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1: /* java.lang.StrictMath -- common mathematical functions, strict Java 2: Copyright (C) 1998, 2001, 2002, 2003 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: * Take <em>e</em><sup>a</sup>. The opposite of <code>log()</code>. If the 637: * argument is NaN, the result is NaN; if the argument is positive infinity, 638: * the result is positive infinity; and if the argument is negative 639: * infinity, the result is positive zero. 640: * 641: * @param x the number to raise to the power 642: * @return the number raised to the power of <em>e</em> 643: * @see #log(double) 644: * @see #pow(double, double) 645: */ 646: public static double exp(double x) 647: { 648: if (x != x) 649: return x; 650: if (x > EXP_LIMIT_H) 651: return Double.POSITIVE_INFINITY; 652: if (x < EXP_LIMIT_L) 653: return 0; 654: 655: // Argument reduction. 656: double hi; 657: double lo; 658: int k; 659: double t = abs(x); 660: if (t > 0.5 * LN2) 661: { 662: if (t < 1.5 * LN2) 663: { 664: hi = t - LN2_H; 665: lo = LN2_L; 666: k = 1; 667: } 668: else 669: { 670: k = (int) (INV_LN2 * t + 0.5); 671: hi = t - k * LN2_H; 672: lo = k * LN2_L; 673: } 674: if (x < 0) 675: { 676: hi = -hi; 677: lo = -lo; 678: k = -k; 679: } 680: x = hi - lo; 681: } 682: else if (t < 1 / TWO_28) 683: return 1; 684: else 685: lo = hi = k = 0; 686: 687: // Now x is in primary range. 688: t = x * x; 689: double c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 690: if (k == 0) 691: return 1 - (x * c / (c - 2) - x); 692: double y = 1 - (lo - x * c / (2 - c) - hi); 693: return scale(y, k); 694: } 695: 696: /** 697: * Take ln(a) (the natural log). The opposite of <code>exp()</code>. If the 698: * argument is NaN or negative, the result is NaN; if the argument is 699: * positive infinity, the result is positive infinity; and if the argument 700: * is either zero, the result is negative infinity. 701: * 702: * <p>Note that the way to get log<sub>b</sub>(a) is to do this: 703: * <code>ln(a) / ln(b)</code>. 704: * 705: * @param x the number to take the natural log of 706: * @return the natural log of <code>a</code> 707: * @see #exp(double) 708: */ 709: public static double log(double x) 710: { 711: if (x == 0) 712: return Double.NEGATIVE_INFINITY; 713: if (x < 0) 714: return Double.NaN; 715: if (! (x < Double.POSITIVE_INFINITY)) 716: return x; 717: 718: // Normalize x. 719: long bits = Double.doubleToLongBits(x); 720: int exp = (int) (bits >> 52); 721: if (exp == 0) // Subnormal x. 722: { 723: x *= TWO_54; 724: bits = Double.doubleToLongBits(x); 725: exp = (int) (bits >> 52) - 54; 726: } 727: exp -= 1023; // Unbias exponent. 728: bits = (bits & 0x000fffffffffffffL) | 0x3ff0000000000000L; 729: x = Double.longBitsToDouble(bits); 730: if (x >= SQRT_2) 731: { 732: x *= 0.5; 733: exp++; 734: } 735: x--; 736: if (abs(x) < 1 / TWO_20) 737: { 738: if (x == 0) 739: return exp * LN2_H + exp * LN2_L; 740: double r = x * x * (0.5 - 1 / 3.0 * x); 741: if (exp == 0) 742: return x - r; 743: return exp * LN2_H - ((r - exp * LN2_L) - x); 744: } 745: double s = x / (2 + x); 746: double z = s * s; 747: double w = z * z; 748: double t1 = w * (LG2 + w * (LG4 + w * LG6)); 749: double t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7))); 750: double r = t2 + t1; 751: if (bits >= 0x3ff6174a00000000L && bits < 0x3ff6b85200000000L) 752: { 753: double h = 0.5 * x * x; // Need more accuracy for x near sqrt(2). 754: if (exp == 0) 755: return x - (h - s * (h + r)); 756: return exp * LN2_H - ((h - (s * (h + r) + exp * LN2_L)) - x); 757: } 758: if (exp == 0) 759: return x - s * (x - r); 760: return exp * LN2_H - ((s * (x - r) - exp * LN2_L) - x); 761: } 762: 763: /** 764: * Take a square root. If the argument is NaN or negative, the result is 765: * NaN; if the argument is positive infinity, the result is positive 766: * infinity; and if the result is either zero, the result is the same. 767: * 768: * <p>For other roots, use pow(x, 1/rootNumber). 769: * 770: * @param x the numeric argument 771: * @return the square root of the argument 772: * @see #pow(double, double) 773: */ 774: public static double sqrt(double x) 775: { 776: if (x < 0) 777: return Double.NaN; 778: if (x == 0 || ! (x < Double.POSITIVE_INFINITY)) 779: return x; 780: 781: // Normalize x. 782: long bits = Double.doubleToLongBits(x); 783: int exp = (int) (bits >> 52); 784: if (exp == 0) // Subnormal x. 785: { 786: x *= TWO_54; 787: bits = Double.doubleToLongBits(x); 788: exp = (int) (bits >> 52) - 54; 789: } 790: exp -= 1023; // Unbias exponent. 791: bits = (bits & 0x000fffffffffffffL) | 0x0010000000000000L; 792: if ((exp & 1) == 1) // Odd exp, double x to make it even. 793: bits <<= 1; 794: exp >>= 1; 795: 796: // Generate sqrt(x) bit by bit. 797: bits <<= 1; 798: long q = 0; 799: long s = 0; 800: long r = 0x0020000000000000L; // Move r right to left. 801: while (r != 0) 802: { 803: long t = s + r; 804: if (t <= bits) 805: { 806: s = t + r; 807: bits -= t; 808: q += r; 809: } 810: bits <<= 1; 811: r >>= 1; 812: } 813: 814: // Use floating add to round correctly. 