/* specfunc/gamma.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program 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
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_log.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_trig.h>
#include <gsl/gsl_sf_gamma.h>
#include "error.h"
#include "check.h"
#include "chebyshev.h"
#include "cheb_eval.c"
#define LogRootTwoPi_ 0.9189385332046727418
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
static struct {int n; double f; long i; } fact_table[GSL_SF_FACT_NMAX + 1] = {
{ 0, 1.0, 1L },
{ 1, 1.0, 1L },
{ 2, 2.0, 2L },
{ 3, 6.0, 6L },
{ 4, 24.0, 24L },
{ 5, 120.0, 120L },
{ 6, 720.0, 720L },
{ 7, 5040.0, 5040L },
{ 8, 40320.0, 40320L },
{ 9, 362880.0, 362880L },
{ 10, 3628800.0, 3628800L },
{ 11, 39916800.0, 39916800L },
{ 12, 479001600.0, 479001600L },
{ 13, 6227020800.0, 0 },
{ 14, 87178291200.0, 0 },
{ 15, 1307674368000.0, 0 },
{ 16, 20922789888000.0, 0 },
{ 17, 355687428096000.0, 0 },
{ 18, 6402373705728000.0, 0 },
{ 19, 121645100408832000.0, 0 },
{ 20, 2432902008176640000.0, 0 },
{ 21, 51090942171709440000.0, 0 },
{ 22, 1124000727777607680000.0, 0 },
{ 23, 25852016738884976640000.0, 0 },
{ 24, 620448401733239439360000.0, 0 },
{ 25, 15511210043330985984000000.0, 0 },
{ 26, 403291461126605635584000000.0, 0 },
{ 27, 10888869450418352160768000000.0, 0 },
{ 28, 304888344611713860501504000000.0, 0 },
{ 29, 8841761993739701954543616000000.0, 0 },
{ 30, 265252859812191058636308480000000.0, 0 },
{ 31, 8222838654177922817725562880000000.0, 0 },
{ 32, 263130836933693530167218012160000000.0, 0 },
{ 33, 8683317618811886495518194401280000000.0, 0 },
{ 34, 2.95232799039604140847618609644e38, 0 },
{ 35, 1.03331479663861449296666513375e40, 0 },
{ 36, 3.71993326789901217467999448151e41, 0 },
{ 37, 1.37637530912263450463159795816e43, 0 },
{ 38, 5.23022617466601111760007224100e44, 0 },
{ 39, 2.03978820811974433586402817399e46, 0 },
{ 40, 8.15915283247897734345611269600e47, 0 },
{ 41, 3.34525266131638071081700620534e49, 0 },
{ 42, 1.40500611775287989854314260624e51, 0 },
{ 43, 6.04152630633738356373551320685e52, 0 },
{ 44, 2.65827157478844876804362581101e54, 0 },
{ 45, 1.19622220865480194561963161496e56, 0 },
{ 46, 5.50262215981208894985030542880e57, 0 },
{ 47, 2.58623241511168180642964355154e59, 0 },
{ 48, 1.24139155925360726708622890474e61, 0 },
{ 49, 6.08281864034267560872252163321e62, 0 },
{ 50, 3.04140932017133780436126081661e64, 0 },
{ 51, 1.55111875328738228022424301647e66, 0 },
{ 52, 8.06581751709438785716606368564e67, 0 },
{ 53, 4.27488328406002556429801375339e69, 0 },
{ 54, 2.30843697339241380472092742683e71, 0 },
{ 55, 1.26964033536582759259651008476e73, 0 },
{ 56, 7.10998587804863451854045647464e74, 0 },
{ 57, 4.05269195048772167556806019054e76, 0 },
{ 58, 2.35056133128287857182947491052e78, 0 },
{ 59, 1.38683118545689835737939019720e80, 0 },
{ 60, 8.32098711274139014427634118320e81, 0 },
{ 61, 5.07580213877224798800856812177e83, 0 },
{ 62, 3.14699732603879375256531223550e85, 0 },
{ 63, 1.982608315404440064116146708360e87, 0 },
{ 64, 1.268869321858841641034333893350e89, 0 },
{ 65, 8.247650592082470666723170306800e90, 0 },
{ 66, 5.443449390774430640037292402480e92, 0 },
{ 67, 3.647111091818868528824985909660e94, 0 },
{ 68, 2.480035542436830599600990418570e96, 0 },
{ 69, 1.711224524281413113724683388810e98, 0 },
{ 70, 1.197857166996989179607278372170e100, 0 },
{ 71, 8.504785885678623175211676442400e101, 0 },
{ 72, 6.123445837688608686152407038530e103, 0 },
{ 73, 4.470115461512684340891257138130e105, 0 },
{ 74, 3.307885441519386412259530282210e107, 0 },
{ 75, 2.480914081139539809194647711660e109, 0 },
{ 76, 1.885494701666050254987932260860e111, 0 },
{ 77, 1.451830920282858696340707840860e113, 0 },
{ 78, 1.132428117820629783145752115870e115, 0 },
{ 79, 8.946182130782975286851441715400e116, 0 },
{ 80, 7.156945704626380229481153372320e118, 0 },
{ 81, 5.797126020747367985879734231580e120, 0 },
{ 82, 4.753643337012841748421382069890e122, 0 },
{ 83, 3.945523969720658651189747118010e124, 0 },
{ 84, 3.314240134565353266999387579130e126, 0 },
{ 85, 2.817104114380550276949479442260e128, 0 },
{ 86, 2.422709538367273238176552320340e130, 0 },
{ 87, 2.107757298379527717213600518700e132, 0 },
{ 88, 1.854826422573984391147968456460e134, 0 },
{ 89, 1.650795516090846108121691926250e136, 0 },
{ 90, 1.485715964481761497309522733620e138, 0 },
{ 91, 1.352001527678402962551665687590e140, 0 },
{ 92, 1.243841405464130725547532432590e142, 0 },
{ 93, 1.156772507081641574759205162310e144, 0 },
{ 94, 1.087366156656743080273652852570e146, 0 },
{ 95, 1.032997848823905926259970209940e148, 0 },
{ 96, 9.916779348709496892095714015400e149, 0 },
{ 97, 9.619275968248211985332842594960e151, 0 },
{ 98, 9.426890448883247745626185743100e153, 0 },
{ 99, 9.332621544394415268169923885600e155, 0 },
{ 100, 9.33262154439441526816992388563e157, 0 },
{ 101, 9.42594775983835942085162312450e159, 0 },
{ 102, 9.61446671503512660926865558700e161, 0 },
{ 103, 9.90290071648618040754671525458e163, 0 },
{ 104, 1.02990167451456276238485838648e166, 0 },
{ 105, 1.08139675824029090050410130580e168, 0 },
{ 106, 1.146280563734708354534347384148e170, 0 },
{ 107, 1.226520203196137939351751701040e172, 0 },
{ 108, 1.324641819451828974499891837120e174, 0 },
{ 109, 1.443859583202493582204882102460e176, 0 },
{ 110, 1.588245541522742940425370312710e178, 0 },
{ 111, 1.762952551090244663872161047110e180, 0 },
{ 112, 1.974506857221074023536820372760e182, 0 },
{ 113, 2.231192748659813646596607021220e184, 0 },
{ 114, 2.543559733472187557120132004190e186, 0 },
{ 115, 2.925093693493015690688151804820e188, 0 },
{ 116, 3.393108684451898201198256093590e190, 0 },
{ 117, 3.96993716080872089540195962950e192, 0 },
{ 118, 4.68452584975429065657431236281e194, 0 },
{ 119, 5.57458576120760588132343171174e196, 0 },
{ 120, 6.68950291344912705758811805409e198, 0 },
{ 121, 8.09429852527344373968162284545e200, 0 },
{ 122, 9.87504420083360136241157987140e202, 0 },
{ 123, 1.21463043670253296757662432419e205, 0 },
{ 124, 1.50614174151114087979501416199e207, 0 },
{ 125, 1.88267717688892609974376770249e209, 0 },
{ 126, 2.37217324288004688567714730514e211, 0 },
{ 127, 3.01266001845765954480997707753e213, 0 },
{ 128, 3.85620482362580421735677065923e215, 0 },
{ 129, 4.97450422247728744039023415041e217, 0 },
{ 130, 6.46685548922047367250730439554e219, 0 },
{ 131, 8.47158069087882051098456875820e221, 0 },
{ 132, 1.11824865119600430744996307608e224, 0 },
{ 133, 1.48727070609068572890845089118e226, 0 },
{ 134, 1.99294274616151887673732419418e228, 0 },
{ 135, 2.69047270731805048359538766215e230, 0 },
{ 136, 3.65904288195254865768972722052e232, 0 },
