1 /*
2 * Copyright (c) 1994, 2019, 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}, {@code toIntExact},
96 * {@code incrementExact}, {@code decrementExact} and {@code negateExact}
97 * throw an {@code ArithmeticException} when the results overflow.
98 * For the arithmetic operations divide and absolute value, overflow
99 * occurs only with a specific minimum or maximum value and
100 * should be checked against 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 @HotSpotIntrinsicCandidate
444 public static double ceil(double a) {
445 return StrictMath.ceil(a); // default impl. delegates to StrictMath
446 }
447
448 /**
449 * Returns the largest (closest to positive infinity)
450 * {@code double} value that is less than or equal to the
451 * argument and is equal to a mathematical integer. Special cases:
452 * <ul><li>If the argument value is already equal to a
453 * mathematical integer, then the result is the same as the
454 * argument. <li>If the argument is NaN or an infinity or
455 * positive zero or negative zero, then the result is the same as
456 * the argument.</ul>
457 *
458 * @param a a value.
459 * @return the largest (closest to positive infinity)
460 * floating-point value that less than or equal to the argument
461 * and is equal to a mathematical integer.
462 */
463 @HotSpotIntrinsicCandidate
464 public static double floor(double a) {
465 return StrictMath.floor(a); // default impl. delegates to StrictMath
466 }
467
468 /**
469 * Returns the {@code double} value that is closest in value
470 * to the argument and is equal to a mathematical integer. If two
471 * {@code double} values that are mathematical integers are
472 * equally close, the result is the integer value that is
473 * even. Special cases:
474 * <ul><li>If the argument value is already equal to a mathematical
475 * integer, then the result is the same as the argument.
476 * <li>If the argument is NaN or an infinity or positive zero or negative
477 * zero, then the result is the same as the argument.</ul>
478 *
479 * @param a a {@code double} value.
480 * @return the closest floating-point value to {@code a} that is
481 * equal to a mathematical integer.
482 */
483 @HotSpotIntrinsicCandidate
484 public static double rint(double a) {
485 return StrictMath.rint(a); // default impl. delegates to StrictMath
486 }
487
488 /**
489 * Returns the angle <i>theta</i> from the conversion of rectangular
490 * coordinates ({@code x}, {@code y}) to polar
491 * coordinates (r, <i>theta</i>).
492 * This method computes the phase <i>theta</i> by computing an arc tangent
493 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
494 * cases:
495 * <ul><li>If either argument is NaN, then the result is NaN.
496 * <li>If the first argument is positive zero and the second argument
497 * is positive, or the first argument is positive and finite and the
498 * second argument is positive infinity, then the result is positive
499 * zero.
500 * <li>If the first argument is negative zero and the second argument
501 * is positive, or the first argument is negative and finite and the
502 * second argument is positive infinity, then the result is negative zero.
503 * <li>If the first argument is positive zero and the second argument
504 * is negative, or the first argument is positive and finite and the
505 * second argument is negative infinity, then the result is the
506 * {@code double} value closest to <i>pi</i>.
507 * <li>If the first argument is negative zero and the second argument
508 * is negative, or the first argument is negative and finite and the
509 * second argument is negative infinity, then the result is the
510 * {@code double} value closest to -<i>pi</i>.
511 * <li>If the first argument is positive and the second argument is
512 * positive zero or negative zero, or the first argument is positive
513 * infinity and the second argument is finite, then the result is the
514 * {@code double} value closest to <i>pi</i>/2.
515 * <li>If the first argument is negative and the second argument is
516 * positive zero or negative zero, or the first argument is negative
517 * infinity and the second argument is finite, then the result is the
518 * {@code double} value closest to -<i>pi</i>/2.
519 * <li>If both arguments are positive infinity, then the result is the
520 * {@code double} value closest to <i>pi</i>/4.
521 * <li>If the first argument is positive infinity and the second argument
522 * is negative infinity, then the result is the {@code double}
523 * value closest to 3*<i>pi</i>/4.
524 * <li>If the first argument is negative infinity and the second argument
525 * is positive infinity, then the result is the {@code double} value
526 * closest to -<i>pi</i>/4.
527 * <li>If both arguments are negative infinity, then the result is the
528 * {@code double} value closest to -3*<i>pi</i>/4.</ul>
529 *
530 * <p>The computed result must be within 2 ulps of the exact result.
531 * Results must be semi-monotonic.
532 *
533 * @param y the ordinate coordinate
534 * @param x the abscissa coordinate
535 * @return the <i>theta</i> component of the point
536 * (<i>r</i>, <i>theta</i>)
537 * in polar coordinates that corresponds to the point
538 * (<i>x</i>, <i>y</i>) in Cartesian coordinates.
539 */
540 @HotSpotIntrinsicCandidate
541 public static double atan2(double y, double x) {
542 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
543 }
544
545 /**
546 * Returns the value of the first argument raised to the power of the
547 * second argument. Special cases:
548 *
549 * <ul><li>If the second argument is positive or negative zero, then the
550 * result is 1.0.
551 * <li>If the second argument is 1.0, then the result is the same as the
552 * first argument.
553 * <li>If the second argument is NaN, then the result is NaN.
554 * <li>If the first argument is NaN and the second argument is nonzero,
555 * then the result is NaN.
556 *
557 * <li>If
558 * <ul>
559 * <li>the absolute value of the first argument is greater than 1
560 * and the second argument is positive infinity, or
561 * <li>the absolute value of the first argument is less than 1 and
562 * the second argument is negative infinity,
563 * </ul>
564 * then the result is positive infinity.
565 *
566 * <li>If
567 * <ul>
568 * <li>the absolute value of the first argument is greater than 1 and
569 * the second argument is negative infinity, or
570 * <li>the absolute value of the
571 * first argument is less than 1 and the second argument is positive
572 * infinity,
573 * </ul>
574 * then the result is positive zero.
575 *
576 * <li>If the absolute value of the first argument equals 1 and the
577 * second argument is infinite, then the result is NaN.
578 *
579 * <li>If
580 * <ul>
581 * <li>the first argument is positive zero and the second argument
582 * is greater than zero, or
583 * <li>the first argument is positive infinity and the second
584 * argument is less than zero,
585 * </ul>
586 * then the result is positive zero.
587 *
588 * <li>If
589 * <ul>
590 * <li>the first argument is positive zero and the second argument
591 * is less than zero, or
592 * <li>the first argument is positive infinity and the second
593 * argument is greater than zero,
594 * </ul>
595 * then the result is positive infinity.
596 *
597 * <li>If
598 * <ul>
599 * <li>the first argument is negative zero and the second argument
600 * is greater than zero but not a finite odd integer, or
601 * <li>the first argument is negative infinity and the second
602 * argument is less than zero but not a finite odd integer,
603 * </ul>
604 * then the result is positive zero.
605 *
606 * <li>If
607 * <ul>
608 * <li>the first argument is negative zero and the second argument
609 * is a positive finite odd integer, or
610 * <li>the first argument is negative infinity and the second
611 * argument is a negative finite odd integer,
612 * </ul>
613 * then the result is negative zero.
614 *
615 * <li>If
616 * <ul>
617 * <li>the first argument is negative zero and the second argument
618 * is less than zero but not a finite odd integer, or
619 * <li>the first argument is negative infinity and the second
620 * argument is greater than zero but not a finite odd integer,
621 * </ul>
622 * then the result is positive infinity.
623 *
624 * <li>If
625 * <ul>
626 * <li>the first argument is negative zero and the second argument
627 * is a negative finite odd integer, or
628 * <li>the first argument is negative infinity and the second
629 * argument is a positive finite odd integer,
630 * </ul>
631 * then the result is negative infinity.
632 *
633 * <li>If the first argument is finite and less than zero
634 * <ul>
635 * <li> if the second argument is a finite even integer, the
636 * result is equal to the result of raising the absolute value of
637 * the first argument to the power of the second argument
638 *
639 * <li>if the second argument is a finite odd integer, the result
640 * is equal to the negative of the result of raising the absolute
641 * value of the first argument to the power of the second
642 * argument
643 *
644 * <li>if the second argument is finite and not an integer, then
645 * the result is NaN.
646 * </ul>
647 *
648 * <li>If both arguments are integers, then the result is exactly equal
649 * to the mathematical result of raising the first argument to the power
650 * of the second argument if that result can in fact be represented
651 * exactly as a {@code double} value.</ul>
652 *
653 * <p>(In the foregoing descriptions, a floating-point value is
654 * considered to be an integer if and only if it is finite and a
655 * fixed point of the method {@link #ceil ceil} or,
656 * equivalently, a fixed point of the method {@link #floor
657 * floor}. A value is a fixed point of a one-argument
658 * method if and only if the result of applying the method to the
659 * value is equal to the value.)
660 *
661 * <p>The computed result must be within 1 ulp of the exact result.
662 * Results must be semi-monotonic.
663 *
664 * @param a the base.
665 * @param b the exponent.
666 * @return the value {@code a}<sup>{@code b}</sup>.
667 */
668 @HotSpotIntrinsicCandidate
669 public static double pow(double a, double b) {
670 return StrictMath.pow(a, b); // default impl. delegates to StrictMath
671 }
672
673 /**
674 * Returns the closest {@code int} to the argument, with ties
675 * rounding to positive infinity.
676 *
677 * <p>
678 * Special cases:
679 * <ul><li>If the argument is NaN, the result is 0.
680 * <li>If the argument is negative infinity or any value less than or
681 * equal to the value of {@code Integer.MIN_VALUE}, the result is
682 * equal to the value of {@code Integer.MIN_VALUE}.
683 * <li>If the argument is positive infinity or any value greater than or
684 * equal to the value of {@code Integer.MAX_VALUE}, the result is
685 * equal to the value of {@code Integer.MAX_VALUE}.</ul>
686 *
687 * @param a a floating-point value to be rounded to an integer.
688 * @return the value of the argument rounded to the nearest
689 * {@code int} value.
