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 * @return a pseudorandom {@code double} greater than or equal
769 * to {@code 0.0} and less than {@code 1.0}.
770 * @see Random#nextDouble()
771 */
772 public static double random() {
773 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
774 }
775
776 /**
777 * Returns the sum of its arguments,
778 * throwing an exception if the result overflows an {@code int}.
779 *
780 * @param x the first value
781 * @param y the second value
782 * @return the result
783 * @throws ArithmeticException if the result overflows an int
784 * @since 1.8
785 */
786 public static int addExact(int x, int y) {
787 int r = x + y;
788 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
789 if (((x ^ r) & (y ^ r)) < 0) {
790 throw new ArithmeticException("integer overflow");
791 }
792 return r;
793 }
794
795 /**
796 * Returns the sum of its arguments,
797 * throwing an exception if the result overflows a {@code long}.
798 *
799 * @param x the first value
800 * @param y the second value
801 * @return the result
802 * @throws ArithmeticException if the result overflows a long
803 * @since 1.8
804 */
805 public static long addExact(long x, long y) {
806 long r = x + y;
807 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
808 if (((x ^ r) & (y ^ r)) < 0) {
809 throw new ArithmeticException("long overflow");
810 }
811 return r;
812 }
813
814 /**
815 * Returns the difference of the arguments,
816 * throwing an exception if the result overflows an {@code int}.
817 *
818 * @param x the first value
819 * @param y the second value to subtract from the first
820 * @return the result
821 * @throws ArithmeticException if the result overflows an int
822 * @since 1.8
823 */
824 public static int subtractExact(int x, int y) {
825 int r = x - y;
826 // HD 2-12 Overflow iff the arguments have different signs and
827 // the sign of the result is different than the sign of x
828 if (((x ^ y) & (x ^ r)) < 0) {
829 throw new ArithmeticException("integer overflow");
830 }
831 return r;
832 }
833
834 /**
835 * Returns the difference of the arguments,
836 * throwing an exception if the result overflows a {@code long}.
837 *
838 * @param x the first value
839 * @param y the second value to subtract from the first
840 * @return the result
841 * @throws ArithmeticException if the result overflows a long
842 * @since 1.8
843 */
844 public static long subtractExact(long x, long y) {
845 long r = x - y;
846 // HD 2-12 Overflow iff the arguments have different signs and
847 // the sign of the result is different than the sign of x
848 if (((x ^ y) & (x ^ r)) < 0) {
849 throw new ArithmeticException("long overflow");
850 }
851 return r;
852 }
853
854 /**
855 * Returns the product of the arguments,
856 * throwing an exception if the result overflows an {@code int}.
857 *
858 * @param x the first value
859 * @param y the second value
860 * @return the result
861 * @throws ArithmeticException if the result overflows an int
862 * @since 1.8
863 */
864 public static int multiplyExact(int x, int y) {
865 long r = (long)x * (long)y;
866 if ((int)r != r) {
867 throw new ArithmeticException("integer overflow");
868 }
869 return (int)r;
870 }
871
872 /**
873 * Returns the product of the arguments,
874 * throwing an exception if the result overflows a {@code long}.
875 *
876 * @param x the first value
877 * @param y the second value
878 * @return the result
879 * @throws ArithmeticException if the result overflows a long
880 * @since 1.8
881 */
882 public static long multiplyExact(long x, long y) {
883 long r = x * y;
884 long ax = Math.abs(x);
885 long ay = Math.abs(y);
886 if (((ax | ay) >>> 31 != 0)) {
887 // Some bits greater than 2^31 that might cause overflow
888 // Check the result using the divide operator
889 // and check for the special case of Long.MIN_VALUE * -1
890 if (((y != 0) && (r / y != x)) ||
891 (x == Long.MIN_VALUE && y == -1)) {
892 throw new ArithmeticException("long overflow");
893 }
894 }
895 return r;
896 }
897
898 /**
899 * Returns the argument incremented by one, throwing an exception if the
900 * result overflows an {@code int}.
901 *
902 * @param a the value to increment
903 * @return the result
904 * @throws ArithmeticException if the result overflows an int
905 * @since 1.8
906 */
907 public static int incrementExact(int a) {
908 if (a == Integer.MAX_VALUE) {
909 throw new ArithmeticException("integer overflow");
910 }
911
912 return a + 1;
913 }
914
915 /**
916 * Returns the argument incremented by one, throwing an exception if the
917 * result overflows a {@code long}.
918 *
919 * @param a the value to increment
920 * @return the result
921 * @throws ArithmeticException if the result overflows a long
922 * @since 1.8
923 */
924 public static long incrementExact(long a) {
925 if (a == Long.MAX_VALUE) {
926 throw new ArithmeticException("long overflow");
927 }
928
929 return a + 1L;
930 }
931
932 /**
933 * Returns the argument decremented by one, throwing an exception if the
934 * result overflows an {@code int}.
935 *
936 * @param a the value to decrement
937 * @return the result
938 * @throws ArithmeticException if the result overflows an int
939 * @since 1.8
940 */
941 public static int decrementExact(int a) {
942 if (a == Integer.MIN_VALUE) {
943 throw new ArithmeticException("integer overflow");
944 }
945
946 return a - 1;
947 }
948
949 /**
950 * Returns the argument decremented by one, throwing an exception if the
951 * result overflows a {@code long}.
952 *
953 * @param a the value to decrement
954 * @return the result
955 * @throws ArithmeticException if the result overflows a long
956 * @since 1.8
957 */
958 public static long decrementExact(long a) {
959 if (a == Long.MIN_VALUE) {
960 throw new ArithmeticException("long overflow");
961 }
962
963 return a - 1L;
964 }
965
966 /**
967 * Returns the negation of the argument, throwing an exception if the
968 * result overflows an {@code int}.
969 *
970 * @param a the value to negate
971 * @return the result
972 * @throws ArithmeticException if the result overflows an int
973 * @since 1.8
974 */
975 public static int negateExact(int a) {
976 if (a == Integer.MIN_VALUE) {
977 throw new ArithmeticException("integer overflow");
978 }
979
980 return -a;
981 }
982
983 /**
984 * Returns the negation of the argument, throwing an exception if the
985 * result overflows a {@code long}.
986 *
987 * @param a the value to negate
988 * @return the result
989 * @throws ArithmeticException if the result overflows a long
990 * @since 1.8
991 */
992 public static long negateExact(long a) {
993 if (a == Long.MIN_VALUE) {
994 throw new ArithmeticException("long overflow");
995 }
996
997 return -a;
998 }
999
1000 /**
1001 * Returns the value of the {@code long} argument;
1002 * throwing an exception if the value overflows an {@code int}.
