1 2 /* 3 * Copyright (c) 1998, 2004, Oracle and/or its affiliates. All rights reserved. 4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 5 * 6 * This code is free software; you can redistribute it and/or modify it 7 * under the terms of the GNU General Public License version 2 only, as 8 * published by the Free Software Foundation. Oracle designates this 9 * particular file as subject to the "Classpath" exception as provided 10 * by Oracle in the LICENSE file that accompanied this code. 11 * 12 * This code is distributed in the hope that it will be useful, but WITHOUT 13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 15 * version 2 for more details (a copy is included in the LICENSE file that 16 * accompanied this code). 17 * 18 * You should have received a copy of the GNU General Public License version 19 * 2 along with this work; if not, write to the Free Software Foundation, 20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 21 * 22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 23 * or visit www.oracle.com if you need additional information or have any 24 * questions. 25 */ 26 27 /* __ieee754_pow(x,y) return x**y 28 * 29 * n 30 * Method: Let x = 2 * (1+f) 31 * 1. Compute and return log2(x) in two pieces: 32 * log2(x) = w1 + w2, 33 * where w1 has 53-24 = 29 bit trailing zeros. 34 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 35 * arithmetic, where |y'|<=0.5. 36 * 3. Return x**y = 2**n*exp(y'*log2) 37 * 38 * Special cases: 39 * 1. (anything) ** 0 is 1 40 * 2. (anything) ** 1 is itself 41 * 3. (anything) ** NAN is NAN 42 * 4. NAN ** (anything except 0) is NAN 43 * 5. +-(|x| > 1) ** +INF is +INF 44 * 6. +-(|x| > 1) ** -INF is +0 45 * 7. +-(|x| < 1) ** +INF is +0 46 * 8. +-(|x| < 1) ** -INF is +INF 47 * 9. +-1 ** +-INF is NAN 48 * 10. +0 ** (+anything except 0, NAN) is +0 49 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 50 * 12. +0 ** (-anything except 0, NAN) is +INF 51 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 52 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 53 * 15. +INF ** (+anything except 0,NAN) is +INF 54 * 16. +INF ** (-anything except 0,NAN) is +0 55 * 17. -INF ** (anything) = -0 ** (-anything) 56 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 57 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 58 * 59 * Accuracy: 60 * pow(x,y) returns x**y nearly rounded. In particular 61 * pow(integer,integer) 62 * always returns the correct integer provided it is 63 * representable. 64 * 65 * Constants : 66 * The hexadecimal values are the intended ones for the following 67 * constants. The decimal values may be used, provided that the 68 * compiler will convert from decimal to binary accurately enough 69 * to produce the hexadecimal values shown. 70 */ 71 72 #include "fdlibm.h" 73 74 #ifdef __STDC__ 75 static const double 76 #else 77 static double 78 #endif 79 bp[] = {1.0, 1.5,}, 80 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 81 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 82 zero = 0.0, 83 one = 1.0, 84 two = 2.0, 85 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 86 huge = 1.0e300, 87 tiny = 1.0e-300, 88 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 89 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 90 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 91 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 92 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 93 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 94 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 95 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 96 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 97 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 98 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 99 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 100 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 101 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 102 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 103 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 104 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 105 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 106 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 107 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 108 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 109 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 110 111 #ifdef __STDC__ 112 double __ieee754_pow(double x, double y) 113 #else 114 double __ieee754_pow(x,y) 115 double x, y; 116 #endif 117 { 118 double z,ax,z_h,z_l,p_h,p_l; 119 double y1,t1,t2,r,s,t,u,v,w; 120 int i0,i1,i,j,k,yisint,n; 121 int hx,hy,ix,iy; 122 unsigned lx,ly; 123 124 i0 = ((*(int*)&one)>>29)^1; i1=1-i0; 125 hx = __HI(x); lx = __LO(x); 126 hy = __HI(y); ly = __LO(y); 127 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 128 129 /* y==zero: x**0 = 1 */ 130 if((iy|ly)==0) return one; 131 132 /* +-NaN return x+y */ 133 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 134 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 135 return x+y; 136 137 /* determine if y is an odd int when x < 0 138 * yisint = 0 ... y is not an integer 139 * yisint = 1 ... y is an odd int 140 * yisint = 2 ... y is an even int 141 */ 142 yisint = 0; 143 if(hx<0) { 144 if(iy>=0x43400000) yisint = 2; /* even integer y */ 145 else if(iy>=0x3ff00000) { 146 k = (iy>>20)-0x3ff; /* exponent */ 147 if(k>20) { 148 j = ly>>(52-k); 149 if((j<<(52-k))==ly) yisint = 2-(j&1); 150 } else if(ly==0) { 151 j = iy>>(20-k); 152 if((j<<(20-k))==iy) yisint = 2-(j&1); 153 } 154 } 155 } 156 157 /* special value of y */ 158 if(ly==0) { 159 if (iy==0x7ff00000) { /* y is +-inf */ 160 if(((ix-0x3ff00000)|lx)==0) 161 return y - y; /* inf**+-1 is NaN */ 162 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 163 return (hy>=0)? y: zero; 164 else /* (|x|<1)**-,+inf = inf,0 */ 165 return (hy<0)?