#include <math.h>
#include "point.h"

static double **coefTh1[40];
static double allCoefTh1[] = {
    1.00088825943439219,
    0.00350107303029312819,
    0.00335010385761378912,
    -0.000188574853838318111,
    1.80723526491795172e-05,
    -1.52300302879811959e-06,
    1.3129822585088852e-07,
    -1.12937155928579476e-08,
    9.72321031784848601e-10,
    -8.37416203363771493e-11,
    7.21448163414644855e-12,
    -6.21683137860164987e-13,
    // piece[0][0]  [0,0.0625)  12 terms

    1.00756753171781299,
    0.00970094640359244094,
    0.00287934019102495213,
    -0.000129236599618845049,
    1.20903753378213849e-05,
    -9.26915138187676703e-07,
    7.37033611209207105e-08,
    -5.83357757825111023e-09,
    4.62306355554221893e-10,
    -3.66499820781233703e-11,
    2.90641428215566319e-12,
    -2.3053997651445757e-13,
    // piece[0][1]  [0.0625,0.125)  12 terms

    1.02002981347114385,
    0.0151160486929011444,
    0.00255589828249223449,
    -8.88512327073214371e-05,
    8.38546078437613817e-06,
    -5.85290111535721182e-07,
    4.32528500444972434e-08,
    -3.16949541550342358e-09,
    2.32687035458907836e-10,
    -1.70874582768814512e-11,
    1.25525418593527551e-12,
    -9.22348551879609083e-14,
    // piece[0][2]  [0.125,0.1875)  12 terms

    1.03762074968462636,
    0.0199921460022982958,
    0.00233439263322172833,
    -6.03964341235711189e-05,
    6.01124190894104949e-06,
    -3.81176455729835385e-07,
    2.63727699228456351e-08,
    -1.79833054954992328e-09,
    1.22976677537832355e-10,
    -8.41097530989072206e-12,
    5.75480922673267777e-13,
    -3.9384754552118075e-14,
    // piece[0][3]  [0.1875,0.25)  12 terms

    1.05989254664101962,
    0.0245020276135001804,
    0.00218582000880160809,
    -3.96924739400777134e-05,
    4.44566726256720343e-06,
    -2.54749120643877666e-07,
    1.66261447555007528e-08,
    -1.05939460756047093e-09,
    6.78060730159330861e-11,
    -4.33962358019622712e-12,
    2.77852472024172447e-13,
    // piece[0][4]  [0.25,0.3125)  11 terms

    1.08654490833800099,
    0.0287711919798240359,
    0.00209109799817309245,
    -2.41583887630629996e-05,
    3.38848764504761745e-06,
    -1.73926576350088395e-07,
    1.07954926461445996e-08,
    -6.44910638506968624e-10,
    3.87992884478786685e-11,
    -2.33318994972962871e-12,
    1.40372888227574276e-13,
    // piece[0][5]  [0.3125,0.375)  11 terms

    1.11738626467548419,
    0.0328936576768730088,
    0.00203736388952570216,
    -1.2148373638968931e-05,
    2.66066417102275463e-06,
    -1.20786127769174014e-07,
    7.19750816142557751e-09,
    -4.04098252634702952e-10,
    2.29407086368956504e-11,
    -1.30089672188894238e-12,
    7.38127447914128273e-14,
    // piece[0][6]  [0.375,0.4375)  11 terms

    1.15230768456140686,
    0.0369420195899262879,
    0.00201577496752979035,
    -2.58258788879900804e-06,
    2.15200887938194115e-06,
    -8.49495376340929582e-08,
    4.91564883732868957e-09,
    -2.59762225050252025e-10,
    1.39653806472240792e-11,
    -7.49169656722782623e-13,
    4.02193640620783792e-14,
    // piece[0][7]  [0.4375,0.5)  11 terms

    1.19126496825945694,
    0.0409740328300665507,
    0.00202015840724801415,
    5.26589717614509482e-06,
    1.79279521379981606e-06,
    -6.02137634337859139e-08,
    3.43295935966514131e-09,
    -1.70813448005705326e-10,
    8.72667300550160948e-12,
    -4.44105730567590608e-13,
    2.26242458745239184e-14,
    // piece[0][8]  [0.5,0.5625)  11 terms

    1.23426616124584787,
    0.045037036916221726,
    0.0020461588690502794,
    1.18980750391455163e-05,
    1.53780162388618342e-06,
    -4.27651416389211423e-08,
    2.4485752561501875e-09,
    -1.14611512889946224e-10,
    5.5832422956260879e-12,
    -2.70211964536666079e-13,
    // piece[0][9]  [0.5625,0.625)  10 terms

    1.28136275248193066,
    0.049171000191756388,
    0.00209068672144123403,
    1.76668816574827894e-05,
    1.35705247691823928e-06,
    -3.0198887307263794e-08,
    1.78241076717065143e-09,
    -7.8289039267722863e-11,
    3.64945371197070789e-12,
    -1.6832724771845922e-13,
    // piece[0][10]  [0.625,0.6875)  10 terms

    1.33264343483798564,
    0.0534106616692784919,
    0.00215155288095013157,
    2.28262018988038529e-05,
    1.23028986108770376e-06,
    -2.09635254498780207e-08,
    1.32391470348719564e-09,
    -5.43261951125679678e-11,
    2.43264726283801412e-12,
    -1.07126961565043691e-13,
    // piece[0][11]  [0.6875,0.75)  10 terms

    1.38822968618836717,
    0.0577870699604393009,
    0.00222722237288865707,
    2.75624307308986122e-05,
    1.14358725416706674e-06,
    -1.40370540572112548e-08,
    1.00365207668865033e-09,
    -3.82159183012859533e-11,
    1.65109969437394998e-12,
    -6.95211766772293181e-14,
    // piece[0][12]  [0.75,0.8125)  10 terms

    1.44827267146964678,
    0.0623287119291679459,
    0.00231664511416656074,
    3.20152316468112072e-05,
    1.08722660443552517e-06,
    -8.73328614554360638e-09,
    7.77096520889803501e-10,
    -2.71940790285933697e-11,
    1.13961581511005387e-12,
    -4.59286341321312413e-14,
    // piece[0][13]  [0.8125,0.875)  10 terms

    1.5129511229889443,
    0.0670623575733705068,
    0.00241913795158008732,
    3.62914555132213488e-05,
    1.05433918302985807e-06,
    -4.58331382824893791e-09,
    6.15148592512570743e-10,
    -1.95294872475868482e-11,
    7.99082986093695793e-13,
    -3.08424061421661017e-14,
    // piece[0][14]  [0.875,0.9375)  10 terms

    1.58246996032166587,
    0.0720137058435451061,
    0.00253430135850503,
    4.04746408418995107e-05,
    1.04001859866795251e-06,
    -1.26139583077902923e-09,
    4.98458544365371405e-10,
    -1.41167842796401839e-11,
    5.68772703733334446e-13,
    -2.10242798173384631e-14,
    // piece[0][15]  [0.9375,1)  10 terms


    1.69633456002226013,
    0.159807687450696928,
    0.0109218827644016782,
    0.000373741911366385,
    1.67343478506258597e-05,
    8.49014787741797366e-08,
    2.43745201445725437e-08,
    -1.11587740776264862e-09,
    8.98369551388854376e-11,
    -6.21394678079168353e-12,
    4.43547665679947713e-13,
    // piece[1][0]  [1,1.125)  11 terms

    1.86745471474073454,
    0.182840179715259038,
    0.0121447080948658657,
    0.000441981309629684016,
    1.74910067265646101e-05,
    2.12061902763811694e-07,
    1.86372660265580618e-08,
    -5.76514261539156225e-10,
    4.96086313519699278e-11,
    -3.11233251027927225e-12,
    2.08717359775181724e-13,
    // piece[1][1]  [1.125,1.25)  11 terms

    2.06289930503656205,
    0.208526672329099799,
    0.0135779874210192643,
    0.000514421062485083988,
    1.88138173685141038e-05,
    3.14211499491492764e-07,
    1.57674748739820023e-08,
    -2.70567204886352245e-10,
    2.90969615760705783e-11,
    -1.61974237482491931e-12,
    1.03243335438556261e-13,
    // piece[1][2]  [1.25,1.375)  11 terms

    2.28553752940251664,
    0.237302829615608529,
    0.0152375072186155761,
    0.000593125831386955818,
    2.06137695903026287e-05,
    4.04583131716159014e-07,
    1.45712200081417461e-08,
    -8.55438613003413143e-11,
    1.82802866585454409e-11,
    -8.66188716812030264e-13,
    5.34409688236750282e-14,
    // piece[1][3]  [1.375,1.5)  11 terms

    2.53869202492399015,
    0.269641786508482417,
    0.0171448304164255778,
    0.000679916128306087353,
    2.2853440278619748e-05,
    4.91140755960259522e-07,
    1.44214455984479018e-08,
    3.5035041609504709e-11,
    1.24567837940335522e-11,
    -4.68518700039668787e-13,
    // piece[1][4]  [1.5,1.625)  10 terms

    2.82618191702673549,
    0.30606515206077517,
    0.0193268282411373886,
    0.000776531613568737385,
    2.55275104998069793e-05,
    5.79050331650117927e-07,
    1.49815416914940249e-08,
    1.20834562021589564e-10,
    9.31972746357268247e-12,
    -2.49414135555373565e-13,
    // piece[1][5]  [1.625,1.75)  10 terms

    3.15237655061451028,
    0.347153499484973371,
    0.0218156061614749146,
    0.000884736521712294792,
    2.86523386217268732e-05,
    6.7197134875752382e-07,
    1.60704681085564254e-08,
    1.88131317057038787e-10,
    7.69288198011966008e-12,
    -1.23116819352023646e-13,
    // piece[1][6]  [1.75,1.875)  10 terms

    3.52225973335881948,
    0.393556988384545559,
    0.024648694690988994,
    0.00100639400519479563,
    3.22603618642694107e-05,
    7.72762449291079255e-07,
    1.75942890966781384e-08,
    2.46271559441743704e-10,
    6.95661093084486902e-12,
    -4.647006727925234e-14,
    // piece[1][7]  [1.875,2)  10 terms


    4.17162074979924569,
    0.94950425929138782,
    0.118561763471149828,
    0.00974862006039329231,
    0.000618902597519358053,
    3.02093755762488882e-05,
    1.31909933334940936e-06,
    4.19962844081521154e-08,
    1.74921330974434273e-09,
    1.11049992480767352e-11,
    3.51155204753201808e-12,
    -2.00532981284016196e-13,
    // piece[2][0]  [2,2.25)  12 terms

