Notebooks · lib/nprandom/ziggurat_tables

The ziggurat tables of NumPy's normal draws

Generated by lib/nprandom/make_tables.py from numpy/random/src/distributions/ ziggurat_constants.h, NumPy v1.24.4 (commit 9315a9072b2636f75c831b4eca9f42a5f67ca2fb). Do not edit; rerun the script. npr_zig_ki is ki_double (uint64 values below 2^52), npr_zig_wi is wi_double, npr_zig_fi is fi_double; each double is the shortest decimal that reads back as the header's double.

Used by lib/nprandom/generator.

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[1]lines 14–260

NPR_ZIG_R = 3.654152885361009
NPR_ZIG_INV_R = 0.2736612373297583

npr_zig_ki = [
    4208095142473578, 0, 3387314423973544, 3838760076542274,
    4030768804392682, 4136731738896254, 4203757248105145, 4249917568205994,
    4283617341590296, 4309289223136604, 4329489775174550, 4345795907393188,
    4359232558744730, 4370494503737299, 4380069246215646, 4388308869042394,
    4395473957549321, 4401761481783924, 4407323076021240, 4412277362218204,
    4416718463613199, 4420722014516422, 4424349484777079, 4427651345409294,
    4430669422005229, 4433438668975191, 4435988524278344, 4438343955930065,
    4440526279077425, 4442553800234660, 4444442329865861, 4446205593658138,
    4447855565093316, 4449402736340121, 4450856340408624, 4452224534496486,
    4453514552210512, 4454732830656798, 4455885117109368, 4456976558985043,
    4458011780094444, 4458994945550386, 4459929817254120, 4460819801517196,
    4461667990089170, 4462477195632268, 4463249982500384, 4463988693531856,
    4464695473445501, 4465372289331869, 4466020948651920, 4466643115089764,
    4467240322552142, 4467813987562542, 4468365420260672, 4468895834186994,
    4469406355006040, 4469898028300364, 4470371826548633, 4470828655385770,
    4471269359229841, 4471694726349190, 4472105493433674, 4472502349725738,
    4472885940759935, 4473256871753524, 4473615710685532, 4473962991097124,
    4474299214642296, 4474624853414418, 4474940352071305, 4475246129778808,
    4475542581990776, 4475830082081194, 4476108982842610, 4476379617863426,
    4476642302795321, 4476897336520866, 4477145002230339, 4477385568415884,
    4477619289790266, 4477846408136804, 4478067153096380, 4478281742896886,
    4478490385029917, 4478693276879082, 4478890606303906, 4479082552182886,
    4479269284918997, 4479450966910588, 4479627752990372, 4479799790834988,
    4479967221347354, 4480130179013872, 4480288792238368, 4480443183654460,
    4480593470417939, 4480739764480586, 4480882172846772, 4481020797814010,
    4481155737198612, 4481287084547452, 4481414929336784, 4481539357158974,
    4481660449897960, 4481778285894165, 4481892940099539, 4482004484223382,
    4482112986869492, 4482218513665204, 4482321127382802, 4482420888053758,
    4482517853076245, 4482612077316275, 4482703613202871, 4482792510817576,
    4482878817978627, 4482962580320076, 4483043841366126, 4483122642600925,
    4483199023534056, 4483273021761922, 4483344673025224, 4483414011262724,
    4483481068661428, 4483545875703378, 4483608461209170, 4483668852378323,
    4483727074826624, 4483783152620564, 4483837108308932, 4483888962951686,
    4483938736146144, 4483986446050596, 4484032109405372, 4484075741551420,
    4484117356446452, 4484156966678662, 4484194583478081, 4484230216725550,
    4484263874959345, 4484295565379450, 4484325293849474, 4484353064896186,
    4484378881706674, 4484402746123075, 4484424658634833, 4484444618368474,
    4484462623074794, 4484478669113436, 4484492751434740, 4484504863558830,
    4484514997551788, 4484523143998833, 4484529291974394, 4484533429008906,
    4484535541052219, 4484535612433424, 4484533625816926, 4484529562154580,
    4484523400633636, 4484515118620291, 4484504691598554, 4484492093104164,
    4484477294653230, 4484460265665252, 4484440973380154, 4484419382768918,
    4484395456437370, 4484369154522621, 4484340434581640, 4484309251471359,
    4484275557219678, 4484239300886654, 4484200428415112, 4484158882469814,
    4484114602264271, 4484067523374160, 4484017577536216, 4483964692431365,
