1 2 /* 3 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. 4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 5 * 6 * This code is free software; you can redistribute it and/or modify it 7 * under the terms of the GNU General Public License version 2 only, as 8 * published by the Free Software Foundation. Oracle designates this 9 * particular file as subject to the "Classpath" exception as provided 10 * by Oracle in the LICENSE file that accompanied this code. 11 * 12 * This code is distributed in the hope that it will be useful, but WITHOUT 13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 15 * version 2 for more details (a copy is included in the LICENSE file that 16 * accompanied this code). 17 * 18 * You should have received a copy of the GNU General Public License version 19 * 2 along with this work; if not, write to the Free Software Foundation, 20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 21 * 22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 23 * or visit www.oracle.com if you need additional information or have any 24 * questions. 25 */ 26 27 #include "fdlibm.h" 28 29 /* cbrt(x) 30 * Return cube root of x 31 */ 32 #ifdef __STDC__ 33 static const unsigned 34 #else 35 static unsigned 36 #endif 37 B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 38 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 39 40 #ifdef __STDC__ 41 static const double 42 #else 43 static double 44 #endif 45 C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 46 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 47 E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 48 F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 49 G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 50 51 #ifdef __STDC__ 52 double cbrt(double x) 53 #else 54 double cbrt(x) 55 double x; 56 #endif 57 { 58 int hx; 59 double r,s,t=0.0,w; 60 unsigned sign; 61 62 63 hx = __HI(x); /* high word of x */ 64 sign=hx&0x80000000; /* sign= sign(x) */ 65 hx ^=sign; 66 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ 67 if((hx|__LO(x))==0) 68 return(x); /* cbrt(0) is itself */ 69 70 __HI(x) = hx; /* x <- |x| */ 71 /* rough cbrt to 5 bits */ 72 if(hx<0x00100000) /* subnormal number */ 73 {__HI(t)=0x43500000; /* set t= 2**54 */ 74 t*=x; __HI(t)=__HI(t)/3+B2; 75 } 76 else 77 __HI(t)=hx/3+B1; 78 79 80 /* new cbrt to 23 bits, may be implemented in single precision */ 81 r=t*t/x; 82 s=C+r*t; 83 t*=G+F/(s+E+D/s); 84 85 /* chopped to 20 bits and make it larger than cbrt(x) */ 86 __LO(t)=0; __HI(t)+=0x00000001; 87 88 89 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 90 s=t*t; /* t*t is exact */ 91 r=x/s; 92 w=t+t; 93 r=(r-t)/(w+r); /* r-s is exact */ 94 t=t+t*r; 95 96 /* retore the sign bit */ 97 __HI(t) |= sign; 98 return(t); 99 } | 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 30 * C to Java. 31 * 32 * <p>The C version of fdlibm relied on the idiom of pointer aliasing 33 * a 64-bit double floating-point value as a two-element array of 34 * 32-bit integers and reading and writing the two halves of the 35 * double independently. This coding pattern was problematic to C 36 * optimizers and not directly expressible in Java. Therefore, rather 37 * than a memory level overlay, if portions of a double need to be 38 * operated on as integer values, the standard library methods for 39 * bitwise floating-point to integer conversion, 40 * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly 41 * or indirectly used. 42 * 43 * <p>The C version of fdlibm also took some pains to signal the 44 * correct IEEE 754 exceptional conditions divide by zero, invalid, 45 * overflow and underflow. For example, overflow would be signaled by 46 * {@code huge * huge} where {@code huge} was a large constant that 47 * would overflow when squared. Since IEEE floating-point exceptional 48 * handling is not supported natively in the JVM, such coding patterns 49 * have been omitted from this port. For example, rather than {@code 50 * return huge * huge}, this port will use {@code return INFINITY}. 