690 * @see java.lang.Integer#MAX_VALUE
691 * @see java.lang.Integer#MIN_VALUE
692 */
693 public static int round(float a) {
694 int intBits = Float.floatToRawIntBits(a);
695 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
696 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
697 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
698 + FloatConsts.EXP_BIAS) - biasedExp;
699 if ((shift & -32) == 0) { // shift >= 0 && shift < 32
700 // a is a finite number such that pow(2,-32) <= ulp(a) < 1
701 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
702 | (FloatConsts.SIGNIF_BIT_MASK + 1));
703 if (intBits < 0) {
704 r = -r;
705 }
706 // In the comments below each Java expression evaluates to the value
707 // the corresponding mathematical expression:
708 // (r) evaluates to a / ulp(a)
709 // (r >> shift) evaluates to floor(a * 2)
710 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
711 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
712 return ((r >> shift) + 1) >> 1;
713 } else {
714 // a is either
715 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
716 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
717 // - an infinity or NaN
718 return (int) a;
719 }
720 }
721
722 /**
723 * Returns the closest {@code long} to the argument, with ties
724 * rounding to positive infinity.
725 *
726 * <p>Special cases:
727 * <ul><li>If the argument is NaN, the result is 0.
728 * <li>If the argument is negative infinity or any value less than or
729 * equal to the value of {@code Long.MIN_VALUE}, the result is
730 * equal to the value of {@code Long.MIN_VALUE}.
731 * <li>If the argument is positive infinity or any value greater than or
732 * equal to the value of {@code Long.MAX_VALUE}, the result is
733 * equal to the value of {@code Long.MAX_VALUE}.</ul>
734 *
735 * @param a a floating-point value to be rounded to a
736 * {@code long}.
737 * @return the value of the argument rounded to the nearest
738 * {@code long} value.
739 * @see java.lang.Long#MAX_VALUE
740 * @see java.lang.Long#MIN_VALUE
741 */
742 public static long round(double a) {
743 long longBits = Double.doubleToRawLongBits(a);
744 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
745 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
746 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
747 + DoubleConsts.EXP_BIAS) - biasedExp;
748 if ((shift & -64) == 0) { // shift >= 0 && shift < 64
749 // a is a finite number such that pow(2,-64) <= ulp(a) < 1
750 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
751 | (DoubleConsts.SIGNIF_BIT_MASK + 1));
752 if (longBits < 0) {
753 r = -r;
754 }
755 // In the comments below each Java expression evaluates to the value
756 // the corresponding mathematical expression:
757 // (r) evaluates to a / ulp(a)
758 // (r >> shift) evaluates to floor(a * 2)
759 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
760 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
761 return ((r >> shift) + 1) >> 1;
762 } else {
763 // a is either
764 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
765 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
766 // - an infinity or NaN
767 return (long) a;
768 }
769 }
770
771 private static final class RandomNumberGeneratorHolder {
772 static final Random randomNumberGenerator = new Random();
773 }
774
775 /**
776 * Returns a {@code double} value with a positive sign, greater
777 * than or equal to {@code 0.0} and less than {@code 1.0}.
778 * Returned values are chosen pseudorandomly with (approximately)
779 * uniform distribution from that range.
780 *
781 * <p>When this method is first called, it creates a single new
782 * pseudorandom-number generator, exactly as if by the expression
783 *
784 * <blockquote>{@code new java.util.Random()}</blockquote>
785 *
786 * This new pseudorandom-number generator is used thereafter for
787 * all calls to this method and is used nowhere else.
788 *
789 * <p>This method is properly synchronized to allow correct use by
790 * more than one thread. However, if many threads need to generate
791 * pseudorandom numbers at a great rate, it may reduce contention
792 * for each thread to have its own pseudorandom-number generator.
793 *
794 * @apiNote
795 * As the largest {@code double} value less than {@code 1.0}
796 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
797 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
798 *
799 * <blockquote><pre>{@code
800 * double f = Math.random()/Math.nextDown(1.0);
801 * double x = x1*(1.0 - f) + x2*f;
802 * }</pre></blockquote>
803 *
804 * @return a pseudorandom {@code double} greater than or equal
805 * to {@code 0.0} and less than {@code 1.0}.
806 * @see #nextDown(double)
807 * @see Random#nextDouble()
808 */
809 public static double random() {
810 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
811 }
812
813 /**
814 * Returns the sum of its arguments,
815 * throwing an exception if the result overflows an {@code int}.
816 *
817 * @param x the first value
818 * @param y the second value
819 * @return the result
820 * @throws ArithmeticException if the result overflows an int
821 * @since 1.8
822 */
823 @HotSpotIntrinsicCandidate
824 public static int addExact(int x, int y) {
825 int r = x + y;
826 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
827 if (((x ^ r) & (y ^ r)) < 0) {
828 throw new ArithmeticException("integer overflow");
829 }
830 return r;
831 }
832
833 /**
834 * Returns the sum of its arguments,
835 * throwing an exception if the result overflows a {@code long}.
836 *
837 * @param x the first value
838 * @param y the second value
839 * @return the result
840 * @throws ArithmeticException if the result overflows a long
841 * @since 1.8
842 */
843 @HotSpotIntrinsicCandidate
844 public static long addExact(long x, long y) {
845 long r = x + y;
846 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
847 if (((x ^ r) & (y ^ r)) < 0) {
848 throw new ArithmeticException("long overflow");
849 }
850 return r;
851 }
852
853 /**
854 * Returns the difference of the arguments,
855 * throwing an exception if the result overflows an {@code int}.
856 *
857 * @param x the first value
858 * @param y the second value to subtract from the first
859 * @return the result
860 * @throws ArithmeticException if the result overflows an int
861 * @since 1.8
862 */
863 @HotSpotIntrinsicCandidate
864 public static int subtractExact(int x, int y) {
865 int r = x - y;
866 // HD 2-12 Overflow iff the arguments have different signs and
867 // the sign of the result is different from the sign of x
868 if (((x ^ y) & (x ^ r)) < 0) {
869 throw new ArithmeticException("integer overflow");
870 }
871 return r;
872 }
873
874 /**
875 * Returns the difference of the arguments,
876 * throwing an exception if the result overflows a {@code long}.
877 *
878 * @param x the first value
879 * @param y the second value to subtract from the first
880 * @return the result
881 * @throws ArithmeticException if the result overflows a long
882 * @since 1.8
883 */
884 @HotSpotIntrinsicCandidate
885 public static long subtractExact(long x, long y) {
886 long r = x - y;
887 // HD 2-12 Overflow iff the arguments have different signs and
888 // the sign of the result is different from the sign of x
889 if (((x ^ y) & (x ^ r)) < 0) {
890 throw new ArithmeticException("long overflow");
891 }
892 return r;
893 }
894
895 /**
896 * Returns the product of the arguments,
897 * throwing an exception if the result overflows an {@code int}.
898 *
899 * @param x the first value
900 * @param y the second value
901 * @return the result
902 * @throws ArithmeticException if the result overflows an int
903 * @since 1.8
904 */
905 @HotSpotIntrinsicCandidate
906 public static int multiplyExact(int x, int y) {
907 long r = (long)x * (long)y;
908 if ((int)r != r) {
909 throw new ArithmeticException("integer overflow");
910 }
911 return (int)r;
912 }
913
914 /**
915 * Returns the product of the arguments, throwing an exception if the result
916 * overflows a {@code long}.
917 *
918 * @param x the first value
919 * @param y the second value
920 * @return the result
921 * @throws ArithmeticException if the result overflows a long
922 * @since 9
923 */
924 public static long multiplyExact(long x, int y) {
925 return multiplyExact(x, (long)y);
926 }
927
928 /**
929 * Returns the product of the arguments,
930 * throwing an exception if the result overflows a {@code long}.
931 *
932 * @param x the first value
933 * @param y the second value
934 * @return the result
935 * @throws ArithmeticException if the result overflows a long
936 * @since 1.8
937 */
938 @HotSpotIntrinsicCandidate
939 public static long multiplyExact(long x, long y) {
940 long r = x * y;
941 long ax = Math.abs(x);
942 long ay = Math.abs(y);
943 if (((ax | ay) >>> 31 != 0)) {
944 // Some bits greater than 2^31 that might cause overflow
945 // Check the result using the divide operator
946 // and check for the special case of Long.MIN_VALUE * -1
947 if (((y != 0) && (r / y != x)) ||
948 (x == Long.MIN_VALUE && y == -1)) {
949 throw new ArithmeticException("long overflow");
950 }
951 }
952 return r;
953 }
954
955 /**
956 * Returns the argument incremented by one, throwing an exception if the
957 * result overflows an {@code int}.
958 * The overflow only occurs for {@linkplain Integer#MAX_VALUE the maximum value}.
959 *
960 * @param a the value to increment
961 * @return the result
962 * @throws ArithmeticException if the result overflows an int
963 * @since 1.8
964 */
965 @HotSpotIntrinsicCandidate
966 public static int incrementExact(int a) {
967 if (a == Integer.MAX_VALUE) {
968 throw new ArithmeticException("integer overflow");
969 }
970
971 return a + 1;
972 }
973
974 /**
975 * Returns the argument incremented by one, throwing an exception if the
976 * result overflows a {@code long}.
977 * The overflow only occurs for {@linkplain Long#MAX_VALUE the maximum value}.
978 *
979 * @param a the value to increment
980 * @return the result
981 * @throws ArithmeticException if the result overflows a long
982 * @since 1.8
983 */
984 @HotSpotIntrinsicCandidate
985 public static long incrementExact(long a) {
986 if (a == Long.MAX_VALUE) {
987 throw new ArithmeticException("long overflow");
988 }
989
990 return a + 1L;
991 }
992
993 /**
994 * Returns the argument decremented by one, throwing an exception if the
995 * result overflows an {@code int}.
996 * The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
997 *
998 * @param a the value to decrement
999 * @return the result
1000 * @throws ArithmeticException if the result overflows an int
1001 * @since 1.8
1002 */
1003 @HotSpotIntrinsicCandidate
1004 public static int decrementExact(int a) {
1005 if (a == Integer.MIN_VALUE) {
1006 throw new ArithmeticException("integer overflow");
1007 }
1008
1009 return a - 1;
1010 }
1011
1012 /**
1013 * Returns the argument decremented by one, throwing an exception if the
1014 * result overflows a {@code long}.
