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 JDK1.0
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 * Returns the trigonometric sine of an angle. Special cases:
127 * <ul><li>If the argument is NaN or an infinity, then the
128 * result is NaN.
129 * <li>If the argument is zero, then the result is a zero with the
130 * same sign as the argument.</ul>
131 *
132 * <p>The computed result must be within 1 ulp of the exact result.
133 * Results must be semi-monotonic.
134 *
135 * @param a an angle, in radians.
136 * @return the sine of the argument.
137 */
138 public static double sin(double a) {
139 return StrictMath.sin(a); // default impl. delegates to StrictMath
140 }
141
142 /**
143 * Returns the trigonometric cosine of an angle. Special cases:
144 * <ul><li>If the argument is NaN or an infinity, then the
145 * result is NaN.</ul>
146 *
147 * <p>The computed result must be within 1 ulp of the exact result.
148 * Results must be semi-monotonic.
149 *
150 * @param a an angle, in radians.
151 * @return the cosine of the argument.
152 */
153 public static double cos(double a) {
154 return StrictMath.cos(a); // default impl. delegates to StrictMath
155 }
156
157 /**
158 * Returns the trigonometric tangent of an angle. Special cases:
159 * <ul><li>If the argument is NaN or an infinity, then the result
160 * is NaN.
161 * <li>If the argument is zero, then the result is a zero with the
162 * same sign as the argument.</ul>
163 *
164 * <p>The computed result must be within 1 ulp of the exact result.
165 * Results must be semi-monotonic.
166 *
167 * @param a an angle, in radians.
168 * @return the tangent of the argument.
169 */
170 public static double tan(double a) {
171 return StrictMath.tan(a); // default impl. delegates to StrictMath
172 }
173
174 /**
175 * Returns the arc sine of a value; the returned angle is in the
176 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
177 * <ul><li>If the argument is NaN or its absolute value is greater
178 * than 1, then the result is NaN.
179 * <li>If the argument is zero, then the result is a zero with the
180 * same sign as the argument.</ul>
181 *
182 * <p>The computed result must be within 1 ulp of the exact result.
183 * Results must be semi-monotonic.
184 *
185 * @param a the value whose arc sine is to be returned.
186 * @return the arc sine of the argument.
187 */
188 public static double asin(double a) {
189 return StrictMath.asin(a); // default impl. delegates to StrictMath
190 }
191
192 /**
193 * Returns the arc cosine of a value; the returned angle is in the
194 * range 0.0 through <i>pi</i>. Special case:
195 * <ul><li>If the argument is NaN or its absolute value is greater
196 * than 1, then the result is NaN.</ul>
197 *
198 * <p>The computed result must be within 1 ulp of the exact result.
199 * Results must be semi-monotonic.
200 *
201 * @param a the value whose arc cosine is to be returned.
202 * @return the arc cosine of the argument.
203 */
204 public static double acos(double a) {
205 return StrictMath.acos(a); // default impl. delegates to StrictMath
206 }
207
208 /**
209 * Returns the arc tangent of a value; the returned angle is in the
210 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
211 * <ul><li>If the argument is NaN, then the result is NaN.
212 * <li>If the argument is zero, then the result is a zero with the
213 * same sign as the argument.</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 tangent is to be returned.
219 * @return the arc tangent of the argument.
220 */
221 public static double atan(double a) {
222 return StrictMath.atan(a); // default impl. delegates to StrictMath
223 }
224
225 /**
226 * Converts an angle measured in degrees to an approximately
227 * equivalent angle measured in radians. The conversion from
228 * degrees to radians is generally inexact.
229 *
230 * @param angdeg an angle, in degrees
231 * @return the measurement of the angle {@code angdeg}
232 * in radians.
233 * @since 1.2
234 */
235 public static double toRadians(double angdeg) {
236 return angdeg / 180.0 * PI;
237 }
238
239 /**
240 * Converts an angle measured in radians to an approximately
241 * equivalent angle measured in degrees. The conversion from
242 * radians to degrees is generally inexact; users should
243 * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
244 * equal {@code 0.0}.
245 *
246 * @param angrad an angle, in radians
247 * @return the measurement of the angle {@code angrad}
248 * in degrees.
249 * @since 1.2
250 */
251 public static double toDegrees(double angrad) {
252 return angrad * 180.0 / PI;
253 }
254
255 /**
256 * Returns Euler's number <i>e</i> raised to the power of a
257 * {@code double} value. Special cases:
258 * <ul><li>If the argument is NaN, the result is NaN.
259 * <li>If the argument is positive infinity, then the result is
260 * positive infinity.
261 * <li>If the argument is negative infinity, then the result is
262 * positive zero.</ul>
263 *
264 * <p>The computed result must be within 1 ulp of the exact result.
265 * Results must be semi-monotonic.
266 *
267 * @param a the exponent to raise <i>e</i> to.
268 * @return the value <i>e</i><sup>{@code a}</sup>,
269 * where <i>e</i> is the base of the natural logarithms.
270 */
271 public static double exp(double a) {
272 return StrictMath.exp(a); // default impl. delegates to StrictMath
273 }
274
275 /**
276 * Returns the natural logarithm (base <i>e</i>) of a {@code double}
277 * value. Special cases:
278 * <ul><li>If the argument is NaN or less than zero, then the result
279 * is NaN.
280 * <li>If the argument is positive infinity, then the result is
281 * positive infinity.
282 * <li>If the argument is positive zero or negative zero, then the
283 * result is negative infinity.</ul>
284 *
285 * <p>The computed result must be within 1 ulp of the exact result.
286 * Results must be semi-monotonic.
287 *
288 * @param a a value
289 * @return the value ln {@code a}, the natural logarithm of
290 * {@code a}.
291 */
292 public static double log(double a) {
293 return StrictMath.log(a); // default impl. delegates to StrictMath
294 }
295
296 /**
297 * Returns the base 10 logarithm of a {@code double} value.
298 * Special cases:
299 *
300 * <ul><li>If the argument is NaN or less than zero, then the result
301 * is NaN.
302 * <li>If the argument is positive infinity, then the result is
303 * positive infinity.
304 * <li>If the argument is positive zero or negative zero, then the
305 * result is negative infinity.
306 * <li> If the argument is equal to 10<sup><i>n</i></sup> for
307 * integer <i>n</i>, then the result is <i>n</i>.
308 * </ul>
309 *
310 * <p>The computed result must be within 1 ulp of the exact result.
311 * Results must be semi-monotonic.
312 *
313 * @param a a value
314 * @return the base 10 logarithm of {@code a}.
315 * @since 1.5
316 */
317 public static double log10(double a) {
318 return StrictMath.log10(a); // default impl. delegates to StrictMath
319 }
320
321 /**
322 * Returns the correctly rounded positive square root of a
323 * {@code double} value.
324 * Special cases:
325 * <ul><li>If the argument is NaN or less than zero, then the result
326 * is NaN.
327 * <li>If the argument is positive infinity, then the result is positive
328 * infinity.
329 * <li>If the argument is positive zero or negative zero, then the
330 * result is the same as the argument.</ul>
331 * Otherwise, the result is the {@code double} value closest to
332 * the true mathematical square root of the argument value.
333 *
334 * @param a a value.
335 * @return the positive square root of {@code a}.
336 * If the argument is NaN or less than zero, the result is NaN.
337 */
338 public static double sqrt(double a) {
339 return StrictMath.sqrt(a); // default impl. delegates to StrictMath
340 // Note that hardware sqrt instructions
341 // frequently can be directly used by JITs
342 // and should be much faster than doing
343 // Math.sqrt in software.
344 }
345
346
347 /**
348 * Returns the cube root of a {@code double} value. For
349 * positive finite {@code x}, {@code cbrt(-x) ==
350 * -cbrt(x)}; that is, the cube root of a negative value is
351 * the negative of the cube root of that value's magnitude.
352 *
353 * Special cases:
354 *
355 * <ul>
356 *
357 * <li>If the argument is NaN, then the result is NaN.
358 *
359 * <li>If the argument is infinite, then the result is an infinity
360 * with the same sign as the argument.
361 *
362 * <li>If the argument is zero, then the result is a zero with the
363 * same sign as the argument.
364 *
365 * </ul>
366 *
367 * <p>The computed result must be within 1 ulp of the exact result.
368 *
369 * @param a a value.
370 * @return the cube root of {@code a}.
371 * @since 1.5
372 */
373 public static double cbrt(double a) {
374 return StrictMath.cbrt(a);
375 }
376
377 /**
378 * Computes the remainder operation on two arguments as prescribed
379 * by the IEEE 754 standard.
380 * The remainder value is mathematically equal to
381 * <code>f1 - f2</code> × <i>n</i>,
382 * where <i>n</i> is the mathematical integer closest to the exact
383 * mathematical value of the quotient {@code f1/f2}, and if two
384 * mathematical integers are equally close to {@code f1/f2},
385 * then <i>n</i> is the integer that is even. If the remainder is
386 * zero, its sign is the same as the sign of the first argument.
387 * Special cases:
388 * <ul><li>If either argument is NaN, or the first argument is infinite,
389 * or the second argument is positive zero or negative zero, then the
390 * result is NaN.
391 * <li>If the first argument is finite and the second argument is
392 * infinite, then the result is the same as the first argument.</ul>
393 *
394 * @param f1 the dividend.
395 * @param f2 the divisor.
396 * @return the remainder when {@code f1} is divided by
397 * {@code f2}.
