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 */ 1083 public static long multiplyFull(int x, int y) { 1084 return (long)x * (long)y; 1085 } 1086 1087 /** 1088 * Returns as a {@code long} the most significant 64 bits of the 128-bit 1089 * product of two 64-bit factors. 1090 * 1091 * @param x the first value 1092 * @param y the second value 1093 * @return the result 1094 */ 1095 public static long multiplyHigh(long x, long y) { 1096 if (x < 0 || y < 0) { 1097 // Use technique from section 8-2 of Henry S. Warren, Jr., 1098 // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174. 1099 long x1 = x >> 32; 1100 long x2 = x & 0xFFFFFFFFL; 1101 long y1 = y >> 32; 1102 long y2 = y & 0xFFFFFFFFL; 1103 long z2 = x2 * y2; 1104 long t = x1 * y2 + (z2 >>> 32); 1105 long z1 = t & 0xFFFFFFFFL; 1106 long z0 = t >> 32; 1107 z1 += x2 * y1; 1108 return x1 * y1 + z0 + (z1 >> 32); 1109 } else { 1110 // Use Karatsuba technique with two base 2^32 digits. 1111 long x1 = x >>> 32; 1112 long y1 = y >>> 32; 1113 long x2 = x & 0xFFFFFFFFL; 1114 long y2 = y & 0xFFFFFFFFL; 1115 long A = x1 * y1; 1116 long B = x2 * y2; 1117 long C = (x1 + x2) * (y1 + y2); 1118 long K = C - A - B; 1119 return (((B >>> 32) + K) >>> 32) + A; 1120 } 1121 } 1122 1123 /** 1124 * Returns the largest (closest to positive infinity) 1125 * {@code int} value that is less than or equal to the algebraic quotient. 1126 * There is one special case, if the dividend is the 1127 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1}, 1128 * then integer overflow occurs and 1129 * the result is equal to {@code Integer.MIN_VALUE}. 1130 * <p> 1131 * Normal integer division operates under the round to zero rounding mode 1132 * (truncation). This operation instead acts under the round toward 1133 * negative infinity (floor) rounding mode. 1134 * The floor rounding mode gives different results from truncation 1135 * when the exact result is negative. 1136 * <ul> 1137 * <li>If the signs of the arguments are the same, the results of 1138 * {@code floorDiv} and the {@code /} operator are the same. <br> 1139 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li> 1140 * <li>If the signs of the arguments are different, the quotient is negative and 1141 * {@code floorDiv} returns the integer less than or equal to the quotient 1142 * and the {@code /} operator returns the integer closest to zero.<br> 1143 * For example, {@code floorDiv(-4, 3) == -2}, 1144 * whereas {@code (-4 / 3) == -1}. 1145 * </li> 1146 * </ul> 1147 * 1148 * @param x the dividend 1149 * @param y the divisor 1150 * @return the largest (closest to positive infinity) 1151 * {@code int} value that is less than or equal to the algebraic quotient. 1152 * @throws ArithmeticException if the divisor {@code y} is zero 1153 * @see #floorMod(int, int) 1154 * @see #floor(double) 1155 * @since 1.8 1156 */ 1157 public static int floorDiv(int x, int y) { 1158 int r = x / y; 1159 // if the signs are different and modulo not zero, round down 1160 if ((x ^ y) < 0 && (r * y != x)) { 1161 r--; 1162 } 1163 return r; 1164 } 1165 1166 /** 1167 * Returns the largest (closest to positive infinity) 1168 * {@code long} value that is less than or equal to the algebraic quotient. 1169 * There is one special case, if the dividend is the 1170 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1171 * then integer overflow occurs and 1172 * the result is equal to {@code Long.MIN_VALUE}. 1173 * <p> 1174 * Normal integer division operates under the round to zero rounding mode 1175 * (truncation). This operation instead acts under the round toward 1176 * negative infinity (floor) rounding mode. 1177 * The floor rounding mode gives different results from truncation 1178 * when the exact result is negative. 1179 * <p> 1180 * For examples, see {@link #floorDiv(int, int)}. 1181 * 1182 * @param x the dividend 1183 * @param y the divisor 1184 * @return the largest (closest to positive infinity) 1185 * {@code int} value that is less than or equal to the algebraic quotient. 1186 * @throws ArithmeticException if the divisor {@code y} is zero 1187 * @see #floorMod(long, int) 1188 * @see #floor(double) 1189 * @since 9 1190 */ 1191 public static long floorDiv(long x, int y) { 1192 return floorDiv(x, (long)y); 1193 } 1194 1195 /** 1196 * Returns the largest (closest to positive infinity) 1197 * {@code long} value that is less than or equal to the algebraic quotient. 1198 * There is one special case, if the dividend is the 1199 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1200 * then integer overflow occurs and 1201 * the result is equal to {@code Long.MIN_VALUE}. 1202 * <p> 1203 * Normal integer division operates under the round to zero rounding mode 1204 * (truncation). This operation instead acts under the round toward 1205 * negative infinity (floor) rounding mode. 1206 * The floor rounding mode gives different results from truncation 1207 * when the exact result is negative. 1208 * <p> 1209 * For examples, see {@link #floorDiv(int, int)}. 1210 * 1211 * @param x the dividend 1212 * @param y the divisor 1213 * @return the largest (closest to positive infinity) 1214 * {@code long} value that is less than or equal to the algebraic quotient. 1215 * @throws ArithmeticException if the divisor {@code y} is zero 1216 * @see #floorMod(long, long) 1217 * @see #floor(double) 1218 * @since 1.8 1219 */ 1220 public static long floorDiv(long x, long y) { 1221 long r = x / y; 1222 // if the signs are different and modulo not zero, round down 1223 if ((x ^ y) < 0 && (r * y != x)) { 1224 r--; 1225 } 1226 return r; 1227 } 1228 1229 /** 1230 * Returns the floor modulus of the {@code int} arguments. 1231 * <p> 1232 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1233 * has the same sign as the divisor {@code y}, and 1234 * is in the range of {@code -abs(y) < r < +abs(y)}. 1235 * 1236 * <p> 1237 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1238 * <ul> 1239 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1240 * </ul> 1241 * <p> 1242 * The difference in values between {@code floorMod} and 1243 * the {@code %} operator is due to the difference between 1244 * {@code floorDiv} that returns the integer less than or equal to the quotient 1245 * and the {@code /} operator that returns the integer closest to zero. 1246 * <p> 1247 * Examples: 1248 * <ul> 1249 * <li>If the signs of the arguments are the same, the results 1250 * of {@code floorMod} and the {@code %} operator are the same. <br> 1251 * <ul> 1252 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li> 1253 * </ul> 1254 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br> 1255 * <ul> 1256 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li> 1257 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li> 1258 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li> 1259 * </ul> 1260 * </li> 1261 * </ul> 1262 * <p> 1263 * If the signs of arguments are unknown and a positive modulus 1264 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}. 1265 * 1266 * @param x the dividend 1267 * @param y the divisor 1268 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1269 * @throws ArithmeticException if the divisor {@code y} is zero 1270 * @see #floorDiv(int, int) 1271 * @since 1.8 1272 */ 1273 public static int floorMod(int x, int y) { 1274 return x - floorDiv(x, y) * y; 1275 } 1276 1277 /** 1278 * Returns the floor modulus of the {@code long} and {@int} arguments. 1279 * <p> 1280 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1281 * has the same sign as the divisor {@code y}, and 1282 * is in the range of {@code -abs(y) < r < +abs(y)}. 