1015 * The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
1016 *
1017 * @param a the value to decrement
1018 * @return the result
1019 * @throws ArithmeticException if the result overflows a long
1020 * @since 1.8
1021 */
1022 @HotSpotIntrinsicCandidate
1023 public static long decrementExact(long a) {
1024 if (a == Long.MIN_VALUE) {
1025 throw new ArithmeticException("long overflow");
1026 }
1027
1028 return a - 1L;
1029 }
1030
1031 /**
1032 * Returns the negation of the argument, throwing an exception if the
1033 * result overflows an {@code int}.
1034 * The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
1035 *
1036 * @param a the value to negate
1037 * @return the result
1038 * @throws ArithmeticException if the result overflows an int
1039 * @since 1.8
1040 */
1041 @HotSpotIntrinsicCandidate
1042 public static int negateExact(int a) {
1043 if (a == Integer.MIN_VALUE) {
1044 throw new ArithmeticException("integer overflow");
1045 }
1046
1047 return -a;
1048 }
1049
1050 /**
1051 * Returns the negation of the argument, throwing an exception if the
1052 * result overflows a {@code long}.
1053 * The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
1054 *
1055 * @param a the value to negate
1056 * @return the result
1057 * @throws ArithmeticException if the result overflows a long
1058 * @since 1.8
1059 */
1060 @HotSpotIntrinsicCandidate
1061 public static long negateExact(long a) {
1062 if (a == Long.MIN_VALUE) {
1063 throw new ArithmeticException("long overflow");
1064 }
1065
1066 return -a;
1067 }
1068
1069 /**
1070 * Returns the value of the {@code long} argument,
1071 * throwing an exception if the value overflows an {@code int}.
1072 *
1073 * @param value the long value
1074 * @return the argument as an int
1075 * @throws ArithmeticException if the {@code argument} overflows an int
1076 * @since 1.8
1077 */
1078 public static int toIntExact(long value) {
1079 if ((int)value != value) {
1080 throw new ArithmeticException("integer overflow");
1081 }
1082 return (int)value;
1083 }
1084
1085 /**
1086 * Returns the exact mathematical product of the arguments.
1087 *
1088 * @param x the first value
1089 * @param y the second value
1090 * @return the result
1091 * @since 9
1092 */
1093 public static long multiplyFull(int x, int y) {
1094 return (long)x * (long)y;
1095 }
1096
1097 /**
1098 * Returns as a {@code long} the most significant 64 bits of the 128-bit
1099 * product of two 64-bit factors.
1100 *
1101 * @param x the first value
1102 * @param y the second value
1103 * @return the result
1104 * @since 9
1105 */
1106 @HotSpotIntrinsicCandidate
1107 public static long multiplyHigh(long x, long y) {
1108 if (x < 0 || y < 0) {
1109 // Use technique from section 8-2 of Henry S. Warren, Jr.,
1110 // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174.
1111 long x1 = x >> 32;
1112 long x2 = x & 0xFFFFFFFFL;
1113 long y1 = y >> 32;
1114 long y2 = y & 0xFFFFFFFFL;
1115 long z2 = x2 * y2;
1116 long t = x1 * y2 + (z2 >>> 32);
1117 long z1 = t & 0xFFFFFFFFL;
1118 long z0 = t >> 32;
1119 z1 += x2 * y1;
1120 return x1 * y1 + z0 + (z1 >> 32);
1121 } else {
1122 // Use Karatsuba technique with two base 2^32 digits.
1123 long x1 = x >>> 32;
1124 long y1 = y >>> 32;
1125 long x2 = x & 0xFFFFFFFFL;
1126 long y2 = y & 0xFFFFFFFFL;
1127 long A = x1 * y1;
1128 long B = x2 * y2;
1129 long C = (x1 + x2) * (y1 + y2);
1130 long K = C - A - B;
1131 return (((B >>> 32) + K) >>> 32) + A;
1132 }
1133 }
1134
1135 /**
1136 * Returns the largest (closest to positive infinity)
1137 * {@code int} value that is less than or equal to the algebraic quotient.
1138 * There is one special case, if the dividend is the
1139 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1140 * then integer overflow occurs and
1141 * the result is equal to {@code Integer.MIN_VALUE}.
1142 * <p>
1143 * Normal integer division operates under the round to zero rounding mode
1144 * (truncation). This operation instead acts under the round toward
1145 * negative infinity (floor) rounding mode.
1146 * The floor rounding mode gives different results from truncation
1147 * when the exact result is negative.
1148 * <ul>
1149 * <li>If the signs of the arguments are the same, the results of
1150 * {@code floorDiv} and the {@code /} operator are the same. <br>
1151 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1152 * <li>If the signs of the arguments are different, the quotient is negative and
1153 * {@code floorDiv} returns the integer less than or equal to the quotient
1154 * and the {@code /} operator returns the integer closest to zero.<br>
1155 * For example, {@code floorDiv(-4, 3) == -2},
1156 * whereas {@code (-4 / 3) == -1}.
1157 * </li>
1158 * </ul>
1159 *
1160 * @param x the dividend
1161 * @param y the divisor
1162 * @return the largest (closest to positive infinity)
1163 * {@code int} value that is less than or equal to the algebraic quotient.
1164 * @throws ArithmeticException if the divisor {@code y} is zero
1165 * @see #floorMod(int, int)
1166 * @see #floor(double)
1167 * @since 1.8
1168 */
1169 public static int floorDiv(int x, int y) {
1170 int r = x / y;
1171 // if the signs are different and modulo not zero, round down
1172 if ((x ^ y) < 0 && (r * y != x)) {
1173 r--;
1174 }
1175 return r;
1176 }
1177
1178 /**
1179 * Returns the largest (closest to positive infinity)
1180 * {@code long} value that is less than or equal to the algebraic quotient.
1181 * There is one special case, if the dividend is the
1182 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1183 * then integer overflow occurs and
1184 * the result is equal to {@code Long.MIN_VALUE}.
1185 * <p>
1186 * Normal integer division operates under the round to zero rounding mode
1187 * (truncation). This operation instead acts under the round toward
1188 * negative infinity (floor) rounding mode.
1189 * The floor rounding mode gives different results from truncation
1190 * when the exact result is negative.
1191 * <p>
1192 * For examples, see {@link #floorDiv(int, int)}.
1193 *
1194 * @param x the dividend
1195 * @param y the divisor
1196 * @return the largest (closest to positive infinity)
1197 * {@code int} value that is less than or equal to the algebraic quotient.
1198 * @throws ArithmeticException if the divisor {@code y} is zero
1199 * @see #floorMod(long, int)
1200 * @see #floor(double)
1201 * @since 9
1202 */
1203 public static long floorDiv(long x, int y) {
1204 return floorDiv(x, (long)y);
1205 }
1206
1207 /**
1208 * Returns the largest (closest to positive infinity)
1209 * {@code long} value that is less than or equal to the algebraic quotient.
1210 * There is one special case, if the dividend is the
1211 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1212 * then integer overflow occurs and
1213 * the result is equal to {@code Long.MIN_VALUE}.
1214 * <p>
1215 * Normal integer division operates under the round to zero rounding mode
1216 * (truncation). This operation instead acts under the round toward
1217 * negative infinity (floor) rounding mode.
1218 * The floor rounding mode gives different results from truncation
1219 * when the exact result is negative.
1220 * <p>
1221 * For examples, see {@link #floorDiv(int, int)}.
1222 *
1223 * @param x the dividend
1224 * @param y the divisor
1225 * @return the largest (closest to positive infinity)
1226 * {@code long} value that is less than or equal to the algebraic quotient.
1227 * @throws ArithmeticException if the divisor {@code y} is zero
1228 * @see #floorMod(long, long)
1229 * @see #floor(double)
1230 * @since 1.8
1231 */
1232 public static long floorDiv(long x, long y) {
1233 long r = x / y;
1234 // if the signs are different and modulo not zero, round down
1235 if ((x ^ y) < 0 && (r * y != x)) {
1236 r--;
1237 }
1238 return r;
1239 }
1240
1241 /**
1242 * Returns the floor modulus of the {@code int} arguments.
1243 * <p>
1244 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1245 * has the same sign as the divisor {@code y}, and
1246 * is in the range of {@code -abs(y) < r < +abs(y)}.
1247 *
1248 * <p>
1249 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1250 * <ul>
1251 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1252 * </ul>
1253 * <p>
1254 * The difference in values between {@code floorMod} and
1255 * the {@code %} operator is due to the difference between
1256 * {@code floorDiv} that returns the integer less than or equal to the quotient
1257 * and the {@code /} operator that returns the integer closest to zero.
1258 * <p>
1259 * Examples:
1260 * <ul>
1261 * <li>If the signs of the arguments are the same, the results
1262 * of {@code floorMod} and the {@code %} operator are the same.<br>
1263 * <ul>
1264 * <li>{@code floorMod(+4, +3) == +1}; and {@code (+4 % +3) == +1}</li>
1265 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1}</li>
1266 * </ul>
1267 * <li>If the signs of the arguments are different, the results
1268 * differ from the {@code %} operator.<br>
1269 * <ul>
1270 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1}</li>
1271 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1}</li>
1272 * </ul>
1273 * </li>
1274 * </ul>
1275 * <p>
1276 * If the signs of arguments are unknown and a positive modulus
1277 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1278 *
1279 * @param x the dividend
1280 * @param y the divisor
1281 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1282 * @throws ArithmeticException if the divisor {@code y} is zero
1283 * @see #floorDiv(int, int)
1284 * @since 1.8
1285 */
1286 public static int floorMod(int x, int y) {
1287 int mod = x % y;
1288 // if the signs are different and modulo not zero, adjust result
1289 if ((mod ^ y) < 0 && mod != 0) {
1290 mod += y;
1291 }
1292 return mod;
1293 }
1294
1295 /**
1296 * Returns the floor modulus of the {@code long} and {@code int} arguments.
