1 /*
   2  * Copyright (c) 1994, 2016, Oracle and/or its affiliates. All rights reserved.
   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
   6  * under the terms of the GNU General Public License version 2 only, as
   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
  16  *
  17  * You should have received a copy of the GNU General Public License version
  18  * 2 along with this work; if not, write to the Free Software Foundation,
  19  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  20  *
  21  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  22  * or visit www.oracle.com if you need additional information or have any
  23  * questions.
  24  */
  25 
  26 package java.lang;
  27 
  28 import java.math.BigDecimal;
  29 import java.util.Random;
  30 import jdk.internal.math.FloatConsts;
  31 import jdk.internal.math.DoubleConsts;
  32 import jdk.internal.HotSpotIntrinsicCandidate;
  33 
  34 /**
  35  * The class {@code Math} contains methods for performing basic
  36  * numeric operations such as the elementary exponential, logarithm,
  37  * square root, and trigonometric functions.
  38  *
  39  * <p>Unlike some of the numeric methods of class
  40  * {@code StrictMath}, all implementations of the equivalent
  41  * functions of class {@code Math} are not defined to return the
  42  * bit-for-bit same results.  This relaxation permits
  43  * better-performing implementations where strict reproducibility is
  44  * not required.
  45  *
  46  * <p>By default many of the {@code Math} methods simply call
  47  * the equivalent method in {@code StrictMath} for their
  48  * implementation.  Code generators are encouraged to use
  49  * platform-specific native libraries or microprocessor instructions,
  50  * where available, to provide higher-performance implementations of
  51  * {@code Math} methods.  Such higher-performance
  52  * implementations still must conform to the specification for
  53  * {@code Math}.
  54  *
  55  * <p>The quality of implementation specifications concern two
  56  * properties, accuracy of the returned result and monotonicity of the
  57  * method.  Accuracy of the floating-point {@code Math} methods is
  58  * measured in terms of <i>ulps</i>, units in the last place.  For a
  59  * given floating-point format, an {@linkplain #ulp(double) ulp} of a
  60  * specific real number value is the distance between the two
  61  * floating-point values bracketing that numerical value.  When
  62  * discussing the accuracy of a method as a whole rather than at a
  63  * specific argument, the number of ulps cited is for the worst-case
  64  * error at any argument.  If a method always has an error less than
  65  * 0.5 ulps, the method always returns the floating-point number
  66  * nearest the exact result; such a method is <i>correctly
  67  * rounded</i>.  A correctly rounded method is generally the best a
  68  * floating-point approximation can be; however, it is impractical for
  69  * many floating-point methods to be correctly rounded.  Instead, for
  70  * the {@code Math} class, a larger error bound of 1 or 2 ulps is
  71  * allowed for certain methods.  Informally, with a 1 ulp error bound,
  72  * when the exact result is a representable number, the exact result
  73  * should be returned as the computed result; otherwise, either of the
  74  * two floating-point values which bracket the exact result may be
  75  * returned.  For exact results large in magnitude, one of the
  76  * endpoints of the bracket may be infinite.  Besides accuracy at
  77  * individual arguments, maintaining proper relations between the
  78  * method at different arguments is also important.  Therefore, most
  79  * methods with more than 0.5 ulp errors are required to be
  80  * <i>semi-monotonic</i>: whenever the mathematical function is
  81  * non-decreasing, so is the floating-point approximation, likewise,
  82  * whenever the mathematical function is non-increasing, so is the
  83  * floating-point approximation.  Not all approximations that have 1
  84  * ulp accuracy will automatically meet the monotonicity requirements.
  85  *
  86  * <p>
  87  * The platform uses signed two's complement integer arithmetic with
  88  * int and long primitive types.  The developer should choose
  89  * the primitive type to ensure that arithmetic operations consistently
  90  * produce correct results, which in some cases means the operations
  91  * will not overflow the range of values of the computation.
  92  * The best practice is to choose the primitive type and algorithm to avoid
  93  * overflow. In cases where the size is {@code int} or {@code long} and
  94  * overflow errors need to be detected, the methods {@code addExact},
  95  * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
  96  * throw an {@code ArithmeticException} when the results overflow.
  97  * For other arithmetic operations such as divide, absolute value,
  98  * increment, decrement, and negation overflow occurs only with
  99  * a specific minimum or maximum value and should be checked against
 100  * the minimum or maximum as appropriate.
 101  *
 102  * @author  unascribed
 103  * @author  Joseph D. Darcy
 104  * @since   1.0
 105  */
 106 
 107 public final class Math {
 108 
 109     /**
 110      * Don't let anyone instantiate this class.
 111      */
 112     private Math() {}
 113 
 114     /**
 115      * The {@code double} value that is closer than any other to
 116      * <i>e</i>, the base of the natural logarithms.
 117      */
 118     public static final double E = 2.7182818284590452354;
 119 
 120     /**
 121      * The {@code double} value that is closer than any other to
 122      * <i>pi</i>, the ratio of the circumference of a circle to its
 123      * diameter.
 124      */
 125     public static final double PI = 3.14159265358979323846;
 126 
 127     /**
 128      * Constant by which to multiply an angular value in degrees to obtain an
 129      * angular value in radians.
 130      */
 131     private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
 132 
 133     /**
 134      * Constant by which to multiply an angular value in radians to obtain an
 135      * angular value in degrees.
 136      */
 137     private static final double RADIANS_TO_DEGREES = 57.29577951308232;
 138 
 139     /**
 140      * Returns the trigonometric sine of an angle.  Special cases:
 141      * <ul><li>If the argument is NaN or an infinity, then the
 142      * result is NaN.
 143      * <li>If the argument is zero, then the result is a zero with the
 144      * same sign as the argument.</ul>
 145      *
 146      * <p>The computed result must be within 1 ulp of the exact result.
 147      * Results must be semi-monotonic.
 148      *
 149      * @param   a   an angle, in radians.
 150      * @return  the sine of the argument.
 151      */
 152     @HotSpotIntrinsicCandidate
 153     public static double sin(double a) {
 154         return StrictMath.sin(a); // default impl. delegates to StrictMath
 155     }
 156 
 157     /**
 158      * Returns the trigonometric cosine of an angle. Special cases:
 159      * <ul><li>If the argument is NaN or an infinity, then the
 160      * result is NaN.</ul>
 161      *
 162      * <p>The computed result must be within 1 ulp of the exact result.
 163      * Results must be semi-monotonic.
 164      *
 165      * @param   a   an angle, in radians.
 166      * @return  the cosine of the argument.
 167      */
 168     @HotSpotIntrinsicCandidate
 169     public static double cos(double a) {
 170         return StrictMath.cos(a); // default impl. delegates to StrictMath
 171     }
 172 
 173     /**
 174      * Returns the trigonometric tangent of an angle.  Special cases:
 175      * <ul><li>If the argument is NaN or an infinity, then the result
 176      * is NaN.
 177      * <li>If the argument is zero, then the result is a zero with the
 178      * same sign as the argument.</ul>
 179      *
 180      * <p>The computed result must be within 1 ulp of the exact result.
 181      * Results must be semi-monotonic.
 182      *
 183      * @param   a   an angle, in radians.
 184      * @return  the tangent of the argument.
 185      */
 186     @HotSpotIntrinsicCandidate
 187     public static double tan(double a) {
 188         return StrictMath.tan(a); // default impl. delegates to StrictMath
 189     }
 190 
 191     /**
 192      * Returns the arc sine of a value; the returned angle is in the
 193      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 194      * <ul><li>If the argument is NaN or its absolute value is greater
 195      * than 1, then the result is NaN.
 196      * <li>If the argument is zero, then the result is a zero with the
 197      * same sign as the argument.</ul>
 198      *
 199      * <p>The computed result must be within 1 ulp of the exact result.
 200      * Results must be semi-monotonic.
 201      *
 202      * @param   a   the value whose arc sine is to be returned.
 203      * @return  the arc sine of the argument.
 204      */
 205     public static double asin(double a) {
 206         return StrictMath.asin(a); // default impl. delegates to StrictMath
 207     }
 208 
 209     /**
 210      * Returns the arc cosine of a value; the returned angle is in the
 211      * range 0.0 through <i>pi</i>.  Special case:
 212      * <ul><li>If the argument is NaN or its absolute value is greater
 213      * than 1, then the result is NaN.</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 cosine is to be returned.
 219      * @return  the arc cosine of the argument.
 220      */
 221     public static double acos(double a) {
 222         return StrictMath.acos(a); // default impl. delegates to StrictMath
 223     }
 224 
 225     /**
 226      * Returns the arc tangent of a value; the returned angle is in the
 227      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 228      * <ul><li>If the argument is NaN, then the result is NaN.
 229      * <li>If the argument is zero, then the result is a zero with the
 230      * same sign as the argument.</ul>
 231      *
 232      * <p>The computed result must be within 1 ulp of the exact result.
 233      * Results must be semi-monotonic.
 234      *
 235      * @param   a   the value whose arc tangent is to be returned.
 236      * @return  the arc tangent of the argument.
 237      */
 238     public static double atan(double a) {
 239         return StrictMath.atan(a); // default impl. delegates to StrictMath
 240     }
 241 
 242     /**
 243      * Converts an angle measured in degrees to an approximately
 244      * equivalent angle measured in radians.  The conversion from
 245      * degrees to radians is generally inexact.
 246      *
 247      * @param   angdeg   an angle, in degrees
 248      * @return  the measurement of the angle {@code angdeg}
 249      *          in radians.
 250      * @since   1.2
 251      */
 252     public static double toRadians(double angdeg) {
 253         return angdeg * DEGREES_TO_RADIANS;
 254     }
 255 
 256     /**
 257      * Converts an angle measured in radians to an approximately
 258      * equivalent angle measured in degrees.  The conversion from
 259      * radians to degrees is generally inexact; users should
 260      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
 261      * equal {@code 0.0}.
 262      *
 263      * @param   angrad   an angle, in radians
 264      * @return  the measurement of the angle {@code angrad}
 265      *          in degrees.
 266      * @since   1.2
 267      */
 268     public static double toDegrees(double angrad) {
 269         return angrad * RADIANS_TO_DEGREES;
 270     }
 271 
 272     /**
 273      * Returns Euler's number <i>e</i> raised to the power of a
 274      * {@code double} value.  Special cases:
 275      * <ul><li>If the argument is NaN, the result is NaN.
 276      * <li>If the argument is positive infinity, then the result is
 277      * positive infinity.
 278      * <li>If the argument is negative infinity, then the result is
 279      * positive zero.</ul>
 280      *
 281      * <p>The computed result must be within 1 ulp of the exact result.
 282      * Results must be semi-monotonic.
 283      *
 284      * @param   a   the exponent to raise <i>e</i> to.
 285      * @return  the value <i>e</i><sup>{@code a}</sup>,
 286      *          where <i>e</i> is the base of the natural logarithms.
 287      */
 288     @HotSpotIntrinsicCandidate
 289     public static double exp(double a) {
 290         return StrictMath.exp(a); // default impl. delegates to StrictMath
 291     }
 292 
 293     /**
 294      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
 295      * value.  Special cases:
 296      * <ul><li>If the argument is NaN or less than zero, then the result
 297      * is NaN.
 298      * <li>If the argument is positive infinity, then the result is
 299      * positive infinity.
 300      * <li>If the argument is positive zero or negative zero, then the
 301      * result is negative infinity.</ul>
 302      *
 303      * <p>The computed result must be within 1 ulp of the exact result.
 304      * Results must be semi-monotonic.
 305      *
 306      * @param   a   a value
 307      * @return  the value ln&nbsp;{@code a}, the natural logarithm of
 308      *          {@code a}.