1283 * 1284 * <p> 1285 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1286 * <ul> 1287 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1288 * </ul> 1289 * <p> 1290 * For examples, see {@link #floorMod(int, int)}. 1291 * 1292 * @param x the dividend 1293 * @param y the divisor 1294 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1295 * @throws ArithmeticException if the divisor {@code y} is zero 1296 * @see #floorDiv(long, int) 1297 * @since 9 1298 */ 1299 public static int floorMod(long x, int y) { 1300 // Result cannot overflow the range of int. 1301 return (int)(x - floorDiv(x, y) * y); 1302 } 1303 1304 /** 1305 * Returns the floor modulus of the {@code long} arguments. 1306 * <p> 1307 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1308 * has the same sign as the divisor {@code y}, and 1309 * is in the range of {@code -abs(y) < r < +abs(y)}. 1310 * 1311 * <p> 1312 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1313 * <ul> 1314 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1315 * </ul> 1316 * <p> 1317 * For examples, see {@link #floorMod(int, int)}. 1318 * 1319 * @param x the dividend 1320 * @param y the divisor 1321 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1322 * @throws ArithmeticException if the divisor {@code y} is zero 1323 * @see #floorDiv(long, long) 1324 * @since 1.8 1325 */ 1326 public static long floorMod(long x, long y) { 1327 return x - floorDiv(x, y) * y; 1328 } 1329 1330 /** 1331 * Returns the absolute value of an {@code int} value. 1332 * If the argument is not negative, the argument is returned. 1333 * If the argument is negative, the negation of the argument is returned. 1334 * 1335 * <p>Note that if the argument is equal to the value of 1336 * {@link Integer#MIN_VALUE}, the most negative representable 1337 * {@code int} value, the result is that same value, which is 1338 * negative. 1339 * 1340 * @param a the argument whose absolute value is to be determined 1341 * @return the absolute value of the argument. 1342 */ 1343 public static int abs(int a) { 1344 return (a < 0) ? -a : a; 1345 } 1346 1347 /** 1348 * Returns the absolute value of a {@code long} value. 1349 * If the argument is not negative, the argument is returned. 1350 * If the argument is negative, the negation of the argument is returned. 1351 * 1352 * <p>Note that if the argument is equal to the value of 1353 * {@link Long#MIN_VALUE}, the most negative representable 1354 * {@code long} value, the result is that same value, which 1355 * is negative. 1356 * 1357 * @param a the argument whose absolute value is to be determined 1358 * @return the absolute value of the argument. 1359 */ 1360 public static long abs(long a) { 1361 return (a < 0) ? -a : a; 1362 } 1363 1364 /** 1365 * Returns the absolute value of a {@code float} value. 1366 * If the argument is not negative, the argument is returned. 1367 * If the argument is negative, the negation of the argument is returned. 1368 * Special cases: 1369 * <ul><li>If the argument is positive zero or negative zero, the 1370 * result is positive zero. 1371 * <li>If the argument is infinite, the result is positive infinity. 1372 * <li>If the argument is NaN, the result is NaN.</ul> 1373 * 1374 * @apiNote As implied by the above, one valid implementation of 1375 * this method is given by the expression below which computes a 1376 * {@code float} with the same exponent and significand as the 1377 * argument but with a guaranteed zero sign bit indicating a 1378 * positive value:<br> 1379 * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))} 1380 * 1381 * @param a the argument whose absolute value is to be determined 1382 * @return the absolute value of the argument. 1383 */ 1384 public static float abs(float a) { 1385 return (a <= 0.0F) ? 0.0F - a : a; 1386 } 1387 1388 /** 1389 * Returns the absolute value of a {@code double} value. 1390 * If the argument is not negative, the argument is returned. 1391 * If the argument is negative, the negation of the argument is returned. 1392 * Special cases: 1393 * <ul><li>If the argument is positive zero or negative zero, the result 1394 * is positive zero. 1395 * <li>If the argument is infinite, the result is positive infinity. 1396 * <li>If the argument is NaN, the result is NaN.</ul> 1397 * 1398 * @apiNote As implied by the above, one valid implementation of 1399 * this method is given by the expression below which computes a 1400 * {@code double} with the same exponent and significand as the 1401 * argument but with a guaranteed zero sign bit indicating a 1402 * positive value:<br> 1403 * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)} 1404 * 1405 * @param a the argument whose absolute value is to be determined 1406 * @return the absolute value of the argument. 1407 */ 1408 @HotSpotIntrinsicCandidate 1409 public static double abs(double a) { 1410 return (a <= 0.0D) ? 0.0D - a : a; 1411 } 1412 1413 /** 1414 * Returns the greater of two {@code int} values. That is, the 1415 * result is the argument closer to the value of 1416 * {@link Integer#MAX_VALUE}. If the arguments have the same value, 1417 * the result is that same value. 1418 * 1419 * @param a an argument. 1420 * @param b another argument. 1421 * @return the larger of {@code a} and {@code b}. 1422 */ 1423 @HotSpotIntrinsicCandidate 1424 public static int max(int a, int b) { 1425 return (a >= b) ? a : b; 1426 } 1427 1428 /** 1429 * Returns the greater of two {@code long} values. That is, the 1430 * result is the argument closer to the value of 1431 * {@link Long#MAX_VALUE}. If the arguments have the same value, 1432 * the result is that same value. 1433 * 1434 * @param a an argument. 1435 * @param b another argument. 1436 * @return the larger of {@code a} and {@code b}. 1437 */ 1438 public static long max(long a, long b) { 1439 return (a >= b) ? a : b; 1440 } 1441 1442 // Use raw bit-wise conversions on guaranteed non-NaN arguments. 1443 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f); 1444 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d); 1445 1446 /** 1447 * Returns the greater of two {@code float} values. That is, 1448 * the result is the argument closer to positive infinity. If the 1449 * arguments have the same value, the result is that same 1450 * value. If either value is NaN, then the result is NaN. Unlike 1451 * the numerical comparison operators, this method considers 1452 * negative zero to be strictly smaller than positive zero. If one 1453 * argument is positive zero and the other negative zero, the 1454 * result is positive zero. 1455 * 1456 * @param a an argument. 1457 * @param b another argument. 1458 * @return the larger of {@code a} and {@code b}. 1459 */ 1460 public static float max(float a, float b) { 1461 if (a != a) 1462 return a; // a is NaN 1463 if ((a == 0.0f) && 1464 (b == 0.0f) && 1465 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) { 1466 // Raw conversion ok since NaN can't map to -0.0. 1467 return b; 1468 } 1469 return (a >= b) ? a : b; 1470 } 1471 1472 /** 1473 * Returns the greater of two {@code double} values. That 1474 * is, the result is the argument closer to positive infinity. If 1475 * the arguments have the same value, the result is that same 1476 * value. If either value is NaN, then the result is NaN. Unlike 1477 * the numerical comparison operators, this method considers 1478 * negative zero to be strictly smaller than positive zero. If one 1479 * argument is positive zero and the other negative zero, the 1480 * result is positive zero. 1481 * 1482 * @param a an argument. 