1 /* 2 * Copyright (c) 1994, 2016, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package java.lang; 27 28 import java.math.BigDecimal; 29 import java.util.Random; 30 import jdk.internal.math.FloatConsts; 31 import jdk.internal.math.DoubleConsts; 32 import jdk.internal.HotSpotIntrinsicCandidate; 33 34 /** 35 * The class {@code Math} contains methods for performing basic 36 * numeric operations such as the elementary exponential, logarithm, 37 * square root, and trigonometric functions. 38 * 39 * <p>Unlike some of the numeric methods of class 40 * {@code StrictMath}, all implementations of the equivalent 41 * functions of class {@code Math} are not defined to return the 42 * bit-for-bit same results. This relaxation permits 43 * better-performing implementations where strict reproducibility is 44 * not required. 45 * 46 * <p>By default many of the {@code Math} methods simply call 47 * the equivalent method in {@code StrictMath} for their 48 * implementation. Code generators are encouraged to use 49 * platform-specific native libraries or microprocessor instructions, 50 * where available, to provide higher-performance implementations of 51 * {@code Math} methods. Such higher-performance 52 * implementations still must conform to the specification for 53 * {@code Math}. 54 * 55 * <p>The quality of implementation specifications concern two 56 * properties, accuracy of the returned result and monotonicity of the 57 * method. Accuracy of the floating-point {@code Math} methods is 58 * measured in terms of <i>ulps</i>, units in the last place. For a 59 * given floating-point format, an {@linkplain #ulp(double) ulp} of a 60 * specific real number value is the distance between the two 61 * floating-point values bracketing that numerical value. When 62 * discussing the accuracy of a method as a whole rather than at a 63 * specific argument, the number of ulps cited is for the worst-case 64 * error at any argument. If a method always has an error less than 65 * 0.5 ulps, the method always returns the floating-point number 66 * nearest the exact result; such a method is <i>correctly 67 * rounded</i>. A correctly rounded method is generally the best a 68 * floating-point approximation can be; however, it is impractical for 69 * many floating-point methods to be correctly rounded. Instead, for 70 * the {@code Math} class, a larger error bound of 1 or 2 ulps is 71 * allowed for certain methods. Informally, with a 1 ulp error bound, 72 * when the exact result is a representable number, the exact result 73 * should be returned as the computed result; otherwise, either of the 74 * two floating-point values which bracket the exact result may be 75 * returned. For exact results large in magnitude, one of the 76 * endpoints of the bracket may be infinite. Besides accuracy at 77 * individual arguments, maintaining proper relations between the 78 * method at different arguments is also important. Therefore, most 79 * methods with more than 0.5 ulp errors are required to be 80 * <i>semi-monotonic</i>: whenever the mathematical function is 81 * non-decreasing, so is the floating-point approximation, likewise, 82 * whenever the mathematical function is non-increasing, so is the 83 * floating-point approximation. Not all approximations that have 1 84 * ulp accuracy will automatically meet the monotonicity requirements. 85 * 86 * <p> 87 * The platform uses signed two's complement integer arithmetic with 88 * int and long primitive types. The developer should choose 89 * the primitive type to ensure that arithmetic operations consistently 90 * produce correct results, which in some cases means the operations 91 * will not overflow the range of values of the computation. 92 * The best practice is to choose the primitive type and algorithm to avoid 93 * overflow. In cases where the size is {@code int} or {@code long} and 94 * overflow errors need to be detected, the methods {@code addExact}, 95 * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact} 96 * throw an {@code ArithmeticException} when the results overflow. 97 * For other arithmetic operations such as divide, absolute value, 98 * increment by one, decrement by one, and negation, overflow occurs only with 99 * a specific minimum or maximum value and should be checked against 100 * the minimum or maximum as appropriate. 101 * 102 * @author unascribed 103 * @author Joseph D. Darcy 104 * @since 1.0 105 */ 106 107 public final class Math { 108 109 /** 110 * Don't let anyone instantiate this class. 111 */ 112 private Math() {} 113 114 /** 115 * The {@code double} value that is closer than any other to 116 * <i>e</i>, the base of the natural logarithms. 117 */ 118 public static final double E = 2.7182818284590452354; 119 120 /** 121 * The {@code double} value that is closer than any other to 122 * <i>pi</i>, the ratio of the circumference of a circle to its 123 * diameter. 124 */ 125 public static final double PI = 3.14159265358979323846; 126 127 /** 128 * Constant by which to multiply an angular value in degrees to obtain an 129 * angular value in radians. 130 */ 131 private static final double DEGREES_TO_RADIANS = 0.017453292519943295; 132 133 /** 134 * Constant by which to multiply an angular value in radians to obtain an 135 * angular value in degrees. 136 */ 137 private static final double RADIANS_TO_DEGREES = 57.29577951308232; 138 139 /** 140 * Returns the trigonometric sine of an angle. Special cases: 141 * <ul><li>If the argument is NaN or an infinity, then the 142 * result is NaN. 143 * <li>If the argument is zero, then the result is a zero with the 144 * same sign as the argument.</ul> 145 * 146 * <p>The computed result must be within 1 ulp of the exact result. 147 * Results must be semi-monotonic. 148 * 149 * @param a an angle, in radians. 150 * @return the sine of the argument. 151 */ 152 @HotSpotIntrinsicCandidate 153 public static double sin(double a) { 154 return StrictMath.sin(a); // default impl. delegates to StrictMath 155 } 156 157 /** 158 * Returns the trigonometric cosine of an angle. Special cases: 159 * <ul><li>If the argument is NaN or an infinity, then the 160 * result is NaN.</ul> 161 * 162 * <p>The computed result must be within 1 ulp of the exact result. 163 * Results must be semi-monotonic. 164 * 165 * @param a an angle, in radians. 166 * @return the cosine of the argument. 167 */ 168 @HotSpotIntrinsicCandidate 169 public static double cos(double a) { 170 return StrictMath.cos(a); // default impl. delegates to StrictMath 171 } 172 173 /** 174 * Returns the trigonometric tangent of an angle. Special cases: 175 * <ul><li>If the argument is NaN or an infinity, then the result 176 * is NaN. 177 * <li>If the argument is zero, then the result is a zero with the 178 * same sign as the argument.</ul> 179 * 180 * <p>The computed result must be within 1 ulp of the exact result. 181 * Results must be semi-monotonic. 182 * 183 * @param a an angle, in radians. 184 * @return the tangent of the argument. 185 */ 186 @HotSpotIntrinsicCandidate 187 public static double tan(double a) { 188 return StrictMath.tan(a); // default impl. delegates to StrictMath 189 } 190 191 /** 192 * Returns the arc sine of a value; the returned angle is in the 193 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: 194 * <ul><li>If the argument is NaN or its absolute value is greater 195 * than 1, then the result is NaN. 196 * <li>If the argument is zero, then the result is a zero with the 197 * same sign as the argument.</ul> 198 * 199 * <p>The computed result must be within 1 ulp of the exact result. 200 * Results must be semi-monotonic. 201 * 202 * @param a the value whose arc sine is to be returned. 203 * @return the arc sine of the argument. 204 */ 205 public static double asin(double a) { 206 return StrictMath.asin(a); // default impl. delegates to StrictMath 207 } 208 209 /** 210 * Returns the arc cosine of a value; the returned angle is in the 211 * range 0.0 through <i>pi</i>. Special case: 212 * <ul><li>If the argument is NaN or its absolute value is greater 213 * than 1, then the result is NaN.</ul> 214 * 215 * <p>The computed result must be within 1 ulp of the exact result. 216 * Results must be semi-monotonic. 217 * 218 * @param a the value whose arc cosine is to be returned. 219 * @return the arc cosine of the argument. 220 */ 221 public static double acos(double a) { 222 return StrictMath.acos(a); // default impl. delegates to StrictMath 223 } 224 225 /** 226 * Returns the arc tangent of a value; the returned angle is in the 227 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: 228 * <ul><li>If the argument is NaN, then the result is NaN. 229 * <li>If the argument is zero, then the result is a zero with the 230 * same sign as the argument.</ul> 231 * 232 * <p>The computed result must be within 1 ulp of the exact result. 233 * Results must be semi-monotonic. 234 * 235 * @param a the value whose arc tangent is to be returned. 236 * @return the arc tangent of the argument. 237 */ 238 public static double atan(double a) { 239 return StrictMath.atan(a); // default impl. delegates to StrictMath 240 } 241 242 /** 243 * Converts an angle measured in degrees to an approximately 244 * equivalent angle measured in radians. The conversion from 245 * degrees to radians is generally inexact. 246 * 247 * @param angdeg an angle, in degrees 248 * @return the measurement of the angle {@code angdeg} 249 * in radians. 250 * @since 1.2 251 */ 252 public static double toRadians(double angdeg) { 253 return angdeg * DEGREES_TO_RADIANS; 254 } 255 256 /** 257 * Converts an angle measured in radians to an approximately 258 * equivalent angle measured in degrees. The conversion from 259 * radians to degrees is generally inexact; users should 260 * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly 261 * equal {@code 0.0}. 262 * 263 * @param angrad an angle, in radians 264 * @return the measurement of the angle {@code angrad} 265 * in degrees. 266 * @since 1.2 267 */ 268 public static double toDegrees(double angrad) { 269 return angrad * RADIANS_TO_DEGREES; 270 } 271 272 /** 273 * Returns Euler's number <i>e</i> raised to the power of a 274 * {@code double} value. Special cases: 275 * <ul><li>If the argument is NaN, the result is NaN. 276 * <li>If the argument is positive infinity, then the result is 277 * positive infinity. 278 * <li>If the argument is negative infinity, then the result is 279 * positive zero.</ul> 280 * 281 * <p>The computed result must be within 1 ulp of the exact result. 282 * Results must be semi-monotonic. 283 * 284 * @param a the exponent to raise <i>e</i> to. 285 * @return the value <i>e</i><sup>{@code a}</sup>, 286 * where <i>e</i> is the base of the natural logarithms. 287 */ 288 @HotSpotIntrinsicCandidate 289 public static double exp(double a) { 290 return StrictMath.exp(a); // default impl. delegates to StrictMath 291 } 292 293 /** 294 * Returns the natural logarithm (base <i>e</i>) of a {@code double} 295 * value. Special cases: 296 * <ul><li>If the argument is NaN or less than zero, then the result 297 * is NaN. 298 * <li>If the argument is positive infinity, then the result is 299 * positive infinity. 300 * <li>If the argument is positive zero or negative zero, then the 301 * result is negative infinity.</ul> 302 * 303 * <p>The computed result must be within 1 ulp of the exact result. 304 * Results must be semi-monotonic. 305 * 306 * @param a a value 307 * @return the value ln {@code a}, the natural logarithm of 308 * {@code a}. 309 */ 310 @HotSpotIntrinsicCandidate 311 public static double log(double a) { 312 return StrictMath.log(a); // default impl. delegates to StrictMath 313 } 314 315 /** 316 * Returns the base 10 logarithm of a {@code double} value. 317 * Special cases: 318 * 319 * <ul><li>If the argument is NaN or less than zero, then the result 320 * is NaN. 321 * <li>If the argument is positive infinity, then the result is 322 * positive infinity. 323 * <li>If the argument is positive zero or negative zero, then the 324 * result is negative infinity. 325 * <li> If the argument is equal to 10<sup><i>n</i></sup> for 326 * integer <i>n</i>, then the result is <i>n</i>. 327 * </ul> 328 * 329 * <p>The computed result must be within 1 ulp of the exact result. 330 * Results must be semi-monotonic. 331 * 332 * @param a a value 333 * @return the base 10 logarithm of {@code a}. 334 * @since 1.5 335 */ 336 @HotSpotIntrinsicCandidate 337 public static double log10(double a) { 338 return StrictMath.log10(a); // default impl. delegates to StrictMath 339 } 340 341 /** 342 * Returns the correctly rounded positive square root of a 343 * {@code double} value. 