 309      */
 310     @HotSpotIntrinsicCandidate
 311     public static double log(double a) {
 312         return StrictMath.log(a); // default impl. delegates to StrictMath
 313     }
 314 
 315     /**
 316      * Returns the base 10 logarithm of a {@code double} value.
 317      * Special cases:
 318      *
 319      * <ul><li>If the argument is NaN or less than zero, then the result
 320      * is NaN.
 321      * <li>If the argument is positive infinity, then the result is
 322      * positive infinity.
 323      * <li>If the argument is positive zero or negative zero, then the
 324      * result is negative infinity.
 325      * <li> If the argument is equal to 10<sup><i>n</i></sup> for
 326      * integer <i>n</i>, then the result is <i>n</i>.
 327      * </ul>
 328      *
 329      * <p>The computed result must be within 1 ulp of the exact result.
 330      * Results must be semi-monotonic.
 331      *
 332      * @param   a   a value
 333      * @return  the base 10 logarithm of  {@code a}.
 334      * @since 1.5
 335      */
 336     @HotSpotIntrinsicCandidate
 337     public static double log10(double a) {
 338         return StrictMath.log10(a); // default impl. delegates to StrictMath
 339     }
 340 
 341     /**
 342      * Returns the correctly rounded positive square root of a
 343      * {@code double} value.
 344      * Special cases:
 345      * <ul><li>If the argument is NaN or less than zero, then the result
 346      * is NaN.
 347      * <li>If the argument is positive infinity, then the result is positive
 348      * infinity.
 349      * <li>If the argument is positive zero or negative zero, then the
 350      * result is the same as the argument.</ul>
 351      * Otherwise, the result is the {@code double} value closest to
 352      * the true mathematical square root of the argument value.
 353      *
 354      * @param   a   a value.
 355      * @return  the positive square root of {@code a}.
 356      *          If the argument is NaN or less than zero, the result is NaN.
 357      */
 358     @HotSpotIntrinsicCandidate
 359     public static double sqrt(double a) {
 360         return StrictMath.sqrt(a); // default impl. delegates to StrictMath
 361                                    // Note that hardware sqrt instructions
 362                                    // frequently can be directly used by JITs
 363                                    // and should be much faster than doing
 364                                    // Math.sqrt in software.
 365     }
 366 
 367 
 368     /**
 369      * Returns the cube root of a {@code double} value.  For
 370      * positive finite {@code x}, {@code cbrt(-x) ==
 371      * -cbrt(x)}; that is, the cube root of a negative value is
 372      * the negative of the cube root of that value's magnitude.
 373      *
 374      * Special cases:
 375      *
 376      * <ul>
 377      *
 378      * <li>If the argument is NaN, then the result is NaN.
 379      *
 380      * <li>If the argument is infinite, then the result is an infinity
 381      * with the same sign as the argument.
 382      *
 383      * <li>If the argument is zero, then the result is a zero with the
 384      * same sign as the argument.
 385      *
 386      * </ul>
 387      *
 388      * <p>The computed result must be within 1 ulp of the exact result.
 389      *
 390      * @param   a   a value.
 391      * @return  the cube root of {@code a}.
 392      * @since 1.5
 393      */
 394     public static double cbrt(double a) {
 395         return StrictMath.cbrt(a);
 396     }
 397 
 398     /**
 399      * Computes the remainder operation on two arguments as prescribed
 400      * by the IEEE 754 standard.
 401      * The remainder value is mathematically equal to
 402      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
 403      * where <i>n</i> is the mathematical integer closest to the exact
 404      * mathematical value of the quotient {@code f1/f2}, and if two
 405      * mathematical integers are equally close to {@code f1/f2},
 406      * then <i>n</i> is the integer that is even. If the remainder is
 407      * zero, its sign is the same as the sign of the first argument.
 408      * Special cases:
 409      * <ul><li>If either argument is NaN, or the first argument is infinite,
 410      * or the second argument is positive zero or negative zero, then the
 411      * result is NaN.
 412      * <li>If the first argument is finite and the second argument is
 413      * infinite, then the result is the same as the first argument.</ul>
 414      *
 415      * @param   f1   the dividend.
 416      * @param   f2   the divisor.
 417      * @return  the remainder when {@code f1} is divided by
 418      *          {@code f2}.
 419      */
 420     public static double IEEEremainder(double f1, double f2) {
 421         return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
 422     }
 423 
 424     /**
 425      * Returns the smallest (closest to negative infinity)
 426      * {@code double} value that is greater than or equal to the
 427      * argument and is equal to a mathematical integer. Special cases:
 428      * <ul><li>If the argument value is already equal to a
 429      * mathematical integer, then the result is the same as the
 430      * argument.  <li>If the argument is NaN or an infinity or
 431      * positive zero or negative zero, then the result is the same as
 432      * the argument.  <li>If the argument value is less than zero but
 433      * greater than -1.0, then the result is negative zero.</ul> Note
 434      * that the value of {@code Math.ceil(x)} is exactly the
 435      * value of {@code -Math.floor(-x)}.
 436      *
 437      *
 438      * @param   a   a value.
 439      * @return  the smallest (closest to negative infinity)
 440      *          floating-point value that is greater than or equal to
 441      *          the argument and is equal to a mathematical integer.
 442      */
 443     public static double ceil(double a) {
 444         return StrictMath.ceil(a); // default impl. delegates to StrictMath
 445     }
 446 
 447     /**
 448      * Returns the largest (closest to positive infinity)
 449      * {@code double} value that is less than or equal to the
 450      * argument and is equal to a mathematical integer. Special cases:
 451      * <ul><li>If the argument value is already equal to a
 452      * mathematical integer, then the result is the same as the
 453      * argument.  <li>If the argument is NaN or an infinity or
 454      * positive zero or negative zero, then the result is the same as
 455      * the argument.</ul>
 456      *
 457      * @param   a   a value.
 458      * @return  the largest (closest to positive infinity)
 459      *          floating-point value that less than or equal to the argument
 460      *          and is equal to a mathematical integer.
 461      */
 462     public static double floor(double a) {
 463         return StrictMath.floor(a); // default impl. delegates to StrictMath
 464     }
 465 
 466     /**
 467      * Returns the {@code double} value that is closest in value
 468      * to the argument and is equal to a mathematical integer. If two
 469      * {@code double} values that are mathematical integers are
 470      * equally close, the result is the integer value that is
 471      * even. Special cases:
 472      * <ul><li>If the argument value is already equal to a mathematical
 473      * integer, then the result is the same as the argument.
 474      * <li>If the argument is NaN or an infinity or positive zero or negative
 475      * zero, then the result is the same as the argument.</ul>
 476      *
 477      * @param   a   a {@code double} value.
 478      * @return  the closest floating-point value to {@code a} that is
 479      *          equal to a mathematical integer.
 480      */
 481     public static double rint(double a) {
 482         return StrictMath.rint(a); // default impl. delegates to StrictMath
 483     }
 484 
 485     /**
 486      * Returns the angle <i>theta</i> from the conversion of rectangular
 487      * coordinates ({@code x},&nbsp;{@code y}) to polar
 488      * coordinates (r,&nbsp;<i>theta</i>).
 489      * This method computes the phase <i>theta</i> by computing an arc tangent
 490      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
 491      * cases:
 492      * <ul><li>If either argument is NaN, then the result is NaN.
 493      * <li>If the first argument is positive zero and the second argument
 494      * is positive, or the first argument is positive and finite and the
 495      * second argument is positive infinity, then the result is positive
 496      * zero.
 497      * <li>If the first argument is negative zero and the second argument
 498      * is positive, or the first argument is negative and finite and the
 499      * second argument is positive infinity, then the result is negative zero.
 500      * <li>If the first argument is positive zero and the second argument
 501      * is negative, or the first argument is positive and finite and the
 502      * second argument is negative infinity, then the result is the
 503      * {@code double} value closest to <i>pi</i>.
 504      * <li>If the first argument is negative zero and the second argument
 505      * is negative, or the first argument is negative and finite and the
 506      * second argument is negative infinity, then the result is the
 507      * {@code double} value closest to -<i>pi</i>.
 508      * <li>If the first argument is positive and the second argument is
 509      * positive zero or negative zero, or the first argument is positive
 510      * infinity and the second argument is finite, then the result is the
 511      * {@code double} value closest to <i>pi</i>/2.
 512      * <li>If the first argument is negative and the second argument is
 513      * positive zero or negative zero, or the first argument is negative
 514      * infinity and the second argument is finite, then the result is the
 515      * {@code double} value closest to -<i>pi</i>/2.
 516      * <li>If both arguments are positive infinity, then the result is the
 517      * {@code double} value closest to <i>pi</i>/4.
 518      * <li>If the first argument is positive infinity and the second argument
 519      * is negative infinity, then the result is the {@code double}
 520      * value closest to 3*<i>pi</i>/4.
 521      * <li>If the first argument is negative infinity and the second argument
 522      * is positive infinity, then the result is the {@code double} value
 523      * closest to -<i>pi</i>/4.
 524      * <li>If both arguments are negative infinity, then the result is the
 525      * {@code double} value closest to -3*<i>pi</i>/4.</ul>
 526      *
 527      * <p>The computed result must be within 2 ulps of the exact result.
 528      * Results must be semi-monotonic.
 529      *
 530      * @param   y   the ordinate coordinate
 531      * @param   x   the abscissa coordinate
 532      * @return  the <i>theta</i> component of the point
 533      *          (<i>r</i>,&nbsp;<i>theta</i>)
 534      *          in polar coordinates that corresponds to the point
 535      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
 536      */
 537     @HotSpotIntrinsicCandidate
 538     public static double atan2(double y, double x) {
 539         return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
 540     }
 541 
 542     /**
 543      * Returns the value of the first argument raised to the power of the
 544      * second argument. Special cases:
 545      *
 546      * <ul><li>If the second argument is positive or negative zero, then the
 547      * result is 1.0.
 548      * <li>If the second argument is 1.0, then the result is the same as the
 549      * first argument.
 550      * <li>If the second argument is NaN, then the result is NaN.
 551      * <li>If the first argument is NaN and the second argument is nonzero,
 552      * then the result is NaN.
 553      *
 554      * <li>If
 555      * <ul>
 556      * <li>the absolute value of the first argument is greater than 1
 557      * and the second argument is positive infinity, or
 558      * <li>the absolute value of the first argument is less than 1 and
 559      * the second argument is negative infinity,
 560      * </ul>
 561      * then the result is positive infinity.
 562      *
 563      * <li>If
 564      * <ul>
 565      * <li>the absolute value of the first argument is greater than 1 and
 566      * the second argument is negative infinity, or
 567      * <li>the absolute value of the
 568      * first argument is less than 1 and the second argument is positive
 569      * infinity,
 570      * </ul>
 571      * then the result is positive zero.
 572      *
 573      * <li>If the absolute value of the first argument equals 1 and the
 574      * second argument is infinite, then the result is NaN.
 575      *
 576      * <li>If
 577      * <ul>
 578      * <li>the first argument is positive zero and the second argument
 579      * is greater than zero, or
 580      * <li>the first argument is positive infinity and the second
 581      * argument is less than zero,
 582      * </ul>
 583      * then the result is positive zero.
 584      *
 585      * <li>If
 586      * <ul>
 587      * <li>the first argument is positive zero and the second argument
 588      * is less than zero, or
 589      * <li>the first argument is positive infinity and the second
 590      * argument is greater than zero,
 591      * </ul>
 592      * then the result is positive infinity.