    5.25008586745453787,
    1.21850852637960005,
    0.151843850931135393,
    0.0125542753439178822,
    0.000791330359666138914,
    3.91060550317726176e-05,
    1.6636449926108685e-06,
    5.67584599268578171e-08,
    1.98224713901201417e-09,
    3.86475990122963117e-11,
    2.26422685530946367e-12,
    // piece[2][1]  [2.25,2.5)  11 terms

    6.63382467895045558,
    1.56323030139458163,
    0.194671923267465824,
    0.0161460312957017321,
    0.00101394594433914099,
    5.0396279850251408e-05,
    2.1201609583211179e-06,
    7.42672945328718751e-08,
    2.42641539313627837e-09,
    5.97194982003826642e-11,
    2.05350081999857335e-12,
    // piece[2][2]  [2.5,2.75)  11 terms

    8.40893947404884301,
    2.00533326779946863,
    0.249731087829110382,
    0.0207509215897222279,
    0.00130050692152905065,
    6.48207850582738205e-05,
    2.71342455300125039e-06,
    9.60790248321603629e-08,
    3.05859709455915921e-09,
    8.1158265419702881e-11,
    2.28664429165888407e-12,
    // piece[2][3]  [2.75,3)  11 terms


    10.6861228916193891,
    2.57259131816122411,
    0.320477909673182848,
    0.026658966762570617,
    0.00166889980120084975,
    8.3301092886356329e-05,
    3.4789335484382794e-06,
    1.23756713676000907e-07,
    3.89850475943559134e-09,
    1.06316012092707055e-10,
    2.78297915302174635e-12,
    // piece[3][0]  [3,3.25)  11 terms

    13.6076068938083754,
    3.30063791453616329,
    0.411356115705072689,
    0.0342417146424460334,
    0.00214220764848982795,
    0.000107006025430112712,
    4.46393073983138439e-06,
    1.5912565805486396e-07,
    4.99053549382331315e-09,
    1.37556236670556147e-10,
    3.50256152900872368e-12,
    // piece[3][1]  [3.25,3.5)  11 terms

    17.3560964805540472,
    4.2352070892569742,
    0.528075011226492141,
    0.0439757449614679233,
    0.00275013356580870857,
    0.000137428969895591371,
    5.72987233876247556e-06,
    2.04448764742674727e-07,
    6.39959159793671368e-09,
    1.77166572606015244e-10,
    4.46292496132859085e-12,
    // piece[3][2]  [3.5,3.75)  11 terms

    22.1662478294371112,
    5.43500788891944353,
    0.677967764486859581,
    0.0564726958887869779,
    0.00353085347270260625,
    0.00017648350052511762,
    7.35606952980145291e-06,
    2.62593685140695823e-07,
    8.21247485023255952e-09,
    2.27777792283180997e-10,
    5.71300118374846437e-12,
    // piece[3][3]  [3.75,4)  11 terms


    32.0540860952870617,
    15.8051858322840548,
    3.94528391708071346,
    0.657403848612676311,
    0.0821905378738998355,
    0.00821775292271599914,
    0.000684916492747229886,
    4.89139097021761143e-05,
    3.05793434590421566e-06,
    1.69803680170759202e-07,
    8.49837069926534792e-09,
    3.854827035876118e-10,
    1.61404890445744108e-11,
    // piece[4][0]  [4,4.5)  13 terms

    52.5531050511022286,
    26.0422941058509778,
    6.50420929872188669,
    1.08394044100641462,
    0.135502407704877842,
    0.0135494168722798663,
    0.00112917831988014646,
    8.06512485207698786e-05,
    5.04105016685982205e-06,
    2.8002729527600805e-07,
    1.40042507841712155e-08,
    6.36292927691533613e-10,
    2.65357466251248162e-11,
    // piece[4][1]  [4.5,5)  13 terms


    86.3338158865726655,
    42.9196733661747558,
    10.7233224851192954,
    1.7871582133881696,
    0.223401282284675345,
    0.0223395907017134437,
    0.00186166999995480325,
    0.000132974017029224931,
    8.31103953790920885e-06,
    4.61711709627740434e-07,
    2.30866723892769053e-08,
    1.04930039038415112e-09,
    4.37286940614920472e-11,
    // piece[5][0]  [5,5.5)  13 terms

    142.01171426519096,
    70.7452677840222464,
    17.6795686607292311,
    2.94655389579607396,
    0.368323542212774802,
    0.0368319984902905698,
    0.0030693574691234792,
    0.000219238362687523261,
    1.37024833855469104e-05,
    7.61243432749672242e-07,
    3.80626032524657839e-08,
    1.73008314517461611e-09,
    7.20896148301763797e-11,
    // piece[5][1]  [5.5,6)  13 terms


    233.791563245867849,
    116.621588639142516,
    29.1485488412039295,
    4.85806469481091163,
    0.607260931749279309,
    0.0607258566097950217,
    0.00506050411930269745,
    0.000361463648054747145,
    2.25915276981555152e-05,
    1.25508210124639197e-06,
    6.27542868904765516e-08,
    2.8524538978947873e-09,
    1.18853303595622498e-10,
    // piece[6][0]  [6,6.5)  13 terms

    385.09319808949175,
    192.258639374649789,
    48.0577461154043704,
    8.0096068791079631,
    1.00120273353534306,
    0.100120116103534398,
    0.00834335374526304699,
    0.000595953222880577687,
    3.72471075477176096e-05,
    2.06928223325238242e-06,
    1.03464194868117485e-07,
    4.70291242383160221e-09,
    1.95955091154045203e-10,
    // piece[6][1]  [6.5,7)  13 terms


    634.529492040021523,
    316.962913241701017,
    79.2337719801021478,
    13.2056173067728011,
    1.65070339223326412,
    0.165070235025584239,
    0.0137558601018832865,
    0.000982561022464204176,
    6.14100843483745949e-05,
    3.41167043329665898e-06,
    1.70583564172978919e-07,
    7.75379600896332809e-09,
    3.23074992921365971e-10,
    // piece[7][0]  [7,7.5)  13 terms

    1045.76237161740892,
    522.565410734934176,
    130.634368714626675,
    21.7723874330284168,
    2.72154923162779383,
    0.272154854421242653,
    0.022679575994340867,
    0.00161996943538404714,
    0.000101248103470642499,
    5.62489404125543127e-06,
    2.81244726493410655e-07,
    1.27838500928441737e-08,
    5.32660484770158859e-10,
    // piece[7][1]  [7.5,8)  13 terms


    1723.75264929905688,
    861.546562472516598,
    215.379638784779218,
    35.8966017207959851,
    4.48707573692504091,
    0.448707528559049329,
    0.0373922972295373315,
    0.00267087818626718376,
    0.000166929895979784656,
    9.27388270869217225e-06,
    4.63694150881473318e-07,
    2.10770062642991307e-08,
    8.78208622063888694e-10,
    // piece[8][0]  [8,8.5)  13 terms

    2841.55147540751159,
    1420.43195950953532,
    355.10097653771021,
    59.1834930398476047,
    7.39793696793513877,
    0.73979366730107099,
    0.06164947437789077,
    0.00440353375879892327,
    0.000275220866260315227,
    1.52900478511677585e-05,
    7.64502402914291362e-07,
    3.47501088636324155e-08,
    1.44792121612688287e-09,
    // piece[8][1]  [8.5,9)  13 terms


    4684.47197944965046,
    2341.87817691808732,
    585.46252350241457,
    97.5770852955261603,
    12.1971358799989673,
    1.21971356881486392,
    0.101642798783047869,
    0.00726019982967682914,
    0.000453762493633113196,
    2.52090272359875288e-05,
    1.26045136893101438e-06,
    5.72932438042689837e-08,
    2.38721849945902583e-09,
    // piece[9][0]  [9,9.5)  13 terms

    7722.9159979048527,
    3861.08613961089259,
    965.264509446574948,
    160.877416991543896,
    20.1096772641614336,
    2.01096771398818852,
    0.167580643735760842,
    0.011970045925966389,
    0.00074812787324362261,
    4.15626594962887559e-05,
    2.07813297975984736e-06,
    9.44605898242561843e-08,
    3.93585791423892809e-09,
    // piece[9][1]  [9.5,10)  13 terms


    12732.445051488894,
    6365.83661224575189,
    1591.45212458613378,
    265.242019967686133,
    33.1552525858458011,
    3.31552525056451053,
    0.276293771468236915,
    0.0197352693543166001,
    0.00123345433655789325,
    6.85252408328959693e-05,
    3.42626204506290616e-06,
    1.55739183750649959e-07,
    6.48913265967061685e-09,
    // piece[10][0]  [10,10.5)  13 terms

    20991.7439207340249,
    10495.4719878504275,
    2623.86096656451082,
    437.310160587343034,
    54.6637701308746031,
    5.46637700792983416,
    0.455531417708372299,
    0.0325379583840169281,
    0.00203362240026755186,
    0.000112979022178595756,
    5.64895111127415729e-06,
    2.56770504976601477e-07,
    1.0698771043113868e-08,
    // piece[10][1]  [10.5,11)  13 terms


    34609.0074047694725,
    17304.0896677579003,
    4326.01538531956521,
    721.002563898289736,
    90.1253205239195125,
    9.01253204908554828,
    0.751044337669619644,
    0.0536460241038171984,
    0.00335287650732185799,
    0.00018627091703440874,
    9.31354585331383e-06,
    4.23342993275845685e-07,
    1.76392913882269636e-08,
    // piece[11][0]  [11,11.5)  13 terms

    57060.0611135826621,
    28529.6024580911362,
    7132.39358212808202,
    1188.73226348426465,
    148.591532958834989,
    14.8591532937698947,
    1.23826277463892254,
    0.0884473410356310802,
    0.00552795881527214095,
    0.000307108823044818606,
    1.53554411533141604e-05,
    6.97974597838936578e-07,
    2.90822749111850753e-08,
    // piece[11][1]  [11.5,12)  13 terms


    94075.5726648057171,
    47037.3441683855344,
    11759.3290092112857,
    1959.88816807304374,
    244.986021023920756,
    24.4986021010444546,
    2.04155017518839399,
    0.145825012506998086,
    0.00911406328204230828,
    0.000506336848985331173,
    2.53168424499833931e-05,
    1.15076556588177991e-06,
    4.79485652459349558e-08,
    // piece[12][0]  [12,12.5)  13 terms

    155103.815655705839,
    77551.4515977318806,
    19387.8558662380201,
    3231.30931095833739,
    403.913663879161447,
    40.3913663870589303,
    3.36594719898637118,
    0.240424799923443927,
    0.0150265499954452715,
    0.000834808333069154648,
    4.17404166539329165e-05,
    1.89729166606994827e-06,
    7.90538194201759672e-08,
    // piece[12][1]  [12.5,13)  13 terms


