1 2 /* @(#)e_pow.c 5.1 93/09/24 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunPro, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 /* __ieee754_pow(x,y) return x**y 15 * 16 * n 17 * Method: Let x = 2 * (1+f) 18 * 1. Compute and return log2(x) in two pieces: 19 * log2(x) = w1 + w2, 20 * where w1 has 53-24 = 29 bit trailing zeros. 21 * 2. Perform y*log2(x) = n+y' by simulating multi-precision 22 * arithmetic, where |y'|<=0.5. 23 * 3. Return x**y = 2**n*exp(y'*log2) 24 * 25 * Special cases: 26 * 1. (anything) ** 0 is 1 27 * 2. (anything) ** 1 is itself 28 * 3. (anything) ** NAN is NAN 29 * 4. NAN ** (anything except 0) is NAN 30 * 5. +-(|x| > 1) ** +INF is +INF 31 * 6. +-(|x| > 1) ** -INF is +0 32 * 7. +-(|x| < 1) ** +INF is +0 33 * 8. +-(|x| < 1) ** -INF is +INF 34 * 9. +-1 ** +-INF is NAN 35 * 10. +0 ** (+anything except 0, NAN) is +0 36 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 37 * 12. +0 ** (-anything except 0, NAN) is +INF 38 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 39 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 40 * 15. +INF ** (+anything except 0,NAN) is +INF 41 * 16. +INF ** (-anything except 0,NAN) is +0 42 * 17. -INF ** (anything) = -0 ** (-anything) 43 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 44 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 45 * 46 * Accuracy: 47 * pow(x,y) returns x**y nearly rounded. In particular 48 * pow(integer,integer) 49 * always returns the correct integer provided it is 50 * representable. 51 * 52 * Constants : 53 * The hexadecimal values are the intended ones for the following 54 * constants. The decimal values may be used, provided that the 55 * compiler will convert from decimal to binary accurately enough 56 * to produce the hexadecimal values shown. 57 */ 58 59 #include "fdlibm.h" 60 61 #ifndef _DOUBLE_IS_32BITS 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)*(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 __int32_t i,j,k,yisint,n; 110 __int32_t hx,hy,ix,iy; 111 __uint32_t lx,ly; 112 113 EXTRACT_WORDS(hx,lx,x); 114 EXTRACT_WORDS(hy,ly,y); 115 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 116 117 /* y==zero: x**0 = 1 */ 118 if((iy|ly)==0) return one; 119 120 /* +-NaN return x+y */ 121 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 122 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 123 return x+y; 124 125 /* determine if y is an odd int when x < 0 126 * yisint = 0 ... y is not an integer 127 * yisint = 1 ... y is an odd int 128 * yisint = 2 ... y is an even int 129 */ 130 yisint = 0; 131 if(hx<0) { 132 if(iy>=0x43400000) yisint = 2; /* even integer y */ 133 else if(iy>=0x3ff00000) { 134 k = (iy>>20)-0x3ff; /* exponent */ 135 if(k>20) { 136 j = ly>>(52-k); 137 if((j<<(52-k))==ly) yisint = 2-(j&1); 138 } else if(ly==0) { 139 j = iy>>(20-k); 140 if((j<<(20-k))==iy) yisint = 2-(j&1); 141 } 142 } 143 } 144 145 /* special value of y */ 146 if(ly==0) { 147 if (iy==0x7ff00000) { /* y is +-inf */ 148 if(((ix-0x3ff00000)|lx)==0) 149 return y - y; /* inf**+-1 is NaN */ 150 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 151 return (hy>=0)? y: zero; 152 else /* (|x|<1)**-,+inf = inf,0 */ 153 return (hy<0)?-y: zero; 154 } 155 if(iy==0x3ff00000) { /* y is +-1 */ 156 if(hy<0) return one/x; else return x; 157 } 158 if(hy==0x40000000) return x*x; /* y is 2 */ 159 if(hy==0x3fe00000) { /* y is 0.5 */ 160 if(hx>=0) /* x >= +0 */ 161 return __ieee754_sqrt(x); 162 } 163 } 164 165 ax = fabs(x); 166 /* special value of x */ 167 if(lx==0) { 168 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 169 z = ax; /*x is +-0,+-inf,+-1*/ 170 if(hy<0) z = one/z; /* z = (1/|x|) */ 171 if(hx<0) { 172 if(((ix-0x3ff00000)|yisint)==0) { 173 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 174 } else if(yisint==1) 175 z = -z; /* (x<0)**odd = -(|x|**odd) */ 176 } 177 return z; 178 } 179 } 180 181 /* (x<0)**(non-int) is NaN */ 182 /* REDHAT LOCAL: This used to be 183 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x); 184 but ANSI C says a right