e_pow.c 9.7 KB

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  1. /* @(#)e_pow.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_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
  14. #endif
  15. /* __ieee754_pow(x,y) return x**y
  16. *
  17. * n
  18. * Method: Let x = 2 * (1+f)
  19. * 1. Compute and return log2(x) in two pieces:
  20. * log2(x) = w1 + w2,
  21. * where w1 has 53-24 = 29 bit trailing zeros.
  22. * 2. Perform y*log2(x) = n+y' by simulating muti-precision
  23. * arithmetic, where |y'|<=0.5.
  24. * 3. Return x**y = 2**n*exp(y'*log2)
  25. *
  26. * Special cases:
  27. * 1. (anything) ** 0 is 1
  28. * 2. (anything) ** 1 is itself
  29. * 3. (anything) ** NAN is NAN
  30. * 4. NAN ** (anything except 0) is NAN
  31. * 5. +-(|x| > 1) ** +INF is +INF
  32. * 6. +-(|x| > 1) ** -INF is +0
  33. * 7. +-(|x| < 1) ** +INF is +0
  34. * 8. +-(|x| < 1) ** -INF is +INF
  35. * 9. +-1 ** +-INF is NAN
  36. * 10. +0 ** (+anything except 0, NAN) is +0
  37. * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
  38. * 12. +0 ** (-anything except 0, NAN) is +INF
  39. * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
  40. * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  41. * 15. +INF ** (+anything except 0,NAN) is +INF
  42. * 16. +INF ** (-anything except 0,NAN) is +0
  43. * 17. -INF ** (anything) = -0 ** (-anything)
  44. * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  45. * 19. (-anything except 0 and inf) ** (non-integer) is NAN
  46. *
  47. * Accuracy:
  48. * pow(x,y) returns x**y nearly rounded. In particular
  49. * pow(integer,integer)
  50. * always returns the correct integer provided it is
  51. * representable.
  52. *
  53. * Constants :
  54. * The hexadecimal values are the intended ones for the following
  55. * constants. The decimal values may be used, provided that the
  56. * compiler will convert from decimal to binary accurately enough
  57. * to produce the hexadecimal values shown.
  58. */
  59. #include "math.h"
  60. #include "math_private.h"
  61. #ifdef __STDC__
  62. static const double
  63. #else
  64. static double
  65. #endif
  66. bp[] = {1.0, 1.5,},
  67. dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  68. dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  69. zero = 0.0,
  70. one = 1.0,
  71. two = 2.0,
  72. two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
  73. huge = 1.0e300,
  74. tiny = 1.0e-300,
  75. /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  76. L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  77. L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  78. L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  79. L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  80. L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  81. L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  82. P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  83. P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  84. P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  85. P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  86. P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  87. lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  88. lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  89. lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  90. ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  91. cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  92. cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  93. cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  94. ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  95. ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  96. ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  97. #ifdef __STDC__
  98. double attribute_hidden __ieee754_pow(double x, double y)
  99. #else
  100. double attribute_hidden __ieee754_pow(x,y)
  101. double x, y;
  102. #endif
  103. {
  104. double z,ax,z_h,z_l,p_h,p_l;
  105. double y1,t1,t2,r,s,t,u,v,w;
  106. int32_t i,j,k,yisint,n;
  107. int32_t hx,hy,ix,iy;
  108. u_int32_t lx,ly;
  109. EXTRACT_WORDS(hx,lx,x);
  110. EXTRACT_WORDS(hy,ly,y);
  111. ix = hx&0x7fffffff; iy = hy&0x7fffffff;
  112. /* y==zero: x**0 = 1 */
  113. if((iy|ly)==0) return one;
  114. /* +-NaN return x+y */
  115. if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
  116. iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
  117. return x+y;
  118. /* determine if y is an odd int when x < 0
  119. * yisint = 0 ... y is not an integer
  120. * yisint = 1 ... y is an odd int
  121. * yisint = 2 ... y is an even int
  122. */
  123. yisint = 0;
  124. if(hx<0) {
  125. if(iy>=0x43400000) yisint = 2; /* even integer y */
  126. else if(iy>=0x3ff00000) {
  127. k = (iy>>20)-0x3ff; /* exponent */
  128. if(k>20) {
  129. j = ly>>(52-k);
  130. if((j<<(52-k))==ly) yisint = 2-(j&1);
  131. } else if(ly==0) {
  132. j = iy>>(20-k);
  133. if((j<<(20-k))==iy) yisint = 2-(j&1);
  134. }
  135. }
  136. }
  137. /* special value of y */
  138. if(ly==0) {
  139. if (iy==0x7ff00000) { /* y is +-inf */
  140. if(((ix-0x3ff00000)|lx)==0)