1003 *
1004 * @param value the long value
1005 * @return the argument as an int
1006 * @throws ArithmeticException if the {@code argument} overflows an int
1007 * @since 1.8
1008 */
1009 public static int toIntExact(long value) {
1010 if ((int)value != value) {
1011 throw new ArithmeticException("integer overflow");
1012 }
1013 return (int)value;
1014 }
1015
1016 /**
1017 * Returns the largest (closest to positive infinity)
1018 * {@code int} value that is less than or equal to the algebraic quotient.
1019 * There is one special case, if the dividend is the
1020 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1021 * then integer overflow occurs and
1022 * the result is equal to the {@code Integer.MIN_VALUE}.
1023 * <p>
1024 * Normal integer division operates under the round to zero rounding mode
1025 * (truncation). This operation instead acts under the round toward
1026 * negative infinity (floor) rounding mode.
1027 * The floor rounding mode gives different results than truncation
1028 * when the exact result is negative.
1029 * <ul>
1030 * <li>If the signs of the arguments are the same, the results of
1031 * {@code floorDiv} and the {@code /} operator are the same. <br>
1032 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1033 * <li>If the signs of the arguments are different, the quotient is negative and
1034 * {@code floorDiv} returns the integer less than or equal to the quotient
1035 * and the {@code /} operator returns the integer closest to zero.<br>
1036 * For example, {@code floorDiv(-4, 3) == -2},
1037 * whereas {@code (-4 / 3) == -1}.
1038 * </li>
1039 * </ul>
1040 * <p>
1041 *
1042 * @param x the dividend
1043 * @param y the divisor
1044 * @return the largest (closest to positive infinity)
1045 * {@code int} value that is less than or equal to the algebraic quotient.
1046 * @throws ArithmeticException if the divisor {@code y} is zero
1047 * @see #floorMod(int, int)
1048 * @see #floor(double)
1049 * @since 1.8
1050 */
1051 public static int floorDiv(int x, int y) {
1052 int r = x / y;
1053 // if the signs are different and modulo not zero, round down
1054 if ((x ^ y) < 0 && (r * y != x)) {
1055 r--;
1056 }
1057 return r;
1058 }
1059
1060 /**
1061 * Returns the largest (closest to positive infinity)
1062 * {@code long} value that is less than or equal to the algebraic quotient.
1063 * There is one special case, if the dividend is the
1064 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1065 * then integer overflow occurs and
1066 * the result is equal to the {@code Long.MIN_VALUE}.
1067 * <p>
1068 * Normal integer division operates under the round to zero rounding mode
1069 * (truncation). This operation instead acts under the round toward
1070 * negative infinity (floor) rounding mode.
1071 * The floor rounding mode gives different results than truncation
1072 * when the exact result is negative.
1073 * <p>
1074 * For examples, see {@link #floorDiv(int, int)}.
1075 *
1076 * @param x the dividend
1077 * @param y the divisor
1078 * @return the largest (closest to positive infinity)
1079 * {@code long} value that is less than or equal to the algebraic quotient.
1080 * @throws ArithmeticException if the divisor {@code y} is zero
1081 * @see #floorMod(long, long)
1082 * @see #floor(double)
1083 * @since 1.8
1084 */
1085 public static long floorDiv(long x, long y) {
1086 long r = x / y;
1087 // if the signs are different and modulo not zero, round down
1088 if ((x ^ y) < 0 && (r * y != x)) {
1089 r--;
1090 }
1091 return r;
1092 }
1093
1094 /**
1095 * Returns the floor modulus of the {@code int} arguments.
1096 * <p>
1097 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1098 * has the same sign as the divisor {@code y}, and
1099 * is in the range of {@code -abs(y) < r < +abs(y)}.
1100 *
1101 * <p>
1102 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1103 * <ul>
1104 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1105 * </ul>
1106 * <p>
1107 * The difference in values between {@code floorMod} and
1108 * the {@code %} operator is due to the difference between
1109 * {@code floorDiv} that returns the integer less than or equal to the quotient
1110 * and the {@code /} operator that returns the integer closest to zero.
1111 * <p>
1112 * Examples:
1113 * <ul>
1114 * <li>If the signs of the arguments are the same, the results
1115 * of {@code floorMod} and the {@code %} operator are the same. <br>
1116 * <ul>
1117 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li>
1118 * </ul>
1119 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1120 * <ul>
1121 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li>
1122 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li>
1123 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li>
1124 * </ul>
1125 * </li>
1126 * </ul>
1127 * <p>
1128 * If the signs of arguments are unknown and a positive modulus
1129 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1130 *
1131 * @param x the dividend
1132 * @param y the divisor
1133 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1134 * @throws ArithmeticException if the divisor {@code y} is zero
1135 * @see #floorDiv(int, int)
1136 * @since 1.8
1137 */
1138 public static int floorMod(int x, int y) {
1139 int r = x - floorDiv(x, y) * y;
1140 return r;
1141 }
1142
1143 /**
1144 * Returns the floor modulus of the {@code long} arguments.
1145 * <p>
1146 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1147 * has the same sign as the divisor {@code y}, and
1148 * is in the range of {@code -abs(y) < r < +abs(y)}.
1149 *
1150 * <p>
1151 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1152 * <ul>
1153 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1154 * </ul>
1155 * <p>
1156 * For examples, see {@link #floorMod(int, int)}.
1157 *
1158 * @param x the dividend
1159 * @param y the divisor
1160 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1161 * @throws ArithmeticException if the divisor {@code y} is zero
1162 * @see #floorDiv(long, long)
1163 * @since 1.8
1164 */
1165 public static long floorMod(long x, long y) {
1166 return x - floorDiv(x, y) * y;
1167 }
1168
1169 /**
1170 * Returns the absolute value of an {@code int} value.
1171 * If the argument is not negative, the argument is returned.
1172 * If the argument is negative, the negation of the argument is returned.
1173 *
1174 * <p>Note that if the argument is equal to the value of
1175 * {@link Integer#MIN_VALUE}, the most negative representable
1176 * {@code int} value, the result is that same value, which is
1177 * negative.
1178 *
1179 * @param a the argument whose absolute value is to be determined
1180 * @return the absolute value of the argument.
1181 */
1182 public static int abs(int a) {
1183 return (a < 0) ? -a : a;
1184 }
1185
1186 /**
1187 * Returns the absolute value of a {@code long} value.
1188 * If the argument is not negative, the argument is returned.
1189 * If the argument is negative, the negation of the argument is returned.