815: if (bits != 0) 816: q += q & 1; 817: return Double.longBitsToDouble((q >> 1) + ((exp + 1022L) << 52)); 818: } 819: 820: /** 821: * Raise a number to a power. Special cases:<ul> 822: * <li>If the second argument is positive or negative zero, then the result 823: * is 1.0.</li> 824: * <li>If the second argument is 1.0, then the result is the same as the 825: * first argument.</li> 826: * <li>If the second argument is NaN, then the result is NaN.</li> 827: * <li>If the first argument is NaN and the second argument is nonzero, 828: * then the result is NaN.</li> 829: * <li>If the absolute value of the first argument is greater than 1 and 830: * the second argument is positive infinity, or the absolute value of the 831: * first argument is less than 1 and the second argument is negative 832: * infinity, then the result is positive infinity.</li> 833: * <li>If the absolute value of the first argument is greater than 1 and 834: * the second argument is negative infinity, or the absolute value of the 835: * first argument is less than 1 and the second argument is positive 836: * infinity, then the result is positive zero.</li> 837: * <li>If the absolute value of the first argument equals 1 and the second 838: * argument is infinite, then the result is NaN.</li> 839: * <li>If the first argument is positive zero and the second argument is 840: * greater than zero, or the first argument is positive infinity and the 841: * second argument is less than zero, then the result is positive zero.</li> 842: * <li>If the first argument is positive zero and the second argument is 843: * less than zero, or the first argument is positive infinity and the 844: * second argument is greater than zero, then the result is positive 845: * infinity.</li> 846: * <li>If the first argument is negative zero and the second argument is 847: * greater than zero but not a finite odd integer, or the first argument is 848: * negative infinity and the second argument is less than zero but not a 849: * finite odd integer, then the result is positive zero.</li> 850: * <li>If the first argument is negative zero and the second argument is a 851: * positive finite odd integer, or the first argument is negative infinity 852: * and the second argument is a negative finite odd integer, then the result 853: * is negative zero.</li> 854: * <li>If the first argument is negative zero and the second argument is 855: * less than zero but not a finite odd integer, or the first argument is 856: * negative infinity and the second argument is greater than zero but not a 857: * finite odd integer, then the result is positive infinity.</li> 858: * <li>If the first argument is negative zero and the second argument is a 859: * negative finite odd integer, or the first argument is negative infinity 860: * and the second argument is a positive finite odd integer, then the result 861: * is negative infinity.</li> 862: * <li>If the first argument is less than zero and the second argument is a 863: * finite even integer, then the result is equal to the result of raising 864: * the absolute value of the first argument to the power of the second 865: * argument.</li> 866: * <li>If the first argument is less than zero and the second argument is a 867: * finite odd integer, then the result is equal to the negative of the 868: * result of raising the absolute value of the first argument to the power 869: * of the second argument.</li> 870: * <li>If the first argument is finite and less than zero and the second 871: * argument is finite and not an integer, then the result is NaN.</li> 872: * <li>If both arguments are integers, then the result is exactly equal to 873: * the mathematical result of raising the first argument to the power of 874: * the second argument if that result can in fact be represented exactly as 875: * a double value.</li> 876: * 877: * </ul><p>(In the foregoing descriptions, a floating-point value is 878: * considered to be an integer if and only if it is a fixed point of the 879: * method {@link #ceil(double)} or, equivalently, a fixed point of the 880: * method {@link #floor(double)}. A value is a fixed point of a one-argument 881: * method if and only if the result of applying the method to the value is 882: * equal to the value.) 883: * 884: * @param x the number to raise 885: * @param y the power to raise it to 886: * @return x<sup>y</sup> 887: */ 888: public static double pow(double x, double y) 889: { 890: // Special cases first. 891: if (y == 0) 892: return 1; 893: if (y == 1) 894: return x; 895: if (y == -1) 896: return 1 / x; 897: if (x != x || y != y) 898: return Double.NaN; 899: 900: // When x < 0, yisint tells if y is not an integer (0), even(1), 901: // or odd (2). 902: int yisint = 0; 903: if (x < 0 && floor(y) == y) 904: yisint = (y % 2 == 0) ? 2 : 1; 905: double ax = abs(x); 906: double ay = abs(y); 907: 908: // More special cases, of y. 909: if (ay == Double.POSITIVE_INFINITY) 910: { 911: if (ax == 1) 912: return Double.NaN; 913: if (ax > 1) 914: return y > 0 ? y : 0; 915: return y < 0 ? -y : 0; 916: } 917: if (y == 2) 918: return x * x; 919: if (y == 0.5) 920: return sqrt(x); 921: 922: // More special cases, of x. 923: if (x == 0 || ax == Double.POSITIVE_INFINITY || ax == 1) 924: { 925: if (y < 0) 926: ax = 1 / ax; 927: if (x < 0) 928: { 929: if (x == -1 && yisint == 0) 930: ax = Double.NaN; 931: else if (yisint == 1) 932: ax = -ax; 933: } 934: return ax; 935: } 936: if (x < 0 && yisint == 0) 937: return Double.NaN; 938: 939: // Now we can start! 