1 /*
   2  * Copyright (c) 1994, 2013, Oracle and/or its affiliates. All rights reserved.
   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
   6  * under the terms of the GNU General Public License version 2 only, as
   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
  16  *
  17  * You should have received a copy of the GNU General Public License version
  18  * 2 along with this work; if not, write to the Free Software Foundation,
  19  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  20  *
  21  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  22  * or visit www.oracle.com if you need additional information or have any
  23  * questions.
  24  */
  25 
  26 package java.lang;
  27 import java.util.Random;
  28 
  29 import sun.misc.FloatConsts;
  30 import sun.misc.DoubleConsts;
  31 
  32 /**
  33  * The class {@code Math} contains methods for performing basic
  34  * numeric operations such as the elementary exponential, logarithm,
  35  * square root, and trigonometric functions.
  36  *
  37  * <p>Unlike some of the numeric methods of class
  38  * {@code StrictMath}, all implementations of the equivalent
  39  * functions of class {@code Math} are not defined to return the
  40  * bit-for-bit same results.  This relaxation permits
  41  * better-performing implementations where strict reproducibility is
  42  * not required.
  43  *
  44  * <p>By default many of the {@code Math} methods simply call
  45  * the equivalent method in {@code StrictMath} for their
  46  * implementation.  Code generators are encouraged to use
  47  * platform-specific native libraries or microprocessor instructions,
  48  * where available, to provide higher-performance implementations of
  49  * {@code Math} methods.  Such higher-performance
  50  * implementations still must conform to the specification for
  51  * {@code Math}.
  52  *
  53  * <p>The quality of implementation specifications concern two
  54  * properties, accuracy of the returned result and monotonicity of the
  55  * method.  Accuracy of the floating-point {@code Math} methods is
  56  * measured in terms of <i>ulps</i>, units in the last place.  For a
  57  * given floating-point format, an {@linkplain #ulp(double) ulp} of a
  58  * specific real number value is the distance between the two
  59  * floating-point values bracketing that numerical value.  When
  60  * discussing the accuracy of a method as a whole rather than at a
  61  * specific argument, the number of ulps cited is for the worst-case
  62  * error at any argument.  If a method always has an error less than
  63  * 0.5 ulps, the method always returns the floating-point number
  64  * nearest the exact result; such a method is <i>correctly
  65  * rounded</i>.  A correctly rounded method is generally the best a
  66  * floating-point approximation can be; however, it is impractical for
  67  * many floating-point methods to be correctly rounded.  Instead, for
  68  * the {@code Math} class, a larger error bound of 1 or 2 ulps is
  69  * allowed for certain methods.  Informally, with a 1 ulp error bound,
  70  * when the exact result is a representable number, the exact result
  71  * should be returned as the computed result; otherwise, either of the
  72  * two floating-point values which bracket the exact result may be
  73  * returned.  For exact results large in magnitude, one of the
  74  * endpoints of the bracket may be infinite.  Besides accuracy at
  75  * individual arguments, maintaining proper relations between the
  76  * method at different arguments is also important.  Therefore, most
  77  * methods with more than 0.5 ulp errors are required to be
  78  * <i>semi-monotonic</i>: whenever the mathematical function is
  79  * non-decreasing, so is the floating-point approximation, likewise,
  80  * whenever the mathematical function is non-increasing, so is the
  81  * floating-point approximation.  Not all approximations that have 1
  82  * ulp accuracy will automatically meet the monotonicity requirements.
  83  *
  84  * <p>
  85  * The platform uses signed two's complement integer arithmetic with
  86  * int and long primitive types.  The developer should choose
  87  * the primitive type to ensure that arithmetic operations consistently
  88  * produce correct results, which in some cases means the operations
  89  * will not overflow the range of values of the computation.
  90  * The best practice is to choose the primitive type and algorithm to avoid
  91  * overflow. In cases where the size is {@code int} or {@code long} and
  92  * overflow errors need to be detected, the methods {@code addExact},
  93  * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
  94  * throw an {@code ArithmeticException} when the results overflow.
  95  * For other arithmetic operations such as divide, absolute value,
  96  * increment, decrement, and negation overflow occurs only with
  97  * a specific minimum or maximum value and should be checked against
  98  * the minimum or maximum as appropriate.
  99  *
 100  * @author  unascribed
 101  * @author  Joseph D. Darcy
 102  * @since   1.0
 103  */
 104 
 105 public final class Math {
 106 
 107     /**
 108      * Don't let anyone instantiate this class.
 109      */
 110     private Math() {}
 111 
 112     /**
 113      * The {@code double} value that is closer than any other to
 114      * <i>e</i>, the base of the natural logarithms.
 115      */
 116     public static final double E = 2.7182818284590452354;
 117 
 118     /**
 119      * The {@code double} value that is closer than any other to
 120      * <i>pi</i>, the ratio of the circumference of a circle to its
 121      * diameter.
 122      */
 123     public static final double PI = 3.14159265358979323846;
 124 
 125     /**
 126      * Returns the trigonometric sine of an angle.  Special cases:
 127      * <ul><li>If the argument is NaN or an infinity, then the
 128      * result is NaN.
 129      * <li>If the argument is zero, then the result is a zero with the
 130      * same sign as the argument.</ul>
 131      *
 132      * <p>The computed result must be within 1 ulp of the exact result.
 133      * Results must be semi-monotonic.
 134      *
 135      * @param   a   an angle, in radians.
 136      * @return  the sine of the argument.
 137      */
 138     public static double sin(double a) {
 139         return StrictMath.sin(a); // default impl. delegates to StrictMath
 140     }
 141 
 142     /**
 143      * Returns the trigonometric cosine of an angle. Special cases:
 144      * <ul><li>If the argument is NaN or an infinity, then the
 145      * result is NaN.</ul>
 146      *
 147      * <p>The computed result must be within 1 ulp of the exact result.
 148      * Results must be semi-monotonic.
 149      *
 150      * @param   a   an angle, in radians.
 151      * @return  the cosine of the argument.
 152      */
 153     public static double cos(double a) {
 154         return StrictMath.cos(a); // default impl. delegates to StrictMath
 155     }
 156 
 157     /**
 158      * Returns the trigonometric tangent of an angle.  Special cases:
 159      * <ul><li>If the argument is NaN or an infinity, then the result
 160      * is NaN.
 161      * <li>If the argument is zero, then the result is a zero with the
 162      * same sign as the argument.</ul>
 163      *
 164      * <p>The computed result must be within 1 ulp of the exact result.
 165      * Results must be semi-monotonic.
 166      *
 167      * @param   a   an angle, in radians.
 168      * @return  the tangent of the argument.
 169      */
 170     public static double tan(double a) {
 171         return StrictMath.tan(a); // default impl. delegates to StrictMath
 172     }
 173 
 174     /**
 175      * Returns the arc sine of a value; the returned angle is in the
 176      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 177      * <ul><li>If the argument is NaN or its absolute value is greater
 178      * than 1, then the result is NaN.
 179      * <li>If the argument is zero, then the result is a zero with the
 180      * same sign as the argument.</ul>
 181      *
 182      * <p>The computed result must be within 1 ulp of the exact result.
 183      * Results must be semi-monotonic.
 184      *
 185      * @param   a   the value whose arc sine is to be returned.
 186      * @return  the arc sine of the argument.
 187      */
 188     public static double asin(double a) {
 189         return StrictMath.asin(a); // default impl. delegates to StrictMath
 190     }
 191 
 192     /**
 193      * Returns the arc cosine of a value; the returned angle is in the
 194      * range 0.0 through <i>pi</i>.  Special case:
 195      * <ul><li>If the argument is NaN or its absolute value is greater
 196      * than 1, then the result is NaN.</ul>
 197      *
 198      * <p>The computed result must be within 1 ulp of the exact result.
 199      * Results must be semi-monotonic.
 200      *
 201      * @param   a   the value whose arc cosine is to be returned.
 202      * @return  the arc cosine of the argument.
 203      */
 204     public static double acos(double a) {
 205         return StrictMath.acos(a); // default impl. delegates to StrictMath
 206     }
 207 
 208     /**
 209      * Returns the arc tangent of a value; the returned angle is in the
 210      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 211      * <ul><li>If the argument is NaN, then the result is NaN.
 212      * <li>If the argument is zero, then the result is a zero with the
 213      * same sign as the argument.</ul>
 214      *
 215      * <p>The computed result must be within 1 ulp of the exact result.
 216      * Results must be semi-monotonic.
 217      *
 218      * @param   a   the value whose arc tangent is to be returned.
 219      * @return  the arc tangent of the argument.
 220      */
 221     public static double atan(double a) {
 222         return StrictMath.atan(a); // default impl. delegates to StrictMath
 223     }
 224 
 225     /**
 226      * Converts an angle measured in degrees to an approximately
 227      * equivalent angle measured in radians.  The conversion from
 228      * degrees to radians is generally inexact.
 229      *
 230      * @param   angdeg   an angle, in degrees
 231      * @return  the measurement of the angle {@code angdeg}
 232      *          in radians.
 233      * @since   1.2
 234      */
 235     public static double toRadians(double angdeg) {
 236         return angdeg / 180.0 * PI;
 237     }
 238 
 239     /**
 240      * Converts an angle measured in radians to an approximately
 241      * equivalent angle measured in degrees.  The conversion from
 242      * radians to degrees is generally inexact; users should
 243      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
 244      * equal {@code 0.0}.
 245      *
 246      * @param   angrad   an angle, in radians
 247      * @return  the measurement of the angle {@code angrad}
 248      *          in degrees.
 249      * @since   1.2
 250      */
 251     public static double toDegrees(double angrad) {
 252         return angrad * 180.0 / PI;
 253     }
 254 
 255     /**
 256      * Returns Euler's number <i>e</i> raised to the power of a
 257      * {@code double} value.  Special cases:
 258      * <ul><li>If the argument is NaN, the result is NaN.
 259      * <li>If the argument is positive infinity, then the result is
 260      * positive infinity.
 261      * <li>If the argument is negative infinity, then the result is
 262      * positive zero.</ul>
 263      *
 264      * <p>The computed result must be within 1 ulp of the exact result.
 265      * Results must be semi-monotonic.
 266      *
 267      * @param   a   the exponent to raise <i>e</i> to.
 268      * @return  the value <i>e</i><sup>{@code a}</sup>,
 269      *          where <i>e</i> is the base of the natural logarithms.
 270      */
 271     public static double exp(double a) {
 272         return StrictMath.exp(a); // default impl. delegates to StrictMath
 273     }
 274 
 275     /**
 276      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
 277      * value.  Special cases:
 278      * <ul><li>If the argument is NaN or less than zero, then the result
 279      * is NaN.
