1 /*
   2  * Copyright (c) 1994, 2016, 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 
  28 import java.util.Random;
  29 import jdk.internal.math.FloatConsts;
  30 import jdk.internal.math.DoubleConsts;
  31 import jdk.internal.HotSpotIntrinsicCandidate;
  32 
  33 /**
  34  * The class {@code Math} contains methods for performing basic
  35  * numeric operations such as the elementary exponential, logarithm,
  36  * square root, and trigonometric functions.
  37  *
  38  * <p>Unlike some of the numeric methods of class
  39  * {@code StrictMath}, all implementations of the equivalent
  40  * functions of class {@code Math} are not defined to return the
  41  * bit-for-bit same results.  This relaxation permits
  42  * better-performing implementations where strict reproducibility is
  43  * not required.
  44  *
  45  * <p>By default many of the {@code Math} methods simply call
  46  * the equivalent method in {@code StrictMath} for their
  47  * implementation.  Code generators are encouraged to use
  48  * platform-specific native libraries or microprocessor instructions,
  49  * where available, to provide higher-performance implementations of
  50  * {@code Math} methods.  Such higher-performance
  51  * implementations still must conform to the specification for
  52  * {@code Math}.
  53  *
  54  * <p>The quality of implementation specifications concern two
  55  * properties, accuracy of the returned result and monotonicity of the
  56  * method.  Accuracy of the floating-point {@code Math} methods is
  57  * measured in terms of <i>ulps</i>, units in the last place.  For a
  58  * given floating-point format, an {@linkplain #ulp(double) ulp} of a
  59  * specific real number value is the distance between the two
  60  * floating-point values bracketing that numerical value.  When
  61  * discussing the accuracy of a method as a whole rather than at a
  62  * specific argument, the number of ulps cited is for the worst-case
  63  * error at any argument.  If a method always has an error less than
  64  * 0.5 ulps, the method always returns the floating-point number
  65  * nearest the exact result; such a method is <i>correctly
  66  * rounded</i>.  A correctly rounded method is generally the best a
  67  * floating-point approximation can be; however, it is impractical for
  68  * many floating-point methods to be correctly rounded.  Instead, for
  69  * the {@code Math} class, a larger error bound of 1 or 2 ulps is
  70  * allowed for certain methods.  Informally, with a 1 ulp error bound,
  71  * when the exact result is a representable number, the exact result
  72  * should be returned as the computed result; otherwise, either of the
  73  * two floating-point values which bracket the exact result may be
  74  * returned.  For exact results large in magnitude, one of the
  75  * endpoints of the bracket may be infinite.  Besides accuracy at
  76  * individual arguments, maintaining proper relations between the
  77  * method at different arguments is also important.  Therefore, most
  78  * methods with more than 0.5 ulp errors are required to be
  79  * <i>semi-monotonic</i>: whenever the mathematical function is
  80  * non-decreasing, so is the floating-point approximation, likewise,
  81  * whenever the mathematical function is non-increasing, so is the
  82  * floating-point approximation.  Not all approximations that have 1
  83  * ulp accuracy will automatically meet the monotonicity requirements.
  84  *
  85  * <p>
  86  * The platform uses signed two's complement integer arithmetic with
  87  * int and long primitive types.  The developer should choose
  88  * the primitive type to ensure that arithmetic operations consistently
  89  * produce correct results, which in some cases means the operations
  90  * will not overflow the range of values of the computation.
  91  * The best practice is to choose the primitive type and algorithm to avoid
  92  * overflow. In cases where the size is {@code int} or {@code long} and
  93  * overflow errors need to be detected, the methods {@code addExact},
  94  * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
  95  * throw an {@code ArithmeticException} when the results overflow.
  96  * For other arithmetic operations such as divide, absolute value,
  97  * increment, decrement, and negation overflow occurs only with
  98  * a specific minimum or maximum value and should be checked against
  99  * the minimum or maximum as appropriate.
 100  *
 101  * @author  unascribed
 102  * @author  Joseph D. Darcy
 103  * @since   1.0
 104  */
 105 
 106 public final class Math {
 107 
 108     /**
 109      * Don't let anyone instantiate this class.
 110      */
 111     private Math() {}
 112 
 113     /**
 114      * The {@code double} value that is closer than any other to
 115      * <i>e</i>, the base of the natural logarithms.
 116      */
 117     public static final double E = 2.7182818284590452354;
 118 
 119     /**
 120      * The {@code double} value that is closer than any other to
 121      * <i>pi</i>, the ratio of the circumference of a circle to its
 122      * diameter.
 123      */
 124     public static final double PI = 3.14159265358979323846;
 125 
 126     /**
 127      * Constant by which to multiply an angular value in degrees to obtain an
 128      * angular value in radians.
 129      */
 130     private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
 131 
 132     /**
 133      * Constant by which to multiply an angular value in radians to obtain an
 134      * angular value in degrees.
 135      */
 136     private static final double RADIANS_TO_DEGREES = 57.29577951308232;
 137 
 138     /**
 139      * Returns the trigonometric sine of an angle.  Special cases:
 140      * <ul><li>If the argument is NaN or an infinity, then the
 141      * result is NaN.
 142      * <li>If the argument is zero, then the result is a zero with the
 143      * same sign as the argument.</ul>
 144      *
 145      * <p>The computed result must be within 1 ulp of the exact result.
 146      * Results must be semi-monotonic.
 147      *
 148      * @param   a   an angle, in radians.
 149      * @return  the sine of the argument.
 150      */
 151     @HotSpotIntrinsicCandidate
 152     public static double sin(double a) {
 153         return StrictMath.sin(a); // default impl. delegates to StrictMath
 154     }
 155 
 156     /**
 157      * Returns the trigonometric cosine of an angle. Special cases:
 158      * <ul><li>If the argument is NaN or an infinity, then the
 159      * result is NaN.</ul>
 160      *
 161      * <p>The computed result must be within 1 ulp of the exact result.
 162      * Results must be semi-monotonic.
 163      *
 164      * @param   a   an angle, in radians.
 165      * @return  the cosine of the argument.
 166      */
 167     @HotSpotIntrinsicCandidate
 168     public static double cos(double a) {
 169         return StrictMath.cos(a); // default impl. delegates to StrictMath
 170     }
 171 
 172     /**
 173      * Returns the trigonometric tangent of an angle.  Special cases:
 174      * <ul><li>If the argument is NaN or an infinity, then the result
 175      * is NaN.
 176      * <li>If the argument is zero, then the result is a zero with the
 177      * same sign as the argument.</ul>
 178      *
 179      * <p>The computed result must be within 1 ulp of the exact result.
 180      * Results must be semi-monotonic.
 181      *
 182      * @param   a   an angle, in radians.
 183      * @return  the tangent of the argument.
 184      */
 185     @HotSpotIntrinsicCandidate
 186     public static double tan(double a) {
 187         return StrictMath.tan(a); // default impl. delegates to StrictMath
 188     }
 189 
 190     /**
 191      * Returns the arc sine of a value; the returned angle is in the
 192      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 193      * <ul><li>If the argument is NaN or its absolute value is greater
 194      * than 1, then the result is NaN.
 195      * <li>If the argument is zero, then the result is a zero with the
 196      * same sign as the argument.</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 sine is to be returned.
 202      * @return  the arc sine of the argument.
 203      */
 204     public static double asin(double a) {
 205         return StrictMath.asin(a); // default impl. delegates to StrictMath
 206     }
 207 
 208     /**
 209      * Returns the arc cosine of a value; the returned angle is in the
 210      * range 0.0 through <i>pi</i>.  Special case:
 211      * <ul><li>If the argument is NaN or its absolute value is greater
 212      * than 1, then the result is NaN.</ul>
 213      *
 214      * <p>The computed result must be within 1 ulp of the exact result.
 215      * Results must be semi-monotonic.
 216      *
 217      * @param   a   the value whose arc cosine is to be returned.
 218      * @return  the arc cosine of the argument.
 219      */
 220     public static double acos(double a) {
 221         return StrictMath.acos(a); // default impl. delegates to StrictMath
 222     }
 223 
 224     /**
 225      * Returns the arc tangent of a value; the returned angle is in the
 226      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 227      * <ul><li>If the argument is NaN, then the result is NaN.
 228      * <li>If the argument is zero, then the result is a zero with the
 229      * same sign as the argument.</ul>
 230      *
 231      * <p>The computed result must be within 1 ulp of the exact result.
 232      * Results must be semi-monotonic.
 233      *
 234      * @param   a   the value whose arc tangent is to be returned.
 235      * @return  the arc tangent of the argument.
 236      */
 237     public static double atan(double a) {
 238         return StrictMath.atan(a); // default impl. delegates to StrictMath
 239     }
 240 
 241     /**
 242      * Converts an angle measured in degrees to an approximately
 243      * equivalent angle measured in radians.  The conversion from
 244      * degrees to radians is generally inexact.
 245      *
 246      * @param   angdeg   an angle, in degrees
 247      * @return  the measurement of the angle {@code angdeg}
 248      *          in radians.
 249      * @since   1.2
 250      */
 251     public static double toRadians(double angdeg) {
 252         return angdeg * DEGREES_TO_RADIANS;
 253     }
 254 
 255     /**
 256      * Converts an angle measured in radians to an approximately
 257      * equivalent angle measured in degrees.  The conversion from
 258      * radians to degrees is generally inexact; users should
 259      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
 260      * equal {@code 0.0}.
 261      *
 262      * @param   angrad   an angle, in radians
 263      * @return  the measurement of the angle {@code angrad}
 264      *          in degrees.
 265      * @since   1.2
 266      */
 267     public static double toDegrees(double angrad) {
 268         return angrad * RADIANS_TO_DEGREES;
 269     }
 270 
 271     /**
 272      * Returns Euler's number <i>e</i> raised to the power of a
 273      * {@code double} value.  Special cases:
 274      * <ul><li>If the argument is NaN, the result is NaN.
 275      * <li>If the argument is positive infinity, then the result is
 276      * positive infinity.
 277      * <li>If the argument is negative infinity, then the result is
 278      * positive zero.</ul>
 279      *
 280      * <p>The computed result must be within 1 ulp of the exact result.
 281      * Results must be semi-monotonic.
 282      *
 283      * @param   a   the exponent to raise <i>e</i> to.
 284      * @return  the value <i>e</i><sup>{@code a}</sup>,
 285      *          where <i>e</i> is the base of the natural logarithms.
 286      */
 287     @HotSpotIntrinsicCandidate
 288     public static double exp(double a) {
 289         return StrictMath.exp(a); // default impl. delegates to StrictMath
 290     }
 291 
 292     /**
 293      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
 294      * value.  Special cases:
 295      * <ul><li>If the argument is NaN or less than zero, then the result
 296      * is NaN.
