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 13 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 /* __ieee754_pow(x,y) return x**y 18 * 19 * n 20 * Method: Let x = 2 * (1+f) 21 * 1. Compute and return log2(x) in two pieces: 22 * log2(x) = w1 + w2, 23 * where w1 has 53-24 = 29 bit trailing zeros. 24 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 25 * arithmetic, where |y'|<=0.5. 26 * 3. Return x**y = 2**n*exp(y'*log2) 27 * 28 * Special cases: 29 * 1. (anything) ** 0 is 1 30 * 2. (anything) ** 1 is itself 31 * 3. (anything) ** NAN is NAN 32 * 4. NAN ** (anything except 0) is NAN 33 * 5. +-(|x| > 1) ** +INF is +INF 34 * 6. +-(|x| > 1) ** -INF is +0 35 * 7. +-(|x| < 1) ** +INF is +0 36 * 8. +-(|x| < 1) ** -INF is +INF 37 * 9. +-1 ** +-INF is NAN 38 * 10. +0 ** (+anything except 0, NAN) is +0 39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 40 * 12. +0 ** (-anything except 0, NAN) is +INF 41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 43 * 15. +INF ** (+anything except 0,NAN) is +INF 44 * 16. +INF ** (-anything except 0,NAN) is +0 45 * 17. -INF ** (anything) = -0 ** (-anything) 46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 48 * 49 * Accuracy: 50 * pow(x,y) returns x**y nearly rounded. In particular 51 * pow(integer,integer) 52 * always returns the correct integer provided it is 53 * representable. 54 * 55 * Constants : 56 * The hexadecimal values are the intended ones for the following 57 * constants. The decimal values may be used, provided that the 58 * compiler will convert from decimal to binary accurately enough 59 * to produce the hexadecimal values shown. 60 */ 61 62 #include "math.h" 63 #include "math_private.h" 64 65 #ifdef __STDC__ 66 static const double 67 #else 68 static double 69 #endif 70 bp[] = {1.0, 1.5,}, 71 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 72 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 73 zero = 0.0, 74 one = 1.0, 75 two = 2.0, 76 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 77 huge = 1.0e300, 78 tiny = 1.0e-300, 79 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 80 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 81 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 82 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 83 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 84 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 85 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 86 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 87 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 88 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 89 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 90 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 91 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 92 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 93 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 94 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 95 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 96 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 97 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 98 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 99 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 100 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 101 102 #ifdef __STDC__ 103 double __ieee754_pow(double x, double y) 104 #else 105 double __ieee754_pow(x,y) 106 double x, y; 107 #endif 108 { 109 double z,ax,z_h,z_l,p_h,p_l; 110 double y1,t1,t2,r,s,t,u,v,w; 111 int32_t i,j,k,yisint,n; 112 int32_t hx,hy,ix,iy; 113 u_int32_t lx,ly; 114 115 EXTRACT_WORDS(hx,lx,x); 116 EXTRACT_WORDS(hy,ly,y); 117 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 118 119 /* y==zero: x**0 = 1 */ 120 if((iy|ly)==0) return one; 121 122 /* +-NaN return x+y */ 123 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 124 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 125 return x+y; 126 127 /* determine if y is an odd int when x < 0 128 * yisint = 0 ... y is not an integer 129 * yisint = 1 ... y is an odd int 130 * yisint = 2 ... y is an even int 131 */ 132 yisint = 0; 133 if(hx<0) { 134 if(iy>=0x43400000) yisint = 2; /* even integer y */ 135 else if(iy>=0x3ff00000) { 136 k = (iy>>20)-0x3ff; /* exponent */ 137 if(k>20) { 138 j = ly>>(52-k); 139 if((j<<(52-k))==ly) yisint = 2-(j&1); 140 } else if(ly==0) { 141 j = iy>>(20-k); 142 if((j<<(20-k))==iy) yisint = 2-(j&1); 143 } 144 } 145 } 146 147 /* special value of y */ 148 if(ly==0) { 149 if (iy==0x7ff00000) { /* y is +-inf */ 150 if(((ix-0x3ff00000)|lx)==0) 151 return y - y; /* inf**+-1 is NaN */ 152 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 153 return (hy>=0)? y: zero; 154 else /* (|x|<1)**-,+inf = inf,0 */ 155 return (hy<0)?