    4483908791450714, 4483849793442887, 4483787612441036, 4483722157367660,
    4483653331715198, 4483581033200083, 4483505153387764, 4483425577285833,
    4483342182902157, 4483254840764470, 4483163413397547, 4483067754753536,
    4482967709590562, 4482863112794072, 4482753788634692, 4482639549955636,
    4482520197281720, 4482395517841076, 4482265284489409, 4482129254525304,
    4481987168383486, 4481838748191074, 4481683696169781, 4481521692864464,
    4481352395175570, 4481175434169564, 4480990412637506, 4480796902367134,
    4480594441088331, 4480382529045225, 4480160625140311, 4479928142586662,
    4479684443993061, 4479428835793398, 4479160561915451, 4478878796564388,
    4478582635972392, 4478271088936406, 4477943065929958, 4477597366530538,
    4477232664848704, 4476847492576192, 4476440219183781, 4476009028690434,
    4475551892286424, 4475066535915646, 4474550401693506, 4474000601739904,
    4473413862618200, 4472786458058295, 4472114126959004, 4471391972746494,
    4470614338917719, 4469774653883156, 4468865235838896, 4467877045039530,
    4466799366045354, 4465619395558397, 4464321701199635, 4462887501169282,
    4461293691124341, 4459511507635972, 4457504658253067, 4455226650325010,
    4452616884242348, 4449594783440798, 4446050695647666, 4441831266659618,
    4436714892174061, 4430368316897338, 4422264825074740, 4411517007702132,
    4396496531309976, 4373832704204284, 4335125104963628, 4251099761679434
]

npr_zig_wi = [
    8.683627060801306e-16, 4.779330175727737e-17, 6.354352417405262e-17,
    7.454870481247696e-17, 8.3293668157931e-17, 9.068060405059482e-17,
    9.714860076567762e-17, 1.0294750314241019e-16, 1.0823430288447684e-16,
    1.131147019610903e-16, 1.176635945702292e-16, 1.2193617278714363e-16,
    1.2597439914637093e-16, 1.2981099886264032e-16, 1.3347203736824123e-16,
    1.3697864842571203e-16, 1.4034823001242382e-16, 1.4359529452056943e-16,
    1.4673208742364422e-16, 1.4976904668391037e-16, 1.5271515003596198e-16,
    1.5557818169460764e-16, 1.5836494009290885e-16, 1.6108140175274928e-16,
    1.6373285203969853e-16, 1.6632399058420835e-16, 1.6885901708676596e-16,
    1.713417017655966e-16, 1.737754436586486e-16, 1.7616331923000996e-16,
    1.7850812316976727e-16, 1.8081240285799152e-16, 1.830784876482675e-16,
    1.853085138861802e-16, 1.8750444639373882e-16, 1.896680970077476e-16,
    1.918011406483862e-16, 1.9390512930625104e-16, 1.9598150426628824e-16,
    1.9803160683128174e-16, 2.000566877627333e-16, 2.0205791562071654e-16,
    2.0403638415480212e-16, 2.0599311887403706e-16, 2.079290829041402e-16,
    2.0984518222370352e-16, 2.1174227035760342e-16, 2.1362115259449868e-16,
    2.1548258978581458e-16, 2.1732730177564367e-16, 2.191559705042727e-16,
    2.2096924282235318e-16, 2.2276773304789553e-16, 2.2455202529414355e-16,
    2.263226755928568e-16, 2.280802138345017e-16, 2.2982514554424684e-16,
    2.3155795351040804e-16, 2.3327909928004356e-16, 2.3498902453470955e-16,
    2.3668815235791604e-16, 2.3837688840454243e-16, 2.4005562198135063e-16,
    2.4172472704675025e-16, 2.433845631371103e-16, 2.4503547622614954e-16,
    2.466777995232705e-16, 2.4831185421610877e-16, 2.4993795016204524e-16,
    2.515563865329658e-16, 2.5316745241713583e-16, 2.547714273816944e-16,
    2.563685819989397e-16, 2.579591783392867e-16, 2.5954347043351707e-16,
    2.6112170470670194e-16, 2.6269412038597256e-16, 2.6426094988411895e-16,
    2.658224191608307e-16, 2.6737874806323633e-16, 2.689301506472616e-16,
    2.704768354811995e-16, 2.720190059327732e-16, 2.735568604408679e-16,
    2.7509059277301666e-16, 2.7662039226963903e-16, 2.781464440759544e-16,
    2.79668929362423e-16, 2.8118802553450207e-16, 2.827039064324479e-16,
    2.842167425218406e-16, 2.8572670107546015e-16, 2.87233946347098e-16,
    2.887386397378482e-16, 2.9024093995538423e-16, 2.9174100316669455e-16,
    2.9323898314471816e-16, 2.947350314092935e-16, 2.9622929736280665e-16,
    2.977219284209029e-16, 2.992130701386013e-16, 3.007028663321331e-16,
