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| 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 #include "libc.h" |
| 83 |
| 84 static const double |
| 85 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ |
| 86 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ |
| 87 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ |
| 88 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ |
| 89 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ |
| 90 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ |
| 91 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ |
| 92 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ |
| 93 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ |
| 94 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ |
| 95 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ |
| 96 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ |
| 97 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ |
| 98 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ |
| 99 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ |
| 100 /* tt = -(tail of tf) */ |
| 101 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ |
| 102 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ |
| 103 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ |
| 104 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ |
| 105 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ |
| 106 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ |
| 107 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ |
| 108 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ |
| 109 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ |
| 110 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ |
| 111 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ |
| 112 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ |
| 113 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ |
| 114 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ |
| 115 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ |
| 116 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ |
| 117 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
| 118 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ |
| 119 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ |
| 120 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ |
| 121 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ |
| 122 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ |
| 123 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ |
| 124 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ |
| 125 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ |
| 126 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ |
| 127 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ |
| 128 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
| 129 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ |
| 130 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ |
| 131 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
| 132 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ |
| 133 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ |
| 134 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ |
| 135 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ |
| 136 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ |
| 137 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ |
| 138 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ |
| 139 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ |
| 140 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ |
| 141 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ |
| 142 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ |
| 143 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ |
| 144 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ |
| 145 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ |
| 146 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ |
| 147 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
| 148 |
| 149 /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */ |
| 150 static double sin_pi(double x) |
| 151 { |
| 152 int n; |
| 153 |
| 154 /* spurious inexact if odd int */ |
| 155 x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */ |
| 156 |
| 157 n = (int)(x*4.0); |
| 158 n = (n+1)/2; |
| 159 x -= n*0.5f; |
| 160 x *= pi; |
| 161 |
| 162 switch (n) { |
| 163 default: /* case 4: */ |
| 164 case 0: return __sin(x, 0.0, 0); |
| 165 case 1: return __cos(x, 0.0); |
| 166 case 2: return __sin(-x, 0.0, 0); |
| 167 case 3: return -__cos(x, 0.0); |
| 168 } |
| 169 } |
| 170 |
| 171 double __lgamma_r(double x, int *signgamp) |
| 172 { |
| 173 union {double f; uint64_t i;} u = {x}; |
| 174 double_t t,y,z,nadj,p,p1,p2,p3,q,r,w; |
| 175 uint32_t ix; |
| 176 int sign,i; |
| 177 |
| 178 /* purge off +-inf, NaN, +-0, tiny and negative arguments */ |
| 179 *signgamp = 1; |
| 180 sign = u.i>>63; |
| 181 ix = u.i>>32 & 0x7fffffff; |
| 182 if (ix >= 0x7ff00000) |
| 183 return x*x; |
| 184 if (ix < (0x3ff-70)<<20) { /* |x|<2**-70, return -log(|x|) */ |
| 185 if(sign) { |
| 186 x = -x; |
| 187 *signgamp = -1; |
| 188 } |
| 189 return -log(x); |
| 190 } |
| 191 if (sign) { |
| 192 x = -x; |
| 193 t = sin_pi(x); |
| 194 if (t == 0.0) /* -integer */ |
| 195 return 1.0/(x-x); |
| 196 if (t > 0.0) |
| 197 *signgamp = -1; |
| 198 else |
| 199 t = -t; |
| 200 nadj = log(pi/(t*x)); |
| 201 } |
| 202 |
| 203 /* purge off 1 and 2 */ |
| 204 if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0) |
| 205 r = 0; |
| 206 /* for x < 2.0 */ |
| 207 else if (ix < 0x40000000) { |
| 208 if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ |
| 209 r = -log(x); |
| 210 if (ix >= 0x3FE76944) { |
| 211 y = 1.0 - x; |
| 212 i = 0; |
| 213 } else if (ix >= 0x3FCDA661) { |
| 214 y = x - (tc-1.0); |
| 215 i = 1; |
| 216 } else { |
| 217 y = x; |
| 218 i = 2; |
| 219 } |
| 220 } else { |
| 221 r = 0.0; |
| 222 if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */ |
| 223 y = 2.0 - x; |
| 224 i = 0; |
| 225 } else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */ |
| 226 y = x - tc; |
| 227 i = 1; |
| 228 } else { |
| 229 y = x - 1.0; |
| 230 i = 2; |
| 231 } |
| 232 } |
| 233 switch (i) { |
| 234 case 0: |
| 235 z = y*y; |
| 236 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); |
| 237 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); |
| 238 p = y*p1+p2; |
| 239 r += (p-0.5*y); |
| 240 break; |
| 241 case 1: |
| 242 z = y*y; |
| 243 w = z*y; |
| 244 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp
*/ |
| 245 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); |
| 246 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); |
| 247 p = z*p1-(tt-w*(p2+y*p3)); |
| 248 r += tf + p; |
| 249 break; |
| 250 case 2: |
| 251 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); |
| 252 p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); |
| 253 r += -0.5*y + p1/p2; |
| 254 } |
| 255 } else if (ix < 0x40200000) { /* x < 8.0 */ |
| 256 i = (int)x; |
| 257 y = x - (double)i; |
| 258 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); |
| 259 q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); |
| 260 r = 0.5*y+p/q; |
| 261 z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ |
| 262 switch (i) { |
| 263 case 7: z *= y + 6.0; /* FALLTHRU */ |
| 264 case 6: z *= y + 5.0; /* FALLTHRU */ |
| 265 case 5: z *= y + 4.0; /* FALLTHRU */ |
| 266 case 4: z *= y + 3.0; /* FALLTHRU */ |
| 267 case 3: z *= y + 2.0; /* FALLTHRU */ |
| 268 r += log(z); |
| 269 break; |
| 270 } |
| 271 } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */ |
| 272 t = log(x); |
| 273 z = 1.0/x; |
| 274 y = z*z; |
| 275 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); |
| 276 r = (x-0.5)*(t-1.0)+w; |
| 277 } else /* 2**58 <= x <= inf */ |
| 278 r = x*(log(x)-1.0); |
| 279 if (sign) |
| 280 r = nadj - r; |
| 281 return r; |
| 282 } |
| 283 |
| 284 weak_alias(__lgamma_r, lgamma_r); |
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