1/* 2 * Copyright (c) 1998, 2017, 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 26package 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 */ 60class 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) | 83 (low & 0x0000_0000_FFFF_FFFFL)); 84 } 85 86 /** 87 * Return the high-order 32 bits of the double argument as an int. 88 */ 89 private static int __HI(double x) { 90 long transducer = Double.doubleToRawLongBits(x); 91 return (int)(transducer >> 32); 92 } 93 94 /** 95 * Return a double with its high-order bits of the second argument 96 * and the low-order bits of the first argument.. 97 */ 98 private static double __HI(double x, int high) { 99 long transX = Double.doubleToRawLongBits(x); 100 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) | 101 ( ((long)high)) << 32 ); 102 } 103 104 /** 105 * cbrt(x) 106 * Return cube root of x 107 */ 108 public static class Cbrt { 109 // unsigned 110 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */ 111 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 112 113 private static final double C = 0x1.15f15f15f15f1p-1; // 19/35 ~= 5.42857142857142815906e-01 114 private static final double D = -0x1.691de2532c834p-1; // -864/1225 ~= 7.05306122448979611050e-01 115 private static final double E = 0x1.6a0ea0ea0ea0fp0; // 99/70 ~= 1.41428571428571436819e+00 116 private static final double F = 0x1.9b6db6db6db6ep0; // 45/28 ~= 1.60714285714285720630e+00 117 private static final double G = 0x1.6db6db6db6db7p-2; // 5/14 ~= 3.57142857142857150787e-01 118 119 private Cbrt() { 120 throw new UnsupportedOperationException(); 121 } 122 123 public static strictfp double compute(double x) { 124 double t = 0.0; 125 double sign; 126 127 if (x == 0.0 || !Double.isFinite(x)) 128 return x; // Handles signed zeros properly 129 130 sign = (x < 0.0) ? -1.0: 1.0; 131 132 x = Math.abs(x); // x <- |x| 133 134 // Rough cbrt to 5 bits 135 if (x < 0x1.0p-1022) { // subnormal number 136 t = 0x1.0p54; // set t= 2**54 137 t *= x; 138 t = __HI(t, __HI(t)/3 + B2); 139 } else { 140 int hx = __HI(x); // high word of x 141 t = __HI(t, hx/3 + B1); 142 } 143 144 // New cbrt to 23 bits, may be implemented in single precision 145 double r, s, w; 146 r = t * t/x; 147 s = C + r*t; 148 t *= G + F/(s + E + D/s); 149 150 // Chopped to 20 bits and make it larger than cbrt(x) 151 t = __LO(t, 0); 152 t = __HI(t, __HI(t) + 0x00000001); 153 154 // One step newton iteration to 53 bits with error less than 0.667 ulps 155 s = t * t; // t*t is exact 156 r = x / s; 157 w = t + t; 158 r = (r - t)/(w + r); // r-s is exact 159 t = t + t*r; 160 161 // Restore the original sign bit 162 return sign * t; 163 } 164 } 165 166 /** 167 * hypot(x,y) 168 * 169 * Method : 170 * If (assume round-to-nearest) z = x*x + y*y 171 * has error less than sqrt(2)/2 ulp, than 172 * sqrt(z) has error less than 1 ulp (exercise). 173 * 174 * So, compute sqrt(x*x + y*y) with some care as 175 * follows to get the error below 1 ulp: 176 * 177 * Assume x > y > 0; 178 * (if possible, set rounding to round-to-nearest) 179 * 1. if x > 2y use 180 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y 181 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else 182 * 2. if x <= 2y use 183 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) 184 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, 185 * y1= y with lower 32 bits chopped, y2 = y - y1. 186 * 187 * NOTE: scaling may be necessary if some argument is too 188 * large or too tiny 189 * 190 * Special cases: 191 * hypot(x,y) is INF if x or y is +INF or -INF; else 192 * hypot(x,y) is NAN if x or y is NAN. 193 * 194 * Accuracy: 195 * hypot(x,y) returns sqrt(x^2 + y^2) with error less 196 * than 1 ulp (unit in the last place) 197 */ 198 public static class Hypot { 199 public static final double TWO_MINUS_600 = 0x1.0p-600; 200 public static final double TWO_PLUS_600 = 0x1.0p+600; 201 202 private Hypot() { 203 throw new UnsupportedOperationException(); 204 } 205 206 public static strictfp double compute(double x, double y) { 207 double a = Math.abs(x); 208 double b = Math.abs(y); 209 210 if (!Double.isFinite(a) || !Double.isFinite(b)) { 211 if (a == INFINITY || b == INFINITY) 212 return INFINITY; 213 else 214 return a + b; // Propagate NaN significand bits 215 } 216 217 if (b > a) { 218 double tmp = a; 219 a = b; 220 b = tmp; 221 } 222 assert a >= b; 223 224 // Doing bitwise conversion after screening for NaN allows 225 // the code to not worry about the possibility of 226 // "negative" NaN values. 