    3.0219145919680615e-16, 3.036789894211802e-16, 3.051655962978219e-16,
    3.0665141783089545e-16, 3.081365908408297e-16, 3.0962125106629225e-16,
    3.111055332636893e-16, 3.125895713043999e-16, 3.140734982699446e-16,
    3.1555744654528006e-16, 3.1704154791040285e-16, 3.1852593363044065e-16,
    3.2001073454440114e-16, 3.214960811527447e-16, 3.2298210370394156e-16,
    3.244689322801698e-16, 3.2595669688230784e-16, 3.2744552751437067e-16,
    3.2893555426753697e-16, 3.3042690740391284e-16, 3.3191971744017523e-16,
    3.3341411523123725e-16, 3.3491023205407785e-16, 3.364081996918765e-16,
    3.37908150518595e-16, 3.394102175841489e-16, 3.409145347003126e-16,
    3.424212365275018e-16, 3.4393045866258313e-16, 3.454423377278584e-16,
    3.4695701146137835e-16, 3.4847461880874137e-16, 3.499953000165381e-16,
    3.5151919672760744e-16, 3.53046452078274e-16, 3.5457721079774357e-16,
    3.5611161930983884e-16, 3.5764982583726505e-16, 3.59191980508603e-16,
    3.6073823546823514e-16, 3.6228874498941915e-16, 3.6384366559073444e-16,
    3.65403156156137e-16, 3.669673780588701e-16, 3.685364952894914e-16,
    3.7011067458828983e-16, 3.716900855823823e-16, 3.7327490092779435e-16,
    3.7486529645684887e-16, 3.7646145133120287e-16, 3.7806354820089604e-16,
    3.7967177336979443e-16, 3.8128631696783774e-16, 3.829073731305243e-16,
    3.8453514018609596e-16, 3.8616982085091493e-16, 3.878116224335587e-16,
    3.894607570481926e-16, 3.9111744183782054e-16, 3.9278189920805415e-16,
    3.944543570720877e-16, 3.9613504910761354e-16, 3.9782421502646826e-16,
    3.995221008578565e-16, 4.012289592460629e-16, 4.029450497636328e-16,
    4.04670639241075e-16, 4.0640600211422504e-16, 4.0815142079049387e-16,
    4.0990718603532664e-16, 4.1167359738030257e-16, 4.134509635544236e-16,
    4.1523960294026883e-16, 4.170398440568316e-16, 4.1885202607101123e-16,
    4.206764993399015e-16, 4.2251362598620494e-16, 4.243637805093078e-16,
    4.262273504347798e-16, 4.2810473700531167e-16, 4.2999635591638323e-16,
    4.3190263810026294e-16, 4.338240305622791e-16, 4.357609972736849e-16,
    4.3771402012585875e-16, 4.3968359995105214e-16, 4.4167025761542035e-16,
    4.4367453519065673e-16, 4.456969972112043e-16, 4.477382320247534e-16,
    4.49798853244555e-16, 4.518795013130059e-16, 4.539808451870034e-16,
    4.561035841567422e-16, 4.582484498109567e-16, 4.604162081631153e-16,
    4.626076619547846e-16, 4.648236531543207e-16, 4.670650656712631e-16,
    4.693328283093329e-16, 4.716279179838351e-16, 4.739513632325867e-16,
    4.763042480533137e-16, 4.786877161048723e-16, 4.811029753147417e-16,
    4.835513029411525e-16, 4.860340511450812e-16, 4.885526531353603e-16,
    4.91108629959527e-16, 4.937035980240335e-16, 4.963392774403987e-16,
    4.990175013091822e-16, 5.017402260718089e-16, 5.045095430818727e-16,
    5.073276915733542e-16, 5.101970732341562e-16, 5.131202686306784e-16,
    5.161000557743228e-16, 5.191394311757699e-16, 5.222416338000234e-16,
    5.254101724177597e-16, 5.286488569504945e-16, 5.3196183453384e-16,
    5.353536311816497e-16, 5.388292001334053e-16, 5.423939782201712e-16,
    5.46053951907478e-16, 5.498157350892814e-16, 5.536866612467876e-16,
    5.576748932926576e-16, 5.617895553555417e-16, 5.660408920082422e-16,
    5.704404621291389e-16, 5.750013768919895e-16, 5.797385945724594e-16,
    5.846692893455479e-16, 5.898133176477899e-16, 5.951938149641444e-16,
    6.008379696271908e-16, 6.067780409333449e-16, 6.130527208725282e-16,
    6.197089894581626e-16, 6.268046963301284e-16, 6.344122407127506e-16,
    6.426239659548055e-16, 6.515603317344994e-16, 6.613827885097664e-16,
    6.723150462505587e-16, 6.846803417564259e-16, 6.98971833638762e-16,
    7.159994934830664e-16, 7.372424301798799e-16, 7.658936370805573e-16,
    8.113849337656484e-16
]

npr_zig_fi = [
    1.0, 0.9771017012676716, 0.9598790918001067,
    0.9451989534422996, 0.9320600759592305, 0.919991505039347,
    0.9087264400521309, 0.8980959218983434, 0.8879846607558334,