    255722.359737977473,
    127860.70957226734,
    31965.17035967642,
    5327.52839322813816,
    665.941049159441832,
    66.5941049154000808,
    5.54950874299131947,
    0.396393481639573886,
    0.0247745926026218533,
    0.00137636625569396184,
    6.88183127850109766e-05,
    3.12810512657943525e-06,
    1.30337713607882462e-07,
    // piece[13][0]  [13,13.5)  13 terms

    421614.275342489651,
    210806.653307610017,
    52701.6562933889654,
    8783.60938219922718,
    1097.95117277864347,
    109.79511727751968,
    9.14959310648625213,
    0.653542364747315419,
    0.0408463977968027206,
    0.00226924432203989487,
    0.000113462216102199563,
    5.15737345918294845e-06,
    2.14890560799563666e-07,
    // piece[13][1]  [13.5,14)  13 terms


    695123.786985700368,
    347561.395062104915,
    86890.3417319352448,
    14481.7236219689257,
    1810.21545274847313,
    181.021545274629347,
    15.0851287729024595,
    1.07750919806337575,
    0.0673443248790222271,
    0.00374135138216486656,
    0.000187067569108376706,
    8.50307132310280207e-06,
    3.54294638462800135e-07,
    // piece[14][0]  [14,14.5)  13 terms

    1146064.71831931896,
    573031.846661679796,
    143257.954631780303,
    23876.325771950651,
    2984.54072149531521,
    298.454072149393937,
    24.871172679126726,
    1.7765123342226472,
    0.111032020888954586,
    0.00616844560493996948,
    0.000308422280247084961,
    1.40191945566822551e-05,
    5.8413310652854895e-07,
    // piece[14][1]  [14.5,15)  13 terms


    1889540.60538690304,
    944769.77612815972,
    236192.436998369754,
    39365.4061663869725,
    4920.6757707993047,
    492.067577079843716,
    41.0056314233269887,
    2.92897367309434697,
    0.183060854568421638,
    0.0101700474760221728,
    0.000508502373801164478,
    2.31137442636870645e-05,
    9.63072677653707872e-07,
    // piece[15][0]  [15,15.5)  13 terms

    3115325.09639631491,
    1557662.00756550464,
    389415.494857686863,
    64902.5824762761404,
    8112.82280953510235,
    811.282280953455711,
    67.6068567461255299,
    4.82906119615154594,
    0.301816324759487498,
    0.0167675735977485044,
    0.000838378679887461084,
    3.81081218130649671e-05,
    1.58783840887775955e-06,
    // piece[15][1]  [15.5,16)  13 terms


    5136302.04176624957,
    2568150.46618308034,
    642037.609512068797,
    107006.268252008333,
    13375.783531501409,
    1337.57835315010675,
    111.464862762511544,
    7.96177591160779219,
    0.497610994475497059,
    0.0276450552486382159,
    0.00138225276243193382,
    6.28296710196324279e-05,
    2.61790295915138532e-06,
    // piece[16][0]  [16,16.5)  13 terms

    8468329.70094058663,
    4234164.28170283884,
    1058541.06339200097,
    176423.510565331526,
    22052.9388206666699,
    2205.29388206664544,
    183.77449017222213,
    13.1267492980157563,
    0.820421831125991097,
    0.0455789906181102919,
    0.00227894953090552934,
    0.000103588615041159818,
    4.31619229338168169e-06,
    // piece[16][1]  [16.5,17)  13 terms


    13961914.5589308105,
    6980956.69663052727,
    1745239.16712391842,
    290873.194520651828,
    36359.1493150816241,
    3635.91493150814904,
    302.992910959013443,
    21.6423507827866075,
    1.35264692392416697,
    0.0751470513291201819,
    0.0037573525664560182,
    0.000170788753020727728,
    7.11619804253033584e-06,
    // piece[17][0]  [17,17.5)  13 terms

    23019304.7484437563,
    11509651.777319571,
    2877412.93729617633,
    479568.82288269524,
    59946.1028603369996,
    5994.61028603369141,
    499.550857169474909,
    35.6822040835338825,
    2.23013775522086988,
    0.123896541956714873,
    0.00619482709783574989,
    0.000281583049901624734,
    1.17326270792343729e-05,
    // piece[17][1]  [17.5,18)  13 terms


    37952416.592672728,
    18976207.6853666231,
    4744051.91430793703,
    790675.319051322411,
    98834.4148814153596,
    9883.4414881415305,
    823.620124011794587,
    58.8300088579853053,
    3.67687555362408292,
    0.204270864090226739,
    0.0102135432045113404,
    0.000464251963841424422,
    1.93438318267260244e-05,
    // piece[18][0]  [18,18.5)  13 terms

    62572955.7097453624,
    31286477.2298355028,
    7821619.30042515602,
    1303603.21673752577,
    162950.40209219075,
    16295.0402092190725,
    1357.92001743492301,
    96.9942869596373356,
    6.06214293497733436,
    0.336785718609851847,
    0.0168392859304925951,
    0.000765422087749663307,
    3.1892586989569309e-05,
    // piece[18][1]  [18.5,19)  13 terms


    103165362.229915887,
    51582680.4758533239,
    12895670.1119296104,
    2149278.35198826808,
    268659.793998533569,
    26865.9793998533532,
    2238.83161665444641,
    159.916544046746139,
    9.99478400292163549,
    0.555265777940090799,
    0.0277632888970045406,
    0.00126196767713657003,
    5.25819865473570823e-05,
    // piece[19][0]  [19,19.5)  13 terms

    170090926.270375878,
    85045462.482015878,
    21261365.6134702489,
    3543560.93557837466,
    442945.11694729689,
    44294.5116947296847,
    3691.20930789414069,
    263.657807706724327,
    16.4786129816702704,
    0.915478498981681654,
    0.0457739249490840827,
    0.00208063295223109455,
    8.66930396762956198e-05,
    // piece[19][1]  [19.5,20)  13 terms


    280432527.239234209,
    140216262.952377588,
    35054065.7310606763,
    5842344.28851011302,
    730293.036063764128,
    73029.3036063764157,
    6085.77530053136798,
    434.698235752240578,
    27.1686397345150361,
    1.5093688741397242,
    0.0754684437069862157,
    0.00343038380486300953,
    0.000142932658535958721,
    // piece[20][0]  [20,20.5)  13 terms

    462355071.781445086,
    231177535.209415615,
    57794383.795320183,
    9632397.29922002926,
    1204049.66240250366,
    120404.966240250375,
    10033.7471866875312,
    716.696227620537911,
    44.7935142262836195,
    2.48852856812686785,
    0.124426428406343395,
    0.0056557467457428812,
    0.000235656114405953401,
    // piece[20][1]  [20.5,21)  13 terms


    762294640.569828033,
    381147319.589539587,
    95286829.8903511763,
    15881138.3150585312,
    1985142.2893823164,
    198514.228938231623,
    16542.8524115193031,
    1181.63231510852165,
    73.8520196942826033,
    4.10288998301570018,
    0.205144499150785004,
    0.00932474996139931803,
    0.000388531248391638269,
    // piece[21][0]  [21,21.5)  13 terms

    1256811387.53760743,
    628405693.059361935,
    157101423.257806748,
    26183570.5429677926,
    3272946.31787097408,
    327294.631787097431,
    27274.5526489247823,
    1948.18233206605601,
    121.7613957541285,
    6.76452198634047175,
    0.338226099317023632,
    0.0153739136053192543,
    0.000640579733554969,
    // piece[21][1]  [21.5,22)  13 terms


    2072131666.96256685,
    1036065832.75777411,
    259016458.182409823,
    43169409.6970683038,
    5396176.21213353798,
    539617.621213353821,
    44968.1351011128136,
    3212.00965007948662,
    200.750603129967914,
    11.1528112849982186,
    0.557640564249910931,
    0.0253472983749959517,
    0.00105613743229149792,
    // piece[22][0]  [22,22.5)  13 terms

    3416367554.06541777,
    1708183776.29513216,
    427045944.066749334,
    71174324.0111248791,
    8896790.50139060989,
    889679.050139061059,
    74139.9208449217549,
    5295.70863178012587,
    330.981789486257867,
    18.3878771936809926,
    0.919393859684049519,
    0.0417906299856386151,
    0.00174127624940160911,
    // piece[22][1]  [22.5,23)  13 terms


    5632637853.95209217,
    2816318926.22440195,
    704079731.549066782,
    117346621.924844459,
    14668327.7406055573,
    1466832.77406055573,
    122236.064505046321,
    8731.14746464616474,
    545.696716540385296,
    30.3164842522436295,
    1.51582421261218148,
    0.0689011005732809712,
    0.00287087919055337409,
    // piece[23][0]  [23,23.5)  13 terms

    9286649838.97795296,
    4643324918.72326469,
    1160831229.67378259,
    193471871.612297088,
    24183983.951537136,
    2418398.39515371341,
    201533.199596142804,
    14395.2285425816281,
    899.701783911351754,
    49.9834324395195395,
    2.49917162197597698,
    0.113598710089817145,
    0.00473327958707571439,
    // piece[23][1]  [23.5,24)  13 terms


    15311097122.0650368,
    7655548560.25273895,
    1913887140.05615091,
    318981190.009358525,
    39872648.7511698157,
    3987264.8751169811,
    332272.072926415072,
    23733.7194947439348,
    1483.35746842149592,
    82.4087482456386624,
    4.12043741228193294,
    0.187292609649178793,
    0.0078038587353824497,
    // piece[24][0]  [24,24.5)  13 terms

    25243731501.8840523,
    12621865750.14818,
    3155466437.53001118,
    525911072.921668589,
    65738884.1152085736,
    6573888.41152085736,
    547824.03429340478,
    39130.2881638146282,
    2445.64301023841426,
    135.869056124356348,
    6.79345280621781722,
    0.308793309373537106,
    0.0128663878905640461,
    // piece[24][1]  [24.5,25)  13 terms


    41619877077.9607925,
    20809938538.1724815,
    5202484634.53608704,
    867080772.422681093,
    108385096.552835137,
    10838509.6552835144,
    903209.137940292829,
    64514.9384243066306,
    4032.18365151916441,
    224.010202862175817,
    11.2005101431087901,
    0.509114097414035904,
    0.0212130873922514972,
    // piece[25][0]  [25,25.5)  13 terms

    68619576621.302063,
    34309788309.829052,
    8577447077.45022964,
    1429574512.90837145,
    178696814.113546431,
    17869681.4113546424,
    1489140.11761288694,
    106367.151258063357,
    6647.94695362895982,
    369.330386312720009,
    18.466519315635999,
    0.839387241619818147,
    0.0349744684008257561,
    // piece[25][1]  [25.5,26)  13 terms


