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1: /* java.lang.StrictMath -- common mathematical functions, strict Java 2: Copyright (C) 1998, 2001, 2002, 2003, 2006 Free Software Foundation, Inc. 3: 4: This file is part of GNU Classpath. 5: 6: GNU Classpath is free software; you can redistribute it and/or modify 7: it under the terms of the GNU General Public License as published by 8: the Free Software Foundation; either version 2, or (at your option) 9: any later version. 10: 11: GNU Classpath is distributed in the hope that it will be useful, but 12: WITHOUT ANY WARRANTY; without even the implied warranty of 13: MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 14: General Public License for more details. 15: 16: You should have received a copy of the GNU General Public License 17: along with GNU Classpath; see the file COPYING. If not, write to the 18: Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 19: 02110-1301 USA. 20: 21: Linking this library statically or dynamically with other modules is 22: making a combined work based on this library. Thus, the terms and 23: conditions of the GNU General Public License cover the whole 24: combination. 25: 26: As a special exception, the copyright holders of this library give you 27: permission to link this library with independent modules to produce an 28: executable, regardless of the license terms of these independent 29: modules, and to copy and distribute the resulting executable under 30: terms of your choice, provided that you also meet, for each linked 31: independent module, the terms and conditions of the license of that 32: module. An independent module is a module which is not derived from 33: or based on this library. If you modify this library, you may extend 34: this exception to your version of the library, but you are not 35: obligated to do so. If you do not wish to do so, delete this 36: exception statement from your version. */ 37: 38: /* 39: * Some of the algorithms in this class are in the public domain, as part 40: * of fdlibm (freely-distributable math library), available at 41: * http://www.netlib.org/fdlibm/, and carry the following copyright: 42: * ==================================================== 43: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 44: * 45: * Developed at SunSoft, a Sun Microsystems, Inc. business. 46: * Permission to use, copy, modify, and distribute this 47: * software is freely granted, provided that this notice 48: * is preserved. 49: * ==================================================== 50: */ 51: 52: package java.lang; 53: 54: import gnu.classpath.Configuration; 55: 56: import java.util.Random; 57: 58: /** 59: * Helper class containing useful mathematical functions and constants. 60: * This class mirrors {@link Math}, but is 100% portable, because it uses 61: * no native methods whatsoever. Also, these algorithms are all accurate 62: * to less than 1 ulp, and execute in <code>strictfp</code> mode, while 63: * Math is allowed to vary in its results for some functions. Unfortunately, 64: * this usually means StrictMath has less efficiency and speed, as Math can 65: * use native methods. 66: * 67: * <p>The source of the various algorithms used is the fdlibm library, at:<br> 68: * <a href="http://www.netlib.org/fdlibm/">http://www.netlib.org/fdlibm/</a> 69: * 70: * Note that angles are specified in radians. Conversion functions are 71: * provided for your convenience. 72: * 73: * @author Eric Blake (ebb9@email.byu.edu) 74: * @since 1.3 75: */ 76: public final strictfp class StrictMath 77: { 78: /** 79: * StrictMath is non-instantiable. 80: */ 81: private StrictMath() 82: { 83: } 84: 85: /** 86: * A random number generator, initialized on first use. 87: * 88: * @see #random() 89: */ 90: private static Random rand; 91: 92: /** 93: * The most accurate approximation to the mathematical constant <em>e</em>: 94: * <code>2.718281828459045</code>. Used in natural log and exp. 95: * 96: * @see #log(double) 97: * @see #exp(double) 98: */ 99: public static final double E 100: = 2.718281828459045; // Long bits 0x4005bf0z8b145769L. 101: 102: /** 103: * The most accurate approximation to the mathematical constant <em>pi</em>: 104: * <code>3.141592653589793</code>. This is the ratio of a circle's diameter 105: * to its circumference. 106: */ 107: public static final double PI 108: = 3.141592653589793; // Long bits 0x400921fb54442d18L. 109: 110: /** 111: * Take the absolute value of the argument. (Absolute value means make 112: * it positive.) 113: * 114: * <p>Note that the the largest negative value (Integer.MIN_VALUE) cannot 115: * be made positive. In this case, because of the rules of negation in 116: * a computer, MIN_VALUE is what will be returned. 117: * This is a <em>negative</em> value. You have been warned. 118: * 119: * @param i the number to take the absolute value of 120: * @return the absolute value 121: * @see Integer#MIN_VALUE 122: */ 123: public static int abs(int i) 124: { 125: return (i < 0) ? -i : i; 126: } 127: 128: /** 129: * Take the absolute value of the argument. (Absolute value means make 130: * it positive.) 131: * 132: * <p>Note that the the largest negative value (Long.MIN_VALUE) cannot 133: * be made positive. In this case, because of the rules of negation in 134: * a computer, MIN_VALUE is what will be returned. 135: * This is a <em>negative</em> value. You have been warned. 136: * 137: * @param l the number to take the absolute value of 138: * @return the absolute value 139: * @see Long#MIN_VALUE 140: */ 141: public static long abs(long l) 142: { 143: return (l < 0) ? -l : l; 144: } 145: 146: /** 147: * Take the absolute value of the argument. (Absolute value means make 148: * it positive.) 149: * 150: * @param f the number to take the absolute value of 151: * @return the absolute value 152: */ 153: public static float abs(float f) 154: { 155: return (f <= 0) ? 0 - f : f; 156: } 157: 158: /** 159: * Take the absolute value of the argument. (Absolute value means make 160: * it positive.) 161: * 162: * @param d the number to take the absolute value of 163: * @return the absolute value 164: */ 165: public static double abs(double d) 166: { 167: return (d <= 0) ? 0 - d : d; 168: } 169: 170: /** 171: * Return whichever argument is smaller. 172: * 173: * @param a the first number 174: * @param b a second number 175: * @return the smaller of the two numbers 176: */ 177: public static int min(int a, int b) 178: { 179: return (a < b) ? a : b; 180: } 181: 182: /** 183: * Return whichever argument is smaller. 184: * 185: * @param a the first number 186: * @param b a second number 187: * @return the smaller of the two numbers 188: */ 189: public static long min(long a, long b) 190: { 191: return (a < b) ? a : b; 192: } 193: 194: /** 195: * Return whichever argument is smaller. If either argument is NaN, the 196: * result is NaN, and when comparing 0 and -0, -0 is always smaller. 197: * 198: * @param a the first number 199: * @param b a second number 200: * @return the smaller of the two numbers 201: */ 202: public static float min(float a, float b) 203: { 204: // this check for NaN, from JLS 15.21.1, saves a method call 205: if (a != a) 206: return a; 207: // no need to check if b is NaN; < will work correctly 208: // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special 209: if (a == 0 && b == 0) 210: return -(-a - b); 211: return (a < b) ? a : b; 212: } 213: 214: /** 215: * Return whichever argument is smaller. If either argument is NaN, the 216: * result is NaN, and when comparing 0 and -0, -0 is always smaller. 217: * 218: * @param a the first number 219: * @param b a second number 220: * @return the smaller of the two numbers 221: */ 222: public static double min(double a, double b) 223: { 224: // this check for NaN, from JLS 15.21.1, saves a method call 225: if (a != a) 226: return a; 227: // no need to check if b is NaN; < will work correctly 228: // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special 229: if (a == 0 && b == 0) 230: return -(-a - b); 231: return (a < b) ? a : b; 232: } 233: 234: /** 235: * Return whichever argument is larger. 236: * 237: * @param a the first number 238: * @param b a second number 239: * @return the larger of the two numbers 240: */ 241: public static int max(int a, int b) 242: { 243: return (a > b) ? a : b; 244: } 245: 246: /** 247: * Return whichever argument is larger. 248: * 249: * @param a the first number 250: * @param b a second number 251: * @return the larger of the two numbers 252: */ 253: public static long max(long a, long b) 254: { 255: return (a > b) ? a : b; 256: } 257: 258: /** 259: * Return whichever argument is larger. If either argument is NaN, the 260: * result is NaN, and when comparing 0 and -0, 0 is always larger. 261: * 262: * @param a the first number 263: * @param b a second number 264: * @return the larger of the two numbers 265: */ 266: public static float max(float a, float b) 267: { 268: // this check for NaN, from JLS 15.21.1, saves a method call 269: if (a != a) 270: return a; 271: // no need to check if b is NaN; > will work correctly 272: // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special 273: if (a == 0 && b == 0) 274: return a - -b; 275: return (a > b) ? a : b; 276: } 277: 278: /** 279: * Return whichever argument is larger. If either argument is NaN, the 280: * result is NaN, and when comparing 0 and -0, 0 is always larger. 281: * 282: * @param a the first number 283: * @param b a second number 284: * @return the larger of the two numbers 285: */ 286: public static double max(double a, double b) 287: { 288: // this check for NaN, from JLS 15.21.1, saves a method call 289: if (a != a) 290: return a; 291: // no need to check if b is NaN; > will work correctly 292: // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special 293: if (a == 0 && b == 0) 294: return a - -b; 295: return (a > b) ? a : b; 296: } 297: 298: /** 299: * The trigonometric function <em>sin</em>. The sine of NaN or infinity is 300: * NaN, and the sine of 0 retains its sign. 301: * 302: * @param a the angle (in radians) 303: * @return sin(a) 304: */ 305: public static double sin(double a) 306: { 307: if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY)) 308: return Double.NaN; 309: 310: if (abs(a) <= PI / 4) 311: return sin(a, 0); 312: 313: // Argument reduction needed. 