398 */
399 public static double IEEEremainder(double f1, double f2) {
400 return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
401 }
402
403 /**
404 * Returns the smallest (closest to negative infinity)
405 * {@code double} value that is greater than or equal to the
406 * argument and is equal to a mathematical integer. Special cases:
407 * <ul><li>If the argument value is already equal to a
408 * mathematical integer, then the result is the same as the
409 * argument. <li>If the argument is NaN or an infinity or
410 * positive zero or negative zero, then the result is the same as
411 * the argument. <li>If the argument value is less than zero but
412 * greater than -1.0, then the result is negative zero.</ul> Note
413 * that the value of {@code Math.ceil(x)} is exactly the
414 * value of {@code -Math.floor(-x)}.
415 *
416 *
417 * @param a a value.
418 * @return the smallest (closest to negative infinity)
419 * floating-point value that is greater than or equal to
420 * the argument and is equal to a mathematical integer.
421 */
422 public static double ceil(double a) {
423 return StrictMath.ceil(a); // default impl. delegates to StrictMath
424 }
425
426 /**
427 * Returns the largest (closest to positive infinity)
428 * {@code double} value that is less than or equal to the
429 * argument and is equal to a mathematical integer. Special cases:
430 * <ul><li>If the argument value is already equal to a
431 * mathematical integer, then the result is the same as the
432 * argument. <li>If the argument is NaN or an infinity or
433 * positive zero or negative zero, then the result is the same as
434 * the argument.</ul>
435 *
436 * @param a a value.
437 * @return the largest (closest to positive infinity)
438 * floating-point value that less than or equal to the argument
439 * and is equal to a mathematical integer.
440 */
441 public static double floor(double a) {
442 return StrictMath.floor(a); // default impl. delegates to StrictMath
443 }
444
445 /**
446 * Returns the {@code double} value that is closest in value
447 * to the argument and is equal to a mathematical integer. If two
448 * {@code double} values that are mathematical integers are
449 * equally close, the result is the integer value that is
450 * even. Special cases:
451 * <ul><li>If the argument value is already equal to a mathematical
452 * integer, then the result is the same as the argument.
453 * <li>If the argument is NaN or an infinity or positive zero or negative
454 * zero, then the result is the same as the argument.</ul>
455 *
456 * @param a a {@code double} value.
457 * @return the closest floating-point value to {@code a} that is
458 * equal to a mathematical integer.
459 */
460 public static double rint(double a) {
461 return StrictMath.rint(a); // default impl. delegates to StrictMath
462 }
463
464 /**
465 * Returns the angle <i>theta</i> from the conversion of rectangular
466 * coordinates ({@code x}, {@code y}) to polar
467 * coordinates (r, <i>theta</i>).
468 * This method computes the phase <i>theta</i> by computing an arc tangent
469 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
470 * cases:
471 * <ul><li>If either argument is NaN, then the result is NaN.
472 * <li>If the first argument is positive zero and the second argument
473 * is positive, or the first argument is positive and finite and the
474 * second argument is positive infinity, then the result is positive
475 * zero.
476 * <li>If the first argument is negative zero and the second argument
477 * is positive, or the first argument is negative and finite and the
478 * second argument is positive infinity, then the result is negative zero.
479 * <li>If the first argument is positive zero and the second argument
480 * is negative, or the first argument is positive and finite and the
481 * second argument is negative infinity, then the result is the
482 * {@code double} value closest to <i>pi</i>.
483 * <li>If the first argument is negative zero and the second argument
484 * is negative, or the first argument is negative and finite and the
485 * second argument is negative infinity, then the result is the
486 * {@code double} value closest to -<i>pi</i>.
487 * <li>If the first argument is positive and the second argument is
488 * positive zero or negative zero, or the first argument is positive
489 * infinity and the second argument is finite, then the result is the
490 * {@code double} value closest to <i>pi</i>/2.
491 * <li>If the first argument is negative and the second argument is
492 * positive zero or negative zero, or the first argument is negative
493 * infinity and the second argument is finite, then the result is the
494 * {@code double} value closest to -<i>pi</i>/2.
495 * <li>If both arguments are positive infinity, then the result is the
496 * {@code double} value closest to <i>pi</i>/4.
497 * <li>If the first argument is positive infinity and the second argument
498 * is negative infinity, then the result is the {@code double}
499 * value closest to 3*<i>pi</i>/4.
500 * <li>If the first argument is negative infinity and the second argument
501 * is positive infinity, then the result is the {@code double} value
502 * closest to -<i>pi</i>/4.
503 * <li>If both arguments are negative infinity, then the result is the
504 * {@code double} value closest to -3*<i>pi</i>/4.</ul>
505 *
506 * <p>The computed result must be within 2 ulps of the exact result.
507 * Results must be semi-monotonic.
508 *
509 * @param y the ordinate coordinate
510 * @param x the abscissa coordinate
511 * @return the <i>theta</i> component of the point
512 * (<i>r</i>, <i>theta</i>)
513 * in polar coordinates that corresponds to the point
514 * (<i>x</i>, <i>y</i>) in Cartesian coordinates.
515 */
516 public static double atan2(double y, double x) {
517 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
518 }
519
520 /**
521 * Returns the value of the first argument raised to the power of the
522 * second argument. Special cases:
523 *
524 * <ul><li>If the second argument is positive or negative zero, then the
525 * result is 1.0.
526 * <li>If the second argument is 1.0, then the result is the same as the
527 * first argument.
528 * <li>If the second argument is NaN, then the result is NaN.
529 * <li>If the first argument is NaN and the second argument is nonzero,
530 * then the result is NaN.
531 *
532 * <li>If
533 * <ul>
534 * <li>the absolute value of the first argument is greater than 1
535 * and the second argument is positive infinity, or
536 * <li>the absolute value of the first argument is less than 1 and
537 * the second argument is negative infinity,
538 * </ul>
539 * then the result is positive infinity.
540 *
541 * <li>If
542 * <ul>
543 * <li>the absolute value of the first argument is greater than 1 and
544 * the second argument is negative infinity, or
545 * <li>the absolute value of the
546 * first argument is less than 1 and the second argument is positive
547 * infinity,
548 * </ul>
549 * then the result is positive zero.
550 *
551 * <li>If the absolute value of the first argument equals 1 and the
552 * second argument is infinite, then the result is NaN.
553 *
554 * <li>If
555 * <ul>
556 * <li>the first argument is positive zero and the second argument
557 * is greater than zero, or
558 * <li>the first argument is positive infinity and the second
559 * argument is less than zero,
560 * </ul>
561 * then the result is positive zero.
562 *
563 * <li>If
564 * <ul>
565 * <li>the first argument is positive zero and the second argument
566 * is less than zero, or
567 * <li>the first argument is positive infinity and the second
568 * argument is greater than zero,
569 * </ul>
570 * then the result is positive infinity.
571 *
572 * <li>If
573 * <ul>
574 * <li>the first argument is negative zero and the second argument
575 * is greater than zero but not a finite odd integer, or
576 * <li>the first argument is negative infinity and the second
577 * argument is less than zero but not a finite odd integer,
578 * </ul>
579 * then the result is positive zero.
580 *
581 * <li>If
582 * <ul>
583 * <li>the first argument is negative zero and the second argument
584 * is a positive finite odd integer, or
585 * <li>the first argument is negative infinity and the second
586 * argument is a negative finite odd integer,
587 * </ul>
588 * then the result is negative zero.
589 *
590 * <li>If
591 * <ul>
592 * <li>the first argument is negative zero and the second argument
593 * is less than zero but not a finite odd integer, or
594 * <li>the first argument is negative infinity and the second
595 * argument is greater than zero but not a finite odd integer,
596 * </ul>
597 * then the result is positive infinity.
598 *
599 * <li>If
600 * <ul>
601 * <li>the first argument is negative zero and the second argument
602 * is a negative finite odd integer, or
603 * <li>the first argument is negative infinity and the second
604 * argument is a positive finite odd integer,
605 * </ul>
606 * then the result is negative infinity.
607 *
608 * <li>If the first argument is finite and less than zero
609 * <ul>
610 * <li> if the second argument is a finite even integer, the
611 * result is equal to the result of raising the absolute value of
612 * the first argument to the power of the second argument
613 *
614 * <li>if the second argument is a finite odd integer, the result
615 * is equal to the negative of the result of raising the absolute
616 * value of the first argument to the power of the second
617 * argument
618 *
619 * <li>if the second argument is finite and not an integer, then
620 * the result is NaN.
621 * </ul>
622 *
623 * <li>If both arguments are integers, then the result is exactly equal
624 * to the mathematical result of raising the first argument to the power
625 * of the second argument if that result can in fact be represented
626 * exactly as a {@code double} value.</ul>
627 *
628 * <p>(In the foregoing descriptions, a floating-point value is
629 * considered to be an integer if and only if it is finite and a
630 * fixed point of the method {@link #ceil ceil} or,
631 * equivalently, a fixed point of the method {@link #floor
632 * floor}. A value is a fixed point of a one-argument
633 * method if and only if the result of applying the method to the
634 * value is equal to the value.)
635 *
636 * <p>The computed result must be within 1 ulp of the exact result.
637 * Results must be semi-monotonic.
638 *
639 * @param a the base.
640 * @param b the exponent.
641 * @return the value {@code a}<sup>{@code b}</sup>.
642 */
643 public static double pow(double a, double b) {
644 return StrictMath.pow(a, b); // default impl. delegates to StrictMath
645 }
646
647 /**
648 * Returns the closest {@code int} to the argument, with ties
649 * rounding to positive infinity.
