1 /* 2 * Copyright (c) 1994, 2017, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package java.lang; 27 28 import java.math.BigDecimal; 29 import java.util.Random; 30 import jdk.internal.math.FloatConsts; 31 import jdk.internal.math.DoubleConsts; 32 import jdk.internal.HotSpotIntrinsicCandidate; 33 34 /** 35 * The class {@code Math} contains methods for performing basic 36 * numeric operations such as the elementary exponential, logarithm, 37 * square root, and trigonometric functions. 38 * 39 * <p>Unlike some of the numeric methods of class 40 * {@code StrictMath}, all implementations of the equivalent 41 * functions of class {@code Math} are not defined to return the 42 * bit-for-bit same results. This relaxation permits 43 * better-performing implementations where strict reproducibility is 44 * not required. 45 * 46 * <p>By default many of the {@code Math} methods simply call 47 * the equivalent method in {@code StrictMath} for their 48 * implementation. Code generators are encouraged to use 49 * platform-specific native libraries or microprocessor instructions, 50 * where available, to provide higher-performance implementations of 51 * {@code Math} methods. Such higher-performance 52 * implementations still must conform to the specification for 53 * {@code Math}. 54 * 55 * <p>The quality of implementation specifications concern two 56 * properties, accuracy of the returned result and monotonicity of the 57 * method. Accuracy of the floating-point {@code Math} methods is 58 * measured in terms of <i>ulps</i>, units in the last place. For a 59 * given floating-point format, an {@linkplain #ulp(double) ulp} of a 60 * specific real number value is the distance between the two 61 * floating-point values bracketing that numerical value. When 62 * discussing the accuracy of a method as a whole rather than at a 63 * specific argument, the number of ulps cited is for the worst-case 64 * error at any argument. If a method always has an error less than 65 * 0.5 ulps, the method always returns the floating-point number 66 * nearest the exact result; such a method is <i>correctly 67 * rounded</i>. A correctly rounded method is generally the best a 68 * floating-point approximation can be; however, it is impractical for 69 * many floating-point methods to be correctly rounded. Instead, for 70 * the {@code Math} class, a larger error bound of 1 or 2 ulps is 71 * allowed for certain methods. Informally, with a 1 ulp error bound, 72 * when the exact result is a representable number, the exact result 73 * should be returned as the computed result; otherwise, either of the 74 * two floating-point values which bracket the exact result may be 75 * returned. For exact results large in magnitude, one of the 76 * endpoints of the bracket may be infinite. Besides accuracy at 77 * individual arguments, maintaining proper relations between the 78 * method at different arguments is also important. Therefore, most 79 * methods with more than 0.5 ulp errors are required to be 80 * <i>semi-monotonic</i>: whenever the mathematical function is 81 * non-decreasing, so is the floating-point approximation, likewise, 82 * whenever the mathematical function is non-increasing, so is the 83 * floating-point approximation. Not all approximations that have 1 84 * ulp accuracy will automatically meet the monotonicity requirements. 85 * 86 * <p> 87 * The platform uses signed two's complement integer arithmetic with 88 * int and long primitive types. The developer should choose 89 * the primitive type to ensure that arithmetic operations consistently 90 * produce correct results, which in some cases means the operations 91 * will not overflow the range of values of the computation. 92 * The best practice is to choose the primitive type and algorithm to avoid 93 * overflow. In cases where the size is {@code int} or {@code long} and 94 * overflow errors need to be detected, the methods {@code addExact}, 95 * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact} 96 * throw an {@code ArithmeticException} when the results overflow. 97 * For other arithmetic operations such as divide, absolute value, 98 * increment by one, decrement by one, and negation, overflow occurs only with 99 * a specific minimum or maximum value and should be checked against 100 * the minimum or maximum as appropriate. 101 * 102 * @author unascribed 103 * @author Joseph D. Darcy 104 * @since 1.0 105 */ 106 107 public final class Math { 108 109 /** 110 * Don't let anyone instantiate this class. 111 */ 112 private Math() {} 113 114 /** 115 * The {@code double} value that is closer than any other to 116 * <i>e</i>, the base of the natural logarithms. 117 */ 118 public static final double E = 2.7182818284590452354; 119 120 /** 121 * The {@code double} value that is closer than any other to 122 * <i>pi</i>, the ratio of the circumference of a circle to its 123 * diameter. 124 */ 125 public static final double PI = 3.14159265358979323846; 126 127 /** 128 * Constant by which to multiply an angular value in degrees to obtain an 129 * angular value in radians. 130 */ 131 private static final double DEGREES_TO_RADIANS = 0.017453292519943295; 132 133 /** 134 * Constant by which to multiply an angular value in radians to obtain an 135 * angular value in degrees. 136 */ 137 private static final double RADIANS_TO_DEGREES = 57.29577951308232; 138 139 /** 140 * Returns the trigonometric sine of an angle. Special cases: 141 * <ul><li>If the argument is NaN or an infinity, then the 142 * result is NaN. 143 * <li>If the argument is zero, then the result is a zero with the 144 * same sign as the argument.</ul> 145 * 146 * <p>The computed result must be within 1 ulp of the exact result. 147 * Results must be semi-monotonic. 148 * 149 * @param a an angle, in radians. 150 * @return the sine of the argument. 151 */ 152 @HotSpotIntrinsicCandidate 153 public static double sin(double a) { 154 return StrictMath.sin(a); // default impl. delegates to StrictMath 155 } 156 157 /** 158 * Returns the trigonometric cosine of an angle. Special cases: 159 * <ul><li>If the argument is NaN or an infinity, then the 160 * result is NaN.</ul> 161 * 162 * <p>The computed result must be within 1 ulp of the exact result. 163 * Results must be semi-monotonic. 164 * 165 * @param a an angle, in radians. 166 * @return the cosine of the argument. 167 */ 168 @HotSpotIntrinsicCandidate 169 public static double cos(double a) { 170 return StrictMath.cos(a); // default impl. delegates to StrictMath 171 } 172 173 /** 174 * Returns the trigonometric tangent of an angle. Special cases: 175 * <ul><li>If the argument is NaN or an infinity, then the result 176 * is NaN. 177 * <li>If the argument is zero, then the result is a zero with the 178 * same sign as the argument.</ul> 179 * 180 * <p>The computed result must be within 1 ulp of the exact result. 181 * Results must be semi-monotonic. 182 * 183 * @param a an angle, in radians. 184 * @return the tangent of the argument. 185 */ 186 @HotSpotIntrinsicCandidate 187 public static double tan(double a) { 188 return StrictMath.tan(a); // default impl. delegates to StrictMath 189 } 190 191 /** 192 * Returns the arc sine of a value; the returned angle is in the 193 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: 194 * <ul><li>If the argument is NaN or its absolute value is greater 195 * than 1, then the result is NaN. 196 * <li>If the argument is zero, then the result is a zero with the 197 * same sign as the argument.</ul> 198 * 199 * <p>The computed result must be within 1 ulp of the exact result. 200 * Results must be semi-monotonic. 201 * 202 * @param a the value whose arc sine is to be returned. 203 * @return the arc sine of the argument. 204 */ 205 public static double asin(double a) { 206 return StrictMath.asin(a); // default impl. delegates to StrictMath 207 } 208 209 /** 210 * Returns the arc cosine of a value; the returned angle is in the 211 * range 0.0 through <i>pi</i>. Special case: 212 * <ul><li>If the argument is NaN or its absolute value is greater 213 * than 1, then the result is NaN.</ul> 214 * 215 * <p>The computed result must be within 1 ulp of the exact result. 216 * Results must be semi-monotonic. 217 * 218 * @param a the value whose arc cosine is to be returned. 219 * @return the arc cosine of the argument. 220 */ 221 public static double acos(double a) { 222 return StrictMath.acos(a); // default impl. delegates to StrictMath 223 } 224 225 /** 226 * Returns the arc tangent of a value; the returned angle is in the 227 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: 228 * <ul><li>If the argument is NaN, then the result is NaN. 229 * <li>If the argument is zero, then the result is a zero with the 230 * same sign as the argument.</ul> 231 * 232 * <p>The computed result must be within 1 ulp of the exact result. 233 * Results must be semi-monotonic. 234 * 235 * @param a the value whose arc tangent is to be returned. 236 * @return the arc tangent of the argument. 237 */ 238 public static double atan(double a) { 239 return StrictMath.atan(a); // default impl. delegates to StrictMath 240 } 241 242 /** 243 * Converts an angle measured in degrees to an approximately 244 * equivalent angle measured in radians. The conversion from 245 * degrees to radians is generally inexact. 246 * 247 * @param angdeg an angle, in degrees 248 * @return the measurement of the angle {@code angdeg} 249 * in radians. 250 * @since 1.2 251 */ 252 public static double toRadians(double angdeg) { 253 return angdeg * DEGREES_TO_RADIANS; 254 } 255 256 /** 257 * Converts an angle measured in radians to an approximately 258 * equivalent angle measured in degrees. The conversion from 259 * radians to degrees is generally inexact; users should 260 * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly 261 * equal {@code 0.0}. 262 * 263 * @param angrad an angle, in radians 264 * @return the measurement of the angle {@code angrad} 265 * in degrees. 266 * @since 1.2 267 */ 268 public static double toDegrees(double angrad) { 269 return angrad * RADIANS_TO_DEGREES; 270 } 271 272 /** 273 * Returns Euler's number <i>e</i> raised to the power of a 274 * {@code double} value. Special cases: 275 * <ul><li>If the argument is NaN, the result is NaN. 276 * <li>If the argument is positive infinity, then the result is 277 * positive infinity. 278 * <li>If the argument is negative infinity, then the result is 279 * positive zero.</ul> 280 * 281 * <p>The computed result must be within 1 ulp of the exact result. 282 * Results must be semi-monotonic. 283 * 284 * @param a the exponent to raise <i>e</i> to. 285 * @return the value <i>e</i><sup>{@code a}</sup>, 286 * where <i>e</i> is the base of the natural logarithms. 287 */ 288 @HotSpotIntrinsicCandidate 289 public static double exp(double a) { 290 return StrictMath.exp(a); // default impl. delegates to StrictMath 291 } 292 293 /** 294 * Returns the natural logarithm (base <i>e</i>) of a {@code double} 295 * value. Special cases: 296 * <ul><li>If the argument is NaN or less than zero, then the result 297 * is NaN. 298 * <li>If the argument is positive infinity, then the result is 299 * positive infinity. 300 * <li>If the argument is positive zero or negative zero, then the 301 * result is negative infinity.</ul> 302 * 303 * <p>The computed result must be within 1 ulp of the exact result. 304 * Results must be semi-monotonic. 305 * 306 * @param a a value 307 * @return the value ln {@code a}, the natural logarithm of 308 * {@code a}. 309 */ 310 @HotSpotIntrinsicCandidate 311 public static double log(double a) { 312 return StrictMath.log(a); // default impl. delegates to StrictMath 313 } 314 315 /** 316 * Returns the base 10 logarithm of a {@code double} value. 317 * Special cases: 318 * 319 * <ul><li>If the argument is NaN or less than zero, then the result 320 * is NaN. 321 * <li>If the argument is positive infinity, then the result is 322 * positive infinity. 323 * <li>If the argument is positive zero or negative zero, then the 324 * result is negative infinity. 325 * <li> If the argument is equal to 10<sup><i>n</i></sup> for 326 * integer <i>n</i>, then the result is <i>n</i>. 327 * </ul> 328 * 329 * <p>The computed result must be within 1 ulp of the exact result. 330 * Results must be semi-monotonic. 331 * 332 * @param a a value 333 * @return the base 10 logarithm of {@code a}. 334 * @since 1.5 335 */ 336 @HotSpotIntrinsicCandidate 337 public static double log10(double a) { 338 return StrictMath.log10(a); // default impl. delegates to StrictMath 339 } 340 341 /** 342 * Returns the correctly rounded positive square root of a 343 * {@code double} value. 