1 /*
2 * Copyright (c) 1994, 2012, 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
18 * 2 along with this work; if not, write to the Free Software Foundation,
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 up.
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 if (a != 0x1.fffffep-2f) // greatest float value less than 0.5
669 return (int)floor(a + 0.5f);
670 else
671 return 0;
672 }
673
674 /**
675 * Returns the closest {@code long} to the argument, with ties
676 * rounding up.
677 *
678 * <p>Special cases:
679 * <ul><li>If the argument is NaN, the result is 0.
680 * <li>If the argument is negative infinity or any value less than or
681 * equal to the value of {@code Long.MIN_VALUE}, the result is
682 * equal to the value of {@code Long.MIN_VALUE}.
683 * <li>If the argument is positive infinity or any value greater than or
684 * equal to the value of {@code Long.MAX_VALUE}, the result is
685 * equal to the value of {@code Long.MAX_VALUE}.</ul>
686 *
687 * @param a a floating-point value to be rounded to a
688 * {@code long}.
689 * @return the value of the argument rounded to the nearest
690 * {@code long} value.
691 * @see java.lang.Long#MAX_VALUE
692 * @see java.lang.Long#MIN_VALUE
693 */
694 public static long round(double a) {
695 if (a != 0x1.fffffffffffffp-2) // greatest double value less than 0.5
696 return (long)floor(a + 0.5d);
697 else
698 return 0;
699 }
700
701 private static Random randomNumberGenerator;
702
703 private static synchronized Random initRNG() {
704 Random rnd = randomNumberGenerator;
705 return (rnd == null) ? (randomNumberGenerator = new Random()) : rnd;
706 }
707
708 /**
709 * Returns a {@code double} value with a positive sign, greater
710 * than or equal to {@code 0.0} and less than {@code 1.0}.
711 * Returned values are chosen pseudorandomly with (approximately)
712 * uniform distribution from that range.
713 *
714 * <p>When this method is first called, it creates a single new
715 * pseudorandom-number generator, exactly as if by the expression
716 *
717 * <blockquote>{@code new java.util.Random()}</blockquote>
718 *
719 * This new pseudorandom-number generator is used thereafter for
720 * all calls to this method and is used nowhere else.
721 *
722 * <p>This method is properly synchronized to allow correct use by
723 * more than one thread. However, if many threads need to generate
724 * pseudorandom numbers at a great rate, it may reduce contention
725 * for each thread to have its own pseudorandom-number generator.
726 *
727 * @return a pseudorandom {@code double} greater than or equal
728 * to {@code 0.0} and less than {@code 1.0}.
729 * @see Random#nextDouble()
730 */
731 public static double random() {
732 Random rnd = randomNumberGenerator;
733 if (rnd == null) rnd = initRNG();
734 return rnd.nextDouble();
735 }
736
737 /**
738 * Returns the sum of its arguments,
739 * throwing an exception if the result overflows an {@code int}.
740 *
741 * @param x the first value
742 * @param y the second value
743 * @return the result
744 * @throws ArithmeticException if the result overflows an int
745 * @since 1.8
746 */
747 public static int addExact(int x, int y) {
748 int r = x + y;
749 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
750 if (((x ^ r) & (y ^ r)) < 0) {
751 throw new ArithmeticException("integer overflow");
752 }
753 return r;
754 }
755
756 /**
757 * Returns the sum of its arguments,
758 * throwing an exception if the result overflows a {@code long}.
759 *
760 * @param x the first value
761 * @param y the second value
762 * @return the result
763 * @throws ArithmeticException if the result overflows a long
764 * @since 1.8
765 */
766 public static long addExact(long x, long y) {
767 long r = x + y;
768 // HD 2-12 Overflow iff both arguments have the opposite sign of the result
769 if (((x ^ r) & (y ^ r)) < 0) {
770 throw new ArithmeticException("long overflow");
771 }
772 return r;
773 }
774
775 /**
776 * Returns the difference of the arguments,
777 * throwing an exception if the result overflows an {@code int}.
778 *
779 * @param x the first value
780 * @param y the second value to subtract from the first
781 * @return the result
782 * @throws ArithmeticException if the result overflows an int
783 * @since 1.8
784 */
785 public static int subtractExact(int x, int y) {
786 int r = x - y;
787 // HD 2-12 Overflow iff the arguments have different signs and
788 // the sign of the result is different than the sign of x
789 if (((x ^ y) & (x ^ r)) < 0) {
790 throw new ArithmeticException("integer overflow");
791 }
792 return r;
793 }
794
795 /**
796 * Returns the difference of the 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 to subtract from the first
801 * @return the result
802 * @throws ArithmeticException if the result overflows a long
803 * @since 1.8
804 */
805 public static long subtractExact(long x, long y) {
806 long r = x - y;
807 // HD 2-12 Overflow iff the arguments have different signs and
808 // the sign of the result is different than the sign of x
809 if (((x ^ y) & (x ^ r)) < 0) {
810 throw new ArithmeticException("long overflow");
811 }
812 return r;
813 }
814
815 /**
816 * Returns the product of the arguments,
817 * throwing an exception if the result overflows an {@code int}.
818 *
819 * @param x the first value
820 * @param y the second value
821 * @return the result
822 * @throws ArithmeticException if the result overflows an int
823 * @since 1.8
824 */
825 public static int multiplyExact(int x, int y) {
826 long r = (long)x * (long)y;
827 if ((int)r != r) {
828 throw new ArithmeticException("long overflow");
829 }
830 return (int)r;
831 }
832
833 /**
834 * Returns the product of the arguments,
835 * throwing an exception if the result overflows a {@code long}.
836 *
837 * @param x the first value
838 * @param y the second value
839 * @return the result
840 * @throws ArithmeticException if the result overflows a long
841 * @since 1.8
842 */
843 public static long multiplyExact(long x, long y) {
844 long r = x * y;
845 long ax = Math.abs(x);
846 long ay = Math.abs(y);
847 if (((ax | ay) >>> 31 != 0)) {
848 // Some bits greater than 2^31 that might cause overflow
849 // Check the result using the divide operator
850 // and check for the special case of Long.MIN_VALUE * -1
851 if (((y != 0) && (r / y != x)) ||
852 (x == Long.MIN_VALUE && y == -1)) {
853 throw new ArithmeticException("long overflow");
854 }
855 }
856 return r;
857 }
858
859 /**
860 * Returns the value of the {@code long} argument;
861 * throwing an exception if the value overflows an {@code int}.
862 *
863 * @param value the long value
864 * @return the argument as an int
865 * @throws ArithmeticException if the {@code argument} overflows an int
866 * @since 1.8
867 */
868 public static int toIntExact(long value) {
869 if ((int)value != value) {
870 throw new ArithmeticException("integer overflow");
871 }
872 return (int)value;
873 }
874
875 /**
876 * Returns the largest (closest to positive infinity)
877 * {@code int} value that is less than or equal to the algebraic quotient.
