1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunSoft, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* lgamma_r(x, signgamp) 13 * Reentrant version of the logarithm of the Gamma function 14 * with user provide pointer for the sign of Gamma(x). 15 * 16 * Method: 17 * 1. Argument Reduction for 0 < x <= 8 18 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 19 * reduce x to a number in [1.5,2.5] by 20 * lgamma(1+s) = log(s) + lgamma(s) 21 * for example, 22 * lgamma(7.3) = log(6.3) + lgamma(6.3) 23 * = log(6.3*5.3) + lgamma(5.3) 24 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 25 * 2. Polynomial approximation of lgamma around its 26 * minimum ymin=1.461632144968362245 to maintain monotonicity. 27 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 28 * Let z = x-ymin; 29 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 30 * where 31 * poly(z) is a 14 degree polynomial. 32 * 2. Rational approximation in the primary interval [2,3] 33 * We use the following approximation: 34 * s = x-2.0; 35 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 36 * with accuracy 37 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 38 * Our algorithms are based on the following observation 39 * 40 * zeta(2)-1 2 zeta(3)-1 3 41 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 42 * 2 3 43 * 44 * where Euler = 0.5771... is the Euler constant, which is very 45 * close to 0.5. 46 * 47 * 3. For x>=8, we have 48 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 49 * (better formula: 50 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 51 * Let z = 1/x, then we approximation 52 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 53 * by 54 * 3 5 11 55 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 56 * where 57 * |w - f(z)| < 2**-58.74 58 * 59 * 4. For negative x, since (G is gamma function) 60 * -x*G(-x)*G(x) = pi/sin(pi*x), 61 * we have 62 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 63 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 64 * Hence, for x<0, signgam = sign(sin(pi*x)) and 65 * lgamma(x) = log(|Gamma(x)|) 66 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 67 * Note: one should avoid compute pi*(-x) directly in the 68 * computation of sin(pi*(-x)). 69 * 70 * 5. Special Cases 71 * lgamma(2+s) ~ s*(1-Euler) for tiny s 72 * lgamma(1) = lgamma(2) = 0 73 * lgamma(x) ~ -log(|x|) for tiny x 74 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 75 * lgamma(inf) = inf 76 * lgamma(-inf) = inf (bug for bug compatible with C99!?) 77 */ 78 79 #include <float.h> 80 81 #include "math.h" 82 #include "math_private.h" 83 84 static const volatile double vzero = 0; 85 86 static const double 87 zero= 0.00000000000000000000e+00, 88 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 89 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 90 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 91 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 92 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 93 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 94 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 95 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 96 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 97 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 98 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 99 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 100 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 101 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 102 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 103 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 104 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 105 /* tt = -(tail of tf) */ 106 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 107 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 108 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 109 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 110 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 111 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 112 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 113 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 114 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 115 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 116 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 117 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 118 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 119 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 120 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 121 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 122 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 123 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 124 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 125 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 126 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 127 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 128 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 129 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 130 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 131 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 132 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 133 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 134 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 135 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 136 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 137 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 138 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 139 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 140 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 141 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 142 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 143 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 144 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 145 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 146 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 147 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 148 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 149 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 150 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 151 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 152 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 153 154 /* 155 * Compute sin(pi*x) without actually doing the pi*x multiplication. 156 * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is 157 * the precision of x. 158 */ 159 static double 160 sin_pi(double x) 161 { 162 volatile double vz; 163 double y,z; 164 int n; 165 166 y = -x; 167 168 vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ 169 z = vz-0x1p52; /* rint(y) for the above range */ 170 if (z == y) 171 return zero; 172 173 vz = y+0x1p50; 174 GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ 175 z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ 176 if (z > y) { 177 z -= 0.25; /* adjust to round down */ 178 n--; 179 } 180 n &= 7; /* octant of y mod 2 */ 181 y = y - z + n * 0.25; /* y mod 2 */ 182 183 switch (n) { 184 case 0: y = __kernel_sin(pi*y,zero,0); break; 185 case 1: 186 case 2: y = __kernel_cos(pi*(0.5-y),zero); break; 187 case 3: 188 case 4: y = __kernel_sin(pi*(one-y),zero,0); break; 189 case 5: 190 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; 191 default: y = __kernel_sin(pi*(y-2.0),zero,0); break; 192 } 193 return -y; 194 } 195 196 197 double 198 lgamma_r(double x, int *signgamp) 199 { 200 double nadj,p,p1,p2,p3,q,r,t,w,y,z; 201 int32_t hx; 202 int i,ix,lx; 203 204 EXTRACT_WORDS(hx,lx,x); 205 206 /* purge +-Inf and NaNs */ 207 *signgamp = 1; 208 ix = hx&0x7fffffff; 209 if(ix>=0x7ff00000) return x*x; 210 211 /* purge +-0 and tiny arguments */ 212 *signgamp = 1-2*((uint32_t)hx>>31); 213 if(ix<0x3c700000) { /* |x|<2**-56, return -log(|x|) */ 214 if((ix|lx)==0) 215 return one/vzero; 216 return -log(fabs(x)); 217 } 218 219 /* purge negative integers and start evaluation for other x < 0 */ 220 if(hx<0) { 221 *signgamp = 1; 222 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ 223 return one/vzero; 224 t = sin_pi(x); 225 if(t==zero) return one/vzero; /* -integer */ 226 nadj = log(pi/fabs(t*x)); 227 if(t<zero) *signgamp = -1; 228 x = -x; 229 } 230 231 /* purge 1 and 2 */ 232 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; 233 /* for x < 2.0 */ 234 else if(ix<0x40000000) { 235 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 236 r = -log(x); 237 if(ix>=0x3FE76944) {y = one-x; i= 0;} 238 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} 239 else {y = x; i=2;} 240 } else { 241 r = zero; 242 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ 243 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ 244 else {y=x-one;i=2;} 245 } 246 switch(i) { 247 case 0: 248 z = y*y; 249 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 250 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 251 p = y*p1+p2; 252 r += p-y/2; break; 253 case 1: 254 z = y*y; 255 w = z*y; 256 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 257 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 258 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 259 p = z*p1-(tt-w*(p2+y*p3)); 260 r += tf + p; break; 261 case 2: 262 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 263 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 264 r += p1/p2-y/2; 265 } 266 } 267 /* x < 8.0 */ 268 else if(ix<0x40200000) { 269 i = x; 270 y = x-i; 271 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 272 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 273 r = y/2+p/q; 274 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 275 switch(i) { 276 case 7: z *= (y+6); /* FALLTHRU */ 277 case 6: z *= (y+5); /* FALLTHRU */ 278 case 5: z *= (y+4); /* FALLTHRU */ 279 case 4: z *= (y+3); /* FALLTHRU */ 280 case 3: z *= (y+2); /* FALLTHRU */ 281 r += log(z); break; 282 } 283 /* 8.0 <= x < 2**56 */ 284 } else if (ix < 0x43700000) { 285 t = log(x); 286 z = one/x; 287 y = z*z; 288 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 289 r = (x-half)*(t-one)+w; 290 } else 291 /* 2**56 <= x <= inf */ 292 r = x*(log(x)-one); 293 if(hx<0) r = nadj - r; 294 return r; 295 } 296 297 #if (LDBL_MANT_DIG == 53) 298 __weak_reference(lgamma_r, lgammal_r); 299 #endif 300