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 *
1052 * @param x the dividend
1053 * @param y the divisor
1054 * @return the largest (closest to positive infinity)
1055 * {@code int} value that is less than or equal to the algebraic quotient.
1056 * @throws ArithmeticException if the divisor {@code y} is zero
1057 * @see #floorMod(int, int)
1058 * @see #floor(double)
1059 * @since 1.8
1060 */
1061 public static int floorDiv(int x, int y) {
1062 int r = x / y;
1063 // if the signs are different and modulo not zero, round down
1064 if ((x ^ y) < 0 && (r * y != x)) {
1065 r--;
1066 }
1067 return r;
1068 }
1069
1070 /**
1071 * Returns the largest (closest to positive infinity)
1072 * {@code long} value that is less than or equal to the algebraic quotient.
1073 * There is one special case, if the dividend is the
1074 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1075 * then integer overflow occurs and
1076 * the result is equal to the {@code Long.MIN_VALUE}.
1077 * <p>
1078 * Normal integer division operates under the round to zero rounding mode
1079 * (truncation). This operation instead acts under the round toward
1080 * negative infinity (floor) rounding mode.
1081 * The floor rounding mode gives different results than truncation
1082 * when the exact result is negative.
1083 * <p>
1084 * For examples, see {@link #floorDiv(int, int)}.
1085 *
1086 * @param x the dividend
1087 * @param y the divisor
1088 * @return the largest (closest to positive infinity)
1089 * {@code long} value that is less than or equal to the algebraic quotient.
1090 * @throws ArithmeticException if the divisor {@code y} is zero
1091 * @see #floorMod(long, long)
1092 * @see #floor(double)
1093 * @since 1.8
1094 */
1095 public static long floorDiv(long x, long y) {
1096 long r = x / y;
1097 // if the signs are different and modulo not zero, round down
1098 if ((x ^ y) < 0 && (r * y != x)) {
1099 r--;
1100 }
1101 return r;
1102 }
1103
1104 /**
1105 * Returns the floor modulus of the {@code int} arguments.
1106 * <p>
1107 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1108 * has the same sign as the divisor {@code y}, and
1109 * is in the range of {@code -abs(y) < r < +abs(y)}.
1110 *
1111 * <p>
1112 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1113 * <ul>
1114 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1115 * </ul>
1116 * <p>
1117 * The difference in values between {@code floorMod} and
1118 * the {@code %} operator is due to the difference between
1119 * {@code floorDiv} that returns the integer less than or equal to the quotient
1120 * and the {@code /} operator that returns the integer closest to zero.
1121 * <p>
1122 * Examples:
1123 * <ul>
1124 * <li>If the signs of the arguments are the same, the results
1125 * of {@code floorMod} and the {@code %} operator are the same. <br>
1126 * <ul>
1127 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li>
1128 * </ul>
1129 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1130 * <ul>
1131 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li>
1132 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li>
1133 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li>
1134 * </ul>
1135 * </li>
1136 * </ul>
1137 * <p>
1138 * If the signs of arguments are unknown and a positive modulus
1139 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1140 *
1141 * @param x the dividend
1142 * @param y the divisor
1143 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1144 * @throws ArithmeticException if the divisor {@code y} is zero
1145 * @see #floorDiv(int, int)
1146 * @since 1.8
1147 */
1148 public static int floorMod(int x, int y) {
1149 int r = x - floorDiv(x, y) * y;
1150 return r;
1151 }
1152
1153 /**
1154 * Returns the floor modulus of the {@code long} arguments.
1155 * <p>
1156 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1157 * has the same sign as the divisor {@code y}, and
1158 * is in the range of {@code -abs(y) < r < +abs(y)}.
1159 *
1160 * <p>
1161 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1162 * <ul>
1163 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1164 * </ul>
1165 * <p>
1166 * For examples, see {@link #floorMod(int, int)}.
1167 *
1168 * @param x the dividend
1169 * @param y the divisor
1170 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1171 * @throws ArithmeticException if the divisor {@code y} is zero
1172 * @see #floorDiv(long, long)
1173 * @since 1.8
1174 */
1175 public static long floorMod(long x, long y) {
1176 return x - floorDiv(x, y) * y;
1177 }
1178
1179 /**
1180 * Returns the absolute value of an {@code int} value.
1181 * If the argument is not negative, the argument is returned.
1182 * If the argument is negative, the negation of the argument is returned.
1183 *
1184 * <p>Note that if the argument is equal to the value of
1185 * {@link Integer#MIN_VALUE}, the most negative representable
1186 * {@code int} value, the result is that same value, which is
1187 * negative.
1188 *
1189 * @param a the argument whose absolute value is to be determined
1190 * @return the absolute value of the argument.
1191 */
1192 public static int abs(int a) {
1193 return (a < 0) ? -a : a;
1194 }
1195
1196 /**
1197 * Returns the absolute value of a {@code long} value.
1198 * If the argument is not negative, the argument is returned.
1199 * If the argument is negative, the negation of the argument is returned.
1200 *
1201 * <p>Note that if the argument is equal to the value of
1202 * {@link Long#MIN_VALUE}, the most negative representable
1203 * {@code long} value, the result is that same value, which
1204 * is negative.
1205 *
1206 * @param a the argument whose absolute value is to be determined
1207 * @return the absolute value of the argument.
1208 */
1209 public static long abs(long a) {
1210 return (a < 0) ? -a : a;
1211 }
1212
1213 /**
1214 * Returns the absolute value of a {@code float} value.
1215 * If the argument is not negative, the argument is returned.
1216 * If the argument is negative, the negation of the argument is returned.