227 228 // Note: the ha and hb variables are the high-order 229 // 32-bits of a and b stored as integer values. The ha and 230 // hb values are used first for a rough magnitude 231 // comparison of a and b and second for simulating higher 232 // precision by allowing a and b, respectively, to be 233 // decomposed into non-overlapping portions. Both of these 234 // uses could be eliminated. The magnitude comparison 235 // could be eliminated by extracting and comparing the 236 // exponents of a and b or just be performing a 237 // floating-point divide. Splitting a floating-point 238 // number into non-overlapping portions can be 239 // accomplished by judicious use of multiplies and 240 // additions. For details see T. J. Dekker, A Floating 241 // Point Technique for Extending the Available Precision , 242 // Numerische Mathematik, vol. 18, 1971, pp.224-242 and 243 // subsequent work. 244 245 int ha = __HI(a); // high word of a 246 int hb = __HI(b); // high word of b 247 248 if ((ha - hb) > 0x3c00000) { 249 return a + b; // x / y > 2**60 250 } 251 252 int k = 0; 253 if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500 254 // scale a and b by 2**-600 255 ha -= 0x25800000; 256 hb -= 0x25800000; 257 a = a * TWO_MINUS_600; 258 b = b * TWO_MINUS_600; 259 k += 600; 260 } 261 double t1, t2; 262 if (b < 0x1.0p-500) { // b < 2**-500 263 if (b < Double.MIN_NORMAL) { // subnormal b or 0 */ 264 if (b == 0.0) 265 return a; 266 t1 = 0x1.0p1022; // t1 = 2^1022 267 b *= t1; 268 a *= t1; 269 k -= 1022; 270 } else { // scale a and b by 2^600 271 ha += 0x25800000; // a *= 2^600 272 hb += 0x25800000; // b *= 2^600 273 a = a * TWO_PLUS_600; 274 b = b * TWO_PLUS_600; 275 k -= 600; 276 } 277 } 278 // medium size a and b 279 double w = a - b; 280 if (w > b) { 281 t1 = 0; 282 t1 = __HI(t1, ha); 283 t2 = a - t1; 284 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); 285 } else { 286 double y1, y2; 287 a = a + a; 288 y1 = 0; 289 y1 = __HI(y1, hb); 290 y2 = b - y1; 291 t1 = 0; 292 t1 = __HI(t1, ha + 0x00100000); 293 t2 = a - t1; 294 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); 295 } 296 if (k != 0) { 297 return Math.powerOfTwoD(k) * w; 298 } else 299 return w; 300 } 301 } 302 303 /** 304 * Compute x**y 305 * n 306 * Method: Let x = 2 * (1+f) 307 * 1. Compute and return log2(x) in two pieces: 308 * log2(x) = w1 + w2, 309 * where w1 has 53 - 24 = 29 bit trailing zeros. 310 * 2. Perform y*log2(x) = n+y' by simulating multi-precision 311 * arithmetic, where |y'| <= 0.5. 312 * 3. Return x**y = 2**n*exp(y'*log2) 313 * 314 * Special cases: 315 * 1. (anything) ** 0 is 1 316 * 2. (anything) ** 1 is itself 317 * 3. (anything) ** NAN is NAN 318 * 4. NAN ** (anything except 0) is NAN 319 * 5. +-(|x| > 1) ** +INF is +INF 320 * 6. +-(|x| > 1) ** -INF is +0 321 * 7. +-(|x| < 1) ** +INF is +0 322 * 8. +-(|x| < 1) ** -INF is +INF 323 * 9. +-1 ** +-INF is NAN 324 * 10. +0 ** (+anything except 0, NAN) is +0 325 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 326 * 12. +0 ** (-anything except 0, NAN) is +INF 327 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 328 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 329 * 15. +INF ** (+anything except 0,NAN) is +INF 330 * 16. +INF ** (-anything except 0,NAN) is +0 331 * 17. -INF ** (anything) = -0 ** (-anything) 332 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 333 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 334 * 335 * Accuracy: 336 * pow(x,y) returns x**y nearly rounded. In particular 337 * pow(integer,integer) 338 * always returns the correct integer provided it is 339 * representable. 