{ 137, 5.01288874827499166103492629211e234, 0 },
{ 138, 6.91778647261948849222819828311e236, 0 },
{ 139, 9.61572319694108900419719561353e238, 0 },
{ 140, 1.34620124757175246058760738589e241, 0 },
{ 141, 1.89814375907617096942852641411e243, 0 },
{ 142, 2.69536413788816277658850750804e245, 0 },
{ 143, 3.85437071718007277052156573649e247, 0 },
{ 144, 5.55029383273930478955105466055e249, 0 },
{ 145, 8.04792605747199194484902925780e251, 0 },
{ 146, 1.17499720439091082394795827164e254, 0 },
{ 147, 1.72724589045463891120349865931e256, 0 },
{ 148, 2.55632391787286558858117801578e258, 0 },
{ 149, 3.80892263763056972698595524351e260, 0 },
{ 150, 5.71338395644585459047893286526e262, 0 },
{ 151, 8.62720977423324043162318862650e264, 0 },
{ 152, 1.31133588568345254560672467123e267, 0 },
{ 153, 2.00634390509568239477828874699e269, 0 },
{ 154, 3.08976961384735088795856467036e271, 0 },
{ 155, 4.78914290146339387633577523906e273, 0 },
{ 156, 7.47106292628289444708380937294e275, 0 },
{ 157, 1.17295687942641442819215807155e278, 0 },
{ 158, 1.85327186949373479654360975305e280, 0 },
{ 159, 2.94670227249503832650433950735e282, 0 },
{ 160, 4.71472363599206132240694321176e284, 0 },
{ 161, 7.59070505394721872907517857094e286, 0 },
{ 162, 1.22969421873944943411017892849e289, 0 },
{ 163, 2.00440157654530257759959165344e291, 0 },
{ 164, 3.28721858553429622726333031164e293, 0 },
{ 165, 5.42391066613158877498449501421e295, 0 },
{ 166, 9.00369170577843736647426172359e297, 0 },
{ 167, 1.50361651486499904020120170784e300, 0 },
{ 168, 2.52607574497319838753801886917e302, 0 },
{ 169, 4.26906800900470527493925188890e304, 0 },
{ 170, 7.25741561530799896739672821113e306, 0 },
/*
{ 171, 1.24101807021766782342484052410e309, 0 },
{ 172, 2.13455108077438865629072570146e311, 0 },
{ 173, 3.69277336973969237538295546352e313, 0 },
{ 174, 6.42542566334706473316634250653e315, 0 },
{ 175, 1.12444949108573632830410993864e318, 0 },
{ 176, 1.97903110431089593781523349201e320, 0 },
{ 177, 3.50288505463028580993296328086e322, 0 },
{ 178, 6.23513539724190874168067463993e324, 0 },
{ 179, 1.11608923610630166476084076055e327, 0 },
{ 180, 2.00896062499134299656951336898e329, 0 },
{ 181, 3.63621873123433082379081919786e331, 0 },
{ 182, 6.61791809084648209929929094011e333, 0 },
{ 183, 1.21107901062490622417177024204e336, 0 },
{ 184, 2.22838537954982745247605724535e338, 0 },
{ 185, 4.12251295216718078708070590390e340, 0 },
{ 186, 7.66787409103095626397011298130e342, 0 },
{ 187, 1.43389245502278882136241112750e345, 0 },
{ 188, 2.69571781544284298416133291969e347, 0 },
{ 189, 5.09490667118697324006491921822e349, 0 },
{ 190, 9.68032267525524915612334651460e351, 0 },
{ 191, 1.84894163097375258881955918429e354, 0 },
{ 192, 3.54996793146960497053355363384e356, 0 },
{ 193, 6.85143810773633759312975851330e358, 0 },
{ 194, 1.32917899290084949306717315158e361, 0 },
{ 195, 2.59189903615665651148098764559e363, 0 },
{ 196, 5.08012211086704676250273578535e365, 0 },
{ 197, 1.00078405584080821221303894971e368, 0 },
{ 198, 1.98155243056480026018181712043e370, 0 },
{ 199, 3.94328933682395251776181606966e372, 0 },
{ 200, 7.88657867364790503552363213932e374, 0 }
*/
};
static struct {int n; double f; long i; } doub_fact_table[GSL_SF_DOUBLEFACT_NMAX + 1] = {
{ 0, 1.000000000000000000000000000, 1L },
{ 1, 1.000000000000000000000000000, 1L },
{ 2, 2.000000000000000000000000000, 2L },
{ 3, 3.000000000000000000000000000, 3L },
{ 4, 8.000000000000000000000000000, 8L },
{ 5, 15.00000000000000000000000000, 15L },
{ 6, 48.00000000000000000000000000, 48L },
{ 7, 105.0000000000000000000000000, 105L },
{ 8, 384.0000000000000000000000000, 384L },
{ 9, 945.0000000000000000000000000, 945L },
{ 10, 3840.000000000000000000000000, 3840L },
{ 11, 10395.00000000000000000000000, 10395L },
{ 12, 46080.00000000000000000000000, 46080L },
{ 13, 135135.0000000000000000000000, 135135L },
{ 14, 645120.00000000000000000000000, 645120L },
{ 15, 2.02702500000000000000000000000e6, 2027025L },
{ 16, 1.03219200000000000000000000000e7, 10321920L },
{ 17, 3.4459425000000000000000000000e7, 34459425L },
{ 18, 1.85794560000000000000000000000e8, 185794560L },
{ 19, 6.5472907500000000000000000000e8, 0 },
{ 20, 3.7158912000000000000000000000e9, 0 },
{ 21, 1.37493105750000000000000000000e10, 0 },
{ 22, 8.1749606400000000000000000000e10, 0 },
{ 23, 3.1623414322500000000000000000e11, 0 },
{ 24, 1.96199055360000000000000000000e12, 0 },
{ 25, 7.9058535806250000000000000000e12, 0 },
{ 26, 5.1011754393600000000000000000e13, 0 },
{ 27, 2.13458046676875000000000000000e14, 0 },
{ 28, 1.42832912302080000000000000000e15, 0 },
{ 29, 6.1902833536293750000000000000e15, 0 },
{ 30, 4.2849873690624000000000000000e16, 0 },
{ 31, 1.91898783962510625000000000000e17, 0 },
{ 32, 1.37119595809996800000000000000e18, 0 },
{ 33, 6.3326598707628506250000000000e18, 0 },
{ 34, 4.6620662575398912000000000000e19, 0 },
{ 35, 2.21643095476699771875000000000e20, 0 },
{ 36, 1.67834385271436083200000000000e21, 0 },
{ 37, 8.2007945326378915593750000000e21, 0 },
{ 38, 6.3777066403145711616000000000e22, 0 },
{ 39, 3.1983098677287777081562500000e23, 0 },
{ 40, 2.55108265612582846464000000000e24, 0 },
{ 41, 1.31130704576879886034406250000e25, 0 },
{ 42, 1.07145471557284795514880000000e26, 0 },
{ 43, 5.6386202968058350994794687500e26, 0 },
{ 44, 4.7144007485205310026547200000e27, 0 },
{ 45, 2.53737913356262579476576093750e28, 0 },
{ 46, 2.16862434431944426122117120000e29, 0 },
{ 47, 1.19256819277443412353990764062e30, 0 },
{ 48, 1.04093968527333324538616217600e31, 0 },
{ 49, 5.8435841445947272053455474391e31, 0 },
{ 50, 5.2046984263666662269308108800e32, 0 },
{ 51, 2.98022791374331087472622919392e33, 0 },
{ 52, 2.70644318171066643800402165760e34, 0 },
{ 53, 1.57952079428395476360490147278e35, 0 },
{ 54, 1.46147931812375987652217169510e36, 0 },
{ 55, 8.6873643685617511998269581003e36, 0 },
{ 56, 8.1842841814930553085241614926e37, 0 },
{ 57, 4.9517976900801981839013661172e38, 0 },
{ 58, 4.7468848252659720789440136657e39, 0 },
{ 59, 2.92156063714731692850180600912e40, 0 },
{ 60, 2.84813089515958324736640819942e41, 0 },
{ 61, 1.78215198865986332638610166557e42, 0 },
{ 62, 1.76584115499894161336717308364e43, 0 },
{ 63, 1.12275575285571389562324404931e44, 0 },
{ 64, 1.13013833919932263255499077353e45, 0 },
{ 65, 7.2979123935621403215510863205e45, 0 },
{ 66, 7.4589130387155293748629391053e46, 0 },
{ 67, 4.8896013036866340154392278347e47, 0 },
{ 68, 5.0720608663265599749067985916e48, 0 },
{ 69, 3.3738248995437774706530672060e49, 0 },
{ 70, 3.5504426064285919824347590141e50, 0 },
{ 71, 2.39541567867608200416367771623e51, 0 },
