1 /*
2 * Copyright (c) 1994, 2017, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26 package java.lang;
27
28 import java.math.BigDecimal;
29 import java.util.Random;
30 import jdk.internal.math.FloatConsts;
31 import jdk.internal.math.DoubleConsts;
32 import jdk.internal.HotSpotIntrinsicCandidate;
33
34 /**
35 * The class {@code Math} contains methods for performing basic
36 * numeric operations such as the elementary exponential, logarithm,
37 * square root, and trigonometric functions.
38 *
39 * <p>Unlike some of the numeric methods of class
40 * {@code StrictMath}, all implementations of the equivalent
41 * functions of class {@code Math} are not defined to return the
42 * bit-for-bit same results. This relaxation permits
43 * better-performing implementations where strict reproducibility is
44 * not required.
45 *
46 * <p>By default many of the {@code Math} methods simply call
47 * the equivalent method in {@code StrictMath} for their
48 * implementation. Code generators are encouraged to use
49 * platform-specific native libraries or microprocessor instructions,
50 * where available, to provide higher-performance implementations of
51 * {@code Math} methods. Such higher-performance
52 * implementations still must conform to the specification for
53 * {@code Math}.
54 *
55 * <p>The quality of implementation specifications concern two
56 * properties, accuracy of the returned result and monotonicity of the
57 * method. Accuracy of the floating-point {@code Math} methods is
58 * measured in terms of <i>ulps</i>, units in the last place. For a
59 * given floating-point format, an {@linkplain #ulp(double) ulp} of a
60 * specific real number value is the distance between the two
61 * floating-point values bracketing that numerical value. When
62 * discussing the accuracy of a method as a whole rather than at a
63 * specific argument, the number of ulps cited is for the worst-case
64 * error at any argument. If a method always has an error less than
65 * 0.5 ulps, the method always returns the floating-point number
66 * nearest the exact result; such a method is <i>correctly
67 * rounded</i>. A correctly rounded method is generally the best a
68 * floating-point approximation can be; however, it is impractical for
69 * many floating-point methods to be correctly rounded. Instead, for
70 * the {@code Math} class, a larger error bound of 1 or 2 ulps is
71 * allowed for certain methods. Informally, with a 1 ulp error bound,
72 * when the exact result is a representable number, the exact result
73 * should be returned as the computed result; otherwise, either of the
74 * two floating-point values which bracket the exact result may be
75 * returned. For exact results large in magnitude, one of the
76 * endpoints of the bracket may be infinite. Besides accuracy at
77 * individual arguments, maintaining proper relations between the
78 * method at different arguments is also important. Therefore, most
79 * methods with more than 0.5 ulp errors are required to be
80 * <i>semi-monotonic</i>: whenever the mathematical function is
81 * non-decreasing, so is the floating-point approximation, likewise,
82 * whenever the mathematical function is non-increasing, so is the
83 * floating-point approximation. Not all approximations that have 1
84 * ulp accuracy will automatically meet the monotonicity requirements.
85 *
86 * <p>
87 * The platform uses signed two's complement integer arithmetic with
88 * int and long primitive types. The developer should choose
89 * the primitive type to ensure that arithmetic operations consistently
90 * produce correct results, which in some cases means the operations
91 * will not overflow the range of values of the computation.
92 * The best practice is to choose the primitive type and algorithm to avoid
93 * overflow. In cases where the size is {@code int} or {@code long} and
94 * overflow errors need to be detected, the methods {@code addExact},
95 * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
96 * throw an {@code ArithmeticException} when the results overflow.
97 * For other arithmetic operations such as divide, absolute value,
98 * increment by one, decrement by one, and negation, overflow occurs only with
99 * a specific minimum or maximum value and should be checked against
100 * the minimum or maximum as appropriate.
101 *
102 * @author unascribed
103 * @author Joseph D. Darcy
104 * @since 1.0
105 */
106
107 public final class Math {
108
109 /**
110 * Don't let anyone instantiate this class.
111 */
112 private Math() {}
113
114 /**
115 * The {@code double} value that is closer than any other to
116 * <i>e</i>, the base of the natural logarithms.
117 */
118 public static final double E = 2.7182818284590452354;
119
120 /**
121 * The {@code double} value that is closer than any other to
122 * <i>pi</i>, the ratio of the circumference of a circle to its
123 * diameter.
124 */
125 public static final double PI = 3.14159265358979323846;
126
127 /**
128 * Constant by which to multiply an angular value in degrees to obtain an
129 * angular value in radians.
130 */
131 private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
132
133 /**
134 * Constant by which to multiply an angular value in radians to obtain an
135 * angular value in degrees.
136 */
137 private static final double RADIANS_TO_DEGREES = 57.29577951308232;
138
139 /**
140 * Returns the trigonometric sine of an angle. Special cases:
141 * <ul><li>If the argument is NaN or an infinity, then the
142 * result is NaN.
143 * <li>If the argument is zero, then the result is a zero with the
144 * same sign as the argument.</ul>
145 *
146 * <p>The computed result must be within 1 ulp of the exact result.
147 * Results must be semi-monotonic.
148 *
149 * @param a an angle, in radians.
150 * @return the sine of the argument.
151 */
152 @HotSpotIntrinsicCandidate
153 public static double sin(double a) {
154 return StrictMath.sin(a); // default impl. delegates to StrictMath
155 }
156
157 /**
158 * Returns the trigonometric cosine of an angle. Special cases:
159 * <ul><li>If the argument is NaN or an infinity, then the
160 * result is NaN.</ul>
161 *
162 * <p>The computed result must be within 1 ulp of the exact result.
163 * Results must be semi-monotonic.
164 *
165 * @param a an angle, in radians.
166 * @return the cosine of the argument.
167 */
168 @HotSpotIntrinsicCandidate
169 public static double cos(double a) {
170 return StrictMath.cos(a); // default impl. delegates to StrictMath
171 }
172
173 /**
174 * Returns the trigonometric tangent of an angle. Special cases:
175 * <ul><li>If the argument is NaN or an infinity, then the result
176 * is NaN.
177 * <li>If the argument is zero, then the result is a zero with the
178 * same sign as the argument.</ul>
179 *
180 * <p>The computed result must be within 1 ulp of the exact result.
181 * Results must be semi-monotonic.
182 *
183 * @param a an angle, in radians.
184 * @return the tangent of the argument.
185 */
186 @HotSpotIntrinsicCandidate
187 public static double tan(double a) {
188 return StrictMath.tan(a); // default impl. delegates to StrictMath
189 }
190
191 /**
192 * Returns the arc sine of a value; the returned angle is in the
193 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
194 * <ul><li>If the argument is NaN or its absolute value is greater
195 * than 1, then the result is NaN.
196 * <li>If the argument is zero, then the result is a zero with the
197 * same sign as the argument.</ul>
198 *
199 * <p>The computed result must be within 1 ulp of the exact result.
200 * Results must be semi-monotonic.
201 *
202 * @param a the value whose arc sine is to be returned.
203 * @return the arc sine of the argument.
204 */
205 public static double asin(double a) {
206 return StrictMath.asin(a); // default impl. delegates to StrictMath
207 }
208
209 /**
210 * Returns the arc cosine of a value; the returned angle is in the
211 * range 0.0 through <i>pi</i>. Special case:
212 * <ul><li>If the argument is NaN or its absolute value is greater
213 * than 1, then the result is NaN.</ul>
214 *
215 * <p>The computed result must be within 1 ulp of the exact result.
216 * Results must be semi-monotonic.
217 *
218 * @param a the value whose arc cosine is to be returned.
219 * @return the arc cosine of the argument.
220 */
221 public static double acos(double a) {
222 return StrictMath.acos(a); // default impl. delegates to StrictMath
223 }
224
225 /**
226 * Returns the arc tangent of a value; the returned angle is in the
227 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
228 * <ul><li>If the argument is NaN, then the result is NaN.
229 * <li>If the argument is zero, then the result is a zero with the
230 * same sign as the argument.</ul>
231 *
232 * <p>The computed result must be within 1 ulp of the exact result.
233 * Results must be semi-monotonic.
234 *
235 * @param a the value whose arc tangent is to be returned.
236 * @return the arc tangent of the argument.
237 */
238 public static double atan(double a) {
239 return StrictMath.atan(a); // default impl. delegates to StrictMath
240 }
241
242 /**
243 * Converts an angle measured in degrees to an approximately
244 * equivalent angle measured in radians. The conversion from
245 * degrees to radians is generally inexact.
246 *
247 * @param angdeg an angle, in degrees
248 * @return the measurement of the angle {@code angdeg}
249 * in radians.
250 * @since 1.2
251 */
252 public static double toRadians(double angdeg) {
253 return angdeg * DEGREES_TO_RADIANS;
254 }
255
256 /**
257 * Converts an angle measured in radians to an approximately
258 * equivalent angle measured in degrees. The conversion from
259 * radians to degrees is generally inexact; users should
260 * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
261 * equal {@code 0.0}.
262 *
263 * @param angrad an angle, in radians
264 * @return the measurement of the angle {@code angrad}
265 * in degrees.
266 * @since 1.2
267 */
268 public static double toDegrees(double angrad) {
269 return angrad * RADIANS_TO_DEGREES;
270 }
271
272 /**
273 * Returns Euler's number <i>e</i> raised to the power of a
274 * {@code double} value. Special cases:
275 * <ul><li>If the argument is NaN, the result is NaN.
276 * <li>If the argument is positive infinity, then the result is
277 * positive infinity.
278 * <li>If the argument is negative infinity, then the result is
279 * positive zero.</ul>
280 *
281 * <p>The computed result must be within 1 ulp of the exact result.
282 * Results must be semi-monotonic.
283 *
284 * @param a the exponent to raise <i>e</i> to.
285 * @return the value <i>e</i><sup>{@code a}</sup>,
286 * where <i>e</i> is the base of the natural logarithms.
287 */
288 @HotSpotIntrinsicCandidate
289 public static double exp(double a) {
290 return StrictMath.exp(a); // default impl. delegates to StrictMath
291 }
292
293 /**
294 * Returns the natural logarithm (base <i>e</i>) of a {@code double}
295 * value. Special cases:
296 * <ul><li>If the argument is NaN or less than zero, then the result
297 * is NaN.
298 * <li>If the argument is positive infinity, then the result is
299 * positive infinity.
300 * <li>If the argument is positive zero or negative zero, then the
301 * result is negative infinity.</ul>
302 *
303 * <p>The computed result must be within 1 ulp of the exact result.
304 * Results must be semi-monotonic.
305 *
306 * @param a a value
307 * @return the value ln {@code a}, the natural logarithm of
308 * {@code a}.
309 */
310 @HotSpotIntrinsicCandidate
311 public static double log(double a) {
312 return StrictMath.log(a); // default impl. delegates to StrictMath
313 }
314
315 /**
316 * Returns the base 10 logarithm of a {@code double} value.
317 * Special cases:
318 *
319 * <ul><li>If the argument is NaN or less than zero, then the result
320 * is NaN.
321 * <li>If the argument is positive infinity, then the result is
322 * positive infinity.
323 * <li>If the argument is positive zero or negative zero, then the
324 * result is negative infinity.
325 * <li> If the argument is equal to 10<sup><i>n</i></sup> for
326 * integer <i>n</i>, then the result is <i>n</i>.
327 * </ul>
328 *
329 * <p>The computed result must be within 1 ulp of the exact result.
330 * Results must be semi-monotonic.
331 *
332 * @param a a value
333 * @return the base 10 logarithm of {@code a}.
334 * @since 1.5
335 */
336 @HotSpotIntrinsicCandidate
337 public static double log10(double a) {
338 return StrictMath.log10(a); // default impl. delegates to StrictMath
339 }
340
341 /**
342 * Returns the correctly rounded positive square root of a
343 * {@code double} value.
