github.com/afumu/libc@v0.0.6/musl/src/math/lgamma_r.c (about) 1 /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 * 12 */ 13 /* lgamma_r(x, signgamp) 14 * Reentrant version of the logarithm of the Gamma function 15 * with user provide pointer for the sign of Gamma(x). 16 * 17 * Method: 18 * 1. Argument Reduction for 0 < x <= 8 19 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 20 * reduce x to a number in [1.5,2.5] by 21 * lgamma(1+s) = log(s) + lgamma(s) 22 * for example, 23 * lgamma(7.3) = log(6.3) + lgamma(6.3) 24 * = log(6.3*5.3) + lgamma(5.3) 25 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 26 * 2. Polynomial approximation of lgamma around its 27 * minimun ymin=1.461632144968362245 to maintain monotonicity. 28 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 29 * Let z = x-ymin; 30 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 31 * where 32 * poly(z) is a 14 degree polynomial. 33 * 2. Rational approximation in the primary interval [2,3] 34 * We use the following approximation: 35 * s = x-2.0; 36 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 37 * with accuracy 38 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 39 * Our algorithms are based on the following observation 40 * 41 * zeta(2)-1 2 zeta(3)-1 3 42 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 43 * 2 3 44 * 45 * where Euler = 0.5771... is the Euler constant, which is very 46 * close to 0.5. 47 * 48 * 3. For x>=8, we have 49 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 50 * (better formula: 51 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 52 * Let z = 1/x, then we approximation 53 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 54 * by 55 * 3 5 11 56 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 57 * where 58 * |w - f(z)| < 2**-58.74 59 * 60 * 4. For negative x, since (G is gamma function) 61 * -x*G(-x)*G(x) = pi/sin(pi*x), 62 * we have 63 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 64 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 65 * Hence, for x<0, signgam = sign(sin(pi*x)) and 66 * lgamma(x) = log(|Gamma(x)|) 67 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 68 * Note: one should avoid compute pi*(-x) directly in the 69 * computation of sin(pi*(-x)). 70 * 71 * 5. Special Cases 72 * lgamma(2+s) ~ s*(1-Euler) for tiny s 73 * lgamma(1) = lgamma(2) = 0 74 * lgamma(x) ~ -log(|x|) for tiny x 75 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 76 * lgamma(inf) = inf 77 * lgamma(-inf) = inf (bug for bug compatible with C99!?) 78 * 79 */ 80 81 #include "libm.h" 82 83 static const double 84 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 85 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 86 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 87 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 88 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 89 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 90 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 91 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 92 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 93 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 94 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 95 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 96 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 97 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 98 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 99 /* tt = -(tail of tf) */ 100 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 101 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 102 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 103 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 104 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 105 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 106 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 107 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 108 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 109 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 110 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 111 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 112 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 113 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 114 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 115 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 116 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 117 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 118 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 119 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 120 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 121 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 122 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 123 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 124 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 125 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 126 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 127 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 128 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 129 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 130 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 131 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 132 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 133 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 134 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 135 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 136 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 137 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 138 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 139 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 140 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 141 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 142 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 143 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 144 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 145 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 146 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 147 148 /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */ 149 static double sin_pi(double x) 150 { 151 int n; 152 153 /* spurious inexact if odd int */ 154 x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */ 155 156 n = (int)(x*4.0); 157 n = (n+1)/2; 158 x -= n*0.5f; 159 x *= pi; 160 161 switch (n) { 162 default: /* case 4: */ 163 case 0: return __sin(x, 0.0, 0); 164 case 1: return __cos(x, 0.0); 165 case 2: return __sin(-x, 0.0, 0); 166 case 3: return -__cos(x, 0.0); 167 } 168 } 169 170 double __lgamma_r(double x, int *signgamp) 171 { 172 union {double f; uint64_t i;} u = {x}; 173 double_t t,y,z,nadj,p,p1,p2,p3,q,r,w; 174 uint32_t ix; 175 int sign,i; 176 177 /* purge off +-inf, NaN, +-0, tiny and negative arguments */ 178 *signgamp = 1; 179 sign = u.i>>63; 180 ix = u.i>>32 & 0x7fffffff; 181 if (ix >= 0x7ff00000) 182 return x*x; 183 if (ix < (0x3ff-70)<<20) { /* |x|<2**-70, return -log(|x|) */ 184 if(sign) { 185 x = -x; 186 *signgamp = -1; 187 } 188 return -log(x); 189 } 190 if (sign) { 191 x = -x; 192 t = sin_pi(x); 193 if (t == 0.0) /* -integer */ 194 return 1.0/(x-x); 195 if (t > 0.0) 196 *signgamp = -1; 197 else 198 t = -t; 199 nadj = log(pi/(t*x)); 200 } 201 202 /* purge off 1 and 2 */ 203 if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0) 204 r = 0; 205 /* for x < 2.0 */ 206 else if (ix < 0x40000000) { 207 if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 208 r = -log(x); 209 if (ix >= 0x3FE76944) { 210 y = 1.0 - x; 211 i = 0; 212 } else if (ix >= 0x3FCDA661) { 213 y = x - (tc-1.0); 214 i = 1; 215 } else { 216 y = x; 217 i = 2; 218 } 219 } else { 220 r = 0.0; 221 if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */ 222 y = 2.0 - x; 223 i = 0; 224 } else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */ 225 y = x - tc; 226 i = 1; 227 } else { 228 y = x - 1.0; 229 i = 2; 230 } 231 } 232 switch (i) { 233 case 0: 234 z = y*y; 235 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 236 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 237 p = y*p1+p2; 238 r += (p-0.5*y); 239 break; 240 case 1: 241 z = y*y; 242 w = z*y; 243 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 244 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 245 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 246 p = z*p1-(tt-w*(p2+y*p3)); 247 r += tf + p; 248 break; 249 case 2: 250 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 251 p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 252 r += -0.5*y + p1/p2; 253 } 254 } else if (ix < 0x40200000) { /* x < 8.0 */ 255 i = (int)x; 256 y = x - (double)i; 257 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 258 q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 259 r = 0.5*y+p/q; 260 z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ 261 switch (i) { 262 case 7: z *= y + 6.0; /* FALLTHRU */ 263 case 6: z *= y + 5.0; /* FALLTHRU */ 264 case 5: z *= y + 4.0; /* FALLTHRU */ 265 case 4: z *= y + 3.0; /* FALLTHRU */ 266 case 3: z *= y + 2.0; /* FALLTHRU */ 267 r += log(z); 268 break; 269 } 270 } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */ 271 t = log(x); 272 z = 1.0/x; 273 y = z*z; 274 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 275 r = (x-0.5)*(t-1.0)+w; 276 } else /* 2**58 <= x <= inf */ 277 r = x*(log(x)-1.0); 278 if (sign) 279 r = nadj - r; 280 return r; 281 } 282 283 weak_alias(__lgamma_r, lgamma_r);