344 * Special cases: 345 * <ul><li>If the argument is NaN or less than zero, then the result 346 * is NaN. 347 * <li>If the argument is positive infinity, then the result is positive 348 * infinity. 349 * <li>If the argument is positive zero or negative zero, then the 350 * result is the same as the argument.</ul> 351 * Otherwise, the result is the {@code double} value closest to 352 * the true mathematical square root of the argument value. 353 * 354 * @param a a value. 355 * @return the positive square root of {@code a}. 356 * If the argument is NaN or less than zero, the result is NaN. 357 */ 358 @HotSpotIntrinsicCandidate 359 public static double sqrt(double a) { 360 return StrictMath.sqrt(a); // default impl. delegates to StrictMath 361 // Note that hardware sqrt instructions 362 // frequently can be directly used by JITs 363 // and should be much faster than doing 364 // Math.sqrt in software. 365 } 366 367 368 /** 369 * Returns the cube root of a {@code double} value. For 370 * positive finite {@code x}, {@code cbrt(-x) == 371 * -cbrt(x)}; that is, the cube root of a negative value is 372 * the negative of the cube root of that value's magnitude. 373 * 374 * Special cases: 375 * 376 * <ul> 377 * 378 * <li>If the argument is NaN, then the result is NaN. 379 * 380 * <li>If the argument is infinite, then the result is an infinity 381 * with the same sign as the argument. 382 * 383 * <li>If the argument is zero, then the result is a zero with the 384 * same sign as the argument. 385 * 386 * </ul> 387 * 388 * <p>The computed result must be within 1 ulp of the exact result. 389 * 390 * @param a a value. 391 * @return the cube root of {@code a}. 392 * @since 1.5 393 */ 394 public static double cbrt(double a) { 395 return StrictMath.cbrt(a); 396 } 397 398 /** 399 * Computes the remainder operation on two arguments as prescribed 400 * by the IEEE 754 standard. 401 * The remainder value is mathematically equal to 402 * <code>f1 - f2</code> × <i>n</i>, 403 * where <i>n</i> is the mathematical integer closest to the exact 404 * mathematical value of the quotient {@code f1/f2}, and if two 405 * mathematical integers are equally close to {@code f1/f2}, 406 * then <i>n</i> is the integer that is even. If the remainder is 407 * zero, its sign is the same as the sign of the first argument. 408 * Special cases: 409 * <ul><li>If either argument is NaN, or the first argument is infinite, 410 * or the second argument is positive zero or negative zero, then the 411 * result is NaN. 412 * <li>If the first argument is finite and the second argument is 413 * infinite, then the result is the same as the first argument.</ul> 414 * 415 * @param f1 the dividend. 416 * @param f2 the divisor. 417 * @return the remainder when {@code f1} is divided by 418 * {@code f2}. 419 */ 420 public static double IEEEremainder(double f1, double f2) { 421 return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath 422 } 423 424 /** 425 * Returns the smallest (closest to negative infinity) 426 * {@code double} value that is greater than or equal to the 427 * argument and is equal to a mathematical integer. Special cases: 428 * <ul><li>If the argument value is already equal to a 429 * mathematical integer, then the result is the same as the 430 * argument. <li>If the argument is NaN or an infinity or 431 * positive zero or negative zero, then the result is the same as 432 * the argument. <li>If the argument value is less than zero but 433 * greater than -1.0, then the result is negative zero.</ul> Note 434 * that the value of {@code Math.ceil(x)} is exactly the 435 * value of {@code -Math.floor(-x)}. 436 * 437 * 438 * @param a a value. 439 * @return the smallest (closest to negative infinity) 440 * floating-point value that is greater than or equal to 441 * the argument and is equal to a mathematical integer. 442 */ 443 public static double ceil(double a) { 444 return StrictMath.ceil(a); // default impl. delegates to StrictMath 445 } 446 447 /** 448 * Returns the largest (closest to positive infinity) 449 * {@code double} value that is less than or equal to the 450 * argument and is equal to a mathematical integer. Special cases: 451 * <ul><li>If the argument value is already equal to a 452 * mathematical integer, then the result is the same as the 453 * argument. <li>If the argument is NaN or an infinity or 454 * positive zero or negative zero, then the result is the same as 455 * the argument.</ul> 456 * 457 * @param a a value. 458 * @return the largest (closest to positive infinity) 459 * floating-point value that less than or equal to the argument 460 * and is equal to a mathematical integer. 461 */ 462 public static double floor(double a) { 463 return StrictMath.floor(a); // default impl. delegates to StrictMath 464 } 465 466 /** 467 * Returns the {@code double} value that is closest in value 468 * to the argument and is equal to a mathematical integer. If two 469 * {@code double} values that are mathematical integers are 470 * equally close, the result is the integer value that is 471 * even. Special cases: 472 * <ul><li>If the argument value is already equal to a mathematical 473 * integer, then the result is the same as the argument. 474 * <li>If the argument is NaN or an infinity or positive zero or negative 475 * zero, then the result is the same as the argument.</ul> 476 * 477 * @param a a {@code double} value. 478 * @return the closest floating-point value to {@code a} that is 479 * equal to a mathematical integer. 480 */ 481 public static double rint(double a) { 482 return StrictMath.rint(a); // default impl. delegates to StrictMath 483 } 484 485 /** 486 * Returns the angle <i>theta</i> from the conversion of rectangular 487 * coordinates ({@code x}, {@code y}) to polar 488 * coordinates (r, <i>theta</i>). 489 * This method computes the phase <i>theta</i> by computing an arc tangent 490 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special 491 * cases: 492 * <ul><li>If either argument is NaN, then the result is NaN. 493 * <li>If the first argument is positive zero and the second argument 494 * is positive, or the first argument is positive and finite and the 495 * second argument is positive infinity, then the result is positive 496 * zero. 497 * <li>If the first argument is negative zero and the second argument 498 * is positive, or the first argument is negative and finite and the 499 * second argument is positive infinity, then the result is negative zero. 500 * <li>If the first argument is positive zero and the second argument 501 * is negative, or the first argument is positive and finite and the 502 * second argument is negative infinity, then the result is the 503 * {@code double} value closest to <i>pi</i>. 504 * <li>If the first argument is negative zero and the second argument 505 * is negative, or the first argument is negative and finite and the 506 * second argument is negative infinity, then the result is the 507 * {@code double} value closest to -<i>pi</i>. 508 * <li>If the first argument is positive and the second argument is 509 * positive zero or negative zero, or the first argument is positive 510 * infinity and the second argument is finite, then the result is the 511 * {@code double} value closest to <i>pi</i>/2. 512 * <li>If the first argument is negative and the second argument is 513 * positive zero or negative zero, or the first argument is negative 514 * infinity and the second argument is finite, then the result is the 515 * {@code double} value closest to -<i>pi</i>/2. 516 * <li>If both arguments are positive infinity, then the result is the 517 * {@code double} value closest to <i>pi</i>/4. 518 * <li>If the first argument is positive infinity and the second argument 519 * is negative infinity, then the result is the {@code double} 520 * value closest to 3*<i>pi</i>/4. 521 * <li>If the first argument is negative infinity and the second argument 522 * is positive infinity, then the result is the {@code double} value 523 * closest to -<i>pi</i>/4. 524 * <li>If both arguments are negative infinity, then the result is the 525 * {@code double} value closest to -3*<i>pi</i>/4.</ul> 526 * 527 * <p>The computed result must be within 2 ulps of the exact result. 528 * Results must be semi-monotonic. 529 * 530 * @param y the ordinate coordinate 531 * @param x the abscissa coordinate 532 * @return the <i>theta</i> component of the point 533 * (<i>r</i>, <i>theta</i>) 534 * in polar coordinates that corresponds to the point 535 * (<i>x</i>, <i>y</i>) in Cartesian coordinates. 536 */ 537 @HotSpotIntrinsicCandidate 538 public static double atan2(double y, double x) { 539 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath 540 } 541 542 /** 543 * Returns the value of the first argument raised to the power of the 544 * second argument. Special cases: 545 * 546 * <ul><li>If the second argument is positive or negative zero, then the 547 * result is 1.0. 548 * <li>If the second argument is 1.0, then the result is the same as the 549 * first argument. 550 * <li>If the second argument is NaN, then the result is NaN. 551 * <li>If the first argument is NaN and the second argument is nonzero, 552 * then the result is NaN. 553 * 554 * <li>If 555 * <ul> 556 * <li>the absolute value of the first argument is greater than 1 557 * and the second argument is positive infinity, or 558 * <li>the absolute value of the first argument is less than 1 and 559 * the second argument is negative infinity, 560 * </ul> 561 * then the result is positive infinity. 562 * 563 * <li>If 564 * <ul> 565 * <li>the absolute value of the first argument is greater than 1 and 566 * the second argument is negative infinity, or 567 * <li>the absolute value of the 568 * first argument is less than 1 and the second argument is positive 569 * infinity, 570 * </ul> 571 * then the result is positive zero. 572 * 573 * <li>If the absolute value of the first argument equals 1 and the 574 * second argument is infinite, then the result is NaN. 575 * 576 * <li>If 577 * <ul> 578 * <li>the first argument is positive zero and the second argument 579 * is greater than zero, or 580 * <li>the first argument is positive infinity and the second 581 * argument is less than zero, 582 * </ul> 583 * then the result is positive zero. 584 * 585 * <li>If 586 * <ul> 587 * <li>the first argument is positive zero and the second argument 588 * is less than zero, or 589 * <li>the first argument is positive infinity and the second 590 * argument is greater than zero, 591 * </ul> 592 * then the result is positive infinity. 593 * 594 * <li>If 595 * <ul> 596 * <li>the first argument is negative zero and the second argument 597 * is greater than zero but not a finite odd integer, or 598 * <li>the first argument is negative infinity and the second 599 * argument is less than zero but not a finite odd integer, 600 * </ul> 601 * then the result is positive zero. 602 * 603 * <li>If 604 * <ul> 605 * <li>the first argument is negative zero and the second argument 606 * is a positive finite odd integer, or 607 * <li>the first argument is negative infinity and the second 608 * argument is a negative finite odd integer, 609 * </ul> 610 * then the result is negative zero. 611 * 612 * <li>If 613 * <ul> 614 * <li>the first argument is negative zero and the second argument 615 * is less than zero but not a finite odd integer, or 616 * <li>the first argument is negative infinity and the second 617 * argument is greater than zero but not a finite odd integer, 618 * </ul> 619 * then the result is positive infinity. 620 * 621 * <li>If 622 * <ul> 623 * <li>the first argument is negative zero and the second argument 624 * is a negative finite odd integer, or 625 * <li>the first argument is negative infinity and the second 626 * argument is a positive finite odd integer, 627 * </ul> 628 * then the result is negative infinity. 629 * 630 * <li>If the first argument is finite and less than zero 631 * <ul> 632 * <li> if the second argument is a finite even integer, the 633 * result is equal to the result of raising the absolute value of 634 * the first argument to the power of the second argument 635 * 636 * <li>if the second argument is a finite odd integer, the result 637 * is equal to the negative of the result of raising the absolute 638 * value of the first argument to the power of the second 639 * argument 640 * 641 * <li>if the second argument is finite and not an integer, then 642 * the result is NaN. 643 * </ul> 644 * 645 * <li>If both arguments are integers, then the result is exactly equal 646 * to the mathematical result of raising the first argument to the power 647 * of the second argument if that result can in fact be represented 648 * exactly as a {@code double} value.