650 *
651 * <p>
652 * Special cases:
653 * <ul><li>If the argument is NaN, the result is 0.
654 * <li>If the argument is negative infinity or any value less than or
655 * equal to the value of {@code Integer.MIN_VALUE}, the result is
656 * equal to the value of {@code Integer.MIN_VALUE}.
657 * <li>If the argument is positive infinity or any value greater than or
658 * equal to the value of {@code Integer.MAX_VALUE}, the result is
659 * equal to the value of {@code Integer.MAX_VALUE}.</ul>
660 *
661 * @param a a floating-point value to be rounded to an integer.
662 * @return the value of the argument rounded to the nearest
663 * {@code int} value.
664 * @see java.lang.Integer#MAX_VALUE
665 * @see java.lang.Integer#MIN_VALUE
666 */
667 public static int round(float a) {
668 int intBits = Float.floatToRawIntBits(a);
669 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
670 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
671 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
672 + FloatConsts.EXP_BIAS) - biasedExp;
673 if ((shift & -32) == 0) { // shift >= 0 && shift < 32
674 // a is a finite number such that pow(2,-32) <= ulp(a) < 1
675 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
676 | (FloatConsts.SIGNIF_BIT_MASK + 1));
677 if (intBits < 0) {
678 r = -r;
679 }
680 // In the comments below each Java expression evaluates to the value
681 // the corresponding mathematical expression:
682 // (r) evaluates to a / ulp(a)
683 // (r >> shift) evaluates to floor(a * 2)
684 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
685 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
686 return ((r >> shift) + 1) >> 1;
687 } else {
688 // a is either
689 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
690 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
691 // - an infinity or NaN
692 return (int) a;
693 }
694 }
695
696 /**
697 * Returns the closest {@code long} to the argument, with ties
698 * rounding to positive infinity.
699 *
700 * <p>Special cases:
701 * <ul><li>If the argument is NaN, the result is 0.
702 * <li>If the argument is negative infinity or any value less than or
703 * equal to the value of {@code Long.MIN_VALUE}, the result is
704 * equal to the value of {@code Long.MIN_VALUE}.
705 * <li>If the argument is positive infinity or any value greater than or
706 * equal to the value of {@code Long.MAX_VALUE}, the result is
707 * equal to the value of {@code Long.MAX_VALUE}.</ul>
708 *
709 * @param a a floating-point value to be rounded to a
710 * {@code long}.
711 * @return the value of the argument rounded to the nearest
712 * {@code long} value.
713 * @see java.lang.Long#MAX_VALUE
714 * @see java.lang.Long#MIN_VALUE
715 */
716 public static long round(double a) {
717 long longBits = Double.doubleToRawLongBits(a);
718 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
719 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
720 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
721 + DoubleConsts.EXP_BIAS) - biasedExp;
722 if ((shift & -64) == 0) { // shift >= 0 && shift < 64
723 // a is a finite number such that pow(2,-64) <= ulp(a) < 1
724 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
725 | (DoubleConsts.SIGNIF_BIT_MASK + 1));
726 if (longBits < 0) {
727 r = -r;
728 }
729 // In the comments below each Java expression evaluates to the value
730 // the corresponding mathematical expression:
731 // (r) evaluates to a / ulp(a)
732 // (r >> shift) evaluates to floor(a * 2)
733 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
734 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
735 return ((r >> shift) + 1) >> 1;
736 } else {
737 // a is either
738 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
739 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
740 // - an infinity or NaN
741 return (long) a;
742 }
743 }
744
745 private static final class RandomNumberGeneratorHolder {
746 static final Random randomNumberGenerator = new Random();
747 }
748
749 /**
750 * Returns a {@code double} value with a positive sign, greater
751 * than or equal to {@code 0.0} and less than {@code 1.0}.
752 * Returned values are chosen pseudorandomly with (approximately)
753 * uniform distribution from that range.
754 *
755 * <p>When this method is first called, it creates a single new
756 * pseudorandom-number generator, exactly as if by the expression
757 *
758 * <blockquote>{@code new java.util.Random()}</blockquote>
759 *
760 * This new pseudorandom-number generator is used thereafter for
761 * all calls to this method and is used nowhere else.
762 *
763 * <p>This method is properly synchronized to allow correct use by
764 * more than one thread. However, if many threads need to generate
765 * pseudorandom numbers at a great rate, it may reduce contention
766 * for each thread to have its own pseudorandom-number generator.
767 *
768 * @apiNote
769 * As the largest {@code double} value less than {@code 1.0}
770 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
771 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
772 *
773 * <blockquote><pre>{@code
774 * double f = Math.random()/Math.nextDown(1.0);
775 * double x = x1*(1.0 - f) + x2*f;
776 * }</pre></blockquote>
777 *
778 * @return a pseudorandom {@code double} greater than or equal
779 * to {@code 0.0} and less than {@code 1.0}.
780 * @see #nextDown(double)
781 * @see Random#nextDouble()
782 */
783 public static double random() {
784 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
785 }
786
787 /**
788 * Returns the sum of its arguments,
789 * throwing an exception if the result overflows an {@code int}.
790 *
791 * @param x the first value
792 * @param y the second value
793 * @return the result
794 * @throws ArithmeticException if the result overflows an int
795 * @since 1.8
796 */
797 public static int addExact(int x, int y) {
798 int r = x + y;
799 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
800 if (((x ^ r) & (y ^ r)) < 0) {
801 throw new ArithmeticException("integer overflow");
802 }
803 return r;
804 }
805
806 /**
807 * Returns the sum of its arguments,
808 * throwing an exception if the result overflows a {@code long}.
809 *
810 * @param x the first value
811 * @param y the second value
812 * @return the result
813 * @throws ArithmeticException if the result overflows a long
814 * @since 1.8
815 */
816 public static long addExact(long x, long y) {
817 long r = x + y;
818 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
819 if (((x ^ r) & (y ^ r)) < 0) {
820 throw new ArithmeticException("long overflow");
821 }
822 return r;
823 }
824
825 /**
826 * Returns the difference of the arguments,
827 * throwing an exception if the result overflows an {@code int}.
828 *
829 * @param x the first value
830 * @param y the second value to subtract from the first
831 * @return the result
832 * @throws ArithmeticException if the result overflows an int
833 * @since 1.8
834 */
835 public static int subtractExact(int x, int y) {
836 int r = x - y;
837 // HD 2-12 Overflow iff the arguments have different signs and
838 // the sign of the result is different than the sign of x
839 if (((x ^ y) & (x ^ r)) < 0) {
840 throw new ArithmeticException("integer overflow");
841 }
842 return r;
843 }
844
845 /**
846 * Returns the difference of the arguments,
847 * throwing an exception if the result overflows a {@code long}.
848 *
849 * @param x the first value
850 * @param y the second value to subtract from the first
851 * @return the result
852 * @throws ArithmeticException if the result overflows a long
853 * @since 1.8
854 */
855 public static long subtractExact(long x, long y) {
856 long r = x - y;
857 // HD 2-12 Overflow iff the arguments have different signs and
858 // the sign of the result is different than the sign of x
859 if (((x ^ y) & (x ^ r)) < 0) {
860 throw new ArithmeticException("long overflow");
861 }
862 return r;
863 }
864
865 /**
866 * Returns the product of the arguments,
867 * throwing an exception if the result overflows an {@code int}.
868 *
869 * @param x the first value
870 * @param y the second value
871 * @return the result
872 * @throws ArithmeticException if the result overflows an int
873 * @since 1.8
874 */
875 public static int multiplyExact(int x, int y) {
876 long r = (long)x * (long)y;
877 if ((int)r != r) {
878 throw new ArithmeticException("integer overflow");
879 }
880 return (int)r;
881 }
882
883 /**
884 * Returns the product of the arguments,
885 * throwing an exception if the result overflows a {@code long}.
886 *
887 * @param x the first value
888 * @param y the second value
889 * @return the result
890 * @throws ArithmeticException if the result overflows a long
891 * @since 1.8
892 */
893 public static long multiplyExact(long x, long y) {
894 long r = x * y;
895 long ax = Math.abs(x);
896 long ay = Math.abs(y);
897 if (((ax | ay) >>> 31 != 0)) {
898 // Some bits greater than 2^31 that might cause overflow
899 // Check the result using the divide operator
900 // and check for the special case of Long.MIN_VALUE * -1
901 if (((y != 0) && (r / y != x)) ||
902 (x == Long.MIN_VALUE && y == -1)) {
903 throw new ArithmeticException("long overflow");
904 }
905 }
906 return r;
907 }
908
909 /**
910 * Returns the argument incremented by one, throwing an exception if the
911 * result overflows an {@code int}.
912 *
913 * @param a the value to increment
914 * @return the result
915 * @throws ArithmeticException if the result overflows an int
916 * @since 1.8
917 */
918 public static int incrementExact(int a) {
919 if (a == Integer.MAX_VALUE) {
920 throw new ArithmeticException("integer overflow");
921 }
922
923 return a + 1;
924 }
925
926 /**
927 * Returns the argument incremented by one, throwing an exception if the
928 * result overflows a {@code long}.
929 *
930 * @param a the value to increment
931 * @return the result
932 * @throws ArithmeticException if the result overflows a long
933 * @since 1.8
934 */
935 public static long incrementExact(long a) {
936 if (a == Long.MAX_VALUE) {
937 throw new ArithmeticException("long overflow");
938 }
939
940 return a + 1L;
941 }
942
943 /**
944 * Returns the argument decremented by one, throwing an exception if the
945 * result overflows an {@code int}.
