1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)e_pow.c 1.3 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* __ieee754_pow(x,y) return x**y 16 * 17 * n 18 * Method: Let x = 2 * (1+f) 19 * 1. Compute and return log2(x) in two pieces: 20 * log2(x) = w1 + w2, 21 * where w1 has 53-24 = 29 bit trailing zeros. 22 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 23 * arithmetic, where |y'|<=0.5. 24 * 3. Return x**y = 2**n*exp(y'*log2) 25 * 26 * Special cases: 27 * 1. (anything) ** 0 is 1 28 * 2. (anything) ** 1 is itself 29 * 3. (anything) ** NAN is NAN 30 * 4. NAN ** (anything except 0) is NAN 31 * 5. +-(|x| > 1) ** +INF is +INF 32 * 6. +-(|x| > 1) ** -INF is +0 33 * 7. +-(|x| < 1) ** +INF is +0 34 * 8. +-(|x| < 1) ** -INF is +INF 35 * 9. +-1 ** +-INF is NAN 36 * 10. +0 ** (+anything except 0, NAN) is +0 37 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 38 * 12. +0 ** (-anything except 0, NAN) is +INF 39 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 40 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 41 * 15. +INF ** (+anything except 0,NAN) is +INF 42 * 16. +INF ** (-anything except 0,NAN) is +0 43 * 17. -INF ** (anything) = -0 ** (-anything) 44 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 45 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 46 * 47 * Accuracy: 48 * pow(x,y) returns x**y nearly rounded. In particular 49 * pow(integer,integer) 50 * always returns the correct integer provided it is 51 * representable. 52 * 53 * Constants : 54 * The hexadecimal values are the intended ones for the following 55 * constants. The decimal values may be used, provided that the 56 * compiler will convert from decimal to binary accurately enough 57 * to produce the hexadecimal values shown. 58 */ 59 60 #include "fdlibm.h" 61 62 static const double 63 bp[] = {1.0, 1.5,}, 64 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 65 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 66 zero = 0.0, 67 one = 1.0, 68 two = 2.0, 69 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 70 Huge = 1.0e300, 71 tiny = 1.0e-300, 72 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 73 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 74 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 75 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 76 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 77 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 78 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 79 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 80 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 81 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 82 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 83 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 84 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 85 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 86 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 87 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 88 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 89 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 90 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 91 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 92 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 93 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 94 95 double __ieee754_pow(double x, double y) 96 { 97 double z,ax,z_h,z_l,p_h,p_l; 98 double y1,t1,t2,r,s,t,u,v,w; 99 int i,j,k,yisint,n; 100 int hx,hy,ix,iy; 101 unsigned lx,ly; 102 103 hx = __HI(x); lx = __LO(x); 104 hy = __HI(y); ly = __LO(y); 105 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 106 107 /* y==zero: x**0 = 1 */ 108 if((iy|ly)==0) return one; 109 110 /* +-NaN return x+y */ 111 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 112 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 113 return x+y; 114 115 /* determine if y is an odd int when x < 0 116 * yisint = 0 ... y is not an integer 117 * yisint = 1 ... y is an odd int 118 * yisint = 2 ... y is an even int 119 */ 120 yisint = 0; 121 if(hx<0) { 122 if(iy>=0x43400000) yisint = 2; /* even integer y */ 123 else if(iy>=0x3ff00000) { 124 k = (iy>>20)-0x3ff; /* exponent */ 125 if(k>20) { 126 j = ly>>(52-k); 127 if((j<<(52-k))==ly) yisint = 2-(j&1); 128 } else if(ly==0) { 129 j = iy>>(20-k); 130 if((j<<(20-k))==iy) yisint = 2-(j&1); 131 } 132 } 133 } 134 135 /* special value of y */ 136 if(ly==0) { 137 if (iy==0x7ff00000) { /* y is +-inf */ 138 if(((ix-0x3ff00000)|lx)==0) 139 return y - y; /* inf**+-1 is NaN */ 140 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 141 return (hy>=0)? y: zero; 142 else /* (|x|<1)**-,+inf = inf,0 */ 143 return (hy<0)?