    113134555560.903107,
    56567277779.6155014,
    14141819444.896843,
    2356969907.48280716,
    294621238.435350895,
    29462123.843535088,
    2455176.98696125718,
    175369.784782946954,
    10960.6115489341846,
    608.922863829676885,
    30.446143191483845,
    1.383915599612902,
    0.0576631499838709191,
    // piece[26][0]  [26,26.5)  13 terms

    186527348203.373322,
    93263674100.8365479,
    23315918525.2021027,
    3885986420.86701679,
    485748302.608377099,
    48574830.2608377114,
    4047902.5217364761,
    289135.894409748318,
    18070.9934006092699,
    1003.94407781162602,
    50.1972038905812994,
    2.28169108593551373,
    0.0950704619139797341,
    // piece[26][1]  [26.5,27)  13 terms


    307531606549.07959,
    153765803273.675598,
    38441450818.4118652,
    6406908469.73531151,
    800863558.716913939,
    80086355.8716913909,
    6673862.98930761591,
    476704.499236258271,
    29794.0312022661419,
    1655.22395568145225,
    82.7611977840726212,
    3.76187262654875543,
    0.156744692772864819,
    // piece[27][0]  [27,27.5)  13 terms

    507033901128.920715,
    253516950563.582123,
    63379237640.8884964,
    10563206273.4814167,
    1320400784.18517709,
    132040078.418517694,
    11003339.8682098091,
    785952.847729271976,
    49122.0529830794985,
    2729.00294350441663,
    136.450147175220849,
    6.20227941705549224,
    0.258428309043978843,
    // piece[27][1]  [27.5,28)  13 terms


    835957577756.169434,
    417978788877.192444,
    104494697219.291077,
    17415782869.8818436,
    2176972858.73523045,
    217697285.873523057,
    18141440.489460256,
    1295817.17781858961,
    80988.5736136618507,
    4499.3652007589917,
    224.968260037949591,
    10.2258300017249812,
    0.426076250071874219,
    // piece[28][0]  [28,28.5)  13 terms

    1378261039848.38672,
    689130519923.286987,
    172282629980.814728,
    28713771663.46912,
    3589221457.93364,
    358922145.793363988,
    29910178.8161136657,
    2136441.34400811931,
    133527.584000507457,
    7418.19911113930266,
    370.909955556965144,
    16.8595434344075059,
    0.702480976433646043,
    // piece[28][1]  [28.5,29)  13 terms


    2272368292974.12793,
    1136184146486.14355,
    284046036621.52887,
    47341006103.5881424,
    5917625762.9485178,
    591762576.29485178,
    49313548.0245709792,
    3522396.28746935586,
    220149.767966834741,
    12230.5426648241519,
    611.527133241207594,
    27.7966878746003445,
    1.15819532810834769,
    // piece[29][0]  [29,29.5)  13 terms

    3746501939489.78271,
    1873250969743.95679,
    468312742435.982178,
    78052123739.330368,
    9756515467.41629601,
    975651546.741629481,
    81304295.5618024617,
    5807449.68298588973,
    362965.605186618108,
    20164.7558437010084,
    1008.23779218505035,
    45.8289905538659283,
    1.90954127307774701,
    // piece[29][1]  [29.5,30)  13 terms


    6176937438354.86816,
    3088468719176.48535,
    772117179794.11438,
    128686196632.352386,
    16085774579.0440483,
    1608577457.90440488,
    134048121.49203375,
    9574865.82085955329,
    598429.113803722081,
    33246.0618779845609,
    1662.30309389922786,
    75.559231540873995,
    3.14830131420308312,
    // piece[30][0]  [30,30.5)  13 terms

    10184048142398.3945,
    5092024071198.23438,
    1273006017799.55151,
    212167669633.258575,
    26520958704.1573219,
    2652095870.41573238,
    221007989.201311022,
    15786284.942950787,
    986642.808934424189,
    54813.4893852457899,
    2740.67446926228968,
    124.576112239194984,
    5.19067134329979041,
    // piece[30][1]  [30.5,31)  13 terms


    16790656794205.1016,
    8395328397101.57422,
    2098832099275.38647,
    349805349879.231079,
    43725668734.9038849,
    4372566873.49038887,
    364380572.790865719,
    26027183.7707761228,
    1626698.98567350768,
    90372.1658707504248,
    4518.6082935375216,
    205.391286069887343,
    8.55797025291197144,
    // piece[31][0]  [31,31.5)  13 terms

    27683113005630.3008,
    13841556502814.1602,
    3460389125703.53271,
    576731520950.588745,
    72091440118.8235931,
    7209144011.88236046,
    600762000.990196705,
    42911571.4992997646,
    2681973.21870623529,
    148998.512150346389,
    7449.9256075173198,
    338.632982159878168,
    14.1097075899949242,
    // piece[31][1]  [31.5,32)  13 terms


    45641737251576.7422,
    22820868625787.3633,
    5705217156446.83398,
    950869526074.47229,
    118858690759.309052,
    11885869075.9309044,
    990489089.660908699,
    70749220.6900649071,
    4421826.29312905669,
    245657.016284947604,
    12282.8508142473802,
    558.311400647608139,
    23.2629750269836748,
    // piece[32][0]  [32,32.5)  13 terms

    75250503038379.6562,
    37625251519188.8125,
    9406312879797.19531,
    1567718813299.53271,
    195964851662.441589,
    19596485166.2441597,
    1633040430.52034664,
    116645745.037167609,
    7290359.06482297555,
    405019.94804572087,
    20250.9974022860442,
    920.499881922092868,
    38.3541617467538742,
    // piece[32][1]  [32.5,33)  13 terms


    124067104990259.844,
    62033552495128.8906,
    15508388123782.2148,
    2584731353963.70264,
    323091419245.46283,
    32309141924.5462799,
    2692428493.71219015,
    192316320.979442149,
    12019770.0612151343,
    667765.003400840797,
    33388.2501700420398,
    1517.64773500191086,
    63.2353222917462858,
    // piece[33][0]  [33,33.5)  13 terms

    204552074991626.062,
    102276037495811.984,
    25569009373952.9883,
    4261501562325.49805,
    532687695290.687256,
    53268769529.0687256,
    4439064127.4223938,
    317076009.101599574,
    19817250.5688499734,
    1100958.36493610963,
    55047.9182468054787,
    2502.17810212752192,
    104.257420921980085,
    // piece[33][1]  [33.5,34)  13 terms


    337249357004540.25,
    168624678502269.062,
    42156169625567.2578,
    7026028270927.87695,
    878253533865.984619,
    87825353386.598465,
    7318779448.88320541,
    522769960.63451463,
    32673122.5396571644,
    1815173.47442539805,
    90758.6737212699081,
    4125.39426005772293,
    171.891427502405122,
    // piece[34][0]  [34,34.5)  13 terms

    556030188423325.438,
    278015094211661.625,
    69503773552915.3984,
    11583962258819.2324,
    1447995282352.40405,
    144799528235.240417,
    12066627352.9367008,
    861901953.781193018,
    53868872.1113245636,
    2992715.11729580909,
    149635.755864790437,
    6801.62526658138358,
    283.401052774224354,
    // piece[34][1]  [34.5,35)  13 terms


    916738798804935.375,
    458369399402466.562,
    114592349850616.641,
    19098724975102.7734,
    2387340621887.84668,
    238734062188.784668,
    19894505182.3987236,
    1421036084.45705152,
    88814755.2785657197,
    4934153.07103142887,
    246707.653551571449,
    11213.9842523441566,
    467.249343847673231,
    // piece[35][0]  [35,35.5)  13 terms

    1511446757265780.75,
    755723378632889.25,
    188930844658222.312,
    31488474109703.7188,
    3936059263712.96484,
    393605926371.296448,
    32800493864.274704,
    2342892418.87676477,
    146430776.179797798,
    8135043.12109987717,
    406752.156054993859,
    18488.7343661360865,
    770.36393192233686,
    // piece[35][1]  [35.5,36)  13 terms


    2491954418234824.5,
    1245977209117411.25,
    311494302279352.812,
    51915717046558.7969,
    6489464630819.84961,
    648946463081.984985,
    54078871923.4987488,
    3862776565.96419621,
    241423535.372762263,
    13412418.6318201255,
    670620.931591006345,
    30482.769617773014,
    1270.11540074054233,
    // piece[36][0]  [36,36.5)  13 terms

    4108538254958917,
    2054269127479457.5,
    513567281869864.375,
    85594546978310.7344,
    10699318372288.8418,
    1069931837228.88416,
    89160986435.7403412,
    6368641888.26716709,
    398040118.016697943,
    22113339.8898165524,
    1105666.99449082767,
    50257.5906586739875,
    2094.06627744474918,
    // piece[36][1]  [36.5,37)  13 terms


    6773834412435952,
    3386917206217974.5,
    846729301554493.625,
    141121550259082.281,
    17640193782385.2852,
    1764019378238.52856,
    147001614853.210693,
    10500115346.6579075,
    656257209.166119218,
    36458733.8425621763,
    1822936.69212810905,
    82860.7587330958631,
    3452.53161387899399,
    // piece[37][0]  [37,37.5)  13 terms

    11168164879983656,
    5584082439991827,
    1396020609997956.75,
    232670101666326.125,
    29083762708290.7656,
    2908376270829.07666,
    242364689235.756378,
    17311763516.8397408,
    1081985219.8024838,
    60110289.9890268818,
    3005514.4994513439,
    136614.295429606544,
    5692.26230956693962,
    // piece[37][1]  [37.5,38)  13 terms


    18413190992315196,
    9206595496157598,
    2301648874039399.5,
    383608145673233.25,
    47951018209154.1562,
    4795101820915.41504,
    399591818409.61792,
    28542272743.5441399,
    1783892046.47150874,
    99105113.6928615868,
    4955255.68464307953,
    225238.894756503607,
    9384.95394818765089,
    // piece[38][0]  [38,38.5)  13 terms

    30358219650494064,
    15179109825247030,
    3794777456311757.5,
    632462909385293,
    79057863673161.625,
    7905786367316.16211,
    658815530609.680176,
    47058252186.4057236,
    2941140761.65035772,
    163396708.980575442,
    8169835.44902877137,
    371356.156774035073,
    15473.173198918128,
    // piece[38][1]  [38.5,39)  13 terms


    50052242478356168,
    25026121239178084,
    6256530309794521,
    1042755051632420.12,
    130344381454052.516,
    13034438145405.252,
    1086203178783.771,
    77585941341.697937,
    4849121333.85612106,
    269395629.658673346,
    13469781.4829336684,
    612262.794678803068,
    25510.9497782834624,
    // piece[39][0]  [39,39.5)  13 terms