946 *
947 * @param a the value to decrement
948 * @return the result
949 * @throws ArithmeticException if the result overflows an int
950 * @since 1.8
951 */
952 public static int decrementExact(int a) {
953 if (a == Integer.MIN_VALUE) {
954 throw new ArithmeticException("integer overflow");
955 }
956
957 return a - 1;
958 }
959
960 /**
961 * Returns the argument decremented by one, throwing an exception if the
962 * result overflows a {@code long}.
963 *
964 * @param a the value to decrement
965 * @return the result
966 * @throws ArithmeticException if the result overflows a long
967 * @since 1.8
968 */
969 public static long decrementExact(long a) {
970 if (a == Long.MIN_VALUE) {
971 throw new ArithmeticException("long overflow");
972 }
973
974 return a - 1L;
975 }
976
977 /**
978 * Returns the negation of the argument, throwing an exception if the
979 * result overflows an {@code int}.
980 *
981 * @param a the value to negate
982 * @return the result
983 * @throws ArithmeticException if the result overflows an int
984 * @since 1.8
985 */
986 public static int negateExact(int a) {
987 if (a == Integer.MIN_VALUE) {
988 throw new ArithmeticException("integer overflow");
989 }
990
991 return -a;
992 }
993
994 /**
995 * Returns the negation of the argument, throwing an exception if the
996 * result overflows a {@code long}.
997 *
998 * @param a the value to negate
999 * @return the result
1000 * @throws ArithmeticException if the result overflows a long
1001 * @since 1.8
1002 */
1003 public static long negateExact(long a) {
1004 if (a == Long.MIN_VALUE) {
1005 throw new ArithmeticException("long overflow");
1006 }
1007
1008 return -a;
1009 }
1010
1011 /**
1012 * Returns the value of the {@code long} argument;
1013 * throwing an exception if the value overflows an {@code int}.
1014 *
1015 * @param value the long value
1016 * @return the argument as an int
1017 * @throws ArithmeticException if the {@code argument} overflows an int
1018 * @since 1.8
1019 */
1020 public static int toIntExact(long value) {
1021 if ((int)value != value) {
1022 throw new ArithmeticException("integer overflow");
1023 }
1024 return (int)value;
1025 }
1026
1027 /**
1028 * Returns the largest (closest to positive infinity)
1029 * {@code int} value that is less than or equal to the algebraic quotient.
1030 * There is one special case, if the dividend is the
1031 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1032 * then integer overflow occurs and
1033 * the result is equal to the {@code Integer.MIN_VALUE}.
1034 * <p>
1035 * Normal integer division operates under the round to zero rounding mode
1036 * (truncation). This operation instead acts under the round toward
1037 * negative infinity (floor) rounding mode.
1038 * The floor rounding mode gives different results than truncation
1039 * when the exact result is negative.
1040 * <ul>
1041 * <li>If the signs of the arguments are the same, the results of
1042 * {@code floorDiv} and the {@code /} operator are the same. <br>
1043 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1044 * <li>If the signs of the arguments are different, the quotient is negative and
1045 * {@code floorDiv} returns the integer less than or equal to the quotient
1046 * and the {@code /} operator returns the integer closest to zero.<br>
1047 * For example, {@code floorDiv(-4, 3) == -2},
1048 * whereas {@code (-4 / 3) == -1}.
1049 * </li>
1050 * </ul>
1051 * <p>
1052 *
1053 * @param x the dividend
1054 * @param y the divisor
1055 * @return the largest (closest to positive infinity)
1056 * {@code int} value that is less than or equal to the algebraic quotient.
1057 * @throws ArithmeticException if the divisor {@code y} is zero
1058 * @see #floorMod(int, int)
1059 * @see #floor(double)
1060 * @since 1.8
1061 */
1062 public static int floorDiv(int x, int y) {
1063 int r = x / y;
1064 // if the signs are different and modulo not zero, round down
1065 if ((x ^ y) < 0 && (r * y != x)) {
1066 r--;
1067 }
1068 return r;
1069 }
1070
1071 /**
1072 * Returns the largest (closest to positive infinity)
1073 * {@code long} value that is less than or equal to the algebraic quotient.
1074 * There is one special case, if the dividend is the
1075 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1076 * then integer overflow occurs and
1077 * the result is equal to the {@code Long.MIN_VALUE}.
1078 * <p>
1079 * Normal integer division operates under the round to zero rounding mode
1080 * (truncation). This operation instead acts under the round toward
1081 * negative infinity (floor) rounding mode.
1082 * The floor rounding mode gives different results than truncation
1083 * when the exact result is negative.
1084 * <p>
1085 * For examples, see {@link #floorDiv(int, int)}.
1086 *
1087 * @param x the dividend
1088 * @param y the divisor
1089 * @return the largest (closest to positive infinity)
1090 * {@code long} value that is less than or equal to the algebraic quotient.
1091 * @throws ArithmeticException if the divisor {@code y} is zero
1092 * @see #floorMod(long, long)
1093 * @see #floor(double)
1094 * @since 1.8
1095 */
1096 public static long floorDiv(long x, long y) {
1097 long r = x / y;
1098 // if the signs are different and modulo not zero, round down
1099 if ((x ^ y) < 0 && (r * y != x)) {
1100 r--;
1101 }
1102 return r;
1103 }
1104
1105 /**
1106 * Returns the floor modulus of the {@code int} arguments.
1107 * <p>
1108 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1109 * has the same sign as the divisor {@code y}, and
1110 * is in the range of {@code -abs(y) < r < +abs(y)}.
1111 *
1112 * <p>
1113 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1114 * <ul>
1115 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1116 * </ul>
1117 * <p>
1118 * The difference in values between {@code floorMod} and
1119 * the {@code %} operator is due to the difference between
1120 * {@code floorDiv} that returns the integer less than or equal to the quotient
1121 * and the {@code /} operator that returns the integer closest to zero.
1122 * <p>
1123 * Examples:
1124 * <ul>
1125 * <li>If the signs of the arguments are the same, the results
1126 * of {@code floorMod} and the {@code %} operator are the same. <br>
1127 * <ul>
1128 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li>
1129 * </ul>
1130 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1131 * <ul>
1132 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li>
1133 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li>
1134 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li>
1135 * </ul>
1136 * </li>
1137 * </ul>
1138 * <p>
1139 * If the signs of arguments are unknown and a positive modulus
1140 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1141 *
1142 * @param x the dividend
1143 * @param y the divisor
1144 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1145 * @throws ArithmeticException if the divisor {@code y} is zero
1146 * @see #floorDiv(int, int)
1147 * @since 1.8
1148 */
1149 public static int floorMod(int x, int y) {
1150 int r = x - floorDiv(x, y) * y;
1151 return r;
1152 }
1153
1154 /**
1155 * Returns the floor modulus of the {@code long} arguments.
1156 * <p>
1157 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1158 * has the same sign as the divisor {@code y}, and
1159 * is in the range of {@code -abs(y) < r < +abs(y)}.
1160 *
1161 * <p>
1162 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1163 * <ul>
1164 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1165 * </ul>
1166 * <p>
1167 * For examples, see {@link #floorMod(int, int)}.
1168 *
1169 * @param x the dividend
1170 * @param y the divisor
1171 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1172 * @throws ArithmeticException if the divisor {@code y} is zero
1173 * @see #floorDiv(long, long)
1174 * @since 1.8
1175 */
1176 public static long floorMod(long x, long y) {
1177 return x - floorDiv(x, y) * y;
1178 }
1179
1180 /**
1181 * Returns the absolute value of an {@code int} value.
1182 * If the argument is not negative, the argument is returned.
1183 * If the argument is negative, the negation of the argument is returned.
1184 *
1185 * <p>Note that if the argument is equal to the value of
1186 * {@link Integer#MIN_VALUE}, the most negative representable
1187 * {@code int} value, the result is that same value, which is
1188 * negative.
1189 *
1190 * @param a the argument whose absolute value is to be determined
1191 * @return the absolute value of the argument.
1192 */
1193 public static int abs(int a) {
1194 return (a < 0) ? -a : a;
1195 }
1196
1197 /**
1198 * Returns the absolute value of a {@code long} value.
1199 * If the argument is not negative, the argument is returned.
1200 * If the argument is negative, the negation of the argument is returned.
1201 *
1202 * <p>Note that if the argument is equal to the value of
1203 * {@link Long#MIN_VALUE}, the most negative representable
1204 * {@code long} value, the result is that same value, which
1205 * is negative.
1206 *
1207 * @param a the argument whose absolute value is to be determined
1208 * @return the absolute value of the argument.
1209 */
1210 public static long abs(long a) {
1211 return (a < 0) ? -a : a;
1212 }
1213
1214 /**
1215 * Returns the absolute value of a {@code float} value.
1216 * If the argument is not negative, the argument is returned.
1217 * If the argument is negative, the negation of the argument is returned.
1218 * Special cases:
1219 * <ul><li>If the argument is positive zero or negative zero, the
1220 * result is positive zero.
1221 * <li>If the argument is infinite, the result is positive infinity.
1222 * <li>If the argument is NaN, the result is NaN.</ul>
1223 * In other words, the result is the same as the value of the expression:
1224 * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1225 *
1226 * @param a the argument whose absolute value is to be determined
1227 * @return the absolute value of the argument.
1228 */
1229 public static float abs(float a) {
1230 return (a <= 0.0F) ? 0.0F - a : a;
1231 }
1232
1233 /**
1234 * Returns the absolute value of a {@code double} value.