 297      * <li>If the argument is positive infinity, then the result is
 298      * positive infinity.
 299      * <li>If the argument is positive zero or negative zero, then the
 300      * result is negative infinity.</ul>
 301      *
 302      * <p>The computed result must be within 1 ulp of the exact result.
 303      * Results must be semi-monotonic.
 304      *
 305      * @param   a   a value
 306      * @return  the value ln&nbsp;{@code a}, the natural logarithm of
 307      *          {@code a}.
 308      */
 309     @HotSpotIntrinsicCandidate
 310     public static double log(double a) {
 311         return StrictMath.log(a); // default impl. delegates to StrictMath
 312     }
 313 
 314     /**
 315      * Returns the base 10 logarithm of a {@code double} value.
 316      * Special cases:
 317      *
 318      * <ul><li>If the argument is NaN or less than zero, then the result
 319      * is NaN.
 320      * <li>If the argument is positive infinity, then the result is
 321      * positive infinity.
 322      * <li>If the argument is positive zero or negative zero, then the
 323      * result is negative infinity.
 324      * <li> If the argument is equal to 10<sup><i>n</i></sup> for
 325      * integer <i>n</i>, then the result is <i>n</i>.
 326      * </ul>
 327      *
 328      * <p>The computed result must be within 1 ulp of the exact result.
 329      * Results must be semi-monotonic.
 330      *
 331      * @param   a   a value
 332      * @return  the base 10 logarithm of  {@code a}.
 333      * @since 1.5
 334      */
 335     @HotSpotIntrinsicCandidate
 336     public static double log10(double a) {
 337         return StrictMath.log10(a); // default impl. delegates to StrictMath
 338     }
 339 
 340     /**
 341      * Returns the correctly rounded positive square root of a
 342      * {@code double} value.
 343      * Special cases:
 344      * <ul><li>If the argument is NaN or less than zero, then the result
 345      * is NaN.
 346      * <li>If the argument is positive infinity, then the result is positive
 347      * infinity.
 348      * <li>If the argument is positive zero or negative zero, then the
 349      * result is the same as the argument.</ul>
 350      * Otherwise, the result is the {@code double} value closest to
 351      * the true mathematical square root of the argument value.
 352      *
 353      * @param   a   a value.
 354      * @return  the positive square root of {@code a}.
 355      *          If the argument is NaN or less than zero, the result is NaN.
 356      */
 357     @HotSpotIntrinsicCandidate
 358     public static double sqrt(double a) {
 359         return StrictMath.sqrt(a); // default impl. delegates to StrictMath
 360                                    // Note that hardware sqrt instructions
 361                                    // frequently can be directly used by JITs
 362                                    // and should be much faster than doing
 363                                    // Math.sqrt in software.
 364     }
 365 
 366 
 367     /**
 368      * Returns the cube root of a {@code double} value.  For
 369      * positive finite {@code x}, {@code cbrt(-x) ==
 370      * -cbrt(x)}; that is, the cube root of a negative value is
 371      * the negative of the cube root of that value's magnitude.
 372      *
 373      * Special cases:
 374      *
 375      * <ul>
 376      *
 377      * <li>If the argument is NaN, then the result is NaN.
 378      *
 379      * <li>If the argument is infinite, then the result is an infinity
 380      * with the same sign as the argument.
 381      *
 382      * <li>If the argument is zero, then the result is a zero with the
 383      * same sign as the argument.
 384      *
 385      * </ul>
 386      *
 387      * <p>The computed result must be within 1 ulp of the exact result.
 388      *
 389      * @param   a   a value.
 390      * @return  the cube root of {@code a}.
 391      * @since 1.5
 392      */
 393     public static double cbrt(double a) {
 394         return StrictMath.cbrt(a);
 395     }
 396 
 397     /**
 398      * Computes the remainder operation on two arguments as prescribed
 399      * by the IEEE 754 standard.
 400      * The remainder value is mathematically equal to
 401      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
 402      * where <i>n</i> is the mathematical integer closest to the exact
 403      * mathematical value of the quotient {@code f1/f2}, and if two
 404      * mathematical integers are equally close to {@code f1/f2},
 405      * then <i>n</i> is the integer that is even. If the remainder is
 406      * zero, its sign is the same as the sign of the first argument.
 407      * Special cases:
 408      * <ul><li>If either argument is NaN, or the first argument is infinite,
 409      * or the second argument is positive zero or negative zero, then the
 410      * result is NaN.
 411      * <li>If the first argument is finite and the second argument is
 412      * infinite, then the result is the same as the first argument.</ul>
 413      *
 414      * @param   f1   the dividend.
 415      * @param   f2   the divisor.
 416      * @return  the remainder when {@code f1} is divided by
 417      *          {@code f2}.
 418      */
 419     public static double IEEEremainder(double f1, double f2) {
 420         return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
 421     }
 422 
 423     /**
 424      * Returns the smallest (closest to negative infinity)
 425      * {@code double} value that is greater than or equal to the
 426      * argument and is equal to a mathematical integer. Special cases:
 427      * <ul><li>If the argument value is already equal to a
 428      * mathematical integer, then the result is the same as the
 429      * argument.  <li>If the argument is NaN or an infinity or
 430      * positive zero or negative zero, then the result is the same as
 431      * the argument.  <li>If the argument value is less than zero but
 432      * greater than -1.0, then the result is negative zero.</ul> Note
 433      * that the value of {@code Math.ceil(x)} is exactly the
 434      * value of {@code -Math.floor(-x)}.
 435      *
 436      *
 437      * @param   a   a value.
 438      * @return  the smallest (closest to negative infinity)
 439      *          floating-point value that is greater than or equal to
 440      *          the argument and is equal to a mathematical integer.
 441      */
 442     public static double ceil(double a) {
 443         return StrictMath.ceil(a); // default impl. delegates to StrictMath
 444     }
 445 
 446     /**
 447      * Returns the largest (closest to positive infinity)
 448      * {@code double} value that is less than or equal to the
 449      * argument and is equal to a mathematical integer. Special cases:
 450      * <ul><li>If the argument value is already equal to a
 451      * mathematical integer, then the result is the same as the
 452      * argument.  <li>If the argument is NaN or an infinity or
 453      * positive zero or negative zero, then the result is the same as
 454      * the argument.</ul>
 455      *
 456      * @param   a   a value.
 457      * @return  the largest (closest to positive infinity)
 458      *          floating-point value that less than or equal to the argument
 459      *          and is equal to a mathematical integer.
 460      */
 461     public static double floor(double a) {
 462         return StrictMath.floor(a); // default impl. delegates to StrictMath
 463     }
 464 
 465     /**
 466      * Returns the {@code double} value that is closest in value
 467      * to the argument and is equal to a mathematical integer. If two
 468      * {@code double} values that are mathematical integers are
 469      * equally close, the result is the integer value that is
 470      * even. Special cases:
 471      * <ul><li>If the argument value is already equal to a mathematical
 472      * integer, then the result is the same as the argument.
 473      * <li>If the argument is NaN or an infinity or positive zero or negative
 474      * zero, then the result is the same as the argument.</ul>
 475      *
 476      * @param   a   a {@code double} value.
 477      * @return  the closest floating-point value to {@code a} that is
 478      *          equal to a mathematical integer.
 479      */
 480     public static double rint(double a) {
 481         return StrictMath.rint(a); // default impl. delegates to StrictMath
 482     }
 483 
 484     /**
 485      * Returns the angle <i>theta</i> from the conversion of rectangular
 486      * coordinates ({@code x},&nbsp;{@code y}) to polar
 487      * coordinates (r,&nbsp;<i>theta</i>).
 488      * This method computes the phase <i>theta</i> by computing an arc tangent
 489      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
 490      * cases:
 491      * <ul><li>If either argument is NaN, then the result is NaN.
 492      * <li>If the first argument is positive zero and the second argument
 493      * is positive, or the first argument is positive and finite and the
 494      * second argument is positive infinity, then the result is positive
 495      * zero.
 496      * <li>If the first argument is negative zero and the second argument
 497      * is positive, or the first argument is negative and finite and the
 498      * second argument is positive infinity, then the result is negative zero.
 499      * <li>If the first argument is positive zero and the second argument
 500      * is negative, or the first argument is positive and finite and the
 501      * second argument is negative infinity, then the result is the
 502      * {@code double} value closest to <i>pi</i>.
 503      * <li>If the first argument is negative zero and the second argument
 504      * is negative, or the first argument is negative and finite and the
 505      * second argument is negative infinity, then the result is the
 506      * {@code double} value closest to -<i>pi</i>.
 507      * <li>If the first argument is positive and the second argument is
 508      * positive zero or negative zero, or the first argument is positive
 509      * infinity and the second argument is finite, then the result is the
 510      * {@code double} value closest to <i>pi</i>/2.
 511      * <li>If the first argument is negative and the second argument is
 512      * positive zero or negative zero, or the first argument is negative
 513      * infinity and the second argument is finite, then the result is the
 514      * {@code double} value closest to -<i>pi</i>/2.
 515      * <li>If both arguments are positive infinity, then the result is the
 516      * {@code double} value closest to <i>pi</i>/4.
 517      * <li>If the first argument is positive infinity and the second argument
 518      * is negative infinity, then the result is the {@code double}
 519      * value closest to 3*<i>pi</i>/4.
 520      * <li>If the first argument is negative infinity and the second argument
 521      * is positive infinity, then the result is the {@code double} value
 522      * closest to -<i>pi</i>/4.
 523      * <li>If both arguments are negative infinity, then the result is the
 524      * {@code double} value closest to -3*<i>pi</i>/4.</ul>
 525      *
 526      * <p>The computed result must be within 2 ulps of the exact result.
 527      * Results must be semi-monotonic.
 528      *
 529      * @param   y   the ordinate coordinate
 530      * @param   x   the abscissa coordinate
 531      * @return  the <i>theta</i> component of the point
 532      *          (<i>r</i>,&nbsp;<i>theta</i>)
 533      *          in polar coordinates that corresponds to the point
 534      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
 535      */
 536     @HotSpotIntrinsicCandidate
 537     public static double atan2(double y, double x) {
 538         return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
 539     }
 540 
 541     /**
 542      * Returns the value of the first argument raised to the power of the
 543      * second argument. Special cases:
 544      *
 545      * <ul><li>If the second argument is positive or negative zero, then the
 546      * result is 1.0.
 547      * <li>If the second argument is 1.0, then the result is the same as the
 548      * first argument.
 549      * <li>If the second argument is NaN, then the result is NaN.