344 * Special cases: 345 * <ul><li>If the argument is NaN or less than zero, then the result 346 * is NaN. 347 * <li>If the argument is positive infinity, then the result is positive 348 * infinity. 349 * <li>If the argument is positive zero or negative zero, then the 350 * result is the same as the argument.</ul> 351 * Otherwise, the result is the {@code double} value closest to 352 * the true mathematical square root of the argument value. 353 * 354 * @param a a value. 355 * @return the positive square root of {@code a}. 356 * If the argument is NaN or less than zero, the result is NaN. 357 */ 358 @HotSpotIntrinsicCandidate 359 public static double sqrt(double a) { 360 return StrictMath.sqrt(a); // default impl. delegates to StrictMath 361 // Note that hardware sqrt instructions 362 // frequently can be directly used by JITs 363 // and should be much faster than doing 364 // Math.sqrt in software. 365 } 366 367 368 /** 369 * Returns the cube root of a {@code double} value. For 370 * positive finite {@code x}, {@code cbrt(-x) == 371 * -cbrt(x)}; that is, the cube root of a negative value is 372 * the negative of the cube root of that value's magnitude. 373 * 374 * Special cases: 375 * 376 * <ul> 377 * 378 * <li>If the argument is NaN, then the result is NaN. 379 * 380 * <li>If the argument is infinite, then the result is an infinity 381 * with the same sign as the argument. 382 * 383 * <li>If the argument is zero, then the result is a zero with the 384 * same sign as the argument. 385 * 386 * </ul> 387 * 388 * <p>The computed result must be within 1 ulp of the exact result. 389 * 390 * @param a a value. 391 * @return the cube root of {@code a}. 392 * @since 1.5 393 */ 394 public static double cbrt(double a) { 395 return StrictMath.cbrt(a); 396 } 397 398 /** 399 * Computes the remainder operation on two arguments as prescribed 400 * by the IEEE 754 standard. 401 * The remainder value is mathematically equal to 402 * <code>f1 - f2</code> × <i>n</i>, 403 * where <i>n</i> is the mathematical integer closest to the exact 404 * mathematical value of the quotient {@code f1/f2}, and if two 405 * mathematical integers are equally close to {@code f1/f2}, 406 * then <i>n</i> is the integer that is even. If the remainder is 407 * zero, its sign is the same as the sign of the first argument. 408 * Special cases: 409 * <ul><li>If either argument is NaN, or the first argument is infinite, 410 * or the second argument is positive zero or negative zero, then the 411 * result is NaN. 412 * <li>If the first argument is finite and the second argument is 413 * infinite, then the result is the same as the first argument.</ul> 414 * 415 * @param f1 the dividend. 416 * @param f2 the divisor. 417 * @return the remainder when {@code f1} is divided by 418 * {@code f2}. 419 */ 420 public static double IEEEremainder(double f1, double f2) { 421 return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath 422 } 423 424 /** 425 * Returns the smallest (closest to negative infinity) 426 * {@code double} value that is greater than or equal to the 427 * argument and is equal to a mathematical integer. Special cases: 428 * <ul><li>If the argument value is already equal to a 429 * mathematical integer, then the result is the same as the 430 * argument. <li>If the argument is NaN or an infinity or 431 * positive zero or negative zero, then the result is the same as 432 * the argument. <li>If the argument value is less than zero but 433 * greater than -1.0, then the result is negative zero.</ul> Note 434 * that the value of {@code Math.ceil(x)} is exactly the 435 * value of {@code -Math.floor(-x)}. 436 * 437 * 438 * @param a a value. 439 * @return the smallest (closest to negative infinity) 440 * floating-point value that is greater than or equal to 441 * the argument and is equal to a mathematical integer. 442 */ 443 public static double ceil(double a) { 444 return StrictMath.ceil(a); // default impl. delegates to StrictMath 445 } 446 447 /** 448 * Returns the largest (closest to positive infinity) 449 * {@code double} value that is less than or equal to the 450 * argument and is equal to a mathematical integer. Special cases: 451 * <ul><li>If the argument value is already equal to a 452 * mathematical integer, then the result is the same as the 453 * argument. <li>If the argument is NaN or an infinity or 454 * positive zero or negative zero, then the result is the same as 455 * the argument.</ul> 456 * 457 * @param a a value. 458 * @return the largest (closest to positive infinity) 459 * floating-point value that less than or equal to the argument 460 * and is equal to a mathematical integer. 461 */ 462 public static double floor(double a) { 463 return StrictMath.floor(a); // default impl. delegates to StrictMath 464 } 465 466 /** 467 * Returns the {@code double} value that is closest in value 468 * to the argument and is equal to a mathematical integer. If two 469 * {@code double} values that are mathematical integers are 470 * equally close, the result is the integer value that is 471 * even. Special cases: 472 * <ul><li>If the argument value is already equal to a mathematical 473 * integer, then the result is the same as the argument. 474 * <li>If the argument is NaN or an infinity or positive zero or negative 475 * zero, then the result is the same as the argument.</ul> 476 * 477 * @param a a {@code double} value. 478 * @return the closest floating-point value to {@code a} that is 479 * equal to a mathematical integer. 480 */ 481 public static double rint(double a) { 482 return StrictMath.rint(a); // default impl. delegates to StrictMath 483 } 484 485 /** 486 * Returns the angle <i>theta</i> from the conversion of rectangular 487 * coordinates ({@code x}, {@code y}) to polar 488 * coordinates (r, <i>theta</i>). 489 * This method computes the phase <i>theta</i> by computing an arc tangent 490 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special 491 * cases: 492 * <ul><li>If either argument is NaN, then the result is NaN. 493 * <li>If the first argument is positive zero and the second argument 494 * is positive, or the first argument is positive and finite and the 495 * second argument is positive infinity, then the result is positive 496 * zero. 497 * <li>If the first argument is negative zero and the second argument 498 * is positive, or the first argument is negative and finite and the 499 * second argument is positive infinity, then the result is negative zero. 500 * <li>If the first argument is positive zero and the second argument 501 * is negative, or the first argument is positive and finite and the 502 * second argument is negative infinity, then the result is the 503 * {@code double} value closest to <i>pi</i>. 504 * <li>If the first argument is negative zero and the second argument 505 * is negative, or the first argument is negative and finite and the 506 * second argument is negative infinity, then the result is the 507 * {@code double} value closest to -<i>pi</i>. 508 * <li>If the first argument is positive and the second argument is 509 * positive zero or negative zero, or the first argument is positive 510 * infinity and the second argument is finite, then the result is the 511 * {@code double} value closest to <i>pi</i>/2. 512 * <li>If the first argument is negative and the second argument is 513 * positive zero or negative zero, or the first argument is negative 514 * infinity and the second argument is finite, then the result is the 515 * {@code double} value closest to -<i>pi</i>/2. 516 * <li>If both arguments are positive infinity, then the result is the 517 * {@code double} value closest to <i>pi</i>/4. 518 * <li>If the first argument is positive infinity and the second argument 519 * is negative infinity, then the result is the {@code double} 520 * value closest to 3*<i>pi</i>/4. 521 * <li>If the first argument is negative infinity and the second argument 522 * is positive infinity, then the result is the {@code double} value 523 * closest to -<i>pi</i>/4. 524 * <li>If both arguments are negative infinity, then the result is the 525 * {@code double} value closest to -3*<i>pi</i>/4.</ul> 526 * 527 * <p>The computed result must be within 2 ulps of the exact result. 528 * Results must be semi-monotonic. 529 * 530 * @param y the ordinate coordinate 531 * @param x the abscissa coordinate 532 * @return the <i>theta</i> component of the point 533 * (<i>r</i>, <i>theta</i>) 534 * in polar coordinates that corresponds to the point 535 * (<i>x</i>, <i>y</i>) in Cartesian coordinates. 536 */ 537 @HotSpotIntrinsicCandidate 538 public static double atan2(double y, double x) { 539 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath 540 } 541 542 /** 543 * Returns the value of the first argument raised to the power of the 544 * second argument. Special cases: 545 * 546 * <ul><li>If the second argument is positive or negative zero, then the 547 * result is 1.0. 548 * <li>If the second argument is 1.0, then the result is the same as the 549 * first argument. 550 * <li>If the second argument is NaN, then the result is NaN. 551 * <li>If the first argument is NaN and the second argument is nonzero, 552 * then the result is NaN. 553 * 554 * <li>If 555 * <ul> 556 * <li>the absolute value of the first argument is greater than 1 557 * and the second argument is positive infinity, or 558 * <li>the absolute value of the first argument is less than 1 and 559 * the second argument is negative infinity, 560 * </ul> 561 * then the result is positive infinity. 562 * 563 * <li>If 564 * <ul> 565 * <li>the absolute value of the first argument is greater than 1 and 566 * the second argument is negative infinity, or 567 * <li>the absolute value of the 568 * first argument is less than 1 and the second argument is positive 569 * infinity, 570 * </ul> 571 * then the result is positive zero. 572 * 573 * <li>If the absolute value of the first argument equals 1 and the 574 * second argument is infinite, then the result is NaN. 575 * 576 * <li>If 577 * <ul> 578 * <li>the first argument is positive zero and the second argument 579 * is greater than zero, or 580 * <li>the first argument is positive infinity and the second 581 * argument is less than zero, 582 * </ul> 583 * then the result is positive zero. 584 * 585 * <li>If 586 * <ul> 587 * <li>the first argument is positive zero and the second argument 588 * is less than zero, or 589 * <li>the first argument is positive infinity and the second 590 * argument is greater than zero, 591 * </ul> 592 * then the result is positive infinity. 593 * 594 * <li>If 595 * <ul> 596 * <li>the first argument is negative zero and the second argument 597 * is greater than zero but not a finite odd integer, or 598 * <li>the first argument is negative infinity and the second 599 * argument is less than zero but not a finite odd integer, 600 * </ul> 601 * then the result is positive zero. 602 * 603 * <li>If 604 * <ul> 605 * <li>the first argument is negative zero and the second argument 606 * is a positive finite odd integer, or 607 * <li>the first argument is negative infinity and the second 608 * argument is a negative finite odd integer, 609 * </ul> 610 * then the result is negative zero. 611 * 612 * <li>If 613 * <ul> 614 * <li>the first argument is negative zero and the second argument 615 * is less than zero but not a finite odd integer, or 616 * <li>the first argument is negative infinity and the second 617 * argument is greater than zero but not a finite odd integer, 618 * </ul> 619 * then the result is positive infinity. 620 * 621 * <li>If 622 * <ul> 623 * <li>the first argument is negative zero and the second argument 624 * is a negative finite odd integer, or 625 * <li>the first argument is negative infinity and the second 626 * argument is a positive finite odd integer, 627 * </ul> 628 * then the result is negative infinity. 629 * 630 * <li>If the first argument is finite and less than zero 631 * <ul> 632 * <li> if the second argument is a finite even integer, the 633 * result is equal to the result of raising the absolute value of 634 * the first argument to the power of the second argument 635 * 636 * <li>if the second argument is a finite odd integer, the result 637 * is equal to the negative of the result of raising the absolute 638 * value of the first argument to the power of the second 639 * argument 640 * 641 * <li>if the second argument is finite and not an integer, then 642 * the result is NaN. 643 * </ul> 644 * 645 * <li>If both arguments are integers, then the result is exactly equal 646 * to the mathematical result of raising the first argument to the power 647 * of the second argument if that result can in fact be represented 648 * exactly as a {@code double} value.