math/tools/lanczos_generator.cpp

//  (C) Copyright John Maddock 2006.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

#include <boost/math/constants/constants.hpp>
#include <boost/math/tools/precision.hpp>
#include <boost/math/tools/polynomial.hpp>
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/numeric/ublas/io.hpp>
#include <boost/numeric/ublas/operation.hpp>
#include <boost/cstdint.hpp>
#include <boost/bind.hpp>
#include <boost/math/special_functions/log1p.hpp>
#include <boost/math/special_functions/relative_difference.hpp>
#include <boost/lexical_cast.hpp>
#include <boost/array.hpp>
#include <boost/random.hpp>
#include <boost/chrono.hpp>
#include "mp_t.hpp"

//
// Define MP_TYPE
// to a specific precision multiprecision type if you want to
// create a lanczos approximation for that type.
//

#define MP_TYPE boost::multiprecision::number<boost::multiprecision::cpp_bin_float<100> >

//
// this is a sort of recursive include, since this file
// is used to create the contents of gamma.hpp.
// However, we will be using the "generic" version of gamma
// which doesn't use a lanczos approximation.
//
#include <boost/math/special_functions/gamma.hpp>

#include <iostream>
#include <iomanip>
#include <vector>

using boost::numeric::ublas::matrix;
using boost::numeric::ublas::vector;

struct lanczos_spot_data
{
   int N;
   double g;
   double err;
};

lanczos_spot_data sweet_spots[] = {
// a few selected spots from Pugh's thesis:
3, 2.03209, 6.4e-7,
4, 3.655180, 8.5e-8,
9, 8.406094, 6.9e-15,
10, 9.656578, 2.1e-16,
11, 10.900511, 6.1e-18,
12, 12.066012, 1.1e-19,
13, 13.144565, 5.2e-21,
14, 13.726821, 4.0e-22,
15, 14.977863, 1.2e-23,
20, 20.298892, 7.8e-31,
21, 21.508926, 2.1e-32,
22, 22.618910, 1.8e-34,
23, 23.118012, 5.2e-35,

// some more we've found, these are all the points where the first
// negleted term from the Lanczos series changes sign, there is one 
// point just above that point, and one just below:

