1 /*
2 * Copyright (c) 1994, 2011, 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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20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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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 * @author unascribed
85 * @author Joseph D. Darcy
86 * @since JDK1.0
87 */
88
89 public final class Math {
90
91 /**
92 * Don't let anyone instantiate this class.
93 */
94 private Math() {}
95
96 /**
97 * The {@code double} value that is closer than any other to
98 * <i>e</i>, the base of the natural logarithms.
99 */
100 public static final double E = 2.7182818284590452354;
101
102 /**
103 * The {@code double} value that is closer than any other to
104 * <i>pi</i>, the ratio of the circumference of a circle to its
105 * diameter.
106 */
107 public static final double PI = 3.14159265358979323846;
108
109 /**
110 * Returns the trigonometric sine of an angle. Special cases:
111 * <ul><li>If the argument is NaN or an infinity, then the
112 * result is NaN.
113 * <li>If the argument is zero, then the result is a zero with the
114 * same sign as the argument.</ul>
115 *
116 * <p>The computed result must be within 1 ulp of the exact result.
117 * Results must be semi-monotonic.
118 *
119 * @param a an angle, in radians.
120 * @return the sine of the argument.
121 */
122 public static double sin(double a) {
123 return StrictMath.sin(a); // default impl. delegates to StrictMath
124 }
125
126 /**
127 * Returns the trigonometric cosine of an angle. Special cases:
128 * <ul><li>If the argument is NaN or an infinity, then the
129 * result is NaN.</ul>
130 *
131 * <p>The computed result must be within 1 ulp of the exact result.
132 * Results must be semi-monotonic.
133 *
134 * @param a an angle, in radians.
135 * @return the cosine of the argument.
136 */
137 public static double cos(double a) {
138 return StrictMath.cos(a); // default impl. delegates to StrictMath
139 }
140
141 /**
142 * Returns the trigonometric tangent of an angle. Special cases:
143 * <ul><li>If the argument is NaN or an infinity, then the result
144 * is NaN.
145 * <li>If the argument is zero, then the result is a zero with the
146 * same sign as the argument.</ul>
147 *
148 * <p>The computed result must be within 1 ulp of the exact result.
149 * Results must be semi-monotonic.
150 *
151 * @param a an angle, in radians.
152 * @return the tangent of the argument.
153 */
154 public static double tan(double a) {
155 return StrictMath.tan(a); // default impl. delegates to StrictMath
156 }
157
158 /**
159 * Returns the arc sine of a value; the returned angle is in the
160 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
161 * <ul><li>If the argument is NaN or its absolute value is greater
162 * than 1, then the result is NaN.
163 * <li>If the argument is zero, then the result is a zero with the
164 * same sign as the argument.</ul>
165 *
166 * <p>The computed result must be within 1 ulp of the exact result.
167 * Results must be semi-monotonic.
168 *
169 * @param a the value whose arc sine is to be returned.
170 * @return the arc sine of the argument.
171 */
172 public static double asin(double a) {
173 return StrictMath.asin(a); // default impl. delegates to StrictMath
174 }
175
176 /**
177 * Returns the arc cosine of a value; the returned angle is in the
178 * range 0.0 through <i>pi</i>. Special case:
179 * <ul><li>If the argument is NaN or its absolute value is greater
180 * than 1, then the result is NaN.</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 cosine is to be returned.
186 * @return the arc cosine of the argument.
187 */
188 public static double acos(double a) {
189 return StrictMath.acos(a); // default impl. delegates to StrictMath
190 }
191
192 /**
193 * Returns the arc tangent of a value; the returned angle is in the
194 * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
195 * <ul><li>If the argument is NaN, then the result is NaN.
196 * <li>If the argument is zero, then the result is a zero with the
197 * same sign as the argument.</ul>
198 *
199 * <p>The computed result must be within 1 ulp of the exact result.
200 * Results must be semi-monotonic.
201 *
202 * @param a the value whose arc tangent is to be returned.
203 * @return the arc tangent of the argument.
204 */
205 public static double atan(double a) {
206 return StrictMath.atan(a); // default impl. delegates to StrictMath
207 }
208
209 /**
210 * Converts an angle measured in degrees to an approximately
211 * equivalent angle measured in radians. The conversion from
212 * degrees to radians is generally inexact.
213 *
214 * @param angdeg an angle, in degrees
215 * @return the measurement of the angle {@code angdeg}
216 * in radians.
217 * @since 1.2
218 */
219 public static double toRadians(double angdeg) {
220 return angdeg / 180.0 * PI;
221 }
222
223 /**
224 * Converts an angle measured in radians to an approximately
225 * equivalent angle measured in degrees. The conversion from
226 * radians to degrees is generally inexact; users should
227 * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
228 * equal {@code 0.0}.
229 *
230 * @param angrad an angle, in radians
231 * @return the measurement of the angle {@code angrad}
232 * in degrees.
233 * @since 1.2
234 */
235 public static double toDegrees(double angrad) {
236 return angrad * 180.0 / PI;
237 }
238
239 /**
240 * Returns Euler's number <i>e</i> raised to the power of a
241 * {@code double} value. Special cases:
242 * <ul><li>If the argument is NaN, the result is NaN.
243 * <li>If the argument is positive infinity, then the result is
244 * positive infinity.
245 * <li>If the argument is negative infinity, then the result is
246 * positive zero.</ul>
247 *
248 * <p>The computed result must be within 1 ulp of the exact result.
249 * Results must be semi-monotonic.
250 *
251 * @param a the exponent to raise <i>e</i> to.
252 * @return the value <i>e</i><sup>{@code a}</sup>,
253 * where <i>e</i> is the base of the natural logarithms.
254 */
255 public static double exp(double a) {
256 return StrictMath.exp(a); // default impl. delegates to StrictMath
257 }
258
259 /**
260 * Returns the natural logarithm (base <i>e</i>) of a {@code double}
261 * value. Special cases:
262 * <ul><li>If the argument is NaN or less than zero, then the result
263 * is NaN.
264 * <li>If the argument is positive infinity, then the result is
265 * positive infinity.
266 * <li>If the argument is positive zero or negative zero, then the
267 * result is negative infinity.</ul>
268 *
269 * <p>The computed result must be within 1 ulp of the exact result.
270 * Results must be semi-monotonic.
271 *
272 * @param a a value
273 * @return the value ln {@code a}, the natural logarithm of
274 * {@code a}.
275 */
276 public static double log(double a) {
277 return StrictMath.log(a); // default impl. delegates to StrictMath
278 }
279
280 /**
281 * Returns the base 10 logarithm of a {@code double} value.
282 * Special cases:
283 *
284 * <ul><li>If the argument is NaN or less than zero, then the result
285 * is NaN.
286 * <li>If the argument is positive infinity, then the result is
287 * positive infinity.
288 * <li>If the argument is positive zero or negative zero, then the
289 * result is negative infinity.
290 * <li> If the argument is equal to 10<sup><i>n</i></sup> for
291 * integer <i>n</i>, then the result is <i>n</i>.
292 * </ul>
293 *
294 * <p>The computed result must be within 1 ulp of the exact result.
295 * Results must be semi-monotonic.
296 *
297 * @param a a value
298 * @return the base 10 logarithm of {@code a}.
299 * @since 1.5
300 */
301 public static double log10(double a) {
302 return StrictMath.log10(a); // default impl. delegates to StrictMath
303 }
304
305 /**
306 * Returns the correctly rounded positive square root of a
307 * {@code double} value.
308 * Special cases:
309 * <ul><li>If the argument is NaN or less than zero, then the result
310 * is NaN.
311 * <li>If the argument is positive infinity, then the result is positive
312 * infinity.