878 * There is one special case, if the dividend is the
879 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
880 * then integer overflow occurs and
881 * the result is equal to the {@code Integer.MIN_VALUE}.
882 * <p>
883 * Normal integer division operates under the round to zero rounding mode
884 * (truncation). This operation instead acts under the round toward
885 * negative infinity (floor) rounding mode.
886 * The floor rounding mode gives different results than truncation
887 * when the exact result is negative.
888 * <ul>
889 * <li>If the signs of the arguments are the same, the results of
890 * {@code floorDiv} and the {@code /} operator are the same. <br>
891 * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
892 * <li>If the signs of the arguments are different, the quotient is negative and
893 * {@code floorDiv} returns the integer less than or equal to the quotient
894 * and the {@code /} operator returns the integer closest to zero.<br>
895 * For example, {@code floorDiv(-4, 3) == -2},
896 * whereas {@code (-4 / 3) == -1}.
897 * </li>
898 * </ul>
899 * <p>
900 *
901 * @param x the dividend
902 * @param y the divisor
903 * @return the largest (closest to positive infinity)
904 * {@code int} value that is less than or equal to the algebraic quotient.
905 * @throws ArithmeticException if the divisor {@code y} is zero
906 * @see #floorMod(int, int)
907 * @see #floor(double)
908 * @since 1.8
909 */
910 public static int floorDiv(int x, int y) {
911 int r = x / y;
912 // if the signs are different and modulo not zero, round down
913 if ((x ^ y) < 0 && (r * y != x)) {
914 r--;
915 }
916 return r;
917 }
918
919 /**
920 * Returns the largest (closest to positive infinity)
921 * {@code long} value that is less than or equal to the algebraic quotient.
922 * There is one special case, if the dividend is the
923 * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
924 * then integer overflow occurs and
925 * the result is equal to the {@code Long.MIN_VALUE}.
926 * <p>
927 * Normal integer division operates under the round to zero rounding mode
928 * (truncation). This operation instead acts under the round toward
929 * negative infinity (floor) rounding mode.
930 * The floor rounding mode gives different results than truncation
931 * when the exact result is negative.
932 * <p>
933 * For examples, see {@link #floorDiv(int, int)}.
934 *
935 * @param x the dividend
936 * @param y the divisor
937 * @return the largest (closest to positive infinity)
938 * {@code long} value that is less than or equal to the algebraic quotient.
939 * @throws ArithmeticException if the divisor {@code y} is zero
940 * @see #floorMod(long, long)
941 * @see #floor(double)
942 * @since 1.8
943 */
944 public static long floorDiv(long x, long y) {
945 long r = x / y;
946 // if the signs are different and modulo not zero, round down
947 if ((x ^ y) < 0 && (r * y != x)) {
948 r--;
949 }
950 return r;
951 }
952
953 /**
954 * Returns the floor modulus of the {@code int} arguments.
955 * <p>
956 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
957 * has the same sign as the divisor {@code y}, and
958 * is in the range of {@code -abs(y) < r < +abs(y)}.
959 *
960 * <p>
961 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
962 * <ul>
963 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
964 * </ul>
965 * <p>
966 * The difference in values between {@code floorMod} and
967 * the {@code %} operator is due to the difference between
968 * {@code floorDiv} that returns the integer less than or equal to the quotient
969 * and the {@code /} operator that returns the integer closest to zero.
970 * <p>
971 * Examples:
972 * <ul>
973 * <li>If the signs of the arguments are the same, the results
974 * of {@code floorMod} and the {@code %} operator are the same. <br>
975 * <ul>
976 * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li>
977 * </ul>
978 * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
979 * <ul>
980 * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li>
981 * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li>
982 * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li>
983 * </ul>
984 * </li>
985 * </ul>
986 * <p>
987 * If the signs of arguments are unknown and a positive modulus
988 * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
989 *
990 * @param x the dividend
991 * @param y the divisor
992 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
993 * @throws ArithmeticException if the divisor {@code y} is zero
994 * @see #floorDiv(int, int)
995 * @since 1.8
996 */
997 public static int floorMod(int x, int y) {
998 int r = x - floorDiv(x, y) * y;
999 return r;
1000 }
1001
1002 /**
1003 * Returns the floor modulus of the {@code long} arguments.
1004 * <p>
1005 * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1006 * has the same sign as the divisor {@code y}, and
1007 * is in the range of {@code -abs(y) < r < +abs(y)}.
1008 *
1009 * <p>
1010 * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1011 * <ul>
1012 * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1013 * </ul>
1014 * <p>
1015 * For examples, see {@link #floorMod(int, int)}.
1016 *
1017 * @param x the dividend
1018 * @param y the divisor
1019 * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1020 * @throws ArithmeticException if the divisor {@code y} is zero
1021 * @see #floorDiv(long, long)
1022 * @since 1.8
1023 */
1024 public static long floorMod(long x, long y) {
1025 return x - floorDiv(x, y) * y;
1026 }
1027
1028 /**
1029 * Returns the absolute value of an {@code int} value.
1030 * If the argument is not negative, the argument is returned.
1031 * If the argument is negative, the negation of the argument is returned.
1032 *
1033 * <p>Note that if the argument is equal to the value of
1034 * {@link Integer#MIN_VALUE}, the most negative representable
1035 * {@code int} value, the result is that same value, which is
1036 * negative.
1037 *
1038 * @param a the argument whose absolute value is to be determined
1039 * @return the absolute value of the argument.
1040 */
1041 public static int abs(int a) {
1042 return (a < 0) ? -a : a;
1043 }
1044
1045 /**
1046 * Returns the absolute value of a {@code long} value.
1047 * If the argument is not negative, the argument is returned.
1048 * If the argument is negative, the negation of the argument is returned.
1049 *
1050 * <p>Note that if the argument is equal to the value of
1051 * {@link Long#MIN_VALUE}, the most negative representable
1052 * {@code long} value, the result is that same value, which
1053 * is negative.
1054 *
1055 * @param a the argument whose absolute value is to be determined
1056 * @return the absolute value of the argument.
1057 */
1058 public static long abs(long a) {
1059 return (a < 0) ? -a : a;
1060 }
1061
1062 /**
1063 * Returns the absolute value of a {@code float} value.
1064 * If the argument is not negative, the argument is returned.
1065 * If the argument is negative, the negation of the argument is returned.
1066 * Special cases:
1067 * <ul><li>If the argument is positive zero or negative zero, the
1068 * result is positive zero.
1069 * <li>If the argument is infinite, the result is positive infinity.