3, 0.58613894134759903, 0.00036580426080686315,
3, 0.58613894879817963, 0.00036580416320668711,
3, 1.249951496720314, 4.2305536798953929e-005,
3, 1.2499515041708946, 4.2305534015367749e-005,
3, 1.8384982049465179, 1.164778449938813e-005,
3, 1.8384982123970985, 1.1647786529958048e-005,
3, 2.3276706635951996, 5.3732582230468998e-006,
3, 2.3276706710457802, 5.3732600294516588e-006,
3, 2.6038404405117035, 7.4060771171390379e-007,
3, 2.6038404479622841, 7.406058543248471e-007,
4, 0.62704569846391678, 0.00012800533557793425,
4, 0.62704570591449738, 0.00012800531551557324,
4, 1.3344274088740349, 9.0166140418663663e-006,
4, 1.3344274163246155, 9.0166116338646694e-006,
4, 1.9880444705486298, 1.4955557778725745e-006,
4, 1.9880444779992104, 1.4955557079994109e-006,
4, 2.5835030898451805, 4.5877676859600527e-007,
4, 2.5835030972957611, 4.5877667876325684e-007,
4, 3.1086832508444786, 2.1441725587707906e-007,
4, 3.1086832582950592, 2.1441723688775765e-007,
4, 3.5476611852645874, 1.0813760990515544e-007,
4, 3.547661192715168, 1.0813766234070327e-007,
4, 3.6492579728364944, 1.047642251124206e-007,
4, 3.649257980287075, 1.0476415796074403e-007,
5, 0.65435918420553207, 5.5274429026631278e-005,
5, 0.65435919165611267, 5.527441185903385e-005,
5, 1.389187254011631, 2.6307833416554052e-006,
5, 1.3891872614622116, 2.6307826816540255e-006,
5, 2.0801975876092911, 2.9014694764756466e-007,
5, 2.0801975950598717, 2.9014696142909125e-007,
5, 2.7272208333015442, 5.9176544536206063e-008,
5, 2.7272208407521248, 5.9176528019009275e-008,
5, 3.3269794136285782, 1.9379184595472579e-008,
5, 3.3269794210791588, 1.9379189057179851e-008,
5, 3.8722873777151108, 9.0123694745032944e-009,
5, 3.8722873851656914, 9.0123661007867372e-009,
5, 4.3516943231225014, 4.9816097608842378e-009,
5, 4.351694330573082, 4.9816087871575021e-009,
6, 0.67425807565450668, 2.7371903086326379e-005,
6, 0.67425808310508728, 2.7371893712684938e-005,
6, 1.4284561350941658, 9.4138428977455083e-007,
6, 1.4284561425447464, 9.4138417896969612e-007,
6, 2.1447814926505089, 7.4109049995920916e-008,
6, 2.1447815001010895, 7.4109044064461768e-008,
6, 2.8244904428720474, 1.0708333024058614e-008,
6, 2.824490450322628, 1.0708333661155651e-008,
6, 3.4669468700885773, 2.4940116660002578e-009,
6, 3.4669468775391579, 2.4940118560592828e-009,
6, 4.0696377158164978, 8.5267622527507512e-010,
6, 4.0696377232670784, 8.5267624048985822e-010,
6, 4.6278460025787354, 3.9323405231792876e-010,
6, 4.6278460100293159, 3.9323393550021793e-010,
6, 5.1345816180109978, 2.1943462473419721e-010,
6, 5.1345816254615784, 2.1943458105218707e-010,
6, 5.5917193591594696, 1.242916896389048e-010,
6, 5.5917193666100502, 1.2429175679293363e-010,
7, 0.68959406018257141, 1.4934177419305624e-005,
7, 0.68959406763315201, 1.4934171696393048e-005,
7, 1.4584139510989189, 3.8891482410396904e-007,
7, 1.4584139585494995, 3.8891475436265016e-007,
7, 2.1934122517704964, 2.2977989848570384e-008,
7, 2.193412259221077, 2.2977982819591052e-008,
7, 2.8964230641722679, 2.4742239281355561e-009,
7, 2.8964230716228485, 2.4742234478616864e-009,
7, 3.5678729563951492, 4.2816138965285379e-010,
7, 3.5678729638457298, 4.2816128067858218e-010,
7, 4.2069913893938065, 1.0931003472338519e-010,
7, 4.2069913968443871, 1.0931004659777642e-010,
7, 4.8118086308240891, 3.8518213120095777e-011,
7, 4.8118086382746696, 3.8518212172324096e-011,
7, 5.3790060579776764, 1.7613602961109952e-011,
7, 5.379006065428257, 1.7613603275796266e-011,
7, 5.9039439857006073, 9.7098281187573355e-012,
7, 5.9039439931511879, 9.7098271522398387e-012,
7, 6.3838147521018982, 5.7878801570305524e-012,
7, 6.3838147595524788, 5.7878771973838702e-012,
7, 6.7833591029047966, 2.829092368392454e-012,
7, 6.7833591103553772, 2.8290901754246269e-012,
8, 0.70188850909471512, 8.7610752024780369e-006,
8, 0.70188851654529572, 8.7610704217796693e-006,
8, 1.4822541922330856, 1.7881701068328982e-007,
8, 1.4822541996836662, 1.7881694190606741e-007,
8, 2.231784276664257, 8.2272386690167009e-009,
8, 2.2317842841148376, 8.2272367814664617e-009,
8, 2.9525627419352531, 6.8591536474980462e-010,
8, 2.9525627493858337, 6.8591515664364383e-010,
8, 3.6455128863453865, 9.1549235855884194e-011,
8, 3.6455128937959671, 9.1549217088498416e-011,
8, 4.3106606528162956, 1.8015466799942756e-011,
8, 4.3106606602668762, 1.8015467001952356e-011,
8, 4.9472371935844421, 4.918149362493385e-012,
8, 4.9472372010350227, 4.9181495178774722e-012,
8, 5.553666315972805, 1.7720036810370511e-012,
8, 5.5536663234233856, 1.7720035779693502e-012,
8, 6.1274929493665695, 8.0403304333886761e-013,
8, 6.1274929568171501, 8.0403274526493467e-013,
8, 6.6654317826032639, 4.3551924163930176e-013,
8, 6.6654317900538445, 4.3551918580885179e-013,
8, 7.1646013185381889, 2.6142608803934048e-013,
8, 7.1646013259887695, 2.6142610866873453e-013,
8, 7.6312204822897911, 1.5767278184484503e-013,
8, 7.6312204897403717, 1.576728292094547e-013,
8, 7.8779716044664383, 6.0579161144149095e-014,
8, 7.8779716119170189, 6.0579145495868576e-014,
9, 0.7120363786816597, 5.4383680639900953e-006,
9, 0.7120363861322403, 5.4383647430909934e-006,
9, 1.5018193200230598, 8.9331552269793478e-008,
9, 1.5018193274736404, 8.9331529217486569e-008,
9, 2.2630845010280609, 3.2922940247769628e-009,
9, 2.2630845084786415, 3.2922936681951672e-009,
9, 2.9980199560523033, 2.1887825488935961e-010,
9, 2.9980199635028839, 2.1887818854849753e-010,
9, 3.7078000977635384, 2.3213138957857667e-011,
9, 3.707800105214119, 2.3213132917173771e-011,
9, 4.3928612694144249, 3.6226509291513855e-012,
9, 4.3928612768650055, 3.6226507116549233e-012,
9, 5.0530400574207306, 7.8472620106742312e-013,
9, 5.0530400648713112, 7.8472599177745781e-013,
9, 5.6876183524727821, 2.2542091218064693e-013,
9, 5.6876183599233627, 2.2542085472082343e-013,
9, 6.2953110262751579, 8.261026286619692e-014,
9, 6.2953110337257385, 8.2610267677539022e-014,
9, 6.8742315545678139, 3.7226229880661886e-014,
9, 6.8742315620183945, 3.722621661830273e-014,
9, 7.4219388365745544, 1.9804198141916339e-014,
9, 7.421938844025135, 1.9804197841080089e-014,
9, 7.936030775308609, 1.1800366216622659e-014,
9, 7.9360307827591896, 1.1800362381207757e-014,
9, 8.4175376817584038, 7.341772297716249e-015,
9, 8.4175376892089844, 7.3417744970482286e-015,
10, 0.72060277312994003, 3.5324438420494327e-006,
10, 0.72060278058052063, 3.5324425213780505e-006,
10, 1.5182576552033424, 4.7690979892410133e-008,
10, 1.518257662653923, 4.7690968887170632e-008,
10, 2.2892609387636185, 1.4396365431590603e-009,
10, 2.2892609462141991, 1.4396363283470758e-009,
10, 3.0358340814709663, 7.8118059457005561e-011,
10, 3.0358340889215469, 7.8118048721726738e-011,
10, 3.7592824399471283, 6.7417155809106958e-012,
10, 3.7592824473977089, 6.7417147580480395e-012,
10, 4.4602697342634201, 8.5434024476488055e-013,
10, 4.4602697417140007, 8.5434000006256964e-013,
10, 5.1389683187007904, 1.5015237157428729e-013,
10, 5.138968326151371, 1.5015232503714518e-013,
10, 5.795127771794796, 3.5035278065891386e-014,
10, 5.7951277792453766, 3.5035266685032581e-014,
10, 6.4280949085950851, 1.0474518584578706e-014,
