1 2 #ifndef lint 3 static char sccsid[] = "@(#)e_pow.c 1.5 04/04/22 SMI"; 4 #endif 5 6 /* 7 * ==================================================== 8 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 9 * 10 * Permission to use, copy, modify, and distribute this 11 * software is freely granted, provided that this notice 12 * is preserved. 13 * ==================================================== 14 */ 15 16 /* __ieee754_pow(x,y) return x**y 17 * 18 * n 19 * Method: Let x = 2 * (1+f) 20 * 1. Compute and return log2(x) in two pieces: 21 * log2(x) = w1 + w2, 22 * where w1 has 53-24 = 29 bit trailing zeros. 23 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 24 * arithmetic, where |y'|<=0.5. 25 * 3. Return x**y = 2**n*ieee_exp(y'*log2) 26 * 27 * Special cases: 28 * 1. (anything) ** 0 is 1 29 * 2. (anything) ** 1 is itself 30 * 3. (anything) ** NAN is NAN 31 * 4. NAN ** (anything except 0) is NAN 32 * 5. +-(|x| > 1) ** +INF is +INF 33 * 6. +-(|x| > 1) ** -INF is +0 34 * 7. +-(|x| < 1) ** +INF is +0 35 * 8. +-(|x| < 1) ** -INF is +INF 36 * 9. +-1 ** +-INF is NAN 37 * 10. +0 ** (+anything except 0, NAN) is +0 38 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 39 * 12. +0 ** (-anything except 0, NAN) is +INF 40 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 41 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 42 * 15. +INF ** (+anything except 0,NAN) is +INF 43 * 16. +INF ** (-anything except 0,NAN) is +0 44 * 17. -INF ** (anything) = -0 ** (-anything) 45 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 46 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 47 * 48 * Accuracy: 49 * pow(x,y) returns x**y nearly rounded. In particular 50 * pow(integer,integer) 51 * always returns the correct integer provided it is 52 * representable. 53 * 54 * Constants : 55 * The hexadecimal values are the intended ones for the following 56 * constants. The decimal values may be used, provided that the 57 * compiler will convert from decimal to binary accurately enough 58 * to produce the hexadecimal values shown. 59 */ 60 61 #include "fdlibm.h" 62 63 #ifdef __STDC__ 64 static const double 65 #else 66 static double 67 #endif 68 bp[] = {1.0, 1.5,}, 69 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 70 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 71 zero = 0.0, 72 one = 1.0, 73 two = 2.0, 74 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 75 huge = 1.0e300, 76 tiny = 1.0e-300, 77 /* poly coefs for (3/2)*(ieee_log(x)-2s-2/3*s**3 */ 78 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 79 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 80 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 81 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 82 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 83 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 84 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 85 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 86 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 87 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 88 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 89 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 90 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 91 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 92 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 93 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 94 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 95 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 96 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 97 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 98 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 99 100 #ifdef __STDC__ __ieee754_pow(double x,double y)101 double __ieee754_pow(double x, double y) 102 #else 103 double __ieee754_pow(x,y) 104 double x, y; 105 #endif 106 { 107 double z,ax,z_h,z_l,p_h,p_l; 108 double y1,t1,t2,r,s,t,u,v,w; 109 int i0,i1,i,j,k,yisint,n; 110 int hx,hy,ix,iy; 111 unsigned lx,ly; 112 113 i0 = ((*(int*)&one)>>29)^1; i1=1-i0; 114 hx = __HI(x); lx = __LO(x); 115 hy = __HI(y); ly = __LO(y); 116 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 117 118 /* y==zero: x**0 = 1 */ 119 if((iy|ly)==0) return one; 120 121 /* +-NaN return x+y */ 122 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 123 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 124 return x+y; 125 126 /* determine if y is an odd int when x < 0 127 * yisint = 0 ... y is not an integer 128 * yisint = 1 ... y is an odd int 129 * yisint = 2 ... y is an even int 130 */ 131 yisint = 0; 132 if(hx<0) { 133 if(iy>=0x43400000) yisint = 2; /* even integer y */ 134 else if(iy>=0x3ff00000) { 135 k = (iy>>20)-0x3ff; /* exponent */ 136 if(k>20) { 137 j = ly>>(52-k); 138 if((j<<(52-k))==ly) yisint = 2-(j&1); 139 } else if(ly==0) { 140 j = iy>>(20-k); 141 if((j<<(20-k))==iy) yisint = 2-(j&1); 142 } 143 } 144 } 145 146 /* special value of y */ 147 if(ly==0) { 148 if (iy==0x7ff00000) { /* y is +-inf */ 149 if(((ix-0x3ff00000)|lx)==0) 150 return y - y; /* inf**+-1 is NaN */ 151 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 152 return (hy>=0)? y: zero; 153 else /* (|x|<1)**-,+inf = inf,0 */ 154 return (hy<0)?