1483 * @param b another argument. 1484 * @return the larger of {@code a} and {@code b}. 1485 */ 1486 public static double max(double a, double b) { 1487 if (a != a) 1488 return a; // a is NaN 1489 if ((a == 0.0d) && 1490 (b == 0.0d) && 1491 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) { 1492 // Raw conversion ok since NaN can't map to -0.0. 1493 return b; 1494 } 1495 return (a >= b) ? a : b; 1496 } 1497 1498 /** 1499 * Returns the smaller of two {@code int} values. That is, 1500 * the result the argument closer to the value of 1501 * {@link Integer#MIN_VALUE}. If the arguments have the same 1502 * value, the result is that same value. 1503 * 1504 * @param a an argument. 1505 * @param b another argument. 1506 * @return the smaller of {@code a} and {@code b}. 1507 */ 1508 @HotSpotIntrinsicCandidate 1509 public static int min(int a, int b) { 1510 return (a <= b) ? a : b; 1511 } 1512 1513 /** 1514 * Returns the smaller of two {@code long} values. That is, 1515 * the result is the argument closer to the value of 1516 * {@link Long#MIN_VALUE}. If the arguments have the same 1517 * value, the result is that same value. 1518 * 1519 * @param a an argument. 1520 * @param b another argument. 1521 * @return the smaller of {@code a} and {@code b}. 1522 */ 1523 public static long min(long a, long b) { 1524 return (a <= b) ? a : b; 1525 } 1526 1527 /** 1528 * Returns the smaller of two {@code float} values. That is, 1529 * the result is the value closer to negative infinity. If the 1530 * arguments have the same value, the result is that same 1531 * value. If either value is NaN, then the result is NaN. Unlike 1532 * the numerical comparison operators, this method considers 1533 * negative zero to be strictly smaller than positive zero. If 1534 * one argument is positive zero and the other is negative zero, 1535 * the result is negative zero. 1536 * 1537 * @param a an argument. 1538 * @param b another argument. 1539 * @return the smaller of {@code a} and {@code b}. 1540 */ 1541 public static float min(float a, float b) { 1542 if (a != a) 1543 return a; // a is NaN 1544 if ((a == 0.0f) && 1545 (b == 0.0f) && 1546 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) { 1547 // Raw conversion ok since NaN can't map to -0.0. 1548 return b; 1549 } 1550 return (a <= b) ? a : b; 1551 } 1552 1553 /** 1554 * Returns the smaller of two {@code double} values. That 1555 * is, the result is the value closer to negative infinity. If the 1556 * arguments have the same value, the result is that same 1557 * value. If either value is NaN, then the result is NaN. Unlike 1558 * the numerical comparison operators, this method considers 1559 * negative zero to be strictly smaller than positive zero. If one 1560 * argument is positive zero and the other is negative zero, the 1561 * result is negative zero. 1562 * 1563 * @param a an argument. 1564 * @param b another argument. 1565 * @return the smaller of {@code a} and {@code b}. 1566 */ 1567 public static double min(double a, double b) { 1568 if (a != a) 1569 return a; // a is NaN 1570 if ((a == 0.0d) && 1571 (b == 0.0d) && 1572 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) { 1573 // Raw conversion ok since NaN can't map to -0.0. 1574 return b; 1575 } 1576 return (a <= b) ? a : b; 1577 } 1578 1579 /** 1580 * Returns the fused multiply add of the three arguments; that is, 1581 * returns the exact product of the first two arguments summed 1582 * with the third argument and then rounded once to the nearest 1583 * {@code double}. 1584 * 1585 * The rounding is done using the {@linkplain 1586 * java.math.RoundingMode#HALF_EVEN round to nearest even 1587 * rounding mode}. 1588 * 1589 * In contrast, if {@code a * b + c} is evaluated as a regular 1590 * floating-point expression, two rounding errors are involved, 1591 * the first for the multiply operation, the second for the 1592 * addition operation. 1593 * 1594 * <p>Special cases: 1595 * <ul> 1596 * <li> If any argument is NaN, the result is NaN. 1597 * 1598 * <li> If one of the first two arguments is infinite and the 1599 * other is zero, the result is NaN. 1600 * 1601 * <li> If the exact product of the first two arguments is infinite 1602 * (in other words, at least one of the arguments is infinite and 1603 * the other is neither zero nor NaN) and the third argument is an 1604 * infinity of the opposite sign, the result is NaN. 1605 * 1606 * </ul> 1607 * 1608 * <p>Note that {@code fma(a, 1.0, c)} returns the same 1609 * result as ({@code a + c}). However, 1610 * {@code fma(a, b, +0.0)} does <em>not</em> always return the 1611 * same result as ({@code a * b}) since 1612 * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while 1613 * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is 1614 * equivalent to ({@code a * b}) however. 1615 * 1616 * @apiNote This method corresponds to the fusedMultiplyAdd 1617 * operation defined in IEEE 754-2008. 1618 * 1619 * @param a a value 1620 * @param b a value 1621 * @param c a value 1622 * 1623 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1624 * computed, as if with unlimited range and precision, and rounded 1625 * once to the nearest {@code double} value 1626 * 1627 * @since 9 1628 */ 1629 @HotSpotIntrinsicCandidate 1630 public static double fma(double a, double b, double c) { 1631 /* 1632 * Infinity and NaN arithmetic is not quite the same with two 1633 * roundings as opposed to just one so the simple expression 1634 * "a * b + c" cannot always be used to compute the correct 1635 * result. With two roundings, the product can overflow and 1636 * if the addend is infinite, a spurious NaN can be produced 1637 * if the infinity from the overflow and the infinite addend 1638 * have opposite signs. 1639 */ 1640 1641 // First, screen for and handle non-finite input values whose 1642 // arithmetic is not supported by BigDecimal. 1643 if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) { 1644 return Double.NaN; 1645 } else { // All inputs non-NaN 1646 boolean infiniteA = Double.isInfinite(a); 1647 boolean infiniteB = Double.isInfinite(b); 1648 boolean infiniteC = Double.isInfinite(c); 1649 double result; 1650 1651 if (infiniteA || infiniteB || infiniteC) { 1652 if (infiniteA && b == 0.0 || 1653 infiniteB && a == 0.0 ) { 1654 return Double.NaN; 1655 } 1656 // Store product in a double field to cause an 1657 // overflow even if non-strictfp evaluation is being 1658 // used. 1659 double product = a * b; 1660 if (Double.isInfinite(product) && !infiniteA && !infiniteB) { 1661 // Intermediate overflow; might cause a 1662 // spurious NaN if added to infinite c. 1663 assert Double.isInfinite(c); 1664 return c; 1665 } else { 1666 result = product + c; 1667 assert !Double.isFinite(result); 1668 return result; 1669 } 1670 } else { // All inputs finite 1671 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b)); 1672 if (c == 0.0) { // Positive or negative zero 1673 // If the product is an exact zero, use a 1674 // floating-point expression to compute the sign 1675 // of the zero final result. The product is an 1676 // exact zero if and only if at least one of a and 1677 // b is zero. 1678 if (a == 0.0 || b == 0.0) { 1679 return a * b + c; 1680 } else { 1681 // The sign of a zero addend doesn't matter if 1682 // the product is nonzero. The sign of a zero 1683 // addend is not factored in the result if the 1684 // exact product is nonzero but underflows to 1685 // zero; see IEEE-754 2008 section 6.3 "The 1686 // sign bit". 