    82522196820306320,
    41261098410153160,
    10315274602538290,
    1719212433756381.5,
    214901554219547.688,
    21490155421954.7695,
    1790846285162.89746,
    127917591797.349823,
    7994849487.33436394,
    444158304.851909101,
    22207915.2425954528,
    1009450.69284524792,
    42060.4455352186633,
    // piece[39][1]  [39.5,40)  13 terms


};  // coefTh1

static int *nTermsTh1[40];
static int allTermsTh1[] = {
    12,
    12,
    12,
    12,
    11,
    11,
    11,
    11,
    11,
    10,
    10,
    10,
    10,
    10,
    10,
    10,

    11,
    11,
    11,
    11,
    10,
    10,
    10,
    10,

    12,
    11,
    11,
    11,

    11,
    11,
    11,
    11,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

    13,
    13,

}; // allTermsTh1

static int nPiecesTh1[40] = {
    16,
    8,
    4,
    4,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
};  // nPiecesTh1

static bool isInitTh1 = false;
static int finishTh1 = 40;

void InitTh1()
{
    for (int iInt = 0, walkTerm = 0, walkCoef = 0;
         iInt < finishTh1;
         walkTerm += nPiecesTh1[iInt], ++iInt)
    {
        nTermsTh1[iInt] = &allTermsTh1[walkTerm];
        coefTh1[iInt] = new double*[nPiecesTh1[iInt]];
        for (int iPiece = 0;  iPiece < nPiecesTh1[iInt];  walkCoef += nTermsTh1[iInt][iPiece], ++iPiece)
        {
            coefTh1[iInt][iPiece] = &allCoefTh1[walkCoef];
        }
    }
    isInitTh1 = true;
}

double Th1(double x)
{
    if (!isInitTh1)
        InitTh1();

    // Transform the domain to approach pole at 1 slower.
    if (x < 0)
        return nan("");
    x = -log(1-x);
    if (x < 0 || x >= finishTh1)
        return nan("");
    int iInt = static_cast<int>(x);
    x -= iInt;

    // The 0.5 adjusts the range to [-0.5,0.5) instead of [0,1), so c[0] is the middle value.
    // The number of pieces per int was adjusted manually.
    x *= nPiecesTh1[iInt];
    int iPiece = static_cast<int>(x);
    x = x - iPiece - 0.5;

    // start with the tiny terms and build up, to avoid rounding errors
    double *c = coefTh1[iInt][iPiece];
    int iTerm = nTermsTh1[iInt][iPiece]-1;
    double y = c[iTerm];
    while (iTerm--)
        y = y * x + c[iTerm];
    return y;
}

#ifdef TESTFUNC
#include <stdio.h>
#include "bigFloat.h"
int main(int argc, char** argv)
{
    double iterations = 137;
    for (int i=0; i<iterations; ++i)
    {
        double x = i/(iterations-1);
        double z = Th1(x);
        printf("%.18g => %.18g\n", x, z);
    }
}
#endif /* TESTFUNC */



static double **coefTh1c[40];
static double allCoefTh1c[] = {
    0.977789271469070909,
    -0.0420730795388624332,
    0.00449554747866532951,
    -0.000384959160613044767,
    3.33992694687933178e-05,
    -2.89442520332287692e-06,
    2.50538991853442986e-07,
    -2.16709928537345231e-08,
    1.87355119024981365e-09,
    -1.61917199353604629e-10,
    1.39894386561881291e-11,
    -1.20841167166720394e-12,
    1.04365404702559669e-13,
    // piece[0][0]  [0,0.0625)  13 terms

    0.939857515685036393,
    -0.0341163719156094211,
    0.0035154715553593909,
    -0.00027596122187881916,
    2.20400755163108753e-05,
    -1.75872531307433394e-06,
    1.40162936316516613e-07,
    -1.11621573988260155e-08,
    8.88463779869676594e-10,
    -7.06912971650126927e-11,
    5.62298560227797416e-12,
    -4.47168359490218892e-13,
    // piece[0][1]  [0.0625,0.125)  12 terms

    0.909001065276870857,
    -0.0278331764293187379,
    0.00280413169111353019,
    -0.000202931165760703475,
    1.5012551890790412e-05,
    -1.11007080837446009e-06,
    8.19695842832967455e-08,
    -6.04816166142130545e-09,
    4.46031169651120964e-10,
    -3.28803425211413093e-11,
    2.42313512623939497e-12,
    -1.7853284206930806e-13,
    // piece[0][2]  [0.125,0.1875)  12 terms

    0.883783068190817755,
    -0.0227787539157385116,
    0.00227542673466240393,
    -0.000152531480498655696,
    1.05075910386018032e-05,
    -7.23892429375015396e-07,
    4.97964333991419562e-08,
    -3.42277546324715507e-09,
    2.35139407429184162e-10,
    -1.6147194193945897e-11,
    1.10849743557895311e-12,
    -7.60795500998630457e-14,
    // piece[0][3]  [0.1875,0.25)  12 terms

    0.863137039821539664,
    -0.0186468074206330645,
    0.00187432003194167463,
    -0.000116851976147835465,
    7.52991349233570364e-06,
    -4.85610942451327154e-07,
    3.12664553241349247e-08,
    -2.01145102919042505e-09,
    1.29331383350385136e-10,
    -8.31225815199974438e-12,
    5.34067652232736897e-13,
    -3.43056672421603982e-14,
    // piece[0][4]  [0.25,0.3125)  12 terms

    0.846254774135775545,
    -0.01522085720253262,
    0.00156451757570473591,
    -9.10269000797795947e-05,
    5.50856250734502333e-06,
    -3.33937153920334312e-07,
    2.02074697625326435e-08,
    -1.22174404640568415e-09,
    7.38261239817545448e-11,
    -4.45920301592552934e-12,
    2.69254575129135399e-13,
    // piece[0][5]  [0.3125,0.375)  11 terms

    0.832512601289567833,
    -0.0123444249414914921,
    0.00132142825227481208,
    -7.19668437173389227e-05,
    4.10388533788899524e-06,
    -2.3471550680320429e-07,
    1.33976685992655055e-08,
    -7.64033240354222061e-10,
    4.35467483669809576e-11,
    -2.48092720927837629e-12,
    1.41295330467125143e-13,
    // piece[0][6]  [0.375,0.4375)  11 terms

    0.821421519601299832,
    -0.0099021516391221559,
    0.00112798993435274603,
    -5.76549992834081169e-05,
    3.10729457159955214e-06,
    -1.68215950657993914e-07,
    9.08668470013397942e-09,
    -4.90349474632696348e-10,
    2.64463777643775503e-11,
    -1.42573840242090706e-12,
    7.68362416211312727e-14,
    // piece[0][7]  [0.4375,0.5)  11 terms

    0.812592650597296684,
    -0.00780749738260717169,
    0.000972113239772848313,
    -4.67420392229391334e-05,
    2.38703933248857213e-06,
    -1.22674214797298176e-07,
    6.28938518518327416e-09,
    -3.22092792839573929e-10,
    1.64859661941357714e-11,
    -8.4345199598993214e-13,
    4.31376373471681825e-14,
    // piece[0][8]  [0.5,0.5625)  11 terms

    0.805712794763333018,
    -0.00599452662810770476,
    0.00084507062584980011,
    -3.83052523850202452e-05,
    1.85779191422284454e-06,
    -9.08749790709285948e-08,
    4.43362772789117928e-09,
    -2.16040347468495981e-10,
    1.05213531470807007e-11,
    -5.12178742213565353e-13,
    2.49240849616877861e-14,
    // piece[0][9]  [0.5625,0.625)  11 terms

    0.800526804653245661,
    -0.00441229917151990218,
    0.000740455115176016834,
    -3.17011744006564668e-05,
    1.4630368440169047e-06,
    -6.82796347918310382e-08,
    3.17764399329514841e-09,
    -1.47674979096201187e-10,
    6.85914022210715868e-12,
    -3.18454819020402107e-13,
    1.47799081741133735e-14,
    // piece[0][10]  [0.625,0.6875)  11 terms

    0.796824657216234078,
    -0.00302096363088097571,
    0.000653491761396491071,
    -2.64730800882236087e-05,
    1.16457761243260464e-06,
    -5.19675277091620403e-08,
    2.31203887231084892e-09,
    -1.02696064561685034e-10,
    4.55902650043292021e-12,
    -2.02306469540045413e-13,
    // piece[0][11]  [0.6875,0.75)  10 terms

    0.794431827090454235,
    -0.0017889876874436785,
    0.000580572956247815751,
    -2.22915591737585834e-05,
    9.36121597393778434e-07,
    -4.00200063137693915e-08,
    1.70554117992942885e-09,
    -7.25485792072272626e-11,
    3.08426104041913232e-12,
    -1.31068628511922851e-13,
    // piece[0][12]  [0.75,0.8125)  10 terms

    0.793202018537626841,
    -0.000691162317156627831,
    0.0005189389497243193,
    -1.89155388839185819e-05,
    7.59265962782343729e-07,
    -3.1152774128072204e-08,
    1.27411927645818202e-09,
    -5.19955698284990829e-11,
    2.12065318170013234e-12,
    -8.64583345180027257e-14,
    // piece[0][13]  [0.8125,0.875)  10 terms

    0.79301160896866052,
    0.000292857562588238357,
    0.000466454477406980854,
    -1.61662283390401374e-05,
    6.20932283632458602e-07,
    -2.44912766417840588e-08,
    9.62944729896654499e-10,
    -3.77631336404580307e-11,
    1.48001757435603485e-12,
    -5.7984356584104706e-14,
    // piece[0][14]  [0.875,0.9375)  10 terms

    0.793755352147929583,
    0.00117963462980289705,
    0.000421450163920954689,
    -1.39093965171428218e-05,
    5.11695106075248863e-07,
    -1.94305253318055475e-08,
    7.35611410337030068e-10,
    -2.77647224847351818e-11,
    1.04724087439667634e-12,
    -3.94874503407431629e-14,
    // piece[0][15]  [0.9375,1)  10 terms


    0.796428573584223698,
    0.00471308836180385059,
    0.00146064691001273261,
    -8.9850144831858463e-05,
    6.20559913719868182e-06,
    -4.46290753518500194e-07,
    3.20172164734636085e-08,
    -2.28811548916467675e-09,
    1.63388471627300993e-10,
    -1.16640929874559586e-11,
    8.32402010867154173e-13,
    -5.9385272310958271e-14,
    // piece[1][0]  [1,1.125)  12 terms

