/**************************************************************************** * libc/math/lib_gamma.c * * Ported to NuttX from FreeBSD by Alan Carvalho de Assis: * * 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. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * 3. Neither the name NuttX nor the names of its contributors may be * used to endorse or promote products derived from this software * without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE * POSSIBILITY OF SUCH DAMAGE. * ****************************************************************************/ /* 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(neg.integer) = inf and raise divide-by-zero * lgamma(inf) = inf * lgamma(-inf) = inf (bug for bug compatible with C99!?) */ /**************************************************************************** * Included Files ****************************************************************************/ #include #include #include #include #ifdef CONFIG_HAVE_DOUBLE /**************************************************************************** * Private Data ****************************************************************************/ static int g_signgam = 0; static const double g_pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ static const double g_a0 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ static const double g_a1 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ static const double g_a2 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ static const double g_a3 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ static const double g_a4 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ static const double g_a5 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ static const double g_a6 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ static const double g_a7 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ static const double g_a8 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ static const double g_a9 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ static const double g_a10 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ static const double g_a11 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ static const double g_tc = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ static const double g_tf = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ /* tt = -(tail of tf) */ static const double g_tt = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ static const double g_t0 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ static const double g_t1 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ static const double g_t2 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ static const double g_t3 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ static const double g_t4 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ static const double g_t5 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ static const double g_t6 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ static const double g_t7 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ static const double g_t8 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ static const double g_t9 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ static const double g_t10 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ static const double g_t11 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ static const double g_t12 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ static const double g_t13 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ static const double g_t14 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ static const double g_u0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ static const double g_u1 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ static const double g_u2 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ static const double g_u3 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ static const double g_u4 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ static const double g_u5 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ static const double g_v1 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ static const double g_v2 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ static const double g_v3 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ static const double g_v4 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ static const double g_v5 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ static const double g_s0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ static const double g_s1 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ static const double g_s2 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ static const double g_s3 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ static const double g_s4 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ static const double g_s5 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ static const double g_s6 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ static const double g_r1 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ static const double g_r2 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ static const double g_r3 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ static const double g_r4 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ static const double g_r5 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ static const double g_r6 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ static const double g_w0 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ static const double g_w1 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ static const double g_w2 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ static const double g_w3 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ static const double g_w4 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ static const double g_w5 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ static const double g_w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ /**************************************************************************** * Private Functions ****************************************************************************/ /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */ static double sin_pi(double x) { int n; /* spurious inexact if odd int */ x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ n = (int)(x * 4.0); n = (n + 1) / 2; x -= n * 0.5f; x *= g_pi; switch (n) { default: /* case 4: */ case 0: return __sin(x, 0.0, 0); case 1: return __cos(x, 0.0); case 2: return __sin(-x, 0.0, 0); case 3: return -__cos(x, 0.0); } } /**************************************************************************** * Public Functions ****************************************************************************/ double lgamma_r(double x, int *signgamp) { union { double f; uint64_t i; } u; u.f = x; double t; double y; double z; double nadj = 0.0; double p; double p1; double p2; double p3; double q; double r; double w; uint32_t ix; int sign; int i; /* purge off +-inf, NaN, +-0, tiny and negative arguments */ *signgamp = 1; sign = u.i >> 63; ix = u.i >> 32 & 0x7fffffff; if (ix >= 0x7ff00000) { return x * x; } /* |x|<2**-70, return -log(|x|) */ if (ix < (0x3ff - 70) << 20) { if (sign) { x = -x; *signgamp = -1; } return -log(x); } if (sign) { x = -x; t = sin_pi(x); if (t == 0.0) { /* -integer */ return 1.0 / (x - x); } if (t > 0.0) { *signgamp = -1; } else { t = -t; } nadj = log(g_pi / (t * x)); } /* purge off 1 and 2 */ if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t) u.i == 0) { r = 0; } else /* for x < 2.0 */ { if (ix < 0x40000000) { if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ r = -log(x); if (ix >= 0x3FE76944) { y = 1.0 - x; i = 0; } else { if (ix >= 0x3FCDA661) { y = x - (g_tc - 1.0); i = 1; } else { y = x; i = 2; } } } else { r = 0.0; if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */ y = 2.0 - x; i = 0; } else { if (ix >= 0x3FF3B4C4) { /* [1.23,1.73] */ y = x - g_tc; i = 1; } else { y = x - 1.0; i = 2; } } } switch (i) { case 0: z = y*y; p1 = g_a0+z*(g_a2+z*(g_a4+z*(g_a6+z*(g_a8+z*g_a10)))); p2 = z*(g_a1+z*(g_a3+z*(g_a5+z*(g_a7+z*(g_a9+z*g_a11))))); p = y*p1+p2; r += (p-0.5*y); break; case 1: z = y*y; w = z*y; p1 = g_t0+w*(g_t3+w*(g_t6+w*(g_t9+w*g_t12))); /* parallel comp */ p2 = g_t1+w*(g_t4+w*(g_t7+w*(g_t10+w*g_t13))); p3 = g_t2+w*(g_t5+w*(g_t8+w*(g_t11+w*g_t14))); p = z*p1-(g_tt-w*(p2+y*p3)); r += g_tf + p; break; case 2: p1 = y*(g_u0+y*(g_u1+y*(g_u2+y*(g_u3+y*(g_u4+y*g_u5))))); p2 = 1.0+y*(g_v1+y*(g_v2+y*(g_v3+y*(g_v4+y*g_v5)))); r += -0.5*y + p1/p2; } } else { if (ix < 0x40200000) { /* x < 8.0 */ i = (int)x; y = x - (double)i; p = y*(g_s0+y*(g_s1+y*(g_s2+y*(g_s3+y*(g_s4+y*(g_s5+y*g_s6)))))); q = 1.0+y*(g_r1+y*(g_r2+y*(g_r3+y*(g_r4+y*(g_r5+y*g_r6))))); r = 0.5*y+p/q; z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ switch (i) { case 7: z *= y + 6.0; /* FALLTHRU */ case 6: z *= y + 5.0; /* FALLTHRU */ case 5: z *= y + 4.0; /* FALLTHRU */ case 4: z *= y + 3.0; /* FALLTHRU */ case 3: z *= y + 2.0; /* FALLTHRU */ r += log(z); break; } } else { if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */ t = log(x); z = 1.0 / x; y = z * z; w = g_w0+z*(g_w1+y*(g_w2+y*(g_w3+y*(g_w4+y*(g_w5+y*g_w6))))); r = (x-0.5)*(t-1.0)+w; } else { /* 2**58 <= x <= inf */ r = x * (log(x) - 1.0); } } } } if (sign) { r = nadj - r; } return r; } double lgamma(double x) { return lgamma_r(x, &g_signgam); } #endif