1297 * <p>
1298 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1299 * has the same sign as the divisor {@code y}, and
1300 * is in the range of {@code -abs(y) < r < +abs(y)}.
1301 *
1302 * <p>
1303 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1304 * <ul>
1305 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1306 * </ul>
1307 * <p>
1308 * For examples, see {@link #floorMod(int, int)}.
1309 *
1310 * @param x the dividend
1311 * @param y the divisor
1312 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1313 * @throws ArithmeticException if the divisor {@code y} is zero
1314 * @see #floorDiv(long, int)
1315 * @since 9
1316 */
1317 public static int floorMod(long x, int y) {
1318 // Result cannot overflow the range of int.
1319 return (int)floorMod(x, (long)y);
1320 }
1321
1322 /**
1323 * Returns the floor modulus of the {@code long} arguments.
1324 * <p>
1325 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1326 * has the same sign as the divisor {@code y}, and
1327 * is in the range of {@code -abs(y) < r < +abs(y)}.
1328 *
1329 * <p>
1330 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1331 * <ul>
1332 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1333 * </ul>
1334 * <p>
1335 * For examples, see {@link #floorMod(int, int)}.
1336 *
1337 * @param x the dividend
1338 * @param y the divisor
1339 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1340 * @throws ArithmeticException if the divisor {@code y} is zero
1341 * @see #floorDiv(long, long)
1342 * @since 1.8
1343 */
1344 public static long floorMod(long x, long y) {
1345 long mod = x % y;
1346 // if the signs are different and modulo not zero, adjust result
1347 if ((x ^ y) < 0 && mod != 0) {
1348 mod += y;
1349 }
1350 return mod;
1351 }
1352
1353 /**
1354 * Returns the absolute value of an {@code int} value.
1355 * If the argument is not negative, the argument is returned.
1356 * If the argument is negative, the negation of the argument is returned.
1357 *
1358 * <p>Note that if the argument is equal to the value of
1359 * {@link Integer#MIN_VALUE}, the most negative representable
1360 * {@code int} value, the result is that same value, which is
1361 * negative.
1362 *
1363 * @param a the argument whose absolute value is to be determined
1364 * @return the absolute value of the argument.
1365 */
1366 @HotSpotIntrinsicCandidate
1367 public static int abs(int a) {
1368 return (a < 0) ? -a : a;
1369 }
1370
1371 /**
1372 * Returns the absolute value of a {@code long} value.
1373 * If the argument is not negative, the argument is returned.
1374 * If the argument is negative, the negation of the argument is returned.
1375 *
1376 * <p>Note that if the argument is equal to the value of
1377 * {@link Long#MIN_VALUE}, the most negative representable
1378 * {@code long} value, the result is that same value, which
1379 * is negative.
1380 *
1381 * @param a the argument whose absolute value is to be determined
1382 * @return the absolute value of the argument.
1383 */
1384 @HotSpotIntrinsicCandidate
1385 public static long abs(long a) {
1386 return (a < 0) ? -a : a;
1387 }
1388
1389 /**
1390 * Returns the absolute value of a {@code float} value.
1391 * If the argument is not negative, the argument is returned.
1392 * If the argument is negative, the negation of the argument is returned.
1393 * Special cases:
1394 * <ul><li>If the argument is positive zero or negative zero, the
1395 * result is positive zero.
1396 * <li>If the argument is infinite, the result is positive infinity.
1397 * <li>If the argument is NaN, the result is NaN.</ul>
1398 *
1399 * @apiNote As implied by the above, one valid implementation of
1400 * this method is given by the expression below which computes a
1401 * {@code float} with the same exponent and significand as the
1402 * argument but with a guaranteed zero sign bit indicating a
1403 * positive value:<br>
1404 * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
1405 *
1406 * @param a the argument whose absolute value is to be determined
1407 * @return the absolute value of the argument.
1408 */
1409 @HotSpotIntrinsicCandidate
1410 public static float abs(float a) {
1411 return (a <= 0.0F) ? 0.0F - a : a;
1412 }
1413
1414 /**
1415 * Returns the absolute value of a {@code double} value.
1416 * If the argument is not negative, the argument is returned.
1417 * If the argument is negative, the negation of the argument is returned.
1418 * Special cases:
1419 * <ul><li>If the argument is positive zero or negative zero, the result
1420 * is positive zero.
1421 * <li>If the argument is infinite, the result is positive infinity.
1422 * <li>If the argument is NaN, the result is NaN.</ul>
1423 *
1424 * @apiNote As implied by the above, one valid implementation of
1425 * this method is given by the expression below which computes a
1426 * {@code double} with the same exponent and significand as the
1427 * argument but with a guaranteed zero sign bit indicating a
1428 * positive value:<br>
1429 * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
1430 *
1431 * @param a the argument whose absolute value is to be determined
1432 * @return the absolute value of the argument.
1433 */
1434 @HotSpotIntrinsicCandidate
1435 public static double abs(double a) {
1436 return (a <= 0.0D) ? 0.0D - a : a;
1437 }
1438
1439 /**
1440 * Returns the greater of two {@code int} values. That is, the
1441 * result is the argument closer to the value of
1442 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1443 * the result is that same value.
1444 *
1445 * @param a an argument.
1446 * @param b another argument.
1447 * @return the larger of {@code a} and {@code b}.
1448 */
1449 @HotSpotIntrinsicCandidate
1450 public static int max(int a, int b) {
1451 return (a >= b) ? a : b;
1452 }
1453
1454 /**
1455 * Returns the greater of two {@code long} values. That is, the
1456 * result is the argument closer to the value of
1457 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1458 * the result is that same value.
1459 *
1460 * @param a an argument.
1461 * @param b another argument.
1462 * @return the larger of {@code a} and {@code b}.
1463 */
1464 public static long max(long a, long b) {
1465 return (a >= b) ? a : b;
1466 }
1467
1468 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1469 private static final long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1470 private static final long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1471
1472 /**
1473 * Returns the greater of two {@code float} values. That is,
1474 * the result is the argument closer to positive infinity. If the
1475 * arguments have the same value, the result is that same
1476 * value. If either value is NaN, then the result is NaN. Unlike
1477 * the numerical comparison operators, this method considers
1478 * negative zero to be strictly smaller than positive zero. If one
1479 * argument is positive zero and the other negative zero, the
1480 * result is positive zero.
1481 *
1482 * @param a an argument.
1483 * @param b another argument.
1484 * @return the larger of {@code a} and {@code b}.
1485 */
1486 @HotSpotIntrinsicCandidate
1487 public static float max(float a, float b) {
1488 if (a != a)
1489 return a; // a is NaN
1490 if ((a == 0.0f) &&
1491 (b == 0.0f) &&
1492 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1493 // Raw conversion ok since NaN can't map to -0.0.
1494 return b;
1495 }
1496 return (a >= b) ? a : b;
1497 }
1498
1499 /**
1500 * Returns the greater of two {@code double} values. That
1501 * is, the result is the argument closer to positive infinity. If
1502 * the arguments have the same value, the result is that same
1503 * value. If either value is NaN, then the result is NaN. Unlike
1504 * the numerical comparison operators, this method considers
1505 * negative zero to be strictly smaller than positive zero. If one
1506 * argument is positive zero and the other negative zero, the
1507 * result is positive zero.
1508 *
1509 * @param a an argument.
1510 * @param b another argument.
1511 * @return the larger of {@code a} and {@code b}.
1512 */
1513 @HotSpotIntrinsicCandidate
1514 public static double max(double a, double b) {
1515 if (a != a)
1516 return a; // a is NaN
1517 if ((a == 0.0d) &&
1518 (b == 0.0d) &&
1519 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1520 // Raw conversion ok since NaN can't map to -0.0.
1521 return b;
1522 }
1523 return (a >= b) ? a : b;
1524 }
1525
1526 /**
1527 * Returns the smaller of two {@code int} values. That is,
1528 * the result the argument closer to the value of
1529 * {@link Integer#MIN_VALUE}. If the arguments have the same
1530 * value, the result is that same value.
1531 *
1532 * @param a an argument.
1533 * @param b another argument.
1534 * @return the smaller of {@code a} and {@code b}.
1535 */
1536 @HotSpotIntrinsicCandidate
1537 public static int min(int a, int b) {
1538 return (a <= b) ? a : b;
1539 }
1540
1541 /**
1542 * Returns the smaller of two {@code long} values. That is,
1543 * the result is the argument closer to the value of
1544 * {@link Long#MIN_VALUE}. If the arguments have the same
1545 * value, the result is that same value.
1546 *
1547 * @param a an argument.
1548 * @param b another argument.
1549 * @return the smaller of {@code a} and {@code b}.
1550 */
1551 public static long min(long a, long b) {
1552 return (a <= b) ? a : b;
1553 }
1554
1555 /**
1556 * Returns the smaller of two {@code float} values. That is,
1557 * the result is the value closer to negative infinity. If the
1558 * arguments have the same value, the result is that same
1559 * value. If either value is NaN, then the result is NaN. Unlike
1560 * the numerical comparison operators, this method considers
1561 * negative zero to be strictly smaller than positive zero. If
1562 * one argument is positive zero and the other is negative zero,
1563 * the result is negative zero.
1564 *
1565 * @param a an argument.
1566 * @param b another argument.
1567 * @return the smaller of {@code a} and {@code b}.
1568 */
1569 @HotSpotIntrinsicCandidate
1570 public static float min(float a, float b) {
1571 if (a != a)
1572 return a; // a is NaN
1573 if ((a == 0.0f) &&
1574 (b == 0.0f) &&
1575 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1576 // Raw conversion ok since NaN can't map to -0.0.
1577 return b;
1578 }
1579 return (a <= b) ? a : b;
1580 }
1581
1582 /**
1583 * Returns the smaller of two {@code double} values. That
1584 * is, the result is the value closer to negative infinity. If the
1585 * arguments have the same value, the result is that same
1586 * value. If either value is NaN, then the result is NaN. Unlike
1587 * the numerical comparison operators, this method considers
1588 * negative zero to be strictly smaller than positive zero. If one
1589 * argument is positive zero and the other is negative zero, the
1590 * result is negative zero.