    0.8783096558089174, 0.869008688036857, 0.8600336211963315,
    0.851346258458678, 0.8429156531122042, 0.8347162929868834,
    0.8267268339462214, 0.8189291916037024, 0.8113078743126563,
    0.8038494831709643, 0.796542330422959, 0.7893761435660246,
    0.7823418326548025, 0.7754313049811872, 0.7686373157984863,
    0.7619533468367954, 0.7553735065070961, 0.7488924472191568,
    0.742505296340151, 0.7362075981268627, 0.7299952645614762,
    0.7238645334686302, 0.717811932630722, 0.7118342488782484,
    0.7059285013327543, 0.7000919181365116, 0.6943219161261167,
    0.6886160830046718, 0.6829721616449949, 0.6773880362187735,
    0.6718617198970821, 0.6663913439087501, 0.6609751477766631,
    0.6556114705796973, 0.6502987431108167, 0.6450354808208223,
    0.6398202774530566, 0.6346517992876236, 0.6295287799248367,
    0.6244500155470265, 0.6194143606058343, 0.6144207238889139,
    0.6094680649257734, 0.6045553906974678, 0.5996817526191253,
    0.5948462437679874, 0.590047996332826, 0.5852861792633715,
    0.5805599961007909, 0.5758686829723537, 0.5712115067352532,
    0.5665877632561644, 0.5619967758145243, 0.557437893618766,
    0.5529104904258323, 0.5484139632552658, 0.5439477311900263,
    0.5395112342569521, 0.5351039323804576, 0.5307253044036621,
    0.5263748471716845, 0.5220520746723218, 0.5177565172297564,
    0.513487720747327, 0.5092452459957479, 0.5050286679434681,
    0.5008375751261487, 0.4966715690524897, 0.49253026364386854,
    0.48841328470545803, 0.4843202694266833, 0.48025086590904675,
    0.47620473271950586, 0.4721815384677302, 0.4681809614056936,
    0.46420268904817436, 0.46024641781284287, 0.45631185267871643,
    0.4523987068618485, 0.44850670150720306, 0.4446355653957394,
    0.440785034665804, 0.43695485254798555, 0.43314476911265226,
    0.4293545410294414, 0.42558393133802197, 0.4218327092294959,
    0.4181006498378482, 0.4143875340408911, 0.41069314827018816,
    0.40701728432947337, 0.4033597392211145, 0.3997203149801972,
    0.39609881851583245, 0.3924950614593156, 0.3889088600187887,
    0.3853400348400773, 0.38178841087339366, 0.3782538172456192,
    0.37473608713789114, 0.3712350576682395, 0.3677505697790326,
    0.36428246812900406, 0.36083060098964803, 0.3573948201457805,
    0.3539749808000768, 0.3505709414814061, 0.34718256395679364,
    0.3438097131468507, 0.34045225704452187, 0.33711006663700605,
    0.33378301583071845, 0.3304709813791636, 0.3271738428136014,
    0.3238914823763911, 0.32062378495690536, 0.3173706380299136,
    0.3141319315963372, 0.3109075581262865, 0.30769741250429206,
    0.30450139197665, 0.30131939610080305, 0.2981513266966855,
    0.2949970877999618, 0.2918565856170952, 0.2887297284821829,
    0.28561642681550176, 0.2825165930837076, 0.27943014176163794,
    0.2763569892956683, 0.27329705406857707, 0.27025025636587546,
    0.26721651834356147, 0.2641957639972612, 0.2611879191327212,
    0.25819291133761924, 0.25521066995466196, 0.2522411260559422,
    0.24928421241852852, 0.24633986350126383, 0.2434080154227503,
    0.2404886059405006, 0.2375815744312381, 0.23468686187233,
    0.23180441082433872, 0.22893416541468034, 0.22607607132238028,
    0.22323007576391748, 0.220396127480152, 0.21757417672433113,
    0.21476417525117358, 0.21196607630703018, 0.20917983462112508,
    0.2064054063978808, 0.2036427493103349, 0.2008918224946566,
    0.19815258654577514, 0.1954250035141343, 0.19270903690358918,
    0.19000465167046499, 0.1873118142238003, 0.18463049242679927,
    0.18196065559952251, 0.17930227452284758, 0.17665532144373486,
    0.17401977008183855, 0.17139559563750575, 0.1687827748012113,
    0.1661812857644819, 0.16359110823236558, 0.161012223437511,
    0.15844461415592428, 0.1558882647244792, 0.15334316106026286,
    0.15080929068184568, 0.14828664273257455, 0.14577520800599403,
    0.14327497897351346, 0.1407859498144447, 0.13830811644855073,
    0.13584147657125376, 0.13338602969166916, 0.13094177717364436,
    0.12850872227999957, 0.1260868702201859, 0.12367622820159657,