313 * <li>If the argument is positive zero or negative zero, then the
314 * result is the same as the argument.</ul>
315 * Otherwise, the result is the {@code double} value closest to
316 * the true mathematical square root of the argument value.
317 *
318 * @param a a value.
319 * @return the positive square root of {@code a}.
320 * If the argument is NaN or less than zero, the result is NaN.
321 */
322 public static double sqrt(double a) {
323 return StrictMath.sqrt(a); // default impl. delegates to StrictMath
324 // Note that hardware sqrt instructions
325 // frequently can be directly used by JITs
326 // and should be much faster than doing
327 // Math.sqrt in software.
328 }
329
330
331 /**
332 * Returns the cube root of a {@code double} value. For
333 * positive finite {@code x}, {@code cbrt(-x) ==
334 * -cbrt(x)}; that is, the cube root of a negative value is
335 * the negative of the cube root of that value's magnitude.
336 *
337 * Special cases:
338 *
339 * <ul>
340 *
341 * <li>If the argument is NaN, then the result is NaN.
342 *
343 * <li>If the argument is infinite, then the result is an infinity
344 * with the same sign as the argument.
345 *
346 * <li>If the argument is zero, then the result is a zero with the
347 * same sign as the argument.
348 *
349 * </ul>
350 *
351 * <p>The computed result must be within 1 ulp of the exact result.
352 *
353 * @param a a value.
354 * @return the cube root of {@code a}.
355 * @since 1.5
356 */
357 public static double cbrt(double a) {
358 return StrictMath.cbrt(a);
359 }
360
361 /**
362 * Computes the remainder operation on two arguments as prescribed
363 * by the IEEE 754 standard.
364 * The remainder value is mathematically equal to
365 * <code>f1 - f2</code> × <i>n</i>,
366 * where <i>n</i> is the mathematical integer closest to the exact
367 * mathematical value of the quotient {@code f1/f2}, and if two
368 * mathematical integers are equally close to {@code f1/f2},
369 * then <i>n</i> is the integer that is even. If the remainder is
370 * zero, its sign is the same as the sign of the first argument.
371 * Special cases:
372 * <ul><li>If either argument is NaN, or the first argument is infinite,
373 * or the second argument is positive zero or negative zero, then the
374 * result is NaN.
375 * <li>If the first argument is finite and the second argument is
376 * infinite, then the result is the same as the first argument.</ul>
377 *
378 * @param f1 the dividend.
379 * @param f2 the divisor.
380 * @return the remainder when {@code f1} is divided by
381 * {@code f2}.
382 */
383 public static double IEEEremainder(double f1, double f2) {
384 return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
385 }
386
387 /**
388 * Returns the smallest (closest to negative infinity)
389 * {@code double} value that is greater than or equal to the
390 * argument and is equal to a mathematical integer. Special cases:
391 * <ul><li>If the argument value is already equal to a
392 * mathematical integer, then the result is the same as the
393 * argument. <li>If the argument is NaN or an infinity or
394 * positive zero or negative zero, then the result is the same as
395 * the argument. <li>If the argument value is less than zero but
396 * greater than -1.0, then the result is negative zero.</ul> Note
397 * that the value of {@code Math.ceil(x)} is exactly the
398 * value of {@code -Math.floor(-x)}.
399 *
400 *
401 * @param a a value.
402 * @return the smallest (closest to negative infinity)
403 * floating-point value that is greater than or equal to
404 * the argument and is equal to a mathematical integer.
405 */
406 public static double ceil(double a) {
407 return StrictMath.ceil(a); // default impl. delegates to StrictMath
408 }
409
410 /**
411 * Returns the largest (closest to positive infinity)
412 * {@code double} value that is less than or equal to the
413 * argument and is equal to a mathematical integer. Special cases:
414 * <ul><li>If the argument value is already equal to a
415 * mathematical integer, then the result is the same as the
416 * argument. <li>If the argument is NaN or an infinity or
417 * positive zero or negative zero, then the result is the same as
418 * the argument.</ul>
419 *
420 * @param a a value.
421 * @return the largest (closest to positive infinity)
422 * floating-point value that less than or equal to the argument
423 * and is equal to a mathematical integer.
424 */
425 public static double floor(double a) {
426 return StrictMath.floor(a); // default impl. delegates to StrictMath
427 }
428
429 /**
430 * Returns the {@code double} value that is closest in value
431 * to the argument and is equal to a mathematical integer. If two
432 * {@code double} values that are mathematical integers are
433 * equally close, the result is the integer value that is
434 * even. Special cases:
435 * <ul><li>If the argument value is already equal to a mathematical
436 * integer, then the result is the same as the argument.
437 * <li>If the argument is NaN or an infinity or positive zero or negative
438 * zero, then the result is the same as the argument.</ul>
439 *
440 * @param a a {@code double} value.
441 * @return the closest floating-point value to {@code a} that is
442 * equal to a mathematical integer.
443 */
444 public static double rint(double a) {
445 return StrictMath.rint(a); // default impl. delegates to StrictMath
446 }
447
448 /**
449 * Returns the angle <i>theta</i> from the conversion of rectangular
450 * coordinates ({@code x}, {@code y}) to polar
451 * coordinates (r, <i>theta</i>).
452 * This method computes the phase <i>theta</i> by computing an arc tangent
453 * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
454 * cases:
455 * <ul><li>If either argument is NaN, then the result is NaN.
456 * <li>If the first argument is positive zero and the second argument
457 * is positive, or the first argument is positive and finite and the
458 * second argument is positive infinity, then the result is positive
459 * zero.
460 * <li>If the first argument is negative zero and the second argument
461 * is positive, or the first argument is negative and finite and the
462 * second argument is positive infinity, then the result is negative zero.
463 * <li>If the first argument is positive zero and the second argument
464 * is negative, or the first argument is positive and finite and the
465 * second argument is negative infinity, then the result is the
466 * {@code double} value closest to <i>pi</i>.
467 * <li>If the first argument is negative zero and the second argument
468 * is negative, or the first argument is negative and finite and the
469 * second argument is negative infinity, then the result is the
470 * {@code double} value closest to -<i>pi</i>.
471 * <li>If the first argument is positive and the second argument is
472 * positive zero or negative zero, or the first argument is positive
473 * infinity and the second argument is finite, then the result is the
474 * {@code double} value closest to <i>pi</i>/2.
475 * <li>If the first argument is negative and the second argument is
476 * positive zero or negative zero, or the first argument is negative
477 * infinity and the second argument is finite, then the result is the
478 * {@code double} value closest to -<i>pi</i>/2.
479 * <li>If both arguments are positive infinity, then the result is the
480 * {@code double} value closest to <i>pi</i>/4.
481 * <li>If the first argument is positive infinity and the second argument
482 * is negative infinity, then the result is the {@code double}
483 * value closest to 3*<i>pi</i>/4.
484 * <li>If the first argument is negative infinity and the second argument
485 * is positive infinity, then the result is the {@code double} value
486 * closest to -<i>pi</i>/4.
487 * <li>If both arguments are negative infinity, then the result is the
488 * {@code double} value closest to -3*<i>pi</i>/4.</ul>
489 *
490 * <p>The computed result must be within 2 ulps of the exact result.
491 * Results must be semi-monotonic.
492 *
493 * @param y the ordinate coordinate
494 * @param x the abscissa coordinate
495 * @return the <i>theta</i> component of the point
496 * (<i>r</i>, <i>theta</i>)
497 * in polar coordinates that corresponds to the point
498 * (<i>x</i>, <i>y</i>) in Cartesian coordinates.
499 */
500 public static double atan2(double y, double x) {
501 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
502 }
503
504 /**
505 * Returns the value of the first argument raised to the power of the
506 * second argument. Special cases:
507 *
508 * <ul><li>If the second argument is positive or negative zero, then the
509 * result is 1.0.
