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 #if defined(LIBM_SCCS) && !defined(lint) 14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc 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_libm.h" 63 #include "math_private.h" 64 65 libm_hidden_proto(scalbn) 66 libm_hidden_proto(fabs) 67 #ifdef __STDC__ 68 static const double 69 #else 70 static double 71 #endif 72 bp[] = { 1.0, 1.5, }, dp_h[] = { 73 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 74 75 dp_l[] = { 76 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 77 78 zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 79 huge_val = 1.0e300, tiny = 1.0e-300, 80 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 81 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 82 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 83 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 84 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 85 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 86 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 87 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 88 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 89 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 90 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 91 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 92 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 93 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 94 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 95 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 96 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 97 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 98 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */ 99 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 100 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2 */ 101 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */ 102 103 #ifdef __STDC__ __ieee754_pow(double x,double y)104 double attribute_hidden __ieee754_pow(double x, double y) 105 #else 106 double attribute_hidden __ieee754_pow(x, y) 107 double x, y; 108 #endif 109 { 110 double z, ax, z_h, z_l, p_h, p_l; 111 double y1, t1, t2, r, s, t, u, v, w; 112 int32_t i, j, k, yisint, n; 113 int32_t hx, hy, ix, iy; 114 u_int32_t lx, ly; 115 116 EXTRACT_WORDS(hx, lx, x); 117 EXTRACT_WORDS(hy, ly, y); 118 ix = hx & 0x7fffffff; 119 iy = hy & 0x7fffffff; 120 121 /* y==zero: x**0 = 1 */ 122 if ((iy | ly) == 0) 123 return one; 124 125 /* +-NaN return x+y */ 126 if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || 127 iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) 128 return x + y; 129 130 /* determine if y is an odd int when x < 0 131 * yisint = 0 ... y is not an integer 132 * yisint = 1 ... y is an odd int 133 * yisint = 2 ... y is an even int 134 */ 135 yisint = 0; 136 if (hx < 0) { 137 if (iy >= 0x43400000) 138 yisint = 2; /* even integer y */ 139 else if (iy >= 0x3ff00000) { 140 k = (iy >> 20) - 0x3ff; /* exponent */ 141 if (k > 20) { 142 j = ly >> (52 - k); 143 if ((j << (52 - k)) == ly) 144 yisint = 2 - (j & 1); 145 } else if (ly == 0) { 146 j = iy >> (20 - k); 147 if ((j << (20 - k)) == iy) 148 yisint = 2 - (j & 1); 149 } 150 } 151 } 152 153 /* special value of y */ 154 if (ly == 0) { 155 if (iy == 0x7ff00000) { /* y is +-inf */ 156 if (((ix - 0x3ff00000) | lx) == 0) 157 return y - y; /* inf**+-1 is NaN */ 158 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ 159 return (hy >= 0) ? y : zero; 160 else /* (|x|<1)**-,+inf = inf,0 */ 161 return (hy < 0) ? -y : zero; 162 } 163 if (iy == 0x3ff00000) { /* y is +-1 */ 164 if (hy < 0) 165 return one / x; 166 else 167 return x; 168 } 169 if (hy == 0x40000000) 170 return x * x; /* y is 2 */ 171 if (hy == 0x3fe00000) { /* y is 0.5 */ 172 if (hx >= 0) /* x >= +0 */ 173 return __ieee754_sqrt(x); 174 } 175 } 176 177 ax = fabs(x); 178 /* special value of x */ 179 if (lx == 0) { 180 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { 181 z = ax; /* x is +-0,+-inf,+-1 */ 182 if (hy < 0) 183 z = one / z; /* z = (1/|x|) */ 184 if (hx < 0) { 185 if (((ix - 0x3ff00000) | yisint) == 0) { 186 z = (z - z) / (z - z); /* (-1)**non-int is NaN */ 187 } else if (yisint == 1) 188 z = -z; /* (x<0)**odd = -(|x|**odd) */ 189 } 190 return z; 191 } 192 } 193 194 /* (x<0)**(non-int) is NaN */ 195 