-y: zero; 166 } 167 if(iy==0x3ff00000) { /* y is +-1 */ 168 if(hy<0) return one/x; else return x; 169 } 170 if(hy==0x40000000) return x*x; /* y is 2 */ 171 if(hy==0x3fe00000) { /* y is 0.5 */ 172 if(hx>=0) /* x >= +0 */ 173 return sqrt(x); 174 } 175 } 176 177 ax = fabs(x); 178 /* special value of x */ 179 if(lx==0) { 180 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 181 z = ax; /*x is +-0,+-inf,+-1*/ 182 if(hy<0) z = one/z; /* z = (1/|x|) */ 183 if(hx<0) { 184 if(((ix-0x3ff00000)|yisint)==0) { 185 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 186 } else if(yisint==1) 187 z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */ 188 } 189 return z; 190 } 191 } 192 193 n = (hx>>31)+1; 194 195 /* (x<0)**(non-int) is NaN */ 196 if((n|yisint)==0) return (x-x)/(x-x); 197 198 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 199 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 200 201 /* |y| is huge */ 202 if(iy>0x41e00000) { /* if |y| > 2**31 */ 203 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 204 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 205 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 206 } 207 /* over/underflow if x is not close to one */ 208 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 209 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 210 /* now |1-x| is tiny <= 2**-20, suffice to compute 211 log(x) by x-x^2/2+x^3/3-x^4/4 */ 212 t = ax-one; /* t has 20 trailing zeros */ 213 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 214 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 215 v = t*ivln2_l-w*ivln2; 216 t1 = u+v; 217 __LO(t1) = 0; 218 t2 = v-(t1-u); 219 } else { 220 double ss,s2,s_h,s_l,t_h,t_l; 221 n = 0; 222 /* take care subnormal number */ 223 if(ix<0x00100000) 224 {ax *= two53; n -= 53; ix = __HI(ax); } 225 n += ((ix)>>20)-0x3ff; 226 j = ix&0x000fffff; 227 /* determine interval */ 228 ix = j|0x3ff00000; /* normalize ix */ 229 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 230 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 231 else {k=0;n+=1;ix -= 0x00100000;} 232 __HI(ax) = ix; 233 234 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 235 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 236 v = one/(ax+bp[k]); 237 ss = u*v; 238 s_h = ss; 239 __LO(s_h) = 0; 240 /* t_h=ax+bp[k] High */ 241 t_h = zero; 242 __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 243 t_l = ax - (t_h-bp[k]); 244 s_l = v*((u-s_h*t_h)-s_h*t_l); 245 /* compute log(ax) */ 246 s2 = ss*ss; 247 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 248 r += s_l*(s_h+ss); 249 s2 = s_h*s_h; 250 t_h = 3.0+s2+r; 251 __LO(t_h) = 0; 252 t_l = r-((t_h-3.0)-s2); 253 /* u+v = ss*(1+...) */ 254 u = s_h*t_h; 255 v = s_l*t_h+t_l*ss; 256 /* 2/(3log2)*(ss+...) */ 257 p_h = u+v; 258 __LO(p_h) = 0; 259 p_l = v-(p_h-u); 260 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 261 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 262 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 263 t = (double)n; 264 t1 = (((z_h+z_l)+dp_h[k])+t); 265 __LO(t1) = 0; 266 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 267 } 268 269 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 270 y1 = y; 271 __LO(y1) = 0; 272 p_l = (y-y1)*t1+y*t2; 273 p_h = y1*t1; 274 z = p_l+p_h; 275 j = __HI(z); 276 i = __LO(z); 277 if (j>=0x40900000) { /* z >= 1024 */ 278 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 279 return s*huge*huge; /* overflow */ 280 else { 281 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 282 } 283 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 284 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 285 return s*tiny*tiny; /* underflow */ 286 else { 287 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 288 } 289 } 290 /* 291 * compute 2**(p_h+p_l) 292 */ 293 i = j&0x7fffffff; 294 k = (i>>20)-0x3ff; 295 n = 0; 296 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 297 n = j+(0x00100000>>(k+1)); 298 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 299 t = zero; 300 __HI(t) = (n&~(0x000fffff>>k)); 301 n = ((n&0x000fffff)|0x00100000)>>(20-k); 302 if(j<0) n = -n; 303 p_h -= t; 304 } 305 t = p_l+p_h; 306 __LO(t) = 0; 307 u = t*lg2_h; 308 v = (p_l-(t-p_h))*lg2+t*lg2_l; 309 z = u+v; 310 w = v-(z-u); 311 t = z*z; 312 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 313 r = (z*t1)/(t1-two)-(w+z*w); 314 z = one-(r-z); 315 j = __HI(z); 316 j += (n<<20); 317 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 318 else __HI(z) += (n<<20); 319 return s*z; 320 }