1687 return product.doubleValue(); 1688 } 1689 } else { 1690 return product.add(new BigDecimal(c)).doubleValue(); 1691 } 1692 } 1693 } 1694 } 1695 1696 /** 1697 * Returns the fused multiply add of the three arguments; that is, 1698 * returns the exact product of the first two arguments summed 1699 * with the third argument and then rounded once to the nearest 1700 * {@code float}. 1701 * 1702 * The rounding is done using the {@linkplain 1703 * java.math.RoundingMode#HALF_EVEN round to nearest even 1704 * rounding mode}. 1705 * 1706 * In contrast, if {@code a * b + c} is evaluated as a regular 1707 * floating-point expression, two rounding errors are involved, 1708 * the first for the multiply operation, the second for the 1709 * addition operation. 1710 * 1711 * <p>Special cases: 1712 * <ul> 1713 * <li> If any argument is NaN, the result is NaN. 1714 * 1715 * <li> If one of the first two arguments is infinite and the 1716 * other is zero, the result is NaN. 1717 * 1718 * <li> If the exact product of the first two arguments is infinite 1719 * (in other words, at least one of the arguments is infinite and 1720 * the other is neither zero nor NaN) and the third argument is an 1721 * infinity of the opposite sign, the result is NaN. 1722 * 1723 * </ul> 1724 * 1725 * <p>Note that {@code fma(a, 1.0f, c)} returns the same 1726 * result as ({@code a + c}). However, 1727 * {@code fma(a, b, +0.0f)} does <em>not</em> always return the 1728 * same result as ({@code a * b}) since 1729 * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while 1730 * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is 1731 * equivalent to ({@code a * b}) however. 1732 * 1733 * @apiNote This method corresponds to the fusedMultiplyAdd 1734 * operation defined in IEEE 754-2008. 1735 * 1736 * @param a a value 1737 * @param b a value 1738 * @param c a value 1739 * 1740 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1741 * computed, as if with unlimited range and precision, and rounded 1742 * once to the nearest {@code float} value 1743 * 1744 * @since 9 1745 */ 1746 @HotSpotIntrinsicCandidate 1747 public static float fma(float a, float b, float c) { 1748 /* 1749 * Since the double format has more than twice the precision 1750 * of the float format, the multiply of a * b is exact in 1751 * double. The add of c to the product then incurs one 1752 * rounding error. Since the double format moreover has more 1753 * than (2p + 2) precision bits compared to the p bits of the 1754 * float format, the two roundings of (a * b + c), first to 1755 * the double format and then secondarily to the float format, 1756 * are equivalent to rounding the intermediate result directly 1757 * to the float format. 1758 * 1759 * In terms of strictfp vs default-fp concerns related to 1760 * overflow and underflow, since 1761 * 1762 * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE 1763 * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE 1764 * 1765 * neither the multiply nor add will overflow or underflow in 1766 * double. Therefore, it is not necessary for this method to 1767 * be declared strictfp to have reproducible 1768 * behavior. However, it is necessary to explicitly store down 1769 * to a float variable to avoid returning a value in the float 1770 * extended value set. 1771 */ 1772 float result = (float)(((double) a * (double) b ) + (double) c); 1773 return result; 1774 } 1775 1776 /** 1777 * Returns the size of an ulp of the argument. An ulp, unit in 1778 * the last place, of a {@code double} value is the positive 1779 * distance between this floating-point value and the {@code 1780 * double} value next larger in magnitude. Note that for non-NaN 1781 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1782 * 1783 * <p>Special Cases: 1784 * <ul> 1785 * <li> If the argument is NaN, then the result is NaN. 1786 * <li> If the argument is positive or negative infinity, then the 1787 * result is positive infinity. 1788 * <li> If the argument is positive or negative zero, then the result is 1789 * {@code Double.MIN_VALUE}. 1790 * <li> If the argument is ±{@code Double.MAX_VALUE}, then 1791 * the result is equal to 2<sup>971</sup>. 1792 * </ul> 1793 * 1794 * @param d the floating-point value whose ulp is to be returned 1795 * @return the size of an ulp of the argument 1796 * @author Joseph D. Darcy 1797 * @since 1.5 1798 */ 1799 public static double ulp(double d) { 1800 int exp = getExponent(d); 1801 1802 switch(exp) { 1803 case Double.MAX_EXPONENT + 1: // NaN or infinity 1804 return Math.abs(d); 1805 1806 case Double.MIN_EXPONENT - 1: // zero or subnormal 1807 return Double.MIN_VALUE; 1808 1809 default: 1810 assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT; 1811 1812 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1813 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1); 1814 if (exp >= Double.MIN_EXPONENT) { 1815 return powerOfTwoD(exp); 1816 } 1817 else { 1818 // return a subnormal result; left shift integer 1819 // representation of Double.MIN_VALUE appropriate 1820 // number of positions 1821 return Double.longBitsToDouble(1L << 1822 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) )); 1823 } 1824 } 1825 } 1826 1827 /** 1828 * Returns the size of an ulp of the argument. An ulp, unit in 1829 * the last place, of a {@code float} value is the positive 1830 * distance between this floating-point value and the {@code 1831 * float} value next larger in magnitude. Note that for non-NaN 1832 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1833 * 1834 * <p>Special Cases: 1835 * <ul> 1836 * <li> If the argument is NaN, then the result is NaN. 1837 * <li> If the argument is positive or negative infinity, then the 1838 * result is positive infinity. 1839 * <li> If the argument is positive or negative zero, then the result is 1840 * {@code Float.MIN_VALUE}. 1841 * <li> If the argument is ±{@code Float.MAX_VALUE}, then 1842 * the result is equal to 2<sup>104</sup>. 1843 * </ul> 1844 * 1845 * @param f the floating-point value whose ulp is to be returned 1846 * @return the size of an ulp of the argument 1847 * @author Joseph D. Darcy 1848 * @since 1.5 1849 */ 1850 public static float ulp(float f) { 1851 int exp = getExponent(f); 1852 1853 switch(exp) { 1854 case Float.MAX_EXPONENT+1: // NaN or infinity 1855 return Math.abs(f); 1856 1857 case Float.MIN_EXPONENT-1: // zero or subnormal 1858 return Float.MIN_VALUE; 1859 1860 default: 1861 assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT; 1862 1863 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1864 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1); 1865 if (exp >= Float.MIN_EXPONENT) { 1866 return powerOfTwoF(exp); 1867 } else { 1868 // return a subnormal result; left shift integer 1869 // representation of FloatConsts.MIN_VALUE appropriate 1870 // number of positions 1871 return Float.intBitsToFloat(1 << 1872 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) )); 1873 } 1874 } 1875 } 1876 1877 /** 1878 * Returns the signum function of the argument; zero if the argument 1879 * is zero, 1.0 if the argument is greater than zero, -1.0 if the 1880 * argument is less than zero. 1881 * 1882 * <p>Special Cases: 1883 * <ul> 1884 * <li> If the argument is NaN, then the result is NaN. 1885 * <li> If the argument is positive zero or negative zero, then the 1886 * result is the same as the argument. 