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author Shinji KONO Mon, 25 May 2020 18:13:55 +0900 1830386684a0
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/*                                                      lgammal
*
*      Natural logarithm of gamma function
*
*
*
* SYNOPSIS:
*
* long double x, y, lgammal();
* extern int sgngam;
*
* y = lgammal(x);
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of the absolute
* value of the gamma function of the argument.
* The sign (+1 or -1) of the gamma function is returned in a
* global (extern) variable named sgngam.
*
* The positive domain is partitioned into numerous segments for approximation.
* For x > 10,
*   log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2)
* Near the minimum at x = x0 = 1.46... the approximation is
*   log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z)
* for small z.
* Elsewhere between 0 and 10,
*   log gamma(n + z) = log gamma(n) + z P(z)/Q(z)
* for various selected n and small z.
*
* The cosecant reflection formula is employed for negative arguments.
*
*
*
* ACCURACY:
*
*
* arithmetic      domain        # trials     peak         rms
*                                            Relative error:
*    IEEE         10, 30         100000     3.9e-34     9.8e-35
*    IEEE          0, 10         100000     3.8e-34     5.3e-35
*                                            Absolute error:
*    IEEE         -10, 0         100000     8.0e-34     8.0e-35
*    IEEE         -30, -10       100000     4.4e-34     1.0e-34
*    IEEE        -100, 100       100000                 1.0e-34
*
* The absolute error criterion is the same as relative error
* when the function magnitude is greater than one but it is absolute
* when the magnitude is less than one.
*
*/

/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
version 2.1 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see

#ifdef HAVE_MATH_H_SIGNGAM
# include <math.h>
#endif
__float128
lgammaq (__float128 x)
{
#ifndef HAVE_MATH_H_SIGNGAM
int signgam;
#endif
}

static const __float128 PIL = 3.1415926535897932384626433832795028841972E0Q;
static const __float128 MAXLGM = 1.0485738685148938358098967157129705071571E4928Q;
static const __float128 one = 1;
static const __float128 huge = FLT128_MAX;

/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
1/x <= 0.0741 (x >= 13.495...)
Peak relative error 1.5e-36  */
static const __float128 ls2pi = 9.1893853320467274178032973640561763986140E-1Q;
#define NRASY 12
static const __float128 RASY[NRASY + 1] =
{
8.333333333333333333333333333310437112111E-2Q,
-2.777777777777777777777774789556228296902E-3Q,
7.936507936507936507795933938448586499183E-4Q,
-5.952380952380952041799269756378148574045E-4Q,
8.417508417507928904209891117498524452523E-4Q,
-1.917526917481263997778542329739806086290E-3Q,
6.410256381217852504446848671499409919280E-3Q,
-2.955064066900961649768101034477363301626E-2Q,
1.796402955865634243663453415388336954675E-1Q,
-1.391522089007758553455753477688592767741E0Q,
1.326130089598399157988112385013829305510E1Q,
-1.420412699593782497803472576479997819149E2Q,
1.218058922427762808938869872528846787020E3Q
};

/* log gamma(x+13) = log gamma(13) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
12.5 <= x+13 <= 13.5
Peak relative error 1.1e-36  */
static const __float128 lgam13a = 1.9987213134765625E1Q;
static const __float128 lgam13b = 1.3608962611495173623870550785125024484248E-6Q;
#define NRN13 7
static const __float128 RN13[NRN13 + 1] =
{
8.591478354823578150238226576156275285700E11Q,
2.347931159756482741018258864137297157668E11Q,
2.555408396679352028680662433943000804616E10Q,
1.408581709264464345480765758902967123937E9Q,
4.126759849752613822953004114044451046321E7Q,
6.133298899622688505854211579222889943778E5Q,
3.929248056293651597987893340755876578072E3Q,
6.850783280018706668924952057996075215223E0Q
};
#define NRD13 6
static const __float128 RD13[NRD13 + 1] =
{
3.401225382297342302296607039352935541669E11Q,
8.756765276918037910363513243563234551784E10Q,
8.873913342866613213078554180987647243903E9Q,
4.483797255342763263361893016049310017973E8Q,
1.178186288833066430952276702931512870676E7Q,
1.519928623743264797939103740132278337476E5Q,
7.989298844938119228411117593338850892311E2Q
/* 1.0E0L */
};

