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  1.  
  2. /* @(#)er_lgamma.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, 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. /* __ieee754_lgamma_r(x, signgamp)
  16.  * Reentrant version of the logarithm of the Gamma function
  17.  * with user provide pointer for the sign of Gamma(x).
  18.  *
  19.  * Method:
  20.  *   1. Argument Reduction for 0 < x <= 8
  21.  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  22.  *      reduce x to a number in [1.5,2.5] by
  23.  *              lgamma(1+s) = log(s) + lgamma(s)
  24.  *      for example,
  25.  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  26.  *                          = log(6.3*5.3) + lgamma(5.3)
  27.  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  28.  *   2. Polynomial approximation of lgamma around its
  29.  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  30.  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  31.  *              Let z = x-ymin;
  32.  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  33.  *      where
  34.  *              poly(z) is a 14 degree polynomial.
  35.  *   2. Rational approximation in the primary interval [2,3]
  36.  *      We use the following approximation:
  37.  *              s = x-2.0;
  38.  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  39.  *      with accuracy
  40.  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  41.  *      Our algorithms are based on the following observation
  42.  *
  43.  *                             zeta(2)-1    2    zeta(3)-1    3
  44.  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  45.  *                                 2                 3
  46.  *
  47.  *      where Euler = 0.5771... is the Euler constant, which is very
  48.  *      close to 0.5.
  49.  *
  50.  *   3. For x>=8, we have
  51.  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  52.  *      (better formula:
  53.  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  54.  *      Let z = 1/x, then we approximation
  55.  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  56.  *      by
  57.  *                                  3       5             11
  58.  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  59.  *      where
  60.  *              |w - f(z)| < 2**-58.74
  61.  *             
  62.  *   4. For negative x, since (G is gamma function)
  63.  *              -x*G(-x)*G(x) = pi/sin(pi*x),
  64.  *      we have
  65.  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  66.  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  67.  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
  68.  *              lgamma(x) = log(|Gamma(x)|)
  69.  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  70.  *      Note: one should avoid compute pi*(-x) directly in the
  71.  *            computation of sin(pi*(-x)).
  72.  *             
  73.  *   5. Special Cases
  74.  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  75.  *              lgamma(1)=lgamma(2)=0
  76.  *              lgamma(x) ~ -log(x) for tiny x
  77.  *              lgamma(0) = lgamma(inf) = inf
  78.  *              lgamma(-integer) = +-inf
  79.  *     
  80.  */
  81.  
  82. #include "fdlibm.h"
  83.  
  84. #ifdef __STDC__
  85. static const double
  86. #else
  87. static double
  88. #endif
  89. two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
  90. half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  91. one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  92. pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  93. a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  94. a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  95. a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
  96. a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
  97. a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
  98. a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
  99. a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
  100. a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
  101. a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
  102. a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
  103. a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
  104. a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  105. tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
  106. tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
  107. /* tt = -(tail of tf) */
  108. tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
  109. t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
  110. t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
  111. t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
  112. t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
  113. t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
  114. t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
  115. t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
  116. t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
  117. t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
  118. t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
  119. t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
  120. t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
  121. t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
  122. t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
  123. t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
  124. u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  125. u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
  126. u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
  127. u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
  128. u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
  129. u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
  130. v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
  131. v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
  132. v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
  133. v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
  134. v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
  135. s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  136. s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
  137. s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
  138. s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
  139. s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
  140. s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
  141. s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
  142. r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
  143. r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
  144. r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
  145. r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
  146. r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
  147. r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
  148. w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
  149. w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
  150. w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
  151. w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
  152. w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
  153. w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
  154. w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
  155.  
  156. #ifdef __STDC__
  157. static const double zero=  0.00000000000000000000e+00;
  158. #else
  159. static double zero=  0.00000000000000000000e+00;
  160. #endif
  161.  
  162. #ifdef __STDC__
  163.         static double sin_pi(double x)
  164. #else
  165.         static double sin_pi(x)
  166.         double x;
  167. #endif
  168. {
  169.         double y,z;
  170.         __int32_t n,ix;
  171.  
  172.         GET_HIGH_WORD(ix,x);
  173.         ix &= 0x7fffffff;
  174.  
  175.         if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
  176.         y = -x;         /* x is assume negative */
  177.  