1070 * <li>If the argument is NaN, the result is NaN.</ul>
1071 * In other words, the result is the same as the value of the expression:
1072 * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1073 *
1074 * @param a the argument whose absolute value is to be determined
1075 * @return the absolute value of the argument.
1076 */
1077 public static float abs(float a) {
1078 return (a <= 0.0F) ? 0.0F - a : a;
1079 }
1080
1081 /**
1082 * Returns the absolute value of a {@code double} value.
1083 * If the argument is not negative, the argument is returned.
1084 * If the argument is negative, the negation of the argument is returned.
1085 * Special cases:
1086 * <ul><li>If the argument is positive zero or negative zero, the result
1087 * is positive zero.
1088 * <li>If the argument is infinite, the result is positive infinity.
1089 * <li>If the argument is NaN, the result is NaN.</ul>
1090 * In other words, the result is the same as the value of the expression:
1091 * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1092 *
1093 * @param a the argument whose absolute value is to be determined
1094 * @return the absolute value of the argument.
1095 */
1096 public static double abs(double a) {
1097 return (a <= 0.0D) ? 0.0D - a : a;
1098 }
1099
1100 /**
1101 * Returns the greater of two {@code int} values. That is, the
1102 * result is the argument closer to the value of
1103 * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1104 * the result is that same value.
1105 *
1106 * @param a an argument.
1107 * @param b another argument.
1108 * @return the larger of {@code a} and {@code b}.
1109 */
1110 public static int max(int a, int b) {
1111 return (a >= b) ? a : b;
1112 }
1113
1114 /**
1115 * Returns the greater of two {@code long} values. That is, the
1116 * result is the argument closer to the value of
1117 * {@link Long#MAX_VALUE}. If the arguments have the same value,
1118 * the result is that same value.
1119 *
1120 * @param a an argument.
1121 * @param b another argument.
1122 * @return the larger of {@code a} and {@code b}.
1123 */
1124 public static long max(long a, long b) {
1125 return (a >= b) ? a : b;
1126 }
1127
1128 // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1129 private static long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
1130 private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1131
1132 /**
1133 * Returns the greater of two {@code float} values. That is,
1134 * the result is the argument closer to positive infinity. If the
1135 * arguments have the same value, the result is that same
1136 * value. If either value is NaN, then the result is NaN. Unlike
1137 * the numerical comparison operators, this method considers
1138 * negative zero to be strictly smaller than positive zero. If one
1139 * argument is positive zero and the other negative zero, the
1140 * result is positive zero.
1141 *
1142 * @param a an argument.
1143 * @param b another argument.
1144 * @return the larger of {@code a} and {@code b}.
1145 */
1146 public static float max(float a, float b) {
1147 if (a != a)
1148 return a; // a is NaN
1149 if ((a == 0.0f) &&
1150 (b == 0.0f) &&
1151 (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1152 // Raw conversion ok since NaN can't map to -0.0.
1153 return b;
1154 }
1155 return (a >= b) ? a : b;
1156 }
1157
1158 /**
1159 * Returns the greater of two {@code double} values. That
1160 * is, the result is the argument closer to positive infinity. If
1161 * the arguments have the same value, the result is that same
1162 * value. If either value is NaN, then the result is NaN. Unlike
1163 * the numerical comparison operators, this method considers
1164 * negative zero to be strictly smaller than positive zero. If one
1165 * argument is positive zero and the other negative zero, the
1166 * result is positive zero.
1167 *
1168 * @param a an argument.
1169 * @param b another argument.
1170 * @return the larger of {@code a} and {@code b}.
1171 */
1172 public static double max(double a, double b) {
1173 if (a != a)
1174 return a; // a is NaN
1175 if ((a == 0.0d) &&
1176 (b == 0.0d) &&
1177 (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1178 // Raw conversion ok since NaN can't map to -0.0.
1179 return b;
1180 }
1181 return (a >= b) ? a : b;
1182 }
1183
1184 /**
1185 * Returns the smaller of two {@code int} values. That is,
1186 * the result the argument closer to the value of
1187 * {@link Integer#MIN_VALUE}. If the arguments have the same
1188 * value, the result is that same value.
1189 *
1190 * @param a an argument.
1191 * @param b another argument.
1192 * @return the smaller of {@code a} and {@code b}.
1193 */
1194 public static int min(int a, int b) {
1195 return (a <= b) ? a : b;
1196 }
1197
1198 /**
1199 * Returns the smaller of two {@code long} values. That is,
1200 * the result is the argument closer to the value of
1201 * {@link Long#MIN_VALUE}. If the arguments have the same
1202 * value, the result is that same value.
1203 *
1204 * @param a an argument.
1205 * @param b another argument.
1206 * @return the smaller of {@code a} and {@code b}.
1207 */
1208 public static long min(long a, long b) {
1209 return (a <= b) ? a : b;
1210 }
1211
1212 /**
1213 * Returns the smaller of two {@code float} values. That is,
1214 * the result is the value closer to negative infinity. If the
1215 * arguments have the same value, the result is that same
1216 * value. If either value is NaN, then the result is NaN. Unlike
1217 * the numerical comparison operators, this method considers
1218 * negative zero to be strictly smaller than positive zero. If
1219 * one argument is positive zero and the other is negative zero,
1220 * the result is negative zero.
1221 *
1222 * @param a an argument.
1223 * @param b another argument.
1224 * @return the smaller of {@code a} and {@code b}.
1225 */
1226 public static float min(float a, float b) {
1227 if (a != a)
1228 return a; // a is NaN
1229 if ((a == 0.0f) &&
1230 (b == 0.0f) &&
1231 (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1232 // Raw conversion ok since NaN can't map to -0.0.
1233 return b;
1234 }
1235 return (a <= b) ? a : b;
1236 }
1237
1238 /**
1239 * Returns the smaller of two {@code double} values. That
1240 * is, the result is the value closer to negative infinity. If the
1241 * arguments have the same value, the result is that same
1242 * value. If either value is NaN, then the result is NaN. Unlike
1243 * the numerical comparison operators, this method considers
1244 * negative zero to be strictly smaller than positive zero. If one
1245 * argument is positive zero and the other is negative zero, the
1246 * result is negative zero.
1247 *
1248 * @param a an argument.
1249 * @param b another argument.
1250 * @return the smaller of {@code a} and {@code b}.
1251 */
1252 public static double min(double a, double b) {
1253 if (a != a)
1254 return a; // a is NaN
1255 if ((a == 0.0d) &&
1256 (b == 0.0d) &&
1257 (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1258 // Raw conversion ok since NaN can't map to -0.0.