51 * 52 * <p>Various comparison and arithmetic operations in fdlibm could be 53 * done either based on the integer view of a value or directly on the 54 * floating-point representation. Which idiom is faster may depend on 55 * platform specific factors. However, for code clarity if no other 56 * reason, this port will favor expressing the semantics of those 57 * operations in terms of floating-point operations when convenient to 58 * do so. 59 */ 60 class FdLibm { 61 // Constants used by multiple algorithms 62 private static final double INFINITY = Double.POSITIVE_INFINITY; 63 64 private FdLibm() { 65 throw new UnsupportedOperationException("No FdLibm instances for you."); 66 } 67 68 /** 69 * Return the low-order 32 bits of the double argument as an int. 70 */ 71 private static int __LO(double x) { 72 long transducer = Double.doubleToRawLongBits(x); 73 return (int)transducer; 74 } 75 76 /** 77 * Return a double with its low-order bits of the second argument 78 * and the high-order bits of the first argument.. 79 */ 80 private static double __LO(double x, int low) { 81 long transX = Double.doubleToRawLongBits(x); 82 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low ); 83 } 84 85 /** 86 * Return the high-order 32 bits of the double argument as an int. 87 */ 88 private static int __HI(double x) { 89 long transducer = Double.doubleToRawLongBits(x); 90 return (int)(transducer >> 32); 91 } 92 93 /** 94 * Return a double with its high-order bits of the second argument 95 * and the low-order bits of the first argument.. 96 */ 97 private static double __HI(double x, int high) { 98 long transX = Double.doubleToRawLongBits(x); 99 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); 100 } 101 102 /** 103 * cbrt(x) 104 * Return cube root of x 105 */ 106 public static class Cbrt { 107 // unsigned 108 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */ 109 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 110 111 private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 112 private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 113 private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 114 private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 115 private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 116 117 public static strictfp double compute(double x) { 118 int hx; 119 double t = 0.0; 120 double sign; 121 122 if (!Double.isFinite(x) || x == 0.0) 123 return x; // Handles signed zeros properly 124 125 sign = (x < 0.0) ? -1.0: 1.0; 126 127 x = Math.abs(x); // x <- |x| 128 hx = __HI(x); // high word of x 129 130 // Rough cbrt to 5 bits 131 if (hx < 0x00100000) { // subnormal number 132 t = __HI(t, 0x43500000); // set t= 2**54 133 t *= x; 134 t = __HI(t, __HI(t)/3 + B2); 135 } else { 136 t = __HI(t, hx/3 + B1); 137 } 138 139 // New cbrt to 23 bits, may be implemented in single precision 140 double r, s, w; 141 r = t * t/x; 142 s = C + r*t; 143 t *= G + F/(s + E + D/s); 144 145 // Chopped to 20 bits and make it larger than cbrt(x) 146 t = __LO(t, 0); 147 t = __HI(t, __HI(t) + 0x00000001); 148 149 // One step newton iteration to 53 bits with error less than 0.667 ulps 150 s = t * t; // t*t is exact 151 r = x / s; 152 w = t + t; 153 r = (r - t)/(w + r); // r-s is exact 154 t = t + t*r; 155 156 // Restore the original sign bit 157 return sign * t; 158 } 159 } 160 161 /** 162 * hypot(x,y) 163 * 164 * Method : 165 * If (assume round-to-nearest) z = x*x + y*y 166 * has error less than sqrt(2)/2 ulp, than 167 * sqrt(z) has error less than 1 ulp (exercise). 168 * 169 * So, compute sqrt(x*x + y*y) with some care as 170 * follows to get the error below 1 ulp: 171 * 172 * Assume x > y > 0; 173 * (if possible, set rounding to round-to-nearest) 174 * 1. if x > 2y use 175 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y 176 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else 177 * 2. if x <= 2y use 178 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) 179 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, 180 * y1= y with lower 32 bits chopped, y2 = y - y1. 