10, 6.4280949160456657, 1.047451900091486e-014,
10, 7.0368044599890709, 3.8902615249132166e-015,
10, 7.0368044674396515, 3.8902605386615012e-015,
10, 7.6197631657123566, 1.742442940023884e-015,
10, 7.6197631731629372, 1.7424426824540998e-015,
10, 8.1750892177224159, 9.1144555287546294e-016,
10, 8.1750892251729965, 9.1144560389833963e-016,
10, 8.7008367404341698, 5.3541847847332351e-016,
10, 8.7008367478847504, 5.3541844723127514e-016,
10, 9.1965094059705734, 3.3560389199894529e-016,
10, 9.196509413421154, 3.3560370421140035e-016,
10, 9.6678015515208244, 2.1059571242602905e-016,
10, 9.667801558971405, 2.1059579862533338e-016,
11, 0.72796426713466644, 2.3816013324880167e-006,
11, 0.72796427458524704, 2.3816003939892356e-006,
11, 1.5323273167014122, 2.6884539057733283e-008,
11, 1.5323273241519928, 2.6884532967611841e-008,
11, 2.3115825653076172, 6.7682186491374862e-010,
11, 2.3115825727581978, 6.768217495558193e-010,
11, 3.0679489448666573, 3.0544688315708309e-011,
11, 3.0679489523172379, 3.054468452617125e-011,
11, 3.8027990683913231, 2.186949692679853e-012,
11, 3.8027990758419037, 2.1869496552307528e-012,
11, 4.5169280469417572, 2.2947351040888336e-013,
11, 4.5169280543923378, 2.2947348773371414e-013,
11, 5.2107065990567207, 3.3353640359739479e-014,
11, 5.2107066065073013, 3.335363154144172e-014,
11, 5.8841593414545059, 6.4351159890193404e-015,
11, 5.8841593489050865, 6.435115249418266e-015,
11, 6.5370001494884491, 1.5930548620019077e-015,
11, 6.5370001569390297, 1.59305482454352e-015,
11, 7.1686412021517754, 4.9183710858836462e-016,
11, 7.168641209602356, 4.9183713116224844e-016,
11, 7.7781866490840912, 1.8464804238726119e-016,
11, 7.7781866565346718, 1.8464801709489452e-016,
11, 8.3644260391592979, 8.2263320775918765e-017,
11, 8.3644260466098785, 8.2263297787354186e-017,
11, 8.9258662387728691, 4.2375346461043962e-017,
11, 8.9258662462234497, 4.2375329267920249e-017,
11, 9.4609249979257584, 2.4482616150935944e-017,
11, 9.460925005376339, 2.4482619419420328e-017,
11, 9.968697115778923, 1.5254006021629585e-017,
11, 9.9686971232295036, 1.5254009931381535e-017,
11, 10.451380550861359, 9.7234265908062383e-018,
11, 10.451380558311939, 9.7234241296957717e-018,
11, 10.909898526966572, 6.3147967493977272e-018,
11, 10.909898534417152, 6.3147961332742783e-018,
12, 0.73438271880149841, 1.6566052109275983e-006,
12, 0.73438272625207901, 1.6566045155548952e-006,
12, 1.5445517003536224, 1.5860666784152459e-008,
12, 1.544551707804203, 1.5860658867525684e-008,
12, 2.33091701567173, 3.380059232629492e-010,
12, 2.3309170231223106, 3.3800585031437757e-010,
12, 3.095676377415657, 1.2884951580384992e-011,
12, 3.0956763848662376, 1.2884947453406877e-011,
12, 3.8402339816093445, 7.7768150022424858e-013,
12, 3.8402339890599251, 7.776814490345326e-013,
12, 4.5654622465372086, 6.8670097741098332e-014,
12, 4.5654622539877892, 6.8670083429297199e-014,
12, 5.2718541026115417, 8.38897478481124e-015,
12, 5.2718541100621223, 8.3889742372884196e-015,
12, 5.9596048966050148, 1.3595098759195538e-015,
12, 5.9596049040555954, 1.3595098086325987e-015,
12, 6.6286550611257553, 2.827608125932321e-016,
12, 6.6286550685763359, 2.8276074709416868e-016,
12, 7.2787091657519341, 7.3455425966950088e-017,
12, 7.2787091732025146, 7.3455407058460195e-017,
12, 7.9092400223016739, 2.3285371485286439e-017,
12, 7.9092400297522545, 2.3285373002934362e-017,
12, 8.5194846615195274, 8.8196897130872609e-018,
12, 8.519484668970108, 8.8196870510698361e-018,
12, 9.1084426045417786, 3.9108299199270105e-018,
12, 9.1084426119923592, 3.9108294386613317e-018,
12, 9.6749012768268585, 1.9869557945423399e-018,
12, 9.6749012842774391, 1.9869552330025425e-018,
12, 10.217559166252613, 1.1287442348688164e-018,
12, 10.217559173703194, 1.1287443889939456e-018,
12, 10.735452570021152, 6.9443548691210419e-019,
12, 10.735452577471733, 6.944357314519752e-019,
12, 11.229217991232872, 4.4666546191302752e-019,
12, 11.229217998683453, 4.4666540313320006e-019,
12, 11.70328252017498, 2.9048073087968165e-019,
12, 11.703282527625561, 2.9048072811110897e-019,
12, 12.068153090775013, 1.1197702118472231e-019,
12, 12.068153098225594, 1.1197723483874788e-019,
13, 0.74004656821489334, 1.1833107184182451e-006,
13, 0.74004657566547394, 1.1833097788321784e-006,
13, 1.5553053691983223, 9.7255030271643972e-009,
13, 1.5553053766489029, 9.7254989421162363e-009,
13, 2.3478806391358376, 1.7766237223855854e-010,
13, 2.3478806465864182, 1.7766232949140296e-010,
13, 3.1199393272399902, 5.7956070959333847e-012,
13, 3.1199393346905708, 5.7956067491263655e-012,
13, 3.8728964179754257, 2.9884476000191141e-013,
13, 3.8728964254260063, 2.9884474795319482e-013,
13, 4.6076695621013641, 2.2511499024927438e-014,
13, 4.6076695695519447, 2.2511492228003123e-014,
13, 5.3248289301991463, 2.3433299337790734e-015,
13, 5.3248289376497269, 2.3433296697120652e-015,
13, 6.024680033326149, 3.2334220967999432e-016,
13, 6.0246800407767296, 3.2334220842696106e-016,
13, 6.7073097303509712, 5.7246438539858037e-017,
13, 6.7073097378015518, 5.7246428548181273e-017,
13, 7.3726105690002441, 1.2665024613881524e-017,
13, 7.3726105764508247, 1.2665026479377931e-017,
13, 8.0202914103865623, 3.4243381685006812e-018,
13, 8.0202914178371429, 3.4243378550191225e-018,
13, 8.6498793810606003, 1.1097375058286432e-018,
13, 8.6498793885111809, 1.1097372565619852e-018,
13, 9.2607174664735794, 4.2345371873704953e-019,
13, 9.26071747392416, 4.234535720073341e-019,
13, 9.8519651964306831, 1.8698553775808502e-019,
13, 9.8519652038812637, 1.8698556036241362e-019,
13, 10.4226189032197, 9.3834845010114218e-020,
13, 10.42261891067028, 9.3834849389696859e-020,
13, 10.971594020724297, 5.2445253216776838e-020,
13, 10.971594028174877, 5.244524330346984e-020,
13, 11.497980736196041, 3.1824489086388546e-020,
13, 11.497980743646622, 3.1824504032698097e-020,
13, 12.001743614673615, 2.0418153275918044e-020,
13, 12.001743622124195, 2.0418144824511038e-020,
13, 12.485165499150753, 1.322118298528861e-020,
13, 12.485165506601334, 1.3221186083014843e-020,
13, 12.954543322324753, 8.948147552026443e-021,
13, 12.954543329775333, 8.9481497369716256e-021,
13, 13.141756847500801, 6.2261736884111559e-021,
13, 13.141756854951382, 6.22616137965588e-021,
14, 0.74509535729885101, 8.6479103419423106e-007,
14, 0.74509536474943161, 8.6479038219181041e-007,
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56, 42.943740234375, 1.04758637844824e-94,
56, 43.542080078125, 4.09126499345346e-95,
56, 43.54212890625, 4.13674598350119e-95,
56, 44.135830078125, 1.65424896423927e-95,
56, 44.13587890625, 1.67288546036148e-95,
56, 44.724990234375, 6.99446261996147e-96,
56, 44.7250390625, 7.07318147969286e-96,
56, 45.3094140625, 3.01976769844332e-96,
56, 45.309462890625, 3.05402248905396e-96,
56, 45.889052734375, 1.32918182143927e-96,
56, 45.8891015625, 1.34452414236527e-96,
56, 46.463955078125, 6.09433185040725e-97,
56, 46.46400390625, 6.16497223474698e-97,
56, 47.033974609375, 2.83946834327271e-97,
56, 47.0340234375, 2.87288014917526e-97,
56, 47.599111328125, 1.35701697542241e-97,