 280      * <li>If the argument is positive infinity, then the result is
 281      * positive infinity.
 282      * <li>If the argument is positive zero or negative zero, then the
 283      * result is negative infinity.</ul>
 284      *
 285      * <p>The computed result must be within 1 ulp of the exact result.
 286      * Results must be semi-monotonic.
 287      *
 288      * @param   a   a value
 289      * @return  the value ln&nbsp;{@code a}, the natural logarithm of
 290      *          {@code a}.
 291      */
 292     public static double log(double a) {
 293         return StrictMath.log(a); // default impl. delegates to StrictMath
 294     }
 295 
 296     /**
 297      * Returns the base 10 logarithm of a {@code double} value.
 298      * Special cases:
 299      *
 300      * <ul><li>If the argument is NaN or less than zero, then the result
 301      * is NaN.
 302      * <li>If the argument is positive infinity, then the result is
 303      * positive infinity.
 304      * <li>If the argument is positive zero or negative zero, then the
 305      * result is negative infinity.
 306      * <li> If the argument is equal to 10<sup><i>n</i></sup> for
 307      * integer <i>n</i>, then the result is <i>n</i>.
 308      * </ul>
 309      *
 310      * <p>The computed result must be within 1 ulp of the exact result.
 311      * Results must be semi-monotonic.
 312      *
 313      * @param   a   a value
 314      * @return  the base 10 logarithm of  {@code a}.
 315      * @since 1.5
 316      */
 317     public static double log10(double a) {
 318         return StrictMath.log10(a); // default impl. delegates to StrictMath
 319     }
 320 
 321     /**
 322      * Returns the correctly rounded positive square root of a
 323      * {@code double} value.
 324      * Special cases:
 325      * <ul><li>If the argument is NaN or less than zero, then the result
 326      * is NaN.
 327      * <li>If the argument is positive infinity, then the result is positive
 328      * infinity.
 329      * <li>If the argument is positive zero or negative zero, then the
 330      * result is the same as the argument.</ul>
 331      * Otherwise, the result is the {@code double} value closest to
 332      * the true mathematical square root of the argument value.
 333      *
 334      * @param   a   a value.
 335      * @return  the positive square root of {@code a}.
 336      *          If the argument is NaN or less than zero, the result is NaN.
 337      */
 338     public static double sqrt(double a) {
 339         return StrictMath.sqrt(a); // default impl. delegates to StrictMath
 340                                    // Note that hardware sqrt instructions
 341                                    // frequently can be directly used by JITs
 342                                    // and should be much faster than doing
 343                                    // Math.sqrt in software.
 344     }
 345 
 346 
 347     /**
 348      * Returns the cube root of a {@code double} value.  For
 349      * positive finite {@code x}, {@code cbrt(-x) ==
 350      * -cbrt(x)}; that is, the cube root of a negative value is
 351      * the negative of the cube root of that value's magnitude.
 352      *
 353      * Special cases:
 354      *
 355      * <ul>
 356      *
 357      * <li>If the argument is NaN, then the result is NaN.
 358      *
 359      * <li>If the argument is infinite, then the result is an infinity
 360      * with the same sign as the argument.
 361      *
 362      * <li>If the argument is zero, then the result is a zero with the
 363      * same sign as the argument.
 364      *
 365      * </ul>
 366      *
 367      * <p>The computed result must be within 1 ulp of the exact result.
 368      *
 369      * @param   a   a value.
 370      * @return  the cube root of {@code a}.
 371      * @since 1.5
 372      */
 373     public static double cbrt(double a) {
 374         return StrictMath.cbrt(a);
 375     }
 376 
 377     /**
 378      * Computes the remainder operation on two arguments as prescribed
 379      * by the IEEE 754 standard.
 380      * The remainder value is mathematically equal to
 381      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
 382      * where <i>n</i> is the mathematical integer closest to the exact
 383      * mathematical value of the quotient {@code f1/f2}, and if two
 384      * mathematical integers are equally close to {@code f1/f2},
 385      * then <i>n</i> is the integer that is even. If the remainder is
 386      * zero, its sign is the same as the sign of the first argument.
 387      * Special cases:
 388      * <ul><li>If either argument is NaN, or the first argument is infinite,
 389      * or the second argument is positive zero or negative zero, then the
 390      * result is NaN.
 391      * <li>If the first argument is finite and the second argument is
 392      * infinite, then the result is the same as the first argument.</ul>
 393      *
 394      * @param   f1   the dividend.
 395      * @param   f2   the divisor.
 396      * @return  the remainder when {@code f1} is divided by
 397      *          {@code f2}.
 398      */
 399     public static double IEEEremainder(double f1, double f2) {
 400         return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
 401     }
 402 
 403     /**
 404      * Returns the smallest (closest to negative infinity)
 405      * {@code double} value that is greater than or equal to the
 406      * argument and is equal to a mathematical integer. Special cases:
 407      * <ul><li>If the argument value is already equal to a
 408      * mathematical integer, then the result is the same as the
 409      * argument.  <li>If the argument is NaN or an infinity or
 410      * positive zero or negative zero, then the result is the same as
 411      * the argument.  <li>If the argument value is less than zero but
 412      * greater than -1.0, then the result is negative zero.</ul> Note
 413      * that the value of {@code Math.ceil(x)} is exactly the
 414      * value of {@code -Math.floor(-x)}.
 415      *
 416      *
 417      * @param   a   a value.
 418      * @return  the smallest (closest to negative infinity)
 419      *          floating-point value that is greater than or equal to
 420      *          the argument and is equal to a mathematical integer.
 421      */
 422     public static double ceil(double a) {
 423         return StrictMath.ceil(a); // default impl. delegates to StrictMath
 424     }
 425 
 426     /**
 427      * Returns the largest (closest to positive infinity)
 428      * {@code double} value that is less than or equal to the
 429      * argument and is equal to a mathematical integer. Special cases:
 430      * <ul><li>If the argument value is already equal to a
 431      * mathematical integer, then the result is the same as the
 432      * argument.  <li>If the argument is NaN or an infinity or
 433      * positive zero or negative zero, then the result is the same as
 434      * the argument.</ul>
 435      *
 436      * @param   a   a value.
 437      * @return  the largest (closest to positive infinity)
 438      *          floating-point value that less than or equal to the argument
 439      *          and is equal to a mathematical integer.
 440      */
 441     public static double floor(double a) {
 442         return StrictMath.floor(a); // default impl. delegates to StrictMath
 443     }
 444 
 445     /**
 446      * Returns the {@code double} value that is closest in value
 447      * to the argument and is equal to a mathematical integer. If two
 448      * {@code double} values that are mathematical integers are
 449      * equally close, the result is the integer value that is
 450      * even. Special cases:
 451      * <ul><li>If the argument value is already equal to a mathematical
 452      * integer, then the result is the same as the argument.
 453      * <li>If the argument is NaN or an infinity or positive zero or negative
 454      * zero, then the result is the same as the argument.</ul>
 455      *
 456      * @param   a   a {@code double} value.
 457      * @return  the closest floating-point value to {@code a} that is
 458      *          equal to a mathematical integer.
 459      */
 460     public static double rint(double a) {
 461         return StrictMath.rint(a); // default impl. delegates to StrictMath
 462     }
 463 
 464     /**
 465      * Returns the angle <i>theta</i> from the conversion of rectangular
 466      * coordinates ({@code x},&nbsp;{@code y}) to polar
 467      * coordinates (r,&nbsp;<i>theta</i>).
 468      * This method computes the phase <i>theta</i> by computing an arc tangent
 469      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
 470      * cases:
 471      * <ul><li>If either argument is NaN, then the result is NaN.
 472      * <li>If the first argument is positive zero and the second argument
 473      * is positive, or the first argument is positive and finite and the
 474      * second argument is positive infinity, then the result is positive
 475      * zero.
 476      * <li>If the first argument is negative zero and the second argument
 477      * is positive, or the first argument is negative and finite and the
 478      * second argument is positive infinity, then the result is negative zero.
 479      * <li>If the first argument is positive zero and the second argument
 480      * is negative, or the first argument is positive and finite and the
 481      * second argument is negative infinity, then the result is the
 482      * {@code double} value closest to <i>pi</i>.
 483      * <li>If the first argument is negative zero and the second argument
 484      * is negative, or the first argument is negative and finite and the
 485      * second argument is negative infinity, then the result is the
 486      * {@code double} value closest to -<i>pi</i>.
 487      * <li>If the first argument is positive and the second argument is
 488      * positive zero or negative zero, or the first argument is positive
 489      * infinity and the second argument is finite, then the result is the
 490      * {@code double} value closest to <i>pi</i>/2.
 491      * <li>If the first argument is negative and the second argument is
 492      * positive zero or negative zero, or the first argument is negative
 493      * infinity and the second argument is finite, then the result is the
 494      * {@code double} value closest to -<i>pi</i>/2.
 495      * <li>If both arguments are positive infinity, then the result is the
 496      * {@code double} value closest to <i>pi</i>/4.
 497      * <li>If the first argument is positive infinity and the second argument
 498      * is negative infinity, then the result is the {@code double}
 499      * value closest to 3*<i>pi</i>/4.
 500      * <li>If the first argument is negative infinity and the second argument
 501      * is positive infinity, then the result is the {@code double} value
 502      * closest to -<i>pi</i>/4.
 503      * <li>If both arguments are negative infinity, then the result is the
 504      * {@code double} value closest to -3*<i>pi</i>/4.</ul>
 505      *
 506      * <p>The computed result must be within 2 ulps of the exact result.
 507      * Results must be semi-monotonic.
 508      *
 509      * @param   y   the ordinate coordinate
 510      * @param   x   the abscissa coordinate
 511      * @return  the <i>theta</i> component of the point
 512      *          (<i>r</i>,&nbsp;<i>theta</i>)
 513      *          in polar coordinates that corresponds to the point
 514      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
 515      */
 516     public static double atan2(double y, double x) {
 517         return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
 518     }
 519 
 520     /**
 521      * Returns the value of the first argument raised to the power of the
 522      * second argument. Special cases:
 523      *
 524      * <ul><li>If the second argument is positive or negative zero, then the
 525      * result is 1.0.
 526      * <li>If the second argument is 1.0, then the result is the same as the
 527      * first argument.
 528      * <li>If the second argument is NaN, then the result is NaN.
 529      * <li>If the first argument is NaN and the second argument is nonzero,
 530      * then the result is NaN.
