// Special functions -*- C++ -*-

// Copyright (C) 2006-2016 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library 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, 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 General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.

/** @file bits/sf_gamma.tcc
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{cmath}
 */

//
// ISO C++ 14882 TR29124: Mathematical Special Functions
//

// Written by Edward Smith-Rowland.
//
// References:
// (1) Handbook of Mathematical Functions,
//     ed. Milton Abramowitz and Irene A. Stegun,
//     Dover Publications,
//     Section 6, pp. 253-266
// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
//     W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
//     2nd ed, pp. 213-216
// (4) Gamma, Exploring Euler's Constant, Julian Havil,
//     Princeton, 2003.

#ifndef _GLIBCXX_BITS_SF_GAMMA_TCC
#define _GLIBCXX_BITS_SF_GAMMA_TCC 1

#pragma GCC system_header

#include <array>
#include <ext/math_const.h>

namespace std _GLIBCXX_VISIBILITY(default)
{
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

template<typename _Tp>
  struct _Factorial_table
  {
    int __n;
    _Tp __factorial;
    _Tp __log_factorial;
  };

template<typename _Tp>
  constexpr std::size_t _S_num_factorials = 0;

template<>
  constexpr std::size_t _S_num_factorials<float> = 35;
template<>
  constexpr std::size_t _S_num_factorials<double> = 171;
template<>
  constexpr std::size_t _S_num_factorials<long double> = 171;
#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_FLOAT128)
template<>
  constexpr std::size_t _S_num_factorials<__float128> = 171;
#endif

constexpr _Factorial_table<long double>
_S_factorial_table[171]
{
  {   0,  1.0L,  				     0.0L},
  {   1,  1.0L,  				     0.0L},
  {   2,  2.0L,  				     6.931471805599453094172321214581766e-01L},
  {   3,  6.0L,					     1.791759469228055000812477358380702e+00L},
  {   4,  24.0L,  				     3.178053830347945619646941601297055e+00L},
  {   5,  120.0L,				     4.787491742782045994247700934523243e+00L},
  {   6,  720.0L,				     6.579251212010100995060178292903945e+00L},
  {   7,  5040.0L,				     8.525161361065414300165531036347125e+00L},
  {   8,  40320.0L,				     1.060460290274525022841722740072165e+01L},
  {   9,  362880.0L,				     1.280182748008146961120771787456671e+01L},
  {  10,  3628800.0L,				     1.510441257307551529522570932925107e+01L},
  {  11,  39916800.0L,				     1.750230784587388583928765290721620e+01L},
  {  12,  479001600.0L,				     1.998721449566188614951736238705508e+01L},
  {  13,  6227020800.0L,			     2.255216385312342288557084982862040e+01L},
  {  14,  87178291200.0L,			     2.519122118273868150009343469352175e+01L},
  {  15,  1307674368000.0L,			     2.789927138384089156608943926367047e+01L},
  {  16,  20922789888000.0L,			     3.067186010608067280375836774950317e+01L},
  {  17,  355687428096000.0L,			     3.350507345013688888400790236737630e+01L},
  {  18,  6402373705728000.0L,			     3.639544520803305357621562496267953e+01L},
  {  19,  121645100408832000.0L,		     3.933988418719949403622465239456738e+01L},
  {  20,  2432902008176640000.0L,		     4.233561646075348502965987597070992e+01L},
  {  21,  51090942171709440000.0L,		     4.538013889847690802616047395107563e+01L},
  {  22,  1124000727777607680000.0L,		     4.847118135183522387963964965049893e+01L},
  {  23,  25852016738884976640000.0L,		     5.160667556776437357044640248230913e+01L},
  {  24,  620448401733239439360000.0L,		     5.478472939811231919009334408360618e+01L},
  {  25,  15511210043330985984000000.0L,	     5.800360522298051993929486275005856e+01L},
  {  26,  403291461126605635584000000.0L,	     6.126170176100200198476558231308206e+01L},
  {  27,  10888869450418352160768000000.0L,	     6.455753862700633105895131802384963e+01L},
  {  28,  304888344611713860501504000000.0L,	     6.788974313718153498289113501020917e+01L},
  {  29,  8841761993739701954543616000000.0L,	     7.125703896716800901007440704257107e+01L},
  {  30,  265252859812191058636308480000000.0L,	     7.465823634883016438548764373417796e+01L},
  {  31,  8222838654177922817725562880000000.0L,     7.809222355331531063141680805872032e+01L},
  {  32,  263130836933693530167218012160000000.0L,   8.155795945611503717850296866601120e+01L},
  {  33,  8683317618811886495518194401280000000.0L,  8.505446701758151741396015748089886e+01L},
  {  34,  2.952327990396041408476186096435200e+38L,  8.858082754219767880362692422023016e+01L},
  {  35,  1.033314796638614492966665133752320e+40L,  9.213617560368709248333303629689953e+01L},
  {  36,  3.719933267899012174679994481508352e+41L,  9.571969454214320248495799101366093e+01L},
  {  37,  1.376375309122634504631597958158090e+43L,  9.933061245478742692932608668469238e+01L},
  {  38,  5.230226174666011117600072241000743e+44L,  1.029681986145138126987523462380384e+02L},
  {  39,  2.039788208119744335864028173990290e+46L,  1.066317602606434591262010789165263e+02L},
  {  40,  8.159152832478977343456112695961160e+47L,  1.103206397147573954290535346141270e+02L},
  {  41,  3.345252661316380710817006205344076e+49L,  1.140342117814617032329202979871644e+02L},
  {  42,  1.405006117752879898543142606244512e+51L,  1.177718813997450715388381280889883e+02L},
  {  43,  6.041526306337383563735513206851401e+52L,  1.215330815154386339623109706023341e+02L},
  {  44,  2.658271574788448768043625811014616e+54L,  1.253172711493568951252073784232156e+02L},
  {  45,  1.196222208654801945619631614956577e+56L,  1.291239336391272148825986282302868e+02L},
  {  46,  5.502622159812088949850305428800256e+57L,  1.329525750356163098828226131835552e+02L},
  {  47,  2.586232415111681806429643551536120e+59L,  1.368027226373263684696435638533274e+02L},
  {  48,  1.241391559253607267086228904737338e+61L,  1.406739236482342593987077375760826e+02L},
  {  49,  6.082818640342675608722521633212954e+62L,  1.445657439463448860089184430629690e+02L},
  {  50,  3.041409320171337804361260816606477e+64L,  1.484777669517730320675371938508795e+02L},
  {  51,  1.551118753287382280224243016469303e+66L,  1.524095925844973578391819737056752e+02L},
  {  52,  8.065817517094387857166063685640377e+67L,  1.563608363030787851940699253901568e+02L},
  {  53,  4.274883284060025564298013753389399e+69L,  1.603311282166309070282143945291859e+02L},
  {  54,  2.308436973392413804720927426830276e+71L,  1.643201122631951814118173623614117e+02L},
  {  55,  1.269640335365827592596510084756652e+73L,  1.683274454484276523304800652726030e+02L},
  {  56,  7.109985878048634518540456474637249e+74L,  1.723527971391628015638371143804207e+02L},
  {  57,  4.052691950487721675568060190543232e+76L,  1.763958484069973517152413870492311e+02L},
  {  58,  2.350561331282878571829474910515074e+78L,  1.804562914175437710518418912030512e+02L},
  {  59,  1.386831185456898357379390197203894e+80L,  1.845338288614494905024579415767709e+02L},
  {  60,  8.320987112741390144276341183223363e+81L,  1.886281734236715911872884103898359e+02L},
  {  61,  5.075802138772247988008568121766251e+83L,  1.927390472878449024360397994932615e+02L},
  {  62,  3.146997326038793752565312235495076e+85L,  1.968661816728899939913861959392621e+02L},
  {  63,  1.982608315404440064116146708361898e+87L,  2.010093163992815266792820391565503e+02L},
  {  64,  1.268869321858841641034333893351615e+89L,  2.051681994826411985357854318852994e+02L},
  {  65,  8.247650592082470666723170306785496e+90L,  2.093425867525368356464396786600909e+02L},
  {  66,  5.443449390774430640037292402478427e+92L,  2.135322414945632611913140995964367e+02L},
  {  67,  3.647111091818868528824985909660546e+94L,  2.177369341139542272509841715928004e+02L},
  {  68,  2.480035542436830599600990418569171e+96L,  2.219564418191303339500681704535899e+02L},
  {  69,  1.711224524281413113724683388812728e+98L,  2.261905483237275933322701685223226e+02L},
  {  70,  1.197857166996989179607278372168910e+100L,  2.304390435657769523213935127204502e+02L},
  {  71,  8.504785885678623175211676442399260e+101L,  2.347017234428182677427229672529632e+02L},
  {  72,  6.123445837688608686152407038527467e+103L,  2.389783895618343230537651540911828e+02L},
  {  73,  4.470115461512684340891257138125051e+105L,  2.432688490029827141828572629486213e+02L},
  {  74,  3.307885441519386412259530282212537e+107L,  2.475729140961868839366425907411109e+02L},
  {  75,  2.480914081139539809194647711659403e+109L,  2.518904022097231943772393546444858e+02L},
  {  76,  1.885494701666050254987932260861146e+111L,  2.562211355500095254560828463192901e+02L},
  {  77,  1.451830920282858696340707840863083e+113L,  2.605649409718632093052501426406984e+02L},
  {  78,  1.132428117820629783145752115873204e+115L,  2.649216497985528010421161074406444e+02L},
  {  79,  8.946182130782975286851441715398315e+116L,  2.692910976510198225362890529821258e+02L},