340 */ 341 public static class Pow { 342 private Pow() { 343 throw new UnsupportedOperationException(); 344 } 345 346 public static strictfp double compute(final double x, final double y) { 347 double z; 348 double r, s, t, u, v, w; 349 int i, j, k, n; 350 351 // y == zero: x**0 = 1 352 if (y == 0.0) 353 return 1.0; 354 355 // +/-NaN return x + y to propagate NaN significands 356 if (Double.isNaN(x) || Double.isNaN(y)) 357 return x + y; 358 359 final double y_abs = Math.abs(y); 360 double x_abs = Math.abs(x); 361 // Special values of y 362 if (y == 2.0) { 363 return x * x; 364 } else if (y == 0.5) { 365 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later 366 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 367 } else if (y_abs == 1.0) { // y is +/-1 368 return (y == 1.0) ? x : 1.0 / x; 369 } else if (y_abs == INFINITY) { // y is +/-infinity 370 if (x_abs == 1.0) 371 return y - y; // inf**+/-1 is NaN 372 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 373 return (y >= 0) ? y : 0.0; 374 else // (|x| < 1)**-/+inf = inf, 0 375 return (y < 0) ? -y : 0.0; 376 } 377 378 final int hx = __HI(x); 379 int ix = hx & 0x7fffffff; 380 381 /* 382 * When x < 0, determine if y is an odd integer: 383 * y_is_int = 0 ... y is not an integer 384 * y_is_int = 1 ... y is an odd int 385 * y_is_int = 2 ... y is an even int 386 */ 387 int y_is_int = 0; 388 if (hx < 0) { 389 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15 390 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0 391 else if (y_abs >= 1.0) { // |y| >= 1.0 392 long y_abs_as_long = (long) y_abs; 393 if ( ((double) y_abs_as_long) == y_abs) { 394 y_is_int = 2 - (int)(y_abs_as_long & 0x1L); 395 } 396 } 397 } 398 399 // Special value of x 400 if (x_abs == 0.0 || 401 x_abs == INFINITY || 402 x_abs == 1.0) { 403 z = x_abs; // x is +/-0, +/-inf, +/-1 404 if (y < 0.0) 405 z = 1.0/z; // z = (1/|x|) 406 if (hx < 0) { 407 if (((ix - 0x3ff00000) | y_is_int) == 0) { 408 z = (z-z)/(z-z); // (-1)**non-int is NaN 409 } else if (y_is_int == 1) 410 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) 411 } 412 return z; 413 } 414 415 n = (hx >> 31) + 1; 416 417 // (x < 0)**(non-int) is NaN 418 if ((n | y_is_int) == 0) 419 return (x-x)/(x-x); 420 421 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1 422 if ( (n | (y_is_int - 1)) == 0) 423 s = -1.0; // (-ve)**(odd int) 424 425 double p_h, p_l, t1, t2; 426 // |y| is huge 427 if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31 428 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2 429 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 430 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail 431 432 // Over/underflow if x is not close to one 433 if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418 434 return (y < 0.0) ? s * INFINITY : s * 0.0; 435 if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0 436 return (y > 0.0) ? s * INFINITY : s * 0.0; 437 /* 438 * now |1-x| is tiny <= 2**-20, sufficient to compute 439 * log(x) by x - x^2/2 + x^3/3 - x^4/4 440 */ 441 t = x_abs - 1.0; // t has 20 trailing zeros 442 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 443 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits 444 v = t * INV_LN2_L - w * INV_LN2; 445 t1 = u + v; 446 t1 =__LO(t1, 0); 447 t2 = v - (t1 - u); 448 } else { 449 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) 450 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp 451 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H 452 453 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; 454 n = 0; 455 // Take care of subnormal numbers 456 if (ix < 0x00100000) { 457 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0 458 n -= 53; 459 ix = __HI(x_abs); 460 } 461 n += ((ix) >> 20) - 0x3ff; 462 j = ix & 0x000fffff; 463 // Determine interval 464 ix = j | 0x3ff00000; // Normalize ix 465 if (j <= 0x3988E) 466 k = 0; // |x| <sqrt(3/2) 467 else if (j < 0xBB67A) 468 k = 1; // |x| <sqrt(3) 469 else { 470 k = 0; 471 n += 1; 472 ix -= 0x00100000; 473 } 474 x_abs = __HI(x_abs, ix); 475 476 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) 477 478 final double BP[] = {1.0, 479 1.5}; 480 final double DP_H[] = {0.0, 481 0x1.2b80_34p-1}; // 5.84962487220764160156e-01 482 final double DP_L[] = {0.0, 483 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08 484 485 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 486 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01 487 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01 488 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01 489 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01 