181 * 182 * NOTE: scaling may be necessary if some argument is too 183 * large or too tiny 184 * 185 * Special cases: 186 * hypot(x,y) is INF if x or y is +INF or -INF; else 187 * hypot(x,y) is NAN if x or y is NAN. 188 * 189 * Accuracy: 190 * hypot(x,y) returns sqrt(x^2 + y^2) with error less 191 * than 1 ulp (unit in the last place) 192 */ 193 public static class Hypot { 194 public static final double TWO_MINUS_600 = 0x1.0p-600; 195 public static final double TWO_PLUS_600 = 0x1.0p+600; 196 197 public static strictfp double compute(double x, double y) { 198 double a = Math.abs(x); 199 double b = Math.abs(y); 200 201 if (!Double.isFinite(a) || !Double.isFinite(b)) { 202 if (a == INFINITY || b == INFINITY) 203 return INFINITY; 204 else 205 return a + b; // Propagate NaN significand bits 206 } 207 208 if (b > a) { 209 double tmp = a; 210 a = b; 211 b = tmp; 212 } 213 assert a >= b; 214 215 // Doing bitwise conversion after screening for NaN allows 216 // the code to not worry about the possibility of 217 // "negative" NaN values. 218 219 // Note: the ha and hb variables are the high-order 220 // 32-bits of a and b stored as integer values. The ha and 221 // hb values are used first for a rough magnitude 222 // comparison of a and b and second for simulating higher 223 // precision by allowing a and b, respectively, to be 224 // decomposed into non-overlapping portions. Both of these 225 // uses could be eliminated. The magnitude comparison 226 // could be eliminated by extracting and comparing the 227 // exponents of a and b or just be performing a 228 // floating-point divide. Splitting a floating-point 229 // number into non-overlapping portions can be 230 // accomplished by judicious use of multiplies and 231 // additions. For details see T. J. Dekker, A Floating 232 // Point Technique for Extending the Available Precision , 233 // Numerische Mathematik, vol. 18, 1971, pp.224-242 and 234 // subsequent work. 235 236 int ha = __HI(a); // high word of a 237 int hb = __HI(b); // high word of b 238 239 if ((ha - hb) > 0x3c00000) { 240 return a + b; // x / y > 2**60 241 } 242 243 int k = 0; 244 if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500 245 // scale a and b by 2**-600 246 ha -= 0x25800000; 247 hb -= 0x25800000; 248 a = a * TWO_MINUS_600; 249 b = b * TWO_MINUS_600; 250 k += 600; 251 } 252 double t1, t2; 253 if (b < 0x1.0p-500) { // b < 2**-500 254 if (b < Double.MIN_NORMAL) { // subnormal b or 0 */ 255 if (b == 0.0) 256 return a; 257 t1 = 0x1.0p1022; // t1 = 2^1022 258 b *= t1; 259 a *= t1; 260 k -= 1022; 261 } else { // scale a and b by 2^600 262 ha += 0x25800000; // a *= 2^600 263 hb += 0x25800000; // b *= 2^600 264 a = a * TWO_PLUS_600; 265 b = b * TWO_PLUS_600; 266 k -= 600; 267 } 268 } 269 // medium size a and b 270 double w = a - b; 271 if (w > b) { 272 t1 = 0; 273 t1 = __HI(t1, ha); 274 t2 = a - t1; 275 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); 276 } else { 277 double y1, y2; 278 a = a + a; 279 y1 = 0; 280 y1 = __HI(y1, hb); 281 y2 = b - y1; 282 t1 = 0; 283 t1 = __HI(t1, ha + 0x00100000); 284 t2 = a - t1; 285 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); 286 } 287 if (k != 0) { 288 return Math.powerOfTwoD(k) * w; 289 } else 290 return w; 291 } 292 } 293 294 /** 295 * Compute x**y 296 * n 297 * Method: Let x = 2 * (1+f) 298 * 1. Compute and return log2(x) in two pieces: 299 * log2(x) = w1 + w2, 300 * where w1 has 53 - 24 = 29 bit trailing zeros. 301 * 2. Perform y*log2(x) = n+y' by simulating multi-precision 302 * arithmetic, where |y'| <= 0.5. 303 * 3. Return x**y = 2**n*exp(y'*log2) 304 * 305 * Special cases: 306 * 1. (anything) ** 0 is 1 307 * 2. (anything) ** 1 is itself 308 * 3. (anything) ** NAN is NAN 309 * 4. NAN ** (anything except 0) is NAN 310 * 5. +-(|x| > 1) ** +INF is +INF 311 * 6. +-(|x| > 1) ** -INF is +0 312 * 7. +-(|x| < 1) ** +INF is +0 313 * 8. +-(|x| < 1) ** -INF is +INF 314 * 9. +-1 ** +-INF is NAN 315 * 10. +0 ** (+anything except 0, NAN) is +0 316 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 317 * 12. +0 ** (-anything except 0, NAN) is +INF 318 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 319 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 320 * 15. +INF ** (+anything except 0,NAN) is +INF 321 * 16. +INF ** (-anything except 0,NAN) is +0 322 * 17. -INF ** (anything) = -0 ** (-anything) 323 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 324 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 325 * 326 * Accuracy: 327 * pow(x,y) returns x**y nearly rounded. In particular 328 * pow(integer,integer) 329 * always returns the correct integer provided it is 330 * representable. 