</ul> 649 * 650 * <p>(In the foregoing descriptions, a floating-point value is 651 * considered to be an integer if and only if it is finite and a 652 * fixed point of the method {@link #ceil ceil} or, 653 * equivalently, a fixed point of the method {@link #floor 654 * floor}. A value is a fixed point of a one-argument 655 * method if and only if the result of applying the method to the 656 * value is equal to the value.) 657 * 658 * <p>The computed result must be within 1 ulp of the exact result. 659 * Results must be semi-monotonic. 660 * 661 * @param a the base. 662 * @param b the exponent. 663 * @return the value {@code a}<sup>{@code b}</sup>. 664 */ 665 @HotSpotIntrinsicCandidate 666 public static double pow(double a, double b) { 667 return StrictMath.pow(a, b); // default impl. delegates to StrictMath 668 } 669 670 /** 671 * Returns the closest {@code int} to the argument, with ties 672 * rounding to positive infinity. 673 * 674 * <p> 675 * Special cases: 676 * <ul><li>If the argument is NaN, the result is 0. 677 * <li>If the argument is negative infinity or any value less than or 678 * equal to the value of {@code Integer.MIN_VALUE}, the result is 679 * equal to the value of {@code Integer.MIN_VALUE}. 680 * <li>If the argument is positive infinity or any value greater than or 681 * equal to the value of {@code Integer.MAX_VALUE}, the result is 682 * equal to the value of {@code Integer.MAX_VALUE}.</ul> 683 * 684 * @param a a floating-point value to be rounded to an integer. 685 * @return the value of the argument rounded to the nearest 686 * {@code int} value. 687 * @see java.lang.Integer#MAX_VALUE 688 * @see java.lang.Integer#MIN_VALUE 689 */ 690 public static int round(float a) { 691 int intBits = Float.floatToRawIntBits(a); 692 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK) 693 >> (FloatConsts.SIGNIFICAND_WIDTH - 1); 694 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2 695 + FloatConsts.EXP_BIAS) - biasedExp; 696 if ((shift & -32) == 0) { // shift >= 0 && shift < 32 697 // a is a finite number such that pow(2,-32) <= ulp(a) < 1 698 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK) 699 | (FloatConsts.SIGNIF_BIT_MASK + 1)); 700 if (intBits < 0) { 701 r = -r; 702 } 703 // In the comments below each Java expression evaluates to the value 704 // the corresponding mathematical expression: 705 // (r) evaluates to a / ulp(a) 706 // (r >> shift) evaluates to floor(a * 2) 707 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) 708 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) 709 return ((r >> shift) + 1) >> 1; 710 } else { 711 // a is either 712 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2 713 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer 714 // - an infinity or NaN 715 return (int) a; 716 } 717 } 718 719 /** 720 * Returns the closest {@code long} to the argument, with ties 721 * rounding to positive infinity. 722 * 723 * <p>Special cases: 724 * <ul><li>If the argument is NaN, the result is 0. 725 * <li>If the argument is negative infinity or any value less than or 726 * equal to the value of {@code Long.MIN_VALUE}, the result is 727 * equal to the value of {@code Long.MIN_VALUE}. 728 * <li>If the argument is positive infinity or any value greater than or 729 * equal to the value of {@code Long.MAX_VALUE}, the result is 730 * equal to the value of {@code Long.MAX_VALUE}.</ul> 731 * 732 * @param a a floating-point value to be rounded to a 733 * {@code long}. 734 * @return the value of the argument rounded to the nearest 735 * {@code long} value. 736 * @see java.lang.Long#MAX_VALUE 737 * @see java.lang.Long#MIN_VALUE 738 */ 739 public static long round(double a) { 740 long longBits = Double.doubleToRawLongBits(a); 741 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK) 742 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1); 743 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2 744 + DoubleConsts.EXP_BIAS) - biasedExp; 745 if ((shift & -64) == 0) { // shift >= 0 && shift < 64 746 // a is a finite number such that pow(2,-64) <= ulp(a) < 1 747 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK) 748 | (DoubleConsts.SIGNIF_BIT_MASK + 1)); 749 if (longBits < 0) { 750 r = -r; 751 } 752 // In the comments below each Java expression evaluates to the value 753 // the corresponding mathematical expression: 754 // (r) evaluates to a / ulp(a) 755 // (r >> shift) evaluates to floor(a * 2) 756 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) 757 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) 758 return ((r >> shift) + 1) >> 1; 759 } else { 760 // a is either 761 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2 762 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer 763 // - an infinity or NaN 764 return (long) a; 765 } 766 } 767 768 private static final class RandomNumberGeneratorHolder { 769 static final Random randomNumberGenerator = new Random(); 770 } 771 772 /** 773 * Returns a {@code double} value with a positive sign, greater 774 * than or equal to {@code 0.0} and less than {@code 1.0}. 775 * Returned values are chosen pseudorandomly with (approximately) 776 * uniform distribution from that range. 777 * 778 * <p>When this method is first called, it creates a single new 779 * pseudorandom-number generator, exactly as if by the expression 780 * 781 * <blockquote>{@code new java.util.Random()}</blockquote> 782 * 783 * This new pseudorandom-number generator is used thereafter for 784 * all calls to this method and is used nowhere else. 785 * 786 * <p>This method is properly synchronized to allow correct use by 787 * more than one thread. However, if many threads need to generate 788 * pseudorandom numbers at a great rate, it may reduce contention 789 * for each thread to have its own pseudorandom-number generator. 790 * 791 * @apiNote 792 * As the largest {@code double} value less than {@code 1.0} 793 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range 794 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements 795 * 796 * <blockquote><pre>{@code 797 * double f = Math.random()/Math.nextDown(1.0); 798 * double x = x1*(1.0 - f) + x2*f; 799 * }</pre></blockquote> 800 * 801 * @return a pseudorandom {@code double} greater than or equal 802 * to {@code 0.0} and less than {@code 1.0}. 803 * @see #nextDown(double) 804 * @see Random#nextDouble() 805 */ 806 public static double random() { 807 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble(); 808 } 809 810 /** 811 * Returns the sum of its arguments, 812 * throwing an exception if the result overflows an {@code int}. 813 * 814 * @param x the first value 815 * @param y the second value 816 * @return the result 817 * @throws ArithmeticException if the result overflows an int 818 * @since 1.8 819 */ 820 @HotSpotIntrinsicCandidate 821 public static int addExact(int x, int y) { 822 int r = x + y; 823 // HD 2-12 Overflow iff both arguments have the opposite sign of the result 824 if (((x ^ r) & (y ^ r)) < 0) { 825 throw new ArithmeticException("integer overflow"); 826 } 827 return r; 828 } 829 830 /** 831 * Returns the sum of its arguments, 832 * throwing an exception if the result overflows a {@code long}. 833 * 834 * @param x the first value 835 * @param y the second value 836 * @return the result 837 * @throws ArithmeticException if the result overflows a long 838 * @since 1.8 839 */ 840 @HotSpotIntrinsicCandidate 841 public static long addExact(long x, long y) { 842 long r = x + y; 843 // HD 2-12 Overflow iff both arguments have the opposite sign of the result 844 if (((x ^ r) & (y ^ r)) < 0) { 845 throw new ArithmeticException("long overflow"); 846 } 847 return r; 848 } 849 850 /** 851 * Returns the difference of the arguments, 852 * throwing an exception if the result overflows an {@code int}. 853 * 854 * @param x the first value 855 * @param y the second value to subtract from the first 856 * @return the result 857 * @throws ArithmeticException if the result overflows an int 858 * @since 1.8 859 */ 860 @HotSpotIntrinsicCandidate 861 public static int subtractExact(int x, int y) { 862 int r = x - y; 863 // HD 2-12 Overflow iff the arguments have different signs and 864 // the sign of the result is different from the sign of x 865 if (((x ^ y) & (x ^ r)) < 0) { 866 throw new ArithmeticException("integer overflow"); 867 } 868 return r; 869 } 870 871 /** 872 * Returns the difference of the arguments, 873 * throwing an exception if the result overflows a {@code long}. 874 * 875 * @param x the first value 876 * @param y the second value to subtract from the first 877 * @return the result 878 * @throws ArithmeticException if the result overflows a long 879 * @since 1.8 880 */ 881 @HotSpotIntrinsicCandidate 882 public static long subtractExact(long x, long y) { 883 long r = x - y; 884 // HD 2-12 Overflow iff the arguments have different signs and 885 // the sign of the result is different from the sign of x 886 if (((x ^ y) & (x ^ r)) < 0) { 887 throw new ArithmeticException("long overflow"); 888 } 889 return r; 890 } 891 892 /** 893 * Returns the product of the arguments, 894 * throwing an exception if the result overflows an {@code int}. 895 * 896 * @param x the first value 897 * @param y the second value 898 * @return the result 899 * @throws ArithmeticException if the result overflows an int 900 * @since 1.8 901 */ 902 @HotSpotIntrinsicCandidate 903 public static int multiplyExact(int x, int y) { 904 long r = (long)x * (long)y; 905 if ((int)r != r) { 906 throw new ArithmeticException("integer overflow"); 907 } 908 return (int)r; 909 } 910 911 /** 912 * Returns the product of the arguments, throwing an exception if the result 913 * overflows a {@code long}. 914 * 915 * @param x the first value 916 * @param y the second value 917 * @return the result 918 * @throws ArithmeticException if the result overflows a long 919 * @since 9 920 */ 921 public static long multiplyExact(long x, int y) { 922 return multiplyExact(x, (long)y); 923 } 924 925 /** 926 * Returns the product of the arguments, 927 * throwing an exception if the result overflows a {@code long}. 928 * 929 * @param x the first value 930 * @param y the second value 931 * @return the result 932 * @throws ArithmeticException if the result overflows a long 933 * @since 1.8 934 */ 935 @HotSpotIntrinsicCandidate 936 public static long multiplyExact(long x, long y) { 937 long r = x * y; 938 long ax = Math.abs(x); 939 long ay = Math.abs(y); 940 if (((ax | ay) >>> 31 != 0)) { 941 // Some bits greater than 2^31 that might cause overflow 942 // Check the result using the divide operator 943 // and check for the special case of Long.MIN_VALUE * -1 944 if (((y != 0) && (r / y != x)) || 945 (x == Long.MIN_VALUE && y == -1)) { 946 throw new ArithmeticException("long overflow"); 947 } 948 } 949 return r; 950 } 951 952 /** 953 * Returns the argument incremented by one, throwing an exception if the 954 * result overflows an {@code int}. 955 * 956 * @param a the value to increment 957 * @return the result 958 * @throws ArithmeticException if the result overflows an int 959 * @since 1.8 960 */ 961 @HotSpotIntrinsicCandidate 962 public static int incrementExact(int a) { 963 if (a == Integer.MAX_VALUE) { 964 throw new ArithmeticException("integer overflow"); 965 } 966 967 return a + 1; 968 } 969 970 /** 971 * Returns the argument incremented by one, throwing an exception if the 972 * result overflows a {@code long}. 973 * 974 * @param a the value to increment 975 * @return the result 976 * @throws ArithmeticException if the result overflows a long 977 * @since 1.8 978 */ 979 @HotSpotIntrinsicCandidate 980 public static long incrementExact(long a) { 981 if (a == Long.MAX_VALUE) { 982 throw new ArithmeticException("long overflow"); 983 } 984 985 return a + 1L; 986 } 987 988 /** 989 * Returns the argument decremented by one, throwing an exception if the 990 * result overflows an {@code int}. 991 * 992 * @param a the value to decrement 993 * @return the result 994 * @throws ArithmeticException if the result overflows an int 995 * @since 1.8 996 */ 997 @HotSpotIntrinsicCandidate 998 public static int decrementExact(int a) { 999 if (a == Integer.MIN_VALUE) { 1000 throw new ArithmeticException("integer overflow"); 1001 } 1002 1003 return a - 1; 1004 } 1005 1006 /** 1007 * Returns the argument decremented by one, throwing an exception if the 1008 * result overflows a {@code long}. 