  {  80,  7.156945704626380229481153372318652e+118L,  2.736731242856937041485587408011847e+02L},
  {  81,  5.797126020747367985879734231578109e+120L,  2.780675734403661429141397217488748e+02L},
  {  82,  4.753643337012841748421382069894049e+122L,  2.824742926876303960274237172433704e+02L},
  {  83,  3.945523969720658651189747118012061e+124L,  2.868931332954269939508991894666617e+02L},
  {  84,  3.314240134565353266999387579130131e+126L,  2.913239500942703075662342516899438e+02L},
  {  85,  2.817104114380550276949479442260611e+128L,  2.957666013507606240210845456410431e+02L},
  {  86,  2.422709538367273238176552320344126e+130L,  3.002209486470141317539746202758472e+02L},
  {  87,  2.107757298379527717213600518699389e+132L,  3.046868567656687154725531375451316e+02L},
  {  88,  1.854826422573984391147968456455462e+134L,  3.091641935801469219448667774874712e+02L},
  {  89,  1.650795516090846108121691926245361e+136L,  3.136528299498790617831845930281411e+02L},
  {  90,  1.485715964481761497309522733620825e+138L,  3.181526396202093268499930749566705e+02L},
  {  91,  1.352001527678402962551665687594951e+140L,  3.226634991267261768911519151416790e+02L},
  {  92,  1.243841405464130725547532432587355e+142L,  3.271852877037752172007931322164055e+02L},
  {  93,  1.156772507081641574759205162306240e+144L,  3.317178871969284731381175417778704e+02L},
  {  94,  1.087366156656743080273652852567865e+146L,  3.362611819791984770343557245691008e+02L},
  {  95,  1.032997848823905926259970209939472e+148L,  3.408150588707990178689655113342148e+02L},
  {  96,  9.916779348709496892095714015418933e+149L,  3.453794070622668541074469171784282e+02L},
  {  97,  9.619275968248211985332842594956366e+151L,  3.499541180407702369295636388001322e+02L},
  {  98,  9.426890448883247745626185743057238e+153L,  3.545390855194408088491915764084767e+02L},
  {  99,  9.332621544394415268169923885626665e+155L,  3.591342053695753987760440104602869e+02L},
  { 100,  9.332621544394415268169923885626666e+157L,  3.637393755555634901440799933696556e+02L},
  { 101,  9.425947759838359420851623124482932e+159L,  3.683544960724047495949641916365686e+02L},
  { 102,  9.614466715035126609268655586972591e+161L,  3.729794688856890206760262036128225e+02L},
  { 103,  9.902900716486180407546715254581770e+163L,  3.776141978739186564467948059278759e+02L},
  { 104,  1.029901674514562762384858386476504e+166L,  3.822585887730600291110999897338157e+02L},
  { 105,  1.081396758240290900504101305800329e+168L,  3.869125491232175524822013470474076e+02L},
  { 106,  1.146280563734708354534347384148349e+170L,  3.915759882173296196257630483078949e+02L},
  { 107,  1.226520203196137939351751701038734e+172L,  3.962488170517915257990674471249182e+02L},
  { 108,  1.324641819451828974499891837121832e+174L,  4.009309482789157454920876470786021e+02L},
  { 109,  1.443859583202493582204882102462797e+176L,  4.056222961611448891924649635308113e+02L},
  { 110,  1.588245541522742940425370312709077e+178L,  4.103227765269373054205448985634608e+02L},
  { 111,  1.762952551090244663872161047107075e+180L,  4.150323067282496395563082394714147e+02L},
  { 112,  1.974506857221074023536820372759924e+182L,  4.197508055995447340990825207006906e+02L},
  { 113,  2.231192748659813646596607021218714e+184L,  4.244781934182570746676646521943066e+02L},
  { 114,  2.543559733472187557120132004189334e+186L,  4.292143918666515701284861569845752e+02L},
  { 115,  2.925093693493015690688151804817735e+188L,  4.339593239950148201938936691496116e+02L},
  { 116,  3.393108684451898201198256093588572e+190L,  4.387129141861211848399114054248899e+02L},
  { 117,  3.969937160808720895401959629498629e+192L,  4.434750881209189409587553833403002e+02L},
  { 118,  4.684525849754290656574312362808383e+194L,  4.482457727453846057187886658354781e+02L},
  { 119,  5.574585761207605881323431711741975e+196L,  4.530248962384961351041435531967944e+02L},
  { 120,  6.689502913449127057588118054090371e+198L,  4.578123879812781810983912541313177e+02L},
  { 121,  8.094298525273443739681622845449348e+200L,  4.626081785268749221865151412872479e+02L},
  { 122,  9.875044200833601362411579871448205e+202L,  4.674121995716081787446837625121317e+02L},
  { 123,  1.214630436702532967576624324188129e+205L,  4.722243839269805962399457711220916e+02L},
  { 124,  1.506141741511140879795014161993280e+207L,  4.770446654925856331047093996895503e+02L},
  { 125,  1.882677176888926099743767702491600e+209L,  4.818729792298879342285116776892289e+02L},
  { 126,  2.372173242880046885677147305139416e+211L,  4.867092611368394122258247530279753e+02L},
  { 127,  3.012660018457659544809977077527059e+213L,  4.915534482232980034988721938356916e+02L},
  { 128,  3.856204823625804217356770659234635e+215L,  4.964054784872176206647928186858988e+02L},
  { 129,  4.974504222477287440390234150412680e+217L,  5.012652908915792927796609064361672e+02L},
  { 130,  6.466855489220473672507304395536484e+219L,  5.061328253420348751997323853324169e+02L},
  { 131,  8.471580690878820510984568758152794e+221L,  5.110080226652360267438818093435864e+02L},
  { 132,  1.118248651196004307449963076076169e+224L,  5.158908245878223975981734624013904e+02L},
  { 133,  1.487270706090685728908450891181305e+226L,  5.207811737160441513632878425767215e+02L},
  { 134,  1.992942746161518876737324194182948e+228L,  5.256790135159950627323751466945434e+02L},
  { 135,  2.690472707318050483595387662146980e+230L,  5.305842882944334921811616417385371e+02L},
  { 136,  3.659042881952548657689727220519892e+232L,  5.354969431801695441896628727207848e+02L},
  { 137,  5.012888748274991661034926292112252e+234L,  5.404169241059976691049780628487646e+02L},
  { 138,  6.917786472619488492228198283114908e+236L,  5.453441777911548737965972930389555e+02L},
  { 139,  9.615723196941089004197195613529722e+238L,  5.502786517242855655537860779578311e+02L},
  { 140,  1.346201247571752460587607385894161e+241L,  5.552202941468948698523266542774168e+02L},
  { 141,  1.898143759076170969428526414110767e+243L,  5.601690540372730381305428501841115e+02L},
  { 142,  2.695364137888162776588507508037289e+245L,  5.651248810948742988612895368380828e+02L},
  { 143,  3.854370717180072770521565736493323e+247L,  5.700877257251342061414049678576132e+02L},
  { 144,  5.550293832739304789551054660550385e+249L,  5.750575390247102067618643868172909e+02L},
  { 145,  8.047926057471991944849029257798059e+251L,  5.800342727671307811636484181828790e+02L},
  { 146,  1.174997204390910823947958271638517e+254L,  5.850178793888391176021577591617758e+02L},
  { 147,  1.727245890454638911203498659308619e+256L,  5.900083119756178539037637098855847e+02L},
  { 148,  2.556323917872865588581178015776756e+258L,  5.950055242493819689669662697995324e+02L},
  { 149,  3.808922637630569726985955243507367e+260L,  6.000094705553274281079586980746365e+02L},
  { 150,  5.713383956445854590478932865261050e+262L,  6.050201058494236838579726940994696e+02L},
  { 151,  8.627209774233240431623188626544186e+264L,  6.100373856862386081867689303989541e+02L},
  { 152,  1.311335885683452545606724671234716e+267L,  6.150612662070848845750296541952165e+02L},
  { 153,  2.006343905095682394778288746989116e+269L,  6.200917041284773200380696792869347e+02L},
  { 154,  3.089769613847350887958564670363239e+271L,  6.251286567308909491966542077298012e+02L},
  { 155,  4.789142901463393876335775239063020e+273L,  6.301720818478101958171841313875697e+02L},
  { 156,  7.471062926282894447083809372938311e+275L,  6.352219378550597328634673283089739e+02L},
  { 157,  1.172956879426414428192158071551315e+278L,  6.402781836604080409208917735453774e+02L},
  { 158,  1.853271869493734796543609753051077e+280L,  6.453407786934350077244819512083170e+02L},
  { 159,  2.946702272495038326504339507351213e+282L,  6.504096828956552392500216655842685e+02L},
  { 160,  4.714723635992061322406943211761940e+284L,  6.554848567108890661717085855247856e+02L},
  { 161,  7.590705053947218729075178570936724e+286L,  6.605662610758735291676206911000390e+02L},
  { 162,  1.229694218739449434110178928491749e+289L,  6.656538574111059132426189041691872e+02L},
  { 163,  2.004401576545302577599591653441551e+291L,  6.707476076119126755766838365353080e+02L},
  { 164,  3.287218585534296227263330311644144e+293L,  6.758474740397368739993850641512618e+02L},
  { 165,  5.423910666131588774984495014212838e+295L,  6.809534195136374546094430122993756e+02L},
  { 166,  9.003691705778437366474261723593311e+297L,  6.860654073019939978423357166441252e+02L},
  { 167,  1.503616514864999040201201707840083e+300L,  6.911834011144107529495954674208699e+02L},
  { 168,  2.526075744973198387538018869171339e+302L,  6.963073650938140118743477617656102e+02L},
  { 169,  4.269068009004705274939251888899563e+304L,  7.014372638087370853464547366487408e+02L},
  { 170,  7.257415615307998967396728211129257e+306L,  7.065730622457873471107222627212983e+02L}
};

