11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* @(#)e_lgamma_r.c 1.3 95/01/18 */ 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Developed at SunSoft, a Sun Microsystems, Inc. business. 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Permission to use, copy, modify, and distribute this 8488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * software is freely granted, provided that this notice 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is preserved. 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 13a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* __ieee754_lgamma_r(x, signgamp) 17488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * Reentrant version of the logarithm of the Gamma function 18488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * with user provide pointer for the sign of Gamma(x). 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Method: 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1. Argument Reduction for 0 < x <= 8 22488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * reduce x to a number in [1.5,2.5] by 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(1+s) = log(s) + lgamma(s) 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * for example, 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(7.3) = log(6.3) + lgamma(6.3) 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = log(6.3*5.3) + lgamma(5.3) 281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. Polynomial approximation of lgamma around its 301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * minimun ymin=1.461632144968362245 to maintain monotonicity. 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Let z = x-ymin; 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * poly(z) is a 14 degree polynomial. 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. Rational approximation in the primary interval [2,3] 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * We use the following approximation: 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * s = x-2.0; 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(x) = 0.5*s + s*P(s)/Q(s) 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * with accuracy 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Our algorithms are based on the following observation 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * zeta(2)-1 2 zeta(3)-1 3 451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2 3 471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where Euler = 0.5771... is the Euler constant, which is very 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * close to 0.5. 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3. For x>=8, we have 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * (better formula: 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Let z = 1/x, then we approximation 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * by 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3 5 11 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 60488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * where 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |w - f(z)| < 2**-58.74 62488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 4. For negative x, since (G is gamma function) 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * -x*G(-x)*G(x) = pi/sin(pi*x), 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * we have 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 68488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * Hence, for x<0, signgam = sign(sin(pi*x)) and 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(x) = log(|Gamma(x)|) 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 71488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * Note: one should avoid compute pi*(-x) directly in the 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * computation of sin(pi*(-x)). 73488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes * 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 5. Special Cases 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * lgamma(2+s) ~ s*(1-Euler) for tiny s 76a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * lgamma(1) = lgamma(2) = 0 77a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * lgamma(x) ~ -log(|x|) for tiny x 78a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 79a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * lgamma(inf) = inf 80a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * lgamma(-inf) = inf (bug for bug compatible with C99!?) 811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 83755318548d787c7ca36901be064c69efba41c6cfElliott Hughes#include <float.h> 84755318548d787c7ca36901be064c69efba41c6cfElliott Hughes 851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 88460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughesstatic const volatile double vzero = 0; 89460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes 901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double 91460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hugheszero= 0.00000000000000000000e+00, 921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecthalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 1031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecta11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 1081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* tt = -(tail of tf) */ 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 1131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 1141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 1151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 1161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 1171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 1181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 1191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 1201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 1211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 1221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 1231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 1241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 1251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectt14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 1261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 1281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 1291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 1301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 1311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectu5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 1321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectv1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 1331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectv2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 1341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectv3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 1351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectv4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 1361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectv5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 1371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 1391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 1401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 1411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 1421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 1431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projects6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 1441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 1451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 1461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 1471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 1481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 1491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectr6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 1501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 1511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 1521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 1531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 1541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 1551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 1561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectw6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 1571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 158460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes/* 159460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes * Compute sin(pi*x) without actually doing the pi*x multiplication. 