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 __FBSDID("$FreeBSD$"); 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 multi-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 except 1 ** NAN = 1 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 1 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 <float.h> 61 #include "math.h" 62 #include "math_private.h" 63 64 static const double 65 bp[] = {1.0, 1.5,}, 66 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 67 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 68 zero = 0.0, 69 half = 0.5, 70 qrtr = 0.25, 71 thrd = 3.3333333333333331e-01, /* 0x3fd55555, 0x55555555 */ 72 one = 1.0, 73 two = 2.0, 74 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 75 huge = 1.0e300, 76 tiny = 1.0e-300, 77 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 78 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 79 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 80 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 81 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 82 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 83 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 84 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 85 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 86 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 87 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 88 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 89 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 90 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 91 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 92 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 93 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 94 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 95 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 96 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 97 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 98 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 99 100 double 101 __ieee754_pow(double x, double y) 102 { 103 double z,ax,z_h,z_l,p_h,p_l; 104 double y1,t1,t2,r,s,t,u,v,w; 105 int32_t i,j,k,yisint,n; 106 int32_t hx,hy,ix,iy; 107 u_int32_t lx,ly; 108 109 EXTRACT_WORDS(hx,lx,x); 110 EXTRACT_WORDS(hy,ly,y); 111 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 112 113 /* y==zero: x**0 = 1 */ 114 if((iy|ly)==0) return one; 115 116 /* x==1: 1**y = 1, even if y is NaN */ 117 if (hx==0x3ff00000 && lx == 0) return one; 118 119 /* y!=zero: result is NaN if either arg is NaN */ 120 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 121 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 122 return nan_mix(x, y); 123 124 /* determine if y is an odd int when x < 0 125 * yisint = 0 ... y is not an integer 126 * yisint = 1 ... y is an odd int 127 * yisint = 2 ... y is an even int 128 */ 129 yisint = 0; 130 if(hx<0) { 131 if(iy>=0x43400000) yisint = 2; /* even integer y */ 132 else if(iy>=0x3ff00000) { 133 k = (iy>>20)-0x3ff; /* exponent */ 134 if(k>20) { 135 j = ly>>(52-k); 136 if(((u_int32_t)j<<(52-k))==ly) yisint = 2-(j&1); 137 } else if(ly==0) { 138 j = iy>>(20-k); 139 if((j<<(20-k))==iy) yisint = 2-(j&1); 140 } 141 } 142 } 143 144 /* special value of y */ 145 if(ly==0) { 146 if (iy==0x7ff00000) { /* y is +-inf */ 147 if(((ix-0x3ff00000)|lx)==0) 148 return one; /* (-1)**+-inf is 1 */ 149 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 150 return (hy>=0)? y: zero; 151 else /* (|x|<1)**-,+inf = inf,0 */ 152 return (hy<0)?-y: zero; 153 } 154 if(iy==0x3ff00000) { /* y is +-1 */ 155 if(hy<0) return one/x; else return x; 156 } 157 if(hy==0x40000000) return x*x; /* y is 2 */ 158 if(hy==0x3fe00000) { /* y is 0.5 */ 159 if(hx>=0) /* x >= +0 */ 160 return sqrt(x); 161 } 162 } 163 164 ax = fabs(x); 165 /* special value of x */ 166 if(lx==0) { 167 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 168 z = ax; /*x is +-0,+-inf,+-1*/ 169 if(hy<0) z = one/z; /* z = (1/|x|) */ 170 if(hx<0) { 171 if(((ix-0x3ff00000)|yisint)==0) { 172 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 173 } else if(yisint==1) 174 z = -z; /* (x<0)**odd = -(|x|**odd) */ 175 } 176 return z; 