344 * Special cases:
345 * <ul><li>If the argument is NaN or less than zero, then the result
346 * is NaN.
347 * <li>If the argument is positive infinity, then the result is positive
348 * infinity.
349 * <li>If the argument is positive zero or negative zero, then the
350 * result is the same as the argument.</ul>
351 * Otherwise, the result is the {@code double} value closest to
352 * the true mathematical square root of the argument value.
353 *
354 * @param a a value.
355 * @return the positive square root of {@code a}.
356 * If the argument is NaN or less than zero, the result is NaN.
357 */
358 @HotSpotIntrinsicCandidate
359 public static double sqrt(double a) {
360 return StrictMath.sqrt(a); // default impl. delegates to StrictMath
361 // Note that hardware sqrt instructions
362 // frequently can be directly used by JITs
363 // and should be much faster than doing
364 // Math.sqrt in software.
365 }
366
367
368 /**
369 * Returns the cube root of a {@code double} value. For
370 * positive finite {@code x}, {@code cbrt(-x) ==
371 * -cbrt(x)}; that is, the cube root of a negative value is
372 * the negative of the cube root of that value's magnitude.
373 *
374 * Special cases:
375 *
376 * <ul>
377 *
378 * <li>If the argument is NaN, then the result is NaN.
379 *
380 * <li>If the argument is infinite, then the result is an infinity
381 * with the same sign as the argument.
382 *
383 * <li>If the argument is zero, then the result is a zero with the
384 * same sign as the argument.
385 *
386 * </ul>
387 *
388 * <p>The computed result must be within 1 ulp of the exact result.
389 *
390 * @param a a value.
391 * @return the cube root of {@code a}.
392 * @since 1.5
393 */
394 public static double cbrt(double a) {
395 return StrictMath.cbrt(a);
396 }
397
398 /**
399 * Computes the remainder operation on two arguments as prescribed
400 * by the IEEE 754 standard.
401 * The remainder value is mathematically equal to
402 * <code>f1 - f2</code> × <i>n</i>,
403 * where <i>n</i> is the mathematical integer closest to the exact
404 * mathematical value of the quotient {@code f1/f2}, and if two
405 * mathematical integers are equally close to {@code f1/f2},
406 * then <i>n</i> is the integer that is even. If the remainder is
407 * zero, its sign is the same as the sign of the first argument.
408 * Special cases:
409 * <ul><li>If either argument is NaN, or the first argument is infinite,
410 * or the second argument is positive zero or negative zero, then the
411 * result is NaN.
412 * <li>If the first argument is finite and the second argument is
413 * infinite, then the result is the same as the first argument.</ul>
414 *
415 * @param f1 the dividend.
416 * @param f2 the divisor.
417 * @return the remainder when {@code f1} is divided by
418 * {@code f2}.
419 */
420 public static double IEEEremainder(double f1, double f2) {
421 return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
422 }
423
424 /**
425 * Returns the smallest (closest to negative infinity)
426 * {@code double} value that is greater than or equal to the
427 * argument and is equal to a mathematical integer. Special cases:
428 * <ul><li>If the argument value is already equal to a
429 * mathematical integer, then the result is the same as the
430 * argument. <li>If the argument is NaN or an infinity or
431 * positive zero or negative zero, then the result is the same as
432 * the argument. <li>If the argument value is less than zero but
433 * greater than -1.0, then the result is negative zero.</ul> Note
434 * that the value of {@code Math.ceil(x)} is exactly the
435 * value of {@code -Math.floor(-x)}.
436 *
437 *
438 * @param a a value.
439 * @return the smallest (closest to negative infinity)
440 * floating-point value that is greater than or equal to
441 * the argument and is equal to a mathematical integer.
442 */
443 public static double ceil(double a) {
444 return StrictMath.ceil(a); // default impl. delegates to StrictMath
445 }
446
447 /**
448 * Returns the largest (closest to positive infinity)
449 * {@code double} value that is less than or equal to the
450 * argument and is equal to a mathematical integer. Special cases:
451 * <ul><li>If the argument value is already equal to a
452 * mathematical integer, then the result is the same as the
453 * argument. <li>If the argument is NaN or an infinity or
454 * positive zero or negative zero, then the result is the same as
455 * the argument.</ul>
456 *
457 * @param a a value.
458 * @return the largest (closest to positive infinity)
459 * floating-point value that less than or equal to the argument
460 * and is equal to a mathematical integer.
461 */
462 public static double floor(double a) {
463 return StrictMath.floor(a); // default impl. delegates to StrictMath
464 }
465
466 /**
467 * Returns the {@code double} value that is closest in value
468 * to the argument and is equal to a mathematical integer. If two
469 * {@code double} values that are mathematical integers are
470 * equally close, the result is the integer value that is
471 * even. Special cases:
472 * <ul><li>If the argument value is already equal to a mathematical
473 * integer, then the result is the same as the argument.
474 * <li>If the argument is NaN or an infinity or positive zero or negative
475 * zero, then the result is the same as the argument.</ul>
476 *
477 * @param a a {@code double} value.
478 * @return the closest floating-point value to {@code a} that is
479 * equal to a mathematical integer.
480 */
481 public static double rint(double a) {
482 return StrictMath.rint(a); // default impl. delegates to StrictMath
483 }
484
485 /**
486 * Returns the angle <i>theta</i> from the conversion of rectangular
487 * coordinates ({@code x}, {@code y}) to polar
488 * coordinates (r, <i>theta</i>).
489 * This method computes the phase <i>theta</i> by computing an arc tangent
490 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
491 * cases:
492 * <ul><li>If either argument is NaN, then the result is NaN.
493 * <li>If the first argument is positive zero and the second argument
494 * is positive, or the first argument is positive and finite and the
495 * second argument is positive infinity, then the result is positive
496 * zero.
497 * <li>If the first argument is negative zero and the second argument
498 * is positive, or the first argument is negative and finite and the
499 * second argument is positive infinity, then the result is negative zero.
500 * <li>If the first argument is positive zero and the second argument
501 * is negative, or the first argument is positive and finite and the
502 * second argument is negative infinity, then the result is the
503 * {@code double} value closest to <i>pi</i>.
504 * <li>If the first argument is negative zero and the second argument
505 * is negative, or the first argument is negative and finite and the
506 * second argument is negative infinity, then the result is the
507 * {@code double} value closest to -<i>pi</i>.
508 * <li>If the first argument is positive and the second argument is
509 * positive zero or negative zero, or the first argument is positive
510 * infinity and the second argument is finite, then the result is the
511 * {@code double} value closest to <i>pi</i>/2.
512 * <li>If the first argument is negative and the second argument is
513 * positive zero or negative zero, or the first argument is negative
514 * infinity and the second argument is finite, then the result is the
515 * {@code double} value closest to -<i>pi</i>/2.
516 * <li>If both arguments are positive infinity, then the result is the
517 * {@code double} value closest to <i>pi</i>/4.
518 * <li>If the first argument is positive infinity and the second argument
519 * is negative infinity, then the result is the {@code double}
520 * value closest to 3*<i>pi</i>/4.
521 * <li>If the first argument is negative infinity and the second argument
522 * is positive infinity, then the result is the {@code double} value
523 * closest to -<i>pi</i>/4.
524 * <li>If both arguments are negative infinity, then the result is the
525 * {@code double} value closest to -3*<i>pi</i>/4.</ul>
526 *
527 * <p>The computed result must be within 2 ulps of the exact result.
528 * Results must be semi-monotonic.
529 *
530 * @param y the ordinate coordinate
531 * @param x the abscissa coordinate
532 * @return the <i>theta</i> component of the point
533 * (<i>r</i>, <i>theta</i>)
534 * in polar coordinates that corresponds to the point
535 * (<i>x</i>, <i>y</i>) in Cartesian coordinates.
536 */
537 @HotSpotIntrinsicCandidate
538 public static double atan2(double y, double x) {
539 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
540 }
541
542 /**
543 * Returns the value of the first argument raised to the power of the
544 * second argument. Special cases:
545 *
546 * <ul><li>If the second argument is positive or negative zero, then the
547 * result is 1.0.
548 * <li>If the second argument is 1.0, then the result is the same as the
549 * first argument.
550 * <li>If the second argument is NaN, then the result is NaN.
551 * <li>If the first argument is NaN and the second argument is nonzero,
552 * then the result is NaN.
553 *
554 * <li>If
555 * <ul>
556 * <li>the absolute value of the first argument is greater than 1
557 * and the second argument is positive infinity, or
558 * <li>the absolute value of the first argument is less than 1 and
559 * the second argument is negative infinity,
560 * </ul>
561 * then the result is positive infinity.
562 *
563 * <li>If
564 * <ul>
565 * <li>the absolute value of the first argument is greater than 1 and
566 * the second argument is negative infinity, or
567 * <li>the absolute value of the
568 * first argument is less than 1 and the second argument is positive
569 * infinity,
570 * </ul>
571 * then the result is positive zero.
572 *
573 * <li>If the absolute value of the first argument equals 1 and the
574 * second argument is infinite, then the result is NaN.
575 *
576 * <li>If
577 * <ul>
578 * <li>the first argument is positive zero and the second argument
579 * is greater than zero, or
580 * <li>the first argument is positive infinity and the second
581 * argument is less than zero,
582 * </ul>
583 * then the result is positive zero.
584 *
585 * <li>If
586 * <ul>
587 * <li>the first argument is positive zero and the second argument
588 * is less than zero, or
589 * <li>the first argument is positive infinity and the second
590 * argument is greater than zero,
591 * </ul>
592 * then the result is positive infinity.
593 *
594 * <li>If
595 * <ul>
596 * <li>the first argument is negative zero and the second argument
597 * is greater than zero but not a finite odd integer, or
598 * <li>the first argument is negative infinity and the second
599 * argument is less than zero but not a finite odd integer,
600 * </ul>
601 * then the result is positive zero.
602 *
603 * <li>If
604 * <ul>
605 * <li>the first argument is negative zero and the second argument
606 * is a positive finite odd integer, or
607 * <li>the first argument is negative infinity and the second
608 * argument is a negative finite odd integer,
609 * </ul>
610 * then the result is negative zero.
611 *
612 * <li>If
613 * <ul>
614 * <li>the first argument is negative zero and the second argument
615 * is less than zero but not a finite odd integer, or
616 * <li>the first argument is negative infinity and the second
617 * argument is greater than zero but not a finite odd integer,
618 * </ul>
619 * then the result is positive infinity.
620 *
621 * <li>If
622 * <ul>
623 * <li>the first argument is negative zero and the second argument
624 * is a negative finite odd integer, or
625 * <li>the first argument is negative infinity and the second
626 * argument is a positive finite odd integer,
627 * </ul>
628 * then the result is negative infinity.
629 *
630 * <li>If the first argument is finite and less than zero
631 * <ul>
632 * <li> if the second argument is a finite even integer, the
633 * result is equal to the result of raising the absolute value of
634 * the first argument to the power of the second argument
635 *
636 * <li>if the second argument is a finite odd integer, the result
637 * is equal to the negative of the result of raising the absolute
638 * value of the first argument to the power of the second
639 * argument
640 *
641 * <li>if the second argument is finite and not an integer, then
642 * the result is NaN.
643 * </ul>
644 *
645 * <li>If both arguments are integers, then the result is exactly equal
646 * to the mathematical result of raising the first argument to the power
647 * of the second argument if that result can in fact be represented
648 * exactly as a {@code double} value.</ul>
649 *
650 * <p>(In the foregoing descriptions, a floating-point value is
651 * considered to be an integer if and only if it is finite and a
652 * fixed point of the method {@link #ceil ceil} or,
653 * equivalently, a fixed point of the method {@link #floor
654 * floor}. A value is a fixed point of a one-argument
655 * method if and only if the result of applying the method to the
656 * value is equal to the value.)