1259 return b;
1260 }
1261 return (a <= b) ? a : b;
1262 }
1263
1264 /**
1265 * Returns the size of an ulp of the argument. An ulp, unit in
1266 * the last place, of a {@code double} value is the positive
1267 * distance between this floating-point value and the {@code
1268 * double} value next larger in magnitude. Note that for non-NaN
1269 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1270 *
1271 * <p>Special Cases:
1272 * <ul>
1273 * <li> If the argument is NaN, then the result is NaN.
1274 * <li> If the argument is positive or negative infinity, then the
1275 * result is positive infinity.
1276 * <li> If the argument is positive or negative zero, then the result is
1277 * {@code Double.MIN_VALUE}.
1278 * <li> If the argument is ±{@code Double.MAX_VALUE}, then
1279 * the result is equal to 2<sup>971</sup>.
1280 * </ul>
1281 *
1282 * @param d the floating-point value whose ulp is to be returned
1283 * @return the size of an ulp of the argument
1284 * @author Joseph D. Darcy
1285 * @since 1.5
1286 */
1287 public static double ulp(double d) {
1288 int exp = getExponent(d);
1289
1290 switch(exp) {
1291 case DoubleConsts.MAX_EXPONENT+1: // NaN or infinity
1292 return Math.abs(d);
1293
1294 case DoubleConsts.MIN_EXPONENT-1: // zero or subnormal
1295 return Double.MIN_VALUE;
1296
1297 default:
1298 assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
1299
1300 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1301 exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1302 if (exp >= DoubleConsts.MIN_EXPONENT) {
1303 return powerOfTwoD(exp);
1304 }
1305 else {
1306 // return a subnormal result; left shift integer
1307 // representation of Double.MIN_VALUE appropriate
1308 // number of positions
1309 return Double.longBitsToDouble(1L <<
1310 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1311 }
1312 }
1313 }
1314
1315 /**
1316 * Returns the size of an ulp of the argument. An ulp, unit in
1317 * the last place, of a {@code float} value is the positive
1318 * distance between this floating-point value and the {@code
1319 * float} value next larger in magnitude. Note that for non-NaN
1320 * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1321 *
1322 * <p>Special Cases:
1323 * <ul>
1324 * <li> If the argument is NaN, then the result is NaN.
1325 * <li> If the argument is positive or negative infinity, then the
1326 * result is positive infinity.
1327 * <li> If the argument is positive or negative zero, then the result is
1328 * {@code Float.MIN_VALUE}.
1329 * <li> If the argument is ±{@code Float.MAX_VALUE}, then
1330 * the result is equal to 2<sup>104</sup>.
1331 * </ul>
1332 *
1333 * @param f the floating-point value whose ulp is to be returned
1334 * @return the size of an ulp of the argument
1335 * @author Joseph D. Darcy
1336 * @since 1.5
1337 */
1338 public static float ulp(float f) {
1339 int exp = getExponent(f);
1340
1341 switch(exp) {
1342 case FloatConsts.MAX_EXPONENT+1: // NaN or infinity
1343 return Math.abs(f);
1344
1345 case FloatConsts.MIN_EXPONENT-1: // zero or subnormal
1346 return FloatConsts.MIN_VALUE;
1347
1348 default:
1349 assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1350
1351 // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1352 exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1353 if (exp >= FloatConsts.MIN_EXPONENT) {
1354 return powerOfTwoF(exp);
1355 }
1356 else {
1357 // return a subnormal result; left shift integer
1358 // representation of FloatConsts.MIN_VALUE appropriate
1359 // number of positions
1360 return Float.intBitsToFloat(1 <<
1361 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1362 }
1363 }
1364 }
1365
1366 /**
1367 * Returns the signum function of the argument; zero if the argument
1368 * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1369 * argument is less than zero.
1370 *
1371 * <p>Special Cases:
1372 * <ul>
1373 * <li> If the argument is NaN, then the result is NaN.
1374 * <li> If the argument is positive zero or negative zero, then the
1375 * result is the same as the argument.
1376 * </ul>
1377 *
1378 * @param d the floating-point value whose signum is to be returned
1379 * @return the signum function of the argument
1380 * @author Joseph D. Darcy
1381 * @since 1.5
1382 */
1383 public static double signum(double d) {
1384 return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1385 }
1386
1387 /**
1388 * Returns the signum function of the argument; zero if the argument
1389 * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1390 * argument is less than zero.
1391 *
1392 * <p>Special Cases:
1393 * <ul>
1394 * <li> If the argument is NaN, then the result is NaN.
1395 * <li> If the argument is positive zero or negative zero, then the
1396 * result is the same as the argument.
1397 * </ul>
1398 *
1399 * @param f the floating-point value whose signum is to be returned
1400 * @return the signum function of the argument
1401 * @author Joseph D. Darcy
1402 * @since 1.5
1403 */
1404 public static float signum(float f) {
1405 return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1406 }
1407
1408 /**
1409 * Returns the hyperbolic sine of a {@code double} value.
1410 * The hyperbolic sine of <i>x</i> is defined to be
1411 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
1412 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1413 *
1414 * <p>Special cases:
1415 * <ul>
1416 *
1417 * <li>If the argument is NaN, then the result is NaN.
1418 *
1419 * <li>If the argument is infinite, then the result is an infinity
1420 * with the same sign as the argument.
1421 *
1422 * <li>If the argument is zero, then the result is a zero with the
1423 * same sign as the argument.
1424 *
1425 * </ul>
1426 *
1427 * <p>The computed result must be within 2.5 ulps of the exact result.
1428 *
1429 * @param x The number whose hyperbolic sine is to be returned.
1430 * @return The hyperbolic sine of {@code x}.
1431 * @since 1.5
1432 */
1433 public static double sinh(double x) {
1434 return StrictMath.sinh(x);
1435 }
1436
1437 /**
1438 * Returns the hyperbolic cosine of a {@code double} value.
1439 * The hyperbolic cosine of <i>x</i> is defined to be
1440 * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
1441 * where <i>e</i> is {@linkplain Math#E Euler's number}.
1442 *
1443 * <p>Special cases:
1444 * <ul>
1445 *
1446 * <li>If the argument is NaN, then the result is NaN.
1447 *
1448 * <li>If the argument is infinite, then the result is positive
1449 * infinity.
1450 *
1451 * <li>If the argument is zero, then the result is {@code 1.0}.
1452 *
1453 * </ul>
1454 *
1455 * <p>The computed result must be within 2.5 ulps of the exact result.
1456 *
1457 * @param x The number whose hyperbolic cosine is to be returned.
1458 * @return The hyperbolic cosine of {@code x}.
1459 * @since 1.5
1460 */
1461 public static double cosh(double x) {
1462 return StrictMath.cosh(x);
1463 }
1464
1465 /**
1466 * Returns the hyperbolic tangent of a {@code double} value.