 531      *
 532      * <li>If
 533      * <ul>
 534      * <li>the absolute value of the first argument is greater than 1
 535      * and the second argument is positive infinity, or
 536      * <li>the absolute value of the first argument is less than 1 and
 537      * the second argument is negative infinity,
 538      * </ul>
 539      * then the result is positive infinity.
 540      *
 541      * <li>If
 542      * <ul>
 543      * <li>the absolute value of the first argument is greater than 1 and
 544      * the second argument is negative infinity, or
 545      * <li>the absolute value of the
 546      * first argument is less than 1 and the second argument is positive
 547      * infinity,
 548      * </ul>
 549      * then the result is positive zero.
 550      *
 551      * <li>If the absolute value of the first argument equals 1 and the
 552      * second argument is infinite, then the result is NaN.
 553      *
 554      * <li>If
 555      * <ul>
 556      * <li>the first argument is positive zero and the second argument
 557      * is greater than zero, or
 558      * <li>the first argument is positive infinity and the second
 559      * argument is less than zero,
 560      * </ul>
 561      * then the result is positive zero.
 562      *
 563      * <li>If
 564      * <ul>
 565      * <li>the first argument is positive zero and the second argument
 566      * is less than zero, or
 567      * <li>the first argument is positive infinity and the second
 568      * argument is greater than zero,
 569      * </ul>
 570      * then the result is positive infinity.
 571      *
 572      * <li>If
 573      * <ul>
 574      * <li>the first argument is negative zero and the second argument
 575      * is greater than zero but not a finite odd integer, or
 576      * <li>the first argument is negative infinity and the second
 577      * argument is less than zero but not a finite odd integer,
 578      * </ul>
 579      * then the result is positive zero.
 580      *
 581      * <li>If
 582      * <ul>
 583      * <li>the first argument is negative zero and the second argument
 584      * is a positive finite odd integer, or
 585      * <li>the first argument is negative infinity and the second
 586      * argument is a negative finite odd integer,
 587      * </ul>
 588      * then the result is negative zero.
 589      *
 590      * <li>If
 591      * <ul>
 592      * <li>the first argument is negative zero and the second argument
 593      * is less than zero but not a finite odd integer, or
 594      * <li>the first argument is negative infinity and the second
 595      * argument is greater than zero but not a finite odd integer,
 596      * </ul>
 597      * then the result is positive infinity.
 598      *
 599      * <li>If
 600      * <ul>
 601      * <li>the first argument is negative zero and the second argument
 602      * is a negative finite odd integer, or
 603      * <li>the first argument is negative infinity and the second
 604      * argument is a positive finite odd integer,
 605      * </ul>
 606      * then the result is negative infinity.
 607      *
 608      * <li>If the first argument is finite and less than zero
 609      * <ul>
 610      * <li> if the second argument is a finite even integer, the
 611      * result is equal to the result of raising the absolute value of
 612      * the first argument to the power of the second argument
 613      *
 614      * <li>if the second argument is a finite odd integer, the result
 615      * is equal to the negative of the result of raising the absolute
 616      * value of the first argument to the power of the second
 617      * argument
 618      *
 619      * <li>if the second argument is finite and not an integer, then
 620      * the result is NaN.
 621      * </ul>
 622      *
 623      * <li>If both arguments are integers, then the result is exactly equal
 624      * to the mathematical result of raising the first argument to the power
 625      * of the second argument if that result can in fact be represented
 626      * exactly as a {@code double} value.</ul>
 627      *
 628      * <p>(In the foregoing descriptions, a floating-point value is
 629      * considered to be an integer if and only if it is finite and a
 630      * fixed point of the method {@link #ceil ceil} or,
 631      * equivalently, a fixed point of the method {@link #floor
 632      * floor}. A value is a fixed point of a one-argument
 633      * method if and only if the result of applying the method to the
 634      * value is equal to the value.)
 635      *
 636      * <p>The computed result must be within 1 ulp of the exact result.
 637      * Results must be semi-monotonic.
 638      *
 639      * @param   a   the base.
 640      * @param   b   the exponent.
 641      * @return  the value {@code a}<sup>{@code b}</sup>.
 642      */
 643     public static double pow(double a, double b) {
 644         return StrictMath.pow(a, b); // default impl. delegates to StrictMath
 645     }
 646 
 647     /**
 648      * Returns the closest {@code int} to the argument, with ties
 649      * rounding to positive infinity.
 650      *
 651      * <p>
 652      * Special cases:
 653      * <ul><li>If the argument is NaN, the result is 0.
 654      * <li>If the argument is negative infinity or any value less than or
 655      * equal to the value of {@code Integer.MIN_VALUE}, the result is
 656      * equal to the value of {@code Integer.MIN_VALUE}.
 657      * <li>If the argument is positive infinity or any value greater than or
 658      * equal to the value of {@code Integer.MAX_VALUE}, the result is
 659      * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 660      *
 661      * @param   a   a floating-point value to be rounded to an integer.
 662      * @return  the value of the argument rounded to the nearest
 663      *          {@code int} value.
 664      * @see     java.lang.Integer#MAX_VALUE
 665      * @see     java.lang.Integer#MIN_VALUE
 666      */
 667     public static int round(float a) {
 668         int intBits = Float.floatToRawIntBits(a);
 669         int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
 670                 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
 671         int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
 672                 + FloatConsts.EXP_BIAS) - biasedExp;
 673         if ((shift & -32) == 0) { // shift >= 0 && shift < 32
 674             // a is a finite number such that pow(2,-32) <= ulp(a) < 1
 675             int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
 676                     | (FloatConsts.SIGNIF_BIT_MASK + 1));
 677             if (intBits < 0) {
 678                 r = -r;
 679             }
 680             // In the comments below each Java expression evaluates to the value
 681             // the corresponding mathematical expression:
 682             // (r) evaluates to a / ulp(a)
 683             // (r >> shift) evaluates to floor(a * 2)
 684             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 685             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 686             return ((r >> shift) + 1) >> 1;
 687         } else {
 688             // a is either
 689             // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
 690             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 691             // - an infinity or NaN
 692             return (int) a;
 693         }
 694     }
 695 
 696     /**
 697      * Returns the closest {@code long} to the argument, with ties
 698      * rounding to positive infinity.
 699      *
 700      * <p>Special cases:
 701      * <ul><li>If the argument is NaN, the result is 0.
 702      * <li>If the argument is negative infinity or any value less than or
 703      * equal to the value of {@code Long.MIN_VALUE}, the result is
 704      * equal to the value of {@code Long.MIN_VALUE}.
 705      * <li>If the argument is positive infinity or any value greater than or
 706      * equal to the value of {@code Long.MAX_VALUE}, the result is
 707      * equal to the value of {@code Long.MAX_VALUE}.</ul>
 708      *
 709      * @param   a   a floating-point value to be rounded to a
 710      *          {@code long}.
 711      * @return  the value of the argument rounded to the nearest
 712      *          {@code long} value.
 713      * @see     java.lang.Long#MAX_VALUE
 714      * @see     java.lang.Long#MIN_VALUE
 715      */
 716     public static long round(double a) {
 717         long longBits = Double.doubleToRawLongBits(a);
 718         long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
 719                 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
 720         long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
 721                 + DoubleConsts.EXP_BIAS) - biasedExp;
 722         if ((shift & -64) == 0) { // shift >= 0 && shift < 64
 723             // a is a finite number such that pow(2,-64) <= ulp(a) < 1
 724             long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
 725                     | (DoubleConsts.SIGNIF_BIT_MASK + 1));
 726             if (longBits < 0) {
 727                 r = -r;
 728             }
 729             // In the comments below each Java expression evaluates to the value
 730             // the corresponding mathematical expression:
 731             // (r) evaluates to a / ulp(a)
 732             // (r >> shift) evaluates to floor(a * 2)
 733             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 734             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 735             return ((r >> shift) + 1) >> 1;
 736         } else {
 737             // a is either
 738             // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
 739             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 740             // - an infinity or NaN
 741             return (long) a;
 742         }
 743     }
 744 
 745     private static final class RandomNumberGeneratorHolder {
 746         static final Random randomNumberGenerator = new Random();
 747     }
 748 
 749     /**
 750      * Returns a {@code double} value with a positive sign, greater
 751      * than or equal to {@code 0.0} and less than {@code 1.0}.
 752      * Returned values are chosen pseudorandomly with (approximately)
 753      * uniform distribution from that range.
 754      *
 755      * <p>When this method is first called, it creates a single new
 756      * pseudorandom-number generator, exactly as if by the expression
 757      *
 758      * <blockquote>{@code new java.util.Random()}</blockquote>
 759      *
 760      * This new pseudorandom-number generator is used thereafter for
 761      * all calls to this method and is used nowhere else.
 762      *
 763      * <p>This method is properly synchronized to allow correct use by
 764      * more than one thread. However, if many threads need to generate
 765      * pseudorandom numbers at a great rate, it may reduce contention
 766      * for each thread to have its own pseudorandom-number generator.
 767      *
 768      * @apiNote
 769      * As the largest {@code double} value less than {@code 1.0}
 770      * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
 771      * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
 772      *
 773      * <blockquote><pre>{@code
 774      * double f = Math.random()/Math.nextDown(1.0);
 775      * double x = x1*(1.0 - f) + x2*f;
 776      * }</pre></blockquote>
 777      *
 778      * @return  a pseudorandom {@code double} greater than or equal
 779      * to {@code 0.0} and less than {@code 1.0}.
 780      * @see #nextDown(double)
 781      * @see Random#nextDouble()
 782      */
 783     public static double random() {
 784         return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 785     }
 786 
 787     /**
 788      * Returns the sum of its arguments,
 789      * throwing an exception if the result overflows an {@code int}.
 790      *
 791      * @param x the first value
 792      * @param y the second value
 793      * @return the result
 794      * @throws ArithmeticException if the result overflows an int
 795      * @since 1.8
 796      */
 797     public static int addExact(int x, int y) {
 798         int r = x + y;
 799         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 800         if (((x ^ r) & (y ^ r)) < 0) {
 801             throw new ArithmeticException("integer overflow");
 802         }
 803         return r;
 804     }
 805 
 806     /**
 807      * Returns the sum of its arguments,
 808      * throwing an exception if the result overflows a {@code long}.
 809      *
 810      * @param x the first value
 811      * @param y the second value
 812      * @return the result
 813      * @throws ArithmeticException if the result overflows a long
 814      * @since 1.8
 815      */
 816     public static long addExact(long x, long y) {
 817         long r = x + y;
 818         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 819         if (((x ^ r) & (y ^ r)) < 0) {
 820             throw new ArithmeticException("long overflow");
 821         }
 822         return r;
 823     }
 824 
 825     /**
 826      * Returns the difference of the arguments,
 827      * throwing an exception if the result overflows an {@code int}.