1009 * 1010 * @param a the value to decrement 1011 * @return the result 1012 * @throws ArithmeticException if the result overflows a long 1013 * @since 1.8 1014 */ 1015 @HotSpotIntrinsicCandidate 1016 public static long decrementExact(long a) { 1017 if (a == Long.MIN_VALUE) { 1018 throw new ArithmeticException("long overflow"); 1019 } 1020 1021 return a - 1L; 1022 } 1023 1024 /** 1025 * Returns the negation of the argument, throwing an exception if the 1026 * result overflows an {@code int}. 1027 * 1028 * @param a the value to negate 1029 * @return the result 1030 * @throws ArithmeticException if the result overflows an int 1031 * @since 1.8 1032 */ 1033 @HotSpotIntrinsicCandidate 1034 public static int negateExact(int a) { 1035 if (a == Integer.MIN_VALUE) { 1036 throw new ArithmeticException("integer overflow"); 1037 } 1038 1039 return -a; 1040 } 1041 1042 /** 1043 * Returns the negation of the argument, throwing an exception if the 1044 * result overflows a {@code long}. 1045 * 1046 * @param a the value to negate 1047 * @return the result 1048 * @throws ArithmeticException if the result overflows a long 1049 * @since 1.8 1050 */ 1051 @HotSpotIntrinsicCandidate 1052 public static long negateExact(long a) { 1053 if (a == Long.MIN_VALUE) { 1054 throw new ArithmeticException("long overflow"); 1055 } 1056 1057 return -a; 1058 } 1059 1060 /** 1061 * Returns the value of the {@code long} argument; 1062 * throwing an exception if the value overflows an {@code int}. 1063 * 1064 * @param value the long value 1065 * @return the argument as an int 1066 * @throws ArithmeticException if the {@code argument} overflows an int 1067 * @since 1.8 1068 */ 1069 public static int toIntExact(long value) { 1070 if ((int)value != value) { 1071 throw new ArithmeticException("integer overflow"); 1072 } 1073 return (int)value; 1074 } 1075 1076 /** 1077 * Returns the exact mathematical product of the arguments. 1078 * 1079 * @param x the first value 1080 * @param y the second value 1081 * @return the result 1082 * @since 9 1083 */ 1084 public static long multiplyFull(int x, int y) { 1085 return (long)x * (long)y; 1086 } 1087 1088 /** 1089 * Returns as a {@code long} the most significant 64 bits of the 128-bit 1090 * product of two 64-bit factors. 1091 * 1092 * @param x the first value 1093 * @param y the second value 1094 * @return the result 1095 * @since 9 1096 */ 1097 public static long multiplyHigh(long x, long y) { 1098 if (x < 0 || y < 0) { 1099 // Use technique from section 8-2 of Henry S. Warren, Jr., 1100 // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174. 1101 long x1 = x >> 32; 1102 long x2 = x & 0xFFFFFFFFL; 1103 long y1 = y >> 32; 1104 long y2 = y & 0xFFFFFFFFL; 1105 long z2 = x2 * y2; 1106 long t = x1 * y2 + (z2 >>> 32); 1107 long z1 = t & 0xFFFFFFFFL; 1108 long z0 = t >> 32; 1109 z1 += x2 * y1; 1110 return x1 * y1 + z0 + (z1 >> 32); 1111 } else { 1112 // Use Karatsuba technique with two base 2^32 digits. 1113 long x1 = x >>> 32; 1114 long y1 = y >>> 32; 1115 long x2 = x & 0xFFFFFFFFL; 1116 long y2 = y & 0xFFFFFFFFL; 1117 long A = x1 * y1; 1118 long B = x2 * y2; 1119 long C = (x1 + x2) * (y1 + y2); 1120 long K = C - A - B; 1121 return (((B >>> 32) + K) >>> 32) + A; 1122 } 1123 } 1124 1125 /** 1126 * Returns the largest (closest to positive infinity) 1127 * {@code int} value that is less than or equal to the algebraic quotient. 1128 * There is one special case, if the dividend is the 1129 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1}, 1130 * then integer overflow occurs and 1131 * the result is equal to {@code Integer.MIN_VALUE}. 1132 * <p> 1133 * Normal integer division operates under the round to zero rounding mode 1134 * (truncation). This operation instead acts under the round toward 1135 * negative infinity (floor) rounding mode. 1136 * The floor rounding mode gives different results from truncation 1137 * when the exact result is negative. 1138 * <ul> 1139 * <li>If the signs of the arguments are the same, the results of 1140 * {@code floorDiv} and the {@code /} operator are the same. <br> 1141 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li> 1142 * <li>If the signs of the arguments are different, the quotient is negative and 1143 * {@code floorDiv} returns the integer less than or equal to the quotient 1144 * and the {@code /} operator returns the integer closest to zero.<br> 1145 * For example, {@code floorDiv(-4, 3) == -2}, 1146 * whereas {@code (-4 / 3) == -1}. 1147 * </li> 1148 * </ul> 1149 * 1150 * @param x the dividend 1151 * @param y the divisor 1152 * @return the largest (closest to positive infinity) 1153 * {@code int} value that is less than or equal to the algebraic quotient. 1154 * @throws ArithmeticException if the divisor {@code y} is zero 1155 * @see #floorMod(int, int) 1156 * @see #floor(double) 1157 * @since 1.8 1158 */ 1159 public static int floorDiv(int x, int y) { 1160 int r = x / y; 1161 // if the signs are different and modulo not zero, round down 1162 if ((x ^ y) < 0 && (r * y != x)) { 1163 r--; 1164 } 1165 return r; 1166 } 1167 1168 /** 1169 * Returns the largest (closest to positive infinity) 1170 * {@code long} value that is less than or equal to the algebraic quotient. 1171 * There is one special case, if the dividend is the 1172 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1173 * then integer overflow occurs and 1174 * the result is equal to {@code Long.MIN_VALUE}. 1175 * <p> 1176 * Normal integer division operates under the round to zero rounding mode 1177 * (truncation). This operation instead acts under the round toward 1178 * negative infinity (floor) rounding mode. 1179 * The floor rounding mode gives different results from truncation 1180 * when the exact result is negative. 1181 * <p> 1182 * For examples, see {@link #floorDiv(int, int)}. 1183 * 1184 * @param x the dividend 1185 * @param y the divisor 1186 * @return the largest (closest to positive infinity) 1187 * {@code int} value that is less than or equal to the algebraic quotient. 1188 * @throws ArithmeticException if the divisor {@code y} is zero 1189 * @see #floorMod(long, int) 1190 * @see #floor(double) 1191 * @since 9 1192 */ 1193 public static long floorDiv(long x, int y) { 1194 return floorDiv(x, (long)y); 1195 } 1196 1197 /** 1198 * Returns the largest (closest to positive infinity) 1199 * {@code long} value that is less than or equal to the algebraic quotient. 1200 * There is one special case, if the dividend is the 1201 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1202 * then integer overflow occurs and 1203 * the result is equal to {@code Long.MIN_VALUE}. 1204 * <p> 1205 * Normal integer division operates under the round to zero rounding mode 1206 * (truncation). This operation instead acts under the round toward 1207 * negative infinity (floor) rounding mode. 1208 * The floor rounding mode gives different results from truncation 1209 * when the exact result is negative. 1210 * <p> 1211 * For examples, see {@link #floorDiv(int, int)}. 1212 * 1213 * @param x the dividend 1214 * @param y the divisor 1215 * @return the largest (closest to positive infinity) 1216 * {@code long} value that is less than or equal to the algebraic quotient. 1217 * @throws ArithmeticException if the divisor {@code y} is zero 1218 * @see #floorMod(long, long) 1219 * @see #floor(double) 1220 * @since 1.8 1221 */ 1222 public static long floorDiv(long x, long y) { 1223 long r = x / y; 1224 // if the signs are different and modulo not zero, round down 1225 if ((x ^ y) < 0 && (r * y != x)) { 1226 r--; 1227 } 1228 return r; 1229 } 1230 1231 /** 1232 * Returns the floor modulus of the {@code int} arguments. 1233 * <p> 1234 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1235 * has the same sign as the divisor {@code y}, and 1236 * is in the range of {@code -abs(y) < r < +abs(y)}. 1237 * 1238 * <p> 1239 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1240 * <ul> 1241 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1242 * </ul> 1243 * <p> 1244 * The difference in values between {@code floorMod} and 1245 * the {@code %} operator is due to the difference between 1246 * {@code floorDiv} that returns the integer less than or equal to the quotient 1247 * and the {@code /} operator that returns the integer closest to zero. 1248 * <p> 1249 * Examples: 1250 * <ul> 1251 * <li>If the signs of the arguments are the same, the results 1252 * of {@code floorMod} and the {@code %} operator are the same. <br> 1253 * <ul> 1254 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li> 1255 * </ul> 1256 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br> 1257 * <ul> 1258 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li> 1259 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li> 1260 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li> 1261 * </ul> 1262 * </li> 1263 * </ul> 1264 * <p> 1265 * If the signs of arguments are unknown and a positive modulus 1266 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}. 1267 * 1268 * @param x the dividend 1269 * @param y the divisor 1270 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1271 * @throws ArithmeticException if the divisor {@code y} is zero 1272 * @see #floorDiv(int, int) 1273 * @since 1.8 1274 */ 1275 public static int floorMod(int x, int y) { 1276 return x - floorDiv(x, y) * y; 1277 } 1278 1279 /** 1280 * Returns the floor modulus of the {@code long} and {@int} arguments. 1281 * <p> 1282 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1283 * has the same sign as the divisor {@code y}, and 1284 * is in the range of {@code -abs(y) < r < +abs(y)}. 1285 * 1286 * <p> 1287 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1288 * <ul> 1289 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1290 * </ul> 1291 * <p> 1292 * For examples, see {@link #floorMod(int, int)}. 1293 * 1294 * @param x the dividend 1295 * @param y the divisor 1296 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1297 * @throws ArithmeticException if the divisor {@code y} is zero 1298 * @see #floorDiv(long, int) 1299 * @since 9 1300 */ 1301 public static int floorMod(long x, int y) { 1302 // Result cannot overflow the range of int. 1303 return (int)(x - floorDiv(x, y) * y); 1304 } 1305 1306 /** 1307 * Returns the floor modulus of the {@code long} arguments. 1308 * <p> 1309 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1310 * has the same sign as the divisor {@code y}, and 1311 * is in the range of {@code -abs(y) < r < +abs(y)}. 1312 * 1313 * <p> 1314 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1315 * <ul> 1316 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1317 * </ul> 1318 * <p> 1319 * For examples, see {@link #floorMod(int, int)}. 1320 * 1321 * @param x the dividend 1322 * @param y the divisor 1323 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1324 * @throws ArithmeticException if the divisor {@code y} is zero 1325 * @see #floorDiv(long, long) 1326 * @since 1.8 1327 */ 1328 public static long floorMod(long x, long y) { 1329 return x - floorDiv(x, y) * y; 1330 } 1331 1332 /** 1333 * Returns the absolute value of an {@code int} value. 1334 * If the argument is not negative, the argument is returned. 1335 * If the argument is negative, the negation of the argument is returned. 1336 * 1337 * <p>Note that if the argument is equal to the value of 1338 * {@link Integer#MIN_VALUE}, the most negative representable 1339 * {@code int} value, the result is that same value, which is 1340 * negative. 1341 * 1342 * @param a the argument whose absolute value is to be determined 1343 * @return the absolute value of the argument. 1344 */ 1345 public static int abs(int a) { 1346 return (a < 0) ? -a : a; 1347 } 1348 1349 /** 1350 * Returns the absolute value of a {@code long} value. 1351 * If the argument is not negative, the argument is returned. 1352 * If the argument is negative, the negation of the argument is returned. 1353 * 1354 * <p>Note that if the argument is equal to the value of 1355 * {@link Long#MIN_VALUE}, the most negative representable 1356 * {@code long} value, the result is that same value, which 1357 * is negative. 1358 * 1359 * @param a the argument whose absolute value is to be determined 1360 * @return the absolute value of the argument. 