1467 * The hyperbolic tangent of <i>x</i> is defined to be
1468 * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
1469 * in other words, {@linkplain Math#sinh
1470 * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
1471 * that the absolute value of the exact tanh is always less than
1472 * 1.
1473 *
1474 * <p>Special cases:
1475 * <ul>
1476 *
1477 * <li>If the argument is NaN, then the result is NaN.
1478 *
1479 * <li>If the argument is zero, then the result is a zero with the
1480 * same sign as the argument.
1481 *
1482 * <li>If the argument is positive infinity, then the result is
1483 * {@code +1.0}.
1484 *
1485 * <li>If the argument is negative infinity, then the result is
1486 * {@code -1.0}.
1487 *
1488 * </ul>
1489 *
1490 * <p>The computed result must be within 2.5 ulps of the exact result.
1491 * The result of {@code tanh} for any finite input must have
1492 * an absolute value less than or equal to 1. Note that once the
1493 * exact result of tanh is within 1/2 of an ulp of the limit value
1494 * of ±1, correctly signed ±{@code 1.0} should
1495 * be returned.
1496 *
1497 * @param x The number whose hyperbolic tangent is to be returned.
1498 * @return The hyperbolic tangent of {@code x}.
1499 * @since 1.5
1500 */
1501 public static double tanh(double x) {
1502 return StrictMath.tanh(x);
1503 }
1504
1505 /**
1506 * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1507 * without intermediate overflow or underflow.
1508 *
1509 * <p>Special cases:
1510 * <ul>
1511 *
1512 * <li> If either argument is infinite, then the result
1513 * is positive infinity.
1514 *
1515 * <li> If either argument is NaN and neither argument is infinite,
1516 * then the result is NaN.
1517 *
1518 * </ul>
1519 *
1520 * <p>The computed result must be within 1 ulp of the exact
1521 * result. If one parameter is held constant, the results must be
1522 * semi-monotonic in the other parameter.
1523 *
1524 * @param x a value
1525 * @param y a value
1526 * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
1527 * without intermediate overflow or underflow
1528 * @since 1.5
1529 */
1530 public static double hypot(double x, double y) {
1531 return StrictMath.hypot(x, y);
1532 }
1533
1534 /**
1535 * Returns <i>e</i><sup>x</sup> -1. Note that for values of
1536 * <i>x</i> near 0, the exact sum of
1537 * {@code expm1(x)} + 1 is much closer to the true
1538 * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1539 *
1540 * <p>Special cases:
1541 * <ul>
1542 * <li>If the argument is NaN, the result is NaN.
1543 *
1544 * <li>If the argument is positive infinity, then the result is
1545 * positive infinity.
1546 *
1547 * <li>If the argument is negative infinity, then the result is
1548 * -1.0.
1549 *
1550 * <li>If the argument is zero, then the result is a zero with the
1551 * same sign as the argument.
1552 *
1553 * </ul>
1554 *
1555 * <p>The computed result must be within 1 ulp of the exact result.
1556 * Results must be semi-monotonic. The result of
1557 * {@code expm1} for any finite input must be greater than or
1558 * equal to {@code -1.0}. Note that once the exact result of
1559 * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2
1560 * ulp of the limit value -1, {@code -1.0} should be
1561 * returned.
1562 *
1563 * @param x the exponent to raise <i>e</i> to in the computation of
1564 * <i>e</i><sup>{@code x}</sup> -1.
1565 * @return the value <i>e</i><sup>{@code x}</sup> - 1.
1566 * @since 1.5
1567 */
1568 public static double expm1(double x) {
1569 return StrictMath.expm1(x);
1570 }
1571
1572 /**
1573 * Returns the natural logarithm of the sum of the argument and 1.
1574 * Note that for small values {@code x}, the result of
1575 * {@code log1p(x)} is much closer to the true result of ln(1
1576 * + {@code x}) than the floating-point evaluation of
1577 * {@code log(1.0+x)}.
1578 *
1579 * <p>Special cases:
1580 *
1581 * <ul>
1582 *
1583 * <li>If the argument is NaN or less than -1, then the result is
1584 * NaN.
1585 *
1586 * <li>If the argument is positive infinity, then the result is
1587 * positive infinity.
1588 *
1589 * <li>If the argument is negative one, then the result is
1590 * negative infinity.
1591 *
1592 * <li>If the argument is zero, then the result is a zero with the
1593 * same sign as the argument.
1594 *
1595 * </ul>
1596 *
1597 * <p>The computed result must be within 1 ulp of the exact result.
1598 * Results must be semi-monotonic.
1599 *
1600 * @param x a value
1601 * @return the value ln({@code x} + 1), the natural
1602 * log of {@code x} + 1
1603 * @since 1.5
1604 */
1605 public static double log1p(double x) {
1606 return StrictMath.log1p(x);
1607 }
1608
1609 /**
1610 * Returns the first floating-point argument with the sign of the
1611 * second floating-point argument. Note that unlike the {@link
1612 * StrictMath#copySign(double, double) StrictMath.copySign}
1613 * method, this method does not require NaN {@code sign}
1614 * arguments to be treated as positive values; implementations are
1615 * permitted to treat some NaN arguments as positive and other NaN
1616 * arguments as negative to allow greater performance.
1617 *
1618 * @param magnitude the parameter providing the magnitude of the result
1619 * @param sign the parameter providing the sign of the result
1620 * @return a value with the magnitude of {@code magnitude}
1621 * and the sign of {@code sign}.
1622 * @since 1.6
1623 */
1624 public static double copySign(double magnitude, double sign) {
1625 return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1626 (DoubleConsts.SIGN_BIT_MASK)) |
1627 (Double.doubleToRawLongBits(magnitude) &
1628 (DoubleConsts.EXP_BIT_MASK |
1629 DoubleConsts.SIGNIF_BIT_MASK)));
1630 }
1631
1632 /**
1633 * Returns the first floating-point argument with the sign of the
1634 * second floating-point argument. Note that unlike the {@link
1635 * StrictMath#copySign(float, float) StrictMath.copySign}
1636 * method, this method does not require NaN {@code sign}
1637 * arguments to be treated as positive values; implementations are
1638 * permitted to treat some NaN arguments as positive and other NaN
1639 * arguments as negative to allow greater performance.
1640 *
1641 * @param magnitude the parameter providing the magnitude of the result
1642 * @param sign the parameter providing the sign of the result
1643 * @return a value with the magnitude of {@code magnitude}
1644 * and the sign of {@code sign}.
1645 * @since 1.6
1646 */
1647 public static float copySign(float magnitude, float sign) {
1648 return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1649 (FloatConsts.SIGN_BIT_MASK)) |
1650 (Float.floatToRawIntBits(magnitude) &
1651 (FloatConsts.EXP_BIT_MASK |
1652 FloatConsts.SIGNIF_BIT_MASK)));
1653 }
1654
1655 /**
1656 * Returns the unbiased exponent used in the representation of a
1657 * {@code float}. Special cases:
1658 *
1659 * <ul>
1660 * <li>If the argument is NaN or infinite, then the result is
1661 * {@link Float#MAX_EXPONENT} + 1.