 828      *
 829      * @param x the first value
 830      * @param y the second value to subtract from the first
 831      * @return the result
 832      * @throws ArithmeticException if the result overflows an int
 833      * @since 1.8
 834      */
 835     public static int subtractExact(int x, int y) {
 836         int r = x - y;
 837         // HD 2-12 Overflow iff the arguments have different signs and
 838         // the sign of the result is different than the sign of x
 839         if (((x ^ y) & (x ^ r)) < 0) {
 840             throw new ArithmeticException("integer overflow");
 841         }
 842         return r;
 843     }
 844 
 845     /**
 846      * Returns the difference of the arguments,
 847      * throwing an exception if the result overflows a {@code long}.
 848      *
 849      * @param x the first value
 850      * @param y the second value to subtract from the first
 851      * @return the result
 852      * @throws ArithmeticException if the result overflows a long
 853      * @since 1.8
 854      */
 855     public static long subtractExact(long x, long y) {
 856         long r = x - y;
 857         // HD 2-12 Overflow iff the arguments have different signs and
 858         // the sign of the result is different than the sign of x
 859         if (((x ^ y) & (x ^ r)) < 0) {
 860             throw new ArithmeticException("long overflow");
 861         }
 862         return r;
 863     }
 864 
 865     /**
 866      * Returns the product of the arguments,
 867      * throwing an exception if the result overflows an {@code int}.
 868      *
 869      * @param x the first value
 870      * @param y the second value
 871      * @return the result
 872      * @throws ArithmeticException if the result overflows an int
 873      * @since 1.8
 874      */
 875     public static int multiplyExact(int x, int y) {
 876         long r = (long)x * (long)y;
 877         if ((int)r != r) {
 878             throw new ArithmeticException("integer overflow");
 879         }
 880         return (int)r;
 881     }
 882 
 883     /**
 884      * Returns the product of the arguments,
 885      * throwing an exception if the result overflows a {@code long}.
 886      *
 887      * @param x the first value
 888      * @param y the second value
 889      * @return the result
 890      * @throws ArithmeticException if the result overflows a long
 891      * @since 1.8
 892      */
 893     public static long multiplyExact(long x, long y) {
 894         long r = x * y;
 895         long ax = Math.abs(x);
 896         long ay = Math.abs(y);
 897         if (((ax | ay) >>> 31 != 0)) {
 898             // Some bits greater than 2^31 that might cause overflow
 899             // Check the result using the divide operator
 900             // and check for the special case of Long.MIN_VALUE * -1
 901            if (((y != 0) && (r / y != x)) ||
 902                (x == Long.MIN_VALUE && y == -1)) {
 903                 throw new ArithmeticException("long overflow");
 904             }
 905         }
 906         return r;
 907     }
 908 
 909     /**
 910      * Returns the argument incremented by one, throwing an exception if the
 911      * result overflows an {@code int}.
 912      *
 913      * @param a the value to increment
 914      * @return the result
 915      * @throws ArithmeticException if the result overflows an int
 916      * @since 1.8
 917      */
 918     public static int incrementExact(int a) {
 919         if (a == Integer.MAX_VALUE) {
 920             throw new ArithmeticException("integer overflow");
 921         }
 922 
 923         return a + 1;
 924     }
 925 
 926     /**
 927      * Returns the argument incremented by one, throwing an exception if the
 928      * result overflows a {@code long}.
 929      *
 930      * @param a the value to increment
 931      * @return the result
 932      * @throws ArithmeticException if the result overflows a long
 933      * @since 1.8
 934      */
 935     public static long incrementExact(long a) {
 936         if (a == Long.MAX_VALUE) {
 937             throw new ArithmeticException("long overflow");
 938         }
 939 
 940         return a + 1L;
 941     }
 942 
 943     /**
 944      * Returns the argument decremented by one, throwing an exception if the
 945      * result overflows an {@code int}.
 946      *
 947      * @param a the value to decrement
 948      * @return the result
 949      * @throws ArithmeticException if the result overflows an int
 950      * @since 1.8
 951      */
 952     public static int decrementExact(int a) {
 953         if (a == Integer.MIN_VALUE) {
 954             throw new ArithmeticException("integer overflow");
 955         }
 956 
 957         return a - 1;
 958     }
 959 
 960     /**
 961      * Returns the argument decremented by one, throwing an exception if the
 962      * result overflows a {@code long}.
 963      *
 964      * @param a the value to decrement
 965      * @return the result
 966      * @throws ArithmeticException if the result overflows a long
 967      * @since 1.8
 968      */
 969     public static long decrementExact(long a) {
 970         if (a == Long.MIN_VALUE) {
 971             throw new ArithmeticException("long overflow");
 972         }
 973 
 974         return a - 1L;
 975     }
 976 
 977     /**
 978      * Returns the negation of the argument, throwing an exception if the
 979      * result overflows an {@code int}.
 980      *
 981      * @param a the value to negate
 982      * @return the result
 983      * @throws ArithmeticException if the result overflows an int
 984      * @since 1.8
 985      */
 986     public static int negateExact(int a) {
 987         if (a == Integer.MIN_VALUE) {
 988             throw new ArithmeticException("integer overflow");
 989         }
 990 
 991         return -a;
 992     }
 993 
 994     /**
 995      * Returns the negation of the argument, throwing an exception if the
 996      * result overflows a {@code long}.
 997      *
 998      * @param a the value to negate
 999      * @return the result
1000      * @throws ArithmeticException if the result overflows a long
1001      * @since 1.8
1002      */
1003     public static long negateExact(long a) {
1004         if (a == Long.MIN_VALUE) {
1005             throw new ArithmeticException("long overflow");
1006         }
1007 
1008         return -a;
1009     }
1010 
1011     /**
1012      * Returns the value of the {@code long} argument;
1013      * throwing an exception if the value overflows an {@code int}.
1014      *
1015      * @param value the long value
1016      * @return the argument as an int
1017      * @throws ArithmeticException if the {@code argument} overflows an int
1018      * @since 1.8
1019      */
1020     public static int toIntExact(long value) {
1021         if ((int)value != value) {
1022             throw new ArithmeticException("integer overflow");
1023         }
1024         return (int)value;
1025     }
1026 
1027     /**
1028      * Returns the largest (closest to positive infinity)
1029      * {@code int} value that is less than or equal to the algebraic quotient.
1030      * There is one special case, if the dividend is the
1031      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1032      * then integer overflow occurs and
1033      * the result is equal to the {@code Integer.MIN_VALUE}.
1034      * <p>
1035      * Normal integer division operates under the round to zero rounding mode
1036      * (truncation).  This operation instead acts under the round toward
1037      * negative infinity (floor) rounding mode.
1038      * The floor rounding mode gives different results than truncation
1039      * when the exact result is negative.
1040      * <ul>
1041      *   <li>If the signs of the arguments are the same, the results of
1042      *       {@code floorDiv} and the {@code /} operator are the same.  <br>
1043      *       For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1044      *   <li>If the signs of the arguments are different,  the quotient is negative and
1045      *       {@code floorDiv} returns the integer less than or equal to the quotient
1046      *       and the {@code /} operator returns the integer closest to zero.<br>
1047      *       For example, {@code floorDiv(-4, 3) == -2},
1048      *       whereas {@code (-4 / 3) == -1}.
1049      *   </li>
1050      * </ul>
1051      *
1052      * @param x the dividend
1053      * @param y the divisor
1054      * @return the largest (closest to positive infinity)
1055      * {@code int} value that is less than or equal to the algebraic quotient.
1056      * @throws ArithmeticException if the divisor {@code y} is zero
1057      * @see #floorMod(int, int)
1058      * @see #floor(double)
1059      * @since 1.8
1060      */
1061     public static int floorDiv(int x, int y) {
1062         int r = x / y;
1063         // if the signs are different and modulo not zero, round down
1064         if ((x ^ y) < 0 && (r * y != x)) {
1065             r--;
1066         }
1067         return r;
1068     }
1069 
1070     /**
1071      * Returns the largest (closest to positive infinity)
1072      * {@code long} value that is less than or equal to the algebraic quotient.
1073      * There is one special case, if the dividend is the
1074      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1075      * then integer overflow occurs and
1076      * the result is equal to the {@code Long.MIN_VALUE}.
1077      * <p>
1078      * Normal integer division operates under the round to zero rounding mode
1079      * (truncation).  This operation instead acts under the round toward
1080      * negative infinity (floor) rounding mode.
1081      * The floor rounding mode gives different results than truncation
1082      * when the exact result is negative.
1083      * <p>
1084      * For examples, see {@link #floorDiv(int, int)}.
1085      *
1086      * @param x the dividend
1087      * @param y the divisor
1088      * @return the largest (closest to positive infinity)
1089      * {@code long} value that is less than or equal to the algebraic quotient.
1090      * @throws ArithmeticException if the divisor {@code y} is zero
1091      * @see #floorMod(long, long)
1092      * @see #floor(double)
1093      * @since 1.8
1094      */
1095     public static long floorDiv(long x, long y) {
1096         long r = x / y;
1097         // if the signs are different and modulo not zero, round down
1098         if ((x ^ y) < 0 && (r * y != x)) {
1099             r--;
1100         }
1101         return r;
1102     }
1103 
1104     /**
1105      * Returns the floor modulus of the {@code int} arguments.
1106      * <p>
1107      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1108      * has the same sign as the divisor {@code y}, and
1109      * is in the range of {@code -abs(y) < r < +abs(y)}.
1110      *
1111      * <p>
1112      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1113      * <ul>
1114      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1115      * </ul>
1116      * <p>
1117      * The difference in values between {@code floorMod} and
1118      * the {@code %} operator is due to the difference between
1119      * {@code floorDiv} that returns the integer less than or equal to the quotient
1120      * and the {@code /} operator that returns the integer closest to zero.