 593      *
 594      * <li>If
 595      * <ul>
 596      * <li>the first argument is negative zero and the second argument
 597      * is greater than zero but not a finite odd integer, or
 598      * <li>the first argument is negative infinity and the second
 599      * argument is less than zero but not a finite odd integer,
 600      * </ul>
 601      * then the result is positive zero.
 602      *
 603      * <li>If
 604      * <ul>
 605      * <li>the first argument is negative zero and the second argument
 606      * is a positive finite odd integer, or
 607      * <li>the first argument is negative infinity and the second
 608      * argument is a negative finite odd integer,
 609      * </ul>
 610      * then the result is negative zero.
 611      *
 612      * <li>If
 613      * <ul>
 614      * <li>the first argument is negative zero and the second argument
 615      * is less than zero but not a finite odd integer, or
 616      * <li>the first argument is negative infinity and the second
 617      * argument is greater than zero but not a finite odd integer,
 618      * </ul>
 619      * then the result is positive infinity.
 620      *
 621      * <li>If
 622      * <ul>
 623      * <li>the first argument is negative zero and the second argument
 624      * is a negative finite odd integer, or
 625      * <li>the first argument is negative infinity and the second
 626      * argument is a positive finite odd integer,
 627      * </ul>
 628      * then the result is negative infinity.
 629      *
 630      * <li>If the first argument is finite and less than zero
 631      * <ul>
 632      * <li> if the second argument is a finite even integer, the
 633      * result is equal to the result of raising the absolute value of
 634      * the first argument to the power of the second argument
 635      *
 636      * <li>if the second argument is a finite odd integer, the result
 637      * is equal to the negative of the result of raising the absolute
 638      * value of the first argument to the power of the second
 639      * argument
 640      *
 641      * <li>if the second argument is finite and not an integer, then
 642      * the result is NaN.
 643      * </ul>
 644      *
 645      * <li>If both arguments are integers, then the result is exactly equal
 646      * to the mathematical result of raising the first argument to the power
 647      * of the second argument if that result can in fact be represented
 648      * exactly as a {@code double} value.</ul>
 649      *
 650      * <p>(In the foregoing descriptions, a floating-point value is
 651      * considered to be an integer if and only if it is finite and a
 652      * fixed point of the method {@link #ceil ceil} or,
 653      * equivalently, a fixed point of the method {@link #floor
 654      * floor}. A value is a fixed point of a one-argument
 655      * method if and only if the result of applying the method to the
 656      * value is equal to the value.)
 657      *
 658      * <p>The computed result must be within 1 ulp of the exact result.
 659      * Results must be semi-monotonic.
 660      *
 661      * @param   a   the base.
 662      * @param   b   the exponent.
 663      * @return  the value {@code a}<sup>{@code b}</sup>.
 664      */
 665     @HotSpotIntrinsicCandidate
 666     public static double pow(double a, double b) {
 667         return StrictMath.pow(a, b); // default impl. delegates to StrictMath
 668     }
 669 
 670     /**
 671      * Returns the closest {@code int} to the argument, with ties
 672      * rounding to positive infinity.
 673      *
 674      * <p>
 675      * Special cases:
 676      * <ul><li>If the argument is NaN, the result is 0.
 677      * <li>If the argument is negative infinity or any value less than or
 678      * equal to the value of {@code Integer.MIN_VALUE}, the result is
 679      * equal to the value of {@code Integer.MIN_VALUE}.
 680      * <li>If the argument is positive infinity or any value greater than or
 681      * equal to the value of {@code Integer.MAX_VALUE}, the result is
 682      * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 683      *
 684      * @param   a   a floating-point value to be rounded to an integer.
 685      * @return  the value of the argument rounded to the nearest
 686      *          {@code int} value.
 687      * @see     java.lang.Integer#MAX_VALUE
 688      * @see     java.lang.Integer#MIN_VALUE
 689      */
 690     public static int round(float a) {
 691         int intBits = Float.floatToRawIntBits(a);
 692         int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
 693                 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
 694         int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
 695                 + FloatConsts.EXP_BIAS) - biasedExp;
 696         if ((shift & -32) == 0) { // shift >= 0 && shift < 32
 697             // a is a finite number such that pow(2,-32) <= ulp(a) < 1
 698             int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
 699                     | (FloatConsts.SIGNIF_BIT_MASK + 1));
 700             if (intBits < 0) {
 701                 r = -r;
 702             }
 703             // In the comments below each Java expression evaluates to the value
 704             // the corresponding mathematical expression:
 705             // (r) evaluates to a / ulp(a)
 706             // (r >> shift) evaluates to floor(a * 2)
 707             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 708             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 709             return ((r >> shift) + 1) >> 1;
 710         } else {
 711             // a is either
 712             // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
 713             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 714             // - an infinity or NaN
 715             return (int) a;
 716         }
 717     }
 718 
 719     /**
 720      * Returns the closest {@code long} to the argument, with ties
 721      * rounding to positive infinity.
 722      *
 723      * <p>Special cases:
 724      * <ul><li>If the argument is NaN, the result is 0.
 725      * <li>If the argument is negative infinity or any value less than or
 726      * equal to the value of {@code Long.MIN_VALUE}, the result is
 727      * equal to the value of {@code Long.MIN_VALUE}.
 728      * <li>If the argument is positive infinity or any value greater than or
 729      * equal to the value of {@code Long.MAX_VALUE}, the result is
 730      * equal to the value of {@code Long.MAX_VALUE}.</ul>
 731      *
 732      * @param   a   a floating-point value to be rounded to a
 733      *          {@code long}.
 734      * @return  the value of the argument rounded to the nearest
 735      *          {@code long} value.
 736      * @see     java.lang.Long#MAX_VALUE
 737      * @see     java.lang.Long#MIN_VALUE
 738      */
 739     public static long round(double a) {
 740         long longBits = Double.doubleToRawLongBits(a);
 741         long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
 742                 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
 743         long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
 744                 + DoubleConsts.EXP_BIAS) - biasedExp;
 745         if ((shift & -64) == 0) { // shift >= 0 && shift < 64
 746             // a is a finite number such that pow(2,-64) <= ulp(a) < 1
 747             long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
 748                     | (DoubleConsts.SIGNIF_BIT_MASK + 1));
 749             if (longBits < 0) {
 750                 r = -r;
 751             }
 752             // In the comments below each Java expression evaluates to the value
 753             // the corresponding mathematical expression:
 754             // (r) evaluates to a / ulp(a)
 755             // (r >> shift) evaluates to floor(a * 2)
 756             // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 757             // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 758             return ((r >> shift) + 1) >> 1;
 759         } else {
 760             // a is either
 761             // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
 762             // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 763             // - an infinity or NaN
 764             return (long) a;
 765         }
 766     }
 767 
 768     private static final class RandomNumberGeneratorHolder {
 769         static final Random randomNumberGenerator = new Random();
 770     }
 771 
 772     /**
 773      * Returns a {@code double} value with a positive sign, greater
 774      * than or equal to {@code 0.0} and less than {@code 1.0}.
 775      * Returned values are chosen pseudorandomly with (approximately)
 776      * uniform distribution from that range.
 777      *
 778      * <p>When this method is first called, it creates a single new
 779      * pseudorandom-number generator, exactly as if by the expression
 780      *
 781      * <blockquote>{@code new java.util.Random()}</blockquote>
 782      *
 783      * This new pseudorandom-number generator is used thereafter for
 784      * all calls to this method and is used nowhere else.
 785      *
 786      * <p>This method is properly synchronized to allow correct use by
 787      * more than one thread. However, if many threads need to generate
 788      * pseudorandom numbers at a great rate, it may reduce contention
 789      * for each thread to have its own pseudorandom-number generator.
 790      *
 791      * @apiNote
 792      * As the largest {@code double} value less than {@code 1.0}
 793      * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
 794      * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
 795      *
 796      * <blockquote><pre>{@code
 797      * double f = Math.random()/Math.nextDown(1.0);
 798      * double x = x1*(1.0 - f) + x2*f;
 799      * }</pre></blockquote>
 800      *
 801      * @return  a pseudorandom {@code double} greater than or equal
 802      * to {@code 0.0} and less than {@code 1.0}.
 803      * @see #nextDown(double)
 804      * @see Random#nextDouble()
 805      */
 806     public static double random() {
 807         return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 808     }
 809 
 810     /**
 811      * Returns the sum of its arguments,
 812      * throwing an exception if the result overflows an {@code int}.
 813      *
 814      * @param x the first value
 815      * @param y the second value
 816      * @return the result
 817      * @throws ArithmeticException if the result overflows an int
 818      * @since 1.8
 819      */
 820     @HotSpotIntrinsicCandidate
 821     public static int addExact(int x, int y) {
 822         int r = x + y;
 823         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 824         if (((x ^ r) & (y ^ r)) < 0) {
 825             throw new ArithmeticException("integer overflow");
 826         }
 827         return r;
 828     }
 829 
 830     /**
 831      * Returns the sum of its arguments,
 832      * throwing an exception if the result overflows a {@code long}.
 833      *
 834      * @param x the first value
 835      * @param y the second value
 836      * @return the result
 837      * @throws ArithmeticException if the result overflows a long
 838      * @since 1.8
 839      */
 840     @HotSpotIntrinsicCandidate
 841     public static long addExact(long x, long y) {
 842         long r = x + y;
 843         // HD 2-12 Overflow iff both arguments have the opposite sign of the result
 844         if (((x ^ r) & (y ^ r)) < 0) {
 845             throw new ArithmeticException("long overflow");
 846         }
 847         return r;
 848     }
 849 
 850     /**
 851      * Returns the difference of the arguments,
 852      * throwing an exception if the result overflows an {@code int}.
 853      *
 854      * @param x the first value
 855      * @param y the second value to subtract from the first
 856      * @return the result
 857      * @throws ArithmeticException if the result overflows an int
 858      * @since 1.8
 859      */
 860     @HotSpotIntrinsicCandidate
 861     public static int subtractExact(int x, int y) {
 862         int r = x - y;
 863         // HD 2-12 Overflow iff the arguments have different signs and
 864         // the sign of the result is different than the sign of x
 865         if (((x ^ y) & (x ^ r)) < 0) {
 866             throw new ArithmeticException("integer overflow");
 867         }
 868         return r;
 869     }
 870 
 871     /**
 872      * Returns the difference of the arguments,
 873      * throwing an exception if the result overflows a {@code long}.
 874      *
 875      * @param x the first value
 876      * @param y the second value to subtract from the first
 877      * @return the result
 878      * @throws ArithmeticException if the result overflows a long
 879      * @since 1.8
 880      */
 881     @HotSpotIntrinsicCandidate
 882     public static long subtractExact(long x, long y) {
 883         long r = x - y;
 884         // HD 2-12 Overflow iff the arguments have different signs and
 885         // the sign of the result is different than the sign of x
 886         if (((x ^ y) & (x ^ r)) < 0) {
 887             throw new ArithmeticException("long overflow");
 888         }
 889         return r;
 890     }
 891 
 892     /**
 893      * Returns the product of the arguments,
 894      * throwing an exception if the result overflows an {@code int}.
 895      *
 896      * @param x the first value
 897      * @param y the second value
 898      * @return the result
 899      * @throws ArithmeticException if the result overflows an int
 900      * @since 1.8
 901      */
 902     @HotSpotIntrinsicCandidate
 903     public static int multiplyExact(int x, int y) {
 904         long r = (long)x * (long)y;
 905         if ((int)r != r) {
 906             throw new ArithmeticException("integer overflow");
 907         }
 908         return (int)r;
 909     }
 910 
 911     /**
 912      * Returns the product of the arguments,
 913      * throwing an exception if the result overflows a {@code long}.
