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