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