template<typename _Tp>
  constexpr std::size_t _S_num_double_factorials = 0;

template<>
  constexpr std::size_t _S_num_double_factorials<float> = 57;
template<>
  constexpr std::size_t _S_num_double_factorials<double> = 301;
template<>
  constexpr std::size_t _S_num_double_factorials<long double> = 301;
#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_FLOAT128)
template<>
  constexpr std::size_t _S_num_double_factorials<__float128> = 301;
#endif

constexpr _Factorial_table<long double>
_S_double_factorial_table[301]
{
  {   0,  1.0L,                                      0.0L},
  {   1,  1.0L,                                      0.0L},
  {   2,  2.0L,                                      6.931471805599453094172321214581766e-01L},
  {   3,  3.0L,                                      1.098612288668109691395245236922526e+00L},
  {   4,  8.0L,                                      2.079441541679835928251696364374530e+00L},
  {   5,  15.0L,                                     2.708050201102210065996004570148713e+00L},
  {   6,  48.0L,                                     3.871201010907890929064173722755232e+00L},
  {   7,  105.0L,                                    4.653960350157523371101357313591893e+00L},
  {   8,  384.0L,                                    5.950642552587726857315870087129762e+00L},
  {   9,  945.0L,                                    6.851184927493742753891847787436945e+00L},
  {  10,  3840.03L,                                  8.253227645581772541333861541814127e+00L},
  {  11,  10395.0L,                                  9.249080200292113297953791365402074e+00L},
  {  12,  46080.0L,                                  1.073813429536977285156357102165301e+01L},
  {  13,  135135.0L,                                 1.181402955775365003400727880696739e+01L},
  {  14,  645120.0L,                                 1.337719162498503146608615588655436e+01L},
  {  15,  2027025.0L,                                1.452207975885586010000328337711611e+01L},
  {  16,  10321920.0L,                               1.614978034722481270375508437238707e+01L},
  {  17,  34459425.0L,                               1.735529310291207618025281799498923e+01L},
  {  18,  185794560.0L,                              1.904015210512097739596280696769029e+01L},
  {  19,  654729075.0L,                              2.029973208207851664026184542687709e+01L},
  {  20,  3715891200.0L,                             2.203588437867496838939803054383284e+01L},
  {  21,  13749310575.0L,                            2.334425451980193963676244340724279e+01L},
  {  22,  81749606400.0L,                            2.512692683203328424287720624325614e+01L},
  {  23,  316234143225.0L,                           2.647974873573108932756919623905299e+01L},
  {  24,  1961990553600.0L,                          2.830498066238122986252414784455320e+01L},
  {  25,  7905853580625.0L,                          2.969862456059929007677071490550536e+01L},
  {  26,  51011754393600.0L,                         3.156307720040271190799486740757669e+01L},
  {  27,  213458046676875.0L,                        3.299446142660361915095645061627294e+01L},
  {  28,  1428329123020800.0L,                       3.489528171057791583193468439393622e+01L},
  {  29,  6190283353629375.0L,                       3.636175725659009317813972264863485e+01L},
  {  30,  42849873690624000.0L,                      3.829647909224007120734792108554311e+01L},
  {  31,  191898783962510625.0L,                     3.979574446107523942406888697317721e+01L},
  {  32,  1371195958099968000.0L,                    4.176221499503979775443408169283400e+01L},
  {  33,  6332659870762850625.0L,                    4.329225202254171965952607578806486e+01L},
  {  34,  46620662575398912000.0L,                   4.528857551965595914410084843216530e+01L},
  {  35,  221643095476699771875.0L,                  4.684760008403113333923218786473423e+01L},
  {  36,  1678343852714360832000.0L,                 4.887209445811206914572580314892671e+01L},
  {  37,  8200794532637891559375.0L,                 5.045851799667535778360028353576568e+01L},
  {  38,  63777066403145711616000.0L,                5.250968061783845491515206270227273e+01L},
  {  39,  319830986772877770815625.0L,               5.412207964280500421104901621425352e+01L},
  {  40,  2551082656125828464640000.0L,              5.619856007195239121800451839987345e+01L},
  {  41,  13113070457687988603440625.0L,             5.783565170950931201491577958729093e+01L},
  {  42,  107145471557284795514880000.0L,            5.993622969023575952392234850169733e+01L},
  {  43,  563862029680583509947946875.0L,            6.159685182520287443838862210063677e+01L},
  {  44,  4714400748520531002654720000.0L,           6.372041932415402068681875632257881e+01L},
  {  45,  25373791335626257947657609375.0L,          6.540351431497319419577987190770802e+01L},
  {  46,  216862434431944426122117120000.0L,         6.754906072064311568704274127584719e+01L},
  {  47,  1192568192774434123539907640625.0L,        6.925366191668325278260082257748019e+01L},
  {  48,  10409396852733332453861621760000.0L,       7.142026173155100661610691499860242e+01L},
  {  49,  58435841445947272053455474390625.0L,       7.314548221479387939281152806436655e+01L},
  {  50,  520469842636666622693081088000000.0L,      7.533228473697915267472566578651297e+01L},
  {  51,  2980227913743310874726229193921875.0L,     7.707730784751820516445630791916221e+01L},
  {  52,  2.706443181710666438004021657600000e+34L,  7.928352845556058002961361747099465e+01L},
  {  53,  1.579520794283954763604901472778594e+35L,  8.104759976107032699860077705819126e+01L},
  {  54,  1.461479318123759876522171695104000e+36L,  8.327251250212485441321658530322040e+01L},
  {  55,  8.687364368561751199826958100282266e+36L,  8.505493294630279791726347996938258e+01L},
  {  56,  8.184284181493055308524161492582400e+37L,  8.729786419286000364657363441103811e+01L},
  {  57,  4.951797690080198183901366117160891e+38L,  8.909798421413734806866775263819296e+01L},
  {  58,  4.746884825265972078944013665697792e+39L,  9.135830720340642298317413856485819e+01L},
  {  59,  2.921560637147316928501806009124926e+40L,  9.317552165804306751928380301191265e+01L},
  {  60,  2.848130895159583247366408199418675e+41L,  9.545265176562852366800460737792326e+01L},
  {  61,  1.782151988659863326386101665566205e+42L,  9.728639552221637876803519211533827e+01L},
  {  62,  1.765841154998941613367173083639579e+43L,  9.957978615067361522335100382392380e+01L},
  {  63,  1.122755752855713895623244049306709e+44L,  1.014295302486079114559310353326265e+02L},
  {  64,  1.130138339199322632554990773529330e+45L,  1.037386692340332870798543965526729e+02L},
  {  65,  7.297912393562140321551086320493608e+45L,  1.056039175185035485665852821074180e+02L},
  {  66,  7.458913038715529374862939105293580e+46L,  1.079283239760597126247288174890187e+02L},
  {  67,  4.889601303686634015439227834730717e+47L,  1.098086101378945146262553541037817e+02L},
  {  68,  5.072060866326559974906798591599634e+48L,  1.121478316812358193238128163498082e+02L},
  {  69,  3.373824899543777470653067205964195e+49L,  1.140427166424917740084573521725145e+02L},
  {  70,  3.550442606428591982434759014119744e+50L,  1.163963269232851783129361605479357e+02L},
  {  71,  2.395415678676082004163677716234578e+51L,  1.183053965195330894297868067050275e+02L},
  {  72,  2.556318676628586227353026490166216e+52L,  1.206729930423012336239783473861553e+02L},
  {  73,  1.748653445433539863039484732851242e+53L,  1.225958559606814805588789155624660e+02L},
  {  74,  1.891675820705153808241239602723000e+54L,  1.249770581355054033777636751786449e+02L},
  {  75,  1.311490084075154897279613549638432e+55L,  1.269133440742177909994756794658409e+02L},
  {  76,  1.437673623735916894263342098069480e+56L,  1.293077914757917344566071668534491e+02L},
  {  77,  1.009847364737869270905302433221592e+57L,  1.312571494960714748486429757872492e+02L},
  {  78,  1.121385426514015177525406836494194e+58L,  1.336645003024813261934731316533951e+02L},
  {  79,  7.977794181429167240151889222450580e+58L,  1.356265973485384963428159213287306e+02L},
  {  80,  8.971083412112121420203254691953555e+59L,  1.380465269371552078057428194724540e+02L},
  {  81,  6.462013286957625464523030270184970e+60L,  1.400210465032109351083969022764207e+02L},
  {  82,  7.356288397931939564566668847401915e+61L,  1.424532461844194609190268149669496e+02L},
  {  83,  5.363471028174829135554115124253525e+62L,  1.444398871110075330318723744997121e+02L},
  {  84,  6.179282254262829234236001831817608e+63L,  1.468840629832627745343618771902317e+02L},
  {  85,  4.558950373948604765220997855615496e+64L,  1.488825383674978494867226684508114e+02L},
  {  86,  5.314182738666033141442961575363143e+65L,  1.513384102795162822672519518250357e+02L},
  {  87,  3.966286825335286145742268134385482e+66L,  1.533484464861524332053011857200959e+02L},
  {  88,  4.676480810026109164469806186319566e+67L,  1.558157470939944887395655917673754e+02L},
  {  89,  3.529995274548404669710618639603079e+68L,  1.578370828558845730436190012607657e+02L},
  {  90,  4.208832729023498248022825567687610e+69L,  1.603155567643247538063740736959048e+02L},
  {  91,  3.212295699839048249436662962038802e+70L,  1.623479423624014230847778414457742e+02L},
  {  92,  3.872126110701618388180999522272601e+71L,  1.648373453413737941160152907706313e+02L},
  {  93,  2.987435000850314871976096554696086e+72L,  1.668805418555546790221022510072391e+02L},
  {  94,  3.639798544059521284890139550936245e+73L,  1.693806401236437980122534735618617e+02L},
  {  95,  2.838063250807799128377291726961281e+74L,  1.714344187471552198567120377723531e+02L},
  {  96,  3.494206602297140433494533968898795e+75L,  1.739449883151116342507348794060751e+02L},
  {  97,  2.752921353283565154525972975152443e+76L,  1.760091297256586026788287593940571e+02L},
  {  98,  3.424322470251197624824643289520819e+77L,  1.785299557937822061703628170144196e+02L},
  {  99,  2.725392139750729502980713245400919e+78L,  1.806042495757931926056811934458673e+02L},
  { 100,  3.424322470251197624824643289520819e+79L,  1.831351259797702975383987999237884e+02L},
  { 101,  2.752646061148236798010520377854928e+80L,  1.852193700926344520565653917127803e+02L},
  { 102,  3.492808919656221577321136155311235e+81L,  1.877600987930545686194608119000422e+02L},
  { 103,  2.835225442982683901950835989190576e+82L,  1.898540990808640878273339940278337e+02L},
  { 104,  3.632521276442470440413981601523684e+83L,  1.924044896921959412837659957059820e+02L},
  { 105,  2.976986715131818097048377788650104e+84L,  1.945080594310216111984353513414256e+02L},
  { 106,  3.850472553029018666838820497615105e+85L,  1.970679287863080084273276969664693e+02L},
  { 107,  3.185375785191045363841764233855612e+86L,  1.991808882654835173717397501584489e+02L},
  { 108,  4.158510357271340160185926137424314e+87L,  2.017500600134322281203478969201532e+02L},
  { 109,  3.472059605858239446587523014902617e+88L,  2.038722361477126610721170666106581e+02L},
  { 110,  4.574361392998474176204518751166745e+89L,  2.064505403792246443484278319528027e+02L},
  { 111,  3.853986162502645785712150546541905e+90L,  2.085817663490249952078804075186121e+02L},
  { 112,  5.123284760158291077349061001306754e+91L,  2.111690392505197388912021131820786e+02L},
  { 113,  4.355004363627989737854730117592352e+92L,  2.133091541677373357764625390122281e+02L},
  { 114,  5.840544626580451828177929541489699e+93L,  2.159052376989142343520236179723471e+02L},
  { 115,  5.008255018172188198532939635231205e+94L,  2.180540862961005858418700511772645e+02L},
  { 116,  6.775031766833324120686398268128051e+95L,  2.206588278900205989980413542476254e+02L},
  { 117,  5.859658371261460192283539373220509e+96L,  2.228162602308983419607140290926748e+02L},
  { 118,  7.994537484863322462409949956391101e+97L,  2.254295125144862637580746367428033e+02L},
  { 119,  6.972993461801137628817411854132406e+98L,  2.275953837240098713460689164539912e+02L},
  { 120,  9.593444981835986954891939947669321e+99L,  2.302170042572683097523223376773265e+02L},
  { 121,  8.437322088779376530869068343500211e+100L,  2.323911742696066124341928036099214e+02L},
  { 122,  1.170400287783990408496816673615657e+102L,  2.350210253020015663104909589022103e+02L},
  { 123,  1.037790616919863313296895406250526e+103L,  2.372033586249790299294548122198813e+02L},
  { 124,  1.451296356852148106536052675283415e+104L,  2.398413068676066031752545874696690e+02L},
  { 125,  1.297238271149829141621119257813157e+105L,  2.420316723622813310532570902195599e+02L},
  { 126,  1.828633409633706614235426370857103e+106L,  2.446775887745580811725676628084154e+02L},
  { 127,  1.647492604360283009858821457422710e+107L,  2.468758594487399223263045310272762e+02L},
  { 128,  2.340650764331144466221345754697092e+108L,  2.495296190384776983384882876586226e+02L},
  { 129,  2.125265459624765082717879680075296e+109L,  2.517356718531015944411726187775446e+02L},
  { 130,  3.042845993630487806087749481106219e+110L,  2.543971534889332807585597665548723e+02L},
  { 131,  2.784097752108442258360422380898638e+111L,  2.566108691763027459853220427887141e+02L},
  { 132,  4.016556711592243904035829315060209e+112L,  2.592799554115196516128514196126764e+02L},
  { 133,  3.702850010304228203619361766595188e+113L,  2.615012183045244997504364229640451e+02L},
  { 134,  5.382185993533606831408011282180680e+114L,  2.641777952114705629819387237304983e+02L},
  { 135,  4.998847513910708074886138384903503e+115L,  2.664064930829629291992229180080389e+02L},
  { 136,  7.319772951205705290714895343765725e+116L,  2.690904500972066149904399547127459e+02L},
  { 137,  6.848421094057670062594009587317800e+117L,  2.713264740087910541145381081360187e+02L},
  { 138,  1.010128667266387330118655557439670e+119L,  2.740177037823638196820591849029368e+02L},
  { 139,  9.519305320740161387005673326371742e+119L,  2.762609479419217458717268930548943e+02L},
  { 140,  1.414180134172942262166117780415538e+121L,  2.789593462049731239805997612225225e+02L},
  { 141,  1.342222050224362755567799939018416e+122L,  2.812097078322999141499430889615890e+02L},
  { 142,  2.008135790525578012275887248190064e+123L,  2.839151732625743847113464478764937e+02L},
  { 143,  1.919377531820838740461953912796334e+124L,  2.861725524625598214300585199811194e+02L},
  { 144,  2.891715538356832337677277637393692e+125L,  2.888849865621503853318058668361715e+02L},
  { 145,  2.783097421140216173669833173554685e+126L,  2.911492862049803958318425513467075e+02L},
  { 146,  4.221904686000975213008825350594791e+127L,  2.938685931838587217703152078150682e+02L},
