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 by one, decrement by one, 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 from 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 from 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, throwing an exception if the result
 913      * 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 9
 920      */
 921     public static long multiplyExact(long x, int y) {
 922         return multiplyExact(x, (long)y);
 923     }
 924 
 925     /**
 926      * Returns the product of the arguments,
 927      * throwing an exception if the result overflows a {@code long}.
 928      *
 929      * @param x the first value
 930      * @param y the second value
 931      * @return the result
 932      * @throws ArithmeticException if the result overflows a long
 933      * @since 1.8
 934      */
 935     @HotSpotIntrinsicCandidate
 936     public static long multiplyExact(long x, long y) {
 937         long r = x * y;
 938         long ax = Math.abs(x);
 939         long ay = Math.abs(y);
 940         if (((ax | ay) >>> 31 != 0)) {
 941             // Some bits greater than 2^31 that might cause overflow
 942             // Check the result using the divide operator
 943             // and check for the special case of Long.MIN_VALUE * -1
 944            if (((y != 0) && (r / y != x)) ||
 945                (x == Long.MIN_VALUE && y == -1)) {
 946                 throw new ArithmeticException("long overflow");
 947             }
 948         }
 949         return r;
 950     }
 951 
 952     /**
 953      * Returns the argument incremented by one, throwing an exception if the
 954      * result overflows an {@code int}.
 955      *
 956      * @param a the value to increment
 957      * @return the result
 958      * @throws ArithmeticException if the result overflows an int
 959      * @since 1.8
 960      */
 961     @HotSpotIntrinsicCandidate
 962     public static int incrementExact(int a) {
 963         if (a == Integer.MAX_VALUE) {
 964             throw new ArithmeticException("integer overflow");
 965         }
 966 
 967         return a + 1;
 968     }
 969 
 970     /**
 971      * Returns the argument incremented by one, throwing an exception if the
 972      * result overflows a {@code long}.
 973      *
 974      * @param a the value to increment
 975      * @return the result
 976      * @throws ArithmeticException if the result overflows a long
 977      * @since 1.8
 978      */
 979     @HotSpotIntrinsicCandidate
 980     public static long incrementExact(long a) {
 981         if (a == Long.MAX_VALUE) {
 982             throw new ArithmeticException("long overflow");
 983         }
 984 
 985         return a + 1L;
 986     }
 987 
 988     /**
 989      * Returns the argument decremented by one, throwing an exception if the
 990      * result overflows an {@code int}.
 991      *
 992      * @param a the value to decrement
 993      * @return the result
 994      * @throws ArithmeticException if the result overflows an int
 995      * @since 1.8
 996      */
 997     @HotSpotIntrinsicCandidate
 998     public static int decrementExact(int a) {
 999         if (a == Integer.MIN_VALUE) {
1000             throw new ArithmeticException("integer overflow");
1001         }
1002 
1003         return a - 1;
1004     }
1005 
1006     /**
1007      * Returns the argument decremented by one, throwing an exception if the
1008      * result overflows a {@code long}.
1009      *
1010      * @param a the value to decrement
1011      * @return the result
1012      * @throws ArithmeticException if the result overflows a long
1013      * @since 1.8
1014      */
1015     @HotSpotIntrinsicCandidate
1016     public static long decrementExact(long a) {
1017         if (a == Long.MIN_VALUE) {
1018             throw new ArithmeticException("long overflow");
1019         }
1020 
1021         return a - 1L;
1022     }
1023 
1024     /**
1025      * Returns the negation of the argument, throwing an exception if the
1026      * result overflows an {@code int}.
1027      *
1028      * @param a the value to negate
1029      * @return the result
1030      * @throws ArithmeticException if the result overflows an int
1031      * @since 1.8
1032      */
1033     @HotSpotIntrinsicCandidate
1034     public static int negateExact(int a) {
1035         if (a == Integer.MIN_VALUE) {
1036             throw new ArithmeticException("integer overflow");
1037         }
1038 
1039         return -a;
1040     }
1041 
1042     /**
1043      * Returns the negation of the argument, throwing an exception if the
1044      * result overflows a {@code long}.
1045      *
1046      * @param a the value to negate
1047      * @return the result
1048      * @throws ArithmeticException if the result overflows a long
1049      * @since 1.8
1050      */
1051     @HotSpotIntrinsicCandidate
1052     public static long negateExact(long a) {
1053         if (a == Long.MIN_VALUE) {
1054             throw new ArithmeticException("long overflow");
1055         }
1056 
1057         return -a;
1058     }
1059 
1060     /**
1061      * Returns the value of the {@code long} argument;
1062      * throwing an exception if the value overflows an {@code int}.
1063      *
1064      * @param value the long value
1065      * @return the argument as an int
1066      * @throws ArithmeticException if the {@code argument} overflows an int
1067      * @since 1.8
1068      */
1069     public static int toIntExact(long value) {
1070         if ((int)value != value) {
1071             throw new ArithmeticException("integer overflow");
1072         }
1073         return (int)value;
1074     }
1075 
1076     /**
1077      * Returns the exact mathematical product of the arguments.
1078      *
1079      * @param x the first value
1080      * @param y the second value
1081      * @return the result
1082      * @since 9
1083      */
1084     public static long multiplyFull(int x, int y) {
1085         return (long)x * (long)y;
1086     }
1087 
1088     /**
1089      * Returns as a {@code long} the most significant 64 bits of the 128-bit
1090      * product of two 64-bit factors.
1091      *
1092      * @param x the first value
1093      * @param y the second value
1094      * @return the result
1095      * @since 9
1096      */
1097     public static long multiplyHigh(long x, long y) {
1098         if (x < 0 || y < 0) {
1099             // Use technique from section 8-2 of Henry S. Warren, Jr.,
1100             // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174.
1101             long x1 = x >> 32;
1102             long x2 = x & 0xFFFFFFFFL;
1103             long y1 = y >> 32;
1104             long y2 = y & 0xFFFFFFFFL;
1105             long z2 = x2 * y2;
1106             long t = x1 * y2 + (z2 >>> 32);
1107             long z1 = t & 0xFFFFFFFFL;
1108             long z0 = t >> 32;
1109             z1 += x2 * y1;
1110             return x1 * y1 + z0 + (z1 >> 32);
1111         } else {
1112             // Use Karatsuba technique with two base 2^32 digits.
1113             long x1 = x >>> 32;
1114             long y1 = y >>> 32;
1115             long x2 = x & 0xFFFFFFFFL;
1116             long y2 = y & 0xFFFFFFFFL;
1117             long A = x1 * y1;
1118             long B = x2 * y2;
1119             long C = (x1 + x2) * (y1 + y2);
1120             long K = C - A - B;
1121             return (((B >>> 32) + K) >>> 32) + A;
1122         }
1123     }
1124 
1125     /**
1126      * Returns the largest (closest to positive infinity)
1127      * {@code int} value that is less than or equal to the algebraic quotient.
1128      * There is one special case, if the dividend is the
1129      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
1130      * then integer overflow occurs and
1131      * the result is equal to {@code Integer.MIN_VALUE}.
1132      * <p>
1133      * Normal integer division operates under the round to zero rounding mode
1134      * (truncation).  This operation instead acts under the round toward
1135      * negative infinity (floor) rounding mode.
1136      * The floor rounding mode gives different results from truncation
1137      * when the exact result is negative.
1138      * <ul>
1139      *   <li>If the signs of the arguments are the same, the results of
1140      *       {@code floorDiv} and the {@code /} operator are the same.  <br>
1141      *       For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
1142      *   <li>If the signs of the arguments are different,  the quotient is negative and
1143      *       {@code floorDiv} returns the integer less than or equal to the quotient
1144      *       and the {@code /} operator returns the integer closest to zero.<br>
1145      *       For example, {@code floorDiv(-4, 3) == -2},
1146      *       whereas {@code (-4 / 3) == -1}.