 914      *
 915      * @param x the first value
 916      * @param y the second value
 917      * @return the result
 918      * @throws ArithmeticException if the result overflows a long
 919      * @since 1.8
 920      */
 921     @HotSpotIntrinsicCandidate
 922     public static long multiplyExact(long x, long y) {
 923         long r = x * y;
 924         long ax = Math.abs(x);
 925         long ay = Math.abs(y);
 926         if (((ax | ay) >>> 31 != 0)) {
 927             // Some bits greater than 2^31 that might cause overflow
 928             // Check the result using the divide operator
 929             // and check for the special case of Long.MIN_VALUE * -1
 930            if (((y != 0) && (r / y != x)) ||
 931                (x == Long.MIN_VALUE && y == -1)) {
 932                 throw new ArithmeticException("long overflow");
 933             }
 934         }
 935         return r;
 936     }
 937 
 938     /**
 939      * Returns the argument incremented by one, throwing an exception if the
 940      * result overflows an {@code int}.
 941      *
 942      * @param a the value to increment
 943      * @return the result
 944      * @throws ArithmeticException if the result overflows an int
 945      * @since 1.8
 946      */
 947     @HotSpotIntrinsicCandidate
 948     public static int incrementExact(int a) {
 949         if (a == Integer.MAX_VALUE) {
 950             throw new ArithmeticException("integer overflow");
 951         }
 952 
 953         return a + 1;
 954     }
 955 
 956     /**
 957      * Returns the argument incremented by one, throwing an exception if the
 958      * result overflows a {@code long}.
 959      *
 960      * @param a the value to increment
 961      * @return the result
 962      * @throws ArithmeticException if the result overflows a long
 963      * @since 1.8
 964      */
 965     @HotSpotIntrinsicCandidate
 966     public static long incrementExact(long a) {
 967         if (a == Long.MAX_VALUE) {
 968             throw new ArithmeticException("long overflow");
 969         }
 970 
 971         return a + 1L;
 972     }
 973 
 974     /**
 975      * Returns the argument decremented by one, throwing an exception if the
 976      * result overflows an {@code int}.
 977      *
 978      * @param a the value to decrement
 979      * @return the result
 980      * @throws ArithmeticException if the result overflows an int
 981      * @since 1.8
 982      */
 983     @HotSpotIntrinsicCandidate
 984     public static int decrementExact(int a) {
 985         if (a == Integer.MIN_VALUE) {
 986             throw new ArithmeticException("integer overflow");
 987         }
 988 
 989         return a - 1;
 990     }
 991 
 992     /**
 993      * Returns the argument decremented by one, throwing an exception if the
 994      * result overflows a {@code long}.
 995      *
 996      * @param a the value to decrement
 997      * @return the result
 998      * @throws ArithmeticException if the result overflows a long
 999      * @since 1.8
1000      */
1001     @HotSpotIntrinsicCandidate
1002     public static long decrementExact(long a) {
1003         if (a == Long.MIN_VALUE) {
1004             throw new ArithmeticException("long overflow");
1005         }
1006 
1007         return a - 1L;
1008     }
1009 
1010     /**
1011      * Returns the negation of the argument, throwing an exception if the
1012      * result overflows an {@code int}.
1013      *
1014      * @param a the value to negate
1015      * @return the result
1016      * @throws ArithmeticException if the result overflows an int
1017      * @since 1.8
1018      */
1019     @HotSpotIntrinsicCandidate
1020     public static int negateExact(int a) {
1021         if (a == Integer.MIN_VALUE) {
1022             throw new ArithmeticException("integer overflow");
1023         }
1024 
1025         return -a;
1026     }
1027 
1028     /**
1029      * Returns the negation of the argument, throwing an exception if the
1030      * result overflows a {@code long}.
1031      *
1032      * @param a the value to negate
1033      * @return the result
1034      * @throws ArithmeticException if the result overflows a long
1035      * @since 1.8
1036      */
1037     @HotSpotIntrinsicCandidate
1038     public static long negateExact(long a) {
1039         if (a == Long.MIN_VALUE) {
1040             throw new ArithmeticException("long overflow");
1041         }
1042 
1043         return -a;
1044     }
1045 
1046     /**
1047      * Returns the value of the {@code long} argument;
1048      * throwing an exception if the value overflows an {@code int}.
1049      *
1050      * @param value the long value
1051      * @return the argument as an int
1052      * @throws ArithmeticException if the {@code argument} overflows an int
1053      * @since 1.8
1054      */
1055     public static int toIntExact(long value) {
1056         if ((int)value != value) {
1057             throw new ArithmeticException("integer overflow");
1058         }
1059         return (int)value;
1060     }
1061 
1062     /**
1063      * Returns the largest (closest to positive infinity)
1064      * {@code int} value that is less than or equal to the algebraic quotient.
1065      * There is one special case, if the dividend is the
1066      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1067      * then integer overflow occurs and
1068      * the result is equal to the {@code Integer.MIN_VALUE}.
1069      * <p>
1070      * Normal integer division operates under the round to zero rounding mode
1071      * (truncation).  This operation instead acts under the round toward
1072      * negative infinity (floor) rounding mode.
1073      * The floor rounding mode gives different results than truncation
1074      * when the exact result is negative.
1075      * <ul>
1076      *   <li>If the signs of the arguments are the same, the results of
1077      *       {@code floorDiv} and the {@code /} operator are the same.  <br>
1078      *       For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1079      *   <li>If the signs of the arguments are different,  the quotient is negative and
1080      *       {@code floorDiv} returns the integer less than or equal to the quotient
1081      *       and the {@code /} operator returns the integer closest to zero.<br>
1082      *       For example, {@code floorDiv(-4, 3) == -2},
1083      *       whereas {@code (-4 / 3) == -1}.
1084      *   </li>
1085      * </ul>
1086      *
1087      * @param x the dividend
1088      * @param y the divisor
1089      * @return the largest (closest to positive infinity)
1090      * {@code int} value that is less than or equal to the algebraic quotient.
1091      * @throws ArithmeticException if the divisor {@code y} is zero
1092      * @see #floorMod(int, int)
1093      * @see #floor(double)
1094      * @since 1.8
1095      */
1096     public static int floorDiv(int x, int y) {
1097         int r = x / y;
1098         // if the signs are different and modulo not zero, round down
1099         if ((x ^ y) < 0 && (r * y != x)) {
1100             r--;
1101         }
1102         return r;
1103     }
1104 
1105     /**
1106      * Returns the largest (closest to positive infinity)
1107      * {@code long} value that is less than or equal to the algebraic quotient.
1108      * There is one special case, if the dividend is the
1109      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1110      * then integer overflow occurs and
1111      * the result is equal to the {@code Long.MIN_VALUE}.
1112      * <p>
1113      * Normal integer division operates under the round to zero rounding mode
1114      * (truncation).  This operation instead acts under the round toward
1115      * negative infinity (floor) rounding mode.
1116      * The floor rounding mode gives different results than truncation
1117      * when the exact result is negative.
1118      * <p>
1119      * For examples, see {@link #floorDiv(int, int)}.
1120      *
1121      * @param x the dividend
1122      * @param y the divisor
1123      * @return the largest (closest to positive infinity)
1124      * {@code long} value that is less than or equal to the algebraic quotient.
1125      * @throws ArithmeticException if the divisor {@code y} is zero
1126      * @see #floorMod(long, long)
1127      * @see #floor(double)
1128      * @since 1.8
1129      */
1130     public static long floorDiv(long x, long y) {
1131         long r = x / y;
1132         // if the signs are different and modulo not zero, round down
1133         if ((x ^ y) < 0 && (r * y != x)) {
1134             r--;
1135         }
1136         return r;
1137     }
1138 
1139     /**
1140      * Returns the floor modulus of the {@code int} arguments.
1141      * <p>
1142      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1143      * has the same sign as the divisor {@code y}, and
1144      * is in the range of {@code -abs(y) < r < +abs(y)}.
1145      *
1146      * <p>
1147      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1148      * <ul>
1149      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1150      * </ul>
1151      * <p>
1152      * The difference in values between {@code floorMod} and
1153      * the {@code %} operator is due to the difference between
1154      * {@code floorDiv} that returns the integer less than or equal to the quotient
1155      * and the {@code /} operator that returns the integer closest to zero.
1156      * <p>
1157      * Examples:
1158      * <ul>
1159      *   <li>If the signs of the arguments are the same, the results
1160      *       of {@code floorMod} and the {@code %} operator are the same.  <br>
1161      *       <ul>
1162      *       <li>{@code floorMod(4, 3) == 1}; &nbsp; and {@code (4 % 3) == 1}</li>
1163      *       </ul>
1164      *   <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1165      *      <ul>
1166      *      <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1} </li>
1167      *      <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1} </li>
1168      *      <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1 } </li>
1169      *      </ul>
1170      *   </li>
1171      * </ul>
1172      * <p>
1173      * If the signs of arguments are unknown and a positive modulus
1174      * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1175      *
1176      * @param x the dividend
1177      * @param y the divisor
1178      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1179      * @throws ArithmeticException if the divisor {@code y} is zero
1180      * @see #floorDiv(int, int)
1181      * @since 1.8
1182      */
1183     public static int floorMod(int x, int y) {
1184         int r = x - floorDiv(x, y) * y;
1185         return r;
1186     }
1187 
1188     /**
1189      * Returns the floor modulus of the {@code long} arguments.
1190      * <p>
1191      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1192      * has the same sign as the divisor {@code y}, and
1193      * is in the range of {@code -abs(y) < r < +abs(y)}.
1194      *
1195      * <p>
1196      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1197      * <ul>
1198      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1199      * </ul>
1200      * <p>
1201      * For examples, see {@link #floorMod(int, int)}.
1202      *
1203      * @param x the dividend
1204      * @param y the divisor
1205      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1206      * @throws ArithmeticException if the divisor {@code y} is zero
1207      * @see #floorDiv(long, long)
1208      * @since 1.8
1209      */
1210     public static long floorMod(long x, long y) {
1211         return x - floorDiv(x, y) * y;
1212     }
1213 
1214     /**
1215      * Returns the absolute value of an {@code int} value.
1216      * If the argument is not negative, the argument is returned.
1217      * If the argument is negative, the negation of the argument is returned.
1218      *
1219      * <p>Note that if the argument is equal to the value of
1220      * {@link Integer#MIN_VALUE}, the most negative representable
1221      * {@code int} value, the result is that same value, which is
1222      * negative.
1223      *
1224      * @param   a   the argument whose absolute value is to be determined
1225      * @return  the absolute value of the argument.
1226      */
1227     public static int abs(int a) {
1228         return (a < 0) ? -a : a;
1229     }
1230 
1231     /**
1232      * Returns the absolute value of a {@code long} value.
1233      * If the argument is not negative, the argument is returned.
1234      * If the argument is negative, the negation of the argument is returned.
1235      *
1236      * <p>Note that if the argument is equal to the value of
1237      * {@link Long#MIN_VALUE}, the most negative representable
1238      * {@code long} value, the result is that same value, which
1239      * is negative.
1240      *
1241      * @param   a   the argument whose absolute value is to be determined
1242      * @return  the absolute value of the argument.
1243      */
1244     public static long abs(long a) {
1245         return (a < 0) ? -a : a;
1246     }
1247 
1248     /**
1249      * Returns the absolute value of a {@code float} value.
1250      * If the argument is not negative, the argument is returned.
1251      * If the argument is negative, the negation of the argument is returned.
1252      * Special cases:
1253      * <ul><li>If the argument is positive zero or negative zero, the
1254      * result is positive zero.
1255      * <li>If the argument is infinite, the result is positive infinity.