  { 147,  4.091153209076117775294654765125387e+128L,  2.961397187917591321334485020705164e+02L},
  { 148,  6.248418935281443315253061518880290e+129L,  2.988658054576228368335177677290160e+02L},
  { 149,  6.095818281523415485189035600036826e+130L,  3.011436650977045912744409303456206e+02L},
  { 150,  9.372628402922164972879592278320435e+131L,  3.038764407517190925835317637538491e+02L},
  { 151,  9.204685605100357382635443756055608e+132L,  3.061609449345195156032371666451050e+02L},
  { 152,  1.424639517244169075877698026304706e+134L,  3.089003212725653689717924875501115e+02L},
  { 153,  1.408316897580354679543222894676508e+135L,  3.111913828559119510662771917368232e+02L},
  { 154,  2.193944856556020376851654960509247e+136L,  3.139372738749789981303770159929780e+02L},
  { 155,  2.182891191249549753291995486748587e+137L,  3.162348079728311976868071153945917e+02L},
  { 156,  3.422553976227391787888581738394426e+138L,  3.189871298822285351766602129143822e+02L},
  { 157,  3.427139170261793112668432914195282e+139L,  3.212910537781795057442315606309952e+02L},
  { 158,  5.407635282439279024863959146663193e+140L,  3.240497249152555019802503905773217e+02L},
  { 159,  5.449151280716251049142808333570499e+141L,  3.263599579803997372697712750069468e+02L},
  { 160,  8.652216451902846439782334634661109e+142L,  3.291248987304893289019373105178388e+02L},
  { 161,  8.773133561953164189119921417048503e+143L,  3.314413623453842002656833805822002e+02L},
  { 162,  1.401659065208261123244738210815100e+145L,  3.342124950657217129769355235869871e+02L},
  { 163,  1.430020770598365762826547190978906e+146L,  3.365351125461909625997483129483210e+02L},
  { 164,  2.298720866941548242121370665736764e+147L,  3.393123614935459113996367512029408e+02L},
  { 165,  2.359534271487303508663802865115195e+148L,  3.416410580200915432098062610964348e+02L},
  { 166,  3.815876639122970081921475305123027e+149L,  3.444243492819024546325294555476904e+02L},
  { 167,  3.940422233383796859468550784742375e+150L,  3.467590518325082983170660118731796e+02L},
  { 168,  6.410672753726589737628078512606686e+151L,  3.495483132613057135572817498924306e+02L},
  { 169,  6.659313574418616692501850826214614e+152L,  3.518889505474313717891729867563102e+02L},
  { 170,  1.089814368133520255396773347143137e+154L,  3.546841116983559753215492759649881e+02L},
  { 171,  1.138742621225583454417816491282699e+155L,  3.570306141039340316319725046620431e+02L},
  { 172,  1.874480713189654839282450157086195e+156L,  3.598316061751694283638565827212503e+02L},
  { 173,  1.970024734720259376142822529919069e+157L,  3.621839056984318105789060050504841e+02L},
  { 174,  3.261596440949999420351463273329979e+158L,  3.649906614743839573918523321119929e+02L},
  { 175,  3.447543285760453908249939427358371e+159L,  3.673486916723553246332128764603796e+02L},
  { 176,  5.740409736071998979818575361060763e+160L,  3.701611454694221091735832041757908e+02L},
  { 177,  6.102151615796003417602392786424317e+161L,  3.725248414049291537752241720710219e+02L},
  { 178,  1.021792933020815818407706414268816e+163L,  3.753429290197141943213182518379188e+02L},
  { 179,  1.092285139227484611750828308769953e+164L,  3.777122272107699087714019765010433e+02L},
  { 180,  1.839227279437468473133871545683868e+165L,  3.805358858706044046975439658879064e+02L},
  { 181,  1.977036102001747147268999238873614e+166L,  3.829107242420357345182410635396763e+02L},
  { 182,  3.347393648576192621103646213144640e+167L,  3.857398925576812000481200381943731e+02L},
  { 183,  3.617976066663197279502268607138714e+168L,  3.881202103948771554583876978800244e+02L},
  { 184,  6.159204313380194422830709032186138e+169L,  3.909548283152901856671784873905578e+02L},
  { 185,  6.693255723326914967079196923206621e+170L,  3.933405662199554802773565528842820e+02L},
  { 186,  1.145612002288716162646511879986622e+172L,  3.961805749890033869139201290734808e+02L},
  { 187,  1.251638820262133098843809824639638e+173L,  3.985716748368100669016680310801203e+02L},
  { 188,  2.153750564302786385775442334374848e+174L,  4.014170169518333361195755439861694e+02L},
  { 189,  2.365597370295431556814800568568916e+175L,  4.038134218518697092809591195343311e+02L},
  { 190,  4.092126072175294132973340435312212e+176L,  4.066640410239938222636025628727416e+02L},
  { 191,  4.518290977264274273516269085966630e+177L,  4.090656952799163391538090694865334e+02L},
  { 192,  7.856882058576564735308813635799447e+178L,  4.119215363960216038115012008384132e+02L},
  { 193,  8.720301586120049347886399335915597e+179L,  4.143283854688212247056640005399165e+02L},
  { 194,  1.524235119363853558649909845345093e+181L,  4.171893945550849319430351545815753e+02L},
  { 195,  1.700458809293409622837847870503541e+182L,  4.196013850273849715077134925516305e+02L},
  { 196,  2.987500833953152974953823296876382e+183L,  4.224675092143154491720803243113780e+02L},
  { 197,  3.349903854308016956990560304891976e+184L,  4.248845887561229600144932898807934e+02L},
  { 198,  5.915251651227242890408570127815236e+185L,  4.277557762450099844083499904846464e+02L},
  { 199,  6.666308670072953744411215006735033e+186L,  4.301778935808474524099034111726619e+02L},
  { 200,  1.183050330245448578081714025563047e+188L,  4.330540936115580210858032055154733e+02L},
  { 201,  1.339928042684663702626654216353742e+189L,  4.354811984889065281609687284059481e+02L},
  { 202,  2.389761667095806127725062331637355e+190L,  4.383623613089592258461046359038445e+02L},
  { 203,  2.720053926649867316332108059198095e+191L,  4.407944044679483154932573531817532e+02L},
  { 204,  4.875113800875444500559127156540205e+192L,  4.436804813028034422365838800015565e+02L},
  { 205,  5.576110549632227998480821521356096e+193L,  4.461174144470867236717248758880168e+02L},
  { 206,  1.004273442980341567115180194247282e+195L,  4.490083574715930233167697144380681e+02L},
  { 207,  1.154254883773871195685530054920712e+196L,  4.514501332403520927453221191936721e+02L},
  { 208,  2.088888761399110459599574804034347e+197L,  4.543458955512943412904921303654661e+02L},
  { 209,  2.412392707087390798982757814784288e+198L,  4.567924674923169037493930902035250e+02L},
  { 210,  4.386666398938131965159107088472129e+199L,  4.596930030820118099710107198005162e+02L},
  { 211,  5.090148611954394585853618989194847e+200L,  4.621443256257929702451350464689793e+02L},
  { 212,  9.299732765748839766137307027560914e+201L,  4.650495893566838224239896531824616e+02L},
  { 213,  1.084201654346286046786820844698502e+203L,  4.675056177915023953578597462384149e+02L},
  { 214,  1.990142811870251709953383703898036e+204L,  4.704155653717056739067112841209430e+02L},
  { 215,  2.331033556844515000591664816101780e+205L,  4.728762558196300581559333480849869e+02L},
  { 216,  4.298708473639743693499308800419757e+206L,  4.757908437793898389091487161960852e+02L},
  { 217,  5.058342818352597551283912650940863e+207L,  4.782561531731705177069678651529724e+02L},
  { 218,  9.371184472534641251828493184915070e+208L,  4.811753388421789279189432647697525e+02L},
  { 219,  1.107777077219218863731176870556049e+210L,  4.836452249029870185274552192473335e+02L},
  { 220,  2.061660583957621075402268500681315e+211L,  4.865689663885312894564404319238602e+02L},
  { 221,  2.448187340654473688845900883928868e+212L,  4.890433876045047713437582413067720e+02L},
  { 222,  4.576886496385918787393036071512520e+213L,  4.919716437704035689016210049532724e+02L},
  { 223,  5.459457769659476326126358971161377e+214L,  4.944505593759648901023943465164951e+02L},
  { 224,  1.025222575190445808376040080018804e+216L,  4.973832898222586087538125183040064e+02L},
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  { 227,  2.788418055853577533569037844520673e+219L,  5.052916097956507129093997372336261e+02L},
  { 228,  5.282766885441329161400059324320896e+220L,  5.082331704504853354020506188308074e+02L},
  { 229,  6.385477347904692551873096663952341e+221L,  5.107253317992049525578409728375586e+02L},
  { 230,  1.215036383651505707122013644593806e+223L,  5.136712497594085307768753631173019e+02L},
  { 231,  1.475045267365983979482685329372991e+224L,  5.161677495097267460984035143958895e+02L},
  { 232,  2.818884410071493240523071655457630e+225L,  5.191179871310748407323103315140384e+02L},
  { 233,  3.436855472962742672194656817439068e+226L,  5.216187879632924466637057392705584e+02L},
  { 234,  6.596189519567294182823987673770854e+227L,  5.245733082464325421605715415509069e+02L},
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  { 237,  1.914156655666599531278814114472689e+231L,  5.325464336185717368106956400519607e+02L},
  { 238,  3.704947729350557796608577396603613e+232L,  5.355094107251296269247941756503175e+02L},
  { 239,  4.574834407043172879756365733589726e+233L,  5.380228971705032474776588036462226e+02L},
  { 240,  8.891874550441338711860585751848673e+234L,  5.409900496484716182284591087062989e+02L},
  { 241,  1.102535092097404664021284141795124e+236L,  5.435076941039939024731252149649334e+02L},
  { 242,  2.151833641206803968270261751947379e+237L,  5.464789873746283046260002279836872e+02L},
  { 243,  2.679160273796693333571720464562151e+238L,  5.490007555473344509301014411495460e+02L},
  { 244,  5.250474084544601682579438674751604e+239L,  5.519761555999215064935860813300292e+02L},
  { 245,  6.563942670801898667250715138177270e+240L,  5.545020137578791779149129059696586e+02L},
  { 246,  1.291616624797972013914541913988895e+242L,  5.574814871358538692982653220614473e+02L},
  { 247,  1.621293839688068970810926639129786e+243L,  5.600114020945071551109754208431118e+02L},
  { 248,  3.203209229498970594508063946692458e+244L,  5.629949158820188514724461827503642e+02L},
  { 249,  4.037021660823291737319207331433167e+245L,  5.655288549909718627258461383033256e+02L},
  { 250,  8.008023073747426486270159866731147e+246L,  5.685163767998810979056656928715010e+02L},
  { 251,  1.013292436866646226067121040189725e+248L,  5.710543079301036466120647235423339e+02L},
  { 252,  2.018021814584351474540080286416249e+249L,  5.740458058873925212123960003317055e+02L},
  { 253,  2.563629865272614951949816231680004e+250L,  5.765876974188311668469334199521092e+02L},
  { 254,  5.125775409044252745331803927497273e+251L,  5.795831401544110577948606732608800e+02L},
  { 255,  6.537256156445168127472031390784010e+252L,  5.821289609639895929931789591401310e+02L},
  { 256,  1.312198504715328702804941805439302e+254L,  5.851283175988906202701985302325455e+02L},
  { 257,  1.680074832206408208760312067431491e+255L,  5.876780370488848127915307534548923e+02L},
  { 258,  3.385472142165548053236749858033399e+256L,  5.906812771838122376944838501042720e+02L},
  { 259,  4.351393815414597260689208254647560e+257L,  5.932348651105843505410042018693669e+02L},
  { 260,  8.802227569630424938415549630886837e+258L,  5.962419588148277654239725611219798e+02L},
  { 261,  1.135713785823209885039883354463013e+260L,  5.987993855179070439509779643755739e+02L},
  { 262,  2.306183623243171333864874003292351e+261L,  6.018103033185888622775392172546075e+02L},
  { 263,  2.986927256715041997654893222237725e+262L,  6.043715395500848085020638906397789e+02L},
  { 264,  6.088324765361972321403267368691808e+263L,  6.073862524217351784412481024338698e+02L},
  { 265,  7.915357230294861293785467038929971e+264L,  6.099512693760710307108091191120342e+02L},
  { 266,  1.619494387586284637493269120072021e+266L,  6.129697487305168775157797147306590e+02L},
  { 267,  2.113400380488727965440719699394302e+267L,  6.155385180344712802405221798896266e+02L},
  { 268,  4.340244958731242828481961241793015e+268L,  6.185607357110277341942842509699390e+02L},
  { 269,  5.685047023514678227035535991370674e+269L,  6.211332294140731193467421330701218e+02L},
  { 270,  1.171866138857435563690129535284114e+271L,  6.241591576700261089524879781353910e+02L},
  { 271,  1.540647743372477799526630253661453e+272L,  6.267353482349528202468390031433277e+02L},
  { 272,  3.187475897692224733237152335972790e+273L,  6.297649597363221062704064412390968e+02L},
  { 273,  4.205968339406864392707700592495766e+274L,  6.323448200301377799793930885652588e+02L},
  { 274,  8.733683959676695769069797400565445e+275L,  6.353780878427101764951388634885349e+02L},
  { 275,  1.156641293336887707994617662936336e+277L,  6.379615911278043512726565508096763e+02L},
  { 276,  2.410496772870768032263264082556063e+278L,  6.409984887084273264961753258001839e+02L},
  { 277,  3.203896382543178951145090926333650e+279L,  6.435856086339916897725259483592008e+02L},
  { 278,  6.701181028580735129691874149505854e+280L,  6.466261098221179635627813428405177e+02L},
  { 279,  8.938870907295469273694803684470883e+281L,  6.492168204158130554012456031575882e+02L},
  { 280,  1.876330688002605836313724761861639e+283L,  6.522608994252872131707391512815615e+02L},
  { 281,  2.511822724950026865908239835336318e+284L,  6.548551750851468011663522414822814e+02L},
  { 282,  5.291252540167348458404703828449822e+285L,  6.579028064962253267583725793097144e+02L},
  { 283,  7.108458311608576030520318734001781e+286L,  6.605006219827900387909982286290766e+02L},
  { 284,  1.502715721407526962186935887279750e+288L,  6.635517807343865327985364980851438e+02L},
  { 285,  2.025910618808444168698290839190508e+289L,  6.661531111630586893170032606311131e+02L},
  { 286,  4.297766963225527111854636637620083e+290L,  6.692077725452063853880691612261325e+02L},
  { 287,  5.814363475980234764164094708476757e+291L,  6.718125933788183104259753767475937e+02L},
  { 288,  1.237756885408951808214135351634584e+293L,  6.748707330253423313179458123072684e+02L},
  { 289,  1.680351044558287846843423370749783e+294L,  6.774790200669307425864744459833399e+02L},
  { 290,  3.589494967685960243820992519740294e+295L,  6.805406139483228510291470757943147e+02L},
  { 291,  4.889821539664617634314362008881867e+296L,  6.831523433341022350999864128419664e+02L},
  { 292,  1.048132530564300391195729815764166e+298L,  6.862173677505911327770736488946695e+02L},
  { 293,  1.432717711121732966854108068602387e+299L,  6.888325159431193024059262531047789e+02L},
  { 294,  3.081509639859043150115445658346647e+300L,  6.919009475179298143880968317399366e+02L},
  { 295,  4.226517247809112252219618802377042e+301L,  6.945194912994591222311430628117248e+02L},
  { 296,  9.121268533982767724341719148706075e+302L,  6.975913069722538747607166237753426e+02L},
  { 297,  1.255275622599306338909226784305981e+304L,  7.002132234382618218493907421004576e+02L},
  { 298,  2.718138023126864781853832306314410e+305L,  7.032884004587592792111262841719049e+02L},
  { 299,  3.753274111571925953338588085074884e+306L,  7.059136670116525082762509823738331e+02L},
  { 300,  8.154414069380594345561496918943231e+307L,  7.089921829334154802705575123181962e+02L}
};