/* log gamma(x+12) = log gamma(12) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
11.5 <= x+12 <= 12.5
Peak relative error 4.1e-36  */
static const __float128 lgam12a = 1.75023040771484375E1Q;
static const __float128 lgam12b = 3.7687254483392876529072161996717039575982E-6Q;
#define NRN12 7
static const __float128 RN12[NRN12 + 1] =
{
4.709859662695606986110997348630997559137E11Q,
1.398713878079497115037857470168777995230E11Q,
1.654654931821564315970930093932954900867E10Q,
9.916279414876676861193649489207282144036E8Q,
3.159604070526036074112008954113411389879E7Q,
5.109099197547205212294747623977502492861E5Q,
3.563054878276102790183396740969279826988E3Q,
6.769610657004672719224614163196946862747E0Q
};
#define NRD12 6
static const __float128 RD12[NRD12 + 1] =
{
1.928167007860968063912467318985802726613E11Q,
5.383198282277806237247492369072266389233E10Q,
5.915693215338294477444809323037871058363E9Q,
3.241438287570196713148310560147925781342E8Q,
9.236680081763754597872713592701048455890E6Q,
1.292246897881650919242713651166596478850E5Q,
7.366532445427159272584194816076600211171E2Q
/* 1.0E0L */
};

/* log gamma(x+11) = log gamma(11) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
10.5 <= x+11 <= 11.5
Peak relative error 1.8e-35  */
static const __float128 lgam11a = 1.5104400634765625E1Q;
static const __float128 lgam11b = 1.1938309890295225709329251070371882250744E-5Q;
#define NRN11 7
static const __float128 RN11[NRN11 + 1] =
{
2.446960438029415837384622675816736622795E11Q,
7.955444974446413315803799763901729640350E10Q,
1.030555327949159293591618473447420338444E10Q,
6.765022131195302709153994345470493334946E8Q,
2.361892792609204855279723576041468347494E7Q,
4.186623629779479136428005806072176490125E5Q,
3.202506022088912768601325534149383594049E3Q,
6.681356101133728289358838690666225691363E0Q
};
#define NRD11 6
static const __float128 RD11[NRD11 + 1] =
{
1.040483786179428590683912396379079477432E11Q,
3.172251138489229497223696648369823779729E10Q,
3.806961885984850433709295832245848084614E9Q,
2.278070344022934913730015420611609620171E8Q,
7.089478198662651683977290023829391596481E6Q,
1.083246385105903533237139380509590158658E5Q,
6.744420991491385145885727942219463243597E2Q
/* 1.0E0L */
};

/* log gamma(x+10) = log gamma(10) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
9.5 <= x+10 <= 10.5
Peak relative error 5.4e-37  */
static const __float128 lgam10a = 1.280181884765625E1Q;
static const __float128 lgam10b = 8.6324252196112077178745667061642811492557E-6Q;
#define NRN10 7
static const __float128 RN10[NRN10 + 1] =
{
-1.239059737177249934158597996648808363783E14Q,
-4.725899566371458992365624673357356908719E13Q,
-7.283906268647083312042059082837754850808E12Q,
-5.802855515464011422171165179767478794637E11Q,
-2.532349691157548788382820303182745897298E10Q,
-5.884260178023777312587193693477072061820E8Q,
-6.437774864512125749845840472131829114906E6Q,
-2.350975266781548931856017239843273049384E4Q
};
#define NRD10 7
static const __float128 RD10[NRD10 + 1] =
{
-5.502645997581822567468347817182347679552E13Q,
-1.970266640239849804162284805400136473801E13Q,
-2.819677689615038489384974042561531409392E12Q,
-2.056105863694742752589691183194061265094E11Q,
-8.053670086493258693186307810815819662078E9Q,
-1.632090155573373286153427982504851867131E8Q,
-1.483575879240631280658077826889223634921E6Q,
-4.002806669713232271615885826373550502510E3Q
/* 1.0E0L */
};

