e_lgamma_r.c 12 KB

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  1. /*
  2. * ====================================================
  3. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  4. *
  5. * Developed at SunPro, a Sun Microsystems, Inc. business.
  6. * Permission to use, copy, modify, and distribute this
  7. * software is freely granted, provided that this notice
  8. * is preserved.
  9. * ====================================================
  10. */
  11. /* __ieee754_lgamma_r(x, signgamp)
  12. * Reentrant version of the logarithm of the Gamma function
  13. * with user provide pointer for the sign of Gamma(x).
  14. *
  15. * Method:
  16. * 1. Argument Reduction for 0 < x <= 8
  17. * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  18. * reduce x to a number in [1.5,2.5] by
  19. * lgamma(1+s) = log(s) + lgamma(s)
  20. * for example,
  21. * lgamma(7.3) = log(6.3) + lgamma(6.3)
  22. * = log(6.3*5.3) + lgamma(5.3)
  23. * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  24. * 2. Polynomial approximation of lgamma around its
  25. * minimun ymin=1.461632144968362245 to maintain monotonicity.
  26. * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  27. * Let z = x-ymin;
  28. * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  29. * where
  30. * poly(z) is a 14 degree polynomial.
  31. * 2. Rational approximation in the primary interval [2,3]
  32. * We use the following approximation:
  33. * s = x-2.0;
  34. * lgamma(x) = 0.5*s + s*P(s)/Q(s)
  35. * with accuracy
  36. * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  37. * Our algorithms are based on the following observation
  38. *
  39. * zeta(2)-1 2 zeta(3)-1 3
  40. * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
  41. * 2 3
  42. *
  43. * where Euler = 0.5771... is the Euler constant, which is very
  44. * close to 0.5.
  45. *
  46. * 3. For x>=8, we have
  47. * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  48. * (better formula:
  49. * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  50. * Let z = 1/x, then we approximation
  51. * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  52. * by
  53. * 3 5 11
  54. * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
  55. * where
  56. * |w - f(z)| < 2**-58.74
  57. *
  58. * 4. For negative x, since (G is gamma function)
  59. * -x*G(-x)*G(x) = pi/sin(pi*x),
  60. * we have
  61. * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  62. * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  63. * Hence, for x<0, signgam = sign(sin(pi*x)) and
  64. * lgamma(x) = log(|Gamma(x)|)
  65. * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  66. * Note: one should avoid compute pi*(-x) directly in the
  67. * computation of sin(pi*(-x)).
  68. *
  69. * 5. Special Cases
  70. * lgamma(2+s) ~ s*(1-Euler) for tiny s
  71. * lgamma(1)=lgamma(2)=0
  72. * lgamma(x) ~ -log(x) for tiny x
  73. * lgamma(0) = lgamma(inf) = inf
  74. * lgamma(-integer) = +-inf
  75. *
  76. */
  77. #include "math.h"
  78. #include "math_private.h"
  79. static const double
  80. two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
  81. half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  82. one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  83. pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  84. a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  85. a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  86. a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
  87. a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
  88. a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
  89. a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
  90. a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
  91. a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
  92. a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
  93. a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
  94. a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
  95. a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  96. tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
  97. tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
  98. /* tt = -(tail of tf) */
  99. tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
  100. t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
  101. t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
  102. t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
  103. t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
  104. t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
  105. t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
  106. t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
  107. t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
  108. t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
  109. t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
  110. t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
  111. t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
  112. t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
  113. t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
  114. t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
  115. u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  116. u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
  117. u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
  118. u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
  119. u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
  120. u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
  121. v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
  122. v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
  123. v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
  124. v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
  125. v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
  126. s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
  127. s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
  128. s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
  129. s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
  130. s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
  131. s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
  132. s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
  133. r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
  134. r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
  135. r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
  136. r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
  137. r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
  138. r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