657 *
658 * <p>The computed result must be within 1 ulp of the exact result.
659 * Results must be semi-monotonic.
660 *
661 * @param a the base.
662 * @param b the exponent.
663 * @return the value {@code a}<sup>{@code b}</sup>.
664 */
665 @HotSpotIntrinsicCandidate
666 public static double pow(double a, double b) {
667 return StrictMath.pow(a, b); // default impl. delegates to StrictMath
668 }
669
670 /**
671 * Returns the closest {@code int} to the argument, with ties
672 * rounding to positive infinity.
673 *
674 * <p>
675 * Special cases:
676 * <ul><li>If the argument is NaN, the result is 0.
677 * <li>If the argument is negative infinity or any value less than or
678 * equal to the value of {@code Integer.MIN_VALUE}, the result is
679 * equal to the value of {@code Integer.MIN_VALUE}.
680 * <li>If the argument is positive infinity or any value greater than or
681 * equal to the value of {@code Integer.MAX_VALUE}, the result is
682 * equal to the value of {@code Integer.MAX_VALUE}.</ul>
683 *
684 * @param a a floating-point value to be rounded to an integer.
685 * @return the value of the argument rounded to the nearest
686 * {@code int} value.
687 * @see java.lang.Integer#MAX_VALUE
688 * @see java.lang.Integer#MIN_VALUE
689 */
690 public static int round(float a) {
691 int intBits = Float.floatToRawIntBits(a);
692 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
693 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
694 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
695 + FloatConsts.EXP_BIAS) - biasedExp;
696 if ((shift & -32) == 0) { // shift >= 0 && shift < 32
697 // a is a finite number such that pow(2,-32) <= ulp(a) < 1
698 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
699 | (FloatConsts.SIGNIF_BIT_MASK + 1));
700 if (intBits < 0) {
701 r = -r;
702 }
703 // In the comments below each Java expression evaluates to the value
704 // the corresponding mathematical expression:
705 // (r) evaluates to a / ulp(a)
706 // (r >> shift) evaluates to floor(a * 2)
707 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
708 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
709 return ((r >> shift) + 1) >> 1;
710 } else {
711 // a is either
712 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
713 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
714 // - an infinity or NaN
715 return (int) a;
716 }
717 }
718
719 /**
720 * Returns the closest {@code long} to the argument, with ties
721 * rounding to positive infinity.
722 *
723 * <p>Special cases:
724 * <ul><li>If the argument is NaN, the result is 0.
725 * <li>If the argument is negative infinity or any value less than or
726 * equal to the value of {@code Long.MIN_VALUE}, the result is
727 * equal to the value of {@code Long.MIN_VALUE}.
728 * <li>If the argument is positive infinity or any value greater than or
729 * equal to the value of {@code Long.MAX_VALUE}, the result is
730 * equal to the value of {@code Long.MAX_VALUE}.</ul>
731 *
732 * @param a a floating-point value to be rounded to a
733 * {@code long}.
734 * @return the value of the argument rounded to the nearest
735 * {@code long} value.
736 * @see java.lang.Long#MAX_VALUE
737 * @see java.lang.Long#MIN_VALUE
738 */
739 public static long round(double a) {
740 long longBits = Double.doubleToRawLongBits(a);
741 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
742 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
743 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
744 + DoubleConsts.EXP_BIAS) - biasedExp;
745 if ((shift & -64) == 0) { // shift >= 0 && shift < 64
746 // a is a finite number such that pow(2,-64) <= ulp(a) < 1
747 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
748 | (DoubleConsts.SIGNIF_BIT_MASK + 1));
749 if (longBits < 0) {
750 r = -r;
751 }
752 // In the comments below each Java expression evaluates to the value
753 // the corresponding mathematical expression:
754 // (r) evaluates to a / ulp(a)
755 // (r >> shift) evaluates to floor(a * 2)
756 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
757 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
758 return ((r >> shift) + 1) >> 1;
759 } else {
760 // a is either
761 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
762 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
763 // - an infinity or NaN
764 return (long) a;
765 }
766 }
767
768 private static final class RandomNumberGeneratorHolder {
769 static final Random randomNumberGenerator = new Random();
770 }
771
772 /**
773 * Returns a {@code double} value with a positive sign, greater
774 * than or equal to {@code 0.0} and less than {@code 1.0}.
775 * Returned values are chosen pseudorandomly with (approximately)
776 * uniform distribution from that range.
777 *
778 * <p>When this method is first called, it creates a single new
779 * pseudorandom-number generator, exactly as if by the expression
780 *
781 * <blockquote>{@code new java.util.Random()}</blockquote>
782 *
783 * This new pseudorandom-number generator is used thereafter for
784 * all calls to this method and is used nowhere else.
785 *
786 * <p>This method is properly synchronized to allow correct use by
787 * more than one thread. However, if many threads need to generate
788 * pseudorandom numbers at a great rate, it may reduce contention
789 * for each thread to have its own pseudorandom-number generator.
790 *
791 * @apiNote
792 * As the largest {@code double} value less than {@code 1.0}
793 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
794 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
795 *
796 * <blockquote><pre>{@code
797 * double f = Math.random()/Math.nextDown(1.0);
798 * double x = x1*(1.0 - f) + x2*f;
799 * }</pre></blockquote>
800 *
801 * @return a pseudorandom {@code double} greater than or equal
802 * to {@code 0.0} and less than {@code 1.0}.
803 * @see #nextDown(double)
804 * @see Random#nextDouble()
805 */
806 public static double random() {
807 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
808 }
809
810 /**
811 * Returns the sum of its arguments,
812 * throwing an exception if the result overflows an {@code int}.
813 *
814 * @param x the first value
815 * @param y the second value
816 * @return the result
817 * @throws ArithmeticException if the result overflows an int
818 * @since 1.8
819 */
820 @HotSpotIntrinsicCandidate
821 public static int addExact(int x, int y) {
822 int r = x + y;
823 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
824 if (((x ^ r) & (y ^ r)) < 0) {
825 throw new ArithmeticException("integer overflow");
826 }
827 return r;
828 }
829
830 /**
831 * Returns the sum of its arguments,
832 * throwing an exception if the result overflows a {@code long}.
833 *
834 * @param x the first value
835 * @param y the second value
836 * @return the result
837 * @throws ArithmeticException if the result overflows a long
838 * @since 1.8
839 */
840 @HotSpotIntrinsicCandidate
841 public static long addExact(long x, long y) {
842 long r = x + y;
843 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
844 if (((x ^ r) & (y ^ r)) < 0) {
845 throw new ArithmeticException("long overflow");
846 }
847 return r;
848 }
849
850 /**
851 * Returns the difference of the arguments,
852 * throwing an exception if the result overflows an {@code int}.
853 *
854 * @param x the first value
855 * @param y the second value to subtract from the first
856 * @return the result
857 * @throws ArithmeticException if the result overflows an int
858 * @since 1.8
859 */
860 @HotSpotIntrinsicCandidate
861 public static int subtractExact(int x, int y) {
862 int r = x - y;
863 // HD 2-12 Overflow iff the arguments have different signs and
864 // the sign of the result is different from the sign of x
865 if (((x ^ y) & (x ^ r)) < 0) {
866 throw new ArithmeticException("integer overflow");
867 }
868 return r;
869 }
870
871 /**
872 * Returns the difference of the arguments,
873 * throwing an exception if the result overflows a {@code long}.
874 *
875 * @param x the first value
876 * @param y the second value to subtract from the first
877 * @return the result
878 * @throws ArithmeticException if the result overflows a long
879 * @since 1.8
880 */
881 @HotSpotIntrinsicCandidate
882 public static long subtractExact(long x, long y) {
883 long r = x - y;
884 // HD 2-12 Overflow iff the arguments have different signs and
885 // the sign of the result is different from the sign of x
886 if (((x ^ y) & (x ^ r)) < 0) {
887 throw new ArithmeticException("long overflow");
888 }
889 return r;
890 }
891
892 /**
893 * Returns the product of the arguments,
894 * throwing an exception if the result overflows an {@code int}.
895 *
896 * @param x the first value
897 * @param y the second value
898 * @return the result
899 * @throws ArithmeticException if the result overflows an int
900 * @since 1.8
901 */
902 @HotSpotIntrinsicCandidate
903 public static int multiplyExact(int x, int y) {
904 long r = (long)x * (long)y;
905 if ((int)r != r) {
906 throw new ArithmeticException("integer overflow");
907 }
908 return (int)r;
909 }
910
911 /**
912 * Returns the product of the arguments, throwing an exception if the result
913 * overflows a {@code long}.
914 *
915 * @param x the first value
916 * @param y the second value
917 * @return the result
918 * @throws ArithmeticException if the result overflows a long
919 * @since 9
920 */
921 public static long multiplyExact(long x, int y) {
922 return multiplyExact(x, (long)y);
923 }
924
925 /**
926 * Returns the product of the arguments,
927 * throwing an exception if the result overflows a {@code long}.
928 *
929 * @param x the first value
930 * @param y the second value
931 * @return the result
932 * @throws ArithmeticException if the result overflows a long
933 * @since 1.8
934 */
935 @HotSpotIntrinsicCandidate
936 public static long multiplyExact(long x, long y) {
937 long r = x * y;
938 long ax = Math.abs(x);
939 long ay = Math.abs(y);
940 if (((ax | ay) >>> 31 != 0)) {
941 // Some bits greater than 2^31 that might cause overflow
942 // Check the result using the divide operator
943 // and check for the special case of Long.MIN_VALUE * -1
944 if (((y != 0) && (r / y != x)) ||
945 (x == Long.MIN_VALUE && y == -1)) {
946 throw new ArithmeticException("long overflow");
947 }
948 }
949 return r;
950 }
951
952 /**
953 * Returns the argument incremented by one, throwing an exception if the
954 * result overflows an {@code int}.
955 *
956 * @param a the value to increment
957 * @return the result
958 * @throws ArithmeticException if the result overflows an int
959 * @since 1.8
960 */
961 @HotSpotIntrinsicCandidate
962 public static int incrementExact(int a) {
963 if (a == Integer.MAX_VALUE) {
964 throw new ArithmeticException("integer overflow");
965 }
966
967 return a + 1;
968 }
969
970 /**
971 * Returns the argument incremented by one, throwing an exception if the
972 * result overflows a {@code long}.
973 *
974 * @param a the value to increment
975 * @return the result
976 * @throws ArithmeticException if the result overflows a long
977 * @since 1.8
978 */
979 @HotSpotIntrinsicCandidate
980 public static long incrementExact(long a) {
981 if (a == Long.MAX_VALUE) {
982 throw new ArithmeticException("long overflow");
983 }
984
985 return a + 1L;
986 }
987
988 /**
989 * Returns the argument decremented by one, throwing an exception if the
990 * result overflows an {@code int}.
991 *
992 * @param a the value to decrement
993 * @return the result
994 * @throws ArithmeticException if the result overflows an int
995 * @since 1.8
996 */
997 @HotSpotIntrinsicCandidate
998 public static int decrementExact(int a) {
999 if (a == Integer.MIN_VALUE) {
1000 throw new ArithmeticException("integer overflow");
1001 }
1002
1003 return a - 1;
1004 }
1005
1006 /**
1007 * Returns the argument decremented by one, throwing an exception if the
1008 * result overflows a {@code long}.
1009 *
1010 * @param a the value to decrement
1011 * @return the result
1012 * @throws ArithmeticException if the result overflows a long
1013 * @since 1.8
1014 */
1015 @HotSpotIntrinsicCandidate
1016 public static long decrementExact(long a) {
1017 if (a == Long.MIN_VALUE) {
1018 throw new ArithmeticException("long overflow");
1019 }
1020
1021 return a - 1L;
1022 }
1023
1024 /**
1025 * Returns the negation of the argument, throwing an exception if the
1026 * result overflows an {@code int}.