331 */ 332 public static class Pow { 333 public static strictfp double compute(final double x, final double y) { 334 double z; 335 double r, s, t, u, v, w; 336 int i, j, k, n; 337 338 // y == zero: x**0 = 1 339 if (y == 0.0) 340 return 1.0; 341 342 // +/-NaN return x + y to propagate NaN significands 343 if (Double.isNaN(x) || Double.isNaN(y)) 344 return x + y; 345 346 final double y_abs = Math.abs(y); 347 double x_abs = Math.abs(x); 348 // Special values of y 349 if (y == 2.0) { 350 return x * x; 351 } else if (y == 0.5) { 352 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later 353 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 354 } else if (y_abs == 1.0) { // y is +/-1 355 return (y == 1.0) ? x : 1.0 / x; 356 } else if (y_abs == INFINITY) { // y is +/-infinity 357 if (x_abs == 1.0) 358 return y - y; // inf**+/-1 is NaN 359 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 360 return (y >= 0) ? y : 0.0; 361 else // (|x| < 1)**-/+inf = inf, 0 362 return (y < 0) ? -y : 0.0; 363 } 364 365 final int hx = __HI(x); 366 int ix = hx & 0x7fffffff; 367 368 /* 369 * When x < 0, determine if y is an odd integer: 370 * y_is_int = 0 ... y is not an integer 371 * y_is_int = 1 ... y is an odd int 372 * y_is_int = 2 ... y is an even int 373 */ 374 int y_is_int = 0; 375 if (hx < 0) { 376 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15 377 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0 378 else if (y_abs >= 1.0) { // |y| >= 1.0 379 long y_abs_as_long = (long) y_abs; 380 if ( ((double) y_abs_as_long) == y_abs) { 381 y_is_int = 2 - (int)(y_abs_as_long & 0x1L); 382 } 383 } 384 } 385 386 // Special value of x 387 if (x_abs == 0.0 || 388 x_abs == INFINITY || 389 x_abs == 1.0) { 390 z = x_abs; // x is +/-0, +/-inf, +/-1 391 if (y < 0.0) 392 z = 1.0/z; // z = (1/|x|) 393 if (hx < 0) { 394 if (((ix - 0x3ff00000) | y_is_int) == 0) { 395 z = (z-z)/(z-z); // (-1)**non-int is NaN 396 } else if (y_is_int == 1) 397 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) 398 } 399 return z; 400 } 401 402 n = (hx >> 31) + 1; 403 404 // (x < 0)**(non-int) is NaN 405 if ((n | y_is_int) == 0) 406 return (x-x)/(x-x); 407 408 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1 409 if ( (n | (y_is_int - 1)) == 0) 410 s = -1.0; // (-ve)**(odd int) 411 412 double p_h, p_l, t1, t2; 413 // |y| is huge 414 if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31 415 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2 416 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 417 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail 418 419 // Over/underflow if x is not close to one 420 if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418 421 return (y < 0.0) ? s * INFINITY : s * 0.0; 422 if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0 423 return (y > 0.0) ? s * INFINITY : s * 0.0; 424 /* 425 * now |1-x| is tiny <= 2**-20, sufficient to compute 426 * log(x) by x - x^2/2 + x^3/3 - x^4/4 427 */ 428 t = x_abs - 1.0; // t has 20 trailing zeros 429 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 430 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits 431 v = t * INV_LN2_L - w * INV_LN2; 432 t1 = u + v; 433 t1 =__LO(t1, 0); 434 t2 = v - (t1 - u); 435 } else { 436 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) 437 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp 438 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H 439 440 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; 441 n = 0; 442 // Take care of subnormal numbers 443 if (ix < 0x00100000) { 444 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0 445 n -= 53; 446 ix = __HI(x_abs); 447 } 448 n += ((ix) >> 20) - 0x3ff; 449 j = ix & 0x000fffff; 450 // Determine interval 451 ix = j | 