 550      * <li>If the first argument is NaN and the second argument is nonzero,
 551      * then the result is NaN.
 552      *
 553      * <li>If
 554      * <ul>
 555      * <li>the absolute value of the first argument is greater than 1
 556      * and the second argument is positive infinity, or
 557      * <li>the absolute value of the first argument is less than 1 and
 558      * the second argument is negative infinity,
 559      * </ul>
 560      * then the result is positive infinity.
 561      *
 562      * <li>If
 563      * <ul>
 564      * <li>the absolute value of the first argument is greater than 1 and
 565      * the second argument is negative infinity, or
 566      * <li>the absolute value of the
 567      * first argument is less than 1 and the second argument is positive
 568      * infinity,
 569      * </ul>
 570      * then the result is positive zero.
 571      *
 572      * <li>If the absolute value of the first argument equals 1 and the
 573      * second argument is infinite, then the result is NaN.
 574      *
 575      * <li>If
 576      * <ul>
 577      * <li>the first argument is positive zero and the second argument
 578      * is greater than zero, or
 579      * <li>the first argument is positive infinity and the second
 580      * argument is less than zero,
 581      * </ul>
 582      * then the result is positive zero.
 583      *
 584      * <li>If
 585      * <ul>
 586      * <li>the first argument is positive zero and the second argument
 587      * is less than zero, or
 588      * <li>the first argument is positive infinity and the second
 589      * argument is greater than zero,
 590      * </ul>
 591      * then the result is positive infinity.
 592      *
 593      * <li>If
 594      * <ul>
 595      * <li>the first argument is negative zero and the second argument
 596      * is greater than zero but not a finite odd integer, or
 597      * <li>the first argument is negative infinity and the second
 598      * argument is less than zero but not a finite odd integer,
 599      * </ul>
 600      * then the result is positive zero.
 601      *
 602      * <li>If
 603      * <ul>
 604      * <li>the first argument is negative zero and the second argument
 605      * is a positive finite odd integer, or
 606      * <li>the first argument is negative infinity and the second
 607      * argument is a negative finite odd integer,
 608      * </ul>
 609      * then the result is negative zero.
 610      *
 611      * <li>If
 612      * <ul>
 613      * <li>the first argument is negative zero and the second argument
 614      * is less than zero but not a finite odd integer, or
 615      * <li>the first argument is negative infinity and the second
 616      * argument is greater than zero but not a finite odd integer,
 617      * </ul>
 618      * then the result is positive infinity.
 619      *
 620      * <li>If
 621      * <ul>
 622      * <li>the first argument is negative zero and the second argument
 623      * is a negative finite odd integer, or
 624      * <li>the first argument is negative infinity and the second
 625      * argument is a positive finite odd integer,
 626      * </ul>
 627      * then the result is negative infinity.
 628      *
 629      * <li>If the first argument is finite and less than zero
 630      * <ul>
 631      * <li> if the second argument is a finite even integer, the
 632      * result is equal to the result of raising the absolute value of
 633      * the first argument to the power of the second argument
 634      *
 635      * <li>if the second argument is a finite odd integer, the result
 636      * is equal to the negative of the result of raising the absolute
 637      * value of the first argument to the power of the second
 638      * argument
 639      *
 640      * <li>if the second argument is finite and not an integer, then
 641      * the result is NaN.
 642      * </ul>
 643      *
 644      * <li>If both arguments are integers, then the result is exactly equal
 645      * to the mathematical result of raising the first argument to the power
 646      * of the second argument if that result can in fact be represented
 647      * exactly as a {@code double} value.</ul>
 648      *
 649      * <p>(In the foregoing descriptions, a floating-point value is
 650      * considered to be an integer if and only if it is finite and a
 651      * fixed point of the method {@link #ceil ceil} or,
 652      * equivalently, a fixed point of the method {@link #floor
 653      * floor}. A value is a fixed point of a one-argument
 654      * method if and only if the result of applying the method to the
 655      * value is equal to the value.)
 656      *
 657      * <p>The computed result must be within 1 ulp of the exact result.
 658      * Results must be semi-monotonic.
 659      *
 660      * @param   a   the base.
 661      * @param   b   the exponent.
 662      * @return  the value {@code a}<sup>{@code b}</sup>.
 663      */
 664     @HotSpotIntrinsicCandidate
 665     public static double pow(double a, double b) {
 666         return StrictMath.pow(a, b); // default impl. delegates to StrictMath
 667     }
 668 
 669     /**
 670      * Returns the closest {@code int} to the argument, with ties
 671      * rounding to positive infinity.
 672      *
 673      * <p>
 674      * Special cases:
 675      * <ul><li>If the argument is NaN, the result is 0.
 676      * <li>If the argument is negative infinity or any value less than or
 677      * equal to the value of {@code Integer.MIN_VALUE}, the result is
 678      * equal to the value of {@code Integer.MIN_VALUE}.
 679      * <li>If the argument is positive infinity or any value greater than or
 680      * equal to the value of {@code Integer.MAX_VALUE}, the result is
 681      * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 682      *
 683      * @param   a   a floating-point value to be rounded to an integer.
 684      * @return  the value of the argument rounded to the nearest
 685      *          {@code int} value.
 686      * @see     java.lang.Integer#MAX_VALUE
 687      * @see     java.lang.Integer#MIN_VALUE
 688      */
 689     public static int round(float a) {
 690         int intBits = Float.floatToRawIntBits(a);
 691         int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
 692                 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
 693         int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
 694                 + FloatConsts.EXP_BIAS) - biasedExp;
 695         if ((shift & -32) == 0) { // shift >= 0 && shift < 32
 696             // a is a finite number such that pow(2,-32) <= ulp(a) < 1
 697             int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
 698                     | (FloatConsts.SIGNIF_BIT_MASK + 1));
 699             if (intBits < 0) {
 700                 r = -r;
 701             }
 702             // In the comments below each Java expression evaluates to the value
 703             // the corresponding mathematical expression:
 704             // (r) evaluates to a / ulp(a)
 705             // (r >> shift) evaluates to floor(a * 2)
 706             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 707             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 708             return ((r >> shift) + 1) >> 1;
 709         } else {
 710             // a is either
 711             // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
 712             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 713             // - an infinity or NaN
 714             return (int) a;
 715         }
 716     }
 717 
 718     /**
 719      * Returns the closest {@code long} to the argument, with ties
 720      * rounding to positive infinity.
 721      *
 722      * <p>Special cases:
 723      * <ul><li>If the argument is NaN, the result is 0.
 724      * <li>If the argument is negative infinity or any value less than or
 725      * equal to the value of {@code Long.MIN_VALUE}, the result is
 726      * equal to the value of {@code Long.MIN_VALUE}.
 727      * <li>If the argument is positive infinity or any value greater than or
 728      * equal to the value of {@code Long.MAX_VALUE}, the result is
 729      * equal to the value of {@code Long.MAX_VALUE}.</ul>
 730      *
 731      * @param   a   a floating-point value to be rounded to a
 732      *          {@code long}.
 733      * @return  the value of the argument rounded to the nearest
 734      *          {@code long} value.
 735      * @see     java.lang.Long#MAX_VALUE
 736      * @see     java.lang.Long#MIN_VALUE
 737      */
 738     public static long round(double a) {
 739         long longBits = Double.doubleToRawLongBits(a);
 740         long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
 741                 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
 742         long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
 743                 + DoubleConsts.EXP_BIAS) - biasedExp;
 744         if ((shift & -64) == 0) { // shift >= 0 && shift < 64
 745             // a is a finite number such that pow(2,-64) <= ulp(a) < 1
 746             long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
 747                     | (DoubleConsts.SIGNIF_BIT_MASK + 1));
 748             if (longBits < 0) {
 749                 r = -r;
 750             }
 751             // In the comments below each Java expression evaluates to the value
 752             // the corresponding mathematical expression:
 753             // (r) evaluates to a / ulp(a)
 754             // (r >> shift) evaluates to floor(a * 2)
 755             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 756             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 757             return ((r >> shift) + 1) >> 1;
 758         } else {
 759             // a is either
 760             // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
 761             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 762             // - an infinity or NaN
 763             return (long) a;
 764         }
 765     }
 766 
 767     private static final class RandomNumberGeneratorHolder {
 768         static final Random randomNumberGenerator = new Random();
 769     }
 770 
 771     /**
 772      * Returns a {@code double} value with a positive sign, greater
 773      * than or equal to {@code 0.0} and less than {@code 1.0}.
 774      * Returned values are chosen pseudorandomly with (approximately)
 775      * uniform distribution from that range.
 776      *
 777      * <p>When this method is first called, it creates a single new
 778      * pseudorandom-number generator, exactly as if by the expression
 779      *
 780      * <blockquote>{@code new java.util.Random()}</blockquote>
 781      *
 782      * This new pseudorandom-number generator is used thereafter for
 783      * all calls to this method and is used nowhere else.
 784      *
 785      * <p>This method is properly synchronized to allow correct use by
 786      * more than one thread. However, if many threads need to generate
 787      * pseudorandom numbers at a great rate, it may reduce contention
 788      * for each thread to have its own pseudorandom-number generator.
 789      *
 790      * @apiNote
 791      * As the largest {@code double} value less than {@code 1.0}
 792      * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
 793      * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
 794      *
 795      * <blockquote><pre>{@code
 796      * double f = Math.random()/Math.nextDown(1.0);
 797      * double x = x1*(1.0 - f) + x2*f;
 798      * }</pre></blockquote>
 799      *
 800      * @return  a pseudorandom {@code double} greater than or equal
 801      * to {@code 0.0} and less than {@code 1.0}.
 802      * @see #nextDown(double)
 803      * @see Random#nextDouble()
 804      */
 805     public static double random() {
 806         return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 807     }
 808 
 809     /**
 810      * Returns the sum of its arguments,
 811      * throwing an exception if the result overflows an {@code int}.
 812      *
 813      * @param x the first value
 814      * @param y the second value
 815      * @return the result
 816      * @throws ArithmeticException if the result overflows an int
 817      * @since 1.8
 818      */
 819     @HotSpotIntrinsicCandidate
 820     public static int addExact(int x, int y) {
 821         int r = x + y;
 822         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 823         if (((x ^ r) & (y ^ r)) < 0) {
 824             throw new ArithmeticException("integer overflow");
 825         }
 826         return r;
 827     }
 828 
 829     /**
 830      * Returns the sum of its arguments,
 831      * throwing an exception if the result overflows a {@code long}.