940: double t; 941: double t1; 942: double t2; 943: double u; 944: double v; 945: double w; 946: if (ay > TWO_31) 947: { 948: if (ay > TWO_64) // Automatic over/underflow. 949: return ((ax < 1) ? y < 0 : y > 0) ? Double.POSITIVE_INFINITY : 0; 950: // Over/underflow if x is not close to one. 951: if (ax < 0.9999995231628418) 952: return y < 0 ? Double.POSITIVE_INFINITY : 0; 953: if (ax >= 1.0000009536743164) 954: return y > 0 ? Double.POSITIVE_INFINITY : 0; 955: // Now |1-x| is <= 2**-20, sufficient to compute 956: // log(x) by x-x^2/2+x^3/3-x^4/4. 957: t = x - 1; 958: w = t * t * (0.5 - t * (1 / 3.0 - t * 0.25)); 959: u = INV_LN2_H * t; 960: v = t * INV_LN2_L - w * INV_LN2; 961: t1 = (float) (u + v); 962: t2 = v - (t1 - u); 963: } 964: else 965: { 966: long bits = Double.doubleToLongBits(ax); 967: int exp = (int) (bits >> 52); 968: if (exp == 0) // Subnormal x. 969: { 970: ax *= TWO_54; 971: bits = Double.doubleToLongBits(ax); 972: exp = (int) (bits >> 52) - 54; 973: } 974: exp -= 1023; // Unbias exponent. 975: ax = Double.longBitsToDouble((bits & 0x000fffffffffffffL) 976: | 0x3ff0000000000000L); 977: boolean k; 978: if (ax < SQRT_1_5) // |x|<sqrt(3/2). 979: k = false; 980: else if (ax < SQRT_3) // |x|<sqrt(3). 981: k = true; 982: else 983: { 984: k = false; 985: ax *= 0.5; 986: exp++; 987: } 988: 989: // Compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5). 990: u = ax - (k ? 1.5 : 1); 991: v = 1 / (ax + (k ? 1.5 : 1)); 992: double s = u * v; 993: double s_h = (float) s; 994: double t_h = (float) (ax + (k ? 1.5 : 1)); 995: double t_l = ax - (t_h - (k ? 1.5 : 1)); 996: double s_l = v * ((u - s_h * t_h) - s_h * t_l); 997: // Compute log(ax). 998: double s2 = s * s; 999: double r = s_l * (s_h + s) + s2 * s2 1000: * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 1001: s2 = s_h * s_h; 1002: t_h = (float) (3.0 + s2 + r); 1003: t_l = r - (t_h - 3.0 - s2); 1004: // u+v = s*(1+...). 1005: u = s_h * t_h; 1006: v = s_l * t_h + t_l * s; 1007: // 2/(3log2)*(s+...). 1008: double p_h = (float) (u + v); 1009: double p_l = v - (p_h - u); 1010: double z_h = CP_H * p_h; 1011: double z_l = CP_L * p_h + p_l * CP + (k ? DP_L : 0); 1012: // log2(ax) = (s+..)*2/(3*log2) = exp + dp_h + z_h + z_l. 1013: t = exp; 1014: t1 = (float) (z_h + z_l + (k ? DP_H : 0) + t); 1015: t2 = z_l - (t1 - t - (k ? DP_H : 0) - z_h); 1016: } 1017: 1018: // Split up y into y1+y2 and compute (y1+y2)*(t1+t2). 1019: boolean negative = x < 0 && yisint == 1; 1020: double y1 = (float) y; 1021: double p_l = (y - y1) * t1 + y * t2; 1022: double p_h = y1 * t1; 1023: double z = p_l + p_h; 1024: if (z >= 1024) // Detect overflow. 1025: { 1026: if (z > 1024 || p_l + OVT > z - p_h) 1027: return negative ? Double.NEGATIVE_INFINITY 1028: : Double.POSITIVE_INFINITY; 1029: } 1030: else if (z <= -1075) // Detect underflow. 1031: { 1032: if (z < -1075 || p_l <= z - p_h) 1033: return negative ? -0.0 : 0; 1034: } 1035: 1036: // Compute 2**(p_h+p_l). 1037: int n = round((float) z); 1038: p_h -= n; 1039: t = (float) (p_l + p_h); 1040: u = t * LN2_H; 1041: v = (p_l - (t - p_h)) * LN2 + t * LN2_L; 1042: z = u + v; 1043: w = v - (z - u); 1044: t = z * z; 1045: t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 1046: double r = (z * t1) / (t1 - 2) - (w + z * w); 1047: z = scale(1 - (r - z), n); 1048: return negative ? -z : z; 1049: } 1050: 1051: /** 1052: * Get the IEEE 754 floating point remainder on two numbers. This is the 1053: * value of <code>x - y * <em>n</em></code>, where <em>n</em> is the closest 1054: * double to <code>x / y</code> (ties go to the even n); for a zero 1055: * remainder, the sign is that of <code>x</code>. If either argument is NaN, 1056: * the first argument is infinite, or the second argument is zero, the result 1057: * is NaN; if x is finite but y is infinite, the result is x. 1058: * 1059: * @param x the dividend (the top half) 1060: * @param y the divisor (the bottom half) 1061: * @return the IEEE 754-defined floating point remainder of x/y 1062: * @see #rint(double) 1063: */ 1064: public static double IEEEremainder(double x, double y) 1065: { 1066: // Purge off exception values. 1067: if (x == Double.NEGATIVE_INFINITY || ! (x < Double.POSITIVE_INFINITY) 1068: || y == 0 || y != y) 1069: return Double.NaN; 1070: 1071: boolean negative = x < 0; 1072: x = abs(x); 1073: y = abs(y); 1074: if (x == y || x == 0) 1075: return 0 * x; // Get correct sign. 1076: 1077: // Achieve x < 2y, then take first shot at remainder. 1078: if (y < TWO_1023) 1079: x %= y + y; 1080: 1081: // Now adjust x to get correct precision. 1082: if (y < 4 / TWO_1023) 1083: { 1084: if (x + x > y) 1085: { 1086: x -= y; 1087: if (x + x >= y) 1088: x -= y; 1089: } 1090: } 1091: else 1092: { 1093: y *= 0.5; 1094: if (x > y) 1095: { 1096: x -= y; 1097: if (x >= y) 1098: x -= y; 1099: } 1100: } 1101: return negative ? -x : x; 1102: } 1103: 1104: /** 1105: * Take the nearest integer that is that is greater than or equal to the 1106: * argument. If the argument is NaN, infinite, or zero, the result is the 1107: * same; if the argument is between -1 and 0, the result is negative zero. 1108: * Note that <code>Math.ceil(x) == -Math.floor(-x)</code>. 