1887 * </ul> 1888 * 1889 * @param d the floating-point value whose signum is to be returned 1890 * @return the signum function of the argument 1891 * @author Joseph D. Darcy 1892 * @since 1.5 1893 */ 1894 public static double signum(double d) { 1895 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d); 1896 } 1897 1898 /** 1899 * Returns the signum function of the argument; zero if the argument 1900 * is zero, 1.0f if the argument is greater than zero, -1.0f if the 1901 * argument is less than zero. 1902 * 1903 * <p>Special Cases: 1904 * <ul> 1905 * <li> If the argument is NaN, then the result is NaN. 1906 * <li> If the argument is positive zero or negative zero, then the 1907 * result is the same as the argument. 1908 * </ul> 1909 * 1910 * @param f the floating-point value whose signum is to be returned 1911 * @return the signum function of the argument 1912 * @author Joseph D. Darcy 1913 * @since 1.5 1914 */ 1915 public static float signum(float f) { 1916 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f); 1917 } 1918 1919 /** 1920 * Returns the hyperbolic sine of a {@code double} value. 1921 * The hyperbolic sine of <i>x</i> is defined to be 1922 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 1923 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1924 * 1925 * <p>Special cases: 1926 * <ul> 1927 * 1928 * <li>If the argument is NaN, then the result is NaN. 1929 * 1930 * <li>If the argument is infinite, then the result is an infinity 1931 * with the same sign as the argument. 1932 * 1933 * <li>If the argument is zero, then the result is a zero with the 1934 * same sign as the argument. 1935 * 1936 * </ul> 1937 * 1938 * <p>The computed result must be within 2.5 ulps of the exact result. 1939 * 1940 * @param x The number whose hyperbolic sine is to be returned. 1941 * @return The hyperbolic sine of {@code x}. 1942 * @since 1.5 1943 */ 1944 public static double sinh(double x) { 1945 return StrictMath.sinh(x); 1946 } 1947 1948 /** 1949 * Returns the hyperbolic cosine of a {@code double} value. 1950 * The hyperbolic cosine of <i>x</i> is defined to be 1951 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 1952 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1953 * 1954 * <p>Special cases: 1955 * <ul> 1956 * 1957 * <li>If the argument is NaN, then the result is NaN. 1958 * 1959 * <li>If the argument is infinite, then the result is positive 1960 * infinity. 1961 * 1962 * <li>If the argument is zero, then the result is {@code 1.0}. 1963 * 1964 * </ul> 1965 * 1966 * <p>The computed result must be within 2.5 ulps of the exact result. 1967 * 1968 * @param x The number whose hyperbolic cosine is to be returned. 1969 * @return The hyperbolic cosine of {@code x}. 1970 * @since 1.5 1971 */ 1972 public static double cosh(double x) { 1973 return StrictMath.cosh(x); 1974 } 1975 1976 /** 1977 * Returns the hyperbolic tangent of a {@code double} value. 1978 * The hyperbolic tangent of <i>x</i> is defined to be 1979 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), 1980 * in other words, {@linkplain Math#sinh 1981 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note 1982 * that the absolute value of the exact tanh is always less than 1983 * 1. 1984 * 1985 * <p>Special cases: 1986 * <ul> 1987 * 1988 * <li>If the argument is NaN, then the result is NaN. 1989 * 1990 * <li>If the argument is zero, then the result is a zero with the 1991 * same sign as the argument. 1992 * 1993 * <li>If the argument is positive infinity, then the result is 1994 * {@code +1.0}. 1995 * 1996 * <li>If the argument is negative infinity, then the result is 1997 * {@code -1.0}. 1998 * 1999 * </ul> 2000 * 2001 * <p>The computed result must be within 2.5 ulps of the exact result. 2002 * The result of {@code tanh} for any finite input must have 2003 * an absolute value less than or equal to 1. Note that once the 2004 * exact result of tanh is within 1/2 of an ulp of the limit value 2005 * of ±1, correctly signed ±{@code 1.0} should 2006 * be returned. 2007 * 2008 * @param x The number whose hyperbolic tangent is to be returned. 2009 * @return The hyperbolic tangent of {@code x}. 2010 * @since 1.5 2011 */ 2012 public static double tanh(double x) { 2013 return StrictMath.tanh(x); 2014 } 2015 2016 /** 2017 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2018 * without intermediate overflow or underflow. 2019 * 2020 * <p>Special cases: 2021 * <ul> 2022 * 2023 * <li> If either argument is infinite, then the result 2024 * is positive infinity. 2025 * 2026 * <li> If either argument is NaN and neither argument is infinite, 2027 * then the result is NaN. 2028 * 2029 * </ul> 2030 * 2031 * <p>The computed result must be within 1 ulp of the exact 2032 * result. If one parameter is held constant, the results must be 2033 * semi-monotonic in the other parameter. 2034 * 2035 * @param x a value 2036 * @param y a value 2037 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2038 * without intermediate overflow or underflow 2039 * @since 1.5 2040 */ 2041 public static double hypot(double x, double y) { 2042 return StrictMath.hypot(x, y); 2043 } 2044 2045 /** 2046 * Returns <i>e</i><sup>x</sup> -1. Note that for values of 2047 * <i>x</i> near 0, the exact sum of 2048 * {@code expm1(x)} + 1 is much closer to the true 2049 * result of <i>e</i><sup>x</sup> than {@code exp(x)}. 2050 * 2051 * <p>Special cases: 2052 * <ul> 2053 * <li>If the argument is NaN, the result is NaN. 2054 * 2055 * <li>If the argument is positive infinity, then the result is 2056 * positive infinity. 2057 * 2058 * <li>If the argument is negative infinity, then the result is 2059 * -1.0. 2060 * 2061 * <li>If the argument is zero, then the result is a zero with the 2062 * same sign as the argument. 2063 * 2064 * </ul> 2065 * 2066 * <p>The computed result must be within 1 ulp of the exact result. 2067 * Results must be semi-monotonic. The result of 2068 * {@code expm1} for any finite input must be greater than or 2069 * equal to {@code -1.0}. Note that once the exact result of 2070 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 2071 * ulp of the limit value -1, {@code -1.0} should be 2072 * returned. 2073 * 2074 * @param x the exponent to raise <i>e</i> to in the computation of 2075 * <i>e</i><sup>{@code x}</sup> -1. 2076 * @return the value <i>e</i><sup>{@code x}</sup> - 1. 2077 * @since 1.5 2078 */ 2079 public static double expm1(double x) { 2080 return StrictMath.expm1(x); 2081 } 2082 2083 /** 2084 * Returns the natural logarithm of the sum of the argument and 1. 2085 * Note that for small values {@code x}, the result of 2086 * {@code log1p(x)} is much closer to the true result of ln(1 2087 * + {@code x}) than the floating-point evaluation of 2088 * {@code log(1.0+x)}. 2089 * 2090 * <p>Special cases: 2091 * 2092 * <ul> 2093 * 2094 * <li>If the argument is NaN or less than -1, then the result is 2095 * NaN. 2096 * 2097 * <li>If the argument is positive infinity, then the result is 2098 * positive infinity. 2099 * 2100 * <li>If the argument is negative one, then the result is 2101 * negative infinity. 2102 * 2103 * <li>If the argument is zero, then the result is a zero with the 2104 * same sign as the argument. 2105 * 2106 * </ul> 2107 * 2108 * <p>The computed result must be within 1 ulp of the exact result. 2109 * Results must be semi-monotonic. 2110 * 2111 * @param x a value 2112 * @return the value ln({@code x} + 1), the natural 2113 * log of {@code x} + 1 2114 * @since 1.5 2115 */ 2116 public static double log1p(double x) { 2117 return StrictMath.log1p(x); 2118 } 2119 2120 /** 2121 * Returns the first floating-point argument with the sign of the 2122 * second floating-point argument. Note that unlike the {@link 2123 * StrictMath#copySign(double, double) StrictMath.copySign} 2124 * method, this method does not require NaN {@code sign} 2125 * arguments to be treated as positive values; implementations are 2126 * permitted to treat some NaN arguments as positive and other NaN 2127 * arguments as negative to allow greater performance. 