314: double[] y = new double[2]; 315: int n = remPiOver2(a, y); 316: switch (n & 3) 317: { 318: case 0: 319: return sin(y[0], y[1]); 320: case 1: 321: return cos(y[0], y[1]); 322: case 2: 323: return -sin(y[0], y[1]); 324: default: 325: return -cos(y[0], y[1]); 326: } 327: } 328: 329: /** 330: * The trigonometric function <em>cos</em>. The cosine of NaN or infinity is 331: * NaN. 332: * 333: * @param a the angle (in radians). 334: * @return cos(a). 335: */ 336: public static double cos(double a) 337: { 338: if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY)) 339: return Double.NaN; 340: 341: if (abs(a) <= PI / 4) 342: return cos(a, 0); 343: 344: // Argument reduction needed. 345: double[] y = new double[2]; 346: int n = remPiOver2(a, y); 347: switch (n & 3) 348: { 349: case 0: 350: return cos(y[0], y[1]); 351: case 1: 352: return -sin(y[0], y[1]); 353: case 2: 354: return -cos(y[0], y[1]); 355: default: 356: return sin(y[0], y[1]); 357: } 358: } 359: 360: /** 361: * The trigonometric function <em>tan</em>. The tangent of NaN or infinity 362: * is NaN, and the tangent of 0 retains its sign. 363: * 364: * @param a the angle (in radians) 365: * @return tan(a) 366: */ 367: public static double tan(double a) 368: { 369: if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY)) 370: return Double.NaN; 371: 372: if (abs(a) <= PI / 4) 373: return tan(a, 0, false); 374: 375: // Argument reduction needed. 376: double[] y = new double[2]; 377: int n = remPiOver2(a, y); 378: return tan(y[0], y[1], (n & 1) == 1); 379: } 380: 381: /** 382: * The trigonometric function <em>arcsin</em>. The range of angles returned 383: * is -pi/2 to pi/2 radians (-90 to 90 degrees). If the argument is NaN or 384: * its absolute value is beyond 1, the result is NaN; and the arcsine of 385: * 0 retains its sign. 386: * 387: * @param x the sin to turn back into an angle 388: * @return arcsin(x) 389: */ 390: public static double asin(double x) 391: { 392: boolean negative = x < 0; 393: if (negative) 394: x = -x; 395: if (! (x <= 1)) 396: return Double.NaN; 397: if (x == 1) 398: return negative ? -PI / 2 : PI / 2; 399: if (x < 0.5) 400: { 401: if (x < 1 / TWO_27) 402: return negative ? -x : x; 403: double t = x * x; 404: double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t 405: * (PS4 + t * PS5))))); 406: double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4))); 407: return negative ? -x - x * (p / q) : x + x * (p / q); 408: } 409: double w = 1 - x; // 1>|x|>=0.5. 410: double t = w * 0.5; 411: double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t 412: * (PS4 + t * PS5))))); 413: double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4))); 414: double s = sqrt(t); 415: if (x >= 0.975) 416: { 417: w = p / q; 418: t = PI / 2 - (2 * (s + s * w) - PI_L / 2); 419: } 420: else 421: { 422: w = (float) s; 423: double c = (t - w * w) / (s + w); 424: p = 2 * s * (p / q) - (PI_L / 2 - 2 * c); 425: q = PI / 4 - 2 * w; 426: t = PI / 4 - (p - q); 427: } 428: return negative ? -t : t; 429: } 430: 431: /** 432: * The trigonometric function <em>arccos</em>. The range of angles returned 433: * is 0 to pi radians (0 to 180 degrees). If the argument is NaN or 434: * its absolute value is beyond 1, the result is NaN. 435: * 436: * @param x the cos to turn back into an angle 437: * @return arccos(x) 438: */ 439: public static double acos(double x) 440: { 441: boolean negative = x < 0; 442: if (negative) 443: x = -x; 444: if (! (x <= 1)) 445: return Double.NaN; 446: if (x == 1) 447: return negative ? PI : 0; 448: if (x < 0.5) 449: { 450: if (x < 1 / TWO_57) 451: return PI / 2; 452: double z = x * x; 453: double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z 454: * (PS4 + z * PS5))))); 455: double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4))); 456: double r = x - (PI_L / 2 - x * (p / q)); 457: return negative ? PI / 2 + r : PI / 2 - r; 458: } 459: if (negative) // x<=-0.5. 460: { 461: double z = (1 + x) * 0.5; 462: double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z 463: * (PS4 + z * PS5))))); 464: double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4))); 465: double s = sqrt(z); 466: double w = p / q * s - PI_L / 2; 467: return PI - 2 * (s + w); 468: } 469: double z = (1 - x) * 0.5; // x>0.5. 470: double s = sqrt(z); 471: double df = (float) s; 472: double c = (z - df * df) / (s + df); 473: double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z 474: * (PS4 + z * PS5))))); 475: double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4))); 476: double w = p / q * s + c; 477: return 2 * (df + w); 478: } 479: 480: /** 481: * The trigonometric function <em>arcsin</em>. The range of angles returned 482: * is -pi/2 to pi/2 radians (-90 to 90 degrees). If the argument is NaN, the 483: * result is NaN; and the arctangent of 0 retains its sign. 484: * 485: * @param x the tan to turn back into an angle 486: * @return arcsin(x) 487: * @see #atan2(double, double) 488: */ 489: public static double atan(double x) 490: { 491: double lo; 492: double hi; 493: boolean negative = x < 0; 494: if (negative) 495: x = -x; 496: if (x >= TWO_66) 497: return negative ? -PI / 2 : PI / 2; 498: if (! (x >= 0.4375)) // |x|<7/16, or NaN. 499: { 500: if (! (x >= 1 / TWO_29)) // Small, or NaN. 501: return negative ? -x : x; 502: lo = hi = 0; 503: } 504: else if (x < 1.1875) 505: { 506: if (x < 0.6875) // 7/16<=|x|<11/16. 507: { 508: x = (2 * x - 1) / (2 + x); 509: hi = ATAN_0_5H; 510: lo = ATAN_0_5L; 511: } 512: else // 11/16<=|x|<19/16. 513: { 514: x = (x - 1) / (x + 1); 515: hi = PI / 4; 516: lo = PI_L / 4; 517: } 518: } 519: else if (x < 2.4375) // 19/16<=|x|<39/16. 520: { 521: x = (x - 1.5) / (1 + 1.5 * x); 522: hi = ATAN_1_5H; 523: lo = ATAN_1_5L; 524: } 525: else // 39/16<=|x|<2**66. 526: { 527: x = -1 / x; 528: hi = PI / 2; 529: lo = PI_L / 2; 530: } 531: 532: // Break sum from i=0 to 10 ATi*z**(i+1) into odd and even poly. 533: double z = x * x; 534: double w = z * z; 535: double s1 = z * (AT0 + w * (AT2 + w * (AT4 + w * (AT6 + w 536: * (AT8 + w * AT10))))); 537: double s2 = w * (AT1 + w * (AT3 + w * (AT5 + w * (AT7 + w * AT9)))); 538: if (hi == 0) 539: return negative ? x * (s1 + s2) - x : x - x * (s1 + s2); 540: z = hi - ((x * (s1 + s2) - lo) - x); 541: return negative ? -z : z; 542: } 543: 544: /** 545: * A special version of the trigonometric function <em>arctan</em>, for 546: * converting rectangular coordinates <em>(x, y)</em> to polar 547: * <em>(r, theta)</em>. This computes the arctangent of x/y in the range 548: * of -pi to pi radians (-180 to 180 degrees). Special cases:<ul> 549: * <li>If either argument is NaN, the result is NaN.</li> 550: * <li>If the first argument is positive zero and the second argument is 551: * positive, or the first argument is positive and finite and the second 552: * argument is positive infinity, then the result is positive zero.</li> 553: * <li>If the first argument is negative zero and the second argument is 554: * positive, or the first argument is negative and finite and the second 555: * argument is positive infinity, then the result is negative zero.</li> 556: * <li>If the first argument is positive zero and the second argument is 557: * negative, or the first argument is positive and finite and the second 558: * argument is negative infinity, then the result is the double value 559: * closest to pi.</li> 560: * <li>If the first argument is negative zero and the second argument is 561: * negative, or the first argument is negative and finite and the second 562: * argument is negative infinity, then the result is the double value 563: * closest to -pi.</li> 564: * <li>If the first argument is positive and the second argument is 565: * positive zero or negative zero, or the first argument is positive 566: * infinity and the second argument is finite, then the result is the 567: * double value closest to pi/2.</li> 568: * <li>If the first argument is negative and the second argument is 569: * positive zero or negative zero, or the first argument is negative 570: * infinity and the second argument is finite, then the result is the 571: * double value closest to -pi/2.</li> 572: * <li>If both arguments are positive infinity, then the result is the 573: * double value closest to pi/4.</li> 574: * <li>If the first argument is positive infinity and the second argument 575: * is negative infinity, then the result is the double value closest to 576: * 3*pi/4.</li> 577: * <li>If the first argument is negative infinity and the second argument 578: * is positive infinity, then the result is the double value closest to 579: * -pi/4.</li> 580: * <li>If both arguments are negative infinity, then the result is the 581: * double value closest to -3*pi/4.</li> 582: * 583: * </ul><p>This returns theta, the angle of the point. To get r, albeit 584: * slightly inaccurately, use sqrt(x*x+y*y). 