</ul> 649 * 650 * <p>(In the foregoing descriptions, a floating-point value is 651 * considered to be an integer if and only if it is finite and a 652 * fixed point of the method {@link #ceil ceil} or, 653 * equivalently, a fixed point of the method {@link #floor 654 * floor}. A value is a fixed point of a one-argument 655 * method if and only if the result of applying the method to the 656 * value is equal to the value.) 657 * 658 * <p>The computed result must be within 1 ulp of the exact result. 659 * Results must be semi-monotonic. 660 * 661 * @param a the base. 662 * @param b the exponent. 663 * @return the value {@code a}<sup>{@code b}</sup>. 664 */ 665 @HotSpotIntrinsicCandidate 666 public static double pow(double a, double b) { 667 return StrictMath.pow(a, b); // default impl. delegates to StrictMath 668 } 669 670 /** 671 * Returns the closest {@code int} to the argument, with ties 672 * rounding to positive infinity. 673 * 674 * <p> 675 * Special cases: 676 * <ul><li>If the argument is NaN, the result is 0. 677 * <li>If the argument is negative infinity or any value less than or 678 * equal to the value of {@code Integer.MIN_VALUE}, the result is 679 * equal to the value of {@code Integer.MIN_VALUE}. 680 * <li>If the argument is positive infinity or any value greater than or 681 * equal to the value of {@code Integer.MAX_VALUE}, the result is 682 * equal to the value of {@code Integer.MAX_VALUE}.</ul> 683 * 684 * @param a a floating-point value to be rounded to an integer. 685 * @return the value of the argument rounded to the nearest 686 * {@code int} value. 687 * @see java.lang.Integer#MAX_VALUE 688 * @see java.lang.Integer#MIN_VALUE 689 */ 690 public static int round(float a) { 691 int intBits = Float.floatToRawIntBits(a); 692 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK) 693 >> (FloatConsts.SIGNIFICAND_WIDTH - 1); 694 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2 695 + FloatConsts.EXP_BIAS) - biasedExp; 696 if ((shift & -32) == 0) { // shift >= 0 && shift < 32 697 // a is a finite number such that pow(2,-32) <= ulp(a) < 1 698 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK) 699 | (FloatConsts.SIGNIF_BIT_MASK + 1)); 700 if (intBits < 0) { 701 r = -r; 702 } 703 // In the comments below each Java expression evaluates to the value 704 // the corresponding mathematical expression: 705 // (r) evaluates to a / ulp(a) 706 // (r >> shift) evaluates to floor(a * 2) 707 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) 708 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) 709 return ((r >> shift) + 1) >> 1; 710 } else { 711 // a is either 712 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2 713 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer 714 // - an infinity or NaN 715 return (int) a; 716 } 717 } 718 719 /** 720 * Returns the closest {@code long} to the argument, with ties 721 * rounding to positive infinity. 722 * 723 * <p>Special cases: 724 * <ul><li>If the argument is NaN, the result is 0. 725 * <li>If the argument is negative infinity or any value less than or 726 * equal to the value of {@code Long.MIN_VALUE}, the result is 727 * equal to the value of {@code Long.MIN_VALUE}. 728 * <li>If the argument is positive infinity or any value greater than or 729 * equal to the value of {@code Long.MAX_VALUE}, the result is 730 * equal to the value of {@code Long.MAX_VALUE}.</ul> 731 * 732 * @param a a floating-point value to be rounded to a 733 * {@code long}. 734 * @return the value of the argument rounded to the nearest 735 * {@code long} value. 736 * @see java.lang.Long#MAX_VALUE 737 * @see java.lang.Long#MIN_VALUE 738 */ 739 public static long round(double a) { 740 long longBits = Double.doubleToRawLongBits(a); 741 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK) 742 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1); 743 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2 744 + DoubleConsts.EXP_BIAS) - biasedExp; 745 if ((shift & -64) == 0) { // shift >= 0 && shift < 64 746 // a is a finite number such that pow(2,-64) <= ulp(a) < 1 747 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK) 748 | (DoubleConsts.SIGNIF_BIT_MASK + 1)); 749 if (longBits < 0) { 750 r = -r; 751 } 752 // In the comments below each Java expression evaluates to the value 753 // the corresponding mathematical expression: 754 // (r) evaluates to a / ulp(a) 755 // (r >> shift) evaluates to floor(a * 2) 756 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) 757 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) 758 return ((r >> shift) + 1) >> 1; 759 } else { 760 // a is either 761 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2 762 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer 763 // - an infinity or NaN 764 return (long) a; 765 } 766 } 767 768 private static final class RandomNumberGeneratorHolder { 769 static final Random randomNumberGenerator = new Random(); 770 } 771 772 /** 773 * Returns a {@code double} value with a positive sign, greater 774 * than or equal to {@code 0.0} and less than {@code 1.0}. 775 * Returned values are chosen pseudorandomly with (approximately) 776 * uniform distribution from that range. 777 * 778 * <p>When this method is first called, it creates a single new 779 * pseudorandom-number generator, exactly as if by the expression 780 * 781 * <blockquote>{@code new java.util.Random()}</blockquote> 782 * 783 * This new pseudorandom-number generator is used thereafter for 784 * all calls to this method and is used nowhere else. 785 * 786 * <p>This method is properly synchronized to allow correct use by 787 * more than one thread. However, if many threads need to generate 788 * pseudorandom numbers at a great rate, it may reduce contention 789 * for each thread to have its own pseudorandom-number generator. 790 * 791 * @apiNote 792 * As the largest {@code double} value less than {@code 1.0} 793 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range 794 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements 795 * 796 * <blockquote><pre>{@code 797 * double f = Math.random()/Math.nextDown(1.0); 798 * double x = x1*(1.0 - f) + x2*f; 799 * }</pre></blockquote> 800 * 801 * @return a pseudorandom {@code double} greater than or equal 802 * to {@code 0.0} and less than {@code 1.0}. 803 * @see #nextDown(double) 804 * @see Random#nextDouble() 805 */ 806 public static double random() { 807 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble(); 808 } 809 810 /** 811 * Returns the sum of its arguments, 812 * throwing an exception if the result overflows an {@code int}. 813 * 814 * @param x the first value 815 * @param y the second value 816 * @return the result 817 * @throws ArithmeticException if the result overflows an int 818 * @since 1.8 819 */ 820 @HotSpotIntrinsicCandidate 821 public static int addExact(int x, int y) { 822 int r = x + y; 823 // HD 2-12 Overflow iff both arguments have the opposite sign of the result 824 if (((x ^ r) & (y ^ r)) < 0) { 825 throw new ArithmeticException("integer overflow"); 826 } 827 return r; 828 } 829 830 /** 831 * Returns the sum of its arguments, 832 * throwing an exception if the result overflows a {@code long}. 833 * 834 * @param x the first value 835 * @param y the second value 836 * @return the result 837 * @throws ArithmeticException if the result overflows a long 838 * @since 1.8 839 */ 840 @HotSpotIntrinsicCandidate 841 public static long addExact(long x, long y) { 842 long r = x + y; 843 // HD 2-12 Overflow iff both arguments have the opposite sign of the result 844 if (((x ^ r) & (y ^ r)) < 0) { 845 throw new ArithmeticException("long overflow"); 846 } 847 return r; 848 } 849 850 /** 851 * Returns the difference of the arguments, 852 * throwing an exception if the result overflows an {@code int}. 853 * 854 * @param x the first value 855 * @param y the second value to subtract from the first 856 * @return the result 857 * @throws ArithmeticException if the result overflows an int 858 * @since 1.8 859 */ 860 @HotSpotIntrinsicCandidate 861 public static int subtractExact(int x, int y) { 862 int r = x - y; 863 // HD 2-12 Overflow iff the arguments have different signs and 864 // the sign of the result is different from the sign of x 865 if (((x ^ y) & (x ^ r)) < 0) { 866 throw new ArithmeticException("integer overflow"); 867 } 868 return r; 869 } 870 871 /** 872 * Returns the difference of the arguments, 873 * throwing an exception if the result overflows a {@code long}. 874 * 875 * @param x the first value 876 * @param y the second value to subtract from the first 877 * @return the result 878 * @throws ArithmeticException if the result overflows a long 879 * @since 1.8 880 */ 881 @HotSpotIntrinsicCandidate 882 public static long subtractExact(long x, long y) { 883 long r = x - y; 884 // HD 2-12 Overflow iff the arguments have different signs and 885 // the sign of the result is different from the sign of x 886 if (((x ^ y) & (x ^ r)) < 0) { 887 throw new ArithmeticException("long overflow"); 888 } 889 return r; 890 } 891 892 /** 893 * Returns the product of the arguments, 894 * throwing an exception if the result overflows an {@code int}. 895 * 896 * @param x the first value 897 * @param y the second value 898 * @return the result 899 * @throws ArithmeticException if the result overflows an int 900 * @since 1.8 901 */ 902 @HotSpotIntrinsicCandidate 903 public static int multiplyExact(int x, int y) { 904 long r = (long)x * (long)y; 905 if ((int)r != r) { 906 throw new ArithmeticException("integer overflow"); 907 } 908 return (int)r; 909 } 910 911 /** 912 * Returns the product of the arguments, throwing an exception if the result 913 * overflows a {@code long}. 914 * 915 * @param x the first value 916 * @param y the second value 917 * @return the result 918 * @throws ArithmeticException if the result overflows a long 919 * @since 9 920 */ 921 public static long multiplyExact(long x, int y) { 922 return multiplyExact(x, (long)y); 923 } 924 925 /** 926 * Returns the product of the arguments, 927 * throwing an exception if the result overflows a {@code long}. 928 * 929 * @param x the first value 930 * @param y the second value 931 * @return the result 932 * @throws ArithmeticException if the result overflows a long 933 * @since 1.8 934 */ 935 @HotSpotIntrinsicCandidate 936 public static long multiplyExact(long x, long y) { 937 long r = x * y; 938 long ax = Math.abs(x); 939 long ay = Math.abs(y); 940 if (((ax | ay) >>> 31 != 0)) { 941 // Some bits greater than 2^31 that might cause overflow 942 // Check the result using the divide operator 943 // and check for the special case of Long.MIN_VALUE * -1 944 if (((y != 0) && (r / y != x)) || 945 (x == Long.MIN_VALUE && y == -1)) { 946 throw new ArithmeticException("long overflow"); 947 } 948 } 949 return r; 950 } 951 952 /** 953 * Returns the argument incremented by one, throwing an exception if the 954 * result overflows an {@code int}. 955 * 956 * @param a the value to increment 957 * @return the result 958 * @throws ArithmeticException if the result overflows an int 959 * @since 1.8 960 */ 961 @HotSpotIntrinsicCandidate 962 public static int incrementExact(int a) { 963 if (a == Integer.MAX_VALUE) { 964 throw new ArithmeticException("integer overflow"); 965 } 966 967 return a + 1; 968 } 969 970 /** 971 * Returns the argument incremented by one, throwing an exception if the 972 * result overflows a {@code long}. 973 * 974 * @param a the value to increment 975 * @return the result 976 * @throws ArithmeticException if the result overflows a long 977 * @since 1.8 978 */ 979 @HotSpotIntrinsicCandidate 980 public static long incrementExact(long a) { 981 if (a == Long.MAX_VALUE) { 982 throw new ArithmeticException("long overflow"); 983 } 984 985 return a + 1L; 986 } 987 988 /** 989 * Returns the argument decremented by one, throwing an exception if the 990 * result overflows an {@code int}. 991 * 992 * @param a the value to decrement 993 * @return the result 994 * @throws ArithmeticException if the result overflows an int 995 * @since 1.8 996 */ 997 @HotSpotIntrinsicCandidate 998 public static int decrementExact(int a) { 999 if (a == Integer.MIN_VALUE) { 1000 throw new ArithmeticException("integer overflow"); 1001 } 1002 1003 return a - 1; 1004 } 1005 1006 /** 1007 * Returns the argument decremented by one, throwing an exception if the 1008 * result overflows a {@code long}. 