  141. return y - y; /* inf**+-1 is NaN */
  142. else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
  143. return (hy>=0)? y: zero;
  144. else /* (|x|<1)**-,+inf = inf,0 */
  145. return (hy<0)?-y: zero;
  146. }
  147. if(iy==0x3ff00000) { /* y is +-1 */
  148. if(hy<0) return one/x; else return x;
  149. }
  150. if(hy==0x40000000) return x*x; /* y is 2 */
  151. if(hy==0x3fe00000) { /* y is 0.5 */
  152. if(hx>=0) /* x >= +0 */
  153. return __ieee754_sqrt(x);
  154. }
  155. }
  156. ax = fabs(x);
  157. /* special value of x */
  158. if(lx==0) {
  159. if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
  160. z = ax; /*x is +-0,+-inf,+-1*/
  161. if(hy<0) z = one/z; /* z = (1/|x|) */
  162. if(hx<0) {
  163. if(((ix-0x3ff00000)|yisint)==0) {
  164. z = (z-z)/(z-z); /* (-1)**non-int is NaN */
  165. } else if(yisint==1)
  166. z = -z; /* (x<0)**odd = -(|x|**odd) */
  167. }
  168. return z;
  169. }
  170. }
  171. /* (x<0)**(non-int) is NaN */
  172. if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
  173. /* |y| is huge */
  174. if(iy>0x41e00000) { /* if |y| > 2**31 */
  175. if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
  176. if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  177. if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  178. }
  179. /* over/underflow if x is not close to one */
  180. if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  181. if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  182. /* now |1-x| is tiny <= 2**-20, suffice to compute
  183. log(x) by x-x^2/2+x^3/3-x^4/4 */
  184. t = x-1; /* t has 20 trailing zeros */
  185. w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
  186. u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
  187. v = t*ivln2_l-w*ivln2;
  188. t1 = u+v;
  189. SET_LOW_WORD(t1,0);
  190. t2 = v-(t1-u);
  191. } else {
  192. double s2,s_h,s_l,t_h,t_l;
  193. n = 0;
  194. /* take care subnormal number */
  195. if(ix<0x00100000)
  196. {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
  197. n += ((ix)>>20)-0x3ff;
  198. j = ix&0x000fffff;
  199. /* determine interval */
  200. ix = j|0x3ff00000; /* normalize ix */
  201. if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
  202. else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
  203. else {k=0;n+=1;ix -= 0x00100000;}
  204. SET_HIGH_WORD(ax,ix);
  205. /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
  206. u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
  207. v = one/(ax+bp[k]);
  208. s = u*v;
  209. s_h = s;
  210. SET_LOW_WORD(s_h,0);
  211. /* t_h=ax+bp[k] High */
  212. t_h = zero;
  213. SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
  214. t_l = ax - (t_h-bp[k]);
  215. s_l = v*((u-s_h*t_h)-s_h*t_l);
  216. /* compute log(ax) */
  217. s2 = s*s;
  218. r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  219. r += s_l*(s_h+s);
  220. s2 = s_h*s_h;
  221. t_h = 3.0+s2+r;
  222. SET_LOW_WORD(t_h,0);
  223. t_l = r-((t_h-3.0)-s2);
  224. /* u+v = s*(1+...) */
  225. u = s_h*t_h;
  226. v = s_l*t_h+t_l*s;
  227. /* 2/(3log2)*(s+...) */
  228. p_h = u+v;
  229. SET_LOW_WORD(p_h,0);
  230. p_l = v-(p_h-u);
  231. z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
  232. z_l = cp_l*p_h+p_l*cp+dp_l[k];
  233. /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
  234. t = (double)n;
  235. t1 = (((z_h+z_l)+dp_h[k])+t);
  236. SET_LOW_WORD(t1,0);
  237. t2 = z_l-(((t1-t)-dp_h[k])-z_h);
  238. }
  239. s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  240. if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
  241. s = -one;/* (-ve)**(odd int) */
  242. /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
  243. y1 = y;
  244. SET_LOW_WORD(y1,0);
  245. p_l = (y-y1)*t1+y*t2;
  246. p_h = y1*t1;
  247. z = p_l+p_h;
  248. EXTRACT_WORDS(j,i,z);
  249. if (j>=0x40900000) { /* z >= 1024 */
  250. if(((j-0x40900000)|i)!=0) /* if z > 1024 */
  251. return s*huge*huge; /* overflow */
  252. else {
  253. if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
  254. }
  255. } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
  256. if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
  257. return s*tiny*tiny; /* underflow */
  258. else {
  259. if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
  260. }
  261. }
  262. /*
  263. * compute 2**(p_h+p_l)
  264. */
  265. i = j&0x7fffffff;
  266. k = (i>>20)-0x3ff;
  267. n = 0;
  268. if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
  269. n = j+(0x00100000>>(k+1));
  270. k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
  271. t = zero;
  272. SET_HIGH_WORD(t,n&~(0x000fffff>>k));
  273. n = ((n&0x000fffff)|0x00100000)>>(20-k);
  274. if(j<0) n = -n;
  275. p_h -= t;
  276. }
  277. t = p_l+p_h;
  278. SET_LOW_WORD(t,0);
  279. u = t*lg2_h;
  280. v = (p_l-(t-p_h))*lg2+t*lg2_l;
  281. z = u+v;
  282. w = v-(z-u);
  283. t = z*z;
  284. t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  285. r = (z*t1)/(t1-two)-(w+z*w);
  286. z = one-(r-z);
  287. GET_HIGH_WORD(j,z);
  288. j += (n<<20);
  289. if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
  290. else SET_HIGH_WORD(z,j);
  291. return s*z;
  292. }