1217 * Special cases:
1218 * <ul><li>If the argument is positive zero or negative zero, the
1219 * result is positive zero.
1220 * <li>If the argument is infinite, the result is positive infinity.
1221 * <li>If the argument is NaN, the result is NaN.</ul>
1222 * In other words, the result is the same as the value of the expression:
1223 * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1224 *
1225 * @param a the argument whose absolute value is to be determined
1226 * @return the absolute value of the argument.
1227 */
1228 public static float abs(float a) {
1229 return (a <= 0.0F) ? 0.0F - a : a;
1230 }
1231
1232 /**
1233 * Returns the absolute value of a {@code double} value.
1234 * If the argument is not negative, the argument is returned.
1235 * If the argument is negative, the negation of the argument is returned.
1236 * Special cases:
1237 * <ul><li>If the argument is positive zero or negative zero, the result
1238 * is positive zero.
1239 * <li>If the argument is infinite, the result is positive infinity.
1240 * <li>If the argument is NaN, the result is NaN.</ul>
1241 * In other words, the result is the same as the value of the expression:
1242 * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1243 *
1244 * @param a the argument whose absolute value is to be determined
1245 * @return the absolute value of the argument.
1246 */
1247 public static double abs(double a) {
1248 return (a <= 0.0D) ? 0.0D - a : a;
1249 }
1250
1251 /**
1252 * Returns the greater of two {@code int} values. That is, the
1253 * result is the argument closer to the value of
1254 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1255 * the result is that same value.
1256 *
1257 * @param a an argument.
1258 * @param b another argument.
1259 * @return the larger of {@code a} and {@code b}.
1260 */
1261 public static int max(int a, int b) {
1262 return (a >= b) ? a : b;
1263 }
1264
1265 /**
1266 * Returns the greater of two {@code long} values. That is, the
1267 * result is the argument closer to the value of
1268 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1269 * the result is that same value.
1270 *
1271 * @param a an argument.
1272 * @param b another argument.
1273 * @return the larger of {@code a} and {@code b}.
1274 */
1275 public static long max(long a, long b) {
1276 return (a >= b) ? a : b;
1277 }
1278
1279 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1280 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1281 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1282
1283 /**
1284 * Returns the greater of two {@code float} values. That is,
1285 * the result is the argument closer to positive infinity. If the
1286 * arguments have the same value, the result is that same
1287 * value. If either value is NaN, then the result is NaN. Unlike
1288 * the numerical comparison operators, this method considers
1289 * negative zero to be strictly smaller than positive zero. If one
1290 * argument is positive zero and the other negative zero, the
1291 * result is positive zero.
1292 *
1293 * @param a an argument.
1294 * @param b another argument.
1295 * @return the larger of {@code a} and {@code b}.
1296 */
1297 public static float max(float a, float b) {
1298 if (a != a)
1299 return a; // a is NaN
1300 if ((a == 0.0f) &&
1301 (b == 0.0f) &&
1302 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1303 // Raw conversion ok since NaN can't map to -0.0.
1304 return b;
1305 }
1306 return (a >= b) ? a : b;
1307 }
1308
1309 /**
1310 * Returns the greater of two {@code double} values. That
1311 * is, the result is the argument closer to positive infinity. If
1312 * the arguments have the same value, the result is that same
1313 * value. If either value is NaN, then the result is NaN. Unlike
1314 * the numerical comparison operators, this method considers
1315 * negative zero to be strictly smaller than positive zero. If one
1316 * argument is positive zero and the other negative zero, the
1317 * result is positive zero.
1318 *
1319 * @param a an argument.
1320 * @param b another argument.
1321 * @return the larger of {@code a} and {@code b}.
1322 */
1323 public static double max(double a, double b) {
1324 if (a != a)
1325 return a; // a is NaN
1326 if ((a == 0.0d) &&
1327 (b == 0.0d) &&
1328 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1329 // Raw conversion ok since NaN can't map to -0.0.
1330 return b;
1331 }
1332 return (a >= b) ? a : b;
1333 }
1334
1335 /**
1336 * Returns the smaller of two {@code int} values. That is,
1337 * the result the argument closer to the value of
1338 * {@link Integer#MIN_VALUE}. If the arguments have the same
1339 * value, the result is that same value.
1340 *
1341 * @param a an argument.
1342 * @param b another argument.
1343 * @return the smaller of {@code a} and {@code b}.
1344 */
1345 public static int min(int a, int b) {
1346 return (a <= b) ? a : b;
1347 }
1348
1349 /**
1350 * Returns the smaller of two {@code long} values. That is,
1351 * the result is the argument closer to the value of
1352 * {@link Long#MIN_VALUE}. If the arguments have the same
1353 * value, the result is that same value.
1354 *
1355 * @param a an argument.
1356 * @param b another argument.
1357 * @return the smaller of {@code a} and {@code b}.
1358 */
1359 public static long min(long a, long b) {
1360 return (a <= b) ? a : b;
1361 }
1362
1363 /**
1364 * Returns the smaller of two {@code float} values. That is,
1365 * the result is the value closer to negative infinity. If the
1366 * arguments have the same value, the result is that same
1367 * value. If either value is NaN, then the result is NaN. Unlike
1368 * the numerical comparison operators, this method considers
1369 * negative zero to be strictly smaller than positive zero. If
1370 * one argument is positive zero and the other is negative zero,
1371 * the result is negative zero.
1372 *
1373 * @param a an argument.
1374 * @param b another argument.
1375 * @return the smaller of {@code a} and {@code b}.