{ 72, 2.55631867662858622735302649017e52, 0 },
{ 73, 1.74865344543353986303948473285e53, 0 },
{ 74, 1.89167582070515380824123960272e54, 0 },
{ 75, 1.31149008407515489727961354964e55, 0 },
{ 76, 1.43767362373591689426334209807e56, 0 },
{ 77, 1.00984736473786927090530243322e57, 0 },
{ 78, 1.12138542651401517752540683649e58, 0 },
{ 79, 7.9777941814291672401518892225e58, 0 },
{ 80, 8.9710834121121214202032546920e59, 0 },
{ 81, 6.4620132869576254645230302702e60, 0 },
{ 82, 7.3562883979319395645666688474e61, 0 },
{ 83, 5.3634710281748291355541151243e62, 0 },
{ 84, 6.1792822542628292342360018318e63, 0 },
{ 85, 4.5589503739486047652209978556e64, 0 },
{ 86, 5.3141827386660331414429615754e65, 0 },
{ 87, 3.9662868253352861457422681344e66, 0 },
{ 88, 4.6764808100261091644698061863e67, 0 },
{ 89, 3.5299952745484046697106186396e68, 0 },
{ 90, 4.2088327290234982480228255677e69, 0 },
{ 91, 3.2122956998390482494366629620e70, 0 },
{ 92, 3.8721261107016183881809995223e71, 0 },
{ 93, 2.98743500085031487197609655470e72, 0 },
{ 94, 3.6397985440595212848901395509e73, 0 },
{ 95, 2.83806325080779912837729172696e74, 0 },
{ 96, 3.4942066022971404334945339689e75, 0 },
{ 97, 2.75292135328356515452597297515e76, 0 },
{ 98, 3.4243224702511976248246432895e77, 0 },
{ 99, 2.72539213975072950298071324540e78, 0 },
{ 100, 3.4243224702511976248246432895e79, 0 },
{ 101, 2.75264606114823679801052037785e80, 0 },
{ 102, 3.4928089196562215773211361553e81, 0 },
{ 103, 2.83522544298268390195083598919e82, 0 },
{ 104, 3.6325212764424704404139816015e83, 0 },
{ 105, 2.97698671513181809704837778865e84, 0 },
{ 106, 3.8504725530290186668388204976e85, 0 },
{ 107, 3.1853757851910453638417642339e86, 0 },
{ 108, 4.1585103572713401601859261374e87, 0 },
{ 109, 3.4720596058582394465875230149e88, 0 },
{ 110, 4.5743613929984741762045187512e89, 0 },
{ 111, 3.8539861625026457857121505465e90, 0 },
{ 112, 5.1232847601582910773490610013e91, 0 },
{ 113, 4.3550043636279897378547301176e92, 0 },
{ 114, 5.8405446265804518281779295415e93, 0 },
{ 115, 5.0082550181721881985329396352e94, 0 },
{ 116, 6.7750317668333241206863982681e95, 0 },
{ 117, 5.8596583712614601922835393732e96, 0 },
{ 118, 7.9945374848633224624099499564e97, 0 },
{ 119, 6.9729934618011376288174118541e98, 0 },
{ 120, 9.5934449818359869548919399477e99, 0 },
{ 121, 8.4373220887793765308690683435e100, 0 },
{ 122, 1.17040028778399040849681667362e102, 0 },
{ 123, 1.03779061691986331329689540625e103, 0 },
{ 124, 1.45129635685214810653605267528e104, 0 },
{ 125, 1.29723827114982914162111925781e105, 0 },
{ 126, 1.82863340963370661423542637086e106, 0 },
{ 127, 1.64749260436028300985882145742e107, 0 },
{ 128, 2.34065076433114446622134575470e108, 0 },
{ 129, 2.12526545962476508271787968008e109, 0 },
{ 130, 3.04284599363048780608774948111e110, 0 },
{ 131, 2.78409775210844225836042238090e111, 0 },
{ 132, 4.0165567115922439040358293151e112, 0 },
{ 133, 3.7028500103042282036193617666e113, 0 },
{ 134, 5.3821859935336068314080112822e114, 0 },
{ 135, 4.9988475139107080748861383849e115, 0 },
{ 136, 7.3197729512057052907148953438e116, 0 },
{ 137, 6.8484210940576700625940095873e117, 0 },
{ 138, 1.01012866726638733011865555744e119, 0 },
{ 139, 9.5193053207401613870056733264e119, 0 },
{ 140, 1.41418013417294226216611778042e121, 0 },
{ 141, 1.34222205022436275556779993902e122, 0 },
{ 142, 2.00813579052557801227588724819e123, 0 },
{ 143, 1.91937753182083874046195391280e124, 0 },
{ 144, 2.89171553835683233767727763739e125, 0 },
{ 145, 2.78309742114021617366983317355e126, 0 },
{ 146, 4.2219046860009752130088253506e127, 0 },
{ 147, 4.0911532090761177752946547651e128, 0 },
{ 148, 6.2484189352814433152530615189e129, 0 },
{ 149, 6.0958182815234154851890356000e130, 0 },
{ 150, 9.3726284029221649728795922783e131, 0 },
{ 151, 9.2046856051003573826354437561e132, 0 },
{ 152, 1.42463951724416907587769802630e134, 0 },
{ 153, 1.40831689758035467954322289468e135, 0 },
{ 154, 2.19394485655602037685165496051e136, 0 },
{ 155, 2.18289119124954975329199548675e137, 0 },
{ 156, 3.4225539762273917878885817384e138, 0 },
{ 157, 3.4271391702617931126684329142e139, 0 },
{ 158, 5.4076352824392790248639591467e140, 0 },
{ 159, 5.4491512807162510491428083336e141, 0 },
{ 160, 8.6522164519028464397823346347e142, 0 },
{ 161, 8.7731335619531641891199214170e143, 0 },
{ 162, 1.40165906520826112324473821082e145, 0 },
{ 163, 1.43002077059836576282654719098e146, 0 },
{ 164, 2.29872086694154824212137066574e147, 0 },
{ 165, 2.35953427148730350866380286512e148, 0 },
{ 166, 3.8158766391229700819214753051e149, 0 },
{ 167, 3.9404222333837968594685507847e150, 0 },
{ 168, 6.4106727537265897376280785126e151, 0 },
{ 169, 6.6593135744186166925018508262e152, 0 },
{ 170, 1.08981436813352025539677334714e154, 0 },
{ 171, 1.13874262122558345441781649128e155, 0 },
{ 172, 1.87448071318965483928245015709e156, 0 },
{ 173, 1.97002473472025937614282252992e157, 0 },
{ 174, 3.2615964409499994203514632733e158, 0 },
{ 175, 3.4475432857604539082499394274e159, 0 },
{ 176, 5.7404097360719989798185753611e160, 0 },
{ 177, 6.1021516157960034176023927864e161, 0 },
{ 178, 1.02179293302081581840770641427e163, 0 },
{ 179, 1.09228513922748461175082830877e164, 0 },
{ 180, 1.83922727943746847313387154568e165, 0 },
{ 181, 1.97703610200174714726899923887e166, 0 },
{ 182, 3.3473936485761926211036462131e167, 0 },
{ 183, 3.6179760666631972795022686071e168, 0 },
{ 184, 6.1592043133801944228307090322e169, 0 },
{ 185, 6.6932557233269149670791969232e170, 0 },
{ 186, 1.14561200228871616264651187999e172, 0 },
{ 187, 1.25163882026213309884380982464e173, 0 },
{ 188, 2.15375056430278638577544233437e174, 0 },
{ 189, 2.36559737029543155681480056857e175, 0 },
{ 190, 4.0921260721752941329733404353e176, 0 },
{ 191, 4.5182909772642742735162690860e177, 0 },
{ 192, 7.8568820585765647353088136358e178, 0 },
{ 193, 8.7203015861200493478863993359e179, 0 },
{ 194, 1.52423511936385355864990984535e181, 0 },
{ 195, 1.70045880929340962283784787050e182, 0 },
{ 196, 2.98750083395315297495382329688e183, 0 },
{ 197, 3.3499038543080169569905603049e184, 0 },
{ 198, 5.9152516512272428904085701278e185, 0 },
{ 199, 6.6663086700729537444112150067e186, 0 },
{ 200, 1.18305033024544857808171402556e188, 0 },
{ 201, 1.33992804268466370262665421635e189, 0 },
{ 202, 2.38976166709580612772506233164e190, 0 },
{ 203, 2.72005392664986731633210805920e191, 0 },
{ 204, 4.8751138008754445005591271565e192, 0 },
{ 205, 5.5761105496322279984808215214e193, 0 },
{ 206, 1.00427344298034156711518019425e195, 0 },
{ 207, 1.15425488377387119568553005492e196, 0 },
{ 208, 2.08888876139911045959957480403e197, 0 },
{ 209, 2.41239270708739079898275781478e198, 0 },