1591 *
1592 * @param a an argument.
1593 * @param b another argument.
1594 * @return the smaller of {@code a} and {@code b}.
1595 */
1596 @HotSpotIntrinsicCandidate
1597 public static double min(double a, double b) {
1598 if (a != a)
1599 return a; // a is NaN
1600 if ((a == 0.0d) &&
1601 (b == 0.0d) &&
1602 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1603 // Raw conversion ok since NaN can't map to -0.0.
1604 return b;
1605 }
1606 return (a <= b) ? a : b;
1607 }
1608
1609 /**
1610 * Returns the fused multiply add of the three arguments; that is,
1611 * returns the exact product of the first two arguments summed
1612 * with the third argument and then rounded once to the nearest
1613 * {@code double}.
1614 *
1615 * The rounding is done using the {@linkplain
1616 * java.math.RoundingMode#HALF_EVEN round to nearest even
1617 * rounding mode}.
1618 *
1619 * In contrast, if {@code a * b + c} is evaluated as a regular
1620 * floating-point expression, two rounding errors are involved,
1621 * the first for the multiply operation, the second for the
1622 * addition operation.
1623 *
1624 * <p>Special cases:
1625 * <ul>
1626 * <li> If any argument is NaN, the result is NaN.
1627 *
1628 * <li> If one of the first two arguments is infinite and the
1629 * other is zero, the result is NaN.
1630 *
1631 * <li> If the exact product of the first two arguments is infinite
1632 * (in other words, at least one of the arguments is infinite and
1633 * the other is neither zero nor NaN) and the third argument is an
1634 * infinity of the opposite sign, the result is NaN.
1635 *
1636 * </ul>
1637 *
1638 * <p>Note that {@code fma(a, 1.0, c)} returns the same
1639 * result as ({@code a + c}). However,
1640 * {@code fma(a, b, +0.0)} does <em>not</em> always return the
1641 * same result as ({@code a * b}) since
1642 * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while
1643 * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is
1644 * equivalent to ({@code a * b}) however.
1645 *
1646 * @apiNote This method corresponds to the fusedMultiplyAdd
1647 * operation defined in IEEE 754-2008.
1648 *
1649 * @param a a value
1650 * @param b a value
1651 * @param c a value
1652 *
1653 * @return (<i>a</i> × <i>b</i> + <i>c</i>)
1654 * computed, as if with unlimited range and precision, and rounded
1655 * once to the nearest {@code double} value
1656 *
1657 * @since 9
1658 */
1659 @HotSpotIntrinsicCandidate
1660 public static double fma(double a, double b, double c) {
1661 /*
1662 * Infinity and NaN arithmetic is not quite the same with two
1663 * roundings as opposed to just one so the simple expression
1664 * "a * b + c" cannot always be used to compute the correct
1665 * result. With two roundings, the product can overflow and
1666 * if the addend is infinite, a spurious NaN can be produced
1667 * if the infinity from the overflow and the infinite addend
1668 * have opposite signs.
1669 */
1670
1671 // First, screen for and handle non-finite input values whose
1672 // arithmetic is not supported by BigDecimal.
1673 if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
1674 return Double.NaN;
1675 } else { // All inputs non-NaN
1676 boolean infiniteA = Double.isInfinite(a);
1677 boolean infiniteB = Double.isInfinite(b);
1678 boolean infiniteC = Double.isInfinite(c);
1679 double result;
1680
1681 if (infiniteA || infiniteB || infiniteC) {
1682 if (infiniteA && b == 0.0 ||
1683 infiniteB && a == 0.0 ) {
1684 return Double.NaN;
1685 }
1686 // Store product in a double field to cause an
1687 // overflow even if non-strictfp evaluation is being
1688 // used.
1689 double product = a * b;
1690 if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
1691 // Intermediate overflow; might cause a
1692 // spurious NaN if added to infinite c.
1693 assert Double.isInfinite(c);
1694 return c;
1695 } else {
1696 result = product + c;
1697 assert !Double.isFinite(result);
1698 return result;
1699 }
1700 } else { // All inputs finite
1701 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
1702 if (c == 0.0) { // Positive or negative zero
1703 // If the product is an exact zero, use a
1704 // floating-point expression to compute the sign
1705 // of the zero final result. The product is an
1706 // exact zero if and only if at least one of a and
1707 // b is zero.
1708 if (a == 0.0 || b == 0.0) {
1709 return a * b + c;
1710 } else {
1711 // The sign of a zero addend doesn't matter if
1712 // the product is nonzero. The sign of a zero
1713 // addend is not factored in the result if the
1714 // exact product is nonzero but underflows to
1715 // zero; see IEEE-754 2008 section 6.3 "The
1716 // sign bit".
1717 return product.doubleValue();
1718 }
1719 } else {
1720 return product.add(new BigDecimal(c)).doubleValue();
1721 }
1722 }
1723 }
1724 }
1725
1726 /**
1727 * Returns the fused multiply add of the three arguments; that is,
1728 * returns the exact product of the first two arguments summed
1729 * with the third argument and then rounded once to the nearest
1730 * {@code float}.
1731 *
1732 * The rounding is done using the {@linkplain
1733 * java.math.RoundingMode#HALF_EVEN round to nearest even
1734 * rounding mode}.
1735 *
1736 * In contrast, if {@code a * b + c} is evaluated as a regular
1737 * floating-point expression, two rounding errors are involved,
1738 * the first for the multiply operation, the second for the
1739 * addition operation.
1740 *
1741 * <p>Special cases:
1742 * <ul>
1743 * <li> If any argument is NaN, the result is NaN.
1744 *
1745 * <li> If one of the first two arguments is infinite and the
1746 * other is zero, the result is NaN.
1747 *
1748 * <li> If the exact product of the first two arguments is infinite
1749 * (in other words, at least one of the arguments is infinite and
1750 * the other is neither zero nor NaN) and the third argument is an
1751 * infinity of the opposite sign, the result is NaN.
1752 *
1753 * </ul>
1754 *
1755 * <p>Note that {@code fma(a, 1.0f, c)} returns the same
1756 * result as ({@code a + c}). However,
1757 * {@code fma(a, b, +0.0f)} does <em>not</em> always return the
1758 * same result as ({@code a * b}) since
1759 * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
1760 * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
1761 * equivalent to ({@code a * b}) however.
1762 *
1763 * @apiNote This method corresponds to the fusedMultiplyAdd
1764 * operation defined in IEEE 754-2008.
1765 *
1766 * @param a a value
1767 * @param b a value
1768 * @param c a value
1769 *
1770 * @return (<i>a</i> × <i>b</i> + <i>c</i>)
1771 * computed, as if with unlimited range and precision, and rounded
1772 * once to the nearest {@code float} value
1773 *
1774 * @since 9
1775 */
1776 @HotSpotIntrinsicCandidate
1777 public static float fma(float a, float b, float c) {
1778 /*
1779 * Since the double format has more than twice the precision
1780 * of the float format, the multiply of a * b is exact in
1781 * double. The add of c to the product then incurs one
1782 * rounding error. Since the double format moreover has more
1783 * than (2p + 2) precision bits compared to the p bits of the
1784 * float format, the two roundings of (a * b + c), first to
1785 * the double format and then secondarily to the float format,
1786 * are equivalent to rounding the intermediate result directly
1787 * to the float format.
1788 *
1789 * In terms of strictfp vs default-fp concerns related to
1790 * overflow and underflow, since
1791 *
1792 * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE
1793 * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE
1794 *
1795 * neither the multiply nor add will overflow or underflow in
1796 * double. Therefore, it is not necessary for this method to
1797 * be declared strictfp to have reproducible
1798 * behavior. However, it is necessary to explicitly store down
1799 * to a float variable to avoid returning a value in the float
1800 * extended value set.
1801 */
1802 float result = (float)(((double) a * (double) b ) + (double) c);
1803 return result;
1804 }
1805
1806 /**
1807 * Returns the size of an ulp of the argument. An ulp, unit in
1808 * the last place, of a {@code double} value is the positive
1809 * distance between this floating-point value and the {@code
1810 * double} value next larger in magnitude. Note that for non-NaN
1811 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1812 *
1813 * <p>Special Cases:
1814 * <ul>
1815 * <li> If the argument is NaN, then the result is NaN.
1816 * <li> If the argument is positive or negative infinity, then the
1817 * result is positive infinity.
1818 * <li> If the argument is positive or negative zero, then the result is
1819 * {@code Double.MIN_VALUE}.
1820 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1821 * the result is equal to 2<sup>971</sup>.
1822 * </ul>
1823 *
1824 * @param d the floating-point value whose ulp is to be returned
1825 * @return the size of an ulp of the argument
1826 * @author Joseph D. Darcy
1827 * @since 1.5
1828 */
1829 public static double ulp(double d) {
1830 int exp = getExponent(d);
1831
1832 switch(exp) {
1833 case Double.MAX_EXPONENT + 1: // NaN or infinity
1834 return Math.abs(d);
1835
1836 case Double.MIN_EXPONENT - 1: // zero or subnormal
1837 return Double.MIN_VALUE;
1838
1839 default:
1840 assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
1841
1842 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1843 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1844 if (exp >= Double.MIN_EXPONENT) {
1845 return powerOfTwoD(exp);
1846 }
1847 else {
1848 // return a subnormal result; left shift integer
1849 // representation of Double.MIN_VALUE appropriate
1850 // number of positions
1851 return Double.longBitsToDouble(1L <<
1852 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1853 }
1854 }
1855 }
1856
1857 /**
1858 * Returns the size of an ulp of the argument. An ulp, unit in
1859 * the last place, of a {@code float} value is the positive
1860 * distance between this floating-point value and the {@code
1861 * float} value next larger in magnitude. Note that for non-NaN
1862 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1863 *
1864 * <p>Special Cases:
1865 * <ul>
1866 * <li> If the argument is NaN, then the result is NaN.