 832      *
 833      * @param x the first value
 834      * @param y the second value
 835      * @return the result
 836      * @throws ArithmeticException if the result overflows a long
 837      * @since 1.8
 838      */
 839     @HotSpotIntrinsicCandidate
 840     public static long addExact(long x, long y) {
 841         long r = x + y;
 842         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 843         if (((x ^ r) & (y ^ r)) < 0) {
 844             throw new ArithmeticException("long overflow");
 845         }
 846         return r;
 847     }
 848 
 849     /**
 850      * Returns the difference of the arguments,
 851      * throwing an exception if the result overflows an {@code int}.
 852      *
 853      * @param x the first value
 854      * @param y the second value to subtract from the first
 855      * @return the result
 856      * @throws ArithmeticException if the result overflows an int
 857      * @since 1.8
 858      */
 859     @HotSpotIntrinsicCandidate
 860     public static int subtractExact(int x, int y) {
 861         int r = x - y;
 862         // HD 2-12 Overflow iff the arguments have different signs and
 863         // the sign of the result is different than the sign of x
 864         if (((x ^ y) & (x ^ r)) < 0) {
 865             throw new ArithmeticException("integer overflow");
 866         }
 867         return r;
 868     }
 869 
 870     /**
 871      * Returns the difference of the arguments,
 872      * throwing an exception if the result overflows a {@code long}.
 873      *
 874      * @param x the first value
 875      * @param y the second value to subtract from the first
 876      * @return the result
 877      * @throws ArithmeticException if the result overflows a long
 878      * @since 1.8
 879      */
 880     @HotSpotIntrinsicCandidate
 881     public static long subtractExact(long x, long y) {
 882         long r = x - y;
 883         // HD 2-12 Overflow iff the arguments have different signs and
 884         // the sign of the result is different than the sign of x
 885         if (((x ^ y) & (x ^ r)) < 0) {
 886             throw new ArithmeticException("long overflow");
 887         }
 888         return r;
 889     }
 890 
 891     /**
 892      * Returns the product of the arguments,
 893      * throwing an exception if the result overflows an {@code int}.
 894      *
 895      * @param x the first value
 896      * @param y the second value
 897      * @return the result
 898      * @throws ArithmeticException if the result overflows an int
 899      * @since 1.8
 900      */
 901     @HotSpotIntrinsicCandidate
 902     public static int multiplyExact(int x, int y) {
 903         long r = (long)x * (long)y;
 904         if ((int)r != r) {
 905             throw new ArithmeticException("integer overflow");
 906         }
 907         return (int)r;
 908     }
 909 
 910     /**
 911      * Returns the product of the arguments,
 912      * throwing an exception if the result overflows a {@code long}.
 913      *
 914      * @param x the first value
 915      * @param y the second value
 916      * @return the result
 917      * @throws ArithmeticException if the result overflows a long
 918      * @since 1.8
 919      */
 920     @HotSpotIntrinsicCandidate
 921     public static long multiplyExact(long x, long y) {
 922         long r = x * y;
 923         long ax = Math.abs(x);
 924         long ay = Math.abs(y);
 925         if (((ax | ay) >>> 31 != 0)) {
 926             // Some bits greater than 2^31 that might cause overflow
 927             // Check the result using the divide operator
 928             // and check for the special case of Long.MIN_VALUE * -1
 929            if (((y != 0) && (r / y != x)) ||
 930                (x == Long.MIN_VALUE && y == -1)) {
 931                 throw new ArithmeticException("long overflow");
 932             }
 933         }
 934         return r;
 935     }
 936 
 937     /**
 938      * Returns the argument incremented by one, throwing an exception if the
 939      * result overflows an {@code int}.
 940      *
 941      * @param a the value to increment
 942      * @return the result
 943      * @throws ArithmeticException if the result overflows an int
 944      * @since 1.8
 945      */
 946     @HotSpotIntrinsicCandidate
 947     public static int incrementExact(int a) {
 948         if (a == Integer.MAX_VALUE) {
 949             throw new ArithmeticException("integer overflow");
 950         }
 951 
 952         return a + 1;
 953     }
 954 
 955     /**
 956      * Returns the argument incremented by one, throwing an exception if the
 957      * result overflows a {@code long}.
 958      *
 959      * @param a the value to increment
 960      * @return the result
 961      * @throws ArithmeticException if the result overflows a long
 962      * @since 1.8
 963      */
 964     @HotSpotIntrinsicCandidate
 965     public static long incrementExact(long a) {
 966         if (a == Long.MAX_VALUE) {
 967             throw new ArithmeticException("long overflow");
 968         }
 969 
 970         return a + 1L;
 971     }
 972 
 973     /**
 974      * Returns the argument decremented by one, throwing an exception if the
 975      * result overflows an {@code int}.
 976      *
 977      * @param a the value to decrement
 978      * @return the result
 979      * @throws ArithmeticException if the result overflows an int
 980      * @since 1.8
 981      */
 982     @HotSpotIntrinsicCandidate
 983     public static int decrementExact(int a) {
 984         if (a == Integer.MIN_VALUE) {
 985             throw new ArithmeticException("integer overflow");
 986         }
 987 
 988         return a - 1;
 989     }
 990 
 991     /**
 992      * Returns the argument decremented by one, throwing an exception if the
 993      * result overflows a {@code long}.
 994      *
 995      * @param a the value to decrement
 996      * @return the result
 997      * @throws ArithmeticException if the result overflows a long
 998      * @since 1.8
 999      */
1000     @HotSpotIntrinsicCandidate
1001     public static long decrementExact(long a) {
1002         if (a == Long.MIN_VALUE) {
1003             throw new ArithmeticException("long overflow");
1004         }
1005 
1006         return a - 1L;
1007     }
1008 
1009     /**
1010      * Returns the negation of the argument, throwing an exception if the
1011      * result overflows an {@code int}.
1012      *
1013      * @param a the value to negate
1014      * @return the result
1015      * @throws ArithmeticException if the result overflows an int
1016      * @since 1.8
1017      */
1018     @HotSpotIntrinsicCandidate
1019     public static int negateExact(int a) {
1020         if (a == Integer.MIN_VALUE) {
1021             throw new ArithmeticException("integer overflow");
1022         }
1023 
1024         return -a;
1025     }
1026 
1027     /**
1028      * Returns the negation of the argument, throwing an exception if the
1029      * result overflows a {@code long}.
1030      *
1031      * @param a the value to negate
1032      * @return the result
1033      * @throws ArithmeticException if the result overflows a long
1034      * @since 1.8
1035      */
1036     @HotSpotIntrinsicCandidate
1037     public static long negateExact(long a) {
1038         if (a == Long.MIN_VALUE) {
1039             throw new ArithmeticException("long overflow");
1040         }
1041 
1042         return -a;
1043     }
1044 
1045     /**
1046      * Returns the value of the {@code long} argument;
1047      * throwing an exception if the value overflows an {@code int}.
1048      *
1049      * @param value the long value
1050      * @return the argument as an int
1051      * @throws ArithmeticException if the {@code argument} overflows an int
1052      * @since 1.8
1053      */
1054     public static int toIntExact(long value) {
1055         if ((int)value != value) {
1056             throw new ArithmeticException("integer overflow");
1057         }
1058         return (int)value;
1059     }
1060 
1061     /**
1062      * Returns the largest (closest to positive infinity)
1063      * {@code int} value that is less than or equal to the algebraic quotient.
1064      * There is one special case, if the dividend is the
1065      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1066      * then integer overflow occurs and
1067      * the result is equal to the {@code Integer.MIN_VALUE}.
1068      * <p>
1069      * Normal integer division operates under the round to zero rounding mode
1070      * (truncation).  This operation instead acts under the round toward
1071      * negative infinity (floor) rounding mode.
1072      * The floor rounding mode gives different results than truncation
1073      * when the exact result is negative.
1074      * <ul>
1075      *   <li>If the signs of the arguments are the same, the results of
1076      *       {@code floorDiv} and the {@code /} operator are the same.  <br>
1077      *       For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1078      *   <li>If the signs of the arguments are different,  the quotient is negative and
1079      *       {@code floorDiv} returns the integer less than or equal to the quotient
1080      *       and the {@code /} operator returns the integer closest to zero.<br>
1081      *       For example, {@code floorDiv(-4, 3) == -2},
1082      *       whereas {@code (-4 / 3) == -1}.
1083      *   </li>
1084      * </ul>
1085      *
1086      * @param x the dividend
1087      * @param y the divisor
1088      * @return the largest (closest to positive infinity)
1089      * {@code int} value that is less than or equal to the algebraic quotient.
1090      * @throws ArithmeticException if the divisor {@code y} is zero
1091      * @see #floorMod(int, int)
1092      * @see #floor(double)
1093      * @since 1.8
1094      */
1095     public static int floorDiv(int x, int y) {
1096         int r = x / y;
1097         // if the signs are different and modulo not zero, round down
1098         if ((x ^ y) < 0 && (r * y != x)) {
1099             r--;
1100         }
1101         return r;
1102     }
1103 
1104     /**
1105      * Returns the largest (closest to positive infinity)
1106      * {@code long} value that is less than or equal to the algebraic quotient.
1107      * There is one special case, if the dividend is the
1108      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1109      * then integer overflow occurs and
1110      * the result is equal to the {@code Long.MIN_VALUE}.
1111      * <p>
1112      * Normal integer division operates under the round to zero rounding mode
1113      * (truncation).  This operation instead acts under the round toward
1114      * negative infinity (floor) rounding mode.
1115      * The floor rounding mode gives different results than truncation
1116      * when the exact result is negative.
1117      * <p>
1118      * For examples, see {@link #floorDiv(int, int)}.
1119      *
1120      * @param x the dividend
1121      * @param y the divisor
1122      * @return the largest (closest to positive infinity)
1123      * {@code long} value that is less than or equal to the algebraic quotient.
1124      * @throws ArithmeticException if the divisor {@code y} is zero
1125      * @see #floorMod(long, long)
1126      * @see #floor(double)
1127      * @since 1.8
1128      */
1129     public static long floorDiv(long x, long y) {
1130         long r = x / y;
1131         // if the signs are different and modulo not zero, round down
1132         if ((x ^ y) < 0 && (r * y != x)) {
1133             r--;
1134         }
1135         return r;
1136     }
1137 
1138     /**
1139      * Returns the floor modulus of the {@code int} arguments.
1140      * <p>
1141      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1142      * has the same sign as the divisor {@code y}, and
1143      * is in the range of {@code -abs(y) < r < +abs(y)}.
1144      *
1145      * <p>
1146      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1147      * <ul>
1148      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1149      * </ul>
1150      * <p>
1151      * The difference in values between {@code floorMod} and
1152      * the {@code %} operator is due to the difference between
1153      * {@code floorDiv} that returns the integer less than or equal to the quotient
1154      * and the {@code /} operator that returns the integer closest to zero.