1147      *   </li>
1148      * </ul>
1149      *
1150      * @param x the dividend
1151      * @param y the divisor
1152      * @return the largest (closest to positive infinity)
1153      * {@code int} value that is less than or equal to the algebraic quotient.
1154      * @throws ArithmeticException if the divisor {@code y} is zero
1155      * @see #floorMod(int, int)
1156      * @see #floor(double)
1157      * @since 1.8
1158      */
1159     public static int floorDiv(int x, int y) {
1160         int r = x / y;
1161         // if the signs are different and modulo not zero, round down
1162         if ((x ^ y) < 0 && (r * y != x)) {
1163             r--;
1164         }
1165         return r;
1166     }
1167 
1168     /**
1169      * Returns the largest (closest to positive infinity)
1170      * {@code long} value that is less than or equal to the algebraic quotient.
1171      * There is one special case, if the dividend is the
1172      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1173      * then integer overflow occurs and
1174      * the result is equal to {@code Long.MIN_VALUE}.
1175      * <p>
1176      * Normal integer division operates under the round to zero rounding mode
1177      * (truncation).  This operation instead acts under the round toward
1178      * negative infinity (floor) rounding mode.
1179      * The floor rounding mode gives different results from truncation
1180      * when the exact result is negative.
1181      * <p>
1182      * For examples, see {@link #floorDiv(int, int)}.
1183      *
1184      * @param x the dividend
1185      * @param y the divisor
1186      * @return the largest (closest to positive infinity)
1187      * {@code int} value that is less than or equal to the algebraic quotient.
1188      * @throws ArithmeticException if the divisor {@code y} is zero
1189      * @see #floorMod(long, int)
1190      * @see #floor(double)
1191      * @since 9
1192      */
1193     public static long floorDiv(long x, int y) {
1194         return floorDiv(x, (long)y);
1195     }
1196 
1197     /**
1198      * Returns the largest (closest to positive infinity)
1199      * {@code long} value that is less than or equal to the algebraic quotient.
1200      * There is one special case, if the dividend is the
1201      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
1202      * then integer overflow occurs and
1203      * the result is equal to {@code Long.MIN_VALUE}.
1204      * <p>
1205      * Normal integer division operates under the round to zero rounding mode
1206      * (truncation).  This operation instead acts under the round toward
1207      * negative infinity (floor) rounding mode.
1208      * The floor rounding mode gives different results from truncation
1209      * when the exact result is negative.
1210      * <p>
1211      * For examples, see {@link #floorDiv(int, int)}.
1212      *
1213      * @param x the dividend
1214      * @param y the divisor
1215      * @return the largest (closest to positive infinity)
1216      * {@code long} value that is less than or equal to the algebraic quotient.
1217      * @throws ArithmeticException if the divisor {@code y} is zero
1218      * @see #floorMod(long, long)
1219      * @see #floor(double)
1220      * @since 1.8
1221      */
1222     public static long floorDiv(long x, long y) {
1223         long r = x / y;
1224         // if the signs are different and modulo not zero, round down
1225         if ((x ^ y) < 0 && (r * y != x)) {
1226             r--;
1227         }
1228         return r;
1229     }
1230 
1231     /**
1232      * Returns the floor modulus of the {@code int} arguments.
1233      * <p>
1234      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1235      * has the same sign as the divisor {@code y}, and
1236      * is in the range of {@code -abs(y) < r < +abs(y)}.
1237      *
1238      * <p>
1239      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1240      * <ul>
1241      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1242      * </ul>
1243      * <p>
1244      * The difference in values between {@code floorMod} and
1245      * the {@code %} operator is due to the difference between
1246      * {@code floorDiv} that returns the integer less than or equal to the quotient
1247      * and the {@code /} operator that returns the integer closest to zero.
1248      * <p>
1249      * Examples:
1250      * <ul>
1251      *   <li>If the signs of the arguments are the same, the results
1252      *       of {@code floorMod} and the {@code %} operator are the same.  <br>
1253      *       <ul>
1254      *       <li>{@code floorMod(4, 3) == 1}; &nbsp; and {@code (4 % 3) == 1}</li>
1255      *       </ul>
1256      *   <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
1257      *      <ul>
1258      *      <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1} </li>
1259      *      <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1} </li>
1260      *      <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1 } </li>
1261      *      </ul>
1262      *   </li>
1263      * </ul>
1264      * <p>
1265      * If the signs of arguments are unknown and a positive modulus
1266      * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
1267      *
1268      * @param x the dividend
1269      * @param y the divisor
1270      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1271      * @throws ArithmeticException if the divisor {@code y} is zero
1272      * @see #floorDiv(int, int)
1273      * @since 1.8
1274      */
1275     public static int floorMod(int x, int y) {
1276         return x - floorDiv(x, y) * y;
1277     }
1278 
1279     /**
1280      * Returns the floor modulus of the {@code long} and {@int} arguments.
1281      * <p>
1282      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1283      * has the same sign as the divisor {@code y}, and
1284      * is in the range of {@code -abs(y) < r < +abs(y)}.
1285      *
1286      * <p>
1287      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1288      * <ul>
1289      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1290      * </ul>
1291      * <p>
1292      * For examples, see {@link #floorMod(int, int)}.
1293      *
1294      * @param x the dividend
1295      * @param y the divisor
1296      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1297      * @throws ArithmeticException if the divisor {@code y} is zero
1298      * @see #floorDiv(long, int)
1299      * @since 9
1300      */
1301     public static int floorMod(long x, int y) {
1302         // Result cannot overflow the range of int.
1303         return (int)(x - floorDiv(x, y) * y);
1304     }
1305 
1306     /**
1307      * Returns the floor modulus of the {@code long} arguments.
1308      * <p>
1309      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
1310      * has the same sign as the divisor {@code y}, and
1311      * is in the range of {@code -abs(y) < r < +abs(y)}.
1312      *
1313      * <p>
1314      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
1315      * <ul>
1316      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
1317      * </ul>
1318      * <p>
1319      * For examples, see {@link #floorMod(int, int)}.
1320      *
1321      * @param x the dividend
1322      * @param y the divisor
1323      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
1324      * @throws ArithmeticException if the divisor {@code y} is zero
1325      * @see #floorDiv(long, long)
1326      * @since 1.8
1327      */
1328     public static long floorMod(long x, long y) {
1329         return x - floorDiv(x, y) * y;
1330     }
1331 
1332     /**
1333      * Returns the absolute value of an {@code int} value.
1334      * If the argument is not negative, the argument is returned.
1335      * If the argument is negative, the negation of the argument is returned.
1336      *
1337      * <p>Note that if the argument is equal to the value of
1338      * {@link Integer#MIN_VALUE}, the most negative representable
1339      * {@code int} value, the result is that same value, which is
1340      * negative.
1341      *
1342      * @param   a   the argument whose absolute value is to be determined
1343      * @return  the absolute value of the argument.
1344      */
1345     public static int abs(int a) {
1346         return (a < 0) ? -a : a;
1347     }
1348 
1349     /**
1350      * Returns the absolute value of a {@code long} value.
1351      * If the argument is not negative, the argument is returned.
1352      * If the argument is negative, the negation of the argument is returned.
1353      *
1354      * <p>Note that if the argument is equal to the value of
1355      * {@link Long#MIN_VALUE}, the most negative representable
1356      * {@code long} value, the result is that same value, which
1357      * is negative.
1358      *
1359      * @param   a   the argument whose absolute value is to be determined
1360      * @return  the absolute value of the argument.
1361      */
1362     public static long abs(long a) {
1363         return (a < 0) ? -a : a;
1364     }
1365 
1366     /**
1367      * Returns the absolute value of a {@code float} value.
1368      * If the argument is not negative, the argument is returned.
1369      * If the argument is negative, the negation of the argument is returned.
1370      * Special cases:
1371      * <ul><li>If the argument is positive zero or negative zero, the
1372      * result is positive zero.
1373      * <li>If the argument is infinite, the result is positive infinity.