585: * 586: * @param y the y position 587: * @param x the x position 588: * @return <em>theta</em> in the conversion of (x, y) to (r, theta) 589: * @see #atan(double) 590: */ 591: public static double atan2(double y, double x) 592: { 593: if (x != x || y != y) 594: return Double.NaN; 595: if (x == 1) 596: return atan(y); 597: if (x == Double.POSITIVE_INFINITY) 598: { 599: if (y == Double.POSITIVE_INFINITY) 600: return PI / 4; 601: if (y == Double.NEGATIVE_INFINITY) 602: return -PI / 4; 603: return 0 * y; 604: } 605: if (x == Double.NEGATIVE_INFINITY) 606: { 607: if (y == Double.POSITIVE_INFINITY) 608: return 3 * PI / 4; 609: if (y == Double.NEGATIVE_INFINITY) 610: return -3 * PI / 4; 611: return (1 / (0 * y) == Double.POSITIVE_INFINITY) ? PI : -PI; 612: } 613: if (y == 0) 614: { 615: if (1 / (0 * x) == Double.POSITIVE_INFINITY) 616: return y; 617: return (1 / y == Double.POSITIVE_INFINITY) ? PI : -PI; 618: } 619: if (y == Double.POSITIVE_INFINITY || y == Double.NEGATIVE_INFINITY 620: || x == 0) 621: return y < 0 ? -PI / 2 : PI / 2; 622: 623: double z = abs(y / x); // Safe to do y/x. 624: if (z > TWO_60) 625: z = PI / 2 + 0.5 * PI_L; 626: else if (x < 0 && z < 1 / TWO_60) 627: z = 0; 628: else 629: z = atan(z); 630: if (x > 0) 631: return y > 0 ? z : -z; 632: return y > 0 ? PI - (z - PI_L) : z - PI_L - PI; 633: } 634: 635: /** 636: * Returns the hyperbolic cosine of <code>x</code>, which is defined as 637: * (exp(x) + exp(-x)) / 2. 638: * 639: * Special cases: 640: * <ul> 641: * <li>If the argument is NaN, the result is NaN</li> 642: * <li>If the argument is positive infinity, the result is positive 643: * infinity.</li> 644: * <li>If the argument is negative infinity, the result is positive 645: * infinity.</li> 646: * <li>If the argument is zero, the result is one.</li> 647: * </ul> 648: * 649: * @param x the argument to <em>cosh</em> 650: * @return the hyperbolic cosine of <code>x</code> 651: * 652: * @since 1.5 653: */ 654: public static double cosh(double x) 655: { 656: // Method : 657: // mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 658: // 1. Replace x by |x| (cosh(x) = cosh(-x)). 659: // 2. 660: // [ exp(x) - 1 ]^2 661: // 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- 662: // 2*exp(x) 663: // 664: // exp(x) + 1/exp(x) 665: // ln2/2 <= x <= 22 : cosh(x) := ------------------ 666: // 2 667: // 22 <= x <= lnovft : cosh(x) := exp(x)/2 668: // lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) 669: // ln2ovft < x : cosh(x) := +inf (overflow) 670: 671: double t, w; 672: long bits; 673: int hx; 674: int lx; 675: 676: // handle special cases 677: if (x != x) 678: return Double.NaN; 679: if (x == Double.POSITIVE_INFINITY) 680: return Double.POSITIVE_INFINITY; 681: if (x == Double.NEGATIVE_INFINITY) 682: return Double.POSITIVE_INFINITY; 683: 684: bits = Double.doubleToLongBits(x); 685: hx = getHighDWord(bits) & 0x7fffffff; // ignore sign 686: lx = getLowDWord(bits); 687: 688: // |x| in [0, 0.5 * ln(2)], return 1 + expm1(|x|)^2 / (2 * exp(|x|)) 689: if (hx < 0x3fd62e43) 690: { 691: t = expm1(abs(x)); 692: w = 1.0 + t; 693: 694: // for tiny arguments return 1. 695: if (hx < 0x3c800000) 696: return w; 697: 698: return 1.0 + (t * t) / (w + w); 699: } 700: 701: // |x| in [0.5 * ln(2), 22], return exp(|x|)/2 + 1 / (2 * exp(|x|)) 702: if (hx < 0x40360000) 703: { 704: t = exp(abs(x)); 705: 706: return 0.5 * t + 0.5 / t; 707: } 708: 709: // |x| in [22, log(Double.MAX_VALUE)], return 0.5 * exp(|x|) 710: if (hx < 0x40862e42) 711: return 0.5 * exp(abs(x)); 712: 713: // |x| in [log(Double.MAX_VALUE), overflowthreshold], 714: // return exp(x/2)/2 * exp(x/2) 715: 716: // we need to force an unsigned <= compare, thus can not use lx. 717: if ((hx < 0x408633ce) 718: || ((hx == 0x408633ce) 719: && ((bits & 0x00000000ffffffffL) <= 0x8fb9f87dL))) 720: { 721: w = exp(0.5 * abs(x)); 722: t = 0.5 * w; 723: 724: return t * w; 725: } 726: 727: // |x| > overflowthreshold 728: return Double.POSITIVE_INFINITY; 729: } 730: 731: /** 732: * Returns the lower two words of a long. This is intended to be 733: * used like this: 734: * <code>getLowDWord(Double.doubleToLongBits(x))</code>. 735: */ 736: private static int getLowDWord(long x) 737: { 738: return (int) (x & 0x00000000ffffffffL); 739: } 740: 741: /** 742: * Returns the higher two words of a long. This is intended to be 743: * used like this: 744: * <code>getHighDWord(Double.doubleToLongBits(x))</code>. 745: */ 746: private static int getHighDWord(long x) 747: { 748: return (int) ((x & 0xffffffff00000000L) >> 32); 749: } 750: 751: /** 752: * Returns a double with the IEEE754 bit pattern given in the lower 753: * and higher two words <code>lowDWord</code> and <code>highDWord</code>. 754: */ 755: private static double buildDouble(int lowDWord, int highDWord) 756: { 757: return Double.longBitsToDouble((((long) highDWord & 0xffffffffL) << 32) 758: | ((long) lowDWord & 0xffffffffL)); 759: } 760: 761: /** 762: * Returns the cube root of <code>x</code>. The sign of the cube root 763: * is equal to the sign of <code>x</code>. 764: * 765: * Special cases: 766: * <ul> 767: * <li>If the argument is NaN, the result is NaN</li> 768: * <li>If the argument is positive infinity, the result is positive 769: * infinity.</li> 770: * <li>If the argument is negative infinity, the result is negative 771: * infinity.</li> 772: * <li>If the argument is zero, the result is zero with the same 773: * sign as the argument.</li> 774: * </ul> 775: * 776: * @param x the number to take the cube root of 777: * @return the cube root of <code>x</code> 778: * @see #sqrt(double) 779: * 780: * @since 1.5 781: */ 782: public static double cbrt(double x) 783: { 784: boolean negative = (x < 0); 785: double r; 786: double s; 787: double t; 788: double w; 789: 790: long bits; 791: int l; 792: int h; 793: 794: // handle the special cases 795: if (x != x) 796: return Double.NaN; 797: if (x == Double.POSITIVE_INFINITY) 798: return Double.POSITIVE_INFINITY; 799: if (x == Double.NEGATIVE_INFINITY) 800: return Double.NEGATIVE_INFINITY; 801: if (x == 0) 802: return x; 803: 804: x = abs(x); 805: bits = Double.doubleToLongBits(x); 806: 807: if (bits < 0x0010000000000000L) // subnormal number 808: { 809: t = TWO_54; 810: t *= x; 811: 812: // __HI(t)=__HI(t)/3+B2; 813: bits = Double.doubleToLongBits(t); 814: h = getHighDWord(bits); 815: l = getLowDWord(bits); 816: 817: h = h / 3 + CBRT_B2; 818: 819: t = buildDouble(l, h); 820: } 821: else 822: { 823: // __HI(t)=__HI(x)/3+B1; 824: h = getHighDWord(bits); 825: l = 0; 826: 827: h = h / 3 + CBRT_B1; 828: t = buildDouble(l, h); 829: } 830: 831: // new cbrt to 23 bits 832: r = t * t / x; 833: s = CBRT_C + r * t; 834: t *= CBRT_G + CBRT_F / (s + CBRT_E + CBRT_D / s); 835: 836: // chopped to 20 bits and make it larger than cbrt(x) 837: bits = Double.doubleToLongBits(t); 838: h = getHighDWord(bits); 839: 840: // __LO(t)=0; 841: // __HI(t)+=0x00000001; 842: l = 0; 843: h += 1; 844: t = buildDouble(l, h); 845: 846: // one step newton iteration to 53 bits with error less than 0.667 ulps 847: s = t * t; // t * t is exact 848: r = x / s; 849: w = t + t; 850: r = (r - t) / (w + r); // r - s is exact 851: t = t + t * r; 852: 853: return negative ? -t : t; 854: } 855: 856: /** 857: * Take <em>e</em><sup>a</sup>. The opposite of <code>log()</code>. If the 858: * argument is NaN, the result is NaN; if the argument is positive infinity, 859: * the result is positive infinity; and if the argument is negative 860: * infinity, the result is positive zero. 861: * 862: * @param x the number to raise to the power 863: * @return the number raised to the power of <em>e</em> 864: * @see #log(double) 865: * @see #pow(double, double) 866: */ 867: public static double exp(double x) 868: { 869: if (x != x) 870: return x; 871: if (x > EXP_LIMIT_H) 872: return Double.POSITIVE_INFINITY; 873: if (x < EXP_LIMIT_L) 874: return 0; 875: 876: // Argument reduction. 877: double hi; 878: double lo; 879: int k; 880: double t = abs(x); 881: if (t > 0.5 * LN2) 882: { 883: if (t < 1.5 * LN2) 884: { 885: hi = t - LN2_H; 886: lo = LN2_L; 887: k = 1; 888: } 889: else 890: { 891: k = (int) (INV_LN2 * t + 0.5); 892: hi = t - k * LN2_H; 893: lo = k * LN2_L; 894: } 895: if (x < 0) 896: { 897: hi = -hi; 898: lo = -lo; 899: k = -k; 900: } 901: x = hi - lo; 902: } 903: else if (t < 1 / TWO_28) 904: return 1; 905: else 906: lo = hi = k = 0; 907: 908: // Now x is in primary range. 909: t = x * x; 910: double c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 911: if (k == 0) 912: return 1 - (x * c / (c - 2) - x); 913: double y = 1 - (lo - x * c / (2 - c) - hi); 914: return scale(y, k); 915: } 916: 917: /** 918: * Returns <em>e</em><sup>x</sup> - 1. 919: * Special cases: 920: * <ul> 921: * <li>If the argument is NaN, the result is NaN.</li> 922: * <li>If the argument is positive infinity, the result is positive 923: * infinity</li> 924: * <li>If the argument is negative infinity, the result is -1.</li> 925: * <li>If the argument is zero, the result is zero.</li> 926: * </ul> 927: * 928: * @param x the argument to <em>e</em><sup>x</sup> - 1. 929: * @return <em>e</em> raised to the power <code>x</code> minus one. 930: * @see #exp(double) 931: */ 932: public static double expm1(double x) 933: { 934: // Method 935: // 1. Argument reduction: 936: // Given x, find r and integer k such that 937: // 938: // x = k * ln(2) + r, |r| <= 0.5 * ln(2) 939: // 940: // Here a correction term c will be computed to compensate 941: // the error in r when rounded to a floating-point number. 942: // 943: // 2. Approximating expm1(r) by a special rational function on 944: // the interval [0, 0.5 * ln(2)]: 945: // Since 946: // r*(exp(r)+1)/(exp(r)-1) = 2 + r^2/6 - r^4/360 + ... 947: // we define R1(r*r) by 948: // r*(exp(r)+1)/(exp(r)-1) = 2 + r^2/6 * R1(r*r) 949: // That is, 950: // R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 951: // = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 952: // = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... 