1256      * <li>If the argument is NaN, the result is NaN.</ul>
1257      * In other words, the result is the same as the value of the expression:
1258      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
1259      *
1260      * @param   a   the argument whose absolute value is to be determined
1261      * @return  the absolute value of the argument.
1262      */
1263     public static float abs(float a) {
1264         return (a <= 0.0F) ? 0.0F - a : a;
1265     }
1266 
1267     /**
1268      * Returns the absolute value of a {@code double} value.
1269      * If the argument is not negative, the argument is returned.
1270      * If the argument is negative, the negation of the argument is returned.
1271      * Special cases:
1272      * <ul><li>If the argument is positive zero or negative zero, the result
1273      * is positive zero.
1274      * <li>If the argument is infinite, the result is positive infinity.
1275      * <li>If the argument is NaN, the result is NaN.</ul>
1276      * In other words, the result is the same as the value of the expression:
1277      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
1278      *
1279      * @param   a   the argument whose absolute value is to be determined
1280      * @return  the absolute value of the argument.
1281      */
1282     @HotSpotIntrinsicCandidate
1283     public static double abs(double a) {
1284         return (a <= 0.0D) ? 0.0D - a : a;
1285     }
1286 
1287     /**
1288      * Returns the greater of two {@code int} values. That is, the
1289      * result is the argument closer to the value of
1290      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1291      * the result is that same value.
1292      *
1293      * @param   a   an argument.
1294      * @param   b   another argument.
1295      * @return  the larger of {@code a} and {@code b}.
1296      */
1297     @HotSpotIntrinsicCandidate
1298     public static int max(int a, int b) {
1299         return (a >= b) ? a : b;
1300     }
1301 
1302     /**
1303      * Returns the greater of two {@code long} values. That is, the
1304      * result is the argument closer to the value of
1305      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1306      * the result is that same value.
1307      *
1308      * @param   a   an argument.
1309      * @param   b   another argument.
1310      * @return  the larger of {@code a} and {@code b}.
1311      */
1312     public static long max(long a, long b) {
1313         return (a >= b) ? a : b;
1314     }
1315 
1316     // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1317     private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
1318     private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1319 
1320     /**
1321      * Returns the greater of two {@code float} values.  That is,
1322      * the result is the argument closer to positive infinity. If the
1323      * arguments have the same value, the result is that same
1324      * value. If either value is NaN, then the result is NaN.  Unlike
1325      * the numerical comparison operators, this method considers
1326      * negative zero to be strictly smaller than positive zero. If one
1327      * argument is positive zero and the other negative zero, the
1328      * result is positive zero.
1329      *
1330      * @param   a   an argument.
1331      * @param   b   another argument.
1332      * @return  the larger of {@code a} and {@code b}.
1333      */
1334     public static float max(float a, float b) {
1335         if (a != a)
1336             return a;   // a is NaN
1337         if ((a == 0.0f) &&
1338             (b == 0.0f) &&
1339             (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1340             // Raw conversion ok since NaN can't map to -0.0.
1341             return b;
1342         }
1343         return (a >= b) ? a : b;
1344     }
1345 
1346     /**
1347      * Returns the greater of two {@code double} values.  That
1348      * is, the result is the argument closer to positive infinity. If
1349      * the arguments have the same value, the result is that same
1350      * value. If either value is NaN, then the result is NaN.  Unlike
1351      * the numerical comparison operators, this method considers
1352      * negative zero to be strictly smaller than positive zero. If one
1353      * argument is positive zero and the other negative zero, the
1354      * result is positive zero.
1355      *
1356      * @param   a   an argument.
1357      * @param   b   another argument.
1358      * @return  the larger of {@code a} and {@code b}.
1359      */
1360     public static double max(double a, double b) {
1361         if (a != a)
1362             return a;   // a is NaN
1363         if ((a == 0.0d) &&
1364             (b == 0.0d) &&
1365             (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1366             // Raw conversion ok since NaN can't map to -0.0.
1367             return b;
1368         }
1369         return (a >= b) ? a : b;
1370     }
1371 
1372     /**
1373      * Returns the smaller of two {@code int} values. That is,
1374      * the result the argument closer to the value of
1375      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1376      * value, the result is that same value.
1377      *
1378      * @param   a   an argument.
1379      * @param   b   another argument.
1380      * @return  the smaller of {@code a} and {@code b}.
1381      */
1382     @HotSpotIntrinsicCandidate
1383     public static int min(int a, int b) {
1384         return (a <= b) ? a : b;
1385     }
1386 
1387     /**
1388      * Returns the smaller of two {@code long} values. That is,
1389      * the result is the argument closer to the value of
1390      * {@link Long#MIN_VALUE}. If the arguments have the same
1391      * value, the result is that same value.
1392      *
1393      * @param   a   an argument.
1394      * @param   b   another argument.
1395      * @return  the smaller of {@code a} and {@code b}.
1396      */
1397     public static long min(long a, long b) {
1398         return (a <= b) ? a : b;
1399     }
1400 
1401     /**
1402      * Returns the smaller of two {@code float} values.  That is,
1403      * the result is the value closer to negative infinity. If the
1404      * arguments have the same value, the result is that same
1405      * value. If either value is NaN, then the result is NaN.  Unlike
1406      * the numerical comparison operators, this method considers
1407      * negative zero to be strictly smaller than positive zero.  If
1408      * one argument is positive zero and the other is negative zero,
1409      * the result is negative zero.
1410      *
1411      * @param   a   an argument.
1412      * @param   b   another argument.
1413      * @return  the smaller of {@code a} and {@code b}.
1414      */
1415     public static float min(float a, float b) {
1416         if (a != a)
1417             return a;   // a is NaN
1418         if ((a == 0.0f) &&
1419             (b == 0.0f) &&
1420             (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1421             // Raw conversion ok since NaN can't map to -0.0.
1422             return b;
1423         }
1424         return (a <= b) ? a : b;
1425     }
1426 
1427     /**
1428      * Returns the smaller of two {@code double} values.  That
1429      * is, the result is the value closer to negative infinity. If the
1430      * arguments have the same value, the result is that same
1431      * value. If either value is NaN, then the result is NaN.  Unlike
1432      * the numerical comparison operators, this method considers
1433      * negative zero to be strictly smaller than positive zero. If one
1434      * argument is positive zero and the other is negative zero, the
1435      * result is negative zero.
1436      *
1437      * @param   a   an argument.
1438      * @param   b   another argument.
1439      * @return  the smaller of {@code a} and {@code b}.
1440      */
1441     public static double min(double a, double b) {
1442         if (a != a)
1443             return a;   // a is NaN
1444         if ((a == 0.0d) &&
1445             (b == 0.0d) &&
1446             (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1447             // Raw conversion ok since NaN can't map to -0.0.
1448             return b;
1449         }
1450         return (a <= b) ? a : b;
1451     }
1452 
1453     /**
1454      * Returns the fused multiply add of the three arguments; that is,
1455      * returns the exact product of the first two arguments summed
1456      * with the third argument and then rounded once to the nearest
1457      * {@code double}.
1458      *
1459      * The rounding is done using the {@linkplain
1460      * java.math.RoundingMode#HALF_EVEN round to nearest even
1461      * rounding mode}.
1462      *
1463      * In contrast, if {@code a * b + c} is evaluated as a regular
1464      * floating-point expression, two rounding errors are involved,
1465      * the first for the multiply operation, the second for the
1466      * addition operation.
1467      *
1468      * <p>Special cases:
1469      * <ul>
1470      * <li> If any argument is NaN, the result is NaN.
1471      *
1472      * <li> If one of the first two arguments is infinite and the
1473      * other is zero, the result is NaN.
1474      *
1475      * <li> If the exact product of the first two arguments is infinite
1476      * (in other words, at least one of the arguments is infinite and
1477      * the other is neither zero nor NaN) and the third argument is an
1478      * infinity of the opposite sign, the result is NaN.
1479      *
1480      * </ul>
1481      *
1482      * <p>Note that {@code fusedMac(a, 1.0, c)} returns the same
1483      * result as ({@code a + c}).  However,
1484      * {@code fusedMac(a, b, +0.0)} does <em>not</em> always return the
1485      * same result as ({@code a * b}) since
1486      * {@code fusedMac(-0.0, +0.0, +0.0)} is {@code +0.0} while
1487      * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fusedMac(a, b, -0.0)} is
1488      * equivalent to ({@code a * b}) however.
1489      *
1490      * @apiNote This method corresponds to the fusedMultiplyAdd
1491      * operation defined in IEEE 754-2008.
1492      *
1493      * @param a a value
1494      * @param b a value
1495      * @param c a value
1496      *
1497      * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
1498      * computed, as if with unlimited range and precision, and rounded
1499      * once to the nearest {@code double} value
1500      */
1501     // @HotSpotIntrinsicCandidate
1502     public static double fma(double a, double b, double c) {
1503         /*
1504          * Infinity and NaN arithmetic is not quite the same with two
1505          * roundings as opposed to just one so the simple expression
1506          * "a * b + c" cannot always be used to compute the correct
1507          * result.  With two roundings, the product can overflow and
1508          * if the addend is infinite, a spurious NaN can be produced
1509          * if the infinity from the overflow and the infinite addend
1510          * have opposite signs.
1511          */
1512 
1513         // First, screen for and handle non-finite input values whose
1514         // arithmetic is not supported by BigDecimal.
1515         if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
1516             return Double.NaN;
1517         } else { // All inputs non-NaN 
1518             boolean infiniteA = Double.isInfinite(a);
1519             boolean infiniteB = Double.isInfinite(b);
1520             boolean infiniteC = Double.isInfinite(c);
1521             double result;
1522 
1523             if (infiniteA || infiniteB || infiniteC) {
1524                 if (infiniteA && b == 0.0 ||
1525                     infiniteB && a == 0.0 ) {
1526                     return Double.NaN;
1527                 }
1528                 // Store product in a double field to cause an
1529                 // overflow even if non-strictfp evaluation is being
1530                 // used.
1531                 double product = a * b;
1532                 if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
1533                     // Intermediate overflow; might cause a
1534                     // spurious NaN if added to infinite c.
1535                     assert Double.isInfinite(c);
1536                     return c;
1537                 } else {
1538                     result = product + c;
1539                     assert !Double.isFinite(result);
1540                     return result;
1541                 }
1542             } else { // All inputs finite
1543                 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
1544                 if (c == 0.0) { // Positive or negative zero
1545                     double tmp = product.doubleValue();
1546 
1547                     // If the product is an exact zero, use a
1548                     // floating-point expression to compute the sign
1549                     // of the zero final result.
1550                     if (tmp == 0.0 && a == 0.0 || b == 0.0) {
1551                         return a * b + c;
1552                     } else {
1553                         // The sign of a zero addend doesn't matter if
1554                         // the product is nonzero. The sign of a zero
1555                         // addend is not factored in the result if the
1556                         // exact product is nonzero but underflows to
1557                         // zero; see IEEE 754 2008 section 6.3 "The
1558                         // sign bit".
1559                         return tmp;
1560                     }
1561                 } else {
1562                     return product.add(new BigDecimal(c)).doubleValue();
1563                 }
1564             }
1565         }
1566     }
1567 
1568     /**
1569      * Returns the fused multiply add of the three arguments; that is,
1570      * returns the exact product of the first two arguments summed
1571      * with the third argument and then rounded once to the nearest
1572      * {@code float}.
1573      *
1574      * The rounding is done using the {@linkplain
1575      * java.math.RoundingMode#HALF_EVEN round to nearest even
1576      * rounding mode}.
1577      *
1578      * In contrast, if {@code a * b + c} is evaluated as a regular
1579      * floating-point expression, two rounding errors are involved,
1580      * the first for the multiply operation, the second for the
1581      * addition operation.