490 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01 491 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01 492 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5 493 v = 1.0 / (x_abs + BP[k]); 494 ss = u * v; 495 s_h = ss; 496 s_h = __LO(s_h, 0); 497 // t_h=x_abs + BP[k] High 498 t_h = 0.0; 499 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) ); 500 t_l = x_abs - (t_h - BP[k]); 501 s_l = v * ((u - s_h * t_h) - s_h * t_l); 502 // Compute log(x_abs) 503 s2 = ss * ss; 504 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 505 r += s_l * (s_h + ss); 506 s2 = s_h * s_h; 507 t_h = 3.0 + s2 + r; 508 t_h = __LO(t_h, 0); 509 t_l = r - ((t_h - 3.0) - s2); 510 // u+v = ss*(1+...) 511 u = s_h * t_h; 512 v = s_l * t_h + t_l * ss; 513 // 2/(3log2)*(ss + ...) 514 p_h = u + v; 515 p_h = __LO(p_h, 0); 516 p_l = v - (p_h - u); 517 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2) 518 z_l = CP_L * p_h + p_l * CP + DP_L[k]; 519 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l 520 t = (double)n; 521 t1 = (((z_h + z_l) + DP_H[k]) + t); 522 t1 = __LO(t1, 0); 523 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h); 524 } 525 526 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2) 527 double y1 = y; 528 y1 = __LO(y1, 0); 529 p_l = (y - y1) * t1 + y * t2; 530 p_h = y1 * t1; 531 z = p_l + p_h; 532 j = __HI(z); 533 i = __LO(z); 534 if (j >= 0x40900000) { // z >= 1024 535 if (((j - 0x40900000) | i)!=0) // if z > 1024 536 return s * INFINITY; // Overflow 537 else { 538 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) 539 if (p_l + OVT > z - p_h) 540 return s * INFINITY; // Overflow 541 } 542 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 543 if (((j - 0xc090cc00) | i)!=0) // z < -1075 544 return s * 0.0; // Underflow 545 else { 546 if (p_l <= z - p_h) 547 return s * 0.0; // Underflow 548 } 549 } 550 /* 551 * Compute 2**(p_h+p_l) 552 */ 553 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 554 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01 555 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03 556 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05 557 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06 558 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08 559 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01 560 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01 561 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09 562 i = j & 0x7fffffff; 563 k = (i >> 20) - 0x3ff; 564 n = 0; 565 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5] 566 n = j + (0x00100000 >> (k + 1)); 567 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n 568 t = 0.0; 569 t = __HI(t, (n & ~(0x000fffff >> k)) ); 570 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 571 if (j < 0) 572 n = -n; 573 p_h -= t; 574 } 575 t = p_l + p_h; 576 t = __LO(t, 0); 577 u = t * LG2_H; 578 v = (p_l - (t - p_h)) * LG2 + t * LG2_L; 579 z = u + v; 580 w = v - (z - u); 581 t = z * z; 582 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 583 r = (z * t1)/(t1 - 2.0) - (w + z * w); 584 z = 1.0 - (r - z); 585 j = __HI(z); 586 j += (n << 20); 587 if ((j >> 20) <= 0) 588 z = Math.scalb(z, n); // subnormal output 589 else { 590 int z_hi = __HI(z); 591 z_hi += (n << 20); 592 z = __HI(z, z_hi); 593 } 594 return s * z; 595 } 596 } 597 598 /** 599 * Returns the exponential of x. 600 * 601 * Method 602 * 1. Argument reduction: 603 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 604 * Given x, find r and integer k such that 605 * 606 * x = k*ln2 + r, |r| <= 0.5*ln2. 607 * 608 * Here r will be represented as r = hi-lo for better 609 * accuracy. 610 * 611 * 2. Approximation of exp(r) by a special rational function on 612 * the interval [0,0.34658]: 613 * Write 614 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 615 * We use a special Reme algorithm on [0,0.34658] to generate 616 * a polynomial of degree 5 to approximate R. The maximum error 617 * of this polynomial approximation is bounded by 2**-59. In 618 * other words, 619 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 620 * (where z=r*r, and the values of P1 to P5 are listed below) 621 * and 622 * | 5 | -59 623 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 624 * | | 625 * The computation of exp(r) thus becomes 626 * 2*r 627 * exp(r) = 1 + ------- 628 * R - r 629 * r*R1(r) 630 * = 1 + r + ----------- (for better accuracy) 631 * 2 - R1(r) 632 * where 633 * 2 4 10 634 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 635 * 636 * 3. Scale back to obtain exp(x): 637 * From step 1, we have 638 * exp(x) = 2^k * exp(r) 639 * 640 * Special cases: 641 * exp(INF) is INF, exp(NaN) is NaN; 642 * exp(-INF) is 0, and 643 * for finite argument, only exp(0)=1 is exact. 