510 * <li>If the second argument is 1.0, then the result is the same as the
511 * first argument.
512 * <li>If the second argument is NaN, then the result is NaN.
513 * <li>If the first argument is NaN and the second argument is nonzero,
514 * then the result is NaN.
515 *
516 * <li>If
517 * <ul>
518 * <li>the absolute value of the first argument is greater than 1
519 * and the second argument is positive infinity, or
520 * <li>the absolute value of the first argument is less than 1 and
521 * the second argument is negative infinity,
522 * </ul>
523 * then the result is positive infinity.
524 *
525 * <li>If
526 * <ul>
527 * <li>the absolute value of the first argument is greater than 1 and
528 * the second argument is negative infinity, or
529 * <li>the absolute value of the
530 * first argument is less than 1 and the second argument is positive
531 * infinity,
532 * </ul>
533 * then the result is positive zero.
534 *
535 * <li>If the absolute value of the first argument equals 1 and the
536 * second argument is infinite, then the result is NaN.
537 *
538 * <li>If
539 * <ul>
540 * <li>the first argument is positive zero and the second argument
541 * is greater than zero, or
542 * <li>the first argument is positive infinity and the second
543 * argument is less than zero,
544 * </ul>
545 * then the result is positive zero.
546 *
547 * <li>If
548 * <ul>
549 * <li>the first argument is positive zero and the second argument
550 * is less than zero, or
551 * <li>the first argument is positive infinity and the second
552 * argument is greater than zero,
553 * </ul>
554 * then the result is positive infinity.
555 *
556 * <li>If
557 * <ul>
558 * <li>the first argument is negative zero and the second argument
559 * is greater than zero but not a finite odd integer, or
560 * <li>the first argument is negative infinity and the second
561 * argument is less than zero but not a finite odd integer,
562 * </ul>
563 * then the result is positive zero.
564 *
565 * <li>If
566 * <ul>
567 * <li>the first argument is negative zero and the second argument
568 * is a positive finite odd integer, or
569 * <li>the first argument is negative infinity and the second
570 * argument is a negative finite odd integer,
571 * </ul>
572 * then the result is negative zero.
573 *
574 * <li>If
575 * <ul>
576 * <li>the first argument is negative zero and the second argument
577 * is less than zero but not a finite odd integer, or
578 * <li>the first argument is negative infinity and the second
579 * argument is greater than zero but not a finite odd integer,
580 * </ul>
581 * then the result is positive infinity.
582 *
583 * <li>If
584 * <ul>
585 * <li>the first argument is negative zero and the second argument
586 * is a negative finite odd integer, or
587 * <li>the first argument is negative infinity and the second
588 * argument is a positive finite odd integer,
589 * </ul>
590 * then the result is negative infinity.
591 *
592 * <li>If the first argument is finite and less than zero
593 * <ul>
594 * <li> if the second argument is a finite even integer, the
595 * result is equal to the result of raising the absolute value of
596 * the first argument to the power of the second argument
597 *
598 * <li>if the second argument is a finite odd integer, the result
599 * is equal to the negative of the result of raising the absolute
600 * value of the first argument to the power of the second
601 * argument
602 *
603 * <li>if the second argument is finite and not an integer, then
604 * the result is NaN.
605 * </ul>
606 *
607 * <li>If both arguments are integers, then the result is exactly equal
608 * to the mathematical result of raising the first argument to the power
609 * of the second argument if that result can in fact be represented
610 * exactly as a {@code double} value.</ul>
611 *
612 * <p>(In the foregoing descriptions, a floating-point value is
613 * considered to be an integer if and only if it is finite and a
614 * fixed point of the method {@link #ceil ceil} or,
615 * equivalently, a fixed point of the method {@link #floor
616 * floor}. A value is a fixed point of a one-argument
617 * method if and only if the result of applying the method to the
618 * value is equal to the value.)
619 *
620 * <p>The computed result must be within 1 ulp of the exact result.
621 * Results must be semi-monotonic.
622 *
623 * @param a the base.
624 * @param b the exponent.
625 * @return the value {@code a}<sup>{@code b}</sup>.
626 */
627 public static double pow(double a, double b) {
628 return StrictMath.pow(a, b); // default impl. delegates to StrictMath
629 }
630
631 /**
632 * Returns the closest {@code int} to the argument, with ties
633 * rounding up.
634 *
635 * <p>
636 * Special cases:
637 * <ul><li>If the argument is NaN, the result is 0.
638 * <li>If the argument is negative infinity or any value less than or
639 * equal to the value of {@code Integer.MIN_VALUE}, the result is
640 * equal to the value of {@code Integer.MIN_VALUE}.
641 * <li>If the argument is positive infinity or any value greater than or
642 * equal to the value of {@code Integer.MAX_VALUE}, the result is
643 * equal to the value of {@code Integer.MAX_VALUE}.</ul>
644 *
645 * @param a a floating-point value to be rounded to an integer.
646 * @return the value of the argument rounded to the nearest
647 * {@code int} value.
648 * @see java.lang.Integer#MAX_VALUE
649 * @see java.lang.Integer#MIN_VALUE
650 */
651 public static int round(float a) {
652 if (a != 0x1.fffffep-2f) // greatest float value less than 0.5
653 return (int)floor(a + 0.5f);
654 else
655 return 0;
656 }
657
658 /**
659 * Returns the closest {@code long} to the argument, with ties
660 * rounding up.
661 *
662 * <p>Special cases:
663 * <ul><li>If the argument is NaN, the result is 0.
664 * <li>If the argument is negative infinity or any value less than or
665 * equal to the value of {@code Long.MIN_VALUE}, the result is
666 * equal to the value of {@code Long.MIN_VALUE}.
667 * <li>If the argument is positive infinity or any value greater than or
668 * equal to the value of {@code Long.MAX_VALUE}, the result is
669 * equal to the value of {@code Long.MAX_VALUE}.</ul>
670 *
671 * @param a a floating-point value to be rounded to a
672 * {@code long}.
673 * @return the value of the argument rounded to the nearest
674 * {@code long} value.
675 * @see java.lang.Long#MAX_VALUE
676 * @see java.lang.Long#MIN_VALUE
677 */
678 public static long round(double a) {
679 if (a != 0x1.fffffffffffffp-2) // greatest double value less than 0.5
680 return (long)floor(a + 0.5d);
681 else
682 return 0;
683 }
684
685 private static Random randomNumberGenerator;
686
687 private static synchronized Random initRNG() {
688 Random rnd = randomNumberGenerator;
689 return (rnd == null) ? (randomNumberGenerator = new Random()) : rnd;
690 }
691
692 /**
693 * Returns a {@code double} value with a positive sign, greater
694 * than or equal to {@code 0.0} and less than {@code 1.0}.
695 * Returned values are chosen pseudorandomly with (approximately)
696 * uniform distribution from that range.
697 *
698 * <p>When this method is first called, it creates a single new
699 * pseudorandom-number generator, exactly as if by the expression
700 *
701 * <blockquote>{@code new java.util.Random()}</blockquote>
702 *
703 * This new pseudorandom-number generator is used thereafter for
704 * all calls to this method and is used nowhere else.
705 *
706 * <p>This method is properly synchronized to allow correct use by
707 * more than one thread. However, if many threads need to generate
708 * pseudorandom numbers at a great rate, it may reduce contention
709 * for each thread to have its own pseudorandom-number generator.
710 *
711 * @return a pseudorandom {@code double} greater than or equal
712 * to {@code 0.0} and less than {@code 1.0}.
