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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_pow.c,v 1.11 1999/07/02 15:37:40 simonb Exp $"); 16 #endif 17 18 /* __ieee754_pow(x,y) return x**y 19 * 20 * n 21 * Method: Let x = 2 * (1+f) 22 * 1. Compute and return log2(x) in two pieces: 23 * log2(x) = w1 + w2, 24 * where w1 has 53-24 = 29 bit trailing zeros. 25 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 26 * arithmetic, where |y'|<=0.5. 27 * 3. Return x**y = 2**n*exp(y'*log2) 28 * 29 * Special cases: 30 * 1. (anything) ** 0 is 1 31 * 2. (anything) ** 1 is itself 32 * 3. (anything) ** NAN is NAN 33 * 4. NAN ** (anything except 0) is NAN 34 * 5. +-(|x| > 1) ** +INF is +INF 35 * 6. +-(|x| > 1) ** -INF is +0 36 * 7. +-(|x| < 1) ** +INF is +0 37 * 8. +-(|x| < 1) ** -INF is +INF 38 * 9. +-1 ** +-INF is NAN 39 * 10. +0 ** (+anything except 0, NAN) is +0 40 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 41 * 12. +0 ** (-anything except 0, NAN) is +INF 42 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 43 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 44 * 15. +INF ** (+anything except 0,NAN) is +INF 45 * 16. +INF ** (-anything except 0,NAN) is +0 46 * 17. -INF ** (anything) = -0 ** (-anything) 47 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 48 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 49 * 50 * Accuracy: 51 * pow(x,y) returns x**y nearly rounded. In particular 52 * pow(integer,integer) 53 * always returns the correct integer provided it is 54 * representable. 55 * 56 * Constants : 57 * The hexadecimal values are the intended ones for the following 58 * constants. The decimal values may be used, provided that the 59 * compiler will convert from decimal to binary accurately enough 60 * to produce the hexadecimal values shown. 61 */ 62 63 #include "math.h" 64 #include "math_private.h" 65 66 #ifdef __STDC__ 67 static const double 68 #else 69 static double 70 #endif 71 bp[] = {1.0, 1.5,}, 72 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 73 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 74 zero = 0.0, 75 one = 1.0, 76 two = 2.0, 77 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 78 huge = 1.0e300, 79 tiny = 1.0e-300, 80 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 81 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 82 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 83 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 84 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 85 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 86 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 87 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 88 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 89 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 90 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 91 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 92 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 93 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 94 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 95 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 96 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 97 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 98 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 99 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 100 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 101 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 102 103 #ifdef __STDC__ 104 double __ieee754_pow(double x, double y) 105 #else 106 double __ieee754_pow(x,y) 107 double x, y; 108 #endif 109 { 110 double z,ax,z_h,z_l,p_h,p_l; 111 double y1,t1,t2,r,s,t,u,v,w; 112 int32_t i,j,k,yisint,n; 113 int32_t hx,hy,ix,iy; 114 u_int32_t lx,ly; 115 116 EXTRACT_WORDS(hx,lx,x); 117 EXTRACT_WORDS(hy,ly,y); 118 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 119 120 /* y==zero: x**0 = 1 */ 121 if((iy|ly)==0) return one; 122 123 /* +-NaN return x+y */ 124 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 125 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 126 return x+y; 127 128 /* determine if y is an odd int when x < 0 129 * yisint = 0 ... y is not an integer 130 * yisint = 1 ... y is an odd int 131 * yisint = 2 ... y is an even int 132 */ 133 yisint = 0; 134 if(hx<0) { 135 if(iy>=0x43400000) yisint = 2; /* even integer y */ 136 else if(iy>=0x3ff00000) { 137 k = (iy>>20)-0x3ff; /* exponent */ 138 if(k>20) { 139 j = ly>>(52-k); 140 if((j<<(52-k))==ly) yisint = 2-(j&1); 141 } else if(ly==0) { 142 j = iy>>(20-k); 143 if((j<<(20-k))==iy) yisint = 2-(j&1); 144 } 145 } 146 } 147 148 /* special value of y */ 149 if(ly==0) { 150 if (iy==0x7ff00000) { /* y is +-inf */ 151 if(((ix-0x3ff00000)|lx)==0) 152 return y - y; /* inf**+-1 is NaN */ 153 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 154 return (hy>=0)? y: zero; 155 else /* (|x|<1)**-,+inf = inf,0 */ 156 return (hy<0)?