1374      * <li>If the argument is NaN, the result is NaN.</ul>
1375      *
1376      * @apiNote As implied by the above, one valid implementation of
1377      * this method is given by the expression below which computes a
1378      * {@code float} with the same exponent and significand as the
1379      * argument but with a guaranteed zero sign bit indicating a
1380      * positive value:<br>
1381      * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
1382      *
1383      * @param   a   the argument whose absolute value is to be determined
1384      * @return  the absolute value of the argument.
1385      */
1386     public static float abs(float a) {
1387         return (a <= 0.0F) ? 0.0F - a : a;
1388     }
1389 
1390     /**
1391      * Returns the absolute value of a {@code double} value.
1392      * If the argument is not negative, the argument is returned.
1393      * If the argument is negative, the negation of the argument is returned.
1394      * Special cases:
1395      * <ul><li>If the argument is positive zero or negative zero, the result
1396      * is positive zero.
1397      * <li>If the argument is infinite, the result is positive infinity.
1398      * <li>If the argument is NaN, the result is NaN.</ul>
1399      *
1400      * @apiNote As implied by the above, one valid implementation of
1401      * this method is given by the expression below which computes a
1402      * {@code double} with the same exponent and significand as the
1403      * argument but with a guaranteed zero sign bit indicating a
1404      * positive value:<br>
1405      * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
1406      *
1407      * @param   a   the argument whose absolute value is to be determined
1408      * @return  the absolute value of the argument.
1409      */
1410     @HotSpotIntrinsicCandidate
1411     public static double abs(double a) {
1412         return (a <= 0.0D) ? 0.0D - a : a;
1413     }
1414 
1415     /**
1416      * Returns the greater of two {@code int} values. That is, the
1417      * result is the argument closer to the value of
1418      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1419      * the result is that same value.
1420      *
1421      * @param   a   an argument.
1422      * @param   b   another argument.
1423      * @return  the larger of {@code a} and {@code b}.
1424      */
1425     @HotSpotIntrinsicCandidate
1426     public static int max(int a, int b) {
1427         return (a >= b) ? a : b;
1428     }
1429 
1430     /**
1431      * Returns the greater of two {@code long} values. That is, the
1432      * result is the argument closer to the value of
1433      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1434      * the result is that same value.
1435      *
1436      * @param   a   an argument.
1437      * @param   b   another argument.
1438      * @return  the larger of {@code a} and {@code b}.
1439      */
1440     public static long max(long a, long b) {
1441         return (a >= b) ? a : b;
1442     }
1443 
1444     // Use raw bit-wise conversions on guaranteed non-NaN arguments.
1445     private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
1446     private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
1447 
1448     /**
1449      * Returns the greater of two {@code float} values.  That is,
1450      * the result is the argument closer to positive infinity. If the
1451      * arguments have the same value, the result is that same
1452      * value. If either value is NaN, then the result is NaN.  Unlike
1453      * the numerical comparison operators, this method considers
1454      * negative zero to be strictly smaller than positive zero. If one
1455      * argument is positive zero and the other negative zero, the
1456      * result is positive zero.
1457      *
1458      * @param   a   an argument.
1459      * @param   b   another argument.
1460      * @return  the larger of {@code a} and {@code b}.
1461      */
1462     public static float max(float a, float b) {
1463         if (a != a)
1464             return a;   // a is NaN
1465         if ((a == 0.0f) &&
1466             (b == 0.0f) &&
1467             (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
1468             // Raw conversion ok since NaN can't map to -0.0.
1469             return b;
1470         }
1471         return (a >= b) ? a : b;
1472     }
1473 
1474     /**
1475      * Returns the greater of two {@code double} values.  That
1476      * is, the result is the argument closer to positive infinity. If
1477      * the arguments have the same value, the result is that same
1478      * value. If either value is NaN, then the result is NaN.  Unlike
1479      * the numerical comparison operators, this method considers
1480      * negative zero to be strictly smaller than positive zero. If one
1481      * argument is positive zero and the other negative zero, the
1482      * result is positive zero.
1483      *
1484      * @param   a   an argument.
1485      * @param   b   another argument.
1486      * @return  the larger of {@code a} and {@code b}.
1487      */
1488     public static double max(double a, double b) {
1489         if (a != a)
1490             return a;   // a is NaN
1491         if ((a == 0.0d) &&
1492             (b == 0.0d) &&
1493             (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
1494             // Raw conversion ok since NaN can't map to -0.0.
1495             return b;
1496         }
1497         return (a >= b) ? a : b;
1498     }
1499 
1500     /**
1501      * Returns the smaller of two {@code int} values. That is,
1502      * the result the argument closer to the value of
1503      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1504      * value, the result is that same value.
1505      *
1506      * @param   a   an argument.
1507      * @param   b   another argument.
1508      * @return  the smaller of {@code a} and {@code b}.
1509      */
1510     @HotSpotIntrinsicCandidate
1511     public static int min(int a, int b) {
1512         return (a <= b) ? a : b;
1513     }
1514 
1515     /**
1516      * Returns the smaller of two {@code long} values. That is,
1517      * the result is the argument closer to the value of
1518      * {@link Long#MIN_VALUE}. If the arguments have the same
1519      * value, the result is that same value.
1520      *
1521      * @param   a   an argument.
1522      * @param   b   another argument.
1523      * @return  the smaller of {@code a} and {@code b}.
1524      */
1525     public static long min(long a, long b) {
1526         return (a <= b) ? a : b;
1527     }
1528 
1529     /**
1530      * Returns the smaller of two {@code float} values.  That is,
1531      * the result is the value closer to negative infinity. If the
1532      * arguments have the same value, the result is that same
1533      * value. If either value is NaN, then the result is NaN.  Unlike
1534      * the numerical comparison operators, this method considers
1535      * negative zero to be strictly smaller than positive zero.  If
1536      * one argument is positive zero and the other is negative zero,
1537      * the result is negative zero.
1538      *
1539      * @param   a   an argument.
1540      * @param   b   another argument.
1541      * @return  the smaller of {@code a} and {@code b}.
1542      */
1543     public static float min(float a, float b) {
1544         if (a != a)
1545             return a;   // a is NaN
1546         if ((a == 0.0f) &&
1547             (b == 0.0f) &&
1548             (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
1549             // Raw conversion ok since NaN can't map to -0.0.
1550             return b;
1551         }
1552         return (a <= b) ? a : b;
1553     }
1554 
1555     /**
1556      * Returns the smaller of two {@code double} values.  That
1557      * is, the result is the value closer to negative infinity. If the
1558      * arguments have the same value, the result is that same
1559      * value. If either value is NaN, then the result is NaN.  Unlike
1560      * the numerical comparison operators, this method considers
1561      * negative zero to be strictly smaller than positive zero. If one
1562      * argument is positive zero and the other is negative zero, the
1563      * result is negative zero.
1564      *
1565      * @param   a   an argument.
1566      * @param   b   another argument.
1567      * @return  the smaller of {@code a} and {@code b}.
1568      */
1569     public static double min(double a, double b) {
1570         if (a != a)
1571             return a;   // a is NaN
1572         if ((a == 0.0d) &&
1573             (b == 0.0d) &&
1574             (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
1575             // Raw conversion ok since NaN can't map to -0.0.
1576             return b;
1577         }
1578         return (a <= b) ? a : b;
1579     }
1580 
1581     /**
1582      * Returns the fused multiply add of the three arguments; that is,
1583      * returns the exact product of the first two arguments summed
1584      * with the third argument and then rounded once to the nearest
1585      * {@code double}.
1586      *
1587      * The rounding is done using the {@linkplain
1588      * java.math.RoundingMode#HALF_EVEN round to nearest even
1589      * rounding mode}.
1590      *
1591      * In contrast, if {@code a * b + c} is evaluated as a regular
1592      * floating-point expression, two rounding errors are involved,
1593      * the first for the multiply operation, the second for the
1594      * addition operation.
1595      *
1596      * <p>Special cases:
1597      * <ul>
1598      * <li> If any argument is NaN, the result is NaN.