1009 * 1010 * @param a the value to decrement 1011 * @return the result 1012 * @throws ArithmeticException if the result overflows a long 1013 * @since 1.8 1014 */ 1015 @HotSpotIntrinsicCandidate 1016 public static long decrementExact(long a) { 1017 if (a == Long.MIN_VALUE) { 1018 throw new ArithmeticException("long overflow"); 1019 } 1020 1021 return a - 1L; 1022 } 1023 1024 /** 1025 * Returns the negation of the argument, throwing an exception if the 1026 * result overflows an {@code int}. 1027 * 1028 * @param a the value to negate 1029 * @return the result 1030 * @throws ArithmeticException if the result overflows an int 1031 * @since 1.8 1032 */ 1033 @HotSpotIntrinsicCandidate 1034 public static int negateExact(int a) { 1035 if (a == Integer.MIN_VALUE) { 1036 throw new ArithmeticException("integer overflow"); 1037 } 1038 1039 return -a; 1040 } 1041 1042 /** 1043 * Returns the negation of the argument, throwing an exception if the 1044 * result overflows a {@code long}. 1045 * 1046 * @param a the value to negate 1047 * @return the result 1048 * @throws ArithmeticException if the result overflows a long 1049 * @since 1.8 1050 */ 1051 @HotSpotIntrinsicCandidate 1052 public static long negateExact(long a) { 1053 if (a == Long.MIN_VALUE) { 1054 throw new ArithmeticException("long overflow"); 1055 } 1056 1057 return -a; 1058 } 1059 1060 /** 1061 * Returns the value of the {@code long} argument; 1062 * throwing an exception if the value overflows an {@code int}. 1063 * 1064 * @param value the long value 1065 * @return the argument as an int 1066 * @throws ArithmeticException if the {@code argument} overflows an int 1067 * @since 1.8 1068 */ 1069 public static int toIntExact(long value) { 1070 if ((int)value != value) { 1071 throw new ArithmeticException("integer overflow"); 1072 } 1073 return (int)value; 1074 } 1075 1076 /** 1077 * Returns the exact mathematical product of the arguments. 1078 * 1079 * @param x the first value 1080 * @param y the second value 1081 * @return the result 1082 * @since 9 1083 */ 1084 public static long multiplyFull(int x, int y) { 1085 return (long)x * (long)y; 1086 } 1087 1088 /** 1089 * Returns as a {@code long} the most significant 64 bits of the 128-bit 1090 * product of two 64-bit factors. 1091 * 1092 * @param x the first value 1093 * @param y the second value 1094 * @return the result 1095 * @since 9 1096 */ 1097 @HotSpotIntrinsicCandidate 1098 public static long multiplyHigh(long x, long y) { 1099 if (x < 0 || y < 0) { 1100 // Use technique from section 8-2 of Henry S. Warren, Jr., 1101 // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174. 1102 long x1 = x >> 32; 1103 long x2 = x & 0xFFFFFFFFL; 1104 long y1 = y >> 32; 1105 long y2 = y & 0xFFFFFFFFL; 1106 long z2 = x2 * y2; 1107 long t = x1 * y2 + (z2 >>> 32); 1108 long z1 = t & 0xFFFFFFFFL; 1109 long z0 = t >> 32; 1110 z1 += x2 * y1; 1111 return x1 * y1 + z0 + (z1 >> 32); 1112 } else { 1113 // Use Karatsuba technique with two base 2^32 digits. 1114 long x1 = x >>> 32; 1115 long y1 = y >>> 32; 1116 long x2 = x & 0xFFFFFFFFL; 1117 long y2 = y & 0xFFFFFFFFL; 1118 long A = x1 * y1; 1119 long B = x2 * y2; 1120 long C = (x1 + x2) * (y1 + y2); 1121 long K = C - A - B; 1122 return (((B >>> 32) + K) >>> 32) + A; 1123 } 1124 } 1125 1126 /** 1127 * Returns the largest (closest to positive infinity) 1128 * {@code int} value that is less than or equal to the algebraic quotient. 1129 * There is one special case, if the dividend is the 1130 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1}, 1131 * then integer overflow occurs and 1132 * the result is equal to {@code Integer.MIN_VALUE}. 1133 * <p> 1134 * Normal integer division operates under the round to zero rounding mode 1135 * (truncation). This operation instead acts under the round toward 1136 * negative infinity (floor) rounding mode. 1137 * The floor rounding mode gives different results from truncation 1138 * when the exact result is negative. 1139 * <ul> 1140 * <li>If the signs of the arguments are the same, the results of 1141 * {@code floorDiv} and the {@code /} operator are the same. <br> 1142 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li> 1143 * <li>If the signs of the arguments are different, the quotient is negative and 1144 * {@code floorDiv} returns the integer less than or equal to the quotient 1145 * and the {@code /} operator returns the integer closest to zero.<br> 1146 * For example, {@code floorDiv(-4, 3) == -2}, 1147 * whereas {@code (-4 / 3) == -1}. 1148 * </li> 1149 * </ul> 1150 * 1151 * @param x the dividend 1152 * @param y the divisor 1153 * @return the largest (closest to positive infinity) 1154 * {@code int} value that is less than or equal to the algebraic quotient. 1155 * @throws ArithmeticException if the divisor {@code y} is zero 1156 * @see #floorMod(int, int) 1157 * @see #floor(double) 1158 * @since 1.8 1159 */ 1160 public static int floorDiv(int x, int y) { 1161 int r = x / y; 1162 // if the signs are different and modulo not zero, round down 1163 if ((x ^ y) < 0 && (r * y != x)) { 1164 r--; 1165 } 1166 return r; 1167 } 1168 1169 /** 1170 * Returns the largest (closest to positive infinity) 1171 * {@code long} value that is less than or equal to the algebraic quotient. 1172 * There is one special case, if the dividend is the 1173 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1174 * then integer overflow occurs and 1175 * the result is equal to {@code Long.MIN_VALUE}. 1176 * <p> 1177 * Normal integer division operates under the round to zero rounding mode 1178 * (truncation). This operation instead acts under the round toward 1179 * negative infinity (floor) rounding mode. 1180 * The floor rounding mode gives different results from truncation 1181 * when the exact result is negative. 1182 * <p> 1183 * For examples, see {@link #floorDiv(int, int)}. 1184 * 1185 * @param x the dividend 1186 * @param y the divisor 1187 * @return the largest (closest to positive infinity) 1188 * {@code int} value that is less than or equal to the algebraic quotient. 1189 * @throws ArithmeticException if the divisor {@code y} is zero 1190 * @see #floorMod(long, int) 1191 * @see #floor(double) 1192 * @since 9 1193 */ 1194 public static long floorDiv(long x, int y) { 1195 return floorDiv(x, (long)y); 1196 } 1197 1198 /** 1199 * Returns the largest (closest to positive infinity) 1200 * {@code long} value that is less than or equal to the algebraic quotient. 1201 * There is one special case, if the dividend is the 1202 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, 1203 * then integer overflow occurs and 1204 * the result is equal to {@code Long.MIN_VALUE}. 1205 * <p> 1206 * Normal integer division operates under the round to zero rounding mode 1207 * (truncation). This operation instead acts under the round toward 1208 * negative infinity (floor) rounding mode. 1209 * The floor rounding mode gives different results from truncation 1210 * when the exact result is negative. 1211 * <p> 1212 * For examples, see {@link #floorDiv(int, int)}. 1213 * 1214 * @param x the dividend 1215 * @param y the divisor 1216 * @return the largest (closest to positive infinity) 1217 * {@code long} value that is less than or equal to the algebraic quotient. 1218 * @throws ArithmeticException if the divisor {@code y} is zero 1219 * @see #floorMod(long, long) 1220 * @see #floor(double) 1221 * @since 1.8 1222 */ 1223 public static long floorDiv(long x, long y) { 1224 long r = x / y; 1225 // if the signs are different and modulo not zero, round down 1226 if ((x ^ y) < 0 && (r * y != x)) { 1227 r--; 1228 } 1229 return r; 1230 } 1231 1232 /** 1233 * Returns the floor modulus of the {@code int} arguments. 1234 * <p> 1235 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1236 * has the same sign as the divisor {@code y}, and 1237 * is in the range of {@code -abs(y) < r < +abs(y)}. 1238 * 1239 * <p> 1240 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1241 * <ul> 1242 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1243 * </ul> 1244 * <p> 1245 * The difference in values between {@code floorMod} and 1246 * the {@code %} operator is due to the difference between 1247 * {@code floorDiv} that returns the integer less than or equal to the quotient 1248 * and the {@code /} operator that returns the integer closest to zero. 1249 * <p> 1250 * Examples: 1251 * <ul> 1252 * <li>If the signs of the arguments are the same, the results 1253 * of {@code floorMod} and the {@code %} operator are the same. <br> 1254 * <ul> 1255 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li> 1256 * </ul> 1257 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br> 1258 * <ul> 1259 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li> 1260 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li> 1261 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li> 1262 * </ul> 1263 * </li> 1264 * </ul> 1265 * <p> 1266 * If the signs of arguments are unknown and a positive modulus 1267 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}. 1268 * 1269 * @param x the dividend 1270 * @param y the divisor 1271 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1272 * @throws ArithmeticException if the divisor {@code y} is zero 1273 * @see #floorDiv(int, int) 1274 * @since 1.8 1275 */ 1276 public static int floorMod(int x, int y) { 1277 int mod = x % y; 1278 // if the signs are different and modulo not zero, adjust result 1279 if ((mod ^ y) < 0 && mod != 0) { 1280 mod += y; 1281 } 1282 return mod; 1283 } 1284 1285 /** 1286 * Returns the floor modulus of the {@code long} and {@code int} arguments. 1287 * <p> 1288 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1289 * has the same sign as the divisor {@code y}, and 1290 * is in the range of {@code -abs(y) < r < +abs(y)}. 1291 * 1292 * <p> 1293 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1294 * <ul> 1295 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1296 * </ul> 1297 * <p> 1298 * For examples, see {@link #floorMod(int, int)}. 1299 * 1300 * @param x the dividend 1301 * @param y the divisor 1302 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1303 * @throws ArithmeticException if the divisor {@code y} is zero 1304 * @see #floorDiv(long, int) 1305 * @since 9 1306 */ 1307 public static int floorMod(long x, int y) { 1308 // Result cannot overflow the range of int. 1309 return (int)floorMod(x, (long)y); 1310 } 1311 1312 /** 1313 * Returns the floor modulus of the {@code long} arguments. 1314 * <p> 1315 * The floor modulus is {@code x - (floorDiv(x, y) * y)}, 1316 * has the same sign as the divisor {@code y}, and 1317 * is in the range of {@code -abs(y) < r < +abs(y)}. 1318 * 1319 * <p> 1320 * The relationship between {@code floorDiv} and {@code floorMod} is such that: 1321 * <ul> 1322 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} 1323 * </ul> 1324 * <p> 1325 * For examples, see {@link #floorMod(int, int)}. 1326 * 1327 * @param x the dividend 1328 * @param y the divisor 1329 * @return the floor modulus {@code x - (floorDiv(x, y) * y)} 1330 * @throws ArithmeticException if the divisor {@code y} is zero 1331 * @see #floorDiv(long, long) 1332 * @since 1.8 1333 */ 1334 public static long floorMod(long x, long y) { 1335 long mod = x % y; 1336 // if the signs are different and modulo not zero, adjust result 1337 if ((x ^ y) < 0 && mod != 0) { 1338 mod += y; 1339 } 1340 return mod; 1341 } 1342 1343 /** 1344 * Returns the absolute value of an {@code int} value. 1345 * If the argument is not negative, the argument is returned. 1346 * If the argument is negative, the negation of the argument is returned. 1347 * 1348 * <p>Note that if the argument is equal to the value of 1349 * {@link Integer#MIN_VALUE}, the most negative representable 1350 * {@code int} value, the result is that same value, which is 1351 * negative. 1352 * 1353 * @param a the argument whose absolute value is to be determined 1354 * @return the absolute value of the argument. 1355 */ 1356 @HotSpotIntrinsicCandidate 1357 public static int abs(int a) { 1358 return (a < 0) ? -a : a; 1359 } 1360 1361 /** 1362 * Returns the absolute value of a {@code long} value. 1363 * If the argument is not negative, the argument is returned. 1364 * If the argument is negative, the negation of the argument is returned. 1365 * 1366 * <p>Note that if the argument is equal to the value of 1367 * {@link Long#MIN_VALUE}, the most negative representable 1368 * {@code long} value, the result is that same value, which 1369 * is negative. 1370 * 1371 * @param a the argument whose absolute value is to be determined 1372 * @return the absolute value of the argument. 