713 * @see Random#nextDouble()
714 */
715 public static double random() {
716 Random rnd = randomNumberGenerator;
717 if (rnd == null) rnd = initRNG();
718 return rnd.nextDouble();
719 }
720
721 /**
722 * Returns the absolute value of an {@code int} value.
723 * If the argument is not negative, the argument is returned.
724 * If the argument is negative, the negation of the argument is returned.
725 *
726 * <p>Note that if the argument is equal to the value of
727 * {@link Integer#MIN_VALUE}, the most negative representable
728 * {@code int} value, the result is that same value, which is
729 * negative.
730 *
731 * @param a the argument whose absolute value is to be determined
732 * @return the absolute value of the argument.
733 */
734 public static int abs(int a) {
735 return (a < 0) ? -a : a;
736 }
737
738 /**
739 * Returns the absolute value of a {@code long} value.
740 * If the argument is not negative, the argument is returned.
741 * If the argument is negative, the negation of the argument is returned.
742 *
743 * <p>Note that if the argument is equal to the value of
744 * {@link Long#MIN_VALUE}, the most negative representable
745 * {@code long} value, the result is that same value, which
746 * is negative.
747 *
748 * @param a the argument whose absolute value is to be determined
749 * @return the absolute value of the argument.
750 */
751 public static long abs(long a) {
752 return (a < 0) ? -a : a;
753 }
754
755 /**
756 * Returns the absolute value of a {@code float} value.
757 * If the argument is not negative, the argument is returned.
758 * If the argument is negative, the negation of the argument is returned.
759 * Special cases:
760 * <ul><li>If the argument is positive zero or negative zero, the
761 * result is positive zero.
762 * <li>If the argument is infinite, the result is positive infinity.
763 * <li>If the argument is NaN, the result is NaN.</ul>
764 * In other words, the result is the same as the value of the expression:
765 * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
766 *
767 * @param a the argument whose absolute value is to be determined
768 * @return the absolute value of the argument.
769 */
770 public static float abs(float a) {
771 return (a <= 0.0F) ? 0.0F - a : a;
772 }
773
774 /**
775 * Returns the absolute value of a {@code double} value.
776 * If the argument is not negative, the argument is returned.
777 * If the argument is negative, the negation of the argument is returned.
778 * Special cases:
779 * <ul><li>If the argument is positive zero or negative zero, the result
780 * is positive zero.
781 * <li>If the argument is infinite, the result is positive infinity.
782 * <li>If the argument is NaN, the result is NaN.</ul>
783 * In other words, the result is the same as the value of the expression:
784 * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
785 *
786 * @param a the argument whose absolute value is to be determined
787 * @return the absolute value of the argument.
788 */
789 public static double abs(double a) {
790 return (a <= 0.0D) ? 0.0D - a : a;
791 }
792
793 /**
794 * Returns the greater of two {@code int} values. That is, the
795 * result is the argument closer to the value of
796 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
797 * the result is that same value.
798 *
799 * @param a an argument.
800 * @param b another argument.
801 * @return the larger of {@code a} and {@code b}.
802 */
803 public static int max(int a, int b) {
804 return (a >= b) ? a : b;
805 }
806
807 /**
808 * Returns the greater of two {@code long} values. That is, the
809 * result is the argument closer to the value of
810 * {@link Long#MAX_VALUE}. If the arguments have the same value,
811 * the result is that same value.
812 *
813 * @param a an argument.
814 * @param b another argument.
815 * @return the larger of {@code a} and {@code b}.
816 */
817 public static long max(long a, long b) {
818 return (a >= b) ? a : b;
819 }
820
821 private static long negativeZeroFloatBits = Float.floatToIntBits(-0.0f);
822 private static long negativeZeroDoubleBits = Double.doubleToLongBits(-0.0d);
823
824 /**
825 * Returns the greater of two {@code float} values. That is,
826 * the result is the argument closer to positive infinity. If the
827 * arguments have the same value, the result is that same
828 * value. If either value is NaN, then the result is NaN. Unlike
829 * the numerical comparison operators, this method considers
830 * negative zero to be strictly smaller than positive zero. If one
831 * argument is positive zero and the other negative zero, the
832 * result is positive zero.
833 *
834 * @param a an argument.
835 * @param b another argument.
836 * @return the larger of {@code a} and {@code b}.
837 */
838 public static float max(float a, float b) {
839 if (a != a) return a; // a is NaN
840 if ((a == 0.0f) && (b == 0.0f)
841 && (Float.floatToIntBits(a) == negativeZeroFloatBits)) {
842 return b;
843 }
844 return (a >= b) ? a : b;
845 }
846
847 /**
848 * Returns the greater of two {@code double} values. That
849 * is, the result is the argument closer to positive infinity. If
850 * the arguments have the same value, the result is that same
851 * value. If either value is NaN, then the result is NaN. Unlike
852 * the numerical comparison operators, this method considers
853 * negative zero to be strictly smaller than positive zero. If one
854 * argument is positive zero and the other negative zero, the
855 * result is positive zero.
856 *
857 * @param a an argument.
858 * @param b another argument.
859 * @return the larger of {@code a} and {@code b}.
860 */
861 public static double max(double a, double b) {
862 if (a != a) return a; // a is NaN
863 if ((a == 0.0d) && (b == 0.0d)
864 && (Double.doubleToLongBits(a) == negativeZeroDoubleBits)) {
865 return b;
866 }
867 return (a >= b) ? a : b;
868 }
869
870 /**
871 * Returns the smaller of two {@code int} values. That is,
872 * the result the argument closer to the value of
873 * {@link Integer#MIN_VALUE}. If the arguments have the same
874 * value, the result is that same value.
875 *
876 * @param a an argument.
877 * @param b another argument.
878 * @return the smaller of {@code a} and {@code b}.
879 */
880 public static int min(int a, int b) {
881 return (a <= b) ? a : b;
882 }
883
884 /**
885 * Returns the smaller of two {@code long} values. That is,
886 * the result is the argument closer to the value of
887 * {@link Long#MIN_VALUE}. If the arguments have the same
888 * value, the result is that same value.
889 *
890 * @param a an argument.
891 * @param b another argument.
892 * @return the smaller of {@code a} and {@code b}.
893 */
894 public static long min(long a, long b) {
895 return (a <= b) ? a : b;
896 }
897
898 /**
899 * Returns the smaller of two {@code float} values. That is,
900 * the result is the value closer to negative infinity. If the
901 * arguments have the same value, the result is that same
902 * value. If either value is NaN, then the result is NaN. Unlike
903 * the numerical comparison operators, this method considers
904 * negative zero to be strictly smaller than positive zero. If
905 * one argument is positive zero and the other is negative zero,
906 * the result is negative zero.
907 *
908 * @param a an argument.
909 * @param b another argument.
910 * @return the smaller of {@code a} and {@code b}.
911 */
912 public static float min(float a, float b) {
913 if (a != a) return a; // a is NaN
914 if ((a == 0.0f) && (b == 0.0f)
915 && (Float.floatToIntBits(b) == negativeZeroFloatBits)) {
916 return b;
917 }
918 return (a <= b) ? a : b;
919 }
920
921 /**
922 * Returns the smaller of two {@code double} values. That
923 * is, the result is the value closer to negative infinity. If the
924 * arguments have the same value, the result is that same
925 * value. If either value is NaN, then the result is NaN. Unlike
926 * the numerical comparison operators, this method considers
927 * negative zero to be strictly smaller than positive zero. If one
928 * argument is positive zero and the other is negative zero, the
929 * result is negative zero.
930 *
931 * @param a an argument.
932 * @param b another argument.
933 * @return the smaller of {@code a} and {@code b}.