644 * 645 * Accuracy: 646 * according to an error analysis, the error is always less than 647 * 1 ulp (unit in the last place). 648 * 649 * Misc. info. 650 * For IEEE double 651 * if x > 7.09782712893383973096e+02 then exp(x) overflow 652 * if x < -7.45133219101941108420e+02 then exp(x) underflow 653 * 654 * Constants: 655 * The hexadecimal values are the intended ones for the following 656 * constants. The decimal values may be used, provided that the 657 * compiler will convert from decimal to binary accurately enough 658 * to produce the hexadecimal values shown. 659 */ 660 static class Exp { 661 private static final double one = 1.0; 662 private static final double[] half = {0.5, -0.5,}; 663 private static final double huge = 1.0e+300; 664 private static final double twom1000= 0x1.0p-1000; // 9.33263618503218878990e-302 = 2^-1000 665 private static final double o_threshold= 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02 666 private static final double u_threshold= -0x1.74910d52d3051p9; // -7.45133219101941108420e+02; 667 private static final double[] ln2HI ={ 0x1.62e42feep-1, // 6.93147180369123816490e-01 668 -0x1.62e42feep-1}; // -6.93147180369123816490e-01 669 private static final double[] ln2LO ={ 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10 670 -0x1.a39ef35793c76p-33}; // -1.90821492927058770002e-10 671 private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 672 673 private static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01 674 private static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03 675 private static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05 676 private static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06 677 private static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08 678 679 private Exp() { 680 throw new UnsupportedOperationException(); 681 } 682 683 // should be able to forgo strictfp due to controlled over/underflow 684 public static strictfp double compute(double x) { 685 double y; 686 double hi = 0.0; 687 double lo = 0.0; 688 double c; 689 double t; 690 int k = 0; 691 int xsb; 692 /*unsigned*/ int hx; 693 694 hx = __HI(x); /* high word of x */ 695 xsb = (hx >> 31) & 1; /* sign bit of x */ 696 hx &= 0x7fffffff; /* high word of |x| */ 697 698 /* filter out non-finite argument */ 699 if (hx >= 0x40862E42) { /* if |x| >= 709.78... */ 700 if (hx >= 0x7ff00000) { 701 if (((hx & 0xfffff) | __LO(x)) != 0) 702 return x + x; /* NaN */ 703 else 704 return (xsb == 0) ? x : 0.0; /* exp(+-inf) = {inf, 0} */ 705 } 706 if (x > o_threshold) 707 return huge * huge; /* overflow */ 708 if (x < u_threshold) // unsigned compare needed here? 709 return twom1000 * twom1000; /* underflow */ 710 } 711 712 /* argument reduction */ 713 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 714 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 715 hi = x - ln2HI[xsb]; 716 lo=ln2LO[xsb]; 717 k = 1 - xsb - xsb; 718 } else { 719 k = (int)(invln2 * x + half[xsb]); 720 t = k; 721 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 722 lo = t*ln2LO[0]; 723 } 724 x = hi - lo; 725 } else if (hx < 0x3e300000) { /* when |x|<2**-28 */ 726 if (huge + x > one) 727 return one + x; /* trigger inexact */ 728 } else { 729 k = 0; 730 } 731 732 /* x is now in primary range */ 733 t = x * x; 734 c = x - t*(P1 + t*(P2 + t*(P3 + t*(P4 + t*P5)))); 735 if (k == 0) 736 return one - ((x*c)/(c - 2.0) - x); 737 else 738 y = one - ((lo - (x*c)/(2.0 - c)) - hi); 739 740 if(k >= -1021) { 741 y = __HI(y, __HI(y) + (k << 20)); /* add k to y's exponent */ 742 return y; 743 } else { 744 y = __HI(y, __HI(y) + ((k + 1000) << 20)); /* add k to y's exponent */ 745 return y * twom1000; 746 } 747 } 748 } 749} 750