1662 * <li>If the argument is zero or subnormal, then the result is
1663 * {@link Float#MIN_EXPONENT} -1.
1664 * </ul>
1665 * @param f a {@code float} value
1666 * @return the unbiased exponent of the argument
1667 * @since 1.6
1668 */
1669 public static int getExponent(float f) {
1670 /*
1671 * Bitwise convert f to integer, mask out exponent bits, shift
1672 * to the right and then subtract out float's bias adjust to
1673 * get true exponent value
1674 */
1675 return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1676 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1677 }
1678
1679 /**
1680 * Returns the unbiased exponent used in the representation of a
1681 * {@code double}. Special cases:
1682 *
1683 * <ul>
1684 * <li>If the argument is NaN or infinite, then the result is
1685 * {@link Double#MAX_EXPONENT} + 1.
1686 * <li>If the argument is zero or subnormal, then the result is
1687 * {@link Double#MIN_EXPONENT} -1.
1688 * </ul>
1689 * @param d a {@code double} value
1690 * @return the unbiased exponent of the argument
1691 * @since 1.6
1692 */
1693 public static int getExponent(double d) {
1694 /*
1695 * Bitwise convert d to long, mask out exponent bits, shift
1696 * to the right and then subtract out double's bias adjust to
1697 * get true exponent value.
1698 */
1699 return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1700 (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1701 }
1702
1703 /**
1704 * Returns the floating-point number adjacent to the first
1705 * argument in the direction of the second argument. If both
1706 * arguments compare as equal the second argument is returned.
1707 *
1708 * <p>
1709 * Special cases:
1710 * <ul>
1711 * <li> If either argument is a NaN, then NaN is returned.
1712 *
1713 * <li> If both arguments are signed zeros, {@code direction}
1714 * is returned unchanged (as implied by the requirement of
1715 * returning the second argument if the arguments compare as
1716 * equal).
1717 *
1718 * <li> If {@code start} is
1719 * ±{@link Double#MIN_VALUE} and {@code direction}
1720 * has a value such that the result should have a smaller
1721 * magnitude, then a zero with the same sign as {@code start}
1722 * is returned.
1723 *
1724 * <li> If {@code start} is infinite and
1725 * {@code direction} has a value such that the result should
1726 * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1727 * same sign as {@code start} is returned.
1728 *
1729 * <li> If {@code start} is equal to ±
1730 * {@link Double#MAX_VALUE} and {@code direction} has a
1731 * value such that the result should have a larger magnitude, an
1732 * infinity with same sign as {@code start} is returned.
1733 * </ul>
1734 *
1735 * @param start starting floating-point value
1736 * @param direction value indicating which of
1737 * {@code start}'s neighbors or {@code start} should
1738 * be returned
1739 * @return The floating-point number adjacent to {@code start} in the
1740 * direction of {@code direction}.
1741 * @since 1.6
1742 */
1743 public static double nextAfter(double start, double direction) {
1744 /*
1745 * The cases:
1746 *
1747 * nextAfter(+infinity, 0) == MAX_VALUE
1748 * nextAfter(+infinity, +infinity) == +infinity
1749 * nextAfter(-infinity, 0) == -MAX_VALUE
1750 * nextAfter(-infinity, -infinity) == -infinity
1751 *
1752 * are naturally handled without any additional testing
1753 */
1754
1755 // First check for NaN values
1756 if (Double.isNaN(start) || Double.isNaN(direction)) {
1757 // return a NaN derived from the input NaN(s)
1758 return start + direction;
1759 } else if (start == direction) {
1760 return direction;
1761 } else { // start > direction or start < direction
1762 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1763 // then bitwise convert start to integer.
1764 long transducer = Double.doubleToRawLongBits(start + 0.0d);
1765
1766 /*
1767 * IEEE 754 floating-point numbers are lexicographically
1768 * ordered if treated as signed- magnitude integers .
1769 * Since Java's integers are two's complement,
1770 * incrementing" the two's complement representation of a
1771 * logically negative floating-point value *decrements*
1772 * the signed-magnitude representation. Therefore, when
1773 * the integer representation of a floating-point values
1774 * is less than zero, the adjustment to the representation
1775 * is in the opposite direction than would be expected at
1776 * first .
1777 */
1778 if (direction > start) { // Calculate next greater value
1779 transducer = transducer + (transducer >= 0L ? 1L:-1L);
1780 } else { // Calculate next lesser value
1781 assert direction < start;
1782 if (transducer > 0L)
1783 --transducer;
1784 else
1785 if (transducer < 0L )
1786 ++transducer;
1787 /*
1788 * transducer==0, the result is -MIN_VALUE
1789 *
1790 * The transition from zero (implicitly
1791 * positive) to the smallest negative
1792 * signed magnitude value must be done
1793 * explicitly.
1794 */
1795 else
1796 transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
1797 }
1798
1799 return Double.longBitsToDouble(transducer);
1800 }
1801 }
1802
1803 /**
1804 * Returns the floating-point number adjacent to the first
1805 * argument in the direction of the second argument. If both
1806 * arguments compare as equal a value equivalent to the second argument
1807 * is returned.
1808 *
1809 * <p>
1810 * Special cases:
1811 * <ul>
1812 * <li> If either argument is a NaN, then NaN is returned.
1813 *
1814 * <li> If both arguments are signed zeros, a value equivalent
1815 * to {@code direction} is returned.
1816 *
1817 * <li> If {@code start} is
1818 * ±{@link Float#MIN_VALUE} and {@code direction}
1819 * has a value such that the result should have a smaller
1820 * magnitude, then a zero with the same sign as {@code start}
1821 * is returned.
1822 *
1823 * <li> If {@code start} is infinite and
1824 * {@code direction} has a value such that the result should
1825 * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1826 * same sign as {@code start} is returned.
1827 *
1828 * <li> If {@code start} is equal to ±
1829 * {@link Float#MAX_VALUE} and {@code direction} has a
1830 * value such that the result should have a larger magnitude, an
1831 * infinity with same sign as {@code start} is returned.
1832 * </ul>
1833 *
1834 * @param start starting floating-point value
1835 * @param direction value indicating which of
1836 * {@code start}'s neighbors or {@code start} should
1837 * be returned
1838 * @return The floating-point number adjacent to {@code start} in the
1839 * direction of {@code direction}.
