/*===-------------------------------------------------------------------------- * ROCm Device Libraries * * This file is distributed under the University of Illinois Open Source * License. See LICENSE.TXT for details. *===------------------------------------------------------------------------*/ #include "mathD.h" // This lgamma routine began with Sun's lgamma code from netlib. // Their original copyright notice follows. /* @(#)e_lgamma_r.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * */ /* __ieee754_lgamma_r(x, signgamp) * Reentrant version of the logarithm of the Gamma function * with user provide pointer for the sign of Gamma(x). * * Method: * 1. Argument Reduction for 0 < x <= 8 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may * reduce x to a number in [1.5,2.5] by * lgamma(1+s) = log(s) + lgamma(s) * for example, * lgamma(7.3) = log(6.3) + lgamma(6.3) * = log(6.3*5.3) + lgamma(5.3) * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) * 2. Polynomial approximation of lgamma around its * minimun ymin=1.461632144968362245 to maintain monotonicity. * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use * Let z = x-ymin; * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) * where * poly(z) is a 14 degree polynomial. * 2. Rational approximation in the primary interval [2,3] * We use the following approximation: * s = x-2.0; * lgamma(x) = 0.5*s + s*P(s)/Q(s) * with accuracy * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 * Our algorithms are based on the following observation * * zeta(2)-1 2 zeta(3)-1 3 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... * 2 3 * * where Euler = 0.5771... is the Euler constant, which is very * close to 0.5. * * 3. For x>=8, we have * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... * (better formula: * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) * Let z = 1/x, then we approximation * f(z) = lgamma(x) - (x-0.5)(log(x)-1) * by * 3 5 11 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z * where * |w - f(z)| < 2**-58.74 * * 4. For negative x, since (G is gamma function) * -x*G(-x)*G(x) = pi/sin(pi*x), * we have * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 * Hence, for x<0, signgam = sign(sin(pi*x)) and * lgamma(x) = log(|Gamma(x)|) * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); * Note: one should avoid compute pi*(-x) directly in the * computation of sin(pi*(-x)). * * 5. Special Cases * lgamma(2+s) ~ s*(1-Euler) for tiny s * lgamma(1)=lgamma(2)=0 * lgamma(x) ~ -log(x) for tiny x * lgamma(0) = lgamma(inf) = inf * lgamma(-integer) = +-inf * */ struct ret_t { double result; int signp; }; static struct ret_t MATH_MANGLE(lgamma_r_impl)(double x) { const double two52= 4.50359962737049600000e+15; const double pi = 3.14159265358979311600e+00; const double a0 = 7.72156649015328655494e-02; const double a1 = 3.22467033424113591611e-01; const double a2 = 6.73523010531292681824e-02; const double a3 = 2.05808084325167332806e-02; const double a4 = 7.38555086081402883957e-03; const double a5 = 2.89051383673415629091e-03; const double a6 = 1.19270763183362067845e-03; const double a7 = 5.10069792153511336608e-04; const double a8 = 2.20862790713908385557e-04; const double a9 = 1.08011567247583939954e-04; const double a10 = 2.52144565451257326939e-05; const double a11 = 4.48640949618915160150e-05; const double tc = 1.46163214496836224576e+00; const double tf = -1.21486290535849611461e-01; const double tt = -3.63867699703950536541e-18; const double t0 = 4.83836122723810047042e-01; const double t1 = -1.47587722994593911752e-01; const double t2 = 6.46249402391333854778e-02; const double t3 = -3.27885410759859649565e-02; const double t4 = 1.79706750811820387126e-02; const double t5 = -1.03142241298341437450e-02; const double t6 = 6.10053870246291332635e-03; const double t7 = -3.68452016781138256760e-03; const double t8 = 2.25964780900612472250e-03; const double t9 = -1.40346469989232843813e-03; const double t10 = 8.81081882437654011382e-04; const double t11 = -5.38595305356740546715e-04; const double t12 = 3.15632070903625950361e-04; const double t13 = -3.12754168375120860518e-04; const double t14 = 3.35529192635519073543e-04; const double u0 = -7.72156649015328655494e-02; const double u1 = 6.32827064025093366517e-01; const double u2 = 1.45492250137234768737e+00; const double u3 = 9.77717527963372745603e-01; const double u4 = 2.28963728064692451092e-01; const double u5 = 1.33810918536787660377e-02; const double v1 = 2.45597793713041134822e+00; const double v2 = 2.12848976379893395361e+00; const double v3 = 7.69285150456672783825e-01; const double v4 = 1.04222645593369134254e-01; const double v5 = 3.21709242282423911810e-03; const double