1867 * <li> If the argument is positive or negative infinity, then the
1868 * result is positive infinity.
1869 * <li> If the argument is positive or negative zero, then the result is
1870 * {@code Float.MIN_VALUE}.
1871 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1872 * the result is equal to 2<sup>104</sup>.
1873 * </ul>
1874 *
1875 * @param f the floating-point value whose ulp is to be returned
1876 * @return the size of an ulp of the argument
1877 * @author Joseph D. Darcy
1878 * @since 1.5
1879 */
1880 public static float ulp(float f) {
1881 int exp = getExponent(f);
1882
1883 switch(exp) {
1884 case Float.MAX_EXPONENT+1: // NaN or infinity
1885 return Math.abs(f);
1886
1887 case Float.MIN_EXPONENT-1: // zero or subnormal
1888 return Float.MIN_VALUE;
1889
1890 default:
1891 assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
1892
1893 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1894 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1895 if (exp >= Float.MIN_EXPONENT) {
1896 return powerOfTwoF(exp);
1897 } else {
1898 // return a subnormal result; left shift integer
1899 // representation of FloatConsts.MIN_VALUE appropriate
1900 // number of positions
1901 return Float.intBitsToFloat(1 <<
1902 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1903 }
1904 }
1905 }
1906
1907 /**
1908 * Returns the signum function of the argument; zero if the argument
1909 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1910 * argument is less than zero.
1911 *
1912 * <p>Special Cases:
1913 * <ul>
1914 * <li> If the argument is NaN, then the result is NaN.
1915 * <li> If the argument is positive zero or negative zero, then the
1916 * result is the same as the argument.
1917 * </ul>
1918 *
1919 * @param d the floating-point value whose signum is to be returned
1920 * @return the signum function of the argument
1921 * @author Joseph D. Darcy
1922 * @since 1.5
1923 */
1924 public static double signum(double d) {
1925 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1926 }
1927
1928 /**
1929 * Returns the signum function of the argument; zero if the argument
1930 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1931 * argument is less than zero.
1932 *
1933 * <p>Special Cases:
1934 * <ul>
1935 * <li> If the argument is NaN, then the result is NaN.
1936 * <li> If the argument is positive zero or negative zero, then the
1937 * result is the same as the argument.
1938 * </ul>
1939 *
1940 * @param f the floating-point value whose signum is to be returned
1941 * @return the signum function of the argument
1942 * @author Joseph D. Darcy
1943 * @since 1.5
1944 */
1945 public static float signum(float f) {
1946 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1947 }
1948
1949 /**
1950 * Returns the hyperbolic sine of a {@code double} value.
1951 * The hyperbolic sine of <i>x</i> is defined to be
1952 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1953 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1954 *
1955 * <p>Special cases:
1956 * <ul>
1957 *
1958 * <li>If the argument is NaN, then the result is NaN.
1959 *
1960 * <li>If the argument is infinite, then the result is an infinity
1961 * with the same sign as the argument.
1962 *
1963 * <li>If the argument is zero, then the result is a zero with the
1964 * same sign as the argument.
1965 *
1966 * </ul>
1967 *
1968 * <p>The computed result must be within 2.5 ulps of the exact result.
1969 *
1970 * @param x The number whose hyperbolic sine is to be returned.
1971 * @return The hyperbolic sine of {@code x}.
1972 * @since 1.5
1973 */
1974 public static double sinh(double x) {
1975 return StrictMath.sinh(x);
1976 }
1977
1978 /**
1979 * Returns the hyperbolic cosine of a {@code double} value.
1980 * The hyperbolic cosine of <i>x</i> is defined to be
1981 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1982 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1983 *
1984 * <p>Special cases:
1985 * <ul>
1986 *
1987 * <li>If the argument is NaN, then the result is NaN.
1988 *
1989 * <li>If the argument is infinite, then the result is positive
1990 * infinity.
1991 *
1992 * <li>If the argument is zero, then the result is {@code 1.0}.
1993 *
1994 * </ul>
1995 *
1996 * <p>The computed result must be within 2.5 ulps of the exact result.
1997 *
1998 * @param x The number whose hyperbolic cosine is to be returned.
1999 * @return The hyperbolic cosine of {@code x}.
2000 * @since 1.5
2001 */
2002 public static double cosh(double x) {
2003 return StrictMath.cosh(x);
2004 }
2005
2006 /**
2007 * Returns the hyperbolic tangent of a {@code double} value.
2008 * The hyperbolic tangent of <i>x</i> is defined to be
2009 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
2010 * in other words, {@linkplain Math#sinh
2011 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
2012 * that the absolute value of the exact tanh is always less than
2013 * 1.
2014 *
2015 * <p>Special cases:
2016 * <ul>
2017 *
2018 * <li>If the argument is NaN, then the result is NaN.
2019 *
2020 * <li>If the argument is zero, then the result is a zero with the
2021 * same sign as the argument.
2022 *
2023 * <li>If the argument is positive infinity, then the result is
2024 * {@code +1.0}.
2025 *
2026 * <li>If the argument is negative infinity, then the result is
2027 * {@code -1.0}.
2028 *
2029 * </ul>
2030 *
2031 * <p>The computed result must be within 2.5 ulps of the exact result.
2032 * The result of {@code tanh} for any finite input must have
2033 * an absolute value less than or equal to 1. Note that once the
2034 * exact result of tanh is within 1/2 of an ulp of the limit value
2035 * of ±1, correctly signed ±{@code 1.0} should
2036 * be returned.
2037 *
2038 * @param x The number whose hyperbolic tangent is to be returned.
2039 * @return The hyperbolic tangent of {@code x}.
2040 * @since 1.5
2041 */
2042 public static double tanh(double x) {
2043 return StrictMath.tanh(x);
2044 }
2045
2046 /**
2047 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
2048 * without intermediate overflow or underflow.
2049 *
2050 * <p>Special cases:
2051 * <ul>
2052 *
2053 * <li> If either argument is infinite, then the result
2054 * is positive infinity.
2055 *
2056 * <li> If either argument is NaN and neither argument is infinite,
2057 * then the result is NaN.
2058 *
2059 * </ul>
2060 *
2061 * <p>The computed result must be within 1 ulp of the exact
2062 * result. If one parameter is held constant, the results must be
2063 * semi-monotonic in the other parameter.
2064 *
2065 * @param x a value
2066 * @param y a value
2067 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
2068 * without intermediate overflow or underflow
2069 * @since 1.5
2070 */
2071 public static double hypot(double x, double y) {
2072 return StrictMath.hypot(x, y);
2073 }
2074
2075 /**
2076 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
2077 * <i>x</i> near 0, the exact sum of
2078 * {@code expm1(x)} + 1 is much closer to the true
2079 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
2080 *
2081 * <p>Special cases:
2082 * <ul>
2083 * <li>If the argument is NaN, the result is NaN.
2084 *
2085 * <li>If the argument is positive infinity, then the result is
2086 * positive infinity.
2087 *
2088 * <li>If the argument is negative infinity, then the result is
2089 * -1.0.
2090 *
2091 * <li>If the argument is zero, then the result is a zero with the
2092 * same sign as the argument.
2093 *
2094 * </ul>
2095 *
2096 * <p>The computed result must be within 1 ulp of the exact result.
2097 * Results must be semi-monotonic. The result of
2098 * {@code expm1} for any finite input must be greater than or
2099 * equal to {@code -1.0}. Note that once the exact result of
2100 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
2101 * ulp of the limit value -1, {@code -1.0} should be
2102 * returned.
2103 *
2104 * @param x the exponent to raise <i>e</i> to in the computation of
2105 * <i>e</i><sup>{@code x}</sup> -1.
2106 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
2107 * @since 1.5
2108 */
2109 public static double expm1(double x) {
2110 return StrictMath.expm1(x);
2111 }
2112
2113 /**
2114 * Returns the natural logarithm of the sum of the argument and 1.
2115 * Note that for small values {@code x}, the result of
2116 * {@code log1p(x)} is much closer to the true result of ln(1
2117 * + {@code x}) than the floating-point evaluation of
2118 * {@code log(1.0+x)}.
2119 *
2120 * <p>Special cases:
2121 *
2122 * <ul>
2123 *
2124 * <li>If the argument is NaN or less than -1, then the result is
2125 * NaN.
2126 *
2127 * <li>If the argument is positive infinity, then the result is
2128 * positive infinity.
2129 *
2130 * <li>If the argument is negative one, then the result is
2131 * negative infinity.
2132 *
2133 * <li>If the argument is zero, then the result is a zero with the
2134 * same sign as the argument.
2135 *
2136 * </ul>
2137 *
2138 * <p>The computed result must be within 1 ulp of the exact result.
2139 * Results must be semi-monotonic.
2140 *
2141 * @param x a value
2142 * @return the value ln({@code x} + 1), the natural
2143 * log of {@code x} + 1
2144 * @since 1.5
2145 */
2146 public static double log1p(double x) {
2147 return StrictMath.log1p(x);
2148 }
2149
2150 /**
2151 * Returns the first floating-point argument with the sign of the
2152 * second floating-point argument. Note that unlike the {@link
2153 * StrictMath#copySign(double, double) StrictMath.copySign}
2154 * method, this method does not require NaN {@code sign}
2155 * arguments to be treated as positive values; implementations are
2156 * permitted to treat some NaN arguments as positive and other NaN
2157 * arguments as negative to allow greater performance.
2158 *
2159 * @param magnitude the parameter providing the magnitude of the result
2160 * @param sign the parameter providing the sign of the result
2161 * @return a value with the magnitude of {@code magnitude}
2162 * and the sign of {@code sign}.