1361 */ 1362 public static long abs(long a) { 1363 return (a < 0) ? -a : a; 1364 } 1365 1366 /** 1367 * Returns the absolute value of a {@code float} value. 1368 * If the argument is not negative, the argument is returned. 1369 * If the argument is negative, the negation of the argument is returned. 1370 * Special cases: 1371 * <ul><li>If the argument is positive zero or negative zero, the 1372 * result is positive zero. 1373 * <li>If the argument is infinite, the result is positive infinity. 1374 * <li>If the argument is NaN, the result is NaN.</ul> 1375 * 1376 * @apiNote As implied by the above, one valid implementation of 1377 * this method is given by the expression below which computes a 1378 * {@code float} with the same exponent and significand as the 1379 * argument but with a guaranteed zero sign bit indicating a 1380 * positive value:<br> 1381 * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))} 1382 * 1383 * @param a the argument whose absolute value is to be determined 1384 * @return the absolute value of the argument. 1385 */ 1386 public static float abs(float a) { 1387 return (a <= 0.0F) ? 0.0F - a : a; 1388 } 1389 1390 /** 1391 * Returns the absolute value of a {@code double} value. 1392 * If the argument is not negative, the argument is returned. 1393 * If the argument is negative, the negation of the argument is returned. 1394 * Special cases: 1395 * <ul><li>If the argument is positive zero or negative zero, the result 1396 * is positive zero. 1397 * <li>If the argument is infinite, the result is positive infinity. 1398 * <li>If the argument is NaN, the result is NaN.</ul> 1399 * 1400 * @apiNote As implied by the above, one valid implementation of 1401 * this method is given by the expression below which computes a 1402 * {@code double} with the same exponent and significand as the 1403 * argument but with a guaranteed zero sign bit indicating a 1404 * positive value:<br> 1405 * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)} 1406 * 1407 * @param a the argument whose absolute value is to be determined 1408 * @return the absolute value of the argument. 1409 */ 1410 @HotSpotIntrinsicCandidate 1411 public static double abs(double a) { 1412 return (a <= 0.0D) ? 0.0D - a : a; 1413 } 1414 1415 /** 1416 * Returns the greater of two {@code int} values. That is, the 1417 * result is the argument closer to the value of 1418 * {@link Integer#MAX_VALUE}. If the arguments have the same value, 1419 * the result is that same value. 1420 * 1421 * @param a an argument. 1422 * @param b another argument. 1423 * @return the larger of {@code a} and {@code b}. 1424 */ 1425 @HotSpotIntrinsicCandidate 1426 public static int max(int a, int b) { 1427 return (a >= b) ? a : b; 1428 } 1429 1430 /** 1431 * Returns the greater of two {@code long} values. That is, the 1432 * result is the argument closer to the value of 1433 * {@link Long#MAX_VALUE}. If the arguments have the same value, 1434 * the result is that same value. 1435 * 1436 * @param a an argument. 1437 * @param b another argument. 1438 * @return the larger of {@code a} and {@code b}. 1439 */ 1440 public static long max(long a, long b) { 1441 return (a >= b) ? a : b; 1442 } 1443 1444 // Use raw bit-wise conversions on guaranteed non-NaN arguments. 1445 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f); 1446 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d); 1447 1448 /** 1449 * Returns the greater of two {@code float} values. That is, 1450 * the result is the argument closer to positive infinity. If the 1451 * arguments have the same value, the result is that same 1452 * value. If either value is NaN, then the result is NaN. Unlike 1453 * the numerical comparison operators, this method considers 1454 * negative zero to be strictly smaller than positive zero. If one 1455 * argument is positive zero and the other negative zero, the 1456 * result is positive zero. 1457 * 1458 * @param a an argument. 1459 * @param b another argument. 1460 * @return the larger of {@code a} and {@code b}. 1461 */ 1462 public static float max(float a, float b) { 1463 if (a != a) 1464 return a; // a is NaN 1465 if ((a == 0.0f) && 1466 (b == 0.0f) && 1467 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) { 1468 // Raw conversion ok since NaN can't map to -0.0. 1469 return b; 1470 } 1471 return (a >= b) ? a : b; 1472 } 1473 1474 /** 1475 * Returns the greater of two {@code double} values. That 1476 * is, the result is the argument closer to positive infinity. If 1477 * the arguments have the same value, the result is that same 1478 * value. If either value is NaN, then the result is NaN. Unlike 1479 * the numerical comparison operators, this method considers 1480 * negative zero to be strictly smaller than positive zero. If one 1481 * argument is positive zero and the other negative zero, the 1482 * result is positive zero. 1483 * 1484 * @param a an argument. 1485 * @param b another argument. 1486 * @return the larger of {@code a} and {@code b}. 1487 */ 1488 public static double max(double a, double b) { 1489 if (a != a) 1490 return a; // a is NaN 1491 if ((a == 0.0d) && 1492 (b == 0.0d) && 1493 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) { 1494 // Raw conversion ok since NaN can't map to -0.0. 1495 return b; 1496 } 1497 return (a >= b) ? a : b; 1498 } 1499 1500 /** 1501 * Returns the smaller of two {@code int} values. That is, 1502 * the result the argument closer to the value of 1503 * {@link Integer#MIN_VALUE}. If the arguments have the same 1504 * value, the result is that same value. 1505 * 1506 * @param a an argument. 1507 * @param b another argument. 1508 * @return the smaller of {@code a} and {@code b}. 1509 */ 1510 @HotSpotIntrinsicCandidate 1511 public static int min(int a, int b) { 1512 return (a <= b) ? a : b; 1513 } 1514 1515 /** 1516 * Returns the smaller of two {@code long} values. That is, 1517 * the result is the argument closer to the value of 1518 * {@link Long#MIN_VALUE}. If the arguments have the same 1519 * value, the result is that same value. 1520 * 1521 * @param a an argument. 1522 * @param b another argument. 1523 * @return the smaller of {@code a} and {@code b}. 1524 */ 1525 public static long min(long a, long b) { 1526 return (a <= b) ? a : b; 1527 } 1528 1529 /** 1530 * Returns the smaller of two {@code float} values. That is, 1531 * the result is the value closer to negative infinity. If the 1532 * arguments have the same value, the result is that same 1533 * value. If either value is NaN, then the result is NaN. Unlike 1534 * the numerical comparison operators, this method considers 1535 * negative zero to be strictly smaller than positive zero. If 1536 * one argument is positive zero and the other is negative zero, 1537 * the result is negative zero. 1538 * 1539 * @param a an argument. 1540 * @param b another argument. 1541 * @return the smaller of {@code a} and {@code b}. 1542 */ 1543 public static float min(float a, float b) { 1544 if (a != a) 1545 return a; // a is NaN 1546 if ((a == 0.0f) && 1547 (b == 0.0f) && 1548 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) { 1549 // Raw conversion ok since NaN can't map to -0.0. 1550 return b; 1551 } 1552 return (a <= b) ? a : b; 1553 } 1554 1555 /** 1556 * Returns the smaller of two {@code double} values. That 1557 * is, the result is the value closer to negative infinity. If the 1558 * arguments have the same value, the result is that same 1559 * value. If either value is NaN, then the result is NaN. Unlike 1560 * the numerical comparison operators, this method considers 1561 * negative zero to be strictly smaller than positive zero. If one 1562 * argument is positive zero and the other is negative zero, the 1563 * result is negative zero. 1564 * 1565 * @param a an argument. 1566 * @param b another argument. 1567 * @return the smaller of {@code a} and {@code b}. 1568 */ 1569 public static double min(double a, double b) { 1570 if (a != a) 1571 return a; // a is NaN 1572 if ((a == 0.0d) && 1573 (b == 0.0d) && 1574 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) { 1575 // Raw conversion ok since NaN can't map to -0.0. 1576 return b; 1577 } 1578 return (a <= b) ? a : b; 1579 } 1580 1581 /** 1582 * Returns the fused multiply add of the three arguments; that is, 1583 * returns the exact product of the first two arguments summed 1584 * with the third argument and then rounded once to the nearest 1585 * {@code double}. 1586 * 1587 * The rounding is done using the {@linkplain 1588 * java.math.RoundingMode#HALF_EVEN round to nearest even 1589 * rounding mode}. 1590 * 1591 * In contrast, if {@code a * b + c} is evaluated as a regular 1592 * floating-point expression, two rounding errors are involved, 1593 * the first for the multiply operation, the second for the 1594 * addition operation. 1595 * 1596 * <p>Special cases: 1597 * <ul> 1598 * <li> If any argument is NaN, the result is NaN. 1599 * 1600 * <li> If one of the first two arguments is infinite and the 1601 * other is zero, the result is NaN. 1602 * 1603 * <li> If the exact product of the first two arguments is infinite 1604 * (in other words, at least one of the arguments is infinite and 1605 * the other is neither zero nor NaN) and the third argument is an 1606 * infinity of the opposite sign, the result is NaN. 1607 * 1608 * </ul> 1609 * 1610 * <p>Note that {@code fma(a, 1.0, c)} returns the same 1611 * result as ({@code a + c}). However, 1612 * {@code fma(a, b, +0.0)} does <em>not</em> always return the 1613 * same result as ({@code a * b}) since 1614 * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while 1615 * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is 1616 * equivalent to ({@code a * b}) however. 1617 * 1618 * @apiNote This method corresponds to the fusedMultiplyAdd 1619 * operation defined in IEEE 754-2008. 1620 * 1621 * @param a a value 1622 * @param b a value 1623 * @param c a value 1624 * 1625 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1626 * computed, as if with unlimited range and precision, and rounded 1627 * once to the nearest {@code double} value 1628 * 1629 * @since 9 1630 */ 1631 @HotSpotIntrinsicCandidate 1632 public static double fma(double a, double b, double c) { 1633 /* 1634 * Infinity and NaN arithmetic is not quite the same with two 1635 * roundings as opposed to just one so the simple expression 1636 * "a * b + c" cannot always be used to compute the correct 1637 * result. With two roundings, the product can overflow and 1638 * if the addend is infinite, a spurious NaN can be produced 1639 * if the infinity from the overflow and the infinite addend 1640 * have opposite signs. 1641 */ 1642 1643 // First, screen for and handle non-finite input values whose 1644 // arithmetic is not supported by BigDecimal. 1645 if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) { 1646 return Double.NaN; 1647 } else { // All inputs non-NaN 1648 boolean infiniteA = Double.isInfinite(a); 1649 boolean infiniteB = Double.isInfinite(b); 1650 boolean infiniteC = Double.isInfinite(c); 1651 double result; 1652 1653 if (infiniteA || infiniteB || infiniteC) { 1654 if (infiniteA && b == 0.0 || 1655 infiniteB && a == 0.0 ) { 1656 return Double.NaN; 1657 } 1658 // Store product in a double field to cause an 1659 // overflow even if non-strictfp evaluation is being 1660 // used. 1661 double product = a * b; 1662 if (Double.isInfinite(product) && !infiniteA && !infiniteB) { 1663 // Intermediate overflow; might cause a 1664 // spurious NaN if added to infinite c. 1665 assert Double.isInfinite(c); 1666 return c; 1667 } else { 1668 result = product + c; 1669 assert !Double.isFinite(result); 1670 return result; 1671 } 1672 } else { // All inputs finite 1673 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b)); 1674 if (c == 0.0) { // Positive or negative zero 1675 // If the product is an exact zero, use a 1676 // floating-point expression to compute the sign 1677 // of the zero final result. The product is an 1678 // exact zero if and only if at least one of a and 1679 // b is zero. 1680 if (a == 0.0 || b == 0.0) { 1681 return a * b + c; 1682 } else { 1683 // The sign of a zero addend doesn't matter if 1684 // the product is nonzero. The sign of a zero 1685 // addend is not factored in the result if the 1686 // exact product is nonzero but underflows to 1687 // zero; see IEEE-754 2008 section 6.3 "The 1688 // sign bit". 1689 return product.doubleValue(); 1690 } 1691 } else { 1692 return product.add(new BigDecimal(c)).doubleValue(); 1693 } 1694 } 1695 } 1696 } 1697 1698 /** 1699 * Returns the fused multiply add of the three arguments; that is, 1700 * returns the exact product of the first two arguments summed 1701 * with the third argument and then rounded once to the nearest 1702 * {@code float}. 