1599      *
1600      * <li> If one of the first two arguments is infinite and the
1601      * other is zero, the result is NaN.
1602      *
1603      * <li> If the exact product of the first two arguments is infinite
1604      * (in other words, at least one of the arguments is infinite and
1605      * the other is neither zero nor NaN) and the third argument is an
1606      * infinity of the opposite sign, the result is NaN.
1607      *
1608      * </ul>
1609      *
1610      * <p>Note that {@code fma(a, 1.0, c)} returns the same
1611      * result as ({@code a + c}).  However,
1612      * {@code fma(a, b, +0.0)} does <em>not</em> always return the
1613      * same result as ({@code a * b}) since
1614      * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while
1615      * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is
1616      * equivalent to ({@code a * b}) however.
1617      *
1618      * @apiNote This method corresponds to the fusedMultiplyAdd
1619      * operation defined in IEEE 754-2008.
1620      *
1621      * @param a a value
1622      * @param b a value
1623      * @param c a value
1624      *
1625      * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
1626      * computed, as if with unlimited range and precision, and rounded
1627      * once to the nearest {@code double} value
1628      *
1629      * @since 9
1630      */
1631     @HotSpotIntrinsicCandidate
1632     public static double fma(double a, double b, double c) {
1633         /*
1634          * Infinity and NaN arithmetic is not quite the same with two
1635          * roundings as opposed to just one so the simple expression
1636          * "a * b + c" cannot always be used to compute the correct
1637          * result.  With two roundings, the product can overflow and
1638          * if the addend is infinite, a spurious NaN can be produced
1639          * if the infinity from the overflow and the infinite addend
1640          * have opposite signs.
1641          */
1642 
1643         // First, screen for and handle non-finite input values whose
1644         // arithmetic is not supported by BigDecimal.
1645         if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
1646             return Double.NaN;
1647         } else { // All inputs non-NaN
1648             boolean infiniteA = Double.isInfinite(a);
1649             boolean infiniteB = Double.isInfinite(b);
1650             boolean infiniteC = Double.isInfinite(c);
1651             double result;
1652 
1653             if (infiniteA || infiniteB || infiniteC) {
1654                 if (infiniteA && b == 0.0 ||
1655                     infiniteB && a == 0.0 ) {
1656                     return Double.NaN;
1657                 }
1658                 // Store product in a double field to cause an
1659                 // overflow even if non-strictfp evaluation is being
1660                 // used.
1661                 double product = a * b;
1662                 if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
1663                     // Intermediate overflow; might cause a
1664                     // spurious NaN if added to infinite c.
1665                     assert Double.isInfinite(c);
1666                     return c;
1667                 } else {
1668                     result = product + c;
1669                     assert !Double.isFinite(result);
1670                     return result;
1671                 }
1672             } else { // All inputs finite
1673                 BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
1674                 if (c == 0.0) { // Positive or negative zero
1675                     // If the product is an exact zero, use a
1676                     // floating-point expression to compute the sign
1677                     // of the zero final result. The product is an
1678                     // exact zero if and only if at least one of a and
1679                     // b is zero.
1680                     if (a == 0.0 || b == 0.0) {
1681                         return a * b + c;
1682                     } else {
1683                         // The sign of a zero addend doesn't matter if
1684                         // the product is nonzero. The sign of a zero
1685                         // addend is not factored in the result if the
1686                         // exact product is nonzero but underflows to
1687                         // zero; see IEEE-754 2008 section 6.3 "The
1688                         // sign bit".
1689                         return product.doubleValue();
1690                     }
1691                 } else {
1692                     return product.add(new BigDecimal(c)).doubleValue();
1693                 }
1694             }
1695         }
1696     }
1697 
1698     /**
1699      * Returns the fused multiply add of the three arguments; that is,
1700      * returns the exact product of the first two arguments summed
1701      * with the third argument and then rounded once to the nearest
1702      * {@code float}.
1703      *
1704      * The rounding is done using the {@linkplain
1705      * java.math.RoundingMode#HALF_EVEN round to nearest even
1706      * rounding mode}.
1707      *
1708      * In contrast, if {@code a * b + c} is evaluated as a regular
1709      * floating-point expression, two rounding errors are involved,
1710      * the first for the multiply operation, the second for the
1711      * addition operation.
1712      *
1713      * <p>Special cases:
1714      * <ul>
1715      * <li> If any argument is NaN, the result is NaN.
1716      *
1717      * <li> If one of the first two arguments is infinite and the
1718      * other is zero, the result is NaN.
1719      *
1720      * <li> If the exact product of the first two arguments is infinite
1721      * (in other words, at least one of the arguments is infinite and
1722      * the other is neither zero nor NaN) and the third argument is an
1723      * infinity of the opposite sign, the result is NaN.
1724      *
1725      * </ul>
1726      *
1727      * <p>Note that {@code fma(a, 1.0f, c)} returns the same
1728      * result as ({@code a + c}).  However,
1729      * {@code fma(a, b, +0.0f)} does <em>not</em> always return the
1730      * same result as ({@code a * b}) since
1731      * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
1732      * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
1733      * equivalent to ({@code a * b}) however.
1734      *
1735      * @apiNote This method corresponds to the fusedMultiplyAdd
1736      * operation defined in IEEE 754-2008.
1737      *
1738      * @param a a value
1739      * @param b a value
1740      * @param c a value
1741      *
1742      * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
1743      * computed, as if with unlimited range and precision, and rounded
1744      * once to the nearest {@code float} value
1745      *
1746      * @since 9
1747      */
1748     @HotSpotIntrinsicCandidate
1749     public static float fma(float a, float b, float c) {
1750         /*
1751          *  Since the double format has more than twice the precision
1752          *  of the float format, the multiply of a * b is exact in
1753          *  double. The add of c to the product then incurs one
1754          *  rounding error. Since the double format moreover has more
1755          *  than (2p + 2) precision bits compared to the p bits of the
1756          *  float format, the two roundings of (a * b + c), first to
1757          *  the double format and then secondarily to the float format,
1758          *  are equivalent to rounding the intermediate result directly
1759          *  to the float format.
1760          *
1761          * In terms of strictfp vs default-fp concerns related to
1762          * overflow and underflow, since
1763          *
1764          * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE
1765          * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE
1766          *
1767          * neither the multiply nor add will overflow or underflow in
1768          * double. Therefore, it is not necessary for this method to
1769          * be declared strictfp to have reproducible
1770          * behavior. However, it is necessary to explicitly store down
1771          * to a float variable to avoid returning a value in the float
1772          * extended value set.
1773          */
1774         float result = (float)(((double) a * (double) b ) + (double) c);
1775         return result;
1776     }
1777 
1778     /**
1779      * Returns the size of an ulp of the argument.  An ulp, unit in
1780      * the last place, of a {@code double} value is the positive
1781      * distance between this floating-point value and the {@code
1782      * double} value next larger in magnitude.  Note that for non-NaN
1783      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1784      *
1785      * <p>Special Cases:
1786      * <ul>
1787      * <li> If the argument is NaN, then the result is NaN.
1788      * <li> If the argument is positive or negative infinity, then the
1789      * result is positive infinity.
1790      * <li> If the argument is positive or negative zero, then the result is
1791      * {@code Double.MIN_VALUE}.
1792      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1793      * the result is equal to 2<sup>971</sup>.
