,
+1.16372485877757751381357359909218536e+00L,
+1.18920711500272106671749997056047593e+00L,
+1.21524735998046887811652025133879836e+00L,
+1.24185781207348404859367746872659561e+00L,
+1.26905095719173322255441908103233805e+00L,
+1.29683955465100966593375411779245118e+00L,
+1.32523664315974129462953709549872168e+00L,
+1.35425554693689272829801474014070273e+00L,
+1.38390988196383195487265952726519287e+00L,
+1.41421356237309504880168872420969798e+00L,
+1.44518080697704662003700624147167095e+00L,
+1.47682614593949931138690748037404985e+00L,
+1.50916442759342273976601955103319352e+00L,
+1.54221082540794082361229186209073479e+00L,
+1.57598084510788648645527016018190504e+00L,
+1.61049033194925430817952066735740067e+00L,
+1.64575547815396484451875672472582254e+00L,
+1.68179283050742908606225095246642969e+00L,
+1.71861929812247791562934437645631244e+00L,
+1.75625216037329948311216061937531314e+00L,
+1.79470907500310718642770324212778174e+00L,
+1.83400808640934246348708318958828892e+00L,
+1.87416763411029990132999894995444645e+00L,
+1.91520656139714729387261127029583086e+00L,
+1.95714412417540026901832225162687149e+00L,
#endif
};
static const long double S_trail[] = {
#if defined(__x86)
+0.0000000000000000000000000e+00L,
+2.6327965667180882569382524e-20L,
+8.3765863521895191129661899e-20L,
+3.9798705777454504249209575e-20L,
+1.0668046596651558640993042e-19L,
+1.9376009847285360448117114e-20L,
+6.7081819456112953751277576e-21L,
+1.9711680502629186462729727e-20L,
+2.9932584438449523689104569e-20L,
+6.8887754153039109411061914e-20L,
+6.8002718741225378942847820e-20L,
+6.5846917376975403439742349e-20L,
+1.2171958727511372194876001e-20L,
+3.5625253228704087115438260e-20L,
+3.1129551559077560956309179e-20L,
+5.7519192396164779846216492e-20L,
+3.7900651177865141593101239e-20L,
+1.1659262405698741798080115e-20L,
+7.1364385105284695967172478e-20L,
+5.2631003710812203588788949e-20L,
+2.6328853788732632868460580e-20L,
+5.4583950085438242788190141e-20L,
+9.5803254376938269960718656e-20L,
+7.6837733983874245823512279e-21L,
+2.4415965910835093824202087e-20L,
+2.6052966871016580981769728e-20L,
+2.6876456344632553875309579e-21L,
+1.2861930155613700201703279e-20L,
+8.8166633394037485606572294e-20L,
+2.9788615389580190940837037e-20L,
+5.2352341619805098677422139e-20L,
+5.2578463064010463732242363e-20L,
#else
+0.00000000000000000000000000000000000e+00L,
+1.80506787420330954745573333054573786e-35L,
-9.37452029228042742195756741973083214e-35L,
-1.59696844729275877071290963023149997e-35L,
+9.11249341012502297851168610167248666e-35L,
-6.50422820697854828723037477525938871e-35L,
-8.14846884452585113732569176748815532e-35L,
-5.06621457672180031337233074514290335e-35L,
-1.35983097468881697374987563824591912e-35L,
+9.49742763556319647030771056643324660e-35L,
-3.28317052317699860161506596533391526e-36L,
-5.01723570938719041029018653045842895e-35L,
-2.39147479768910917162283430160264014e-35L,
-8.35057135763390881529889073794408385e-36L,
+7.03675688907326504242173719067187644e-35L,
-5.18248485306464645753689301856695619e-35L,
+9.42224254862183206569211673639406488e-35L,
-3.96750082539886230916730613021641828e-35L,
+7.14352899156330061452327361509276724e-35L,
+1.15987125286798512424651783410044433e-35L,
+4.69693347835811549530973921320187447e-35L,
-3.38651317599500471079924198499981917e-35L,
-8.58731877429824706886865593510387445e-35L,
-9.60595154874935050318549936224606909e-35L,
+9.60973393212801278450755869714178581e-35L,
+6.37839792144002843924476144978084855e-35L,
+7.79243078569586424945646112516927770e-35L,
+7.36133776758845652413193083663393220e-35L,
-6.47299514791334723003521457561217053e-35L,
+8.58747441795369869427879806229522962e-35L,
+2.37181542282517483569165122830269098e-35L,
-3.02689168209611877300459737342190031e-37L,
#endif
};
/* INDENT ON */
/* INDENT OFF */
/*
* return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula
* log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
* = L1 + L2 + L3,
*/
/* INDENT ON */
static struct LDouble
large_gam(long double x, int *m) {
long double z, t1, t2, t3, z2, t5, w, y, u, r, v;
long double t24 = 16777216.0L, p24 = 1.0L / 16777216.0L;
int n2, j2, k, ix, j, i;
struct LDouble zz;
long double u2, ss_h, ss_l, r_h, w_h, w_l, t4;
/* INDENT OFF */
/*
* compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details)
*
* log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
* j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
* T1(n) = T1[2n,2n+1] = n*log(2)-1,
* T2(j) = T2[2j,2j+1] = log(z[j]),
* T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + ... + T3[6]s^15
* Note
* (1) the leading entries are truncated to 24 binary point.