1027 *
1028 * @param a the value to negate
1029 * @return the result
1030 * @throws ArithmeticException if the result overflows an int
1031 * @since 1.8
1032 */
1033 @HotSpotIntrinsicCandidate
1034 public static int negateExact(int a) {
1035 if (a == Integer.MIN_VALUE) {
1036 throw new ArithmeticException("integer overflow");
1037 }
1038
1039 return -a;
1040 }
1041
1042 /**
1043 * Returns the negation of the argument, throwing an exception if the
1044 * result overflows a {@code long}.
1045 *
1046 * @param a the value to negate
1047 * @return the result
1048 * @throws ArithmeticException if the result overflows a long
1049 * @since 1.8
1050 */
1051 @HotSpotIntrinsicCandidate
1052 public static long negateExact(long a) {
1053 if (a == Long.MIN_VALUE) {
1054 throw new ArithmeticException("long overflow");
1055 }
1056
1057 return -a;
1058 }
1059
1060 /**
1061 * Returns the value of the {@code long} argument;
1062 * throwing an exception if the value overflows an {@code int}.
1063 *
1064 * @param value the long value
1065 * @return the argument as an int
1066 * @throws ArithmeticException if the {@code argument} overflows an int
1067 * @since 1.8
1068 */
1069 public static int toIntExact(long value) {
1070 if ((int)value != value) {
1071 throw new ArithmeticException("integer overflow");
1072 }
1073 return (int)value;
1074 }
1075
1076 /**
1077 * Returns the exact mathematical product of the arguments.
1078 *
1079 * @param x the first value
1080 * @param y the second value
1081 * @return the result
1082 * @since 9
1083 */
1084 public static long multiplyFull(int x, int y) {
1085 return (long)x * (long)y;
1086 }
1087
1088 /**
1089 * Returns as a {@code long} the most significant 64 bits of the 128-bit
1090 * product of two 64-bit factors.
1091 *
1092 * @param x the first value
1093 * @param y the second value
1094 * @return the result
1095 * @since 9
1096 */
1097 @HotSpotIntrinsicCandidate
1098 public static long multiplyHigh(long x, long y) {
1099 if (x < 0 || y < 0) {
1100 // Use technique from section 8-2 of Henry S. Warren, Jr.,
1101 // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174.
1102 long x1 = x >> 32;
1103 long x2 = x & 0xFFFFFFFFL;
1104 long y1 = y >> 32;
1105 long y2 = y & 0xFFFFFFFFL;
1106 long z2 = x2 * y2;
1107 long t = x1 * y2 + (z2 >>> 32);
1108 long z1 = t & 0xFFFFFFFFL;
1109 long z0 = t >> 32;
1110 z1 += x2 * y1;
1111 return x1 * y1 + z0 + (z1 >> 32);
1112 } else {
1113 // Use Karatsuba technique with two base 2^32 digits.
1114 long x1 = x >>> 32;
1115 long y1 = y >>> 32;
1116 long x2 = x & 0xFFFFFFFFL;
1117 long y2 = y & 0xFFFFFFFFL;
1118 long A = x1 * y1;
1119 long B = x2 * y2;
1120 long C = (x1 + x2) * (y1 + y2);
1121 long K = C - A - B;
1122 return (((B >>> 32) + K) >>> 32) + A;
1123 }
1124 }
1125
1126 /**
1127 * Returns the largest (closest to positive infinity)
1128 * {@code int} value that is less than or equal to the algebraic quotient.
1129 * There is one special case, if the dividend is the
1130 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1131 * then integer overflow occurs and
1132 * the result is equal to {@code Integer.MIN_VALUE}.
1133 * <p>
1134 * Normal integer division operates under the round to zero rounding mode
1135 * (truncation). This operation instead acts under the round toward
1136 * negative infinity (floor) rounding mode.
1137 * The floor rounding mode gives different results from truncation
1138 * when the exact result is negative.
1139 * <ul>
1140 * <li>If the signs of the arguments are the same, the results of
1141 * {@code floorDiv} and the {@code /} operator are the same. <br>
1142 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1143 * <li>If the signs of the arguments are different, the quotient is negative and
1144 * {@code floorDiv} returns the integer less than or equal to the quotient
1145 * and the {@code /} operator returns the integer closest to zero.<br>
1146 * For example, {@code floorDiv(-4, 3) == -2},
1147 * whereas {@code (-4 / 3) == -1}.
1148 * </li>
1149 * </ul>
1150 *
1151 * @param x the dividend
1152 * @param y the divisor
1153 * @return the largest (closest to positive infinity)
1154 * {@code int} value that is less than or equal to the algebraic quotient.
1155 * @throws ArithmeticException if the divisor {@code y} is zero
1156 * @see #floorMod(int, int)
1157 * @see #floor(double)
1158 * @since 1.8
1159 */
1160 public static int floorDiv(int x, int y) {
1161 int r = x / y;
1162 // if the signs are different and modulo not zero, round down
1163 if ((x ^ y) < 0 && (r * y != x)) {
1164 r--;
1165 }
1166 return r;
1167 }
1168
1169 /**
1170 * Returns the largest (closest to positive infinity)
1171 * {@code long} value that is less than or equal to the algebraic quotient.
1172 * There is one special case, if the dividend is the
1173 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1174 * then integer overflow occurs and
1175 * the result is equal to {@code Long.MIN_VALUE}.
1176 * <p>
1177 * Normal integer division operates under the round to zero rounding mode
1178 * (truncation). This operation instead acts under the round toward
1179 * negative infinity (floor) rounding mode.
1180 * The floor rounding mode gives different results from truncation
1181 * when the exact result is negative.
1182 * <p>
1183 * For examples, see {@link #floorDiv(int, int)}.
1184 *
1185 * @param x the dividend
1186 * @param y the divisor
1187 * @return the largest (closest to positive infinity)
1188 * {@code int} value that is less than or equal to the algebraic quotient.
1189 * @throws ArithmeticException if the divisor {@code y} is zero
1190 * @see #floorMod(long, int)
1191 * @see #floor(double)
1192 * @since 9
1193 */
1194 public static long floorDiv(long x, int y) {
1195 return floorDiv(x, (long)y);
1196 }
1197
1198 /**
1199 * Returns the largest (closest to positive infinity)
1200 * {@code long} value that is less than or equal to the algebraic quotient.
1201 * There is one special case, if the dividend is the
1202 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1203 * then integer overflow occurs and
1204 * the result is equal to {@code Long.MIN_VALUE}.
1205 * <p>
1206 * Normal integer division operates under the round to zero rounding mode
1207 * (truncation). This operation instead acts under the round toward
1208 * negative infinity (floor) rounding mode.
1209 * The floor rounding mode gives different results from truncation
1210 * when the exact result is negative.
1211 * <p>
1212 * For examples, see {@link #floorDiv(int, int)}.
1213 *
1214 * @param x the dividend
1215 * @param y the divisor
1216 * @return the largest (closest to positive infinity)
1217 * {@code long} value that is less than or equal to the algebraic quotient.
1218 * @throws ArithmeticException if the divisor {@code y} is zero
1219 * @see #floorMod(long, long)
1220 * @see #floor(double)
1221 * @since 1.8
1222 */
1223 public static long floorDiv(long x, long y) {
1224 long r = x / y;
1225 // if the signs are different and modulo not zero, round down
1226 if ((x ^ y) < 0 && (r * y != x)) {
1227 r--;
1228 }
1229 return r;
1230 }
1231
1232 /**
1233 * Returns the floor modulus of the {@code int} arguments.
1234 * <p>
1235 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1236 * has the same sign as the divisor {@code y}, and
1237 * is in the range of {@code -abs(y) < r < +abs(y)}.
1238 *
1239 * <p>
1240 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1241 * <ul>
1242 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1243 * </ul>
1244 * <p>
1245 * The difference in values between {@code floorMod} and
1246 * the {@code %} operator is due to the difference between
1247 * {@code floorDiv} that returns the integer less than or equal to the quotient
1248 * and the {@code /} operator that returns the integer closest to zero.
1249 * <p>
1250 * Examples:
1251 * <ul>
1252 * <li>If the signs of the arguments are the same, the results
1253 * of {@code floorMod} and the {@code %} operator are the same. <br>
1254 * <ul>
1255 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li>
1256 * </ul>
1257 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1258 * <ul>
1259 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li>
1260 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li>
1261 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li>
1262 * </ul>
1263 * </li>
1264 * </ul>
1265 * <p>
1266 * If the signs of arguments are unknown and a positive modulus
1267 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1268 *
1269 * @param x the dividend
1270 * @param y the divisor
1271 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1272 * @throws ArithmeticException if the divisor {@code y} is zero
1273 * @see #floorDiv(int, int)
1274 * @since 1.8
1275 */
1276 public static int floorMod(int x, int y) {
1277 int mod = x % y;
1278 // if the signs are different and modulo not zero, adjust result
1279 if ((mod ^ y) < 0 && mod != 0) {
1280 mod += y;
1281 }
1282 return mod;
1283 }
1284
1285 /**
1286 * Returns the floor modulus of the {@code long} and {@code int} arguments.
1287 * <p>
1288 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1289 * has the same sign as the divisor {@code y}, and
1290 * is in the range of {@code -abs(y) < r < +abs(y)}.
1291 *
1292 * <p>
1293 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1294 * <ul>
1295 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1296 * </ul>
1297 * <p>
1298 * For examples, see {@link #floorMod(int, int)}.
1299 *
1300 * @param x the dividend
1301 * @param y the divisor
1302 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1303 * @throws ArithmeticException if the divisor {@code y} is zero
1304 * @see #floorDiv(long, int)
1305 * @since 9
1306 */
1307 public static int floorMod(long x, int y) {
1308 // Result cannot overflow the range of int.
1309 return (int)floorMod(x, (long)y);
1310 }
1311
1312 /**
1313 * Returns the floor modulus of the {@code long} arguments.
1314 * <p>
1315 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1316 * has the same sign as the divisor {@code y}, and
1317 * is in the range of {@code -abs(y) < r < +abs(y)}.
1318 *
1319 * <p>
1320 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1321 * <ul>
1322 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1323 * </ul>
1324 * <p>
1325 * For examples, see {@link #floorMod(int, int)}.
1326 *
1327 * @param x the dividend
1328 * @param y the divisor
1329 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1330 * @throws ArithmeticException if the divisor {@code y} is zero
1331 * @see #floorDiv(long, long)
1332 * @since 1.8
1333 */
1334 public static long floorMod(long x, long y) {
1335 long mod = x % y;
1336 // if the signs are different and modulo not zero, adjust result
1337 if ((x ^ y) < 0 && mod != 0) {
1338 mod += y;
1339 }
1340 return mod;
1341 }
1342
1343 /**
1344 * Returns the absolute value of an {@code int} value.
1345 * If the argument is not negative, the argument is returned.
1346 * If the argument is negative, the negation of the argument is returned.
1347 *
1348 * <p>Note that if the argument is equal to the value of
1349 * {@link Integer#MIN_VALUE}, the most negative representable
1350 * {@code int} value, the result is that same value, which is
1351 * negative.
1352 *
1353 * @param a the argument whose absolute value is to be determined
1354 * @return the absolute value of the argument.
1355 */
1356 @HotSpotIntrinsicCandidate
1357 public static int abs(int a) {
1358 return (a < 0) ? -a : a;
1359 }
1360
1361 /**
1362 * Returns the absolute value of a {@code long} value.
1363 * If the argument is not negative, the argument is returned.
1364 * If the argument is negative, the negation of the argument is returned.
1365 *
1366 * <p>Note that if the argument is equal to the value of
1367 * {@link Long#MIN_VALUE}, the most negative representable
1368 * {@code long} value, the result is that same value, which
1369 * is negative.
