__lgammal.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* 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.
*/
/*
* long double __k_lgammal(long double x, int *signgamlp);
* K.C. Ng, August, 1989.
*
* We choose [1.5,2.5] to be the primary interval. Our algorithms
* are mainly derived from
*
*
* zeta(2)-1 2 zeta(3)-1 3
* lgamma(2+s) = s*(1-euler) + --------- * s - --------- * s + ...
* 2 3
*
*
* Note 1. Since gamma(1+s)=s*gamma(s), hence
* lgamma(1+s) = log(s) + lgamma(s), or
* lgamma(s) = lgamma(1+s) - log(s).
* When s is really tiny (like roundoff), lgamma(1+s) ~ s(1-enler)
* Hence lgamma(s) ~ -log(s) for tiny s
*
*/
#include "libm.h"
#include "longdouble.h"
static long double neg(long double, int *);
static long double poly(long double, const long double *, int);
static long double polytail(long double);
static long double primary(long double);
static const long double
c0 = 0.0L,
ch = 0.5L,
c1 = 1.0L,
c2 = 2.0L,
c3 = 3.0L,
c4 = 4.0L,
c5 = 5.0L,
c6 = 6.0L,
pi = 3.1415926535897932384626433832795028841971L,
tiny = 1.0e-40L;
long double
__k_lgammal(long double x, int *signgamlp) {
long double t,y;
int i;
/* purge off +-inf, NaN and negative arguments */
if (!finitel(x)) return x*x;
*signgamlp = 1;
if (signbitl(x)) return (neg(x,signgamlp));
/* for x < 8.0 */
if (x<8.0L) {
y = anintl(x);
i = (int) y;
switch(i) {
case 0:
if (x<1.0e-40L) return -logl(x); else
return (primary(x)-log1pl(x))-logl(x);
case 1:
return primary(x-y)-logl(x);
case 2:
return primary(x-y);
case 3:
return primary(x-y)+logl(x-c1);
case 4:
return primary(x-y)+logl((x-c1)*(x-c2));
case 5:
return primary(x-y)+logl((x-c1)*(x-c2)*(x-c3));
case 6:
return primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4));
case 7:
return primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5));
case 8:
return primary(x-y)+
logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)*(x-c6));
}
}
/* 8.0 <= x < 1.0e40 */
if (x < 1.0e40L) {
t = logl(x);
return x*(t-c1)-(ch*t-polytail(c1/x));
}
/* 1.0e40 <= x <= inf */
return x*(logl(x)-c1);
}
static const long double an1[] = { /* 20 terms */
-0.0772156649015328606065120900824024309741L,
3.224670334241132182362075833230130289059e-0001L,
-6.735230105319809513324605383668929964120e-0002L,
2.058080842778454787900092432928910226297e-0002L,
-7.385551028673985266273054086081102125704e-0003L,
2.890510330741523285758867304409628648727e-0003L,
-1.192753911703260976581414338096267498555e-0003L,
5.096695247430424562831956662855697824035e-0004L,
-2.231547584535777978926798502084300123638e-0004L,
9.945751278186384670278268034322157947635e-0005L,
-4.492623673665547726647838474125147631082e-0005L,
2.050721280617796810096993154281561168706e-0005L,
-9.439487785617396552092393234044767313568e-0006L,
4.374872903516051510689234173139793159340e-0006L,
-2.039156676413643091040459825776029327487e-0006L,
9.555777181318621470466563543806211523634e-0007L,
-4.468344919709630637558538313482398989638e-0007L,
2.216738086090045781773004477831059444178e-0007L,
-7.472783403418388455860445842543843485916e-0008L,
8.777317930927149922056782132706238921648e-0008L,
};
static const long double an2[] = { /* 20 terms */
-.0772156649015328606062692723698127607018L,
3.224670334241132182635552349060279118047e-0001L,
-6.735230105319809367555642883133994818325e-0002L,
2.058080842778459676880822202762143671813e-0002L,
-7.385551028672828216011343150077846918930e-0003L,
2.890510330762060607399561536905727853178e-0003L,
-1.192753911419623262328187532759756368041e-0003L,
5.096695278636456678258091134532258618614e-0004L,
-2.231547306817535743052975194022893369135e-0004L,
9.945771461633313282744264853986643877087e-0005L,
-4.492503279458972037926876061257489481619e-0005L,
2.051311416812082875492678651369394595613e-0005L,
-9.415778282365955203915850761537462941165e-0006L,
4.452428829045147098722932981088650055919e-0006L,
-1.835024727987632579886951760650722695781e-0006L,
1.379783080658545009579060714946381462565e-0006L,
2.282637532109775156769736768748402175238e-0007L,
1.002577375515900191362119718128149880168e-0006L,
5.177028794262638311939991106423220002463e-0007L,
3.127947245174847104122426445937830555755e-0007L,
};
static const long double an3[] = { /* 20 terms */
-.0772156649015328227870646417729220690875L,