1376 */
1377 public static float min(float a, float b) {
1378 if (a != a)
1379 return a; // a is NaN
1380 if ((a == 0.0f) &&
1381 (b == 0.0f) &&
1382 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1383 // Raw conversion ok since NaN can't map to -0.0.
1384 return b;
1385 }
1386 return (a <= b) ? a : b;
1387 }
1388
1389 /**
1390 * Returns the smaller of two {@code double} values. That
1391 * is, the result is the value closer to negative infinity. If the
1392 * arguments have the same value, the result is that same
1393 * value. If either value is NaN, then the result is NaN. Unlike
1394 * the numerical comparison operators, this method considers
1395 * negative zero to be strictly smaller than positive zero. If one
1396 * argument is positive zero and the other is negative zero, the
1397 * result is negative zero.
1398 *
1399 * @param a an argument.
1400 * @param b another argument.
1401 * @return the smaller of {@code a} and {@code b}.
1402 */
1403 public static double min(double a, double b) {
1404 if (a != a)
1405 return a; // a is NaN
1406 if ((a == 0.0d) &&
1407 (b == 0.0d) &&
1408 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1409 // Raw conversion ok since NaN can't map to -0.0.
1410 return b;
1411 }
1412 return (a <= b) ? a : b;
1413 }
1414
1415 /**
1416 * Returns the size of an ulp of the argument. An ulp, unit in
1417 * the last place, of a {@code double} value is the positive
1418 * distance between this floating-point value and the {@code
1419 * double} value next larger in magnitude. Note that for non-NaN
1420 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1421 *
1422 * <p>Special Cases:
1423 * <ul>
1424 * <li> If the argument is NaN, then the result is NaN.
1425 * <li> If the argument is positive or negative infinity, then the
1426 * result is positive infinity.
1427 * <li> If the argument is positive or negative zero, then the result is
1428 * {@code Double.MIN_VALUE}.
1429 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1430 * the result is equal to 2<sup>971</sup>.
1431 * </ul>
1432 *
1433 * @param d the floating-point value whose ulp is to be returned
1434 * @return the size of an ulp of the argument
1435 * @author Joseph D. Darcy
1436 * @since 1.5
1437 */
1438 public static double ulp(double d) {
1439 int exp = getExponent(d);
1440
1441 switch(exp) {
1442 case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
1443 return Math.abs(d);
1444
1445 case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
1446 return Double.MIN_VALUE;
1447
1448 default:
1449 assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
1450
1451 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1452 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1453 if (exp >= DoubleConsts.MIN_EXPONENT) {
1454 return powerOfTwoD(exp);
1455 }
1456 else {
1457 // return a subnormal result; left shift integer
1458 // representation of Double.MIN_VALUE appropriate
1459 // number of positions
1460 return Double.longBitsToDouble(1L <<
1461 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1462 }
1463 }
1464 }
1465
1466 /**
1467 * Returns the size of an ulp of the argument. An ulp, unit in
1468 * the last place, of a {@code float} value is the positive
1469 * distance between this floating-point value and the {@code
1470 * float} value next larger in magnitude. Note that for non-NaN
1471 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1472 *
1473 * <p>Special Cases:
1474 * <ul>
1475 * <li> If the argument is NaN, then the result is NaN.
1476 * <li> If the argument is positive or negative infinity, then the
1477 * result is positive infinity.
1478 * <li> If the argument is positive or negative zero, then the result is
1479 * {@code Float.MIN_VALUE}.
1480 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1481 * the result is equal to 2<sup>104</sup>.
1482 * </ul>
1483 *
1484 * @param f the floating-point value whose ulp is to be returned
1485 * @return the size of an ulp of the argument
1486 * @author Joseph D. Darcy
1487 * @since 1.5
1488 */
1489 public static float ulp(float f) {
1490 int exp = getExponent(f);
1491
1492 switch(exp) {
1493 case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
1494 return Math.abs(f);
1495
1496 case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
1497 return FloatConsts.MIN_VALUE;
1498
1499 default:
1500 assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1501
1502 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1503 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1504 if (exp >= FloatConsts.MIN_EXPONENT) {
1505 return powerOfTwoF(exp);
1506 }
1507 else {
1508 // return a subnormal result; left shift integer
1509 // representation of FloatConsts.MIN_VALUE appropriate
1510 // number of positions
1511 return Float.intBitsToFloat(1 <<
1512 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1513 }
1514 }
1515 }
1516
1517 /**
1518 * Returns the signum function of the argument; zero if the argument
1519 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1520 * argument is less than zero.
1521 *
1522 * <p>Special Cases:
1523 * <ul>
1524 * <li> If the argument is NaN, then the result is NaN.
1525 * <li> If the argument is positive zero or negative zero, then the
1526 * result is the same as the argument.
1527 * </ul>
1528 *
1529 * @param d the floating-point value whose signum is to be returned
1530 * @return the signum function of the argument
1531 * @author Joseph D. Darcy
1532 * @since 1.5
1533 */
1534 public static double signum(double d) {
1535 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1536 }
1537
1538 /**
1539 * Returns the signum function of the argument; zero if the argument
1540 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1541 * argument is less than zero.
1542 *
1543 * <p>Special Cases:
1544 * <ul>
1545 * <li> If the argument is NaN, then the result is NaN.
1546 * <li> If the argument is positive zero or negative zero, then the
1547 * result is the same as the argument.
1548 * </ul>
1549 *
1550 * @param f the floating-point value whose signum is to be returned
1551 * @return the signum function of the argument
1552 * @author Joseph D. Darcy
1553 * @since 1.5
1554 */
1555 public static float signum(float f) {
1556 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1557 }
1558
1559 /**
1560 * Returns the hyperbolic sine of a {@code double} value.