1235 * If the argument is not negative, the argument is returned.
1236 * If the argument is negative, the negation of the argument is returned.
1237 * Special cases:
1238 * <ul><li>If the argument is positive zero or negative zero, the result
1239 * is positive zero.
1240 * <li>If the argument is infinite, the result is positive infinity.
1241 * <li>If the argument is NaN, the result is NaN.</ul>
1242 * In other words, the result is the same as the value of the expression:
1243 * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1244 *
1245 * @param a the argument whose absolute value is to be determined
1246 * @return the absolute value of the argument.
1247 */
1248 public static double abs(double a) {
1249 return (a <= 0.0D) ? 0.0D - a : a;
1250 }
1251
1252 /**
1253 * Returns the greater of two {@code int} values. That is, the
1254 * result is the argument closer to the value of
1255 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1256 * the result is that same value.
1257 *
1258 * @param a an argument.
1259 * @param b another argument.
1260 * @return the larger of {@code a} and {@code b}.
1261 */
1262 public static int max(int a, int b) {
1263 return (a >= b) ? a : b;
1264 }
1265
1266 /**
1267 * Returns the greater of two {@code long} values. That is, the
1268 * result is the argument closer to the value of
1269 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1270 * the result is that same value.
1271 *
1272 * @param a an argument.
1273 * @param b another argument.
1274 * @return the larger of {@code a} and {@code b}.
1275 */
1276 public static long max(long a, long b) {
1277 return (a >= b) ? a : b;
1278 }
1279
1280 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1281 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1282 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1283
1284 /**
1285 * Returns the greater of two {@code float} values. That is,
1286 * the result is the argument closer to positive infinity. If the
1287 * arguments have the same value, the result is that same
1288 * value. If either value is NaN, then the result is NaN. Unlike
1289 * the numerical comparison operators, this method considers
1290 * negative zero to be strictly smaller than positive zero. If one
1291 * argument is positive zero and the other negative zero, the
1292 * result is positive zero.
1293 *
1294 * @param a an argument.
1295 * @param b another argument.
1296 * @return the larger of {@code a} and {@code b}.
1297 */
1298 public static float max(float a, float b) {
1299 if (a != a)
1300 return a; // a is NaN
1301 if ((a == 0.0f) &&
1302 (b == 0.0f) &&
1303 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1304 // Raw conversion ok since NaN can't map to -0.0.
1305 return b;
1306 }
1307 return (a >= b) ? a : b;
1308 }
1309
1310 /**
1311 * Returns the greater of two {@code double} values. That
1312 * is, the result is the argument closer to positive infinity. If
1313 * the arguments have the same value, the result is that same
1314 * value. If either value is NaN, then the result is NaN. Unlike
1315 * the numerical comparison operators, this method considers
1316 * negative zero to be strictly smaller than positive zero. If one
1317 * argument is positive zero and the other negative zero, the
1318 * result is positive zero.
1319 *
1320 * @param a an argument.
1321 * @param b another argument.
1322 * @return the larger of {@code a} and {@code b}.
1323 */
1324 public static double max(double a, double b) {
1325 if (a != a)
1326 return a; // a is NaN
1327 if ((a == 0.0d) &&
1328 (b == 0.0d) &&
1329 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1330 // Raw conversion ok since NaN can't map to -0.0.
1331 return b;
1332 }
1333 return (a >= b) ? a : b;
1334 }
1335
1336 /**
1337 * Returns the smaller of two {@code int} values. That is,
1338 * the result the argument closer to the value of
1339 * {@link Integer#MIN_VALUE}. If the arguments have the same
1340 * value, the result is that same value.
1341 *
1342 * @param a an argument.
1343 * @param b another argument.
1344 * @return the smaller of {@code a} and {@code b}.
1345 */
1346 public static int min(int a, int b) {
1347 return (a <= b) ? a : b;
1348 }
1349
1350 /**
1351 * Returns the smaller of two {@code long} values. That is,
1352 * the result is the argument closer to the value of
1353 * {@link Long#MIN_VALUE}. If the arguments have the same
1354 * value, the result is that same value.
1355 *
1356 * @param a an argument.
1357 * @param b another argument.
1358 * @return the smaller of {@code a} and {@code b}.
1359 */
1360 public static long min(long a, long b) {
1361 return (a <= b) ? a : b;
1362 }
1363
1364 /**
1365 * Returns the smaller of two {@code float} values. That is,
1366 * the result is the value closer to negative infinity. If the
1367 * arguments have the same value, the result is that same
1368 * value. If either value is NaN, then the result is NaN. Unlike
1369 * the numerical comparison operators, this method considers
1370 * negative zero to be strictly smaller than positive zero. If
1371 * one argument is positive zero and the other is negative zero,
1372 * the result is negative zero.
1373 *
1374 * @param a an argument.
1375 * @param b another argument.
1376 * @return the smaller of {@code a} and {@code b}.
1377 */
1378 public static float min(float a, float b) {
1379 if (a != a)
1380 return a; // a is NaN
1381 if ((a == 0.0f) &&
1382 (b == 0.0f) &&
1383 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1384 // Raw conversion ok since NaN can't map to -0.0.
1385 return b;
1386 }
1387 return (a <= b) ? a : b;
1388 }
1389
1390 /**
1391 * Returns the smaller of two {@code double} values. That
1392 * is, the result is the value closer to negative infinity. If the
1393 * arguments have the same value, the result is that same
1394 * value. If either value is NaN, then the result is NaN. Unlike
1395 * the numerical comparison operators, this method considers
1396 * negative zero to be strictly smaller than positive zero. If one
1397 * argument is positive zero and the other is negative zero, the
1398 * result is negative zero.
1399 *
1400 * @param a an argument.
1401 * @param b another argument.
1402 * @return the smaller of {@code a} and {@code b}.
1403 */
1404 public static double min(double a, double b) {
1405 if (a != a)
1406 return a; // a is NaN
1407 if ((a == 0.0d) &&
1408 (b == 0.0d) &&
1409 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1410 // Raw conversion ok since NaN can't map to -0.0.
1411 return b;
1412 }
1413 return (a <= b) ? a : b;
1414 }
1415
1416 /**
1417 * Returns the size of an ulp of the argument. An ulp, unit in
1418 * the last place, of a {@code double} value is the positive
1419 * distance between this floating-point value and the {@code
1420 * double} value next larger in magnitude. Note that for non-NaN
1421 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1422 *
1423 * <p>Special Cases:
1424 * <ul>
1425 * <li> If the argument is NaN, then the result is NaN.
1426 * <li> If the argument is positive or negative infinity, then the
1427 * result is positive infinity.
1428 * <li> If the argument is positive or negative zero, then the result is
1429 * {@code Double.MIN_VALUE}.
1430 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1431 * the result is equal to 2<sup>971</sup>.
1432 * </ul>
1433 *
1434 * @param d the floating-point value whose ulp is to be returned
1435 * @return the size of an ulp of the argument
1436 * @author Joseph D. Darcy
1437 * @since 1.5
1438 */
1439 public static double ulp(double d) {
1440 int exp = getExponent(d);
1441
1442 switch(exp) {
1443 case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
1444 return Math.abs(d);
1445
1446 case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
1447 return Double.MIN_VALUE;
1448
1449 default:
1450 assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
1451
1452 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1453 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1454 if (exp >= DoubleConsts.MIN_EXPONENT) {
1455 return powerOfTwoD(exp);
1456 }
1457 else {
1458 // return a subnormal result; left shift integer
1459 // representation of Double.MIN_VALUE appropriate
1460 // number of positions
1461 return Double.longBitsToDouble(1L <<
1462 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1463 }
1464 }
1465 }
1466
1467 /**
1468 * Returns the size of an ulp of the argument. An ulp, unit in
1469 * the last place, of a {@code float} value is the positive
1470 * distance between this floating-point value and the {@code
1471 * float} value next larger in magnitude. Note that for non-NaN
1472 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1473 *
1474 * <p>Special Cases:
1475 * <ul>
1476 * <li> If the argument is NaN, then the result is NaN.
1477 * <li> If the argument is positive or negative infinity, then the
1478 * result is positive infinity.
1479 * <li> If the argument is positive or negative zero, then the result is
1480 * {@code Float.MIN_VALUE}.
1481 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1482 * the result is equal to 2<sup>104</sup>.
1483 * </ul>
1484 *
1485 * @param f the floating-point value whose ulp is to be returned
1486 * @return the size of an ulp of the argument
1487 * @author Joseph D. Darcy
1488 * @since 1.5
1489 */
1490 public static float ulp(float f) {
1491 int exp = getExponent(f);
1492
1493 switch(exp) {
1494 case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
1495 return Math.abs(f);
1496
1497 case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
1498 return FloatConsts.MIN_VALUE;
1499
1500 default:
1501 assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1502
1503 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1504 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1505 if (exp >= FloatConsts.MIN_EXPONENT) {
1506 return powerOfTwoF(exp);
1507 }
1508 else {
1509 // return a subnormal result; left shift integer
1510 // representation of FloatConsts.MIN_VALUE appropriate
1511 // number of positions
1512 return Float.intBitsToFloat(1 <<
1513 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1514 }
1515 }
1516 }
1517
1518 /**
1519 * Returns the signum function of the argument; zero if the argument
1520 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1521 * argument is less than zero.
1522 *
1523 * <p>Special Cases:
1524 * <ul>
1525 * <li> If the argument is NaN, then the result is NaN.