1155      * <p>
1156      * Examples:
1157      * <ul>
1158      *   <li>If the signs of the arguments are the same, the results
1159      *       of {@code floorMod} and the {@code %} operator are the same.  <br>
1160      *       <ul>
1161      *       <li>{@code floorMod(4, 3) == 1}; &nbsp; and {@code (4 % 3) == 1}</li>
1162      *       </ul>
1163      *   <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1164      *      <ul>
1165      *      <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1} </li>
1166      *      <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1} </li>
1167      *      <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1 } </li>
1168      *      </ul>
1169      *   </li>
1170      * </ul>
1171      * <p>
1172      * If the signs of arguments are unknown and a positive modulus
1173      * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1174      *
1175      * @param x the dividend
1176      * @param y the divisor
1177      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1178      * @throws ArithmeticException if the divisor {@code y} is zero
1179      * @see #floorDiv(int, int)
1180      * @since 1.8
1181      */
1182     public static int floorMod(int x, int y) {
1183         int r = x - floorDiv(x, y) * y;
1184         return r;
1185     }
1186 
1187     /**
1188      * Returns the floor modulus of the {@code long} arguments.
1189      * <p>
1190      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1191      * has the same sign as the divisor {@code y}, and
1192      * is in the range of {@code -abs(y) < r < +abs(y)}.
1193      *
1194      * <p>
1195      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1196      * <ul>
1197      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1198      * </ul>
1199      * <p>
1200      * For examples, see {@link #floorMod(int, int)}.
1201      *
1202      * @param x the dividend
1203      * @param y the divisor
1204      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1205      * @throws ArithmeticException if the divisor {@code y} is zero
1206      * @see #floorDiv(long, long)
1207      * @since 1.8
1208      */
1209     public static long floorMod(long x, long y) {
1210         return x - floorDiv(x, y) * y;
1211     }
1212 
1213     /**
1214      * Returns the absolute value of an {@code int} value.
1215      * If the argument is not negative, the argument is returned.
1216      * If the argument is negative, the negation of the argument is returned.
1217      *
1218      * <p>Note that if the argument is equal to the value of
1219      * {@link Integer#MIN_VALUE}, the most negative representable
1220      * {@code int} value, the result is that same value, which is
1221      * negative.
1222      *
1223      * @param   a   the argument whose absolute value is to be determined
1224      * @return  the absolute value of the argument.
1225      */
1226     public static int abs(int a) {
1227         return (a < 0) ? -a : a;
1228     }
1229 
1230     /**
1231      * Returns the absolute value of a {@code long} value.
1232      * If the argument is not negative, the argument is returned.
1233      * If the argument is negative, the negation of the argument is returned.
1234      *
1235      * <p>Note that if the argument is equal to the value of
1236      * {@link Long#MIN_VALUE}, the most negative representable
1237      * {@code long} value, the result is that same value, which
1238      * is negative.
1239      *
1240      * @param   a   the argument whose absolute value is to be determined
1241      * @return  the absolute value of the argument.
1242      */
1243     public static long abs(long a) {
1244         return (a < 0) ? -a : a;
1245     }
1246 
1247     /**
1248      * Returns the absolute value of a {@code float} value.
1249      * If the argument is not negative, the argument is returned.
1250      * If the argument is negative, the negation of the argument is returned.
1251      * Special cases:
1252      * <ul><li>If the argument is positive zero or negative zero, the
1253      * result is positive zero.
1254      * <li>If the argument is infinite, the result is positive infinity.
1255      * <li>If the argument is NaN, the result is NaN.</ul>
1256      * In other words, the result is the same as the value of the expression:
1257      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1258      *
1259      * @param   a   the argument whose absolute value is to be determined
1260      * @return  the absolute value of the argument.
1261      */
1262     public static float abs(float a) {
1263         return (a <= 0.0F) ? 0.0F - a : a;
1264     }
1265 
1266     /**
1267      * Returns the absolute value of a {@code double} value.
1268      * If the argument is not negative, the argument is returned.
1269      * If the argument is negative, the negation of the argument is returned.
1270      * Special cases:
1271      * <ul><li>If the argument is positive zero or negative zero, the result
1272      * is positive zero.
1273      * <li>If the argument is infinite, the result is positive infinity.
1274      * <li>If the argument is NaN, the result is NaN.</ul>
1275      * In other words, the result is the same as the value of the expression:
1276      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1277      *
1278      * @param   a   the argument whose absolute value is to be determined
1279      * @return  the absolute value of the argument.
1280      */
1281     @HotSpotIntrinsicCandidate
1282     public static double abs(double a) {
1283         return (a <= 0.0D) ? 0.0D - a : a;
1284     }
1285 
1286     /**
1287      * Returns the greater of two {@code int} values. That is, the
1288      * result is the argument closer to the value of
1289      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1290      * the result is that same value.
1291      *
1292      * @param   a   an argument.
1293      * @param   b   another argument.
1294      * @return  the larger of {@code a} and {@code b}.
1295      */
1296     @HotSpotIntrinsicCandidate
1297     public static int max(int a, int b) {
1298         return (a >= b) ? a : b;
1299     }
1300 
1301     /**
1302      * Returns the greater of two {@code long} values. That is, the
1303      * result is the argument closer to the value of
1304      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1305      * the result is that same value.
1306      *
1307      * @param   a   an argument.
1308      * @param   b   another argument.
1309      * @return  the larger of {@code a} and {@code b}.
1310      */
1311     public static long max(long a, long b) {
1312         return (a >= b) ? a : b;
1313     }
1314 
1315     // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1316     private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
1317     private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1318 
1319     /**
1320      * Returns the greater of two {@code float} values.  That is,
1321      * the result is the argument closer to positive infinity. If the
1322      * arguments have the same value, the result is that same
1323      * value. If either value is NaN, then the result is NaN.  Unlike
1324      * the numerical comparison operators, this method considers
1325      * negative zero to be strictly smaller than positive zero. If one
1326      * argument is positive zero and the other negative zero, the
1327      * result is positive zero.
1328      *
1329      * @param   a   an argument.
1330      * @param   b   another argument.
1331      * @return  the larger of {@code a} and {@code b}.
1332      */
1333     public static float max(float a, float b) {
1334         if (a != a)
1335             return a;   // a is NaN
1336         if ((a == 0.0f) &&
1337             (b == 0.0f) &&
1338             (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1339             // Raw conversion ok since NaN can't map to -0.0.
1340             return b;
1341         }
1342         return (a >= b) ? a : b;
1343     }
1344 
1345     /**
1346      * Returns the greater of two {@code double} values.  That
1347      * is, the result is the argument closer to positive infinity. If
1348      * the arguments have the same value, the result is that same
1349      * value. If either value is NaN, then the result is NaN.  Unlike
1350      * the numerical comparison operators, this method considers
1351      * negative zero to be strictly smaller than positive zero. If one
1352      * argument is positive zero and the other negative zero, the
1353      * result is positive zero.
1354      *
1355      * @param   a   an argument.
1356      * @param   b   another argument.
1357      * @return  the larger of {@code a} and {@code b}.
1358      */
1359     public static double max(double a, double b) {
1360         if (a != a)
1361             return a;   // a is NaN
1362         if ((a == 0.0d) &&
1363             (b == 0.0d) &&
1364             (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1365             // Raw conversion ok since NaN can't map to -0.0.
1366             return b;
1367         }
1368         return (a >= b) ? a : b;
1369     }
1370 
1371     /**
1372      * Returns the smaller of two {@code int} values. That is,
1373      * the result the argument closer to the value of
1374      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1375      * value, the result is that same value.
1376      *
1377      * @param   a   an argument.
1378      * @param   b   another argument.
1379      * @return  the smaller of {@code a} and {@code b}.
1380      */
1381     @HotSpotIntrinsicCandidate
1382     public static int min(int a, int b) {
1383         return (a <= b) ? a : b;
1384     }
1385 
1386     /**
1387      * Returns the smaller of two {@code long} values. That is,
1388      * the result is the argument closer to the value of
1389      * {@link Long#MIN_VALUE}. If the arguments have the same
1390      * value, the result is that same value.
1391      *
1392      * @param   a   an argument.
1393      * @param   b   another argument.
1394      * @return  the smaller of {@code a} and {@code b}.
1395      */
1396     public static long min(long a, long b) {
1397         return (a <= b) ? a : b;
1398     }
1399 
1400     /**
1401      * Returns the smaller of two {@code float} values.  That is,
1402      * the result is the value closer to negative infinity. If the
1403      * arguments have the same value, the result is that same
1404      * value. If either value is NaN, then the result is NaN.  Unlike
1405      * the numerical comparison operators, this method considers
1406      * negative zero to be strictly smaller than positive zero.  If
1407      * one argument is positive zero and the other is negative zero,
1408      * the result is negative zero.
1409      *
1410      * @param   a   an argument.
1411      * @param   b   another argument.
1412      * @return  the smaller of {@code a} and {@code b}.
1413      */
1414     public static float min(float a, float b) {
1415         if (a != a)
1416             return a;   // a is NaN
1417         if ((a == 0.0f) &&
1418             (b == 0.0f) &&
1419             (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1420             // Raw conversion ok since NaN can't map to -0.0.
1421             return b;
1422         }
1423         return (a <= b) ? a : b;
1424     }
1425 
1426     /**
1427      * Returns the smaller of two {@code double} values.  That
1428      * is, the result is the value closer to negative infinity. If the
1429      * arguments have the same value, the result is that same
1430      * value. If either value is NaN, then the result is NaN.  Unlike
1431      * the numerical comparison operators, this method considers
1432      * negative zero to be strictly smaller than positive zero. If one
1433      * argument is positive zero and the other is negative zero, the
1434      * result is negative zero.
1435      *
1436      * @param   a   an argument.
1437      * @param   b   another argument.
1438      * @return  the smaller of {@code a} and {@code b}.
1439      */
1440     public static double min(double a, double b) {
1441         if (a != a)
1442             return a;   // a is NaN
1443         if ((a == 0.0d) &&
1444             (b == 0.0d) &&
1445             (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1446             // Raw conversion ok since NaN can't map to -0.0.
1447             return b;
1448         }
1449         return (a <= b) ? a : b;
1450     }
1451 
1452     /**
1453      * Returns the size of an ulp of the argument.  An ulp, unit in
1454      * the last place, of a {@code double} value is the positive
1455      * distance between this floating-point value and the {@code
1456      * double} value next larger in magnitude.  Note that for non-NaN
1457      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1458      *
1459      * <p>Special Cases:
1460      * <ul>
1461      * <li> If the argument is NaN, then the result is NaN.
1462      * <li> If the argument is positive or negative infinity, then the
1463      * result is positive infinity.
