compiler/stdc/math/e_lgamma_r.c

   1
   2 /* @(#)e_lgamma_r.c 1.3 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  *
  13  */
  14
  15 #ifndef lint
  16 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_lgamma_r.c,v 1.11 2011/10/15 07:00:28 das Exp $";
  17 #endif
  18
  19 /* __ieee754_lgamma_r(x, signgamp)
  20  * Reentrant version of the logarithm of the Gamma function
  21  * with user provide pointer for the sign of Gamma(x).
  22  *
  23  * Method:
  24  *   1. Argument Reduction for 0 < x <= 8
  25  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  26  *      reduce x to a number in [1.5,2.5] by
  27  *              lgamma(1+s) = log(s) + lgamma(s)
  28  *      for example,
  29  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  30  *                          = log(6.3*5.3) + lgamma(5.3)
  31  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  32  *   2. Polynomial approximation of lgamma around its
  33  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  34  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  35  *              Let z = x-ymin;
  36  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  37  *      where
  38  *              poly(z) is a 14 degree polynomial.
  39  *   2. Rational approximation in the primary interval [2,3]
  40  *      We use the following approximation:
  41  *              s = x-2.0;
  42  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  43  *      with accuracy
  44  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  45  *      Our algorithms are based on the following observation
  46  *
  47  *                             zeta(2)-1    2    zeta(3)-1    3
  48  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  49  *                                 2                 3
  50  *
  51  *      where Euler = 0.5771... is the Euler constant, which is very
  52  *      close to 0.5.
  53  *
  54  *   3. For x>=8, we have
  55  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  56  *      (better formula:
  57  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  58  *      Let z = 1/x, then we approximation
  59  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  60  *      by
  61  *                                  3       5             11
  62  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  63  *      where
  64  *              |w - f(z)| < 2**-58.74
  65  *
  66  *   4. For negative x, since (G is gamma function)
  67  *              -x*G(-x)*G(x) = pi/sin(pi*x),
  68  *      we have
  69  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  70  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  71  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
  72  *              lgamma(x) = log(|Gamma(x)|)
  73  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  74  *      Note: one should avoid compute pi*(-x) directly in the
  75  *            computation of sin(pi*(-x)).
  76  *
  77  *   5. Special Cases
  78  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  79  *              lgamma(1) = lgamma(2) = 0
  80  *              lgamma(x) ~ -log(|x|) for tiny x
  81  *              lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
  82  *              lgamma(inf) = inf
  83  *              lgamma(-inf) = inf (bug for bug compatible with C99!?)
  84  */
  85
  86 #include <float.h>
  87 #include "math.h"
  88 #include "math_private.h"
  89
  90 static const volatile double vzero __attribute__ ((__section__(".rodata,\"a\" " SECTIONCOMMENT))) = 0.0;
  91
  92 static const double
  93 zero=  0.00000000000000000000e+00,
  94 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  95 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  96 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  97 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  98 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  99 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
 100 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
 101 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
 102 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
 103 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
 104 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
 105 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
 106 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
 107 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
 108 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
 109 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
 110 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
 111 /* tt = -(tail of tf) */
 112 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
 113 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
 114 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
 115 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
 116 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
 117 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
 118 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
 119 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
 120 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
 121 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
 122 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
 123 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
 124 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
 125 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
 126 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
 127 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
 128 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 129 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
 130 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
 131 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
 132 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
 133 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
 134 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
 135 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
 136 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
 137 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
 138 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
 139 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 140 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
 141 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
 142 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
 143 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
 144 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
 145 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
 146 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
 147 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
 148 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
 149 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
 150 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
 151 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
 152 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
 153 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
 154 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
 155 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
 156 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 157 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 158 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 159
 160 /*
 161  * Compute sin(pi*x) without actually doing the pi*x multiplication.
 162  * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is
 163  * the precision of x.
