1 /*
   2  * Copyright (c) 1999, 2013, Oracle and/or its affiliates. All rights reserved.
   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
   6  * under the terms of the GNU General Public License version 2 only, as
   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
  16  *
  17  * You should have received a copy of the GNU General Public License version
  18  * 2 along with this work; if not, write to the Free Software Foundation,
  19  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
  20  *
  21  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
  22  * or visit www.oracle.com if you need additional information or have any
  23  * questions.
  24  */
  25 
  26 package java.lang;
  27 import java.util.Random;
  28 import sun.misc.DoubleConsts;
  29 
  30 /**
  31  * The class {@code StrictMath} contains methods for performing basic
  32  * numeric operations such as the elementary exponential, logarithm,
  33  * square root, and trigonometric functions.
  34  *
  35  * <p>To help ensure portability of Java programs, the definitions of
  36  * some of the numeric functions in this package require that they
  37  * produce the same results as certain published algorithms. These
  38  * algorithms are available from the well-known network library
  39  * {@code netlib} as the package "Freely Distributable Math
  40  * Library," <a
  41  * href="ftp://ftp.netlib.org/fdlibm.tar">{@code fdlibm}</a>. These
  42  * algorithms, which are written in the C programming language, are
  43  * then to be understood as executed with all floating-point
  44  * operations following the rules of Java floating-point arithmetic.
  45  *
  46  * <p>The Java math library is defined with respect to
  47  * {@code fdlibm} version 5.3. Where {@code fdlibm} provides
  48  * more than one definition for a function (such as
  49  * {@code acos}), use the "IEEE 754 core function" version
  50  * (residing in a file whose name begins with the letter
  51  * {@code e}).  The methods which require {@code fdlibm}
  52  * semantics are {@code sin}, {@code cos}, {@code tan},
  53  * {@code asin}, {@code acos}, {@code atan},
  54  * {@code exp}, {@code log}, {@code log10},
  55  * {@code cbrt}, {@code atan2}, {@code pow},
  56  * {@code sinh}, {@code cosh}, {@code tanh},
  57  * {@code hypot}, {@code expm1}, and {@code log1p}.
  58  *
  59  * <p>
  60  * The platform uses signed two's complement integer arithmetic with
  61  * int and long primitive types.  The developer should choose
  62  * the primitive type to ensure that arithmetic operations consistently
  63  * produce correct results, which in some cases means the operations
  64  * will not overflow the range of values of the computation.
  65  * The best practice is to choose the primitive type and algorithm to avoid
  66  * overflow. In cases where the size is {@code int} or {@code long} and
  67  * overflow errors need to be detected, the methods {@code addExact},
  68  * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
  69  * throw an {@code ArithmeticException} when the results overflow.
  70  * For other arithmetic operations such as divide, absolute value,
  71  * increment, decrement, and negation overflow occurs only with
  72  * a specific minimum or maximum value and should be checked against
  73  * the minimum or maximum as appropriate.
  74  *
  75  * @author  unascribed
  76  * @author  Joseph D. Darcy
  77  * @since   1.3
  78  */
  79 
  80 public final class StrictMath {
  81 
  82     /**
  83      * Don't let anyone instantiate this class.
  84      */
  85     private StrictMath() {}
  86 
  87     /**
  88      * The {@code double} value that is closer than any other to
  89      * <i>e</i>, the base of the natural logarithms.
  90      */
  91     public static final double E = 2.7182818284590452354;
  92 
  93     /**
  94      * The {@code double} value that is closer than any other to
  95      * <i>pi</i>, the ratio of the circumference of a circle to its
  96      * diameter.
  97      */
  98     public static final double PI = 3.14159265358979323846;
  99 
 100     /**
 101      * Returns the trigonometric sine of an angle. Special cases:
 102      * <ul><li>If the argument is NaN or an infinity, then the
 103      * result is NaN.
 104      * <li>If the argument is zero, then the result is a zero with the
 105      * same sign as the argument.</ul>
 106      *
 107      * @param   a   an angle, in radians.
 108      * @return  the sine of the argument.
 109      */
 110     public static native double sin(double a);
 111 
 112     /**
 113      * Returns the trigonometric cosine of an angle. Special cases:
 114      * <ul><li>If the argument is NaN or an infinity, then the
 115      * result is NaN.</ul>
 116      *
 117      * @param   a   an angle, in radians.
 118      * @return  the cosine of the argument.
 119      */
 120     public static native double cos(double a);
 121 
 122     /**
 123      * Returns the trigonometric tangent of an angle. Special cases:
 124      * <ul><li>If the argument is NaN or an infinity, then the result
 125      * is NaN.
 126      * <li>If the argument is zero, then the result is a zero with the
 127      * same sign as the argument.</ul>
 128      *
 129      * @param   a   an angle, in radians.
 130      * @return  the tangent of the argument.
 131      */
 132     public static native double tan(double a);
 133 
 134     /**
 135      * Returns the arc sine of a value; the returned angle is in the
 136      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 137      * <ul><li>If the argument is NaN or its absolute value is greater
 138      * than 1, then the result is NaN.