1703 * 1704 * The rounding is done using the {@linkplain 1705 * java.math.RoundingMode#HALF_EVEN round to nearest even 1706 * rounding mode}. 1707 * 1708 * In contrast, if {@code a * b + c} is evaluated as a regular 1709 * floating-point expression, two rounding errors are involved, 1710 * the first for the multiply operation, the second for the 1711 * addition operation. 1712 * 1713 * <p>Special cases: 1714 * <ul> 1715 * <li> If any argument is NaN, the result is NaN. 1716 * 1717 * <li> If one of the first two arguments is infinite and the 1718 * other is zero, the result is NaN. 1719 * 1720 * <li> If the exact product of the first two arguments is infinite 1721 * (in other words, at least one of the arguments is infinite and 1722 * the other is neither zero nor NaN) and the third argument is an 1723 * infinity of the opposite sign, the result is NaN. 1724 * 1725 * </ul> 1726 * 1727 * <p>Note that {@code fma(a, 1.0f, c)} returns the same 1728 * result as ({@code a + c}). However, 1729 * {@code fma(a, b, +0.0f)} does <em>not</em> always return the 1730 * same result as ({@code a * b}) since 1731 * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while 1732 * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is 1733 * equivalent to ({@code a * b}) however. 1734 * 1735 * @apiNote This method corresponds to the fusedMultiplyAdd 1736 * operation defined in IEEE 754-2008. 1737 * 1738 * @param a a value 1739 * @param b a value 1740 * @param c a value 1741 * 1742 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1743 * computed, as if with unlimited range and precision, and rounded 1744 * once to the nearest {@code float} value 1745 * 1746 * @since 9 1747 */ 1748 @HotSpotIntrinsicCandidate 1749 public static float fma(float a, float b, float c) { 1750 /* 1751 * Since the double format has more than twice the precision 1752 * of the float format, the multiply of a * b is exact in 1753 * double. The add of c to the product then incurs one 1754 * rounding error. Since the double format moreover has more 1755 * than (2p + 2) precision bits compared to the p bits of the 1756 * float format, the two roundings of (a * b + c), first to 1757 * the double format and then secondarily to the float format, 1758 * are equivalent to rounding the intermediate result directly 1759 * to the float format. 1760 * 1761 * In terms of strictfp vs default-fp concerns related to 1762 * overflow and underflow, since 1763 * 1764 * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE 1765 * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE 1766 * 1767 * neither the multiply nor add will overflow or underflow in 1768 * double. Therefore, it is not necessary for this method to 1769 * be declared strictfp to have reproducible 1770 * behavior. However, it is necessary to explicitly store down 1771 * to a float variable to avoid returning a value in the float 1772 * extended value set. 1773 */ 1774 float result = (float)(((double) a * (double) b ) + (double) c); 1775 return result; 1776 } 1777 1778 /** 1779 * Returns the size of an ulp of the argument. An ulp, unit in 1780 * the last place, of a {@code double} value is the positive 1781 * distance between this floating-point value and the {@code 1782 * double} value next larger in magnitude. Note that for non-NaN 1783 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1784 * 1785 * <p>Special Cases: 1786 * <ul> 1787 * <li> If the argument is NaN, then the result is NaN. 1788 * <li> If the argument is positive or negative infinity, then the 1789 * result is positive infinity. 1790 * <li> If the argument is positive or negative zero, then the result is 1791 * {@code Double.MIN_VALUE}. 1792 * <li> If the argument is ±{@code Double.MAX_VALUE}, then 1793 * the result is equal to 2<sup>971</sup>. 1794 * </ul> 1795 * 1796 * @param d the floating-point value whose ulp is to be returned 1797 * @return the size of an ulp of the argument 1798 * @author Joseph D. Darcy 1799 * @since 1.5 1800 */ 1801 public static double ulp(double d) { 1802 int exp = getExponent(d); 1803 1804 switch(exp) { 1805 case Double.MAX_EXPONENT + 1: // NaN or infinity 1806 return Math.abs(d); 1807 1808 case Double.MIN_EXPONENT - 1: // zero or subnormal 1809 return Double.MIN_VALUE; 1810 1811 default: 1812 assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT; 1813 1814 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1815 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1); 1816 if (exp >= Double.MIN_EXPONENT) { 1817 return powerOfTwoD(exp); 1818 } 1819 else { 1820 // return a subnormal result; left shift integer 1821 // representation of Double.MIN_VALUE appropriate 1822 // number of positions 1823 return Double.longBitsToDouble(1L << 1824 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) )); 1825 } 1826 } 1827 } 1828 1829 /** 1830 * Returns the size of an ulp of the argument. An ulp, unit in 1831 * the last place, of a {@code float} value is the positive 1832 * distance between this floating-point value and the {@code 1833 * float} value next larger in magnitude. Note that for non-NaN 1834 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1835 * 1836 * <p>Special Cases: 1837 * <ul> 1838 * <li> If the argument is NaN, then the result is NaN. 1839 * <li> If the argument is positive or negative infinity, then the 1840 * result is positive infinity. 1841 * <li> If the argument is positive or negative zero, then the result is 1842 * {@code Float.MIN_VALUE}. 1843 * <li> If the argument is ±{@code Float.MAX_VALUE}, then 1844 * the result is equal to 2<sup>104</sup>. 1845 * </ul> 1846 * 1847 * @param f the floating-point value whose ulp is to be returned 1848 * @return the size of an ulp of the argument 1849 * @author Joseph D. Darcy 1850 * @since 1.5 1851 */ 1852 public static float ulp(float f) { 1853 int exp = getExponent(f); 1854 1855 switch(exp) { 1856 case Float.MAX_EXPONENT+1: // NaN or infinity 1857 return Math.abs(f); 1858 1859 case Float.MIN_EXPONENT-1: // zero or subnormal 1860 return Float.MIN_VALUE; 1861 1862 default: 1863 assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT; 1864 1865 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1866 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1); 1867 if (exp >= Float.MIN_EXPONENT) { 1868 return powerOfTwoF(exp); 1869 } else { 1870 // return a subnormal result; left shift integer 1871 // representation of FloatConsts.MIN_VALUE appropriate 1872 // number of positions 1873 return Float.intBitsToFloat(1 << 1874 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) )); 1875 } 1876 } 1877 } 1878 1879 /** 1880 * Returns the signum function of the argument; zero if the argument 1881 * is zero, 1.0 if the argument is greater than zero, -1.0 if the 1882 * argument is less than zero. 1883 * 1884 * <p>Special Cases: 1885 * <ul> 1886 * <li> If the argument is NaN, then the result is NaN. 1887 * <li> If the argument is positive zero or negative zero, then the 1888 * result is the same as the argument. 1889 * </ul> 1890 * 1891 * @param d the floating-point value whose signum is to be returned 1892 * @return the signum function of the argument 1893 * @author Joseph D. Darcy 1894 * @since 1.5 1895 */ 1896 public static double signum(double d) { 1897 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d); 1898 } 1899 1900 /** 1901 * Returns the signum function of the argument; zero if the argument 1902 * is zero, 1.0f if the argument is greater than zero, -1.0f if the 1903 * argument is less than zero. 1904 * 1905 * <p>Special Cases: 1906 * <ul> 1907 * <li> If the argument is NaN, then the result is NaN. 1908 * <li> If the argument is positive zero or negative zero, then the 1909 * result is the same as the argument. 1910 * </ul> 1911 * 1912 * @param f the floating-point value whose signum is to be returned 1913 * @return the signum function of the argument 1914 * @author Joseph D. Darcy 1915 * @since 1.5 1916 */ 1917 public static float signum(float f) { 1918 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f); 1919 } 1920 1921 /** 1922 * Returns the hyperbolic sine of a {@code double} value. 1923 * The hyperbolic sine of <i>x</i> is defined to be 1924 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 1925 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1926 * 1927 * <p>Special cases: 1928 * <ul> 1929 * 1930 * <li>If the argument is NaN, then the result is NaN. 1931 * 1932 * <li>If the argument is infinite, then the result is an infinity 1933 * with the same sign as the argument. 1934 * 1935 * <li>If the argument is zero, then the result is a zero with the 1936 * same sign as the argument. 1937 * 1938 * </ul> 1939 * 1940 * <p>The computed result must be within 2.5 ulps of the exact result. 1941 * 1942 * @param x The number whose hyperbolic sine is to be returned. 1943 * @return The hyperbolic sine of {@code x}. 1944 * @since 1.5 1945 */ 1946 public static double sinh(double x) { 1947 return StrictMath.sinh(x); 1948 } 1949 1950 /** 1951 * Returns the hyperbolic cosine of a {@code double} value. 1952 * The hyperbolic cosine of <i>x</i> is defined to be 1953 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 1954 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1955 * 1956 * <p>Special cases: 1957 * <ul> 1958 * 1959 * <li>If the argument is NaN, then the result is NaN. 1960 * 1961 * <li>If the argument is infinite, then the result is positive 1962 * infinity. 1963 * 1964 * <li>If the argument is zero, then the result is {@code 1.0}. 1965 * 1966 * </ul> 1967 * 1968 * <p>The computed result must be within 2.5 ulps of the exact result. 1969 * 1970 * @param x The number whose hyperbolic cosine is to be returned. 1971 * @return The hyperbolic cosine of {@code x}. 1972 * @since 1.5 1973 */ 1974 public static double cosh(double x) { 1975 return StrictMath.cosh(x); 1976 } 1977 1978 /** 1979 * Returns the hyperbolic tangent of a {@code double} value. 1980 * The hyperbolic tangent of <i>x</i> is defined to be 1981 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), 1982 * in other words, {@linkplain Math#sinh 1983 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note 1984 * that the absolute value of the exact tanh is always less than 1985 * 1. 1986 * 1987 * <p>Special cases: 1988 * <ul> 1989 * 1990 * <li>If the argument is NaN, then the result is NaN. 1991 * 1992 * <li>If the argument is zero, then the result is a zero with the 1993 * same sign as the argument. 1994 * 1995 * <li>If the argument is positive infinity, then the result is 1996 * {@code +1.0}. 1997 * 1998 * <li>If the argument is negative infinity, then the result is 1999 * {@code -1.0}. 2000 * 2001 * </ul> 2002 * 2003 * <p>The computed result must be within 2.5 ulps of the exact result. 2004 * The result of {@code tanh} for any finite input must have 2005 * an absolute value less than or equal to 1. Note that once the 2006 * exact result of tanh is within 1/2 of an ulp of the limit value 2007 * of ±1, correctly signed ±{@code 1.0} should 2008 * be returned. 2009 * 2010 * @param x The number whose hyperbolic tangent is to be returned. 2011 * @return The hyperbolic tangent of {@code x}. 2012 * @since 1.5 2013 */ 2014 public static double tanh(double x) { 2015 return StrictMath.tanh(x); 2016 } 2017 2018 /** 2019 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2020 * without intermediate overflow or underflow. 2021 * 2022 * <p>Special cases: 2023 * <ul> 2024 * 2025 * <li> If either argument is infinite, then the result 2026 * is positive infinity. 2027 * 2028 * <li> If either argument is NaN and neither argument is infinite, 2029 * then the result is NaN. 2030 * 2031 * </ul> 2032 * 2033 * <p>The computed result must be within 1 ulp of the exact 2034 * result. If one parameter is held constant, the results must be 2035 * semi-monotonic in the other parameter. 2036 * 2037 * @param x a value 2038 * @param y a value 2039 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2040 * without intermediate overflow or underflow 2041 * @since 1.5 2042 */ 2043 public static double hypot(double x, double y) { 2044 return StrictMath.hypot(x, y); 2045 } 2046 2047 /** 2048 * Returns <i>e</i><sup>x</sup> -1. Note that for values of 2049 * <i>x</i> near 0, the exact sum of 2050 * {@code expm1(x)} + 1 is much closer to the true 2051 * result of <i>e</i><sup>x</sup> than {@code exp(x)}. 