* (2) Remez error for T3(s) is bounded by 2**(-72.4)
* 2**(-24)
* _________V___________________
* T1(n): |_________|___________________|
* _______ ______________________
* T2(j): |_______|______________________|
* ____ _______________________
* 2s: |____|_______________________|
* __________________________
* + T3(s)-2s: |__________________________|
* -------------------------------------------
* [leading] + [Trailing]
*/
/* INDENT ON */
ix = H0_WORD(x);
n2 = (ix >> 16) - 0x3fff; /* exponent of x, range:3-10 */
y = scalbnl(x, -n2); /* y = scale x to [1,2] */
n2 += n2; /* 2n */
j = (ix >> 10) & 0x3f; /* j */
z = 1.0078125L + (long double) j * 0.015625L; /* z[j]=1+j/64+1/128 */
j2 = j + j;
t1 = y + z;
t2 = y - z;
r = one / t1;
u = r * t2; /* u = (y-z)/(y+z) */
t1 = CHOPPED(t1);
t4 = T2[j2 + 1] + T1[n2 + 1];
z2 = u * u;
k = H0_WORD(u) & 0x7fffffff;
t3 = T2[j2] + T1[n2];
for (t5 = T3[6], i = 5; i >= 0; i--)
t5 = z2 * t5 + T3[i];
if ((k >> 16) < 0x3fec) { /* |u|<2**-19 */
t2 = t4 + u * (two + z2 * t5);
} else {
t5 = t4 + (u * z2) * t5;
u2 = u + u;
v = (long double) ((int) (u2 * t24)) * p24;
t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z)));
t3 += v;
}
ss_h = CHOPPED((t2 + t3));
ss_l = t2 - (ss_h - t3);
/* INDENT OFF */
/*
* compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
* where ss = log(x) - 1 in already in extra precision
*/
/* INDENT ON */
z = one / x;
r = x - half;
r_h = CHOPPED((r));
w_h = r_h * ss_h + hln2pim1_h;
z2 = z * z;
w = (r - r_h) * ss_h + r * ss_l;
t1 = GP[19];
for (i = 18; i > 0; i--)
t1 = z2 * t1 + GP[i];
w += hln2pim1_l;
w_l = z * (GP[0] + z2 * t1) + w;
k = (int) ((w_h + w_l) * invln2_32 + half);
/* compute the exponential of w_h+w_l */
j = k & 0x1f;
*m = k >> 5;
t3 = (long double) k;
/* perform w - k*ln2_32 (represent as w_h - w_l) */
t1 = w_h - t3 * ln2_32hi;
t2 = t3 * ln2_32lo;
w = t2 - w_l;
w_h = t1 - w;
w_l = w - (t1 - w_h);
/* compute exp(w_h-w_l) */
z = w_h - w_l;
for (t1 = Et[10], i = 9; i >= 0; i--)
t1 = z * t1 + Et[i];
t3 = w_h - (w_l - (z * z) * t1); /* t3 = expm1(z) */
zz.l = S_trail[j] * (one + t3) + S[j] * t3;
zz.h = S[j];
return (zz);
}
/* INDENT OFF */
/*
* kpsin(x)= sin(pi*x)/pi
* 3 5 7 9 11 27
* = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x + ... + ks[12]*x
*/
static const long double ks[] = {
-1.64493406684822643647241516664602518705158902870e+0000L,
+8.11742425283353643637002772405874238094995726160e-0001L,
-1.90751824122084213696472111835337366232282723933e-0001L,
+2.61478478176548005046532613563241288115395517084e-0002L,
-2.34608103545582363750893072647117829448016479971e-0003L,
+1.48428793031071003684606647212534027556262040158e-0004L,
-6.97587366165638046518462722252768122615952898698e-0006L,
+2.53121740413702536928659271747187500934840057929e-0007L,
-7.30471182221385990397683641695766121301933621956e-0009L,
+1.71653847451163495739958249695549313987973589884e-0010L,
-3.34813314714560776122245796929054813458341420565e-0012L,
+5.50724992262622033449487808306969135431411753047e-0014L,
-7.67678132753577998601234393215802221104236979928e-0016L,
};
/* INDENT ON */
/*
* assume x is not tiny and positive
*/
static struct LDouble
kpsin(long double x) {
long double z, t1, t2;
struct LDouble xx;
int i;
z = x * x;
xx.h = x;
for (t2 = ks[12], i = 11; i > 0; i--)
t2 = z * t2 + ks[i];