    0.1212768054847903, 0.11888861344291006, 0.11651166562561087,
    0.11414597782783849, 0.11179156816383809, 0.1094484571468118,
    0.1071166677746838, 0.10479622562248707, 0.10248715894193525,
    0.10018949876881002, 0.09790327903886246, 0.095628536713009,
    0.09336531191269101, 0.09111364806637376, 0.08887359206827589,
    0.08664519445055807, 0.08442850957035347, 0.0822235958132029,
    0.08003051581466307, 0.07784933670209612, 0.07568013035892718,
    0.07352297371398132, 0.0713779490588904, 0.06924514439700676,
    0.0671246538277885, 0.0650165779712429, 0.06292102443775814,
    0.06083810834953988, 0.05876795292093374, 0.0567106901062029,
    0.05466646132488892, 0.05263541827679219, 0.05061772386094778,
    0.04861355321586854, 0.04662309490193038, 0.044646552251294463,
    0.04268414491647446, 0.04073611065594094, 0.03880270740452615,
    0.036884215688567305, 0.034980941461716125, 0.03309321945857858,
    0.0312214171919203, 0.02936593975813336, 0.027527235669603113,
    0.02570580400854891, 0.02390220330579588, 0.02211706270730885,
    0.02035109623004451, 0.018605121275724622, 0.016880083152543142,
    0.01517708830793531, 0.013497450601739867, 0.011842757857907879,
    0.010214971439701459, 0.008616582769398726, 0.007050875471373222,
    0.0055224032992509916, 0.0040379725933630236, 0.0026090727461021593,
    0.001260285930498598
]

NPR_ZIG_R = 3.654152885361009
NPR_ZIG_INV_R = 0.2736612373297583

npr_zig_ki = [
    4208095142473578, 0, 3387314423973544, 3838760076542274,
    4030768804392682, 4136731738896254, 4203757248105145, 4249917568205994,
    4283617341590296, 4309289223136604, 4329489775174550, 4345795907393188,
    4359232558744730, 4370494503737299, 4380069246215646, 4388308869042394,
    4395473957549321, 4401761481783924, 4407323076021240, 4412277362218204,
    4416718463613199, 4420722014516422, 4424349484777079, 4427651345409294,
    4430669422005229, 4433438668975191, 4435988524278344, 4438343955930065,
    4440526279077425, 4442553800234660, 4444442329865861, 4446205593658138,
    4447855565093316, 4449402736340121, 4450856340408624, 4452224534496486,
    4453514552210512, 4454732830656798, 4455885117109368, 4456976558985043,
    4458011780094444, 4458994945550386, 4459929817254120, 4460819801517196,
    4461667990089170, 4462477195632268, 4463249982500384, 4463988693531856,
    4464695473445501, 4465372289331869, 4466020948651920, 4466643115089764,
    4467240322552142, 4467813987562542, 4468365420260672, 4468895834186994,
    4469406355006040, 4469898028300364, 4470371826548633, 4470828655385770,
    4471269359229841, 4471694726349190, 4472105493433674, 4472502349725738,
    4472885940759935, 4473256871753524, 4473615710685532, 4473962991097124,
    4474299214642296, 4474624853414418, 4474940352071305, 4475246129778808,
    4475542581990776, 4475830082081194, 4476108982842610, 4476379617863426,
    4476642302795321, 4476897336520866, 4477145002230339, 4477385568415884,
    4477619289790266, 4477846408136804, 4478067153096380, 4478281742896886,
    4478490385029917, 4478693276879082, 4478890606303906, 4479082552182886,
    4479269284918997, 4479450966910588, 4479627752990372, 4479799790834988,
    4479967221347354, 4480130179013872, 4480288792238368, 4480443183654460,
    4480593470417939, 4480739764480586, 4480882172846772, 4481020797814010,
    4481155737198612, 4481287084547452, 4481414929336784, 4481539357158974,
    4481660449897960, 4481778285894165, 4481892940099539, 4482004484223382,
    4482112986869492, 4482218513665204, 4482321127382802, 4482420888053758,
    4482517853076245, 4482612077316275, 4482703613202871, 4482792510817576,
    4482878817978627, 4482962580320076, 4483043841366126, 4483122642600925,
    4483199023534056, 4483273021761922, 4483344673025224, 4483414011262724,
    4483481068661428, 4483545875703378, 4483608461209170, 4483668852378323,
    4483727074826624, 4483783152620564, 4483837108308932, 4483888962951686,
    4483938736146144, 4483986446050596, 4484032109405372, 4484075741551420,
