e_pow.c 10 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_pow(x,y) return x**y
  12. *
  13. * n
  14. * Method: Let x = 2 * (1+f)
  15. * 1. Compute and return log2(x) in two pieces:
  16. * log2(x) = w1 + w2,
  17. * where w1 has 53-24 = 29 bit trailing zeros.
  18. * 2. Perform y*log2(x) = n+y' by simulating muti-precision
  19. * arithmetic, where |y'|<=0.5.
  20. * 3. Return x**y = 2**n*exp(y'*log2)
  21. *
  22. * Special cases:
  23. * 1. +-1 ** anything is 1.0
  24. * 2. +-1 ** +-INF is 1.0
  25. * 3. (anything) ** 0 is 1
  26. * 4. (anything) ** 1 is itself
  27. * 5. (anything) ** NAN is NAN
  28. * 6. NAN ** (anything except 0) is NAN
  29. * 7. +-(|x| > 1) ** +INF is +INF
  30. * 8. +-(|x| > 1) ** -INF is +0
  31. * 9. +-(|x| < 1) ** +INF is +0
  32. * 10 +-(|x| < 1) ** -INF is +INF
  33. * 11. +0 ** (+anything except 0, NAN) is +0
  34. * 12. -0 ** (+anything except 0, NAN, odd integer) is +0
  35. * 13. +0 ** (-anything except 0, NAN) is +INF
  36. * 14. -0 ** (-anything except 0, NAN, odd integer) is +INF
  37. * 15. -0 ** (odd integer) = -( +0 ** (odd integer) )
  38. * 16. +INF ** (+anything except 0,NAN) is +INF
  39. * 17. +INF ** (-anything except 0,NAN) is +0
  40. * 18. -INF ** (anything) = -0 ** (-anything)
  41. * 19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  42. * 20. (-anything except 0 and inf) ** (non-integer) is NAN
  43. *
  44. * Accuracy:
  45. * pow(x,y) returns x**y nearly rounded. In particular
  46. * pow(integer,integer)
  47. * always returns the correct integer provided it is
  48. * representable.
  49. *
  50. * Constants :
  51. * The hexadecimal values are the intended ones for the following
  52. * constants. The decimal values may be used, provided that the
  53. * compiler will convert from decimal to binary accurately enough
  54. * to produce the hexadecimal values shown.
  55. */
  56. #include "math.h"
  57. #include "math_private.h"
  58. static const double
  59. bp[] = {1.0, 1.5,},
  60. dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  61. dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  62. zero = 0.0,
  63. one = 1.0,
  64. two = 2.0,
  65. two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
  66. huge = 1.0e300,
  67. tiny = 1.0e-300,
  68. /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  69. L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  70. L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  71. L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  72. L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  73. L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  74. L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  75. P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  76. P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  77. P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  78. P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  79. P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  80. lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  81. lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  82. lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  83. ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  84. cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  85. cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  86. cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  87. ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  88. ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  89. ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  90. double attribute_hidden __ieee754_pow(double x, double y)
  91. {
  92. double z,ax,z_h,z_l,p_h,p_l;
  93. double y1,t1,t2,r,s,t,u,v,w;
  94. int32_t i,j,k,yisint,n;