953: // We use a special Remes algorithm on [0, 0.347] to generate 954: // a polynomial of degree 5 in r*r to approximate R1. The 955: // maximum error of this polynomial approximation is bounded 956: // by 2**-61. In other words, 957: // R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 958: // where Q1 = -1.6666666666666567384E-2, 959: // Q2 = 3.9682539681370365873E-4, 960: // Q3 = -9.9206344733435987357E-6, 961: // Q4 = 2.5051361420808517002E-7, 962: // Q5 = -6.2843505682382617102E-9; 963: // (where z=r*r, and Q1 to Q5 are called EXPM1_Qx in the source) 964: // with error bounded by 965: // | 5 | -61 966: // | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 967: // | | 968: // 969: // expm1(r) = exp(r)-1 is then computed by the following 970: // specific way which minimize the accumulation rounding error: 971: // 2 3 972: // r r [ 3 - (R1 + R1*r/2) ] 973: // expm1(r) = r + --- + --- * [--------------------] 974: // 2 2 [ 6 - r*(3 - R1*r/2) ] 975: // 976: // To compensate the error in the argument reduction, we use 977: // expm1(r+c) = expm1(r) + c + expm1(r)*c 978: // ~ expm1(r) + c + r*c 979: // Thus c+r*c will be added in as the correction terms for 980: // expm1(r+c). Now rearrange the term to avoid optimization 981: // screw up: 982: // ( 2 2 ) 983: // ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 984: // expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 985: // ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 986: // ( ) 987: // 988: // = r - E 989: // 3. Scale back to obtain expm1(x): 990: // From step 1, we have 991: // expm1(x) = either 2^k*[expm1(r)+1] - 1 992: // = or 2^k*[expm1(r) + (1-2^-k)] 993: // 4. Implementation notes: 994: // (A). To save one multiplication, we scale the coefficient Qi 995: // to Qi*2^i, and replace z by (x^2)/2. 996: // (B). To achieve maximum accuracy, we compute expm1(x) by 997: // (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 998: // (ii) if k=0, return r-E 999: // (iii) if k=-1, return 0.5*(r-E)-0.5 1000: // (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 1001: // else return 1.0+2.0*(r-E); 1002: // (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) 1003: // (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else 1004: // (vii) return 2^k(1-((E+2^-k)-r)) 1005: 1006: boolean negative = (x < 0); 1007: double y, hi, lo, c, t, e, hxs, hfx, r1; 1008: int k; 1009: 1010: long bits; 1011: int h_bits; 1012: int l_bits; 1013: 1014: c = 0.0; 1015: y = abs(x); 1016: 1017: bits = Double.doubleToLongBits(y); 1018: h_bits = getHighDWord(bits); 1019: l_bits = getLowDWord(bits); 1020: 1021: // handle special cases and large arguments 1022: if (h_bits >= 0x4043687a) // if |x| >= 56 * ln(2) 1023: { 1024: if (h_bits >= 0x40862e42) // if |x| >= EXP_LIMIT_H 1025: { 1026: if (h_bits >= 0x7ff00000) 1027: { 1028: if (((h_bits & 0x000fffff) | (l_bits & 0xffffffff)) != 0) 1029: return Double.NaN; // exp(NaN) = NaN 1030: else 1031: return negative ? -1.0 : x; // exp({+-inf}) = {+inf, -1} 1032: } 1033: 1034: if (x > EXP_LIMIT_H) 1035: return Double.POSITIVE_INFINITY; // overflow 1036: } 1037: 1038: if (negative) // x <= -56 * ln(2) 1039: return -1.0; 1040: } 1041: 1042: // argument reduction 1043: if (h_bits > 0x3fd62e42) // |x| > 0.5 * ln(2) 1044: { 1045: if (h_bits < 0x3ff0a2b2) // |x| < 1.5 * ln(2) 1046: { 1047: if (negative) 1048: { 1049: hi = x + LN2_H; 1050: lo = -LN2_L; 1051: k = -1; 1052: } 1053: else 1054: { 1055: hi = x - LN2_H; 1056: lo = LN2_L; 1057: k = 1; 1058: } 1059: } 1060: else 1061: { 1062: k = (int) (INV_LN2 * x + (negative ? - 0.5 : 0.5)); 1063: t = k; 1064: hi = x - t * LN2_H; 1065: lo = t * LN2_L; 1066: } 1067: 1068: x = hi - lo; 1069: c = (hi - x) - lo; 1070: 1071: } 1072: else if (h_bits < 0x3c900000) // |x| < 2^-54 return x 1073: return x; 1074: else 1075: k = 0; 1076: 1077: // x is now in primary range 1078: hfx = 0.5 * x; 1079: hxs = x * hfx; 1080: r1 = 1.0 + hxs * (EXPM1_Q1 1081: + hxs * (EXPM1_Q2 1082: + hxs * (EXPM1_Q3 1083: + hxs * (EXPM1_Q4 1084: + hxs * EXPM1_Q5)))); 1085: t = 3.0 - r1 * hfx; 1086: e = hxs * ((r1 - t) / (6.0 - x * t)); 1087: 1088: if (k == 0) 1089: { 1090: return x - (x * e - hxs); // c == 0 1091: } 1092: else 1093: { 1094: e = x * (e - c) - c; 1095: e -= hxs; 1096: 1097: if (k == -1) 1098: return 0.5 * (x - e) - 0.5; 1099: 1100: if (k == 1) 1101: { 1102: if (x < - 0.25) 1103: return -2.0 * (e - (x + 0.5)); 1104: else 1105: return 1.0 + 2.0 * (x - e); 1106: } 1107: 1108: if (k <= -2 || k > 56) // sufficient to return exp(x) - 1 1109: { 1110: y = 1.0 - (e - x); 1111: 1112: bits = Double.doubleToLongBits(y); 1113: h_bits = getHighDWord(bits); 1114: l_bits = getLowDWord(bits); 1115: 1116: h_bits += (k << 20); // add k to y's exponent 1117: 1118: y = buildDouble(l_bits, h_bits); 1119: 1120: return y - 1.0; 1121: } 1122: 1123: t = 1.0; 1124: if (k < 20) 1125: { 1126: bits = Double.doubleToLongBits(t); 1127: h_bits = 0x3ff00000 - (0x00200000 >> k); 1128: l_bits = getLowDWord(bits); 1129: 1130: t = buildDouble(l_bits, h_bits); // t = 1 - 2^(-k) 1131: y = t - (e - x); 1132: 1133: bits = Double.doubleToLongBits(y); 1134: h_bits = getHighDWord(bits); 1135: l_bits = getLowDWord(bits); 1136: 1137: h_bits += (k << 20); // add k to y's exponent 1138: 1139: y = buildDouble(l_bits, h_bits); 1140: } 1141: else 1142: { 1143: bits = Double.doubleToLongBits(t); 1144: h_bits = (0x000003ff - k) << 20; 1145: l_bits = getLowDWord(bits); 1146: 1147: t = buildDouble(l_bits, h_bits); // t = 2^(-k) 1148: 1149: y = x - (e + t); 1150: y += 1.0; 1151: 1152: bits = Double.doubleToLongBits(y); 1153: h_bits = getHighDWord(bits); 1154: l_bits = getLowDWord(bits); 1155: 1156: h_bits += (k << 20); // add k to y's exponent 1157: 1158: y = buildDouble(l_bits, h_bits); 1159: } 1160: } 1161: 1162: return y; 1163: } 1164: 1165: /** 1166: * Take ln(a) (the natural log). The opposite of <code>exp()</code>. If the 1167: * argument is NaN or negative, the result is NaN; if the argument is 1168: * positive infinity, the result is positive infinity; and if the argument 1169: * is either zero, the result is negative infinity. 1170: * 1171: * <p>Note that the way to get log<sub>b</sub>(a) is to do this: 1172: * <code>ln(a) / ln(b)</code>. 1173: * 1174: * @param x the number to take the natural log of 1175: * @return the natural log of <code>a</code> 1176: * @see #exp(double) 1177: */ 1178: public static double log(double x) 1179: { 1180: if (x == 0) 1181: return Double.NEGATIVE_INFINITY; 1182: if (x < 0) 1183: return Double.NaN; 1184: if (! (x < Double.POSITIVE_INFINITY)) 1185: return x; 1186: 1187: // Normalize x. 1188: long bits = Double.doubleToLongBits(x); 1189: int exp = (int) (bits >> 52); 1190: if (exp == 0) // Subnormal x. 1191: { 1192: x *= TWO_54; 1193: bits = Double.doubleToLongBits(x); 1194: exp = (int) (bits >> 52) - 54; 1195: } 1196: exp -= 1023; // Unbias exponent. 1197: bits = (bits & 0x000fffffffffffffL) | 0x3ff0000000000000L; 1198: x = Double.longBitsToDouble(bits); 1199: if (x >= SQRT_2) 1200: { 1201: x *= 0.5; 1202: exp++; 1203: } 1204: x--; 1205: if (abs(x) < 1 / TWO_20) 1206: { 1207: if (x == 0) 1208: return exp * LN2_H + exp * LN2_L; 1209: double r = x * x * (0.5 - 1 / 3.0 * x); 1210: if (exp == 0) 1211: return x - r; 1212: return exp * LN2_H - ((r - exp * LN2_L) - x); 1213: } 1214: double s = x / (2 + x); 1215: double z = s * s; 1216: double w = z * z; 1217: double t1 = w * (LG2 + w * (LG4 + w * LG6)); 1218: double t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7))); 1219: double r = t2 + t1; 1220: if (bits >= 0x3ff6174a00000000L && bits < 0x3ff6b85200000000L) 1221: { 1222: double h = 0.5 * x * x; // Need more accuracy for x near sqrt(2). 1223: if (exp == 0) 1224: return x - (h - s * (h + r)); 1225: return exp * LN2_H - ((h - (s * (h + r) + exp * LN2_L)) - x); 1226: } 1227: if (exp == 0) 1228: return x - s * (x - r); 1229: return exp * LN2_H - ((s * (x - r) - exp * LN2_L) - x); 1230: } 1231: 1232: /** 1233: * Take a square root. If the argument is NaN or negative, the result is 1234: * NaN; if the argument is positive infinity, the result is positive 1235: * infinity; and if the result is either zero, the result is the same. 1236: * 1237: * <p>For other roots, use pow(x, 1/rootNumber). 1238: * 1239: * @param x the numeric argument 1240: * @return the square root of the argument 1241: * @see #pow(double, double) 1242: */ 1243: public static double sqrt(double x) 1244: { 1245: if (x < 0) 1246: return Double.NaN; 1247: if (x == 0 || ! (x < Double.POSITIVE_INFINITY)) 1248: return x; 1249: 1250: // Normalize x. 1251: long bits = Double.doubleToLongBits(x); 1252: int exp = (int) (bits >> 52); 1253: if (exp == 0) // Subnormal x. 1254: { 1255: x *= TWO_54; 1256: bits = Double.doubleToLongBits(x); 1257: exp = (int) (bits >> 52) - 54; 1258: } 1259: exp -= 1023; // Unbias exponent. 1260: bits = (bits & 0x000fffffffffffffL) | 0x0010000000000000L; 1261: if ((exp & 1) == 1) // Odd exp, double x to make it even. 1262: bits <<= 1; 1263: exp >>= 1; 1264: 1265: // Generate sqrt(x) bit by bit. 1266: bits <<= 1; 1267: long q = 0; 1268: long s = 0; 1269: long r = 0x0020000000000000L; // Move r right to left. 1270: while (r != 0) 1271: { 1272: long t = s + r; 1273: if (t <= bits) 1274: { 1275: s = t + r; 1276: bits -= t; 1277: q += r; 1278: } 1279: bits <<= 1; 1280: r >>= 1; 1281: } 1282: 1283: // Use floating add to round correctly. 1284: if (bits != 0) 1285: q += q & 1; 1286: return Double.longBitsToDouble((q >> 1) + ((exp + 1022L) << 52)); 1287: } 1288: 1289: /** 1290: * Raise a number to a power. Special cases:<ul> 1291: * <li>If the second argument is positive or negative zero, then the result 1292: * is 1.0.</li> 1293: * <li>If the second argument is 1.0, then the result is the same as the 1294: * first argument.</li> 1295: * <li>If the second argument is NaN, then the result is NaN.</li> 1296: * <li>If the first argument is NaN and the second argument is nonzero, 1297: * then the result is NaN.</li> 1298: * <li>If the absolute value of the first argument is greater than 1 and 1299: * the second argument is positive infinity, or the absolute value of the 1300: * first argument is less than 1 and the second argument is negative 1301: * infinity, then the result is positive infinity.</li> 1302: * <li>If the absolute value of the first argument is greater than 1 and 1303: * the second argument is negative infinity, or the absolute value of the 1304: * first argument is less than 1 and the second argument is positive 1305: * infinity, then the result is positive zero.