s0 = -7.72156649015328655494e-02; const double s1 = 2.14982415960608852501e-01; const double s2 = 3.25778796408930981787e-01; const double s3 = 1.46350472652464452805e-01; const double s4 = 2.66422703033638609560e-02; const double s5 = 1.84028451407337715652e-03; const double s6 = 3.19475326584100867617e-05; const double r1 = 1.39200533467621045958e+00; const double r2 = 7.21935547567138069525e-01; const double r3 = 1.71933865632803078993e-01; const double r4 = 1.86459191715652901344e-02; const double r5 = 7.77942496381893596434e-04; const double r6 = 7.32668430744625636189e-06; const double w0 = 4.18938533204672725052e-01; const double w1 = 8.33333333333329678849e-02; const double w2 = -2.77777777728775536470e-03; const double w3 = 7.93650558643019558500e-04; const double w4 = -5.95187557450339963135e-04; const double w5 = 8.36339918996282139126e-04; const double w6 = -1.63092934096575273989e-03; const double z1 = -0x1.2788cfc6fb619p-1; const double z2 = 0x1.a51a6625307d3p-1; const double z3 = -0x1.9a4d55beab2d7p-2; const double z4 = 0x1.151322ac7d848p-2; const double z5 = -0x1.a8b9c17aa6149p-3; double ax = BUILTIN_ABS_F64(x); uint hax = AS_UINT2(ax).hi; double ret; if (hax < 0x3f700000) { // ax < 0x1.0p-8 ret = MATH_MAD(ax, MATH_MAD(ax, MATH_MAD(ax, MATH_MAD(ax, MATH_MAD(ax, z5, z4), z3), z2), z1), -MATH_MANGLE(log)(ax)); } else if (hax < 0x40000000) { // ax < 2.0 int i; bool c; double y, t; if (hax <= 0x3feccccc) { // |x| < 0.9 : lgamma(x) = lgamma(x+1)-log(x) ret = -MATH_MANGLE(log)(ax); y = 1.0 - ax; i = 0; c = hax < 0x3FE76944; // x < 0.7316 t = ax - (tc - 1.0); y = c ? t : y; i = c ? 1 : i; c = hax < 0x3FCDA661; // x < .2316 y = c ? ax : y; i = c ? 2 : i; } else { ret = 0.0; y = 2.0 - ax; i = 0; c = hax < 0x3FFBB4C3; // x < 1.7316 t = ax - tc; y = c ? t : y; i = c ? 1 : i; c = hax < 0x3FF3B4C4; // x < 1.2316 t = ax - 1.0; y = c ? t : y; i = c ? 2 : i; } double w, z, p, p1, p2, p3; switch(i) { case 0: z = y*y; p1 = MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, a10, a8), a6), a4), a2), a0); p2 = z * MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, MATH_MAD(z, a11, a9), a7), a5), a3), a1); p = MATH_MAD(y, p1, p2); ret += MATH_MAD(y, -0.5, p); break; case 1: z = y*y; w = z*y; p1 = MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, t12, t9), t6), t3), t0); p2 = MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, t13, t10), t7), t4), t1); p3 = MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, MATH_MAD(w, t14, t11), t8), t5), t2); p = MATH_MAD(z, p1, -MATH_MAD(w, -MATH_MAD(y, p3,p2), tt)); ret += tf + p; break; case 2: p1 = y * MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, u5, u4), u3), u2), u1), u0); p2 = MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, v5, v4), v3), v2), v1), 1.0); ret += MATH_MAD(y, -0.5, MATH_DIV(p1, p2)); break; } } else if (hax < 0x40200000) { // 2 < ax < 8 int i = (int)ax; double y = ax - (double)i; double p = y * MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, s6, s5), s4), s3), s2), s1), s0); double q = MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, r6, r5), r4), r3), r2), r1), 1.0); ret = MATH_MAD(y, 0.5, MATH_DIV(p, q)); double y2 = y + 2.0; double y3 = y + 3.0; double y4 = y + 4.0; double y5 = y + 5.0; double y6 = y + 6.0; double z = 1.0; z *= i > 2 ? y2 : 1.0; z *= i > 3 ? y3 : 1.0; z *= i > 4 ? y4 : 1.0; z *= i > 5 ? y5 : 1.0; z *= i > 6 ? y6 : 1.0; ret += MATH_MANGLE(log)(z); } else if (hax < 0x43900000) { // 8 <= ax < 2^58 double z = MATH_RCP(ax); double y = z*z; double w = MATH_MAD(z, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, MATH_MAD(y, w6, w5), w4), w3), w2), w1), w0); ret = MATH_MAD(ax - 0.5, MATH_MANGLE(log)(ax) - 1.0, w); } else { // 2^58 <= ax <= Inf ret = MATH_MAD(ax, MATH_MANGLE(log)(ax), -ax); } int s = 0; if (x >= 0.0) { ret = (x == 1.0 | x == 2.0) ? 0.0 : ret; s = x == 0.0 ? 0 : 1; } else if (hax < 0x43300000) { // x > -0x1.0p+52 if (hax > 0x3cd00000) { // x < -0x1.0p-50 double t = MATH_MANGLE(sinpi)(x); double negadj = MATH_MANGLE(log)(MATH_DIV(pi, BUILTIN_ABS_F64(t * x))); ret = negadj - ret; bool z = BUILTIN_FRACTION_F64(x) == 0.0; ret = z ? AS_DOUBLE(PINFBITPATT_DP64) : ret; s = t < 0.0 ? -1 : 1; s = z ? 0 : s; } else { s = -1; } } if (!FINITE_ONLY_OPT()) { // Handle negative integer, Inf, NaN ret = BUILTIN_CLASS_F64(ax, CLASS_NZER|CLASS_PZER|CLASS_PINF) | (x < 0.0f & hax >= 0x43300000) ? AS_DOUBLE(PINFBITPATT_DP64) : ret; ret = BUILTIN_ISNAN_F64(x) ? x : ret; } struct ret_t result; result.result = ret; result.signp = s; return result; } double MATH_MANGLE(lgamma_r)(double x, __private int *signp) { struct ret_t ret = MATH_MANGLE(lgamma_r_impl)(x); *signp = ret.signp; return ret.result; }