56, 47.59916015625, 1.37323222143338e-97,
56, 48.159365234375, 6.71405638326024e-98,
56, 48.1594140625, 6.79470565448062e-98,
56, 48.714638671875, 3.38572542994683e-98,
56, 48.7146875, 3.42679289110478e-98,
56, 49.2648828125, 1.73113405012133e-98,
56, 49.264931640625, 1.75251860795891e-98,
56, 49.81009765625, 9.02288638273931e-99,
56, 49.810146484375, 9.13660050992263e-99,
56, 50.35033203125, 4.87338593256003e-99,
56, 50.350380859375, 4.93504167152289e-99,
56, 50.885439453125, 2.62897432578778e-99,
56, 50.88548828125, 2.66302222150824e-99,
56, 51.41546875, 1.43071360755971e-99,
56, 51.41556640625, 1.46894783889895e-99,
56, 51.94056640625, 8.12763956845104e-100,
56, 51.9406640625, 8.34551910155042e-100,
56, 52.4605859375, 4.58251489759803e-100,
56, 52.46068359375, 4.70835817217817e-100,
56, 52.97572265625, 2.66457725640481e-100,
56, 52.9758203125, 2.73809355167495e-100,
56, 53.48587890625, 1.52026667140348e-100,
56, 53.4859765625, 1.5636427659068e-100,
56, 53.99134765625, 8.99584141358291e-101,
56, 53.9914453125, 9.25372909114439e-101,
56, 54.49212890625, 5.32825924303324e-101,
56, 54.4922265625, 5.48250901435612e-101,
56, 54.9883203125, 3.11278188657358e-101,
56, 54.98841796875, 3.20543677910749e-101,
56, 55.48021484375, 1.86269305932801e-101,
56, 55.4803125, 1.91846988478933e-101,
57, 38.0996875, 2.20476439543612e-92,
57, 38.099736328125, 2.22917432379422e-92,
57, 38.737333984375, 6.34433356523382e-93,
57, 38.7373828125, 6.41479399056003e-93,
57, 39.370830078125, 1.91018618887989e-93,
57, 39.37087890625, 1.93131173925967e-93,
57, 40.000078125, 5.92216057314857e-94,
57, 40.000126953125, 5.98791393861005e-94,
57, 40.625078125, 1.90309216004904e-94,
57, 40.625126953125, 1.92431835469877e-94,
57, 41.245830078125, 6.38398110903664e-95,
57, 41.24587890625, 6.45498354446764e-95,
57, 41.862236328125, 2.20268568579443e-95,
57, 41.86228515625, 2.22728247906098e-95,
57, 42.474248046875, 7.79068157690964e-96,
57, 42.474296875, 7.87885966415e-96,
57, 43.0819140625, 2.88050550788237e-96,
57, 43.081962890625, 2.91318272595493e-96,
57, 43.68513671875, 1.09806112159568e-96,
57, 43.685185546875, 1.11057160166479e-96,
57, 44.2838671875, 4.30562812876992e-97,
57, 44.283916015625, 4.35507143677268e-97,
57, 44.87810546875, 1.7530232640245e-97,
57, 44.878154296875, 1.7731758707715e-97,
57, 45.46775390625, 7.31286187931292e-98,
57, 45.467802734375, 7.39751767287751e-98,
57, 46.052763671875, 3.11976304411238e-98,
57, 46.0528125, 3.15638274069021e-98,
57, 46.633134765625, 1.37464423777275e-98,
57, 46.63318359375, 1.39093953120416e-98,
57, 47.208818359375, 6.24536630012012e-99,
57, 47.2088671875, 6.31989226309693e-99,
57, 47.779716796875, 2.88489889714976e-99,
57, 47.779765625, 2.91990162728214e-99,
57, 48.34587890625, 1.38459867556409e-99,
57, 48.345927734375, 1.40146037072031e-99,
57, 48.907158203125, 6.71827295305273e-100,
57, 48.90720703125, 6.80152139672612e-100,
57, 49.463603515625, 3.36417686184697e-100,
57, 49.46365234375, 3.40624440071944e-100,
57, 50.0151171875, 1.70838155932608e-100,
57, 50.015166015625, 1.73011766047011e-100,
57, 50.561748046875, 8.96377195112949e-101,
57, 50.561796875, 9.07844496707364e-101,
57, 51.1033984375, 4.7573624982879e-101,
57, 51.103447265625, 4.81906515869594e-101,
57, 51.64001953125, 2.52428955803692e-101,
57, 51.6401171875, 2.59192149716193e-101,
57, 52.1717578125, 1.39187870966602e-101,
57, 52.17185546875, 1.42956730803746e-101,
57, 52.698515625, 7.7317500805067e-102,
57, 52.69861328125, 7.94501950652444e-102,
57, 53.220390625, 4.4082211286455e-102,
57, 53.22048828125, 4.53055204196072e-102,
57, 53.73728515625, 2.47396166798242e-102,
57, 53.7373828125, 2.54499159923448e-102,
57, 54.2494921875, 1.45907071692507e-102,
57, 54.24958984375, 1.50072529856044e-102,
57, 54.75681640625, 8.32179730748228e-103,
57, 54.7569140625, 8.56822893627666e-103,
57, 55.2596484375, 4.99089804791844e-103,
57, 55.25974609375, 5.1376157827554e-103,
57, 55.757890625, 2.93846786006917e-103,
57, 55.75798828125, 3.02626005750667e-103,
57, 56.25173828125, 1.71972275925121e-103,
57, 56.2518359375, 1.77242201439855e-103,
57, 56.74138671875, 1.00746867642176e-103,
57, 56.741484375, 1.03914712466117e-103,
58, 38.841474609375, 2.28962685408854e-94,
58, 38.8415234375, 2.31583571384595e-94,
58, 39.478828125, 6.65279878387051e-95,
58, 39.478876953125, 6.72851011961114e-95,
58, 40.11203125, 1.98889585849889e-95,
58, 40.112080078125, 2.0116031211386e-95,
58, 40.741083984375, 6.15547181773793e-96,
58, 40.7411328125, 6.22611457465467e-96,
58, 41.365986328125, 1.98535700187755e-96,
58, 41.36603515625, 2.00813333128919e-96,
58, 41.986640625, 6.56800883846017e-97,
58, 41.986689453125, 6.64407409629831e-97,
58, 42.603095703125, 2.27151430941761e-97,
58, 42.60314453125, 2.29779962665845e-97,
58, 43.21525390625, 8.08887967205788e-98,
58, 43.215302734375, 8.18281575271904e-98,
58, 43.82306640625, 2.95592121751972e-98,
58, 43.823115234375, 2.99061314771485e-98,
58, 44.42658203125, 1.13134888356557e-98,
58, 44.426630859375, 1.14457524376905e-98,
58, 45.025654296875, 4.41845285569886e-99,
58, 45.025703125, 4.47048673389212e-99,
58, 45.620283203125, 1.77714624291313e-99,
58, 45.62033203125, 1.79825016216838e-99,
58, 46.210419921875, 7.34443777835212e-100,
58, 46.21046875, 7.43260970647061e-100,
58, 46.796064453125, 3.14964704226534e-100,
58, 46.79611328125, 3.18755872885081e-100,
58, 47.377119140625, 1.38201826867972e-100,
58, 47.37716796875, 1.39878290995527e-100,
58, 47.95353515625, 6.19027397147059e-101,
58, 47.953583984375, 6.26644970781979e-101,
58, 48.5253125, 2.85831172869441e-101,
58, 48.525361328125, 2.89384123773075e-101,
58, 49.09240234375, 1.35697005525987e-101,
58, 49.092451171875, 1.37396450788758e-101,
58, 49.65470703125, 6.51873789254857e-102,
58, 49.654755859375, 6.60202992320111e-102,
58, 50.212275390625, 3.23653359270438e-102,
58, 50.21232421875, 3.2783090214757e-102,
58, 50.76505859375, 1.65267733466975e-102,
58, 50.765107421875, 1.67409628228237e-102,
58, 51.31291015625, 8.39865374766313e-103,
58, 51.3130078125, 8.62296916999248e-103,
58, 51.8559765625, 4.43898857454993e-103,
58, 51.85607421875, 4.55876608676698e-103,
58, 52.39416015625, 2.38284566979309e-103,
58, 52.3942578125, 2.44798937063255e-103,
58, 52.9274609375, 1.29896683738751e-103,
58, 52.92755859375, 1.33500351596253e-103,
58, 53.45587890625, 7.16661264811987e-104,
58, 53.4559765625, 7.36908508006679e-104,
58, 53.9794140625, 3.97139434881222e-104,
58, 53.97951171875, 4.08676376045491e-104,
58, 54.49826171875, 2.31430475778334e-104,
58, 54.498359375, 2.38084236991499e-104,
58, 55.0122265625, 1.31587661343037e-104,
58, 55.01232421875, 1.354675556019e-104,
58, 55.52150390625, 7.55721877752536e-105,
58, 55.5216015625, 7.78550839762316e-105,
58, 56.02619140625, 4.38540257957703e-105,
58, 56.0262890625, 4.52070576457663e-105,
58, 56.526484375, 2.63611015718921e-105,
58, 56.52658203125, 2.71673162725835e-105,
58, 57.02228515625, 1.52363825877265e-105,
58, 57.0223828125, 1.57187635540744e-105,
58, 57.513984375, 9.12003014967339e-106,
58, 57.51408203125, 9.40921222802051e-106,
};