/* log gamma(x+9) = log gamma(9) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
8.5 <= x+9 <= 9.5
Peak relative error 3.6e-36  */
static const __float128 lgam9a = 1.06045989990234375E1Q;
static const __float128 lgam9b = 3.9037218127284172274007216547549861681400E-6Q;
#define NRN9 7
static const __float128 RN9[NRN9 + 1] =
{
-4.936332264202687973364500998984608306189E13Q,
-2.101372682623700967335206138517766274855E13Q,
-3.615893404644823888655732817505129444195E12Q,
-3.217104993800878891194322691860075472926E11Q,
-1.568465330337375725685439173603032921399E10Q,
-4.073317518162025744377629219101510217761E8Q,
-4.983232096406156139324846656819246974500E6Q,
-2.036280038903695980912289722995505277253E4Q
};
#define NRD9 7
static const __float128 RD9[NRD9 + 1] =
{
-2.306006080437656357167128541231915480393E13Q,
-9.183606842453274924895648863832233799950E12Q,
-1.461857965935942962087907301194381010380E12Q,
-1.185728254682789754150068652663124298303E11Q,
-5.166285094703468567389566085480783070037E9Q,
-1.164573656694603024184768200787835094317E8Q,
-1.177343939483908678474886454113163527909E6Q,
-3.529391059783109732159524500029157638736E3Q
/* 1.0E0L */
};

/* log gamma(x+8) = log gamma(8) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
7.5 <= x+8 <= 8.5
Peak relative error 2.4e-37  */
static const __float128 lgam8a = 8.525146484375E0Q;
static const __float128 lgam8b = 1.4876690414300165531036347125050759667737E-5Q;
#define NRN8 8
static const __float128 RN8[NRN8 + 1] =
{
6.600775438203423546565361176829139703289E11Q,
3.406361267593790705240802723914281025800E11Q,
7.222460928505293914746983300555538432830E10Q,
8.102984106025088123058747466840656458342E9Q,
5.157620015986282905232150979772409345927E8Q,
1.851445288272645829028129389609068641517E7Q,
3.489261702223124354745894067468953756656E5Q,
2.892095396706665774434217489775617756014E3Q,
6.596977510622195827183948478627058738034E0Q
};
#define NRD8 7
static const __float128 RD8[NRD8 + 1] =
{
3.274776546520735414638114828622673016920E11Q,
1.581811207929065544043963828487733970107E11Q,
3.108725655667825188135393076860104546416E10Q,
3.193055010502912617128480163681842165730E9Q,
1.830871482669835106357529710116211541839E8Q,
5.790862854275238129848491555068073485086E6Q,
9.305213264307921522842678835618803553589E4Q,
6.216974105861848386918949336819572333622E2Q
/* 1.0E0L */
};

/* log gamma(x+7) = log gamma(7) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
6.5 <= x+7 <= 7.5
Peak relative error 3.2e-36  */
static const __float128 lgam7a = 6.5792388916015625E0Q;
static const __float128 lgam7b = 1.2320408538495060178292903945321122583007E-5Q;
#define NRN7 8
static const __float128 RN7[NRN7 + 1] =
{
2.065019306969459407636744543358209942213E11Q,
1.226919919023736909889724951708796532847E11Q,
2.996157990374348596472241776917953749106E10Q,
3.873001919306801037344727168434909521030E9Q,
2.841575255593761593270885753992732145094E8Q,
1.176342515359431913664715324652399565551E7Q,
2.558097039684188723597519300356028511547E5Q,
2.448525238332609439023786244782810774702E3Q,
6.460280377802030953041566617300902020435E0Q
};
#define NRD7 7
static const __float128 RD7[NRD7 + 1] =
{
1.102646614598516998880874785339049304483E11Q,
6.099297512712715445879759589407189290040E10Q,
1.372898136289611312713283201112060238351E10Q,
1.615306270420293159907951633566635172343E9Q,
1.061114435798489135996614242842561967459E8Q,
3.845638971184305248268608902030718674691E6Q,
7.081730675423444975703917836972720495507E4Q,
5.423122582741398226693137276201344096370E2Q
/* 1.0E0L */
};

