e_pow.c (9912B)
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: src/lib/msun/src/e_pow.c,v 1.10 2004/06/01 19:28:38 bde Exp $"; 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 static const double 66 bp[] = {1.0, 1.5,}, 67 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 68 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 69 zero = 0.0, 70 one = 1.0, 71 two = 2.0, 72 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 73 huge = 1.0e300, 74 tiny = 1.0e-300, 75 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 76 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 77 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 78 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 79 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 80 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 81 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 82 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 83 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 84 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 85 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 86 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 87 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 88 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 89 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 90 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 91 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 92 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 93 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 94 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 95 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 96 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 97 98 double 99 __ieee754_pow(double x, double y) 100 { 101 double z,ax,z_h,z_l,p_h,p_l; 102 double y1,t1,t2,r,s,sn,t,u,v,w; 103 int32_t i,j,k,yisint,n; 104 int32_t hx,hy,ix,iy; 105 u_int32_t lx,ly; 106 107 EXTRACT_WORDS(hx,lx,x); 108 EXTRACT_WORDS(hy,ly,y); 109 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 110 111 /* y==zero: x**0 = 1 */ 112 if((iy|ly)==0) return one; 113 114 /* +-NaN return x+y */ 115 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 116 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 117 return x+y; 118 119 /* determine if y is an odd int when x < 0 120 * yisint = 0 ... y is not an integer 121 * yisint = 1 ... y is an odd int 122 * yisint = 2 ... y is an even int 123 */ 124 yisint = 0; 125 if(hx<0) { 126 if(iy>=0x43400000) yisint = 2; /* even integer y */ 127 else if(iy>=0x3ff00000) { 128 k = (iy>>20)-0x3ff; /* exponent */ 129 if(k>20) { 130 j = ly>>(52-k); 131 if((j<<(52-k))==ly) yisint = 2-(j&1); 132 } else if(ly==0) { 133 j = iy>>(20-k); 134 if((j<<(20-k))==iy) yisint = 2-(j&1); 135 } 136 } 137 } 138 139 /* special value of y */ 140 if(ly==0) { 141 if (iy==0x7ff00000) { /* y is +-inf */ 142 if(((ix-0x3ff00000)|lx)==0) 143 return y - y; /* inf**+-1 is NaN */ 144 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 145 return (hy>=0)? y: zero; 146 else /* (|x|<1)**-,+inf = inf,0 */ 147 return (hy<0)?-y: zero; 148 } 149 if(iy==0x3ff00000) { /* y is +-1 */ 150 if(hy<0) return one/x; else return x; 151 } 152 if(hy==0x40000000) return x*x; /* y is 2 */ 153 if(hy==0x3fe00000) { /* y is 0.5 */ 154 if(hx>=0) /* x >= +0 */ 155 return __ieee754_sqrt(x); 156 } 157 } 158 159 ax = fabs(x); 160 /* special value of x */ 161 if(lx==0) { 162 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 163 z = ax; /*x is +-0,+-inf,+-1*/ 164 if(hy<0) z = one/z; /* z = (1/|x|) */ 165 if(hx<0) { 166 if(((ix-0x3ff00000)|yisint)==0) { 167 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 168 } else if(yisint==1) 169 z = -z; /* (x<0)**odd = -(|x|**odd) */ 170 } 171 return z; 172 } 173 } 174 175 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be 176 n = (hx>>31)+1; 177 but ANSI C says a right shift of a signed negative quantity is 178 implementation defined. */ 179 n = ((u_int32_t)hx>>31)-1; 180 181 /* (x<0)**(non-int) is NaN */ 182 if((n|yisint)==0) return (x-x)/(x-x); 183 184 sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 185 if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */ 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)? sn*huge*huge:sn*tiny*tiny; 195 if(ix>0x3ff00000) return (hy>0)? sn*huge*huge:sn*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 = ax-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 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 256 y1 = y; 257 SET_LOW_WORD(y1,0); 258 p_l = (y-y1)*t1+y*t2; 259 p_h = y1*t1; 260 z = p_l+p_h; 261 EXTRACT_WORDS(j,i,z); 262 if (j>=0x40900000) { /* z >= 1024 */ 263 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 264 return sn*huge*huge; /* overflow */ 265 else { 266 if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */ 267 } 268 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 269 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 270 return sn*tiny*tiny; /* underflow */ 271 else { 272 if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */ 273 } 274 } 275 /* 276 * compute 2**(p_h+p_l) 277 */ 278 i = j&0x7fffffff; 279 k = (i>>20)-0x3ff; 280 n = 0; 281 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 282 n = j+(0x00100000>>(k+1)); 283 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 284 t = zero; 285 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 286 n = ((n&0x000fffff)|0x00100000)>>(20-k); 287 if(j<0) n = -n; 288 p_h -= t; 289 } 290 t = p_l+p_h; 291 SET_LOW_WORD(t,0); 292 u = t*lg2_h; 293 v = (p_l-(t-p_h))*lg2+t*lg2_l; 294 z = u+v; 295 w = v-(z-u); 296 t = z*z; 297 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 298 r = (z*t1)/(t1-two)-(w+z*w); 299 z = one-(r-z); 300 GET_HIGH_WORD(j,z); 301 j += (n<<20); 302 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 303 else SET_HIGH_WORD(z,j); 304 return sn*z; 305 }