 164  */
 165 static double
 166 sin_pi(double x)
 167 {
 168         volatile double vz;
 169         double y,z;
 170         int n;
 171
 172         y = -x;
 173
 174         vz = y+0x1p52;                  /* depend on 0 <= y < 0x1p52 */
 175         z = vz-0x1p52;                  /* rint(y) for the above range */
 176         if (z == y)
 177             return zero;
 178
 179         vz = y+0x1p50;
 180         GET_LOW_WORD(n,vz);             /* bits for rounded y (units 0.25) */
 181         z = vz-0x1p50;                  /* y rounded to a multiple of 0.25 */
 182         if (z > y) {
 183             z -= 0.25;                  /* adjust to round down */
 184             n--;
 185         }
 186         n &= 7;                         /* octant of y mod 2 */
 187         y = y - z + n * 0.25;           /* y mod 2 */
 188
 189         switch (n) {
 190             case 0:   y =  __kernel_sin(pi*y,zero,0); break;
 191             case 1:
 192             case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
 193             case 3:
 194             case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
 195             case 5:
 196             case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
 197             default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
 198             }
 199         return -y;
 200 }
 201
 202
 203 double
 204 __ieee754_lgamma_r(double x, int *signgamp)
 205 {
 206         double nadj,p,p1,p2,p3,q,r,t,w,y,z;
 207         int32_t hx;
 208         int i,ix,lx;
 209
 210         EXTRACT_WORDS(hx,lx,x);
 211
 212     /* purge +-Inf and NaNs */
 213         *signgamp = 1;
 214         ix = hx&0x7fffffff;
 215         if(ix>=0x7ff00000) return x*x;
 216
 217     /* purge +-0 and tiny arguments */
 218         *signgamp = 1-2*((uint32_t)hx>>31);
 219         if(ix<0x3c700000) {     /* |x|<2**-56, return -log(|x|) */
 220             if((ix|lx)==0)
 221                 return one/vzero;
 222             return -__ieee754_log(fabs(x));
 223         }
 224     /* purge negative integers and start evaluation for other x < 0 */
 225         if(hx<0) {
 226             *signgamp = 1;
 227             if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
 228                 return one/vzero;
 229             t = sin_pi(x);
 230             if(t==zero) return one/vzero; /* -integer */
 231             nadj = __ieee754_log(pi/fabs(t*x));
 232             if(t<zero) *signgamp = -1;
 233             x = -x;
 234         }
 235
 236     /* purge 1 and 2 */
 237         if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
 238     /* for x < 2.0 */
 239         else if(ix<0x40000000) {
 240             if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */
 241                 r = -__ieee754_log(x);
 242                 if(ix>=0x3FE76944) {y = one-x; i= 0;}
 243                 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
 244                 else {y = x; i=2;}
 245             } else {
 246                 r = zero;
 247                 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
 248                 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
 249                 else {y=x-one;i=2;}
 250             }
 251             switch(i) {
 252               case 0:
 253                 z = y*y;
 254                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
 255                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
 256                 p  = y*p1+p2;
 257                 r  += p-y/2; break;
 258               case 1:
 259                 z = y*y;
 260                 w = z*y;
 261                 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
 262                 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
 263                 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
 264                 p  = z*p1-(tt-w*(p2+y*p3));
 265                 r += tf + p; break;
 266               case 2:
 267                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
 268                 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
 269                 r += p1/p2-y/2;
 270             }
 271         }
 272     /* x < 8.0 */
 273         else if(ix<0x40200000) {
 274             i = x;
 275             y = x-i;
 276             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
 277             q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
 278             r = y/2+p/q;
 279             z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
 280             switch(i) {
 281             case 7: z *= (y+6);         /* FALLTHRU */
 282             case 6: z *= (y+5);         /* FALLTHRU */
 283             case 5: z *= (y+4);         /* FALLTHRU */
 284             case 4: z *= (y+3);         /* FALLTHRU */
 285             case 3: z *= (y+2);         /* FALLTHRU */
 286                     r += __ieee754_log(z); break;
 287             }
 288     /* 8.0 <= x < 2**56 */
 289         } else if (ix < 0x43700000) {
 290             t = __ieee754_log(x);
 291             z = one/x;
 292             y = z*z;
 293             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
 294             r = (x-half)*(t-one)+w;
 295         } else
 296     /* 2**56 <= x <= inf */
 297             r =  x*(__ieee754_log(x)-one);
 298         if(hx<0) r = nadj - r;
 299         return r;
 300 }
 301
 302 #if     LDBL_MANT_DIG == DBL_MANT_DIG
 303 AROS_MAKE_ASM_SYM(typeof(lgammal_r), lgammal_r, AROS_CSYM_FROM_ASM_NAME(lgammal_r), AROS_CSYM_FROM_ASM_NAME(lgamma_r));
 304 AROS_EXPORT_ASM_SYM(AROS_CSYM_FROM_ASM_NAME(lgammal_r));
 305 #endif