-y: zero; 157 } 158 if(iy==0x3ff00000) { /* y is +-1 */ 159 if(hy<0) return one/x; else return x; 160 } 161 if(hy==0x40000000) return x*x; /* y is 2 */ 162 if(hy==0x3fe00000) { /* y is 0.5 */ 163 if(hx>=0) /* x >= +0 */ 164 return __ieee754_sqrt(x); 165 } 166 } 167 168 ax = fabs(x); 169 /* special value of x */ 170 if(lx==0) { 171 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 172 z = ax; /*x is +-0,+-inf,+-1*/ 173 if(hy<0) z = one/z; /* z = (1/|x|) */ 174 if(hx<0) { 175 if(((ix-0x3ff00000)|yisint)==0) { 176 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 177 } else if(yisint==1) 178 z = -z; /* (x<0)**odd = -(|x|**odd) */ 179 } 180 return z; 181 } 182 } 183 184 /* (x<0)**(non-int) is NaN */ 185 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 186 187 /* |y| is huge */ 188 if(iy>0x41e00000) { /* if |y| > 2**31 */ 189 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 190 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 191 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 192 } 193 /* over/underflow if x is not close to one */ 194 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 195 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 196 /* now |1-x| is tiny <= 2**-20, suffice to compute 197 log(x) by x-x^2/2+x^3/3-x^4/4 */ 198 t = x-1; /* t has 20 trailing zeros */ 199 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 200 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 201 v = t*ivln2_l-w*ivln2; 202 t1 = u+v; 203 SET_LOW_WORD(t1,0); 204 t2 = v-(t1-u); 205 } else { 206 double s2,s_h,s_l,t_h,t_l; 207 n = 0; 208 /* take care subnormal number */ 209 if(ix<0x00100000) 210 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 211 n += ((ix)>>20)-0x3ff; 212 j = ix&0x000fffff; 213 /* determine interval */ 214 ix = j|0x3ff00000; /* normalize ix */ 215 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 216 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 217 else {k=0;n+=1;ix -= 0x00100000;} 218 SET_HIGH_WORD(ax,ix); 219 220 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 221 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 222 v = one/(ax+bp[k]); 223 s = u*v; 224 s_h = s; 225 SET_LOW_WORD(s_h,0); 226 /* t_h=ax+bp[k] High */ 227 t_h = zero; 228 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 229 t_l = ax - (t_h-bp[k]); 230 s_l = v*((u-s_h*t_h)-s_h*t_l); 231 /* compute log(ax) */ 232 s2 = s*s; 233 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 234 r += s_l*(s_h+s); 235 s2 = s_h*s_h; 236 t_h = 3.0+s2+r; 237 SET_LOW_WORD(t_h,0); 238 t_l = r-((t_h-3.0)-s2); 239 /* u+v = s*(1+...) */ 240 u = s_h*t_h; 241 v = s_l*t_h+t_l*s; 242 /* 2/(3log2)*(s+...) */ 243 p_h = u+v; 244 SET_LOW_WORD(p_h,0); 245 p_l = v-(p_h-u); 246 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 247 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 248 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 249 t = (double)n; 250 t1 = (((z_h+z_l)+dp_h[k])+t); 251 SET_LOW_WORD(t1,0); 252 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 253 } 254 255 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 256 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 257 s = -one;/* (-ve)**(odd int) */ 258 259 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 260 y1 = y; 261 SET_LOW_WORD(y1,0); 262 p_l = (y-y1)*t1+y*t2; 263 p_h = y1*t1; 264 z = p_l+p_h; 265 EXTRACT_WORDS(j,i,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 SET_HIGH_WORD(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 SET_LOW_WORD(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 GET_HIGH_WORD(j,z); 305 j += (n<<20); 306 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 307 else SET_HIGH_WORD(z,j); 308 return s*z; 309 } 310