1109: * 1110: * @param a the value to act upon 1111: * @return the nearest integer >= <code>a</code> 1112: */ 1113: public static double ceil(double a) 1114: { 1115: return -floor(-a); 1116: } 1117: 1118: /** 1119: * Take the nearest integer that is that is less than or equal to the 1120: * argument. If the argument is NaN, infinite, or zero, the result is the 1121: * same. Note that <code>Math.ceil(x) == -Math.floor(-x)</code>. 1122: * 1123: * @param a the value to act upon 1124: * @return the nearest integer <= <code>a</code> 1125: */ 1126: public static double floor(double a) 1127: { 1128: double x = abs(a); 1129: if (! (x < TWO_52) || (long) a == a) 1130: return a; // No fraction bits; includes NaN and infinity. 1131: if (x < 1) 1132: return a >= 0 ? 0 * a : -1; // Worry about signed zero. 1133: return a < 0 ? (long) a - 1.0 : (long) a; // Cast to long truncates. 1134: } 1135: 1136: /** 1137: * Take the nearest integer to the argument. If it is exactly between 1138: * two integers, the even integer is taken. If the argument is NaN, 1139: * infinite, or zero, the result is the same. 1140: * 1141: * @param a the value to act upon 1142: * @return the nearest integer to <code>a</code> 1143: */ 1144: public static double rint(double a) 1145: { 1146: double x = abs(a); 1147: if (! (x < TWO_52)) 1148: return a; // No fraction bits; includes NaN and infinity. 1149: if (x <= 0.5) 1150: return 0 * a; // Worry about signed zero. 1151: if (x % 2 <= 0.5) 1152: return (long) a; // Catch round down to even. 1153: return (long) (a + (a < 0 ? -0.5 : 0.5)); // Cast to long truncates. 1154: } 1155: 1156: /** 1157: * Take the nearest integer to the argument. This is equivalent to 1158: * <code>(int) Math.floor(f + 0.5f)</code>. If the argument is NaN, the 1159: * result is 0; otherwise if the argument is outside the range of int, the 1160: * result will be Integer.MIN_VALUE or Integer.MAX_VALUE, as appropriate. 1161: * 1162: * @param f the argument to round 1163: * @return the nearest integer to the argument 1164: * @see Integer#MIN_VALUE 1165: * @see Integer#MAX_VALUE 1166: */ 1167: public static int round(float f) 1168: { 1169: return (int) floor(f + 0.5f); 1170: } 1171: 1172: /** 1173: * Take the nearest long to the argument. This is equivalent to 1174: * <code>(long) Math.floor(d + 0.5)</code>. If the argument is NaN, the 1175: * result is 0; otherwise if the argument is outside the range of long, the 1176: * result will be Long.MIN_VALUE or Long.MAX_VALUE, as appropriate. 1177: * 1178: * @param d the argument to round 1179: * @return the nearest long to the argument 1180: * @see Long#MIN_VALUE 1181: * @see Long#MAX_VALUE 1182: */ 1183: public static long round(double d) 1184: { 1185: return (long) floor(d + 0.5); 1186: } 1187: 1188: /** 1189: * Get a random number. This behaves like Random.nextDouble(), seeded by 1190: * System.currentTimeMillis() when first called. In other words, the number 1191: * is from a pseudorandom sequence, and lies in the range [+0.0, 1.0). 1192: * This random sequence is only used by this method, and is threadsafe, 1193: * although you may want your own random number generator if it is shared 1194: * among threads. 1195: * 1196: * @return a random number 1197: * @see Random#nextDouble() 1198: * @see System#currentTimeMillis() 1199: */ 1200: public static synchronized double random() 1201: { 1202: if (rand == null) 1203: rand = new Random(); 1204: return rand.nextDouble(); 1205: } 1206: 1207: /** 1208: * Convert from degrees to radians. The formula for this is 1209: * radians = degrees * (pi/180); however it is not always exact given the 1210: * limitations of floating point numbers. 1211: * 1212: * @param degrees an angle in degrees 1213: * @return the angle in radians 1214: */ 1215: public static double toRadians(double degrees) 1216: { 1217: return (degrees * PI) / 180; 1218: } 1219: 1220: /** 1221: * Convert from radians to degrees. The formula for this is 1222: * degrees = radians * (180/pi); however it is not always exact given the 1223: * limitations of floating point numbers. 1224: * 1225: * @param rads an angle in radians 1226: * @return the angle in degrees 1227: */ 1228: public static double toDegrees(double rads) 1229: { 1230: return (rads * 180) / PI; 1231: } 1232: 1233: /** 1234: * Constants for scaling and comparing doubles by powers of 2. The compiler 1235: * must automatically inline constructs like (1/TWO_54), so we don't list 1236: * negative powers of two here. 1237: */ 1238: private static final double 1239: TWO_16 = 0x10000, // Long bits 0x40f0000000000000L. 1240: TWO_20 = 0x100000, // Long bits 0x4130000000000000L. 1241: TWO_24 = 0x1000000, // Long bits 0x4170000000000000L. 1242: TWO_27 = 0x8000000, // Long bits 0x41a0000000000000L. 1243: TWO_28 = 0x10000000, // Long bits 0x41b0000000000000L. 1244: TWO_29 = 0x20000000, // Long bits 0x41c0000000000000L. 1245: TWO_31 = 0x80000000L, // Long bits 0x41e0000000000000L. 1246: TWO_49 = 0x2000000000000L, // Long bits 0x4300000000000000L. 1247: TWO_52 = 0x10000000000000L, // Long bits 0x4330000000000000L. 1248: TWO_54 = 0x40000000000000L, // Long bits 0x4350000000000000L. 1249: TWO_57 = 0x200000000000000L, // Long bits 0x4380000000000000L. 1250: TWO_60 = 0x1000000000000000L, // Long bits 0x43b0000000000000L. 1251: TWO_64 = 1.8446744073709552e19, // Long bits 0x43f0000000000000L. 1252: TWO_66 = 7.378697629483821e19, // Long bits 0x4410000000000000L. 1253: TWO_1023 = 8.98846567431158e307; // Long bits 0x7fe0000000000000L. 1254: 1255: /** 1256: * Super precision for 2/pi in 24-bit chunks, for use in 1257: * {@link #remPiOver2(double, double[])}. 