1794      * </ul>
1795      *
1796      * @param d the floating-point value whose ulp is to be returned
1797      * @return the size of an ulp of the argument
1798      * @author Joseph D. Darcy
1799      * @since 1.5
1800      */
1801     public static double ulp(double d) {
1802         int exp = getExponent(d);
1803 
1804         switch(exp) {
1805         case Double.MAX_EXPONENT + 1:       // NaN or infinity
1806             return Math.abs(d);
1807 
1808         case Double.MIN_EXPONENT - 1:       // zero or subnormal
1809             return Double.MIN_VALUE;
1810 
1811         default:
1812             assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
1813 
1814             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1815             exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
1816             if (exp >= Double.MIN_EXPONENT) {
1817                 return powerOfTwoD(exp);
1818             }
1819             else {
1820                 // return a subnormal result; left shift integer
1821                 // representation of Double.MIN_VALUE appropriate
1822                 // number of positions
1823                 return Double.longBitsToDouble(1L <<
1824                 (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
1825             }
1826         }
1827     }
1828 
1829     /**
1830      * Returns the size of an ulp of the argument.  An ulp, unit in
1831      * the last place, of a {@code float} value is the positive
1832      * distance between this floating-point value and the {@code
1833      * float} value next larger in magnitude.  Note that for non-NaN
1834      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1835      *
1836      * <p>Special Cases:
1837      * <ul>
1838      * <li> If the argument is NaN, then the result is NaN.
1839      * <li> If the argument is positive or negative infinity, then the
1840      * result is positive infinity.
1841      * <li> If the argument is positive or negative zero, then the result is
1842      * {@code Float.MIN_VALUE}.
1843      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1844      * the result is equal to 2<sup>104</sup>.
1845      * </ul>
1846      *
1847      * @param f the floating-point value whose ulp is to be returned
1848      * @return the size of an ulp of the argument
1849      * @author Joseph D. Darcy
1850      * @since 1.5
1851      */
1852     public static float ulp(float f) {
1853         int exp = getExponent(f);
1854 
1855         switch(exp) {
1856         case Float.MAX_EXPONENT+1:        // NaN or infinity
1857             return Math.abs(f);
1858 
1859         case Float.MIN_EXPONENT-1:        // zero or subnormal
1860             return Float.MIN_VALUE;
1861 
1862         default:
1863             assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
1864 
1865             // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
1866             exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
1867             if (exp >= Float.MIN_EXPONENT) {
1868                 return powerOfTwoF(exp);
1869             } else {
1870                 // return a subnormal result; left shift integer
1871                 // representation of FloatConsts.MIN_VALUE appropriate
1872                 // number of positions
1873                 return Float.intBitsToFloat(1 <<
1874                 (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
1875             }
1876         }
1877     }
1878 
1879     /**
1880      * Returns the signum function of the argument; zero if the argument
1881      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1882      * argument is less than zero.
1883      *
1884      * <p>Special Cases:
1885      * <ul>
1886      * <li> If the argument is NaN, then the result is NaN.
1887      * <li> If the argument is positive zero or negative zero, then the
1888      *      result is the same as the argument.
1889      * </ul>
1890      *
1891      * @param d the floating-point value whose signum is to be returned
1892      * @return the signum function of the argument
1893      * @author Joseph D. Darcy
1894      * @since 1.5
1895      */
1896     public static double signum(double d) {
1897         return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
1898     }
1899 
1900     /**
1901      * Returns the signum function of the argument; zero if the argument
1902      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1903      * argument is less than zero.
1904      *
1905      * <p>Special Cases:
1906      * <ul>
1907      * <li> If the argument is NaN, then the result is NaN.
1908      * <li> If the argument is positive zero or negative zero, then the
1909      *      result is the same as the argument.
1910      * </ul>
1911      *
1912      * @param f the floating-point value whose signum is to be returned
1913      * @return the signum function of the argument
1914      * @author Joseph D. Darcy
1915      * @since 1.5
1916      */
1917     public static float signum(float f) {
1918         return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
1919     }
1920 
1921     /**
1922      * Returns the hyperbolic sine of a {@code double} value.
1923      * The hyperbolic sine of <i>x</i> is defined to be
1924      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1925      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1926      *
1927      * <p>Special cases:
1928      * <ul>
1929      *
1930      * <li>If the argument is NaN, then the result is NaN.
1931      *
1932      * <li>If the argument is infinite, then the result is an infinity
1933      * with the same sign as the argument.
1934      *
1935      * <li>If the argument is zero, then the result is a zero with the
1936      * same sign as the argument.
1937      *
1938      * </ul>
1939      *
1940      * <p>The computed result must be within 2.5 ulps of the exact result.
1941      *
1942      * @param   x The number whose hyperbolic sine is to be returned.
1943      * @return  The hyperbolic sine of {@code x}.
1944      * @since 1.5
1945      */
1946     public static double sinh(double x) {
1947         return StrictMath.sinh(x);
1948     }
1949 
1950     /**
1951      * Returns the hyperbolic cosine of a {@code double} value.
1952      * The hyperbolic cosine of <i>x</i> is defined to be
1953      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1954      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1955      *
1956      * <p>Special cases:
1957      * <ul>
1958      *
1959      * <li>If the argument is NaN, then the result is NaN.
1960      *
1961      * <li>If the argument is infinite, then the result is positive
1962      * infinity.
1963      *
1964      * <li>If the argument is zero, then the result is {@code 1.0}.
1965      *
1966      * </ul>
1967      *
1968      * <p>The computed result must be within 2.5 ulps of the exact result.
1969      *
1970      * @param   x The number whose hyperbolic cosine is to be returned.
1971      * @return  The hyperbolic cosine of {@code x}.
1972      * @since 1.5
1973      */
1974     public static double cosh(double x) {
1975         return StrictMath.cosh(x);
1976     }
1977 
1978     /**
1979      * Returns the hyperbolic tangent of a {@code double} value.
1980      * The hyperbolic tangent of <i>x</i> is defined to be
1981      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1982      * in other words, {@linkplain Math#sinh
1983      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1984      * that the absolute value of the exact tanh is always less than
1985      * 1.
1986      *
1987      * <p>Special cases:
1988      * <ul>
1989      *
1990      * <li>If the argument is NaN, then the result is NaN.
1991      *
1992      * <li>If the argument is zero, then the result is a zero with the
1993      * same sign as the argument.
1994      *
1995      * <li>If the argument is positive infinity, then the result is
1996      * {@code +1.0}.
1997      *
1998      * <li>If the argument is negative infinity, then the result is
1999      * {@code -1.0}.
2000      *
2001      * </ul>
2002      *
2003      * <p>The computed result must be within 2.5 ulps of the exact result.
2004      * The result of {@code tanh} for any finite input must have
2005      * an absolute value less than or equal to 1.  Note that once the
2006      * exact result of tanh is within 1/2 of an ulp of the limit value
2007      * of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
2008      * be returned.
2009      *
2010      * @param   x The number whose hyperbolic tangent is to be returned.
2011      * @return  The hyperbolic tangent of {@code x}.
2012      * @since 1.5
2013      */
2014     public static double tanh(double x) {
2015         return StrictMath.tanh(x);
2016     }
2017 
2018     /**
2019      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
2020      * without intermediate overflow or underflow.
2021      *
2022      * <p>Special cases:
2023      * <ul>
2024      *
2025      * <li> If either argument is infinite, then the result
2026      * is positive infinity.
2027      *
2028      * <li> If either argument is NaN and neither argument is infinite,
2029      * then the result is NaN.
2030      *
2031      * </ul>
2032      *
2033      * <p>The computed result must be within 1 ulp of the exact
2034      * result.  If one parameter is held constant, the results must be
2035      * semi-monotonic in the other parameter.
2036      *
2037      * @param x a value
2038      * @param y a value
2039      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
2040      * without intermediate overflow or underflow
2041      * @since 1.5
2042      */
2043     public static double hypot(double x, double y) {
2044         return StrictMath.hypot(x, y);
2045     }
2046 
2047     /**
2048      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
2049      * <i>x</i> near 0, the exact sum of
2050      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
2051      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
2052      *
2053      * <p>Special cases:
2054      * <ul>
2055      * <li>If the argument is NaN, the result is NaN.
2056      *
2057      * <li>If the argument is positive infinity, then the result is
2058      * positive infinity.
2059      *
2060      * <li>If the argument is negative infinity, then the result is
2061      * -1.0.
2062      *
2063      * <li>If the argument is zero, then the result is a zero with the
2064      * same sign as the argument.
2065      *
2066      * </ul>
2067      *
2068      * <p>The computed result must be within 1 ulp of the exact result.