//
// A bunch of helper functions used for calculating the coefficients:
//
mp_t factorial(unsigned n)
{
   static boost::array<boost::array<mp_t, 1>, 201> result; 
   static bool init = false;
   if(!init)
   {
      unsigned k = 1;
      mp_t fact = 1;
      do{
         result[k-1][0] = fact;
         fact *= k++;
      }
      while(k < result.size());
      init = true;
   }
   //static const int array_size = result.size();
   if(n < result.size())
      return result[n][0];

   unsigned i = (unsigned)result.size()-1;
   mp_t r = result[i][0];
   while(i < n)
      r *= ++i;
   return r;
}

template <class T>
T binomial(int n, int k, T)
{
   T result;
   if(k < 0) 
   {
      result = 0;
   }
   else if(k > n)
   {
      result = 1;
   }
   else
   {
      result = factorial(n);
      result /= factorial(k);
      result /= factorial(n-k);
   }
   return result;
}
//
// Functions for creating the matrices that generate the coefficents.
// See http://my.fit.edu/~gabdo/gamma.txt and http://www.rskey.org/gamma.htm
//
template <class T>
matrix<T> make_B(unsigned n, T)
{
   matrix<T> result(n, n);
   for(unsigned i = 0; i < n; ++i)
   {
      for(unsigned j = 0; j < n; ++j)
      {
         if(i == 0)
            result(i, j) = 1;
         else if(j >= i)
         {
            T r = binomial(i+j-1, j-i, T());
            if((j-i) %2)
               r = -r;
            result(i, j) = r;
         }
         else
            result(i, j) = 0;
      }
   }
   return result;
}

template <class T>
matrix<T> make_C(unsigned n, T)
{
   matrix<T> result(n, n);
   for(unsigned i = 0; i < n; ++i)
   {
      for(unsigned j = 0; j < n; ++j)
      {
         if((i==0) && (j == 0))
            result(i, j) = 0.5;
         else if(j > i)
            result(i, j) = 0;
         else
         {
            T r = 0;
            for(unsigned k = 0; k <= i; ++k)
            {
               r += binomial(2*i, 2*k, T()) * binomial(k, k+j-i, T());
            }
            if((i-j)%2)
               r = -r;
            result(i, j) = r;
         }
      }
   }
   return result;
}

template <class T>
matrix<T> make_D(unsigned n, T)
{
   matrix<T> result(n, n);
   for(unsigned i = 0; i < n; ++i)
   {
      for(unsigned j = 0; j < n; ++j)
      {
         if(i != j)
            result(i, j) = 0;
         else if(i == 0)
            result(i, j) = 1;
         else if(i == 1)
            result(i, j) = -1;
         else
         {
            T r = result(i-1, i-1) * (2 * (2*i - 1));
            r /= (i-1);
            result(i, j) = r;
         }
      }
   }
   return result;
}

template <class T>
matrix<T> make_F(unsigned n, T g)
{
   using namespace std;
   matrix<T> result(n, 1);
   for(unsigned i = 0; i < n; ++i)
   {
      T r = factorial(2 * i); 
      r /= factorial(i);
      r *= exp(double(i) + g + 0.5);
      r /= ldexp(1.0, (2*i)-1);
      r /= pow(g + i + 0.5, (long)i);
      r /= sqrt(g + i + 0.5);
      result(i, 0) = r;
   }
   return result;
}

template <class T>
struct lanczos_info
{
   int n;                   // number of coefficients
   T r;                     // the arbitrary parameter
   std::vector<T> c;        // the coefficients
   T err;                   // error found
};
//
// Create the coefficients, caching previous matrix values as appropriate:
//
template <class T>
lanczos_info<T> generate_lanczos(unsigned n, T g)
{
   static unsigned last_n = 0;
   static matrix<T> last_prefix;

   matrix<T> B, C, D;
   matrix<T> F, P, E, T1, T2;

   if(n == last_n)
   {
      F = make_F(n, g);
      P = prod(last_prefix, F);
   }
   else
   {
      B = make_B(n, g);
      C = make_C(n, g);
      D = make_D(n, g);
      F = make_F(n, g);

      T1 = prod(D, B);
      T2 = prod(T1, C);
      P = prod(T2, F);

      last_n = n;
      last_prefix = T2;
   }

   lanczos_info<T> result;
   result.n = n;
   result.r = g;
   for(unsigned i = 0; i < n; ++i)
      result.c.push_back(P(i, 0));

   return result;
}
//
// Stopwatch for measuring performance:
//
template <class Clock>
struct stopwatch
{
   typedef typename Clock::duration duration;
   stopwatch()
   {
      m_start = Clock::now();
   }
   duration elapsed()
   {
      return Clock::now() - m_start;
   }
   void reset()
   {
      m_start = Clock::now();
   }

private:
   typename Clock::time_point m_start;
};
//
// Returns the factorials and half factorials:
//
template <class T>
std::vector<std::vector<T> > const & get_test_data()
{
   // returns factorials and half factorials in the range [0, 100]:
   static std::vector<std::vector<T> > data;
   if(data.empty())
   {
      mp_t fact = 1;
      int k = 1;
      std::vector<T> item;
      do
      {
         data.push_back(item);
         data.back().push_back(k-1);
         data.back().push_back(static_cast<T>(fact));
         data.back().push_back(static_cast<T>(log(fact)));
         fact = fact * T(k++);
      }while(k < 100);

      fact = 0.5;
      mp_t srpi = sqrt(boost::math::constants::pi<mp_t>());
      mp_t mul = 1.5;
      do{
         data.push_back(item);
         data.back().push_back(static_cast<T>(mul-1));
         data.back().push_back(static_cast<T>(fact*srpi));
         data.back().push_back(static_cast<T>(log(fact*srpi)));
         fact *= mul;
         mul += 1;
      }while(mul < 100);
   }

