1 /* @(#)e_lgamma_r.c 1.3 95/01/18 */ 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 #include <sys/cdefs.h> 14 __FBSDID("$FreeBSD$"); 15 16 /* __ieee754_lgamma_r(x, signgamp) 17 * Reentrant version of the logarithm of the Gamma function 18 * with user provide pointer for the sign of Gamma(x). 19 * 20 * Method: 21 * 1. Argument Reduction for 0 < x <= 8 22 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 23 * reduce x to a number in [1.5,2.5] by 24 * lgamma(1+s) = log(s) + lgamma(s) 25 * for example, 26 * lgamma(7.3) = log(6.3) + lgamma(6.3) 27 * = log(6.3*5.3) + lgamma(5.3) 28 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 29 * 2. Polynomial approximation of lgamma around its 30 * minimun ymin=1.461632144968362245 to maintain monotonicity. 31 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 32 * Let z = x-ymin; 33 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 34 * where 35 * poly(z) is a 14 degree polynomial. 36 * 2. Rational approximation in the primary interval [2,3] 37 * We use the following approximation: 38 * s = x-2.0; 39 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 40 * with accuracy 41 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 42 * Our algorithms are based on the following observation 43 * 44 * zeta(2)-1 2 zeta(3)-1 3 45 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 46 * 2 3 47 * 48 * where Euler = 0.5771... is the Euler constant, which is very 49 * close to 0.5. 50 * 51 * 3. For x>=8, we have 52 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 53 * (better formula: 54 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 55 * Let z = 1/x, then we approximation 56 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 57 * by 58 * 3 5 11 59 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 60 * where 61 * |w - f(z)| < 2**-58.74 62 * 63 * 4. For negative x, since (G is gamma function) 64 * -x*G(-x)*G(x) = pi/sin(pi*x), 65 * we have 66 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 67 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 68 * Hence, for x<0, signgam = sign(sin(pi*x)) and 69 * lgamma(x) = log(|Gamma(x)|) 70 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 71 * Note: one should avoid compute pi*(-x) directly in the 72 * computation of sin(pi*(-x)). 73 * 74 * 5. Special Cases 75 * lgamma(2+s) ~ s*(1-Euler) for tiny s 76 * lgamma(1) = lgamma(2) = 0 77 * lgamma(x) ~ -log(|x|) for tiny x 78 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 79 * lgamma(inf) = inf 80 * lgamma(-inf) = inf (bug for bug compatible with C99!?) 81 */ 82 83 #include <float.h> 84 85 #include "math.h" 86 #include "math_private.h" 87 88 static const volatile double vzero = 0; 89 90 static const double 91 zero= 0.00000000000000000000e+00, 92 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 93 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 94 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 95 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 96 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 97 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 98 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 99 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 100 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 101 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 102 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 103 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 104 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 105 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 106 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 107 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 108 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 109 /* tt = -(tail of tf) */ 110 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 111 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 112 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 113 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 114 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 115 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 116 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 117 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 118 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 119 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 120 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 121 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 122 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 123 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 124 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 125 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 126 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 127 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 128 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 129 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 130 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 131 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 132 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 133 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 134 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 135 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 136 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 137 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 138 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 139 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 140 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 141 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 142 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 143 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 144 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 145 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 146 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 147 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 148 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 149 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 150 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 151 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 152 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 153 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 154 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 155 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 156 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 157 158 /* 159 * Compute sin(pi*x) without actually doing the pi*x multiplication. 160 * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is 161 * the precision of x. 162 */ 163 static double 164 sin_pi(double x) 165 { 166 volatile double vz; 167 double y,z; 168 int n; 169 170 y = -x; 171 172 vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ 173 z = vz-0x1p52; /* rint(y) for the above range */ 174 if (z == y) 175 return zero; 176 177 vz = y+0x1p50; 178 GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ 179 z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ 180 if (z > y) { 181 z -= 0.25; /* adjust to round down */ 182 n--; 183 } 184 n &= 7; /* octant of y mod 2 */ 185 y = y - z + n * 0.25; /* y mod 2 */ 186 187 switch (n) { 188 case 0: y = __kernel_sin(pi*y,zero,0); break; 189 case 1: 190 case 2: y = __kernel_cos(pi*(0.5-y),zero); break; 191 case 3: 192 case 4: y = __kernel_sin(pi*(one-y),zero,0); break; 193 case 5: 194 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; 195 default: y = __kernel_sin(pi*(y-2.0),zero,0); break; 196 } 197 return -y; 198 } 199 200 201 double 202 __ieee754_lgamma_r(double x, int *signgamp) 203 { 204 double t,y,z,nadj,p,p1,p2,p3,q,r,w; 205 int32_t hx; 206 int i,ix,lx; 207 208 EXTRACT_WORDS(hx,lx,x); 209 210 /* purge off +-inf, NaN, +-0, tiny and negative arguments */ 211 *signgamp = 1; 212 ix = hx&0x7fffffff; 213 if(ix>=0x7ff00000) return x*x; 214 if((ix|lx)==0) { 215 if(hx<0) 216 *signgamp = -1; 217 return one/vzero; 218 } 219 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ 220 if(hx<0) { 221 *signgamp = -1; 222 return -__ieee754_log(-x); 223 } else return -__ieee754_log(x); 224 } 225 if(hx<0) { 226 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ 227 return one/vzero; 228 t = sin_pi(x); 229 if(t==zero) return one/vzero; /* -integer */ 230 nadj = __ieee754_log(pi/fabs(t*x)); 231 if(t<zero) *signgamp = -1; 232 x = -x; 233 } 234 235 /* purge off 1 and 2 */ 236 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; 237 /* for x < 2.0 */ 238 else if(ix<0x40000000) { 239 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 240 r = -__ieee754_log(x); 241 if(ix>=0x3FE76944) {y = one-x; i= 0;} 242 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} 243 else {y = x; i=2;} 244 } else { 245 r = zero; 246 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ 247 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ 248 else {y=x-one;i=2;} 249 } 250 switch(i) { 251 case 0: 252 z = y*y; 253 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 254 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 255 p = y*p1+p2; 256 r += (p-y/2); break; 257 case 1: 258 z = y*y; 259 w = z*y; 260 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 261 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 262 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 263 p = z*p1-(tt-w*(p2+y*p3)); 264 r += (tf + p); break; 265 case 2: 266 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 267 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 268 r += (-0.5*y + p1/p2); 269 } 270 } 271 else if(ix<0x40200000) { /* x < 8.0 */ 272 i = (int)x; 273 y = x-(double)i; 274 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 275 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 276 r = half*y+p/q; 277 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 278 switch(i) { 279 case 7: z *= (y+6); /* FALLTHRU */ 280 case 6: z *= (y+5); /* FALLTHRU */ 281 case 5: z *= (y+4); /* FALLTHRU */ 282 case 4: z *= (y+3); /* FALLTHRU */ 283 case 3: z *= (y+2); /* FALLTHRU */ 284 r += __ieee754_log(z); break; 285 } 286 /* 8.0 <= x < 2**58 */ 287 } else if (ix < 0x43900000) { 288 t = __ieee754_log(x); 289 z = one/x; 290 y = z*z; 291 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 292 r = (x-half)*(t-one)+w; 293 } else 294 /* 2**58 <= x <= inf */ 295 r = x*(__ieee754_log(x)-one); 296 if(hx<0) r = nadj - r; 297 return r; 298 } 299 300 #if (LDBL_MANT_DIG == 53) 301 __weak_reference(lgamma_r, lgammal_r); 302 #endif 303