    4484117356446452, 4484156966678662, 4484194583478081, 4484230216725550,
    4484263874959345, 4484295565379450, 4484325293849474, 4484353064896186,
    4484378881706674, 4484402746123075, 4484424658634833, 4484444618368474,
    4484462623074794, 4484478669113436, 4484492751434740, 4484504863558830,
    4484514997551788, 4484523143998833, 4484529291974394, 4484533429008906,
    4484535541052219, 4484535612433424, 4484533625816926, 4484529562154580,
    4484523400633636, 4484515118620291, 4484504691598554, 4484492093104164,
    4484477294653230, 4484460265665252, 4484440973380154, 4484419382768918,
    4484395456437370, 4484369154522621, 4484340434581640, 4484309251471359,
    4484275557219678, 4484239300886654, 4484200428415112, 4484158882469814,
    4484114602264271, 4484067523374160, 4484017577536216, 4483964692431365,
    4483908791450714, 4483849793442887, 4483787612441036, 4483722157367660,
    4483653331715198, 4483581033200083, 4483505153387764, 4483425577285833,
    4483342182902157, 4483254840764470, 4483163413397547, 4483067754753536,
    4482967709590562, 4482863112794072, 4482753788634692, 4482639549955636,
    4482520197281720, 4482395517841076, 4482265284489409, 4482129254525304,
    4481987168383486, 4481838748191074, 4481683696169781, 4481521692864464,
    4481352395175570, 4481175434169564, 4480990412637506, 4480796902367134,
    4480594441088331, 4480382529045225, 4480160625140311, 4479928142586662,
    4479684443993061, 4479428835793398, 4479160561915451, 4478878796564388,
    4478582635972392, 4478271088936406, 4477943065929958, 4477597366530538,
    4477232664848704, 4476847492576192, 4476440219183781, 4476009028690434,
    4475551892286424, 4475066535915646, 4474550401693506, 4474000601739904,
    4473413862618200, 4472786458058295, 4472114126959004, 4471391972746494,
    4470614338917719, 4469774653883156, 4468865235838896, 4467877045039530,
    4466799366045354, 4465619395558397, 4464321701199635, 4462887501169282,
    4461293691124341, 4459511507635972, 4457504658253067, 4455226650325010,
    4452616884242348, 4449594783440798, 4446050695647666, 4441831266659618,
    4436714892174061, 4430368316897338, 4422264825074740, 4411517007702132,
    4396496531309976, 4373832704204284, 4335125104963628, 4251099761679434
]

npr_zig_wi = [
    8.683627060801306e-16, 4.779330175727737e-17, 6.354352417405262e-17,
    7.454870481247696e-17, 8.3293668157931e-17, 9.068060405059482e-17,
    9.714860076567762e-17, 1.0294750314241019e-16, 1.0823430288447684e-16,
    1.131147019610903e-16, 1.176635945702292e-16, 1.2193617278714363e-16,
    1.2597439914637093e-16, 1.2981099886264032e-16, 1.3347203736824123e-16,
    1.3697864842571203e-16, 1.4034823001242382e-16, 1.4359529452056943e-16,
    1.4673208742364422e-16, 1.4976904668391037e-16, 1.5271515003596198e-16,
    1.5557818169460764e-16, 1.5836494009290885e-16, 1.6108140175274928e-16,
    1.6373285203969853e-16, 1.6632399058420835e-16, 1.6885901708676596e-16,
    1.713417017655966e-16, 1.737754436586486e-16, 1.7616331923000996e-16,
    1.7850812316976727e-16, 1.8081240285799152e-16, 1.830784876482675e-16,
    1.853085138861802e-16, 1.8750444639373882e-16, 1.896680970077476e-16,
    1.918011406483862e-16, 1.9390512930625104e-16, 1.9598150426628824e-16,
    1.9803160683128174e-16, 2.000566877627333e-16, 2.0205791562071654e-16,
    2.0403638415480212e-16, 2.0599311887403706e-16, 2.079290829041402e-16,
    2.0984518222370352e-16, 2.1174227035760342e-16, 2.1362115259449868e-16,
    2.1548258978581458e-16, 2.1732730177564367e-16, 2.191559705042727e-16,
    2.2096924282235318e-16, 2.2276773304789553e-16, 2.2455202529414355e-16,
    2.263226755928568e-16, 2.280802138345017e-16, 2.2982514554424684e-16,
    2.3155795351040804e-16, 2.3327909928004356e-16, 2.3498902453470955e-16,
    2.3668815235791604e-16, 2.3837688840454243e-16, 2.4005562198135063e-16,
    2.4172472704675025e-16, 2.433845631371103e-16, 2.4503547622614954e-16,
    2.466777995232705e-16, 2.4831185421610877e-16, 2.4993795016204524e-16,