2128 * 2129 * @param magnitude the parameter providing the magnitude of the result 2130 * @param sign the parameter providing the sign of the result 2131 * @return a value with the magnitude of {@code magnitude} 2132 * and the sign of {@code sign}. 2133 * @since 1.6 2134 */ 2135 public static double copySign(double magnitude, double sign) { 2136 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) & 2137 (DoubleConsts.SIGN_BIT_MASK)) | 2138 (Double.doubleToRawLongBits(magnitude) & 2139 (DoubleConsts.EXP_BIT_MASK | 2140 DoubleConsts.SIGNIF_BIT_MASK))); 2141 } 2142 2143 /** 2144 * Returns the first floating-point argument with the sign of the 2145 * second floating-point argument. Note that unlike the {@link 2146 * StrictMath#copySign(float, float) StrictMath.copySign} 2147 * method, this method does not require NaN {@code sign} 2148 * arguments to be treated as positive values; implementations are 2149 * permitted to treat some NaN arguments as positive and other NaN 2150 * arguments as negative to allow greater performance. 2151 * 2152 * @param magnitude the parameter providing the magnitude of the result 2153 * @param sign the parameter providing the sign of the result 2154 * @return a value with the magnitude of {@code magnitude} 2155 * and the sign of {@code sign}. 2156 * @since 1.6 2157 */ 2158 public static float copySign(float magnitude, float sign) { 2159 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) & 2160 (FloatConsts.SIGN_BIT_MASK)) | 2161 (Float.floatToRawIntBits(magnitude) & 2162 (FloatConsts.EXP_BIT_MASK | 2163 FloatConsts.SIGNIF_BIT_MASK))); 2164 } 2165 2166 /** 2167 * Returns the unbiased exponent used in the representation of a 2168 * {@code float}. Special cases: 2169 * 2170 * <ul> 2171 * <li>If the argument is NaN or infinite, then the result is 2172 * {@link Float#MAX_EXPONENT} + 1. 2173 * <li>If the argument is zero or subnormal, then the result is 2174 * {@link Float#MIN_EXPONENT} -1. 2175 * </ul> 2176 * @param f a {@code float} value 2177 * @return the unbiased exponent of the argument 2178 * @since 1.6 2179 */ 2180 public static int getExponent(float f) { 2181 /* 2182 * Bitwise convert f to integer, mask out exponent bits, shift 2183 * to the right and then subtract out float's bias adjust to 2184 * get true exponent value 2185 */ 2186 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >> 2187 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS; 2188 } 2189 2190 /** 2191 * Returns the unbiased exponent used in the representation of a 2192 * {@code double}. Special cases: 2193 * 2194 * <ul> 2195 * <li>If the argument is NaN or infinite, then the result is 2196 * {@link Double#MAX_EXPONENT} + 1. 2197 * <li>If the argument is zero or subnormal, then the result is 2198 * {@link Double#MIN_EXPONENT} -1. 2199 * </ul> 2200 * @param d a {@code double} value 2201 * @return the unbiased exponent of the argument 2202 * @since 1.6 2203 */ 2204 public static int getExponent(double d) { 2205 /* 2206 * Bitwise convert d to long, mask out exponent bits, shift 2207 * to the right and then subtract out double's bias adjust to 2208 * get true exponent value. 2209 */ 2210 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >> 2211 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS); 2212 } 2213 2214 /** 2215 * Returns the floating-point number adjacent to the first 2216 * argument in the direction of the second argument. If both 2217 * arguments compare as equal the second argument is returned. 2218 * 2219 * <p> 2220 * Special cases: 2221 * <ul> 2222 * <li> If either argument is a NaN, then NaN is returned. 2223 * 2224 * <li> If both arguments are signed zeros, {@code direction} 2225 * is returned unchanged (as implied by the requirement of 2226 * returning the second argument if the arguments compare as 2227 * equal). 2228 * 2229 * <li> If {@code start} is 2230 * ±{@link Double#MIN_VALUE} and {@code direction} 2231 * has a value such that the result should have a smaller 2232 * magnitude, then a zero with the same sign as {@code start} 2233 * is returned. 2234 * 2235 * <li> If {@code start} is infinite and 2236 * {@code direction} has a value such that the result should 2237 * have a smaller magnitude, {@link Double#MAX_VALUE} with the 2238 * same sign as {@code start} is returned. 2239 * 2240 * <li> If {@code start} is equal to ± 2241 * {@link Double#MAX_VALUE} and {@code direction} has a 2242 * value such that the result should have a larger magnitude, an 2243 * infinity with same sign as {@code start} is returned. 2244 * </ul> 2245 * 2246 * @param start starting floating-point value 2247 * @param direction value indicating which of 2248 * {@code start}'s neighbors or {@code start} should 2249 * be returned 2250 * @return The floating-point number adjacent to {@code start} in the 2251 * direction of {@code direction}. 2252 * @since 1.6 2253 */ 2254 public static double nextAfter(double start, double direction) { 2255 /* 2256 * The cases: 2257 * 2258 * nextAfter(+infinity, 0) == MAX_VALUE 2259 * nextAfter(+infinity, +infinity) == +infinity 2260 * nextAfter(-infinity, 0) == -MAX_VALUE 2261 * nextAfter(-infinity, -infinity) == -infinity 2262 * 2263 * are naturally handled without any additional testing 2264 */ 2265 2266 /* 2267 * IEEE 754 floating-point numbers are lexicographically 2268 * ordered if treated as signed-magnitude integers. 2269 * Since Java's integers are two's complement, 2270 * incrementing the two's complement representation of a 2271 * logically negative floating-point value *decrements* 2272 * the signed-magnitude representation. Therefore, when 2273 * the integer representation of a floating-point value 2274 * is negative, the adjustment to the representation is in 2275 * the opposite direction from what would initially be expected. 2276 */ 2277 2278 // Branch to descending case first as it is more costly than ascending 2279 // case due to start != 0.0d conditional. 2280 if (start > direction) { // descending 2281 if (start != 0.0d) { 2282 final long transducer = Double.doubleToRawLongBits(start); 2283 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L)); 2284 } else { // start == 0.0d && direction < 0.0d 2285 return -Double.MIN_VALUE; 2286 } 2287 } else if (start < direction) { // ascending 2288 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2289 // then bitwise convert start to integer. 2290 final long transducer = Double.doubleToRawLongBits(start + 0.0d); 2291 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2292 } else if (start == direction) { 2293 return direction; 2294 } else { // isNaN(start) || isNaN(direction) 2295 return start + direction; 2296 } 2297 } 2298 2299 /** 2300 * Returns the floating-point number adjacent to the first 2301 * argument in the direction of the second argument. If both 2302 * arguments compare as equal a value equivalent to the second argument 2303 * is returned. 2304 * 2305 * <p> 2306 * Special cases: 2307 * <ul> 2308 * <li> If either argument is a NaN, then NaN is returned. 2309 * 2310 * <li> If both arguments are signed zeros, a value equivalent 2311 * to {@code direction} is returned. 2312 * 2313 * <li> If {@code start} is 2314 * ±{@link Float#MIN_VALUE} and {@code direction} 2315 * has a value such that the result should have a smaller 2316 * magnitude, then a zero with the same sign as {@code start} 2317 * is returned. 