t1 = z * x;
t2 *= z * t1;
xx.l = t1 * ks[0] + t2;
return (xx);
}
/* INDENT OFF */
/*
* kpcos(x)= cos(pi*x)/pi
* 2 4 6 8 10 12
* = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x
*
* 2 4 6 8 10 22
* = 1/pi - pi/2*x +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +...+kc[9]*x
*
* -pi/2*x*x = (npi_2_h + npi_2_l) * (x_f+x_l)*(x_f+x_l)
* = npi_2_h*(x_f+x_l)*(x_f+x_l) + npi_2_l*x*x
* = npi_2_h*x_f*x_f + npi_2_h*(x*x-x_f*x_f) + npi_2_l*x*x
* = npi_2_h*x_f*x_f + npi_2_h*(x+x_f)*(x-x_f) + npi_2_l*x*x
* Here x_f = (long double) (float)x
* Note that pi/2(in hex) =
* 1.921FB54442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29
* npi_2_h = -pi/2 chopped to 25 bits = -1.921FB50000000000000000000000000 =
* -1.570796310901641845703125000000000 and
* npi_2_l =
* -0.0000004442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29 =
* -.0000000158932547735281966916397514420985846996875529104874722961539 =
* -1.5893254773528196691639751442098584699687552910487472296153e-8
* 1/pi(in hex) =
* .517CC1B727220A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B
* will be splitted into:
* one_pi_h = 1/pi chopped to 48 bits = .517CC1B727220000000000... and
* one_pi_l = .0000000000000A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B
*/
static const long double
#if defined(__x86)
one_pi_h = 0.3183098861481994390487670898437500L, /* 31 bits */
one_pi_l = 3.559123248900043690127872406891929148e-11L,
#else
one_pi_h = 0.31830988618379052468299050815403461456298828125L,
one_pi_l = 1.46854777018590994109505931010230912897495334688117e-16L,
#endif
npi_2_h = -1.570796310901641845703125000000000L,
npi_2_l = -1.5893254773528196691639751442098584699687552910e-8L;
static const long double kc[] = {
+1.29192819501249250731151312779548918765320728489e+0000L,
-4.25027339979557573976029596929319207009444090366e-0001L,
+7.49080661650990096109672954618317623888421628613e-0002L,
-8.21458866111282287985539464173976555436050215120e-0003L,
+6.14202578809529228503205255165761204750211603402e-0004L,
-3.33073432691149607007217330302595267179545908740e-0005L,
+1.36970959047832085796809745461530865597993680204e-0006L,
-4.41780774262583514450246512727201806217271097336e-0008L,
+1.14741409212381858820016567664488123478660705759e-0009L,
-2.44261236114707374558437500654381006300502749632e-0011L,
};
/* INDENT ON */
/*
* assume x is not tiny and positive
*/
static struct LDouble
kpcos(long double x) {
long double z, t1, t2, t3, t4, x4, x8;
int i;
struct LDouble xx;
z = x * x;
xx.h = one_pi_h;
t1 = (long double) ((float) x);
x4 = z * z;
t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1);
for (i = 8, t3 = kc[9]; i >= 0; i--)
t3 = z * t3 + kc[i];
t3 = one_pi_l + x4 * t3;
t4 = t1 * t1 * npi_2_h;
x8 = t2 + t3;
xx.l = x8 + t4;
return (xx);
}
/* INDENT OFF */
static const long double
/* 0.13486180573279076968979393577465291700642511139552429398233 */
#if defined(__x86)
t0z1 = 0.1348618057327907696779385054997035808810L,
t0z1_l = 1.1855430274949336125392717150257379614654e-20L,
#else
t0z1 = 0.1348618057327907696897939357746529168654L,
t0z1_l = 1.4102088588676879418739164486159514674310e-37L,
#endif
/* 0.46163214496836234126265954232572132846819620400644635129599 */
#if defined(__x86)
t0z2 = 0.4616321449683623412538115843295472018326L,
t0z2_l = 8.84795799617412663558532305039261747030640e-21L,