    0.802518247901194415,
    0.00738759998645425933,
    0.00122430356004655399,
    -6.89221346179818705e-05,
    4.38444419947743757e-06,
    -2.9437246713219369e-07,
    1.97463230457269287e-08,
    -1.31772251709313774e-09,
    8.783508603803499e-11,
    -5.85387268420839905e-12,
    3.90022589927809245e-13,
    -2.59773943266831561e-14,
    // piece[1][1]  [1.125,1.25)  12 terms

    0.811065337895757077,
    0.0096456165246659939,
    0.00104117088426206083,
    -5.39748064124341114e-05,
    3.1681414026049351e-06,
    -1.99298004999844259e-07,
    1.25612471276740852e-08,
    -7.86355333120538697e-10,
    4.9143131200590241e-11,
    -3.07105900554789579e-12,
    1.91875692439584956e-13,
    // piece[1][2]  [1.25,1.375)  11 terms

    0.821701131162814735,
    0.0115777801798345285,
    0.000896435512106126173,
    -4.30690030997848031e-05,
    2.33563739242319231e-06,
    -1.38035568465663854e-07,
    8.21017491236280145e-09,
    -4.84176658805747948e-10,
    2.84786198127158464e-11,
    -1.67516528828642089e-12,
    9.8528387359049149e-14,
    // piece[1][3]  [1.375,1.5)  11 terms

    0.834134483206374289,
    0.0132501426519655227,
    0.000779975697622070706,
    -3.49580868736262869e-05,
    1.75346735780692963e-06,
    -9.75360981746605285e-08,
    5.49592376489691458e-09,
    -3.0650303146256811e-10,
    1.7025960526329924e-11,
    -9.45853585133466665e-13,
    5.2552008973490814e-14,
    // piece[1][4]  [1.5,1.625)  11 terms

    0.848131304605898584,
    0.0147117769337407577,
    0.000684723327349601348,
    -2.88195076949922272e-05,
    1.33858090391890227e-06,
    -7.01508307487968811e-08,
    3.75750079732179199e-09,
    -1.98888611329136979e-10,
    1.04664799098730105e-11,
    -5.5077498174613109e-13,
    2.89949460058650837e-14,
    // piece[1][5]  [1.625,1.75)  11 terms

    0.863500257357922463,
    0.0159997898666503595,
    0.000605647241691137763,
    -2.40979603703540491e-05,
    1.03789703895197556e-06,
    -5.12591013256449516e-08,
    2.61752830695480393e-09,
    -1.31951843374905943e-10,
    6.59738425308052636e-12,
    -3.29722275801568743e-13,
    1.6491193493669638e-14,
    // piece[1][6]  [1.75,1.875)  11 terms

    0.880082585635691084,
    0.0171427005930455713,
    0.000539104817292493442,
    -2.04108873679583485e-05,
    8.16669624316638471e-07,
    -3.79932067422650179e-08,
    1.85401245437872384e-09,
    -8.93065534676869299e-11,
    4.25394029038005964e-12,
    -2.02391020691668851e-13,
    // piece[1][7]  [1.875,2)  10 terms


    0.906944601192596567,
    0.0372647655658498098,
    0.00182850553311686512,
    -0.000130038092859279257,
    9.36125950679585089e-06,
    -7.94740435804969104e-07,
    7.26883016122203154e-08,
    -6.57947234701668353e-09,
    5.85751052122251485e-10,
    -5.19741570050070065e-11,
    4.61959205417608239e-12,
    -4.10923791381609643e-13,
    // piece[2][0]  [2,2.25)  12 terms

    0.945916467364623426,
    0.0405655280231738205,
    0.00148757828024943871,
    -9.9288044707168945e-05,
    6.28290468202003176e-06,
    -4.69526670279057162e-07,
    3.94764073554002045e-08,
    -3.32157260684108118e-09,
    2.72697122781000442e-10,
    -2.21910281705630092e-11,
    1.80907470578623822e-12,
    -1.47801328974979218e-13,
    // piece[2][1]  [2.25,2.5)  12 terms

    0.987876135408364497,
    0.0432658200408195595,
    0.00122324561125138325,
    -7.81648174803850669e-05,
    4.42779160526308856e-06,
    -2.89550240027001365e-07,
    2.23181727575891304e-08,
    -1.76244638992869486e-09,
    1.34913134396328742e-10,
    -1.01441257587244713e-11,
    7.62695841904823187e-13,
    -5.75739757586799438e-14,
    // piece[2][2]  [2.5,2.75)  12 terms

    1.03229119516552492,
    0.045494202793063758,
    0.00101272361105634451,
    -6.29576897037163264e-05,
    3.26143624072167459e-06,
    -1.86206933675070108e-07,
    1.30435992732522585e-08,
    -9.72848702036364197e-10,
    7.02327233448537568e-11,
    -4.91968737057324976e-12,
    3.42754093582808662e-13,
    // piece[2][3]  [2.75,3)  11 terms


    1.07873825124563227,
    0.0473429636287813899,
    0.000841734117832877477,
    -5.15436345088657346e-05,
    2.49636708028668757e-06,
    -1.24985528203440175e-07,
    7.84918508872870044e-09,
    -5.53729949806261887e-10,
    3.81223839863476228e-11,
    -2.51201934695261176e-12,
    1.62978926890827382e-13,
    // piece[3][0]  [3,3.25)  11 terms

    1.12687378407050831,
    0.0488812050044931737,
    0.000700938654739749955,
    -4.26684765267864739e-05,
    1.97217975409315962e-06,
    -8.76637470988301024e-08,
    4.85938296697923734e-09,
    -3.22649689122977239e-10,
    2.13891720817616209e-11,
    -1.34081523993500688e-12,
    8.15258825207213191e-14,
    // piece[3][1]  [3.25,3.5)  11 terms

    1.17641514832607985,
    0.0501625543422333509,
    0.000583956335356144171,
    -3.55694058690521437e-05,
    1.5968029458332298e-06,
    -6.4235768999386645e-08,
    3.1021854245173242e-09,
    -1.9144631777347959e-10,
    1.23020004995399197e-11,
    -7.42695474951210348e-13,
    4.26859971750024056e-14,
    // piece[3][2]  [3.5,3.75)  11 terms

    1.22712762508731599,
    0.0512298421933986073,
    0.000486229409657579041,
    -2.97685774125914955e-05,
    1.31623204784140281e-06,
    -4.9037960872047186e-08,
    2.05208922666995591e-09,
    -1.15348330547828472e-10,
    7.19784434564889633e-12,
    -4.23655487880647328e-13,
    2.32722009898666048e-14,
    // piece[3][3]  [3.75,4)  11 terms


    1.30497224821627578,
    0.105008411159083923,
    0.00147409763341791486,
    -0.000182830104125114438,
    1.61003939064583123e-05,
    -1.11649370039645593e-06,
    7.64976913372952084e-08,
    -7.11710609656695896e-09,
    8.39629961620130788e-10,
    -9.7040014698893459e-11,
    1.02110035949814359e-11,
    -1.01072972504482499e-12,
    9.90390511793780846e-14,
    // piece[4][0]  [4,4.5)  13 terms

    1.41128698093732408,
    0.10746735032281772,
    0.00101206317956960864,
    -0.000128272664639413623,
    1.14646927209584986e-05,
    -7.70000876498117776e-07,
    4.37906950845393047e-08,
    -2.93737093783611187e-09,
    2.977597685979263e-10,
    -3.42066554689931951e-11,
    3.56578723905719569e-12,
    -3.32616644879793595e-13,
    // piece[4][1]  [4.5,5)  12 terms


    1.5196488575870537,
    0.109148911743652016,
    0.000688935758599076246,
    -8.93267166226314801e-05,
    8.18592583231360246e-06,
    -5.55811802150771836e-07,
    2.93113904408726077e-08,
    -1.44915847571759688e-09,
    1.07235850592237606e-10,
    -1.18666104457805209e-11,
    1.28406916012620765e-12,
    -1.1988475540790783e-13,
    // piece[5][0]  [5,5.5)  12 terms

    1.62940503644548684,
    0.110288934242253389,
    0.000464924911428284376,
    -6.16004793473961469e-05,
    5.80206296221076383e-06,
    -4.0559059041874211e-07,
    2.13953719813807171e-08,
    -8.9722159472402763e-10,
    4.25146711528176426e-11,
    -3.88736066462232097e-12,
    4.47825752973596204e-13,
    // piece[5][1]  [5.5,6)  11 terms


    1.7401027121293775,
    0.111055285326807335,
    0.000311183102581901502,
    -4.20495272801589984e-05,
    4.06620509893498355e-06,
    -2.94074044563112047e-07,
    1.60552260529592227e-08,
    -6.55441928032897686e-10,
    2.18847087914480418e-11,
    -1.19980478018185813e-12,
    1.41266974933586907e-13,
    // piece[6][0]  [6,6.5)  11 terms

    1.8514309185831368,
    0.11156638930909997,
    0.000206718650150190082,
    -2.84261438943338722e-05,
    2.81512862421384154e-06,
    -2.10403321155174697e-07,
    1.20026146438657283e-08,
    -5.10870952117195359e-10,
    1.54132065755188478e-11,
    -4.10316159078071662e-13,
    3.61325366986544374e-14,
    // piece[6][1]  [6.5,7)  11 terms


    1.96317821663057401,
    0.111904825235124059,
    0.000136396686109134567,
    -1.90466585184819251e-05,
    1.92630413477190499e-06,
    -1.48297342432708201e-07,
    8.82929813437123041e-09,
    -3.99314216884910833e-10,
    1.2726412154057897e-11,
    -2.36521652480475832e-13,
    // piece[7][0]  [7,7.5)  10 terms

    2.07520217834255938,
    0.112127492641929352,
    8.94559638277328924e-05,
    -1.26611121325037993e-05,
    1.30414277584559001e-06,
    -1.03023329070324591e-07,
    6.37130127871247769e-09,
    -3.05644924819842089e-10,
    1.07128076239501177e-11,
    -2.19875847872202851e-13,
    // piece[7][1]  [7.5,8)  10 terms


    2.18740767303177952,
    0.112273158859660716,
    5.83566938072388895e-05,
    -8.35746443507679357e-06,
    8.74620242499620171e-07,
    -7.0641881816638371e-08,
    4.51325468940253148e-09,
    -2.27866040935858935e-10,
    8.73409936330895936e-12,
    -2.18037506361187783e-13,
    // piece[8][0]  [8,8.5)  10 terms

    2.2997316393930789,
    0.11236797067791772,
    3.78887535260235677e-05,
    -5.48264157479594422e-06,
    5.81718493630287417e-07,
    -4.78856086390906106e-08,
    3.14469016958892676e-09,
    -1.65684701538684969e-10,
    6.83735167393766856e-12,
    -2.00715880627762191e-13,
    // piece[8][1]  [8.5,9)  10 terms