 139      * <li>If the argument is zero, then the result is a zero with the
 140      * same sign as the argument.</ul>
 141      *
 142      * @param   a   the value whose arc sine is to be returned.
 143      * @return  the arc sine of the argument.
 144      */
 145     public static native double asin(double a);
 146 
 147     /**
 148      * Returns the arc cosine of a value; the returned angle is in the
 149      * range 0.0 through <i>pi</i>.  Special case:
 150      * <ul><li>If the argument is NaN or its absolute value is greater
 151      * than 1, then the result is NaN.</ul>
 152      *
 153      * @param   a   the value whose arc cosine is to be returned.
 154      * @return  the arc cosine of the argument.
 155      */
 156     public static native double acos(double a);
 157 
 158     /**
 159      * Returns the arc tangent of a value; the returned angle is in the
 160      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 161      * <ul><li>If the argument is NaN, then the result is NaN.
 162      * <li>If the argument is zero, then the result is a zero with the
 163      * same sign as the argument.</ul>
 164      *
 165      * @param   a   the value whose arc tangent is to be returned.
 166      * @return  the arc tangent of the argument.
 167      */
 168     public static native double atan(double a);
 169 
 170     /**
 171      * Converts an angle measured in degrees to an approximately
 172      * equivalent angle measured in radians.  The conversion from
 173      * degrees to radians is generally inexact.
 174      *
 175      * @param   angdeg   an angle, in degrees
 176      * @return  the measurement of the angle {@code angdeg}
 177      *          in radians.
 178      */
 179     public static strictfp double toRadians(double angdeg) {
 180         // Do not delegate to Math.toRadians(angdeg) because
 181         // this method has the strictfp modifier.
 182         return angdeg / 180.0 * PI;
 183     }
 184 
 185     /**
 186      * Converts an angle measured in radians to an approximately
 187      * equivalent angle measured in degrees.  The conversion from
 188      * radians to degrees is generally inexact; users should
 189      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
 190      * equal {@code 0.0}.
 191      *
 192      * @param   angrad   an angle, in radians
 193      * @return  the measurement of the angle {@code angrad}
 194      *          in degrees.
 195      */
 196     public static strictfp double toDegrees(double angrad) {
 197         // Do not delegate to Math.toDegrees(angrad) because
 198         // this method has the strictfp modifier.
 199         return angrad * 180.0 / PI;
 200     }
 201 
 202     /**
 203      * Returns Euler's number <i>e</i> raised to the power of a
 204      * {@code double} value. Special cases:
 205      * <ul><li>If the argument is NaN, the result is NaN.
 206      * <li>If the argument is positive infinity, then the result is
 207      * positive infinity.
 208      * <li>If the argument is negative infinity, then the result is
 209      * positive zero.</ul>
 210      *
 211      * @param   a   the exponent to raise <i>e</i> to.
 212      * @return  the value <i>e</i><sup>{@code a}</sup>,
 213      *          where <i>e</i> is the base of the natural logarithms.
 214      */
 215     public static native double exp(double a);
 216 
 217     /**
 218      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
 219      * value. Special cases:
 220      * <ul><li>If the argument is NaN or less than zero, then the result
 221      * is NaN.
 222      * <li>If the argument is positive infinity, then the result is
 223      * positive infinity.
 224      * <li>If the argument is positive zero or negative zero, then the
 225      * result is negative infinity.</ul>
 226      *
 227      * @param   a   a value
 228      * @return  the value ln&nbsp;{@code a}, the natural logarithm of
 229      *          {@code a}.
 230      */
 231     public static native double log(double a);
 232 
 233 
 234     /**
 235      * Returns the base 10 logarithm of a {@code double} value.
 236      * Special cases:
 237      *
 238      * <ul><li>If the argument is NaN or less than zero, then the result
 239      * is NaN.
 240      * <li>If the argument is positive infinity, then the result is
 241      * positive infinity.
 242      * <li>If the argument is positive zero or negative zero, then the
 243      * result is negative infinity.
 244      * <li> If the argument is equal to 10<sup><i>n</i></sup> for
 245      * integer <i>n</i>, then the result is <i>n</i>.
 246      * </ul>
 247      *
 248      * @param   a   a value
 249      * @return  the base 10 logarithm of  {@code a}.
 250      * @since 1.5
 251      */
 252     public static native double log10(double a);
 253 
 254     /**
 255      * Returns the correctly rounded positive square root of a
 256      * {@code double} value.
 257      * Special cases:
 258      * <ul><li>If the argument is NaN or less than zero, then the result
 259      * is NaN.
 260      * <li>If the argument is positive infinity, then the result is positive
 261      * infinity.
 262      * <li>If the argument is positive zero or negative zero, then the
 263      * result is the same as the argument.</ul>
 264      * Otherwise, the result is the {@code double} value closest to
 265      * the true mathematical square root of the argument value.
 266      *
 267      * @param   a   a value.
 268      * @return  the positive square root of {@code a}.
 269      */
 270     public static native double sqrt(double a);
 271 
 272     /**
 273      * Returns the cube root of a {@code double} value.  For
 274      * positive finite {@code x}, {@code cbrt(-x) ==
 275      * -cbrt(x)}; that is, the cube root of a negative value is
 276      * the negative of the cube root of that value's magnitude.