   return data;
}

struct sort_on_0
{
   template<class T>
   bool operator ()(const T& a, const T& b)
   {
      return ((a[0]) < (b[0]));
   }
};
//
// Get test data for points near 0, 1, 2 etc.
template <class T>
std::vector<std::vector<T> > const & get_test_data_near_x(T x)
{
   // 
   // this *must* be called with a *very* high precision
   // type T in order to stand any chance of long double
   // precision results:
   //
   using namespace std;
   using namespace boost::math;
   static std::vector<std::vector<T> > data;
   static T last_x(-10000);
   if(last_x != x)
      data.clear();
   if(!data.size())
   {
      last_x = x;
      std::vector<T> item;
      for(int i = -4; i > -20; --i)
      {
         double val = ldexp(2.0, i);
         data.push_back(item);
         data.back().push_back(val+x);
         data.back().push_back(tgamma(T(val)+x));
         data.back().push_back(lgamma(T(val)+x));
         data.push_back(item);
         data.back().push_back(-val+x);
         data.back().push_back(tgamma(T(-val)+x));
         data.back().push_back(lgamma(T(-val)+x));
      }
      T v = x-0.5;
      T interval = T(1)/8;
      while(v <= x+0.5)
      {
         if((floor(v) != v) || (v > 0))
         {
            data.push_back(item);
            data.back().push_back(v);
            data.back().push_back(tgamma(v));
            data.back().push_back(lgamma(v));
         }
         v += interval;
      }
      //
      // sort data:
      //
      std::sort(data.begin(), data.end(), sort_on_0());
   }
   return data;
}

template <class T>
std::vector<std::vector<T> > const & get_test_data_near_1()
{
   return get_test_data_near_x(T(1));
}

template <class T>
std::vector<std::vector<T> > const & get_test_data_near_2()
{
   return get_test_data_near_x(T(2));
}

template <class T>
std::vector<std::vector<T> > const& get_random_test_data()
{
   static std::vector<std::vector<T> > data;
   if (data.empty())
   {
      boost::random::mt19937_64 gen;
      boost::random::uniform_real_distribution<T> dist(1e-30, 100);
      for (unsigned i = 0; i < 100; ++i)
      {
         T value = dist(gen);
         T result = static_cast<T>(boost::math::tgamma(mp_t(value) + 1));
         data.push_back(std::vector<T>({value, result}));
      }
   }
   return data;
}

//
// Converts Lanczos approximation into rational form via 
// polynomial arithmetic:
//
template <class T>
struct lanczos_rational
{
   std::vector<T> num, denom;
   int N;
   T g;

   T gamma(T z)
   {
      using namespace std;
      BOOST_ASSERT(num.size() == denom.size());
      BOOST_ASSERT(num.size() == N);
      T zgh = z + g - T(0.5);
      T prefix = pow(zgh, T(z - 0.5)) / exp(zgh);
      T s1, s2;
      if(z < 1)
      {
         s1 = num[N-1];
         s2 = denom[N-1];
         for(unsigned i = N-2; i >= 0; --i)
         {
            s1 *= z;
            s2 *= z;
            s1 += num[i];
            s2 += num[i];
         }
         s1 /= s2;
         return prefix * s1;
      }
      else
      {
         z = 1/z;
         s1 = num[0];
         s2 = denom[0];
         for(int i = 1; i < N; ++i)
         {
            s1 *= z;
            s2 *= z;
            s1 += num[i];
            s2 += denom[i];
         }
         s1 /= s2;
         return prefix * s1;
      }
   }
   T factorial(T z)
   {
      return this->gamma(z+1);
   }
   template <class U>
   lanczos_rational(const lanczos_info<U>& info)
   {
      U l_denom_coef[2] = { 0, 1 };
      boost::math::tools::polynomial<U> l_denom(l_denom_coef, 1), l_num(info.c[1]);
      for(unsigned i = 2; i < info.c.size(); ++i)
      {
         l_denom_coef[0] = i - 1;
         boost::math::tools::polynomial<U> bot(l_denom_coef, 1), top(info.c[i]);
         l_num *= bot;
         top *= l_denom;
         l_denom *= bot;
         l_num += top;
      }
      l_num += l_denom * info.c[0];
      for(unsigned i = 0; i < info.c.size(); ++i)
      {
         num.push_back(boost::math::tools::real_cast<T>(l_num[i]));
         denom.push_back(boost::math::tools::real_cast<T>(l_denom[i]));
      }
      //std::cout << l_num << std::endl;
      //std::cout << l_denom << std::endl;
      BOOST_ASSERT(num.size() == l_num.degree()+1);
      BOOST_ASSERT(denom.size() == l_denom.degree()+1);
      g = boost::math::tools::real_cast<T>(info.r);
      N = info.n;
      /*
      for(unsigned i = 0; i < num.size(); ++i)
         std::cout << num[i] << " ";
      std::cout << std::endl;
      for(unsigned i = 0; i < denom.size(); ++i)
         std::cout << denom[i] << " ";
      std::cout << std::endl;
      */
   }
};
//
// Code to test an approximation against the factorials and
// half factorials, returns the max error found:
//
template <class T, class R>
T get_max_error(const lanczos_info<T>& dat, R, double* time = nullptr)
{
   R max_error = 0;
   std::vector<std::vector<R> > const & tests1 = get_test_data<R>();
   std::vector<std::vector<R> > const & tests2 = get_random_test_data<R>();

   std::vector<std::vector<R> > tests(tests1.begin(), tests1.end());
   tests.insert(tests.end(), tests2.begin(), tests2.end());

   lanczos_rational<R> rational(dat);

   stopwatch<boost::chrono::high_resolution_clock> w;

   for(unsigned i = 0; i < tests.size(); ++i)
   {
      R input = tests[i][0];
      R expected = tests[i][1];
      if(std::numeric_limits<R>::is_specialized && (boost::math::tools::real_cast<R>(tests[i][1]) > (std::numeric_limits<R>::max)()))
         continue;

      R gamr = rational.factorial(input);
      if(std::numeric_limits<R>::is_specialized && (gamr > (std::numeric_limits<R>::max)()))
         continue;
      R err = boost::math::relative_difference(gamr, expected);
      if(err > max_error)
         max_error = err;
   }

   if (time)
      * time = boost::chrono::duration_cast<boost::chrono::duration<double>>(w.elapsed()).count();

   return max_error;
}

template <class T>
T get_max_error_sterling(double* time = nullptr)
{
   std::vector<std::vector<T> > const& tests1 = get_test_data<T>();
   std::vector<std::vector<T> > const& tests2 = get_random_test_data<T>();

   std::vector<std::vector<T> > tests(tests1.begin(), tests1.end());
   tests.insert(tests.end(), tests2.begin(), tests2.end());