1582      *
1583      * <p>Special cases:
1584      * <ul>
1585      * <li> If any argument is NaN, the result is NaN.
1586      *
1587      * <li> If one of the first two arguments is infinite and the
1588      * other is zero, the result is NaN.
1589      *
1590      * <li> If the exact product of the first two arguments is infinite
1591      * (in other words, at least one of the arguments is infinite and
1592      * the other is neither zero nor NaN) and the third argument is an
1593      * infinity of the opposite sign, the result is NaN.
1594      *
1595      * </ul>
1596      *
1597      * <p>Note that {@code fusedMac(a, 1.0f, c)} returns the same
1598      * result as ({@code a + c}).  However,
1599      * {@code fusedMac(a, b, +0.0f)} does <em>not</em> always return the
1600      * same result as ({@code a * b}) since
1601      * {@code fusedMac(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
1602      * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fusedMac(a, b, -0.0f)} is
1603      * equivalent to ({@code a * b}) however.
1604      *
1605      * @apiNote This method corresponds to the fusedMultiplyAdd
1606      * operation defined in IEEE 754-2008.
1607      *
1608      * @param a a value
1609      * @param b a value
1610      * @param c a value
1611      *
1612      * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
1613      * computed, as if with unlimited range and precision, and rounded
1614      * once to the nearest {@code float} value
1615      */
1616     // @HotSpotIntrinsicCandidate
1617     public static float fma(float a, float b, float c) {
1618         /*
1619          *  Since the double format has more than twice the precision
1620          *  of the float format, the multiply of a * b is exact in
1621          *  double. The add of c to the product then incurs one
1622          *  rounding error. Since the double format moreover has more
1623          *  than (2p + 2) precision bits compared to the p bits of the
1624          *  float format, the two roundings of (a * b + c), first to
1625          *  the double format and then secondarily to the float format,
1626          *  are equivalent to rounding the intermediate result directly
1627          *  to the float format.
1628          *
1629          * In terms of strictfp vs default-fp concerns related to
1630          * overflow and underflow, since
1631          * 
1632          * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE
1633          * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE
1634          *
1635          * neither the multiply nor add will overflow or underflow in
1636          * double. Therefore, it is not necessary for this method to
1637          * be declared strictfp to have reproducible
1638          * behavior. However, it is necessary to explicitly store down
1639          * to a float variable to avoid returning a value in the float
1640          * extended value set.
1641          */
1642         float result = (float)(((double) a * (double) b ) + (double) c);
1643         return result;
1644     }
1645 
1646     /**
1647      * Returns the size of an ulp of the argument.  An ulp, unit in
1648      * the last place, of a {@code double} value is the positive
1649      * distance between this floating-point value and the {@code
1650      * double} value next larger in magnitude.  Note that for non-NaN
1651      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1652      *
1653      * <p>Special Cases:
1654      * <ul>
1655      * <li> If the argument is NaN, then the result is NaN.
1656      * <li> If the argument is positive or negative infinity, then the
1657      * result is positive infinity.
1658      * <li> If the argument is positive or negative zero, then the result is
1659      * {@code Double.MIN_VALUE}.
1660      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1661      * the result is equal to 2<sup>971</sup>.
1662      * </ul>
1663      *
1664      * @param d the floating-point value whose ulp is to be returned
1665      * @return the size of an ulp of the argument
1666      * @author Joseph D. Darcy
1667      * @since 1.5
1668      */
1669     public static double ulp(double d) {
1670         int exp = getExponent(d);
1671 
1672         switch(exp) {
1673         case Double.MAX_EXPONENT + 1:       // NaN or infinity
1674             return Math.abs(d);
1675 
1676         case Double.MIN_EXPONENT - 1:       // zero or subnormal
1677             return Double.MIN_VALUE;
1678 
1679         default:
1680             assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
1681 
1682             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1683             exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1684             if (exp >= Double.MIN_EXPONENT) {
1685                 return powerOfTwoD(exp);
1686             }
1687             else {
1688                 // return a subnormal result; left shift integer
1689                 // representation of Double.MIN_VALUE appropriate
1690                 // number of positions
1691                 return Double.longBitsToDouble(1L <<
1692                 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1693             }
1694         }
1695     }
1696 
1697     /**
1698      * Returns the size of an ulp of the argument.  An ulp, unit in
1699      * the last place, of a {@code float} value is the positive
1700      * distance between this floating-point value and the {@code
1701      * float} value next larger in magnitude.  Note that for non-NaN
1702      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1703      *
1704      * <p>Special Cases:
1705      * <ul>
1706      * <li> If the argument is NaN, then the result is NaN.
1707      * <li> If the argument is positive or negative infinity, then the
1708      * result is positive infinity.
1709      * <li> If the argument is positive or negative zero, then the result is
1710      * {@code Float.MIN_VALUE}.
1711      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1712      * the result is equal to 2<sup>104</sup>.
1713      * </ul>
1714      *
1715      * @param f the floating-point value whose ulp is to be returned
1716      * @return the size of an ulp of the argument
1717      * @author Joseph D. Darcy
1718      * @since 1.5
1719      */
1720     public static float ulp(float f) {
1721         int exp = getExponent(f);
1722 
1723         switch(exp) {
1724         case Float.MAX_EXPONENT+1:        // NaN or infinity
1725             return Math.abs(f);
1726 
1727         case Float.MIN_EXPONENT-1:        // zero or subnormal
1728             return Float.MIN_VALUE;
1729 
1730         default:
1731             assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
1732 
1733             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1734             exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1735             if (exp >= Float.MIN_EXPONENT) {
1736                 return powerOfTwoF(exp);
1737             } else {
1738                 // return a subnormal result; left shift integer
1739                 // representation of FloatConsts.MIN_VALUE appropriate
1740                 // number of positions
1741                 return Float.intBitsToFloat(1 <<
1742                 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1743             }
1744         }
1745     }
1746 
1747     /**
1748      * Returns the signum function of the argument; zero if the argument
1749      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1750      * argument is less than zero.
1751      *
1752      * <p>Special Cases:
1753      * <ul>
1754      * <li> If the argument is NaN, then the result is NaN.
1755      * <li> If the argument is positive zero or negative zero, then the
1756      *      result is the same as the argument.
1757      * </ul>
1758      *
1759      * @param d the floating-point value whose signum is to be returned
1760      * @return the signum function of the argument
1761      * @author Joseph D. Darcy
1762      * @since 1.5
1763      */
1764     public static double signum(double d) {
1765         return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1766     }
1767 
1768     /**
1769      * Returns the signum function of the argument; zero if the argument
1770      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1771      * argument is less than zero.
1772      *
1773      * <p>Special Cases:
1774      * <ul>
1775      * <li> If the argument is NaN, then the result is NaN.
1776      * <li> If the argument is positive zero or negative zero, then the
1777      *      result is the same as the argument.
1778      * </ul>
1779      *
1780      * @param f the floating-point value whose signum is to be returned
1781      * @return the signum function of the argument
1782      * @author Joseph D. Darcy
1783      * @since 1.5
1784      */
1785     public static float signum(float f) {
1786         return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1787     }
1788 
1789     /**
1790      * Returns the hyperbolic sine of a {@code double} value.
1791      * The hyperbolic sine of <i>x</i> is defined to be
1792      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1793      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1794      *
1795      * <p>Special cases:
1796      * <ul>
1797      *
1798      * <li>If the argument is NaN, then the result is NaN.
1799      *
1800      * <li>If the argument is infinite, then the result is an infinity
1801      * with the same sign as the argument.
1802      *
1803      * <li>If the argument is zero, then the result is a zero with the
1804      * same sign as the argument.
1805      *
1806      * </ul>
1807      *
1808      * <p>The computed result must be within 2.5 ulps of the exact result.
1809      *
1810      * @param   x The number whose hyperbolic sine is to be returned.
1811      * @return  The hyperbolic sine of {@code x}.
1812      * @since 1.5
1813      */
1814     public static double sinh(double x) {
1815         return StrictMath.sinh(x);
1816     }
1817 
1818     /**
1819      * Returns the hyperbolic cosine of a {@code double} value.
1820      * The hyperbolic cosine of <i>x</i> is defined to be
1821      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1822      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1823      *
1824      * <p>Special cases:
1825      * <ul>
1826      *
1827      * <li>If the argument is NaN, then the result is NaN.
1828      *
1829      * <li>If the argument is infinite, then the result is positive
1830      * infinity.
1831      *
1832      * <li>If the argument is zero, then the result is {@code 1.0}.
1833      *
1834      * </ul>
1835      *
1836      * <p>The computed result must be within 2.5 ulps of the exact result.
1837      *
1838      * @param   x The number whose hyperbolic cosine is to be returned.
1839      * @return  The hyperbolic cosine of {@code x}.
1840      * @since 1.5
1841      */
1842     public static double cosh(double x) {
1843         return StrictMath.cosh(x);
1844     }
1845 
1846     /**
1847      * Returns the hyperbolic tangent of a {@code double} value.
1848      * The hyperbolic tangent of <i>x</i> is defined to be
1849      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1850      * in other words, {@linkplain Math#sinh
1851      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1852      * that the absolute value of the exact tanh is always less than
1853      * 1.
1854      *
1855      * <p>Special cases:
1856      * <ul>
1857      *
1858      * <li>If the argument is NaN, then the result is NaN.
1859      *
1860      * <li>If the argument is zero, then the result is a zero with the
1861      * same sign as the argument.
1862      *
1863      * <li>If the argument is positive infinity, then the result is
1864      * {@code +1.0}.
1865      *
1866      * <li>If the argument is negative infinity, then the result is
1867      * {@code -1.0}.
1868      *
1869      * </ul>
1870      *
1871      * <p>The computed result must be within 2.5 ulps of the exact result.
1872      * The result of {@code tanh} for any finite input must have
1873      * an absolute value less than or equal to 1.  Note that once the
1874      * exact result of tanh is within 1/2 of an ulp of the limit value
1875      * of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
1876      * be returned.
1877      *
1878      * @param   x The number whose hyperbolic tangent is to be returned.
1879      * @return  The hyperbolic tangent of {@code x}.
1880      * @since 1.5
1881      */
1882     public static double tanh(double x) {
1883         return StrictMath.tanh(x);
1884     }
1885 
1886     /**
1887      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1888      * without intermediate overflow or underflow.
1889      *
1890      * <p>Special cases:
1891      * <ul>
1892      *
1893      * <li> If either argument is infinite, then the result
1894      * is positive infinity.
1895      *
1896      * <li> If either argument is NaN and neither argument is infinite,
1897      * then the result is NaN.
1898      *
1899      * </ul>
1900      *
1901      * <p>The computed result must be within 1 ulp of the exact
1902      * result.  If one parameter is held constant, the results must be
1903      * semi-monotonic in the other parameter.
1904      *
1905      * @param x a value
1906      * @param y a value
1907      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1908      * without intermediate overflow or underflow
1909      * @since 1.5
1910      */
1911     public static double hypot(double x, double y) {
1912         return StrictMath.hypot(x, y);
1913     }
1914 
1915     /**
1916      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
1917      * <i>x</i> near 0, the exact sum of
1918      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
1919      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1920      *
1921      * <p>Special cases:
1922      * <ul>
1923      * <li>If the argument is NaN, the result is NaN.
1924      *
1925      * <li>If the argument is positive infinity, then the result is
1926      * positive infinity.
1927      *
1928      * <li>If the argument is negative infinity, then the result is
1929      * -1.0.
1930      *
1931      * <li>If the argument is zero, then the result is a zero with the
1932      * same sign as the argument.