160460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is 161460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes * the precision of x. 162460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes */ 163460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughesstatic double 164460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughessin_pi(double x) 1651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 166460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes volatile double vz; 1671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double y,z; 168460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes int n; 169460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes 170460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes y = -x; 1711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 172460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ 173460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes z = vz-0x1p52; /* rint(y) for the above range */ 174460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes if (z == y) 175460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes return zero; 1761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 177460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes vz = y+0x1p50; 178460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ 179460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ 180460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes if (z > y) { 181460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes z -= 0.25; /* adjust to round down */ 182460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes n--; 183460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes } 184460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes n &= 7; /* octant of y mod 2 */ 185460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes y = y - z + n * 0.25; /* y mod 2 */ 1861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project switch (n) { 1881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 0: y = __kernel_sin(pi*y,zero,0); break; 189488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes case 1: 1901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 2: y = __kernel_cos(pi*(0.5-y),zero); break; 191488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes case 3: 1921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 4: y = __kernel_sin(pi*(one-y),zero,0); break; 1931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 5: 1941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; 1951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project default: y = __kernel_sin(pi*(y-2.0),zero,0); break; 1961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return -y; 1981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 1991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 2021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project__ieee754_lgamma_r(double x, int *signgamp) 2031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 204488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes double nadj,p,p1,p2,p3,q,r,t,w,y,z; 205a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes int32_t hx; 206460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes int i,ix,lx; 2071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project EXTRACT_WORDS(hx,lx,x); 2091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 210488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* purge +-Inf and NaNs */ 2111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project *signgamp = 1; 2121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx&0x7fffffff; 2131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x7ff00000) return x*x; 214488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes 215488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* purge +-0 and tiny arguments */ 216488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes *signgamp = 1-2*((uint32_t)hx>>31); 217488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes if(ix<0x3c700000) { /* |x|<2**-56, return -log(|x|) */ 218488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes if((ix|lx)==0) 219488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes return one/vzero; 220488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes return -__ieee754_log(fabs(x)); 2211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 222488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes 223488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* purge negative integers and start evaluation for other x < 0 */ 2241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx<0) { 225488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes *signgamp = 1; 2261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ 227460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes return one/vzero; 2281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = sin_pi(x); 229460ad7454a281470a1938103f099a1e3192fb3d6Elliott Hughes if(t==zero) return one/vzero; /* -integer */ 2301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project nadj = __ieee754_log(pi/fabs(t*x)); 2311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(t<zero) *signgamp = -1; 2321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project x = -x; 2331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 235488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* purge 1 and 2 */ 2361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; 2371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* for x < 2.0 */ 2381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else if(ix<0x40000000) { 2391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 2401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = -__ieee754_log(x); 2411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x3FE76944) {y = one-x; i= 0;} 2421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} 2431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else {y = x; i=2;} 2441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 2451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = zero; 2461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ 2471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ 2481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else {y=x-one;i=2;} 2491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project switch(i) { 2511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 0: 2521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = y*y; 2531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 2541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 2551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p = y*p1+p2; 256488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes r += p-y/2; break; 2571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project case 1: 2581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = y*y; 2591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = z*y; 2601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 2611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 2621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 2631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p = z*p1-(tt-w*(p2+y*p3)); 264488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes r += tf + p; break; 265488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes case 2: 2661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 2671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 268488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes r += p1/p2-y/2; 2691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 271488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* x < 8.0 */ 272488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes else if(ix<0x40200000) { 273488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes i = x; 274488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes y = x-i; 2751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 2761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 277488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes r = y/2+p/q; 2781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 2791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project switch(i) { 280755318548d787c7ca36901be064c69efba41c6cfElliott Hughes case 7: z *= (y+6); /* FALLTHRU */ 281755318548d787c7ca36901be064c69efba41c6cfElliott Hughes case 6: z *= (y+5); /* FALLTHRU */ 282755318548d787c7ca36901be064c69efba41c6cfElliott Hughes case 5: z *= (y+4); /* FALLTHRU */ 283755318548d787c7ca36901be064c69efba41c6cfElliott Hughes case 4: z *= (y+3); /* FALLTHRU */ 284755318548d787c7ca36901be064c69efba41c6cfElliott Hughes case 3: z *= (y+2); /* FALLTHRU */ 2851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r += __ieee754_log(z); break; 2861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 287488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* 8.0 <= x < 2**56 */ 288488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes } else if (ix < 0x43700000) { 2891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = __ieee754_log(x); 2901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = one/x; 2911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project y = z*z; 2921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 2931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = (x-half)*(t-one)+w; 294488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes } else 295488268b134723c7a6598338bb253be5f64d53be4Elliott Hughes /* 2**56 <= x <= inf */ 2961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = x*(__ieee754_log(x)-one); 2971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx<0) r = nadj - r; 2981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return r; 2991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 300755318548d787c7ca36901be064c69efba41c6cfElliott Hughes 301755318548d787c7ca36901be064c69efba41c6cfElliott Hughes#if (LDBL_MANT_DIG == 53) 302755318548d787c7ca36901be064c69efba41c6cfElliott Hughes__weak_reference(lgamma_r, lgammal_r); 303755318548d787c7ca36901be064c69efba41c6cfElliott Hughes#endif 304