2163 * @since 1.6
2164 */
2165 public static double copySign(double magnitude, double sign) {
2166 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
2167 (DoubleConsts.SIGN_BIT_MASK)) |
2168 (Double.doubleToRawLongBits(magnitude) &
2169 (DoubleConsts.EXP_BIT_MASK |
2170 DoubleConsts.SIGNIF_BIT_MASK)));
2171 }
2172
2173 /**
2174 * Returns the first floating-point argument with the sign of the
2175 * second floating-point argument. Note that unlike the {@link
2176 * StrictMath#copySign(float, float) StrictMath.copySign}
2177 * method, this method does not require NaN {@code sign}
2178 * arguments to be treated as positive values; implementations are
2179 * permitted to treat some NaN arguments as positive and other NaN
2180 * arguments as negative to allow greater performance.
2181 *
2182 * @param magnitude the parameter providing the magnitude of the result
2183 * @param sign the parameter providing the sign of the result
2184 * @return a value with the magnitude of {@code magnitude}
2185 * and the sign of {@code sign}.
2186 * @since 1.6
2187 */
2188 public static float copySign(float magnitude, float sign) {
2189 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
2190 (FloatConsts.SIGN_BIT_MASK)) |
2191 (Float.floatToRawIntBits(magnitude) &
2192 (FloatConsts.EXP_BIT_MASK |
2193 FloatConsts.SIGNIF_BIT_MASK)));
2194 }
2195
2196 /**
2197 * Returns the unbiased exponent used in the representation of a
2198 * {@code float}. Special cases:
2199 *
2200 * <ul>
2201 * <li>If the argument is NaN or infinite, then the result is
2202 * {@link Float#MAX_EXPONENT} + 1.
2203 * <li>If the argument is zero or subnormal, then the result is
2204 * {@link Float#MIN_EXPONENT} -1.
2205 * </ul>
2206 * @param f a {@code float} value
2207 * @return the unbiased exponent of the argument
2208 * @since 1.6
2209 */
2210 public static int getExponent(float f) {
2211 /*
2212 * Bitwise convert f to integer, mask out exponent bits, shift
2213 * to the right and then subtract out float's bias adjust to
2214 * get true exponent value
2215 */
2216 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
2217 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
2218 }
2219
2220 /**
2221 * Returns the unbiased exponent used in the representation of a
2222 * {@code double}. Special cases:
2223 *
2224 * <ul>
2225 * <li>If the argument is NaN or infinite, then the result is
2226 * {@link Double#MAX_EXPONENT} + 1.
2227 * <li>If the argument is zero or subnormal, then the result is
2228 * {@link Double#MIN_EXPONENT} -1.
2229 * </ul>
2230 * @param d a {@code double} value
2231 * @return the unbiased exponent of the argument
2232 * @since 1.6
2233 */
2234 public static int getExponent(double d) {
2235 /*
2236 * Bitwise convert d to long, mask out exponent bits, shift
2237 * to the right and then subtract out double's bias adjust to
2238 * get true exponent value.
2239 */
2240 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
2241 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
2242 }
2243
2244 /**
2245 * Returns the floating-point number adjacent to the first
2246 * argument in the direction of the second argument. If both
2247 * arguments compare as equal the second argument is returned.
2248 *
2249 * <p>
2250 * Special cases:
2251 * <ul>
2252 * <li> If either argument is a NaN, then NaN is returned.
2253 *
2254 * <li> If both arguments are signed zeros, {@code direction}
2255 * is returned unchanged (as implied by the requirement of
2256 * returning the second argument if the arguments compare as
2257 * equal).
2258 *
2259 * <li> If {@code start} is
2260 * ±{@link Double#MIN_VALUE} and {@code direction}
2261 * has a value such that the result should have a smaller
2262 * magnitude, then a zero with the same sign as {@code start}
2263 * is returned.
2264 *
2265 * <li> If {@code start} is infinite and
2266 * {@code direction} has a value such that the result should
2267 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
2268 * same sign as {@code start} is returned.
2269 *
2270 * <li> If {@code start} is equal to ±
2271 * {@link Double#MAX_VALUE} and {@code direction} has a
2272 * value such that the result should have a larger magnitude, an
2273 * infinity with same sign as {@code start} is returned.
2274 * </ul>
2275 *
2276 * @param start starting floating-point value
2277 * @param direction value indicating which of
2278 * {@code start}'s neighbors or {@code start} should
2279 * be returned
2280 * @return The floating-point number adjacent to {@code start} in the
2281 * direction of {@code direction}.
2282 * @since 1.6
2283 */
2284 public static double nextAfter(double start, double direction) {
2285 /*
2286 * The cases:
2287 *
2288 * nextAfter(+infinity, 0) == MAX_VALUE
2289 * nextAfter(+infinity, +infinity) == +infinity
2290 * nextAfter(-infinity, 0) == -MAX_VALUE
2291 * nextAfter(-infinity, -infinity) == -infinity
2292 *
2293 * are naturally handled without any additional testing
2294 */
2295
2296 /*
2297 * IEEE 754 floating-point numbers are lexicographically
2298 * ordered if treated as signed-magnitude integers.
2299 * Since Java's integers are two's complement,
2300 * incrementing the two's complement representation of a
2301 * logically negative floating-point value *decrements*
2302 * the signed-magnitude representation. Therefore, when
2303 * the integer representation of a floating-point value
2304 * is negative, the adjustment to the representation is in
2305 * the opposite direction from what would initially be expected.
2306 */
2307
2308 // Branch to descending case first as it is more costly than ascending
2309 // case due to start != 0.0d conditional.
2310 if (start > direction) { // descending
2311 if (start != 0.0d) {
2312 final long transducer = Double.doubleToRawLongBits(start);
2313 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
2314 } else { // start == 0.0d && direction < 0.0d
2315 return -Double.MIN_VALUE;
2316 }
2317 } else if (start < direction) { // ascending
2318 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2319 // then bitwise convert start to integer.
2320 final long transducer = Double.doubleToRawLongBits(start + 0.0d);
2321 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2322 } else if (start == direction) {
2323 return direction;
2324 } else { // isNaN(start) || isNaN(direction)
2325 return start + direction;
2326 }
2327 }
2328
2329 /**
2330 * Returns the floating-point number adjacent to the first
2331 * argument in the direction of the second argument. If both
2332 * arguments compare as equal a value equivalent to the second argument
2333 * is returned.
2334 *
2335 * <p>
2336 * Special cases:
2337 * <ul>
2338 * <li> If either argument is a NaN, then NaN is returned.
2339 *
2340 * <li> If both arguments are signed zeros, a value equivalent
2341 * to {@code direction} is returned.
2342 *
2343 * <li> If {@code start} is
2344 * ±{@link Float#MIN_VALUE} and {@code direction}
2345 * has a value such that the result should have a smaller
2346 * magnitude, then a zero with the same sign as {@code start}
2347 * is returned.
2348 *
2349 * <li> If {@code start} is infinite and
2350 * {@code direction} has a value such that the result should
2351 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
2352 * same sign as {@code start} is returned.
2353 *
2354 * <li> If {@code start} is equal to ±
2355 * {@link Float#MAX_VALUE} and {@code direction} has a
2356 * value such that the result should have a larger magnitude, an
2357 * infinity with same sign as {@code start} is returned.
2358 * </ul>
2359 *
2360 * @param start starting floating-point value
2361 * @param direction value indicating which of
2362 * {@code start}'s neighbors or {@code start} should
2363 * be returned
2364 * @return The floating-point number adjacent to {@code start} in the
2365 * direction of {@code direction}.
2366 * @since 1.6
2367 */
2368 public static float nextAfter(float start, double direction) {
2369 /*
2370 * The cases:
2371 *
2372 * nextAfter(+infinity, 0) == MAX_VALUE
2373 * nextAfter(+infinity, +infinity) == +infinity
2374 * nextAfter(-infinity, 0) == -MAX_VALUE
2375 * nextAfter(-infinity, -infinity) == -infinity
2376 *
2377 * are naturally handled without any additional testing
2378 */
2379
2380 /*
2381 * IEEE 754 floating-point numbers are lexicographically
2382 * ordered if treated as signed-magnitude integers.
2383 * Since Java's integers are two's complement,
2384 * incrementing the two's complement representation of a
2385 * logically negative floating-point value *decrements*
2386 * the signed-magnitude representation. Therefore, when
2387 * the integer representation of a floating-point value
2388 * is negative, the adjustment to the representation is in
2389 * the opposite direction from what would initially be expected.
2390 */
2391
2392 // Branch to descending case first as it is more costly than ascending
2393 // case due to start != 0.0f conditional.
2394 if (start > direction) { // descending
2395 if (start != 0.0f) {
2396 final int transducer = Float.floatToRawIntBits(start);
2397 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2398 } else { // start == 0.0f && direction < 0.0f
2399 return -Float.MIN_VALUE;
2400 }
2401 } else if (start < direction) { // ascending
2402 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2403 // then bitwise convert start to integer.
2404 final int transducer = Float.floatToRawIntBits(start + 0.0f);
2405 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2406 } else if (start == direction) {
2407 return (float)direction;
2408 } else { // isNaN(start) || isNaN(direction)
2409 return start + (float)direction;
2410 }
2411 }
2412
2413 /**
2414 * Returns the floating-point value adjacent to {@code d} in
2415 * the direction of positive infinity. This method is
2416 * semantically equivalent to {@code nextAfter(d,
2417 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2418 * implementation may run faster than its equivalent
2419 * {@code nextAfter} call.
2420 *
2421 * <p>Special Cases:
2422 * <ul>
2423 * <li> If the argument is NaN, the result is NaN.
2424 *
2425 * <li> If the argument is positive infinity, the result is
2426 * positive infinity.
2427 *
2428 * <li> If the argument is zero, the result is
2429 * {@link Double#MIN_VALUE}
2430 *
2431 * </ul>
2432 *
2433 * @param d starting floating-point value
2434 * @return The adjacent floating-point value closer to positive
2435 * infinity.
2436 * @since 1.6
2437 */
2438 public static double nextUp(double d) {
2439 // Use a single conditional and handle the likely cases first.