1526 * <li> If the argument is positive zero or negative zero, then the
1527 * result is the same as the argument.
1528 * </ul>
1529 *
1530 * @param d the floating-point value whose signum is to be returned
1531 * @return the signum function of the argument
1532 * @author Joseph D. Darcy
1533 * @since 1.5
1534 */
1535 public static double signum(double d) {
1536 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1537 }
1538
1539 /**
1540 * Returns the signum function of the argument; zero if the argument
1541 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1542 * argument is less than zero.
1543 *
1544 * <p>Special Cases:
1545 * <ul>
1546 * <li> If the argument is NaN, then the result is NaN.
1547 * <li> If the argument is positive zero or negative zero, then the
1548 * result is the same as the argument.
1549 * </ul>
1550 *
1551 * @param f the floating-point value whose signum is to be returned
1552 * @return the signum function of the argument
1553 * @author Joseph D. Darcy
1554 * @since 1.5
1555 */
1556 public static float signum(float f) {
1557 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1558 }
1559
1560 /**
1561 * Returns the hyperbolic sine of a {@code double} value.
1562 * The hyperbolic sine of <i>x</i> is defined to be
1563 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1564 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1565 *
1566 * <p>Special cases:
1567 * <ul>
1568 *
1569 * <li>If the argument is NaN, then the result is NaN.
1570 *
1571 * <li>If the argument is infinite, then the result is an infinity
1572 * with the same sign as the argument.
1573 *
1574 * <li>If the argument is zero, then the result is a zero with the
1575 * same sign as the argument.
1576 *
1577 * </ul>
1578 *
1579 * <p>The computed result must be within 2.5 ulps of the exact result.
1580 *
1581 * @param x The number whose hyperbolic sine is to be returned.
1582 * @return The hyperbolic sine of {@code x}.
1583 * @since 1.5
1584 */
1585 public static double sinh(double x) {
1586 return StrictMath.sinh(x);
1587 }
1588
1589 /**
1590 * Returns the hyperbolic cosine of a {@code double} value.
1591 * The hyperbolic cosine of <i>x</i> is defined to be
1592 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1593 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1594 *
1595 * <p>Special cases:
1596 * <ul>
1597 *
1598 * <li>If the argument is NaN, then the result is NaN.
1599 *
1600 * <li>If the argument is infinite, then the result is positive
1601 * infinity.
1602 *
1603 * <li>If the argument is zero, then the result is {@code 1.0}.
1604 *
1605 * </ul>
1606 *
1607 * <p>The computed result must be within 2.5 ulps of the exact result.
1608 *
1609 * @param x The number whose hyperbolic cosine is to be returned.
1610 * @return The hyperbolic cosine of {@code x}.
1611 * @since 1.5
1612 */
1613 public static double cosh(double x) {
1614 return StrictMath.cosh(x);
1615 }
1616
1617 /**
1618 * Returns the hyperbolic tangent of a {@code double} value.
1619 * The hyperbolic tangent of <i>x</i> is defined to be
1620 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
1621 * in other words, {@linkplain Math#sinh
1622 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
1623 * that the absolute value of the exact tanh is always less than
1624 * 1.
1625 *
1626 * <p>Special cases:
1627 * <ul>
1628 *
1629 * <li>If the argument is NaN, then the result is NaN.
1630 *
1631 * <li>If the argument is zero, then the result is a zero with the
1632 * same sign as the argument.
1633 *
1634 * <li>If the argument is positive infinity, then the result is
1635 * {@code +1.0}.
1636 *
1637 * <li>If the argument is negative infinity, then the result is
1638 * {@code -1.0}.
1639 *
1640 * </ul>
1641 *
1642 * <p>The computed result must be within 2.5 ulps of the exact result.
1643 * The result of {@code tanh} for any finite input must have
1644 * an absolute value less than or equal to 1. Note that once the
1645 * exact result of tanh is within 1/2 of an ulp of the limit value
1646 * of ±1, correctly signed ±{@code 1.0} should
1647 * be returned.
1648 *
1649 * @param x The number whose hyperbolic tangent is to be returned.
1650 * @return The hyperbolic tangent of {@code x}.
1651 * @since 1.5
1652 */
1653 public static double tanh(double x) {
1654 return StrictMath.tanh(x);
1655 }
1656
1657 /**
1658 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1659 * without intermediate overflow or underflow.
1660 *
1661 * <p>Special cases:
1662 * <ul>
1663 *
1664 * <li> If either argument is infinite, then the result
1665 * is positive infinity.
1666 *
1667 * <li> If either argument is NaN and neither argument is infinite,
1668 * then the result is NaN.
1669 *
1670 * </ul>
1671 *
1672 * <p>The computed result must be within 1 ulp of the exact
1673 * result. If one parameter is held constant, the results must be
1674 * semi-monotonic in the other parameter.
1675 *
1676 * @param x a value
1677 * @param y a value
1678 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1679 * without intermediate overflow or underflow
1680 * @since 1.5
1681 */
1682 public static double hypot(double x, double y) {
1683 return StrictMath.hypot(x, y);
1684 }
1685
1686 /**
1687 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
1688 * <i>x</i> near 0, the exact sum of
1689 * {@code expm1(x)} + 1 is much closer to the true
1690 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1691 *
1692 * <p>Special cases:
1693 * <ul>
1694 * <li>If the argument is NaN, the result is NaN.
1695 *
1696 * <li>If the argument is positive infinity, then the result is
1697 * positive infinity.
1698 *
1699 * <li>If the argument is negative infinity, then the result is
1700 * -1.0.
1701 *
1702 * <li>If the argument is zero, then the result is a zero with the
1703 * same sign as the argument.
1704 *
1705 * </ul>
1706 *
1707 * <p>The computed result must be within 1 ulp of the exact result.
1708 * Results must be semi-monotonic. The result of
1709 * {@code expm1} for any finite input must be greater than or
1710 * equal to {@code -1.0}. Note that once the exact result of
1711 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
1712 * ulp of the limit value -1, {@code -1.0} should be
1713 * returned.
1714 *
1715 * @param x the exponent to raise <i>e</i> to in the computation of
1716 * <i>e</i><sup>{@code x}</sup> -1.
1717 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
1718 * @since 1.5
1719 */
1720 public static double expm1(double x) {
1721 return StrictMath.expm1(x);
1722 }
1723
1724 /**
1725 * Returns the natural logarithm of the sum of the argument and 1.
1726 * Note that for small values {@code x}, the result of
1727 * {@code log1p(x)} is much closer to the true result of ln(1
1728 * + {@code x}) than the floating-point evaluation of
1729 * {@code log(1.0+x)}.
1730 *
1731 * <p>Special cases:
1732 *
1733 * <ul>
1734 *
1735 * <li>If the argument is NaN or less than -1, then the result is
1736 * NaN.
1737 *
1738 * <li>If the argument is positive infinity, then the result is
1739 * positive infinity.
1740 *
1741 * <li>If the argument is negative one, then the result is
1742 * negative infinity.
1743 *
1744 * <li>If the argument is zero, then the result is a zero with the
1745 * same sign as the argument.
1746 *
1747 * </ul>
1748 *
1749 * <p>The computed result must be within 1 ulp of the exact result.
1750 * Results must be semi-monotonic.
1751 *
1752 * @param x a value
1753 * @return the value ln({@code x} + 1), the natural
1754 * log of {@code x} + 1
1755 * @since 1.5
1756 */
1757 public static double log1p(double x) {
1758 return StrictMath.log1p(x);
1759 }
1760
1761 /**
1762 * Returns the first floating-point argument with the sign of the
1763 * second floating-point argument. Note that unlike the {@link
1764 * StrictMath#copySign(double, double) StrictMath.copySign}
1765 * method, this method does not require NaN {@code sign}
1766 * arguments to be treated as positive values; implementations are
1767 * permitted to treat some NaN arguments as positive and other NaN
1768 * arguments as negative to allow greater performance.
1769 *
1770 * @param magnitude the parameter providing the magnitude of the result
1771 * @param sign the parameter providing the sign of the result
1772 * @return a value with the magnitude of {@code magnitude}
1773 * and the sign of {@code sign}.
1774 * @since 1.6
1775 */
1776 public static double copySign(double magnitude, double sign) {
1777 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1778 (DoubleConsts.SIGN_BIT_MASK)) |
1779 (Double.doubleToRawLongBits(magnitude) &
1780 (DoubleConsts.EXP_BIT_MASK |
1781 DoubleConsts.SIGNIF_BIT_MASK)));
1782 }
1783
1784 /**
1785 * Returns the first floating-point argument with the sign of the
1786 * second floating-point argument. Note that unlike the {@link
1787 * StrictMath#copySign(float, float) StrictMath.copySign}
1788 * method, this method does not require NaN {@code sign}
1789 * arguments to be treated as positive values; implementations are
1790 * permitted to treat some NaN arguments as positive and other NaN
1791 * arguments as negative to allow greater performance.
1792 *
1793 * @param magnitude the parameter providing the magnitude of the result
1794 * @param sign the parameter providing the sign of the result
1795 * @return a value with the magnitude of {@code magnitude}
1796 * and the sign of {@code sign}.
1797 * @since 1.6
1798 */
1799 public static float copySign(float magnitude, float sign) {
1800 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1801 (FloatConsts.SIGN_BIT_MASK)) |
1802 (Float.floatToRawIntBits(magnitude) &
1803 (FloatConsts.EXP_BIT_MASK |
1804 FloatConsts.SIGNIF_BIT_MASK)));
1805 }
1806
1807 /**
1808 * Returns the unbiased exponent used in the representation of a
1809 * {@code float}. Special cases:
1810 *
1811 * <ul>
1812 * <li>If the argument is NaN or infinite, then the result is
1813 * {@link Float#MAX_EXPONENT} + 1.
