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