  178.     /*
  179.      * argument reduction, make sure inexact flag not raised if input
  180.      * is an integer
  181.      */
  182.         z = floor(y);
  183.         if(z!=y) {                              /* inexact anyway */
  184.             y  *= 0.5;
  185.             y   = 2.0*(y - floor(y));           /* y = |x| mod 2.0 */
  186.             n   = (__int32_t) (y*4.0);
  187.         } else {
  188.             if(ix>=0x43400000) {
  189.                 y = zero; n = 0;                 /* y must be even */
  190.             } else {
  191.                 if(ix<0x43300000) z = y+two52;  /* exact */
  192.                 GET_LOW_WORD(n,z);
  193.                 n &= 1;
  194.                 y  = n;
  195.                 n<<= 2;
  196.             }
  197.         }
  198.         switch (n) {
  199.             case 0:   y =  __kernel_sin(pi*y,zero,0); break;
  200.             case 1:  
  201.             case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
  202.             case 3:  
  203.             case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
  204.             case 5:
  205.             case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
  206.             default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
  207.             }
  208.         return -y;
  209. }
  210.  
  211.  
  212. #ifdef __STDC__
  213.         double __ieee754_lgamma_r(double x, int *signgamp)
  214. #else
  215.         double __ieee754_lgamma_r(x,signgamp)
  216.         double x; int *signgamp;
  217. #endif
  218. {
  219.         double t,y,z,nadj = 0.0,p,p1,p2,p3,q,r,w;
  220.         __int32_t i,hx,lx,ix;
  221.  
  222.         EXTRACT_WORDS(hx,lx,x);
  223.  
  224.     /* purge off +-inf, NaN, +-0, and negative arguments */
  225.         *signgamp = 1;
  226.         ix = hx&0x7fffffff;
  227.         if(ix>=0x7ff00000) return x*x;
  228.         if((ix|lx)==0) return one/zero;
  229.         if(ix<0x3b900000) {     /* |x|<2**-70, return -log(|x|) */
  230.             if(hx<0) {
  231.                 *signgamp = -1;
  232.                 return -__ieee754_log(-x);
  233.             } else return -__ieee754_log(x);
  234.         }
  235.         if(hx<0) {
  236.             if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
  237.                 return one/zero;
  238.             t = sin_pi(x);
  239.             if(t==zero) return one/zero; /* -integer */
  240.             nadj = __ieee754_log(pi/fabs(t*x));
  241.             if(t<zero) *signgamp = -1;
  242.             x = -x;
  243.         }
  244.  
  245.     /* purge off 1 and 2 */
  246.         if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
  247.     /* for x < 2.0 */
  248.         else if(ix<0x40000000) {
  249.             if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */
  250.                 r = -__ieee754_log(x);
  251.                 if(ix>=0x3FE76944) {y = one-x; i= 0;}
  252.                 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
  253.                 else {y = x; i=2;}
  254.             } else {
  255.                 r = zero;
  256.                 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
  257.                 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
  258.                 else {y=x-one;i=2;}
  259.             }
  260.             switch(i) {
  261.               case 0:
  262.                 z = y*y;
  263.                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
  264.                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
  265.                 p  = y*p1+p2;
  266.                 r  += (p-0.5*y); break;
  267.               case 1:
  268.                 z = y*y;
  269.                 w = z*y;
  270.                 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
  271.                 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
  272.                 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
  273.                 p  = z*p1-(tt-w*(p2+y*p3));
  274.                 r += (tf + p); break;
  275.               case 2:  
  276.                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
  277.                 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
  278.                 r += (-0.5*y + p1/p2);
  279.             }
  280.         }
  281.         else if(ix<0x40200000) {                        /* x < 8.0 */
  282.             i = (__int32_t)x;
  283.             t = zero;
  284.             y = x-(double)i;
  285.             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
  286.             q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
  287.             r = half*y+p/q;
  288.             z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
  289.             switch(i) {
  290.             case 7: z *= (y+6.0);       /* FALLTHRU */
  291.             case 6: z *= (y+5.0);       /* FALLTHRU */
  292.             case 5: z *= (y+4.0);       /* FALLTHRU */
  293.             case 4: z *= (y+3.0);       /* FALLTHRU */
  294.             case 3: z *= (y+2.0);       /* FALLTHRU */
  295.                     r += __ieee754_log(z); break;
  296.             }
  297.     /* 8.0 <= x < 2**58 */
  298.         } else if (ix < 0x43900000) {
  299.             t = __ieee754_log(x);
  300.             z = one/x;
  301.             y = z*z;
  302.             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
  303.             r = (x-half)*(t-one)+w;
  304.         } else
  305.     /* 2**58 <= x <= inf */
  306.             r =  x*(__ieee754_log(x)-one);
  307.         if(hx<0) r = nadj - r;
  308.         return r;
  309. }
  310.