    2.515563865329658e-16, 2.5316745241713583e-16, 2.547714273816944e-16,
    2.563685819989397e-16, 2.579591783392867e-16, 2.5954347043351707e-16,
    2.6112170470670194e-16, 2.6269412038597256e-16, 2.6426094988411895e-16,
    2.658224191608307e-16, 2.6737874806323633e-16, 2.689301506472616e-16,
    2.704768354811995e-16, 2.720190059327732e-16, 2.735568604408679e-16,
    2.7509059277301666e-16, 2.7662039226963903e-16, 2.781464440759544e-16,
    2.79668929362423e-16, 2.8118802553450207e-16, 2.827039064324479e-16,
    2.842167425218406e-16, 2.8572670107546015e-16, 2.87233946347098e-16,
    2.887386397378482e-16, 2.9024093995538423e-16, 2.9174100316669455e-16,
    2.9323898314471816e-16, 2.947350314092935e-16, 2.9622929736280665e-16,
    2.977219284209029e-16, 2.992130701386013e-16, 3.007028663321331e-16,
    3.0219145919680615e-16, 3.036789894211802e-16, 3.051655962978219e-16,
    3.0665141783089545e-16, 3.081365908408297e-16, 3.0962125106629225e-16,
    3.111055332636893e-16, 3.125895713043999e-16, 3.140734982699446e-16,
    3.1555744654528006e-16, 3.1704154791040285e-16, 3.1852593363044065e-16,
    3.2001073454440114e-16, 3.214960811527447e-16, 3.2298210370394156e-16,
    3.244689322801698e-16, 3.2595669688230784e-16, 3.2744552751437067e-16,
    3.2893555426753697e-16, 3.3042690740391284e-16, 3.3191971744017523e-16,
    3.3341411523123725e-16, 3.3491023205407785e-16, 3.364081996918765e-16,
    3.37908150518595e-16, 3.394102175841489e-16, 3.409145347003126e-16,
    3.424212365275018e-16, 3.4393045866258313e-16, 3.454423377278584e-16,
    3.4695701146137835e-16, 3.4847461880874137e-16, 3.499953000165381e-16,
    3.5151919672760744e-16, 3.53046452078274e-16, 3.5457721079774357e-16,
    3.5611161930983884e-16, 3.5764982583726505e-16, 3.59191980508603e-16,
    3.6073823546823514e-16, 3.6228874498941915e-16, 3.6384366559073444e-16,
    3.65403156156137e-16, 3.669673780588701e-16, 3.685364952894914e-16,
    3.7011067458828983e-16, 3.716900855823823e-16, 3.7327490092779435e-16,
    3.7486529645684887e-16, 3.7646145133120287e-16, 3.7806354820089604e-16,
    3.7967177336979443e-16, 3.8128631696783774e-16, 3.829073731305243e-16,
    3.8453514018609596e-16, 3.8616982085091493e-16, 3.878116224335587e-16,
    3.894607570481926e-16, 3.9111744183782054e-16, 3.9278189920805415e-16,
    3.944543570720877e-16, 3.9613504910761354e-16, 3.9782421502646826e-16,
    3.995221008578565e-16, 4.012289592460629e-16, 4.029450497636328e-16,
    4.04670639241075e-16, 4.0640600211422504e-16, 4.0815142079049387e-16,
    4.0990718603532664e-16, 4.1167359738030257e-16, 4.134509635544236e-16,
    4.1523960294026883e-16, 4.170398440568316e-16, 4.1885202607101123e-16,
    4.206764993399015e-16, 4.2251362598620494e-16, 4.243637805093078e-16,
    4.262273504347798e-16, 4.2810473700531167e-16, 4.2999635591638323e-16,
    4.3190263810026294e-16, 4.338240305622791e-16, 4.357609972736849e-16,
    4.3771402012585875e-16, 4.3968359995105214e-16, 4.4167025761542035e-16,
    4.4367453519065673e-16, 4.456969972112043e-16, 4.477382320247534e-16,
    4.49798853244555e-16, 4.518795013130059e-16, 4.539808451870034e-16,
    4.561035841567422e-16, 4.582484498109567e-16, 4.604162081631153e-16,
    4.626076619547846e-16, 4.648236531543207e-16, 4.670650656712631e-16,
    4.693328283093329e-16, 4.716279179838351e-16, 4.739513632325867e-16,
    4.763042480533137e-16, 4.786877161048723e-16, 4.811029753147417e-16,
    4.835513029411525e-16, 4.860340511450812e-16, 4.885526531353603e-16,
    4.91108629959527e-16, 4.937035980240335e-16, 4.963392774403987e-16,
    4.990175013091822e-16, 5.017402260718089e-16, 5.045095430818727e-16,
    5.073276915733542e-16, 5.101970732341562e-16, 5.131202686306784e-16,
    5.161000557743228e-16, 5.191394311757699e-16, 5.222416338000234e-16,
    5.254101724177597e-16, 5.286488569504945e-16, 5.3196183453384e-16,
    5.353536311816497e-16, 5.388292001334053e-16, 5.423939782201712e-16,
    5.46053951907478e-16, 5.498157350892814e-16, 5.536866612467876e-16,