1258: */ 1259: private static final int TWO_OVER_PI[] = { 1260: 0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62, 1261: 0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a, 1262: 0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129, 1263: 0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41, 1264: 0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8, 1265: 0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf, 1266: 0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5, 1267: 0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08, 1268: 0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3, 1269: 0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880, 1270: 0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b, 1271: }; 1272: 1273: /** 1274: * Super precision for pi/2 in 24-bit chunks, for use in 1275: * {@link #remPiOver2(double, double[])}. 1276: */ 1277: private static final double PI_OVER_TWO[] = { 1278: 1.570796251296997, // Long bits 0x3ff921fb40000000L. 1279: 7.549789415861596e-8, // Long bits 0x3e74442d00000000L. 1280: 5.390302529957765e-15, // Long bits 0x3cf8469880000000L. 1281: 3.282003415807913e-22, // Long bits 0x3b78cc5160000000L. 1282: 1.270655753080676e-29, // Long bits 0x39f01b8380000000L. 1283: 1.2293330898111133e-36, // Long bits 0x387a252040000000L. 1284: 2.7337005381646456e-44, // Long bits 0x36e3822280000000L. 1285: 2.1674168387780482e-51, // Long bits 0x3569f31d00000000L. 1286: }; 1287: 1288: /** 1289: * More constants related to pi, used in 1290: * {@link #remPiOver2(double, double[])} and elsewhere. 1291: */ 1292: private static final double 1293: PI_L = 1.2246467991473532e-16, // Long bits 0x3ca1a62633145c07L. 1294: PIO2_1 = 1.5707963267341256, // Long bits 0x3ff921fb54400000L. 1295: PIO2_1L = 6.077100506506192e-11, // Long bits 0x3dd0b4611a626331L. 1296: PIO2_2 = 6.077100506303966e-11, // Long bits 0x3dd0b4611a600000L. 1297: PIO2_2L = 2.0222662487959506e-21, // Long bits 0x3ba3198a2e037073L. 1298: PIO2_3 = 2.0222662487111665e-21, // Long bits 0x3ba3198a2e000000L. 1299: PIO2_3L = 8.4784276603689e-32; // Long bits 0x397b839a252049c1L. 1300: 1301: /** 1302: * Natural log and square root constants, for calculation of 1303: * {@link #exp(double)}, {@link #log(double)} and 1304: * {@link #pow(double, double)}. CP is 2/(3*ln(2)). 1305: */ 1306: private static final double 1307: SQRT_1_5 = 1.224744871391589, // Long bits 0x3ff3988e1409212eL. 1308: SQRT_2 = 1.4142135623730951, // Long bits 0x3ff6a09e667f3bcdL. 1309: SQRT_3 = 1.7320508075688772, // Long bits 0x3ffbb67ae8584caaL. 1310: EXP_LIMIT_H = 709.782712893384, // Long bits 0x40862e42fefa39efL. 1311: EXP_LIMIT_L = -745.1332191019411, // Long bits 0xc0874910d52d3051L. 1312: CP = 0.9617966939259756, // Long bits 0x3feec709dc3a03fdL. 1313: CP_H = 0.9617967009544373, // Long bits 0x3feec709e0000000L. 1314: CP_L = -7.028461650952758e-9, // Long bits 0xbe3e2fe0145b01f5L. 1315: LN2 = 0.6931471805599453, // Long bits 0x3fe62e42fefa39efL. 1316: LN2_H = 0.6931471803691238, // Long bits 0x3fe62e42fee00000L. 1317: LN2_L = 1.9082149292705877e-10, // Long bits 0x3dea39ef35793c76L. 1318: INV_LN2 = 1.4426950408889634, // Long bits 0x3ff71547652b82feL. 1319: INV_LN2_H = 1.4426950216293335, // Long bits 0x3ff7154760000000L. 1320: INV_LN2_L = 1.9259629911266175e-8; // Long bits 0x3e54ae0bf85ddf44L. 1321: 1322: /** 1323: * Constants for computing {@link #log(double)}. 1324: */ 1325: private static final double 1326: LG1 = 0.6666666666666735, // Long bits 0x3fe5555555555593L. 1327: LG2 = 0.3999999999940942, // Long bits 0x3fd999999997fa04L. 1328: LG3 = 0.2857142874366239, // Long bits 0x3fd2492494229359L. 1329: LG4 = 0.22222198432149784, // Long bits 0x3fcc71c51d8e78afL. 1330: LG5 = 0.1818357216161805, // Long bits 0x3fc7466496cb03deL. 1331: LG6 = 0.15313837699209373, // Long bits 0x3fc39a09d078c69fL. 1332: LG7 = 0.14798198605116586; // Long bits 0x3fc2f112df3e5244L. 1333: 1334: /** 1335: * Constants for computing {@link #pow(double, double)}. L and P are 1336: * coefficients for series; OVT is -(1024-log2(ovfl+.5ulp)); and DP is ???. 1337: * The P coefficients also calculate {@link #exp(double)}. 1338: */ 1339: private static final double 1340: L1 = 0.5999999999999946, // Long bits 0x3fe3333333333303L. 1341: L2 = 0.4285714285785502, // Long bits 0x3fdb6db6db6fabffL. 