177 } 178 } 179 180 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be 181 n = (hx>>31)+1; 182 but ANSI C says a right shift of a signed negative quantity is 183 implementation defined. */ 184 n = ((u_int32_t)hx>>31)-1; 185 186 /* (x<0)**(non-int) is NaN */ 187 if((n|yisint)==0) return (x-x)/(x-x); 188 189 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 190 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 191 192 /* |y| is huge */ 193 if(iy>0x41e00000) { /* if |y| > 2**31 */ 194 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 195 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 196 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 197 } 198 /* over/underflow if x is not close to one */ 199 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 200 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 201 /* now |1-x| is tiny <= 2**-20, suffice to compute 202 log(x) by x-x^2/2+x^3/3-x^4/4 */ 203 t = ax-one; /* t has 20 trailing zeros */ 204 w = (t*t)*(half-t*(thrd-t*qrtr)); 205 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 206 v = t*ivln2_l-w*ivln2; 207 t1 = u+v; 208 SET_LOW_WORD(t1,0); 209 t2 = v-(t1-u); 210 } else { 211 double ss,s2,s_h,s_l,t_h,t_l; 212 n = 0; 213 /* take care subnormal number */ 214 if(ix<0x00100000) 215 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 216 n += ((ix)>>20)-0x3ff; 217 j = ix&0x000fffff; 218 /* determine interval */ 219 ix = j|0x3ff00000; /* normalize ix */ 220 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 221 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 222 else {k=0;n+=1;ix -= 0x00100000;} 223 SET_HIGH_WORD(ax,ix); 224 225 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 226 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 227 v = one/(ax+bp[k]); 228 ss = u*v; 229 s_h = ss; 230 SET_LOW_WORD(s_h,0); 231 /* t_h=ax+bp[k] High */ 232 t_h = zero; 233 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 234 t_l = ax - (t_h-bp[k]); 235 s_l = v*((u-s_h*t_h)-s_h*t_l); 236 /* compute log(ax) */ 237 s2 = ss*ss; 238 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 239 r += s_l*(s_h+ss); 240 s2 = s_h*s_h; 241 t_h = 3+s2+r; 242 SET_LOW_WORD(t_h,0); 243 t_l = r-((t_h-3)-s2); 244 /* u+v = ss*(1+...) */ 245 u = s_h*t_h; 246 v = s_l*t_h+t_l*ss; 247 /* 2/(3log2)*(ss+...) */ 248 p_h = u+v; 249 SET_LOW_WORD(p_h,0); 250 p_l = v-(p_h-u); 251 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 252 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 253 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 254 t = n; 255 t1 = (((z_h+z_l)+dp_h[k])+t); 256 SET_LOW_WORD(t1,0); 257 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 258 } 259 260 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 261 y1 = y; 262 SET_LOW_WORD(y1,0); 263 p_l = (y-y1)*t1+y*t2; 264 p_h = y1*t1; 265 z = p_l+p_h; 266 EXTRACT_WORDS(j,i,z); 267 if (j>=0x40900000) { /* z >= 1024 */ 268 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 269 return s*huge*huge; /* overflow */ 270 else { 271 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 272 } 273 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 274 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 275 return s*tiny*tiny; /* underflow */ 276 else { 277 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 278 } 279 } 280 /* 281 * compute 2**(p_h+p_l) 282 */ 283 i = j&0x7fffffff; 284 k = (i>>20)-0x3ff; 285 n = 0; 286 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 287 n = j+(0x00100000>>(k+1)); 288 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 289 t = zero; 290 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 291 n = ((n&0x000fffff)|0x00100000)>>(20-k); 292 if(j<0) n = -n; 293 p_h -= t; 294 } 295 t = p_l+p_h; 296 SET_LOW_WORD(t,0); 297 u = t*lg2_h; 298 v = (p_l-(t-p_h))*lg2+t*lg2_l; 299 z = u+v; 300 w = v-(z-u); 301 t = z*z; 302 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 303 r = (z*t1)/(t1-two)-(w+z*w); 304 z = one-(r-z); 305 GET_HIGH_WORD(j,z); 306 j += (n<<20); 307 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 308 else SET_HIGH_WORD(z,j); 309 return s*z; 310 } 311 312 #if (LDBL_MANT_DIG == 53) 313 __weak_reference(pow, powl); 314 #endif 315