2052 * 2053 * <p>Special cases: 2054 * <ul> 2055 * <li>If the argument is NaN, the result is NaN. 2056 * 2057 * <li>If the argument is positive infinity, then the result is 2058 * positive infinity. 2059 * 2060 * <li>If the argument is negative infinity, then the result is 2061 * -1.0. 2062 * 2063 * <li>If the argument is zero, then the result is a zero with the 2064 * same sign as the argument. 2065 * 2066 * </ul> 2067 * 2068 * <p>The computed result must be within 1 ulp of the exact result. 2069 * Results must be semi-monotonic. The result of 2070 * {@code expm1} for any finite input must be greater than or 2071 * equal to {@code -1.0}. Note that once the exact result of 2072 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 2073 * ulp of the limit value -1, {@code -1.0} should be 2074 * returned. 2075 * 2076 * @param x the exponent to raise <i>e</i> to in the computation of 2077 * <i>e</i><sup>{@code x}</sup> -1. 2078 * @return the value <i>e</i><sup>{@code x}</sup> - 1. 2079 * @since 1.5 2080 */ 2081 public static double expm1(double x) { 2082 return StrictMath.expm1(x); 2083 } 2084 2085 /** 2086 * Returns the natural logarithm of the sum of the argument and 1. 2087 * Note that for small values {@code x}, the result of 2088 * {@code log1p(x)} is much closer to the true result of ln(1 2089 * + {@code x}) than the floating-point evaluation of 2090 * {@code log(1.0+x)}. 2091 * 2092 * <p>Special cases: 2093 * 2094 * <ul> 2095 * 2096 * <li>If the argument is NaN or less than -1, then the result is 2097 * NaN. 2098 * 2099 * <li>If the argument is positive infinity, then the result is 2100 * positive infinity. 2101 * 2102 * <li>If the argument is negative one, then the result is 2103 * negative infinity. 2104 * 2105 * <li>If the argument is zero, then the result is a zero with the 2106 * same sign as the argument. 2107 * 2108 * </ul> 2109 * 2110 * <p>The computed result must be within 1 ulp of the exact result. 2111 * Results must be semi-monotonic. 2112 * 2113 * @param x a value 2114 * @return the value ln({@code x} + 1), the natural 2115 * log of {@code x} + 1 2116 * @since 1.5 2117 */ 2118 public static double log1p(double x) { 2119 return StrictMath.log1p(x); 2120 } 2121 2122 /** 2123 * Returns the first floating-point argument with the sign of the 2124 * second floating-point argument. Note that unlike the {@link 2125 * StrictMath#copySign(double, double) StrictMath.copySign} 2126 * method, this method does not require NaN {@code sign} 2127 * arguments to be treated as positive values; implementations are 2128 * permitted to treat some NaN arguments as positive and other NaN 2129 * arguments as negative to allow greater performance. 2130 * 2131 * @param magnitude the parameter providing the magnitude of the result 2132 * @param sign the parameter providing the sign of the result 2133 * @return a value with the magnitude of {@code magnitude} 2134 * and the sign of {@code sign}. 2135 * @since 1.6 2136 */ 2137 public static double copySign(double magnitude, double sign) { 2138 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) & 2139 (DoubleConsts.SIGN_BIT_MASK)) | 2140 (Double.doubleToRawLongBits(magnitude) & 2141 (DoubleConsts.EXP_BIT_MASK | 2142 DoubleConsts.SIGNIF_BIT_MASK))); 2143 } 2144 2145 /** 2146 * Returns the first floating-point argument with the sign of the 2147 * second floating-point argument. Note that unlike the {@link 2148 * StrictMath#copySign(float, float) StrictMath.copySign} 2149 * method, this method does not require NaN {@code sign} 2150 * arguments to be treated as positive values; implementations are 2151 * permitted to treat some NaN arguments as positive and other NaN 2152 * arguments as negative to allow greater performance. 2153 * 2154 * @param magnitude the parameter providing the magnitude of the result 2155 * @param sign the parameter providing the sign of the result 2156 * @return a value with the magnitude of {@code magnitude} 2157 * and the sign of {@code sign}. 2158 * @since 1.6 2159 */ 2160 public static float copySign(float magnitude, float sign) { 2161 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) & 2162 (FloatConsts.SIGN_BIT_MASK)) | 2163 (Float.floatToRawIntBits(magnitude) & 2164 (FloatConsts.EXP_BIT_MASK | 2165 FloatConsts.SIGNIF_BIT_MASK))); 2166 } 2167 2168 /** 2169 * Returns the unbiased exponent used in the representation of a 2170 * {@code float}. Special cases: 2171 * 2172 * <ul> 2173 * <li>If the argument is NaN or infinite, then the result is 2174 * {@link Float#MAX_EXPONENT} + 1. 2175 * <li>If the argument is zero or subnormal, then the result is 2176 * {@link Float#MIN_EXPONENT} -1. 2177 * </ul> 2178 * @param f a {@code float} value 2179 * @return the unbiased exponent of the argument 2180 * @since 1.6 2181 */ 2182 public static int getExponent(float f) { 2183 /* 2184 * Bitwise convert f to integer, mask out exponent bits, shift 2185 * to the right and then subtract out float's bias adjust to 2186 * get true exponent value 2187 */ 2188 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >> 2189 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS; 2190 } 2191 2192 /** 2193 * Returns the unbiased exponent used in the representation of a 2194 * {@code double}. Special cases: 2195 * 2196 * <ul> 2197 * <li>If the argument is NaN or infinite, then the result is 2198 * {@link Double#MAX_EXPONENT} + 1. 2199 * <li>If the argument is zero or subnormal, then the result is 2200 * {@link Double#MIN_EXPONENT} -1. 2201 * </ul> 2202 * @param d a {@code double} value 2203 * @return the unbiased exponent of the argument 2204 * @since 1.6 2205 */ 2206 public static int getExponent(double d) { 2207 /* 2208 * Bitwise convert d to long, mask out exponent bits, shift 2209 * to the right and then subtract out double's bias adjust to 2210 * get true exponent value. 2211 */ 2212 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >> 2213 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS); 2214 } 2215 2216 /** 2217 * Returns the floating-point number adjacent to the first 2218 * argument in the direction of the second argument. If both 2219 * arguments compare as equal the second argument is returned. 2220 * 2221 * <p> 2222 * Special cases: 2223 * <ul> 2224 * <li> If either argument is a NaN, then NaN is returned. 2225 * 2226 * <li> If both arguments are signed zeros, {@code direction} 2227 * is returned unchanged (as implied by the requirement of 2228 * returning the second argument if the arguments compare as 2229 * equal). 2230 * 2231 * <li> If {@code start} is 2232 * ±{@link Double#MIN_VALUE} and {@code direction} 2233 * has a value such that the result should have a smaller 2234 * magnitude, then a zero with the same sign as {@code start} 2235 * is returned. 2236 * 2237 * <li> If {@code start} is infinite and 2238 * {@code direction} has a value such that the result should 2239 * have a smaller magnitude, {@link Double#MAX_VALUE} with the 2240 * same sign as {@code start} is returned. 2241 * 2242 * <li> If {@code start} is equal to ± 2243 * {@link Double#MAX_VALUE} and {@code direction} has a 2244 * value such that the result should have a larger magnitude, an 2245 * infinity with same sign as {@code start} is returned. 2246 * </ul> 2247 * 2248 * @param start starting floating-point value 2249 * @param direction value indicating which of 2250 * {@code start}'s neighbors or {@code start} should 2251 * be returned 2252 * @return The floating-point number adjacent to {@code start} in the 2253 * direction of {@code direction}. 2254 * @since 1.6 2255 */ 2256 public static double nextAfter(double start, double direction) { 2257 /* 2258 * The cases: 2259 * 2260 * nextAfter(+infinity, 0) == MAX_VALUE 2261 * nextAfter(+infinity, +infinity) == +infinity 2262 * nextAfter(-infinity, 0) == -MAX_VALUE 2263 * nextAfter(-infinity, -infinity) == -infinity 2264 * 2265 * are naturally handled without any additional testing 2266 */ 2267 2268 /* 2269 * IEEE 754 floating-point numbers are lexicographically 2270 * ordered if treated as signed-magnitude integers. 2271 * Since Java's integers are two's complement, 2272 * incrementing the two's complement representation of a 2273 * logically negative floating-point value *decrements* 2274 * the signed-magnitude representation. Therefore, when 2275 * the integer representation of a floating-point value 2276 * is negative, the adjustment to the representation is in 2277 * the opposite direction from what would initially be expected. 2278 */ 2279 2280 // Branch to descending case first as it is more costly than ascending 2281 // case due to start != 0.0d conditional. 2282 if (start > direction) { // descending 2283 if (start != 0.0d) { 2284 final long transducer = Double.doubleToRawLongBits(start); 2285 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L)); 2286 } else { // start == 0.0d && direction < 0.0d 2287 return -Double.MIN_VALUE; 2288 } 2289 } else if (start < direction) { // ascending 2290 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2291 // then bitwise convert start to integer. 2292 final long transducer = Double.doubleToRawLongBits(start + 0.0d); 2293 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2294 } else if (start == direction) { 2295 return direction; 2296 } else { // isNaN(start) || isNaN(direction) 2297 return start + direction; 2298 } 2299 } 2300 2301 /** 2302 * Returns the floating-point number adjacent to the first 2303 * argument in the direction of the second argument. If both 2304 * arguments compare as equal a value equivalent to the second argument 2305 * is returned. 2306 * 2307 * <p> 2308 * Special cases: 2309 * <ul> 2310 * <li> If either argument is a NaN, then NaN is returned. 2311 * 2312 * <li> If both arguments are signed zeros, a value equivalent 2313 * to {@code direction} is returned. 2314 * 2315 * <li> If {@code start} is 2316 * ±{@link Float#MIN_VALUE} and {@code direction} 2317 * has a value such that the result should have a smaller 2318 * magnitude, then a zero with the same sign as {@code start} 2319 * is returned. 2320 * 2321 * <li> If {@code start} is infinite and 2322 * {@code direction} has a value such that the result should 2323 * have a smaller magnitude, {@link Float#MAX_VALUE} with the 2324 * same sign as {@code start} is returned. 2325 * 2326 * <li> If {@code start} is equal to ± 2327 * {@link Float#MAX_VALUE} and {@code direction} has a 2328 * value such that the result should have a larger magnitude, an 2329 * infinity with same sign as {@code start} is returned. 2330 * </ul> 2331 * 2332 * @param start starting floating-point value 2333 * @param direction value indicating which of 2334 * {@code start}'s neighbors or {@code start} should 2335 * be returned 2336 * @return The floating-point number adjacent to {@code start} in the 2337 * direction of {@code direction}. 2338 * @since 1.6 2339 */ 2340 public static float nextAfter(float start, double direction) { 2341 /* 2342 * The cases: 2343 * 2344 * nextAfter(+infinity, 0) == MAX_VALUE 2345 * nextAfter(+infinity, +infinity) == +infinity 2346 * nextAfter(-infinity, 0) == -MAX_VALUE 2347 * nextAfter(-infinity, -infinity) == -infinity 2348 * 2349 * are naturally handled without any additional testing 2350 */ 2351 2352 /* 2353 * IEEE 754 floating-point numbers are lexicographically 2354 * ordered if treated as signed-magnitude integers. 2355 * Since Java's integers are two's complement, 2356 * incrementing the two's complement representation of a 2357 * logically negative floating-point value *decrements* 2358 * the signed-magnitude representation. Therefore, when 2359 * the integer representation of a floating-point value 2360 * is negative, the adjustment to the representation is in 2361 * the opposite direction from what would initially be expected. 2362 */ 2363 2364 // Branch to descending case first as it is more costly than ascending 2365 // case due to start != 0.0f conditional. 2366 if (start > direction) { // descending 2367 if (start != 0.0f) { 2368 final int transducer = Float.floatToRawIntBits(start); 2369 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1)); 2370 } else { // start == 0.0f && direction < 0.0f 2371 return -Float.MIN_VALUE; 2372 } 2373 } else if (start < direction) { // ascending 2374 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2375 // then bitwise convert start to integer. 