</li> 1306: * <li>If the absolute value of the first argument equals 1 and the second 1307: * argument is infinite, then the result is NaN.</li> 1308: * <li>If the first argument is positive zero and the second argument is 1309: * greater than zero, or the first argument is positive infinity and the 1310: * second argument is less than zero, then the result is positive zero.</li> 1311: * <li>If the first argument is positive zero and the second argument is 1312: * less than zero, or the first argument is positive infinity and the 1313: * second argument is greater than zero, then the result is positive 1314: * infinity.</li> 1315: * <li>If the first argument is negative zero and the second argument is 1316: * greater than zero but not a finite odd integer, or the first argument is 1317: * negative infinity and the second argument is less than zero but not a 1318: * finite odd integer, then the result is positive zero.</li> 1319: * <li>If the first argument is negative zero and the second argument is a 1320: * positive finite odd integer, or the first argument is negative infinity 1321: * and the second argument is a negative finite odd integer, then the result 1322: * is negative zero.</li> 1323: * <li>If the first argument is negative zero and the second argument is 1324: * less than zero but not a finite odd integer, or the first argument is 1325: * negative infinity and the second argument is greater than zero but not a 1326: * finite odd integer, then the result is positive infinity.</li> 1327: * <li>If the first argument is negative zero and the second argument is a 1328: * negative finite odd integer, or the first argument is negative infinity 1329: * and the second argument is a positive finite odd integer, then the result 1330: * is negative infinity.</li> 1331: * <li>If the first argument is less than zero and the second argument is a 1332: * finite even integer, then the result is equal to the result of raising 1333: * the absolute value of the first argument to the power of the second 1334: * argument.</li> 1335: * <li>If the first argument is less than zero and the second argument is a 1336: * finite odd integer, then the result is equal to the negative of the 1337: * result of raising the absolute value of the first argument to the power 1338: * of the second argument.</li> 1339: * <li>If the first argument is finite and less than zero and the second 1340: * argument is finite and not an integer, then the result is NaN.</li> 1341: * <li>If both arguments are integers, then the result is exactly equal to 1342: * the mathematical result of raising the first argument to the power of 1343: * the second argument if that result can in fact be represented exactly as 1344: * a double value.</li> 1345: * 1346: * </ul><p>(In the foregoing descriptions, a floating-point value is 1347: * considered to be an integer if and only if it is a fixed point of the 1348: * method {@link #ceil(double)} or, equivalently, a fixed point of the 1349: * method {@link #floor(double)}. A value is a fixed point of a one-argument 1350: * method if and only if the result of applying the method to the value is 1351: * equal to the value.) 1352: * 1353: * @param x the number to raise 1354: * @param y the power to raise it to 1355: * @return x<sup>y</sup> 1356: */ 1357: public static double pow(double x, double y) 1358: { 1359: // Special cases first. 1360: if (y == 0) 1361: return 1; 1362: if (y == 1) 1363: return x; 1364: if (y == -1) 1365: return 1 / x; 1366: if (x != x || y != y) 1367: return Double.NaN; 1368: 1369: // When x < 0, yisint tells if y is not an integer (0), even(1), 1370: // or odd (2). 1371: int yisint = 0; 1372: if (x < 0 && floor(y) == y) 1373: yisint = (y % 2 == 0) ? 2 : 1; 1374: double ax = abs(x); 1375: double ay = abs(y); 1376: 1377: // More special cases, of y. 1378: if (ay == Double.POSITIVE_INFINITY) 1379: { 1380: if (ax == 1) 1381: return Double.NaN; 1382: if (ax > 1) 1383: return y > 0 ? y : 0; 1384: return y < 0 ? -y : 0; 1385: } 1386: if (y == 2) 1387: return x * x; 1388: if (y == 0.5) 1389: return sqrt(x); 1390: 1391: // More special cases, of x. 1392: if (x == 0 || ax == Double.POSITIVE_INFINITY || ax == 1) 1393: { 1394: if (y < 0) 1395: ax = 1 / ax; 1396: if (x < 0) 1397: { 1398: if (x == -1 && yisint == 0) 1399: ax = Double.NaN; 1400: else if (yisint == 1) 1401: ax = -ax; 1402: } 1403: return ax; 1404: } 1405: if (x < 0 && yisint == 0) 1406: return Double.NaN; 1407: 1408: // Now we can start! 1409: double t; 1410: double t1; 1411: double t2; 1412: double u; 1413: double v; 1414: double w; 1415: if (ay > TWO_31) 1416: { 1417: if (ay > TWO_64) // Automatic over/underflow. 1418: return ((ax < 1) ? y < 0 : y > 0) ? Double.POSITIVE_INFINITY : 0; 1419: // Over/underflow if x is not close to one. 1420: if (ax < 0.9999995231628418) 1421: return y < 0 ? Double.POSITIVE_INFINITY : 0; 1422: if (ax >= 1.0000009536743164) 1423: return y > 0 ? Double.POSITIVE_INFINITY : 0; 1424: // Now |1-x| is <= 2**-20, sufficient to compute 1425: // log(x) by x-x^2/2+x^3/3-x^4/4. 1426: t = x - 1; 1427: w = t * t * (0.5 - t * (1 / 3.0 - t * 0.25)); 1428: u = INV_LN2_H * t; 1429: v = t * INV_LN2_L - w * INV_LN2; 1430: t1 = (float) (u + v); 1431: t2 = v - (t1 - u); 1432: } 1433: else 1434: { 1435: long bits = Double.doubleToLongBits(ax); 1436: int exp = (int) (bits >> 52); 1437: if (exp == 0) // Subnormal x. 1438: { 1439: ax *= TWO_54; 1440: bits = Double.doubleToLongBits(ax); 1441: exp = (int) (bits >> 52) - 54; 1442: } 1443: exp -= 1023; // Unbias exponent. 1444: ax = Double.longBitsToDouble((bits & 0x000fffffffffffffL) 1445: | 0x3ff0000000000000L); 1446: boolean k; 1447: if (ax < SQRT_1_5) // |x|<sqrt(3/2). 1448: k = false; 1449: else if (ax < SQRT_3) // |x|<sqrt(3). 1450: k = true; 1451: else 1452: { 1453: k = false; 1454: ax *= 0.5; 1455: exp++; 1456: } 1457: 1458: // Compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5). 1459: u = ax - (k ? 1.5 : 1); 1460: v = 1 / (ax + (k ? 1.5 : 1)); 1461: double s = u * v; 1462: double s_h = (float) s; 1463: double t_h = (float) (ax + (k ? 1.5 : 1)); 1464: double t_l = ax - (t_h - (k ? 1.5 : 1)); 1465: double s_l = v * ((u - s_h * t_h) - s_h * t_l); 1466: // Compute log(ax). 1467: double s2 = s * s; 1468: double r = s_l * (s_h + s) + s2 * s2 1469: * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 1470: s2 = s_h * s_h; 1471: t_h = (float) (3.0 + s2 + r); 1472: t_l = r - (t_h - 3.0 - s2); 1473: // u+v = s*(1+...). 1474: u = s_h * t_h; 1475: v = s_l * t_h + t_l * s; 1476: // 2/(3log2)*(s+...). 1477: double p_h = (float) (u + v); 1478: double p_l = v - (p_h - u); 1479: double z_h = CP_H * p_h; 1480: double z_l = CP_L * p_h + p_l * CP + (k ? DP_L : 0); 1481: // log2(ax) = (s+..)*2/(3*log2) = exp + dp_h + z_h + z_l. 1482: t = exp; 1483: t1 = (float) (z_h + z_l + (k ? DP_H : 0) + t); 1484: t2 = z_l - (t1 - t - (k ? DP_H : 0) - z_h); 1485: } 1486: 1487: // Split up y into y1+y2 and compute (y1+y2)*(t1+t2). 1488: boolean negative = x < 0 && yisint == 1; 1489: double y1 = (float) y; 1490: double p_l = (y - y1) * t1 + y * t2; 1491: double p_h = y1 * t1; 1492: double z = p_l + p_h; 1493: if (z >= 1024) // Detect overflow. 1494: { 1495: if (z > 1024 || p_l + OVT > z - p_h) 1496: return negative ? Double.NEGATIVE_INFINITY 1497: : Double.POSITIVE_INFINITY; 1498: } 1499: else if (z <= -1075) // Detect underflow. 1500: { 1501: if (z < -1075 || p_l <= z - p_h) 1502: return negative ? -0.0 : 0; 1503: } 1504: 1505: // Compute 2**(p_h+p_l). 1506: int n = round((float) z); 1507: p_h -= n; 1508: t = (float) (p_l + p_h); 1509: u = t * LN2_H; 1510: v = (p_l - (t - p_h)) * LN2 + t * LN2_L; 1511: z = u + v; 1512: w = v - (z - u); 1513: t = z * z; 1514: t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 1515: double r = (z * t1) / (t1 - 2) - (w + z * w); 1516: z = scale(1 - (r - z), n); 1517: return negative ? -z : z; 1518: } 1519: 1520: /** 1521: * Get the IEEE 754 floating point remainder on two numbers. This is the 1522: * value of <code>x - y * <em>n</em></code>, where <em>n</em> is the closest 1523: * double to <code>x / y</code> (ties go to the even n); for a zero 1524: * remainder, the sign is that of <code>x</code>. If either argument is NaN, 1525: * the first argument is infinite, or the second argument is zero, the result 1526: * is NaN; if x is finite but y is infinite, the result is x. 1527: * 1528: * @param x the dividend (the top half) 1529: * @param y the divisor (the bottom half) 1530: * @return the IEEE 754-defined floating point remainder of x/y 1531: * @see #rint(double) 1532: */ 1533: public static double IEEEremainder(double x, double y) 1534: { 1535: // Purge off exception values. 1536: if (x == Double.NEGATIVE_INFINITY || ! (x < Double.POSITIVE_INFINITY) 1537: || y == 0 || y != y) 1538: return Double.NaN; 1539: 1540: boolean negative = x < 0; 1541: x = abs(x); 1542: y = abs(y); 1543: if (x == y || x == 0) 1544: return 0 * x; // Get correct sign. 1545: 1546: // Achieve x < 2y, then take first shot at remainder. 1547: if (y < TWO_1023) 1548: x %= y + y; 1549: 1550: // Now adjust x to get correct precision. 1551: if (y < 4 / TWO_1023) 1552: { 1553: if (x + x > y) 1554: { 1555: x -= y; 1556: if (x + x >= y) 1557: x -= y; 1558: } 1559: } 1560: else 1561: { 1562: y *= 0.5; 1563: if (x > y) 1564: { 1565: x -= y; 1566: if (x >= y) 1567: x -= y; 1568: } 1569: } 1570: return negative ? -x : x; 1571: } 1572: 1573: /** 1574: * Take the nearest integer that is that is greater than or equal to the 1575: * argument. If the argument is NaN, infinite, or zero, the result is the 1576: * same; if the argument is between -1 and 0, the result is negative zero. 1577: * Note that <code>Math.ceil(x) == -Math.floor(-x)</code>. 