1814 * <li>If the argument is zero or subnormal, then the result is
1815 * {@link Float#MIN_EXPONENT} -1.
1816 * </ul>
1817 * @param f a {@code float} value
1818 * @return the unbiased exponent of the argument
1819 * @since 1.6
1820 */
1821 public static int getExponent(float f) {
1822 /*
1823 * Bitwise convert f to integer, mask out exponent bits, shift
1824 * to the right and then subtract out float's bias adjust to
1825 * get true exponent value
1826 */
1827 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1828 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1829 }
1830
1831 /**
1832 * Returns the unbiased exponent used in the representation of a
1833 * {@code double}. Special cases:
1834 *
1835 * <ul>
1836 * <li>If the argument is NaN or infinite, then the result is
1837 * {@link Double#MAX_EXPONENT} + 1.
1838 * <li>If the argument is zero or subnormal, then the result is
1839 * {@link Double#MIN_EXPONENT} -1.
1840 * </ul>
1841 * @param d a {@code double} value
1842 * @return the unbiased exponent of the argument
1843 * @since 1.6
1844 */
1845 public static int getExponent(double d) {
1846 /*
1847 * Bitwise convert d to long, mask out exponent bits, shift
1848 * to the right and then subtract out double's bias adjust to
1849 * get true exponent value.
1850 */
1851 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1852 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1853 }
1854
1855 /**
1856 * Returns the floating-point number adjacent to the first
1857 * argument in the direction of the second argument. If both
1858 * arguments compare as equal the second argument is returned.
1859 *
1860 * <p>
1861 * Special cases:
1862 * <ul>
1863 * <li> If either argument is a NaN, then NaN is returned.
1864 *
1865 * <li> If both arguments are signed zeros, {@code direction}
1866 * is returned unchanged (as implied by the requirement of
1867 * returning the second argument if the arguments compare as
1868 * equal).
1869 *
1870 * <li> If {@code start} is
1871 * ±{@link Double#MIN_VALUE} and {@code direction}
1872 * has a value such that the result should have a smaller
1873 * magnitude, then a zero with the same sign as {@code start}
1874 * is returned.
1875 *
1876 * <li> If {@code start} is infinite and
1877 * {@code direction} has a value such that the result should
1878 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1879 * same sign as {@code start} is returned.
1880 *
1881 * <li> If {@code start} is equal to ±
1882 * {@link Double#MAX_VALUE} and {@code direction} has a
1883 * value such that the result should have a larger magnitude, an
1884 * infinity with same sign as {@code start} is returned.
1885 * </ul>
1886 *
1887 * @param start starting floating-point value
1888 * @param direction value indicating which of
1889 * {@code start}'s neighbors or {@code start} should
1890 * be returned
1891 * @return The floating-point number adjacent to {@code start} in the
1892 * direction of {@code direction}.
1893 * @since 1.6
1894 */
1895 public static double nextAfter(double start, double direction) {
1896 /*
1897 * The cases:
1898 *
1899 * nextAfter(+infinity, 0) == MAX_VALUE
1900 * nextAfter(+infinity, +infinity) == +infinity
1901 * nextAfter(-infinity, 0) == -MAX_VALUE
1902 * nextAfter(-infinity, -infinity) == -infinity
1903 *
1904 * are naturally handled without any additional testing
1905 */
1906
1907 /*
1908 * IEEE 754 floating-point numbers are lexicographically
1909 * ordered if treated as signed-magnitude integers.
1910 * Since Java's integers are two's complement,
1911 * incrementing the two's complement representation of a
1912 * logically negative floating-point value *decrements*
1913 * the signed-magnitude representation. Therefore, when
1914 * the integer representation of a floating-point value
1915 * is negative, the adjustment to the representation is in
1916 * the opposite direction from what would initially be expected.
1917 */
1918
1919 // Branch to descending case first as it is more costly than ascending
1920 // case due to start != 0.0d conditional.
1921 if (start > direction) { // descending
1922 if (start != 0.0d) {
1923 final long transducer = Double.doubleToRawLongBits(start);
1924 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
1925 } else { // start == 0.0d && direction < 0.0d
1926 return -Double.MIN_VALUE;
1927 }
1928 } else if (start < direction) { // ascending
1929 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1930 // then bitwise convert start to integer.
1931 final long transducer = Double.doubleToRawLongBits(start + 0.0d);
1932 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
1933 } else if (start == direction) {
1934 return direction;
1935 } else { // isNaN(start) || isNaN(direction)
1936 return start + direction;
1937 }
1938 }
1939
1940 /**
1941 * Returns the floating-point number adjacent to the first
1942 * argument in the direction of the second argument. If both
1943 * arguments compare as equal a value equivalent to the second argument
1944 * is returned.
1945 *
1946 * <p>
1947 * Special cases:
1948 * <ul>
1949 * <li> If either argument is a NaN, then NaN is returned.
1950 *
1951 * <li> If both arguments are signed zeros, a value equivalent
1952 * to {@code direction} is returned.
1953 *
1954 * <li> If {@code start} is
1955 * ±{@link Float#MIN_VALUE} and {@code direction}
1956 * has a value such that the result should have a smaller
1957 * magnitude, then a zero with the same sign as {@code start}
1958 * is returned.
1959 *
1960 * <li> If {@code start} is infinite and
1961 * {@code direction} has a value such that the result should
1962 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1963 * same sign as {@code start} is returned.
1964 *
1965 * <li> If {@code start} is equal to ±
1966 * {@link Float#MAX_VALUE} and {@code direction} has a
1967 * value such that the result should have a larger magnitude, an
1968 * infinity with same sign as {@code start} is returned.
1969 * </ul>
1970 *
1971 * @param start starting floating-point value
1972 * @param direction value indicating which of
1973 * {@code start}'s neighbors or {@code start} should
1974 * be returned
1975 * @return The floating-point number adjacent to {@code start} in the
1976 * direction of {@code direction}.
1977 * @since 1.6
1978 */
1979 public static float nextAfter(float start, double direction) {
1980 /*
1981 * The cases:
1982 *
1983 * nextAfter(+infinity, 0) == MAX_VALUE
1984 * nextAfter(+infinity, +infinity) == +infinity
1985 * nextAfter(-infinity, 0) == -MAX_VALUE
1986 * nextAfter(-infinity, -infinity) == -infinity
1987 *
1988 * are naturally handled without any additional testing
1989 */
1990
1991 /*
1992 * IEEE 754 floating-point numbers are lexicographically
1993 * ordered if treated as signed-magnitude integers.
1994 * Since Java's integers are two's complement,
1995 * incrementing the two's complement representation of a
1996 * logically negative floating-point value *decrements*
1997 * the signed-magnitude representation. Therefore, when
1998 * the integer representation of a floating-point value
1999 * is negative, the adjustment to the representation is in
2000 * the opposite direction from what would initially be expected.
2001 */
2002
2003 // Branch to descending case first as it is more costly than ascending
2004 // case due to start != 0.0f conditional.
2005 if (start > direction) { // descending
2006 if (start != 0.0f) {
2007 final int transducer = Float.floatToRawIntBits(start);
2008 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2009 } else { // start == 0.0f && direction < 0.0f
2010 return -Float.MIN_VALUE;
2011 }
2012 } else if (start < direction) { // ascending
2013 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2014 // then bitwise convert start to integer.
2015 final int transducer = Float.floatToRawIntBits(start + 0.0f);
2016 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2017 } else if (start == direction) {
2018 return (float)direction;
2019 } else { // isNaN(start) || isNaN(direction)
2020 return start + (float)direction;
2021 }
2022 }
2023
2024 /**
2025 * Returns the floating-point value adjacent to {@code d} in
2026 * the direction of positive infinity. This method is
2027 * semantically equivalent to {@code nextAfter(d,
2028 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2029 * implementation may run faster than its equivalent
2030 * {@code nextAfter} call.
2031 *
2032 * <p>Special Cases:
2033 * <ul>
2034 * <li> If the argument is NaN, the result is NaN.
2035 *
2036 * <li> If the argument is positive infinity, the result is
2037 * positive infinity.
2038 *
2039 * <li> If the argument is zero, the result is
2040 * {@link Double#MIN_VALUE}
2041 *
2042 * </ul>
2043 *
2044 * @param d starting floating-point value
2045 * @return The adjacent floating-point value closer to positive
2046 * infinity.
2047 * @since 1.6
2048 */
2049 public static double nextUp(double d) {
2050 // Use a single conditional and handle the likely cases first.
2051 if (d < Double.POSITIVE_INFINITY) {
2052 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2053 final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2054 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2055 } else { // d is NaN or +Infinity
2056 return d;
2057 }
2058 }
2059
2060 /**
2061 * Returns the floating-point value adjacent to {@code f} in
2062 * the direction of positive infinity. This method is
2063 * semantically equivalent to {@code nextAfter(f,
2064 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2065 * implementation may run faster than its equivalent
2066 * {@code nextAfter} call.
2067 *
2068 * <p>Special Cases:
2069 * <ul>
2070 * <li> If the argument is NaN, the result is NaN.
2071 *
2072 * <li> If the argument is positive infinity, the result is
2073 * positive infinity.