2069      * Results must be semi-monotonic.  The result of
2070      * {@code expm1} for any finite input must be greater than or
2071      * equal to {@code -1.0}.  Note that once the exact result of
2072      * <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
2073      * ulp of the limit value -1, {@code -1.0} should be
2074      * returned.
2075      *
2076      * @param   x   the exponent to raise <i>e</i> to in the computation of
2077      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
2078      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
2079      * @since 1.5
2080      */
2081     public static double expm1(double x) {
2082         return StrictMath.expm1(x);
2083     }
2084 
2085     /**
2086      * Returns the natural logarithm of the sum of the argument and 1.
2087      * Note that for small values {@code x}, the result of
2088      * {@code log1p(x)} is much closer to the true result of ln(1
2089      * + {@code x}) than the floating-point evaluation of
2090      * {@code log(1.0+x)}.
2091      *
2092      * <p>Special cases:
2093      *
2094      * <ul>
2095      *
2096      * <li>If the argument is NaN or less than -1, then the result is
2097      * NaN.
2098      *
2099      * <li>If the argument is positive infinity, then the result is
2100      * positive infinity.
2101      *
2102      * <li>If the argument is negative one, then the result is
2103      * negative infinity.
2104      *
2105      * <li>If the argument is zero, then the result is a zero with the
2106      * same sign as the argument.
2107      *
2108      * </ul>
2109      *
2110      * <p>The computed result must be within 1 ulp of the exact result.
2111      * Results must be semi-monotonic.
2112      *
2113      * @param   x   a value
2114      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
2115      * log of {@code x}&nbsp;+&nbsp;1
2116      * @since 1.5
2117      */
2118     public static double log1p(double x) {
2119         return StrictMath.log1p(x);
2120     }
2121 
2122     /**
2123      * Returns the first floating-point argument with the sign of the
2124      * second floating-point argument.  Note that unlike the {@link
2125      * StrictMath#copySign(double, double) StrictMath.copySign}
2126      * method, this method does not require NaN {@code sign}
2127      * arguments to be treated as positive values; implementations are
2128      * permitted to treat some NaN arguments as positive and other NaN
2129      * arguments as negative to allow greater performance.
2130      *
2131      * @param magnitude  the parameter providing the magnitude of the result
2132      * @param sign   the parameter providing the sign of the result
2133      * @return a value with the magnitude of {@code magnitude}
2134      * and the sign of {@code sign}.
2135      * @since 1.6
2136      */
2137     public static double copySign(double magnitude, double sign) {
2138         return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
2139                                         (DoubleConsts.SIGN_BIT_MASK)) |
2140                                        (Double.doubleToRawLongBits(magnitude) &
2141                                         (DoubleConsts.EXP_BIT_MASK |
2142                                          DoubleConsts.SIGNIF_BIT_MASK)));
2143     }
2144 
2145     /**
2146      * Returns the first floating-point argument with the sign of the
2147      * second floating-point argument.  Note that unlike the {@link
2148      * StrictMath#copySign(float, float) StrictMath.copySign}
2149      * method, this method does not require NaN {@code sign}
2150      * arguments to be treated as positive values; implementations are
2151      * permitted to treat some NaN arguments as positive and other NaN
2152      * arguments as negative to allow greater performance.
2153      *
2154      * @param magnitude  the parameter providing the magnitude of the result
2155      * @param sign   the parameter providing the sign of the result
2156      * @return a value with the magnitude of {@code magnitude}
2157      * and the sign of {@code sign}.
2158      * @since 1.6
2159      */
2160     public static float copySign(float magnitude, float sign) {
2161         return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
2162                                      (FloatConsts.SIGN_BIT_MASK)) |
2163                                     (Float.floatToRawIntBits(magnitude) &
2164                                      (FloatConsts.EXP_BIT_MASK |
2165                                       FloatConsts.SIGNIF_BIT_MASK)));
2166     }
2167 
2168     /**
2169      * Returns the unbiased exponent used in the representation of a
2170      * {@code float}.  Special cases:
2171      *
2172      * <ul>
2173      * <li>If the argument is NaN or infinite, then the result is
2174      * {@link Float#MAX_EXPONENT} + 1.
2175      * <li>If the argument is zero or subnormal, then the result is
2176      * {@link Float#MIN_EXPONENT} -1.
2177      * </ul>
2178      * @param f a {@code float} value
2179      * @return the unbiased exponent of the argument
2180      * @since 1.6
2181      */
2182     public static int getExponent(float f) {
2183         /*
2184          * Bitwise convert f to integer, mask out exponent bits, shift
2185          * to the right and then subtract out float's bias adjust to
2186          * get true exponent value
2187          */
2188         return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
2189                 (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
2190     }
2191 
2192     /**
2193      * Returns the unbiased exponent used in the representation of a
2194      * {@code double}.  Special cases:
2195      *
2196      * <ul>
2197      * <li>If the argument is NaN or infinite, then the result is
2198      * {@link Double#MAX_EXPONENT} + 1.
2199      * <li>If the argument is zero or subnormal, then the result is
2200      * {@link Double#MIN_EXPONENT} -1.
2201      * </ul>
2202      * @param d a {@code double} value
2203      * @return the unbiased exponent of the argument
2204      * @since 1.6
2205      */
2206     public static int getExponent(double d) {
2207         /*
2208          * Bitwise convert d to long, mask out exponent bits, shift
2209          * to the right and then subtract out double's bias adjust to
2210          * get true exponent value.
2211          */
2212         return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
2213                       (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
2214     }
2215 
2216     /**
2217      * Returns the floating-point number adjacent to the first
2218      * argument in the direction of the second argument.  If both
2219      * arguments compare as equal the second argument is returned.
2220      *
2221      * <p>
2222      * Special cases:
2223      * <ul>
2224      * <li> If either argument is a NaN, then NaN is returned.
2225      *
2226      * <li> If both arguments are signed zeros, {@code direction}
2227      * is returned unchanged (as implied by the requirement of
2228      * returning the second argument if the arguments compare as
2229      * equal).
2230      *
2231      * <li> If {@code start} is
2232      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
2233      * has a value such that the result should have a smaller
2234      * magnitude, then a zero with the same sign as {@code start}
2235      * is returned.
2236      *
2237      * <li> If {@code start} is infinite and
2238      * {@code direction} has a value such that the result should
2239      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
2240      * same sign as {@code start} is returned.
2241      *
2242      * <li> If {@code start} is equal to &plusmn;
2243      * {@link Double#MAX_VALUE} and {@code direction} has a
2244      * value such that the result should have a larger magnitude, an
2245      * infinity with same sign as {@code start} is returned.
2246      * </ul>
2247      *
2248      * @param start  starting floating-point value
2249      * @param direction value indicating which of
2250      * {@code start}'s neighbors or {@code start} should
2251      * be returned
2252      * @return The floating-point number adjacent to {@code start} in the
2253      * direction of {@code direction}.
2254      * @since 1.6
2255      */
2256     public static double nextAfter(double start, double direction) {
2257         /*
2258          * The cases:
2259          *
2260          * nextAfter(+infinity, 0)  == MAX_VALUE
2261          * nextAfter(+infinity, +infinity)  == +infinity
2262          * nextAfter(-infinity, 0)  == -MAX_VALUE
2263          * nextAfter(-infinity, -infinity)  == -infinity
2264          *
2265          * are naturally handled without any additional testing
2266          */
2267 
2268         /*
2269          * IEEE 754 floating-point numbers are lexicographically
2270          * ordered if treated as signed-magnitude integers.
2271          * Since Java's integers are two's complement,
2272          * incrementing the two's complement representation of a
2273          * logically negative floating-point value *decrements*
2274          * the signed-magnitude representation. Therefore, when
2275          * the integer representation of a floating-point value
2276          * is negative, the adjustment to the representation is in
2277          * the opposite direction from what would initially be expected.