1840 * @since 1.6
1841 */
1842 public static float nextAfter(float start, double direction) {
1843 /*
1844 * The cases:
1845 *
1846 * nextAfter(+infinity, 0) == MAX_VALUE
1847 * nextAfter(+infinity, +infinity) == +infinity
1848 * nextAfter(-infinity, 0) == -MAX_VALUE
1849 * nextAfter(-infinity, -infinity) == -infinity
1850 *
1851 * are naturally handled without any additional testing
1852 */
1853
1854 // First check for NaN values
1855 if (Float.isNaN(start) || Double.isNaN(direction)) {
1856 // return a NaN derived from the input NaN(s)
1857 return start + (float)direction;
1858 } else if (start == direction) {
1859 return (float)direction;
1860 } else { // start > direction or start < direction
1861 // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1862 // then bitwise convert start to integer.
1863 int transducer = Float.floatToRawIntBits(start + 0.0f);
1864
1865 /*
1866 * IEEE 754 floating-point numbers are lexicographically
1867 * ordered if treated as signed- magnitude integers .
1868 * Since Java's integers are two's complement,
1869 * incrementing" the two's complement representation of a
1870 * logically negative floating-point value *decrements*
1871 * the signed-magnitude representation. Therefore, when
1872 * the integer representation of a floating-point values
1873 * is less than zero, the adjustment to the representation
1874 * is in the opposite direction than would be expected at
1875 * first.
1876 */
1877 if (direction > start) {// Calculate next greater value
1878 transducer = transducer + (transducer >= 0 ? 1:-1);
1879 } else { // Calculate next lesser value
1880 assert direction < start;
1881 if (transducer > 0)
1882 --transducer;
1883 else
1884 if (transducer < 0 )
1885 ++transducer;
1886 /*
1887 * transducer==0, the result is -MIN_VALUE
1888 *
1889 * The transition from zero (implicitly
1890 * positive) to the smallest negative
1891 * signed magnitude value must be done
1892 * explicitly.
1893 */
1894 else
1895 transducer = FloatConsts.SIGN_BIT_MASK | 1;
1896 }
1897
1898 return Float.intBitsToFloat(transducer);
1899 }
1900 }
1901
1902 /**
1903 * Returns the floating-point value adjacent to {@code d} in
1904 * the direction of positive infinity. This method is
1905 * semantically equivalent to {@code nextAfter(d,
1906 * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
1907 * implementation may run faster than its equivalent
1908 * {@code nextAfter} call.
1909 *
1910 * <p>Special Cases:
1911 * <ul>
1912 * <li> If the argument is NaN, the result is NaN.
1913 *
1914 * <li> If the argument is positive infinity, the result is
1915 * positive infinity.
1916 *
1917 * <li> If the argument is zero, the result is
1918 * {@link Double#MIN_VALUE}
1919 *
1920 * </ul>
1921 *
1922 * @param d starting floating-point value
1923 * @return The adjacent floating-point value closer to positive
1924 * infinity.
1925 * @since 1.6
1926 */
1927 public static double nextUp(double d) {
1928 if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
1929 return d;
1930 else {
1931 d += 0.0d;
1932 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
1933 ((d >= 0.0d)?+1L:-1L));
1934 }
1935 }
1936
1937 /**
1938 * Returns the floating-point value adjacent to {@code f} in
1939 * the direction of positive infinity. This method is
1940 * semantically equivalent to {@code nextAfter(f,
1941 * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
1942 * implementation may run faster than its equivalent
1943 * {@code nextAfter} call.
1944 *
1945 * <p>Special Cases:
1946 * <ul>
1947 * <li> If the argument is NaN, the result is NaN.
1948 *
1949 * <li> If the argument is positive infinity, the result is
1950 * positive infinity.
1951 *
1952 * <li> If the argument is zero, the result is
1953 * {@link Float#MIN_VALUE}
1954 *
1955 * </ul>
1956 *
1957 * @param f starting floating-point value
1958 * @return The adjacent floating-point value closer to positive
1959 * infinity.
1960 * @since 1.6
1961 */
1962 public static float nextUp(float f) {
1963 if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
1964 return f;
1965 else {
1966 f += 0.0f;
1967 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
1968 ((f >= 0.0f)?+1:-1));
1969 }
1970 }
1971
1972 /**
1973 * Returns the floating-point value adjacent to {@code d} in
1974 * the direction of negative infinity. This method is
1975 * semantically equivalent to {@code nextAfter(d,
1976 * Double.NEGATIVE_INFINITY)}; however, a
1977 * {@code nextDown} implementation may run faster than its
1978 * equivalent {@code nextAfter} call.
1979 *
1980 * <p>Special Cases:
1981 * <ul>
1982 * <li> If the argument is NaN, the result is NaN.
1983 *
1984 * <li> If the argument is negative infinity, the result is
1985 * negative infinity.
1986 *
1987 * <li> If the argument is zero, the result is
1988 * {@code -Double.MIN_VALUE}
1989 *
1990 * </ul>
1991 *
1992 * @param d starting floating-point value
1993 * @return The adjacent floating-point value closer to negative
1994 * infinity.
1995 * @since 1.8
1996 */
1997 public static double nextDown(double d) {
1998 if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
1999 return d;
2000 else {
2001 if (d == 0.0)
2002 return -Double.MIN_VALUE;
2003 else
2004 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2005 ((d > 0.0d)?-1L:+1L));
2006 }
2007 }
2008
2009 /**
2010 * Returns the floating-point value adjacent to {@code f} in
2011 * the direction of negative infinity. This method is
2012 * semantically equivalent to {@code nextAfter(f,
2013 * Float.NEGATIVE_INFINITY)}; however, a
2014 * {@code nextDown} implementation may run faster than its
2015 * equivalent {@code nextAfter} call.
2016 *
2017 * <p>Special Cases:
2018 * <ul>
2019 * <li> If the argument is NaN, the result is NaN.
2020 *
2021 * <li> If the argument is negative infinity, the result is
2022 * negative infinity.
2023 *
2024 * <li> If the argument is zero, the result is
2025 * {@code -Float.MIN_VALUE}
2026 *
2027 * </ul>
2028 *
2029 * @param f starting floating-point value
2030 * @return The adjacent floating-point value closer to negative
2031 * infinity.
2032 * @since 1.8
2033 */
2034 public static float nextDown(float f) {
2035 if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2036 return f;
2037 else {
2038 if (f == 0.0f)
2039 return -Float.MIN_VALUE;
2040 else
2041 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2042 ((f > 0.0f)?-1:+1));
2043 }
2044 }
2045
2046 /**
2047 * Returns {@code d} ×
2048 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2049 * by a single correctly rounded floating-point multiply to a
2050 * member of the double value set. See the Java
2051 * Language Specification for a discussion of floating-point
2052 * value sets. If the exponent of the result is between {@link
2053 * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2054 * answer is calculated exactly. If the exponent of the result
2055 * would be larger than {@code Double.MAX_EXPONENT}, an
2056 * infinity is returned. Note that if the result is subnormal,
2057 * precision may be lost; that is, when {@code scalb(x, n)}
2058 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2059 * <i>x</i>. When the result is non-NaN, the result has the same
2060 * sign as {@code d}.