   T max_err = 0;

   stopwatch<boost::chrono::high_resolution_clock> w;

   for (unsigned i = 0; i < tests.size(); ++i)
   {
      T loc = tests[i][0] + 1;
      T found = boost::math::detail::gamma_imp(loc, boost::math::policies::policy<>(), boost::math::lanczos::undefined_lanczos());
      T expected = tests[i][1];
      T err = boost::math::relative_difference(expected, found);
      if (err > max_err)
         max_err = err;
   }

   if(time)
      *time = boost::chrono::duration_cast<boost::chrono::duration<double>>(w.elapsed()).count();

   return max_err;
}

//
// This is what prints the "best" approximation out as code:
//
template <class T>
void print_code(const lanczos_info<T>& l, const char* name, int precision = std::numeric_limits<T>::max_digits10, int precision2 = std::numeric_limits<T>::digits)
{
   std::cout << std::scientific << std::setprecision(precision);
   using namespace std;
   lanczos_info<T> l2(l);
   T factor = exp(l.r);
   for(unsigned k = 0; k < l2.c.size(); ++k)
      l2.c[k] /= factor;

   lanczos_rational<T> rat(l);
   T max_term = 0;
   for(unsigned i = 0; i < rat.denom.size(); ++i)
   {
      if(rat.denom[i] > max_term)
         max_term = rat.denom[i];
   }
   const char* denom_type;
   const char* cast_type;
   const char* suffix_type;
   if(max_term < (std::numeric_limits<boost::uint16_t>::max)())
   {
      denom_type = "boost::uint16_t";
      cast_type = "static_cast<boost::uint16_t>";
      suffix_type = "u";
   }
   else if(max_term < (std::numeric_limits<boost::uint32_t>::max)())
   {
      denom_type = "boost::uint32_t";
      cast_type = "static_cast<boost::uint32_t>";
      suffix_type = "u";
   }
#ifdef BOOST_HAS_LONG_LONG
   else if(max_term < (std::numeric_limits<boost::uint64_t>::max)())
   {
      denom_type = "boost::uint64_t";
      cast_type = "";
      suffix_type = "uLL";
   }
#endif
   else
   {
      denom_type = "T";
      cast_type = "static_cast<T>";
      suffix_type = "L";
   }
   
   std::cout <<
      "//\n"
      "// Lanczos Coefficients for N=" << l.n << " G=" << l.r << "\n"
      "// Max experimental error (with ";
   if(std::strlen(name) == 0)
      std::cout << "arbitary";
   else
      std::cout << name;
   std::cout << " precision arithmetic) " << l.err <<
      "\n// Generated with compiler: " << BOOST_COMPILER << " on " << BOOST_PLATFORM << " at " << __DATE__ << "\n"
      "// Type precision was " << precision2 << " bits or " << precision << " max_digits10\n"
      "//\n"
      "struct lanczos" << l.n << name << " : public mpl::int_<" << precision2 << ">\n"
      "{\n"
      "   template <class T>\n"
      "   static T lanczos_sum(const T& z)\n"
      "   {\n"
      "      static const T num[" << rat.num.size() << "] = {\n";

   for(unsigned i = 0; i < rat.num.size(); ++i)
   {
      if(precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         static_cast<T>(" << rat.num[i] << "L)";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << rat.num[i] << "))";
      if(i != rat.num.size() - 1)
         std::cout << ",";
      std::cout << "\n";
   }
   std::cout << 
      "      };\n"
      "      static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n";
   for(unsigned i = 0; i < rat.denom.size(); ++i)
   {
      if(precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         " << cast_type << "(" << rat.denom[i] << suffix_type << ")";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << rat.denom[i] << "))";
      if(i != rat.denom.size() - 1)
         std::cout << ",";
      std::cout << "\n";
   }
   std::cout << 
      "      };\n"
      "      return boost::math::tools::evaluate_rational(num, denom, z);\n"
      "   }\n\n"
      "   template <class T>\n"
      "   static T lanczos_sum_expG_scaled(const T& z)\n"
      "   {\n"
      "      static const T num[" << rat.num.size() << "] = {\n";

   for(unsigned i = 0; i < rat.num.size(); ++i)
   {
      if (precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         static_cast<T>(" << (rat.num[i]/factor) << "L)";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << (rat.num[i]/factor) << "))";
      if(i != rat.num.size() - 1)
         std::cout << ",";
      std::cout << "\n";
   }
   std::cout << 
      "      };\n"
      "      static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n";
   for(unsigned i = 0; i < rat.denom.size(); ++i)
   {
      if (precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         " << cast_type << "(" << rat.denom[i] << suffix_type << ")";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << rat.denom[i] << "))";
      if(i != rat.denom.size() - 1)
         std::cout << ",";
      std::cout << "\n";
   }
   std::cout << 
      "      };\n"
      "      return boost::math::tools::evaluate_rational(num, denom, z);\n"
      "   }\n\n";

   std::cout <<
      "\n   template<class T>\n"
      "   static T lanczos_sum_near_1(const T& dz)\n"
      "   {\n"
      "      static const T d[" << l2.n-1 << "] = {\n";

   factor = sqrt((l.r + 0.5)/boost::math::constants::e<T>()) / exp(l.r);

   for(int i = 1; i < l.n; ++i)
   {
      if (precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         static_cast<T>(" << l.c[i]*factor << "L)";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << l.c[i] * factor << "))";
      if(i != l.n - 1)
         std::cout << ",";
      std::cout << "\n";
   }
   std::cout << 
      "      };\n"
      "      T result = 0;\n"
      "      for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n"
      "      {\n"
      "         result += (-d[k-1]*dz)/(k*dz + k*k);\n"
      "      }\n"
      "      return result;\n"
      "   }\n";

   std::cout <<
      "\n   template<class T>\n"
      "   static T lanczos_sum_near_2(const T& dz)\n"
      "   {\n"
      "      static const T d[" << l2.n-1 << "] = {\n";

   factor = pow(boost::math::constants::e<T>()/(l.r+1.5), T(1.5)) * exp(l.r);
      //pow((l.r + 1.5)/boost::math::constants::e<T>(), 1.5) / exp(l.r);

   for(int i = 1; i < l.n; ++i)
   {
      if (precision <= std::numeric_limits<long double>::digits10)
         std::cout << "         static_cast<T>(" << l.c[i]/factor << "L),\n";
      else
         std::cout << "         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, " << precision2 << ", " << l.c[i]/factor << ")),\n";
   }
   std::cout << 
      "      };\n"
      "      T result = 0;\n"
      "      T z = dz + 2;\n"
      "      for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n"
      "      {\n"
      "         result += (-d[k-1]*dz)/(z + k*z + k*k - 1);\n"
      "      }\n"
      "      return result;\n"
      "   }\n\n"
      "   static double g(){ return " << l.r << "; }\n"
      "};\n\n";
}
//
// Print out the test values:
//
void print_test_values(const std::vector<std::vector<mp_t> >& v, const char* name, int offset = 1)
{
   std::cout << std::setprecision(110);
   std::cout << 
      "   static const boost::array<boost::array<T, 3>, " << v.size() << "> " << name << " = {\n";
   for(unsigned i = 0; i < v.size(); ++i)
   {
      std::cout << "      SC_(" << (v[i][0] + offset) << "), SC_(" << v[i][1] << "), SC_(" << v[i][2] << "),\n";
   }
   std::cout << "   };\n\n";
}
//
// Get the error for a specific approximation, and print out it's code:
//
void calculate_lanczos_spot(int n, mp_t r, const char* suffix = "")
{
   lanczos_info<mp_t> info = generate_lanczos(n, r);
   // note error is calculated at high precision:
   info.err = get_max_error(info, r);
   print_code(info, suffix);
}
//
// Scan the sweet-spots for the best approximation:
//
template <class T>
void find_best_lanczos(const char* name, T eps, int max_scan = 100)
{
   using namespace std;

   double exec_time;

   std::cout << "Enter value for N, or 0 to scan for best approximation: ";
   int N;
   std::cin >> N;

   if (N == 0)
   {
      T sterling_err = get_max_error_sterling<T>(&exec_time);

      std::cout << "Max error from generic Sterling approximation was: " << static_cast<int>(sterling_err / eps) << "eps (in " << (int)(exec_time * 1000) << "ms)" << std::endl;

      lanczos_info<mp_t> best;
      best.err = 100; // best case had better be better than this!