1121      * <p>
1122      * Examples:
1123      * <ul>
1124      *   <li>If the signs of the arguments are the same, the results
1125      *       of {@code floorMod} and the {@code %} operator are the same.  <br>
1126      *       <ul>
1127      *       <li>{@code floorMod(4, 3) == 1}; &nbsp; and {@code (4 % 3) == 1}</li>
1128      *       </ul>
1129      *   <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1130      *      <ul>
1131      *      <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1} </li>
1132      *      <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1} </li>
1133      *      <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1 } </li>
1134      *      </ul>
1135      *   </li>
1136      * </ul>
1137      * <p>
1138      * If the signs of arguments are unknown and a positive modulus
1139      * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1140      *
1141      * @param x the dividend
1142      * @param y the divisor
1143      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1144      * @throws ArithmeticException if the divisor {@code y} is zero
1145      * @see #floorDiv(int, int)
1146      * @since 1.8
1147      */
1148     public static int floorMod(int x, int y) {
1149         int r = x - floorDiv(x, y) * y;
1150         return r;
1151     }
1152 
1153     /**
1154      * Returns the floor modulus of the {@code long} arguments.
1155      * <p>
1156      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1157      * has the same sign as the divisor {@code y}, and
1158      * is in the range of {@code -abs(y) < r < +abs(y)}.
1159      *
1160      * <p>
1161      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1162      * <ul>
1163      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1164      * </ul>
1165      * <p>
1166      * For examples, see {@link #floorMod(int, int)}.
1167      *
1168      * @param x the dividend
1169      * @param y the divisor
1170      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1171      * @throws ArithmeticException if the divisor {@code y} is zero
1172      * @see #floorDiv(long, long)
1173      * @since 1.8
1174      */
1175     public static long floorMod(long x, long y) {
1176         return x - floorDiv(x, y) * y;
1177     }
1178 
1179     /**
1180      * Returns the absolute value of an {@code int} value.
1181      * If the argument is not negative, the argument is returned.
1182      * If the argument is negative, the negation of the argument is returned.
1183      *
1184      * <p>Note that if the argument is equal to the value of
1185      * {@link Integer#MIN_VALUE}, the most negative representable
1186      * {@code int} value, the result is that same value, which is
1187      * negative.
1188      *
1189      * @param   a   the argument whose absolute value is to be determined
1190      * @return  the absolute value of the argument.
1191      */
1192     public static int abs(int a) {
1193         return (a < 0) ? -a : a;
1194     }
1195 
1196     /**
1197      * Returns the absolute value of a {@code long} value.
1198      * If the argument is not negative, the argument is returned.
1199      * If the argument is negative, the negation of the argument is returned.
1200      *
1201      * <p>Note that if the argument is equal to the value of
1202      * {@link Long#MIN_VALUE}, the most negative representable
1203      * {@code long} value, the result is that same value, which
1204      * is negative.
1205      *
1206      * @param   a   the argument whose absolute value is to be determined
1207      * @return  the absolute value of the argument.
1208      */
1209     public static long abs(long a) {
1210         return (a < 0) ? -a : a;
1211     }
1212 
1213     /**
1214      * Returns the absolute value of a {@code float} value.
1215      * If the argument is not negative, the argument is returned.
1216      * If the argument is negative, the negation of the argument is returned.
1217      * Special cases:
1218      * <ul><li>If the argument is positive zero or negative zero, the
1219      * result is positive zero.
1220      * <li>If the argument is infinite, the result is positive infinity.
1221      * <li>If the argument is NaN, the result is NaN.</ul>
1222      * In other words, the result is the same as the value of the expression:
1223      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1224      *
1225      * @param   a   the argument whose absolute value is to be determined
1226      * @return  the absolute value of the argument.
1227      */
1228     public static float abs(float a) {
1229         return (a <= 0.0F) ? 0.0F - a : a;
1230     }
1231 
1232     /**
1233      * Returns the absolute value of a {@code double} value.
1234      * If the argument is not negative, the argument is returned.
1235      * If the argument is negative, the negation of the argument is returned.
1236      * Special cases:
1237      * <ul><li>If the argument is positive zero or negative zero, the result
1238      * is positive zero.
1239      * <li>If the argument is infinite, the result is positive infinity.
1240      * <li>If the argument is NaN, the result is NaN.</ul>
1241      * In other words, the result is the same as the value of the expression:
1242      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1243      *
1244      * @param   a   the argument whose absolute value is to be determined
1245      * @return  the absolute value of the argument.
1246      */
1247     public static double abs(double a) {
1248         return (a <= 0.0D) ? 0.0D - a : a;
1249     }
1250 
1251     /**
1252      * Returns the greater of two {@code int} values. That is, the
1253      * result is the argument closer to the value of
1254      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1255      * the result is that same value.
1256      *
1257      * @param   a   an argument.
1258      * @param   b   another argument.
1259      * @return  the larger of {@code a} and {@code b}.
1260      */
1261     public static int max(int a, int b) {
1262         return (a >= b) ? a : b;
1263     }
1264 
1265     /**
1266      * Returns the greater of two {@code long} values. That is, the
1267      * result is the argument closer to the value of
1268      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1269      * the result is that same value.
1270      *
1271      * @param   a   an argument.
1272      * @param   b   another argument.
1273      * @return  the larger of {@code a} and {@code b}.
1274      */
1275     public static long max(long a, long b) {
1276         return (a >= b) ? a : b;
1277     }
1278 
1279     // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1280     private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
1281     private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1282 
1283     /**
1284      * Returns the greater of two {@code float} values.  That is,
1285      * the result is the argument closer to positive infinity. If the
1286      * arguments have the same value, the result is that same
1287      * value. If either value is NaN, then the result is NaN.  Unlike
1288      * the numerical comparison operators, this method considers
1289      * negative zero to be strictly smaller than positive zero. If one
1290      * argument is positive zero and the other negative zero, the
1291      * result is positive zero.
1292      *
1293      * @param   a   an argument.
1294      * @param   b   another argument.
1295      * @return  the larger of {@code a} and {@code b}.
1296      */
1297     public static float max(float a, float b) {
1298         if (a != a)
1299             return a;   // a is NaN
1300         if ((a == 0.0f) &&
1301             (b == 0.0f) &&
1302             (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1303             // Raw conversion ok since NaN can't map to -0.0.
1304             return b;
1305         }
1306         return (a >= b) ? a : b;
1307     }
1308 
1309     /**
1310      * Returns the greater of two {@code double} values.  That
1311      * is, the result is the argument closer to positive infinity. If
1312      * the arguments have the same value, the result is that same
1313      * value. If either value is NaN, then the result is NaN.  Unlike
1314      * the numerical comparison operators, this method considers
1315      * negative zero to be strictly smaller than positive zero. If one
1316      * argument is positive zero and the other negative zero, the
1317      * result is positive zero.
1318      *
1319      * @param   a   an argument.
1320      * @param   b   another argument.
1321      * @return  the larger of {@code a} and {@code b}.
1322      */
1323     public static double max(double a, double b) {
1324         if (a != a)
1325             return a;   // a is NaN
1326         if ((a == 0.0d) &&
1327             (b == 0.0d) &&
1328             (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1329             // Raw conversion ok since NaN can't map to -0.0.
1330             return b;
1331         }
1332         return (a >= b) ? a : b;
1333     }
1334 
1335     /**
1336      * Returns the smaller of two {@code int} values. That is,
1337      * the result the argument closer to the value of
1338      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1339      * value, the result is that same value.
1340      *
1341      * @param   a   an argument.
1342      * @param   b   another argument.
1343      * @return  the smaller of {@code a} and {@code b}.
1344      */
1345     public static int min(int a, int b) {
1346         return (a <= b) ? a : b;
1347     }
1348 
1349     /**
1350      * Returns the smaller of two {@code long} values. That is,
1351      * the result is the argument closer to the value of
1352      * {@link Long#MIN_VALUE}. If the arguments have the same
1353      * value, the result is that same value.
1354      *
1355      * @param   a   an argument.
1356      * @param   b   another argument.
1357      * @return  the smaller of {@code a} and {@code b}.
1358      */
1359     public static long min(long a, long b) {
1360         return (a <= b) ? a : b;
1361     }
1362 
1363     /**
1364      * Returns the smaller of two {@code float} values.  That is,
1365      * the result is the value closer to negative infinity. If the
1366      * arguments have the same value, the result is that same
1367      * value. If either value is NaN, then the result is NaN.  Unlike
1368      * the numerical comparison operators, this method considers
1369      * negative zero to be strictly smaller than positive zero.  If
1370      * one argument is positive zero and the other is negative zero,
1371      * the result is negative zero.
1372      *
1373      * @param   a   an argument.
1374      * @param   b   another argument.
1375      * @return  the smaller of {@code a} and {@code b}.
1376      */
1377     public static float min(float a, float b) {
1378         if (a != a)
1379             return a;   // a is NaN
1380         if ((a == 0.0f) &&
1381             (b == 0.0f) &&
1382             (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1383             // Raw conversion ok since NaN can't map to -0.0.
1384             return b;
1385         }
1386         return (a <= b) ? a : b;
1387     }
1388 
1389     /**
1390      * Returns the smaller of two {@code double} values.  That
1391      * is, the result is the value closer to negative infinity. If the
1392      * arguments have the same value, the result is that same
1393      * value. If either value is NaN, then the result is NaN.  Unlike
1394      * the numerical comparison operators, this method considers
1395      * negative zero to be strictly smaller than positive zero. If one
1396      * argument is positive zero and the other is negative zero, the
1397      * result is negative zero.
1398      *
1399      * @param   a   an argument.
1400      * @param   b   another argument.
1401      * @return  the smaller of {@code a} and {@code b}.
1402      */
1403     public static double min(double a, double b) {
1404         if (a != a)
1405             return a;   // a is NaN
1406         if ((a == 0.0d) &&
1407             (b == 0.0d) &&
1408             (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1409             // Raw conversion ok since NaN can't map to -0.0.
1410             return b;
1411         }
1412         return (a <= b) ? a : b;
1413     }
1414 
1415     /**
1416      * Returns the size of an ulp of the argument.  An ulp, unit in
1417      * the last place, of a {@code double} value is the positive
1418      * distance between this floating-point value and the {@code
1419      * double} value next larger in magnitude.  Note that for non-NaN
1420      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1421      *
1422      * <p>Special Cases:
1423      * <ul>
1424      * <li> If the argument is NaN, then the result is NaN.
1425      * <li> If the argument is positive or negative infinity, then the
1426      * result is positive infinity.
1427      * <li> If the argument is positive or negative zero, then the result is
1428      * {@code Double.MIN_VALUE}.
1429      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1430      * the result is equal to 2<sup>971</sup>.