1190 *
1191 * <p>Note that if the argument is equal to the value of
1192 * {@link Long#MIN_VALUE}, the most negative representable
1193 * {@code long} value, the result is that same value, which
1194 * is negative.
1195 *
1196 * @param a the argument whose absolute value is to be determined
1197 * @return the absolute value of the argument.
1198 */
1199 public static long abs(long a) {
1200 return (a < 0) ? -a : a;
1201 }
1202
1203 /**
1204 * Returns the absolute value of a {@code float} value.
1205 * If the argument is not negative, the argument is returned.
1206 * If the argument is negative, the negation of the argument is returned.
1207 * Special cases:
1208 * <ul><li>If the argument is positive zero or negative zero, the
1209 * result is positive zero.
1210 * <li>If the argument is infinite, the result is positive infinity.
1211 * <li>If the argument is NaN, the result is NaN.</ul>
1212 * In other words, the result is the same as the value of the expression:
1213 * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1214 *
1215 * @param a the argument whose absolute value is to be determined
1216 * @return the absolute value of the argument.
1217 */
1218 public static float abs(float a) {
1219 return (a <= 0.0F) ? 0.0F - a : a;
1220 }
1221
1222 /**
1223 * Returns the absolute value of a {@code double} value.
1224 * If the argument is not negative, the argument is returned.
1225 * If the argument is negative, the negation of the argument is returned.
1226 * Special cases:
1227 * <ul><li>If the argument is positive zero or negative zero, the result
1228 * is positive zero.
1229 * <li>If the argument is infinite, the result is positive infinity.
1230 * <li>If the argument is NaN, the result is NaN.</ul>
1231 * In other words, the result is the same as the value of the expression:
1232 * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1233 *
1234 * @param a the argument whose absolute value is to be determined
1235 * @return the absolute value of the argument.
1236 */
1237 public static double abs(double a) {
1238 return (a <= 0.0D) ? 0.0D - a : a;
1239 }
1240
1241 /**
1242 * Returns the greater of two {@code int} values. That is, the
1243 * result is the argument closer to the value of
1244 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1245 * the result is that same value.
1246 *
1247 * @param a an argument.
1248 * @param b another argument.
1249 * @return the larger of {@code a} and {@code b}.
1250 */
1251 public static int max(int a, int b) {
1252 return (a >= b) ? a : b;
1253 }
1254
1255 /**
1256 * Returns the greater of two {@code long} values. That is, the
1257 * result is the argument closer to the value of
1258 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1259 * the result is that same value.
1260 *
1261 * @param a an argument.
1262 * @param b another argument.
1263 * @return the larger of {@code a} and {@code b}.
1264 */
1265 public static long max(long a, long b) {
1266 return (a >= b) ? a : b;
1267 }
1268
1269 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1270 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1271 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1272
1273 /**
1274 * Returns the greater of two {@code float} values. That is,
1275 * the result is the argument closer to positive infinity. If the
1276 * arguments have the same value, the result is that same
1277 * value. If either value is NaN, then the result is NaN. Unlike
1278 * the numerical comparison operators, this method considers
1279 * negative zero to be strictly smaller than positive zero. If one
1280 * argument is positive zero and the other negative zero, the
1281 * result is positive zero.
1282 *
1283 * @param a an argument.
1284 * @param b another argument.
1285 * @return the larger of {@code a} and {@code b}.
1286 */
1287 public static float max(float a, float b) {
1288 if (a != a)
1289 return a; // a is NaN
1290 if ((a == 0.0f) &&
1291 (b == 0.0f) &&
1292 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1293 // Raw conversion ok since NaN can't map to -0.0.
1294 return b;
1295 }
1296 return (a >= b) ? a : b;
1297 }
1298
1299 /**
1300 * Returns the greater of two {@code double} values. That
1301 * is, the result is the argument closer to positive infinity. If
1302 * the arguments have the same value, the result is that same
1303 * value. If either value is NaN, then the result is NaN. Unlike
1304 * the numerical comparison operators, this method considers
1305 * negative zero to be strictly smaller than positive zero. If one
1306 * argument is positive zero and the other negative zero, the
1307 * result is positive zero.
1308 *
1309 * @param a an argument.
1310 * @param b another argument.
1311 * @return the larger of {@code a} and {@code b}.
1312 */
1313 public static double max(double a, double b) {
1314 if (a != a)
1315 return a; // a is NaN
1316 if ((a == 0.0d) &&
1317 (b == 0.0d) &&
1318 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1319 // Raw conversion ok since NaN can't map to -0.0.
1320 return b;
1321 }
1322 return (a >= b) ? a : b;
1323 }
1324
1325 /**
1326 * Returns the smaller of two {@code int} values. That is,
1327 * the result the argument closer to the value of
1328 * {@link Integer#MIN_VALUE}. If the arguments have the same
1329 * value, the result is that same value.
1330 *
1331 * @param a an argument.
1332 * @param b another argument.
1333 * @return the smaller of {@code a} and {@code b}.
1334 */
1335 public static int min(int a, int b) {
1336 return (a <= b) ? a : b;
1337 }
1338
1339 /**
1340 * Returns the smaller of two {@code long} values. That is,
1341 * the result is the argument closer to the value of
1342 * {@link Long#MIN_VALUE}. If the arguments have the same
1343 * value, the result is that same value.
1344 *
1345 * @param a an argument.
1346 * @param b another argument.
1347 * @return the smaller of {@code a} and {@code b}.
1348 */
1349 public static long min(long a, long b) {
1350 return (a <= b) ? a : b;
1351 }
1352
1353 /**
1354 * Returns the smaller of two {@code float} values. That is,
1355 * the result is the value closer to negative infinity. If the
1356 * arguments have the same value, the result is that same
1357 * value. If either value is NaN, then the result is NaN. Unlike
1358 * the numerical comparison operators, this method considers
1359 * negative zero to be strictly smaller than positive zero. If
1360 * one argument is positive zero and the other is negative zero,
1361 * the result is negative zero.
1362 *
1363 * @param a an argument.
1364 * @param b another argument.
1365 * @return the smaller of {@code a} and {@code b}.
1366 */
1367 public static float min(float a, float b) {
1368 if (a != a)
1369 return a; // a is NaN
1370 if ((a == 0.0f) &&
1371 (b == 0.0f) &&
1372 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1373 // Raw conversion ok since NaN can't map to -0.0.
1374 return b;
1375 }
1376 return (a <= b) ? a : b;
1377 }
1378
1379 /**
1380 * Returns the smaller of two {@code double} values. That
1381 * is, the result is the value closer to negative infinity. If the
1382 * arguments have the same value, the result is that same
1383 * value. If either value is NaN, then the result is NaN. Unlike
1384 * the numerical comparison operators, this method considers
1385 * negative zero to be strictly smaller than positive zero. If one
1386 * argument is positive zero and the other is negative zero, the
1387 * result is negative zero.