1933      *
1934      * </ul>
1935      *
1936      * <p>The computed result must be within 1 ulp of the exact result.
1937      * Results must be semi-monotonic.  The result of
1938      * {@code expm1} for any finite input must be greater than or
1939      * equal to {@code -1.0}.  Note that once the exact result of
1940      * <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
1941      * ulp of the limit value -1, {@code -1.0} should be
1942      * returned.
1943      *
1944      * @param   x   the exponent to raise <i>e</i> to in the computation of
1945      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
1946      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
1947      * @since 1.5
1948      */
1949     public static double expm1(double x) {
1950         return StrictMath.expm1(x);
1951     }
1952 
1953     /**
1954      * Returns the natural logarithm of the sum of the argument and 1.
1955      * Note that for small values {@code x}, the result of
1956      * {@code log1p(x)} is much closer to the true result of ln(1
1957      * + {@code x}) than the floating-point evaluation of
1958      * {@code log(1.0+x)}.
1959      *
1960      * <p>Special cases:
1961      *
1962      * <ul>
1963      *
1964      * <li>If the argument is NaN or less than -1, then the result is
1965      * NaN.
1966      *
1967      * <li>If the argument is positive infinity, then the result is
1968      * positive infinity.
1969      *
1970      * <li>If the argument is negative one, then the result is
1971      * negative infinity.
1972      *
1973      * <li>If the argument is zero, then the result is a zero with the
1974      * same sign as the argument.
1975      *
1976      * </ul>
1977      *
1978      * <p>The computed result must be within 1 ulp of the exact result.
1979      * Results must be semi-monotonic.
1980      *
1981      * @param   x   a value
1982      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
1983      * log of {@code x}&nbsp;+&nbsp;1
1984      * @since 1.5
1985      */
1986     public static double log1p(double x) {
1987         return StrictMath.log1p(x);
1988     }
1989 
1990     /**
1991      * Returns the first floating-point argument with the sign of the
1992      * second floating-point argument.  Note that unlike the {@link
1993      * StrictMath#copySign(double, double) StrictMath.copySign}
1994      * method, this method does not require NaN {@code sign}
1995      * arguments to be treated as positive values; implementations are
1996      * permitted to treat some NaN arguments as positive and other NaN
1997      * arguments as negative to allow greater performance.
1998      *
1999      * @param magnitude  the parameter providing the magnitude of the result
2000      * @param sign   the parameter providing the sign of the result
2001      * @return a value with the magnitude of {@code magnitude}
2002      * and the sign of {@code sign}.
2003      * @since 1.6
2004      */
2005     public static double copySign(double magnitude, double sign) {
2006         return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
2007                                         (DoubleConsts.SIGN_BIT_MASK)) |
2008                                        (Double.doubleToRawLongBits(magnitude) &
2009                                         (DoubleConsts.EXP_BIT_MASK |
2010                                          DoubleConsts.SIGNIF_BIT_MASK)));
2011     }
2012 
2013     /**
2014      * Returns the first floating-point argument with the sign of the
2015      * second floating-point argument.  Note that unlike the {@link
2016      * StrictMath#copySign(float, float) StrictMath.copySign}
2017      * method, this method does not require NaN {@code sign}
2018      * arguments to be treated as positive values; implementations are
2019      * permitted to treat some NaN arguments as positive and other NaN
2020      * arguments as negative to allow greater performance.
2021      *
2022      * @param magnitude  the parameter providing the magnitude of the result
2023      * @param sign   the parameter providing the sign of the result
2024      * @return a value with the magnitude of {@code magnitude}
2025      * and the sign of {@code sign}.
2026      * @since 1.6
2027      */
2028     public static float copySign(float magnitude, float sign) {
2029         return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
2030                                      (FloatConsts.SIGN_BIT_MASK)) |
2031                                     (Float.floatToRawIntBits(magnitude) &
2032                                      (FloatConsts.EXP_BIT_MASK |
2033                                       FloatConsts.SIGNIF_BIT_MASK)));
2034     }
2035 
2036     /**
2037      * Returns the unbiased exponent used in the representation of a
2038      * {@code float}.  Special cases:
2039      *
2040      * <ul>
2041      * <li>If the argument is NaN or infinite, then the result is
2042      * {@link Float#MAX_EXPONENT} + 1.
2043      * <li>If the argument is zero or subnormal, then the result is
2044      * {@link Float#MIN_EXPONENT} -1.
2045      * </ul>
2046      * @param f a {@code float} value
2047      * @return the unbiased exponent of the argument
2048      * @since 1.6
2049      */
2050     public static int getExponent(float f) {
2051         /*
2052          * Bitwise convert f to integer, mask out exponent bits, shift
2053          * to the right and then subtract out float's bias adjust to
2054          * get true exponent value
2055          */
2056         return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
2057                 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
2058     }
2059 
2060     /**
2061      * Returns the unbiased exponent used in the representation of a
2062      * {@code double}.  Special cases:
2063      *
2064      * <ul>
2065      * <li>If the argument is NaN or infinite, then the result is
2066      * {@link Double#MAX_EXPONENT} + 1.
2067      * <li>If the argument is zero or subnormal, then the result is
2068      * {@link Double#MIN_EXPONENT} -1.
2069      * </ul>
2070      * @param d a {@code double} value
2071      * @return the unbiased exponent of the argument
2072      * @since 1.6
2073      */
2074     public static int getExponent(double d) {
2075         /*
2076          * Bitwise convert d to long, mask out exponent bits, shift
2077          * to the right and then subtract out double's bias adjust to
2078          * get true exponent value.
2079          */
2080         return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
2081                       (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
2082     }
2083 
2084     /**
2085      * Returns the floating-point number adjacent to the first
2086      * argument in the direction of the second argument.  If both
2087      * arguments compare as equal the second argument is returned.
2088      *
2089      * <p>
2090      * Special cases:
2091      * <ul>
2092      * <li> If either argument is a NaN, then NaN is returned.
2093      *
2094      * <li> If both arguments are signed zeros, {@code direction}
2095      * is returned unchanged (as implied by the requirement of
2096      * returning the second argument if the arguments compare as
2097      * equal).
2098      *
2099      * <li> If {@code start} is
2100      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
2101      * has a value such that the result should have a smaller
2102      * magnitude, then a zero with the same sign as {@code start}
2103      * is returned.
2104      *
2105      * <li> If {@code start} is infinite and
2106      * {@code direction} has a value such that the result should
2107      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
2108      * same sign as {@code start} is returned.
2109      *
2110      * <li> If {@code start} is equal to &plusmn;
2111      * {@link Double#MAX_VALUE} and {@code direction} has a
2112      * value such that the result should have a larger magnitude, an
2113      * infinity with same sign as {@code start} is returned.
2114      * </ul>
2115      *
2116      * @param start  starting floating-point value
2117      * @param direction value indicating which of
2118      * {@code start}'s neighbors or {@code start} should
2119      * be returned
2120      * @return The floating-point number adjacent to {@code start} in the
2121      * direction of {@code direction}.
2122      * @since 1.6
2123      */
2124     public static double nextAfter(double start, double direction) {
2125         /*
2126          * The cases:
2127          *
2128          * nextAfter(+infinity, 0)  == MAX_VALUE
2129          * nextAfter(+infinity, +infinity)  == +infinity
2130          * nextAfter(-infinity, 0)  == -MAX_VALUE
2131          * nextAfter(-infinity, -infinity)  == -infinity
2132          *
2133          * are naturally handled without any additional testing
2134          */
2135 
2136         /*
2137          * IEEE 754 floating-point numbers are lexicographically
2138          * ordered if treated as signed-magnitude integers.
2139          * Since Java's integers are two's complement,
2140          * incrementing the two's complement representation of a
2141          * logically negative floating-point value *decrements*
2142          * the signed-magnitude representation. Therefore, when
2143          * the integer representation of a floating-point value
2144          * is negative, the adjustment to the representation is in
2145          * the opposite direction from what would initially be expected.
2146          */
2147 
2148         // Branch to descending case first as it is more costly than ascending
2149         // case due to start != 0.0d conditional.
2150         if (start > direction) { // descending
2151             if (start != 0.0d) {
2152                 final long transducer = Double.doubleToRawLongBits(start);
2153                 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
2154             } else { // start == 0.0d && direction < 0.0d
2155                 return -Double.MIN_VALUE;
2156             }
2157         } else if (start < direction) { // ascending
2158             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2159             // then bitwise convert start to integer.
2160             final long transducer = Double.doubleToRawLongBits(start + 0.0d);
2161             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2162         } else if (start == direction) {
2163             return direction;
2164         } else { // isNaN(start) || isNaN(direction)
2165             return start + direction;
2166         }
2167     }
2168 
2169     /**
2170      * Returns the floating-point number adjacent to the first
2171      * argument in the direction of the second argument.  If both
2172      * arguments compare as equal a value equivalent to the second argument
2173      * is returned.
2174      *
2175      * <p>
2176      * Special cases:
2177      * <ul>
2178      * <li> If either argument is a NaN, then NaN is returned.
2179      *
2180      * <li> If both arguments are signed zeros, a value equivalent
2181      * to {@code direction} is returned.
2182      *
2183      * <li> If {@code start} is
2184      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
2185      * has a value such that the result should have a smaller
2186      * magnitude, then a zero with the same sign as {@code start}
2187      * is returned.
2188      *
2189      * <li> If {@code start} is infinite and
2190      * {@code direction} has a value such that the result should
2191      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
2192      * same sign as {@code start} is returned.
2193      *
2194      * <li> If {@code start} is equal to &plusmn;
2195      * {@link Float#MAX_VALUE} and {@code direction} has a
2196      * value such that the result should have a larger magnitude, an
2197      * infinity with same sign as {@code start} is returned.
2198      * </ul>
2199      *
2200      * @param start  starting floating-point value
2201      * @param direction value indicating which of
2202      * {@code start}'s neighbors or {@code start} should
2203      * be returned
2204      * @return The floating-point number adjacent to {@code start} in the
2205      * direction of {@code direction}.
2206      * @since 1.6
2207      */
2208     public static float nextAfter(float start, double direction) {
2209         /*
2210          * The cases:
2211          *
2212          * nextAfter(+infinity, 0)  == MAX_VALUE
2213          * nextAfter(+infinity, +infinity)  == +infinity
2214          * nextAfter(-infinity, 0)  == -MAX_VALUE
2215          * nextAfter(-infinity, -infinity)  == -infinity
2216          *
2217          * are naturally handled without any additional testing
2218          */
2219 
2220         /*
2221          * IEEE 754 floating-point numbers are lexicographically
2222          * ordered if treated as signed-magnitude integers.
2223          * Since Java's integers are two's complement,
2224          * incrementing the two's complement representation of a
2225          * logically negative floating-point value *decrements*
2226          * the signed-magnitude representation. Therefore, when
2227          * the integer representation of a floating-point value
2228          * is negative, the adjustment to the representation is in
2229          * the opposite direction from what would initially be expected.
2230          */
2231 
2232         // Branch to descending case first as it is more costly than ascending
2233         // case due to start != 0.0f conditional.
2234         if (start > direction) { // descending
2235             if (start != 0.0f) {
2236                 final int transducer = Float.floatToRawIntBits(start);
2237                 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2238             } else { // start == 0.0f && direction < 0.0f
2239                 return -Float.MIN_VALUE;
2240             }
2241         } else if (start < direction) { // ascending
2242             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2243             // then bitwise convert start to integer.