1370 *
1371 * @param a the argument whose absolute value is to be determined
1372 * @return the absolute value of the argument.
1373 */
1374 @HotSpotIntrinsicCandidate
1375 public static long abs(long a) {
1376 return (a < 0) ? -a : a;
1377 }
1378
1379 /**
1380 * Returns the absolute value of a {@code float} value.
1381 * If the argument is not negative, the argument is returned.
1382 * If the argument is negative, the negation of the argument is returned.
1383 * Special cases:
1384 * <ul><li>If the argument is positive zero or negative zero, the
1385 * result is positive zero.
1386 * <li>If the argument is infinite, the result is positive infinity.
1387 * <li>If the argument is NaN, the result is NaN.</ul>
1388 *
1389 * @apiNote As implied by the above, one valid implementation of
1390 * this method is given by the expression below which computes a
1391 * {@code float} with the same exponent and significand as the
1392 * argument but with a guaranteed zero sign bit indicating a
1393 * positive value:<br>
1394 * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
1395 *
1396 * @param a the argument whose absolute value is to be determined
1397 * @return the absolute value of the argument.
1398 */
1399 @HotSpotIntrinsicCandidate
1400 public static float abs(float a) {
1401 return (a <= 0.0F) ? 0.0F - a : a;
1402 }
1403
1404 /**
1405 * Returns the absolute value of a {@code double} value.
1406 * If the argument is not negative, the argument is returned.
1407 * If the argument is negative, the negation of the argument is returned.
1408 * Special cases:
1409 * <ul><li>If the argument is positive zero or negative zero, the result
1410 * is positive zero.
1411 * <li>If the argument is infinite, the result is positive infinity.
1412 * <li>If the argument is NaN, the result is NaN.</ul>
1413 *
1414 * @apiNote As implied by the above, one valid implementation of
1415 * this method is given by the expression below which computes a
1416 * {@code double} with the same exponent and significand as the
1417 * argument but with a guaranteed zero sign bit indicating a
1418 * positive value:<br>
1419 * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
1420 *
1421 * @param a the argument whose absolute value is to be determined
1422 * @return the absolute value of the argument.
1423 */
1424 @HotSpotIntrinsicCandidate
1425 public static double abs(double a) {
1426 return (a <= 0.0D) ? 0.0D - a : a;
1427 }
1428
1429 /**
1430 * Returns the greater of two {@code int} values. That is, the
1431 * result is the argument closer to the value of
1432 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1433 * the result is that same value.
1434 *
1435 * @param a an argument.
1436 * @param b another argument.
1437 * @return the larger of {@code a} and {@code b}.
1438 */
1439 @HotSpotIntrinsicCandidate
1440 public static int max(int a, int b) {
1441 return (a >= b) ? a : b;
1442 }
1443
1444 /**
1445 * Returns the greater of two {@code long} values. That is, the
1446 * result is the argument closer to the value of
1447 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1448 * the result is that same value.
1449 *
1450 * @param a an argument.
1451 * @param b another argument.
1452 * @return the larger of {@code a} and {@code b}.
1453 */
1454 public static long max(long a, long b) {
1455 return (a >= b) ? a : b;
1456 }
1457
1458 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1459 private static final long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1460 private static final long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1461
1462 /**
1463 * Returns the greater of two {@code float} values. That is,
1464 * the result is the argument closer to positive infinity. If the
1465 * arguments have the same value, the result is that same
1466 * value. If either value is NaN, then the result is NaN. Unlike
1467 * the numerical comparison operators, this method considers
1468 * negative zero to be strictly smaller than positive zero. If one
1469 * argument is positive zero and the other negative zero, the
1470 * result is positive zero.
1471 *
1472 * @param a an argument.
1473 * @param b another argument.
1474 * @return the larger of {@code a} and {@code b}.
1475 */
1476 @HotSpotIntrinsicCandidate
1477 public static float max(float a, float b) {
1478 if (a != a)
1479 return a; // a is NaN
1480 if ((a == 0.0f) &&
1481 (b == 0.0f) &&
1482 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1483 // Raw conversion ok since NaN can't map to -0.0.
1484 return b;
1485 }
1486 return (a >= b) ? a : b;
1487 }
1488
1489 /**
1490 * Returns the greater of two {@code double} values. That
1491 * is, the result is the argument closer to positive infinity. If
1492 * the arguments have the same value, the result is that same
1493 * value. If either value is NaN, then the result is NaN. Unlike
1494 * the numerical comparison operators, this method considers
1495 * negative zero to be strictly smaller than positive zero. If one
1496 * argument is positive zero and the other negative zero, the
1497 * result is positive zero.
1498 *
1499 * @param a an argument.
1500 * @param b another argument.
1501 * @return the larger of {@code a} and {@code b}.
1502 */
1503 @HotSpotIntrinsicCandidate
1504 public static double max(double a, double b) {
1505 if (a != a)
1506 return a; // a is NaN
1507 if ((a == 0.0d) &&
1508 (b == 0.0d) &&
1509 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1510 // Raw conversion ok since NaN can't map to -0.0.
1511 return b;
1512 }
1513 return (a >= b) ? a : b;
1514 }
1515
1516 /**
1517 * Returns the smaller of two {@code int} values. That is,
1518 * the result the argument closer to the value of
1519 * {@link Integer#MIN_VALUE}. If the arguments have the same
1520 * value, the result is that same value.
1521 *
1522 * @param a an argument.
1523 * @param b another argument.
1524 * @return the smaller of {@code a} and {@code b}.
1525 */
1526 @HotSpotIntrinsicCandidate
1527 public static int min(int a, int b) {
1528 return (a <= b) ? a : b;
1529 }
1530
1531 /**
1532 * Returns the smaller of two {@code long} values. That is,
1533 * the result is the argument closer to the value of
1534 * {@link Long#MIN_VALUE}. If the arguments have the same
1535 * value, the result is that same value.
1536 *
1537 * @param a an argument.
1538 * @param b another argument.
1539 * @return the smaller of {@code a} and {@code b}.
1540 */
1541 public static long min(long a, long b) {
1542 return (a <= b) ? a : b;
1543 }
1544
1545 /**
1546 * Returns the smaller of two {@code float} values. That is,
1547 * the result is the value closer to negative infinity. If the
1548 * arguments have the same value, the result is that same
1549 * value. If either value is NaN, then the result is NaN. Unlike
1550 * the numerical comparison operators, this method considers
1551 * negative zero to be strictly smaller than positive zero. If
1552 * one argument is positive zero and the other is negative zero,
1553 * the result is negative zero.
1554 *
1555 * @param a an argument.
1556 * @param b another argument.
1557 * @return the smaller of {@code a} and {@code b}.
1558 */
1559 @HotSpotIntrinsicCandidate
1560 public static float min(float a, float b) {
1561 if (a != a)
1562 return a; // a is NaN
1563 if ((a == 0.0f) &&
1564 (b == 0.0f) &&
1565 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1566 // Raw conversion ok since NaN can't map to -0.0.
1567 return b;
1568 }
1569 return (a <= b) ? a : b;
1570 }
1571
1572 /**
1573 * Returns the smaller of two {@code double} values. That
1574 * is, the result is the value closer to negative infinity. If the
1575 * arguments have the same value, the result is that same
1576 * value. If either value is NaN, then the result is NaN. Unlike
1577 * the numerical comparison operators, this method considers
1578 * negative zero to be strictly smaller than positive zero. If one
1579 * argument is positive zero and the other is negative zero, the
1580 * result is negative zero.
1581 *
1582 * @param a an argument.
1583 * @param b another argument.
1584 * @return the smaller of {@code a} and {@code b}.
1585 */
1586 @HotSpotIntrinsicCandidate
1587 public static double min(double a, double b) {
1588 if (a != a)
1589 return a; // a is NaN
1590 if ((a == 0.0d) &&
1591 (b == 0.0d) &&
1592 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1593 // Raw conversion ok since NaN can't map to -0.0.
1594 return b;
1595 }
1596 return (a <= b) ? a : b;
1597 }
1598
1599 /**
1600 * Returns the fused multiply add of the three arguments; that is,
1601 * returns the exact product of the first two arguments summed
1602 * with the third argument and then rounded once to the nearest
1603 * {@code double}.
1604 *
1605 * The rounding is done using the {@linkplain
1606 * java.math.RoundingMode#HALF_EVEN round to nearest even
1607 * rounding mode}.
1608 *
1609 * In contrast, if {@code a * b + c} is evaluated as a regular
1610 * floating-point expression, two rounding errors are involved,
1611 * the first for the multiply operation, the second for the
1612 * addition operation.
1613 *
1614 * <p>Special cases:
1615 * <ul>
1616 * <li> If any argument is NaN, the result is NaN.
1617 *
1618 * <li> If one of the first two arguments is infinite and the
1619 * other is zero, the result is NaN.
1620 *
1621 * <li> If the exact product of the first two arguments is infinite
1622 * (in other words, at least one of the arguments is infinite and
1623 * the other is neither zero nor NaN) and the third argument is an
1624 * infinity of the opposite sign, the result is NaN.
1625 *
1626 * </ul>
1627 *
1628 * <p>Note that {@code fma(a, 1.0, c)} returns the same
1629 * result as ({@code a + c}). However,
1630 * {@code fma(a, b, +0.0)} does <em>not</em> always return the
1631 * same result as ({@code a * b}) since
1632 * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while
1633 * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is
1634 * equivalent to ({@code a * b}) however.
1635 *
1636 * @apiNote This method corresponds to the fusedMultiplyAdd
1637 * operation defined in IEEE 754-2008.
1638 *
1639 * @param a a value
1640 * @param b a value
1641 * @param c a value
1642 *
1643 * @return (<i>a</i> × <i>b</i> + <i>c</i>)
1644 * computed, as if with unlimited range and precision, and rounded
1645 * once to the nearest {@code double} value
1646 *
1647 * @since 9
1648 */
1649 @HotSpotIntrinsicCandidate
1650 public static double fma(double a, double b, double c) {
1651 /*
1652 * Infinity and NaN arithmetic is not quite the same with two
1653 * roundings as opposed to just one so the simple expression
1654 * "a * b + c" cannot always be used to compute the correct
1655 * result. With two roundings, the product can overflow and
1656 * if the addend is infinite, a spurious NaN can be produced
1657 * if the infinity from the overflow and the infinite addend
1658 * have opposite signs.
1659 */
1660
1661 // First, screen for and handle non-finite input values whose
1662 // arithmetic is not supported by BigDecimal.
1663 if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
1664 return Double.NaN;
1665 } else { // All inputs non-NaN
1666 boolean infiniteA = Double.isInfinite(a);
1667 boolean infiniteB = Double.isInfinite(b);
1668 boolean infiniteC = Double.isInfinite(c);
1669 double result;
1670
1671 if (infiniteA || infiniteB || infiniteC) {
1672 if (infiniteA && b == 0.0 ||
1673 infiniteB && a == 0.0 ) {
1674 return Double.NaN;
1675 }
1676 // Store product in a double field to cause an
1677 // overflow even if non-strictfp evaluation is being
1678 // used.
1679 double product = a * b;
1680 if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
1681 // Intermediate overflow; might cause a
1682 // spurious NaN if added to infinite c.
1683 assert Double.isInfinite(c);
1684 return c;
1685 } else {
1686 result = product + c;
1687 assert !Double.isFinite(result);
1688 return result;
1689 }
1690 } else { // All inputs finite
1691 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
1692 if (c == 0.0) { // Positive or negative zero
1693 // If the product is an exact zero, use a
1694 // floating-point expression to compute the sign
1695 // of the zero final result. The product is an
1696 // exact zero if and only if at least one of a and
1697 // b is zero.