1388 *
1389 * @param a an argument.
1390 * @param b another argument.
1391 * @return the smaller of {@code a} and {@code b}.
1392 */
1393 public static double min(double a, double b) {
1394 if (a != a)
1395 return a; // a is NaN
1396 if ((a == 0.0d) &&
1397 (b == 0.0d) &&
1398 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1399 // Raw conversion ok since NaN can't map to -0.0.
1400 return b;
1401 }
1402 return (a <= b) ? a : b;
1403 }
1404
1405 /**
1406 * Returns the size of an ulp of the argument. An ulp, unit in
1407 * the last place, of a {@code double} value is the positive
1408 * distance between this floating-point value and the {@code
1409 * double} value next larger in magnitude. Note that for non-NaN
1410 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1411 *
1412 * <p>Special Cases:
1413 * <ul>
1414 * <li> If the argument is NaN, then the result is NaN.
1415 * <li> If the argument is positive or negative infinity, then the
1416 * result is positive infinity.
1417 * <li> If the argument is positive or negative zero, then the result is
1418 * {@code Double.MIN_VALUE}.
1419 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1420 * the result is equal to 2<sup>971</sup>.
1421 * </ul>
1422 *
1423 * @param d the floating-point value whose ulp is to be returned
1424 * @return the size of an ulp of the argument
1425 * @author Joseph D. Darcy
1426 * @since 1.5
1427 */
1428 public static double ulp(double d) {
1429 int exp = getExponent(d);
1430
1431 switch(exp) {
1432 case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
1433 return Math.abs(d);
1434
1435 case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
1436 return Double.MIN_VALUE;
1437
1438 default:
1439 assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
1440
1441 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1442 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1443 if (exp >= DoubleConsts.MIN_EXPONENT) {
1444 return powerOfTwoD(exp);
1445 }
1446 else {
1447 // return a subnormal result; left shift integer
1448 // representation of Double.MIN_VALUE appropriate
1449 // number of positions
1450 return Double.longBitsToDouble(1L <<
1451 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1452 }
1453 }
1454 }
1455
1456 /**
1457 * Returns the size of an ulp of the argument. An ulp, unit in
1458 * the last place, of a {@code float} value is the positive
1459 * distance between this floating-point value and the {@code
1460 * float} value next larger in magnitude. Note that for non-NaN
1461 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1462 *
1463 * <p>Special Cases:
1464 * <ul>
1465 * <li> If the argument is NaN, then the result is NaN.
1466 * <li> If the argument is positive or negative infinity, then the
1467 * result is positive infinity.
1468 * <li> If the argument is positive or negative zero, then the result is
1469 * {@code Float.MIN_VALUE}.
1470 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1471 * the result is equal to 2<sup>104</sup>.
1472 * </ul>
1473 *
1474 * @param f the floating-point value whose ulp is to be returned
1475 * @return the size of an ulp of the argument
1476 * @author Joseph D. Darcy
1477 * @since 1.5
1478 */
1479 public static float ulp(float f) {
1480 int exp = getExponent(f);
1481
1482 switch(exp) {
1483 case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
1484 return Math.abs(f);
1485
1486 case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
1487 return FloatConsts.MIN_VALUE;
1488
1489 default:
1490 assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1491
1492 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1493 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1494 if (exp >= FloatConsts.MIN_EXPONENT) {
1495 return powerOfTwoF(exp);
1496 }
1497 else {
1498 // return a subnormal result; left shift integer
1499 // representation of FloatConsts.MIN_VALUE appropriate
1500 // number of positions
1501 return Float.intBitsToFloat(1 <<
1502 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1503 }
1504 }
1505 }
1506
1507 /**
1508 * Returns the signum function of the argument; zero if the argument
1509 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1510 * argument is less than zero.
1511 *
1512 * <p>Special Cases:
1513 * <ul>
1514 * <li> If the argument is NaN, then the result is NaN.
1515 * <li> If the argument is positive zero or negative zero, then the
1516 * result is the same as the argument.
1517 * </ul>
1518 *
1519 * @param d the floating-point value whose signum is to be returned
1520 * @return the signum function of the argument
1521 * @author Joseph D. Darcy
1522 * @since 1.5
1523 */
1524 public static double signum(double d) {
1525 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1526 }
1527
1528 /**
1529 * Returns the signum function of the argument; zero if the argument
1530 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1531 * argument is less than zero.
1532 *
1533 * <p>Special Cases:
1534 * <ul>
1535 * <li> If the argument is NaN, then the result is NaN.
1536 * <li> If the argument is positive zero or negative zero, then the
1537 * result is the same as the argument.
1538 * </ul>
1539 *
1540 * @param f the floating-point value whose signum is to be returned
1541 * @return the signum function of the argument
1542 * @author Joseph D. Darcy
1543 * @since 1.5
1544 */
1545 public static float signum(float f) {
1546 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1547 }
1548
1549 /**
1550 * Returns the hyperbolic sine of a {@code double} value.
1551 * The hyperbolic sine of <i>x</i> is defined to be
1552 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1553 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1554 *
1555 * <p>Special cases:
1556 * <ul>
1557 *
1558 * <li>If the argument is NaN, then the result is NaN.
1559 *
1560 * <li>If the argument is infinite, then the result is an infinity
1561 * with the same sign as the argument.
1562 *
1563 * <li>If the argument is zero, then the result is a zero with the
1564 * same sign as the argument.
1565 *
1566 * </ul>
1567 *
1568 * <p>The computed result must be within 2.5 ulps of the exact result.
1569 *
1570 * @param x The number whose hyperbolic sine is to be returned.
1571 * @return The hyperbolic sine of {@code x}.
1572 * @since 1.5
1573 */
1574 public static double sinh(double x) {
1575 return StrictMath.sinh(x);
1576 }
1577
1578 /**
1579 * Returns the hyperbolic cosine of a {@code double} value.
1580 * The hyperbolic cosine of <i>x</i> is defined to be
1581 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1582 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1583 *
1584 * <p>Special cases:
1585 * <ul>
1586 *
1587 * <li>If the argument is NaN, then the result is NaN.
1588 *
1589 * <li>If the argument is infinite, then the result is positive
1590 * infinity.
1591 *
1592 * <li>If the argument is zero, then the result is {@code 1.0}.
1593 *
1594 * </ul>
1595 *
1596 * <p>The computed result must be within 2.5 ulps of the exact result.