if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) 196 return (x - x) / (x - x); 197 198 /* |y| is huge */ 199 if (iy > 0x41e00000) { /* if |y| > 2**31 */ 200 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */ 201 if (ix <= 0x3fefffff) 202 return (hy < 0) ? huge_val * huge_val : tiny * tiny; 203 if (ix >= 0x3ff00000) 204 return (hy > 0) ? huge_val * huge_val : tiny * tiny; 205 } 206 /* over/underflow if x is not close to one */ 207 if (ix < 0x3fefffff) 208 return (hy < 0) ? huge_val * huge_val : tiny * tiny; 209 if (ix > 0x3ff00000) 210 return (hy > 0) ? huge_val * huge_val : tiny * tiny; 211 /* now |1-x| is tiny <= 2**-20, suffice to compute 212 log(x) by x-x^2/2+x^3/3-x^4/4 */ 213 t = x - 1; /* t has 20 trailing zeros */ 214 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 215 u = ivln2_h * t; /* ivln2_h has 21 sig. bits */ 216 v = t * ivln2_l - w * ivln2; 217 t1 = u + v; 218 SET_LOW_WORD(t1, 0); 219 t2 = v - (t1 - u); 220 } else { 221 double s2, s_h, s_l, t_h, t_l; 222 n = 0; 223 /* take care subnormal number */ 224 if (ix < 0x00100000) { 225 ax *= two53; 226 n -= 53; 227 GET_HIGH_WORD(ix, ax); 228 } 229 n += ((ix) >> 20) - 0x3ff; 230 j = ix & 0x000fffff; 231 /* determine interval */ 232 ix = j | 0x3ff00000; /* normalize ix */ 233 if (j <= 0x3988E) 234 k = 0; /* |x|<sqrt(3/2) */ 235 else if (j < 0xBB67A) 236 k = 1; /* |x|<sqrt(3) */ 237 else { 238 k = 0; 239 n += 1; 240 ix -= 0x00100000; 241 } 242 SET_HIGH_WORD(ax, ix); 243 244 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 245 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 246 v = one / (ax + bp[k]); 247 s = u * v; 248 s_h = s; 249 SET_LOW_WORD(s_h, 0); 250 /* t_h=ax+bp[k] High */ 251 t_h = zero; 252 SET_HIGH_WORD(t_h, 253 ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18)); 254 t_l = ax - (t_h - bp[k]); 255 s_l = v * ((u - s_h * t_h) - s_h * t_l); 256 /* compute log(ax) */ 257 s2 = s * s; 258 r = s2 * s2 * (L1 + 259 s2 * (L2 + 260 s2 * (L3 + 261 s2 * (L4 + s2 * (L5 + s2 * L6))))); 262 r += s_l * (s_h + s); 263 s2 = s_h * s_h; 264 t_h = 3.0 + s2 + r; 265 SET_LOW_WORD(t_h, 0); 266 t_l = r - ((t_h - 3.0) - s2); 267 /* u+v = s*(1+...) */ 268 u = s_h * t_h; 269 v = s_l * t_h + t_l * s; 270 /* 2/(3log2)*(s+...) */ 271 p_h = u + v; 272 SET_LOW_WORD(p_h, 0); 273 p_l = v - (p_h - u); 274 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ 275 z_l = cp_l * p_h + p_l * cp + dp_l[k]; 276 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 277 t = (double) n; 278 t1 = (((z_h + z_l) + dp_h[k]) + t); 279 SET_LOW_WORD(t1, 0); 280 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); 281 } 282 283 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 284 if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0) 285 s = -one; /* (-ve)**(odd int) */ 286 287 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 288 y1 = y; 289 SET_LOW_WORD(y1, 0); 290 p_l = (y - y1) * t1 + y * t2; 291 p_h = y1 * t1; 292 z = p_l + p_h; 293 EXTRACT_WORDS(j, i, z); 294 if (j >= 0x40900000) { /* z >= 1024 */ 295 if (((j - 0x40900000) | i) != 0) /* if z > 1024 */ 296 return s * huge_val * huge_val; /* overflow */ 297 else { 298 if (p_l + ovt > z - p_h) 299 return s * huge_val * huge_val; /* overflow */ 300 } 301 } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ 302 if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ 303 return s * tiny * tiny; /* underflow */ 304 else { 305 if (p_l <= z - p_h) 306 return s * tiny * tiny; /* underflow */ 307 } 308 } 309 /* 310 * compute 2**(p_h+p_l) 311 */ 312 i = j & 0x7fffffff; 313 k = (i >> 20) - 0x3ff; 314 n = 0; 315 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 316 n = j + (0x00100000 >> (k + 1)); 317 k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */ 318 t = zero; 319 SET_HIGH_WORD(t, n & ~(0x000fffff >> k)); 320 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 321 if (j < 0) 322 n = -n; 323 p_h -= t; 324 } 325 t = p_l + p_h; 326 SET_LOW_WORD(t, 0); 327 u = t * lg2_h; 328 v = (p_l - (t - p_h)) * lg2 + t * lg2_l; 329 z = u + v; 330 w = v - (z - u); 331 t = z * z; 332 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 333 r = (z * t1) / (t1 - two) - (w + z * w); 334 z = one - (r - z); 335 GET_HIGH_WORD(j, z); 336 j += (n << 20); 337 if ((j >> 20) <= 0) 338 z = scalbn(z, n); /* subnormal output */ 339 else 340 SET_HIGH_WORD(z, j); 341 return s * z; 342 } 343