/* log gamma(x+6) = log gamma(6) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
5.5 <= x+6 <= 6.5
Peak relative error 6.2e-37  */
static const __float128 lgam6a = 4.7874908447265625E0Q;
static const __float128 lgam6b = 8.9805548349424770093452324304839959231517E-7Q;
#define NRN6 8
static const __float128 RN6[NRN6 + 1] =
{
-3.538412754670746879119162116819571823643E13Q,
-2.613432593406849155765698121483394257148E13Q,
-8.020670732770461579558867891923784753062E12Q,
-1.322227822931250045347591780332435433420E12Q,
-1.262809382777272476572558806855377129513E11Q,
-7.015006277027660872284922325741197022467E9Q,
-2.149320689089020841076532186783055727299E8Q,
-3.167210585700002703820077565539658995316E6Q,
-1.576834867378554185210279285358586385266E4Q
};
#define NRD6 8
static const __float128 RD6[NRD6 + 1] =
{
-2.073955870771283609792355579558899389085E13Q,
-1.421592856111673959642750863283919318175E13Q,
-4.012134994918353924219048850264207074949E12Q,
-6.013361045800992316498238470888523722431E11Q,
-5.145382510136622274784240527039643430628E10Q,
-2.510575820013409711678540476918249524123E9Q,
-6.564058379709759600836745035871373240904E7Q,
-7.861511116647120540275354855221373571536E5Q,
-2.821943442729620524365661338459579270561E3Q
/* 1.0E0L */
};

/* log gamma(x+5) = log gamma(5) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
4.5 <= x+5 <= 5.5
Peak relative error 3.4e-37  */
static const __float128 lgam5a = 3.17803955078125E0Q;
static const __float128 lgam5b = 1.4279566695619646941601297055408873990961E-5Q;
#define NRN5 9
static const __float128 RN5[NRN5 + 1] =
{
2.010952885441805899580403215533972172098E11Q,
1.916132681242540921354921906708215338584E11Q,
7.679102403710581712903937970163206882492E10Q,
1.680514903671382470108010973615268125169E10Q,
2.181011222911537259440775283277711588410E9Q,
1.705361119398837808244780667539728356096E8Q,
7.792391565652481864976147945997033946360E6Q,
1.910741381027985291688667214472560023819E5Q,
2.088138241893612679762260077783794329559E3Q,
6.330318119566998299106803922739066556550E0Q
};
#define NRD5 8
static const __float128 RD5[NRD5 + 1] =
{
1.335189758138651840605141370223112376176E11Q,
1.174130445739492885895466097516530211283E11Q,
4.308006619274572338118732154886328519910E10Q,
8.547402888692578655814445003283720677468E9Q,
9.934628078575618309542580800421370730906E8Q,
6.847107420092173812998096295422311820672E7Q,
2.698552646016599923609773122139463150403E6Q,
5.526516251532464176412113632726150253215E4Q,
4.772343321713697385780533022595450486932E2Q
/* 1.0E0L */
};

/* log gamma(x+4) = log gamma(4) +  x P(x)/Q(x)
-0.5 <= x <= 0.5
3.5 <= x+4 <= 4.5
Peak relative error 6.7e-37  */
static const __float128 lgam4a = 1.791748046875E0Q;
static const __float128 lgam4b = 1.1422353055000812477358380702272722990692E-5Q;
#define NRN4 9
static const __float128 RN4[NRN4 + 1] =
{
-1.026583408246155508572442242188887829208E13Q,
-1.306476685384622809290193031208776258809E13Q,
-7.051088602207062164232806511992978915508E12Q,
-2.100849457735620004967624442027793656108E12Q,
-3.767473790774546963588549871673843260569E11Q,
-4.156387497364909963498394522336575984206E10Q,
-2.764021460668011732047778992419118757746E9Q,
-1.036617204107109779944986471142938641399E8Q,
-1.895730886640349026257780896972598305443E6Q,
-1.180509051468390914200720003907727988201E4Q
};
#define NRD4 9
static const __float128 RD4[NRD4 + 1] =
{
-8.172669122056002077809119378047536240889E12Q,
-9.477592426087986751343695251801814226960E12Q,
-4.629448850139318158743900253637212801682E12Q,
-1.237965465892012573255370078308035272942E12Q,
-1.971624313506929845158062177061297598956E11Q,
-1.905434843346570533229942397763361493610E10Q,
-1.089409357680461419743730978512856675984E9Q,
-3.416703082301143192939774401370222822430E7Q,
-4.981791914177103793218433195857635265295E5Q,
-2.192507743896742751483055798411231453733E3Q
/* 1.0E0L */
};