1597 *
1598 * @param x The number whose hyperbolic cosine is to be returned.
1599 * @return The hyperbolic cosine of {@code x}.
1600 * @since 1.5
1601 */
1602 public static double cosh(double x) {
1603 return StrictMath.cosh(x);
1604 }
1605
1606 /**
1607 * Returns the hyperbolic tangent of a {@code double} value.
1608 * The hyperbolic tangent of <i>x</i> is defined to be
1609 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
1610 * in other words, {@linkplain Math#sinh
1611 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
1612 * that the absolute value of the exact tanh is always less than
1613 * 1.
1614 *
1615 * <p>Special cases:
1616 * <ul>
1617 *
1618 * <li>If the argument is NaN, then the result is NaN.
1619 *
1620 * <li>If the argument is zero, then the result is a zero with the
1621 * same sign as the argument.
1622 *
1623 * <li>If the argument is positive infinity, then the result is
1624 * {@code +1.0}.
1625 *
1626 * <li>If the argument is negative infinity, then the result is
1627 * {@code -1.0}.
1628 *
1629 * </ul>
1630 *
1631 * <p>The computed result must be within 2.5 ulps of the exact result.
1632 * The result of {@code tanh} for any finite input must have
1633 * an absolute value less than or equal to 1. Note that once the
1634 * exact result of tanh is within 1/2 of an ulp of the limit value
1635 * of ±1, correctly signed ±{@code 1.0} should
1636 * be returned.
1637 *
1638 * @param x The number whose hyperbolic tangent is to be returned.
1639 * @return The hyperbolic tangent of {@code x}.
1640 * @since 1.5
1641 */
1642 public static double tanh(double x) {
1643 return StrictMath.tanh(x);
1644 }
1645
1646 /**
1647 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1648 * without intermediate overflow or underflow.
1649 *
1650 * <p>Special cases:
1651 * <ul>
1652 *
1653 * <li> If either argument is infinite, then the result
1654 * is positive infinity.
1655 *
1656 * <li> If either argument is NaN and neither argument is infinite,
1657 * then the result is NaN.
1658 *
1659 * </ul>
1660 *
1661 * <p>The computed result must be within 1 ulp of the exact
1662 * result. If one parameter is held constant, the results must be
1663 * semi-monotonic in the other parameter.
1664 *
1665 * @param x a value
1666 * @param y a value
1667 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1668 * without intermediate overflow or underflow
1669 * @since 1.5
1670 */
1671 public static double hypot(double x, double y) {
1672 return StrictMath.hypot(x, y);
1673 }
1674
1675 /**
1676 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
1677 * <i>x</i> near 0, the exact sum of
1678 * {@code expm1(x)} + 1 is much closer to the true
1679 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1680 *
1681 * <p>Special cases:
1682 * <ul>
1683 * <li>If the argument is NaN, the result is NaN.
1684 *
1685 * <li>If the argument is positive infinity, then the result is
1686 * positive infinity.
1687 *
1688 * <li>If the argument is negative infinity, then the result is
1689 * -1.0.
1690 *
1691 * <li>If the argument is zero, then the result is a zero with the
1692 * same sign as the argument.
1693 *
1694 * </ul>
1695 *
1696 * <p>The computed result must be within 1 ulp of the exact result.
1697 * Results must be semi-monotonic. The result of
1698 * {@code expm1} for any finite input must be greater than or
1699 * equal to {@code -1.0}. Note that once the exact result of
1700 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
1701 * ulp of the limit value -1, {@code -1.0} should be
1702 * returned.
1703 *
1704 * @param x the exponent to raise <i>e</i> to in the computation of
1705 * <i>e</i><sup>{@code x}</sup> -1.
1706 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
1707 * @since 1.5
1708 */
1709 public static double expm1(double x) {
1710 return StrictMath.expm1(x);
1711 }
1712
1713 /**
1714 * Returns the natural logarithm of the sum of the argument and 1.
1715 * Note that for small values {@code x}, the result of
1716 * {@code log1p(x)} is much closer to the true result of ln(1
1717 * + {@code x}) than the floating-point evaluation of
1718 * {@code log(1.0+x)}.
1719 *
1720 * <p>Special cases:
1721 *
1722 * <ul>
1723 *
1724 * <li>If the argument is NaN or less than -1, then the result is
1725 * NaN.
1726 *
1727 * <li>If the argument is positive infinity, then the result is
1728 * positive infinity.
1729 *
1730 * <li>If the argument is negative one, then the result is
1731 * negative infinity.
1732 *
1733 * <li>If the argument is zero, then the result is a zero with the
1734 * same sign as the argument.
1735 *
1736 * </ul>
1737 *
1738 * <p>The computed result must be within 1 ulp of the exact result.
1739 * Results must be semi-monotonic.
1740 *
1741 * @param x a value
1742 * @return the value ln({@code x} + 1), the natural
1743 * log of {@code x} + 1
1744 * @since 1.5
1745 */
1746 public static double log1p(double x) {
1747 return StrictMath.log1p(x);
1748 }
1749
1750 /**
1751 * Returns the first floating-point argument with the sign of the
1752 * second floating-point argument. Note that unlike the {@link
1753 * StrictMath#copySign(double, double) StrictMath.copySign}
1754 * method, this method does not require NaN {@code sign}
1755 * arguments to be treated as positive values; implementations are
1756 * permitted to treat some NaN arguments as positive and other NaN
1757 * arguments as negative to allow greater performance.
1758 *
1759 * @param magnitude the parameter providing the magnitude of the result
1760 * @param sign the parameter providing the sign of the result
1761 * @return a value with the magnitude of {@code magnitude}
1762 * and the sign of {@code sign}.
1763 * @since 1.6
1764 */
1765 public static double copySign(double magnitude, double sign) {
1766 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1767 (DoubleConsts.SIGN_BIT_MASK)) |
1768 (Double.doubleToRawLongBits(magnitude) &
1769 (DoubleConsts.EXP_BIT_MASK |
1770 DoubleConsts.SIGNIF_BIT_MASK)));
1771 }
1772
1773 /**
1774 * Returns the first floating-point argument with the sign of the
1775 * second floating-point argument. Note that unlike the {@link
1776 * StrictMath#copySign(float, float) StrictMath.copySign}
1777 * method, this method does not require NaN {@code sign}
1778 * arguments to be treated as positive values; implementations are
1779 * permitted to treat some NaN arguments as positive and other NaN
1780 * arguments as negative to allow greater performance.
1781 *
1782 * @param magnitude the parameter providing the magnitude of the result
1783 * @param sign the parameter providing the sign of the result
1784 * @return a value with the magnitude of {@code magnitude}
1785 * and the sign of {@code sign}.