1342: L3 = 0.33333332981837743, // Long bits 0x3fd55555518f264dL. 1343: L4 = 0.272728123808534, // Long bits 0x3fd17460a91d4101L. 1344: L5 = 0.23066074577556175, // Long bits 0x3fcd864a93c9db65L. 1345: L6 = 0.20697501780033842, // Long bits 0x3fca7e284a454eefL. 1346: P1 = 0.16666666666666602, // Long bits 0x3fc555555555553eL. 1347: P2 = -2.7777777777015593e-3, // Long bits 0xbf66c16c16bebd93L. 1348: P3 = 6.613756321437934e-5, // Long bits 0x3f11566aaf25de2cL. 1349: P4 = -1.6533902205465252e-6, // Long bits 0xbebbbd41c5d26bf1L. 1350: P5 = 4.1381367970572385e-8, // Long bits 0x3e66376972bea4d0L. 1351: DP_H = 0.5849624872207642, // Long bits 0x3fe2b80340000000L. 1352: DP_L = 1.350039202129749e-8, // Long bits 0x3e4cfdeb43cfd006L. 1353: OVT = 8.008566259537294e-17; // Long bits 0x3c971547652b82feL. 1354: 1355: /** 1356: * Coefficients for computing {@link #sin(double)}. 1357: */ 1358: private static final double 1359: S1 = -0.16666666666666632, // Long bits 0xbfc5555555555549L. 1360: S2 = 8.33333333332249e-3, // Long bits 0x3f8111111110f8a6L. 1361: S3 = -1.984126982985795e-4, // Long bits 0xbf2a01a019c161d5L. 1362: S4 = 2.7557313707070068e-6, // Long bits 0x3ec71de357b1fe7dL. 1363: S5 = -2.5050760253406863e-8, // Long bits 0xbe5ae5e68a2b9cebL. 1364: S6 = 1.58969099521155e-10; // Long bits 0x3de5d93a5acfd57cL. 1365: 1366: /** 1367: * Coefficients for computing {@link #cos(double)}. 1368: */ 1369: private static final double 1370: C1 = 0.0416666666666666, // Long bits 0x3fa555555555554cL. 1371: C2 = -1.388888888887411e-3, // Long bits 0xbf56c16c16c15177L. 1372: C3 = 2.480158728947673e-5, // Long bits 0x3efa01a019cb1590L. 1373: C4 = -2.7557314351390663e-7, // Long bits 0xbe927e4f809c52adL. 1374: C5 = 2.087572321298175e-9, // Long bits 0x3e21ee9ebdb4b1c4L. 1375: C6 = -1.1359647557788195e-11; // Long bits 0xbda8fae9be8838d4L. 1376: 1377: /** 1378: * Coefficients for computing {@link #tan(double)}. 1379: */ 1380: private static final double 1381: T0 = 0.3333333333333341, // Long bits 0x3fd5555555555563L. 1382: T1 = 0.13333333333320124, // Long bits 0x3fc111111110fe7aL. 1383: T2 = 0.05396825397622605, // Long bits 0x3faba1ba1bb341feL. 1384: T3 = 0.021869488294859542, // Long bits 0x3f9664f48406d637L. 1385: T4 = 8.8632398235993e-3, // Long bits 0x3f8226e3e96e8493L. 1386: T5 = 3.5920791075913124e-3, // Long bits 0x3f6d6d22c9560328L. 1387: T6 = 1.4562094543252903e-3, // Long bits 0x3f57dbc8fee08315L. 1388: T7 = 5.880412408202641e-4, // Long bits 0x3f4344d8f2f26501L. 1389: T8 = 2.464631348184699e-4, // Long bits 0x3f3026f71a8d1068L. 1390: T9 = 7.817944429395571e-5, // Long bits 0x3f147e88a03792a6L. 1391: T10 = 7.140724913826082e-5, // Long bits 0x3f12b80f32f0a7e9L. 1392: T11 = -1.8558637485527546e-5, // Long bits 0xbef375cbdb605373L. 1393: T12 = 2.590730518636337e-5; // Long bits 0x3efb2a7074bf7ad4L. 1394: 1395: /** 1396: * Coefficients for computing {@link #asin(double)} and 1397: * {@link #acos(double)}. 1398: */ 1399: private static final double 1400: PS0 = 0.16666666666666666, // Long bits 0x3fc5555555555555L. 1401: PS1 = -0.3255658186224009, // Long bits 0xbfd4d61203eb6f7dL. 1402: PS2 = 0.20121253213486293, // Long bits 0x3fc9c1550e884455L. 1403: PS3 = -0.04005553450067941, // Long bits 0xbfa48228b5688f3bL. 1404: PS4 = 7.915349942898145e-4, // Long bits 0x3f49efe07501b288L. 1405: PS5 = 3.479331075960212e-5, // Long bits 0x3f023de10dfdf709L. 1406: QS1 = -2.403394911734414, // Long bits 0xc0033a271c8a2d4bL. 1407: QS2 = 2.0209457602335057, // Long bits 0x40002ae59c598ac8L. 1408: QS3 = -0.6882839716054533, // Long bits 0xbfe6066c1b8d0159L. 1409: QS4 = 0.07703815055590194; // Long bits 0x3fb3b8c5b12e9282L. 1410: 1411: /** 1412: * Coefficients for computing {@link #atan(double)}. 1413: */ 1414: private static final double 1415: ATAN_0_5H = 0.4636476090008061, // Long bits 0x3fddac670561bb4fL. 1416: ATAN_0_5L = 2.2698777452961687e-17, // Long bits 0x3c7a2b7f222f65e2L. 1417: ATAN_1_5H = 0.982793723247329, // Long bits 0x3fef730bd281f69bL. 1418: ATAN_1_5L = 1.3903311031230998e-17, // Long bits 0x3c7007887af0cbbdL. 1419: AT0 = 0.3333333333333293, // Long bits 0x3fd555555555550dL. 1420: AT1 = -0.19999999999876483, // Long bits 0xbfc999999998ebc4L. 1421: AT2 = 0.14285714272503466, // Long bits 0x3fc24924920083ffL. 1422: AT3 = -0.11111110405462356, // Long bits 0xbfbc71c6fe231671L. 1423: AT4 = 0.09090887133436507, // Long bits 0x3fb745cdc54c206eL. 1424: AT5 = -0.0769187620504483, // Long bits 0xbfb3b0f2af749a6dL. 1425: AT6 = 0.06661073137387531, // Long bits 0x3fb10d66a0d03d51L. 1426: AT7 = -0.058335701337905735, // Long bits 0xbfadde2d52defd9aL. 1427: AT8 = 0.049768779946159324, // Long bits 0x3fa97b4b24760debL. 1428: AT9 = -0.036531572744216916, // Long bits 0xbfa2b4442c6a6c2fL. 1429: AT10 = 0.016285820115365782; // Long bits 0x3f90ad3ae322da11L. 1430: 1431: /** 1432: * Helper function for reducing an angle to a multiple of pi/2 within 1433: * [-pi/4, pi/4]. 1434: * 1435: * @param x the angle; not infinity or NaN, and outside pi/4 1436: * @param y an array of 2 doubles modified to hold the remander x % pi/2 1437: * @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4], 1438: * 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4] 1439: */ 1440: private static int remPiOver2(double x, double[] y) 1441: { 1442: boolean negative = x < 0; 1443: x = abs(x); 1444: double z; 1445: int n; 1446: if (Configuration.DEBUG && (x <= PI / 4 || x != x 1447: || x == Double.POSITIVE_INFINITY)) 1448: throw new InternalError("Assertion failure"); 1449: if (x < 3 * PI / 4) // If |x| is small. 