1373 */ 1374 @HotSpotIntrinsicCandidate 1375 public static long abs(long a) { 1376 return (a < 0) ? -a : a; 1377 } 1378 1379 /** 1380 * Returns the absolute value of a {@code float} value. 1381 * If the argument is not negative, the argument is returned. 1382 * If the argument is negative, the negation of the argument is returned. 1383 * Special cases: 1384 * <ul><li>If the argument is positive zero or negative zero, the 1385 * result is positive zero. 1386 * <li>If the argument is infinite, the result is positive infinity. 1387 * <li>If the argument is NaN, the result is NaN.</ul> 1388 * 1389 * @apiNote As implied by the above, one valid implementation of 1390 * this method is given by the expression below which computes a 1391 * {@code float} with the same exponent and significand as the 1392 * argument but with a guaranteed zero sign bit indicating a 1393 * positive value:<br> 1394 * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))} 1395 * 1396 * @param a the argument whose absolute value is to be determined 1397 * @return the absolute value of the argument. 1398 */ 1399 @HotSpotIntrinsicCandidate 1400 public static float abs(float a) { 1401 return (a <= 0.0F) ? 0.0F - a : a; 1402 } 1403 1404 /** 1405 * Returns the absolute value of a {@code double} value. 1406 * If the argument is not negative, the argument is returned. 1407 * If the argument is negative, the negation of the argument is returned. 1408 * Special cases: 1409 * <ul><li>If the argument is positive zero or negative zero, the result 1410 * is positive zero. 1411 * <li>If the argument is infinite, the result is positive infinity. 1412 * <li>If the argument is NaN, the result is NaN.</ul> 1413 * 1414 * @apiNote As implied by the above, one valid implementation of 1415 * this method is given by the expression below which computes a 1416 * {@code double} with the same exponent and significand as the 1417 * argument but with a guaranteed zero sign bit indicating a 1418 * positive value:<br> 1419 * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)} 1420 * 1421 * @param a the argument whose absolute value is to be determined 1422 * @return the absolute value of the argument. 1423 */ 1424 @HotSpotIntrinsicCandidate 1425 public static double abs(double a) { 1426 return (a <= 0.0D) ? 0.0D - a : a; 1427 } 1428 1429 /** 1430 * Returns the greater of two {@code int} values. That is, the 1431 * result is the argument closer to the value of 1432 * {@link Integer#MAX_VALUE}. If the arguments have the same value, 1433 * the result is that same value. 1434 * 1435 * @param a an argument. 1436 * @param b another argument. 1437 * @return the larger of {@code a} and {@code b}. 1438 */ 1439 @HotSpotIntrinsicCandidate 1440 public static int max(int a, int b) { 1441 return (a >= b) ? a : b; 1442 } 1443 1444 /** 1445 * Returns the greater of two {@code long} values. That is, the 1446 * result is the argument closer to the value of 1447 * {@link Long#MAX_VALUE}. If the arguments have the same value, 1448 * the result is that same value. 1449 * 1450 * @param a an argument. 1451 * @param b another argument. 1452 * @return the larger of {@code a} and {@code b}. 1453 */ 1454 public static long max(long a, long b) { 1455 return (a >= b) ? a : b; 1456 } 1457 1458 // Use raw bit-wise conversions on guaranteed non-NaN arguments. 1459 private static final long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f); 1460 private static final long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d); 1461 1462 /** 1463 * Returns the greater of two {@code float} values. That is, 1464 * the result is the argument closer to positive infinity. If the 1465 * arguments have the same value, the result is that same 1466 * value. If either value is NaN, then the result is NaN. Unlike 1467 * the numerical comparison operators, this method considers 1468 * negative zero to be strictly smaller than positive zero. If one 1469 * argument is positive zero and the other negative zero, the 1470 * result is positive zero. 1471 * 1472 * @param a an argument. 1473 * @param b another argument. 1474 * @return the larger of {@code a} and {@code b}. 1475 */ 1476 @HotSpotIntrinsicCandidate 1477 public static float max(float a, float b) { 1478 if (a != a) 1479 return a; // a is NaN 1480 if ((a == 0.0f) && 1481 (b == 0.0f) && 1482 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) { 1483 // Raw conversion ok since NaN can't map to -0.0. 1484 return b; 1485 } 1486 return (a >= b) ? a : b; 1487 } 1488 1489 /** 1490 * Returns the greater of two {@code double} values. That 1491 * is, the result is the argument closer to positive infinity. If 1492 * the arguments have the same value, the result is that same 1493 * value. If either value is NaN, then the result is NaN. Unlike 1494 * the numerical comparison operators, this method considers 1495 * negative zero to be strictly smaller than positive zero. If one 1496 * argument is positive zero and the other negative zero, the 1497 * result is positive zero. 1498 * 1499 * @param a an argument. 1500 * @param b another argument. 1501 * @return the larger of {@code a} and {@code b}. 1502 */ 1503 @HotSpotIntrinsicCandidate 1504 public static double max(double a, double b) { 1505 if (a != a) 1506 return a; // a is NaN 1507 if ((a == 0.0d) && 1508 (b == 0.0d) && 1509 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) { 1510 // Raw conversion ok since NaN can't map to -0.0. 1511 return b; 1512 } 1513 return (a >= b) ? a : b; 1514 } 1515 1516 /** 1517 * Returns the smaller of two {@code int} values. That is, 1518 * the result the argument closer to the value of 1519 * {@link Integer#MIN_VALUE}. If the arguments have the same 1520 * value, the result is that same value. 1521 * 1522 * @param a an argument. 1523 * @param b another argument. 1524 * @return the smaller of {@code a} and {@code b}. 1525 */ 1526 @HotSpotIntrinsicCandidate 1527 public static int min(int a, int b) { 1528 return (a <= b) ? a : b; 1529 } 1530 1531 /** 1532 * Returns the smaller of two {@code long} values. That is, 1533 * the result is the argument closer to the value of 1534 * {@link Long#MIN_VALUE}. If the arguments have the same 1535 * value, the result is that same value. 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 long min(long a, long b) { 1542 return (a <= b) ? a : b; 1543 } 1544 1545 /** 1546 * Returns the smaller of two {@code float} values. That is, 1547 * the result is the value closer to negative infinity. If the 1548 * arguments have the same value, the result is that same 1549 * value. If either value is NaN, then the result is NaN. Unlike 1550 * the numerical comparison operators, this method considers 1551 * negative zero to be strictly smaller than positive zero. If 1552 * one argument is positive zero and the other is negative zero, 1553 * the result is negative zero. 1554 * 1555 * @param a an argument. 1556 * @param b another argument. 1557 * @return the smaller of {@code a} and {@code b}. 1558 */ 1559 @HotSpotIntrinsicCandidate 1560 public static float min(float a, float b) { 1561 if (a != a) 1562 return a; // a is NaN 1563 if ((a == 0.0f) && 1564 (b == 0.0f) && 1565 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) { 1566 // Raw conversion ok since NaN can't map to -0.0. 1567 return b; 1568 } 1569 return (a <= b) ? a : b; 1570 } 1571 1572 /** 1573 * Returns the smaller of two {@code double} values. That 1574 * is, the result is the value closer to negative infinity. If the 1575 * arguments have the same value, the result is that same 1576 * value. If either value is NaN, then the result is NaN. Unlike 1577 * the numerical comparison operators, this method considers 1578 * negative zero to be strictly smaller than positive zero. If one 1579 * argument is positive zero and the other is negative zero, the 1580 * result is negative zero. 1581 * 1582 * @param a an argument. 1583 * @param b another argument. 1584 * @return the smaller of {@code a} and {@code b}. 1585 */ 1586 @HotSpotIntrinsicCandidate 1587 public static double min(double a, double b) { 1588 if (a != a) 1589 return a; // a is NaN 1590 if ((a == 0.0d) && 1591 (b == 0.0d) && 1592 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) { 1593 // Raw conversion ok since NaN can't map to -0.0. 1594 return b; 1595 } 1596 return (a <= b) ? a : b; 1597 } 1598 1599 /** 1600 * Returns the fused multiply add of the three arguments; that is, 1601 * returns the exact product of the first two arguments summed 1602 * with the third argument and then rounded once to the nearest 1603 * {@code double}. 1604 * 1605 * The rounding is done using the {@linkplain 1606 * java.math.RoundingMode#HALF_EVEN round to nearest even 1607 * rounding mode}. 1608 * 1609 * In contrast, if {@code a * b + c} is evaluated as a regular 1610 * floating-point expression, two rounding errors are involved, 1611 * the first for the multiply operation, the second for the 1612 * addition operation. 1613 * 1614 * <p>Special cases: 1615 * <ul> 1616 * <li> If any argument is NaN, the result is NaN. 1617 * 1618 * <li> If one of the first two arguments is infinite and the 1619 * other is zero, the result is NaN. 1620 * 1621 * <li> If the exact product of the first two arguments is infinite 1622 * (in other words, at least one of the arguments is infinite and 1623 * the other is neither zero nor NaN) and the third argument is an 1624 * infinity of the opposite sign, the result is NaN. 1625 * 1626 * </ul> 1627 * 1628 * <p>Note that {@code fma(a, 1.0, c)} returns the same 1629 * result as ({@code a + c}). However, 1630 * {@code fma(a, b, +0.0)} does <em>not</em> always return the 1631 * same result as ({@code a * b}) since 1632 * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while 1633 * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is 1634 * equivalent to ({@code a * b}) however. 1635 * 1636 * @apiNote This method corresponds to the fusedMultiplyAdd 1637 * operation defined in IEEE 754-2008. 1638 * 1639 * @param a a value 1640 * @param b a value 1641 * @param c a value 1642 * 1643 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1644 * computed, as if with unlimited range and precision, and rounded 1645 * once to the nearest {@code double} value 1646 * 1647 * @since 9 1648 */ 1649 @HotSpotIntrinsicCandidate 1650 public static double fma(double a, double b, double c) { 1651 /* 1652 * Infinity and NaN arithmetic is not quite the same with two 1653 * roundings as opposed to just one so the simple expression 1654 * "a * b + c" cannot always be used to compute the correct 1655 * result. With two roundings, the product can overflow and 1656 * if the addend is infinite, a spurious NaN can be produced 1657 * if the infinity from the overflow and the infinite addend 1658 * have opposite signs. 1659 */ 1660 1661 // First, screen for and handle non-finite input values whose 1662 // arithmetic is not supported by BigDecimal. 1663 if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) { 1664 return Double.NaN; 1665 } else { // All inputs non-NaN 1666 boolean infiniteA = Double.isInfinite(a); 1667 boolean infiniteB = Double.isInfinite(b); 1668 boolean infiniteC = Double.isInfinite(c); 1669 double result; 1670 1671 if (infiniteA || infiniteB || infiniteC) { 1672 if (infiniteA && b == 0.0 || 1673 infiniteB && a == 0.0 ) { 1674 return Double.NaN; 1675 } 1676 // Store product in a double field to cause an 1677 // overflow even if non-strictfp evaluation is being 1678 // used. 1679 double product = a * b; 1680 if (Double.isInfinite(product) && !infiniteA && !infiniteB) { 1681 // Intermediate overflow; might cause a 1682 // spurious NaN if added to infinite c. 1683 assert Double.isInfinite(c); 1684 return c; 1685 } else { 1686 result = product + c; 1687 assert !Double.isFinite(result); 1688 return result; 1689 } 1690 } else { // All inputs finite 1691 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b)); 1692 if (c == 0.0) { // Positive or negative zero 1693 // If the product is an exact zero, use a 1694 // floating-point expression to compute the sign 1695 // of the zero final result. The product is an 1696 // exact zero if and only if at least one of a and 1697 // b is zero. 1698 if (a == 0.0 || b == 0.0) { 1699 return a * b + c; 1700 } else { 1701 // The sign of a zero addend doesn't matter if 1702 // the product is nonzero. The sign of a zero 1703 // addend is not factored in the result if the 1704 // exact product is nonzero but underflows to 1705 // zero; see IEEE-754 2008 section 6.3 "The 1706 // sign bit". 