#else
t0z2 = 0.46163214496836234126265954232572132343318L,
t0z2_l = 5.03501162329616380465302666480916271611101e-36L,
#endif
/* 0.81977310110050060178786870492160699631174407846245179119586 */
#if defined(__x86)
t0z3 = 0.81977310110050060178773362329351925836817L,
t0z3_l = 1.350816280877379435658077052534574556256230e-22L
#else
t0z3 = 0.8197731011005006017878687049216069516957449L,
t0z3_l = 4.461599916947014419045492615933551648857380e-35L
#endif
;
/* INDENT ON */
/*
* gamma(x+i) for 0 <= x < 1
*/
static struct LDouble
gam_n(int i, long double x) {
struct LDouble rr = {0.0L, 0.0L}, yy;
long double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl;
/* compute yy = gamma(x+1) */
if (x > 0.2845L) {
if (x > 0.6374L) {
r1 = x - t0z3;
r2 = CHOPPED((r1 - t0z3_l));
t2 = r1 - r2;
yy = GT3(r2, t2 - t0z3_l);
} else {
r1 = x - t0z2;
r2 = CHOPPED((r1 - t0z2_l));
t2 = r1 - r2;
yy = GT2(r2, t2 - t0z2_l);
}
} else {
r1 = x - t0z1;
r2 = CHOPPED((r1 - t0z1_l));
t2 = r1 - r2;
yy = GT1(r2, t2 - t0z1_l);
}
/* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
switch (i) {
case 0: /* yy/x */
r1 = one / x;
xh = CHOPPED((x)); /* x is not tiny */
rr.h = CHOPPED(((yy.h + yy.l) * r1));
rr.l = r1 * (yy.h - rr.h * xh) - ((r1 * rr.h) * (x - xh) -
r1 * yy.l);
break;
case 1: /* yy */
rr.h = yy.h;
rr.l = yy.l;
break;
case 2: /* (x+1)*yy */
z = x + one; /* may not be exact */
zh = CHOPPED((z));
rr.h = zh * yy.h;
rr.l = z * yy.l + (x - (zh - one)) * yy.h;
break;
case 3: /* (x+2)*(x+1)*yy */
z1 = x + one;
z2 = x + 2.0L;
z = z1 * z2;
xh = CHOPPED((z));
zh = CHOPPED((z1));
xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one));
rr.h = xh * yy.h;
rr.l = z * yy.l + xl * yy.h;
break;
case 4: /* (x+1)*(x+3)*(x+2)*yy */
z1 = x + 2.0L;
z2 = (x + one) * (x + 3.0L);
zh = CHOPPED(z1);
zl = x - (zh - 2.0L);
xh = CHOPPED(z2);
xl = zl * (zh + z1) - (xh - (zh * zh - one));
/* wh+wl=(x+2)*yy */
wh = CHOPPED((z1 * (yy.h + yy.l)));
wl = (zl * yy.h + z1 * yy.l) - (wh - zh * yy.h);
rr.h = xh * wh;
rr.l = z2 * wl + xl * wh;
break;
case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
z1 = x + 2.0L;
z2 = x + 3.0L;
z = z1 * z2;
zh = CHOPPED((z1));
yh = CHOPPED((z));
yl = (x - (zh - 2.0L)) * (z2 + zh) - (yh - zh * (zh + one));
z2 = z - 2.0L;
z *= z2;
xh = CHOPPED((z));
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
rr.h = xh * yy.h;
rr.l = z * yy.l + xl * yy.h;
break;
case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
z1 = x + 2.0L;
z2 = x + 3.0L;
z = z1 * z2;
zh = CHOPPED((z1));
yh = CHOPPED((z));
z1 = x - (zh - 2.0L);
yl = z1 * (z2 + zh) - (yh - zh * (zh + one));
z2 = z - 2.0L;
x5 = x + 5.0L;
z *= z2;
xh = CHOPPED(z);
zh += 3.0;
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
/* xh+xl=(x+1)*...*(x+4) */
/* wh+wl=(x+5)*yy */
wh = CHOPPED((x5 * (yy.h + yy.l)));
wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h);
rr.h = wh * xh;
rr.l = z * wl + xl * wh;
break;
case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
z1 = x + 3.0L;
z2 = x + 4.0L;
z = z2 * z1;
zh = CHOPPED((z1));
yh = CHOPPED((z)); /* yh+yl = (x+3)(x+4) */
yl = (x - (zh - 3.0L)) * (z2 + zh) - (yh - (zh * (zh + one)));
z1 = x + 6.0L;
z2 = z - 2.0L; /* z2 = (x+2)*(x+5) */
z *= z2;
xh = CHOPPED((z));
xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
/* xh+xl=(x+2)*...*(x+5) */