1578: * 1579: * @param a the value to act upon 1580: * @return the nearest integer >= <code>a</code> 1581: */ 1582: public static double ceil(double a) 1583: { 1584: return -floor(-a); 1585: } 1586: 1587: /** 1588: * Take the nearest integer that is that is less than or equal to the 1589: * argument. If the argument is NaN, infinite, or zero, the result is the 1590: * same. Note that <code>Math.ceil(x) == -Math.floor(-x)</code>. 1591: * 1592: * @param a the value to act upon 1593: * @return the nearest integer <= <code>a</code> 1594: */ 1595: public static double floor(double a) 1596: { 1597: double x = abs(a); 1598: if (! (x < TWO_52) || (long) a == a) 1599: return a; // No fraction bits; includes NaN and infinity. 1600: if (x < 1) 1601: return a >= 0 ? 0 * a : -1; // Worry about signed zero. 1602: return a < 0 ? (long) a - 1.0 : (long) a; // Cast to long truncates. 1603: } 1604: 1605: /** 1606: * Take the nearest integer to the argument. If it is exactly between 1607: * two integers, the even integer is taken. If the argument is NaN, 1608: * infinite, or zero, the result is the same. 1609: * 1610: * @param a the value to act upon 1611: * @return the nearest integer to <code>a</code> 1612: */ 1613: public static double rint(double a) 1614: { 1615: double x = abs(a); 1616: if (! (x < TWO_52)) 1617: return a; // No fraction bits; includes NaN and infinity. 1618: if (x <= 0.5) 1619: return 0 * a; // Worry about signed zero. 1620: if (x % 2 <= 0.5) 1621: return (long) a; // Catch round down to even. 1622: return (long) (a + (a < 0 ? -0.5 : 0.5)); // Cast to long truncates. 1623: } 1624: 1625: /** 1626: * Take the nearest integer to the argument. This is equivalent to 1627: * <code>(int) Math.floor(f + 0.5f)</code>. If the argument is NaN, the 1628: * result is 0; otherwise if the argument is outside the range of int, the 1629: * result will be Integer.MIN_VALUE or Integer.MAX_VALUE, as appropriate. 1630: * 1631: * @param f the argument to round 1632: * @return the nearest integer to the argument 1633: * @see Integer#MIN_VALUE 1634: * @see Integer#MAX_VALUE 1635: */ 1636: public static int round(float f) 1637: { 1638: return (int) floor(f + 0.5f); 1639: } 1640: 1641: /** 1642: * Take the nearest long to the argument. This is equivalent to 1643: * <code>(long) Math.floor(d + 0.5)</code>. If the argument is NaN, the 1644: * result is 0; otherwise if the argument is outside the range of long, the 1645: * result will be Long.MIN_VALUE or Long.MAX_VALUE, as appropriate. 1646: * 1647: * @param d the argument to round 1648: * @return the nearest long to the argument 1649: * @see Long#MIN_VALUE 1650: * @see Long#MAX_VALUE 1651: */ 1652: public static long round(double d) 1653: { 1654: return (long) floor(d + 0.5); 1655: } 1656: 1657: /** 1658: * Get a random number. This behaves like Random.nextDouble(), seeded by 1659: * System.currentTimeMillis() when first called. In other words, the number 1660: * is from a pseudorandom sequence, and lies in the range [+0.0, 1.0). 1661: * This random sequence is only used by this method, and is threadsafe, 1662: * although you may want your own random number generator if it is shared 1663: * among threads. 1664: * 1665: * @return a random number 1666: * @see Random#nextDouble() 1667: * @see System#currentTimeMillis() 1668: */ 1669: public static synchronized double random() 1670: { 1671: if (rand == null) 1672: rand = new Random(); 1673: return rand.nextDouble(); 1674: } 1675: 1676: /** 1677: * Convert from degrees to radians. The formula for this is 1678: * radians = degrees * (pi/180); however it is not always exact given the 1679: * limitations of floating point numbers. 1680: * 1681: * @param degrees an angle in degrees 1682: * @return the angle in radians 1683: */ 1684: public static double toRadians(double degrees) 1685: { 1686: return (degrees * PI) / 180; 1687: } 1688: 1689: /** 1690: * Convert from radians to degrees. The formula for this is 1691: * degrees = radians * (180/pi); however it is not always exact given the 1692: * limitations of floating point numbers. 1693: * 1694: * @param rads an angle in radians 1695: * @return the angle in degrees 1696: */ 1697: public static double toDegrees(double rads) 1698: { 1699: return (rads * 180) / PI; 1700: } 1701: 1702: /** 1703: * Constants for scaling and comparing doubles by powers of 2. The compiler 1704: * must automatically inline constructs like (1/TWO_54), so we don't list 1705: * negative powers of two here. 1706: */ 1707: private static final double 1708: TWO_16 = 0x10000, // Long bits 0x40f0000000000000L. 1709: TWO_20 = 0x100000, // Long bits 0x4130000000000000L. 1710: TWO_24 = 0x1000000, // Long bits 0x4170000000000000L. 1711: TWO_27 = 0x8000000, // Long bits 0x41a0000000000000L. 1712: TWO_28 = 0x10000000, // Long bits 0x41b0000000000000L. 1713: TWO_29 = 0x20000000, // Long bits 0x41c0000000000000L. 1714: TWO_31 = 0x80000000L, // Long bits 0x41e0000000000000L. 1715: TWO_49 = 0x2000000000000L, // Long bits 0x4300000000000000L. 1716: TWO_52 = 0x10000000000000L, // Long bits 0x4330000000000000L. 1717: TWO_54 = 0x40000000000000L, // Long bits 0x4350000000000000L. 1718: TWO_57 = 0x200000000000000L, // Long bits 0x4380000000000000L. 1719: TWO_60 = 0x1000000000000000L, // Long bits 0x43b0000000000000L. 1720: TWO_64 = 1.8446744073709552e19, // Long bits 0x43f0000000000000L. 1721: TWO_66 = 7.378697629483821e19, // Long bits 0x4410000000000000L. 1722: TWO_1023 = 8.98846567431158e307; // Long bits 0x7fe0000000000000L. 1723: 1724: /** 1725: * Super precision for 2/pi in 24-bit chunks, for use in 1726: * {@link #remPiOver2(double, double[])}. 1727: */ 1728: private static final int TWO_OVER_PI[] = { 1729: 0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62, 1730: 0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a, 1731: 0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129, 1732: 0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41, 1733: 0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8, 1734: 0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf, 1735: 0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5, 1736: 0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08, 1737: 0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3, 1738: 0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880, 1739: 0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b, 1740: }; 1741: 1742: /** 1743: * Super precision for pi/2 in 24-bit chunks, for use in 1744: * {@link #remPiOver2(double, double[])}. 1745: */ 1746: private static final double PI_OVER_TWO[] = { 1747: 1.570796251296997, // Long bits 0x3ff921fb40000000L. 1748: 7.549789415861596e-8, // Long bits 0x3e74442d00000000L. 1749: 5.390302529957765e-15, // Long bits 0x3cf8469880000000L. 1750: 3.282003415807913e-22, // Long bits 0x3b78cc5160000000L. 1751: 1.270655753080676e-29, // Long bits 0x39f01b8380000000L. 1752: 1.2293330898111133e-36, // Long bits 0x387a252040000000L. 1753: 2.7337005381646456e-44, // Long bits 0x36e3822280000000L. 1754: 2.1674168387780482e-51, // Long bits 0x3569f31d00000000L. 1755: }; 1756: 1757: /** 1758: * More constants related to pi, used in 1759: * {@link #remPiOver2(double, double[])} and elsewhere. 1760: */ 1761: private static final double 1762: PI_L = 1.2246467991473532e-16, // Long bits 0x3ca1a62633145c07L. 1763: PIO2_1 = 1.5707963267341256, // Long bits 0x3ff921fb54400000L. 1764: PIO2_1L = 6.077100506506192e-11, // Long bits 0x3dd0b4611a626331L. 1765: PIO2_2 = 6.077100506303966e-11, // Long bits 0x3dd0b4611a600000L. 1766: PIO2_2L = 2.0222662487959506e-21, // Long bits 0x3ba3198a2e037073L. 1767: PIO2_3 = 2.0222662487111665e-21, // Long bits 0x3ba3198a2e000000L. 1768: PIO2_3L = 8.4784276603689e-32; // Long bits 0x397b839a252049c1L. 1769: 1770: /** 1771: * Natural log and square root constants, for calculation of 1772: * {@link #exp(double)}, {@link #log(double)} and 1773: * {@link #pow(double, double)}. CP is 2/(3*ln(2)). 1774: */ 1775: private static final double 1776: SQRT_1_5 = 1.224744871391589, // Long bits 0x3ff3988e1409212eL. 1777: SQRT_2 = 1.4142135623730951, // Long bits 0x3ff6a09e667f3bcdL. 1778: SQRT_3 = 1.7320508075688772, // Long bits 0x3ffbb67ae8584caaL. 1779: EXP_LIMIT_H = 709.782712893384, // Long bits 0x40862e42fefa39efL. 1780: EXP_LIMIT_L = -745.1332191019411, // Long bits 0xc0874910d52d3051L. 1781: CP = 0.9617966939259756, // Long bits 0x3feec709dc3a03fdL. 1782: CP_H = 0.9617967009544373, // Long bits 0x3feec709e0000000L. 1783: CP_L = -7.028461650952758e-9, // Long bits 0xbe3e2fe0145b01f5L. 1784: LN2 = 0.6931471805599453, // Long bits 0x3fe62e42fefa39efL. 1785: LN2_H = 0.6931471803691238, // Long bits 0x3fe62e42fee00000L. 1786: LN2_L = 1.9082149292705877e-10, // Long bits 0x3dea39ef35793c76L. 1787: INV_LN2 = 1.4426950408889634, // Long bits 0x3ff71547652b82feL. 1788: INV_LN2_H = 1.4426950216293335, // Long bits 0x3ff7154760000000L. 1789: INV_LN2_L = 1.9259629911266175e-8; // Long bits 0x3e54ae0bf85ddf44L. 1790: 1791: /** 1792: * Constants for computing {@link #log(double)}. 1793: */ 1794: private static final double 1795: LG1 = 0.6666666666666735, // Long bits 0x3fe5555555555593L. 1796: LG2 = 0.3999999999940942, // Long bits 0x3fd999999997fa04L. 1797: LG3 = 0.2857142874366239, // Long bits 0x3fd2492494229359L. 1798: LG4 = 0.22222198432149784, // Long bits 0x3fcc71c51d8e78afL. 1799: LG5 = 0.1818357216161805, // Long bits 0x3fc7466496cb03deL. 1800: LG6 = 0.15313837699209373, // Long bits 0x3fc39a09d078c69fL. 1801: LG7 = 0.14798198605116586; // Long bits 0x3fc2f112df3e5244L. 1802: 1803: /** 1804: * Constants for computing {@link #pow(double, double)}. L and P are 1805: * coefficients for series; OVT is -(1024-log2(ovfl+.5ulp)); and DP is ???. 1806: * The P coefficients also calculate {@link #exp(double)}. 1807: */ 1808: private static final double 1809: L1 = 0.5999999999999946, // Long bits 0x3fe3333333333303L. 1810: L2 = 0.4285714285785502, // Long bits 0x3fdb6db6db6fabffL. 