  95. int32_t hx,hy,ix,iy;
  96. u_int32_t lx,ly;
  97. EXTRACT_WORDS(hx,lx,x);
  98. /* x==1: 1**y = 1 (even if y is NaN) */
  99. if (hx==0x3ff00000 && lx==0) {
  100. return x;
  101. }
  102. ix = hx&0x7fffffff;
  103. EXTRACT_WORDS(hy,ly,y);
  104. iy = hy&0x7fffffff;
  105. /* y==zero: x**0 = 1 */
  106. if((iy|ly)==0) return one;
  107. /* +-NaN return x+y */
  108. if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
  109. iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
  110. return x+y;
  111. /* determine if y is an odd int when x < 0
  112. * yisint = 0 ... y is not an integer
  113. * yisint = 1 ... y is an odd int
  114. * yisint = 2 ... y is an even int
  115. */
  116. yisint = 0;
  117. if(hx<0) {
  118. if(iy>=0x43400000) yisint = 2; /* even integer y */
  119. else if(iy>=0x3ff00000) {
  120. k = (iy>>20)-0x3ff; /* exponent */
  121. if(k>20) {
  122. j = ly>>(52-k);
  123. if((j<<(52-k))==ly) yisint = 2-(j&1);
  124. } else if(ly==0) {
  125. j = iy>>(20-k);
  126. if((j<<(20-k))==iy) yisint = 2-(j&1);
  127. }
  128. }
  129. }
  130. /* special value of y */
  131. if(ly==0) {
  132. if (iy==0x7ff00000) { /* y is +-inf */
  133. if (((ix-0x3ff00000)|lx)==0)
  134. return one; /* +-1**+-inf is 1 (yes, weird rule) */
  135. if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
  136. return (hy>=0) ? y : zero;
  137. /* (|x|<1)**-,+inf = inf,0 */
  138. return (hy<0) ? -y : zero;
  139. }
  140. if(iy==0x3ff00000) { /* y is +-1 */
  141. if(hy<0) return one/x; else return x;
  142. }
  143. if(hy==0x40000000) return x*x; /* y is 2 */
  144. if(hy==0x3fe00000) { /* y is 0.5 */
  145. if(hx>=0) /* x >= +0 */
  146. return __ieee754_sqrt(x);
  147. }
  148. }
  149. ax = fabs(x);
  150. /* special value of x */
  151. if(lx==0) {
  152. if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
  153. z = ax; /*x is +-0,+-inf,+-1*/
  154. if(hy<0) z = one/z; /* z = (1/|x|) */
  155. if(hx<0) {
  156. if(((ix-0x3ff00000)|yisint)==0) {
  157. z = (z-z)/(z-z); /* (-1)**non-int is NaN */
  158. } else if(yisint==1)
  159. z = -z; /* (x<0)**odd = -(|x|**odd) */
  160. }
  161. return z;
  162. }
  163. }
  164. /* (x<0)**(non-int) is NaN */
  165. if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
  166. /* |y| is huge */
  167. if(iy>0x41e00000) { /* if |y| > 2**31 */
  168. if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
  169. if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  170. if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  171. }
  172. /* over/underflow if x is not close to one */
  173. if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  174. if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  175. /* now |1-x| is tiny <= 2**-20, suffice to compute
  176. log(x) by x-x^2/2+x^3/3-x^4/4 */
  177. t = x-1; /* t has 20 trailing zeros */
  178. w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
  179. u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
  180. v = t*ivln2_l-w*ivln2;
  181. t1 = u+v;
  182. SET_LOW_WORD(t1,0);
  183. t2 = v-(t1-u);
  184. } else {
  185. double s2,s_h,s_l,t_h,t_l;
  186. n = 0;
  187. /* take care subnormal number */
  188. if(ix<0x00100000)
  189. {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
  190. n += ((ix)>>20)-0x3ff;
  191. j = ix&0x000fffff;
  192. /* determine interval */
  193. ix = j|0x3ff00000; /* normalize ix */
  194. if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
  195. else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
  196. else {k=0;n+=1;ix -= 0x00100000;}
  197. SET_HIGH_WORD(ax,ix);
  198. /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
  199. u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
  200. v = one/(ax+bp[k]);