-y: zero; 144 } 145 if(iy==0x3ff00000) { /* y is +-1 */ 146 if(hy<0) return one/x; else return x; 147 } 148 if(hy==0x40000000) return x*x; /* y is 2 */ 149 if(hy==0x3fe00000) { /* y is 0.5 */ 150 if(hx>=0) /* x >= +0 */ 151 return sqrt(x); 152 } 153 } 154 155 ax = fabs(x); 156 /* special value of x */ 157 if(lx==0) { 158 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 159 z = ax; /*x is +-0,+-inf,+-1*/ 160 if(hy<0) z = one/z; /* z = (1/|x|) */ 161 if(hx<0) { 162 if(((ix-0x3ff00000)|yisint)==0) { 163 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 164 } else if(yisint==1) 165 z = -z; /* (x<0)**odd = -(|x|**odd) */ 166 } 167 return z; 168 } 169 } 170 171 /* (x<0)**(non-int) is NaN */ 172 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x); 173 174 /* |y| is Huge */ 175 if(iy>0x41e00000) { /* if |y| > 2**31 */ 176 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 177 if(ix<=0x3fefffff) return (hy<0)? Huge*Huge:tiny*tiny; 178 if(ix>=0x3ff00000) return (hy>0)? Huge*Huge:tiny*tiny; 179 } 180 /* over/underflow if x is not close to one */ 181 if(ix<0x3fefffff) return (hy<0)? Huge*Huge:tiny*tiny; 182 if(ix>0x3ff00000) return (hy>0)? Huge*Huge:tiny*tiny; 183 /* now |1-x| is tiny <= 2**-20, suffice to compute 184 log(x) by x-x^2/2+x^3/3-x^4/4 */ 185 t = x-1; /* t has 20 trailing zeros */ 186 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 187 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 188 v = t*ivln2_l-w*ivln2; 189 t1 = u+v; 190 __LO(t1) = 0; 191 t2 = v-(t1-u); 192 } else { 193 double s2,s_h,s_l,t_h,t_l; 194 n = 0; 195 /* take care subnormal number */ 196 if(ix<0x00100000) 197 {ax *= two53; n -= 53; ix = __HI(ax); } 198 n += ((ix)>>20)-0x3ff; 199 j = ix&0x000fffff; 200 /* determine interval */ 201 ix = j|0x3ff00000; /* normalize ix */ 202 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 203 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 204 else {k=0;n+=1;ix -= 0x00100000;} 205 __HI(ax) = ix; 206 207 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 208 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 209 v = one/(ax+bp[k]); 210 s = u*v; 211 s_h = s; 212 __LO(s_h) = 0; 213 /* t_h=ax+bp[k] High */ 214 t_h = zero; 215 __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 216 t_l = ax - (t_h-bp[k]); 217 s_l = v*((u-s_h*t_h)-s_h*t_l); 218 /* compute log(ax) */ 219 s2 = s*s; 220 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 221 r += s_l*(s_h+s); 222 s2 = s_h*s_h; 223 t_h = 3.0+s2+r; 224 __LO(t_h) = 0; 225 t_l = r-((t_h-3.0)-s2); 226 /* u+v = s*(1+...) */ 227 u = s_h*t_h; 228 v = s_l*t_h+t_l*s; 229 /* 2/(3log2)*(s+...) */ 230 p_h = u+v; 231 __LO(p_h) = 0; 232 p_l = v-(p_h-u); 233 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 234 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 235 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 236 t = (double)n; 237 t1 = (((z_h+z_l)+dp_h[k])+t); 238 __LO(t1) = 0; 239 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 240 } 241 242 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 243 if((((hx>>31)+1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 244 245 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 246 y1 = y; 247 __LO(y1) = 0; 248 p_l = (y-y1)*t1+y*t2; 249 p_h = y1*t1; 250 z = p_l+p_h; 251 j = __HI(z); 252 i = __LO(z); 253 if (j>=0x40900000) { /* z >= 1024 */ 254 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 255 return s*Huge*Huge; /* overflow */ 256 else { 257 if(p_l+ovt>z-p_h) return s*Huge*Huge; /* overflow */ 258 } 259 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 260 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 261 return s*tiny*tiny; /* underflow */ 262 else { 263 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 264 } 265 } 266 /* 267 * compute 2**(p_h+p_l) 268 */ 269 i = j&0x7fffffff; 270 k = (i>>20)-0x3ff; 271 n = 0; 272 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 273 n = j+(0x00100000>>(k+1)); 274 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 275 t = zero; 276 __HI(t) = (n&~(0x000fffff>>k)); 277 n = ((n&0x000fffff)|0x00100000)>>(20-k); 278 if(j<0) n = -n; 279 p_h -= t; 280 } 281 t = p_l+p_h; 282 __LO(t) = 0; 283 u = t*lg2_h; 284 v = (p_l-(t-p_h))*lg2+t*lg2_l; 285 z = u+v; 286 w = v-(z-u); 287 t = z*z; 288 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 289 r = (z*t1)/(t1-two)-(w+z*w); 290 z = one-(r-z); 291 j = __HI(z); 292 j += (n<<20); 293 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 294 else __HI(z) += (n<<20); 295 return s*z; 296 } 297