1561 * The hyperbolic sine of <i>x</i> is defined to be
1562 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1563 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1564 *
1565 * <p>Special cases:
1566 * <ul>
1567 *
1568 * <li>If the argument is NaN, then the result is NaN.
1569 *
1570 * <li>If the argument is infinite, then the result is an infinity
1571 * with the same sign as the argument.
1572 *
1573 * <li>If the argument is zero, then the result is a zero with the
1574 * same sign as the argument.
1575 *
1576 * </ul>
1577 *
1578 * <p>The computed result must be within 2.5 ulps of the exact result.
1579 *
1580 * @param x The number whose hyperbolic sine is to be returned.
1581 * @return The hyperbolic sine of {@code x}.
1582 * @since 1.5
1583 */
1584 public static double sinh(double x) {
1585 return StrictMath.sinh(x);
1586 }
1587
1588 /**
1589 * Returns the hyperbolic cosine of a {@code double} value.
1590 * The hyperbolic cosine of <i>x</i> is defined to be
1591 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1592 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1593 *
1594 * <p>Special cases:
1595 * <ul>
1596 *
1597 * <li>If the argument is NaN, then the result is NaN.
1598 *
1599 * <li>If the argument is infinite, then the result is positive
1600 * infinity.
1601 *
1602 * <li>If the argument is zero, then the result is {@code 1.0}.
1603 *
1604 * </ul>
1605 *
1606 * <p>The computed result must be within 2.5 ulps of the exact result.
1607 *
1608 * @param x The number whose hyperbolic cosine is to be returned.
1609 * @return The hyperbolic cosine of {@code x}.
1610 * @since 1.5
1611 */
1612 public static double cosh(double x) {
1613 return StrictMath.cosh(x);
1614 }
1615
1616 /**
1617 * Returns the hyperbolic tangent of a {@code double} value.
1618 * The hyperbolic tangent of <i>x</i> is defined to be
1619 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
1620 * in other words, {@linkplain Math#sinh
1621 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
1622 * that the absolute value of the exact tanh is always less than
1623 * 1.
1624 *
1625 * <p>Special cases:
1626 * <ul>
1627 *
1628 * <li>If the argument is NaN, then the result is NaN.
1629 *
1630 * <li>If the argument is zero, then the result is a zero with the
1631 * same sign as the argument.
1632 *
1633 * <li>If the argument is positive infinity, then the result is
1634 * {@code +1.0}.
1635 *
1636 * <li>If the argument is negative infinity, then the result is
1637 * {@code -1.0}.
1638 *
1639 * </ul>
1640 *
1641 * <p>The computed result must be within 2.5 ulps of the exact result.
1642 * The result of {@code tanh} for any finite input must have
1643 * an absolute value less than or equal to 1. Note that once the
1644 * exact result of tanh is within 1/2 of an ulp of the limit value
1645 * of ±1, correctly signed ±{@code 1.0} should
1646 * be returned.
1647 *
1648 * @param x The number whose hyperbolic tangent is to be returned.
1649 * @return The hyperbolic tangent of {@code x}.
1650 * @since 1.5
1651 */
1652 public static double tanh(double x) {
1653 return StrictMath.tanh(x);
1654 }
1655
1656 /**
1657 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1658 * without intermediate overflow or underflow.
1659 *
1660 * <p>Special cases:
1661 * <ul>
1662 *
1663 * <li> If either argument is infinite, then the result
1664 * is positive infinity.
1665 *
1666 * <li> If either argument is NaN and neither argument is infinite,
1667 * then the result is NaN.
1668 *
1669 * </ul>
1670 *
1671 * <p>The computed result must be within 1 ulp of the exact
1672 * result. If one parameter is held constant, the results must be
1673 * semi-monotonic in the other parameter.
1674 *
1675 * @param x a value
1676 * @param y a value
1677 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1678 * without intermediate overflow or underflow
1679 * @since 1.5
1680 */
1681 public static double hypot(double x, double y) {
1682 return StrictMath.hypot(x, y);
1683 }
1684
1685 /**
1686 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
1687 * <i>x</i> near 0, the exact sum of
1688 * {@code expm1(x)} + 1 is much closer to the true
1689 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1690 *
1691 * <p>Special cases:
1692 * <ul>
1693 * <li>If the argument is NaN, the result is NaN.
1694 *
1695 * <li>If the argument is positive infinity, then the result is
1696 * positive infinity.
1697 *
1698 * <li>If the argument is negative infinity, then the result is
1699 * -1.0.
1700 *
1701 * <li>If the argument is zero, then the result is a zero with the
1702 * same sign as the argument.
1703 *
1704 * </ul>
1705 *
1706 * <p>The computed result must be within 1 ulp of the exact result.
1707 * Results must be semi-monotonic. The result of
1708 * {@code expm1} for any finite input must be greater than or
1709 * equal to {@code -1.0}. Note that once the exact result of
1710 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
1711 * ulp of the limit value -1, {@code -1.0} should be
1712 * returned.
1713 *
1714 * @param x the exponent to raise <i>e</i> to in the computation of
1715 * <i>e</i><sup>{@code x}</sup> -1.
1716 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
1717 * @since 1.5
1718 */
1719 public static double expm1(double x) {
1720 return StrictMath.expm1(x);
1721 }
1722
1723 /**
1724 * Returns the natural logarithm of the sum of the argument and 1.