934 */
935 public static double min(double a, double b) {
936 if (a != a) return a; // a is NaN
937 if ((a == 0.0d) && (b == 0.0d)
938 && (Double.doubleToLongBits(b) == negativeZeroDoubleBits)) {
939 return b;
940 }
941 return (a <= b) ? a : b;
942 }
943
944 /**
945 * Returns the size of an ulp of the argument. An ulp, unit in
946 * the last place, of a {@code double} value is the positive
947 * distance between this floating-point value and the {@code
948 * double} value next larger in magnitude. Note that for non-NaN
949 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
950 *
951 * <p>Special Cases:
952 * <ul>
953 * <li> If the argument is NaN, then the result is NaN.
954 * <li> If the argument is positive or negative infinity, then the
955 * result is positive infinity.
956 * <li> If the argument is positive or negative zero, then the result is
957 * {@code Double.MIN_VALUE}.
958 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
959 * the result is equal to 2<sup>971</sup>.
960 * </ul>
961 *
962 * @param d the floating-point value whose ulp is to be returned
963 * @return the size of an ulp of the argument
964 * @author Joseph D. Darcy
965 * @since 1.5
966 */
967 public static double ulp(double d) {
968 int exp = getExponent(d);
969
970 switch(exp) {
971 case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
972 return Math.abs(d);
973
974 case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
975 return Double.MIN_VALUE;
976
977 default:
978 assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
979
980 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
981 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
982 if (exp >= DoubleConsts.MIN_EXPONENT) {
983 return powerOfTwoD(exp);
984 }
985 else {
986 // return a subnormal result; left shift integer
987 // representation of Double.MIN_VALUE appropriate
988 // number of positions
989 return Double.longBitsToDouble(1L <<
990 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
991 }
992 }
993 }
994
995 /**
996 * Returns the size of an ulp of the argument. An ulp, unit in
997 * the last place, of a {@code float} value is the positive
998 * distance between this floating-point value and the {@code
999 * float} value next larger in magnitude. Note that for non-NaN
1000 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1001 *
1002 * <p>Special Cases:
1003 * <ul>
1004 * <li> If the argument is NaN, then the result is NaN.
1005 * <li> If the argument is positive or negative infinity, then the
1006 * result is positive infinity.
1007 * <li> If the argument is positive or negative zero, then the result is
1008 * {@code Float.MIN_VALUE}.
1009 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1010 * the result is equal to 2<sup>104</sup>.
1011 * </ul>
1012 *
1013 * @param f the floating-point value whose ulp is to be returned
1014 * @return the size of an ulp of the argument
1015 * @author Joseph D. Darcy
1016 * @since 1.5
1017 */
1018 public static float ulp(float f) {
1019 int exp = getExponent(f);
1020
1021 switch(exp) {
1022 case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
1023 return Math.abs(f);
1024
1025 case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
1026 return FloatConsts.MIN_VALUE;
1027
1028 default:
1029 assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1030
1031 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1032 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1033 if (exp >= FloatConsts.MIN_EXPONENT) {
1034 return powerOfTwoF(exp);
1035 }
1036 else {
1037 // return a subnormal result; left shift integer
1038 // representation of FloatConsts.MIN_VALUE appropriate
1039 // number of positions
1040 return Float.intBitsToFloat(1 <<
1041 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1042 }
1043 }
1044 }
1045
1046 /**
1047 * Returns the signum function of the argument; zero if the argument
1048 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1049 * argument is less than zero.
1050 *
1051 * <p>Special Cases:
1052 * <ul>
1053 * <li> If the argument is NaN, then the result is NaN.
1054 * <li> If the argument is positive zero or negative zero, then the
1055 * result is the same as the argument.
1056 * </ul>
1057 *
1058 * @param d the floating-point value whose signum is to be returned
1059 * @return the signum function of the argument
1060 * @author Joseph D. Darcy
1061 * @since 1.5
1062 */
1063 public static double signum(double d) {
1064 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1065 }
1066
1067 /**
1068 * Returns the signum function of the argument; zero if the argument
1069 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1070 * argument is less than zero.
1071 *
1072 * <p>Special Cases:
1073 * <ul>
1074 * <li> If the argument is NaN, then the result is NaN.
1075 * <li> If the argument is positive zero or negative zero, then the
1076 * result is the same as the argument.
1077 * </ul>
1078 *
1079 * @param f the floating-point value whose signum is to be returned
1080 * @return the signum function of the argument
1081 * @author Joseph D. Darcy
1082 * @since 1.5
1083 */
1084 public static float signum(float f) {
1085 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1086 }
1087
1088 /**
1089 * Returns the hyperbolic sine of a {@code double} value.
1090 * The hyperbolic sine of <i>x</i> is defined to be
1091 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1092 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1093 *
1094 * <p>Special cases:
1095 * <ul>
1096 *
1097 * <li>If the argument is NaN, then the result is NaN.
1098 *
1099 * <li>If the argument is infinite, then the result is an infinity
1100 * with the same sign as the argument.
1101 *
1102 * <li>If the argument is zero, then the result is a zero with the
1103 * same sign as the argument.
1104 *
1105 * </ul>
1106 *
1107 * <p>The computed result must be within 2.5 ulps of the exact result.
1108 *
1109 * @param x The number whose hyperbolic sine is to be returned.
1110 * @return The hyperbolic sine of {@code x}.
1111 * @since 1.5
1112 */
1113 public static double sinh(double x) {
1114 return StrictMath.sinh(x);
1115 }
1116
1117 /**
1118 * Returns the hyperbolic cosine of a {@code double} value.
1119 * The hyperbolic cosine of <i>x</i> is defined to be
1120 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1121 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1122 *
1123 * <p>Special cases:
1124 * <ul>
1125 *
1126 * <li>If the argument is NaN, then the result is NaN.
1127 *
1128 * <li>If the argument is infinite, then the result is positive
1129 * infinity.
1130 *
1131 * <li>If the argument is zero, then the result is {@code 1.0}.
1132 *
1133 * </ul>
1134 *
1135 * <p>The computed result must be within 2.5 ulps of the exact result.
1136 *
1137 * @param x The number whose hyperbolic cosine is to be returned.
1138 * @return The hyperbolic cosine of {@code x}.
1139 * @since 1.5
1140 */
1141 public static double cosh(double x) {
1142 return StrictMath.cosh(x);
1143 }
1144
1145 /**
1146 * Returns the hyperbolic tangent of a {@code double} value.
1147 * The hyperbolic tangent of <i>x</i> is defined to be
1148 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
1149 * in other words, {@linkplain Math#sinh
1150 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
1151 * that the absolute value of the exact tanh is always less than
1152 * 1.
1153 *
1154 * <p>Special cases:
1155 * <ul>
1156 *
1157 * <li>If the argument is NaN, then the result is NaN.
1158 *
1159 * <li>If the argument is zero, then the result is a zero with the
1160 * same sign as the argument.
1161 *
1162 * <li>If the argument is positive infinity, then the result is
1163 * {@code +1.0}.
1164 *
1165 * <li>If the argument is negative infinity, then the result is
1166 * {@code -1.0}.
1167 *
1168 * </ul>
1169 *
1170 * <p>The computed result must be within 2.5 ulps of the exact result.
1171 * The result of {@code tanh} for any finite input must have
1172 * an absolute value less than or equal to 1. Note that once the
1173 * exact result of tanh is within 1/2 of an ulp of the limit value
1174 * of ±1, correctly signed ±{@code 1.0} should
1175 * be returned.
1176 *
1177 * @param x The number whose hyperbolic tangent is to be returned.
1178 * @return The hyperbolic tangent of {@code x}.
1179 * @since 1.5
1180 */
1181 public static double tanh(double x) {
1182 return StrictMath.tanh(x);
1183 }
1184
1185 /**
1186 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1187 * without intermediate overflow or underflow.