1698 if (a == 0.0 || b == 0.0) {
1699 return a * b + c;
1700 } else {
1701 // The sign of a zero addend doesn't matter if
1702 // the product is nonzero. The sign of a zero
1703 // addend is not factored in the result if the
1704 // exact product is nonzero but underflows to
1705 // zero; see IEEE-754 2008 section 6.3 "The
1706 // sign bit".
1707 return product.doubleValue();
1708 }
1709 } else {
1710 return product.add(new BigDecimal(c)).doubleValue();
1711 }
1712 }
1713 }
1714 }
1715
1716 /**
1717 * Returns the fused multiply add of the three arguments; that is,
1718 * returns the exact product of the first two arguments summed
1719 * with the third argument and then rounded once to the nearest
1720 * {@code float}.
1721 *
1722 * The rounding is done using the {@linkplain
1723 * java.math.RoundingMode#HALF_EVEN round to nearest even
1724 * rounding mode}.
1725 *
1726 * In contrast, if {@code a * b + c} is evaluated as a regular
1727 * floating-point expression, two rounding errors are involved,
1728 * the first for the multiply operation, the second for the
1729 * addition operation.
1730 *
1731 * <p>Special cases:
1732 * <ul>
1733 * <li> If any argument is NaN, the result is NaN.
1734 *
1735 * <li> If one of the first two arguments is infinite and the
1736 * other is zero, the result is NaN.
1737 *
1738 * <li> If the exact product of the first two arguments is infinite
1739 * (in other words, at least one of the arguments is infinite and
1740 * the other is neither zero nor NaN) and the third argument is an
1741 * infinity of the opposite sign, the result is NaN.
1742 *
1743 * </ul>
1744 *
1745 * <p>Note that {@code fma(a, 1.0f, c)} returns the same
1746 * result as ({@code a + c}). However,
1747 * {@code fma(a, b, +0.0f)} does <em>not</em> always return the
1748 * same result as ({@code a * b}) since
1749 * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
1750 * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
1751 * equivalent to ({@code a * b}) however.
1752 *
1753 * @apiNote This method corresponds to the fusedMultiplyAdd
1754 * operation defined in IEEE 754-2008.
1755 *
1756 * @param a a value
1757 * @param b a value
1758 * @param c a value
1759 *
1760 * @return (<i>a</i> × <i>b</i> + <i>c</i>)
1761 * computed, as if with unlimited range and precision, and rounded
1762 * once to the nearest {@code float} value
1763 *
1764 * @since 9
1765 */
1766 @HotSpotIntrinsicCandidate
1767 public static float fma(float a, float b, float c) {
1768 /*
1769 * Since the double format has more than twice the precision
1770 * of the float format, the multiply of a * b is exact in
1771 * double. The add of c to the product then incurs one
1772 * rounding error. Since the double format moreover has more
1773 * than (2p + 2) precision bits compared to the p bits of the
1774 * float format, the two roundings of (a * b + c), first to
1775 * the double format and then secondarily to the float format,
1776 * are equivalent to rounding the intermediate result directly
1777 * to the float format.
1778 *
1779 * In terms of strictfp vs default-fp concerns related to
1780 * overflow and underflow, since
1781 *
1782 * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE
1783 * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE
1784 *
1785 * neither the multiply nor add will overflow or underflow in
1786 * double. Therefore, it is not necessary for this method to
1787 * be declared strictfp to have reproducible
1788 * behavior. However, it is necessary to explicitly store down
1789 * to a float variable to avoid returning a value in the float
1790 * extended value set.
1791 */
1792 float result = (float)(((double) a * (double) b ) + (double) c);
1793 return result;
1794 }
1795
1796 /**
1797 * Returns the size of an ulp of the argument. An ulp, unit in
1798 * the last place, of a {@code double} value is the positive
1799 * distance between this floating-point value and the {@code
1800 * double} value next larger in magnitude. Note that for non-NaN
1801 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1802 *
1803 * <p>Special Cases:
1804 * <ul>
1805 * <li> If the argument is NaN, then the result is NaN.
1806 * <li> If the argument is positive or negative infinity, then the
1807 * result is positive infinity.
1808 * <li> If the argument is positive or negative zero, then the result is
1809 * {@code Double.MIN_VALUE}.
1810 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1811 * the result is equal to 2<sup>971</sup>.
1812 * </ul>
1813 *
1814 * @param d the floating-point value whose ulp is to be returned
1815 * @return the size of an ulp of the argument
1816 * @author Joseph D. Darcy
1817 * @since 1.5
1818 */
1819 public static double ulp(double d) {
1820 int exp = getExponent(d);
1821
1822 switch(exp) {
1823 case Double.MAX_EXPONENT + 1: // NaN or infinity
1824 return Math.abs(d);
1825
1826 case Double.MIN_EXPONENT - 1: // zero or subnormal
1827 return Double.MIN_VALUE;
1828
1829 default:
1830 assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
1831
1832 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1833 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1834 if (exp >= Double.MIN_EXPONENT) {
1835 return powerOfTwoD(exp);
1836 }
1837 else {
1838 // return a subnormal result; left shift integer
1839 // representation of Double.MIN_VALUE appropriate
1840 // number of positions
1841 return Double.longBitsToDouble(1L <<
1842 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1843 }
1844 }
1845 }
1846
1847 /**
1848 * Returns the size of an ulp of the argument. An ulp, unit in
1849 * the last place, of a {@code float} value is the positive
1850 * distance between this floating-point value and the {@code
1851 * float} value next larger in magnitude. Note that for non-NaN
1852 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1853 *
1854 * <p>Special Cases:
1855 * <ul>
1856 * <li> If the argument is NaN, then the result is NaN.
1857 * <li> If the argument is positive or negative infinity, then the
1858 * result is positive infinity.
1859 * <li> If the argument is positive or negative zero, then the result is
1860 * {@code Float.MIN_VALUE}.
1861 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1862 * the result is equal to 2<sup>104</sup>.
1863 * </ul>
1864 *
1865 * @param f the floating-point value whose ulp is to be returned
1866 * @return the size of an ulp of the argument
1867 * @author Joseph D. Darcy
1868 * @since 1.5
1869 */
1870 public static float ulp(float f) {
1871 int exp = getExponent(f);
1872
1873 switch(exp) {
1874 case Float.MAX_EXPONENT+1: // NaN or infinity
1875 return Math.abs(f);
1876
1877 case Float.MIN_EXPONENT-1: // zero or subnormal
1878 return Float.MIN_VALUE;
1879
1880 default:
1881 assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
1882
1883 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1884 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1885 if (exp >= Float.MIN_EXPONENT) {
1886 return powerOfTwoF(exp);
1887 } else {
1888 // return a subnormal result; left shift integer
1889 // representation of FloatConsts.MIN_VALUE appropriate
1890 // number of positions
1891 return Float.intBitsToFloat(1 <<
1892 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1893 }
1894 }
1895 }
1896
1897 /**
1898 * Returns the signum function of the argument; zero if the argument
1899 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1900 * argument is less than zero.
1901 *
1902 * <p>Special Cases:
1903 * <ul>
1904 * <li> If the argument is NaN, then the result is NaN.
1905 * <li> If the argument is positive zero or negative zero, then the
1906 * result is the same as the argument.
1907 * </ul>
1908 *
1909 * @param d the floating-point value whose signum is to be returned
1910 * @return the signum function of the argument
1911 * @author Joseph D. Darcy
1912 * @since 1.5
1913 */
1914 public static double signum(double d) {
1915 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1916 }
1917
1918 /**
1919 * Returns the signum function of the argument; zero if the argument
1920 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1921 * argument is less than zero.
1922 *
1923 * <p>Special Cases:
1924 * <ul>
1925 * <li> If the argument is NaN, then the result is NaN.
1926 * <li> If the argument is positive zero or negative zero, then the
1927 * result is the same as the argument.
1928 * </ul>
1929 *
1930 * @param f the floating-point value whose signum is to be returned
1931 * @return the signum function of the argument
1932 * @author Joseph D. Darcy
1933 * @since 1.5
1934 */
1935 public static float signum(float f) {
1936 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1937 }
1938
1939 /**
1940 * Returns the hyperbolic sine of a {@code double} value.
1941 * The hyperbolic sine of <i>x</i> is defined to be
1942 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1943 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1944 *
1945 * <p>Special cases:
1946 * <ul>
1947 *
1948 * <li>If the argument is NaN, then the result is NaN.
1949 *
1950 * <li>If the argument is infinite, then the result is an infinity
1951 * with the same sign as the argument.
1952 *
1953 * <li>If the argument is zero, then the result is a zero with the
1954 * same sign as the argument.
1955 *
1956 * </ul>
1957 *
1958 * <p>The computed result must be within 2.5 ulps of the exact result.
1959 *
1960 * @param x The number whose hyperbolic sine is to be returned.
1961 * @return The hyperbolic sine of {@code x}.
1962 * @since 1.5
1963 */
1964 public static double sinh(double x) {
1965 return StrictMath.sinh(x);
1966 }
1967
1968 /**
1969 * Returns the hyperbolic cosine of a {@code double} value.
1970 * The hyperbolic cosine of <i>x</i> is defined to be
1971 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1972 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1973 *
1974 * <p>Special cases:
1975 * <ul>
1976 *
1977 * <li>If the argument is NaN, then the result is NaN.
1978 *
1979 * <li>If the argument is infinite, then the result is positive
1980 * infinity.
1981 *
1982 * <li>If the argument is zero, then the result is {@code 1.0}.
1983 *
1984 * </ul>
1985 *
1986 * <p>The computed result must be within 2.5 ulps of the exact result.
1987 *
1988 * @param x The number whose hyperbolic cosine is to be returned.
1989 * @return The hyperbolic cosine of {@code x}.
1990 * @since 1.5
1991 */
1992 public static double cosh(double x) {
1993 return StrictMath.cosh(x);
1994 }
1995
1996 /**
1997 * Returns the hyperbolic tangent of a {@code double} value.
1998 * The hyperbolic tangent of <i>x</i> is defined to be
1999 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
2000 * in other words, {@linkplain Math#sinh
2001 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
2002 * that the absolute value of the exact tanh is always less than
2003 * 1.
2004 *
2005 * <p>Special cases:
2006 * <ul>
2007 *
2008 * <li>If the argument is NaN, then the result is NaN.
2009 *
2010 * <li>If the argument is zero, then the result is a zero with the
2011 * same sign as the argument.
2012 *
2013 * <li>If the argument is positive infinity, then the result is
2014 * {@code +1.0}.
2015 *
2016 * <li>If the argument is negative infinity, then the result is
2017 * {@code -1.0}.
2018 *
2019 * </ul>
2020 *
2021 * <p>The computed result must be within 2.5 ulps of the exact result.
2022 * The result of {@code tanh} for any finite input must have
2023 * an absolute value less than or equal to 1. Note that once the
2024 * exact result of tanh is within 1/2 of an ulp of the limit value
2025 * of ±1, correctly signed ±{@code 1.0} should
2026 * be returned.
2027 *
2028 * @param x The number whose hyperbolic tangent is to be returned.
2029 * @return The hyperbolic tangent of {@code x}.
2030 * @since 1.5
2031 */
2032 public static double tanh(double x) {
2033 return StrictMath.tanh(x);
2034 }
2035
2036 /**
2037 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
2038 * without intermediate overflow or underflow.
2039 *
2040 * <p>Special cases:
2041 * <ul>
2042 *
2043 * <li> If either argument is infinite, then the result
2044 * is positive infinity.
2045 *
2046 * <li> If either argument is NaN and neither argument is infinite,
2047 * then the result is NaN.
2048 *
2049 * </ul>
2050 *
2051 * <p>The computed result must be within 1 ulp of the exact
2052 * result. If one parameter is held constant, the results must be
2053 * semi-monotonic in the other parameter.