1431      * </ul>
1432      *
1433      * @param d the floating-point value whose ulp is to be returned
1434      * @return the size of an ulp of the argument
1435      * @author Joseph D. Darcy
1436      * @since 1.5
1437      */
1438     public static double ulp(double d) {
1439         int exp = getExponent(d);
1440 
1441         switch(exp) {
1442         case DoubleConsts.MAX_EXPONENT+1:       // NaN or infinity
1443             return Math.abs(d);
1444 
1445         case DoubleConsts.MIN_EXPONENT-1:       // zero or subnormal
1446             return Double.MIN_VALUE;
1447 
1448         default:
1449             assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
1450 
1451             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1452             exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1453             if (exp >= DoubleConsts.MIN_EXPONENT) {
1454                 return powerOfTwoD(exp);
1455             }
1456             else {
1457                 // return a subnormal result; left shift integer
1458                 // representation of Double.MIN_VALUE appropriate
1459                 // number of positions
1460                 return Double.longBitsToDouble(1L <<
1461                 (exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1462             }
1463         }
1464     }
1465 
1466     /**
1467      * Returns the size of an ulp of the argument.  An ulp, unit in
1468      * the last place, of a {@code float} value is the positive
1469      * distance between this floating-point value and the {@code
1470      * float} value next larger in magnitude.  Note that for non-NaN
1471      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1472      *
1473      * <p>Special Cases:
1474      * <ul>
1475      * <li> If the argument is NaN, then the result is NaN.
1476      * <li> If the argument is positive or negative infinity, then the
1477      * result is positive infinity.
1478      * <li> If the argument is positive or negative zero, then the result is
1479      * {@code Float.MIN_VALUE}.
1480      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1481      * the result is equal to 2<sup>104</sup>.
1482      * </ul>
1483      *
1484      * @param f the floating-point value whose ulp is to be returned
1485      * @return the size of an ulp of the argument
1486      * @author Joseph D. Darcy
1487      * @since 1.5
1488      */
1489     public static float ulp(float f) {
1490         int exp = getExponent(f);
1491 
1492         switch(exp) {
1493         case FloatConsts.MAX_EXPONENT+1:        // NaN or infinity
1494             return Math.abs(f);
1495 
1496         case FloatConsts.MIN_EXPONENT-1:        // zero or subnormal
1497             return FloatConsts.MIN_VALUE;
1498 
1499         default:
1500             assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
1501 
1502             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1503             exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1504             if (exp >= FloatConsts.MIN_EXPONENT) {
1505                 return powerOfTwoF(exp);
1506             }
1507             else {
1508                 // return a subnormal result; left shift integer
1509                 // representation of FloatConsts.MIN_VALUE appropriate
1510                 // number of positions
1511                 return Float.intBitsToFloat(1 <<
1512                 (exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1513             }
1514         }
1515     }
1516 
1517     /**
1518      * Returns the signum function of the argument; zero if the argument
1519      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1520      * argument is less than zero.
1521      *
1522      * <p>Special Cases:
1523      * <ul>
1524      * <li> If the argument is NaN, then the result is NaN.
1525      * <li> If the argument is positive zero or negative zero, then the
1526      *      result is the same as the argument.
1527      * </ul>
1528      *
1529      * @param d the floating-point value whose signum is to be returned
1530      * @return the signum function of the argument
1531      * @author Joseph D. Darcy
1532      * @since 1.5
1533      */
1534     public static double signum(double d) {
1535         return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1536     }
1537 
1538     /**
1539      * Returns the signum function of the argument; zero if the argument
1540      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1541      * argument is less than zero.
1542      *
1543      * <p>Special Cases:
1544      * <ul>
1545      * <li> If the argument is NaN, then the result is NaN.
1546      * <li> If the argument is positive zero or negative zero, then the
1547      *      result is the same as the argument.
1548      * </ul>
1549      *
1550      * @param f the floating-point value whose signum is to be returned
1551      * @return the signum function of the argument
1552      * @author Joseph D. Darcy
1553      * @since 1.5
1554      */
1555     public static float signum(float f) {
1556         return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1557     }
1558 
1559     /**
1560      * Returns the hyperbolic sine of a {@code double} value.
1561      * The hyperbolic sine of <i>x</i> is defined to be
1562      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1563      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1564      *
1565      * <p>Special cases:
1566      * <ul>
1567      *
1568      * <li>If the argument is NaN, then the result is NaN.
1569      *
1570      * <li>If the argument is infinite, then the result is an infinity
1571      * with the same sign as the argument.
1572      *
1573      * <li>If the argument is zero, then the result is a zero with the
1574      * same sign as the argument.
1575      *
1576      * </ul>
1577      *
1578      * <p>The computed result must be within 2.5 ulps of the exact result.
1579      *
1580      * @param   x The number whose hyperbolic sine is to be returned.
1581      * @return  The hyperbolic sine of {@code x}.
1582      * @since 1.5
1583      */
1584     public static double sinh(double x) {
1585         return StrictMath.sinh(x);
1586     }
1587 
1588     /**
1589      * Returns the hyperbolic cosine of a {@code double} value.
1590      * The hyperbolic cosine of <i>x</i> is defined to be
1591      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1592      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1593      *
1594      * <p>Special cases:
1595      * <ul>
1596      *
1597      * <li>If the argument is NaN, then the result is NaN.
1598      *
1599      * <li>If the argument is infinite, then the result is positive
1600      * infinity.
1601      *
1602      * <li>If the argument is zero, then the result is {@code 1.0}.
1603      *
1604      * </ul>
1605      *
1606      * <p>The computed result must be within 2.5 ulps of the exact result.
1607      *
1608      * @param   x The number whose hyperbolic cosine is to be returned.
1609      * @return  The hyperbolic cosine of {@code x}.
1610      * @since 1.5
1611      */
1612     public static double cosh(double x) {
1613         return StrictMath.cosh(x);
1614     }
1615 
1616     /**
1617      * Returns the hyperbolic tangent of a {@code double} value.
1618      * The hyperbolic tangent of <i>x</i> is defined to be
1619      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1620      * in other words, {@linkplain Math#sinh
1621      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1622      * that the absolute value of the exact tanh is always less than
1623      * 1.
1624      *
1625      * <p>Special cases:
1626      * <ul>
1627      *
1628      * <li>If the argument is NaN, then the result is NaN.
1629      *
1630      * <li>If the argument is zero, then the result is a zero with the
1631      * same sign as the argument.
1632      *
1633      * <li>If the argument is positive infinity, then the result is
1634      * {@code +1.0}.
1635      *
1636      * <li>If the argument is negative infinity, then the result is
1637      * {@code -1.0}.
1638      *
1639      * </ul>
1640      *
1641      * <p>The computed result must be within 2.5 ulps of the exact result.
1642      * The result of {@code tanh} for any finite input must have
1643      * an absolute value less than or equal to 1.  Note that once the
1644      * exact result of tanh is within 1/2 of an ulp of the limit value
1645      * of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
1646      * be returned.
1647      *
1648      * @param   x The number whose hyperbolic tangent is to be returned.
1649      * @return  The hyperbolic tangent of {@code x}.
1650      * @since 1.5
1651      */
1652     public static double tanh(double x) {
1653         return StrictMath.tanh(x);
1654     }
1655 
1656     /**
1657      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1658      * without intermediate overflow or underflow.
1659      *
1660      * <p>Special cases:
1661      * <ul>
1662      *
1663      * <li> If either argument is infinite, then the result
1664      * is positive infinity.
1665      *
1666      * <li> If either argument is NaN and neither argument is infinite,
1667      * then the result is NaN.
1668      *
1669      * </ul>
1670      *
1671      * <p>The computed result must be within 1 ulp of the exact
1672      * result.  If one parameter is held constant, the results must be
1673      * semi-monotonic in the other parameter.
1674      *
1675      * @param x a value
1676      * @param y a value
1677      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1678      * without intermediate overflow or underflow
1679      * @since 1.5
1680      */
1681     public static double hypot(double x, double y) {
1682         return StrictMath.hypot(x, y);
1683     }
1684 
1685     /**
1686      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
1687      * <i>x</i> near 0, the exact sum of
1688      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
1689      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1690      *
1691      * <p>Special cases:
1692      * <ul>
1693      * <li>If the argument is NaN, the result is NaN.
1694      *
1695      * <li>If the argument is positive infinity, then the result is
1696      * positive infinity.
1697      *
1698      * <li>If the argument is negative infinity, then the result is
1699      * -1.0.
1700      *
1701      * <li>If the argument is zero, then the result is a zero with the
1702      * same sign as the argument.
1703      *
1704      * </ul>
1705      *
1706      * <p>The computed result must be within 1 ulp of the exact result.
1707      * Results must be semi-monotonic.  The result of
1708      * {@code expm1} for any finite input must be greater than or
1709      * equal to {@code -1.0}.  Note that once the exact result of
1710      * <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
1711      * ulp of the limit value -1, {@code -1.0} should be
1712      * returned.
1713      *
1714      * @param   x   the exponent to raise <i>e</i> to in the computation of
1715      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
1716      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
1717      * @since 1.5
1718      */
1719     public static double expm1(double x) {
1720         return StrictMath.expm1(x);
1721     }
1722 
1723     /**
1724      * Returns the natural logarithm of the sum of the argument and 1.
1725      * Note that for small values {@code x}, the result of
1726      * {@code log1p(x)} is much closer to the true result of ln(1
1727      * + {@code x}) than the floating-point evaluation of
1728      * {@code log(1.0+x)}.
1729      *
1730      * <p>Special cases:
1731      *
1732      * <ul>
1733      *
1734      * <li>If the argument is NaN or less than -1, then the result is
1735      * NaN.
1736      *
1737      * <li>If the argument is positive infinity, then the result is
1738      * positive infinity.
1739      *
1740      * <li>If the argument is negative one, then the result is
1741      * negative infinity.
1742      *
1743      * <li>If the argument is zero, then the result is a zero with the
1744      * same sign as the argument.
1745      *
1746      * </ul>
1747      *
1748      * <p>The computed result must be within 1 ulp of the exact result.
1749      * Results must be semi-monotonic.
1750      *
1751      * @param   x   a value
1752      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
1753      * log of {@code x}&nbsp;+&nbsp;1
1754      * @since 1.5
1755      */
1756     public static double log1p(double x) {
1757         return StrictMath.log1p(x);
1758     }
1759 
1760     /**
1761      * Returns the first floating-point argument with the sign of the
1762      * second floating-point argument.  Note that unlike the {@link
1763      * StrictMath#copySign(double, double) StrictMath.copySign}
1764      * method, this method does not require NaN {@code sign}
1765      * arguments to be treated as positive values; implementations are
1766      * permitted to treat some NaN arguments as positive and other NaN
1767      * arguments as negative to allow greater performance.
