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 1.12 2002/05/26 22:01:51 wiz 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 static const double 67 bp[] = {1.0, 1.5,}, 68 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 69 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 70 zero = 0.0, 71 one = 1.0, 72 two = 2.0, 73 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 74 huge = 1.0e300, 75 tiny = 1.0e-300, 76 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 77 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 78 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 79 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 80 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 81 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 82 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 83 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 84 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 85 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 86 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 87 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 88 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 89 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 90 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 91 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 92 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 93 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 94 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 95 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 96 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 97 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 98 99 double 100 __ieee754_pow(double x, double y) 101 { 102 double z,ax,z_h,z_l,p_h,p_l; 103 double y1,t1,t2,r,s,t,u,v,w; 104 int32_t i,j,k,yisint,n; 105 int32_t hx,hy,ix,iy; 106 u_int32_t lx,ly; 107 108 EXTRACT_WORDS(hx,lx,x); 109 EXTRACT_WORDS(hy,ly,y); 110 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 111 112 /* y==zero: x**0 = 1 */ 113 if((iy|ly)==0) return one; 114 115 /* +-NaN return x+y */ 116 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 117 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 118 return x+y; 119 120 /* determine if y is an odd int when x < 0 121 * yisint = 0 ... y is not an integer 122 * yisint = 1 ... y is an odd int 123 * yisint = 2 ... y is an even int 124 */ 125 yisint = 0; 126 if(hx<0) { 127 if(iy>=0x43400000) yisint = 2; /* even integer y */ 128 else if(iy>=0x3ff00000) { 129 k = (iy>>20)-0x3ff; /* exponent */ 130 if(k>20) { 131 j = ly>>(52-k); 132 if((j<<(52-k))==ly) yisint = 2-(j&1); 133 } else if(ly==0) { 134 j = iy>>(20-k); 135 if((j<<(20-k))==iy) yisint = 2-(j&1); 136 } 137 } 138 } 139 140 /* special value of y */ 141 if(ly==0) { 142 if (iy==0x7ff00000) { /* y is +-inf */ 143 if(((ix-0x3ff00000)|lx)==0) 144 return y - y; /* inf**+-1 is NaN */ 145 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 146 return (hy>=0)? y: zero; 147 else /* (|x|<1)**-,+inf = inf,0 */ 148 return (hy<0)?-y: zero; 149 } 150 if(iy==0x3ff00000) { /* y is +-1 */ 151 if(hy<0) return one/x; else return x; 152 } 153 if(hy==0x40000000) return x*x; /* y is 2 */ 154 if(hy==0x3fe00000) { /* y is 0.5 */ 155 if(hx>=0) /* x >= +0 */ 156 return __ieee754_sqrt(x); 157 } 158 } 159 160 ax = fabs(x); 161 /* special value of x */ 162 if(lx==0) { 163 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 164 z = ax; /*x is +-0,+-inf,+-1*/ 165 if(hy<0) z = one/z; /* z = (1/|x|) */ 166 if(hx<0) { 167 if(((ix-0x3ff00000)|yisint)==0) { 168 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 169 } else if(yisint==1) 170 z = -z; /* (x<0)**odd = -(|x|**odd) */ 171 } 172 return z; 173 } 174 } 175 176 /* (x<0)**(non-int) is NaN */ 177 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 178 179 /* |y| is huge */ 180 if(iy>0x41e00000) { /* if |y| > 2**31 */ 181 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 182 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 183 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 184 } 185 /* over/underflow if x is not close to one */ 186 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 187 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 188 /* now |1-x| is tiny <= 2**-20, suffice to compute 189 log(x) by x-x^2/2+x^3/3-x^4/4 */ 190 t = x-1; /* t has 20 trailing zeros */ 191 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 192 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 193 v = t*ivln2_l-w*ivln2; 194 t1 = u+v; 195 SET_LOW_WORD(t1,0); 196 t2 = v-(t1-u); 197 } else { 198 double s2,s_h,s_l,t_h,t_l; 199 n = 0; 200 /* take care subnormal number */ 201 if(ix<0x00100000) 202 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 203 n += ((ix)>>20)-0x3ff; 204 j = ix&0x000fffff; 205 /* determine interval */ 206 ix = j|0x3ff00000; /* normalize ix */ 207 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 208 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 209 else {k=0;n+=1;ix -= 0x00100000;} 210 SET_HIGH_WORD(ax,ix); 211 212 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 213 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 214 v = one/(ax+bp[k]); 215 s = u*v; 216 s_h = s; 217 SET_LOW_WORD(s_h,0); 218 /* t_h=ax+bp[k] High */ 219 t_h = zero; 220 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 221 t_l = ax - (t_h-bp[k]); 222 s_l = v*((u-s_h*t_h)-s_h*t_l); 223 /* compute log(ax) */ 224 s2 = s*s; 225 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 226 r += s_l*(s_h+s); 227 s2 = s_h*s_h; 228 t_h = 3.0+s2+r; 229 SET_LOW_WORD(t_h,0); 230 t_l = r-((t_h-3.0)-s2); 231 /* u+v = s*(1+...) */ 232 u = s_h*t_h; 233 v = s_l*t_h+t_l*s; 234 /* 2/(3log2)*(s+...) */ 235 p_h = u+v; 236 SET_LOW_WORD(p_h,0); 237 p_l = v-(p_h-u); 238 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 239 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 240 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 241 t = (double)n; 242 t1 = (((z_h+z_l)+dp_h[k])+t); 243 SET_LOW_WORD(t1,0); 244 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 245 } 246 247 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 248 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 249 s = -one;/* (-ve)**(odd int) */ 250 251 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 252 y1 = y; 253 SET_LOW_WORD(y1,0); 254 p_l = (y-y1)*t1+y*t2; 255 p_h = y1*t1; 256 z = p_l+p_h; 257 EXTRACT_WORDS(j,i,z); 258 if (j>=0x40900000) { /* z >= 1024 */ 259 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 260 return s*huge*huge; /* overflow */ 261 else { 262 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 263 } 264 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 265 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 266 return s*tiny*tiny; /* underflow */ 267 else { 268 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 269 } 270 } 271 /* 272 * compute 2**(p_h+p_l) 273 */ 274 i = j&0x7fffffff; 275 k = (i>>20)-0x3ff; 276 n = 0; 277 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 278 n = j+(0x00100000>>(k+1)); 279 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 280 t = zero; 281 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 282 n = ((n&0x000fffff)|0x00100000)>>(20-k); 283 if(j<0) n = -n; 284 p_h -= t; 285 } 286 t = p_l+p_h; 287 SET_LOW_WORD(t,0); 288 u = t*lg2_h; 289 v = (p_l-(t-p_h))*lg2+t*lg2_l; 290 z = u+v; 291 w = v-(z-u); 292 t = z*z; 293 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 294 r = (z*t1)/(t1-two)-(w+z*w); 295 z = one-(r-z); 296 GET_HIGH_WORD(j,z); 297 j += (n<<20); 298 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 299 else SET_HIGH_WORD(z,j); 300 return s*z; 301 } 302