2054 *
2055 * @param x a value
2056 * @param y a value
2057 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
2058 * without intermediate overflow or underflow
2059 * @since 1.5
2060 */
2061 public static double hypot(double x, double y) {
2062 return StrictMath.hypot(x, y);
2063 }
2064
2065 /**
2066 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
2067 * <i>x</i> near 0, the exact sum of
2068 * {@code expm1(x)} + 1 is much closer to the true
2069 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
2070 *
2071 * <p>Special cases:
2072 * <ul>
2073 * <li>If the argument is NaN, the result is NaN.
2074 *
2075 * <li>If the argument is positive infinity, then the result is
2076 * positive infinity.
2077 *
2078 * <li>If the argument is negative infinity, then the result is
2079 * -1.0.
2080 *
2081 * <li>If the argument is zero, then the result is a zero with the
2082 * same sign as the argument.
2083 *
2084 * </ul>
2085 *
2086 * <p>The computed result must be within 1 ulp of the exact result.
2087 * Results must be semi-monotonic. The result of
2088 * {@code expm1} for any finite input must be greater than or
2089 * equal to {@code -1.0}. Note that once the exact result of
2090 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
2091 * ulp of the limit value -1, {@code -1.0} should be
2092 * returned.
2093 *
2094 * @param x the exponent to raise <i>e</i> to in the computation of
2095 * <i>e</i><sup>{@code x}</sup> -1.
2096 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
2097 * @since 1.5
2098 */
2099 public static double expm1(double x) {
2100 return StrictMath.expm1(x);
2101 }
2102
2103 /**
2104 * Returns the natural logarithm of the sum of the argument and 1.
2105 * Note that for small values {@code x}, the result of
2106 * {@code log1p(x)} is much closer to the true result of ln(1
2107 * + {@code x}) than the floating-point evaluation of
2108 * {@code log(1.0+x)}.
2109 *
2110 * <p>Special cases:
2111 *
2112 * <ul>
2113 *
2114 * <li>If the argument is NaN or less than -1, then the result is
2115 * NaN.
2116 *
2117 * <li>If the argument is positive infinity, then the result is
2118 * positive infinity.
2119 *
2120 * <li>If the argument is negative one, then the result is
2121 * negative infinity.
2122 *
2123 * <li>If the argument is zero, then the result is a zero with the
2124 * same sign as the argument.
2125 *
2126 * </ul>
2127 *
2128 * <p>The computed result must be within 1 ulp of the exact result.
2129 * Results must be semi-monotonic.
2130 *
2131 * @param x a value
2132 * @return the value ln({@code x} + 1), the natural
2133 * log of {@code x} + 1
2134 * @since 1.5
2135 */
2136 public static double log1p(double x) {
2137 return StrictMath.log1p(x);
2138 }
2139
2140 /**
2141 * Returns the first floating-point argument with the sign of the
2142 * second floating-point argument. Note that unlike the {@link
2143 * StrictMath#copySign(double, double) StrictMath.copySign}
2144 * method, this method does not require NaN {@code sign}
2145 * arguments to be treated as positive values; implementations are
2146 * permitted to treat some NaN arguments as positive and other NaN
2147 * arguments as negative to allow greater performance.
2148 *
2149 * @param magnitude the parameter providing the magnitude of the result
2150 * @param sign the parameter providing the sign of the result
2151 * @return a value with the magnitude of {@code magnitude}
2152 * and the sign of {@code sign}.
2153 * @since 1.6
2154 */
2155 public static double copySign(double magnitude, double sign) {
2156 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
2157 (DoubleConsts.SIGN_BIT_MASK)) |
2158 (Double.doubleToRawLongBits(magnitude) &
2159 (DoubleConsts.EXP_BIT_MASK |
2160 DoubleConsts.SIGNIF_BIT_MASK)));
2161 }
2162
2163 /**
2164 * Returns the first floating-point argument with the sign of the
2165 * second floating-point argument. Note that unlike the {@link
2166 * StrictMath#copySign(float, float) StrictMath.copySign}
2167 * method, this method does not require NaN {@code sign}
2168 * arguments to be treated as positive values; implementations are
2169 * permitted to treat some NaN arguments as positive and other NaN
2170 * arguments as negative to allow greater performance.
2171 *
2172 * @param magnitude the parameter providing the magnitude of the result
2173 * @param sign the parameter providing the sign of the result
2174 * @return a value with the magnitude of {@code magnitude}
2175 * and the sign of {@code sign}.
2176 * @since 1.6
2177 */
2178 public static float copySign(float magnitude, float sign) {
2179 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
2180 (FloatConsts.SIGN_BIT_MASK)) |
2181 (Float.floatToRawIntBits(magnitude) &
2182 (FloatConsts.EXP_BIT_MASK |
2183 FloatConsts.SIGNIF_BIT_MASK)));
2184 }
2185
2186 /**
2187 * Returns the unbiased exponent used in the representation of a
2188 * {@code float}. Special cases:
2189 *
2190 * <ul>
2191 * <li>If the argument is NaN or infinite, then the result is
2192 * {@link Float#MAX_EXPONENT} + 1.
2193 * <li>If the argument is zero or subnormal, then the result is
2194 * {@link Float#MIN_EXPONENT} -1.
2195 * </ul>
2196 * @param f a {@code float} value
2197 * @return the unbiased exponent of the argument
2198 * @since 1.6
2199 */
2200 public static int getExponent(float f) {
2201 /*
2202 * Bitwise convert f to integer, mask out exponent bits, shift
2203 * to the right and then subtract out float's bias adjust to
2204 * get true exponent value
2205 */
2206 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
2207 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
2208 }
2209
2210 /**
2211 * Returns the unbiased exponent used in the representation of a
2212 * {@code double}. Special cases:
2213 *
2214 * <ul>
2215 * <li>If the argument is NaN or infinite, then the result is
2216 * {@link Double#MAX_EXPONENT} + 1.
2217 * <li>If the argument is zero or subnormal, then the result is
2218 * {@link Double#MIN_EXPONENT} -1.
2219 * </ul>
2220 * @param d a {@code double} value
2221 * @return the unbiased exponent of the argument
2222 * @since 1.6
2223 */
2224 public static int getExponent(double d) {
2225 /*
2226 * Bitwise convert d to long, mask out exponent bits, shift
2227 * to the right and then subtract out double's bias adjust to
2228 * get true exponent value.
2229 */
2230 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
2231 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
2232 }
2233
2234 /**
2235 * Returns the floating-point number adjacent to the first
2236 * argument in the direction of the second argument. If both
2237 * arguments compare as equal the second argument is returned.
2238 *
2239 * <p>
2240 * Special cases:
2241 * <ul>
2242 * <li> If either argument is a NaN, then NaN is returned.
2243 *
2244 * <li> If both arguments are signed zeros, {@code direction}
2245 * is returned unchanged (as implied by the requirement of
2246 * returning the second argument if the arguments compare as
2247 * equal).
2248 *
2249 * <li> If {@code start} is
2250 * ±{@link Double#MIN_VALUE} and {@code direction}
2251 * has a value such that the result should have a smaller
2252 * magnitude, then a zero with the same sign as {@code start}
2253 * is returned.
2254 *
2255 * <li> If {@code start} is infinite and
2256 * {@code direction} has a value such that the result should
2257 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
2258 * same sign as {@code start} is returned.
2259 *
2260 * <li> If {@code start} is equal to ±
2261 * {@link Double#MAX_VALUE} and {@code direction} has a
2262 * value such that the result should have a larger magnitude, an
2263 * infinity with same sign as {@code start} is returned.
2264 * </ul>
2265 *
2266 * @param start starting floating-point value
2267 * @param direction value indicating which of
2268 * {@code start}'s neighbors or {@code start} should
2269 * be returned
2270 * @return The floating-point number adjacent to {@code start} in the
2271 * direction of {@code direction}.
2272 * @since 1.6
2273 */
2274 public static double nextAfter(double start, double direction) {
2275 /*
2276 * The cases:
2277 *
2278 * nextAfter(+infinity, 0) == MAX_VALUE
2279 * nextAfter(+infinity, +infinity) == +infinity
2280 * nextAfter(-infinity, 0) == -MAX_VALUE
2281 * nextAfter(-infinity, -infinity) == -infinity
2282 *
2283 * are naturally handled without any additional testing
2284 */
2285
2286 /*
2287 * IEEE 754 floating-point numbers are lexicographically
2288 * ordered if treated as signed-magnitude integers.
2289 * Since Java's integers are two's complement,
2290 * incrementing the two's complement representation of a
2291 * logically negative floating-point value *decrements*
2292 * the signed-magnitude representation. Therefore, when
2293 * the integer representation of a floating-point value
2294 * is negative, the adjustment to the representation is in
2295 * the opposite direction from what would initially be expected.
2296 */
2297
2298 // Branch to descending case first as it is more costly than ascending
2299 // case due to start != 0.0d conditional.
2300 if (start > direction) { // descending
2301 if (start != 0.0d) {
2302 final long transducer = Double.doubleToRawLongBits(start);
2303 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
2304 } else { // start == 0.0d && direction < 0.0d
2305 return -Double.MIN_VALUE;
2306 }
2307 } else if (start < direction) { // ascending
2308 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2309 // then bitwise convert start to integer.
2310 final long transducer = Double.doubleToRawLongBits(start + 0.0d);
2311 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2312 } else if (start == direction) {
2313 return direction;
2314 } else { // isNaN(start) || isNaN(direction)
2315 return start + direction;
2316 }
2317 }
2318
2319 /**
2320 * Returns the floating-point number adjacent to the first
2321 * argument in the direction of the second argument. If both
2322 * arguments compare as equal a value equivalent to the second argument
2323 * is returned.
2324 *
2325 * <p>
2326 * Special cases:
2327 * <ul>
2328 * <li> If either argument is a NaN, then NaN is returned.
2329 *
2330 * <li> If both arguments are signed zeros, a value equivalent
2331 * to {@code direction} is returned.
2332 *
2333 * <li> If {@code start} is
2334 * ±{@link Float#MIN_VALUE} and {@code direction}
2335 * has a value such that the result should have a smaller
2336 * magnitude, then a zero with the same sign as {@code start}
2337 * is returned.
2338 *
2339 * <li> If {@code start} is infinite and
2340 * {@code direction} has a value such that the result should
2341 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
2342 * same sign as {@code start} is returned.
2343 *
2344 * <li> If {@code start} is equal to ±
2345 * {@link Float#MAX_VALUE} and {@code direction} has a
2346 * value such that the result should have a larger magnitude, an
2347 * infinity with same sign as {@code start} is returned.
2348 * </ul>
2349 *
2350 * @param start starting floating-point value
2351 * @param direction value indicating which of
2352 * {@code start}'s neighbors or {@code start} should
2353 * be returned
2354 * @return The floating-point number adjacent to {@code start} in the
2355 * direction of {@code direction}.
2356 * @since 1.6
2357 */
2358 public static float nextAfter(float start, double direction) {
2359 /*
2360 * The cases:
2361 *
2362 * nextAfter(+infinity, 0) == MAX_VALUE
2363 * nextAfter(+infinity, +infinity) == +infinity
2364 * nextAfter(-infinity, 0) == -MAX_VALUE
2365 * nextAfter(-infinity, -infinity) == -infinity
2366 *
2367 * are naturally handled without any additional testing
2368 */
2369
2370 /*
2371 * IEEE 754 floating-point numbers are lexicographically
2372 * ordered if treated as signed-magnitude integers.
2373 * Since Java's integers are two's complement,
2374 * incrementing the two's complement representation of a
2375 * logically negative floating-point value *decrements*
2376 * the signed-magnitude representation. Therefore, when
2377 * the integer representation of a floating-point value
2378 * is negative, the adjustment to the representation is in
2379 * the opposite direction from what would initially be expected.