2244             final int transducer = Float.floatToRawIntBits(start + 0.0f);
2245             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2246         } else if (start == direction) {
2247             return (float)direction;
2248         } else { // isNaN(start) || isNaN(direction)
2249             return start + (float)direction;
2250         }
2251     }
2252 
2253     /**
2254      * Returns the floating-point value adjacent to {@code d} in
2255      * the direction of positive infinity.  This method is
2256      * semantically equivalent to {@code nextAfter(d,
2257      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2258      * implementation may run faster than its equivalent
2259      * {@code nextAfter} call.
2260      *
2261      * <p>Special Cases:
2262      * <ul>
2263      * <li> If the argument is NaN, the result is NaN.
2264      *
2265      * <li> If the argument is positive infinity, the result is
2266      * positive infinity.
2267      *
2268      * <li> If the argument is zero, the result is
2269      * {@link Double#MIN_VALUE}
2270      *
2271      * </ul>
2272      *
2273      * @param d starting floating-point value
2274      * @return The adjacent floating-point value closer to positive
2275      * infinity.
2276      * @since 1.6
2277      */
2278     public static double nextUp(double d) {
2279         // Use a single conditional and handle the likely cases first.
2280         if (d < Double.POSITIVE_INFINITY) {
2281             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2282             final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2283             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2284         } else { // d is NaN or +Infinity
2285             return d;
2286         }
2287     }
2288 
2289     /**
2290      * Returns the floating-point value adjacent to {@code f} in
2291      * the direction of positive infinity.  This method is
2292      * semantically equivalent to {@code nextAfter(f,
2293      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2294      * implementation may run faster than its equivalent
2295      * {@code nextAfter} call.
2296      *
2297      * <p>Special Cases:
2298      * <ul>
2299      * <li> If the argument is NaN, the result is NaN.
2300      *
2301      * <li> If the argument is positive infinity, the result is
2302      * positive infinity.
2303      *
2304      * <li> If the argument is zero, the result is
2305      * {@link Float#MIN_VALUE}
2306      *
2307      * </ul>
2308      *
2309      * @param f starting floating-point value
2310      * @return The adjacent floating-point value closer to positive
2311      * infinity.
2312      * @since 1.6
2313      */
2314     public static float nextUp(float f) {
2315         // Use a single conditional and handle the likely cases first.
2316         if (f < Float.POSITIVE_INFINITY) {
2317             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2318             final int transducer = Float.floatToRawIntBits(f + 0.0F);
2319             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2320         } else { // f is NaN or +Infinity
2321             return f;
2322         }
2323     }
2324 
2325     /**
2326      * Returns the floating-point value adjacent to {@code d} in
2327      * the direction of negative infinity.  This method is
2328      * semantically equivalent to {@code nextAfter(d,
2329      * Double.NEGATIVE_INFINITY)}; however, a
2330      * {@code nextDown} implementation may run faster than its
2331      * equivalent {@code nextAfter} call.
2332      *
2333      * <p>Special Cases:
2334      * <ul>
2335      * <li> If the argument is NaN, the result is NaN.
2336      *
2337      * <li> If the argument is negative infinity, the result is
2338      * negative infinity.
2339      *
2340      * <li> If the argument is zero, the result is
2341      * {@code -Double.MIN_VALUE}
2342      *
2343      * </ul>
2344      *
2345      * @param d  starting floating-point value
2346      * @return The adjacent floating-point value closer to negative
2347      * infinity.
2348      * @since 1.8
2349      */
2350     public static double nextDown(double d) {
2351         if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2352             return d;
2353         else {
2354             if (d == 0.0)
2355                 return -Double.MIN_VALUE;
2356             else
2357                 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2358                                                ((d > 0.0d)?-1L:+1L));
2359         }
2360     }
2361 
2362     /**
2363      * Returns the floating-point value adjacent to {@code f} in
2364      * the direction of negative infinity.  This method is
2365      * semantically equivalent to {@code nextAfter(f,
2366      * Float.NEGATIVE_INFINITY)}; however, a
2367      * {@code nextDown} implementation may run faster than its
2368      * equivalent {@code nextAfter} call.
2369      *
2370      * <p>Special Cases:
2371      * <ul>
2372      * <li> If the argument is NaN, the result is NaN.
2373      *
2374      * <li> If the argument is negative infinity, the result is
2375      * negative infinity.
2376      *
2377      * <li> If the argument is zero, the result is
2378      * {@code -Float.MIN_VALUE}
2379      *
2380      * </ul>
2381      *
2382      * @param f  starting floating-point value
2383      * @return The adjacent floating-point value closer to negative
2384      * infinity.
2385      * @since 1.8
2386      */
2387     public static float nextDown(float f) {
2388         if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2389             return f;
2390         else {
2391             if (f == 0.0f)
2392                 return -Float.MIN_VALUE;
2393             else
2394                 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2395                                             ((f > 0.0f)?-1:+1));
2396         }
2397     }
2398 
2399     /**
2400      * Returns {@code d} &times;
2401      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2402      * by a single correctly rounded floating-point multiply to a
2403      * member of the double value set.  See the Java
2404      * Language Specification for a discussion of floating-point
2405      * value sets.  If the exponent of the result is between {@link
2406      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2407      * answer is calculated exactly.  If the exponent of the result
2408      * would be larger than {@code Double.MAX_EXPONENT}, an
2409      * infinity is returned.  Note that if the result is subnormal,
2410      * precision may be lost; that is, when {@code scalb(x, n)}
2411      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2412      * <i>x</i>.  When the result is non-NaN, the result has the same
2413      * sign as {@code d}.
2414      *
2415      * <p>Special cases:
2416      * <ul>
2417      * <li> If the first argument is NaN, NaN is returned.
2418      * <li> If the first argument is infinite, then an infinity of the
2419      * same sign is returned.
2420      * <li> If the first argument is zero, then a zero of the same
2421      * sign is returned.
2422      * </ul>
2423      *
2424      * @param d number to be scaled by a power of two.
2425      * @param scaleFactor power of 2 used to scale {@code d}
2426      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
2427      * @since 1.6
2428      */
2429     public static double scalb(double d, int scaleFactor) {
2430         /*
2431          * This method does not need to be declared strictfp to
2432          * compute the same correct result on all platforms.  When
2433          * scaling up, it does not matter what order the
2434          * multiply-store operations are done; the result will be
2435          * finite or overflow regardless of the operation ordering.
2436          * However, to get the correct result when scaling down, a
2437          * particular ordering must be used.
2438          *
2439          * When scaling down, the multiply-store operations are
2440          * sequenced so that it is not possible for two consecutive
2441          * multiply-stores to return subnormal results.  If one
2442          * multiply-store result is subnormal, the next multiply will
2443          * round it away to zero.  This is done by first multiplying
2444          * by 2 ^ (scaleFactor % n) and then multiplying several
2445          * times by 2^n as needed where n is the exponent of number
2446          * that is a covenient power of two.  In this way, at most one
2447          * real rounding error occurs.  If the double value set is
2448          * being used exclusively, the rounding will occur on a
2449          * multiply.  If the double-extended-exponent value set is
2450          * being used, the products will (perhaps) be exact but the
2451          * stores to d are guaranteed to round to the double value
2452          * set.
2453          *
2454          * It is _not_ a valid implementation to first multiply d by
2455          * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2456          * MIN_EXPONENT) since even in a strictfp program double
2457          * rounding on underflow could occur; e.g. if the scaleFactor
2458          * argument was (MIN_EXPONENT - n) and the exponent of d was a
2459          * little less than -(MIN_EXPONENT - n), meaning the final
2460          * result would be subnormal.
2461          *
2462          * Since exact reproducibility of this method can be achieved
2463          * without any undue performance burden, there is no
2464          * compelling reason to allow double rounding on underflow in
2465          * scalb.
2466          */
2467 
2468         // magnitude of a power of two so large that scaling a finite
2469         // nonzero value by it would be guaranteed to over or
2470         // underflow; due to rounding, scaling down takes an
2471         // additional power of two which is reflected here
2472         final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
2473                               DoubleConsts.SIGNIFICAND_WIDTH + 1;
2474         int exp_adjust = 0;
2475         int scale_increment = 0;
2476         double exp_delta = Double.NaN;
2477 
2478         // Make sure scaling factor is in a reasonable range
2479 
2480         if(scaleFactor < 0) {
2481             scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2482             scale_increment = -512;
2483             exp_delta = twoToTheDoubleScaleDown;
2484         }
2485         else {
2486             scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2487             scale_increment = 512;
2488             exp_delta = twoToTheDoubleScaleUp;
2489         }
2490 
2491         // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2492         // technique from "Hacker's Delight" section 10-2.
2493         int t = (scaleFactor >> 9-1) >>> 32 - 9;
2494         exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2495 
2496         d *= powerOfTwoD(exp_adjust);
2497         scaleFactor -= exp_adjust;
2498 
2499         while(scaleFactor != 0) {
2500             d *= exp_delta;
2501             scaleFactor -= scale_increment;
2502         }
2503         return d;
2504     }
2505 
2506     /**
2507      * Returns {@code f} &times;
2508      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2509      * by a single correctly rounded floating-point multiply to a
2510      * member of the float value set.  See the Java
2511      * Language Specification for a discussion of floating-point
2512      * value sets.  If the exponent of the result is between {@link
2513      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2514      * answer is calculated exactly.  If the exponent of the result
2515      * would be larger than {@code Float.MAX_EXPONENT}, an
2516      * infinity is returned.  Note that if the result is subnormal,
2517      * precision may be lost; that is, when {@code scalb(x, n)}
2518      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2519      * <i>x</i>.  When the result is non-NaN, the result has the same
2520      * sign as {@code f}.
2521      *
2522      * <p>Special cases:
2523      * <ul>
2524      * <li> If the first argument is NaN, NaN is returned.
2525      * <li> If the first argument is infinite, then an infinity of the
2526      * same sign is returned.
2527      * <li> If the first argument is zero, then a zero of the same
2528      * sign is returned.
2529      * </ul>
2530      *
2531      * @param f number to be scaled by a power of two.
2532      * @param scaleFactor power of 2 used to scale {@code f}
2533      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
2534      * @since 1.6
2535      */
2536     public static float scalb(float f, int scaleFactor) {
2537         // magnitude of a power of two so large that scaling a finite
2538         // nonzero value by it would be guaranteed to over or
2539         // underflow; due to rounding, scaling down takes an
2540         // additional power of two which is reflected here
2541         final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
2542                               FloatConsts.SIGNIFICAND_WIDTH + 1;
2543 
2544         // Make sure scaling factor is in a reasonable range
2545         scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2546 
2547         /*
2548          * Since + MAX_SCALE for float fits well within the double
2549          * exponent range and + float -> double conversion is exact
2550          * the multiplication below will be exact. Therefore, the
2551          * rounding that occurs when the double product is cast to
2552          * float will be the correctly rounded float result.  Since
2553          * all operations other than the final multiply will be exact,
2554          * it is not necessary to declare this method strictfp.
2555          */
2556         return (float)((double)f*powerOfTwoD(scaleFactor));
2557     }
2558 
2559     // Constants used in scalb
2560     static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2561     static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2562 
2563     /**
2564      * Returns a floating-point power of two in the normal range.
2565      */
2566     static double powerOfTwoD(int n) {
2567         assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
2568         return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2569                                         (DoubleConsts.SIGNIFICAND_WIDTH-1))
2570                                        & DoubleConsts.EXP_BIT_MASK);
2571     }
2572 
2573     /**
2574      * Returns a floating-point power of two in the normal range.
2575      */
2576     static float powerOfTwoF(int n) {
2577         assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
2578         return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2579                                      (FloatConsts.SIGNIFICAND_WIDTH-1))
2580                                     & FloatConsts.EXP_BIT_MASK);
2581     }
2582 }