1 /* @(#)e_pow.c 1.5 04/04/22 SMI */ 2 /* 3 * ==================================================== 4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 #include <sys/cdefs.h> 13 #if defined(LIBM_SCCS) && !defined(lint) 14 __RCSID("$NetBSD: e_pow.c,v 1.17 2016/08/27 10:01:08 christos 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 multi-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 except 1 ** NAN = 1 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 1 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 "namespace.h" 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 yy1,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 /* x==1: 1**y = 1, even if y is NaN */ 116 if (hx==0x3ff00000 && lx == 0) return one; 117 118 /* y!=zero: result is NaN if either arg is NaN */ 119 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 120 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 121 return (x+0.0)+(y+0.0); 122 123 /* determine if y is an odd int when x < 0 124 * yisint = 0 ... y is not an integer 125 * yisint = 1 ... y is an odd int 126 * yisint = 2 ... y is an even int 127 */ 128 yisint = 0; 129 if(hx<0) { 130 if(iy>=0x43400000) yisint = 2; /* even integer y */ 131 else if(iy>=0x3ff00000) { 132 k = (iy>>20)-0x3ff; /* exponent */ 133 if(k>20) { 134 j = ly>>(52-k); 135 if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1); 136 } else if(ly==0) { 137 j = iy>>(20-k); 138 if((j<<(20-k))==iy) yisint = 2-(j&1); 139 } 140 } 141 } 142 143 /* special value of y */ 144 if(ly==0) { 145 if (iy==0x7ff00000) { /* y is +-inf */ 146 if(((ix-0x3ff00000)|lx)==0) 147 return one; /* (-1)**+-inf is 1 */ 148 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 149 return (hy>=0)? y: zero; 150 else /* (|x|<1)**-,+inf = inf,0 */ 151 return (hy<0)?-y: zero; 152 } 153 if(iy==0x3ff00000) { /* y is +-1 */ 154 if(hy<0) return one/x; else return x; 155 } 156 if(hy==0x40000000) return x*x; /* y is 2 */ 157 if(hy==0x3fe00000) { /* y is 0.5 */ 158 if(hx>=0) /* x >= +0 */ 159 return __ieee754_sqrt(x); 160 } 161 } 162 163 ax = fabs(x); 164 /* special value of x */ 165 if(lx==0) { 166 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 167 z = ax; /*x is +-0,+-inf,+-1*/ 168 if(hy<0) z = one/z; /* z = (1/|x|) */ 169 if(hx<0) { 170 if(((ix-0x3ff00000)|yisint)==0) { 171 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 172 } else if(yisint==1) 173 z = -z; /* (x<0)**odd = -(|x|**odd) */ 174 } 175 return z; 176 } 177 } 178 179 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be 180 n = (hx>>31)+1; 181 but ANSI C says a right shift of a signed negative quantity is 182 implementation defined. */ 183 n = ((u_int32_t)hx>>31)-1; 184 185 /* (x<0)**(non-int) is NaN */ 186 if((n|yisint)==0) return (x-x)/(x-x); 187 188 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 189 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 190 191 /* |y| is huge */ 192 if(iy>0x41e00000) { /* if |y| > 2**31 */ 193 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 194 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 195 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 196 } 197 /* over/underflow if x is not close to one */ 198 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 199 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 200 /* now |1-x| is tiny <= 2**-20, suffice to compute 201 log(x) by x-x^2/2+x^3/3-x^4/4 */ 202 t = ax-one; /* t has 20 trailing zeros */ 203 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 204 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 205 v = t*ivln2_l-w*ivln2; 206 t1 = u+v; 207 SET_LOW_WORD(t1,0); 208 t2 = v-(t1-u); 209 } else { 210 double ss,s2,s_h,s_l,t_h,t_l; 211 n = 0; 212 /* take care subnormal number */ 213 if(ix<0x00100000) 214 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 215 n += ((ix)>>20)-0x3ff; 216 j = ix&0x000fffff; 217 /* determine interval */ 218 ix = j|0x3ff00000; /* normalize ix */ 219 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 220 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 221 else {k=0;n+=1;ix -= 0x00100000;} 222 SET_HIGH_WORD(ax,ix); 223 224 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 225 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 226 v = one/(ax+bp[k]); 227 ss = u*v; 228 s_h = ss; 229 SET_LOW_WORD(s_h,0); 230 /* t_h=ax+bp[k] High */ 231 t_h = zero; 232 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 233 t_l = ax - (t_h-bp[k]); 234 s_l = v*((u-s_h*t_h)-s_h*t_l); 235 /* compute log(ax) */ 236 s2 = ss*ss; 237 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 238 r += s_l*(s_h+ss); 239 s2 = s_h*s_h; 240 t_h = 3.0+s2+r; 241 SET_LOW_WORD(t_h,0); 242 t_l = r-((t_h-3.0)-s2); 243 /* u+v = ss*(1+...) */ 244 u = s_h*t_h; 245 v = s_l*t_h+t_l*ss; 246 /* 2/(3log2)*(ss+...) */ 247 p_h = u+v; 248 SET_LOW_WORD(p_h,0); 249 p_l = v-(p_h-u); 250 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 251 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 252 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 253 t = (double)n; 254 t1 = (((z_h+z_l)+dp_h[k])+t); 255 SET_LOW_WORD(t1,0); 256 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 257 } 258 259 /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ 260 yy1 = y; 261 SET_LOW_WORD(yy1,0); 262 p_l = (y-yy1)*t1+y*t2; 263 p_h = yy1*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
Indexes created Sat Sep 27 22:09:54 GMT 2025