2278          */
2279 
2280         // Branch to descending case first as it is more costly than ascending
2281         // case due to start != 0.0d conditional.
2282         if (start > direction) { // descending
2283             if (start != 0.0d) {
2284                 final long transducer = Double.doubleToRawLongBits(start);
2285                 return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
2286             } else { // start == 0.0d && direction < 0.0d
2287                 return -Double.MIN_VALUE;
2288             }
2289         } else if (start < direction) { // ascending
2290             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2291             // then bitwise convert start to integer.
2292             final long transducer = Double.doubleToRawLongBits(start + 0.0d);
2293             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2294         } else if (start == direction) {
2295             return direction;
2296         } else { // isNaN(start) || isNaN(direction)
2297             return start + direction;
2298         }
2299     }
2300 
2301     /**
2302      * Returns the floating-point number adjacent to the first
2303      * argument in the direction of the second argument.  If both
2304      * arguments compare as equal a value equivalent to the second argument
2305      * is returned.
2306      *
2307      * <p>
2308      * Special cases:
2309      * <ul>
2310      * <li> If either argument is a NaN, then NaN is returned.
2311      *
2312      * <li> If both arguments are signed zeros, a value equivalent
2313      * to {@code direction} is returned.
2314      *
2315      * <li> If {@code start} is
2316      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
2317      * has a value such that the result should have a smaller
2318      * magnitude, then a zero with the same sign as {@code start}
2319      * is returned.
2320      *
2321      * <li> If {@code start} is infinite and
2322      * {@code direction} has a value such that the result should
2323      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
2324      * same sign as {@code start} is returned.
2325      *
2326      * <li> If {@code start} is equal to &plusmn;
2327      * {@link Float#MAX_VALUE} and {@code direction} has a
2328      * value such that the result should have a larger magnitude, an
2329      * infinity with same sign as {@code start} is returned.
2330      * </ul>
2331      *
2332      * @param start  starting floating-point value
2333      * @param direction value indicating which of
2334      * {@code start}'s neighbors or {@code start} should
2335      * be returned
2336      * @return The floating-point number adjacent to {@code start} in the
2337      * direction of {@code direction}.
2338      * @since 1.6
2339      */
2340     public static float nextAfter(float start, double direction) {
2341         /*
2342          * The cases:
2343          *
2344          * nextAfter(+infinity, 0)  == MAX_VALUE
2345          * nextAfter(+infinity, +infinity)  == +infinity
2346          * nextAfter(-infinity, 0)  == -MAX_VALUE
2347          * nextAfter(-infinity, -infinity)  == -infinity
2348          *
2349          * are naturally handled without any additional testing
2350          */
2351 
2352         /*
2353          * IEEE 754 floating-point numbers are lexicographically
2354          * ordered if treated as signed-magnitude integers.
2355          * Since Java's integers are two's complement,
2356          * incrementing the two's complement representation of a
2357          * logically negative floating-point value *decrements*
2358          * the signed-magnitude representation. Therefore, when
2359          * the integer representation of a floating-point value
2360          * is negative, the adjustment to the representation is in
2361          * the opposite direction from what would initially be expected.
2362          */
2363 
2364         // Branch to descending case first as it is more costly than ascending
2365         // case due to start != 0.0f conditional.
2366         if (start > direction) { // descending
2367             if (start != 0.0f) {
2368                 final int transducer = Float.floatToRawIntBits(start);
2369                 return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
2370             } else { // start == 0.0f && direction < 0.0f
2371                 return -Float.MIN_VALUE;
2372             }
2373         } else if (start < direction) { // ascending
2374             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
2375             // then bitwise convert start to integer.
2376             final int transducer = Float.floatToRawIntBits(start + 0.0f);
2377             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2378         } else if (start == direction) {
2379             return (float)direction;
2380         } else { // isNaN(start) || isNaN(direction)
2381             return start + (float)direction;
2382         }
2383     }
2384 
2385     /**
2386      * Returns the floating-point value adjacent to {@code d} in
2387      * the direction of positive infinity.  This method is
2388      * semantically equivalent to {@code nextAfter(d,
2389      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
2390      * implementation may run faster than its equivalent
2391      * {@code nextAfter} call.
2392      *
2393      * <p>Special Cases:
2394      * <ul>
2395      * <li> If the argument is NaN, the result is NaN.
2396      *
2397      * <li> If the argument is positive infinity, the result is
2398      * positive infinity.
2399      *
2400      * <li> If the argument is zero, the result is
2401      * {@link Double#MIN_VALUE}
2402      *
2403      * </ul>
2404      *
2405      * @param d starting floating-point value
2406      * @return The adjacent floating-point value closer to positive
2407      * infinity.
2408      * @since 1.6
2409      */
2410     public static double nextUp(double d) {
2411         // Use a single conditional and handle the likely cases first.
2412         if (d < Double.POSITIVE_INFINITY) {
2413             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2414             final long transducer = Double.doubleToRawLongBits(d + 0.0D);
2415             return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
2416         } else { // d is NaN or +Infinity
2417             return d;
2418         }
2419     }
2420 
2421     /**
2422      * Returns the floating-point value adjacent to {@code f} in
2423      * the direction of positive infinity.  This method is
2424      * semantically equivalent to {@code nextAfter(f,
2425      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
2426      * implementation may run faster than its equivalent
2427      * {@code nextAfter} call.
2428      *
2429      * <p>Special Cases:
2430      * <ul>
2431      * <li> If the argument is NaN, the result is NaN.
2432      *
2433      * <li> If the argument is positive infinity, the result is
2434      * positive infinity.
2435      *
2436      * <li> If the argument is zero, the result is
2437      * {@link Float#MIN_VALUE}
2438      *
2439      * </ul>
2440      *
2441      * @param f starting floating-point value
2442      * @return The adjacent floating-point value closer to positive
2443      * infinity.
2444      * @since 1.6
2445      */
2446     public static float nextUp(float f) {
2447         // Use a single conditional and handle the likely cases first.
2448         if (f < Float.POSITIVE_INFINITY) {
2449             // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
2450             final int transducer = Float.floatToRawIntBits(f + 0.0F);
2451             return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
2452         } else { // f is NaN or +Infinity
2453             return f;
2454         }
2455     }
2456 
2457     /**
2458      * Returns the floating-point value adjacent to {@code d} in
2459      * the direction of negative infinity.  This method is
2460      * semantically equivalent to {@code nextAfter(d,
2461      * Double.NEGATIVE_INFINITY)}; however, a
2462      * {@code nextDown} implementation may run faster than its
2463      * equivalent {@code nextAfter} call.
2464      *
2465      * <p>Special Cases:
2466      * <ul>
2467      * <li> If the argument is NaN, the result is NaN.
2468      *
2469      * <li> If the argument is negative infinity, the result is
2470      * negative infinity.
2471      *
2472      * <li> If the argument is zero, the result is
2473      * {@code -Double.MIN_VALUE}
2474      *
2475      * </ul>
2476      *
2477      * @param d  starting floating-point value
2478      * @return The adjacent floating-point value closer to negative
2479      * infinity.
2480      * @since 1.8
2481      */
2482     public static double nextDown(double d) {
2483         if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
2484             return d;
2485         else {
2486             if (d == 0.0)
2487                 return -Double.MIN_VALUE;
2488             else
2489                 return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
2490                                                ((d > 0.0d)?-1L:+1L));
2491         }
2492     }
2493 
2494     /**
2495      * Returns the floating-point value adjacent to {@code f} in
2496      * the direction of negative infinity.  This method is
2497      * semantically equivalent to {@code nextAfter(f,
2498      * Float.NEGATIVE_INFINITY)}; however, a
2499      * {@code nextDown} implementation may run faster than its
2500      * equivalent {@code nextAfter} call.
2501      *
2502      * <p>Special Cases:
2503      * <ul>
2504      * <li> If the argument is NaN, the result is NaN.
2505      *
2506      * <li> If the argument is negative infinity, the result is
2507      * negative infinity.
