1/** @file 2 Compute the base 10 logrithm of x. 3 4 Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR> 5 This program and the accompanying materials are licensed and made available under 6 the terms and conditions of the BSD License that accompanies this distribution. 7 The full text of the license may be found at 8 http://opensource.org/licenses/bsd-license. 9 10 THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS, 11 WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED. 12 13 * ==================================================== 14 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 15 * 16 * Developed at SunPro, a Sun Microsystems, Inc. business. 17 * Permission to use, copy, modify, and distribute this 18 * software is freely granted, provided that this notice 19 * is preserved. 20 * ==================================================== 21 22 e_pow.c 5.1 93/09/24 23 NetBSD: e_pow.c,v 1.13 2004/06/30 18:43:15 drochner Exp 24**/ 25#include <LibConfig.h> 26#include <sys/EfiCdefs.h> 27 28#if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ 29 // C4723: potential divide by zero. 30 #pragma warning ( disable : 4723 ) 31 // C4756: overflow in constant arithmetic 32 #pragma warning ( disable : 4756 ) 33#endif 34 35/* __ieee754_pow(x,y) return x**y 36 * 37 * n 38 * Method: Let x = 2 * (1+f) 39 * 1. Compute and return log2(x) in two pieces: 40 * log2(x) = w1 + w2, 41 * where w1 has 53-24 = 29 bit trailing zeros. 42 * 2. Perform y*log2(x) = n+y' by simulating multi-precision 43 * arithmetic, where |y'|<=0.5. 44 * 3. Return x**y = 2**n*exp(y'*log2) 45 * 46 * Special cases: 47 * 1. (anything) ** 0 is 1 48 * 2. (anything) ** 1 is itself 49 * 3. (anything) ** NAN is NAN 50 * 4. NAN ** (anything except 0) is NAN 51 * 5. +-(|x| > 1) ** +INF is +INF 52 * 6. +-(|x| > 1) ** -INF is +0 53 * 7. +-(|x| < 1) ** +INF is +0 54 * 8. +-(|x| < 1) ** -INF is +INF 55 * 9. +-1 ** +-INF is NAN 56 * 10. +0 ** (+anything except 0, NAN) is +0 57 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 58 * 12. +0 ** (-anything except 0, NAN) is +INF 59 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 60 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 61 * 15. +INF ** (+anything except 0,NAN) is +INF 62 * 16. +INF ** (-anything except 0,NAN) is +0 63 * 17. -INF ** (anything) = -0 ** (-anything) 64 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 65 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 66 * 67 * Accuracy: 68 * pow(x,y) returns x**y nearly rounded. In particular 69 * pow(integer,integer) 70 * always returns the correct integer provided it is 71 * representable. 72 * 73 * Constants : 74 * The hexadecimal values are the intended ones for the following 75 * constants. The decimal values may be used, provided that the 76 * compiler will convert from decimal to binary accurately enough 77 * to produce the hexadecimal values shown. 78 */ 79 80#include "math.h" 81#include "math_private.h" 82#include <errno.h> 83 84static const double 85bp[] = {1.0, 1.5,}, 86dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 87dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 88zero = 0.0, 89one = 1.0, 90two = 2.0, 91two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 92huge = 1.0e300, 93tiny = 1.0e-300, 94 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 95L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 96L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 97L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 98L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 99L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 100L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 101P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 102P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 103P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 104P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 105P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 106lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 107lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 108lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 109ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 110cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 111cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 112cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 113ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 114ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 115ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 116 117double 118__ieee754_pow(double x, double y) 119{ 120 double z,ax,z_h,z_l,p_h,p_l; 121 double y1,t1,t2,r,s,t,u,v,w; 122 int32_t i,j,k,yisint,n; 123 int32_t hx,hy,ix,iy; 124 u_int32_t lx,ly; 125 126 EXTRACT_WORDS(hx,lx,x); 127 EXTRACT_WORDS(hy,ly,y); 128 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 129 130 /* y==zero: x**0 = 1 */ 131 if((iy|ly)==0) return one; 132 133 /* +-NaN return x+y */ 134 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 135 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 136 return x+y; 137 138 /* determine if y is an odd int when x < 0 139 * yisint = 0 ... y is not an integer 140 * yisint = 1 ... y is an odd int 141 * yisint = 2 ... y is an even int 142 */ 143 yisint = 0; 144 if(hx<0) { 145 if(iy>=0x43400000) yisint = 2; /* even integer y */ 146 else if(iy>=0x3ff00000) { 147 k = (iy>>20)-0x3ff; /* exponent */ 148 if(k>20) { 149 j = ly>>(52-k); 150 if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); 151 } else if(ly==0) { 152 j = iy>>(20-k); 153 if((j<<(20-k))==iy) yisint = 2-(j&1); 154 } 155 } 156 } 157 158 /* special value of y */ 159 if(ly==0) { 160 if (iy==0x7ff00000) { /* y is +-inf */ 161 if(((ix-0x3ff00000)|lx)==0) 162 return y - y; /* inf**+-1 is NaN */ 163 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 164 return (hy>=0)? y: zero; 165 else /* (|x|<1)**-,+inf = inf,0 */ 166 return (hy<0)?