1707 return product.doubleValue(); 1708 } 1709 } else { 1710 return product.add(new BigDecimal(c)).doubleValue(); 1711 } 1712 } 1713 } 1714 } 1715 1716 /** 1717 * Returns the fused multiply add of the three arguments; that is, 1718 * returns the exact product of the first two arguments summed 1719 * with the third argument and then rounded once to the nearest 1720 * {@code float}. 1721 * 1722 * The rounding is done using the {@linkplain 1723 * java.math.RoundingMode#HALF_EVEN round to nearest even 1724 * rounding mode}. 1725 * 1726 * In contrast, if {@code a * b + c} is evaluated as a regular 1727 * floating-point expression, two rounding errors are involved, 1728 * the first for the multiply operation, the second for the 1729 * addition operation. 1730 * 1731 * <p>Special cases: 1732 * <ul> 1733 * <li> If any argument is NaN, the result is NaN. 1734 * 1735 * <li> If one of the first two arguments is infinite and the 1736 * other is zero, the result is NaN. 1737 * 1738 * <li> If the exact product of the first two arguments is infinite 1739 * (in other words, at least one of the arguments is infinite and 1740 * the other is neither zero nor NaN) and the third argument is an 1741 * infinity of the opposite sign, the result is NaN. 1742 * 1743 * </ul> 1744 * 1745 * <p>Note that {@code fma(a, 1.0f, c)} returns the same 1746 * result as ({@code a + c}). However, 1747 * {@code fma(a, b, +0.0f)} does <em>not</em> always return the 1748 * same result as ({@code a * b}) since 1749 * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while 1750 * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is 1751 * equivalent to ({@code a * b}) however. 1752 * 1753 * @apiNote This method corresponds to the fusedMultiplyAdd 1754 * operation defined in IEEE 754-2008. 1755 * 1756 * @param a a value 1757 * @param b a value 1758 * @param c a value 1759 * 1760 * @return (<i>a</i> × <i>b</i> + <i>c</i>) 1761 * computed, as if with unlimited range and precision, and rounded 1762 * once to the nearest {@code float} value 1763 * 1764 * @since 9 1765 */ 1766 @HotSpotIntrinsicCandidate 1767 public static float fma(float a, float b, float c) { 1768 /* 1769 * Since the double format has more than twice the precision 1770 * of the float format, the multiply of a * b is exact in 1771 * double. The add of c to the product then incurs one 1772 * rounding error. Since the double format moreover has more 1773 * than (2p + 2) precision bits compared to the p bits of the 1774 * float format, the two roundings of (a * b + c), first to 1775 * the double format and then secondarily to the float format, 1776 * are equivalent to rounding the intermediate result directly 1777 * to the float format. 1778 * 1779 * In terms of strictfp vs default-fp concerns related to 1780 * overflow and underflow, since 1781 * 1782 * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE 1783 * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE 1784 * 1785 * neither the multiply nor add will overflow or underflow in 1786 * double. Therefore, it is not necessary for this method to 1787 * be declared strictfp to have reproducible 1788 * behavior. However, it is necessary to explicitly store down 1789 * to a float variable to avoid returning a value in the float 1790 * extended value set. 1791 */ 1792 float result = (float)(((double) a * (double) b ) + (double) c); 1793 return result; 1794 } 1795 1796 /** 1797 * Returns the size of an ulp of the argument. An ulp, unit in 1798 * the last place, of a {@code double} value is the positive 1799 * distance between this floating-point value and the {@code 1800 * double} value next larger in magnitude. Note that for non-NaN 1801 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1802 * 1803 * <p>Special Cases: 1804 * <ul> 1805 * <li> If the argument is NaN, then the result is NaN. 1806 * <li> If the argument is positive or negative infinity, then the 1807 * result is positive infinity. 1808 * <li> If the argument is positive or negative zero, then the result is 1809 * {@code Double.MIN_VALUE}. 1810 * <li> If the argument is ±{@code Double.MAX_VALUE}, then 1811 * the result is equal to 2<sup>971</sup>. 1812 * </ul> 1813 * 1814 * @param d the floating-point value whose ulp is to be returned 1815 * @return the size of an ulp of the argument 1816 * @author Joseph D. Darcy 1817 * @since 1.5 1818 */ 1819 public static double ulp(double d) { 1820 int exp = getExponent(d); 1821 1822 switch(exp) { 1823 case Double.MAX_EXPONENT + 1: // NaN or infinity 1824 return Math.abs(d); 1825 1826 case Double.MIN_EXPONENT - 1: // zero or subnormal 1827 return Double.MIN_VALUE; 1828 1829 default: 1830 assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT; 1831 1832 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1833 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1); 1834 if (exp >= Double.MIN_EXPONENT) { 1835 return powerOfTwoD(exp); 1836 } 1837 else { 1838 // return a subnormal result; left shift integer 1839 // representation of Double.MIN_VALUE appropriate 1840 // number of positions 1841 return Double.longBitsToDouble(1L << 1842 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) )); 1843 } 1844 } 1845 } 1846 1847 /** 1848 * Returns the size of an ulp of the argument. An ulp, unit in 1849 * the last place, of a {@code float} value is the positive 1850 * distance between this floating-point value and the {@code 1851 * float} value next larger in magnitude. Note that for non-NaN 1852 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. 1853 * 1854 * <p>Special Cases: 1855 * <ul> 1856 * <li> If the argument is NaN, then the result is NaN. 1857 * <li> If the argument is positive or negative infinity, then the 1858 * result is positive infinity. 1859 * <li> If the argument is positive or negative zero, then the result is 1860 * {@code Float.MIN_VALUE}. 1861 * <li> If the argument is ±{@code Float.MAX_VALUE}, then 1862 * the result is equal to 2<sup>104</sup>. 1863 * </ul> 1864 * 1865 * @param f the floating-point value whose ulp is to be returned 1866 * @return the size of an ulp of the argument 1867 * @author Joseph D. Darcy 1868 * @since 1.5 1869 */ 1870 public static float ulp(float f) { 1871 int exp = getExponent(f); 1872 1873 switch(exp) { 1874 case Float.MAX_EXPONENT+1: // NaN or infinity 1875 return Math.abs(f); 1876 1877 case Float.MIN_EXPONENT-1: // zero or subnormal 1878 return Float.MIN_VALUE; 1879 1880 default: 1881 assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT; 1882 1883 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) 1884 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1); 1885 if (exp >= Float.MIN_EXPONENT) { 1886 return powerOfTwoF(exp); 1887 } else { 1888 // return a subnormal result; left shift integer 1889 // representation of FloatConsts.MIN_VALUE appropriate 1890 // number of positions 1891 return Float.intBitsToFloat(1 << 1892 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) )); 1893 } 1894 } 1895 } 1896 1897 /** 1898 * Returns the signum function of the argument; zero if the argument 1899 * is zero, 1.0 if the argument is greater than zero, -1.0 if the 1900 * argument is less than zero. 1901 * 1902 * <p>Special Cases: 1903 * <ul> 1904 * <li> If the argument is NaN, then the result is NaN. 1905 * <li> If the argument is positive zero or negative zero, then the 1906 * result is the same as the argument. 1907 * </ul> 1908 * 1909 * @param d the floating-point value whose signum is to be returned 1910 * @return the signum function of the argument 1911 * @author Joseph D. Darcy 1912 * @since 1.5 1913 */ 1914 public static double signum(double d) { 1915 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d); 1916 } 1917 1918 /** 1919 * Returns the signum function of the argument; zero if the argument 1920 * is zero, 1.0f if the argument is greater than zero, -1.0f if the 1921 * argument is less than zero. 1922 * 1923 * <p>Special Cases: 1924 * <ul> 1925 * <li> If the argument is NaN, then the result is NaN. 1926 * <li> If the argument is positive zero or negative zero, then the 1927 * result is the same as the argument. 1928 * </ul> 1929 * 1930 * @param f the floating-point value whose signum is to be returned 1931 * @return the signum function of the argument 1932 * @author Joseph D. Darcy 1933 * @since 1.5 1934 */ 1935 public static float signum(float f) { 1936 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f); 1937 } 1938 1939 /** 1940 * Returns the hyperbolic sine of a {@code double} value. 1941 * The hyperbolic sine of <i>x</i> is defined to be 1942 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 1943 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1944 * 1945 * <p>Special cases: 1946 * <ul> 1947 * 1948 * <li>If the argument is NaN, then the result is NaN. 1949 * 1950 * <li>If the argument is infinite, then the result is an infinity 1951 * with the same sign as the argument. 1952 * 1953 * <li>If the argument is zero, then the result is a zero with the 1954 * same sign as the argument. 1955 * 1956 * </ul> 1957 * 1958 * <p>The computed result must be within 2.5 ulps of the exact result. 1959 * 1960 * @param x The number whose hyperbolic sine is to be returned. 1961 * @return The hyperbolic sine of {@code x}. 1962 * @since 1.5 1963 */ 1964 public static double sinh(double x) { 1965 return StrictMath.sinh(x); 1966 } 1967 1968 /** 1969 * Returns the hyperbolic cosine of a {@code double} value. 1970 * The hyperbolic cosine of <i>x</i> is defined to be 1971 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 1972 * where <i>e</i> is {@linkplain Math#E Euler's number}. 1973 * 1974 * <p>Special cases: 1975 * <ul> 1976 * 1977 * <li>If the argument is NaN, then the result is NaN. 1978 * 1979 * <li>If the argument is infinite, then the result is positive 1980 * infinity. 1981 * 1982 * <li>If the argument is zero, then the result is {@code 1.0}. 1983 * 1984 * </ul> 1985 * 1986 * <p>The computed result must be within 2.5 ulps of the exact result. 1987 * 1988 * @param x The number whose hyperbolic cosine is to be returned. 1989 * @return The hyperbolic cosine of {@code x}. 1990 * @since 1.5 1991 */ 1992 public static double cosh(double x) { 1993 return StrictMath.cosh(x); 1994 } 1995 1996 /** 1997 * Returns the hyperbolic tangent of a {@code double} value. 1998 * The hyperbolic tangent of <i>x</i> is defined to be 1999 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), 2000 * in other words, {@linkplain Math#sinh 2001 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note 2002 * that the absolute value of the exact tanh is always less than 2003 * 1. 2004 * 2005 * <p>Special cases: 2006 * <ul> 2007 * 2008 * <li>If the argument is NaN, then the result is NaN. 2009 * 2010 * <li>If the argument is zero, then the result is a zero with the 2011 * same sign as the argument. 2012 * 2013 * <li>If the argument is positive infinity, then the result is 2014 * {@code +1.0}. 2015 * 2016 * <li>If the argument is negative infinity, then the result is 2017 * {@code -1.0}. 2018 * 2019 * </ul> 2020 * 2021 * <p>The computed result must be within 2.5 ulps of the exact result. 2022 * The result of {@code tanh} for any finite input must have 2023 * an absolute value less than or equal to 1. Note that once the 2024 * exact result of tanh is within 1/2 of an ulp of the limit value 2025 * of ±1, correctly signed ±{@code 1.0} should 2026 * be returned. 2027 * 2028 * @param x The number whose hyperbolic tangent is to be returned. 2029 * @return The hyperbolic tangent of {@code x}. 2030 * @since 1.5 2031 */ 2032 public static double tanh(double x) { 2033 return StrictMath.tanh(x); 2034 } 2035 2036 /** 2037 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2038 * without intermediate overflow or underflow. 2039 * 2040 * <p>Special cases: 2041 * <ul> 2042 * 2043 * <li> If either argument is infinite, then the result 2044 * is positive infinity. 2045 * 2046 * <li> If either argument is NaN and neither argument is infinite, 2047 * then the result is NaN. 2048 * 2049 * </ul> 2050 * 2051 * <p>The computed result must be within 1 ulp of the exact 2052 * result. If one parameter is held constant, the results must be 2053 * semi-monotonic in the other parameter. 2054 * 2055 * @param x a value 2056 * @param y a value 2057 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) 2058 * without intermediate overflow or underflow 2059 * @since 1.5 2060 */ 2061 public static double hypot(double x, double y) { 2062 return StrictMath.hypot(x, y); 2063 } 2064 2065 /** 2066 * Returns <i>e</i><sup>x</sup> -1. Note that for values of 2067 * <i>x</i> near 0, the exact sum of 2068 * {@code expm1(x)} + 1 is much closer to the true 2069 * result of <i>e</i><sup>x</sup> than {@code exp(x)}. 