  201. s = u*v;
  202. s_h = s;
  203. SET_LOW_WORD(s_h,0);
  204. /* t_h=ax+bp[k] High */
  205. t_h = zero;
  206. SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
  207. t_l = ax - (t_h-bp[k]);
  208. s_l = v*((u-s_h*t_h)-s_h*t_l);
  209. /* compute log(ax) */
  210. s2 = s*s;
  211. r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  212. r += s_l*(s_h+s);
  213. s2 = s_h*s_h;
  214. t_h = 3.0+s2+r;
  215. SET_LOW_WORD(t_h,0);
  216. t_l = r-((t_h-3.0)-s2);
  217. /* u+v = s*(1+...) */
  218. u = s_h*t_h;
  219. v = s_l*t_h+t_l*s;
  220. /* 2/(3log2)*(s+...) */
  221. p_h = u+v;
  222. SET_LOW_WORD(p_h,0);
  223. p_l = v-(p_h-u);
  224. z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
  225. z_l = cp_l*p_h+p_l*cp+dp_l[k];
  226. /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
  227. t = (double)n;
  228. t1 = (((z_h+z_l)+dp_h[k])+t);
  229. SET_LOW_WORD(t1,0);
  230. t2 = z_l-(((t1-t)-dp_h[k])-z_h);
  231. }
  232. s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  233. if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
  234. s = -one;/* (-ve)**(odd int) */
  235. /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
  236. y1 = y;
  237. SET_LOW_WORD(y1,0);
  238. p_l = (y-y1)*t1+y*t2;
  239. p_h = y1*t1;
  240. z = p_l+p_h;
  241. EXTRACT_WORDS(j,i,z);
  242. if (j>=0x40900000) { /* z >= 1024 */
  243. if(((j-0x40900000)|i)!=0) /* if z > 1024 */
  244. return s*huge*huge; /* overflow */
  245. else {
  246. if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
  247. }
  248. } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
  249. if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
  250. return s*tiny*tiny; /* underflow */
  251. else {
  252. if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
  253. }
  254. }
  255. /*
  256. * compute 2**(p_h+p_l)
  257. */
  258. i = j&0x7fffffff;
  259. k = (i>>20)-0x3ff;
  260. n = 0;
  261. if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
  262. n = j+(0x00100000>>(k+1));
  263. k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
  264. t = zero;
  265. SET_HIGH_WORD(t,n&~(0x000fffff>>k));
  266. n = ((n&0x000fffff)|0x00100000)>>(20-k);
  267. if(j<0) n = -n;
  268. p_h -= t;
  269. }
  270. t = p_l+p_h;
  271. SET_LOW_WORD(t,0);
  272. u = t*lg2_h;
  273. v = (p_l-(t-p_h))*lg2+t*lg2_l;
  274. z = u+v;
  275. w = v-(z-u);
  276. t = z*z;
  277. t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  278. r = (z*t1)/(t1-two)-(w+z*w);
  279. z = one-(r-z);
  280. GET_HIGH_WORD(j,z);
  281. j += (n<<20);
  282. if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
  283. else SET_HIGH_WORD(z,j);
  284. return s*z;
  285. }
  286. /*
  287. * wrapper pow(x,y) return x**y
  288. */
  289. #ifndef _IEEE_LIBM
  290. double pow(double x, double y)
  291. {
  292. double z = __ieee754_pow(x, y);
  293. if (_LIB_VERSION == _IEEE_|| isnan(y))
  294. return z;
  295. if (isnan(x)) {
  296. if (y == 0.0)
  297. return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
  298. return z;
  299. }
  300. if (x == 0.0) {
  301. if (y == 0.0)
  302. return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
  303. if (isfinite(y) && y < 0.0)
  304. return __kernel_standard(x,y,23); /* pow(0.0,negative) */
  305. return z;
  306. }
  307. if (!isfinite(z)) {
  308. if (isfinite(x) && isfinite(y)) {
  309. if (isnan(z))
  310. return __kernel_standard(x, y, 24); /* pow neg**non-int */
  311. return __kernel_standard(x, y, 21); /* pow overflow */
  312. }
  313. }
  314. if (z == 0.0 && isfinite(x) && isfinite(y))
  315. return __kernel_standard(x, y, 22); /* pow underflow */
  316. return z;
  317. }
  318. #else
  319. strong_alias(__ieee754_pow, pow)
  320. #endif
  321. libm_hidden_def(pow)