template<typename _Tp>
  constexpr std::size_t _S_num_neg_double_factorials = 0;

template<>
  constexpr std::size_t _S_num_neg_double_factorials<float> = 27;
template<>
  constexpr std::size_t _S_num_neg_double_factorials<double> = 150;
template<>
  constexpr std::size_t _S_num_neg_double_factorials<long double> = 999;
#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_FLOAT128)
template<>
  constexpr std::size_t _S_num_neg_double_factorials<__float128> = 999;
#endif


constexpr _Factorial_table<long double>
_S_neg_double_factorial_table[999]
{
  {  -1,  1.000000000000000000000000000000000e+00L,  0.000000000000000000000000000000000e+00L},
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  {-1889,  1.003193260327282301570300524508686e-2686L, -6.184740371609307314220318117087064e+03L},
  {-1891, -5.305093920292344270599156660543026e-2690L, -6.192285232677965771714998670515032e+03L},
  {-1893,  2.802479619805781442471820739853685e-2693L, -6.199831150829175094563672653785620e+03L},
  {-1895, -1.478881065860570682043177171426747e-2696L, -6.207378124946691621286090649702500e+03L},
  {-1897,  7.795893863260783774608208600035566e-2700L, -6.214926153916626635491292872519148e+03L},
  {-1899, -4.105262697872977237813695945253063e-2703L, -6.222475236627438921369825319258448e+03L},
  {-1901,  2.159527984152013276072433427276729e-2706L, -6.230025371969927350531266355105530e+03L},
  {-1903, -1.134801883421972294310264544023505e-2709L, -6.237576558837223500022261799071936e+03L},
  {-1905,  5.956965267306941177481703643168004e-2713L, -6.245128796124784301361305244449801e+03L},
  {-1907, -3.123736375095407014935345381839540e-2716L, -6.252682082730384720427531387636673e+03L},
  {-1909,  1.636320783182507603423439173305155e-2719L, -6.260236417554110468041813612691774e+03L},
  {-1911, -8.562641460923639996982936542674802e-2723L, -6.267791799498350741079473050857149e+03L},
  {-1913,  4.476027946117950860942465521523681e-2726L, -6.275348227467790993954914867019734e+03L},
  {-1915, -2.337351407894491311197109932910538e-2729L, -6.282905700369405740319508680890113e+03L},
  {-1917,  1.219275643137449823264011441267887e-2732L, -6.290464217112451384815023871133393e+03L},
  {-1919, -6.353703195088326332798392085814938e-2736L, -6.298023776608459084725917096832194e+03L},
  {-1921,  3.307497759025677424673811601153013e-2739L, -6.305584377771227641374748762943683e+03L},
  {-1923, -1.719967633398688208358716381254817e-2742L, -6.313146019516816421105977414718608e+03L},
  {-1925,  8.934896796876302381084240941583465e-2746L, -6.320708700763538305704346229706468e+03L},
  {-1927, -4.636687491892217115248697945813941e-2749L, -6.328272420431952672095033943749278e+03L},
  {-1929,  2.403674179311672947251787426549477e-2752L, -6.335837177444858401173693757489748e+03L},
  {-1931, -1.244782071109100438763224974909103e-2755L, -6.343402970727286915615448080068424e+03L},
  {-1933,  6.439638236467151778392265778112274e-2759L, -6.350969799206495246512844434000853e+03L},
  {-1935, -3.327978416778889807954659316853889e-2762L, -6.358537661811959128693708526321270e+03L},
  {-1937,  1.718109662766592569930128712882751e-2765L, -6.366106557475366124570754442037939e+03L},
  {-1939, -8.860802799208832232749503418683604e-2769L, -6.373676485130608776375729192330907e+03L},
  {-1941,  4.565070993925209805641166109574242e-2772L, -6.381247443713777786631779506783487e+03L},
  {-1943, -2.349496136863206281853405100141144e-2775L, -6.388819432163155226718632850812213e+03L},
  {-1945,  1.207967165482368268305092596473596e-2778L, -6.396392449419207773386082230379072e+03L},
  {-1947, -6.204248410284377341063649699402135e-2782L, -6.403966494424579973072155469567679e+03L},
  {-1949,  3.183298312100757999519573986353071e-2785L, -6.411541566124087533883234365720056e+03L},
  {-1951, -1.631623942645186058185327517351651e-2788L, -6.419117663464710645094267494111651e+03L},
  {-1953,  8.354449271096702806888517753976706e-2792L, -6.426694785395587324028092501653481e+03L},
  {-1955, -4.273375586238722663370085807660719e-2795L, -6.434272930868006790173749548436390e+03L},
  {-1957,  2.183635966396894564828863468400981e-2798L, -6.441852098835402866404527178183331e+03L},
  {-1959, -1.114668691371564351622697023175590e-2801L, -6.449432288253347407157335374491452e+03L},
  {-1961,  5.684185065637758039891366767851045e-2805L, -6.457013498079543753435847939301513e+03L},
  {-1963, -2.895662285093101395767379912303131e-2808L, -6.464595727273820214500697663042563e+03L},
  {-1965,  1.473619483507939641611898174200066e-2811L, -6.472178974798123576110843091623881e+03L},
  {-1967, -7.491710643151701279165725339095407e-2815L, -6.479763239616512635181055082692017e+03L},
  {-1969,  3.804830189513306896478275946721893e-2818L, -6.487348520695151760721294830700004e+03L},
  {-1971, -1.930405981488232824189891398641244e-2821L, -6.494934817002304480924572675268210e+03L},
  {-1973,  9.784115466235341227520990363108183e-2825L, -6.502522127508327096270688837534246e+03L},
  {-1975, -4.953982514549539862035944487649713e-2828L, -6.510110451185662318514063301742180e+03L},
  {-1977,  2.505808049848022186158798425720644e-2831L, -6.517699787008832935424662420829923e+03L},
  {-1979, -1.266199115638212322465284702233777e-2834L, -6.525290133954435501151824521434509e+03L},
  {-1981,  6.391716888633075832737429087500134e-2838L, -6.532881491001134052081575861324747e+03L},
  {-1983, -3.223256121347995881360276897377778e-2841L, -6.540473857129653848058811796150148e+03L},
  {-1985,  1.623806610250879537209207504976211e-2844L, -6.548067231322775138846495987513552e+03L},
  {-1987, -8.172152039511220620076535002396632e-2848L, -6.555661612565326955694802975274266e+03L},
  {-1989,  4.108673725244454811501525893613188e-2851L, -6.563256999844180927893896487807550e+03L},
  {-1991, -2.063623166873156610498003964647508e-2854L, -6.570853392148245124184797518424148e+03L},
  {-1993,  1.035435608064805123180132445884349e-2857L, -6.578450788468457918903552497634620e+03L},
  {-1995, -5.190153423883734953283871909194733e-2861L, -6.586049187797781882734662882380101e+03L},
  {-1997,  2.598975174703923361684462648570222e-2864L, -6.593648589131197697950483207309507e+03L},
};

  /**
   *  @brief This returns Bernoulli numbers from a table or by summation
   *         for larger values.
   *
   *  Upward recursion is unstable.
   *
   *  @param __n the order n of the Bernoulli number.
   *  @return  The Bernoulli number of order n.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __bernoulli_series(unsigned int __n)
    {
      constexpr unsigned long _S_num_bern_tab = 13;
      constexpr _Tp
      _S_bernoulli_2n[_S_num_bern_tab]
      {
	 _Tp{1ULL},
	 _Tp{1ULL}             / _Tp{6ULL},
	-_Tp{1ULL}             / _Tp{30ULL},
	 _Tp{1ULL}             / _Tp{42ULL},
	-_Tp{1ULL}             / _Tp{30ULL},
	 _Tp{5ULL}             / _Tp{66ULL},
	-_Tp{691ULL}           / _Tp{2730ULL},
	 _Tp{7ULL}             / _Tp{6ULL},
	-_Tp{3617ULL}          / _Tp{510ULL},
	 _Tp{43867ULL}         / _Tp{798ULL},
	-_Tp{174611ULL}        / _Tp{330ULL},
	 _Tp{854513ULL}        / _Tp{138ULL},
	-_Tp{23749461029ULL}   / _Tp{2730ULL}
      };
      constexpr _Tp _S_2pi = __gnu_cxx::__math_constants<_Tp>::__2_pi;

      if (__n == 0)
	return _Tp{1};
      else if (__n == 1)
	return -_Tp{1} / _Tp{2};
      // Take care of the rest of the odd ones.
      else if (__n % 2 == 1)
	return _Tp{0};
      // Take care of some small evens that are painful for the series.
      else if (__n / 2 < _S_num_bern_tab)
	return _S_bernoulli_2n[__n / 2];
      else
	{
	  auto __fact = _Tp{1};
	  if ((__n / 2) % 2 == 0)
	    __fact *= -_Tp{1};
	  for (unsigned int __k = 1; __k <= __n; ++__k)
	    __fact *= __k / _S_2pi;
	  __fact *= _Tp{2};

	 // Riemann zeta function minus-1 for even integer argument.
	  auto __sum = _Tp{0};
	  for (unsigned int __i = 2; __i < 1000; ++__i)
	    {
	      auto __term = std::pow(_Tp(__i), -_Tp(__n));
	      __sum += __term;
	      if (__term < __gnu_cxx::__epsilon<_Tp>() * __sum)
		break;
	    }

	  return __fact + __fact * __sum;
	}
    }


  /**
   *   @brief This returns Bernoulli number @f$ B_n @f$.
   *
   *   @param __n the order n of the Bernoulli number.
   *   @return  The Bernoulli number of order n.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __bernoulli(int __n)
    { return __bernoulli_series<_Tp>(__n); }


  /**
   * @brief This returns Bernoulli number @f$ B_2n @f$ at even integer
   * arguments @f$ 2n @f$.
   *
   * @param __n the half-order n of the Bernoulli number.
   * @return  The Bernoulli number of order 2n.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __bernoulli_2n(int __n)
    { return __bernoulli_series<_Tp>(2 * __n); }


  /**
   *  @brief Return @f$log(\Gamma(x))@f$ by asymptotic expansion
   *         with Bernoulli number coefficients.  This is like
   *         Sterling's approximation.
   *
   *  @param __x The argument of the log of the gamma function.
   *  @return  The logarithm of the gamma function.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_gamma_bernoulli(_Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_eps = _Real{0.01L} * __gnu_cxx::__epsilon<_Real>();
      constexpr auto _S_ln2pi
	= __gnu_cxx::__math_constants<_Real>::__ln_2
	+ __gnu_cxx::__math_constants<_Real>::__ln_pi;

      auto __lg = (__x - _Real{0.5L}) * std::log(__x)
		- __x + _Real{0.5L} * _S_ln2pi;

      const auto __xx = _Real{1} / (__x * __x);
      auto __xk = _Real{1} / __x;
      for ( unsigned int __i = 1; __i < 100; ++__i )
	{
	  const auto __2i = _Tp(2 * __i);
	  const auto __term = __bernoulli<_Tp>(__2i) * __xk
			    / (__2i * (__2i - _Tp{1}));
	  __lg += __term;
	  if (std::abs(__term) < _S_eps * std::abs(__lg))
	    break;
	  __xk *= __xx;
	}

      return __lg;
    }


  /**
   * A struct for Spouge algorithm Chebyshev arrays of coefficients.
   */
  template<typename _Tp>
    struct _GammaSpouge
    {
    };

  template<>
    struct _GammaSpouge<float>
    {
      static constexpr std::array<float, 7>
      _S_cheby
      {
	 2.901419e+03F,
	-5.929168e+03F,
	 4.148274e+03F,
	-1.164761e+03F,
	 1.174135e+02F,
	-2.786588e+00F,
	 3.775392e-03F,
      };
    };

  template<>
    struct _GammaSpouge<double>
    {
      static constexpr std::array<double, 18>
      _S_cheby
      {
	 2.785716565770350e+08,
	-1.693088166941517e+09,
	 4.549688586500031e+09,
	-7.121728036151557e+09,
	 7.202572947273274e+09,
	-4.935548868770376e+09,
	 2.338187776097503e+09,
	-7.678102458920741e+08,
	 1.727524819329867e+08,
	-2.595321377008346e+07,
	 2.494811203993971e+06,
	-1.437252641338402e+05,
	 4.490767356961276e+03,
	-6.505596924745029e+01,
	 3.362323142416327e-01,
	-3.817361443986454e-04,
	 3.273137866873352e-08,
	-7.642333165976788e-15,
      };
    };

  template<>
    struct _GammaSpouge<long double>
    {
      static constexpr std::array<long double, 22>
      _S_cheby
      {
	 1.681473171108908244e+10L,
	-1.269150315503303974e+11L,
	 4.339449429013039995e+11L,
	-8.893680202692714895e+11L,
	 1.218472425867950986e+12L,
	-1.178403473259353616e+12L,
	 8.282455311246278274e+11L,
	-4.292112878930625978e+11L,
	 1.646988347276488710e+11L,
	-4.661514921989111004e+10L,
	 9.619972564515443397e+09L,
	-1.419382551781042824e+09L,
	 1.454145470816386107e+08L,
	-9.923020719435758179e+06L,
	 4.253557563919127284e+05L,
	-1.053371059784341875e+04L,
	 1.332425479537961437e+02L,
	-7.118343974029489132e-01L,
	 1.172051640057979518e-03L,
	-3.323940885824119041e-07L,
	 4.503801674404338524e-12L,
	-5.320477002211632680e-20L,
      };
    };

#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_FLOAT128)
  template<>
    struct _GammaSpouge<__float128>
    {
      static constexpr std::array<__float128, 40>
      _S_cheby
      {
	 1.488707141702962349642653219904701e+18Q,
	-2.109024888057172629401888926607933e+19Q,
	 1.417814156004491800291266261243181e+20Q,
	-6.017978256631943800541949222618531e+20Q,
	 1.810303371559858936645834474459794e+21Q,
	-4.106765117367170106628989406117407e+21Q,
	 7.299495565278381248758472830920901e+21Q,
	-1.042658085606638732437326916809544e+22Q,
	 1.218085331069918351049721248713456e+22Q,
	-1.178425136120009375043789561151657e+22Q,
	 9.524552911654596104822221145923981e+21Q,
	-6.470848515795635226546940452396693e+21Q,
	 3.710085524031367371144056238201665e+21Q,
	-1.799208570324458717498947049552296e+21Q,
	 7.385190850868692826830617550673786e+20Q,
	-2.564085902302497902610094974644620e+20Q,
	 7.515077439232198453710859556732366e+19Q,
	-1.853282816124970754507275065198273e+19Q,
	 3.827869971220176147395256318872661e+18Q,
	-6.582119247090799001499961431273749e+17Q,
	 9.351471019350666459087977506154173e+16Q,
	-1.087561099332606693611359806619008e+16Q,
	 1.023682895113200510083891964414548e+15Q,
	-7.692392155137210713863610487671446e+13Q,
	 4.538847752507923706006970038411543e+12Q,
	-2.061098065296257001203592871224284e+11Q,
	 7.029050529288024691234709461995537e+09Q,
	-1.746978910485928115412705762596253e+08Q,
	 3.048305382188066788914727298031581e+06Q,
	-3.562744619669248206793629956902640e+04Q,
	 2.625939669969919276517969162685481e+02Q,
	-1.127892258247142779807411800956670e+00Q,
	 2.538684230850237447901013114737673e-03Q,
	-2.583211865372192597501314906164821e-06Q,
	 9.590040155479072110068965236664741e-10Q,
	-9.347057516377501180809930331336725e-14Q,
	 1.386220962446620618073769872331415e-18Q,
	-1.138134595144440607620794028391791e-24Q,
	 5.491908932091769529804906638081618e-33Q,
	-1.332629445370080503706686280760692e-46Q,
      };
    };
#endif