1811: L3 = 0.33333332981837743, // Long bits 0x3fd55555518f264dL. 1812: L4 = 0.272728123808534, // Long bits 0x3fd17460a91d4101L. 1813: L5 = 0.23066074577556175, // Long bits 0x3fcd864a93c9db65L. 1814: L6 = 0.20697501780033842, // Long bits 0x3fca7e284a454eefL. 1815: P1 = 0.16666666666666602, // Long bits 0x3fc555555555553eL. 1816: P2 = -2.7777777777015593e-3, // Long bits 0xbf66c16c16bebd93L. 1817: P3 = 6.613756321437934e-5, // Long bits 0x3f11566aaf25de2cL. 1818: P4 = -1.6533902205465252e-6, // Long bits 0xbebbbd41c5d26bf1L. 1819: P5 = 4.1381367970572385e-8, // Long bits 0x3e66376972bea4d0L. 1820: DP_H = 0.5849624872207642, // Long bits 0x3fe2b80340000000L. 1821: DP_L = 1.350039202129749e-8, // Long bits 0x3e4cfdeb43cfd006L. 1822: OVT = 8.008566259537294e-17; // Long bits 0x3c971547652b82feL. 1823: 1824: /** 1825: * Coefficients for computing {@link #sin(double)}. 1826: */ 1827: private static final double 1828: S1 = -0.16666666666666632, // Long bits 0xbfc5555555555549L. 1829: S2 = 8.33333333332249e-3, // Long bits 0x3f8111111110f8a6L. 1830: S3 = -1.984126982985795e-4, // Long bits 0xbf2a01a019c161d5L. 1831: S4 = 2.7557313707070068e-6, // Long bits 0x3ec71de357b1fe7dL. 1832: S5 = -2.5050760253406863e-8, // Long bits 0xbe5ae5e68a2b9cebL. 1833: S6 = 1.58969099521155e-10; // Long bits 0x3de5d93a5acfd57cL. 1834: 1835: /** 1836: * Coefficients for computing {@link #cos(double)}. 1837: */ 1838: private static final double 1839: C1 = 0.0416666666666666, // Long bits 0x3fa555555555554cL. 1840: C2 = -1.388888888887411e-3, // Long bits 0xbf56c16c16c15177L. 1841: C3 = 2.480158728947673e-5, // Long bits 0x3efa01a019cb1590L. 1842: C4 = -2.7557314351390663e-7, // Long bits 0xbe927e4f809c52adL. 1843: C5 = 2.087572321298175e-9, // Long bits 0x3e21ee9ebdb4b1c4L. 1844: C6 = -1.1359647557788195e-11; // Long bits 0xbda8fae9be8838d4L. 1845: 1846: /** 1847: * Coefficients for computing {@link #tan(double)}. 1848: */ 1849: private static final double 1850: T0 = 0.3333333333333341, // Long bits 0x3fd5555555555563L. 1851: T1 = 0.13333333333320124, // Long bits 0x3fc111111110fe7aL. 1852: T2 = 0.05396825397622605, // Long bits 0x3faba1ba1bb341feL. 1853: T3 = 0.021869488294859542, // Long bits 0x3f9664f48406d637L. 1854: T4 = 8.8632398235993e-3, // Long bits 0x3f8226e3e96e8493L. 1855: T5 = 3.5920791075913124e-3, // Long bits 0x3f6d6d22c9560328L. 1856: T6 = 1.4562094543252903e-3, // Long bits 0x3f57dbc8fee08315L. 1857: T7 = 5.880412408202641e-4, // Long bits 0x3f4344d8f2f26501L. 1858: T8 = 2.464631348184699e-4, // Long bits 0x3f3026f71a8d1068L. 1859: T9 = 7.817944429395571e-5, // Long bits 0x3f147e88a03792a6L. 1860: T10 = 7.140724913826082e-5, // Long bits 0x3f12b80f32f0a7e9L. 1861: T11 = -1.8558637485527546e-5, // Long bits 0xbef375cbdb605373L. 1862: T12 = 2.590730518636337e-5; // Long bits 0x3efb2a7074bf7ad4L. 1863: 1864: /** 1865: * Coefficients for computing {@link #asin(double)} and 1866: * {@link #acos(double)}. 1867: */ 1868: private static final double 1869: PS0 = 0.16666666666666666, // Long bits 0x3fc5555555555555L. 1870: PS1 = -0.3255658186224009, // Long bits 0xbfd4d61203eb6f7dL. 1871: PS2 = 0.20121253213486293, // Long bits 0x3fc9c1550e884455L. 1872: PS3 = -0.04005553450067941, // Long bits 0xbfa48228b5688f3bL. 1873: PS4 = 7.915349942898145e-4, // Long bits 0x3f49efe07501b288L. 1874: PS5 = 3.479331075960212e-5, // Long bits 0x3f023de10dfdf709L. 1875: QS1 = -2.403394911734414, // Long bits 0xc0033a271c8a2d4bL. 1876: QS2 = 2.0209457602335057, // Long bits 0x40002ae59c598ac8L. 1877: QS3 = -0.6882839716054533, // Long bits 0xbfe6066c1b8d0159L. 1878: QS4 = 0.07703815055590194; // Long bits 0x3fb3b8c5b12e9282L. 1879: 1880: /** 1881: * Coefficients for computing {@link #atan(double)}. 1882: */ 1883: private static final double 1884: ATAN_0_5H = 0.4636476090008061, // Long bits 0x3fddac670561bb4fL. 1885: ATAN_0_5L = 2.2698777452961687e-17, // Long bits 0x3c7a2b7f222f65e2L. 1886: ATAN_1_5H = 0.982793723247329, // Long bits 0x3fef730bd281f69bL. 1887: ATAN_1_5L = 1.3903311031230998e-17, // Long bits 0x3c7007887af0cbbdL. 1888: AT0 = 0.3333333333333293, // Long bits 0x3fd555555555550dL. 1889: AT1 = -0.19999999999876483, // Long bits 0xbfc999999998ebc4L. 1890: AT2 = 0.14285714272503466, // Long bits 0x3fc24924920083ffL. 1891: AT3 = -0.11111110405462356, // Long bits 0xbfbc71c6fe231671L. 1892: AT4 = 0.09090887133436507, // Long bits 0x3fb745cdc54c206eL. 1893: AT5 = -0.0769187620504483, // Long bits 0xbfb3b0f2af749a6dL. 1894: AT6 = 0.06661073137387531, // Long bits 0x3fb10d66a0d03d51L. 1895: AT7 = -0.058335701337905735, // Long bits 0xbfadde2d52defd9aL. 1896: AT8 = 0.049768779946159324, // Long bits 0x3fa97b4b24760debL. 1897: AT9 = -0.036531572744216916, // Long bits 0xbfa2b4442c6a6c2fL. 1898: AT10 = 0.016285820115365782; // Long bits 0x3f90ad3ae322da11L. 1899: 1900: /** 1901: * Constants for computing {@link #cbrt(double)}. 1902: */ 1903: private static final int 1904: CBRT_B1 = 715094163, // B1 = (682-0.03306235651)*2**20 1905: CBRT_B2 = 696219795; // B2 = (664-0.03306235651)*2**20 1906: 1907: /** 1908: * Constants for computing {@link #cbrt(double)}. 1909: */ 1910: private static final double 1911: CBRT_C = 5.42857142857142815906e-01, // Long bits 0x3fe15f15f15f15f1L 1912: CBRT_D = -7.05306122448979611050e-01, // Long bits 0xbfe691de2532c834L 1913: CBRT_E = 1.41428571428571436819e+00, // Long bits 0x3ff6a0ea0ea0ea0fL 1914: CBRT_F = 1.60714285714285720630e+00, // Long bits 0x3ff9b6db6db6db6eL 1915: CBRT_G = 3.57142857142857150787e-01; // Long bits 0x3fd6db6db6db6db7L 1916: 1917: /** 1918: * Constants for computing {@link #expm1(double)} 1919: */ 1920: private static final double 1921: EXPM1_Q1 = -3.33333333333331316428e-02, // Long bits 0xbfa11111111110f4L 1922: EXPM1_Q2 = 1.58730158725481460165e-03, // Long bits 0x3f5a01a019fe5585L 1923: EXPM1_Q3 = -7.93650757867487942473e-05, // Long bits 0xbf14ce199eaadbb7L 1924: EXPM1_Q4 = 4.00821782732936239552e-06, // Long bits 0x3ed0cfca86e65239L 1925: EXPM1_Q5 = -2.01099218183624371326e-07; // Long bits 0xbe8afdb76e09c32dL 1926: 1927: /** 1928: * Helper function for reducing an angle to a multiple of pi/2 within 1929: * [-pi/4, pi/4]. 1930: * 1931: * @param x the angle; not infinity or NaN, and outside pi/4 1932: * @param y an array of 2 doubles modified to hold the remander x % pi/2 1933: * @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4], 1934: * 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4] 1935: */ 1936: private static int remPiOver2(double x, double[] y) 1937: { 1938: boolean negative = x < 0; 1939: x = abs(x); 1940: double z; 1941: int n; 1942: if (Configuration.DEBUG && (x <= PI / 4 || x != x 1943: || x == Double.POSITIVE_INFINITY)) 1944: throw new InternalError("Assertion failure"); 1945: if (x < 3 * PI / 4) // If |x| is small. 1946: { 1947: z = x - PIO2_1; 1948: if ((float) x != (float) (PI / 2)) // 33+53 bit pi is good enough. 1949: { 1950: y[0] = z - PIO2_1L; 1951: y[1] = z - y[0] - PIO2_1L; 1952: } 1953: else // Near pi/2, use 33+33+53 bit pi. 1954: { 1955: z -= PIO2_2; 1956: y[0] = z - PIO2_2L; 1957: y[1] = z - y[0] - PIO2_2L; 1958: } 1959: n = 1; 1960: } 1961: else if (x <= TWO_20 * PI / 2) // Medium size. 1962: { 1963: n = (int) (2 / PI * x + 0.5); 1964: z = x - n * PIO2_1; 1965: double w = n * PIO2_1L; // First round good to 85 bits. 1966: y[0] = z - w; 1967: if (n >= 32 || (float) x == (float) (w)) 1968: { 1969: if (x / y[0] >= TWO_16) // Second iteration, good to 118 bits. 1970: { 1971: double t = z; 1972: w = n * PIO2_2; 1973: z = t - w; 1974: w = n * PIO2_2L - (t - z - w); 1975: y[0] = z - w; 1976: if (x / y[0] >= TWO_49) // Third iteration, 151 bits accuracy. 1977: { 1978: t = z; 1979: w = n * PIO2_3; 1980: z = t - w; 1981: w = n * PIO2_3L - (t - z - w); 1982: y[0] = z - w; 1983: } 1984: } 1985: } 1986: y[1] = z - y[0] - w; 1987: } 1988: else 1989: { 1990: // All other (large) arguments. 1991: int e0 = (int) (Double.doubleToLongBits(x) >> 52) - 1046; 1992: z = scale(x, -e0); // e0 = ilogb(z) - 23. 1993: double[] tx = new double[3]; 1994: for (int i = 0; i < 2; i++) 1995: { 1996: tx[i] = (int) z; 1997: z = (z - tx[i]) * TWO_24; 1998: } 1999: tx[2] = z; 2000: int nx = 2; 2001: while (tx[nx] == 0) 2002: nx--; 2003: n = remPiOver2(tx, y, e0, nx); 2004: } 2005: if (negative) 2006: { 2007: y[0] = -y[0]; 2008: y[1] = -y[1]; 2009: return -n; 2010: } 2011: return n; 2012: } 2013: 2014: /** 2015: * Helper function for reducing an angle to a multiple of pi/2 within 2016: * [-pi/4, pi/4]. 2017: * 2018: * @param x the positive angle, broken into 24-bit chunks 2019: * @param y an array of 2 doubles modified to hold the remander x % pi/2 2020: * @param e0 the exponent of x[0] 2021: * @param nx the last index used in x 2022: * @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4], 2023: * 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4] 2024: */ 2025: private static int remPiOver2(double[] x, double[] y, int e0, int nx) 2026: { 2027: int i; 2028: int ih; 2029: int n; 2030: double fw; 2031: double z; 2032: int[] iq = new int[20]; 2033: double[] f = new double[20]; 2034: double[] q = new double[20]; 2035: boolean recompute = false; 2036: 2037: // Initialize jk, jz, jv, q0; note that 3>q0. 2038: int jk = 4; 2039: int jz = jk; 2040: int jv = max((e0 - 3) / 24, 0); 2041: int q0 = e0 - 24 * (jv + 1); 2042: 2043: // Set up f[0] to f[nx+jk] where f[nx+jk] = TWO_OVER_PI[jv+jk]. 2044: int j = jv - nx; 2045: int m = nx + jk; 2046: for (i = 0; i <= m; i++, j++) 2047: f[i] = (j < 0) ? 0 : TWO_OVER_PI[j]; 2048: 2049: // Compute q[0],q[1],...q[jk]. 2050: for (i = 0; i <= jk; i++) 2051: { 2052: for (j = 0, fw = 0; j <= nx; j++) 2053: fw += x[j] * f[nx + i - j]; 2054: q[i] = fw; 2055: } 2056: 2057: do 2058: { 2059: // Distill q[] into iq[] reversingly. 2060: for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) 2061: { 2062: fw = (int) (1 / TWO_24 * z); 2063: iq[i] = (int) (z - TWO_24 * fw); 2064: z = q[j - 1] + fw; 2065: } 2066: 2067: // Compute n. 2068: z = scale(z, q0); 2069: z -= 8 * floor(z * 0.125); // Trim off integer >= 8. 