2440 if (d < Double.POSITIVE_INFINITY) {
2441 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2442 final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2443 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2444 } else { // d is NaN or +Infinity
2445 return d;
2446 }
2447 }
2448
2449 /**
2450 * Returns the floating-point value adjacent to {@code f} in
2451 * the direction of positive infinity. This method is
2452 * semantically equivalent to {@code nextAfter(f,
2453 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2454 * implementation may run faster than its equivalent
2455 * {@code nextAfter} call.
2456 *
2457 * <p>Special Cases:
2458 * <ul>
2459 * <li> If the argument is NaN, the result is NaN.
2460 *
2461 * <li> If the argument is positive infinity, the result is
2462 * positive infinity.
2463 *
2464 * <li> If the argument is zero, the result is
2465 * {@link Float#MIN_VALUE}
2466 *
2467 * </ul>
2468 *
2469 * @param f starting floating-point value
2470 * @return The adjacent floating-point value closer to positive
2471 * infinity.
2472 * @since 1.6
2473 */
2474 public static float nextUp(float f) {
2475 // Use a single conditional and handle the likely cases first.
2476 if (f < Float.POSITIVE_INFINITY) {
2477 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2478 final int transducer = Float.floatToRawIntBits(f + 0.0F);
2479 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2480 } else { // f is NaN or +Infinity
2481 return f;
2482 }
2483 }
2484
2485 /**
2486 * Returns the floating-point value adjacent to {@code d} in
2487 * the direction of negative infinity. This method is
2488 * semantically equivalent to {@code nextAfter(d,
2489 * Double.NEGATIVE_INFINITY)}; however, a
2490 * {@code nextDown} implementation may run faster than its
2491 * equivalent {@code nextAfter} call.
2492 *
2493 * <p>Special Cases:
2494 * <ul>
2495 * <li> If the argument is NaN, the result is NaN.
2496 *
2497 * <li> If the argument is negative infinity, the result is
2498 * negative infinity.
2499 *
2500 * <li> If the argument is zero, the result is
2501 * {@code -Double.MIN_VALUE}
2502 *
2503 * </ul>
2504 *
2505 * @param d starting floating-point value
2506 * @return The adjacent floating-point value closer to negative
2507 * infinity.
2508 * @since 1.8
2509 */
2510 public static double nextDown(double d) {
2511 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2512 return d;
2513 else {
2514 if (d == 0.0)
2515 return -Double.MIN_VALUE;
2516 else
2517 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2518 ((d > 0.0d)?-1L:+1L));
2519 }
2520 }
2521
2522 /**
2523 * Returns the floating-point value adjacent to {@code f} in
2524 * the direction of negative infinity. This method is
2525 * semantically equivalent to {@code nextAfter(f,
2526 * Float.NEGATIVE_INFINITY)}; however, a
2527 * {@code nextDown} implementation may run faster than its
2528 * equivalent {@code nextAfter} call.
2529 *
2530 * <p>Special Cases:
2531 * <ul>
2532 * <li> If the argument is NaN, the result is NaN.
2533 *
2534 * <li> If the argument is negative infinity, the result is
2535 * negative infinity.
2536 *
2537 * <li> If the argument is zero, the result is
2538 * {@code -Float.MIN_VALUE}
2539 *
2540 * </ul>
2541 *
2542 * @param f starting floating-point value
2543 * @return The adjacent floating-point value closer to negative
2544 * infinity.
2545 * @since 1.8
2546 */
2547 public static float nextDown(float f) {
2548 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2549 return f;
2550 else {
2551 if (f == 0.0f)
2552 return -Float.MIN_VALUE;
2553 else
2554 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2555 ((f > 0.0f)?-1:+1));
2556 }
2557 }
2558
2559 /**
2560 * Returns {@code d} ×
2561 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2562 * by a single correctly rounded floating-point multiply to a
2563 * member of the double value set. See the Java
2564 * Language Specification for a discussion of floating-point
2565 * value sets. If the exponent of the result is between {@link
2566 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2567 * answer is calculated exactly. If the exponent of the result
2568 * would be larger than {@code Double.MAX_EXPONENT}, an
2569 * infinity is returned. Note that if the result is subnormal,
2570 * precision may be lost; that is, when {@code scalb(x, n)}
2571 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2572 * <i>x</i>. When the result is non-NaN, the result has the same
2573 * sign as {@code d}.
2574 *
2575 * <p>Special cases:
2576 * <ul>
2577 * <li> If the first argument is NaN, NaN is returned.
2578 * <li> If the first argument is infinite, then an infinity of the
2579 * same sign is returned.
2580 * <li> If the first argument is zero, then a zero of the same
2581 * sign is returned.
2582 * </ul>
2583 *
2584 * @param d number to be scaled by a power of two.
2585 * @param scaleFactor power of 2 used to scale {@code d}
2586 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2587 * @since 1.6
2588 */
2589 public static double scalb(double d, int scaleFactor) {
2590 /*
2591 * This method does not need to be declared strictfp to
2592 * compute the same correct result on all platforms. When
2593 * scaling up, it does not matter what order the
2594 * multiply-store operations are done; the result will be
2595 * finite or overflow regardless of the operation ordering.
2596 * However, to get the correct result when scaling down, a
2597 * particular ordering must be used.
2598 *
2599 * When scaling down, the multiply-store operations are
2600 * sequenced so that it is not possible for two consecutive
2601 * multiply-stores to return subnormal results. If one
2602 * multiply-store result is subnormal, the next multiply will
2603 * round it away to zero. This is done by first multiplying
2604 * by 2 ^ (scaleFactor % n) and then multiplying several
2605 * times by 2^n as needed where n is the exponent of number
2606 * that is a covenient power of two. In this way, at most one
2607 * real rounding error occurs. If the double value set is
2608 * being used exclusively, the rounding will occur on a
2609 * multiply. If the double-extended-exponent value set is
2610 * being used, the products will (perhaps) be exact but the
2611 * stores to d are guaranteed to round to the double value
2612 * set.
2613 *
2614 * It is _not_ a valid implementation to first multiply d by
2615 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2616 * MIN_EXPONENT) since even in a strictfp program double
2617 * rounding on underflow could occur; e.g. if the scaleFactor
2618 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2619 * little less than -(MIN_EXPONENT - n), meaning the final
2620 * result would be subnormal.
2621 *
2622 * Since exact reproducibility of this method can be achieved
2623 * without any undue performance burden, there is no
2624 * compelling reason to allow double rounding on underflow in
2625 * scalb.
2626 */
2627
2628 // magnitude of a power of two so large that scaling a finite
2629 // nonzero value by it would be guaranteed to over or
2630 // underflow; due to rounding, scaling down takes an
2631 // additional power of two which is reflected here
2632 final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
2633 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2634 int exp_adjust = 0;
2635 int scale_increment = 0;
2636 double exp_delta = Double.NaN;
2637
2638 // Make sure scaling factor is in a reasonable range
2639
2640 if(scaleFactor < 0) {
2641 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2642 scale_increment = -512;
2643 exp_delta = twoToTheDoubleScaleDown;
2644 }
2645 else {
2646 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2647 scale_increment = 512;
2648 exp_delta = twoToTheDoubleScaleUp;
2649 }
2650
2651 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2652 // technique from "Hacker's Delight" section 10-2.
2653 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2654 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2655
2656 d *= powerOfTwoD(exp_adjust);
2657 scaleFactor -= exp_adjust;
2658
2659 while(scaleFactor != 0) {
2660 d *= exp_delta;
2661 scaleFactor -= scale_increment;
2662 }
2663 return d;
2664 }
2665
2666 /**
2667 * Returns {@code f} ×
2668 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2669 * by a single correctly rounded floating-point multiply to a
2670 * member of the float value set. See the Java
2671 * Language Specification for a discussion of floating-point
2672 * value sets. If the exponent of the result is between {@link
2673 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2674 * answer is calculated exactly. If the exponent of the result
2675 * would be larger than {@code Float.MAX_EXPONENT}, an
2676 * infinity is returned. Note that if the result is subnormal,
2677 * precision may be lost; that is, when {@code scalb(x, n)}
2678 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2679 * <i>x</i>. When the result is non-NaN, the result has the same
2680 * sign as {@code f}.
2681 *
2682 * <p>Special cases:
2683 * <ul>
2684 * <li> If the first argument is NaN, NaN is returned.
2685 * <li> If the first argument is infinite, then an infinity of the
2686 * same sign is returned.
2687 * <li> If the first argument is zero, then a zero of the same
2688 * sign is returned.
2689 * </ul>
2690 *
2691 * @param f number to be scaled by a power of two.
2692 * @param scaleFactor power of 2 used to scale {@code f}
2693 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2694 * @since 1.6
2695 */
2696 public static float scalb(float f, int scaleFactor) {
2697 // magnitude of a power of two so large that scaling a finite
2698 // nonzero value by it would be guaranteed to over or
2699 // underflow; due to rounding, scaling down takes an
2700 // additional power of two which is reflected here
2701 final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
2702 FloatConsts.SIGNIFICAND_WIDTH + 1;
2703
2704 // Make sure scaling factor is in a reasonable range
2705 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2706
2707 /*
2708 * Since + MAX_SCALE for float fits well within the double
2709 * exponent range and + float -> double conversion is exact
2710 * the multiplication below will be exact. Therefore, the
2711 * rounding that occurs when the double product is cast to
2712 * float will be the correctly rounded float result. Since
2713 * all operations other than the final multiply will be exact,
2714 * it is not necessary to declare this method strictfp.
2715 */
2716 return (float)((double)f*powerOfTwoD(scaleFactor));
2717 }
2718
2719 // Constants used in scalb
2720 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2721 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2722
2723 /**
2724 * Returns a floating-point power of two in the normal range.
2725 */
2726 static double powerOfTwoD(int n) {
2727 assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
2728 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2729 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2730 & DoubleConsts.EXP_BIT_MASK);
2731 }
2732
2733 /**
2734 * Returns a floating-point power of two in the normal range.
2735 */
2736 static float powerOfTwoF(int n) {
2737 assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
2738 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2739 (FloatConsts.SIGNIFICAND_WIDTH-1))
2740 & FloatConsts.EXP_BIT_MASK);
2741 }
2742 }