1768      *
1769      * @param magnitude  the parameter providing the magnitude of the result
1770      * @param sign   the parameter providing the sign of the result
1771      * @return a value with the magnitude of {@code magnitude}
1772      * and the sign of {@code sign}.
1773      * @since 1.6
1774      */
1775     public static double copySign(double magnitude, double sign) {
1776         return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
1777                                         (DoubleConsts.SIGN_BIT_MASK)) |
1778                                        (Double.doubleToRawLongBits(magnitude) &
1779                                         (DoubleConsts.EXP_BIT_MASK |
1780                                          DoubleConsts.SIGNIF_BIT_MASK)));
1781     }
1782 
1783     /**
1784      * Returns the first floating-point argument with the sign of the
1785      * second floating-point argument.  Note that unlike the {@link
1786      * StrictMath#copySign(float, float) StrictMath.copySign}
1787      * method, this method does not require NaN {@code sign}
1788      * arguments to be treated as positive values; implementations are
1789      * permitted to treat some NaN arguments as positive and other NaN
1790      * arguments as negative to allow greater performance.
1791      *
1792      * @param magnitude  the parameter providing the magnitude of the result
1793      * @param sign   the parameter providing the sign of the result
1794      * @return a value with the magnitude of {@code magnitude}
1795      * and the sign of {@code sign}.
1796      * @since 1.6
1797      */
1798     public static float copySign(float magnitude, float sign) {
1799         return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
1800                                      (FloatConsts.SIGN_BIT_MASK)) |
1801                                     (Float.floatToRawIntBits(magnitude) &
1802                                      (FloatConsts.EXP_BIT_MASK |
1803                                       FloatConsts.SIGNIF_BIT_MASK)));
1804     }
1805 
1806     /**
1807      * Returns the unbiased exponent used in the representation of a
1808      * {@code float}.  Special cases:
1809      *
1810      * <ul>
1811      * <li>If the argument is NaN or infinite, then the result is
1812      * {@link Float#MAX_EXPONENT} + 1.
1813      * <li>If the argument is zero or subnormal, then the result is
1814      * {@link Float#MIN_EXPONENT} -1.
1815      * </ul>
1816      * @param f a {@code float} value
1817      * @return the unbiased exponent of the argument
1818      * @since 1.6
1819      */
1820     public static int getExponent(float f) {
1821         /*
1822          * Bitwise convert f to integer, mask out exponent bits, shift
1823          * to the right and then subtract out float's bias adjust to
1824          * get true exponent value
1825          */
1826         return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
1827                 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
1828     }
1829 
1830     /**
1831      * Returns the unbiased exponent used in the representation of a
1832      * {@code double}.  Special cases:
1833      *
1834      * <ul>
1835      * <li>If the argument is NaN or infinite, then the result is
1836      * {@link Double#MAX_EXPONENT} + 1.
1837      * <li>If the argument is zero or subnormal, then the result is
1838      * {@link Double#MIN_EXPONENT} -1.
1839      * </ul>
1840      * @param d a {@code double} value
1841      * @return the unbiased exponent of the argument
1842      * @since 1.6
1843      */
1844     public static int getExponent(double d) {
1845         /*
1846          * Bitwise convert d to long, mask out exponent bits, shift
1847          * to the right and then subtract out double's bias adjust to
1848          * get true exponent value.
1849          */
1850         return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
1851                       (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
1852     }
1853 
1854     /**
1855      * Returns the floating-point number adjacent to the first
1856      * argument in the direction of the second argument.  If both
1857      * arguments compare as equal the second argument is returned.
1858      *
1859      * <p>
1860      * Special cases:
1861      * <ul>
1862      * <li> If either argument is a NaN, then NaN is returned.
1863      *
1864      * <li> If both arguments are signed zeros, {@code direction}
1865      * is returned unchanged (as implied by the requirement of
1866      * returning the second argument if the arguments compare as
1867      * equal).
1868      *
1869      * <li> If {@code start} is
1870      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
1871      * has a value such that the result should have a smaller
1872      * magnitude, then a zero with the same sign as {@code start}
1873      * is returned.
1874      *
1875      * <li> If {@code start} is infinite and
1876      * {@code direction} has a value such that the result should
1877      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1878      * same sign as {@code start} is returned.
1879      *
1880      * <li> If {@code start} is equal to &plusmn;
1881      * {@link Double#MAX_VALUE} and {@code direction} has a
1882      * value such that the result should have a larger magnitude, an
1883      * infinity with same sign as {@code start} is returned.
1884      * </ul>
1885      *
1886      * @param start  starting floating-point value
1887      * @param direction value indicating which of
1888      * {@code start}'s neighbors or {@code start} should
1889      * be returned
1890      * @return The floating-point number adjacent to {@code start} in the
1891      * direction of {@code direction}.
1892      * @since 1.6
1893      */
1894     public static double nextAfter(double start, double direction) {
1895         /*
1896          * The cases:
1897          *
1898          * nextAfter(+infinity, 0)  == MAX_VALUE
1899          * nextAfter(+infinity, +infinity)  == +infinity
1900          * nextAfter(-infinity, 0)  == -MAX_VALUE
1901          * nextAfter(-infinity, -infinity)  == -infinity
1902          *
1903          * are naturally handled without any additional testing
1904          */
1905 
1906         /*
1907          * IEEE 754 floating-point numbers are lexicographically
1908          * ordered if treated as signed-magnitude integers.
1909          * Since Java's integers are two's complement,
1910          * incrementing the two's complement representation of a
1911          * logically negative floating-point value *decrements*
1912          * the signed-magnitude representation. Therefore, when
1913          * the integer representation of a floating-point value
1914          * is negative, the adjustment to the representation is in
1915          * the opposite direction from what would initially be expected.
1916          */
1917 
1918         // Branch to descending case first as it is more costly than ascending
1919         // case due to start != 0.0d conditional.
1920         if (start > direction) { // descending
1921             if (start != 0.0d) {
1922                 final long transducer = Double.doubleToRawLongBits(start);
1923                 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
1924             } else { // start == 0.0d && direction < 0.0d
1925                 return -Double.MIN_VALUE;
1926             }
1927         } else if (start < direction) { // ascending
1928             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
1929             // then bitwise convert start to integer.
1930             final long transducer = Double.doubleToRawLongBits(start + 0.0d);
1931             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
1932         } else if (start == direction) {
1933             return direction;
1934         } else { // isNaN(start) || isNaN(direction)
1935             return start + direction;
1936         }
1937     }
1938 
1939     /**
1940      * Returns the floating-point number adjacent to the first
1941      * argument in the direction of the second argument.  If both
1942      * arguments compare as equal a value equivalent to the second argument
1943      * is returned.
1944      *
1945      * <p>
1946      * Special cases:
1947      * <ul>
1948      * <li> If either argument is a NaN, then NaN is returned.
1949      *
1950      * <li> If both arguments are signed zeros, a value equivalent
1951      * to {@code direction} is returned.
1952      *
1953      * <li> If {@code start} is
1954      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
1955      * has a value such that the result should have a smaller
1956      * magnitude, then a zero with the same sign as {@code start}
1957      * is returned.
1958      *
1959      * <li> If {@code start} is infinite and
1960      * {@code direction} has a value such that the result should
1961      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1962      * same sign as {@code start} is returned.
1963      *
1964      * <li> If {@code start} is equal to &plusmn;
1965      * {@link Float#MAX_VALUE} and {@code direction} has a
1966      * value such that the result should have a larger magnitude, an
1967      * infinity with same sign as {@code start} is returned.
1968      * </ul>
1969      *
1970      * @param start  starting floating-point value
1971      * @param direction value indicating which of
1972      * {@code start}'s neighbors or {@code start} should
1973      * be returned
1974      * @return The floating-point number adjacent to {@code start} in the
1975      * direction of {@code direction}.
1976      * @since 1.6
1977      */
1978     public static float nextAfter(float start, double direction) {
1979         /*
1980          * The cases:
1981          *
1982          * nextAfter(+infinity, 0)  == MAX_VALUE
1983          * nextAfter(+infinity, +infinity)  == +infinity
1984          * nextAfter(-infinity, 0)  == -MAX_VALUE
1985          * nextAfter(-infinity, -infinity)  == -infinity
1986          *
1987          * are naturally handled without any additional testing
1988          */
1989 
1990         /*
1991          * IEEE 754 floating-point numbers are lexicographically
1992          * ordered if treated as signed-magnitude integers.
1993          * Since Java's integers are two's complement,
1994          * incrementing the two's complement representation of a
1995          * logically negative floating-point value *decrements*
1996          * the signed-magnitude representation. Therefore, when
1997          * the integer representation of a floating-point value
1998          * is negative, the adjustment to the representation is in
1999          * the opposite direction from what would initially be expected.
2000          */
2001 
2002         // Branch to descending case first as it is more costly than ascending
2003         // case due to start != 0.0f conditional.
2004         if (start > direction) { // descending
2005             if (start != 0.0f) {
2006                 final int transducer = Float.floatToRawIntBits(start);
2007                 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2008             } else { // start == 0.0f && direction < 0.0f
2009                 return -Float.MIN_VALUE;
2010             }
2011         } else if (start < direction) { // ascending
2012             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2013             // then bitwise convert start to integer.
2014             final int transducer = Float.floatToRawIntBits(start + 0.0f);
2015             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2016         } else if (start == direction) {
2017             return (float)direction;
2018         } else { // isNaN(start) || isNaN(direction)
2019             return start + (float)direction;
2020         }
2021     }
2022 
2023     /**
2024      * Returns the floating-point value adjacent to {@code d} in
2025      * the direction of positive infinity.  This method is
2026      * semantically equivalent to {@code nextAfter(d,
2027      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2028      * implementation may run faster than its equivalent
2029      * {@code nextAfter} call.
2030      *
2031      * <p>Special Cases:
2032      * <ul>
2033      * <li> If the argument is NaN, the result is NaN.
2034      *
2035      * <li> If the argument is positive infinity, the result is
2036      * positive infinity.
2037      *
2038      * <li> If the argument is zero, the result is
2039      * {@link Double#MIN_VALUE}
2040      *
2041      * </ul>
2042      *
2043      * @param d starting floating-point value
2044      * @return The adjacent floating-point value closer to positive
2045      * infinity.
2046      * @since 1.6
2047      */
2048     public static double nextUp(double d) {
2049         // Use a single conditional and handle the likely cases first.