1786 * @since 1.6
1787 */
1788 public static float copySign(float magnitude, float sign) {
1789 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1790 (FloatConsts.SIGN_BIT_MASK)) |
1791 (Float.floatToRawIntBits(magnitude) &
1792 (FloatConsts.EXP_BIT_MASK |
1793 FloatConsts.SIGNIF_BIT_MASK)));
1794 }
1795
1796 /**
1797 * Returns the unbiased exponent used in the representation of a
1798 * {@code float}. Special cases:
1799 *
1800 * <ul>
1801 * <li>If the argument is NaN or infinite, then the result is
1802 * {@link Float#MAX_EXPONENT} + 1.
1803 * <li>If the argument is zero or subnormal, then the result is
1804 * {@link Float#MIN_EXPONENT} -1.
1805 * </ul>
1806 * @param f a {@code float} value
1807 * @return the unbiased exponent of the argument
1808 * @since 1.6
1809 */
1810 public static int getExponent(float f) {
1811 /*
1812 * Bitwise convert f to integer, mask out exponent bits, shift
1813 * to the right and then subtract out float's bias adjust to
1814 * get true exponent value
1815 */
1816 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1817 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1818 }
1819
1820 /**
1821 * Returns the unbiased exponent used in the representation of a
1822 * {@code double}. Special cases:
1823 *
1824 * <ul>
1825 * <li>If the argument is NaN or infinite, then the result is
1826 * {@link Double#MAX_EXPONENT} + 1.
1827 * <li>If the argument is zero or subnormal, then the result is
1828 * {@link Double#MIN_EXPONENT} -1.
1829 * </ul>
1830 * @param d a {@code double} value
1831 * @return the unbiased exponent of the argument
1832 * @since 1.6
1833 */
1834 public static int getExponent(double d) {
1835 /*
1836 * Bitwise convert d to long, mask out exponent bits, shift
1837 * to the right and then subtract out double's bias adjust to
1838 * get true exponent value.
1839 */
1840 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1841 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1842 }
1843
1844 /**
1845 * Returns the floating-point number adjacent to the first
1846 * argument in the direction of the second argument. If both
1847 * arguments compare as equal the second argument is returned.
1848 *
1849 * <p>
1850 * Special cases:
1851 * <ul>
1852 * <li> If either argument is a NaN, then NaN is returned.
1853 *
1854 * <li> If both arguments are signed zeros, {@code direction}
1855 * is returned unchanged (as implied by the requirement of
1856 * returning the second argument if the arguments compare as
1857 * equal).
1858 *
1859 * <li> If {@code start} is
1860 * ±{@link Double#MIN_VALUE} and {@code direction}
1861 * has a value such that the result should have a smaller
1862 * magnitude, then a zero with the same sign as {@code start}
1863 * is returned.
1864 *
1865 * <li> If {@code start} is infinite and
1866 * {@code direction} has a value such that the result should
1867 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1868 * same sign as {@code start} is returned.
1869 *
1870 * <li> If {@code start} is equal to ±
1871 * {@link Double#MAX_VALUE} and {@code direction} has a
1872 * value such that the result should have a larger magnitude, an
1873 * infinity with same sign as {@code start} is returned.
1874 * </ul>
1875 *
1876 * @param start starting floating-point value
1877 * @param direction value indicating which of
1878 * {@code start}'s neighbors or {@code start} should
1879 * be returned
1880 * @return The floating-point number adjacent to {@code start} in the
1881 * direction of {@code direction}.
1882 * @since 1.6
1883 */
1884 public static double nextAfter(double start, double direction) {
1885 /*
1886 * The cases:
1887 *
1888 * nextAfter(+infinity, 0) == MAX_VALUE
1889 * nextAfter(+infinity, +infinity) == +infinity
1890 * nextAfter(-infinity, 0) == -MAX_VALUE
1891 * nextAfter(-infinity, -infinity) == -infinity
1892 *
1893 * are naturally handled without any additional testing
1894 */
1895
1896 // First check for NaN values
1897 if (Double.isNaN(start) || Double.isNaN(direction)) {
1898 // return a NaN derived from the input NaN(s)
1899 return start + direction;
1900 } else if (start == direction) {
1901 return direction;
1902 } else { // start > direction or start < direction
1903 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1904 // then bitwise convert start to integer.
1905 long transducer = Double.doubleToRawLongBits(start + 0.0d);
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 values
1915 * is less than zero, the adjustment to the representation
1916 * is in the opposite direction than would be expected at
1917 * first .
1918 */
1919 if (direction > start) { // Calculate next greater value
1920 transducer = transducer + (transducer >= 0L ? 1L:-1L);
1921 } else { // Calculate next lesser value
1922 assert direction < start;
1923 if (transducer > 0L)
1924 --transducer;
1925 else
1926 if (transducer < 0L )
1927 ++transducer;
1928 /*
1929 * transducer==0, the result is -MIN_VALUE
1930 *
1931 * The transition from zero (implicitly
1932 * positive) to the smallest negative
1933 * signed magnitude value must be done
1934 * explicitly.
1935 */
1936 else
1937 transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
1938 }
1939
1940 return Double.longBitsToDouble(transducer);
1941 }
1942 }
1943
1944 /**
1945 * Returns the floating-point number adjacent to the first
1946 * argument in the direction of the second argument. If both
1947 * arguments compare as equal a value equivalent to the second argument
1948 * is returned.
1949 *
1950 * <p>
1951 * Special cases:
1952 * <ul>
1953 * <li> If either argument is a NaN, then NaN is returned.
1954 *
1955 * <li> If both arguments are signed zeros, a value equivalent
1956 * to {@code direction} is returned.
1957 *
1958 * <li> If {@code start} is
1959 * ±{@link Float#MIN_VALUE} and {@code direction}
1960 * has a value such that the result should have a smaller
1961 * magnitude, then a zero with the same sign as {@code start}
1962 * is returned.
1963 *
1964 * <li> If {@code start} is infinite and
1965 * {@code direction} has a value such that the result should
1966 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1967 * same sign as {@code start} is returned.
1968 *
1969 * <li> If {@code start} is equal to ±
1970 * {@link Float#MAX_VALUE} and {@code direction} has a
1971 * value such that the result should have a larger magnitude, an
1972 * infinity with same sign as {@code start} is returned.
1973 * </ul>
1974 *
1975 * @param start starting floating-point value
1976 * @param direction value indicating which of
1977 * {@code start}'s neighbors or {@code start} should
1978 * be returned
1979 * @return The floating-point number adjacent to {@code start} in the
1980 * direction of {@code direction}.