/* log gamma(x+3) = log gamma(3) +  x P(x)/Q(x)
-0.25 <= x <= 0.5
2.75 <= x+3 <= 3.5
Peak relative error 6.0e-37  */
static const __float128 lgam3a = 6.93145751953125E-1Q;
static const __float128 lgam3b = 1.4286068203094172321214581765680755001344E-6Q;

#define NRN3 9
static const __float128 RN3[NRN3 + 1] =
{
-4.813901815114776281494823863935820876670E11Q,
-8.425592975288250400493910291066881992620E11Q,
-6.228685507402467503655405482985516909157E11Q,
-2.531972054436786351403749276956707260499E11Q,
-6.170200796658926701311867484296426831687E10Q,
-9.211477458528156048231908798456365081135E9Q,
-8.251806236175037114064561038908691305583E8Q,
-4.147886355917831049939930101151160447495E7Q,
-1.010851868928346082547075956946476932162E6Q,
-8.333374463411801009783402800801201603736E3Q
};
#define NRD3 9
static const __float128 RD3[NRD3 + 1] =
{
-5.216713843111675050627304523368029262450E11Q,
-8.014292925418308759369583419234079164391E11Q,
-5.180106858220030014546267824392678611990E11Q,
-1.830406975497439003897734969120997840011E11Q,
-3.845274631904879621945745960119924118925E10Q,
-4.891033385370523863288908070309417710903E9Q,
-3.670172254411328640353855768698287474282E8Q,
-1.505316381525727713026364396635522516989E7Q,
-2.856327162923716881454613540575964890347E5Q,
-1.622140448015769906847567212766206894547E3Q
/* 1.0E0L */
};

/* log gamma(x+2.5) = log gamma(2.5) +  x P(x)/Q(x)
-0.125 <= x <= 0.25
2.375 <= x+2.5 <= 2.75  */
static const __float128 lgam2r5a = 2.8466796875E-1Q;
static const __float128 lgam2r5b = 1.4901722919159632494669682701924320137696E-5Q;
#define NRN2r5 8
static const __float128 RN2r5[NRN2r5 + 1] =
{
-4.676454313888335499356699817678862233205E9Q,
-9.361888347911187924389905984624216340639E9Q,
-7.695353600835685037920815799526540237703E9Q,
-3.364370100981509060441853085968900734521E9Q,
-8.449902011848163568670361316804900559863E8Q,
-1.225249050950801905108001246436783022179E8Q,
-9.732972931077110161639900388121650470926E6Q,
-3.695711763932153505623248207576425983573E5Q,
-4.717341584067827676530426007495274711306E3Q
};
#define NRD2r5 8
static const __float128 RD2r5[NRD2r5 + 1] =
{
-6.650657966618993679456019224416926875619E9Q,
-1.099511409330635807899718829033488771623E10Q,
-7.482546968307837168164311101447116903148E9Q,
-2.702967190056506495988922973755870557217E9Q,
-5.570008176482922704972943389590409280950E8Q,
-6.536934032192792470926310043166993233231E7Q,
-4.101991193844953082400035444146067511725E6Q,
-1.174082735875715802334430481065526664020E5Q,
-9.932840389994157592102947657277692978511E2Q
/* 1.0E0L */
};

/* log gamma(x+2) = x P(x)/Q(x)
-0.125 <= x <= +0.375
1.875 <= x+2 <= 2.375
Peak relative error 4.6e-36  */
#define NRN2 9
static const __float128 RN2[NRN2 + 1] =
{
-3.716661929737318153526921358113793421524E9Q,
-1.138816715030710406922819131397532331321E10Q,
-1.421017419363526524544402598734013569950E10Q,
-9.510432842542519665483662502132010331451E9Q,
-3.747528562099410197957514973274474767329E9Q,
-8.923565763363912474488712255317033616626E8Q,
-1.261396653700237624185350402781338231697E8Q,
-9.918402520255661797735331317081425749014E6Q,
-3.753996255897143855113273724233104768831E5Q,
-4.778761333044147141559311805999540765612E3Q
};
#define NRD2 9
static const __float128 RD2[NRD2 + 1] =
{
-8.790916836764308497770359421351673950111E9Q,
-2.023108608053212516399197678553737477486E10Q,
-1.958067901852022239294231785363504458367E10Q,
-1.035515043621003101254252481625188704529E10Q,
-3.253884432621336737640841276619272224476E9Q,
-6.186383531162456814954947669274235815544E8Q,
-6.932557847749518463038934953605969951466E7Q,
-4.240731768287359608773351626528479703758E6Q,
-1.197343995089189188078944689846348116630E5Q,
-1.004622911670588064824904487064114090920E3Q
/* 1.0E0 */
};