{ 210, 4.3866663989381319651591070885e199, 0 },
{ 211, 5.0901486119543945858536189892e200, 0 },
{ 212, 9.2997327657488397661373070276e201, 0 },
{ 213, 1.08420165434628604678682084470e203, 0 },
{ 214, 1.99014281187025170995338370390e204, 0 },
{ 215, 2.33103355684451500059166481610e205, 0 },
{ 216, 4.2987084736397436934993088004e206, 0 },
{ 217, 5.0583428183525975512839126509e207, 0 },
{ 218, 9.3711844725346412518284931849e208, 0 },
{ 219, 1.10777707721921886373117687056e210, 0 },
{ 220, 2.06166058395762107540226850068e211, 0 },
{ 221, 2.44818734065447368884590088393e212, 0 },
{ 222, 4.5768864963859187873930360715e213, 0 },
{ 223, 5.4594577696594763261263589712e214, 0 },
{ 224, 1.02522257519044580837604008002e216, 0 },
{ 225, 1.22837799817338217337843076851e217, 0 },
{ 226, 2.31700301993040752692985058084e218, 0 },
{ 227, 2.78841805585357753356903784452e219, 0 },
{ 228, 5.2827668854413291614000593243e220, 0 },
{ 229, 6.3854773479046925518730966640e221, 0 },
{ 230, 1.21503638365150570712201364459e223, 0 },
{ 231, 1.47504526736598397948268532937e224, 0 },
{ 232, 2.81888441007149324052307165546e225, 0 },
{ 233, 3.4368554729627426721946568174e226, 0 },
{ 234, 6.5961895195672941828239876738e227, 0 },
{ 235, 8.0766103614624452796574435210e228, 0 },
{ 236, 1.55670072661788142714646109101e230, 0 },
{ 237, 1.91415665566659953127881411447e231, 0 },
{ 238, 3.7049477293505577966085773966e232, 0 },
{ 239, 4.5748344070431728797563657336e233, 0 },
{ 240, 8.8918745504413387118605857518e234, 0 },
{ 241, 1.10253509209740466402128414180e236, 0 },
{ 242, 2.15183364120680396827026175195e237, 0 },
{ 243, 2.67916027379669333357172046456e238, 0 },
{ 244, 5.2504740845446016825794386748e239, 0 },
{ 245, 6.5639426708018986672507151382e240, 0 },
{ 246, 1.29161662479797201391454191399e242, 0 },
{ 247, 1.62129383968806897081092663913e243, 0 },
{ 248, 3.2032092294989705945080639467e244, 0 },
{ 249, 4.0370216608232917373192073314e245, 0 },
{ 250, 8.0080230737474264862701598667e246, 0 },
{ 251, 1.01329243686664622606712104019e248, 0 },
{ 252, 2.01802181458435147454008028642e249, 0 },
{ 253, 2.56362986527261495194981623168e250, 0 },
{ 254, 5.1257754090442527453318039275e251, 0 },
{ 255, 6.5372561564451681274720313908e252, 0 },
{ 256, 1.31219850471532870280494180544e254, 0 },
{ 257, 1.68007483220640820876031206743e255, 0 },
{ 258, 3.3854721421655480532367498580e256, 0 },
{ 259, 4.3513938154145972606892082546e257, 0 },
{ 260, 8.8022275696304249384155496309e258, 0 },
{ 261, 1.13571378582320988503988335446e260, 0 },
{ 262, 2.30618362324317133386487400329e261, 0 },
{ 263, 2.98692725671504199765489322224e262, 0 },
{ 264, 6.0883247653619723214032673687e263, 0 },
{ 265, 7.9153572302948612937854670389e264, 0 },
{ 266, 1.61949438758628463749326912007e266, 0 },
{ 267, 2.11340038048872796544071969939e267, 0 },
{ 268, 4.3402449587312428284819612418e268, 0 },
{ 269, 5.6850470235146782270355359914e269, 0 },
{ 270, 1.17186613885743556369012953528e271, 0 },
{ 271, 1.54064774337247779952663025366e272, 0 },
{ 272, 3.1874758976922247332371523360e273, 0 },
{ 273, 4.2059683394068643927077005925e274, 0 },
{ 274, 8.7336839596766957690697974006e275, 0 },
{ 275, 1.15664129333688770799461766294e277, 0 },
{ 276, 2.41049677287076803226326408256e278, 0 },
{ 277, 3.2038963825431789511450909263e279, 0 },
{ 278, 6.7011810285807351296918741495e280, 0 },
{ 279, 8.9388709072954692736948036845e281, 0 },
{ 280, 1.87633068800260583631372476186e283, 0 },
{ 281, 2.51182272495002686590823983534e284, 0 },
{ 282, 5.2912525401673484584047038284e285, 0 },
{ 283, 7.1084583116085760305203187340e286, 0 },
{ 284, 1.50271572140752696218693588728e288, 0 },
{ 285, 2.02591061880844416869829083919e289, 0 },
{ 286, 4.2977669632255271118546366376e290, 0 },
{ 287, 5.8143634759802347641640947085e291, 0 },
{ 288, 1.23775688540895180821413535163e293, 0 },
{ 289, 1.68035104455828784684342337075e294, 0 },
{ 290, 3.5894949676859602438209925197e295, 0 },
{ 291, 4.8898215396646176343143620089e296, 0 },
{ 292, 1.04813253056430039119572981576e298, 0 },
{ 293, 1.43271771112173296685410806860e299, 0 },
{ 294, 3.08150963985904315011544565835e300, 0 },
{ 295, 4.2265172478091122522196188024e301, 0 },
{ 296, 9.1212685339827677243417191487e302, 0 },
{ 297, 1.25527562259930633890922678431e304, 0 },
/*
{ 298, 2.71813802312686478185383230631e305, 0 },
{ 299, 3.7532741115719259533385880851e306, 0 },
{ 300, 8.1544140693805943455614969189e307, }
*/
};
/* Chebyshev coefficients for Gamma*(3/4(t+1)+1/2), -1<t<1
*/
static double gstar_a_data[30] = {
2.16786447866463034423060819465,
-0.05533249018745584258035832802,
0.01800392431460719960888319748,
-0.00580919269468937714480019814,
0.00186523689488400339978881560,
-0.00059746524113955531852595159,
0.00019125169907783353925426722,
-0.00006124996546944685735909697,
0.00001963889633130842586440945,
-6.3067741254637180272515795142e-06,
2.0288698405861392526872789863e-06,
-6.5384896660838465981983750582e-07,
2.1108698058908865476480734911e-07,
-6.8260714912274941677892994580e-08,
2.2108560875880560555583978510e-08,
-7.1710331930255456643627187187e-09,
2.3290892983985406754602564745e-09,
-7.5740371598505586754890405359e-10,
2.4658267222594334398525312084e-10,
-8.0362243171659883803428749516e-11,
2.6215616826341594653521346229e-11,
-8.5596155025948750540420068109e-12,
2.7970831499487963614315315444e-12,
-9.1471771211886202805502562414e-13,
2.9934720198063397094916415927e-13,
-9.8026575909753445931073620469e-14,
3.2116773667767153777571410671e-14,
-1.0518035333878147029650507254e-14,
3.4144405720185253938994854173e-15,
-1.0115153943081187052322643819e-15
};
static cheb_series gstar_a_cs = {
gstar_a_data,
29,
-1, 1,
17
};
/* Chebyshev coefficients for
* x^2(Gamma*(x) - 1 - 1/(12x)), x = 4(t+1)+2, -1 < t < 1
*/
static double gstar_b_data[] = {
0.0057502277273114339831606096782,
0.0004496689534965685038254147807,
-0.0001672763153188717308905047405,
0.0000615137014913154794776670946,
-0.0000223726551711525016380862195,
8.0507405356647954540694800545e-06,
-2.8671077107583395569766746448e-06,
1.0106727053742747568362254106e-06,
-3.5265558477595061262310873482e-07,
1.2179216046419401193247254591e-07,
-4.1619640180795366971160162267e-08,
1.4066283500795206892487241294e-08,
-4.6982570380537099016106141654e-09,
1.5491248664620612686423108936e-09,
-5.0340936319394885789686867772e-10,
1.6084448673736032249959475006e-10,
-5.0349733196835456497619787559e-11,
1.5357154939762136997591808461e-11,
-4.5233809655775649997667176224e-12,
1.2664429179254447281068538964e-12,
-3.2648287937449326771785041692e-13,
7.1528272726086133795579071407e-14,
-9.4831735252566034505739531258e-15,