  /**
   * @brief Return @f$\Gamma(z)@f$ by the Spouge algorithm:
   * @f[
   *   \Gamma(z+1) = (z+a)^{z+1/2}e^{-z-a}\left[ \sqrt{2\pi} +
   *	 \sum_{k=1}^{\lceil a \rceil + 1}\frac{c_k(a)}{z+k}\right]
   * @f]
   * where
   * @f[
   *   c_k(a) = \frac{(-1)^{k-1}}{(k-1)!}(a-k)^{k-1/2}e^{a-k}
   * @f]
   * and the error is bounded by
   * @f[
   *   \epsilon(a) < a^{-1/2}(2\pi)^{-a-1/2}
   * @f]
   * @see Spouge, J.L., Computation of the gamma, digamma,
   *	  and trigamma functions.
   *	  SIAM Journal on Numerical Analysis 31, 3 (1994), pp. 931-944
   *
   * @param __z The argument of the gamma function.
   * @return  The the gamma function.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_gamma1p_spouge(_Tp __z)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_ln_pi = __gnu_cxx::__math_constants<_Real>::__ln_pi;
      constexpr auto _S_sqrt_pi = __gnu_cxx::__math_constants<_Real>::__root_pi;
      constexpr auto _S_sqrt_2 = __gnu_cxx::__math_constants<_Real>::__root_2;
      constexpr auto _S_sqrt_2pi = _S_sqrt_2 * _S_sqrt_pi;
      auto __a = _Real{_GammaSpouge<_Real>::_S_cheby.size() + 1};
      const auto& __c = _GammaSpouge<_Real>::_S_cheby;

      // Reflection; move the transition upwards to prevent instability.
      if (std::real(__z) < _Real{-0.5L})
	return _S_ln_pi - std::log(__sin_pi(__z))
			- __log_gamma1p_spouge(-_Real{1} - __z);
      else
	{
	  _Val __sum = _S_sqrt_2pi;
	  for (int __k = 0; __k < __c.size(); ++__k)
	    __sum += __c[__k] / (__z + _Real(__k + 1));
	  return std::log(__sum)
	       + (__z + _Real{0.5L}) * std::log(__z + __a)
	       - (__z + __a);
	}
    }


  /**
   * A struct for Lanczos algorithm Chebyshev arrays of coefficients.
   */
  template<typename _Tp>
    struct _GammaLanczos
    {
    };

  template<>
    struct _GammaLanczos<float>
    {
      static constexpr float _S_g = 6.5F;
      static constexpr std::array<float, 7>
      _S_cheby
      {
	 3.307139e+02F,
	-2.255998e+02F,
	 6.989520e+01F,
	-9.058929e+00F,
	 4.110107e-01F,
	-4.150391e-03F,
	-3.417969e-03F,
      };
    };

  template<>
    struct _GammaLanczos<double>
    {
      static constexpr double _S_g = 9.5;
      static constexpr std::array<double, 10>
      _S_cheby
      {
	 5.557569219204146e+03,
	-4.248114953727554e+03,
	 1.881719608233706e+03,
	-4.705537221412237e+02,
	 6.325224688788239e+01,
	-4.206901076213398e+00,
	 1.202512485324405e-01,
	-1.141081476816908e-03,
	 2.055079676210880e-06,
	 1.280568540096283e-09,
      };
    };

  template<>
    struct _GammaLanczos<long double>
    {
      static constexpr long double _S_g = 10.5L;
      static constexpr std::array<long double, 11>
      _S_cheby
      {
	 1.440399692024250728e+04L,
	-1.128006201837065341e+04L,
	 5.384108670160999829e+03L,
	-1.536234184127325861e+03L,
	 2.528551924697309561e+02L,
	-2.265389090278717887e+01L,
	 1.006663776178612579e+00L,
	-1.900805731354182626e-02L,
	 1.150508317664389324e-04L,
	-1.208915136885480024e-07L,
	-1.518856151960790157e-10L,
      };
    };

#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_FLOAT128)
  template<>
    struct _GammaLanczos<__float128>
    {
      static constexpr __float128 _S_g = 13.5Q;
      static constexpr std::array<__float128, 14>
      _S_cheby
      {
	 2.564476893267270739326759539239521e+05Q,
	-2.115503710351143292058877626137631e+05Q,
	 1.184019145386031178030546551084647e+05Q,
	-4.454725318050137764345651484051269e+04Q,
	 1.108444304734911694218772523380670e+04Q,
	-1.779039702565864879298077142685014e+03Q,
	 1.775917720477127714789670669062174e+02Q,
	-1.046153541164985987873520383376722e+01Q,
	 3.366596234159295518095091028749941e-01Q,
	-5.260241043537793269590387229607226e-03Q,
	 3.290526994052737279641248387319708e-05Q,
	-5.791840871423216610203896349432783e-08Q,
	 1.318192361117106013253744235732287e-11Q,
	-3.352799410216973507737605805778536e-17Q,
      };
    };
#endif

  /**
   * @brief Return @f$log(\Gamma(x))@f$ by the Lanczos method.
   * This method dominates all others on the positive axis I think.
   *
   * @param __x The argument of the log of the gamma function.
   * @return  The logarithm of the gamma function.
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_gamma1p_lanczos(_Tp __z)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_ln_2 = __gnu_cxx::__math_constants<_Real>::__ln_2;
      constexpr auto _S_ln_pi = __gnu_cxx::__math_constants<_Real>::__ln_pi;
      constexpr auto _S_log_sqrt_2pi = (_S_ln_2 + _S_ln_pi) / _Real{2};
      const auto& __c = _GammaLanczos<_Real>::_S_cheby;
      auto __g =  _GammaLanczos<_Real>::_S_g;
      // Reflection; move the transition upwards to prevent instability.
      if (std::real(__z) < _Real{-0.5L})
	return _S_ln_pi - std::log(__sin_pi(__z))
			- __log_gamma1p_lanczos(-_Real{1} - __z);
      else
        {
	  auto __fact = _Val{1};
	  auto __sum = _Val{0.5L} * __c[0];
	  for (unsigned int __k = 1, __n = __c.size(); __k < __n; ++__k)
	    {
	      __fact *= (__z - _Real(__k - 1)) / (__z + _Real(__k));
	      __sum += __fact * __c[__k];
	    }
	  return _S_log_sqrt_2pi + std::log(__sum)
	       + (__z + _Real{0.5L}) * std::log(__z + __g + _Real{0.5L})
	       - (__z + __g + _Real{0.5L});
	}
    }


  /**
   * @brief Return @f$ log(|\Gamma(x)|) @f$.
   * 	    This will return values even for @f$ x < 0 @f$.
   * 	    To recover the sign of @f$ \Gamma(x) @f$ for
   * 	    any argument use @a __log_gamma_sign.
   *
   * @param __x The argument of the log of the gamma function.
   * @return  The logarithm of the gamma function.
   */
  template<typename _Tp>
    _Tp
    __log_gamma(_Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_eps = _Real{3} * __gnu_cxx::__epsilon<_Real>();
      constexpr auto _S_logpi = __gnu_cxx::__math_constants<_Real>::__ln_pi;
      if (std::real(__x) >= _Real{0.5L})
	return __log_gamma1p_spouge(__x - _Real{1});
      else
	{
	  const auto __sin_fact = std::abs(__sin_pi(__x));
	  if (__sin_fact < _S_eps * std::abs(__x))
	    return __gnu_cxx::__infinity<_Real>();
	  else
	    return _S_logpi - std::log(__sin_fact) - __log_gamma(_Val{1} - __x);
	}
    }

  /**
   * @brief Return @f$ log(\Gamma(x)) @f$ for complex argument.
   *
   * @param __x The complex argument of the log of the gamma function.
   * @return  The complex logarithm of the gamma function.
   */
  template<typename _Tp>
    std::complex<_Tp>
    __log_gamma(std::complex<_Tp> __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      using _Cmplx = std::complex<_Real>;
      constexpr auto _S_eps = _Real{3} * __gnu_cxx::__epsilon<_Real>();
      constexpr auto _S_logpi = __gnu_cxx::__math_constants<_Real>::__ln_pi;
      if (std::real(__x) >= _Real{0.5L})
	return __log_gamma1p_spouge(__x - _Real{1});
      else
	{
	  const auto __sin_fact = __sin_pi(__x);
	  if (std::abs(__sin_fact) < _S_eps * std::abs(__x))
	    return _Cmplx(__gnu_cxx::__quiet_NaN<_Real>(), _Real{0});
	  else
	    return _S_logpi - std::log(__sin_fact) - __log_gamma(_Val{1} - __x);
	}
    }


  /**
   * @brief Return the sign of @f$ \Gamma(x) @f$.
   * At nonpositive integers zero is returned indicating @f$ \Gamma(x) @f$
   * is undefined.
   *
   * @param __x The argument of the gamma function.
   * @return  The sign of the gamma function.
   */
  template<typename _Tp>
    _Tp
    __log_gamma_sign(_Tp __x)
    {
      if (__x >= _Tp{0})
	return _Tp{1};
      else if (__x == std::nearbyint(__x))
	return _Tp{0};
      else
	return (int(-__x) % 2 == 0) ? -_Tp{1} : _Tp{1};
    }

  template<typename _Tp>
    std::complex<_Tp>
    __log_gamma_sign(std::complex<_Tp> __x)
    { return std::complex<_Tp>{1}; }


  /**
   * @brief Return the logarithm of the binomial coefficient.
   * The binomial coefficient is given by:
   * @f[
   *   \binom{n}{k} = \frac{n!}{(n-k)! k!}
   * @f]
   * The binomial coefficients are generated by:
   * @f[
   *   \left(1 + t\right)^n = \sum_{k=0}^n \binom{n}{k} t^k
   * @f]
   *
   * @param __n The first argument of the binomial coefficient.
   * @param __k The second argument of the binomial coefficient.
   * @return  The logarithm of the binomial coefficient.
   */
  template<typename _Tp>
    _Tp
    __log_bincoef(unsigned int __n, unsigned int __k)
    {
      if (__k > __n)
	return -__gnu_cxx::__infinity<_Tp>();
      else if (__k == 0 || __k == __n)
	return _Tp{0};
      else
	return __log_gamma(_Tp(1 + __n))
	     - __log_gamma(_Tp(1 + __k))
	     - __log_gamma(_Tp(1 + __n - __k));
    }


  /**
   * @brief Return the logarithm of the binomial coefficient
   * for non-integral degree.
   * The binomial coefficient is given by:
   * @f[
   *   \binom{\nu}{k} = \frac{\Gamma(\nu+1)}{\Gamma(\nu-k+1) \Gamma(k+1)}
   * @f]
   * The binomial coefficients are generated by:
   * @f[
   *   \left(1 + t\right)^\nu = \sum_{k=0}^\infty \binom{\nu}{k} t^k
   * @f]
   *
   * @param __nu The first argument of the binomial coefficient.
   * @param __k The second argument of the binomial coefficient.
   * @return  The logarithm of the binomial coefficient.
   */
  template<typename _Tp>
    _Tp
    __log_bincoef(_Tp __nu, unsigned int __k)
    {
      auto __n = std::nearbyint(__nu);
      if (__n >= 0 && __nu == __n)
	return __log_bincoef<_Tp>((unsigned int)__n, __k);
      else
	{
	  return __log_gamma(_Tp(1) + __nu)
	       - __log_gamma(_Tp(1 + __k))
	       - __log_gamma(_Tp(1 - __k) + __nu);
	}
    }

  /**
   * @brief Return the sign of @f$ \Gamma(x) @f$.
   * At nonpositive integers zero is returned.
   *
   * @param __x The argument of the gamma function.
   * @return  The sign of the gamma function.
   */
  template<typename _Tp>
    _Tp
    __log_bincoef_sign(_Tp __nu, unsigned int __k)
    {
      auto __n = std::nearbyint(__nu);
      if (__n >= 0 && __nu == __n)
	return _Tp{1};
      else
	{
	  return __log_gamma_sign(_Tp(1) + __nu)
	       * __log_gamma_sign(_Tp(1 + __k))
	       * __log_gamma_sign(_Tp(1 - __k) + __nu);
	}
    }

  template<typename _Tp>
    std::complex<_Tp>
    __log_bincoef_sign(std::complex<_Tp> __nu, unsigned int __k)
    { return std::complex<_Tp>{1}; }


  /**
   * @brief Return the binomial coefficient.
   * The binomial coefficient is given by:
   * @f[
   *   \binom{n}{k} = \frac{n!}{(n-k)! k!}
   * @f]
   * The binomial coefficients are generated by:
   * @f[
   *   \left(1 + t\right)^n = \sum_{k=0}^n \binom{n}{k} t^k
   * @f]
   *
   * @param __n The first argument of the binomial coefficient.
   * @param __k The second argument of the binomial coefficient.
   * @return  The binomial coefficient.
   */
  template<typename _Tp>
    _Tp
    __bincoef(unsigned int __n, unsigned int __k)
    {
      // Max e exponent before overflow.
      constexpr auto __max_bincoeff
                      = std::numeric_limits<_Tp>::max_exponent10
                      * std::log(_Tp(10)) - _Tp(1);

      if (__k > __n)
	return _Tp{0};
      else if (__k == 0 || __k == __n)
	return _Tp{1};
      else
        {
	  const auto __log_coeff = __log_bincoef<_Tp>(__n, __k);
	  if (__log_coeff > __max_bincoeff)
	    return __gnu_cxx::__quiet_NaN<_Tp>();
	  else
	    return std::exp(__log_coeff);
	}
    }


