1 /* 2 * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package java.lang; 27 28 /** 29 * Port of the "Freely Distributable Math Library", version 5.3, from C to Java. 30 */ 31 class FdLibm { 32 /** 33 * Return the low-order 32 bits of the double argument as an int. 34 */ 35 private static int __LO(double x) { 36 long transducer = Double.doubleToLongBits(x); 37 return (int)transducer; 38 } 39 40 /** 41 * Return the a double with its low-order bits reset. 42 */ 43 private static double __LO(double x, int low) { 44 long transX = Double.doubleToLongBits(x); 45 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low ); 46 } 47 48 /** 49 * Return the high-order 32 bits of the double argument as an int. 50 */ 51 private static int __HI(double x) { 52 long transducer = Double.doubleToLongBits(x); 53 return (int)(transducer >> 32); 54 } 55 /** 56 * Return the a double with its high-order bits reset. 57 */ 58 private static double __HI(double x, int high) { 59 long transX = Double.doubleToLongBits(x); 60 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); 61 } 62 63 /** 64 * Compute x**y 65 * n 66 * Method: Let x = 2 * (1+f) 67 * 1. Compute and return log2(x) in two pieces: 68 * log2(x) = w1 + w2, 69 * where w1 has 53 - 24 = 29 bit trailing zeros. 70 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 71 * arithmetic, where |y'| <= 0.5. 72 * 3. Return x**y = 2**n*exp(y'*log2) 73 * 74 * Special cases: 75 * 1. (anything) ** 0 is 1 76 * 2. (anything) ** 1 is itself 77 * 3. (anything) ** NAN is NAN 78 * 4. NAN ** (anything except 0) is NAN 79 * 5. +-(|x| > 1) ** +INF is +INF 80 * 6. +-(|x| > 1) ** -INF is +0 81 * 7. +-(|x| < 1) ** +INF is +0 82 * 8. +-(|x| < 1) ** -INF is +INF 83 * 9. +-1 ** +-INF is NAN 84 * 10. +0 ** (+anything except 0, NAN) is +0 85 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 86 * 12. +0 ** (-anything except 0, NAN) is +INF 87 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 88 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 89 * 15. +INF ** (+anything except 0,NAN) is +INF 90 * 16. +INF ** (-anything except 0,NAN) is +0 91 * 17. -INF ** (anything) = -0 ** (-anything) 92 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 93 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 94 * 95 * Accuracy: 96 * pow(x,y) returns x**y nearly rounded. In particular 97 * pow(integer,integer) 98 * always returns the correct integer provided it is 99 * representable. 100 */ 101 public static class Pow { 102 static final double bp[] = {1.0, 103 1.5}; 104 static final double dp_h[] = {0.0, 105 0x1.2b8034p-1}; // 5.84962487220764160156e-01 106 static final double dp_l[] = {0.0, 107 0x1.cfdeb43cfd006p-27};// 1.35003920212974897128e-08 108 static final double zero = 0.0; 109 static final double one = 1.0; 110 static final double two = 2.0; 111 static final double two53 = 0x1.0p53; // 9007199254740992.0 112 static final double huge = 1.0e300; 113 static final double tiny = 1.0e-300; 114 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 115 static final double L1 = 0x1.3333333333303p-1; // 5.99999999999994648725e-01 116 static final double L2 = 0x1.b6db6db6fabffp-2; // 4.28571428578550184252e-01 117 static final double L3 = 0x1.55555518f264dp-2; // 3.33333329818377432918e-01 118 static final double L4 = 0x1.17460a91d4101p-2; // 2.72728123808534006489e-01 119 static final double L5 = 0x1.d864a93c9db65p-3; // 2.30660745775561754067e-01 120 static final double L6 = 0x1.a7e284a454eefp-3; // 2.06975017800338417784e-01 121 static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01 122 static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03 123 static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05 124 static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06 125 