1188 *
1189 * <p>Special cases:
1190 * <ul>
1191 *
1192 * <li> If either argument is infinite, then the result
1193 * is positive infinity.
1194 *
1195 * <li> If either argument is NaN and neither argument is infinite,
1196 * then the result is NaN.
1197 *
1198 * </ul>
1199 *
1200 * <p>The computed result must be within 1 ulp of the exact
1201 * result. If one parameter is held constant, the results must be
1202 * semi-monotonic in the other parameter.
1203 *
1204 * @param x a value
1205 * @param y a value
1206 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1207 * without intermediate overflow or underflow
1208 * @since 1.5
1209 */
1210 public static double hypot(double x, double y) {
1211 return StrictMath.hypot(x, y);
1212 }
1213
1214 /**
1215 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
1216 * <i>x</i> near 0, the exact sum of
1217 * {@code expm1(x)} + 1 is much closer to the true
1218 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1219 *
1220 * <p>Special cases:
1221 * <ul>
1222 * <li>If the argument is NaN, the result is NaN.
1223 *
1224 * <li>If the argument is positive infinity, then the result is
1225 * positive infinity.
1226 *
1227 * <li>If the argument is negative infinity, then the result is
1228 * -1.0.
1229 *
1230 * <li>If the argument is zero, then the result is a zero with the
1231 * same sign as the argument.
1232 *
1233 * </ul>
1234 *
1235 * <p>The computed result must be within 1 ulp of the exact result.
1236 * Results must be semi-monotonic. The result of
1237 * {@code expm1} for any finite input must be greater than or
1238 * equal to {@code -1.0}. Note that once the exact result of
1239 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
1240 * ulp of the limit value -1, {@code -1.0} should be
1241 * returned.
1242 *
1243 * @param x the exponent to raise <i>e</i> to in the computation of
1244 * <i>e</i><sup>{@code x}</sup> -1.
1245 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
1246 * @since 1.5
1247 */
1248 public static double expm1(double x) {
1249 return StrictMath.expm1(x);
1250 }
1251
1252 /**
1253 * Returns the natural logarithm of the sum of the argument and 1.
1254 * Note that for small values {@code x}, the result of
1255 * {@code log1p(x)} is much closer to the true result of ln(1
1256 * + {@code x}) than the floating-point evaluation of
1257 * {@code log(1.0+x)}.
1258 *
1259 * <p>Special cases:
1260 *
1261 * <ul>
1262 *
1263 * <li>If the argument is NaN or less than -1, then the result is
1264 * NaN.
1265 *
1266 * <li>If the argument is positive infinity, then the result is
1267 * positive infinity.
1268 *
1269 * <li>If the argument is negative one, then the result is
1270 * negative infinity.
1271 *
1272 * <li>If the argument is zero, then the result is a zero with the
1273 * same sign as the argument.
1274 *
1275 * </ul>
1276 *
1277 * <p>The computed result must be within 1 ulp of the exact result.
1278 * Results must be semi-monotonic.
1279 *
1280 * @param x a value
1281 * @return the value ln({@code x} + 1), the natural
1282 * log of {@code x} + 1
1283 * @since 1.5
1284 */
1285 public static double log1p(double x) {
1286 return StrictMath.log1p(x);
1287 }
1288
1289 /**
1290 * Returns the first floating-point argument with the sign of the
1291 * second floating-point argument. Note that unlike the {@link
1292 * StrictMath#copySign(double, double) StrictMath.copySign}
1293 * method, this method does not require NaN {@code sign}
1294 * arguments to be treated as positive values; implementations are
1295 * permitted to treat some NaN arguments as positive and other NaN
1296 * arguments as negative to allow greater performance.
1297 *
1298 * @param magnitude the parameter providing the magnitude of the result
1299 * @param sign the parameter providing the sign of the result
1300 * @return a value with the magnitude of {@code magnitude}
1301 * and the sign of {@code sign}.
1302 * @since 1.6
1303 */
1304 public static double copySign(double magnitude, double sign) {
1305 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1306 (DoubleConsts.SIGN_BIT_MASK)) |
1307 (Double.doubleToRawLongBits(magnitude) &
1308 (DoubleConsts.EXP_BIT_MASK |
1309 DoubleConsts.SIGNIF_BIT_MASK)));
1310 }
1311
1312 /**
1313 * Returns the first floating-point argument with the sign of the
1314 * second floating-point argument. Note that unlike the {@link
1315 * StrictMath#copySign(float, float) StrictMath.copySign}
1316 * method, this method does not require NaN {@code sign}
1317 * arguments to be treated as positive values; implementations are
1318 * permitted to treat some NaN arguments as positive and other NaN
1319 * arguments as negative to allow greater performance.
1320 *
1321 * @param magnitude the parameter providing the magnitude of the result
1322 * @param sign the parameter providing the sign of the result
1323 * @return a value with the magnitude of {@code magnitude}
1324 * and the sign of {@code sign}.
1325 * @since 1.6
1326 */
1327 public static float copySign(float magnitude, float sign) {
1328 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1329 (FloatConsts.SIGN_BIT_MASK)) |
1330 (Float.floatToRawIntBits(magnitude) &
1331 (FloatConsts.EXP_BIT_MASK |
1332 FloatConsts.SIGNIF_BIT_MASK)));
1333 }
1334
1335 /**
1336 * Returns the unbiased exponent used in the representation of a
1337 * {@code float}. Special cases:
1338 *
1339 * <ul>
1340 * <li>If the argument is NaN or infinite, then the result is
1341 * {@link Float#MAX_EXPONENT} + 1.
1342 * <li>If the argument is zero or subnormal, then the result is
1343 * {@link Float#MIN_EXPONENT} -1.
1344 * </ul>
1345 * @param f a {@code float} value
1346 * @return the unbiased exponent of the argument
1347 * @since 1.6
1348 */
1349 public static int getExponent(float f) {
1350 /*
1351 * Bitwise convert f to integer, mask out exponent bits, shift
1352 * to the right and then subtract out float's bias adjust to
1353 * get true exponent value
1354 */
1355 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1356 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1357 }
1358
1359 /**
1360 * Returns the unbiased exponent used in the representation of a
1361 * {@code double}. Special cases:
1362 *
1363 * <ul>
1364 * <li>If the argument is NaN or infinite, then the result is
1365 * {@link Double#MAX_EXPONENT} + 1.
1366 * <li>If the argument is zero or subnormal, then the result is
1367 * {@link Double#MIN_EXPONENT} -1.
1368 * </ul>
1369 * @param d a {@code double} value
1370 * @return the unbiased exponent of the argument
1371 * @since 1.6
1372 */
1373 public static int getExponent(double d) {
1374 /*
1375 * Bitwise convert d to long, mask out exponent bits, shift
1376 * to the right and then subtract out double's bias adjust to
1377 * get true exponent value.
1378 */
1379 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1380 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1381 }
1382
1383 /**
1384 * Returns the floating-point number adjacent to the first
1385 * argument in the direction of the second argument. If both
1386 * arguments compare as equal the second argument is returned.
1387 *
1388 * <p>
1389 * Special cases:
1390 * <ul>
1391 * <li> If either argument is a NaN, then NaN is returned.
1392 *
1393 * <li> If both arguments are signed zeros, {@code direction}
1394 * is returned unchanged (as implied by the requirement of
1395 * returning the second argument if the arguments compare as
1396 * equal).
1397 *
1398 * <li> If {@code start} is
1399 * ±{@link Double#MIN_VALUE} and {@code direction}
1400 * has a value such that the result should have a smaller
1401 * magnitude, then a zero with the same sign as {@code start}
1402 * is returned.