    5.576748932926576e-16, 5.617895553555417e-16, 5.660408920082422e-16,
    5.704404621291389e-16, 5.750013768919895e-16, 5.797385945724594e-16,
    5.846692893455479e-16, 5.898133176477899e-16, 5.951938149641444e-16,
    6.008379696271908e-16, 6.067780409333449e-16, 6.130527208725282e-16,
    6.197089894581626e-16, 6.268046963301284e-16, 6.344122407127506e-16,
    6.426239659548055e-16, 6.515603317344994e-16, 6.613827885097664e-16,
    6.723150462505587e-16, 6.846803417564259e-16, 6.98971833638762e-16,
    7.159994934830664e-16, 7.372424301798799e-16, 7.658936370805573e-16,
    8.113849337656484e-16
]

npr_zig_fi = [
    1.0, 0.9771017012676716, 0.9598790918001067,
    0.9451989534422996, 0.9320600759592305, 0.919991505039347,
    0.9087264400521309, 0.8980959218983434, 0.8879846607558334,
    0.8783096558089174, 0.869008688036857, 0.8600336211963315,
    0.851346258458678, 0.8429156531122042, 0.8347162929868834,
    0.8267268339462214, 0.8189291916037024, 0.8113078743126563,
    0.8038494831709643, 0.796542330422959, 0.7893761435660246,
    0.7823418326548025, 0.7754313049811872, 0.7686373157984863,
    0.7619533468367954, 0.7553735065070961, 0.7488924472191568,
    0.742505296340151, 0.7362075981268627, 0.7299952645614762,
    0.7238645334686302, 0.717811932630722, 0.7118342488782484,
    0.7059285013327543, 0.7000919181365116, 0.6943219161261167,
    0.6886160830046718, 0.6829721616449949, 0.6773880362187735,
    0.6718617198970821, 0.6663913439087501, 0.6609751477766631,
    0.6556114705796973, 0.6502987431108167, 0.6450354808208223,
    0.6398202774530566, 0.6346517992876236, 0.6295287799248367,
    0.6244500155470265, 0.6194143606058343, 0.6144207238889139,
    0.6094680649257734, 0.6045553906974678, 0.5996817526191253,
    0.5948462437679874, 0.590047996332826, 0.5852861792633715,
    0.5805599961007909, 0.5758686829723537, 0.5712115067352532,
    0.5665877632561644, 0.5619967758145243, 0.557437893618766,
    0.5529104904258323, 0.5484139632552658, 0.5439477311900263,
    0.5395112342569521, 0.5351039323804576, 0.5307253044036621,
    0.5263748471716845, 0.5220520746723218, 0.5177565172297564,
    0.513487720747327, 0.5092452459957479, 0.5050286679434681,
    0.5008375751261487, 0.4966715690524897, 0.49253026364386854,
    0.48841328470545803, 0.4843202694266833, 0.48025086590904675,
    0.47620473271950586, 0.4721815384677302, 0.4681809614056936,
    0.46420268904817436, 0.46024641781284287, 0.45631185267871643,
    0.4523987068618485, 0.44850670150720306, 0.4446355653957394,
    0.440785034665804, 0.43695485254798555, 0.43314476911265226,
    0.4293545410294414, 0.42558393133802197, 0.4218327092294959,
    0.4181006498378482, 0.4143875340408911, 0.41069314827018816,
    0.40701728432947337, 0.4033597392211145, 0.3997203149801972,
    0.39609881851583245, 0.3924950614593156, 0.3889088600187887,
    0.3853400348400773, 0.38178841087339366, 0.3782538172456192,
    0.37473608713789114, 0.3712350576682395, 0.3677505697790326,
    0.36428246812900406, 0.36083060098964803, 0.3573948201457805,
    0.3539749808000768, 0.3505709414814061, 0.34718256395679364,
    0.3438097131468507, 0.34045225704452187, 0.33711006663700605,
    0.33378301583071845, 0.3304709813791636, 0.3271738428136014,
    0.3238914823763911, 0.32062378495690536, 0.3173706380299136,
    0.3141319315963372, 0.3109075581262865, 0.30769741250429206,
    0.30450139197665, 0.30131939610080305, 0.2981513266966855,
    0.2949970877999618, 0.2918565856170952, 0.2887297284821829,
    0.28561642681550176, 0.2825165930837076, 0.27943014176163794,
    0.2763569892956683, 0.27329705406857707, 0.27025025636587546,
    0.26721651834356147, 0.2641957639972612, 0.2611879191327212,
    0.25819291133761924, 0.25521066995466196, 0.2522411260559422,
    0.24928421241852852, 0.24633986350126383, 0.2434080154227503,
    0.2404886059405006, 0.2375815744312381, 0.23468686187233,
    0.23180441082433872, 0.22893416541468034, 0.22607607132238028,
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