-y: zero; 155 } 156 if(iy==0x3ff00000) { /* y is +-1 */ 157 if(hy<0) return one/x; else return x; 158 } 159 if(hy==0x40000000) return x*x; /* y is 2 */ 160 if(hy==0x3fe00000) { /* y is 0.5 */ 161 if(hx>=0) /* x >= +0 */ 162 return ieee_sqrt(x); 163 } 164 } 165 166 ax = ieee_fabs(x); 167 /* special value of x */ 168 if(lx==0) { 169 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 170 z = ax; /*x is +-0,+-inf,+-1*/ 171 if(hy<0) z = one/z; /* z = (1/|x|) */ 172 if(hx<0) { 173 if(((ix-0x3ff00000)|yisint)==0) { 174 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 175 } else if(yisint==1) 176 z = -z; /* (x<0)**odd = -(|x|**odd) */ 177 } 178 return z; 179 } 180 } 181 182 n = (hx>>31)+1; 183 184 /* (x<0)**(non-int) is NaN */ 185 if((n|yisint)==0) return (x-x)/(x-x); 186 187 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 188 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 189 190 /* |y| is huge */ 191 if(iy>0x41e00000) { /* if |y| > 2**31 */ 192 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 193 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 194 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 195 } 196 /* over/underflow if x is not close to one */ 197 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 198 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 199 /* now |1-x| is tiny <= 2**-20, suffice to compute 200 ieee_log(x) by x-x^2/2+x^3/3-x^4/4 */ 201 t = ax-one; /* t has 20 trailing zeros */ 202 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 203 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 204 v = t*ivln2_l-w*ivln2; 205 t1 = u+v; 206 __LO(t1) = 0; 207 t2 = v-(t1-u); 208 } else { 209 double ss,s2,s_h,s_l,t_h,t_l; 210 n = 0; 211 /* take care subnormal number */ 212 if(ix<0x00100000) 213 {ax *= two53; n -= 53; ix = __HI(ax); } 214 n += ((ix)>>20)-0x3ff; 215 j = ix&0x000fffff; 216 /* determine interval */ 217 ix = j|0x3ff00000; /* normalize ix */ 218 if(j<=0x3988E) k=0; /* |x|<ieee_sqrt(3/2) */ 219 else if(j<0xBB67A) k=1; /* |x|<ieee_sqrt(3) */ 220 else {k=0;n+=1;ix -= 0x00100000;} 221 __HI(ax) = ix; 222 223 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 224 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 225 v = one/(ax+bp[k]); 226 ss = u*v; 227 s_h = ss; 228 __LO(s_h) = 0; 229 /* t_h=ax+bp[k] High */ 230 t_h = zero; 231 __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 232 t_l = ax - (t_h-bp[k]); 233 s_l = v*((u-s_h*t_h)-s_h*t_l); 234 /* compute ieee_log(ax) */ 235 s2 = ss*ss; 236 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 237 r += s_l*(s_h+ss); 238 s2 = s_h*s_h; 239 t_h = 3.0+s2+r; 240 __LO(t_h) = 0; 241 t_l = r-((t_h-3.0)-s2); 242 /* u+v = ss*(1+...) */ 243 u = s_h*t_h; 244 v = s_l*t_h+t_l*ss; 245 /* 2/(3log2)*(ss+...) */ 246 p_h = u+v; 247 __LO(p_h) = 0; 248 p_l = v-(p_h-u); 249 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 250 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 251 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 252 t = (double)n; 253 t1 = (((z_h+z_l)+dp_h[k])+t); 254 __LO(t1) = 0; 255 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 256 } 257 258 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 259 y1 = y; 260 __LO(y1) = 0; 261 p_l = (y-y1)*t1+y*t2; 262 p_h = y1*t1; 263 z = p_l+p_h; 264 j = __HI(z); 265 i = __LO(z); 266 if (j>=0x40900000) { /* z >= 1024 */ 267 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 268 return s*huge*huge; /* overflow */ 269 else { 270 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 271 } 272 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 273 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 274 return s*tiny*tiny; /* underflow */ 275 else { 276 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 277 } 278 } 279 /* 280 * compute 2**(p_h+p_l) 281 */ 282 i = j&0x7fffffff; 283 k = (i>>20)-0x3ff; 284 n = 0; 285 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 286 n = j+(0x00100000>>(k+1)); 287 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 288 t = zero; 289 __HI(t) = (n&~(0x000fffff>>k)); 290 n = ((n&0x000fffff)|0x00100000)>>(20-k); 291 if(j<0) n = -n; 292 p_h -= t; 293 } 294 t = p_l+p_h; 295 __LO(t) = 0; 296 u = t*lg2_h; 297 v = (p_l-(t-p_h))*lg2+t*lg2_l; 298 z = u+v; 299 w = v-(z-u); 300 t = z*z; 301 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 302 r = (z*t1)/(t1-two)-(w+z*w); 303 z = one-(r-z); 304 j = __HI(z); 305 j += (n<<20); 306 if((j>>20)<=0) z = ieee_scalbn(z,n); /* subnormal output */ 307 else __HI(z) += (n<<20); 308 return s*z; 309 } 310