1464      * <li> If the argument is positive or negative zero, then the result is
1465      * {@code Double.MIN_VALUE}.
1466      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1467      * the result is equal to 2<sup>971</sup>.
1468      * </ul>
1469      *
1470      * @param d the floating-point value whose ulp is to be returned
1471      * @return the size of an ulp of the argument
1472      * @author Joseph D. Darcy
1473      * @since 1.5
1474      */
1475     public static double ulp(double d) {
1476         int exp = getExponent(d);
1477 
1478         switch(exp) {
1479         case Double.MAX_EXPONENT + 1:       // NaN or infinity
1480             return Math.abs(d);
1481 
1482         case Double.MIN_EXPONENT - 1:       // zero or subnormal
1483             return Double.MIN_VALUE;
1484 
1485         default:
1486             assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
1487 
1488             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1489             exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1490             if (exp >= Double.MIN_EXPONENT) {
1491                 return powerOfTwoD(exp);
1492             }
1493             else {
1494                 // return a subnormal result; left shift integer
1495                 // representation of Double.MIN_VALUE appropriate
1496                 // number of positions
1497                 return Double.longBitsToDouble(1L <<
1498                 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1499             }
1500         }
1501     }
1502 
1503     /**
1504      * Returns the size of an ulp of the argument.  An ulp, unit in
1505      * the last place, of a {@code float} value is the positive
1506      * distance between this floating-point value and the {@code
1507      * float} value next larger in magnitude.  Note that for non-NaN
1508      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1509      *
1510      * <p>Special Cases:
1511      * <ul>
1512      * <li> If the argument is NaN, then the result is NaN.
1513      * <li> If the argument is positive or negative infinity, then the
1514      * result is positive infinity.
1515      * <li> If the argument is positive or negative zero, then the result is
1516      * {@code Float.MIN_VALUE}.
1517      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1518      * the result is equal to 2<sup>104</sup>.
1519      * </ul>
1520      *
1521      * @param f the floating-point value whose ulp is to be returned
1522      * @return the size of an ulp of the argument
1523      * @author Joseph D. Darcy
1524      * @since 1.5
1525      */
1526     public static float ulp(float f) {
1527         int exp = getExponent(f);
1528 
1529         switch(exp) {
1530         case Float.MAX_EXPONENT+1:        // NaN or infinity
1531             return Math.abs(f);
1532 
1533         case Float.MIN_EXPONENT-1:        // zero or subnormal
1534             return Float.MIN_VALUE;
1535 
1536         default:
1537             assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
1538 
1539             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1540             exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1541             if (exp >= Float.MIN_EXPONENT) {
1542                 return powerOfTwoF(exp);
1543             } else {

1544                 // return a subnormal result; left shift integer
1545                 // representation of FloatConsts.MIN_VALUE appropriate
1546                 // number of positions
1547                 return Float.intBitsToFloat(1 <<
1548                 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1549             }
1550         }
1551     }
1552 
1553     /**
1554      * Returns the signum function of the argument; zero if the argument
1555      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1556      * argument is less than zero.
1557      *
1558      * <p>Special Cases:
1559      * <ul>
1560      * <li> If the argument is NaN, then the result is NaN.
1561      * <li> If the argument is positive zero or negative zero, then the
1562      *      result is the same as the argument.
1563      * </ul>
1564      *
1565      * @param d the floating-point value whose signum is to be returned
1566      * @return the signum function of the argument
1567      * @author Joseph D. Darcy
1568      * @since 1.5
1569      */
1570     public static double signum(double d) {
1571         return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1572     }
1573 
1574     /**
1575      * Returns the signum function of the argument; zero if the argument
1576      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1577      * argument is less than zero.
1578      *
1579      * <p>Special Cases:
1580      * <ul>
1581      * <li> If the argument is NaN, then the result is NaN.
1582      * <li> If the argument is positive zero or negative zero, then the
1583      *      result is the same as the argument.
1584      * </ul>
1585      *
1586      * @param f the floating-point value whose signum is to be returned
1587      * @return the signum function of the argument
1588      * @author Joseph D. Darcy
1589      * @since 1.5
1590      */
1591     public static float signum(float f) {
1592         return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1593     }
1594 
1595     /**
1596      * Returns the hyperbolic sine of a {@code double} value.
1597      * The hyperbolic sine of <i>x</i> is defined to be
1598      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1599      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1600      *
1601      * <p>Special cases:
1602      * <ul>
1603      *
1604      * <li>If the argument is NaN, then the result is NaN.
1605      *
1606      * <li>If the argument is infinite, then the result is an infinity
1607      * with the same sign as the argument.
1608      *
1609      * <li>If the argument is zero, then the result is a zero with the
1610      * same sign as the argument.
1611      *
1612      * </ul>
1613      *
1614      * <p>The computed result must be within 2.5 ulps of the exact result.
1615      *
1616      * @param   x The number whose hyperbolic sine is to be returned.
1617      * @return  The hyperbolic sine of {@code x}.
1618      * @since 1.5
1619      */
1620     public static double sinh(double x) {
1621         return StrictMath.sinh(x);
1622     }
1623 
1624     /**
1625      * Returns the hyperbolic cosine of a {@code double} value.
1626      * The hyperbolic cosine of <i>x</i> is defined to be
1627      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1628      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1629      *
1630      * <p>Special cases:
1631      * <ul>
1632      *
1633      * <li>If the argument is NaN, then the result is NaN.
1634      *
1635      * <li>If the argument is infinite, then the result is positive
1636      * infinity.
1637      *
1638      * <li>If the argument is zero, then the result is {@code 1.0}.
1639      *
1640      * </ul>
1641      *
1642      * <p>The computed result must be within 2.5 ulps of the exact result.
1643      *
1644      * @param   x The number whose hyperbolic cosine is to be returned.
1645      * @return  The hyperbolic cosine of {@code x}.
1646      * @since 1.5
1647      */
1648     public static double cosh(double x) {
1649         return StrictMath.cosh(x);
1650     }
1651 
1652     /**
1653      * Returns the hyperbolic tangent of a {@code double} value.
1654      * The hyperbolic tangent of <i>x</i> is defined to be
1655      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1656      * in other words, {@linkplain Math#sinh
1657      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1658      * that the absolute value of the exact tanh is always less than
1659      * 1.
1660      *
1661      * <p>Special cases:
1662      * <ul>
1663      *
1664      * <li>If the argument is NaN, then the result is NaN.
1665      *
1666      * <li>If the argument is zero, then the result is a zero with the
1667      * same sign as the argument.
1668      *
1669      * <li>If the argument is positive infinity, then the result is
1670      * {@code +1.0}.
1671      *
1672      * <li>If the argument is negative infinity, then the result is
1673      * {@code -1.0}.
1674      *
1675      * </ul>
1676      *
1677      * <p>The computed result must be within 2.5 ulps of the exact result.
1678      * The result of {@code tanh} for any finite input must have
1679      * an absolute value less than or equal to 1.  Note that once the
1680      * exact result of tanh is within 1/2 of an ulp of the limit value
1681      * of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
1682      * be returned.
1683      *
1684      * @param   x The number whose hyperbolic tangent is to be returned.
1685      * @return  The hyperbolic tangent of {@code x}.
1686      * @since 1.5
1687      */
1688     public static double tanh(double x) {
1689         return StrictMath.tanh(x);
1690     }
1691 
1692     /**
1693      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1694      * without intermediate overflow or underflow.
1695      *
1696      * <p>Special cases:
1697      * <ul>
1698      *
1699      * <li> If either argument is infinite, then the result
1700      * is positive infinity.
1701      *
1702      * <li> If either argument is NaN and neither argument is infinite,
1703      * then the result is NaN.
1704      *
1705      * </ul>
1706      *
1707      * <p>The computed result must be within 1 ulp of the exact
1708      * result.  If one parameter is held constant, the results must be
1709      * semi-monotonic in the other parameter.
1710      *
1711      * @param x a value
1712      * @param y a value
1713      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1714      * without intermediate overflow or underflow
1715      * @since 1.5
1716      */
1717     public static double hypot(double x, double y) {
1718         return StrictMath.hypot(x, y);
1719     }
1720 
1721     /**
1722      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
1723      * <i>x</i> near 0, the exact sum of
1724      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
1725      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1726      *
1727      * <p>Special cases:
1728      * <ul>
1729      * <li>If the argument is NaN, the result is NaN.
1730      *
1731      * <li>If the argument is positive infinity, then the result is
1732      * positive infinity.
1733      *
1734      * <li>If the argument is negative infinity, then the result is
1735      * -1.0.
1736      *
1737      * <li>If the argument is zero, then the result is a zero with the
1738      * same sign as the argument.
1739      *
1740      * </ul>
1741      *
1742      * <p>The computed result must be within 1 ulp of the exact result.
1743      * Results must be semi-monotonic.  The result of
1744      * {@code expm1} for any finite input must be greater than or
1745      * equal to {@code -1.0}.  Note that once the exact result of
1746      * <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
1747      * ulp of the limit value -1, {@code -1.0} should be
1748      * returned.
1749      *
1750      * @param   x   the exponent to raise <i>e</i> to in the computation of
1751      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
1752      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
1753      * @since 1.5
1754      */
1755     public static double expm1(double x) {
1756         return StrictMath.expm1(x);
1757     }
1758 
1759     /**
1760      * Returns the natural logarithm of the sum of the argument and 1.
1761      * Note that for small values {@code x}, the result of
1762      * {@code log1p(x)} is much closer to the true result of ln(1
1763      * + {@code x}) than the floating-point evaluation of
1764      * {@code log(1.0+x)}.
1765      *
1766      * <p>Special cases:
1767      *
1768      * <ul>
1769      *
1770      * <li>If the argument is NaN or less than -1, then the result is
1771      * NaN.
1772      *
1773      * <li>If the argument is positive infinity, then the result is
1774      * positive infinity.
1775      *
1776      * <li>If the argument is negative one, then the result is
1777      * negative infinity.
1778      *
1779      * <li>If the argument is zero, then the result is a zero with the
1780      * same sign as the argument.
1781      *
1782      * </ul>
1783      *
1784      * <p>The computed result must be within 1 ulp of the exact result.
1785      * Results must be semi-monotonic.
1786      *
1787      * @param   x   a value
1788      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
1789      * log of {@code x}&nbsp;+&nbsp;1
1790      * @since 1.5
1791      */
1792     public static double log1p(double x) {
1793         return StrictMath.log1p(x);
1794     }
1795 
1796     /**
1797      * Returns the first floating-point argument with the sign of the
1798      * second floating-point argument.  Note that unlike the {@link
1799      * StrictMath#copySign(double, double) StrictMath.copySign}
1800      * method, this method does not require NaN {@code sign}
1801      * arguments to be treated as positive values; implementations are
1802      * permitted to treat some NaN arguments as positive and other NaN
1803      * arguments as negative to allow greater performance.