-2.3124001991413207293120906691e-15,
2.8406613277170391482590129474e-15,
-1.7245370321618816421281770927e-15,
8.6507923128671112154695006592e-16,
-3.9506563665427555895391869919e-16,
1.6779342132074761078792361165e-16,
-6.0483153034414765129837716260e-17
};
static cheb_series gstar_b_cs = {
gstar_b_data,
29,
-1, 1,
18
};
/* coefficients for gamma=7, kmax=8 Lanczos method */
static double lanczos_7_c[9] = {
0.99999999999980993227684700473478,
676.520368121885098567009190444019,
-1259.13921672240287047156078755283,
771.3234287776530788486528258894,
-176.61502916214059906584551354,
12.507343278686904814458936853,
-0.13857109526572011689554707,
9.984369578019570859563e-6,
1.50563273514931155834e-7
};
/* complex version of Lanczos method; this is not safe for export
* since it becomes bad in the left half-plane
*/
static
int
lngamma_lanczos_complex(double zr, double zi, gsl_sf_result * yr, gsl_sf_result * yi)
{
int k;
gsl_sf_result log1_r, log1_i;
gsl_sf_result logAg_r, logAg_i;
double Ag_r, Ag_i;
double yi_tmp_val, yi_tmp_err;
zr -= 1.0; /* Lanczos writes z! instead of Gamma(z) */
Ag_r = lanczos_7_c[0];
Ag_i = 0.0;
for(k=1; k<=8; k++) {
double R = zr + k;
double I = zi;
double a = lanczos_7_c[k] / (R*R + I*I);
Ag_r += a * R;
Ag_i -= a * I;
}
gsl_sf_complex_log_e(zr + 7.5, zi, &log1_r, &log1_i);
gsl_sf_complex_log_e(Ag_r, Ag_i, &logAg_r, &logAg_i);
/* (z+0.5)*log(z+7.5) - (z+7.5) + LogRootTwoPi_ + log(Ag(z)) */
yr->val = (zr+0.5)*log1_r.val - zi*log1_i.val - (zr+7.5) + LogRootTwoPi_ + logAg_r.val;
yi->val = zi*log1_r.val + (zr+0.5)*log1_i.val - zi + logAg_i.val;
yr->err = 4.0 * GSL_DBL_EPSILON * fabs(yr->val);
yi->err = 4.0 * GSL_DBL_EPSILON * fabs(yi->val);
yi_tmp_val = yi->val;
yi_tmp_err = yi->err;
gsl_sf_angle_restrict_symm_err_e(yi_tmp_val, yi);
yi->err += yi_tmp_err;
return GSL_SUCCESS;
}
/* Lanczos method for real x > 0;
* gamma=7, truncated at 1/(z+8)
* [J. SIAM Numer. Anal, Ser. B, 1 (1964) 86]
*/
static
int
lngamma_lanczos(double x, gsl_sf_result * result)
{
int k;
double Ag;
double term1, term2;
x -= 1.0; /* Lanczos writes z! instead of Gamma(z) */
Ag = lanczos_7_c[0];
for(k=1; k<=8; k++) { Ag += lanczos_7_c[k]/(x+k); }
/* (x+0.5)*log(x+7.5) - (x+7.5) + LogRootTwoPi_ + log(Ag(x)) */
term1 = (x+0.5)*log((x+7.5)/M_E);
term2 = LogRootTwoPi_ + log(Ag);
result->val = term1 + (term2 - 7.0);
result->err = 2.0 * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2) + 7.0);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/* x = eps near zero
* gives double-precision for |eps| < 0.02
*/
static
int
lngamma_sgn_0(double eps, gsl_sf_result * lng, double * sgn)
{
/* calculate series for g(eps) = Gamma(eps) eps - 1/(1+eps) - eps/2 */
const double c1 = -0.07721566490153286061;
const double c2 = -0.01094400467202744461;
const double c3 = 0.09252092391911371098;
const double c4 = -0.01827191316559981266;
const double c5 = 0.01800493109685479790;
const double c6 = -0.00685088537872380685;
const double c7 = 0.00399823955756846603;
const double c8 = -0.00189430621687107802;
const double c9 = 0.00097473237804513221;
const double c10 = -0.00048434392722255893;
const double g6 = c6+eps*(c7+eps*(c8 + eps*(c9 + eps*c10)));
const double g = eps*(c1+eps*(c2+eps*(c3+eps*(c4+eps*(c5+eps*g6)))));
/* calculate Gamma(eps) eps, a positive quantity */
const double gee = g + 1.0/(1.0+eps) + 0.5*eps;
lng->val = log(gee/fabs(eps));
lng->err = 4.0 * GSL_DBL_EPSILON * fabs(lng->val);
*sgn = GSL_SIGN(eps);
return GSL_SUCCESS;
}
/* x near a negative integer
* Calculates sign as well as log(|gamma(x)|).
* x = -N + eps
* assumes N >= 1
*/
static
int
lngamma_sgn_sing(int N, double eps, gsl_sf_result * lng, double * sgn)
{
if(eps == 0.0) {
lng->val = 0.0;
lng->err = 0.0;
*sgn = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else if(N == 1) {
/* calculate series for
* g = eps gamma(-1+eps) + 1 + eps/2 (1+3eps)/(1-eps^2)
* double-precision for |eps| < 0.02
*/
const double c0 = 0.07721566490153286061;
const double c1 = 0.08815966957356030521;
const double c2 = -0.00436125434555340577;
const double c3 = 0.01391065882004640689;
const double c4 = -0.00409427227680839100;
const double c5 = 0.00275661310191541584;
const double c6 = -0.00124162645565305019;
const double c7 = 0.00065267976121802783;
const double c8 = -0.00032205261682710437;
const double c9 = 0.00016229131039545456;
const double g5 = c5 + eps*(c6 + eps*(c7 + eps*(c8 + eps*c9)));
const double g = eps*(c0 + eps*(c1 + eps*(c2 + eps*(c3 + eps*(c4 + eps*g5)))));
/* calculate eps gamma(-1+eps), a negative quantity */
const double gam_e = g - 1.0 - 0.5*eps*(1.0+3.0*eps)/(1.0 - eps*eps);
lng->val = log(fabs(gam_e)/fabs(eps));
lng->err = 2.0 * GSL_DBL_EPSILON * fabs(lng->val);
*sgn = ( eps > 0.0 ? -1.0 : 1.0 );
return GSL_SUCCESS;
}
else {
double g;
/* series for sin(Pi(N+1-eps))/(Pi eps) modulo the sign
* double-precision for |eps| < 0.02
*/
const double cs1 = -1.6449340668482264365;
const double cs2 = 0.8117424252833536436;
const double cs3 = -0.1907518241220842137;
const double cs4 = 0.0261478478176548005;
const double cs5 = -0.0023460810354558236;
const double e2 = eps*eps;
const double sin_ser = 1.0 + e2*(cs1+e2*(cs2+e2*(cs3+e2*(cs4+e2*cs5))));
/* calculate series for ln(gamma(1+N-eps))
* double-precision for |eps| < 0.02
*/
double aeps = fabs(eps);
double c1, c2, c3, c4, c5, c6, c7;
double lng_ser;
gsl_sf_result c0;
gsl_sf_result psi_0;
gsl_sf_result psi_1;
gsl_sf_result psi_2;
gsl_sf_result psi_3;
gsl_sf_result psi_4;
gsl_sf_result psi_5;
gsl_sf_result psi_6;
psi_2.val = 0.0;
psi_3.val = 0.0;
psi_4.val = 0.0;
psi_5.val = 0.0;
psi_6.val = 0.0;
gsl_sf_lnfact_e(N, &c0);
gsl_sf_psi_int_e(N+1, &psi_0);
gsl_sf_psi_1_int_e(N+1, &psi_1);
if(aeps > 0.00001) gsl_sf_psi_n_e(2, N+1.0, &psi_2);
if(aeps > 0.0002) gsl_sf_psi_n_e(3, N+1.0, &psi_3);
if(aeps > 0.001) gsl_sf_psi_n_e(4, N+1.0, &psi_4);
if(aeps > 0.005) gsl_sf_psi_n_e(5, N+1.0, &psi_5);
if(aeps > 0.01) gsl_sf_psi_n_e(6, N+1.0, &psi_6);
c1 = psi_0.val;
c2 = psi_1.val/2.0;
c3 = psi_2.val/6.0;
c4 = psi_3.val/24.0;
c5 = psi_4.val/120.0;
c6 = psi_5.val/720.0;
c7 = psi_6.val/5040.0;
lng_ser = c0.val-eps*(c1-eps*(c2-eps*(c3-eps*(c4-eps*(c5-eps*(c6-eps*c7))))));
/* calculate
* g = ln(|eps gamma(-N+eps)|)
* = -ln(gamma(1+N-eps)) + ln(|eps Pi/sin(Pi(N+1+eps))|)
*/
g = -lng_ser - log(sin_ser);
lng->val = g - log(fabs(eps));
lng->err = c0.err + 2.0 * GSL_DBL_EPSILON * (fabs(g) + fabs(lng->val));
*sgn = ( GSL_IS_ODD(N) ? -1.0 : 1.0 ) * ( eps > 0.0 ? 1.0 : -1.0 );
return GSL_SUCCESS;
}
}
/* This gets bad near the negative half axis. However, this
* region can be avoided by use of the reflection formula, as usual.