2070 * 2071 * <p>Special cases: 2072 * <ul> 2073 * <li>If the argument is NaN, the result is NaN. 2074 * 2075 * <li>If the argument is positive infinity, then the result is 2076 * positive infinity. 2077 * 2078 * <li>If the argument is negative infinity, then the result is 2079 * -1.0. 2080 * 2081 * <li>If the argument is zero, then the result is a zero with the 2082 * same sign as the argument. 2083 * 2084 * </ul> 2085 * 2086 * <p>The computed result must be within 1 ulp of the exact result. 2087 * Results must be semi-monotonic. The result of 2088 * {@code expm1} for any finite input must be greater than or 2089 * equal to {@code -1.0}. Note that once the exact result of 2090 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 2091 * ulp of the limit value -1, {@code -1.0} should be 2092 * returned. 2093 * 2094 * @param x the exponent to raise <i>e</i> to in the computation of 2095 * <i>e</i><sup>{@code x}</sup> -1. 2096 * @return the value <i>e</i><sup>{@code x}</sup> - 1. 2097 * @since 1.5 2098 */ 2099 public static double expm1(double x) { 2100 return StrictMath.expm1(x); 2101 } 2102 2103 /** 2104 * Returns the natural logarithm of the sum of the argument and 1. 2105 * Note that for small values {@code x}, the result of 2106 * {@code log1p(x)} is much closer to the true result of ln(1 2107 * + {@code x}) than the floating-point evaluation of 2108 * {@code log(1.0+x)}. 2109 * 2110 * <p>Special cases: 2111 * 2112 * <ul> 2113 * 2114 * <li>If the argument is NaN or less than -1, then the result is 2115 * NaN. 2116 * 2117 * <li>If the argument is positive infinity, then the result is 2118 * positive infinity. 2119 * 2120 * <li>If the argument is negative one, then the result is 2121 * negative infinity. 2122 * 2123 * <li>If the argument is zero, then the result is a zero with the 2124 * same sign as the argument. 2125 * 2126 * </ul> 2127 * 2128 * <p>The computed result must be within 1 ulp of the exact result. 2129 * Results must be semi-monotonic. 2130 * 2131 * @param x a value 2132 * @return the value ln({@code x} + 1), the natural 2133 * log of {@code x} + 1 2134 * @since 1.5 2135 */ 2136 public static double log1p(double x) { 2137 return StrictMath.log1p(x); 2138 } 2139 2140 /** 2141 * Returns the first floating-point argument with the sign of the 2142 * second floating-point argument. Note that unlike the {@link 2143 * StrictMath#copySign(double, double) StrictMath.copySign} 2144 * method, this method does not require NaN {@code sign} 2145 * arguments to be treated as positive values; implementations are 2146 * permitted to treat some NaN arguments as positive and other NaN 2147 * arguments as negative to allow greater performance. 2148 * 2149 * @param magnitude the parameter providing the magnitude of the result 2150 * @param sign the parameter providing the sign of the result 2151 * @return a value with the magnitude of {@code magnitude} 2152 * and the sign of {@code sign}. 2153 * @since 1.6 2154 */ 2155 public static double copySign(double magnitude, double sign) { 2156 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) & 2157 (DoubleConsts.SIGN_BIT_MASK)) | 2158 (Double.doubleToRawLongBits(magnitude) & 2159 (DoubleConsts.EXP_BIT_MASK | 2160 DoubleConsts.SIGNIF_BIT_MASK))); 2161 } 2162 2163 /** 2164 * Returns the first floating-point argument with the sign of the 2165 * second floating-point argument. Note that unlike the {@link 2166 * StrictMath#copySign(float, float) StrictMath.copySign} 2167 * method, this method does not require NaN {@code sign} 2168 * arguments to be treated as positive values; implementations are 2169 * permitted to treat some NaN arguments as positive and other NaN 2170 * arguments as negative to allow greater performance. 2171 * 2172 * @param magnitude the parameter providing the magnitude of the result 2173 * @param sign the parameter providing the sign of the result 2174 * @return a value with the magnitude of {@code magnitude} 2175 * and the sign of {@code sign}. 2176 * @since 1.6 2177 */ 2178 public static float copySign(float magnitude, float sign) { 2179 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) & 2180 (FloatConsts.SIGN_BIT_MASK)) | 2181 (Float.floatToRawIntBits(magnitude) & 2182 (FloatConsts.EXP_BIT_MASK | 2183 FloatConsts.SIGNIF_BIT_MASK))); 2184 } 2185 2186 /** 2187 * Returns the unbiased exponent used in the representation of a 2188 * {@code float}. Special cases: 2189 * 2190 * <ul> 2191 * <li>If the argument is NaN or infinite, then the result is 2192 * {@link Float#MAX_EXPONENT} + 1. 2193 * <li>If the argument is zero or subnormal, then the result is 2194 * {@link Float#MIN_EXPONENT} -1. 2195 * </ul> 2196 * @param f a {@code float} value 2197 * @return the unbiased exponent of the argument 2198 * @since 1.6 2199 */ 2200 public static int getExponent(float f) { 2201 /* 2202 * Bitwise convert f to integer, mask out exponent bits, shift 2203 * to the right and then subtract out float's bias adjust to 2204 * get true exponent value 2205 */ 2206 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >> 2207 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS; 2208 } 2209 2210 /** 2211 * Returns the unbiased exponent used in the representation of a 2212 * {@code double}. Special cases: 2213 * 2214 * <ul> 2215 * <li>If the argument is NaN or infinite, then the result is 2216 * {@link Double#MAX_EXPONENT} + 1. 2217 * <li>If the argument is zero or subnormal, then the result is 2218 * {@link Double#MIN_EXPONENT} -1. 2219 * </ul> 2220 * @param d a {@code double} value 2221 * @return the unbiased exponent of the argument 2222 * @since 1.6 2223 */ 2224 public static int getExponent(double d) { 2225 /* 2226 * Bitwise convert d to long, mask out exponent bits, shift 2227 * to the right and then subtract out double's bias adjust to 2228 * get true exponent value. 2229 */ 2230 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >> 2231 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS); 2232 } 2233 2234 /** 2235 * Returns the floating-point number adjacent to the first 2236 * argument in the direction of the second argument. If both 2237 * arguments compare as equal the second argument is returned. 2238 * 2239 * <p> 2240 * Special cases: 2241 * <ul> 2242 * <li> If either argument is a NaN, then NaN is returned. 2243 * 2244 * <li> If both arguments are signed zeros, {@code direction} 2245 * is returned unchanged (as implied by the requirement of 2246 * returning the second argument if the arguments compare as 2247 * equal). 2248 * 2249 * <li> If {@code start} is 2250 * ±{@link Double#MIN_VALUE} and {@code direction} 2251 * has a value such that the result should have a smaller 2252 * magnitude, then a zero with the same sign as {@code start} 2253 * is returned. 2254 * 2255 * <li> If {@code start} is infinite and 2256 * {@code direction} has a value such that the result should 2257 * have a smaller magnitude, {@link Double#MAX_VALUE} with the 2258 * same sign as {@code start} is returned. 2259 * 2260 * <li> If {@code start} is equal to ± 2261 * {@link Double#MAX_VALUE} and {@code direction} has a 2262 * value such that the result should have a larger magnitude, an 2263 * infinity with same sign as {@code start} is returned. 2264 * </ul> 2265 * 2266 * @param start starting floating-point value 2267 * @param direction value indicating which of 2268 * {@code start}'s neighbors or {@code start} should 2269 * be returned 2270 * @return The floating-point number adjacent to {@code start} in the 2271 * direction of {@code direction}. 2272 * @since 1.6 2273 */ 2274 public static double nextAfter(double start, double direction) { 2275 /* 2276 * The cases: 2277 * 2278 * nextAfter(+infinity, 0) == MAX_VALUE 2279 * nextAfter(+infinity, +infinity) == +infinity 2280 * nextAfter(-infinity, 0) == -MAX_VALUE 2281 * nextAfter(-infinity, -infinity) == -infinity 2282 * 2283 * are naturally handled without any additional testing 2284 */ 2285 2286 /* 2287 * IEEE 754 floating-point numbers are lexicographically 2288 * ordered if treated as signed-magnitude integers. 2289 * Since Java's integers are two's complement, 2290 * incrementing the two's complement representation of a 2291 * logically negative floating-point value *decrements* 2292 * the signed-magnitude representation. Therefore, when 2293 * the integer representation of a floating-point value 2294 * is negative, the adjustment to the representation is in 2295 * the opposite direction from what would initially be expected. 2296 */ 2297 2298 // Branch to descending case first as it is more costly than ascending 2299 // case due to start != 0.0d conditional. 2300 if (start > direction) { // descending 2301 if (start != 0.0d) { 2302 final long transducer = Double.doubleToRawLongBits(start); 2303 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L)); 2304 } else { // start == 0.0d && direction < 0.0d 2305 return -Double.MIN_VALUE; 2306 } 2307 } else if (start < direction) { // ascending 2308 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2309 // then bitwise convert start to integer. 2310 final long transducer = Double.doubleToRawLongBits(start + 0.0d); 2311 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2312 } else if (start == direction) { 2313 return direction; 2314 } else { // isNaN(start) || isNaN(direction) 2315 return start + direction; 2316 } 2317 } 2318 2319 /** 2320 * Returns the floating-point number adjacent to the first 2321 * argument in the direction of the second argument. If both 2322 * arguments compare as equal a value equivalent to the second argument 2323 * is returned. 2324 * 2325 * <p> 2326 * Special cases: 2327 * <ul> 2328 * <li> If either argument is a NaN, then NaN is returned. 2329 * 2330 * <li> If both arguments are signed zeros, a value equivalent 2331 * to {@code direction} is returned. 2332 * 2333 * <li> If {@code start} is 2334 * ±{@link Float#MIN_VALUE} and {@code direction} 2335 * has a value such that the result should have a smaller 2336 * magnitude, then a zero with the same sign as {@code start} 2337 * is returned. 2338 * 2339 * <li> If {@code start} is infinite and 2340 * {@code direction} has a value such that the result should 2341 * have a smaller magnitude, {@link Float#MAX_VALUE} with the 2342 * same sign as {@code start} is returned. 2343 * 2344 * <li> If {@code start} is equal to ± 2345 * {@link Float#MAX_VALUE} and {@code direction} has a 2346 * value such that the result should have a larger magnitude, an 2347 * infinity with same sign as {@code start} is returned. 2348 * </ul> 2349 * 2350 * @param start starting floating-point value 2351 * @param direction value indicating which of 2352 * {@code start}'s neighbors or {@code start} should 2353 * be returned 2354 * @return The floating-point number adjacent to {@code start} in the 2355 * direction of {@code direction}. 2356 * @since 1.6 2357 */ 2358 public static float nextAfter(float start, double direction) { 2359 /* 2360 * The cases: 2361 * 2362 * nextAfter(+infinity, 0) == MAX_VALUE 2363 * nextAfter(+infinity, +infinity) == +infinity 2364 * nextAfter(-infinity, 0) == -MAX_VALUE 2365 * nextAfter(-infinity, -infinity) == -infinity 2366 * 2367 * are naturally handled without any additional testing 2368 */ 2369 2370 /* 2371 * IEEE 754 floating-point numbers are lexicographically 2372 * ordered if treated as signed-magnitude integers. 2373 * Since Java's integers are two's complement, 2374 * incrementing the two's complement representation of a 2375 * logically negative floating-point value *decrements* 2376 * the signed-magnitude representation. Therefore, when 2377 * the integer representation of a floating-point value 2378 * is negative, the adjustment to the representation is in 2379 * the opposite direction from what would initially be expected. 2380 */ 2381 2382 // Branch to descending case first as it is more costly than ascending 2383 // case due to start != 0.0f conditional. 2384 if (start > direction) { // descending 2385 if (start != 0.0f) { 2386 final int transducer = Float.floatToRawIntBits(start); 2387 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1)); 2388 } else { // start == 0.0f && direction < 0.0f 2389 return -Float.MIN_VALUE; 2390 } 2391 } else if (start < direction) { // ascending 2392 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) 2393 // then bitwise convert start to integer. 