  /**
   *  @brief Return the binomial coefficient for non-integral degree.
   *  The binomial coefficient is given by:
   *  @f[
   *    \binom{\nu}{k} = \frac{\Gamma(\nu+1)}{\Gamma(\nu-k+1) \Gamma(k+1)}
   *  @f]
   *  The binomial coefficients are generated by:
   *  @f[
   *    \left(1 + t\right)^\nu = \sum_{k=0}^\infty \binom{\nu}{k} t^k
   *  @f]
   *
   *  @param __nu The real first argument of the binomial coefficient.
   *  @param __k The second argument of the binomial coefficient.
   *  @return  The binomial coefficient.
   */
  template<typename _Tp>
    _Tp
    __bincoef(_Tp __nu, unsigned int __k)
    {
      // Max e exponent before overflow.
      auto __n = std::nearbyint(__nu);
      if (__n >= 0 && __nu == __n)
	return __bincoef<_Tp>((unsigned int)__n, __k);
      else
	{
	  constexpr auto __max_bincoeff
                	  = std::numeric_limits<_Tp>::max_exponent10
                	  * std::log(_Tp(10)) - _Tp(1);

	  const auto __log_coeff = __log_bincoef(__nu, __k);
	  const auto __sign = __log_bincoef_sign(__nu, __k);
	  if (__log_coeff > __max_bincoeff || __sign == _Tp{0})
	    return __gnu_cxx::__quiet_NaN<_Tp>();
	  else
	    return std::exp(__log_coeff) * __sign;
	}
    }


  /**
   * @brief Return the gamma function @f$ \Gamma(x) @f$.
   * The gamma function is defined by:
   * @f[
   *   \Gamma(a) = \int_0^\infty e^{-t}t^{a-1}dt  (a > 0)
   * @f]
   *
   * @param __x The argument of the gamma function.
   * @return  The gamma function.
   */
  template<typename _Tp>
    _Tp
    __gamma(_Tp __x)
    {
      const auto __sign = __log_gamma_sign(__x);
      if (__sign == _Tp{0})
	return __gnu_cxx::__quiet_NaN<_Tp>();
      else
	return __sign * std::exp(__log_gamma(__x));
    }


  /**
   * @brief Return the incomplete gamma function by series summation.
   */
  template<typename _Tp>
    std::pair<_Tp, _Tp>
    __gamma_series(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_eps = _Real{3} * __gnu_cxx::__epsilon<_Real>();
      const auto _S_itmax = 10 * int(10 + std::sqrt(std::abs(__a)));

      auto __lngam = __log_gamma(__a);

      if (std::real(__x) < _Real{0})
	std::__throw_domain_error(__N("__gamma_series: argument less than 0"));
      else if (__x == _Real{0})
	return std::make_pair(_Val{0}, __lngam);
      else
	{
	  auto __aa = __a;
	  _Val __term, __sum;
	  __term = __sum = _Tp{1} / __a;
	  for (unsigned int __n = 1; __n <= _S_itmax; ++__n)
	    {
	      __aa += _Real{1};
	      __term *= __x / __aa;
	      __sum += __term;
	      if (std::abs(__term) < _S_eps * std::abs(__sum))
		{
		  auto __gamser = std::exp(-__x + __a * std::log(__x) - __lngam)
				* __sum;
		  return std::make_pair(__gamser, __lngam);
		}
	    }
	  std::__throw_logic_error(__N("__gamma_series: "
	  			"a too large, itmax too small in routine."));
	}
    }

  /**
   * @brief Return the incomplete gamma function by continued fraction.
   */
  template<typename _Tp>
    std::pair<_Tp, _Tp>
    __gamma_cont_frac(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_fpmin = _Real{3} * __gnu_cxx::__min<_Real>();
      constexpr auto _S_eps = _Real{3} * __gnu_cxx::__epsilon<_Real>();
      const auto _S_itmax = 10 * int(10 + std::sqrt(std::abs(__a)));

      auto __lngam = __log_gamma(__a);

      auto __b = __x + _Real{1} - __a;
      auto __c = _Real{1} / _S_fpmin;
      auto __d = _Real{1} / __b;
      auto __h = __d;
      for (unsigned int __n = 1; __n <= _S_itmax; ++__n)
	{
	  auto __an = -_Real{__n} * (_Real{__n} - __a);
	  __b += _Real{2};
	  __d = __an * __d + __b;
	  if (std::abs(__d) < _S_fpmin)
	    __d = _S_fpmin;
	  __c = __b + __an / __c;
	  if (std::abs(__c) < _S_fpmin)
	    __c = _S_fpmin;
	  __d = _Real{1} / __d;
	  auto __del = __d * __c;
	  __h *= __del;
	  if (std::abs(__del - _Real{1}) < _S_eps)
	    {
	      auto __gamcf = std::exp(-__x + __a * std::log(__x) - __lngam)
			  * __h;
	      return std::make_pair(__gamcf, __lngam);
	    }
	}
      std::__throw_logic_error(__N("__gamma_cont_fraction: "
				   "a too large, itmax too small in routine."));
    }


  /**
   * @brief  Return the regularized lower incomplete gamma function.
   * The regularized lower incomplete gamma function is defined by
   * @f[
   *   P(a,x) = \frac{\gamma(a,x)}{\Gamma(a)}
   * @f]
   * where @f$ \Gamma(a) @f$ is the gamma function and
   * @f[
   *   \gamma(a,x) = \int_0^x e^{-t}t^{a-1}dt  (a > 0)
   * @f]
   * is the lower incomplete gamma function.
   */
  template<typename _Tp>
    _Tp
    __pgamma(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_NaN = __gnu_cxx::__quiet_NaN<_Real>();

      if (__isnan(__a) || __isnan(__x))
	return _S_NaN;

      if (std::real(__x) < _Real{0} || std::real(__a) <= _Real{0})
	std::__throw_domain_error("pgamma: invalid arguments");

      if (std::real(__x) < std::real(__a + _Real{1}))
	return __gamma_series(__a, __x).first;
      else
	return _Val{1} - __gamma_cont_frac(__a, __x).first;
    }


  /**
   * @brief  Return the regularized upper incomplete gamma function.
   * The regularized upper incomplete gamma function is defined by
   * @f[
   *   Q(a,x) = \frac{\Gamma(a,x)}{\Gamma(a)}
   * @f]
   * where @f$ \Gamma(a) @f$ is the gamma function and
   * @f[
   *   \Gamma(a,x) = \int_x^\infty e^{-t}t^{a-1}dt  (a > 0)
   * @f]
   * is the upper incomplete gamma function.
   */
  template<typename _Tp>
    _Tp
    __qgamma(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_NaN = __gnu_cxx::__quiet_NaN<_Real>();

      if (__isnan(__a) || __isnan(__x))
	return _S_NaN;

      if (std::real(__x) < _Real{0} || std::real(__a) <= _Real{0})
	std::__throw_domain_error("__qgamma: invalid arguments");

      if (std::real(__x) < std::real(__a + _Real{1}))
	return _Val{1} - __gamma_series(__a, __x).first;
      else
	return __gamma_cont_frac(__a, __x).first;
    }


  /**
   * @brief  Return the lower incomplete gamma function.
   * The lower incomplete gamma function is defined by
   * @f[
   *   \gamma(a,x) = \int_0^x e^{-t}t^{a-1}dt  (a > 0)
   * @f]
   */
  template<typename _Tp>
    _Tp
    __tgamma_lower(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_NaN = __gnu_cxx::__quiet_NaN<_Real>();

      if (__isnan(__a) || __isnan(__x))
	return _S_NaN;

      if (std::real(__x) < _Real{0} || std::real(__a) <= _Real{0})
	std::__throw_domain_error("__tgamma_lower: invalid arguments");

      if (std::real(__x) < std::real(__a + _Real{1}))
	{
	  std::pair<_Tp, _Tp> __gp = __gamma_series(__a, __x);
	  return std::exp(__gp.second) * __gp.first;
	}
      else
	{
	  std::pair<_Tp, _Tp> __gp = __gamma_cont_frac(__a, __x);
	  return std::exp(__gp.second) * (_Tp{1} - __gp.first);
	}
    }


  /**
   * @brief  Return the upper incomplete gamma function.
   * The lower incomplete gamma function is defined by
   * @f[
   *   \Gamma(a,x) = \int_x^\infty e^{-t}t^{a-1}dt  (a > 0)
   * @f]
   */
  template<typename _Tp>
    _Tp
    __tgamma(_Tp __a, _Tp __x)
    {
      using _Val = _Tp;
      using _Real = std::__detail::__num_traits_t<_Val>;
      constexpr auto _S_NaN = __gnu_cxx::__quiet_NaN<_Real>();

      if (__isnan(__a) || __isnan(__x))
	return _S_NaN;

      if (std::real(__x) < _Real{0} || std::real(__a) <= _Real{0})
	std::__throw_domain_error("__tgamma: invalid arguments");

      if (std::real(__x) < std::real(__a + _Real{1}))
	{
	  auto __gp = __gamma_series(__a, __x);
	  return std::exp(__gp.second) * (_Tp{1} - __gp.first);
	}
      else
	{
	  auto __gp = __gamma_cont_frac(__a, __x);
	  return std::exp(__gp.second) * __gp.first;
	}
    }


  /**
   * @brief  Return the logarithm of the (upper) Pochhammer symbol
   * or the rising factorial function.
   * The Pochammer symbol is defined for integer order by
   * @f[
   *   (a)_n = \prod_{k=0}^{n-1} (a + k), (a)_0 = 1
   *	     = \Gamma(a + n) / \Gamma(n)
   * @f]
   * Thus this function returns
   * @f[
   *   ln[(a)_n] = \Gamma(a + n) - \Gamma(n), ln[(a)_0] = 0
   * @f]
   * Many notations exist: @f[ a^{\overline{n}} @f],
   *  @f[ \left[ \begin{array}{c}
   *	  a \\
   *	  n \end{array} \right] @f], and others.
   */
  template<typename _Tp>
    _Tp
    __log_pochhammer(_Tp __a, _Tp __n)
    {
      if (__isnan(__n) || __isnan(__a))
	return __gnu_cxx::__quiet_NaN<_Tp>();
      else if (__n == _Tp{0})
	return _Tp{0};
      else
	return __log_gamma(__a + __n) - __log_gamma(__a);
    }


  /**
   * @brief  Return the (upper) Pochhammer function
   * or the rising factorial function.
   * The Pochammer symbol is defined by
   * @f[
   *   (a)_n = \prod_{k=0}^{n-1} (a + k), (a)_0 = 1
   *	     = \Gamma(a + n) / \Gamma(n)
   * @f]
   * Many notations exist: @f[ a^{\overline{n}} @f],
   *  @f[ \left[ \begin{array}{c}
   *	  a \\
   *	  n \end{array} \right] @f], and others.
   */
  template<typename _Tp>
    _Tp
    __pochhammer(_Tp __a, _Tp __n)
    {
      constexpr auto __log10{2.3025850929940456840179914546843642L};
      if (__isnan(__n) || __isnan(__a))
	return __gnu_cxx::__quiet_NaN<_Tp>();
      else if (__n == _Tp{0})
	return _Tp{1};
      else
	{
          _Tp __logpoch = __log_gamma(__a + __n) - __log_gamma(__a);
          if (std::abs(__logpoch)
              > std::numeric_limits<_Tp>::max_digits10 * __log10)
            return __gnu_cxx::__infinity<_Tp>();
          else
            return std::exp(__logpoch);
	}
    }


  /**
   * @brief  Return the logarithm of the lower Pochhammer symbol
   * or the falling factorial function.
   * The lower Pochammer symbol is defined by
   * @f[
   *   (a)_n = \prod_{k=0}^{n-1} (a - k), (a)_0 = 1
   *	     = \Gamma(a + 1) / \Gamma(a - n + 1)
   * @f]
   * In particular, $f[ (n)_n = n! $f].
   * Thus this function returns
   * @f[
   *   ln[(a)_n] = \Gamma(a + 1) - \Gamma(a - n + 1), ln[(a)_0] = 0
   * @f]
   * Many notations exist: @f[ a^{\underline{n}} @f],
   *  @f[ \{ \begin{array}{c}
   *	  a \\
   *	  n \end{array} \} @f], and others.
   */
  template<typename _Tp>
    _Tp
    __log_pochhammer_lower(_Tp __a, _Tp __n)
    {
      if (__isnan(__n) || __isnan(__a))
	return __gnu_cxx::__quiet_NaN<_Tp>();
      else if (__n == _Tp{0})
	return _Tp{0};
      else
	return __log_gamma(__a + _Tp{1}) - __log_gamma(__a - __n + _Tp{1});
    }


