e_lgamma_r.c 11 KB

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  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. #if defined(LIBM_SCCS) && !defined(lint)
  13. static char rcsid[] = "$NetBSD: e_lgamma_r.c,v 1.7 1995/05/10 20:45:42 jtc Exp $";
  14. #endif
  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. #include "math.h"
  82. #include "math_private.h"
  83. #ifdef __STDC__
  84. static const double
  85. #else
  86. static double
  87. #endif
  88. two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
  89. half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  90. one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  91. pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  92. a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  93. a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  94. a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
  95. a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
  96. a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
  97. a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
  98. a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
  99. a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
  100. a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
  101. a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
  102. a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
  103. a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  104. tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
  105. tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
  106. /* tt = -(tail of tf) */
  107. tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
  108. t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
  109. t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
  110. t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
  111. t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
  112. t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
  113. t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
  114. t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
  115. t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
  116. t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
  117. t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
  118. t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
  119. t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
  120. t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
  121. t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
  122. t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
  123. u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  124. u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
  125. u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
  126. u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
  127. u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
  128. u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
  129. v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
  130. v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
  131. v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
  132. v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
  133. v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
  134. s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  135. s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
  136. s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
  137. s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
  138. s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
  139. s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
  140. s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
  141. r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
  142. r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
  143. r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
  144. r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
  145. r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
  146. r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
  147. w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
  148. w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
  149. w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
  150. w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
  151. w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
  152. w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
  153. w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
  154. #ifdef __STDC__
  155. static const double zero= 0.00000000000000000000e+00;
  156. #else
  157. static double zero= 0.00000000000000000000e+00;
  158. #endif
  159. static
  160. #ifdef __GNUC__
  161. __inline__
  162. #endif
  163. #ifdef __STDC__
  164. double sin_pi(double x)
  165. #else
  166. double sin_pi(x)
  167. double x;
  168. #endif
  169. {
  170. double y,z;
  171. int n,ix;
  172. GET_HIGH_WORD(ix,x);
  173. ix &= 0x7fffffff;
  174. if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
  175. y = -x; /* x is assume negative */
  176. /*
  177. * argument reduction, make sure inexact flag not raised if input
  178. * is an integer
  179. */
  180. z = floor(y);
  181. if(z!=y) { /* inexact anyway */
  182. y *= 0.5;
  183. y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
  184. n = (int) (y*4.0);
  185. } else {
  186. if(ix>=0x43400000) {
  187. y = zero; n = 0; /* y must be even */
  188. } else {
  189. if(ix<0x43300000) z = y+two52; /* exact */
  190. GET_LOW_WORD(n,z);
  191. n &= 1;
  192. y = n;
  193. n<<= 2;
  194. }
  195. }
  196. switch (n) {
  197. case 0: y = __kernel_sin(pi*y,zero,0); break;
  198. case 1:
  199. case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
  200. case 3:
  201. case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
  202. case 5:
  203. case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
  204. default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
  205. }
  206. return -y;
  207. }
  208. #ifdef __STDC__
  209. double __ieee754_lgamma_r(double x, int *signgamp)
  210. #else
  211. double __ieee754_lgamma_r(x,signgamp)
  212. double x; int *signgamp;
  213. #endif
  214. {
  215. double t,y,z,nadj,p,p1,p2,p3,q,r,w;
  216. int i,hx,lx,ix;
  217. EXTRACT_WORDS(hx,lx,x);
  218. /* purge off +-inf, NaN, +-0, and negative arguments */
  219. *signgamp = 1;
  220. ix = hx&0x7fffffff;
  221. if(ix>=0x7ff00000) return x*x;
  222. if((ix|lx)==0) return one/zero;
  223. if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
  224. if(hx<0) {
  225. *signgamp = -1;
  226. return -__ieee754_log(-x);
  227. } else return -__ieee754_log(x);
  228. }
  229. if(hx<0) {
  230. if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
  231. return one/zero;
  232. t = sin_pi(x);
  233. if(t==zero) return one/zero; /* -integer */
  234. nadj = __ieee754_log(pi/fabs(t*x));
  235. if(t<zero) *signgamp = -1;
  236. x = -x;
  237. }
  238. /* purge off 1 and 2 */
  239. if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
  240. /* for x < 2.0 */
  241. else if(ix<0x40000000) {
  242. if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
  243. r = -__ieee754_log(x);
  244. if(ix>=0x3FE76944) {y = one-x; i= 0;}
  245. else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
  246. else {y = x; i=2;}
  247. } else {
  248. r = zero;
  249. if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
  250. else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
  251. else {y=x-one;i=2;}
  252. }
  253. switch(i) {
  254. case 0:
  255. z = y*y;
  256. p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
  257. p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
  258. p = y*p1+p2;
  259. r += (p-0.5*y); break;
  260. case 1:
  261. z = y*y;
  262. w = z*y;
  263. p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
  264. p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
  265. p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
  266. p = z*p1-(tt-w*(p2+y*p3));
  267. r += (tf + p); break;
  268. case 2:
  269. p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
  270. p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
  271. r += (-0.5*y + p1/p2);
  272. }
  273. }
  274. else if(ix<0x40200000) { /* x < 8.0 */
  275. i = (int)x;
  276. t = zero;
  277. y = x-(double)i;
  278. p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
  279. q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
  280. r = half*y+p/q;
  281. z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
  282. switch(i) {
  283. case 7: z *= (y+6.0); /* FALLTHRU */
  284. case 6: z *= (y+5.0); /* FALLTHRU */
  285. case 5: z *= (y+4.0); /* FALLTHRU */
  286. case 4: z *= (y+3.0); /* FALLTHRU */
  287. case 3: z *= (y+2.0); /* FALLTHRU */
  288. r += __ieee754_log(z); break;
  289. }
  290. /* 8.0 <= x < 2**58 */
  291. } else if (ix < 0x43900000) {
  292. t = __ieee754_log(x);
  293. z = one/x;
  294. y = z*z;
  295. w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
  296. r = (x-half)*(t-one)+w;
  297. } else
  298. /* 2**58 <= x <= inf */
  299. r = x*(__ieee754_log(x)-one);
  300. if(hx<0) r = nadj - r;
  301. return r;
  302. }