2318 * 2319 * <li> If {@code start} is infinite and 2320 * {@code direction} has a value such that the result should 2321 * have a smaller magnitude, {@link Float#MAX_VALUE} with the 2322 * same sign as {@code start} is returned. 2323 * 2324 * <li> If {@code start} is equal to ± 2325 * {@link Float#MAX_VALUE} and {@code direction} has a 2326 * value such that the result should have a larger magnitude, an 2327 * infinity with same sign as {@code start} is returned. 2328 * </ul> 2329 * 2330 * @param start starting floating-point value 2331 * @param direction value indicating which of 2332 * {@code start}'s neighbors or {@code start} should 2333 * be returned 2334 * @return The floating-point number adjacent to {@code start} in the 2335 * direction of {@code direction}. 2336 * @since 1.6 2337 */ 2338 public static float nextAfter(float start, double direction) { 2339 /* 2340 * The cases: 2341 * 2342 * nextAfter(+infinity, 0) == MAX_VALUE 2343 * nextAfter(+infinity, +infinity) == +infinity 2344 * nextAfter(-infinity, 0) == -MAX_VALUE 2345 * nextAfter(-infinity, -infinity) == -infinity 2346 * 2347 * are naturally handled without any additional testing 2348 */ 2349 2350 /* 2351 * IEEE 754 floating-point numbers are lexicographically 2352 * ordered if treated as signed-magnitude integers. 2353 * Since Java's integers are two's complement, 2354 * incrementing the two's complement representation of a 2355 * logically negative floating-point value *decrements* 2356 * the signed-magnitude representation. Therefore, when 2357 * the integer representation of a floating-point value 2358 * is negative, the adjustment to the representation is in 2359 * the opposite direction from what would initially be expected. 2360 */ 2361 2362 // Branch to descending case first as it is more costly than ascending 2363 // case due to start != 0.0f conditional. 2364 if (start > direction) { // descending 2365 if (start != 0.0f) { 2366 final int transducer = Float.floatToRawIntBits(start); 2367 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1)); 2368 } else { // start == 0.0f && direction < 0.0f 2369 return -Float.MIN_VALUE; 2370 } 2371 } else if (start < direction) { // ascending 2372 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2373 // then bitwise convert start to integer. 2374 final int transducer = Float.floatToRawIntBits(start + 0.0f); 2375 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2376 } else if (start == direction) { 2377 return (float)direction; 2378 } else { // isNaN(start) || isNaN(direction) 2379 return start + (float)direction; 2380 } 2381 } 2382 2383 /** 2384 * Returns the floating-point value adjacent to {@code d} in 2385 * the direction of positive infinity. This method is 2386 * semantically equivalent to {@code nextAfter(d, 2387 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp} 2388 * implementation may run faster than its equivalent 2389 * {@code nextAfter} call. 2390 * 2391 * <p>Special Cases: 2392 * <ul> 2393 * <li> If the argument is NaN, the result is NaN. 2394 * 2395 * <li> If the argument is positive infinity, the result is 2396 * positive infinity. 2397 * 2398 * <li> If the argument is zero, the result is 2399 * {@link Double#MIN_VALUE} 2400 * 2401 * </ul> 2402 * 2403 * @param d starting floating-point value 2404 * @return The adjacent floating-point value closer to positive 2405 * infinity. 2406 * @since 1.6 2407 */ 2408 public static double nextUp(double d) { 2409 // Use a single conditional and handle the likely cases first. 2410 if (d < Double.POSITIVE_INFINITY) { 2411 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2412 final long transducer = Double.doubleToRawLongBits(d + 0.0D); 2413 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2414 } else { // d is NaN or +Infinity 2415 return d; 2416 } 2417 } 2418 2419 /** 2420 * Returns the floating-point value adjacent to {@code f} in 2421 * the direction of positive infinity. This method is 2422 * semantically equivalent to {@code nextAfter(f, 2423 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp} 2424 * implementation may run faster than its equivalent 2425 * {@code nextAfter} call. 2426 * 2427 * <p>Special Cases: 2428 * <ul> 2429 * <li> If the argument is NaN, the result is NaN. 2430 * 2431 * <li> If the argument is positive infinity, the result is 2432 * positive infinity. 2433 * 2434 * <li> If the argument is zero, the result is 2435 * {@link Float#MIN_VALUE} 2436 * 2437 * </ul> 2438 * 2439 * @param f starting floating-point value 2440 * @return The adjacent floating-point value closer to positive 2441 * infinity. 2442 * @since 1.6 2443 */ 2444 public static float nextUp(float f) { 2445 // Use a single conditional and handle the likely cases first. 2446 if (f < Float.POSITIVE_INFINITY) { 2447 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2448 final int transducer = Float.floatToRawIntBits(f + 0.0F); 2449 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2450 } else { // f is NaN or +Infinity 2451 return f; 2452 } 2453 } 2454 2455 /** 2456 * Returns the floating-point value adjacent to {@code d} in 2457 * the direction of negative infinity. This method is 2458 * semantically equivalent to {@code nextAfter(d, 2459 * Double.NEGATIVE_INFINITY)}; however, a 2460 * {@code nextDown} implementation may run faster than its 2461 * equivalent {@code nextAfter} call. 2462 * 2463 * <p>Special Cases: 2464 * <ul> 2465 * <li> If the argument is NaN, the result is NaN. 2466 * 2467 * <li> If the argument is negative infinity, the result is 2468 * negative infinity. 2469 * 2470 * <li> If the argument is zero, the result is 2471 * {@code -Double.MIN_VALUE} 2472 * 2473 * </ul> 2474 * 2475 * @param d starting floating-point value 2476 * @return The adjacent floating-point value closer to negative 2477 * infinity. 2478 * @since 1.8 2479 */ 2480 public static double nextDown(double d) { 2481 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY) 2482 return d; 2483 else { 2484 if (d == 0.0) 2485 return -Double.MIN_VALUE; 2486 else 2487 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) + 2488 ((d > 0.0d)?-1L:+1L)); 2489 } 2490 } 2491 2492 /** 2493 * Returns the floating-point value adjacent to {@code f} in 2494 * the direction of negative infinity. This method is 2495 * semantically equivalent to {@code nextAfter(f, 2496 * Float.NEGATIVE_INFINITY)}; however, a 2497 * {@code nextDown} implementation may run faster than its 2498 * equivalent {@code nextAfter} call. 2499 * 2500 * <p>Special Cases: 2501 * <ul> 2502 * <li> If the argument is NaN, the result is NaN. 2503 * 2504 * <li> If the argument is negative infinity, the result is 2505 * negative infinity. 2506 * 2507 * <li> If the argument is zero, the result is 2508 * {@code -Float.MIN_VALUE} 2509 * 2510 * </ul> 2511 * 2512 * @param f starting floating-point value 2513 * @return The adjacent floating-point value closer to negative 2514 * infinity. 2515 * @since 1.8 2516 */ 2517 public static float nextDown(float f) { 2518 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY) 2519 return f; 2520 else { 2521 if (f == 0.0f) 2522 return -Float.MIN_VALUE; 2523 else 2524 return Float.intBitsToFloat(Float.floatToRawIntBits(f) + 2525 ((f > 0.0f)?