1981 * @since 1.6
1982 */
1983 public static float nextAfter(float start, double direction) {
1984 /*
1985 * The cases:
1986 *
1987 * nextAfter(+infinity, 0) == MAX_VALUE
1988 * nextAfter(+infinity, +infinity) == +infinity
1989 * nextAfter(-infinity, 0) == -MAX_VALUE
1990 * nextAfter(-infinity, -infinity) == -infinity
1991 *
1992 * are naturally handled without any additional testing
1993 */
1994
1995 // First check for NaN values
1996 if (Float.isNaN(start) || Double.isNaN(direction)) {
1997 // return a NaN derived from the input NaN(s)
1998 return start + (float)direction;
1999 } else if (start == direction) {
2000 return (float)direction;
2001 } else { // start > direction or start < direction
2002 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2003 // then bitwise convert start to integer.
2004 int transducer = Float.floatToRawIntBits(start + 0.0f);
2005
2006 /*
2007 * IEEE 754 floating-point numbers are lexicographically
2008 * ordered if treated as signed- magnitude integers .
2009 * Since Java's integers are two's complement,
2010 * incrementing" the two's complement representation of a
2011 * logically negative floating-point value *decrements*
2012 * the signed-magnitude representation. Therefore, when
2013 * the integer representation of a floating-point values
2014 * is less than zero, the adjustment to the representation
2015 * is in the opposite direction than would be expected at
2016 * first.
2017 */
2018 if (direction > start) {// Calculate next greater value
2019 transducer = transducer + (transducer >= 0 ? 1:-1);
2020 } else { // Calculate next lesser value
2021 assert direction < start;
2022 if (transducer > 0)
2023 --transducer;
2024 else
2025 if (transducer < 0 )
2026 ++transducer;
2027 /*
2028 * transducer==0, the result is -MIN_VALUE
2029 *
2030 * The transition from zero (implicitly
2031 * positive) to the smallest negative
2032 * signed magnitude value must be done
2033 * explicitly.
2034 */
2035 else
2036 transducer = FloatConsts.SIGN_BIT_MASK | 1;
2037 }
2038
2039 return Float.intBitsToFloat(transducer);
2040 }
2041 }
2042
2043 /**
2044 * Returns the floating-point value adjacent to {@code d} in
2045 * the direction of positive infinity. This method is
2046 * semantically equivalent to {@code nextAfter(d,
2047 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2048 * implementation may run faster than its equivalent
2049 * {@code nextAfter} call.
2050 *
2051 * <p>Special Cases:
2052 * <ul>
2053 * <li> If the argument is NaN, the result is NaN.
2054 *
2055 * <li> If the argument is positive infinity, the result is
2056 * positive infinity.
2057 *
2058 * <li> If the argument is zero, the result is
2059 * {@link Double#MIN_VALUE}
2060 *
2061 * </ul>
2062 *
2063 * @param d starting floating-point value
2064 * @return The adjacent floating-point value closer to positive
2065 * infinity.
2066 * @since 1.6
2067 */
2068 public static double nextUp(double d) {
2069 if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
2070 return d;
2071 else {
2072 d += 0.0d;
2073 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2074 ((d >= 0.0d)?+1L:-1L));
2075 }
2076 }
2077
2078 /**
2079 * Returns the floating-point value adjacent to {@code f} in
2080 * the direction of positive infinity. This method is
2081 * semantically equivalent to {@code nextAfter(f,
2082 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2083 * implementation may run faster than its equivalent
2084 * {@code nextAfter} call.
2085 *
2086 * <p>Special Cases:
2087 * <ul>
2088 * <li> If the argument is NaN, the result is NaN.
2089 *
2090 * <li> If the argument is positive infinity, the result is
2091 * positive infinity.
2092 *
2093 * <li> If the argument is zero, the result is
2094 * {@link Float#MIN_VALUE}
2095 *
2096 * </ul>
2097 *
2098 * @param f starting floating-point value
2099 * @return The adjacent floating-point value closer to positive
2100 * infinity.
2101 * @since 1.6
2102 */
2103 public static float nextUp(float f) {
2104 if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
2105 return f;
2106 else {
2107 f += 0.0f;
2108 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2109 ((f >= 0.0f)?+1:-1));
2110 }
2111 }
2112
2113 /**
2114 * Returns the floating-point value adjacent to {@code d} in
2115 * the direction of negative infinity. This method is
2116 * semantically equivalent to {@code nextAfter(d,
2117 * Double.NEGATIVE_INFINITY)}; however, a
2118 * {@code nextDown} implementation may run faster than its
2119 * equivalent {@code nextAfter} call.
2120 *
2121 * <p>Special Cases:
2122 * <ul>
2123 * <li> If the argument is NaN, the result is NaN.
2124 *
2125 * <li> If the argument is negative infinity, the result is
2126 * negative infinity.
2127 *
2128 * <li> If the argument is zero, the result is
2129 * {@code -Double.MIN_VALUE}
2130 *
2131 * </ul>
2132 *
2133 * @param d starting floating-point value
2134 * @return The adjacent floating-point value closer to negative
2135 * infinity.
2136 * @since 1.8
2137 */
2138 public static double nextDown(double d) {
2139 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2140 return d;
2141 else {
2142 if (d == 0.0)
2143 return -Double.MIN_VALUE;
2144 else
2145 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2146 ((d > 0.0d)?-1L:+1L));
2147 }
2148 }
2149
2150 /**
2151 * Returns the floating-point value adjacent to {@code f} in
2152 * the direction of negative infinity. This method is
2153 * semantically equivalent to {@code nextAfter(f,
2154 * Float.NEGATIVE_INFINITY)}; however, a
2155 * {@code nextDown} implementation may run faster than its
2156 * equivalent {@code nextAfter} call.
2157 *
2158 * <p>Special Cases:
2159 * <ul>
2160 * <li> If the argument is NaN, the result is NaN.
2161 *
2162 * <li> If the argument is negative infinity, the result is
2163 * negative infinity.
2164 *
2165 * <li> If the argument is zero, the result is
2166 * {@code -Float.MIN_VALUE}
2167 *
2168 * </ul>
2169 *
2170 * @param f starting floating-point value
2171 * @return The adjacent floating-point value closer to negative
2172 * infinity.