1403 *
1404 * <li> If {@code start} is infinite and
1405 * {@code direction} has a value such that the result should
1406 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1407 * same sign as {@code start} is returned.
1408 *
1409 * <li> If {@code start} is equal to ±
1410 * {@link Double#MAX_VALUE} and {@code direction} has a
1411 * value such that the result should have a larger magnitude, an
1412 * infinity with same sign as {@code start} is returned.
1413 * </ul>
1414 *
1415 * @param start starting floating-point value
1416 * @param direction value indicating which of
1417 * {@code start}'s neighbors or {@code start} should
1418 * be returned
1419 * @return The floating-point number adjacent to {@code start} in the
1420 * direction of {@code direction}.
1421 * @since 1.6
1422 */
1423 public static double nextAfter(double start, double direction) {
1424 /*
1425 * The cases:
1426 *
1427 * nextAfter(+infinity, 0) == MAX_VALUE
1428 * nextAfter(+infinity, +infinity) == +infinity
1429 * nextAfter(-infinity, 0) == -MAX_VALUE
1430 * nextAfter(-infinity, -infinity) == -infinity
1431 *
1432 * are naturally handled without any additional testing
1433 */
1434
1435 // First check for NaN values
1436 if (Double.isNaN(start) || Double.isNaN(direction)) {
1437 // return a NaN derived from the input NaN(s)
1438 return start + direction;
1439 } else if (start == direction) {
1440 return direction;
1441 } else { // start > direction or start < direction
1442 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1443 // then bitwise convert start to integer.
1444 long transducer = Double.doubleToRawLongBits(start + 0.0d);
1445
1446 /*
1447 * IEEE 754 floating-point numbers are lexicographically
1448 * ordered if treated as signed- magnitude integers .
1449 * Since Java's integers are two's complement,
1450 * incrementing" the two's complement representation of a
1451 * logically negative floating-point value *decrements*
1452 * the signed-magnitude representation. Therefore, when
1453 * the integer representation of a floating-point values
1454 * is less than zero, the adjustment to the representation
1455 * is in the opposite direction than would be expected at
1456 * first .
1457 */
1458 if (direction > start) { // Calculate next greater value
1459 transducer = transducer + (transducer >= 0L ? 1L:-1L);
1460 } else { // Calculate next lesser value
1461 assert direction < start;
1462 if (transducer > 0L)
1463 --transducer;
1464 else
1465 if (transducer < 0L )
1466 ++transducer;
1467 /*
1468 * transducer==0, the result is -MIN_VALUE
1469 *
1470 * The transition from zero (implicitly
1471 * positive) to the smallest negative
1472 * signed magnitude value must be done
1473 * explicitly.
1474 */
1475 else
1476 transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
1477 }
1478
1479 return Double.longBitsToDouble(transducer);
1480 }
1481 }
1482
1483 /**
1484 * Returns the floating-point number adjacent to the first
1485 * argument in the direction of the second argument. If both
1486 * arguments compare as equal a value equivalent to the second argument
1487 * is returned.
1488 *
1489 * <p>
1490 * Special cases:
1491 * <ul>
1492 * <li> If either argument is a NaN, then NaN is returned.
1493 *
1494 * <li> If both arguments are signed zeros, a value equivalent
1495 * to {@code direction} is returned.
1496 *
1497 * <li> If {@code start} is
1498 * ±{@link Float#MIN_VALUE} and {@code direction}
1499 * has a value such that the result should have a smaller
1500 * magnitude, then a zero with the same sign as {@code start}
1501 * is returned.
1502 *
1503 * <li> If {@code start} is infinite and
1504 * {@code direction} has a value such that the result should
1505 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1506 * same sign as {@code start} is returned.
1507 *
1508 * <li> If {@code start} is equal to ±
1509 * {@link Float#MAX_VALUE} and {@code direction} has a
1510 * value such that the result should have a larger magnitude, an
1511 * infinity with same sign as {@code start} is returned.
1512 * </ul>
1513 *
1514 * @param start starting floating-point value
1515 * @param direction value indicating which of
1516 * {@code start}'s neighbors or {@code start} should
1517 * be returned
1518 * @return The floating-point number adjacent to {@code start} in the
1519 * direction of {@code direction}.
1520 * @since 1.6
1521 */
1522 public static float nextAfter(float start, double direction) {
1523 /*
1524 * The cases:
1525 *
1526 * nextAfter(+infinity, 0) == MAX_VALUE
1527 * nextAfter(+infinity, +infinity) == +infinity
1528 * nextAfter(-infinity, 0) == -MAX_VALUE
1529 * nextAfter(-infinity, -infinity) == -infinity
1530 *
1531 * are naturally handled without any additional testing
1532 */
1533
1534 // First check for NaN values
1535 if (Float.isNaN(start) || Double.isNaN(direction)) {
1536 // return a NaN derived from the input NaN(s)
1537 return start + (float)direction;
1538 } else if (start == direction) {
1539 return (float)direction;
1540 } else { // start > direction or start < direction
1541 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1542 // then bitwise convert start to integer.
1543 int transducer = Float.floatToRawIntBits(start + 0.0f);
1544
1545 /*
1546 * IEEE 754 floating-point numbers are lexicographically
1547 * ordered if treated as signed- magnitude integers .
1548 * Since Java's integers are two's complement,
1549 * incrementing" the two's complement representation of a
1550 * logically negative floating-point value *decrements*
1551 * the signed-magnitude representation. Therefore, when
1552 * the integer representation of a floating-point values
1553 * is less than zero, the adjustment to the representation
1554 * is in the opposite direction than would be expected at
1555 * first.
1556 */
1557 if (direction > start) {// Calculate next greater value
1558 transducer = transducer + (transducer >= 0 ? 1:-1);
1559 } else { // Calculate next lesser value
1560 assert direction < start;
1561 if (transducer > 0)
1562 --transducer;
1563 else
1564 if (transducer < 0 )
1565 ++transducer;
1566 /*
1567 * transducer==0, the result is -MIN_VALUE
1568 *
1569 * The transition from zero (implicitly
1570 * positive) to the smallest negative
1571 * signed magnitude value must be done
1572 * explicitly.
1573 */
1574 else
1575 transducer = FloatConsts.SIGN_BIT_MASK | 1;
1576 }
1577
1578 return Float.intBitsToFloat(transducer);
1579 }
1580 }
1581
1582 /**
1583 * Returns the floating-point value adjacent to {@code d} in
1584 * the direction of positive infinity. This method is
1585 * semantically equivalent to {@code nextAfter(d,
1586 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
1587 * implementation may run faster than its equivalent
1588 * {@code nextAfter} call.
1589 *
1590 * <p>Special Cases:
1591 * <ul>
1592 * <li> If the argument is NaN, the result is NaN.
1593 *
1594 * <li> If the argument is positive infinity, the result is
1595 * positive infinity.
1596 *
1597 * <li> If the argument is zero, the result is
1598 * {@link Double#MIN_VALUE}
1599 *
1600 * </ul>
1601 *
1602 * @param d starting floating-point value
1603 * @return The adjacent floating-point value closer to positive
1604 * infinity.
1605 * @since 1.6
1606 */
1607 public static double nextUp(double d) {
1608 if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
1609 return d;
1610 else {
1611 d += 0.0d;
1612 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
1613 ((d >= 0.0d)?+1L:-1L));
1614 }
1615 }
1616
1617 /**
1618 * Returns the floating-point value adjacent to {@code f} in
1619 * the direction of positive infinity. This method is
1620 * semantically equivalent to {@code nextAfter(f,
1621 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
1622 * implementation may run faster than its equivalent
1623 * {@code nextAfter} call.