static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08 126 static final double lg2 = 0x1.62e42fefa39efp-1; // 6.93147180559945286227e-01 127 static final double lg2_h = 0x1.62e43p-1; // 6.93147182464599609375e-01 128 static final double lg2_l = -0x1.05c610ca86c39p-29; // -1.90465429995776804525e-09 129 static final double ovt = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) 130 static final double cp = 0x1.ec709dc3a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) 131 static final double cp_h = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp 132 static final double cp_l = -0x1.e2fe0145b01f5p-28; // -7.02846165095275826516e-09 = tail of cp_h 133 static final double ivln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 = 1/ln2 134 static final double ivln2_h = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 135 static final double ivln2_l = 0x1.4ae0bf85ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail 136 137 public static double pow(final double x, final double y) { 138 double z; 139 double t1, t2, r, s, t, u, v, w; 140 int i, j, k, n; 141 142 // y == zero: x**0 = 1 143 if (y == 0.0) 144 return 1.0; 145 146 // +/-NaN return x + y to propagate NaN significands 147 if (Double.isNaN(x) || Double.isNaN(y)) 148 return x + y; 149 150 final double y_abs = Math.abs(y); 151 double x_abs = Math.abs(x); 152 // Special values of y 153 if (y == 2.0) { 154 return x * x; 155 } else if (y == 0.5) { 156 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later 157 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 158 } else if (y_abs == 1.0) { // y is +/-1 159 return (y == 1.0) ? x : one / x; 160 } else if (y_abs == Double.POSITIVE_INFINITY) { // y is +/-infinity 161 if (x_abs == 1.0) 162 return y - y; // inf**+/-1 is NaN 163 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 164 return (y >= 0) ? y : zero; 165 else // (|x| < 1)**-, +inf = inf, 0 166 return (y < 0) ? -y : zero; 167 } 168 169 final int hx = __HI(x); // Try to replace with copysign usage 170 // final int lx = __LO(x); 171 // final int hy = __HI(y); 172 final int ly = __LO(y); 173 int ix = hx & 0x7fffffff; 174 final int iy = __HI(y) & 0x7fffffff; // Try to replace with getExponent in yisint 175 176 /* 177 * When x < 0, determine if y is an odd integer: 178 * yisint = 0 ... y is not an integer 179 * yisint = 1 ... y is an odd int 180 * yisint = 2 ... y is an even int 181 */ 182 { 183 int yisint = 0; 184 if (hx < 0) { 185 if (iy >= 0x43400000) 186 yisint = 2; // even integer y 187 else if (iy >= 0x3ff00000) { 188 k = (iy >> 20) - 0x3ff; // exponent 189 if (k > 20) { 190 j = ly >> (52 - k); 191 if ((j << (52 - k) ) == ly) 192 yisint = 2 - (j & 1); 193 } else if (ly == 0) { 194 j = iy >> (20 - k); 195 if ((j << (20 - k)) == iy) { 196 yisint = 2 - (j & 1); 197 } 198 } 199 } 200 } 201 202 // Special value of x 203 if (x_abs == 0.0 || 204 x_abs == Double.POSITIVE_INFINITY || 205 x_abs == 1.0) { 206 z = x_abs; // x is +/-0, +/-inf, +/-1 207 if (y < 0.0) 208 z = one/z; // z = (1/|x|) 209 if (hx < 0) { 210 if (((ix - 0x3ff00000) | yisint) == 0) { 211 z = (z-z)/(z-z); // (-1)**non-int is NaN 212 } else if (yisint == 1) 213 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) 214 } 215 return z; 216 } 217 218 n = (hx >> 31) + 1; 219 220 // (x < 0)**(non-int) is NaN 221 if ((n | yisint) == 0) 222 return (x-x)/(x-x); 223 224 s = one; // s (sign of result -ve**odd) = -1 else = 1 225 if ( (n | (yisint - 1)) == 0) 226 s = -one; // (-ve)**(odd int) 227 } 228 double p_h, p_l; 229 // |y| is huge 230 if (y_abs > 0x1.0p31) { // if |y| > 2**31 231 if (y_abs > 0x1.0p64){ // if |y| > 2**64, must over/underflow 232 if (ix <= 0x3fefffff) 233 return (y < 0.0) ? huge*huge : tiny*tiny; 234 if (ix >= 0x3ff00000) 235 return (y > 0.0) ? huge*huge : tiny*tiny; 236 } 237 // Over/underflow if x is not close to one 238 if (ix < 0x3fefffff) 239 return (y < 0.0) ? s*huge*huge : s*tiny*tiny; 240 if (ix > 0x3ff00000) 241 return (y > 0.0) ? s*huge*huge : s*tiny*tiny; 242 /* 243 * now |1-x| is tiny <= 2**-20, sufficient to compute 244 * log(x) by x - x^2/2 + x^3/3 - x^4/4 245 */ 246 t = x_abs - one; // t has 20 trailing zeros 247 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 248 u = ivln2_h * t; // ivln2_h has 21 sig. bits 249 v = t * ivln2_l - w * ivln2; 250 t1 = u + v; 251 t1 =__LO(t1, 0); 252 t2 = v - (t1 - u); 253 } else { 254 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; 255 n = 0; 256 // Take care of subnormal numbers 257 if (ix < 0x00100000) { 258 x_abs *= two53; 259 n -= 53; 260 ix = __HI(x_abs); 261 } 262 n += ((ix) >> 20) - 0x3ff; 263 j = ix & 0x000fffff; 264 // Determine interval 265 ix = j | 0x3ff00000; // Normalize ix 266 if (j <= 0x3988E) 267 k = 0; // |x| <sqrt(3/2) 268 else if (j < 0xBB67A) 269 k = 1; // |x| <sqrt(3) 270 else { 271 k = 0; 272 n += 1; 273 ix -= 0x00100000; 274 } 275 x_abs = __HI(x_abs, ix); 276 277 // Compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) 278 u = x_abs - bp[k]; // bp[0]=1.0, bp[1]=1.5 279 v = one / (x_abs + bp[k]); 280 ss = u * v; 281 s_h = ss; 282 s_h = __LO(s_h, 0); 283 // t_h=x_abs+bp[k] High 284 t_h = zero; 285 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) ); 286 t_l = x_abs - (t_h - bp[k]); 287 s_l = v * ((u - s_h * t_h) - s_h * t_l); 288 // Compute log(x_abs) 289 s2 = ss * ss; 290 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 291 r += s_l * (s_h + ss); 292 s2 = s_h * s_h; 293 t_h = 3.0 + s2 + r; 294 t_h = __LO(t_h, 0); 295 t_l = r - ((t_h - 3.0) - s2); 296 // u+v = ss*(1+...) 297 u = s_h * t_h; 298 v = s_l * t_h + t_l * ss; 299 // 2/(3log2)*(ss+...) 300 p_h = u + v; 301 p_h = __LO(p_h, 0); 302 p_l = v - (p_h - u); 303 z_h = cp_h * p_h; // cp_h+cp_l = 2/(3*log2) 304 z_l = cp_l * p_h + p_l * cp + dp_l[k]; 305 // log2(x_abs) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l 306 t = (double)n; 307 t1 = (((z_h + z_l) + dp_h[k]) + t); 308 t1 = __LO(t1, 0); 309 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); 310 } 311 312 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2) 313 double y1 = y; 314 y1 = __LO(y1, 0); 315 p_l = (y - y1) * t1 + y * t2; 316 p_h = y1 * t1; 317 z = p_l + p_h; 318 j = __HI(z); 319 i = __LO(z); 320 if (j >= 0x40900000) { // z >= 1024 321 if (((j - 0x40900000) | i)!=0) // if z > 1024 322 return s*huge*huge; // Overflow 323 else { 324 if (p_l + ovt > z - p_h) 325 return s*huge*huge; // Overflow 326 } 327 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 328 if (((j - 0xc090cc00) | i)!=0) // z < -1075 329 return s*tiny*tiny; // Underflow 330 else { 331 if (p_l <= z - p_h) 332 return s*tiny*tiny; // Underflow 333 } 334 } 335 /* 336 * Compute 2**(p_h+p_l) 337 */ 338 i = j & 0x7fffffff; 339 k = (i >> 20) - 0x3ff; 340 n = 0; 341 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z+0.5] 342 n = j + (0x00100000 >> (k + 1)); 343 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n 344 t = zero; 345 t = __HI(t, (n & ~(0x000fffff >> k)) ); 346 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 347 if (j < 0) 348 n = -n; 349 p_h -= t; 350 } 351 t = p_l + p_h; 352 t = __LO(t, 0); 353 u = t * lg2_h; 354 v = (p_l - (t - p_h)) * lg2 + t * lg2_l; 355 z = u + v; 356 w = v - (z - u); 357 t = z * z; 358 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 359 r = (z * t1)/(t1 - two) - (w + z * w); 360 z = one - (r - z); 361 j = __HI(z); 362 j += (n << 20); 363 if ((j >> 20) <= 0) 364 z = Math.scalb(z, n); // subnormal output 365 else { 366 int z_hi = __HI(z); 367 z_hi += (n << 20); 368 z = __HI(z, z_hi); 369 } 370 return s * z; 371 } 372 } 373 }