    2.41213255300147766,
    0.112429405467443427,
    2.44961589895683951e-05,
    -3.57716249800512957e-06,
    3.84115723884903814e-07,
    -3.21385652276564673e-08,
    2.1600565475166998e-09,
    -1.17877979915442271e-10,
    5.1595125551977723e-12,
    -1.70743574197549741e-13,
    // piece[9][0]  [9,9.5)  10 terms

    2.52458323148974184,
    0.112469054242964522,
    1.57780440688127338e-05,
    -2.32273485706280234e-06,
    2.52038379868292755e-07,
    -2.13853899668610415e-08,
    1.46574806789080867e-09,
    -8.23399549369036355e-11,
    3.77653521790700021e-12,
    -1.36485726415371725e-13,
    // piece[9][1]  [9.5,10)  10 terms


    2.63706597308195967,
    0.112494551600227047,
    1.0128573940655832e-05,
    -1.50180174824181579e-06,
    1.64463620349992698e-07,
    -1.41249047706385146e-08,
    9.84350187800750954e-10,
    -5.66411337685307339e-11,
    2.69681853990842256e-12,
    -1.04102122222014163e-13,
    // piece[10][0]  [10,10.5)  10 terms

    2.74956930272339939,
    0.112510896103101693,
    6.48234905589909787e-06,
    -9.67349118577181081e-07,
    1.06798188653725783e-07,
    -9.26962342706674462e-09,
    6.55240649819337244e-10,
    -3.84678164402267402e-11,
    1.88773236975756418e-12,
    -7.66327765150794858e-14,
    // piece[10][1]  [10.5,11)  10 terms


    2.86208581197359013,
    0.112521343275100619,
    4.13746558898462317e-06,
    -6.20994596353326188e-07,
    6.90552926037196551e-08,
    -6.04935060629604903e-09,
    4.32877736411612667e-10,
    -2.58467890429044732e-11,
    1.30012166369174377e-12,
    -5.48935074454314017e-14,
    // piece[11][0]  [11,11.5)  10 terms

    2.97461073513390328,
    0.112528003623171155,
    2.6343049428101174e-06,
    -3.97445599707225896e-07,
    4.44815419822052742e-08,
    -3.92852117056697527e-09,
    2.84118435153283758e-10,
    -1.72102531770887146e-11,
    8.83592503561363527e-13,
    -3.84911780678135611e-14,
    // piece[11][1]  [11.5,12)  10 terms


    3.08714101643719285,
    0.112532239770794462,
    1.67349596087998688e-06,
    -2.53678224067460558e-07,
    2.85556366904238254e-08,
    -2.54025953480540056e-09,
    1.85430287466763974e-10,
    -1.13717831941978681e-11,
    5.93933175128006997e-13,
    -2.65356948790643812e-14,
    // piece[12][0]  [12,12.5)  10 terms

    3.19967470221572814,
    0.112534928286794719,
    1.06095087878931865e-06,
    -1.61516532312062951e-07,
    1.82762140776554509e-08,
    -1.63632882508747509e-09,
    1.20426431138053865e-10,
    -7.46463449337421144e-12,
    3.95568670427124281e-13,
    // piece[12][1]  [12.5,13)  9 terms


    3.31221054669009485,
    0.112536631235482659,
    6.71355374092748571e-07,
    -1.0260698063564679e-07,
    1.16652614268953189e-08,
    -1.05048153103985647e-09,
    7.78723591087180506e-11,
    -4.87211091163724825e-12,
    2.61411302348832045e-13,
    // piece[13][0]  [13,13.5)  9 terms

    3.42474775736200066,
    0.112537707969043152,
    4.24093860269819547e-07,
    -6.50501512387274965e-08,
    7.42728238587494305e-09,
    -6.72335955527862464e-10,
    5.01629845932474034e-11,
    -3.16427839314386847e-12,
    1.71609202368875719e-13,
    // piece[13][1]  [13.5,14)  9 terms


    3.53728583117686224,
    0.112538387633891027,
    2.67474264630698623e-07,
    -4.11629021690701565e-08,
    4.71835783849687606e-09,
    -4.29137743001886411e-10,
    3.2203699006397446e-11,
    -2.04619028094216497e-12,
    1.12013593330386935e-13,
    // piece[14][0]  [14,14.5)  9 terms

    3.64982444944160012,
    0.112538816001205466,
    1.68447366897287627e-07,
    -2.60025413925412271e-08,
    2.99132011808448016e-09,
    -2.73233198670367687e-10,
    2.0611349159810865e-11,
    -1.31811088613370809e-12,
    7.27513545818927849e-14,
    // piece[14][1]  [14.5,15)  9 terms


    3.76236341062508028,
    0.112539085602422748,
    1.05938738532490209e-07,
    -1.63997033545002515e-08,
    1.8928706675629797e-09,
    -1.73578203106859805e-10,
    1.31558360733834863e-11,
    -8.4620946575033824e-13,
    4.70457158812246469e-14,
    // piece[15][0]  [15,15.5)  9 terms

    3.87490258749818528,
    0.112539255057749671,
    6.65418766792231486e-08,
    -1.03280498552976667e-08,
    1.195724085827832e-09,
    -1.10044332421932017e-10,
    8.37637685471823993e-12,
    -5.41602513591422077e-13,
    3.0306138056874852e-14,
    // piece[15][1]  [15.5,16)  9 terms


    3.98744189986330522,
    0.112539361436724869,
    4.17466981216908474e-08,
    -6.49544578826046275e-09,
    7.54138844238940553e-10,
    -6.9634749431438658e-11,
    5.32130969191732652e-12,
    -3.45695707637546734e-13,
    1.94562562654015691e-14,
    // piece[16][0]  [16,16.5)  9 terms

    4.09998129724078098,
    0.11253942814182076,
    2.61619183331498868e-08,
    -4.07989748259151538e-09,
    4.74934510121557255e-10,
    -4.39882037038785869e-11,
    3.37358294536638173e-12,
    -2.20106321025381538e-13,
    // piece[16][1]  [16.5,17)  8 terms


    4.21252074789873365,
    0.112539469924557073,
    1.63782599722799829e-08,
    -2.55962510618376059e-09,
    2.98692567014619972e-10,
    -2.77432072825269204e-11,
    2.13476439456006952e-12,
    -1.3982837407044091e-13,
    // piece[17][0]  [17,17.5)  8 terms

    4.32506023191487809,
    0.112539496070145892,
    1.02434020510973438e-08,
    -1.60407189778330217e-09,
    1.87613780222204025e-10,
    -1.74718602571784203e-11,
    1.34852901354635878e-12,
    -8.86478988660644306e-14,
    // piece[17][1]  [17.5,18)  8 terms


    4.43759973679576003,
    0.112539512415339063,
    6.40064979400074166e-09,
    -1.00420415535578311e-09,
    1.177041738964949e-10,
    -1.09882839538098489e-11,
    8.50512723120311157e-13,
    -5.60953903163051322e-14,
    // piece[18][0]  [18,18.5)  8 terms

    4.55013925471505853,
    0.112539522624636193,
    3.99604370477221643e-09,
    -6.2805615810297837e-10,
    7.37631311983573458e-11,
    -6.90192468761529397e-12,
    5.35628089736570782e-13,
    -3.54354229459178567e-14,
    // piece[18][1]  [18.5,19)  8 terms


    4.66267878077504605,
    0.112539528996079177,
    2.49277859586408548e-09,
    -3.92444573567644385e-10,
    4.61782568081024709e-11,
    -4.33009435306052936e-12,
    3.3686361145354802e-13,
    -2.23490055131507845e-14,
    // piece[19][0]  [19,19.5)  8 terms

    4.77521831191362267,
    0.1125395329692397,
    1.55383097367851497e-09,
    -2.45010385735436277e-10,
    2.88811181300476271e-11,
    -2.7135938653865083e-12,
    2.11589669687634893e-13,
    -1.40747525998240837e-14,
    // piece[19][1]  [19.5,20)  8 terms


    4.8877578462180491,
    0.112539535445004169,
    9.67850256615123831e-10,
    -1.52840332132791385e-10,
    1.804653613944041e-11,
    -1.69880309936714116e-12,
    1.32746052476915737e-13,
    -8.85179909079776651e-15,
    // piece[20][0]  [20,20.5)  8 terms

    5.00029738249453537,
    0.112539536986614216,
    6.02439239906570431e-10,
    -9.52704114587431191e-11,
    1.12667854116682251e-11,
    -1.06247694785912772e-12,
    8.31896142980101458e-14,
    // piece[20][1]  [20.5,21)  7 terms


    5.11283691999860057,
    0.112539537945898882,
    3.74743528027322148e-10,
    -5.93420351611014544e-11,
    7.02835197339575264e-12,
    -6.63894958478945743e-13,
    5.20795605052830055e-14,
    // piece[21][0]  [21,21.5)  7 terms

    5.22537645826631358,
    0.112539538542443363,
    2.32961867764546614e-10,
    -3.69375017832369227e-11,
    4.38100320714432608e-12,
    -4.14481870390881206e-13,
    3.25719518754427907e-14,
    // piece[21][1]  [21.5,22)  7 terms


    5.33791599700877839,
    0.112539538913187284,
    1.44736591060607872e-10,
    -2.29767399318825261e-11,
    2.72883933830162478e-12,
    -2.58558618984872817e-13,
    2.03527754217587135e-14,
    // piece[22][0]  [22,22.5)  7 terms

    5.45045553604621524,
    0.112539539143465955,
    8.98724784116170585e-11,
    -1.42836469494209185e-11,
    1.69856843802870675e-12,
    -1.61168100916547822e-13,
    1.27066131981914459e-14,
    // piece[22][1]  [22.5,23)  7 terms


    5.56299507526681936,
    0.112539539286419035,
    5.57752110123683421e-11,
    -8.87426937079370657e-12,
    1.05658378028588191e-12,
    -1.00388388989613689e-13,
    7.92653334016851334e-15,
    // piece[23][0]  [23,23.5)  7 terms

    5.6755346146011032,
    0.1125395393751151,
    3.45965139296876247e-11,
    -5.5103728550596555e-12,
    6.56832507026454178e-13,
    -6.24868407424383042e-14,
    // piece[23][1]  [23.5,24)  6 terms


    5.78807415400590308,
    0.112539539430119351,
    2.14491393775765667e-11,
    -3.41975890946528006e-12,
    4.08082657049105931e-13,
    -3.88695450510724351e-14,
    // piece[24][0]  [24,24.5)  6 terms

    5.90061369345442355,
    0.112539539464213439,
    1.32917751033291456e-11,
    -2.12122436838489591e-12,
    2.53394110795294803e-13,
    -2.41635175578503531e-14,
    // piece[24][1]  [24.5,25)  6 terms