* Only the first two terms of the series are kept.
*/
#if 0
static
int
lngamma_complex_stirling(const double zr, const double zi, double * lg_r, double * arg)
{
double re_zinv, im_zinv;
double re_zinv2, im_zinv2;
double re_zinv3, im_zinv3;
double re_zhlnz, im_zhlnz;
double r, lnr, theta;
gsl_sf_complex_log_e(zr, zi, &lnr, &theta); /* z = r e^{i theta} */
r = exp(lnr);
re_zinv = (zr/r)/r;
im_zinv = -(zi/r)/r;
re_zinv2 = re_zinv*re_zinv - im_zinv*im_zinv;
re_zinv2 = 2.0*re_zinv*im_zinv;
re_zinv3 = re_zinv2*re_zinv - im_zinv2*im_zinv;
re_zinv3 = re_zinv2*im_zinv + im_zinv2*re_zinv;
re_zhlnz = (zr - 0.5)*lnr - zi*theta;
im_zhlnz = zi*lnr + zr*theta;
*lg_r = re_zhlnz - zr + 0.5*(M_LN2+M_LNPI) + re_zinv/12.0 - re_zinv3/360.0;
*arg = im_zhlnz - zi + 1.0/12.0*im_zinv - im_zinv3/360.0;
return GSL_SUCCESS;
}
#endif /* 0 */
inline
static
int
lngamma_1_pade(const double eps, gsl_sf_result * result)
{
/* Use (2,2) Pade for Log[Gamma[1+eps]]/eps
* plus a correction series.
*/
const double n1 = -1.0017419282349508699871138440;
const double n2 = 1.7364839209922879823280541733;
const double d1 = 1.2433006018858751556055436011;
const double d2 = 5.0456274100274010152489597514;
const double num = (eps + n1) * (eps + n2);
const double den = (eps + d1) * (eps + d2);
const double pade = 2.0816265188662692474880210318 * num / den;
const double c0 = 0.004785324257581753;
const double c1 = -0.01192457083645441;
const double c2 = 0.01931961413960498;
const double c3 = -0.02594027398725020;
const double c4 = 0.03141928755021455;
const double eps5 = eps*eps*eps*eps*eps;
const double corr = eps5 * (c0 + eps*(c1 + eps*(c2 + eps*(c3 + c4*eps))));
result->val = eps * (pade + corr);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
inline
static
int
lngamma_2_pade(const double eps, gsl_sf_result * result)
{
/* Use (2,2) Pade for Log[Gamma[2+eps]]/eps
* plus a correction series.
*/
const double n1 = 1.000895834786669227164446568;
const double n2 = 4.209376735287755081642901277;
const double d1 = 2.618851904903217274682578255;
const double d2 = 10.85766559900983515322922936;
const double num = (eps + n1) * (eps + n2);
const double den = (eps + d1) * (eps + d2);
const double pade = 2.85337998765781918463568869 * num/den;
const double c0 = 0.0001139406357036744;
const double c1 = -0.0001365435269792533;
const double c2 = 0.0001067287169183665;
const double c3 = -0.0000693271800931282;
const double c4 = 0.0000407220927867950;
const double eps5 = eps*eps*eps*eps*eps;
const double corr = eps5 * (c0 + eps*(c1 + eps*(c2 + eps*(c3 + c4*eps))));
result->val = eps * (pade + corr);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/* series for gammastar(x)
* double-precision for x > 10.0
*/
static
int
gammastar_ser(const double x, gsl_sf_result * result)
{
/* Use the Stirling series for the correction to Log(Gamma(x)),
* which is better behaved and easier to compute than the
* regular Stirling series for Gamma(x).
*/
const double y = 1.0/(x*x);
const double c0 = 1.0/12.0;
const double c1 = -1.0/360.0;
const double c2 = 1.0/1260.0;
const double c3 = -1.0/1680.0;
const double c4 = 1.0/1188.0;
const double c5 = -691.0/360360.0;
const double c6 = 1.0/156.0;
const double c7 = -3617.0/122400.0;
const double ser = c0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7))))));
result->val = exp(ser/x);
result->err = 2.0 * GSL_DBL_EPSILON * result->val * GSL_MAX_DBL(1.0, ser/x);
return GSL_SUCCESS;
}
/* Chebyshev expansion for log(gamma(x)/gamma(8))
* 5 < x < 10
* -1 < t < 1
*/
static double gamma_5_10_data[24] = {
-1.5285594096661578881275075214,
4.8259152300595906319768555035,
0.2277712320977614992970601978,
-0.0138867665685617873604917300,
0.0012704876495201082588139723,
-0.0001393841240254993658962470,
0.0000169709242992322702260663,
-2.2108528820210580075775889168e-06,
3.0196602854202309805163918716e-07,
-4.2705675000079118380587357358e-08,
6.2026423818051402794663551945e-09,
-9.1993973208880910416311405656e-10,
1.3875551258028145778301211638e-10,
-2.1218861491906788718519522978e-11,
3.2821736040381439555133562600e-12,
-5.1260001009953791220611135264e-13,
8.0713532554874636696982146610e-14,
-1.2798522376569209083811628061e-14,
2.0417711600852502310258808643e-15,
-3.2745239502992355776882614137e-16,
5.2759418422036579482120897453e-17,
-8.5354147151695233960425725513e-18,
1.3858639703888078291599886143e-18,
-2.2574398807738626571560124396e-19
};
static const cheb_series gamma_5_10_cs = {
gamma_5_10_data,
23,
-1, 1,
11
};
/* gamma(x) for x >= 1/2
* assumes x >= 1/2
*/
static
int
gamma_xgthalf(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x == 0.5) {
result->val = 1.77245385090551602729817;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
} else if (x <= (GSL_SF_FACT_NMAX + 1.0) && x == floor(x)) {
int n = (int) floor (x);
result->val = fact_table[n - 1].f;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(fabs(x - 1.0) < 0.01) {
/* Use series for Gamma[1+eps] - 1/(1+eps).
*/
const double eps = x - 1.0;
const double c1 = 0.4227843350984671394;
const double c2 = -0.01094400467202744461;
const double c3 = 0.09252092391911371098;
const double c4 = -0.018271913165599812664;
const double c5 = 0.018004931096854797895;
const double c6 = -0.006850885378723806846;
const double c7 = 0.003998239557568466030;
result->val = 1.0/x + eps*(c1+eps*(c2+eps*(c3+eps*(c4+eps*(c5+eps*(c6+eps*c7))))));
result->err = GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(fabs(x - 2.0) < 0.01) {
/* Use series for Gamma[1 + eps].