-y: zero; 156 } 157 if(iy==0x3ff00000) { /* y is +-1 */ 158 if(hy<0) return one/x; else return x; 159 } 160 if(hy==0x40000000) return x*x; /* y is 2 */ 161 if(hy==0x3fe00000) { /* y is 0.5 */ 162 if(hx>=0) /* x >= +0 */ 163 return __ieee754_sqrt(x); 164 } 165 } 166 167 ax = fabs(x); 168 /* special value of x */ 169 if(lx==0) { 170 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 171 z = ax; /*x is +-0,+-inf,+-1*/ 172 if(hy<0) z = one/z; /* z = (1/|x|) */ 173 if(hx<0) { 174 if(((ix-0x3ff00000)|yisint)==0) { 175 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 176 } else if(yisint==1) 177 z = -z; /* (x<0)**odd = -(|x|**odd) */ 178 } 179 return z; 180 } 181 } 182 183 /* (x<0)**(non-int) is NaN */ 184 /* CYGNUS LOCAL: This used to be 185 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x); 186 but ANSI C says a right shift of a signed negative quantity is 187 implementation defined. */ 188 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 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)? huge*huge:tiny*tiny; 198 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 199 /* now |1-x| is tiny <= 2**-20, suffice to compute 200 log(x) by x-x^2/2+x^3/3-x^4/4 */ 201 t = x-1; /* 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 SET_LOW_WORD(t1,0); 207 t2 = v-(t1-u); 208 } else { 209 double 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; GET_HIGH_WORD(ix,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|<sqrt(3/2) */ 219 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 220 else {k=0;n+=1;ix -= 0x00100000;} 221 SET_HIGH_WORD(ax,ix); 222 223 /* compute s = 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 s = u*v; 227 s_h = s; 228 SET_LOW_WORD(s_h,0); 229 /* t_h=ax+bp[k] High */ 230 t_h = zero; 231 SET_HIGH_WORD(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 log(ax) */ 235 s2 = s*s; 236 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 237 r += s_l*(s_h+s); 238 s2 = s_h*s_h; 239 t_h = 3.0+s2+r; 240 SET_LOW_WORD(t_h,0); 241 t_l = r-((t_h-3.0)-s2); 242 /* u+v = s*(1+...) */ 243 u = s_h*t_h; 244 v = s_l*t_h+t_l*s; 245 /* 2/(3log2)*(s+...) */ 246 p_h = u+v; 247 SET_LOW_WORD(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) = (s+..)*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 SET_LOW_WORD(t1,0); 255 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 256 } 257 258 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 259 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 260 s = -one;/* (-ve)**(odd int) */ 261 262 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 263 y1 = y; 264 SET_LOW_WORD(y1,0); 265 p_l = (y-y1)*t1+y*t2; 266 p_h = y1*t1; 267 z = p_l+p_h; 268 EXTRACT_WORDS(j,i,z); 269 if (j>=0x40900000) { /* z >= 1024 */ 270 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 271 return s*huge*huge; /* overflow */ 272 else { 273 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 274 } 275 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 276 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 277 return s*tiny*tiny; /* underflow */ 278 else { 279 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 280 } 281 } 282 /* 283 * compute 2**(p_h+p_l) 284 */ 285 i = j&0x7fffffff; 286 k = (i>>20)-0x3ff; 287 n = 0; 288 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 289 n = j+(0x00100000>>(k+1)); 290 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 291 t = zero; 292 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 293 n = ((n&0x000fffff)|0x00100000)>>(20-k); 294 if(j<0) n = -n; 295 p_h -= t; 296 } 297 t = p_l+p_h; 298 SET_LOW_WORD(t,0); 299 u = t*lg2_h; 300 v = (p_l-(t-p_h))*lg2+t*lg2_l; 301 z = u+v; 302 w = v-(z-u); 303 t = z*z; 304 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 305 r = (z*t1)/(t1-two)-(w+z*w); 306 z = one-(r-z); 307 GET_HIGH_WORD(j,z); 308 j += (n<<20); 309 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 310 else SET_HIGH_WORD(z,j); 311 return s*z; 312 } 313