1450: { 1451: z = x - PIO2_1; 1452: if ((float) x != (float) (PI / 2)) // 33+53 bit pi is good enough. 1453: { 1454: y[0] = z - PIO2_1L; 1455: y[1] = z - y[0] - PIO2_1L; 1456: } 1457: else // Near pi/2, use 33+33+53 bit pi. 1458: { 1459: z -= PIO2_2; 1460: y[0] = z - PIO2_2L; 1461: y[1] = z - y[0] - PIO2_2L; 1462: } 1463: n = 1; 1464: } 1465: else if (x <= TWO_20 * PI / 2) // Medium size. 1466: { 1467: n = (int) (2 / PI * x + 0.5); 1468: z = x - n * PIO2_1; 1469: double w = n * PIO2_1L; // First round good to 85 bits. 1470: y[0] = z - w; 1471: if (n >= 32 || (float) x == (float) (w)) 1472: { 1473: if (x / y[0] >= TWO_16) // Second iteration, good to 118 bits. 1474: { 1475: double t = z; 1476: w = n * PIO2_2; 1477: z = t - w; 1478: w = n * PIO2_2L - (t - z - w); 1479: y[0] = z - w; 1480: if (x / y[0] >= TWO_49) // Third iteration, 151 bits accuracy. 1481: { 1482: t = z; 1483: w = n * PIO2_3; 1484: z = t - w; 1485: w = n * PIO2_3L - (t - z - w); 1486: y[0] = z - w; 1487: } 1488: } 1489: } 1490: y[1] = z - y[0] - w; 1491: } 1492: else 1493: { 1494: // All other (large) arguments. 1495: int e0 = (int) (Double.doubleToLongBits(x) >> 52) - 1046; 1496: z = scale(x, -e0); // e0 = ilogb(z) - 23. 1497: double[] tx = new double[3]; 1498: for (int i = 0; i < 2; i++) 1499: { 1500: tx[i] = (int) z; 1501: z = (z - tx[i]) * TWO_24; 1502: } 1503: tx[2] = z; 1504: int nx = 2; 1505: while (tx[nx] == 0) 1506: nx--; 1507: n = remPiOver2(tx, y, e0, nx); 1508: } 1509: if (negative) 1510: { 1511: y[0] = -y[0]; 1512: y[1] = -y[1]; 1513: return -n; 1514: } 1515: return n; 1516: } 1517: 1518: /** 1519: * Helper function for reducing an angle to a multiple of pi/2 within 1520: * [-pi/4, pi/4]. 1521: * 1522: * @param x the positive angle, broken into 24-bit chunks 1523: * @param y an array of 2 doubles modified to hold the remander x % pi/2 1524: * @param e0 the exponent of x[0] 1525: * @param nx the last index used in x 1526: * @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4], 1527: * 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4] 1528: */ 1529: private static int remPiOver2(double[] x, double[] y, int e0, int nx) 1530: { 1531: int i; 1532: int ih; 1533: int n; 1534: double fw; 1535: double z; 1536: int[] iq = new int[20]; 1537: double[] f = new double[20]; 1538: double[] q = new double[20]; 1539: boolean recompute = false; 1540: 1541: // Initialize jk, jz, jv, q0; note that 3>q0. 1542: int jk = 4; 1543: int jz = jk; 1544: int jv = max((e0 - 3) / 24, 0); 1545: int q0 = e0 - 24 * (jv + 1); 1546: 1547: // Set up f[0] to f[nx+jk] where f[nx+jk] = TWO_OVER_PI[jv+jk]. 1548: int j = jv - nx; 1549: int m = nx + jk; 1550: for (i = 0; i <= m; i++, j++) 1551: f[i] = (j < 0) ? 0 : TWO_OVER_PI[j]; 1552: 1553: // Compute q[0],q[1],...q[jk]. 1554: for (i = 0; i <= jk; i++) 1555: { 1556: for (j = 0, fw = 0; j <= nx; j++) 1557: fw += x[j] * f[nx + i - j]; 1558: q[i] = fw; 1559: } 1560: 1561: do 1562: { 1563: // Distill q[] into iq[] reversingly. 1564: for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) 1565: { 1566: fw = (int) (1 / TWO_24 * z); 1567: iq[i] = (int) (z - TWO_24 * fw); 1568: z = q[j - 1] + fw; 1569: } 1570: 1571: // Compute n. 1572: z = scale(z, q0); 1573: z -= 8 * floor(z * 0.125); // Trim off integer >= 8. 1574: n = (int) z; 1575: z -= n; 1576: ih = 0; 1577: if (q0 > 0) // Need iq[jz-1] to determine n. 1578: { 1579: i = iq[jz - 1] >> (24 - q0); 1580: n += i; 1581: iq[jz - 1] -= i << (24 - q0); 1582: ih = iq[jz - 1] >> (23 - q0); 1583: } 1584: else if (q0 == 0) 1585: ih = iq[jz - 1] >> 23; 1586: else if (z >= 0.5) 1587: ih = 2; 1588: 1589: if (ih > 0) // If q > 0.5. 1590: { 1591: n += 1; 1592: int carry = 0; 1593: for (i = 0; i < jz; i++) // Compute 1-q. 1594: { 1595: j = iq[i]; 1596: if (carry == 0) 1597: { 1598: if (j != 0) 1599: { 1600: carry = 1; 1601: iq[i] = 0x1000000 - j; 1602: } 1603: } 1604: else 1605: iq[i] = 0xffffff - j; 1606: } 1607: switch (q0) 1608: { 1609: case 1: // Rare case: chance is 1 in 12 for non-default. 1610: iq[jz - 1] &= 0x7fffff; 1611: break; 1612: case 2: 1613: iq[jz - 1] &= 0x3fffff; 1614: } 1615: if (ih == 2) 1616: { 1617: z = 1 - z; 1618: if (carry != 0) 1619: z -= scale(1, q0); 1620: } 1621: } 1622: 1623: // Check if recomputation is needed. 1624: if (z == 0) 1625: { 1626: j = 0; 1627: for (i = jz - 1; i >= jk; i--) 1628: j |= iq[i]; 1629: if (j == 0) // Need recomputation. 1630: { 1631: int k; 1632: for (k = 1; iq[jk - k] == 0; k++); // k = no. of terms needed. 1633: 1634: for (i = jz + 1; i <= jz + k; i++) // Add q[jz+1] to q[jz+k]. 1635: { 1636: f[nx + i] = TWO_OVER_PI[jv + i]; 1637: for (j = 0, fw = 0; j <= nx; j++) 1638: fw += x[j] * f[nx + i - j]; 1639: q[i] = fw; 1640: } 1641: jz += k; 1642: recompute = true; 1643: } 1644: } 1645: } 1646: while (recompute); 1647: 1648: // Chop off zero terms. 1649: if (z == 0) 1650: { 1651: jz--; 1652: q0 -= 24; 1653: while (iq[jz] == 0) 1654: { 1655: jz--; 1656: q0 -= 24; 1657: } 1658: } 1659: else // Break z into 24-bit if necessary. 1660: { 1661: z = scale(z, -q0); 1662: if (z >= TWO_24) 1663: { 1664: fw = (int) (1 / TWO_24 * z); 1665: iq[jz] = (int) (z - TWO_24 * fw); 1666: jz++; 1667: q0 += 24; 1668: iq[jz] = (int) fw; 1669: } 1670: else 1671: iq[jz] = (int) z; 1672: } 1673: 1674: // Convert integer "bit" chunk to floating-point value. 