1725 * Note that for small values {@code x}, the result of
1726 * {@code log1p(x)} is much closer to the true result of ln(1
1727 * + {@code x}) than the floating-point evaluation of
1728 * {@code log(1.0+x)}.
1729 *
1730 * <p>Special cases:
1731 *
1732 * <ul>
1733 *
1734 * <li>If the argument is NaN or less than -1, then the result is
1735 * NaN.
1736 *
1737 * <li>If the argument is positive infinity, then the result is
1738 * positive infinity.
1739 *
1740 * <li>If the argument is negative one, then the result is
1741 * negative infinity.
1742 *
1743 * <li>If the argument is zero, then the result is a zero with the
1744 * same sign as the argument.
1745 *
1746 * </ul>
1747 *
1748 * <p>The computed result must be within 1 ulp of the exact result.
1749 * Results must be semi-monotonic.
1750 *
1751 * @param x a value
1752 * @return the value ln({@code x} + 1), the natural
1753 * log of {@code x} + 1
1754 * @since 1.5
1755 */
1756 public static double log1p(double x) {
1757 return StrictMath.log1p(x);
1758 }
1759
1760 /**
1761 * Returns the first floating-point argument with the sign of the
1762 * second floating-point argument. Note that unlike the {@link
1763 * StrictMath#copySign(double, double) StrictMath.copySign}
1764 * method, this method does not require NaN {@code sign}
1765 * arguments to be treated as positive values; implementations are
1766 * permitted to treat some NaN arguments as positive and other NaN
1767 * arguments as negative to allow greater performance.
1768 *
1769 * @param magnitude the parameter providing the magnitude of the result
1770 * @param sign the parameter providing the sign of the result
1771 * @return a value with the magnitude of {@code magnitude}
1772 * and the sign of {@code sign}.
1773 * @since 1.6
1774 */
1775 public static double copySign(double magnitude, double sign) {
1776 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1777 (DoubleConsts.SIGN_BIT_MASK)) |
1778 (Double.doubleToRawLongBits(magnitude) &
1779 (DoubleConsts.EXP_BIT_MASK |
1780 DoubleConsts.SIGNIF_BIT_MASK)));
1781 }
1782
1783 /**
1784 * Returns the first floating-point argument with the sign of the
1785 * second floating-point argument. Note that unlike the {@link
1786 * StrictMath#copySign(float, float) StrictMath.copySign}
1787 * method, this method does not require NaN {@code sign}
1788 * arguments to be treated as positive values; implementations are
1789 * permitted to treat some NaN arguments as positive and other NaN
1790 * arguments as negative to allow greater performance.
1791 *
1792 * @param magnitude the parameter providing the magnitude of the result
1793 * @param sign the parameter providing the sign of the result
1794 * @return a value with the magnitude of {@code magnitude}
1795 * and the sign of {@code sign}.
1796 * @since 1.6
1797 */
1798 public static float copySign(float magnitude, float sign) {
1799 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1800 (FloatConsts.SIGN_BIT_MASK)) |
1801 (Float.floatToRawIntBits(magnitude) &
1802 (FloatConsts.EXP_BIT_MASK |
1803 FloatConsts.SIGNIF_BIT_MASK)));
1804 }
1805
1806 /**
1807 * Returns the unbiased exponent used in the representation of a
1808 * {@code float}. Special cases:
1809 *
1810 * <ul>
1811 * <li>If the argument is NaN or infinite, then the result is
1812 * {@link Float#MAX_EXPONENT} + 1.
1813 * <li>If the argument is zero or subnormal, then the result is
1814 * {@link Float#MIN_EXPONENT} -1.
1815 * </ul>
1816 * @param f a {@code float} value
1817 * @return the unbiased exponent of the argument
1818 * @since 1.6
1819 */
1820 public static int getExponent(float f) {
1821 /*
1822 * Bitwise convert f to integer, mask out exponent bits, shift
1823 * to the right and then subtract out float's bias adjust to
1824 * get true exponent value
1825 */
1826 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1827 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1828 }
1829
1830 /**
1831 * Returns the unbiased exponent used in the representation of a
1832 * {@code double}. Special cases:
1833 *
1834 * <ul>
1835 * <li>If the argument is NaN or infinite, then the result is
1836 * {@link Double#MAX_EXPONENT} + 1.
1837 * <li>If the argument is zero or subnormal, then the result is
1838 * {@link Double#MIN_EXPONENT} -1.
1839 * </ul>
1840 * @param d a {@code double} value
1841 * @return the unbiased exponent of the argument
1842 * @since 1.6
1843 */
1844 public static int getExponent(double d) {
1845 /*
1846 * Bitwise convert d to long, mask out exponent bits, shift
1847 * to the right and then subtract out double's bias adjust to
1848 * get true exponent value.
1849 */
1850 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1851 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1852 }
1853
1854 /**
1855 * Returns the floating-point number adjacent to the first
1856 * argument in the direction of the second argument. If both
1857 * arguments compare as equal the second argument is returned.
1858 *
1859 * <p>
1860 * Special cases:
1861 * <ul>
1862 * <li> If either argument is a NaN, then NaN is returned.
1863 *
1864 * <li> If both arguments are signed zeros, {@code direction}
1865 * is returned unchanged (as implied by the requirement of
1866 * returning the second argument if the arguments compare as
1867 * equal).
1868 *
1869 * <li> If {@code start} is
1870 * ±{@link Double#MIN_VALUE} and {@code direction}
1871 * has a value such that the result should have a smaller
1872 * magnitude, then a zero with the same sign as {@code start}
1873 * is returned.
1874 *
1875 * <li> If {@code start} is infinite and
1876 * {@code direction} has a value such that the result should
1877 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1878 * same sign as {@code start} is returned.
