blob: eef62dbc91f6c9d28a890c99895ba9a047fe2636 [file] [log] [blame]
 /* lgammal * * Natural logarithm of gamma function * * * * SYNOPSIS: * * __float128 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 signgam. * * 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 This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either 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, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ #include "quadmath-imp.h" #ifdef HAVE_MATH_H_SIGNGAM #include /* For POSIX's extern int signgam. */ #endif static const __float128 PIQ = 3.1415926535897932384626433832795028841972E0Q; static const __float128 MAXLGM = 1.0485738685148938358098967157129705071571E4928Q; static const __float128 one = 1.0Q; static const __float128 zero = 0.0Q; static const __float128 huge = 1.0e4000Q; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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, /* 1.0E0Q */ }; /* 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 /* 1.0E0Q */ }; /* 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 lgammaq (__float128 x) { __float128 p, q, w, z, nx; int i, nn; #ifndef HAVE_MATH_H_SIGNGAM int signgam; #endif signgam = 1; if (! finiteq (x)) return x * x; if (x == 0.0Q) { if (signbitq (x)) signgam = -1; } if (x < 0.0Q) { q = -x; p = floorq (q); if (p == q) return (one / (p - p)); i = p; if ((i & 1) == 0) signgam = -1; else signgam = 1; z = q - p; if (z > 0.5Q) { p += 1.0Q; z = p - q; } z = q * sinq (PIQ * z); if (z == 0.0Q) return (signgam * huge * huge); w = lgammaq (q); z = logq (PIQ / z) - w; return (z); } if (x < 13.5Q) { p = 0.0Q; nx = floorq (x + 0.5Q); nn = nx; switch (nn) { case 0: /* log gamma (x + 1) = log(x) + log gamma(x) */ 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 + (1.0Q - 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 - 1.0Q; 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 + (1.0Q - 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 - 1.0Q; p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); } p = p - logq (x); } else if (x < 1.0Q) { z = x - 1.0Q; p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9); } else if (x == 1.0Q) p = 0.0Q; else if (x <= 1.125Q) { z = x - 1.0Q; 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 == 2.0Q) p = 0.0Q; else if (x < 2.375Q) { z = x - 2.0Q; 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 - 3.0Q; p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3); p += lgam3b; p += lgam3a; } break; case 4: z = x - 4.0Q; p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4); p += lgam4b; p += lgam4a; break; case 5: z = x - 5.0Q; p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5); p += lgam5b; p += lgam5a; break; case 6: z = x - 6.0Q; p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6); p += lgam6b; p += lgam6a; break; case 7: z = x - 7.0Q; p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7); p += lgam7b; p += lgam7a; break; case 8: z = x - 8.0Q; p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8); p += lgam8b; p += lgam8a; break; case 9: z = x - 9.0Q; p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9); p += lgam9b; p += lgam9a; break; case 10: z = x - 10.0Q; p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10); p += lgam10b; p += lgam10a; break; case 11: z = x - 11.0Q; p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11); p += lgam11b; p += lgam11a; break; case 12: z = x - 12.0Q; p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12); p += lgam12b; p += lgam12a; break; case 13: z = x - 13.0Q; p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13); p += lgam13b; p += lgam13a; break; } return p; } if (x > MAXLGM) return (signgam * huge * huge); q = ls2pi - x; q = (x - 0.5Q) * logq (x) + q; if (x > 1.0e18Q) return (q); p = 1.0Q / (x * x); q += neval (p, RASY, NRASY) / x; return (q); }