1675: fw = scale(1, q0); 1676: for (i = jz; i >= 0; i--) 1677: { 1678: q[i] = fw * iq[i]; 1679: fw *= 1 / TWO_24; 1680: } 1681: 1682: // Compute PI_OVER_TWO[0,...,jk]*q[jz,...,0]. 1683: double[] fq = new double[20]; 1684: for (i = jz; i >= 0; i--) 1685: { 1686: fw = 0; 1687: for (int k = 0; k <= jk && k <= jz - i; k++) 1688: fw += PI_OVER_TWO[k] * q[i + k]; 1689: fq[jz - i] = fw; 1690: } 1691: 1692: // Compress fq[] into y[]. 1693: fw = 0; 1694: for (i = jz; i >= 0; i--) 1695: fw += fq[i]; 1696: y[0] = (ih == 0) ? fw : -fw; 1697: fw = fq[0] - fw; 1698: for (i = 1; i <= jz; i++) 1699: fw += fq[i]; 1700: y[1] = (ih == 0) ? fw : -fw; 1701: return n; 1702: } 1703: 1704: /** 1705: * Helper method for scaling a double by a power of 2. 1706: * 1707: * @param x the double 1708: * @param n the scale; |n| < 2048 1709: * @return x * 2**n 1710: */ 1711: private static double scale(double x, int n) 1712: { 1713: if (Configuration.DEBUG && abs(n) >= 2048) 1714: throw new InternalError("Assertion failure"); 1715: if (x == 0 || x == Double.NEGATIVE_INFINITY 1716: || ! (x < Double.POSITIVE_INFINITY) || n == 0) 1717: return x; 1718: long bits = Double.doubleToLongBits(x); 1719: int exp = (int) (bits >> 52) & 0x7ff; 1720: if (exp == 0) // Subnormal x. 1721: { 1722: x *= TWO_54; 1723: exp = ((int) (Double.doubleToLongBits(x) >> 52) & 0x7ff) - 54; 1724: } 1725: exp += n; 1726: if (exp > 0x7fe) // Overflow. 1727: return Double.POSITIVE_INFINITY * x; 1728: if (exp > 0) // Normal. 1729: return Double.longBitsToDouble((bits & 0x800fffffffffffffL) 1730: | ((long) exp << 52)); 1731: if (exp <= -54) 1732: return 0 * x; // Underflow. 1733: exp += 54; // Subnormal result. 1734: x = Double.longBitsToDouble((bits & 0x800fffffffffffffL) 1735: | ((long) exp << 52)); 1736: return x * (1 / TWO_54); 1737: } 1738: 1739: /** 1740: * Helper trig function; computes sin in range [-pi/4, pi/4]. 1741: * 1742: * @param x angle within about pi/4 1743: * @param y tail of x, created by remPiOver2 1744: * @return sin(x+y) 1745: */ 1746: private static double sin(double x, double y) 1747: { 1748: if (Configuration.DEBUG && abs(x + y) > 0.7854) 1749: throw new InternalError("Assertion failure"); 1750: if (abs(x) < 1 / TWO_27) 1751: return x; // If |x| ~< 2**-27, already know answer. 1752: 1753: double z = x * x; 1754: double v = z * x; 1755: double r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); 1756: if (y == 0) 1757: return x + v * (S1 + z * r); 1758: return x - ((z * (0.5 * y - v * r) - y) - v * S1); 1759: } 1760: 1761: /** 1762: * Helper trig function; computes cos in range [-pi/4, pi/4]. 1763: * 1764: * @param x angle within about pi/4 1765: * @param y tail of x, created by remPiOver2 1766: * @return cos(x+y) 1767: */ 1768: private static double cos(double x, double y) 1769: { 1770: if (Configuration.DEBUG && abs(x + y) > 0.7854) 1771: throw new InternalError("Assertion failure"); 1772: x = abs(x); 1773: if (x < 1 / TWO_27) 1774: return 1; // If |x| ~< 2**-27, already know answer. 1775: 1776: double z = x * x; 1777: double r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); 1778: 1779: if (x < 0.3) 1780: return 1 - (0.5 * z - (z * r - x * y)); 1781: 1782: double qx = (x > 0.78125) ? 0.28125 : (x * 0.25); 1783: return 1 - qx - ((0.5 * z - qx) - (z * r - x * y)); 1784: } 1785: 1786: /** 1787: * Helper trig function; computes tan in range [-pi/4, pi/4]. 1788: * 1789: * @param x angle within about pi/4 1790: * @param y tail of x, created by remPiOver2 1791: * @param invert true iff -1/tan should be returned instead 1792: * @return tan(x+y) 1793: */ 1794: private static double tan(double x, double y, boolean invert) 1795: { 1796: // PI/2 is irrational, so no double is a perfect multiple of it. 1797: if (Configuration.DEBUG && (abs(x + y) > 0.7854 || (x == 0 && invert))) 1798: throw new InternalError("Assertion failure"); 1799: boolean negative = x < 0; 1800: if (negative) 1801: { 1802: x = -x; 1803: y = -y; 1804: } 1805: if (x < 1 / TWO_28) // If |x| ~< 2**-28, already know answer. 1806: return (negative ? -1 : 1) * (invert ? -1 / x : x); 1807: 1808: double z; 1809: double w; 1810: boolean large = x >= 0.6744; 1811: if (large) 1812: { 1813: z = PI / 4 - x; 1814: w = PI_L / 4 - y; 1815: x = z + w; 1816: y = 0; 1817: } 1818: z = x * x; 1819: w = z * z; 1820: // Break x**5*(T1+x**2*T2+...) into 1821: // x**5(T1+x**4*T3+...+x**20*T11) 1822: // + x**5(x**2*(T2+x**4*T4+...+x**22*T12)). 1823: double r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11)))); 1824: double v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12))))); 1825: double s = z * x; 1826: r = y + z * (s * (r + v) + y); 1827: r += T0 * s; 1828: w = x + r; 1829: if (large) 1830: { 1831: v = invert ? -1 : 1; 1832: return (negative ? -1 : 1) * (v - 2 * (x - (w * w / (w + v) - r))); 1833: } 1834: if (! invert) 1835: return w; 1836: 1837: // Compute -1.0/(x+r) accurately. 1838: z = (float) w; 1839: v = r - (z - x); 1840: double a = -1 / w; 1841: double t = (float) a; 1842: return t + a * (1 + t * z + t * v); 1843: } 1844: }
GNU Classpath (0.20) |