1879 *
1880 * <li> If {@code start} is equal to ±
1881 * {@link Double#MAX_VALUE} and {@code direction} has a
1882 * value such that the result should have a larger magnitude, an
1883 * infinity with same sign as {@code start} is returned.
1884 * </ul>
1885 *
1886 * @param start starting floating-point value
1887 * @param direction value indicating which of
1888 * {@code start}'s neighbors or {@code start} should
1889 * be returned
1890 * @return The floating-point number adjacent to {@code start} in the
1891 * direction of {@code direction}.
1892 * @since 1.6
1893 */
1894 public static double nextAfter(double start, double direction) {
1895 /*
1896 * The cases:
1897 *
1898 * nextAfter(+infinity, 0) == MAX_VALUE
1899 * nextAfter(+infinity, +infinity) == +infinity
1900 * nextAfter(-infinity, 0) == -MAX_VALUE
1901 * nextAfter(-infinity, -infinity) == -infinity
1902 *
1903 * are naturally handled without any additional testing
1904 */
1905
1906 /*
1907 * IEEE 754 floating-point numbers are lexicographically
1908 * ordered if treated as signed-magnitude integers.
1909 * Since Java's integers are two's complement,
1910 * incrementing the two's complement representation of a
1911 * logically negative floating-point value *decrements*
1912 * the signed-magnitude representation. Therefore, when
1913 * the integer representation of a floating-point value
1914 * is negative, the adjustment to the representation is in
1915 * the opposite direction from what would initially be expected.
1916 */
1917
1918 // Branch to descending case first as it is more costly than ascending
1919 // case due to start != 0.0d conditional.
1920 if (start > direction) { // descending
1921 if (start != 0.0d) {
1922 final long transducer = Double.doubleToRawLongBits(start);
1923 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
1924 } else { // start == 0.0d && direction < 0.0d
1925 return -Double.MIN_VALUE;
1926 }
1927 } else if (start < direction) { // ascending
1928 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1929 // then bitwise convert start to integer.
1930 final long transducer = Double.doubleToRawLongBits(start + 0.0d);
1931 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
1932 } else if (start == direction) {
1933 return direction;
1934 } else { // isNaN(start) || isNaN(direction)
1935 return start + direction;
1936 }
1937 }
1938
1939 /**
1940 * Returns the floating-point number adjacent to the first
1941 * argument in the direction of the second argument. If both
1942 * arguments compare as equal a value equivalent to the second argument
1943 * is returned.
1944 *
1945 * <p>
1946 * Special cases:
1947 * <ul>
1948 * <li> If either argument is a NaN, then NaN is returned.
1949 *
1950 * <li> If both arguments are signed zeros, a value equivalent
1951 * to {@code direction} is returned.
1952 *
1953 * <li> If {@code start} is
1954 * ±{@link Float#MIN_VALUE} and {@code direction}
1955 * has a value such that the result should have a smaller
1956 * magnitude, then a zero with the same sign as {@code start}
1957 * is returned.
1958 *
1959 * <li> If {@code start} is infinite and
1960 * {@code direction} has a value such that the result should
1961 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1962 * same sign as {@code start} is returned.
1963 *
1964 * <li> If {@code start} is equal to ±
1965 * {@link Float#MAX_VALUE} and {@code direction} has a
1966 * value such that the result should have a larger magnitude, an
1967 * infinity with same sign as {@code start} is returned.
1968 * </ul>
1969 *
1970 * @param start starting floating-point value
1971 * @param direction value indicating which of
1972 * {@code start}'s neighbors or {@code start} should
1973 * be returned
1974 * @return The floating-point number adjacent to {@code start} in the
1975 * direction of {@code direction}.
1976 * @since 1.6
1977 */
1978 public static float nextAfter(float start, double direction) {
1979 /*
1980 * The cases:
1981 *
1982 * nextAfter(+infinity, 0) == MAX_VALUE
1983 * nextAfter(+infinity, +infinity) == +infinity
1984 * nextAfter(-infinity, 0) == -MAX_VALUE
1985 * nextAfter(-infinity, -infinity) == -infinity
1986 *
1987 * are naturally handled without any additional testing
1988 */
1989
1990 /*
1991 * IEEE 754 floating-point numbers are lexicographically
1992 * ordered if treated as signed-magnitude integers.
1993 * Since Java's integers are two's complement,
1994 * incrementing the two's complement representation of a
1995 * logically negative floating-point value *decrements*
1996 * the signed-magnitude representation. Therefore, when
1997 * the integer representation of a floating-point value
1998 * is negative, the adjustment to the representation is in
1999 * the opposite direction from what would initially be expected.
2000 */
2001
2002 // Branch to descending case first as it is more costly than ascending
2003 // case due to start != 0.0f conditional.
2004 if (start > direction) { // descending
2005 if (start != 0.0f) {
2006 final int transducer = Float.floatToRawIntBits(start);
2007 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2008 } else { // start == 0.0f && direction < 0.0f
2009 return -Float.MIN_VALUE;
2010 }
2011 } else if (start < direction) { // ascending
2012 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2013 // then bitwise convert start to integer.
2014 final int transducer = Float.floatToRawIntBits(start + 0.0f);
2015 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2016 } else if (start == direction) {
2017 return (float)direction;
2018 } else { // isNaN(start) || isNaN(direction)
2019 return start + (float)direction;
2020 }
2021 }
2022
2023 /**
2024 * Returns the floating-point value adjacent to {@code d} in
2025 * the direction of positive infinity. This method is
2026 * semantically equivalent to {@code nextAfter(d,
2027 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2028 * implementation may run faster than its equivalent
2029 * {@code nextAfter} call.