2376 final int transducer = Float.floatToRawIntBits(start + 0.0f); 2377 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2378 } else if (start == direction) { 2379 return (float)direction; 2380 } else { // isNaN(start) || isNaN(direction) 2381 return start + (float)direction; 2382 } 2383 } 2384 2385 /** 2386 * Returns the floating-point value adjacent to {@code d} in 2387 * the direction of positive infinity. This method is 2388 * semantically equivalent to {@code nextAfter(d, 2389 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp} 2390 * implementation may run faster than its equivalent 2391 * {@code nextAfter} call. 2392 * 2393 * <p>Special Cases: 2394 * <ul> 2395 * <li> If the argument is NaN, the result is NaN. 2396 * 2397 * <li> If the argument is positive infinity, the result is 2398 * positive infinity. 2399 * 2400 * <li> If the argument is zero, the result is 2401 * {@link Double#MIN_VALUE} 2402 * 2403 * </ul> 2404 * 2405 * @param d starting floating-point value 2406 * @return The adjacent floating-point value closer to positive 2407 * infinity. 2408 * @since 1.6 2409 */ 2410 public static double nextUp(double d) { 2411 // Use a single conditional and handle the likely cases first. 2412 if (d < Double.POSITIVE_INFINITY) { 2413 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2414 final long transducer = Double.doubleToRawLongBits(d + 0.0D); 2415 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2416 } else { // d is NaN or +Infinity 2417 return d; 2418 } 2419 } 2420 2421 /** 2422 * Returns the floating-point value adjacent to {@code f} in 2423 * the direction of positive infinity. This method is 2424 * semantically equivalent to {@code nextAfter(f, 2425 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp} 2426 * implementation may run faster than its equivalent 2427 * {@code nextAfter} call. 2428 * 2429 * <p>Special Cases: 2430 * <ul> 2431 * <li> If the argument is NaN, the result is NaN. 2432 * 2433 * <li> If the argument is positive infinity, the result is 2434 * positive infinity. 2435 * 2436 * <li> If the argument is zero, the result is 2437 * {@link Float#MIN_VALUE} 2438 * 2439 * </ul> 2440 * 2441 * @param f starting floating-point value 2442 * @return The adjacent floating-point value closer to positive 2443 * infinity. 2444 * @since 1.6 2445 */ 2446 public static float nextUp(float f) { 2447 // Use a single conditional and handle the likely cases first. 2448 if (f < Float.POSITIVE_INFINITY) { 2449 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2450 final int transducer = Float.floatToRawIntBits(f + 0.0F); 2451 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2452 } else { // f is NaN or +Infinity 2453 return f; 2454 } 2455 } 2456 2457 /** 2458 * Returns the floating-point value adjacent to {@code d} in 2459 * the direction of negative infinity. This method is 2460 * semantically equivalent to {@code nextAfter(d, 2461 * Double.NEGATIVE_INFINITY)}; however, a 2462 * {@code nextDown} implementation may run faster than its 2463 * equivalent {@code nextAfter} call. 2464 * 2465 * <p>Special Cases: 2466 * <ul> 2467 * <li> If the argument is NaN, the result is NaN. 2468 * 2469 * <li> If the argument is negative infinity, the result is 2470 * negative infinity. 2471 * 2472 * <li> If the argument is zero, the result is 2473 * {@code -Double.MIN_VALUE} 2474 * 2475 * </ul> 2476 * 2477 * @param d starting floating-point value 2478 * @return The adjacent floating-point value closer to negative 2479 * infinity. 2480 * @since 1.8 2481 */ 2482 public static double nextDown(double d) { 2483 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY) 2484 return d; 2485 else { 2486 if (d == 0.0) 2487 return -Double.MIN_VALUE; 2488 else 2489 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) + 2490 ((d > 0.0d)?-1L:+1L)); 2491 } 2492 } 2493 2494 /** 2495 * Returns the floating-point value adjacent to {@code f} in 2496 * the direction of negative infinity. This method is 2497 * semantically equivalent to {@code nextAfter(f, 2498 * Float.NEGATIVE_INFINITY)}; however, a 2499 * {@code nextDown} implementation may run faster than its 2500 * equivalent {@code nextAfter} call. 2501 * 2502 * <p>Special Cases: 2503 * <ul> 2504 * <li> If the argument is NaN, the result is NaN. 2505 * 2506 * <li> If the argument is negative infinity, the result is 2507 * negative infinity. 2508 * 2509 * <li> If the argument is zero, the result is 2510 * {@code -Float.MIN_VALUE} 2511 * 2512 * </ul> 2513 * 2514 * @param f starting floating-point value 2515 * @return The adjacent floating-point value closer to negative 2516 * infinity. 2517 * @since 1.8 2518 */ 2519 public static float nextDown(float f) { 2520 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY) 2521 return f; 2522 else { 2523 if (f == 0.0f) 2524 return -Float.MIN_VALUE; 2525 else 2526 return Float.intBitsToFloat(Float.floatToRawIntBits(f) + 2527 ((f > 0.0f)?-1:+1)); 2528 } 2529 } 2530 2531 /** 2532 * Returns {@code d} × 2533 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2534 * by a single correctly rounded floating-point multiply to a 2535 * member of the double value set. See the Java 2536 * Language Specification for a discussion of floating-point 2537 * value sets. If the exponent of the result is between {@link 2538 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the 2539 * answer is calculated exactly. If the exponent of the result 2540 * would be larger than {@code Double.MAX_EXPONENT}, an 2541 * infinity is returned. Note that if the result is subnormal, 2542 * precision may be lost; that is, when {@code scalb(x, n)} 2543 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2544 * <i>x</i>. When the result is non-NaN, the result has the same 2545 * sign as {@code d}. 2546 * 2547 * <p>Special cases: 2548 * <ul> 2549 * <li> If the first argument is NaN, NaN is returned. 2550 * <li> If the first argument is infinite, then an infinity of the 2551 * same sign is returned. 2552 * <li> If the first argument is zero, then a zero of the same 2553 * sign is returned. 2554 * </ul> 2555 * 2556 * @param d number to be scaled by a power of two. 2557 * @param scaleFactor power of 2 used to scale {@code d} 2558 * @return {@code d} × 2<sup>{@code scaleFactor}</sup> 2559 * @since 1.6 2560 */ 2561 public static double scalb(double d, int scaleFactor) { 2562 /* 2563 * This method does not need to be declared strictfp to 2564 * compute the same correct result on all platforms. When 2565 * scaling up, it does not matter what order the 2566 * multiply-store operations are done; the result will be 2567 * finite or overflow regardless of the operation ordering. 2568 * However, to get the correct result when scaling down, a 2569 * particular ordering must be used. 2570 * 2571 * When scaling down, the multiply-store operations are 2572 * sequenced so that it is not possible for two consecutive 2573 * multiply-stores to return subnormal results. If one 2574 * multiply-store result is subnormal, the next multiply will 2575 * round it away to zero. This is done by first multiplying 2576 * by 2 ^ (scaleFactor % n) and then multiplying several 2577 * times by 2^n as needed where n is the exponent of number 2578 * that is a covenient power of two. In this way, at most one 2579 * real rounding error occurs. If the double value set is 2580 * being used exclusively, the rounding will occur on a 2581 * multiply. If the double-extended-exponent value set is 2582 * being used, the products will (perhaps) be exact but the 2583 * stores to d are guaranteed to round to the double value 2584 * set. 2585 * 2586 * It is _not_ a valid implementation to first multiply d by 2587 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor % 2588 * MIN_EXPONENT) since even in a strictfp program double 2589 * rounding on underflow could occur; e.g. if the scaleFactor 2590 * argument was (MIN_EXPONENT - n) and the exponent of d was a 2591 * little less than -(MIN_EXPONENT - n), meaning the final 2592 * result would be subnormal. 2593 * 2594 * Since exact reproducibility of this method can be achieved 2595 * without any undue performance burden, there is no 2596 * compelling reason to allow double rounding on underflow in 2597 * scalb. 2598 */ 2599 2600 // magnitude of a power of two so large that scaling a finite 2601 // nonzero value by it would be guaranteed to over or 2602 // underflow; due to rounding, scaling down takes an 2603 // additional power of two which is reflected here 2604 final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT + 2605 DoubleConsts.SIGNIFICAND_WIDTH + 1; 2606 int exp_adjust = 0; 2607 int scale_increment = 0; 2608 double exp_delta = Double.NaN; 2609 2610 // Make sure scaling factor is in a reasonable range 2611 2612 if(scaleFactor < 0) { 2613 scaleFactor = Math.max(scaleFactor, -MAX_SCALE); 2614 scale_increment = -512; 2615 exp_delta = twoToTheDoubleScaleDown; 2616 } 2617 else { 2618 scaleFactor = Math.min(scaleFactor, MAX_SCALE); 2619 scale_increment = 512; 2620 exp_delta = twoToTheDoubleScaleUp; 2621 } 2622 2623 // Calculate (scaleFactor % +/-512), 512 = 2^9, using 2624 // technique from "Hacker's Delight" section 10-2. 2625 int t = (scaleFactor >> 9-1) >>> 32 - 9; 2626 exp_adjust = ((scaleFactor + t) & (512 -1)) - t; 2627 2628 d *= powerOfTwoD(exp_adjust); 2629 scaleFactor -= exp_adjust; 2630 2631 while(scaleFactor != 0) { 2632 d *= exp_delta; 2633 scaleFactor -= scale_increment; 2634 } 2635 return d; 2636 } 2637 2638 /** 2639 * Returns {@code f} × 2640 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2641 * by a single correctly rounded floating-point multiply to a 2642 * member of the float value set. See the Java 2643 * Language Specification for a discussion of floating-point 2644 * value sets. If the exponent of the result is between {@link 2645 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the 2646 * answer is calculated exactly. If the exponent of the result 2647 * would be larger than {@code Float.MAX_EXPONENT}, an 2648 * infinity is returned. Note that if the result is subnormal, 2649 * precision may be lost; that is, when {@code scalb(x, n)} 2650 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2651 * <i>x</i>. When the result is non-NaN, the result has the same 2652 * sign as {@code f}. 2653 * 2654 * <p>Special cases: 2655 * <ul> 2656 * <li> If the first argument is NaN, NaN is returned. 2657 * <li> If the first argument is infinite, then an infinity of the 2658 * same sign is returned. 2659 * <li> If the first argument is zero, then a zero of the same 2660 * sign is returned. 2661 * </ul> 2662 * 2663 * @param f number to be scaled by a power of two. 2664 * @param scaleFactor power of 2 used to scale {@code f} 2665 * @return {@code f} × 2<sup>{@code scaleFactor}</sup> 2666 * @since 1.6 2667 */ 2668 public static float scalb(float f, int scaleFactor) { 2669 // magnitude of a power of two so large that scaling a finite 2670 // nonzero value by it would be guaranteed to over or 2671 // underflow; due to rounding, scaling down takes an 2672 // additional power of two which is reflected here 2673 final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT + 2674 FloatConsts.SIGNIFICAND_WIDTH + 1; 2675 2676 // Make sure scaling factor is in a reasonable range 2677 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE); 2678 2679 /* 2680 * Since + MAX_SCALE for float fits well within the double 2681 * exponent range and + float -> double conversion is exact 2682 * the multiplication below will be exact. Therefore, the 2683 * rounding that occurs when the double product is cast to 2684 * float will be the correctly rounded float result. Since 2685 * all operations other than the final multiply will be exact, 2686 * it is not necessary to declare this method strictfp. 2687 */ 2688 return (float)((double)f*powerOfTwoD(scaleFactor)); 2689 } 2690 2691 // Constants used in scalb 2692 static double twoToTheDoubleScaleUp = powerOfTwoD(512); 2693 static double twoToTheDoubleScaleDown = powerOfTwoD(-512); 2694 2695 /** 2696 * Returns a floating-point power of two in the normal range. 2697 */ 2698 static double powerOfTwoD(int n) { 2699 assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT); 2700 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) << 2701 (DoubleConsts.SIGNIFICAND_WIDTH-1)) 2702 & DoubleConsts.EXP_BIT_MASK); 2703 } 2704 2705 /** 2706 * Returns a floating-point power of two in the normal range. 2707 */ 2708 static float powerOfTwoF(int n) { 2709 assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT); 2710 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) << 2711 (FloatConsts.SIGNIFICAND_WIDTH-1)) 2712 & FloatConsts.EXP_BIT_MASK); 2713 } 2714 }