3.224670334241156699881788955959915250365e-0001L,
-6.735230105312273571375431059744975563170e-0002L,
2.058080842924464587662846071337083809005e-0002L,
-7.385551008677271654723604653956131791619e-0003L,
2.890510536479782086197110272583833176602e-0003L,
-1.192752262076857692740571567808259138697e-0003L,
5.096800771149805289371135155128380707889e-0004L,
-2.231000836682831335505058492409860123647e-0004L,
9.968912171073936803871803966360595275047e-0005L,
-4.412020779327746243544387946167256187258e-0005L,
2.281374113541454151067016632998630209049e-0005L,
-4.028361291428629491824694655287954266830e-0006L,
1.470694920619518924598956849226530750139e-0005L,
1.381686137617987197975289545582377713772e-0005L,
2.012493539265777728944759982054970441601e-0005L,
1.723917864208965490251560644681933675799e-0005L,
1.202954035243788300138608765425123713395e-0005L,
5.079851887558623092776296577030850938146e-0006L,
1.220657945824153751555138592006604026282e-0006L,
};
static const long double an4[] = { /* 21 terms */
-.0772156649015732285350261816697540392371L,
3.224670334221752060691751340365212226097e-0001L,
-6.735230109744009693977755991488196368279e-0002L,
2.058080778913037626909954141611580783216e-0002L,
-7.385557567931505621170483708950557506819e-0003L,
2.890459838416254326340844289785254883436e-0003L,
-1.193059036207136762877351596966718455737e-0003L,
5.081914708100372836613371356529568937869e-0004L,
-2.289855016133600313131553005982542045338e-0004L,
8.053454537980585879620331053833498511491e-0005L,
-9.574620532104845821243493405855672438998e-0005L,
-9.269085628207107155601445001196317715686e-0005L,
-2.183276779859490461716196344776208220180e-0004L,
-3.134834305597571096452454999737269668868e-0004L,
-3.973878894951937437018305986901392888619e-0004L,
-3.953352414899222799161275564386488057119e-0004L,
-3.136740932204038779362660900621212816511e-0004L,
-1.884502253819634073946130825196078627664e-0004L,
-8.192655799958926853585332542123631379301e-0005L,
-2.292183750010571062891605074281744854436e-0005L,
-3.223980628729716864927724265781406614294e-0006L,
};
static const long double ap1[] = { /* 19 terms */
-0.0772156649015328606065120900824024296961L,
3.224670334241132182362075833230047956465e-0001L,
-6.735230105319809513324605382963943777301e-0002L,
2.058080842778454787900092126606252375465e-0002L,
-7.385551028673985266272518231365020063941e-0003L,
2.890510330741523285681704570797770736423e-0003L,
-1.192753911703260971285304221165990244515e-0003L,
5.096695247430420878696018188830886972245e-0004L,
-2.231547584535654004647639737841526025095e-0004L,
9.945751278137201960636098805852315982919e-0005L,
-4.492623672777606053587919463929044226280e-0005L,
2.050721258703289487603702670753053765201e-0005L,
-9.439485626565616989352750672499008021041e-0006L,
4.374838162403994645138200419356844574219e-0006L,
-2.038979492862555348577006944451002161496e-0006L,
9.536763152382263548086981191378885102802e-0007L,
-4.426111214332434049863595231916564014913e-0007L,
1.911148847512947464234633846270287546882e-0007L,
-5.788673944861923038157839080272303519671e-0008L,
};
static const long double ap2[] = { /* 19 terms */
-0.077215664901532860606428624449354836087L,
3.224670334241132182271948744265855440139e-0001L,
-6.735230105319809467356126599005051676203e-0002L,
2.058080842778453315716389815213496002588e-0002L,
-7.385551028673653323064118422580096222959e-0003L,
2.890510330735923572088003424849289006039e-0003L,
-1.192753911629952368606185543945790688144e-0003L,
5.096695239806718875364547587043220998766e-0004L,
-2.231547520600616108991867127392089144886e-0004L,
9.945746913898151120612322833059416008973e-0005L,
-4.492599307461977003570224943054585729684e-0005L,
2.050609891889165453592046505651759999090e-0005L,
-9.435329866734193796540515247917165988579e-0006L,
4.362267138522223236241016136585565144581e-0006L,
-2.008556356653246579300491601497510230557e-0006L,
8.961498103387207161105347118042844354395e-0007L,
-3.614187228330216282235692806488341157741e-0007L,
1.136978988247816860500420915014777753153e-0007L,
-2.000532786387196664019286514899782691776e-0008L,
};
static const long double ap3[] = { /* 19 terms */
-0.077215664901532859888521470795348856446L,
3.224670334241131733364048614484228443077e-0001L,
-6.735230105319676541660495145259038151576e-0002L,