  /**
   * @brief  Return the logarithm of the lower Pochhammer symbol
   * or the falling factorial function.
   * The lower Pochammer symbol is defined by
   * @f[
   *   (a)_n = \prod_{k=0}^{n-1} (a - k), (a)_0 = 1
   *	     = \Gamma(a + 1) / \Gamma(a - n + 1)
   * @f]
   * In particular, $f[ (n)_n = n! $f].
   */
  template<typename _Tp>
    _Tp
    __pochhammer_lower(_Tp __a, _Tp __n)
    {
      constexpr auto __log10{2.3025850929940456840179914546843642L};
      if (__isnan(__n) || __isnan(__a))
	return __gnu_cxx::__quiet_NaN<_Tp>();
      else if (__n == _Tp{0})
	return _Tp{1};
      else
	{
          auto __logpoch = __log_gamma(__a + _Tp{1})
			 - __log_gamma(__a - __n + _Tp{1});
	  auto __sign = __log_gamma_sign(__a + _Tp{1})
		      * __log_gamma_sign(__a - __n + _Tp{1});
          if (__sign == _Tp{0})
            return __gnu_cxx::__quiet_NaN<_Tp>();
          else if (__logpoch > __gnu_cxx::__log_max<_Tp>())
            return __sign * __gnu_cxx::__infinity<_Tp>();
          else
            return __sign * std::exp(__logpoch);
	}
    }

  constexpr unsigned long long
  _S_num_harmonic_numer = 29;
  constexpr unsigned long long
  _S_harmonic_numer[_S_num_harmonic_numer]
  {
    1ULL,
    3ULL,
    11ULL,
    25ULL,
    137ULL,
    49ULL,
    363ULL,
    761ULL,
    7129ULL,
    7381ULL,
    83711ULL,
    86021ULL,
    1145993ULL,
    1171733ULL,
    1195757ULL,
    2436559ULL,
    42142223ULL,
    14274301ULL,
    275295799ULL,
    55835135ULL,
    18858053ULL,
    19093197ULL,
    444316699ULL,
    1347822955ULL,
    34052522467ULL,
    34395742267ULL,
    312536252003ULL,
    315404588903ULL,
    9227046511387ULL
  };
  constexpr unsigned long long
  _S_harmonic_denom[_S_num_harmonic_numer]
  {
    1ULL,
    2ULL,
    6ULL,
    12ULL,
    60ULL,
    20ULL,
    140ULL,
    280ULL,
    2520ULL,
    2520ULL,
    27720ULL,
    27720ULL,
    360360ULL,
    360360ULL,
    360360ULL,
    720720ULL,
    12252240ULL,
    4084080ULL,
    77597520ULL,
    15519504ULL,
    5173168ULL,
    5173168ULL,
    118982864ULL,
    356948592ULL,
    8923714800ULL,
    8923714800ULL,
    80313433200ULL,
    80313433200ULL,
    2329089562800ULL
  };

  template<typename _Tp>
    _Tp
    __harmonic_number(unsigned int __n)
    {
      if (__n <= _S_num_harmonic_numer)
	return _Tp(_S_harmonic_numer[__n - 1])
	     / _Tp(_S_harmonic_denom[__n - 1]);
      else
        {
	  unsigned int __k = _S_num_harmonic_numer - 1;
	  auto _H_k = _Tp(_S_harmonic_numer[__k]) / _Tp(_S_harmonic_denom[__k]);
	  for (__k = _S_num_harmonic_numer; __k <= __n; ++__k)
	    _H_k += _Tp{1} / _Tp(__k);
	  return _H_k;
	}
    }


  /**
   * @brief  Return the digamma function of integral argument.
   * The digamma or @f$ \psi(x) @f$ function is defined as the logarithmic
   * derivative of the gamma function:
   * @f[
   *   \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
   * @f]
   * The digamma series for integral argument is given by:
   * @f[
   *   \psi(n) = -\gamma_E + \sum_{k=1}^{\infty} \frac{1}{k}
   * @f]
   * The latter sum is called the harmonic number, @f$ H_n @f$.
   */
  template<typename _Tp>
    _Tp
    __psi(unsigned int __n)
    {
      constexpr _Tp __gamma_E = __gnu_cxx::__math_constants<_Tp>::__gamma_e;
      return -__gamma_E + __harmonic_number<_Tp>(__n);
    }

  /**
   * @brief  Return the digamma function by series expansion.
   * The digamma or @f$ \psi(x) @f$ function is defined by
   * @f[
   *   \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
   * @f]
   *
   * The series is given by:
   * @f[
   *   \psi(x) = -\gamma_E - \frac{1}{x}
   *		\sum_{k=1}^{\infty} \frac{x - 1}{(k + 1)(x + k)}
   * @f]
   */
  template<typename _Tp>
    _Tp
    __psi_series(_Tp __x)
    {
      _Tp __sum = -__gnu_cxx::__math_constants<_Tp>::__gamma_e;
      const unsigned int _S_max_iter = 100000;
      for (unsigned int __k = 0; __k < _S_max_iter; ++__k)
	{
	  const auto __term = (__x - _Tp{1})
			    / (_Tp(__k + 1) * (_Tp(__k) + __x));
	  __sum += __term;
	  if (std::abs(__term) < __gnu_cxx::__epsilon<_Tp>())
	    break;
	}
      return __sum;
    }


  /**
   * @brief  Return the digamma function for large argument.
   * The digamma or @f$ \psi(x) @f$ function is defined by
   * @f[
   *   \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
   * @f]
   *
   * The asymptotic series is given by:
   * @f[
   *   \psi(x) = \ln(x) - \frac{1}{2x}
   *	       - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}}
   * @f]
   */
  template<typename _Tp>
    _Tp
    __psi_asymp(_Tp __x)
    {
      auto __sum = std::log(__x) - _Tp{0.5L} / __x;
      const auto __xx = __x * __x;
      auto __xp = __xx;
      const unsigned int __max_iter = 100;
      for (unsigned int __k = 1; __k < __max_iter; ++__k)
	{
	  const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp);
	  __sum -= __term;
	  if (std::abs(__term / __sum) < __gnu_cxx::__epsilon<_Tp>())
	    break;
	  __xp *= __xx;
	}
      return __sum;
    }


  /**
   * @brief  Return the digamma function.
   * The digamma or @f$ \psi(x) @f$ function is defined by
   * @f[
   *   \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
   * @f]
   * For negative argument the reflection formula is used:
   * @f[
   *   \psi(x) = \psi(1-x) - \pi \cot(\pi x)
   * @f]
   */
  template<typename _Tp>
    _Tp
    __psi(_Tp __x)
    {
      constexpr auto _S_eps = _Tp{4} * __gnu_cxx::__epsilon<_Tp>();
      constexpr auto _S_x_asymp = _Tp{20};
      constexpr auto __gamma_E = __gnu_cxx::__math_constants<_Tp>::__gamma_e;
      constexpr auto __2_ln_2 = 2 * __gnu_cxx::__math_constants<_Tp>::__ln_2;

      const auto __n = std::nearbyint(__x);
      const bool __integral = (std::abs(__x - _Tp{__n}) < _S_eps);
      const auto __m = std::nearbyint(2 * __x);
      const bool __half_integral = !__integral
				&& (std::abs(2 * __x - _Tp{__m}) < _S_eps);
      if (__integral)
	{
	  if (__n <= 0)
	    return __gnu_cxx::__quiet_NaN<_Tp>();
	  else
	    {
	      _Tp __sum = -__gamma_E;
	      for (int __k = 1; __k < __n; ++__k)
		__sum += _Tp{1} / __k;
	      return __sum;
	    }
	}
      if (__half_integral)
	{
	  _Tp __sum = -__gamma_E - __2_ln_2;
	  for (int __k = 1; __k < __m / 2; ++__k)
	    __sum += _Tp{2} / (2 * __k - 1);
	  return __sum;
	}
      else if (__x < _Tp{0})
	{
	  constexpr auto __pi = __gnu_cxx::__math_constants<_Tp>::__pi;
	  return __psi(_Tp{1} - __x) - __pi / std::tan(__pi * __x);
	}
      else if (__x > _S_x_asymp)
	return __psi_asymp(__x);
      else
	{
	  //return __psi_series(__x);
	  // The series does not converge quickly enough.
	  // Reflect to larger argument and use asymptotic expansion.
	  auto __w = _Tp{0};
	  auto __y = __x;
	  while (__y <= _S_x_asymp)
	    {
	      __w += 1 / __y;
	      __y += 1;
	    }
	  return __psi_asymp(__y) - __w;
	}
    }


  /**
   * @brief  Return the polygamma function @f$ \psi^{(n)}(x) @f$.
   *
   * The polygamma function is related to the Hurwitz zeta function:
   * @f[
   *   \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x)
   * @f]
   */
  template<typename _Tp>
    _Tp
    __psi(unsigned int __n, _Tp __x)
    {
      if (__x <= _Tp{0})
	std::__throw_domain_error(__N("__psi: argument out of range"));
      else if (__n == 0)
	return __psi(__x);
      else
	{
	  const auto __hzeta = __hurwitz_zeta(_Tp{__n + 1}, __x);
	  const auto __ln_nfact = __log_gamma(_Tp{__n + 1});
	  auto __result = std::exp(__ln_nfact) * __hzeta;
	  if (__n % 2 == 1)
	    __result = -__result;
	  return __result;
	}
    }

  /**
   * @brief  Return the factorial of the integer n.
   *
   * The factorial is:
   * @f[
   *   n! = 1 2 ... (n-1) n, 0! = 1
   * @f]
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __factorial(unsigned int __n)
    {
      if (__n <= _S_num_factorials<_Tp>)
	return _S_factorial_table[__n].__factorial;
      else
	return __gnu_cxx::__infinity<_Tp>();
    }

  /**
   * @brief  Return the logarithm of the factorial of the integer n.
   *
   * The factorial is:
   * @f[
   *   n! = 1 2 ... (n-1) n, 0! = 1
   * @f]
   */
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_factorial(unsigned int __n)
    {
      if (__n <= _S_num_factorials<_Tp>)
	return _S_factorial_table[__n].__log_factorial;
      else
	return __log_gamma(_Tp(__n + 1));
    }

  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_double_factorial(_Tp __x)
    {
      constexpr auto _S_pi = __gnu_cxx::__math_constants<_Tp>::__pi;
      return (__x / _Tp{2}) * std::log(_Tp{2})
	   + (__cos_pi(__x) - _Tp{1})
		* std::log(_S_pi / 2) / _Tp{4}
	   + __log_gamma(_Tp{1} + __x / _Tp{2});
    }

  /**
   * @brief  Return the double factorial of the integer n.
   *
   * The double factorial is defined for integral n by:
   * @f[
   *   n!! = 1 3 5 ... (n-2) n, n odd
   *   n!! = 2 4 6 ... (n-2) n, n even
   *   -1!! = 1
   *   0!! = 1
   * @f]
   * The double factorial is defined for odd negative integers
   * in the obvious way:
   * @f[
   *   (-2m - 1)!! = 1 / (1 (-1) (-3) ... (-2m + 1) (-2m - 1))
   *	   = \frac{(-1)^m}{(2m-1)!!}
   * @f]
   * for $f[ n = -2m - 1 $f].
   */
  // I must watch neg log double factorial.  Or do the log_t thing.
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __double_factorial(int __n)
    {
      if (__n < 0 && __n % 2 == 1)
	{
	  if (-__n <= _S_num_neg_double_factorials<_Tp>)
	    return _S_neg_double_factorial_table[-(1 + __n) / 2].__factorial;
	  else
	    return std::exp(__log_double_factorial(_Tp(__n)));
	}
      else if (__n <= _S_num_double_factorials<_Tp>)
	return _S_double_factorial_table[__n].__factorial;
      else
	return __gnu_cxx::__quiet_NaN<_Tp>();
    }

  /**
   * @brief  Return the logarithm of the double factorial of the integer n.
   *
   * The double factorial is defined for integral n by:
   * @f[
   *   n!! = 1 3 5 ... (n-2) n, n odd
   *   n!! = 2 4 6 ... (n-2) n, n even
   *   -1!! = 1
   *   0!! = 1
   * @f]
   * The double factorial is defined for odd negative integers
   * in the obvious way:
   * @f[
   *   (-2m - 1)!! = 1 / (1 (-1) (-3) ... (-2m + 1) (-2m - 1))
   *	   = \frac{(-1)^m}{(2m-1)!!}
   * @f]
   * for $f[ n = -2m - 1 $f].
   */
  // I should do a signed version.  Or do the log_t thing.
  template<typename _Tp>
    _GLIBCXX14_CONSTEXPR _Tp
    __log_double_factorial(int __n)
    {
      if (__n < 0 && __n % 2 == 1)
	{
	  if (-__n <= _S_num_neg_double_factorials<_Tp>)
	    return _S_neg_double_factorial_table[-(1 + __n) / 2]
				.__log_factorial;
	  else
	    return __log_double_factorial(_Tp(__n));
	}
      else if (__n <= _S_num_double_factorials<_Tp>)
	return _S_double_factorial_table[__n].__log_factorial;
      else
	return __log_double_factorial(_Tp(__n));
    }

_GLIBCXX_END_NAMESPACE_VERSION
} // namespace __detail
} // namespace std

#endif // _GLIBCXX_BITS_SF_GAMMA_TCC