2394 final int transducer = Float.floatToRawIntBits(start + 0.0f); 2395 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2396 } else if (start == direction) { 2397 return (float)direction; 2398 } else { // isNaN(start) || isNaN(direction) 2399 return start + (float)direction; 2400 } 2401 } 2402 2403 /** 2404 * Returns the floating-point value adjacent to {@code d} in 2405 * the direction of positive infinity. This method is 2406 * semantically equivalent to {@code nextAfter(d, 2407 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp} 2408 * implementation may run faster than its equivalent 2409 * {@code nextAfter} call. 2410 * 2411 * <p>Special Cases: 2412 * <ul> 2413 * <li> If the argument is NaN, the result is NaN. 2414 * 2415 * <li> If the argument is positive infinity, the result is 2416 * positive infinity. 2417 * 2418 * <li> If the argument is zero, the result is 2419 * {@link Double#MIN_VALUE} 2420 * 2421 * </ul> 2422 * 2423 * @param d starting floating-point value 2424 * @return The adjacent floating-point value closer to positive 2425 * infinity. 2426 * @since 1.6 2427 */ 2428 public static double nextUp(double d) { 2429 // Use a single conditional and handle the likely cases first. 2430 if (d < Double.POSITIVE_INFINITY) { 2431 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2432 final long transducer = Double.doubleToRawLongBits(d + 0.0D); 2433 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); 2434 } else { // d is NaN or +Infinity 2435 return d; 2436 } 2437 } 2438 2439 /** 2440 * Returns the floating-point value adjacent to {@code f} in 2441 * the direction of positive infinity. This method is 2442 * semantically equivalent to {@code nextAfter(f, 2443 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp} 2444 * implementation may run faster than its equivalent 2445 * {@code nextAfter} call. 2446 * 2447 * <p>Special Cases: 2448 * <ul> 2449 * <li> If the argument is NaN, the result is NaN. 2450 * 2451 * <li> If the argument is positive infinity, the result is 2452 * positive infinity. 2453 * 2454 * <li> If the argument is zero, the result is 2455 * {@link Float#MIN_VALUE} 2456 * 2457 * </ul> 2458 * 2459 * @param f starting floating-point value 2460 * @return The adjacent floating-point value closer to positive 2461 * infinity. 2462 * @since 1.6 2463 */ 2464 public static float nextUp(float f) { 2465 // Use a single conditional and handle the likely cases first. 2466 if (f < Float.POSITIVE_INFINITY) { 2467 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). 2468 final int transducer = Float.floatToRawIntBits(f + 0.0F); 2469 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); 2470 } else { // f is NaN or +Infinity 2471 return f; 2472 } 2473 } 2474 2475 /** 2476 * Returns the floating-point value adjacent to {@code d} in 2477 * the direction of negative infinity. This method is 2478 * semantically equivalent to {@code nextAfter(d, 2479 * Double.NEGATIVE_INFINITY)}; however, a 2480 * {@code nextDown} implementation may run faster than its 2481 * equivalent {@code nextAfter} call. 2482 * 2483 * <p>Special Cases: 2484 * <ul> 2485 * <li> If the argument is NaN, the result is NaN. 2486 * 2487 * <li> If the argument is negative infinity, the result is 2488 * negative infinity. 2489 * 2490 * <li> If the argument is zero, the result is 2491 * {@code -Double.MIN_VALUE} 2492 * 2493 * </ul> 2494 * 2495 * @param d starting floating-point value 2496 * @return The adjacent floating-point value closer to negative 2497 * infinity. 2498 * @since 1.8 2499 */ 2500 public static double nextDown(double d) { 2501 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY) 2502 return d; 2503 else { 2504 if (d == 0.0) 2505 return -Double.MIN_VALUE; 2506 else 2507 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) + 2508 ((d > 0.0d)?-1L:+1L)); 2509 } 2510 } 2511 2512 /** 2513 * Returns the floating-point value adjacent to {@code f} in 2514 * the direction of negative infinity. This method is 2515 * semantically equivalent to {@code nextAfter(f, 2516 * Float.NEGATIVE_INFINITY)}; however, a 2517 * {@code nextDown} implementation may run faster than its 2518 * equivalent {@code nextAfter} call. 2519 * 2520 * <p>Special Cases: 2521 * <ul> 2522 * <li> If the argument is NaN, the result is NaN. 2523 * 2524 * <li> If the argument is negative infinity, the result is 2525 * negative infinity. 2526 * 2527 * <li> If the argument is zero, the result is 2528 * {@code -Float.MIN_VALUE} 2529 * 2530 * </ul> 2531 * 2532 * @param f starting floating-point value 2533 * @return The adjacent floating-point value closer to negative 2534 * infinity. 2535 * @since 1.8 2536 */ 2537 public static float nextDown(float f) { 2538 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY) 2539 return f; 2540 else { 2541 if (f == 0.0f) 2542 return -Float.MIN_VALUE; 2543 else 2544 return Float.intBitsToFloat(Float.floatToRawIntBits(f) + 2545 ((f > 0.0f)?-1:+1)); 2546 } 2547 } 2548 2549 /** 2550 * Returns {@code d} × 2551 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2552 * by a single correctly rounded floating-point multiply to a 2553 * member of the double value set. See the Java 2554 * Language Specification for a discussion of floating-point 2555 * value sets. If the exponent of the result is between {@link 2556 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the 2557 * answer is calculated exactly. If the exponent of the result 2558 * would be larger than {@code Double.MAX_EXPONENT}, an 2559 * infinity is returned. Note that if the result is subnormal, 2560 * precision may be lost; that is, when {@code scalb(x, n)} 2561 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2562 * <i>x</i>. When the result is non-NaN, the result has the same 2563 * sign as {@code d}. 2564 * 2565 * <p>Special cases: 2566 * <ul> 2567 * <li> If the first argument is NaN, NaN is returned. 2568 * <li> If the first argument is infinite, then an infinity of the 2569 * same sign is returned. 2570 * <li> If the first argument is zero, then a zero of the same 2571 * sign is returned. 2572 * </ul> 2573 * 2574 * @param d number to be scaled by a power of two. 2575 * @param scaleFactor power of 2 used to scale {@code d} 2576 * @return {@code d} × 2<sup>{@code scaleFactor}</sup> 2577 * @since 1.6 2578 */ 2579 public static double scalb(double d, int scaleFactor) { 2580 /* 2581 * This method does not need to be declared strictfp to 2582 * compute the same correct result on all platforms. When 2583 * scaling up, it does not matter what order the 2584 * multiply-store operations are done; the result will be 2585 * finite or overflow regardless of the operation ordering. 2586 * However, to get the correct result when scaling down, a 2587 * particular ordering must be used. 2588 * 2589 * When scaling down, the multiply-store operations are 2590 * sequenced so that it is not possible for two consecutive 2591 * multiply-stores to return subnormal results. If one 2592 * multiply-store result is subnormal, the next multiply will 2593 * round it away to zero. This is done by first multiplying 2594 * by 2 ^ (scaleFactor % n) and then multiplying several 2595 * times by 2^n as needed where n is the exponent of number 2596 * that is a covenient power of two. In this way, at most one 2597 * real rounding error occurs. If the double value set is 2598 * being used exclusively, the rounding will occur on a 2599 * multiply. If the double-extended-exponent value set is 2600 * being used, the products will (perhaps) be exact but the 2601 * stores to d are guaranteed to round to the double value 2602 * set. 2603 * 2604 * It is _not_ a valid implementation to first multiply d by 2605 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor % 2606 * MIN_EXPONENT) since even in a strictfp program double 2607 * rounding on underflow could occur; e.g. if the scaleFactor 2608 * argument was (MIN_EXPONENT - n) and the exponent of d was a 2609 * little less than -(MIN_EXPONENT - n), meaning the final 2610 * result would be subnormal. 2611 * 2612 * Since exact reproducibility of this method can be achieved 2613 * without any undue performance burden, there is no 2614 * compelling reason to allow double rounding on underflow in 2615 * scalb. 2616 */ 2617 2618 // magnitude of a power of two so large that scaling a finite 2619 // nonzero value by it would be guaranteed to over or 2620 // underflow; due to rounding, scaling down takes an 2621 // additional power of two which is reflected here 2622 final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT + 2623 DoubleConsts.SIGNIFICAND_WIDTH + 1; 2624 int exp_adjust = 0; 2625 int scale_increment = 0; 2626 double exp_delta = Double.NaN; 2627 2628 // Make sure scaling factor is in a reasonable range 2629 2630 if(scaleFactor < 0) { 2631 scaleFactor = Math.max(scaleFactor, -MAX_SCALE); 2632 scale_increment = -512; 2633 exp_delta = twoToTheDoubleScaleDown; 2634 } 2635 else { 2636 scaleFactor = Math.min(scaleFactor, MAX_SCALE); 2637 scale_increment = 512; 2638 exp_delta = twoToTheDoubleScaleUp; 2639 } 2640 2641 // Calculate (scaleFactor % +/-512), 512 = 2^9, using 2642 // technique from "Hacker's Delight" section 10-2. 2643 int t = (scaleFactor >> 9-1) >>> 32 - 9; 2644 exp_adjust = ((scaleFactor + t) & (512 -1)) - t; 2645 2646 d *= powerOfTwoD(exp_adjust); 2647 scaleFactor -= exp_adjust; 2648 2649 while(scaleFactor != 0) { 2650 d *= exp_delta; 2651 scaleFactor -= scale_increment; 2652 } 2653 return d; 2654 } 2655 2656 /** 2657 * Returns {@code f} × 2658 * 2<sup>{@code scaleFactor}</sup> rounded as if performed 2659 * by a single correctly rounded floating-point multiply to a 2660 * member of the float value set. See the Java 2661 * Language Specification for a discussion of floating-point 2662 * value sets. If the exponent of the result is between {@link 2663 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the 2664 * answer is calculated exactly. If the exponent of the result 2665 * would be larger than {@code Float.MAX_EXPONENT}, an 2666 * infinity is returned. Note that if the result is subnormal, 2667 * precision may be lost; that is, when {@code scalb(x, n)} 2668 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal 2669 * <i>x</i>. When the result is non-NaN, the result has the same 2670 * sign as {@code f}. 2671 * 2672 * <p>Special cases: 2673 * <ul> 2674 * <li> If the first argument is NaN, NaN is returned. 2675 * <li> If the first argument is infinite, then an infinity of the 2676 * same sign is returned. 2677 * <li> If the first argument is zero, then a zero of the same 2678 * sign is returned. 2679 * </ul> 2680 * 2681 * @param f number to be scaled by a power of two. 2682 * @param scaleFactor power of 2 used to scale {@code f} 2683 * @return {@code f} × 2<sup>{@code scaleFactor}</sup> 2684 * @since 1.6 2685 */ 2686 public static float scalb(float f, int scaleFactor) { 2687 // magnitude of a power of two so large that scaling a finite 2688 // nonzero value by it would be guaranteed to over or 2689 // underflow; due to rounding, scaling down takes an 2690 // additional power of two which is reflected here 2691 final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT + 2692 FloatConsts.SIGNIFICAND_WIDTH + 1; 2693 2694 // Make sure scaling factor is in a reasonable range 2695 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE); 2696 2697 /* 2698 * Since + MAX_SCALE for float fits well within the double 2699 * exponent range and + float -> double conversion is exact 2700 * the multiplication below will be exact. Therefore, the 2701 * rounding that occurs when the double product is cast to 2702 * float will be the correctly rounded float result. Since 2703 * all operations other than the final multiply will be exact, 2704 * it is not necessary to declare this method strictfp. 2705 */ 2706 return (float)((double)f*powerOfTwoD(scaleFactor)); 2707 } 2708 2709 // Constants used in scalb 2710 static double twoToTheDoubleScaleUp = powerOfTwoD(512); 2711 static double twoToTheDoubleScaleDown = powerOfTwoD(-512); 2712 2713 /** 2714 * Returns a floating-point power of two in the normal range. 2715 */ 2716 static double powerOfTwoD(int n) { 2717 assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT); 2718 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) << 2719 (DoubleConsts.SIGNIFICAND_WIDTH-1)) 2720 & DoubleConsts.EXP_BIT_MASK); 2721 } 2722 2723 /** 2724 * Returns a floating-point power of two in the normal range. 2725 */ 2726 static float powerOfTwoF(int n) { 2727 assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT); 2728 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) << 2729 (FloatConsts.SIGNIFICAND_WIDTH-1)) 2730 & FloatConsts.EXP_BIT_MASK); 2731 } 2732 }