2030 *
2031 * <p>Special Cases:
2032 * <ul>
2033 * <li> If the argument is NaN, the result is NaN.
2034 *
2035 * <li> If the argument is positive infinity, the result is
2036 * positive infinity.
2037 *
2038 * <li> If the argument is zero, the result is
2039 * {@link Double#MIN_VALUE}
2040 *
2041 * </ul>
2042 *
2043 * @param d starting floating-point value
2044 * @return The adjacent floating-point value closer to positive
2045 * infinity.
2046 * @since 1.6
2047 */
2048 public static double nextUp(double d) {
2049 // Use a single conditional and handle the likely cases first.
2050 if (d < Double.POSITIVE_INFINITY) {
2051 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2052 final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2053 return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2054 } else { // d is NaN or +Infinity
2055 return d;
2056 }
2057 }
2058
2059 /**
2060 * Returns the floating-point value adjacent to {@code f} in
2061 * the direction of positive infinity. This method is
2062 * semantically equivalent to {@code nextAfter(f,
2063 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2064 * implementation may run faster than its equivalent
2065 * {@code nextAfter} call.
2066 *
2067 * <p>Special Cases:
2068 * <ul>
2069 * <li> If the argument is NaN, the result is NaN.
2070 *
2071 * <li> If the argument is positive infinity, the result is
2072 * positive infinity.
2073 *
2074 * <li> If the argument is zero, the result is
2075 * {@link Float#MIN_VALUE}
2076 *
2077 * </ul>
2078 *
2079 * @param f starting floating-point value
2080 * @return The adjacent floating-point value closer to positive
2081 * infinity.
2082 * @since 1.6
2083 */
2084 public static float nextUp(float f) {
2085 // Use a single conditional and handle the likely cases first.
2086 if (f < Float.POSITIVE_INFINITY) {
2087 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2088 final int transducer = Float.floatToRawIntBits(f + 0.0F);
2089 return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2090 } else { // f is NaN or +Infinity
2091 return f;
2092 }
2093 }
2094
2095 /**
2096 * Returns the floating-point value adjacent to {@code d} in
2097 * the direction of negative infinity. This method is
2098 * semantically equivalent to {@code nextAfter(d,
2099 * Double.NEGATIVE_INFINITY)}; however, a
2100 * {@code nextDown} implementation may run faster than its
2101 * equivalent {@code nextAfter} call.
2102 *
2103 * <p>Special Cases:
2104 * <ul>
2105 * <li> If the argument is NaN, the result is NaN.
2106 *
2107 * <li> If the argument is negative infinity, the result is
2108 * negative infinity.
2109 *
2110 * <li> If the argument is zero, the result is
2111 * {@code -Double.MIN_VALUE}
2112 *
2113 * </ul>
2114 *
2115 * @param d starting floating-point value
2116 * @return The adjacent floating-point value closer to negative
2117 * infinity.
2118 * @since 1.8
2119 */
2120 public static double nextDown(double d) {
2121 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2122 return d;
2123 else {
2124 if (d == 0.0)
2125 return -Double.MIN_VALUE;
2126 else
2127 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2128 ((d > 0.0d)?-1L:+1L));
2129 }
2130 }
2131
2132 /**
2133 * Returns the floating-point value adjacent to {@code f} in
2134 * the direction of negative infinity. This method is
2135 * semantically equivalent to {@code nextAfter(f,
2136 * Float.NEGATIVE_INFINITY)}; however, a
2137 * {@code nextDown} implementation may run faster than its
2138 * equivalent {@code nextAfter} call.
2139 *
2140 * <p>Special Cases:
2141 * <ul>
2142 * <li> If the argument is NaN, the result is NaN.
2143 *
2144 * <li> If the argument is negative infinity, the result is
2145 * negative infinity.
2146 *
2147 * <li> If the argument is zero, the result is
2148 * {@code -Float.MIN_VALUE}
2149 *
2150 * </ul>
2151 *
2152 * @param f starting floating-point value
2153 * @return The adjacent floating-point value closer to negative
2154 * infinity.
2155 * @since 1.8
2156 */
2157 public static float nextDown(float f) {
2158 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2159 return f;
2160 else {
2161 if (f == 0.0f)
2162 return -Float.MIN_VALUE;
2163 else
2164 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2165 ((f > 0.0f)?-1:+1));
2166 }
2167 }
2168
2169 /**
2170 * Returns {@code d} ×
2171 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2172 * by a single correctly rounded floating-point multiply to a
2173 * member of the double value set. See the Java
2174 * Language Specification for a discussion of floating-point
2175 * value sets. If the exponent of the result is between {@link
2176 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2177 * answer is calculated exactly. If the exponent of the result
2178 * would be larger than {@code Double.MAX_EXPONENT}, an
2179 * infinity is returned. Note that if the result is subnormal,
2180 * precision may be lost; that is, when {@code scalb(x, n)}
2181 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2182 * <i>x</i>. When the result is non-NaN, the result has the same
2183 * sign as {@code d}.
2184 *
2185 * <p>Special cases:
2186 * <ul>
2187 * <li> If the first argument is NaN, NaN is returned.
2188 * <li> If the first argument is infinite, then an infinity of the
2189 * same sign is returned.
2190 * <li> If the first argument is zero, then a zero of the same
2191 * sign is returned.
2192 * </ul>
2193 *
2194 * @param d number to be scaled by a power of two.
