src/math/lgamma_r.c

   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);