*/
const double eps = x - 2.0;
const double c1 = 0.4227843350984671394;
const double c2 = 0.4118403304264396948;
const double c3 = 0.08157691924708626638;
const double c4 = 0.07424901075351389832;
const double c5 = -0.00026698206874501476832;
const double c6 = 0.011154045718130991049;
const double c7 = -0.002852645821155340816;
const double c8 = 0.0021039333406973880085;
result->val = 1.0 + eps*(c1+eps*(c2+eps*(c3+eps*(c4+eps*(c5+eps*(c6+eps*(c7+eps*c8)))))));
result->err = GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(x < 5.0) {
/* Exponentiating the logarithm is fine, as
* long as the exponential is not so large
* that it greatly amplifies the error.
*/
gsl_sf_result lg;
lngamma_lanczos(x, &lg);
result->val = exp(lg.val);
result->err = result->val * (lg.err + 2.0 * GSL_DBL_EPSILON);
return GSL_SUCCESS;
}
else if(x < 10.0) {
/* This is a sticky area. The logarithm
* is too large and the gammastar series
* is not good.
*/
const double gamma_8 = 5040.0;
const double t = (2.0*x - 15.0)/5.0;
gsl_sf_result c;
cheb_eval_e(&gamma_5_10_cs, t, &c);
result->val = exp(c.val) * gamma_8;
result->err = result->val * c.err;
result->err += 2.0 * GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x < GSL_SF_GAMMA_XMAX) {
/* We do not want to exponentiate the logarithm
* if x is large because of the inevitable
* inflation of the error. So we carefully
* use pow() and exp() with exact quantities.
*/
double p = pow(x, 0.5*x);
double e = exp(-x);
double q = (p * e) * p;
double pre = M_SQRT2 * M_SQRTPI * q/sqrt(x);
gsl_sf_result gstar;
int stat_gs = gammastar_ser(x, &gstar);
result->val = pre * gstar.val;
result->err = (x + 2.5) * GSL_DBL_EPSILON * result->val;
return stat_gs;
}
else {
OVERFLOW_ERROR(result);
}
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int gsl_sf_lngamma_e(double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(fabs(x - 1.0) < 0.01) {
/* Note that we must amplify the errors
* from the Pade evaluations because of
* the way we must pass the argument, i.e.
* writing (1-x) is a loss of precision
* when x is near 1.
*/
int stat = lngamma_1_pade(x - 1.0, result);
result->err *= 1.0/(GSL_DBL_EPSILON + fabs(x - 1.0));
return stat;
}
else if(fabs(x - 2.0) < 0.01) {
int stat = lngamma_2_pade(x - 2.0, result);
result->err *= 1.0/(GSL_DBL_EPSILON + fabs(x - 2.0));
return stat;
}
else if(x >= 0.5) {
return lngamma_lanczos(x, result);
}
else if(x == 0.0) {
DOMAIN_ERROR(result);
}
else if(fabs(x) < 0.02) {
double sgn;
return lngamma_sgn_0(x, result, &sgn);
}
else if(x > -0.5/(GSL_DBL_EPSILON*M_PI)) {
/* Try to extract a fractional
* part from x.
*/
double z = 1.0 - x;
double s = sin(M_PI*z);
double as = fabs(s);
if(s == 0.0) {
DOMAIN_ERROR(result);
}
else if(as < M_PI*0.015) {
/* x is near a negative integer, -N */
if(x < INT_MIN + 2.0) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EROUND);
}
else {
int N = -(int)(x - 0.5);
double eps = x + N;
double sgn;
return lngamma_sgn_sing(N, eps, result, &sgn);
}
}
else {
gsl_sf_result lg_z;
lngamma_lanczos(z, &lg_z);
result->val = M_LNPI - (log(as) + lg_z.val);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + lg_z.err;
return GSL_SUCCESS;
}
}
else {
/* |x| was too large to extract any fractional part */
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EROUND);
}
}
int gsl_sf_lngamma_sgn_e(double x, gsl_sf_result * result_lg, double * sgn)
{
if(fabs(x - 1.0) < 0.01) {
int stat = lngamma_1_pade(x - 1.0, result_lg);
result_lg->err *= 1.0/(GSL_DBL_EPSILON + fabs(x - 1.0));
*sgn = 1.0;
return stat;
}
else if(fabs(x - 2.0) < 0.01) {
int stat = lngamma_2_pade(x - 2.0, result_lg);
result_lg->err *= 1.0/(GSL_DBL_EPSILON + fabs(x - 2.0));
*sgn = 1.0;
return stat;
}
else if(x >= 0.5) {
*sgn = 1.0;
return lngamma_lanczos(x, result_lg);
}
else if(x == 0.0) {
*sgn = 0.0;
DOMAIN_ERROR(result_lg);
}
else if(fabs(x) < 0.02) {
return lngamma_sgn_0(x, result_lg, sgn);
}
else if(x > -0.5/(GSL_DBL_EPSILON*M_PI)) {
/* Try to extract a fractional
* part from x.
*/
double z = 1.0 - x;
double s = sin(M_PI*x);
double as = fabs(s);
if(s == 0.0) {
*sgn = 0.0;
DOMAIN_ERROR(result_lg);
}
else if(as < M_PI*0.015) {
/* x is near a negative integer, -N */
if(x < INT_MIN + 2.0) {
result_lg->val = 0.0;
result_lg->err = 0.0;
*sgn = 0.0;
GSL_ERROR ("error", GSL_EROUND);
}
else {
int N = -(int)(x - 0.5);
double eps = x + N;
return lngamma_sgn_sing(N, eps, result_lg, sgn);
}
}
else {
gsl_sf_result lg_z;
lngamma_lanczos(z, &lg_z);
*sgn = (s > 0.0 ? 1.0 : -1.0);
result_lg->val = M_LNPI - (log(as) + lg_z.val);
result_lg->err = 2.0 * GSL_DBL_EPSILON * fabs(result_lg->val) + lg_z.err;
return GSL_SUCCESS;
}
}
else {
/* |x| was too large to extract any fractional part */
result_lg->val = 0.0;
result_lg->err = 0.0;
*sgn = 0.0;
GSL_ERROR ("x too large to extract fraction part", GSL_EROUND);
}
}
int
gsl_sf_gamma_e(const double x, gsl_sf_result * result)
{
if(x < 0.5) {
int rint_x = (int)floor(x+0.5);
double f_x = x - rint_x;
double sgn_gamma = ( GSL_IS_EVEN(rint_x) ? 1.0 : -1.0 );
double sin_term = sgn_gamma * sin(M_PI * f_x) / M_PI;
if(sin_term == 0.0) {
DOMAIN_ERROR(result);
}
else if(x > -169.0) {
gsl_sf_result g;
gamma_xgthalf(1.0-x, &g);
if(fabs(sin_term) * g.val * GSL_DBL_MIN < 1.0) {
result->val = 1.0/(sin_term * g.val);
result->err = fabs(g.err/g.val) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
UNDERFLOW_ERROR(result);
}
}
else {
/* It is hard to control it here.
* We can only exponentiate the
* logarithm and eat the loss of
* precision.
*/
gsl_sf_result lng;
double sgn;
int stat_lng = gsl_sf_lngamma_sgn_e(x, &lng, &sgn);
int stat_e = gsl_sf_exp_mult_err_e(lng.val, lng.err, sgn, 0.0, result);
return GSL_ERROR_SELECT_2(stat_e, stat_lng);
}
}
else {
return gamma_xgthalf(x, result);
}
}
int
gsl_sf_gammastar_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 0.5) {
gsl_sf_result lg;
const int stat_lg = gsl_sf_lngamma_e(x, &lg);
const double lx = log(x);
const double c = 0.5*(M_LN2+M_LNPI);
const double lnr_val = lg.val - (x-0.5)*lx + x - c;
const double lnr_err = lg.err + 2.0 * GSL_DBL_EPSILON *((x+0.5)*fabs(lx) + c);
const int stat_e = gsl_sf_exp_err_e(lnr_val, lnr_err, result);
return GSL_ERROR_SELECT_2(stat_lg, stat_e);
}
else if(x < 2.0) {
const double t = 4.0/3.0*(x-0.5) - 1.0;
return cheb_eval_e(&gstar_a_cs, t, result);
}
else if(x < 10.0) {
const double t = 0.25*(x-2.0) - 1.0;
gsl_sf_result c;
cheb_eval_e(&gstar_b_cs, t, &c);
result->val = c.val/(x*x) + 1.0 + 1.0/(12.0*x);
result->err = c.err/(x*x);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 1.0/GSL_ROOT4_DBL_EPSILON) {
return gammastar_ser(x, result);
}
else if(x < 1.0/GSL_DBL_EPSILON) {
/* Use Stirling formula for Gamma(x).