1804      *
1805      * @param magnitude  the parameter providing the magnitude of the result
1806      * @param sign   the parameter providing the sign of the result
1807      * @return a value with the magnitude of {@code magnitude}
1808      * and the sign of {@code sign}.
1809      * @since 1.6
1810      */
1811     public static double copySign(double magnitude, double sign) {
1812         return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1813                                         (DoubleConsts.SIGN_BIT_MASK)) |
1814                                        (Double.doubleToRawLongBits(magnitude) &
1815                                         (DoubleConsts.EXP_BIT_MASK |
1816                                          DoubleConsts.SIGNIF_BIT_MASK)));
1817     }
1818 
1819     /**
1820      * Returns the first floating-point argument with the sign of the
1821      * second floating-point argument.  Note that unlike the {@link
1822      * StrictMath#copySign(float, float) StrictMath.copySign}
1823      * method, this method does not require NaN {@code sign}
1824      * arguments to be treated as positive values; implementations are
1825      * permitted to treat some NaN arguments as positive and other NaN
1826      * arguments as negative to allow greater performance.
1827      *
1828      * @param magnitude  the parameter providing the magnitude of the result
1829      * @param sign   the parameter providing the sign of the result
1830      * @return a value with the magnitude of {@code magnitude}
1831      * and the sign of {@code sign}.
1832      * @since 1.6
1833      */
1834     public static float copySign(float magnitude, float sign) {
1835         return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1836                                      (FloatConsts.SIGN_BIT_MASK)) |
1837                                     (Float.floatToRawIntBits(magnitude) &
1838                                      (FloatConsts.EXP_BIT_MASK |
1839                                       FloatConsts.SIGNIF_BIT_MASK)));
1840     }
1841 
1842     /**
1843      * Returns the unbiased exponent used in the representation of a
1844      * {@code float}.  Special cases:
1845      *
1846      * <ul>
1847      * <li>If the argument is NaN or infinite, then the result is
1848      * {@link Float#MAX_EXPONENT} + 1.
1849      * <li>If the argument is zero or subnormal, then the result is
1850      * {@link Float#MIN_EXPONENT} -1.
1851      * </ul>
1852      * @param f a {@code float} value
1853      * @return the unbiased exponent of the argument
1854      * @since 1.6
1855      */
1856     public static int getExponent(float f) {
1857         /*
1858          * Bitwise convert f to integer, mask out exponent bits, shift
1859          * to the right and then subtract out float's bias adjust to
1860          * get true exponent value
1861          */
1862         return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1863                 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1864     }
1865 
1866     /**
1867      * Returns the unbiased exponent used in the representation of a
1868      * {@code double}.  Special cases:
1869      *
1870      * <ul>
1871      * <li>If the argument is NaN or infinite, then the result is
1872      * {@link Double#MAX_EXPONENT} + 1.
1873      * <li>If the argument is zero or subnormal, then the result is
1874      * {@link Double#MIN_EXPONENT} -1.
1875      * </ul>
1876      * @param d a {@code double} value
1877      * @return the unbiased exponent of the argument
1878      * @since 1.6
1879      */
1880     public static int getExponent(double d) {
1881         /*
1882          * Bitwise convert d to long, mask out exponent bits, shift
1883          * to the right and then subtract out double's bias adjust to
1884          * get true exponent value.
1885          */
1886         return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1887                       (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1888     }
1889 
1890     /**
1891      * Returns the floating-point number adjacent to the first
1892      * argument in the direction of the second argument.  If both
1893      * arguments compare as equal the second argument is returned.
1894      *
1895      * <p>
1896      * Special cases:
1897      * <ul>
1898      * <li> If either argument is a NaN, then NaN is returned.
1899      *
1900      * <li> If both arguments are signed zeros, {@code direction}
1901      * is returned unchanged (as implied by the requirement of
1902      * returning the second argument if the arguments compare as
1903      * equal).
1904      *
1905      * <li> If {@code start} is
1906      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
1907      * has a value such that the result should have a smaller
1908      * magnitude, then a zero with the same sign as {@code start}
1909      * is returned.
1910      *
1911      * <li> If {@code start} is infinite and
1912      * {@code direction} has a value such that the result should
1913      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1914      * same sign as {@code start} is returned.
1915      *
1916      * <li> If {@code start} is equal to &plusmn;
1917      * {@link Double#MAX_VALUE} and {@code direction} has a
1918      * value such that the result should have a larger magnitude, an
1919      * infinity with same sign as {@code start} is returned.
1920      * </ul>
1921      *
1922      * @param start  starting floating-point value
1923      * @param direction value indicating which of
1924      * {@code start}'s neighbors or {@code start} should
1925      * be returned
1926      * @return The floating-point number adjacent to {@code start} in the
1927      * direction of {@code direction}.
1928      * @since 1.6
1929      */
1930     public static double nextAfter(double start, double direction) {
1931         /*
1932          * The cases:
1933          *
1934          * nextAfter(+infinity, 0)  == MAX_VALUE
1935          * nextAfter(+infinity, +infinity)  == +infinity
1936          * nextAfter(-infinity, 0)  == -MAX_VALUE
1937          * nextAfter(-infinity, -infinity)  == -infinity
1938          *
1939          * are naturally handled without any additional testing
1940          */
1941 
1942         /*
1943          * IEEE 754 floating-point numbers are lexicographically
1944          * ordered if treated as signed-magnitude integers.
1945          * Since Java's integers are two's complement,
1946          * incrementing the two's complement representation of a
1947          * logically negative floating-point value *decrements*
1948          * the signed-magnitude representation. Therefore, when
1949          * the integer representation of a floating-point value
1950          * is negative, the adjustment to the representation is in
1951          * the opposite direction from what would initially be expected.
1952          */
1953 
1954         // Branch to descending case first as it is more costly than ascending
1955         // case due to start != 0.0d conditional.
1956         if (start > direction) { // descending
1957             if (start != 0.0d) {
1958                 final long transducer = Double.doubleToRawLongBits(start);
1959                 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
1960             } else { // start == 0.0d && direction < 0.0d
1961                 return -Double.MIN_VALUE;
1962             }
1963         } else if (start < direction) { // ascending
1964             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1965             // then bitwise convert start to integer.
1966             final long transducer = Double.doubleToRawLongBits(start + 0.0d);
1967             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
1968         } else if (start == direction) {
1969             return direction;
1970         } else { // isNaN(start) || isNaN(direction)
1971             return start + direction;
1972         }
1973     }
1974 
1975     /**
1976      * Returns the floating-point number adjacent to the first
1977      * argument in the direction of the second argument.  If both
1978      * arguments compare as equal a value equivalent to the second argument
1979      * is returned.
1980      *
1981      * <p>
1982      * Special cases:
1983      * <ul>
1984      * <li> If either argument is a NaN, then NaN is returned.
1985      *
1986      * <li> If both arguments are signed zeros, a value equivalent
1987      * to {@code direction} is returned.
1988      *
1989      * <li> If {@code start} is
1990      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
1991      * has a value such that the result should have a smaller
1992      * magnitude, then a zero with the same sign as {@code start}
1993      * is returned.
1994      *
1995      * <li> If {@code start} is infinite and
1996      * {@code direction} has a value such that the result should
1997      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1998      * same sign as {@code start} is returned.
1999      *
2000      * <li> If {@code start} is equal to &plusmn;
2001      * {@link Float#MAX_VALUE} and {@code direction} has a
2002      * value such that the result should have a larger magnitude, an
2003      * infinity with same sign as {@code start} is returned.
2004      * </ul>
2005      *
2006      * @param start  starting floating-point value
2007      * @param direction value indicating which of
2008      * {@code start}'s neighbors or {@code start} should
2009      * be returned
2010      * @return The floating-point number adjacent to {@code start} in the
2011      * direction of {@code direction}.
2012      * @since 1.6
2013      */
2014     public static float nextAfter(float start, double direction) {
2015         /*
2016          * The cases:
2017          *
2018          * nextAfter(+infinity, 0)  == MAX_VALUE
2019          * nextAfter(+infinity, +infinity)  == +infinity
2020          * nextAfter(-infinity, 0)  == -MAX_VALUE
2021          * nextAfter(-infinity, -infinity)  == -infinity
2022          *
2023          * are naturally handled without any additional testing
2024          */
2025 
2026         /*
2027          * IEEE 754 floating-point numbers are lexicographically
2028          * ordered if treated as signed-magnitude integers.
2029          * Since Java's integers are two's complement,
2030          * incrementing the two's complement representation of a
2031          * logically negative floating-point value *decrements*
2032          * the signed-magnitude representation. Therefore, when
2033          * the integer representation of a floating-point value
2034          * is negative, the adjustment to the representation is in
2035          * the opposite direction from what would initially be expected.
2036          */
2037 
2038         // Branch to descending case first as it is more costly than ascending
2039         // case due to start != 0.0f conditional.
2040         if (start > direction) { // descending
2041             if (start != 0.0f) {
2042                 final int transducer = Float.floatToRawIntBits(start);
2043                 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2044             } else { // start == 0.0f && direction < 0.0f
2045                 return -Float.MIN_VALUE;
2046             }
2047         } else if (start < direction) { // ascending
2048             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2049             // then bitwise convert start to integer.
2050             final int transducer = Float.floatToRawIntBits(start + 0.0f);
2051             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2052         } else if (start == direction) {
2053             return (float)direction;
2054         } else { // isNaN(start) || isNaN(direction)
2055             return start + (float)direction;
2056         }
2057     }
2058 
2059     /**
2060      * Returns the floating-point value adjacent to {@code d} in
2061      * the direction of positive infinity.  This method is
2062      * semantically equivalent to {@code nextAfter(d,
2063      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2064      * implementation may run faster than its equivalent
2065      * {@code nextAfter} call.
2066      *
2067      * <p>Special Cases:
2068      * <ul>
2069      * <li> If the argument is NaN, the result is NaN.
2070      *
2071      * <li> If the argument is positive infinity, the result is
2072      * positive infinity.
2073      *
2074      * <li> If the argument is zero, the result is
2075      * {@link Double#MIN_VALUE}
2076      *
2077      * </ul>
2078      *
2079      * @param d starting floating-point value
2080      * @return The adjacent floating-point value closer to positive
2081      * infinity.
2082      * @since 1.6
2083      */
2084     public static double nextUp(double d) {
2085         // Use a single conditional and handle the likely cases first.