2508      *
2509      * <li> If the argument is zero, the result is
2510      * {@code -Float.MIN_VALUE}
2511      *
2512      * </ul>
2513      *
2514      * @param f  starting floating-point value
2515      * @return The adjacent floating-point value closer to negative
2516      * infinity.
2517      * @since 1.8
2518      */
2519     public static float nextDown(float f) {
2520         if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
2521             return f;
2522         else {
2523             if (f == 0.0f)
2524                 return -Float.MIN_VALUE;
2525             else
2526                 return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
2527                                             ((f > 0.0f)?-1:+1));
2528         }
2529     }
2530 
2531     /**
2532      * Returns {@code d} &times;
2533      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2534      * by a single correctly rounded floating-point multiply to a
2535      * member of the double value set.  See the Java
2536      * Language Specification for a discussion of floating-point
2537      * value sets.  If the exponent of the result is between {@link
2538      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
2539      * answer is calculated exactly.  If the exponent of the result
2540      * would be larger than {@code Double.MAX_EXPONENT}, an
2541      * infinity is returned.  Note that if the result is subnormal,
2542      * precision may be lost; that is, when {@code scalb(x, n)}
2543      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2544      * <i>x</i>.  When the result is non-NaN, the result has the same
2545      * sign as {@code d}.
2546      *
2547      * <p>Special cases:
2548      * <ul>
2549      * <li> If the first argument is NaN, NaN is returned.
2550      * <li> If the first argument is infinite, then an infinity of the
2551      * same sign is returned.
2552      * <li> If the first argument is zero, then a zero of the same
2553      * sign is returned.
2554      * </ul>
2555      *
2556      * @param d number to be scaled by a power of two.
2557      * @param scaleFactor power of 2 used to scale {@code d}
2558      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
2559      * @since 1.6
2560      */
2561     public static double scalb(double d, int scaleFactor) {
2562         /*
2563          * This method does not need to be declared strictfp to
2564          * compute the same correct result on all platforms.  When
2565          * scaling up, it does not matter what order the
2566          * multiply-store operations are done; the result will be
2567          * finite or overflow regardless of the operation ordering.
2568          * However, to get the correct result when scaling down, a
2569          * particular ordering must be used.
2570          *
2571          * When scaling down, the multiply-store operations are
2572          * sequenced so that it is not possible for two consecutive
2573          * multiply-stores to return subnormal results.  If one
2574          * multiply-store result is subnormal, the next multiply will
2575          * round it away to zero.  This is done by first multiplying
2576          * by 2 ^ (scaleFactor % n) and then multiplying several
2577          * times by 2^n as needed where n is the exponent of number
2578          * that is a covenient power of two.  In this way, at most one
2579          * real rounding error occurs.  If the double value set is
2580          * being used exclusively, the rounding will occur on a
2581          * multiply.  If the double-extended-exponent value set is
2582          * being used, the products will (perhaps) be exact but the
2583          * stores to d are guaranteed to round to the double value
2584          * set.
2585          *
2586          * It is _not_ a valid implementation to first multiply d by
2587          * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
2588          * MIN_EXPONENT) since even in a strictfp program double
2589          * rounding on underflow could occur; e.g. if the scaleFactor
2590          * argument was (MIN_EXPONENT - n) and the exponent of d was a
2591          * little less than -(MIN_EXPONENT - n), meaning the final
2592          * result would be subnormal.
2593          *
2594          * Since exact reproducibility of this method can be achieved
2595          * without any undue performance burden, there is no
2596          * compelling reason to allow double rounding on underflow in
2597          * scalb.
2598          */
2599 
2600         // magnitude of a power of two so large that scaling a finite
2601         // nonzero value by it would be guaranteed to over or
2602         // underflow; due to rounding, scaling down takes an
2603         // additional power of two which is reflected here
2604         final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
2605                               DoubleConsts.SIGNIFICAND_WIDTH + 1;
2606         int exp_adjust = 0;
2607         int scale_increment = 0;
2608         double exp_delta = Double.NaN;
2609 
2610         // Make sure scaling factor is in a reasonable range
2611 
2612         if(scaleFactor < 0) {
2613             scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
2614             scale_increment = -512;
2615             exp_delta = twoToTheDoubleScaleDown;
2616         }
2617         else {
2618             scaleFactor = Math.min(scaleFactor, MAX_SCALE);
2619             scale_increment = 512;
2620             exp_delta = twoToTheDoubleScaleUp;
2621         }
2622 
2623         // Calculate (scaleFactor % +/-512), 512 = 2^9, using
2624         // technique from "Hacker's Delight" section 10-2.
2625         int t = (scaleFactor >> 9-1) >>> 32 - 9;
2626         exp_adjust = ((scaleFactor + t) & (512 -1)) - t;
2627 
2628         d *= powerOfTwoD(exp_adjust);
2629         scaleFactor -= exp_adjust;
2630 
2631         while(scaleFactor != 0) {
2632             d *= exp_delta;
2633             scaleFactor -= scale_increment;
2634         }
2635         return d;
2636     }
2637 
2638     /**
2639      * Returns {@code f} &times;
2640      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
2641      * by a single correctly rounded floating-point multiply to a
2642      * member of the float value set.  See the Java
2643      * Language Specification for a discussion of floating-point
2644      * value sets.  If the exponent of the result is between {@link
2645      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
2646      * answer is calculated exactly.  If the exponent of the result
2647      * would be larger than {@code Float.MAX_EXPONENT}, an
2648      * infinity is returned.  Note that if the result is subnormal,
2649      * precision may be lost; that is, when {@code scalb(x, n)}
2650      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
2651      * <i>x</i>.  When the result is non-NaN, the result has the same
2652      * sign as {@code f}.
2653      *
2654      * <p>Special cases:
2655      * <ul>
2656      * <li> If the first argument is NaN, NaN is returned.
2657      * <li> If the first argument is infinite, then an infinity of the
2658      * same sign is returned.
2659      * <li> If the first argument is zero, then a zero of the same
2660      * sign is returned.
2661      * </ul>
2662      *
2663      * @param f number to be scaled by a power of two.
2664      * @param scaleFactor power of 2 used to scale {@code f}
2665      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
2666      * @since 1.6
2667      */
2668     public static float scalb(float f, int scaleFactor) {
2669         // magnitude of a power of two so large that scaling a finite
2670         // nonzero value by it would be guaranteed to over or
2671         // underflow; due to rounding, scaling down takes an
2672         // additional power of two which is reflected here
2673         final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
2674                               FloatConsts.SIGNIFICAND_WIDTH + 1;
2675 
2676         // Make sure scaling factor is in a reasonable range
2677         scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
2678 
2679         /*
2680          * Since + MAX_SCALE for float fits well within the double
2681          * exponent range and + float -> double conversion is exact
2682          * the multiplication below will be exact. Therefore, the
2683          * rounding that occurs when the double product is cast to
2684          * float will be the correctly rounded float result.  Since
2685          * all operations other than the final multiply will be exact,
2686          * it is not necessary to declare this method strictfp.
2687          */
2688         return (float)((double)f*powerOfTwoD(scaleFactor));
2689     }
2690 
2691     // Constants used in scalb
2692     static double twoToTheDoubleScaleUp = powerOfTwoD(512);
2693     static double twoToTheDoubleScaleDown = powerOfTwoD(-512);
2694 
2695     /**
2696      * Returns a floating-point power of two in the normal range.
2697      */
2698     static double powerOfTwoD(int n) {
2699         assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
2700         return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
2701                                         (DoubleConsts.SIGNIFICAND_WIDTH-1))
2702                                        & DoubleConsts.EXP_BIT_MASK);
2703     }
2704 
2705     /**
2706      * Returns a floating-point power of two in the normal range.
2707      */
2708     static float powerOfTwoF(int n) {
2709         assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
2710         return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
2711                                      (FloatConsts.SIGNIFICAND_WIDTH-1))
2712                                     & FloatConsts.EXP_BIT_MASK);
2713     }
2714 }
--- EOF ---