*/
const double xi = 1.0/x;
result->val = 1.0 + xi/12.0*(1.0 + xi/24.0*(1.0 - xi*(139.0/180.0 + 571.0/8640.0*xi)));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
result->val = 1.0;
result->err = 1.0/x;
return GSL_SUCCESS;
}
}
int
gsl_sf_gammainv_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if (x <= 0.0 && x == floor(x)) { /* negative integer */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
} else if(x < 0.5) {
gsl_sf_result lng;
double sgn;
int stat_lng = gsl_sf_lngamma_sgn_e(x, &lng, &sgn);
if(stat_lng == GSL_EDOM) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(stat_lng != GSL_SUCCESS) {
result->val = 0.0;
result->err = 0.0;
return stat_lng;
}
else {
return gsl_sf_exp_mult_err_e(-lng.val, lng.err, sgn, 0.0, result);
}
}
else {
gsl_sf_result g;
int stat_g = gamma_xgthalf(x, &g);
if(stat_g == GSL_EOVRFLW) {
UNDERFLOW_ERROR(result);
}
else {
result->val = 1.0/g.val;
result->err = fabs(g.err/g.val) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
}
int
gsl_sf_lngamma_complex_e(double zr, double zi, gsl_sf_result * lnr, gsl_sf_result * arg)
{
if(zr <= 0.5) {
/* Transform to right half plane using reflection;
* in fact we do a little better by stopping at 1/2.
*/
double x = 1.0-zr;
double y = -zi;
gsl_sf_result a, b;
gsl_sf_result lnsin_r, lnsin_i;
int stat_l = lngamma_lanczos_complex(x, y, &a, &b);
int stat_s = gsl_sf_complex_logsin_e(M_PI*zr, M_PI*zi, &lnsin_r, &lnsin_i);
if(stat_s == GSL_SUCCESS) {
int stat_r;
lnr->val = M_LNPI - lnsin_r.val - a.val;
lnr->err = lnsin_r.err + a.err + 2.0 * GSL_DBL_EPSILON * fabs(lnr->val);
arg->val = -lnsin_i.val - b.val;
arg->err = lnsin_i.err + b.err + 2.0 * GSL_DBL_EPSILON * fabs(arg->val);
stat_r = gsl_sf_angle_restrict_symm_e(&(arg->val));
return GSL_ERROR_SELECT_2(stat_r, stat_l);
}
else {
DOMAIN_ERROR_2(lnr,arg);
}
}
else {
/* otherwise plain vanilla Lanczos */
return lngamma_lanczos_complex(zr, zi, lnr, arg);
}
}
int gsl_sf_taylorcoeff_e(const int n, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x < 0.0 || n < 0) {
DOMAIN_ERROR(result);
}
else if(n == 0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(n == 1) {
result->val = x;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
const double log2pi = M_LNPI + M_LN2;
const double ln_test = n*(log(x)+1.0) + 1.0 - (n+0.5)*log(n+1.0) + 0.5*log2pi;
if(ln_test < GSL_LOG_DBL_MIN+1.0) {
UNDERFLOW_ERROR(result);
}
else if(ln_test > GSL_LOG_DBL_MAX-1.0) {
OVERFLOW_ERROR(result);
}
else {
double product = 1.0;
int k;
for(k=1; k<=n; k++) {
product *= (x/k);
}
result->val = product;
result->err = n * GSL_DBL_EPSILON * product;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
}
int gsl_sf_fact_e(const unsigned int n, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(n < 18) {
result->val = fact_table[n].f;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(n <= GSL_SF_FACT_NMAX){
result->val = fact_table[n].f;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
int gsl_sf_doublefact_e(const unsigned int n, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(n < 26) {
result->val = doub_fact_table[n].f;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(n <= GSL_SF_DOUBLEFACT_NMAX){
result->val = doub_fact_table[n].f;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
int gsl_sf_lnfact_e(const unsigned int n, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(n <= GSL_SF_FACT_NMAX){
result->val = log(fact_table[n].f);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_lngamma_e(n+1.0, result);
return GSL_SUCCESS;
}
}
int gsl_sf_lndoublefact_e(const unsigned int n, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(n <= GSL_SF_DOUBLEFACT_NMAX){
result->val = log(doub_fact_table[n].f);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(GSL_IS_ODD(n)) {
gsl_sf_result lg;
gsl_sf_lngamma_e(0.5*(n+2.0), &lg);
result->val = 0.5*(n+1.0) * M_LN2 - 0.5*M_LNPI + lg.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + lg.err;
return GSL_SUCCESS;
}
else {
gsl_sf_result lg;
gsl_sf_lngamma_e(0.5*n+1.0, &lg);
result->val = 0.5*n*M_LN2 + lg.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + lg.err;
return GSL_SUCCESS;
}
}
int gsl_sf_lnchoose_e(unsigned int n, unsigned int m, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(m > n) {
DOMAIN_ERROR(result);
}
else if(m == n || m == 0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
gsl_sf_result nf;
gsl_sf_result mf;
gsl_sf_result nmmf;
if(m*2 > n) m = n-m;
gsl_sf_lnfact_e(n, &nf);
gsl_sf_lnfact_e(m, &mf);
gsl_sf_lnfact_e(n-m, &nmmf);
result->val = nf.val - mf.val - nmmf.val;
result->err = nf.err + mf.err + nmmf.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int gsl_sf_choose_e(unsigned int n, unsigned int m, gsl_sf_result * result)
{
if(m > n) {
DOMAIN_ERROR(result);
}
else if(m == n || m == 0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if (n <= GSL_SF_FACT_NMAX) {
result->val = (fact_table[n].f / fact_table[m].f) / fact_table[n-m].f;
result->err = 6.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
} else {
if(m*2 < n) m = n-m;
if (n - m < 64) /* compute product for a manageable number of terms */
{
double prod = 1.0;
unsigned int k;
for(k=n; k>=m+1; k--) {
double tk = (double)k / (double)(k-m);
if(tk > GSL_DBL_MAX/prod) {
OVERFLOW_ERROR(result);
}
prod *= tk;
}
result->val = prod;
result->err = 2.0 * GSL_DBL_EPSILON * prod * fabs(n-m);
return GSL_SUCCESS;
}
else
{
gsl_sf_result lc;
const int stat_lc = gsl_sf_lnchoose_e (n, m, &lc);
const int stat_e = gsl_sf_exp_err_e(lc.val, lc.err, result);
return GSL_ERROR_SELECT_2(stat_lc, stat_e);
}
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_fact(const unsigned int n)
{
EVAL_RESULT(gsl_sf_fact_e(n, &result));
}
double gsl_sf_lnfact(const unsigned int n)
{
EVAL_RESULT(gsl_sf_lnfact_e(n, &result));
}
double gsl_sf_doublefact(const unsigned int n)
{
EVAL_RESULT(gsl_sf_doublefact_e(n, &result));
}
double gsl_sf_lndoublefact(const unsigned int n)
{
EVAL_RESULT(gsl_sf_lndoublefact_e(n, &result));
}
double gsl_sf_lngamma(const double x)
{
EVAL_RESULT(gsl_sf_lngamma_e(x, &result));
}
double gsl_sf_gamma(const double x)
{
EVAL_RESULT(gsl_sf_gamma_e(x, &result));
}
double gsl_sf_gammastar(const double x)
{
EVAL_RESULT(gsl_sf_gammastar_e(x, &result));
}
double gsl_sf_gammainv(const double x)
{
EVAL_RESULT(gsl_sf_gammainv_e(x, &result));
}
double gsl_sf_taylorcoeff(const int n, const double x)
{
EVAL_RESULT(gsl_sf_taylorcoeff_e(n, x, &result));
}
double gsl_sf_choose(unsigned int n, unsigned int m)
{
EVAL_RESULT(gsl_sf_choose_e(n, m, &result));
}
double gsl_sf_lnchoose(unsigned int n, unsigned int m)
{
EVAL_RESULT(gsl_sf_lnchoose_e(n, m, &result));
}