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