2086         if (d < Double.POSITIVE_INFINITY) {
2087             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2088             final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2089             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2090         } else { // d is NaN or +Infinity
2091             return d;
2092         }
2093     }
2094 
2095     /**
2096      * Returns the floating-point value adjacent to {@code f} in
2097      * the direction of positive infinity.  This method is
2098      * semantically equivalent to {@code nextAfter(f,
2099      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2100      * implementation may run faster than its equivalent
2101      * {@code nextAfter} call.
2102      *
2103      * <p>Special Cases:
2104      * <ul>
2105      * <li> If the argument is NaN, the result is NaN.
2106      *
2107      * <li> If the argument is positive infinity, the result is
2108      * positive infinity.
2109      *
2110      * <li> If the argument is zero, the result is
2111      * {@link Float#MIN_VALUE}
2112      *
2113      * </ul>
2114      *
2115      * @param f starting floating-point value
2116      * @return The adjacent floating-point value closer to positive
2117      * infinity.
2118      * @since 1.6
2119      */
2120     public static float nextUp(float f) {
2121         // Use a single conditional and handle the likely cases first.
2122         if (f < Float.POSITIVE_INFINITY) {
2123             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2124             final int transducer = Float.floatToRawIntBits(f + 0.0F);
2125             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2126         } else { // f is NaN or +Infinity
2127             return f;
2128         }
2129     }
2130 
2131     /**
2132      * Returns the floating-point value adjacent to {@code d} in
2133      * the direction of negative infinity.  This method is
2134      * semantically equivalent to {@code nextAfter(d,
2135      * Double.NEGATIVE_INFINITY)}; however, a
2136      * {@code nextDown} implementation may run faster than its
2137      * equivalent {@code nextAfter} call.
2138      *
2139      * <p>Special Cases:
2140      * <ul>
2141      * <li> If the argument is NaN, the result is NaN.
2142      *
2143      * <li> If the argument is negative infinity, the result is
2144      * negative infinity.
2145      *
2146      * <li> If the argument is zero, the result is
2147      * {@code -Double.MIN_VALUE}
2148      *
2149      * </ul>
2150      *
2151      * @param d  starting floating-point value
2152      * @return The adjacent floating-point value closer to negative
2153      * infinity.
2154      * @since 1.8
2155      */
2156     public static double nextDown(double d) {
2157         if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2158             return d;
2159         else {
2160             if (d == 0.0)
2161                 return -Double.MIN_VALUE;
2162             else
2163                 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2164                                                ((d > 0.0d)?-1L:+1L));
2165         }
2166     }
2167 
2168     /**
2169      * Returns the floating-point value adjacent to {@code f} in
2170      * the direction of negative infinity.  This method is
2171      * semantically equivalent to {@code nextAfter(f,
2172      * Float.NEGATIVE_INFINITY)}; however, a
2173      * {@code nextDown} implementation may run faster than its
2174      * equivalent {@code nextAfter} call.
2175      *
2176      * <p>Special Cases:
2177      * <ul>
2178      * <li> If the argument is NaN, the result is NaN.
2179      *
2180      * <li> If the argument is negative infinity, the result is
2181      * negative infinity.
2182      *
2183      * <li> If the argument is zero, the result is
2184      * {@code -Float.MIN_VALUE}
2185      *
2186      * </ul>
2187      *
2188      * @param f  starting floating-point value
2189      * @return The adjacent floating-point value closer to negative
2190      * infinity.
2191      * @since 1.8
2192      */
2193     public static float nextDown(float f) {
2194         if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2195             return f;
2196         else {
2197             if (f == 0.0f)
2198                 return -Float.MIN_VALUE;
2199             else
2200                 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2201                                             ((f > 0.0f)?-1:+1));
2202         }
2203     }
2204 
2205     /**
2206      * Returns {@code d} &times;
2207      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2208      * by a single correctly rounded floating-point multiply to a
2209      * member of the double value set.  See the Java
2210      * Language Specification for a discussion of floating-point
2211      * value sets.  If the exponent of the result is between {@link
2212      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2213      * answer is calculated exactly.  If the exponent of the result
2214      * would be larger than {@code Double.MAX_EXPONENT}, an
2215      * infinity is returned.  Note that if the result is subnormal,
2216      * precision may be lost; that is, when {@code scalb(x, n)}
2217      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2218      * <i>x</i>.  When the result is non-NaN, the result has the same
2219      * sign as {@code d}.
2220      *
2221      * <p>Special cases:
2222      * <ul>
2223      * <li> If the first argument is NaN, NaN is returned.
2224      * <li> If the first argument is infinite, then an infinity of the
2225      * same sign is returned.
2226      * <li> If the first argument is zero, then a zero of the same
2227      * sign is returned.
2228      * </ul>
2229      *
2230      * @param d number to be scaled by a power of two.
2231      * @param scaleFactor power of 2 used to scale {@code d}
2232      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
2233      * @since 1.6
2234      */
2235     public static double scalb(double d, int scaleFactor) {
2236         /*
2237          * This method does not need to be declared strictfp to
2238          * compute the same correct result on all platforms.  When
2239          * scaling up, it does not matter what order the
2240          * multiply-store operations are done; the result will be
2241          * finite or overflow regardless of the operation ordering.
2242          * However, to get the correct result when scaling down, a
2243          * particular ordering must be used.
2244          *
2245          * When scaling down, the multiply-store operations are
2246          * sequenced so that it is not possible for two consecutive
2247          * multiply-stores to return subnormal results.  If one
2248          * multiply-store result is subnormal, the next multiply will
2249          * round it away to zero.  This is done by first multiplying
2250          * by 2 ^ (scaleFactor % n) and then multiplying several
2251          * times by 2^n as needed where n is the exponent of number
2252          * that is a covenient power of two.  In this way, at most one
2253          * real rounding error occurs.  If the double value set is
2254          * being used exclusively, the rounding will occur on a
2255          * multiply.  If the double-extended-exponent value set is
2256          * being used, the products will (perhaps) be exact but the
2257          * stores to d are guaranteed to round to the double value
2258          * set.
2259          *
2260          * It is _not_ a valid implementation to first multiply d by
2261          * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2262          * MIN_EXPONENT) since even in a strictfp program double
2263          * rounding on underflow could occur; e.g. if the scaleFactor
2264          * argument was (MIN_EXPONENT - n) and the exponent of d was a
2265          * little less than -(MIN_EXPONENT - n), meaning the final
2266          * result would be subnormal.
2267          *
2268          * Since exact reproducibility of this method can be achieved
2269          * without any undue performance burden, there is no
2270          * compelling reason to allow double rounding on underflow in
2271          * scalb.
2272          */
2273 
2274         // magnitude of a power of two so large that scaling a finite
2275         // nonzero value by it would be guaranteed to over or
2276         // underflow; due to rounding, scaling down takes an
2277         // additional power of two which is reflected here
2278         final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
2279                               DoubleConsts.SIGNIFICAND_WIDTH + 1;
2280         int exp_adjust = 0;
2281         int scale_increment = 0;
2282         double exp_delta = Double.NaN;
2283 
2284         // Make sure scaling factor is in a reasonable range
2285 
2286         if(scaleFactor < 0) {
2287             scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2288             scale_increment = -512;
2289             exp_delta = twoToTheDoubleScaleDown;
2290         }
2291         else {
2292             scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2293             scale_increment = 512;
2294             exp_delta = twoToTheDoubleScaleUp;
2295         }
2296 
2297         // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2298         // technique from "Hacker's Delight" section 10-2.
2299         int t = (scaleFactor >> 9-1) >>> 32 - 9;
2300         exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2301 
2302         d *= powerOfTwoD(exp_adjust);
2303         scaleFactor -= exp_adjust;
2304 
2305         while(scaleFactor != 0) {
2306             d *= exp_delta;
2307             scaleFactor -= scale_increment;
2308         }
2309         return d;
2310     }
2311 
2312     /**
2313      * Returns {@code f} &times;
2314      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2315      * by a single correctly rounded floating-point multiply to a
2316      * member of the float value set.  See the Java
2317      * Language Specification for a discussion of floating-point
2318      * value sets.  If the exponent of the result is between {@link
2319      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2320      * answer is calculated exactly.  If the exponent of the result
2321      * would be larger than {@code Float.MAX_EXPONENT}, an
2322      * infinity is returned.  Note that if the result is subnormal,
2323      * precision may be lost; that is, when {@code scalb(x, n)}
2324      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2325      * <i>x</i>.  When the result is non-NaN, the result has the same
2326      * sign as {@code f}.
2327      *
2328      * <p>Special cases:
2329      * <ul>
2330      * <li> If the first argument is NaN, NaN is returned.
2331      * <li> If the first argument is infinite, then an infinity of the
2332      * same sign is returned.
2333      * <li> If the first argument is zero, then a zero of the same
2334      * sign is returned.
2335      * </ul>
2336      *
2337      * @param f number to be scaled by a power of two.
2338      * @param scaleFactor power of 2 used to scale {@code f}
2339      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
2340      * @since 1.6
2341      */
2342     public static float scalb(float f, int scaleFactor) {
2343         // magnitude of a power of two so large that scaling a finite
2344         // nonzero value by it would be guaranteed to over or
2345         // underflow; due to rounding, scaling down takes an
2346         // additional power of two which is reflected here
2347         final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
2348                               FloatConsts.SIGNIFICAND_WIDTH + 1;
2349 
2350         // Make sure scaling factor is in a reasonable range
2351         scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2352 
2353         /*
2354          * Since + MAX_SCALE for float fits well within the double
2355          * exponent range and + float -> double conversion is exact
2356          * the multiplication below will be exact. Therefore, the
2357          * rounding that occurs when the double product is cast to
2358          * float will be the correctly rounded float result.  Since
2359          * all operations other than the final multiply will be exact,
2360          * it is not necessary to declare this method strictfp.
2361          */
2362         return (float)((double)f*powerOfTwoD(scaleFactor));
2363     }
2364 
2365     // Constants used in scalb
2366     static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2367     static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2368 
2369     /**
2370      * Returns a floating-point power of two in the normal range.
2371      */
2372     static double powerOfTwoD(int n) {
2373         assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
2374         return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2375                                         (DoubleConsts.SIGNIFICAND_WIDTH-1))
2376                                        & DoubleConsts.EXP_BIT_MASK);
2377     }
2378 
2379     /**
2380      * Returns a floating-point power of two in the normal range.
2381      */
2382     static float powerOfTwoF(int n) {
2383         assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
2384         return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2385                                      (FloatConsts.SIGNIFICAND_WIDTH-1))
2386                                     & FloatConsts.EXP_BIT_MASK);
2387     }
2388 }
--- EOF ---