2.058080842775975461837768839015444273830e-0002L,
-7.385551028347615729728618066663566606906e-0003L,
2.890510327517954083379032008643080256676e-0003L,
-1.192753886919470728001821137439430882603e-0003L,
5.096693728898932234814903769146577482912e-0004L,
-2.231540055048827662528594010961874258037e-0004L,
9.945446210018649311491619999438833843723e-0005L,
-4.491608206598064519190236245753867697750e-0005L,
2.047939071322271016498065052853746466669e-0005L,
-9.376824046522786006677541036631536790762e-0006L,
4.259329829498149111582277209189150127347e-0006L,
-1.866064770421594266702176289764212873428e-0006L,
7.462066721137579592928128104534957135669e-0007L,
-2.483546217529077735074007138457678727371e-0007L,
5.915166576378161473299324673649144297574e-0008L,
-7.334139641706988966966252333759604701905e-0009L,
};
static const long double ap4[] = { /* 19 terms */
-0.0772156649015326785569313252637238673675L,
3.224670334241051435008842685722468344822e-0001L,
-6.735230105302832007479431772160948499254e-0002L,
2.058080842553481183648529360967441889912e-0002L,
-7.385551007602909242024706804659879199244e-0003L,
2.890510182473907253939821312248303471206e-0003L,
-1.192753098427856770847894497586825614450e-0003L,
5.096659636418811568063339214203693550804e-0004L,
-2.231421144004355691166194259675004483639e-0004L,
9.942073842343832132754332881883387625136e-0005L,
-4.483809261973204531263252655050701205397e-0005L,
2.033260142610284888319116654931994447173e-0005L,
-9.153539544026646699870528191410440585796e-0006L,
3.988460469925482725894144688699584997971e-0006L,
-1.609692980087029172567957221850825977621e-0006L,
5.634916377249975825399706694496688803488e-0007L,
-1.560065465929518563549083208482591437696e-0007L,
2.961350193868935325526962209019387821584e-0008L,
-2.834602215195368130104649234505033159842e-0009L,
};
static long double
primary(long double s) { /* assume |s|<=0.5 */
int i;
i = (int) (8.0L * (s + 0.5L));
switch(i) {
case 0: return ch*s+s*poly(s,an4,21);
case 1: return ch*s+s*poly(s,an3,20);
case 2: return ch*s+s*poly(s,an2,20);
case 3: return ch*s+s*poly(s,an1,20);
case 4: return ch*s+s*poly(s,ap1,19);
case 5: return ch*s+s*poly(s,ap2,19);
case 6: return ch*s+s*poly(s,ap3,19);
case 7: return ch*s+s*poly(s,ap4,19);
}
/* NOTREACHED */
return 0.0L;
}
static long double
poly(long double s, const long double *p, int n) {
long double y;
int i;
y = p[n-1];
for (i=n-2;i>=0;i--) y = p[i]+s*y;
return y;
}
static const long double pt[] = {
9.189385332046727417803297364056176804663e-0001L,
8.333333333333333333333333333331286969123e-0002L,
-2.777777777777777777777777553194796036402e-0003L,
7.936507936507936507927283071433584248176e-0004L,
-5.952380952380952362351042163192634108297e-0004L,
8.417508417508395661774286645578379460131e-0004L,
-1.917526917525263651186066417934685675649e-0003L,
6.410256409395203164659292973142293199083e-0003L,
-2.955065327248303301763594514012418438188e-0002L,
1.796442830099067542945998615411893822886e-0001L,
-1.392413465829723742489974310411118662919e+0000L,
1.339984238037267658352656597960492029261e+0001L,
-1.564707657605373662425785904278645727813e+0002L,
2.156323807499211356127813962223067079300e+0003L,
-3.330486427626223184647299834137041307569e+0004L,
5.235535072011889213611369254140123518699e+0005L,
-7.258160984602220710491988573430212593080e+0006L,
7.316526934569686459641438882340322673357e+0007L,
-3.806450279064900548836571789284896711473e+0008L,
};
static long double
polytail(long double s) {
long double t,z;
int i;
z = s*s;
t = pt[18];
for (i=17;i>=1;i--) t = pt[i]+z*t;
return pt[0]+s*t;
}
static long double
neg(long double z, int *signgamlp) {
long double t,p;
/*
* written by K.C. Ng, Feb 2, 1989.
*
* Since
* -z*G(-z)*G(z) = pi/sin(pi*z),
* we have
* G(-z) = -pi/(sin(pi*z)*G(z)*z)
* = pi/(sin(pi*(-z))*G(z)*z)
* Algorithm
* z = |z|
* t = sinpi(z); ...note that when z>2**112, z is an int
* and hence t=0.
*
* if (t == 0.0) return 1.0/0.0;
* if (t< 0.0) *signgamlp = -1; else t= -t;
* if (z<1.0e-40) ...tiny z
* return -log(z);
* else
* return log(pi/(t*z))-lgamma(z);
*
*/
t = sinpil(z); /* t := sin(pi*z) */
if (t == c0) /* return 1.0/0.0 = +INF */
return c1/c0;
z = -z;
if (z<=tiny)
p = -logl(z);
else
p = logl(pi/(fabsl(t)*z))-__k_lgammal(z,signgamlp);
if (t<c0) *signgamlp = -1;
return p;
}