gonzui

    1: /* @(#)er_lgamma.c 5.1 93/09/24 */
    2: /*
    3:  * ====================================================
    4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    5:  *
    6:  * Developed at SunPro, 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(inf) = inf
   76:  *           lgamma(-integer) = +-inf
   77:  *      
   78:  */
   79: 
   80: #include "math.h"
   81: #include "math_private.h"
   82: 
   83: static const double 
   84: two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
   85: half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
   86: one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
   87: pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
   88: a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
   89: a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
   90: a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
   91: a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
   92: a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
   93: a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
   94: a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
   95: a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
   96: a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
   97: a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
   98: a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
   99: a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  100: tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
  101: tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
  102: /* tt = -(tail of tf) */
  103: tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
  104: t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
  105: t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
  106: t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
  107: t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
  108: t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
  109: t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
  110: t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
  111: t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
  112: t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
  113: t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
  114: t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
  115: t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
  116: t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
  117: t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
  118: t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
  119: u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  120: u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
  121: u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
  122: u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
  123: u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
  124: u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
  125: v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
  126: v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
  127: v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
  128: v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
  129: v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
  130: s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  131: s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
  132: s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
  133: s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
  134: s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
  135: s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
  136: s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
  137: r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
  138: r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
  139: r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
  140: r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
  141: r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
  142: r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
  143: w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
  144: w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
  145: w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
  146: w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
  147: w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
  148: w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
  149: w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
  150: 
  151: static const double zero=  0.00000000000000000000e+00;
  152: 
  153: static double
  154: sin_pi(double x)
  155: {
  156:         double y,z;
  157:         int n,ix;
  158: 
  159:         GET_HIGH_WORD(ix,x);
  160:         ix &= 0x7fffffff;
  161: 
  162:         if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
  163:         y = -x;                /* x is assume negative */
  164: 
  165:     /*
  166:      * argument reduction, make sure inexact flag not raised if input
  167:      * is an integer
  168:      */
  169:         z = floor(y);
  170:         if(z!=y) {                             /* inexact anyway */
  171:             y  *= 0.5;
  172:             y   = 2.0*(y - floor(y));          /* y = |x| mod 2.0 */
  173:             n   = (int) (y*4.0);
  174:         } else {
  175:             if(ix>=0x43400000) {
  176:                 y = zero; n = 0;                 /* y must be even */
  177:             } else {
  178:                 if(ix<0x43300000) z = y+two52;  /* exact */
  179:                 GET_LOW_WORD(n,z);
  180:                 n &= 1;
  181:                 y  = n;
  182:                 n<<= 2;
  183:             }
  184:         }
  185:         switch (n) {
  186:             case 0:   y =  __kernel_sin(pi*y,zero,0); break;
  187:             case 1:   
  188:             case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
  189:             case 3:  
  190:             case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
  191:             case 5:
  192:             case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
  193:             default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
  194:             }
  195:         return -y;
  196: }
  197: 
  198: 
  199: double
  200: lgamma_r(double x, int *signgamp)
  201: {
  202:         double t,y,z,nadj,p,p1,p2,p3,q,r,w;
  203:         int i,hx,lx,ix;
  204: 
  205:         EXTRACT_WORDS(hx,lx,x);
  206: 
  207:     /* purge off +-inf, NaN, +-0, and negative arguments */
  208:         *signgamp = 1;
  209:         ix = hx&0x7fffffff;
  210:         if(ix>=0x7ff00000) return x*x;
  211:         if((ix|lx)==0) {
  212:             if(hx<0)
  213:                 *signgamp = -1;
  214:             return one/zero;
  215:         }
  216:         if(ix<0x3b900000) {    /* |x|<2**-70, return -log(|x|) */
  217:             if(hx<0) {
  218:                 *signgamp = -1;
  219:                 return - log(-x);
  220:             } else return - log(x);
  221:         }
  222:         if(hx<0) {
  223:             if(ix>=0x43300000)         /* |x|>=2**52, must be -integer */
  224:                 return one/zero;
  225:             t = sin_pi(x);
  226:             if(t==zero) return one/zero; /* -integer */
  227:             nadj = log(pi/fabs(t*x));
  228:             if(t<zero) *signgamp = -1;
  229:             x = -x;
  230:         }
  231: 
  232:     /* purge off 1 and 2 */
  233:         if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
  234:     /* for x < 2.0 */
  235:         else if(ix<0x40000000) {
  236:             if(ix<=0x3feccccc) {       /* lgamma(x) = lgamma(x+1)-log(x) */
  237:                 r = - log(x);
  238:                 if(ix>=0x3FE76944) {y = one-x; i= 0;}
  239:                 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
  240:                else {y = x; i=2;}
  241:             } else {
  242:                r = zero;
  243:                 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
  244:                 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
  245:                 else {y=x-one;i=2;}
  246:             }
  247:             switch(i) {
  248:               case 0:
  249:                 z = y*y;
  250:                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
  251:                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
  252:                 p  = y*p1+p2;
  253:                 r  += (p-0.5*y); break;
  254:               case 1:
  255:                 z = y*y;
  256:                 w = z*y;
  257:                 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));  /* parallel comp */
  258:                 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
  259:                 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
  260:                 p  = z*p1-(tt-w*(p2+y*p3));
  261:                 r += (tf + p); break;
  262:               case 2:  
  263:                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
  264:                 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
  265:                 r += (-0.5*y + p1/p2);
  266:             }
  267:         }
  268:         else if(ix<0x40200000) {                       /* x < 8.0 */
  269:             i = (int)x;
  270:             t = zero;
  271:             y = x-(double)i;
  272:             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
  273:             q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
  274:             r = half*y+p/q;
  275:             z = one;   /* lgamma(1+s) = log(s) + lgamma(s) */
  276:             switch(i) {
  277:             case 7: z *= (y+6.0);      /* FALLTHRU */
  278:             case 6: z *= (y+5.0);      /* FALLTHRU */
  279:             case 5: z *= (y+4.0);      /* FALLTHRU */
  280:             case 4: z *= (y+3.0);      /* FALLTHRU */
  281:             case 3: z *= (y+2.0);      /* FALLTHRU */
  282:                     r += log(z); break;
  283:             }
  284:     /* 8.0 <= x < 2**58 */
  285:         } else if (ix < 0x43900000) {
  286:             t = log(x);
  287:             z = one/x;
  288:             y = z*z;
  289:             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
  290:             r = (x-half)*(t-one)+w;
  291:         } else 
  292:     /* 2**58 <= x <= inf */
  293:             r =  x*(log(x)-one);
  294:         if(hx<0) r = nadj - r;
  295:         return r;
  296: }
  297: #include <sys/cdefs.h>
  298: #include <float.h>
  299: #if     LDBL_MANT_DIG == 53
  300: #ifdef  lint
  301: #else   /* lint */
  302: __weak_alias(lgammal_r, lgamma_r);
  303: #endif  /* lint */
  304: #endif  /* LDBL_MANT_DIG == 53 */