2074 *
2075 * <li> If the argument is zero, the result is
2076 * {@link Float#MIN_VALUE}
2077 *
2078 * </ul>
2079 *
2080 * @param f starting floating-point value
2081 * @return The adjacent floating-point value closer to positive
2082 * infinity.
2083 * @since 1.6
2084 */
2085 public static float nextUp(float f) {
2086 // Use a single conditional and handle the likely cases first.
2087 if (f < Float.POSITIVE_INFINITY) {
2088 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2089 final int transducer = Float.floatToRawIntBits(f + 0.0F);
2090 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2091 } else { // f is NaN or +Infinity
2092 return f;
2093 }
2094 }
2095
2096 /**
2097 * Returns the floating-point value adjacent to {@code d} in
2098 * the direction of negative infinity. This method is
2099 * semantically equivalent to {@code nextAfter(d,
2100 * Double.NEGATIVE_INFINITY)}; however, a
2101 * {@code nextDown} implementation may run faster than its
2102 * equivalent {@code nextAfter} call.
2103 *
2104 * <p>Special Cases:
2105 * <ul>
2106 * <li> If the argument is NaN, the result is NaN.
2107 *
2108 * <li> If the argument is negative infinity, the result is
2109 * negative infinity.
2110 *
2111 * <li> If the argument is zero, the result is
2112 * {@code -Double.MIN_VALUE}
2113 *
2114 * </ul>
2115 *
2116 * @param d starting floating-point value
2117 * @return The adjacent floating-point value closer to negative
2118 * infinity.
2119 * @since 1.8
2120 */
2121 public static double nextDown(double d) {
2122 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2123 return d;
2124 else {
2125 if (d == 0.0)
2126 return -Double.MIN_VALUE;
2127 else
2128 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2129 ((d > 0.0d)?-1L:+1L));
2130 }
2131 }
2132
2133 /**
2134 * Returns the floating-point value adjacent to {@code f} in
2135 * the direction of negative infinity. This method is
2136 * semantically equivalent to {@code nextAfter(f,
2137 * Float.NEGATIVE_INFINITY)}; however, a
2138 * {@code nextDown} implementation may run faster than its
2139 * equivalent {@code nextAfter} call.
2140 *
2141 * <p>Special Cases:
2142 * <ul>
2143 * <li> If the argument is NaN, the result is NaN.
2144 *
2145 * <li> If the argument is negative infinity, the result is
2146 * negative infinity.
2147 *
2148 * <li> If the argument is zero, the result is
2149 * {@code -Float.MIN_VALUE}
2150 *
2151 * </ul>
2152 *
2153 * @param f starting floating-point value
2154 * @return The adjacent floating-point value closer to negative
2155 * infinity.
2156 * @since 1.8
2157 */
2158 public static float nextDown(float f) {
2159 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2160 return f;
2161 else {
2162 if (f == 0.0f)
2163 return -Float.MIN_VALUE;
2164 else
2165 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2166 ((f > 0.0f)?-1:+1));
2167 }
2168 }
2169
2170 /**
2171 * Returns {@code d} ×
2172 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2173 * by a single correctly rounded floating-point multiply to a
2174 * member of the double value set. See the Java
2175 * Language Specification for a discussion of floating-point
2176 * value sets. If the exponent of the result is between {@link
2177 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2178 * answer is calculated exactly. If the exponent of the result
2179 * would be larger than {@code Double.MAX_EXPONENT}, an
2180 * infinity is returned. Note that if the result is subnormal,
2181 * precision may be lost; that is, when {@code scalb(x, n)}
2182 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2183 * <i>x</i>. When the result is non-NaN, the result has the same
2184 * sign as {@code d}.
2185 *
2186 * <p>Special cases:
2187 * <ul>
2188 * <li> If the first argument is NaN, NaN is returned.
2189 * <li> If the first argument is infinite, then an infinity of the
2190 * same sign is returned.
2191 * <li> If the first argument is zero, then a zero of the same
2192 * sign is returned.
2193 * </ul>
2194 *
2195 * @param d number to be scaled by a power of two.
2196 * @param scaleFactor power of 2 used to scale {@code d}
2197 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2198 * @since 1.6
2199 */
2200 public static double scalb(double d, int scaleFactor) {
2201 /*
2202 * This method does not need to be declared strictfp to
2203 * compute the same correct result on all platforms. When
2204 * scaling up, it does not matter what order the
2205 * multiply-store operations are done; the result will be
2206 * finite or overflow regardless of the operation ordering.
2207 * However, to get the correct result when scaling down, a
2208 * particular ordering must be used.
2209 *
2210 * When scaling down, the multiply-store operations are
2211 * sequenced so that it is not possible for two consecutive
2212 * multiply-stores to return subnormal results. If one
2213 * multiply-store result is subnormal, the next multiply will
2214 * round it away to zero. This is done by first multiplying
2215 * by 2 ^ (scaleFactor % n) and then multiplying several
2216 * times by by 2^n as needed where n is the exponent of number
2217 * that is a covenient power of two. In this way, at most one
2218 * real rounding error occurs. If the double value set is
2219 * being used exclusively, the rounding will occur on a
2220 * multiply. If the double-extended-exponent value set is
2221 * being used, the products will (perhaps) be exact but the
2222 * stores to d are guaranteed to round to the double value
2223 * set.
2224 *
2225 * It is _not_ a valid implementation to first multiply d by
2226 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2227 * MIN_EXPONENT) since even in a strictfp program double
2228 * rounding on underflow could occur; e.g. if the scaleFactor
2229 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2230 * little less than -(MIN_EXPONENT - n), meaning the final
2231 * result would be subnormal.
2232 *
2233 * Since exact reproducibility of this method can be achieved
2234 * without any undue performance burden, there is no
2235 * compelling reason to allow double rounding on underflow in
2236 * scalb.
2237 */
2238
2239 // magnitude of a power of two so large that scaling a finite
2240 // nonzero value by it would be guaranteed to over or
2241 // underflow; due to rounding, scaling down takes takes an
2242 // additional power of two which is reflected here
2243 final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
2244 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2245 int exp_adjust = 0;
2246 int scale_increment = 0;
2247 double exp_delta = Double.NaN;
2248
2249 // Make sure scaling factor is in a reasonable range
2250
2251 if(scaleFactor < 0) {
2252 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2253 scale_increment = -512;
2254 exp_delta = twoToTheDoubleScaleDown;
2255 }
2256 else {
2257 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2258 scale_increment = 512;
2259 exp_delta = twoToTheDoubleScaleUp;
2260 }
2261
2262 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2263 // technique from "Hacker's Delight" section 10-2.
2264 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2265 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2266
2267 d *= powerOfTwoD(exp_adjust);
2268 scaleFactor -= exp_adjust;
2269
2270 while(scaleFactor != 0) {
2271 d *= exp_delta;
2272 scaleFactor -= scale_increment;
2273 }
2274 return d;
2275 }
2276
2277 /**
2278 * Returns {@code f} ×
2279 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2280 * by a single correctly rounded floating-point multiply to a
2281 * member of the float value set. See the Java
2282 * Language Specification for a discussion of floating-point
2283 * value sets. If the exponent of the result is between {@link
2284 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2285 * answer is calculated exactly. If the exponent of the result
2286 * would be larger than {@code Float.MAX_EXPONENT}, an
2287 * infinity is returned. Note that if the result is subnormal,
2288 * precision may be lost; that is, when {@code scalb(x, n)}
2289 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2290 * <i>x</i>. When the result is non-NaN, the result has the same
2291 * sign as {@code f}.
2292 *
2293 * <p>Special cases:
2294 * <ul>
2295 * <li> If the first argument is NaN, NaN is returned.
2296 * <li> If the first argument is infinite, then an infinity of the
2297 * same sign is returned.
2298 * <li> If the first argument is zero, then a zero of the same
2299 * sign is returned.
2300 * </ul>
2301 *
2302 * @param f number to be scaled by a power of two.
2303 * @param scaleFactor power of 2 used to scale {@code f}
2304 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2305 * @since 1.6
2306 */
2307 public static float scalb(float f, int scaleFactor) {
2308 // magnitude of a power of two so large that scaling a finite
2309 // nonzero value by it would be guaranteed to over or
2310 // underflow; due to rounding, scaling down takes takes an
2311 // additional power of two which is reflected here
2312 final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
2313 FloatConsts.SIGNIFICAND_WIDTH + 1;
2314
2315 // Make sure scaling factor is in a reasonable range
2316 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2317
2318 /*
2319 * Since + MAX_SCALE for float fits well within the double
2320 * exponent range and + float -> double conversion is exact
2321 * the multiplication below will be exact. Therefore, the
2322 * rounding that occurs when the double product is cast to
2323 * float will be the correctly rounded float result. Since
2324 * all operations other than the final multiply will be exact,
2325 * it is not necessary to declare this method strictfp.
2326 */
2327 return (float)((double)f*powerOfTwoD(scaleFactor));
2328 }
2329
2330 // Constants used in scalb
2331 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2332 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2333
2334 /**
2335 * Returns a floating-point power of two in the normal range.
2336 */
2337 static double powerOfTwoD(int n) {
2338 assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
2339 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2340 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2341 & DoubleConsts.EXP_BIT_MASK);
2342 }
2343
2344 /**
2345 * Returns a floating-point power of two in the normal range.
2346 */
2347 static float powerOfTwoF(int n) {
2348 assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
2349 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2350 (FloatConsts.SIGNIFICAND_WIDTH-1))
2351 & FloatConsts.EXP_BIT_MASK);
2352 }
2353 }