-1:+1)); 2526 } 2527 } 2528 2529 /** 2530 * Returns {@code d} × 2531 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2532 * by a single correctly rounded floating-point multiply to a 2533 * member of the double value set. See the Java 2534 * Language Specification for a discussion of floating-point 2535 * value sets. If the exponent of the result is between {@link 2536 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the 2537 * answer is calculated exactly. If the exponent of the result 2538 * would be larger than {@code Double.MAX_EXPONENT}, an 2539 * infinity is returned. Note that if the result is subnormal, 2540 * precision may be lost; that is, when {@code scalb(x, n)} 2541 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2542 * <i>x</i>. When the result is non-NaN, the result has the same 2543 * sign as {@code d}. 2544 * 2545 * <p>Special cases: 2546 * <ul> 2547 * <li> If the first argument is NaN, NaN is returned. 2548 * <li> If the first argument is infinite, then an infinity of the 2549 * same sign is returned. 2550 * <li> If the first argument is zero, then a zero of the same 2551 * sign is returned. 2552 * </ul> 2553 * 2554 * @param d number to be scaled by a power of two. 2555 * @param scaleFactor power of 2 used to scale {@code d} 2556 * @return {@code d} × 2<sup>{@code scaleFactor}</sup> 2557 * @since 1.6 2558 */ 2559 public static double scalb(double d, int scaleFactor) { 2560 /* 2561 * This method does not need to be declared strictfp to 2562 * compute the same correct result on all platforms. When 2563 * scaling up, it does not matter what order the 2564 * multiply-store operations are done; the result will be 2565 * finite or overflow regardless of the operation ordering. 2566 * However, to get the correct result when scaling down, a 2567 * particular ordering must be used. 2568 * 2569 * When scaling down, the multiply-store operations are 2570 * sequenced so that it is not possible for two consecutive 2571 * multiply-stores to return subnormal results. If one 2572 * multiply-store result is subnormal, the next multiply will 2573 * round it away to zero. This is done by first multiplying 2574 * by 2 ^ (scaleFactor % n) and then multiplying several 2575 * times by 2^n as needed where n is the exponent of number 2576 * that is a covenient power of two. In this way, at most one 2577 * real rounding error occurs. If the double value set is 2578 * being used exclusively, the rounding will occur on a 2579 * multiply. If the double-extended-exponent value set is 2580 * being used, the products will (perhaps) be exact but the 2581 * stores to d are guaranteed to round to the double value 2582 * set. 2583 * 2584 * It is _not_ a valid implementation to first multiply d by 2585 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor % 2586 * MIN_EXPONENT) since even in a strictfp program double 2587 * rounding on underflow could occur; e.g. if the scaleFactor 2588 * argument was (MIN_EXPONENT - n) and the exponent of d was a 2589 * little less than -(MIN_EXPONENT - n), meaning the final 2590 * result would be subnormal. 2591 * 2592 * Since exact reproducibility of this method can be achieved 2593 * without any undue performance burden, there is no 2594 * compelling reason to allow double rounding on underflow in 2595 * scalb. 2596 */ 2597 2598 // magnitude of a power of two so large that scaling a finite 2599 // nonzero value by it would be guaranteed to over or 2600 // underflow; due to rounding, scaling down takes an 2601 // additional power of two which is reflected here 2602 final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT + 2603 DoubleConsts.SIGNIFICAND_WIDTH + 1; 2604 int exp_adjust = 0; 2605 int scale_increment = 0; 2606 double exp_delta = Double.NaN; 2607 2608 // Make sure scaling factor is in a reasonable range 2609 2610 if(scaleFactor < 0) { 2611 scaleFactor = Math.max(scaleFactor, -MAX_SCALE); 2612 scale_increment = -512; 2613 exp_delta = twoToTheDoubleScaleDown; 2614 } 2615 else { 2616 scaleFactor = Math.min(scaleFactor, MAX_SCALE); 2617 scale_increment = 512; 2618 exp_delta = twoToTheDoubleScaleUp; 2619 } 2620 2621 // Calculate (scaleFactor % +/-512), 512 = 2^9, using 2622 // technique from "Hacker's Delight" section 10-2. 2623 int t = (scaleFactor >> 9-1) >>> 32 - 9; 2624 exp_adjust = ((scaleFactor + t) & (512 -1)) - t; 2625 2626 d *= powerOfTwoD(exp_adjust); 2627 scaleFactor -= exp_adjust; 2628 2629 while(scaleFactor != 0) { 2630 d *= exp_delta; 2631 scaleFactor -= scale_increment; 2632 } 2633 return d; 2634 } 2635 2636 /** 2637 * Returns {@code f} × 2638 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2639 * by a single correctly rounded floating-point multiply to a 2640 * member of the float value set. See the Java 2641 * Language Specification for a discussion of floating-point 2642 * value sets. If the exponent of the result is between {@link 2643 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the 2644 * answer is calculated exactly. If the exponent of the result 2645 * would be larger than {@code Float.MAX_EXPONENT}, an 2646 * infinity is returned. Note that if the result is subnormal, 2647 * precision may be lost; that is, when {@code scalb(x, n)} 2648 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2649 * <i>x</i>. When the result is non-NaN, the result has the same 2650 * sign as {@code f}. 2651 * 2652 * <p>Special cases: 2653 * <ul> 2654 * <li> If the first argument is NaN, NaN is returned. 2655 * <li> If the first argument is infinite, then an infinity of the 2656 * same sign is returned. 2657 * <li> If the first argument is zero, then a zero of the same 2658 * sign is returned. 2659 * </ul> 2660 * 2661 * @param f number to be scaled by a power of two. 2662 * @param scaleFactor power of 2 used to scale {@code f} 2663 * @return {@code f} × 2<sup>{@code scaleFactor}</sup> 2664 * @since 1.6 2665 */ 2666 public static float scalb(float f, int scaleFactor) { 2667 // magnitude of a power of two so large that scaling a finite 2668 // nonzero value by it would be guaranteed to over or 2669 // underflow; due to rounding, scaling down takes an 2670 // additional power of two which is reflected here 2671 final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT + 2672 FloatConsts.SIGNIFICAND_WIDTH + 1; 2673 2674 // Make sure scaling factor is in a reasonable range 2675 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE); 2676 2677 /* 2678 * Since + MAX_SCALE for float fits well within the double 2679 * exponent range and + float -> double conversion is exact 2680 * the multiplication below will be exact. Therefore, the 2681 * rounding that occurs when the double product is cast to 2682 * float will be the correctly rounded float result. Since 2683 * all operations other than the final multiply will be exact, 2684 * it is not necessary to declare this method strictfp. 2685 */ 2686 return (float)((double)f*powerOfTwoD(scaleFactor)); 2687 } 2688 2689 // Constants used in scalb 2690 static double twoToTheDoubleScaleUp = powerOfTwoD(512); 2691 static double twoToTheDoubleScaleDown = powerOfTwoD(-512); 2692 2693 /** 2694 * Returns a floating-point power of two in the normal range. 2695 */ 2696 static double powerOfTwoD(int n) { 2697 assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT); 2698 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) << 2699 (DoubleConsts.SIGNIFICAND_WIDTH-1)) 2700 & DoubleConsts.EXP_BIT_MASK); 2701 } 2702 2703 /** 2704 * Returns a floating-point power of two in the normal range. 2705 */ 2706 static float powerOfTwoF(int n) { 2707 assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT); 2708 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) << 2709 (FloatConsts.SIGNIFICAND_WIDTH-1)) 2710 & FloatConsts.EXP_BIT_MASK); 2711 } 2712 }