    6.01315323293003878,
    0.112539539485336723,
    8.23304083665380868e-12,
    -1.31511623459939627e-12,
    1.57257374559826725e-13,
    -1.50125219808287434e-14,
    // piece[25][0]  [25,25.5)  6 terms

    6.12569277242243704,
    0.112539539498418037,
    5.09741258305557398e-12,
    -8.14961799516812747e-13,
    9.75443544014850325e-14,
    -9.32184838684687099e-15,
    // piece[25][1]  [25.5,26)  6 terms


    6.23823231192522609,
    0.112539539506515671,
    3.15470757154581878e-12,
    -5.04794410349442726e-13,
    6.04755281811355735e-14,
    -5.78517550705084181e-15,
    // piece[26][0]  [26,26.5)  6 terms

    6.35077185143444733,
    0.112539539511526232,
    1.95162043667275352e-12,
    -3.12538882077047235e-13,
    3.74759614222806977e-14,
    -3.58845625856511469e-15,
    // piece[26][1]  [26.5,27)  6 terms


    6.46331139094764673,
    0.112539539514625406,
    1.20688320345070149e-12,
    -1.93425343079151048e-13,
    2.32129356980362993e-14,
    // piece[27][0]  [27,27.5)  5 terms

    6.57585093046330638,
    0.112539539516541609,
    7.46062295549186863e-13,
    -1.19660172603363165e-13,
    1.43720786602845809e-14,
    // piece[27][1]  [27.5,28)  5 terms


    6.68839046998048747,
    0.112539539517725953,
    4.61031794168080458e-13,
    -7.39979197406639551e-14,
    8.89465089001709868e-15,
    // piece[28][0]  [28,28.5)  5 terms

    6.80093000949860826,
    0.112539539518457715,
    2.8479885616495575e-13,
    -4.57434967275784204e-14,
    5.50256467846532235e-15,
    // piece[28][1]  [28.5,29)  5 terms


    6.91346954901731081,
    0.112539539518909687,
    1.75874357429501294e-13,
    -2.82673531334546561e-14,
    3.40278917025178899e-15,
    // piece[29][0]  [29,29.5)  5 terms

    7.02600908853637041,
    0.112539539519188742,
    1.08574736024292212e-13,
    -1.74619406753658459e-14,
    2.10351150166973205e-15,
    // piece[29][1]  [29.5,30)  5 terms


    7.13854862805565293,
    0.112539539519361007,
    6.70072522103341212e-14,
    -1.07834267212563974e-14,
    // piece[30][0]  [30,30.5)  4 terms

    7.25108816757507135,
    0.112539539519467297,
    4.13414925633789057e-14,
    -6.65706886899143278e-15,
    // piece[30][1]  [30.5,31)  4 terms


    7.36362770709457415,
    0.112539539519532869,
    2.549917501837173e-14,
    -4.10843174959496338e-15,
    // piece[31][0]  [31,31.5)  4 terms

    7.47616724661412846,
    0.112539539519573309,
    1.57233777106330427e-14,
    -2.53478086356808097e-15,
    // piece[31][1]  [31.5,32)  4 terms


    7.58870678613371474,
    0.112539539519598233,
    9.69279905539110562e-15,
    -1.56343704268185646e-15,
    // piece[32][0]  [32,32.5)  4 terms

    7.70124632565332146,
    0.11253953951961361,
    5.97365220554157505e-15,
    -9.64051234252516753e-16,
    // piece[32][1]  [32.5,33)  4 terms


    7.8137858651729406,
    0.112539539519623075,
    3.68062492635757454e-15,
    // piece[33][0]  [33,33.5)  3 terms

    7.92632540469256686,
    0.112539539519628917,
    2.2672398943496648e-15,
    // piece[33][1]  [33.5,34)  3 terms


    8.03886494421219666,
    0.112539539519632498,
    1.3962747766233044e-15,
    // piece[34][0]  [34,34.5)  3 terms

    8.15140448373183091,
    0.112539539519634718,
    8.59695977475931737e-16,
    // piece[34][1]  [34.5,35)  3 terms


    8.26394402325146693,
    0.112539539519636078,
    5.29203152195586983e-16,
    // piece[35][0]  [35,35.5)  3 terms

    8.37648356277110295,
    0.112539539519636925,
    // piece[35][1]  [35.5,36)  2 terms


    8.48902310229074075,
    0.112539539519637438,
    // piece[36][0]  [36,36.5)  2 terms

    8.60156264181037855,
    0.112539539519637757,
    // piece[36][1]  [36.5,37)  2 terms


    8.71410218133001635,
    0.112539539519637952,
    // piece[37][0]  [37,37.5)  2 terms

    8.82664172084965415,
    0.112539539519638063,
    // piece[37][1]  [37.5,38)  2 terms


    8.93918126036929195,
    0.112539539519638146,
    // piece[38][0]  [38,38.5)  2 terms

    9.05172079988892975,
    0.112539539519638188,
    // piece[38][1]  [38.5,39)  2 terms


    9.16426033940856755,
    0.112539539519638215,
    // piece[39][0]  [39,39.5)  2 terms

    9.27679987892820712,
    0.112539539519638229,
    // piece[39][1]  [39.5,40)  2 terms


};  // coefTh1c

static int *nTermsTh1c[40];
static int allTermsTh1c[] = {
    13,
    12,
    12,
    12,
    12,
    11,
    11,
    11,
    11,
    11,
    11,
    10,
    10,
    10,
    10,
    10,

    12,
    12,
    11,
    11,
    11,
    11,
    11,
    10,

    12,
    12,
    12,
    11,

    11,
    11,
    11,
    11,

    13,
    12,

    12,
    11,

    11,
    11,

    10,
    10,

    10,
    10,

    10,
    10,

    10,
    10,

    10,
    10,

    10,
    9,

    9,
    9,

    9,
    9,

    9,
    9,

    9,
    8,

    8,
    8,

    8,
    8,

    8,
    8,

    8,
    7,

    7,
    7,

    7,
    7,

    7,
    6,

    6,
    6,

    6,
    6,

    6,
    6,

    5,
    5,

    5,
    5,

    5,
    5,

    4,
    4,

    4,
    4,

    4,
    4,

    3,
    3,

    3,
    3,

    3,
    2,

    2,
    2,

    2,
    2,

    2,
    2,

    2,
    2,

}; // allTermsTh1c

static int nPiecesTh1c[40] = {
    16,
    8,
    4,
    4,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
    2,
};  // nPiecesTh1c

static bool isInitTh1c = false;
static int finishTh1c = 40;

void InitTh1c()
{
    for (int iInt = 0, walkTerm = 0, walkCoef = 0;
         iInt < finishTh1c;
         walkTerm += nPiecesTh1c[iInt], ++iInt)
    {
        nTermsTh1c[iInt] = &allTermsTh1c[walkTerm];
        coefTh1c[iInt] = new double*[nPiecesTh1c[iInt]];
        for (int iPiece = 0;  iPiece < nPiecesTh1c[iInt];  walkCoef += nTermsTh1c[iInt][iPiece], ++iPiece)
        {
            coefTh1c[iInt][iPiece] = &allCoefTh1c[walkCoef];
        }
    }
    isInitTh1c = true;
}

double Th1c(double x)
{
    if (!isInitTh1c)
        InitTh1c();

    // Transform the domain to approach pole at 1 slower.
    if (x < 0)
        return nan("");
    x = -log(1-x);
    if (x < 0 || x >= finishTh1c)
        return nan("");
    int iInt = static_cast<int>(x);
    x -= iInt;

    // The 0.5 adjusts the range to [-0.5,0.5) instead of [0,1), so c[0] is the middle value.
    // The number of pieces per int was adjusted manually.
    x *= nPiecesTh1c[iInt];
    int iPiece = static_cast<int>(x);
    x = x - iPiece - 0.5;

    // start with the tiny terms and build up, to avoid rounding errors
    double *c = coefTh1c[iInt][iPiece];
    int iTerm = nTermsTh1c[iInt][iPiece]-1;
    double y = c[iTerm];
    while (iTerm--)
        y = y * x + c[iTerm];
    return y;
}


// find the attraction of a ring at (nearD,0) to a ring at (farD,z)
Point Attraction(double nearD, double farD, double z)
{
    // scale farD to distance 1.0
    double x2 = nearD/farD;
    double x = sqrt(x2); // distance to the far point if you scale both so the near point is a ring with x=1
    double z2 = z*z/farD; // z position, scaled
    double x2z21 = x2+z2+1; // x*x + z*z + 1, scaled
    double x2z212 = sqrt(x2z21);
    double m = 2*x/x2z21; // argument to the fancy functions
    double th1m = Th1(m);  // integral t=0..2pi of (1-mcos(t))^(-3/2)
    double th1cm = Th1c(m); // integral t=0..2pi of (1-cos(t))*(1-mcos(t))^(-3/2)
    double c = 1/(farD*x2z212*x2z21); // coefficient for the fancy functions

    Point a;
    double dist = sqrt(farD);
    a._x = c*(th1cm+(x-1)*th1m); // acceleration in the x direction (will need to rotate toward actual point)
    a._y = 0; // acceleration in the y direction
    a._z = c*z*th1m/dist; // acceleration in the z direction
    return a;
}

// Calculate the gravitational attraction of a ring through r1 centered on x==0 y==0 on point r2
Point RingAttraction(const Point& r, const Point& p)
{
    Point a;
    double rd = r._x*r._x + r._y*r._y;
    double pd = p._x*p._x + p._y*p._y;
    double z = p._z - r._z;
    if (rd == 0.0)
    {
        // special case, ring is actually a point
        double dist2 = (z*z + pd);
        double dist3 = dist2*sqrt(dist2);
        a._x = -p._x/dist3;
        a._y = -p._y/dist3;
        a._z = -z/dist3;
    }
    else if (rd == pd && r._z == p._z)
    {
        // special case, point is in the ring, the force is a logarithm of the number of moons
        // assume there were 8192 moons
        double dist2 = (z*z + pd);
        double dist3 = dist2*sqrt(dist2);
        double f1024 = -1.12317308649484082;
        double f2048 = -1.23349090217562418;
        double f4096 = -1.34380870615306458;
        double f8192 = -1.45412650720466918;
        double f16384 = -1.56444430752481445;
        a._x = f8192*p._x/dist3;
        a._y = f8192*p._y/dist3;
        a._z = 0.0;
    }
    else
    {
        // scale ring to radius 1
        a = Attraction(pd, rd, z);
        double dist = sqrt(pd);
        a._y = -a._x*p._y/dist;
        a._x = -a._x*p._x/dist;
        a._z = -a._z;
    }
    return a;
}