0x3ff00000; // Normalize ix 452 if (j <= 0x3988E) 453 k = 0; // |x| <sqrt(3/2) 454 else if (j < 0xBB67A) 455 k = 1; // |x| <sqrt(3) 456 else { 457 k = 0; 458 n += 1; 459 ix -= 0x00100000; 460 } 461 x_abs = __HI(x_abs, ix); 462 463 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) 464 465 final double BP[] = {1.0, 466 1.5}; 467 final double DP_H[] = {0.0, 468 0x1.2b80_34p-1}; // 5.84962487220764160156e-01 469 final double DP_L[] = {0.0, 470 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08 471 472 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 473 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01 474 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01 475 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01 476 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01 477 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01 478 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01 479 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5 480 v = 1.0 / (x_abs + BP[k]); 481 ss = u * v; 482 s_h = ss; 483 s_h = __LO(s_h, 0); 484 // t_h=x_abs + BP[k] High 485 t_h = 0.0; 486 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) ); 487 t_l = x_abs - (t_h - BP[k]); 488 s_l = v * ((u - s_h * t_h) - s_h * t_l); 489 // Compute log(x_abs) 490 s2 = ss * ss; 491 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 492 r += s_l * (s_h + ss); 493 s2 = s_h * s_h; 494 t_h = 3.0 + s2 + r; 495 t_h = __LO(t_h, 0); 496 t_l = r - ((t_h - 3.0) - s2); 497 // u+v = ss*(1+...) 498 u = s_h * t_h; 499 v = s_l * t_h + t_l * ss; 500 // 2/(3log2)*(ss + ...) 501 p_h = u + v; 502 p_h = __LO(p_h, 0); 503 p_l = v - (p_h - u); 504 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2) 505 z_l = CP_L * p_h + p_l * CP + DP_L[k]; 506 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l 507 t = (double)n; 508 t1 = (((z_h + z_l) + DP_H[k]) + t); 509 t1 = __LO(t1, 0); 510 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h); 511 } 512 513 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2) 514 double y1 = y; 515 y1 = __LO(y1, 0); 516 p_l = (y - y1) * t1 + y * t2; 517 p_h = y1 * t1; 518 z = p_l + p_h; 519 j = __HI(z); 520 i = __LO(z); 521 if (j >= 0x40900000) { // z >= 1024 522 if (((j - 0x40900000) | i)!=0) // if z > 1024 523 return s * INFINITY; // Overflow 524 else { 525 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) 526 if (p_l + OVT > z - p_h) 527 return s * INFINITY; // Overflow 528 } 529 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 530 if (((j - 0xc090cc00) | i)!=0) // z < -1075 531 return s * 0.0; // Underflow 532 else { 533 if (p_l <= z - p_h) 534 return s * 0.0; // Underflow 535 } 536 } 537 /* 538 * Compute 2**(p_h+p_l) 539 */ 540 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 541 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01 542 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03 543 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05 544 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06 545 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08 546 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01 547 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01 548 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09 549 i = j & 0x7fffffff; 550 k = (i >> 20) - 0x3ff; 551 n = 0; 552 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5] 553 n = j + (0x00100000 >> (k + 1)); 554 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n 555 t = 0.0; 556 t = __HI(t, (n & ~(0x000fffff >> k)) ); 557 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 558 if (j < 0) 559 n = -n; 560 p_h -= t; 561 } 562 t = p_l + p_h; 563 t = __LO(t, 0); 564 u = t * LG2_H; 565 v = (p_l - (t - p_h)) * LG2 + t * LG2_L; 566 z = u + v; 567 w = v - (z - u); 568 t = z * z; 569 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 570 r = (z * t1)/(t1 - 2.0) - (w + z * w); 571 z = 1.0 - (r - z); 572 j = __HI(z); 573 j += (n << 20); 574 if ((j >> 20) <= 0) 575 z = Math.scalb(z, n); // subnormal output 576 else { 577 int z_hi = __HI(z); 578 z_hi += (n << 20); 579 z = __HI(z, z_hi); 580 } 581 return s * z; 582 } 583 } 584 } |