/* log gamma(x+1.75) = log gamma(1.75) +  x P(x)/Q(x)
-0.125 <= x <= +0.125
1.625 <= x+1.75 <= 1.875
Peak relative error 9.2e-37 */
static const __float128 lgam1r75a = -8.441162109375E-2Q;
static const __float128 lgam1r75b = 1.0500073264444042213965868602268256157604E-5Q;
#define NRN1r75 8
static const __float128 RN1r75[NRN1r75 + 1] =
{
-5.221061693929833937710891646275798251513E7Q,
-2.052466337474314812817883030472496436993E8Q,
-2.952718275974940270675670705084125640069E8Q,
-2.132294039648116684922965964126389017840E8Q,
-8.554103077186505960591321962207519908489E7Q,
-1.940250901348870867323943119132071960050E7Q,
-2.379394147112756860769336400290402208435E6Q,
-1.384060879999526222029386539622255797389E5Q,
-2.698453601378319296159355612094598695530E3Q
};
#define NRD1r75 8
static const __float128 RD1r75[NRD1r75 + 1] =
{
-2.109754689501705828789976311354395393605E8Q,
-5.036651829232895725959911504899241062286E8Q,
-4.954234699418689764943486770327295098084E8Q,
-2.589558042412676610775157783898195339410E8Q,
-7.731476117252958268044969614034776883031E7Q,
-1.316721702252481296030801191240867486965E7Q,
-1.201296501404876774861190604303728810836E6Q,
-5.007966406976106636109459072523610273928E4Q,
-6.155817990560743422008969155276229018209E2Q
/* 1.0E0L */
};

/* log gamma(x+x0) = y0 +  x^2 P(x)/Q(x)
-0.0867 <= x <= +0.1634
1.374932... <= x+x0 <= 1.625032...
Peak relative error 4.0e-36  */
static const __float128 x0a = 1.4616241455078125Q;
static const __float128 x0b = 7.9994605498412626595423257213002588621246E-6Q;
static const __float128 y0a = -1.21490478515625E-1Q;
static const __float128 y0b = 4.1879797753919044854428223084178486438269E-6Q;
#define NRN1r5 8
static const __float128 RN1r5[NRN1r5 + 1] =
{
6.827103657233705798067415468881313128066E5Q,
1.910041815932269464714909706705242148108E6Q,
2.194344176925978377083808566251427771951E6Q,
1.332921400100891472195055269688876427962E6Q,
4.589080973377307211815655093824787123508E5Q,
8.900334161263456942727083580232613796141E4Q,
9.053840838306019753209127312097612455236E3Q,
4.053367147553353374151852319743594873771E2Q,
5.040631576303952022968949605613514584950E0Q
};
#define NRD1r5 8
static const __float128 RD1r5[NRD1r5 + 1] =
{
1.411036368843183477558773688484699813355E6Q,
4.378121767236251950226362443134306184849E6Q,
5.682322855631723455425929877581697918168E6Q,
3.999065731556977782435009349967042222375E6Q,
1.653651390456781293163585493620758410333E6Q,
4.067774359067489605179546964969435858311E5Q,
5.741463295366557346748361781768833633256E4Q,
4.226404539738182992856094681115746692030E3Q,
1.316980975410327975566999780608618774469E2Q,
/* 1.0E0L */
};