-y: zero; 167 } 168 if(iy==0x3ff00000) { /* y is +-1 */ 169 if(hy<0) return one/x; else return x; 170 } 171 if(hy==0x40000000) return x*x; /* y is 2 */ 172 if(hy==0x3fe00000) { /* y is 0.5 */ 173 if(hx>=0) /* x >= +0 */ 174 return __ieee754_sqrt(x); 175 } 176 } 177 178 ax = fabs(x); 179 /* special value of x */ 180 if(lx==0) { 181 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 182 z = ax; /*x is +-0,+-inf,+-1*/ 183 if(hy<0) z = one/z; /* z = (1/|x|) */ 184 if(hx<0) { 185 if(((ix-0x3ff00000)|yisint)==0) { 186 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 187 } else if(yisint==1) 188 z = -z; /* (x<0)**odd = -(|x|**odd) */ 189 } 190 return z; 191 } 192 } 193 194 n = (hx>>31)+1; 195 196 /* (x<0)**(non-int) is NaN */ 197 if((n|yisint)==0) { 198 errno = EDOM; 199 return (x-x)/(x-x); 200 } 201 202 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 203 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 204 205 /* |y| is huge */ 206 if(iy>0x41e00000) { /* if |y| > 2**31 */ 207 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 208 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 209 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 210 } 211 /* over/underflow if x is not close to one */ 212 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 213 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 214 /* now |1-x| is tiny <= 2**-20, suffice to compute 215 log(x) by x-x^2/2+x^3/3-x^4/4 */ 216 t = ax-one; /* t has 20 trailing zeros */ 217 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 218 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 219 v = t*ivln2_l-w*ivln2; 220 t1 = u+v; 221 SET_LOW_WORD(t1,0); 222 t2 = v-(t1-u); 223 } else { 224 double ss,s2,s_h,s_l,t_h,t_l; 225 n = 0; 226 /* take care subnormal number */ 227 if(ix<0x00100000) 228 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 229 n += ((ix)>>20)-0x3ff; 230 j = ix&0x000fffff; 231 /* determine interval */ 232 ix = j|0x3ff00000; /* normalize ix */ 233 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 234 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 235 else {k=0;n+=1;ix -= 0x00100000;} 236 SET_HIGH_WORD(ax,ix); 237 238 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 239 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 240 v = one/(ax+bp[k]); 241 ss = u*v; 242 s_h = ss; 243 SET_LOW_WORD(s_h,0); 244 /* t_h=ax+bp[k] High */ 245 t_h = zero; 246 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 247 t_l = ax - (t_h-bp[k]); 248 s_l = v*((u-s_h*t_h)-s_h*t_l); 249 /* compute log(ax) */ 250 s2 = ss*ss; 251 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 252 r += s_l*(s_h+ss); 253 s2 = s_h*s_h; 254 t_h = 3.0+s2+r; 255 SET_LOW_WORD(t_h,0); 256 t_l = r-((t_h-3.0)-s2); 257 /* u+v = ss*(1+...) */ 258 u = s_h*t_h; 259 v = s_l*t_h+t_l*ss; 260 /* 2/(3log2)*(ss+...) */ 261 p_h = u+v; 262 SET_LOW_WORD(p_h,0); 263 p_l = v-(p_h-u); 264 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 265 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 266 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 267 t = (double)n; 268 t1 = (((z_h+z_l)+dp_h[k])+t); 269 SET_LOW_WORD(t1,0); 270 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 271 } 272 273 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 274 y1 = y; 275 SET_LOW_WORD(y1,0); 276 p_l = (y-y1)*t1+y*t2; 277 p_h = y1*t1; 278 z = p_l+p_h; 279 EXTRACT_WORDS(j,i,z); 280 if (j>=0x40900000) { /* z >= 1024 */ 281 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 282 return s*huge*huge; /* overflow */ 283 else { 284 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 285 } 286 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 287 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 288 return s*tiny*tiny; /* underflow */ 289 else { 290 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 291 } 292 } 293 /* 294 * compute 2**(p_h+p_l) 295 */ 296 i = j&0x7fffffff; 297 k = (i>>20)-0x3ff; 298 n = 0; 299 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 300 n = j+(0x00100000>>(k+1)); 301 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 302 t = zero; 303 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 304 n = ((n&0x000fffff)|0x00100000)>>(20-k); 305 if(j<0) n = -n; 306 p_h -= t; 307 } 308 t = p_l+p_h; 309 SET_LOW_WORD(t,0); 310 u = t*lg2_h; 311 v = (p_l-(t-p_h))*lg2+t*lg2_l; 312 z = u+v; 313 w = v-(z-u); 314 t = z*z; 315 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 316 r = (z*t1)/(t1-two)-(w+z*w); 317 z = one-(r-z); 318 GET_HIGH_WORD(j,z); 319 j += (n<<20); 320 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 321 else SET_HIGH_WORD(z,j); 322 return s*z; 323} 324