1624 *
1625 * <p>Special Cases:
1626 * <ul>
1627 * <li> If the argument is NaN, the result is NaN.
1628 *
1629 * <li> If the argument is positive infinity, the result is
1630 * positive infinity.
1631 *
1632 * <li> If the argument is zero, the result is
1633 * {@link Float#MIN_VALUE}
1634 *
1635 * </ul>
1636 *
1637 * @param f starting floating-point value
1638 * @return The adjacent floating-point value closer to positive
1639 * infinity.
1640 * @since 1.6
1641 */
1642 public static float nextUp(float f) {
1643 if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
1644 return f;
1645 else {
1646 f += 0.0f;
1647 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
1648 ((f >= 0.0f)?+1:-1));
1649 }
1650 }
1651
1652
1653 /**
1654 * Return {@code d} ×
1655 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1656 * by a single correctly rounded floating-point multiply to a
1657 * member of the double value set. See the Java
1658 * Language Specification for a discussion of floating-point
1659 * value sets. If the exponent of the result is between {@link
1660 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
1661 * answer is calculated exactly. If the exponent of the result
1662 * would be larger than {@code Double.MAX_EXPONENT}, an
1663 * infinity is returned. Note that if the result is subnormal,
1664 * precision may be lost; that is, when {@code scalb(x, n)}
1665 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1666 * <i>x</i>. When the result is non-NaN, the result has the same
1667 * sign as {@code d}.
1668 *
1669 * <p>Special cases:
1670 * <ul>
1671 * <li> If the first argument is NaN, NaN is returned.
1672 * <li> If the first argument is infinite, then an infinity of the
1673 * same sign is returned.
1674 * <li> If the first argument is zero, then a zero of the same
1675 * sign is returned.
1676 * </ul>
1677 *
1678 * @param d number to be scaled by a power of two.
1679 * @param scaleFactor power of 2 used to scale {@code d}
1680 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
1681 * @since 1.6
1682 */
1683 public static double scalb(double d, int scaleFactor) {
1684 /*
1685 * This method does not need to be declared strictfp to
1686 * compute the same correct result on all platforms. When
1687 * scaling up, it does not matter what order the
1688 * multiply-store operations are done; the result will be
1689 * finite or overflow regardless of the operation ordering.
1690 * However, to get the correct result when scaling down, a
1691 * particular ordering must be used.
1692 *
1693 * When scaling down, the multiply-store operations are
1694 * sequenced so that it is not possible for two consecutive
1695 * multiply-stores to return subnormal results. If one
1696 * multiply-store result is subnormal, the next multiply will
1697 * round it away to zero. This is done by first multiplying
1698 * by 2 ^ (scaleFactor % n) and then multiplying several
1699 * times by by 2^n as needed where n is the exponent of number
1700 * that is a covenient power of two. In this way, at most one
1701 * real rounding error occurs. If the double value set is
1702 * being used exclusively, the rounding will occur on a
1703 * multiply. If the double-extended-exponent value set is
1704 * being used, the products will (perhaps) be exact but the
1705 * stores to d are guaranteed to round to the double value
1706 * set.
1707 *
1708 * It is _not_ a valid implementation to first multiply d by
1709 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
1710 * MIN_EXPONENT) since even in a strictfp program double
1711 * rounding on underflow could occur; e.g. if the scaleFactor
1712 * argument was (MIN_EXPONENT - n) and the exponent of d was a
1713 * little less than -(MIN_EXPONENT - n), meaning the final
1714 * result would be subnormal.
1715 *
1716 * Since exact reproducibility of this method can be achieved
1717 * without any undue performance burden, there is no
1718 * compelling reason to allow double rounding on underflow in
1719 * scalb.
1720 */
1721
1722 // magnitude of a power of two so large that scaling a finite
1723 // nonzero value by it would be guaranteed to over or
1724 // underflow; due to rounding, scaling down takes takes an
1725 // additional power of two which is reflected here
1726 final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
1727 DoubleConsts.SIGNIFICAND_WIDTH + 1;
1728 int exp_adjust = 0;
1729 int scale_increment = 0;
1730 double exp_delta = Double.NaN;
1731
1732 // Make sure scaling factor is in a reasonable range
1733
1734 if(scaleFactor < 0) {
1735 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
1736 scale_increment = -512;
1737 exp_delta = twoToTheDoubleScaleDown;
1738 }
1739 else {
1740 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
1741 scale_increment = 512;
1742 exp_delta = twoToTheDoubleScaleUp;
1743 }
1744
1745 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
1746 // technique from "Hacker's Delight" section 10-2.
1747 int t = (scaleFactor >> 9-1) >>> 32 - 9;
1748 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
1749
1750 d *= powerOfTwoD(exp_adjust);
1751 scaleFactor -= exp_adjust;
1752
1753 while(scaleFactor != 0) {
1754 d *= exp_delta;
1755 scaleFactor -= scale_increment;
1756 }
1757 return d;
1758 }
1759
1760 /**
1761 * Return {@code f} ×
1762 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1763 * by a single correctly rounded floating-point multiply to a
1764 * member of the float value set. See the Java
1765 * Language Specification for a discussion of floating-point
1766 * value sets. If the exponent of the result is between {@link
1767 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
1768 * answer is calculated exactly. If the exponent of the result
1769 * would be larger than {@code Float.MAX_EXPONENT}, an
1770 * infinity is returned. Note that if the result is subnormal,
1771 * precision may be lost; that is, when {@code scalb(x, n)}
1772 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1773 * <i>x</i>. When the result is non-NaN, the result has the same
1774 * sign as {@code f}.
1775 *
1776 * <p>Special cases:
1777 * <ul>
1778 * <li> If the first argument is NaN, NaN is returned.
1779 * <li> If the first argument is infinite, then an infinity of the
1780 * same sign is returned.
1781 * <li> If the first argument is zero, then a zero of the same
1782 * sign is returned.
1783 * </ul>
1784 *
1785 * @param f number to be scaled by a power of two.
1786 * @param scaleFactor power of 2 used to scale {@code f}
1787 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
1788 * @since 1.6
1789 */
1790 public static float scalb(float f, int scaleFactor) {
1791 // magnitude of a power of two so large that scaling a finite
1792 // nonzero value by it would be guaranteed to over or
1793 // underflow; due to rounding, scaling down takes takes an
1794 // additional power of two which is reflected here
1795 final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
1796 FloatConsts.SIGNIFICAND_WIDTH + 1;
1797
1798 // Make sure scaling factor is in a reasonable range
1799 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
1800
1801 /*
1802 * Since + MAX_SCALE for float fits well within the double
1803 * exponent range and + float -> double conversion is exact
1804 * the multiplication below will be exact. Therefore, the
1805 * rounding that occurs when the double product is cast to
1806 * float will be the correctly rounded float result. Since
1807 * all operations other than the final multiply will be exact,
1808 * it is not necessary to declare this method strictfp.
1809 */
1810 return (float)((double)f*powerOfTwoD(scaleFactor));
1811 }
1812
1813 // Constants used in scalb
1814 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
1815 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
1816
1817 /**
1818 * Returns a floating-point power of two in the normal range.
1819 */
1820 static double powerOfTwoD(int n) {
1821 assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
1822 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
1823 (DoubleConsts.SIGNIFICAND_WIDTH-1))
1824 & DoubleConsts.EXP_BIT_MASK);
1825 }
1826
1827 /**
1828 * Returns a floating-point power of two in the normal range.
1829 */
1830 public static float powerOfTwoF(int n) {
1831 assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
1832 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
1833 (FloatConsts.SIGNIFICAND_WIDTH-1))
1834 & FloatConsts.EXP_BIT_MASK);
1835 }
1836 }