/* log gamma(x+1.25) = log gamma(1.25) +  x P(x)/Q(x)
-.125 <= x <= +.125
1.125 <= x+1.25 <= 1.375
Peak relative error = 4.9e-36 */
static const __float128 lgam1r25a = -9.82818603515625E-2Q;
static const __float128 lgam1r25b = 1.0023929749338536146197303364159774377296E-5Q;
#define NRN1r25 9
static const __float128 RN1r25[NRN1r25 + 1] =
{
-9.054787275312026472896002240379580536760E4Q,
-8.685076892989927640126560802094680794471E4Q,
2.797898965448019916967849727279076547109E5Q,
6.175520827134342734546868356396008898299E5Q,
5.179626599589134831538516906517372619641E5Q,
2.253076616239043944538380039205558242161E5Q,
5.312653119599957228630544772499197307195E4Q,
6.434329437514083776052669599834938898255E3Q,
3.385414416983114598582554037612347549220E2Q,
4.907821957946273805080625052510832015792E0Q
};
#define NRD1r25 8
static const __float128 RD1r25[NRD1r25 + 1] =
{
3.980939377333448005389084785896660309000E5Q,
1.429634893085231519692365775184490465542E6Q,
2.145438946455476062850151428438668234336E6Q,
1.743786661358280837020848127465970357893E6Q,
8.316364251289743923178092656080441655273E5Q,
2.355732939106812496699621491135458324294E5Q,
3.822267399625696880571810137601310855419E4Q,
3.228463206479133236028576845538387620856E3Q,
1.152133170470059555646301189220117965514E2Q
/* 1.0E0L */
};

/* log gamma(x + 1) = x P(x)/Q(x)
0.0 <= x <= +0.125
1.0 <= x+1 <= 1.125
Peak relative error 1.1e-35  */
#define NRN1 8
static const __float128 RN1[NRN1 + 1] =
{
-9.987560186094800756471055681088744738818E3Q,
-2.506039379419574361949680225279376329742E4Q,
-1.386770737662176516403363873617457652991E4Q,
1.439445846078103202928677244188837130744E4Q,
2.159612048879650471489449668295139990693E4Q,
1.047439813638144485276023138173676047079E4Q,
2.250316398054332592560412486630769139961E3Q,
1.958510425467720733041971651126443864041E2Q,
4.516830313569454663374271993200291219855E0Q
};
#define NRD1 7
static const __float128 RD1[NRD1 + 1] =
{
1.730299573175751778863269333703788214547E4Q,
6.807080914851328611903744668028014678148E4Q,
1.090071629101496938655806063184092302439E5Q,
9.124354356415154289343303999616003884080E4Q,
4.262071638655772404431164427024003253954E4Q,
1.096981664067373953673982635805821283581E4Q,
1.431229503796575892151252708527595787588E3Q,
7.734110684303689320830401788262295992921E1Q
/* 1.0E0 */
};

/* log gamma(x + 1) = x P(x)/Q(x)
-0.125 <= x <= 0
0.875 <= x+1 <= 1.0
Peak relative error 7.0e-37  */
#define NRNr9 8
static const __float128 RNr9[NRNr9 + 1] =
{
4.441379198241760069548832023257571176884E5Q,
1.273072988367176540909122090089580368732E6Q,
9.732422305818501557502584486510048387724E5Q,
-5.040539994443998275271644292272870348684E5Q,
-1.208719055525609446357448132109723786736E6Q,
-7.434275365370936547146540554419058907156E5Q,
-2.075642969983377738209203358199008185741E5Q,
-2.565534860781128618589288075109372218042E4Q,
-1.032901669542994124131223797515913955938E3Q,
};
#define NRDr9 8
static const __float128 RDr9[NRDr9 + 1] =
{
-7.694488331323118759486182246005193998007E5Q,
-3.301918855321234414232308938454112213751E6Q,
-5.856830900232338906742924836032279404702E6Q,
-5.540672519616151584486240871424021377540E6Q,
-3.006530901041386626148342989181721176919E6Q,
-9.350378280513062139466966374330795935163E5Q,
-1.566179100031063346901755685375732739511E5Q,
-1.205016539620260779274902967231510804992E4Q,
-2.724583156305709733221564484006088794284E2Q
/* 1.0E0 */
};