      std::cout << std::setw(20) << std::right << "N" << std::setw(20) << std::right << "g" << std::setw(20) << std::right << "eps" << std::setw(20) << std::right << "time (ms)\n";

      for (int i = 0; i < sizeof(sweet_spots) / sizeof(sweet_spots[0]); ++i)
      {
         if ((sweet_spots[i].err < eps * 10) && (sweet_spots[i].N < max_scan))
         {
            lanczos_info<mp_t> info = generate_lanczos(sweet_spots[i].N, mp_t(sweet_spots[i].g));
            mp_t err = get_max_error(info, eps, &exec_time);
            if (err / eps < 1000)
            {
               std::cout << std::setprecision(14) << std::fixed << std::setw(20) << std::right << sweet_spots[i].N
                  << std::setw(20) << std::right << sweet_spots[i].g << std::setw(20) << std::right << static_cast<int>(err / eps)
                  << std::setw(20) << std::right << (int)(exec_time * 1000) << std::endl;
            }
            if (err < best.err)
            {
               best = info;
               best.err = err;
            }
         }
      }
      std::cout << std::endl;

      if (best.err < 100)
         print_code(best, name, std::numeric_limits<T>::max_digits10, std::numeric_limits<T>::digits);
      else
         std::cout << "Sorry, no viable approximation was found!!" << std::endl;
   }
   else
   {
      //
      // Test a specific N and g:
      //
      std::cout << "Enter a value for g: ";
      double g;
      std::cin >> g;
      lanczos_info<mp_t> info = generate_lanczos(N, mp_t(g));
      mp_t err = get_max_error(info, eps, &exec_time);

      std::cout << "N                 = " << N << std::endl;
      std::cout << "g                 = " << g << std::endl;
      std::cout << "eps error         = " << static_cast<int>(err / eps) << std::endl;
      std::cout << "Test time (ms)    = " << static_cast<int>(exec_time * 1000) << std::endl;
      print_code(info, name, std::numeric_limits<T>::max_digits10, std::numeric_limits<T>::digits);
   }
}

void scan_for_sweet_spots(int N)
{
   mp_t r = N * 0.66;
   lanczos_info<mp_t> lower, upper(generate_lanczos(N + 1, r));
   r += 0.1;
   while (r < N)
   {
      lower = upper;
      upper = generate_lanczos(N + 1, r);

      if (lower.c.back() * upper.c.back() < 0)
      {
         std::pair<mp_t, mp_t> location = boost::math::tools::bisect([N](const mp_t& pos)
            {
               return generate_lanczos(N + 1, pos).c.back();
            }, r - 0.1, r, boost::math::tools::eps_tolerance<mp_t>(20));
         mp_t err = get_max_error(generate_lanczos(N, location.first), std::numeric_limits<mp_t>::epsilon());
         std::cout << std::setprecision(15) << N << ", " << location.first << ", " << err << std::endl;
         err = get_max_error(generate_lanczos(N, location.second), std::numeric_limits<mp_t>::epsilon());
         std::cout << std::setprecision(15) << N << ", " << location.second << ", " << err << std::endl;
         //std::cout << std::setprecision(15) << N << ", " << location.first << ", " << location.second << std::endl;
      }

      r += 0.1;
   }
}

int main(int argc, char*argv [])
{
   bool test_double(false), test_long(false), test_float(false), test_quad(false), test_mp(false), spots(false), test_data(false), find_sweet(false);

   if(argc < 2)
   {
      std::cout << 
         "Useage:\n"
         "  -float        test type float for the best approximation\n"
         "  -double       test type double for the best approximation\n"
         "  -long-double  test type long double for the best approximation\n"
         "  -quad         test quad precision for the best approximation\n"
         "  -MP           test current multiprecision type for the best approximation\n"
         "  -spots        print out the best cases found in previous runs\n"
         "  -sweet        Scan for more sweet spots for the arbitary parameter G: these "
         "                will need to cut and pasted into the big table at the start of this file"
         "                in order to search for greater precision approximations than otherwise supported."
         "  -data         print out the test data\n" << std::flush;
      return 1;
   }

   for(int i = 1; i < argc; ++i)
   {
      if(std::strcmp(argv[i], "-float") == 0)
         test_float = true;
      else if(std::strcmp(argv[i], "-double") == 0)
         test_double = true;
      else if(std::strcmp(argv[i], "-long-double") == 0)
         test_long = true;
      else if(std::strcmp(argv[i], "-quad-float") == 0)
         test_quad = true;
      else if(std::strcmp(argv[i], "-MP") == 0)
         test_mp = true;
      else if(std::strcmp(argv[i], "-spots") == 0)
         spots = true;
      else if(std::strcmp(argv[i], "-data") == 0)
         test_data = true;
      else if(std::strcmp(argv[i], "-sweet") == 0)
         find_sweet = true;
   }

   if(spots)
   {
      // these are optimal N and R from Pugh:
      calculate_lanczos_spot(61, 63.192152);
      calculate_lanczos_spot(31, 32.080670);
      calculate_lanczos_spot(6, 5.581);
      calculate_lanczos_spot(11, 10.900511);
      calculate_lanczos_spot(13, 13.144565);
      calculate_lanczos_spot(22, 22.61891);
      // these are the best cases we've already determined
      // from previous runs:
      calculate_lanczos_spot(6, 1.428456135094165802001953125);
      calculate_lanczos_spot(13, 6.024680040776729583740234375);
      calculate_lanczos_spot(17, 12.2252227365970611572265625);
      calculate_lanczos_spot(24, 20.3209821879863739013671875);
   }
   if(test_float)
   {
      find_best_lanczos("float", std::numeric_limits<float>::epsilon());
   }
   if(test_double)
   {
      find_best_lanczos("double", std::numeric_limits<double>::epsilon());
   }

   if(test_long)
   {
      find_best_lanczos("long_double", std::numeric_limits<long double>::epsilon());
   }
   if(test_quad)
   {
      find_best_lanczos("quad_float", std::numeric_limits<boost::multiprecision::cpp_bin_float_quad>::epsilon());
   }
#ifdef MP_TYPE
   if(test_mp)
   {
      find_best_lanczos("MP", std::numeric_limits<MP_TYPE>::epsilon());
   }
#endif
   if(test_data)
   {
      std::cout << "Test Data follows:\n\n";

      std::vector<std::vector<mp_t> > const & tests = get_test_data<mp_t>();
      print_test_values(tests, "factorials");
      print_test_values(get_test_data_near_1<mp_t>(), "near_1", 0);
      print_test_values(get_test_data_near_2<mp_t>(), "near_2", 0);
      print_test_values(get_test_data_near_x<mp_t>(mp_t(0)), "near_0", 0);
      print_test_values(get_test_data_near_x<mp_t>(mp_t(-10)), "near_m10", 0);
      print_test_values(get_test_data_near_x<mp_t>(mp_t(-55)), "near_m55", 0);
   }
   if (find_sweet)
   {
      unsigned last_index = sizeof(sweet_spots) / sizeof(sweet_spots[0]) - 1;
      for(unsigned N = sweet_spots[last_index].N + 1; N < sweet_spots[last_index].N + 11; ++N)
         scan_for_sweet_spots(N);
   }
   return 0;
}