2380 */
2381
2382 // Branch to descending case first as it is more costly than ascending
2383 // case due to start != 0.0f conditional.
2384 if (start > direction) { // descending
2385 if (start != 0.0f) {
2386 final int transducer = Float.floatToRawIntBits(start);
2387 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2388 } else { // start == 0.0f && direction < 0.0f
2389 return -Float.MIN_VALUE;
2390 }
2391 } else if (start < direction) { // ascending
2392 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2393 // then bitwise convert start to integer.
2394 final int transducer = Float.floatToRawIntBits(start + 0.0f);
2395 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2396 } else if (start == direction) {
2397 return (float)direction;
2398 } else { // isNaN(start) || isNaN(direction)
2399 return start + (float)direction;
2400 }
2401 }
2402
2403 /**
2404 * Returns the floating-point value adjacent to {@code d} in
2405 * the direction of positive infinity. This method is
2406 * semantically equivalent to {@code nextAfter(d,
2407 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2408 * implementation may run faster than its equivalent
2409 * {@code nextAfter} call.
2410 *
2411 * <p>Special Cases:
2412 * <ul>
2413 * <li> If the argument is NaN, the result is NaN.
2414 *
2415 * <li> If the argument is positive infinity, the result is
2416 * positive infinity.
2417 *
2418 * <li> If the argument is zero, the result is
2419 * {@link Double#MIN_VALUE}
2420 *
2421 * </ul>
2422 *
2423 * @param d starting floating-point value
2424 * @return The adjacent floating-point value closer to positive
2425 * infinity.
2426 * @since 1.6
2427 */
2428 public static double nextUp(double d) {
2429 // Use a single conditional and handle the likely cases first.
2430 if (d < Double.POSITIVE_INFINITY) {
2431 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2432 final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2433 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2434 } else { // d is NaN or +Infinity
2435 return d;
2436 }
2437 }
2438
2439 /**
2440 * Returns the floating-point value adjacent to {@code f} in
2441 * the direction of positive infinity. This method is
2442 * semantically equivalent to {@code nextAfter(f,
2443 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2444 * implementation may run faster than its equivalent
2445 * {@code nextAfter} call.
2446 *
2447 * <p>Special Cases:
2448 * <ul>
2449 * <li> If the argument is NaN, the result is NaN.
2450 *
2451 * <li> If the argument is positive infinity, the result is
2452 * positive infinity.
2453 *
2454 * <li> If the argument is zero, the result is
2455 * {@link Float#MIN_VALUE}
2456 *
2457 * </ul>
2458 *
2459 * @param f starting floating-point value
2460 * @return The adjacent floating-point value closer to positive
2461 * infinity.
2462 * @since 1.6
2463 */
2464 public static float nextUp(float f) {
2465 // Use a single conditional and handle the likely cases first.
2466 if (f < Float.POSITIVE_INFINITY) {
2467 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2468 final int transducer = Float.floatToRawIntBits(f + 0.0F);
2469 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2470 } else { // f is NaN or +Infinity
2471 return f;
2472 }
2473 }
2474
2475 /**
2476 * Returns the floating-point value adjacent to {@code d} in
2477 * the direction of negative infinity. This method is
2478 * semantically equivalent to {@code nextAfter(d,
2479 * Double.NEGATIVE_INFINITY)}; however, a
2480 * {@code nextDown} implementation may run faster than its
2481 * equivalent {@code nextAfter} call.
2482 *
2483 * <p>Special Cases:
2484 * <ul>
2485 * <li> If the argument is NaN, the result is NaN.
2486 *
2487 * <li> If the argument is negative infinity, the result is
2488 * negative infinity.
2489 *
2490 * <li> If the argument is zero, the result is
2491 * {@code -Double.MIN_VALUE}
2492 *
2493 * </ul>
2494 *
2495 * @param d starting floating-point value
2496 * @return The adjacent floating-point value closer to negative
2497 * infinity.
2498 * @since 1.8
2499 */
2500 public static double nextDown(double d) {
2501 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2502 return d;
2503 else {
2504 if (d == 0.0)
2505 return -Double.MIN_VALUE;
2506 else
2507 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2508 ((d > 0.0d)?-1L:+1L));
2509 }
2510 }
2511
2512 /**
2513 * Returns the floating-point value adjacent to {@code f} in
2514 * the direction of negative infinity. This method is
2515 * semantically equivalent to {@code nextAfter(f,
2516 * Float.NEGATIVE_INFINITY)}; however, a
2517 * {@code nextDown} implementation may run faster than its
2518 * equivalent {@code nextAfter} call.
2519 *
2520 * <p>Special Cases:
2521 * <ul>
2522 * <li> If the argument is NaN, the result is NaN.
2523 *
2524 * <li> If the argument is negative infinity, the result is
2525 * negative infinity.
2526 *
2527 * <li> If the argument is zero, the result is
2528 * {@code -Float.MIN_VALUE}
2529 *
2530 * </ul>
2531 *
2532 * @param f starting floating-point value
2533 * @return The adjacent floating-point value closer to negative
2534 * infinity.
2535 * @since 1.8
2536 */
2537 public static float nextDown(float f) {
2538 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2539 return f;
2540 else {
2541 if (f == 0.0f)
2542 return -Float.MIN_VALUE;
2543 else
2544 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2545 ((f > 0.0f)?-1:+1));
2546 }
2547 }
2548
2549 /**
2550 * Returns {@code d} ×
2551 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2552 * by a single correctly rounded floating-point multiply to a
2553 * member of the double value set. See the Java
2554 * Language Specification for a discussion of floating-point
2555 * value sets. If the exponent of the result is between {@link
2556 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2557 * answer is calculated exactly. If the exponent of the result
2558 * would be larger than {@code Double.MAX_EXPONENT}, an
2559 * infinity is returned. Note that if the result is subnormal,
2560 * precision may be lost; that is, when {@code scalb(x, n)}
2561 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2562 * <i>x</i>. When the result is non-NaN, the result has the same
2563 * sign as {@code d}.
2564 *
2565 * <p>Special cases:
2566 * <ul>
2567 * <li> If the first argument is NaN, NaN is returned.
2568 * <li> If the first argument is infinite, then an infinity of the
2569 * same sign is returned.
2570 * <li> If the first argument is zero, then a zero of the same
2571 * sign is returned.
2572 * </ul>
2573 *
2574 * @param d number to be scaled by a power of two.
2575 * @param scaleFactor power of 2 used to scale {@code d}
2576 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2577 * @since 1.6
2578 */
2579 public static double scalb(double d, int scaleFactor) {
2580 /*
2581 * This method does not need to be declared strictfp to
2582 * compute the same correct result on all platforms. When
2583 * scaling up, it does not matter what order the
2584 * multiply-store operations are done; the result will be
2585 * finite or overflow regardless of the operation ordering.
2586 * However, to get the correct result when scaling down, a
2587 * particular ordering must be used.
2588 *
2589 * When scaling down, the multiply-store operations are
2590 * sequenced so that it is not possible for two consecutive
2591 * multiply-stores to return subnormal results. If one
2592 * multiply-store result is subnormal, the next multiply will
2593 * round it away to zero. This is done by first multiplying
2594 * by 2 ^ (scaleFactor % n) and then multiplying several
2595 * times by 2^n as needed where n is the exponent of number
2596 * that is a covenient power of two. In this way, at most one
2597 * real rounding error occurs. If the double value set is
2598 * being used exclusively, the rounding will occur on a
2599 * multiply. If the double-extended-exponent value set is
2600 * being used, the products will (perhaps) be exact but the
2601 * stores to d are guaranteed to round to the double value
2602 * set.
2603 *
2604 * It is _not_ a valid implementation to first multiply d by
2605 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2606 * MIN_EXPONENT) since even in a strictfp program double
2607 * rounding on underflow could occur; e.g. if the scaleFactor
2608 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2609 * little less than -(MIN_EXPONENT - n), meaning the final
2610 * result would be subnormal.
2611 *
2612 * Since exact reproducibility of this method can be achieved
2613 * without any undue performance burden, there is no
2614 * compelling reason to allow double rounding on underflow in
2615 * scalb.
2616 */
2617
2618 // magnitude of a power of two so large that scaling a finite
2619 // nonzero value by it would be guaranteed to over or
2620 // underflow; due to rounding, scaling down takes an
2621 // additional power of two which is reflected here
2622 final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
2623 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2624 int exp_adjust = 0;
2625 int scale_increment = 0;
2626 double exp_delta = Double.NaN;
2627
2628 // Make sure scaling factor is in a reasonable range
2629
2630 if(scaleFactor < 0) {
2631 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2632 scale_increment = -512;
2633 exp_delta = twoToTheDoubleScaleDown;
2634 }
2635 else {
2636 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2637 scale_increment = 512;
2638 exp_delta = twoToTheDoubleScaleUp;
2639 }
2640
2641 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2642 // technique from "Hacker's Delight" section 10-2.
2643 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2644 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2645
2646 d *= powerOfTwoD(exp_adjust);
2647 scaleFactor -= exp_adjust;
2648
2649 while(scaleFactor != 0) {
2650 d *= exp_delta;
2651 scaleFactor -= scale_increment;
2652 }
2653 return d;
2654 }
2655
2656 /**
2657 * Returns {@code f} ×
2658 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2659 * by a single correctly rounded floating-point multiply to a
2660 * member of the float value set. See the Java
2661 * Language Specification for a discussion of floating-point
2662 * value sets. If the exponent of the result is between {@link
2663 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2664 * answer is calculated exactly. If the exponent of the result
2665 * would be larger than {@code Float.MAX_EXPONENT}, an
2666 * infinity is returned. Note that if the result is subnormal,
2667 * precision may be lost; that is, when {@code scalb(x, n)}
2668 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2669 * <i>x</i>. When the result is non-NaN, the result has the same
2670 * sign as {@code f}.
2671 *
2672 * <p>Special cases:
2673 * <ul>
2674 * <li> If the first argument is NaN, NaN is returned.
2675 * <li> If the first argument is infinite, then an infinity of the
2676 * same sign is returned.
2677 * <li> If the first argument is zero, then a zero of the same
2678 * sign is returned.
2679 * </ul>
2680 *
2681 * @param f number to be scaled by a power of two.
2682 * @param scaleFactor power of 2 used to scale {@code f}
2683 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2684 * @since 1.6
2685 */
2686 public static float scalb(float f, int scaleFactor) {
2687 // magnitude of a power of two so large that scaling a finite
2688 // nonzero value by it would be guaranteed to over or
2689 // underflow; due to rounding, scaling down takes an
2690 // additional power of two which is reflected here
2691 final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
2692 FloatConsts.SIGNIFICAND_WIDTH + 1;
2693
2694 // Make sure scaling factor is in a reasonable range
2695 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2696
2697 /*
2698 * Since + MAX_SCALE for float fits well within the double
2699 * exponent range and + float -> double conversion is exact
2700 * the multiplication below will be exact. Therefore, the
2701 * rounding that occurs when the double product is cast to
2702 * float will be the correctly rounded float result. Since
2703 * all operations other than the final multiply will be exact,
2704 * it is not necessary to declare this method strictfp.
2705 */
2706 return (float)((double)f*powerOfTwoD(scaleFactor));
2707 }
2708
2709 // Constants used in scalb
2710 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2711 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2712
2713 /**
2714 * Returns a floating-point power of two in the normal range.
2715 */
2716 static double powerOfTwoD(int n) {
2717 assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
2718 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2719 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2720 & DoubleConsts.EXP_BIT_MASK);
2721 }
2722
2723 /**
2724 * Returns a floating-point power of two in the normal range.
2725 */
2726 static float powerOfTwoF(int n) {
2727 assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
2728 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2729 (FloatConsts.SIGNIFICAND_WIDTH-1))
2730 & FloatConsts.EXP_BIT_MASK);
2731 }
2732 }