shift of a signed negative quantity is 185 implementation defined. */ 186 if(((((__uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 187 188 /* |y| is huge */ 189 if(iy>0x41e00000) { /* if |y| > 2**31 */ 190 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 191 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 192 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 193 } 194 /* over/underflow if x is not close to one */ 195 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 196 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 197 /* now |1-x| is tiny <= 2**-20, suffice to compute 198 log(x) by x-x^2/2+x^3/3-x^4/4 */ 199 t = ax-1; /* t has 20 trailing zeros */ 200 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 201 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 202 v = t*ivln2_l-w*ivln2; 203 t1 = u+v; 204 SET_LOW_WORD(t1,0); 205 t2 = v-(t1-u); 206 } else { 207 double s2,s_h,s_l,t_h,t_l; 208 n = 0; 209 /* take care subnormal number */ 210 if(ix<0x00100000) 211 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 212 n += ((ix)>>20)-0x3ff; 213 j = ix&0x000fffff; 214 /* determine interval */ 215 ix = j|0x3ff00000; /* normalize ix */ 216 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 217 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 218 else {k=0;n+=1;ix -= 0x00100000;} 219 SET_HIGH_WORD(ax,ix); 220 221 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 222 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 223 v = one/(ax+bp[k]); 224 s = u*v; 225 s_h = s; 226 SET_LOW_WORD(s_h,0); 227 /* t_h=ax+bp[k] High */ 228 t_h = zero; 229 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 230 t_l = ax - (t_h-bp[k]); 231 s_l = v*((u-s_h*t_h)-s_h*t_l); 232 /* compute log(ax) */ 233 s2 = s*s; 234 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 235 r += s_l*(s_h+s); 236 s2 = s_h*s_h; 237 t_h = 3.0+s2+r; 238 SET_LOW_WORD(t_h,0); 239 t_l = r-((t_h-3.0)-s2); 240 /* u+v = s*(1+...) */ 241 u = s_h*t_h; 242 v = s_l*t_h+t_l*s; 243 /* 2/(3log2)*(s+...) */ 244 p_h = u+v; 245 SET_LOW_WORD(p_h,0); 246 p_l = v-(p_h-u); 247 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 248 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 249 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 250 t = (double)n; 251 t1 = (((z_h+z_l)+dp_h[k])+t); 252 SET_LOW_WORD(t1,0); 253 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 254 } 255 256 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 257 if(((((__uint32_t)hx>>31)-1)|(yisint-1))==0) 258 s = -one;/* (-ve)**(odd int) */ 259 260 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 261 y1 = y; 262 SET_LOW_WORD(y1,0); 263 p_l = (y-y1)*t1+y*t2; 264 p_h = y1*t1; 265 z = p_l+p_h; 266 EXTRACT_WORDS(j,i,z); 267 if (j>=0x40900000) { /* z >= 1024 */ 268 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 269 return s*huge*huge; /* overflow */ 270 else { 271 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 272 } 273 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 274 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 275 return s*tiny*tiny; /* underflow */ 276 else { 277 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 278 } 279 } 280 /* 281 * compute 2**(p_h+p_l) 282 */ 283 i = j&0x7fffffff; 284 k = (i>>20)-0x3ff; 285 n = 0; 286 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 287 n = j+(0x00100000>>(k+1)); 288 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 289 t = zero; 290 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 291 n = ((n&0x000fffff)|0x00100000)>>(20-k); 292 if(j<0) n = -n; 293 p_h -= t; 294 } 295 t = p_l+p_h; 296 SET_LOW_WORD(t,0); 297 u = t*lg2_h; 298 v = (p_l-(t-p_h))*lg2+t*lg2_l; 299 z = u+v; 300 w = v-(z-u); 301 t = z*z; 302 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 303 r = (z*t1)/(t1-two)-(w+z*w); 304 z = one-(r-z); 305 GET_HIGH_WORD(j,z); 306 j += (n<<20); 307 if((j>>20)<=0) z = scalbn(z,(int)n); /* subnormal output */ 308 else SET_HIGH_WORD(z,j); 309 return s*z; 310 } 311 312 #endif /* defined(_DOUBLE_IS_32BITS) */ 313