2070: n = (int) z; 2071: z -= n; 2072: ih = 0; 2073: if (q0 > 0) // Need iq[jz-1] to determine n. 2074: { 2075: i = iq[jz - 1] >> (24 - q0); 2076: n += i; 2077: iq[jz - 1] -= i << (24 - q0); 2078: ih = iq[jz - 1] >> (23 - q0); 2079: } 2080: else if (q0 == 0) 2081: ih = iq[jz - 1] >> 23; 2082: else if (z >= 0.5) 2083: ih = 2; 2084: 2085: if (ih > 0) // If q > 0.5. 2086: { 2087: n += 1; 2088: int carry = 0; 2089: for (i = 0; i < jz; i++) // Compute 1-q. 2090: { 2091: j = iq[i]; 2092: if (carry == 0) 2093: { 2094: if (j != 0) 2095: { 2096: carry = 1; 2097: iq[i] = 0x1000000 - j; 2098: } 2099: } 2100: else 2101: iq[i] = 0xffffff - j; 2102: } 2103: switch (q0) 2104: { 2105: case 1: // Rare case: chance is 1 in 12 for non-default. 2106: iq[jz - 1] &= 0x7fffff; 2107: break; 2108: case 2: 2109: iq[jz - 1] &= 0x3fffff; 2110: } 2111: if (ih == 2) 2112: { 2113: z = 1 - z; 2114: if (carry != 0) 2115: z -= scale(1, q0); 2116: } 2117: } 2118: 2119: // Check if recomputation is needed. 2120: if (z == 0) 2121: { 2122: j = 0; 2123: for (i = jz - 1; i >= jk; i--) 2124: j |= iq[i]; 2125: if (j == 0) // Need recomputation. 2126: { 2127: int k; 2128: for (k = 1; iq[jk - k] == 0; k++); // k = no. of terms needed. 2129: 2130: for (i = jz + 1; i <= jz + k; i++) // Add q[jz+1] to q[jz+k]. 2131: { 2132: f[nx + i] = TWO_OVER_PI[jv + i]; 2133: for (j = 0, fw = 0; j <= nx; j++) 2134: fw += x[j] * f[nx + i - j]; 2135: q[i] = fw; 2136: } 2137: jz += k; 2138: recompute = true; 2139: } 2140: } 2141: } 2142: while (recompute); 2143: 2144: // Chop off zero terms. 2145: if (z == 0) 2146: { 2147: jz--; 2148: q0 -= 24; 2149: while (iq[jz] == 0) 2150: { 2151: jz--; 2152: q0 -= 24; 2153: } 2154: } 2155: else // Break z into 24-bit if necessary. 2156: { 2157: z = scale(z, -q0); 2158: if (z >= TWO_24) 2159: { 2160: fw = (int) (1 / TWO_24 * z); 2161: iq[jz] = (int) (z - TWO_24 * fw); 2162: jz++; 2163: q0 += 24; 2164: iq[jz] = (int) fw; 2165: } 2166: else 2167: iq[jz] = (int) z; 2168: } 2169: 2170: // Convert integer "bit" chunk to floating-point value. 2171: fw = scale(1, q0); 2172: for (i = jz; i >= 0; i--) 2173: { 2174: q[i] = fw * iq[i]; 2175: fw *= 1 / TWO_24; 2176: } 2177: 2178: // Compute PI_OVER_TWO[0,...,jk]*q[jz,...,0]. 2179: double[] fq = new double[20]; 2180: for (i = jz; i >= 0; i--) 2181: { 2182: fw = 0; 2183: for (int k = 0; k <= jk && k <= jz - i; k++) 2184: fw += PI_OVER_TWO[k] * q[i + k]; 2185: fq[jz - i] = fw; 2186: } 2187: 2188: // Compress fq[] into y[]. 2189: fw = 0; 2190: for (i = jz; i >= 0; i--) 2191: fw += fq[i]; 2192: y[0] = (ih == 0) ? fw : -fw; 2193: fw = fq[0] - fw; 2194: for (i = 1; i <= jz; i++) 2195: fw += fq[i]; 2196: y[1] = (ih == 0) ? fw : -fw; 2197: return n; 2198: } 2199: 2200: /** 2201: * Helper method for scaling a double by a power of 2. 2202: * 2203: * @param x the double 2204: * @param n the scale; |n| < 2048 2205: * @return x * 2**n 2206: */ 2207: private static double scale(double x, int n) 2208: { 2209: if (Configuration.DEBUG && abs(n) >= 2048) 2210: throw new InternalError("Assertion failure"); 2211: if (x == 0 || x == Double.NEGATIVE_INFINITY 2212: || ! (x < Double.POSITIVE_INFINITY) || n == 0) 2213: return x; 2214: long bits = Double.doubleToLongBits(x); 2215: int exp = (int) (bits >> 52) & 0x7ff; 2216: if (exp == 0) // Subnormal x. 2217: { 2218: x *= TWO_54; 2219: exp = ((int) (Double.doubleToLongBits(x) >> 52) & 0x7ff) - 54; 2220: } 2221: exp += n; 2222: if (exp > 0x7fe) // Overflow. 2223: return Double.POSITIVE_INFINITY * x; 2224: if (exp > 0) // Normal. 2225: return Double.longBitsToDouble((bits & 0x800fffffffffffffL) 2226: | ((long) exp << 52)); 2227: if (exp <= -54) 2228: return 0 * x; // Underflow. 2229: exp += 54; // Subnormal result. 2230: x = Double.longBitsToDouble((bits & 0x800fffffffffffffL) 2231: | ((long) exp << 52)); 2232: return x * (1 / TWO_54); 2233: } 2234: 2235: /** 2236: * Helper trig function; computes sin in range [-pi/4, pi/4]. 2237: * 2238: * @param x angle within about pi/4 2239: * @param y tail of x, created by remPiOver2 2240: * @return sin(x+y) 2241: */ 2242: private static double sin(double x, double y) 2243: { 2244: if (Configuration.DEBUG && abs(x + y) > 0.7854) 2245: throw new InternalError("Assertion failure"); 2246: if (abs(x) < 1 / TWO_27) 2247: return x; // If |x| ~< 2**-27, already know answer. 2248: 2249: double z = x * x; 2250: double v = z * x; 2251: double r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); 2252: if (y == 0) 2253: return x + v * (S1 + z * r); 2254: return x - ((z * (0.5 * y - v * r) - y) - v * S1); 2255: } 2256: 2257: /** 2258: * Helper trig function; computes cos in range [-pi/4, pi/4]. 2259: * 2260: * @param x angle within about pi/4 2261: * @param y tail of x, created by remPiOver2 2262: * @return cos(x+y) 2263: */ 2264: private static double cos(double x, double y) 2265: { 2266: if (Configuration.DEBUG && abs(x + y) > 0.7854) 2267: throw new InternalError("Assertion failure"); 2268: x = abs(x); 2269: if (x < 1 / TWO_27) 2270: return 1; // If |x| ~< 2**-27, already know answer. 2271: 2272: double z = x * x; 2273: double r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); 2274: 2275: if (x < 0.3) 2276: return 1 - (0.5 * z - (z * r - x * y)); 2277: 2278: double qx = (x > 0.78125) ? 0.28125 : (x * 0.25); 2279: return 1 - qx - ((0.5 * z - qx) - (z * r - x * y)); 2280: } 2281: 2282: /** 2283: * Helper trig function; computes tan in range [-pi/4, pi/4]. 2284: * 2285: * @param x angle within about pi/4 2286: * @param y tail of x, created by remPiOver2 2287: * @param invert true iff -1/tan should be returned instead 2288: * @return tan(x+y) 2289: */ 2290: private static double tan(double x, double y, boolean invert) 2291: { 2292: // PI/2 is irrational, so no double is a perfect multiple of it. 2293: if (Configuration.DEBUG && (abs(x + y) > 0.7854 || (x == 0 && invert))) 2294: throw new InternalError("Assertion failure"); 2295: boolean negative = x < 0; 2296: if (negative) 2297: { 2298: x = -x; 2299: y = -y; 2300: } 2301: if (x < 1 / TWO_28) // If |x| ~< 2**-28, already know answer. 2302: return (negative ? -1 : 1) * (invert ? -1 / x : x); 2303: 2304: double z; 2305: double w; 2306: boolean large = x >= 0.6744; 2307: if (large) 2308: { 2309: z = PI / 4 - x; 2310: w = PI_L / 4 - y; 2311: x = z + w; 2312: y = 0; 2313: } 2314: z = x * x; 2315: w = z * z; 2316: // Break x**5*(T1+x**2*T2+...) into 2317: // x**5(T1+x**4*T3+...+x**20*T11) 2318: // + x**5(x**2*(T2+x**4*T4+...+x**22*T12)). 2319: double r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11)))); 2320: double v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12))))); 2321: double s = z * x; 2322: r = y + z * (s * (r + v) + y); 2323: r += T0 * s; 2324: w = x + r; 2325: if (large) 2326: { 2327: v = invert ? -1 : 1; 2328: return (negative ? -1 : 1) * (v - 2 * (x - (w * w / (w + v) - r))); 2329: } 2330: if (! invert) 2331: return w; 2332: 2333: // Compute -1.0/(x+r) accurately. 2334: z = (float) w; 2335: v = r - (z - x); 2336: double a = -1 / w; 2337: double t = (float) a; 2338: return t + a * (1 + t * z + t * v); 2339: } 2340: 2341: /** 2342: * <p> 2343: * Returns the sign of the argument as follows: 2344: * </p> 2345: * <ul> 2346: * <li>If <code>a</code> is greater than zero, the result is 1.0.</li> 2347: * <li>If <code>a</code> is less than zero, the result is -1.0.</li> 2348: * <li>If <code>a</code> is <code>NaN</code>, the result is <code>NaN</code>. 2349: * <li>If <code>a</code> is positive or negative zero, the result is the 2350: * same.</li> 2351: * </ul> 2352: * 2353: * @param a the numeric argument. 2354: * @return the sign of the argument. 2355: * @since 1.5. 2356: */ 2357: public static double signum(double a) 2358: { 2359: // There's no difference. 2360: return Math.signum(a); 2361: } 2362: 2363: /** 2364: * <p> 2365: * Returns the sign of the argument as follows: 2366: * </p> 2367: * <ul> 2368: * <li>If <code>a</code> is greater than zero, the result is 1.0f.</li> 2369: * <li>If <code>a</code> is less than zero, the result is -1.0f.</li> 2370: * <li>If <code>a</code> is <code>NaN</code>, the result is <code>NaN</code>. 2371: * <li>If <code>a</code> is positive or negative zero, the result is the 2372: * same.</li> 2373: * </ul> 2374: * 2375: * @param a the numeric argument. 2376: * @return the sign of the argument. 2377: * @since 1.5. 2378: */ 2379: public static float signum(float a) 2380: { 2381: // There's no difference. 2382: return Math.signum(a); 2383: } 2384: 2385: /** 2386: * Return the ulp for the given double argument. The ulp is the 2387: * difference between the argument and the next larger double. Note 2388: * that the sign of the double argument is ignored, that is, 2389: * ulp(x) == ulp(-x). If the argument is a NaN, then NaN is returned. 2390: * If the argument is an infinity, then +Inf is returned. If the 2391: * argument is zero (either positive or negative), then 2392: * {@link Double#MIN_VALUE} is returned. 2393: * @param d the double whose ulp should be returned 2394: * @return the difference between the argument and the next larger double 2395: * @since 1.5 2396: */ 2397: public static double ulp(double d) 2398: { 2399: // There's no difference. 2400: return Math.ulp(d); 2401: } 2402: 2403: /** 2404: * Return the ulp for the given float argument. The ulp is the 2405: * difference between the argument and the next larger float. Note 2406: * that the sign of the float argument is ignored, that is, 2407: * ulp(x) == ulp(-x). If the argument is a NaN, then NaN is returned. 2408: * If the argument is an infinity, then +Inf is returned. If the 2409: * argument is zero (either positive or negative), then 2410: * {@link Float#MIN_VALUE} is returned. 2411: * @param f the float whose ulp should be returned 2412: * @return the difference between the argument and the next larger float 2413: * @since 1.5 2414: */ 2415: public static float ulp(float f) 2416: { 2417: // There's no difference. 2418: return Math.ulp(f); 2419: } 2420: }