2195 * @param scaleFactor power of 2 used to scale {@code d}
2196 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2197 * @since 1.6
2198 */
2199 public static double scalb(double d, int scaleFactor) {
2200 /*
2201 * This method does not need to be declared strictfp to
2202 * compute the same correct result on all platforms. When
2203 * scaling up, it does not matter what order the
2204 * multiply-store operations are done; the result will be
2205 * finite or overflow regardless of the operation ordering.
2206 * However, to get the correct result when scaling down, a
2207 * particular ordering must be used.
2208 *
2209 * When scaling down, the multiply-store operations are
2210 * sequenced so that it is not possible for two consecutive
2211 * multiply-stores to return subnormal results. If one
2212 * multiply-store result is subnormal, the next multiply will
2213 * round it away to zero. This is done by first multiplying
2214 * by 2 ^ (scaleFactor % n) and then multiplying several
2215 * times by by 2^n as needed where n is the exponent of number
2216 * that is a covenient power of two. In this way, at most one
2217 * real rounding error occurs. If the double value set is
2218 * being used exclusively, the rounding will occur on a
2219 * multiply. If the double-extended-exponent value set is
2220 * being used, the products will (perhaps) be exact but the
2221 * stores to d are guaranteed to round to the double value
2222 * set.
2223 *
2224 * It is _not_ a valid implementation to first multiply d by
2225 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2226 * MIN_EXPONENT) since even in a strictfp program double
2227 * rounding on underflow could occur; e.g. if the scaleFactor
2228 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2229 * little less than -(MIN_EXPONENT - n), meaning the final
2230 * result would be subnormal.
2231 *
2232 * Since exact reproducibility of this method can be achieved
2233 * without any undue performance burden, there is no
2234 * compelling reason to allow double rounding on underflow in
2235 * scalb.
2236 */
2237
2238 // magnitude of a power of two so large that scaling a finite
2239 // nonzero value by it would be guaranteed to over or
2240 // underflow; due to rounding, scaling down takes takes an
2241 // additional power of two which is reflected here
2242 final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
2243 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2244 int exp_adjust = 0;
2245 int scale_increment = 0;
2246 double exp_delta = Double.NaN;
2247
2248 // Make sure scaling factor is in a reasonable range
2249
2250 if(scaleFactor < 0) {
2251 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2252 scale_increment = -512;
2253 exp_delta = twoToTheDoubleScaleDown;
2254 }
2255 else {
2256 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2257 scale_increment = 512;
2258 exp_delta = twoToTheDoubleScaleUp;
2259 }
2260
2261 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2262 // technique from "Hacker's Delight" section 10-2.
2263 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2264 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2265
2266 d *= powerOfTwoD(exp_adjust);
2267 scaleFactor -= exp_adjust;
2268
2269 while(scaleFactor != 0) {
2270 d *= exp_delta;
2271 scaleFactor -= scale_increment;
2272 }
2273 return d;
2274 }
2275
2276 /**
2277 * Returns {@code f} ×
2278 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2279 * by a single correctly rounded floating-point multiply to a
2280 * member of the float value set. See the Java
2281 * Language Specification for a discussion of floating-point
2282 * value sets. If the exponent of the result is between {@link
2283 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2284 * answer is calculated exactly. If the exponent of the result
2285 * would be larger than {@code Float.MAX_EXPONENT}, an
2286 * infinity is returned. Note that if the result is subnormal,
2287 * precision may be lost; that is, when {@code scalb(x, n)}
2288 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2289 * <i>x</i>. When the result is non-NaN, the result has the same
2290 * sign as {@code f}.
2291 *
2292 * <p>Special cases:
2293 * <ul>
2294 * <li> If the first argument is NaN, NaN is returned.
2295 * <li> If the first argument is infinite, then an infinity of the
2296 * same sign is returned.
2297 * <li> If the first argument is zero, then a zero of the same
2298 * sign is returned.
2299 * </ul>
2300 *
2301 * @param f number to be scaled by a power of two.
2302 * @param scaleFactor power of 2 used to scale {@code f}
2303 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2304 * @since 1.6
2305 */
2306 public static float scalb(float f, int scaleFactor) {
2307 // magnitude of a power of two so large that scaling a finite
2308 // nonzero value by it would be guaranteed to over or
2309 // underflow; due to rounding, scaling down takes takes an
2310 // additional power of two which is reflected here
2311 final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
2312 FloatConsts.SIGNIFICAND_WIDTH + 1;
2313
2314 // Make sure scaling factor is in a reasonable range
2315 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2316
2317 /*
2318 * Since + MAX_SCALE for float fits well within the double
2319 * exponent range and + float -> double conversion is exact
2320 * the multiplication below will be exact. Therefore, the
2321 * rounding that occurs when the double product is cast to
2322 * float will be the correctly rounded float result. Since
2323 * all operations other than the final multiply will be exact,
2324 * it is not necessary to declare this method strictfp.
2325 */
2326 return (float)((double)f*powerOfTwoD(scaleFactor));
2327 }
2328
2329 // Constants used in scalb
2330 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2331 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2332
2333 /**
2334 * Returns a floating-point power of two in the normal range.
2335 */
2336 static double powerOfTwoD(int n) {
2337 assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
2338 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2339 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2340 & DoubleConsts.EXP_BIT_MASK);
2341 }
2342
2343 /**
2344 * Returns a floating-point power of two in the normal range.
2345 */
2346 static float powerOfTwoF(int n) {
2347 assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
2348 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2349 (FloatConsts.SIGNIFICAND_WIDTH-1))
2350 & FloatConsts.EXP_BIT_MASK);
2351 }
2352 }