2173 * @since 1.8
2174 */
2175 public static float nextDown(float f) {
2176 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2177 return f;
2178 else {
2179 if (f == 0.0f)
2180 return -Float.MIN_VALUE;
2181 else
2182 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2183 ((f > 0.0f)?-1:+1));
2184 }
2185 }
2186
2187 /**
2188 * Returns {@code d} ×
2189 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2190 * by a single correctly rounded floating-point multiply to a
2191 * member of the double value set. See the Java
2192 * Language Specification for a discussion of floating-point
2193 * value sets. If the exponent of the result is between {@link
2194 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2195 * answer is calculated exactly. If the exponent of the result
2196 * would be larger than {@code Double.MAX_EXPONENT}, an
2197 * infinity is returned. Note that if the result is subnormal,
2198 * precision may be lost; that is, when {@code scalb(x, n)}
2199 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2200 * <i>x</i>. When the result is non-NaN, the result has the same
2201 * sign as {@code d}.
2202 *
2203 * <p>Special cases:
2204 * <ul>
2205 * <li> If the first argument is NaN, NaN is returned.
2206 * <li> If the first argument is infinite, then an infinity of the
2207 * same sign is returned.
2208 * <li> If the first argument is zero, then a zero of the same
2209 * sign is returned.
2210 * </ul>
2211 *
2212 * @param d number to be scaled by a power of two.
2213 * @param scaleFactor power of 2 used to scale {@code d}
2214 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2215 * @since 1.6
2216 */
2217 public static double scalb(double d, int scaleFactor) {
2218 /*
2219 * This method does not need to be declared strictfp to
2220 * compute the same correct result on all platforms. When
2221 * scaling up, it does not matter what order the
2222 * multiply-store operations are done; the result will be
2223 * finite or overflow regardless of the operation ordering.
2224 * However, to get the correct result when scaling down, a
2225 * particular ordering must be used.
2226 *
2227 * When scaling down, the multiply-store operations are
2228 * sequenced so that it is not possible for two consecutive
2229 * multiply-stores to return subnormal results. If one
2230 * multiply-store result is subnormal, the next multiply will
2231 * round it away to zero. This is done by first multiplying
2232 * by 2 ^ (scaleFactor % n) and then multiplying several
2233 * times by by 2^n as needed where n is the exponent of number
2234 * that is a covenient power of two. In this way, at most one
2235 * real rounding error occurs. If the double value set is
2236 * being used exclusively, the rounding will occur on a
2237 * multiply. If the double-extended-exponent value set is
2238 * being used, the products will (perhaps) be exact but the
2239 * stores to d are guaranteed to round to the double value
2240 * set.
2241 *
2242 * It is _not_ a valid implementation to first multiply d by
2243 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2244 * MIN_EXPONENT) since even in a strictfp program double
2245 * rounding on underflow could occur; e.g. if the scaleFactor
2246 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2247 * little less than -(MIN_EXPONENT - n), meaning the final
2248 * result would be subnormal.
2249 *
2250 * Since exact reproducibility of this method can be achieved
2251 * without any undue performance burden, there is no
2252 * compelling reason to allow double rounding on underflow in
2253 * scalb.
2254 */
2255
2256 // magnitude of a power of two so large that scaling a finite
2257 // nonzero value by it would be guaranteed to over or
2258 // underflow; due to rounding, scaling down takes takes an
2259 // additional power of two which is reflected here
2260 final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
2261 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2262 int exp_adjust = 0;
2263 int scale_increment = 0;
2264 double exp_delta = Double.NaN;
2265
2266 // Make sure scaling factor is in a reasonable range
2267
2268 if(scaleFactor < 0) {
2269 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2270 scale_increment = -512;
2271 exp_delta = twoToTheDoubleScaleDown;
2272 }
2273 else {
2274 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2275 scale_increment = 512;
2276 exp_delta = twoToTheDoubleScaleUp;
2277 }
2278
2279 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2280 // technique from "Hacker's Delight" section 10-2.
2281 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2282 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2283
2284 d *= powerOfTwoD(exp_adjust);
2285 scaleFactor -= exp_adjust;
2286
2287 while(scaleFactor != 0) {
2288 d *= exp_delta;
2289 scaleFactor -= scale_increment;
2290 }
2291 return d;
2292 }
2293
2294 /**
2295 * Returns {@code f} ×
2296 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2297 * by a single correctly rounded floating-point multiply to a
2298 * member of the float value set. See the Java
2299 * Language Specification for a discussion of floating-point
2300 * value sets. If the exponent of the result is between {@link
2301 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2302 * answer is calculated exactly. If the exponent of the result
2303 * would be larger than {@code Float.MAX_EXPONENT}, an
2304 * infinity is returned. Note that if the result is subnormal,
2305 * precision may be lost; that is, when {@code scalb(x, n)}
2306 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2307 * <i>x</i>. When the result is non-NaN, the result has the same
2308 * sign as {@code f}.
2309 *
2310 * <p>Special cases:
2311 * <ul>
2312 * <li> If the first argument is NaN, NaN is returned.
2313 * <li> If the first argument is infinite, then an infinity of the
2314 * same sign is returned.
2315 * <li> If the first argument is zero, then a zero of the same
2316 * sign is returned.
2317 * </ul>
2318 *
2319 * @param f number to be scaled by a power of two.
2320 * @param scaleFactor power of 2 used to scale {@code f}
2321 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2322 * @since 1.6
2323 */
2324 public static float scalb(float f, int scaleFactor) {
2325 // magnitude of a power of two so large that scaling a finite
2326 // nonzero value by it would be guaranteed to over or
2327 // underflow; due to rounding, scaling down takes takes an
2328 // additional power of two which is reflected here
2329 final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
2330 FloatConsts.SIGNIFICAND_WIDTH + 1;
2331
2332 // Make sure scaling factor is in a reasonable range
2333 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2334
2335 /*
2336 * Since + MAX_SCALE for float fits well within the double
2337 * exponent range and + float -> double conversion is exact
2338 * the multiplication below will be exact. Therefore, the
2339 * rounding that occurs when the double product is cast to
2340 * float will be the correctly rounded float result. Since
2341 * all operations other than the final multiply will be exact,
2342 * it is not necessary to declare this method strictfp.
2343 */
2344 return (float)((double)f*powerOfTwoD(scaleFactor));
2345 }
2346
2347 // Constants used in scalb
2348 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2349 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2350
2351 /**
2352 * Returns a floating-point power of two in the normal range.
2353 */
2354 static double powerOfTwoD(int n) {
2355 assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
2356 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2357 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2358 & DoubleConsts.EXP_BIT_MASK);
2359 }
2360
2361 /**
2362 * Returns a floating-point power of two in the normal range.
2363 */
2364 static float powerOfTwoF(int n) {
2365 assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
2366 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2367 (FloatConsts.SIGNIFICAND_WIDTH-1))
2368 & FloatConsts.EXP_BIT_MASK);
2369 }
2370 }