/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
neval (__float128 x, const __float128 *p, int n)
{
__float128 y;

p += n;
y = *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}

/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
deval (__float128 x, const __float128 *p, int n)
{
__float128 y;

p += n;
y = x + *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}

__float128
{
__float128 p, q, w, z, nx;
int i, nn;

*signgamp = 1;

if (! finiteq (x))
return x * x;

if (x == 0)
{
if (signbitq (x))
*signgamp = -1;
}

if (x < 0)
{
if (x < -2 && x > -50)
q = -x;
p = floorq (q);
if (p == q)
return (one / fabsq (p - p));
__float128 halfp = p * 0.5Q;
if (halfp == floorq (halfp))
*signgamp = -1;
else
*signgamp = 1;
if (q < 0x1p-120Q)
return -logq (q);
z = q - p;
if (z > 0.5Q)
{
p += 1;
z = p - q;
}
z = q * sinq (PIL * z);
z = logq (PIL / z) - w;
return (z);
}

if (x < 13.5Q)
{
p = 0;
nx = floorq (x + 0.5Q);
nn = nx;
switch (nn)
{
case 0:
/* log gamma (x + 1) = log(x) + log gamma(x) */
if (x < 0x1p-120Q)
return -logq (x);
else if (x <= 0.125)
{
p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1);
}
else if (x <= 0.375)
{
z = x - 0.25Q;
p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
p += lgam1r25b;
p += lgam1r25a;
}
else if (x <= 0.625)
{
z = x + (1 - x0a);
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x <= 0.875)
{
z = x - 0.75Q;
p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else
{
z = x - 1;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
p = p - logq (x);
break;

case 1:
if (x < 0.875Q)
{
if (x <= 0.625)
{
z = x + (1 - x0a);
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x <= 0.875)
{
z = x - 0.75Q;
p = z * neval (z, RN1r75, NRN1r75)
/ deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else
{
z = x - 1;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
p = p - logq (x);
}
else if (x < 1)
{
z = x - 1;
p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9);
}
else if (x == 1)
p = 0;
else if (x <= 1.125Q)
{
z = x - 1;
p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1);
}
else if (x <= 1.375)
{
z = x - 1.25Q;
p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
p += lgam1r25b;
p += lgam1r25a;
}
else
{
/* 1.375 <= x+x0 <= 1.625 */
z = x - x0a;
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
break;

case 2:
if (x < 1.625Q)
{
z = x - x0a;
z = z - x0b;
p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
p = p * z * z;
p = p + y0b;
p = p + y0a;
}
else if (x < 1.875Q)
{
z = x - 1.75Q;
p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
p += lgam1r75b;
p += lgam1r75a;
}
else if (x == 2)
p = 0;
else if (x < 2.375Q)
{
z = x - 2;
p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
}
else
{
z = x - 2.5Q;
p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
p += lgam2r5b;
p += lgam2r5a;
}
break;

case 3:
if (x < 2.75)
{
z = x - 2.5Q;
p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
p += lgam2r5b;
p += lgam2r5a;
}
else
{
z = x - 3;
p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3);
p += lgam3b;
p += lgam3a;
}
break;

case 4:
z = x - 4;
p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4);
p += lgam4b;
p += lgam4a;
break;

case 5:
z = x - 5;
p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5);
p += lgam5b;
p += lgam5a;
break;

case 6:
z = x - 6;
p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6);
p += lgam6b;
p += lgam6a;
break;

case 7:
z = x - 7;
p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7);
p += lgam7b;
p += lgam7a;
break;

case 8:
z = x - 8;
p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8);
p += lgam8b;
p += lgam8a;
break;

case 9:
z = x - 9;
p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9);
p += lgam9b;
p += lgam9a;
break;

case 10:
z = x - 10;
p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10);
p += lgam10b;
p += lgam10a;
break;

case 11:
z = x - 11;
p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11);
p += lgam11b;
p += lgam11a;
break;

case 12:
z = x - 12;
p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12);
p += lgam12b;
p += lgam12a;
break;

case 13:
z = x - 13;
p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13);
p += lgam13b;
p += lgam13a;
break;
}
return p;
}

if (x > MAXLGM)
return (*signgamp * huge * huge);

if (x > 0x1p120Q)
return x * (logq (x) - 1);
q = ls2pi - x;
q = (x - 0.5Q) * logq (x) + q;
if (x > 1.0e18Q)
return (q);

p = 1 / (x * x);
q += neval (p, RASY, NRASY) / x;
return (q);
}
```