/* wh+wl=(x+1)(x+6)*yy */
z2 -= 4.0L; /* z2 = (x+1)(x+6) */
wh = CHOPPED((z2 * (yy.h + yy.l)));
wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0L) * yy.h);
rr.h = wh * xh;
rr.l = z * wl + xl * wh;
}
return (rr);
}
long double
tgammal(long double x) {
struct LDouble ss, ww;
long double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5;
int i, j, m, ix, hx, xk;
unsigned lx;
hx = H0_WORD(x);
lx = H3_WORD(x);
ix = hx & 0x7fffffff;
y = x;
if (ix < 0x3f8e0000) { /* x < 2**-113 */
return (one / x);
}
if (ix >= 0x7fff0000)
return (x * ((hx < 0)? zero : x)); /* Inf or NaN */
if (x > overflow) /* overflow threshold */
return (x * 1.0e4932L);
if (hx >= 0x40020000) { /* x >= 8 */
ww = large_gam(x, &m);
w = ww.h + ww.l;
return (scalbnl(w, m));
}
if (hx > 0) { /* 0 < x < 8 */
i = (int) x;
ww = gam_n(i, x - (long double) i);
return (ww.h + ww.l);
}
/* INDENT OFF */
/* negative x */
/*
* compute xk =
* -2 ... x is an even int (-inf is considered an even #)
* -1 ... x is an odd int
* +0 ... x is not an int but chopped to an even int
* +1 ... x is not an int but chopped to an odd int
*/
/* INDENT ON */
xk = 0;
#if defined(__x86)
if (ix >= 0x403e0000) { /* x >= 2**63 } */
if (ix >= 0x403f0000)
xk = -2;
else
xk = -2 + (lx & 1);
#else
if (ix >= 0x406f0000) { /* x >= 2**112 */
if (ix >= 0x40700000)
xk = -2;
else
xk = -2 + (lx & 1);
#endif
} else if (ix >= 0x3fff0000) {
w = -x;
t1 = floorl(w);
t2 = t1 * half;
t3 = floorl(t2);
if (t1 == w) {
if (t2 == t3)
xk = -2;
else
xk = -1;
} else {
if (t2 == t3)
xk = 0;
else
xk = 1;
}
}
if (xk < 0) {
/* return NaN. Ideally gamma(-n)= (-1)**(n+1) * inf */
return (x - x) / (x - x);
}
/*
* negative underflow thresold -(1774+9ulp)
*/
if (x < -1774.0000000000000000000000000000017749370L) {
z = tiny / x;
if (xk == 1)
z = -z;
return (z * tiny);
}
/* INDENT OFF */
/*
* now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x
*/
/*
* First compute ss = -sin(pi*y)/pi so that
* gamma(x) = 1/(ss*gamma(1+y))
*/
/* INDENT ON */
y = -x;
j = (int) y;
z = y - (long double) j;
if (z > 0.3183098861837906715377675L)
if (z > 0.6816901138162093284622325L)
ss = kpsin(one - z);
else
ss = kpcos(0.5L - z);
else
ss = kpsin(z);
if (xk == 0) {
ss.h = -ss.h;
ss.l = -ss.l;
}
/* Then compute ww = gamma(1+y), note that result scale to 2**m */
m = 0;
if (j < 7) {
ww = gam_n(j + 1, z);
} else {
w = y + one;
if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */
ww = large_gam(w, &m);
} else {
t = w - one;
if (t == y) { /* y+one exact */
ww = large_gam(w, &m);
} else { /* use y*gamma(y) */
if (j == 7)
ww = gam_n(j, z);
else
ww = large_gam(y, &m);
t4 = ww.h + ww.l;
t1 = CHOPPED((y));
t2 = CHOPPED((t4));
/* t4 will not be too large */
ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2;
ww.h = t1 * t2;
}
}
}
/* compute 1/(ss*ww) */
t3 = ss.h + ss.l;
t4 = ww.h + ww.l;
t1 = CHOPPED((t3));
t2 = CHOPPED((t4));
z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */
z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */
t3 = t3 * t4; /* t3 = ss*ww */
z3 = one / t3; /* z3 = 1/(ss*ww) */
t5 = t1 * t2;
z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */
t1 = CHOPPED((t3)); /* (t1,z1) = ss*ww */
z1 = z5 - (t1 - t5);
t2 = CHOPPED((z3)); /* leading 1/(ss*ww) */
z2 = z3 * (t2 * z1 - (one - t2 * t1));
z = t2 - z2;
return (scalbnl(z, -m));
}