2050         if (d < Double.POSITIVE_INFINITY) {
2051             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2052             final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2053             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2054         } else { // d is NaN or +Infinity
2055             return d;
2056         }
2057     }
2058 
2059     /**
2060      * Returns the floating-point value adjacent to {@code f} in
2061      * the direction of positive infinity.  This method is
2062      * semantically equivalent to {@code nextAfter(f,
2063      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2064      * implementation may run faster than its equivalent
2065      * {@code nextAfter} call.
2066      *
2067      * <p>Special Cases:
2068      * <ul>
2069      * <li> If the argument is NaN, the result is NaN.
2070      *
2071      * <li> If the argument is positive infinity, the result is
2072      * positive infinity.
2073      *
2074      * <li> If the argument is zero, the result is
2075      * {@link Float#MIN_VALUE}
2076      *
2077      * </ul>
2078      *
2079      * @param f starting floating-point value
2080      * @return The adjacent floating-point value closer to positive
2081      * infinity.
2082      * @since 1.6
2083      */
2084     public static float nextUp(float f) {
2085         // Use a single conditional and handle the likely cases first.
2086         if (f < Float.POSITIVE_INFINITY) {
2087             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2088             final int transducer = Float.floatToRawIntBits(f + 0.0F);
2089             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2090         } else { // f is NaN or +Infinity
2091             return f;
2092         }
2093     }
2094 
2095     /**
2096      * Returns the floating-point value adjacent to {@code d} in
2097      * the direction of negative infinity.  This method is
2098      * semantically equivalent to {@code nextAfter(d,
2099      * Double.NEGATIVE_INFINITY)}; however, a
2100      * {@code nextDown} implementation may run faster than its
2101      * equivalent {@code nextAfter} call.
2102      *
2103      * <p>Special Cases:
2104      * <ul>
2105      * <li> If the argument is NaN, the result is NaN.
2106      *
2107      * <li> If the argument is negative infinity, the result is
2108      * negative infinity.
2109      *
2110      * <li> If the argument is zero, the result is
2111      * {@code -Double.MIN_VALUE}
2112      *
2113      * </ul>
2114      *
2115      * @param d  starting floating-point value
2116      * @return The adjacent floating-point value closer to negative
2117      * infinity.
2118      * @since 1.8
2119      */
2120     public static double nextDown(double d) {
2121         if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2122             return d;
2123         else {
2124             if (d == 0.0)
2125                 return -Double.MIN_VALUE;
2126             else
2127                 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2128                                                ((d > 0.0d)?-1L:+1L));
2129         }
2130     }
2131 
2132     /**
2133      * Returns the floating-point value adjacent to {@code f} in
2134      * the direction of negative infinity.  This method is
2135      * semantically equivalent to {@code nextAfter(f,
2136      * Float.NEGATIVE_INFINITY)}; however, a
2137      * {@code nextDown} implementation may run faster than its
2138      * equivalent {@code nextAfter} call.
2139      *
2140      * <p>Special Cases:
2141      * <ul>
2142      * <li> If the argument is NaN, the result is NaN.
2143      *
2144      * <li> If the argument is negative infinity, the result is
2145      * negative infinity.
2146      *
2147      * <li> If the argument is zero, the result is
2148      * {@code -Float.MIN_VALUE}
2149      *
2150      * </ul>
2151      *
2152      * @param f  starting floating-point value
2153      * @return The adjacent floating-point value closer to negative
2154      * infinity.
2155      * @since 1.8
2156      */
2157     public static float nextDown(float f) {
2158         if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2159             return f;
2160         else {
2161             if (f == 0.0f)
2162                 return -Float.MIN_VALUE;
2163             else
2164                 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2165                                             ((f > 0.0f)?-1:+1));
2166         }
2167     }
2168 
2169     /**
2170      * Returns {@code d} &times;
2171      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2172      * by a single correctly rounded floating-point multiply to a
2173      * member of the double value set.  See the Java
2174      * Language Specification for a discussion of floating-point
2175      * value sets.  If the exponent of the result is between {@link
2176      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2177      * answer is calculated exactly.  If the exponent of the result
2178      * would be larger than {@code Double.MAX_EXPONENT}, an
2179      * infinity is returned.  Note that if the result is subnormal,
2180      * precision may be lost; that is, when {@code scalb(x, n)}
2181      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2182      * <i>x</i>.  When the result is non-NaN, the result has the same
2183      * sign as {@code d}.
2184      *
2185      * <p>Special cases:
2186      * <ul>
2187      * <li> If the first argument is NaN, NaN is returned.
2188      * <li> If the first argument is infinite, then an infinity of the
2189      * same sign is returned.
2190      * <li> If the first argument is zero, then a zero of the same
2191      * sign is returned.
2192      * </ul>
2193      *
2194      * @param d number to be scaled by a power of two.
2195      * @param scaleFactor power of 2 used to scale {@code d}
2196      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
2197      * @since 1.6
2198      */
2199     public static double scalb(double d, int scaleFactor) {
2200         /*
2201          * This method does not need to be declared strictfp to
2202          * compute the same correct result on all platforms.  When
2203          * scaling up, it does not matter what order the
2204          * multiply-store operations are done; the result will be
2205          * finite or overflow regardless of the operation ordering.
2206          * However, to get the correct result when scaling down, a
2207          * particular ordering must be used.
2208          *
2209          * When scaling down, the multiply-store operations are
2210          * sequenced so that it is not possible for two consecutive
2211          * multiply-stores to return subnormal results.  If one
2212          * multiply-store result is subnormal, the next multiply will
2213          * round it away to zero.  This is done by first multiplying
2214          * by 2 ^ (scaleFactor % n) and then multiplying several
2215          * times by by 2^n as needed where n is the exponent of number
2216          * that is a covenient power of two.  In this way, at most one
2217          * real rounding error occurs.  If the double value set is
2218          * being used exclusively, the rounding will occur on a
2219          * multiply.  If the double-extended-exponent value set is
2220          * being used, the products will (perhaps) be exact but the
2221          * stores to d are guaranteed to round to the double value
2222          * set.
2223          *
2224          * It is _not_ a valid implementation to first multiply d by
2225          * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2226          * MIN_EXPONENT) since even in a strictfp program double
2227          * rounding on underflow could occur; e.g. if the scaleFactor
2228          * argument was (MIN_EXPONENT - n) and the exponent of d was a
2229          * little less than -(MIN_EXPONENT - n), meaning the final
2230          * result would be subnormal.
2231          *
2232          * Since exact reproducibility of this method can be achieved
2233          * without any undue performance burden, there is no
2234          * compelling reason to allow double rounding on underflow in
2235          * scalb.
2236          */
2237 
2238         // magnitude of a power of two so large that scaling a finite
2239         // nonzero value by it would be guaranteed to over or
2240         // underflow; due to rounding, scaling down takes takes an
2241         // additional power of two which is reflected here
2242         final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.MIN_EXPONENT +
2243                               DoubleConsts.SIGNIFICAND_WIDTH + 1;
2244         int exp_adjust = 0;
2245         int scale_increment = 0;
2246         double exp_delta = Double.NaN;
2247 
2248         // Make sure scaling factor is in a reasonable range
2249 
2250         if(scaleFactor < 0) {
2251             scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2252             scale_increment = -512;
2253             exp_delta = twoToTheDoubleScaleDown;
2254         }
2255         else {
2256             scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2257             scale_increment = 512;
2258             exp_delta = twoToTheDoubleScaleUp;
2259         }
2260 
2261         // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2262         // technique from "Hacker's Delight" section 10-2.
2263         int t = (scaleFactor >> 9-1) >>> 32 - 9;
2264         exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2265 
2266         d *= powerOfTwoD(exp_adjust);
2267         scaleFactor -= exp_adjust;
2268 
2269         while(scaleFactor != 0) {
2270             d *= exp_delta;
2271             scaleFactor -= scale_increment;
2272         }
2273         return d;
2274     }
2275 
2276     /**
2277      * Returns {@code f} &times;
2278      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2279      * by a single correctly rounded floating-point multiply to a
2280      * member of the float value set.  See the Java
2281      * Language Specification for a discussion of floating-point
2282      * value sets.  If the exponent of the result is between {@link
2283      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2284      * answer is calculated exactly.  If the exponent of the result
2285      * would be larger than {@code Float.MAX_EXPONENT}, an
2286      * infinity is returned.  Note that if the result is subnormal,
2287      * precision may be lost; that is, when {@code scalb(x, n)}
2288      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2289      * <i>x</i>.  When the result is non-NaN, the result has the same
2290      * sign as {@code f}.
2291      *
2292      * <p>Special cases:
2293      * <ul>
2294      * <li> If the first argument is NaN, NaN is returned.
2295      * <li> If the first argument is infinite, then an infinity of the
2296      * same sign is returned.
2297      * <li> If the first argument is zero, then a zero of the same
2298      * sign is returned.
2299      * </ul>
2300      *
2301      * @param f number to be scaled by a power of two.
2302      * @param scaleFactor power of 2 used to scale {@code f}
2303      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
2304      * @since 1.6
2305      */
2306     public static float scalb(float f, int scaleFactor) {
2307         // magnitude of a power of two so large that scaling a finite
2308         // nonzero value by it would be guaranteed to over or
2309         // underflow; due to rounding, scaling down takes takes an
2310         // additional power of two which is reflected here
2311         final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.MIN_EXPONENT +
2312                               FloatConsts.SIGNIFICAND_WIDTH + 1;
2313 
2314         // Make sure scaling factor is in a reasonable range
2315         scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2316 
2317         /*
2318          * Since + MAX_SCALE for float fits well within the double
2319          * exponent range and + float -> double conversion is exact
2320          * the multiplication below will be exact. Therefore, the
2321          * rounding that occurs when the double product is cast to
2322          * float will be the correctly rounded float result.  Since
2323          * all operations other than the final multiply will be exact,
2324          * it is not necessary to declare this method strictfp.
2325          */
2326         return (float)((double)f*powerOfTwoD(scaleFactor));
2327     }
2328 
2329     // Constants used in scalb
2330     static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2331     static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2332 
2333     /**
2334      * Returns a floating-point power of two in the normal range.
2335      */
2336     static double powerOfTwoD(int n) {
2337         assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.MAX_EXPONENT);
2338         return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2339                                         (DoubleConsts.SIGNIFICAND_WIDTH-1))
2340                                        & DoubleConsts.EXP_BIT_MASK);
2341     }
2342 
2343     /**
2344      * Returns a floating-point power of two in the normal range.
2345      */
2346     static float powerOfTwoF(int n) {
2347         assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
2348         return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2349                                      (FloatConsts.SIGNIFICAND_WIDTH-1))
2350                                     & FloatConsts.EXP_BIT_MASK);
2351     }
2352 }