  139. w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
  140. w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
  141. w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
  142. w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
  143. w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
  144. w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
  145. w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
  146. static const double zero= 0.00000000000000000000e+00;
  147. static
  148. #ifdef __GNUC__
  149. __inline__
  150. #endif
  151. double sin_pi(double x)
  152. {
  153. double y,z;
  154. int n,ix;
  155. GET_HIGH_WORD(ix,x);
  156. ix &= 0x7fffffff;
  157. if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
  158. y = -x; /* x is assume negative */
  159. /*
  160. * argument reduction, make sure inexact flag not raised if input
  161. * is an integer
  162. */
  163. z = floor(y);
  164. if(z!=y) { /* inexact anyway */
  165. y *= 0.5;
  166. y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
  167. n = (int) (y*4.0);
  168. } else {
  169. if(ix>=0x43400000) {
  170. y = zero; n = 0; /* y must be even */
  171. } else {
  172. if(ix<0x43300000) z = y+two52; /* exact */
  173. GET_LOW_WORD(n,z);
  174. n &= 1;
  175. y = n;
  176. n<<= 2;
  177. }
  178. }
  179. switch (n) {
  180. case 0: y = __kernel_sin(pi*y,zero,0); break;
  181. case 1:
  182. case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
  183. case 3:
  184. case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
  185. case 5:
  186. case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
  187. default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
  188. }
  189. return -y;
  190. }
  191. double __ieee754_lgamma_r(double x, int *signgamp)
  192. {
  193. double t,y,z,nadj=0,p,p1,p2,p3,q,r,w;
  194. int i,hx,lx,ix;
  195. EXTRACT_WORDS(hx,lx,x);
  196. /* purge off +-inf, NaN, +-0, and negative arguments */
  197. *signgamp = 1;
  198. ix = hx&0x7fffffff;
  199. if(ix>=0x7ff00000) return x*x;
  200. if((ix|lx)==0) {
  201. if (signbit(x))
  202. *signgamp = -1;
  203. return one/zero;
  204. }
  205. if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
  206. if(hx<0) {
  207. *signgamp = -1;
  208. return -__ieee754_log(-x);
  209. } else return -__ieee754_log(x);
  210. }
  211. if(hx<0) {
  212. if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
  213. return one/zero;
  214. t = sin_pi(x);
  215. if(t==zero) return one/zero; /* -integer */
  216. nadj = __ieee754_log(pi/fabs(t*x));
  217. if(t<zero) *signgamp = -1;
  218. x = -x;
  219. }
  220. /* purge off 1 and 2 */
  221. if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
  222. /* for x < 2.0 */
  223. else if(ix<0x40000000) {
  224. if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
  225. r = -__ieee754_log(x);
  226. if(ix>=0x3FE76944) {y = one-x; i= 0;}
  227. else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
  228. else {y = x; i=2;}
  229. } else {
  230. r = zero;
  231. if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
  232. else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
  233. else {y=x-one;i=2;}
  234. }
  235. switch(i) {
  236. case 0:
  237. z = y*y;
  238. p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
  239. p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
  240. p = y*p1+p2;
  241. r += (p-0.5*y); break;
  242. case 1:
  243. z = y*y;
  244. w = z*y;
  245. p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
  246. p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
  247. p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
  248. p = z*p1-(tt-w*(p2+y*p3));
  249. r += (tf + p); break;
  250. case 2:
  251. p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
  252. p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
  253. r += (-0.5*y + p1/p2);
  254. }
  255. }
  256. else if(ix<0x40200000) { /* x < 8.0 */
  257. i = (int)x;
  258. t = zero;
  259. y = x-(double)i;
  260. p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
  261. q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
  262. r = half*y+p/q;
  263. z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
  264. switch(i) {
  265. case 7: z *= (y+6.0); /* FALLTHRU */
  266. case 6: z *= (y+5.0); /* FALLTHRU */
  267. case 5: z *= (y+4.0); /* FALLTHRU */
  268. case 4: z *= (y+3.0); /* FALLTHRU */
  269. case 3: z *= (y+2.0); /* FALLTHRU */
  270. r += __ieee754_log(z); break;
  271. }
  272. /* 8.0 <= x < 2**58 */
  273. } else if (ix < 0x43900000) {
  274. t = __ieee754_log(x);
  275. z = one/x;
  276. y = z*z;
  277. w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
  278. r = (x-half)*(t-one)+w;
  279. } else
  280. /* 2**58 <= x <= inf */
  281. r = x*(__ieee754_log(x)-one);
  282. if(hx<0) r = nadj - r;
  283. return r;
  284. }
  285. strong_alias(__ieee754_lgamma_r, lgamma_r)
  286. libm_hidden_def(lgamma_r)
  287. /* __ieee754_lgamma(x)
  288. * Return the logarithm of the Gamma function of x.
  289. */
  290. double __ieee754_lgamma(double x)
  291. {
  292. return __ieee754_lgamma_r(x, &signgam);
  293. }
  294. strong_alias(__ieee754_lgamma, lgamma);
  295. libm_hidden_def(lgamma)
  296. /* NB: gamma function is an old name for lgamma.
  297. * It is deprecated.
  298. * Some C math libraries redefine it as a "true gamma", i.e.,
  299. * not a ln(|Gamma(x)|) but just Gamma(x), but standards
  300. * introduced tgamma name for that.
  301. */
  302. strong_alias(__ieee754_lgamma_r, gamma_r)
  303. strong_alias(__ieee754_lgamma, gamma)
  304. libm_hidden_def(gamma)
  305. /* double tgamma(double x)
  306. * Return the Gamma function of x.
  307. */
  308. double tgamma(double x)
  309. {
  310. int sign_of_gamma;
  311. int32_t hx;
  312. u_int32_t lx;
  313. /* We don't have a real gamma implementation now. We'll use lgamma
  314. and the exp function. But due to the required boundary
  315. conditions we must check some values separately. */
  316. EXTRACT_WORDS(hx, lx, x);
  317. if (((hx & 0x7fffffff) | lx) == 0) {
  318. /* Return value for x == 0 is Inf with divide by zero exception. */
  319. return 1.0 / x;
  320. }
  321. if (hx < 0 && (u_int32_t)hx < 0xfff00000 && rint(x) == x) {
  322. /* Return value for integer x < 0 is NaN with invalid exception. */
  323. return (x - x) / (x - x);
  324. }
  325. if ((u_int32_t)hx == 0xfff00000 && lx == 0) {
  326. /* x == -Inf. According to ISO this is NaN. */
  327. return x - x;
  328. }
  329. x = exp(lgamma_r(x, &sign_of_gamma));
  330. return sign_of_gamma >= 0 ? x : -x;
  331. }
  332. libm_hidden_def(tgamma)