2061 *
2062 * <p>Special cases:
2063 * <ul>
2064 * <li> If the first argument is NaN, NaN is returned.
2065 * <li> If the first argument is infinite, then an infinity of the
2066 * same sign is returned.
2067 * <li> If the first argument is zero, then a zero of the same
2068 * sign is returned.
2069 * </ul>
2070 *
2071 * @param d number to be scaled by a power of two.
2072 * @param scaleFactor power of 2 used to scale {@code d}
2073 * @return {@code d} × 2<sup>{@code scaleFactor}</sup>
2074 * @since 1.6
2075 */
2076 public static double scalb(double d, int scaleFactor) {
2077 /*
2078 * This method does not need to be declared strictfp to
2079 * compute the same correct result on all platforms. When
2080 * scaling up, it does not matter what order the
2081 * multiply-store operations are done; the result will be
2082 * finite or overflow regardless of the operation ordering.
2083 * However, to get the correct result when scaling down, a
2084 * particular ordering must be used.
2085 *
2086 * When scaling down, the multiply-store operations are
2087 * sequenced so that it is not possible for two consecutive
2088 * multiply-stores to return subnormal results. If one
2089 * multiply-store result is subnormal, the next multiply will
2090 * round it away to zero. This is done by first multiplying
2091 * by 2 ^ (scaleFactor % n) and then multiplying several
2092 * times by by 2^n as needed where n is the exponent of number
2093 * that is a covenient power of two. In this way, at most one
2094 * real rounding error occurs. If the double value set is
2095 * being used exclusively, the rounding will occur on a
2096 * multiply. If the double-extended-exponent value set is
2097 * being used, the products will (perhaps) be exact but the
2098 * stores to d are guaranteed to round to the double value
2099 * set.
2100 *
2101 * It is _not_ a valid implementation to first multiply d by
2102 * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2103 * MIN_EXPONENT) since even in a strictfp program double
2104 * rounding on underflow could occur; e.g. if the scaleFactor
2105 * argument was (MIN_EXPONENT - n) and the exponent of d was a
2106 * little less than -(MIN_EXPONENT - n), meaning the final
2107 * result would be subnormal.
2108 *
2109 * Since exact reproducibility of this method can be achieved
2110 * without any undue performance burden, there is no
2111 * compelling reason to allow double rounding on underflow in
2112 * scalb.
2113 */
2114
2115 // magnitude of a power of two so large that scaling a finite
2116 // nonzero value by it would be guaranteed to over or
2117 // underflow; due to rounding, scaling down takes takes an
2118 // additional power of two which is reflected here
2119 final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
2120 DoubleConsts.SIGNIFICAND_WIDTH + 1;
2121 int exp_adjust = 0;
2122 int scale_increment = 0;
2123 double exp_delta = Double.NaN;
2124
2125 // Make sure scaling factor is in a reasonable range
2126
2127 if(scaleFactor < 0) {
2128 scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2129 scale_increment = -512;
2130 exp_delta = twoToTheDoubleScaleDown;
2131 }
2132 else {
2133 scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2134 scale_increment = 512;
2135 exp_delta = twoToTheDoubleScaleUp;
2136 }
2137
2138 // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2139 // technique from "Hacker's Delight" section 10-2.
2140 int t = (scaleFactor >> 9-1) >>> 32 - 9;
2141 exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2142
2143 d *= powerOfTwoD(exp_adjust);
2144 scaleFactor -= exp_adjust;
2145
2146 while(scaleFactor != 0) {
2147 d *= exp_delta;
2148 scaleFactor -= scale_increment;
2149 }
2150 return d;
2151 }
2152
2153 /**
2154 * Returns {@code f} ×
2155 * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2156 * by a single correctly rounded floating-point multiply to a
2157 * member of the float value set. See the Java
2158 * Language Specification for a discussion of floating-point
2159 * value sets. If the exponent of the result is between {@link
2160 * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2161 * answer is calculated exactly. If the exponent of the result
2162 * would be larger than {@code Float.MAX_EXPONENT}, an
2163 * infinity is returned. Note that if the result is subnormal,
2164 * precision may be lost; that is, when {@code scalb(x, n)}
2165 * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2166 * <i>x</i>. When the result is non-NaN, the result has the same
2167 * sign as {@code f}.
2168 *
2169 * <p>Special cases:
2170 * <ul>
2171 * <li> If the first argument is NaN, NaN is returned.
2172 * <li> If the first argument is infinite, then an infinity of the
2173 * same sign is returned.
2174 * <li> If the first argument is zero, then a zero of the same
2175 * sign is returned.
2176 * </ul>
2177 *
2178 * @param f number to be scaled by a power of two.
2179 * @param scaleFactor power of 2 used to scale {@code f}
2180 * @return {@code f} × 2<sup>{@code scaleFactor}</sup>
2181 * @since 1.6
2182 */
2183 public static float scalb(float f, int scaleFactor) {
2184 // magnitude of a power of two so large that scaling a finite
2185 // nonzero value by it would be guaranteed to over or
2186 // underflow; due to rounding, scaling down takes takes an
2187 // additional power of two which is reflected here
2188 final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
2189 FloatConsts.SIGNIFICAND_WIDTH + 1;
2190
2191 // Make sure scaling factor is in a reasonable range
2192 scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2193
2194 /*
2195 * Since + MAX_SCALE for float fits well within the double
2196 * exponent range and + float -> double conversion is exact
2197 * the multiplication below will be exact. Therefore, the
2198 * rounding that occurs when the double product is cast to
2199 * float will be the correctly rounded float result. Since
2200 * all operations other than the final multiply will be exact,
2201 * it is not necessary to declare this method strictfp.
2202 */
2203 return (float)((double)f*powerOfTwoD(scaleFactor));
2204 }
2205
2206 // Constants used in scalb
2207 static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2208 static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2209
2210 /**
2211 * Returns a floating-point power of two in the normal range.
2212 */
2213 static double powerOfTwoD(int n) {
2214 assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
2215 return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2216 (DoubleConsts.SIGNIFICAND_WIDTH-1))
2217 & DoubleConsts.EXP_BIT_MASK);
2218 }
2219
2220 /**
2221 * Returns a floating-point power of two in the normal range.
2222 */
2223 static float powerOfTwoF(int n) {
2224 assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
2225 return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2226 (FloatConsts.SIGNIFICAND_WIDTH-1))
2227 & FloatConsts.EXP_BIT_MASK);
2228 }
2229 }