 277      * Special cases:
 278      *
 279      * <ul>
 280      *
 281      * <li>If the argument is NaN, then the result is NaN.
 282      *
 283      * <li>If the argument is infinite, then the result is an infinity
 284      * with the same sign as the argument.
 285      *
 286      * <li>If the argument is zero, then the result is a zero with the
 287      * same sign as the argument.
 288      *
 289      * </ul>
 290      *
 291      * @param   a   a value.
 292      * @return  the cube root of {@code a}.
 293      * @since 1.5
 294      */
 295     public static native double cbrt(double a);
 296 
 297     /**
 298      * Computes the remainder operation on two arguments as prescribed
 299      * by the IEEE 754 standard.
 300      * The remainder value is mathematically equal to
 301      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
 302      * where <i>n</i> is the mathematical integer closest to the exact
 303      * mathematical value of the quotient {@code f1/f2}, and if two
 304      * mathematical integers are equally close to {@code f1/f2},
 305      * then <i>n</i> is the integer that is even. If the remainder is
 306      * zero, its sign is the same as the sign of the first argument.
 307      * Special cases:
 308      * <ul><li>If either argument is NaN, or the first argument is infinite,
 309      * or the second argument is positive zero or negative zero, then the
 310      * result is NaN.
 311      * <li>If the first argument is finite and the second argument is
 312      * infinite, then the result is the same as the first argument.</ul>
 313      *
 314      * @param   f1   the dividend.
 315      * @param   f2   the divisor.
 316      * @return  the remainder when {@code f1} is divided by
 317      *          {@code f2}.
 318      */
 319     public static native double IEEEremainder(double f1, double f2);
 320 
 321     /**
 322      * Returns the smallest (closest to negative infinity)
 323      * {@code double} value that is greater than or equal to the
 324      * argument and is equal to a mathematical integer. Special cases:
 325      * <ul><li>If the argument value is already equal to a
 326      * mathematical integer, then the result is the same as the
 327      * argument.  <li>If the argument is NaN or an infinity or
 328      * positive zero or negative zero, then the result is the same as
 329      * the argument.  <li>If the argument value is less than zero but
 330      * greater than -1.0, then the result is negative zero.</ul> Note
 331      * that the value of {@code StrictMath.ceil(x)} is exactly the
 332      * value of {@code -StrictMath.floor(-x)}.
 333      *
 334      * @param   a   a value.
 335      * @return  the smallest (closest to negative infinity)
 336      *          floating-point value that is greater than or equal to
 337      *          the argument and is equal to a mathematical integer.
 338      */
 339     public static double ceil(double a) {
 340         return floorOrCeil(a, -0.0, 1.0, 1.0);
 341     }
 342 
 343     /**
 344      * Returns the largest (closest to positive infinity)
 345      * {@code double} value that is less than or equal to the
 346      * argument and is equal to a mathematical integer. Special cases:
 347      * <ul><li>If the argument value is already equal to a
 348      * mathematical integer, then the result is the same as the
 349      * argument.  <li>If the argument is NaN or an infinity or
 350      * positive zero or negative zero, then the result is the same as
 351      * the argument.</ul>
 352      *
 353      * @param   a   a value.
 354      * @return  the largest (closest to positive infinity)
 355      *          floating-point value that less than or equal to the argument
 356      *          and is equal to a mathematical integer.
 357      */
 358     public static double floor(double a) {
 359         return floorOrCeil(a, -1.0, 0.0, -1.0);
 360     }
 361 
 362     /**
 363      * Internal method to share logic between floor and ceil.
 364      *
 365      * @param a the value to be floored or ceiled
 366      * @param negativeBoundary result for values in (-1, 0)
 367      * @param positiveBoundary result for values in (0, 1)
 368      * @param increment value to add when the argument is non-integral
 369      */
 370     private static double floorOrCeil(double a,
 371                                       double negativeBoundary,
 372                                       double positiveBoundary,
 373                                       double sign) {
 374         int exponent = Math.getExponent(a);
 375 
 376         if (exponent < 0) {
 377             /*
 378              * Absolute value of argument is less than 1.
 379              * floorOrceil(-0.0) => -0.0
 380              * floorOrceil(+0.0) => +0.0
 381              */
 382             return ((a == 0.0) ? a :
 383                     ( (a < 0.0) ?  negativeBoundary : positiveBoundary) );
 384         } else if (exponent >= 52) {
 385             /*
 386              * Infinity, NaN, or a value so large it must be integral.
 387              */
 388             return a;
 389         }
 390         // Else the argument is either an integral value already XOR it
 391         // has to be rounded to one.
 392         assert exponent >= 0 && exponent <= 51;
 393 
 394         long doppel = Double.doubleToRawLongBits(a);
 395         long mask   = DoubleConsts.SIGNIF_BIT_MASK >> exponent;
 396 
 397         if ( (mask & doppel) == 0L )
 398             return a; // integral value
 399         else {
 400             double result = Double.longBitsToDouble(doppel & (~mask));
 401             if (sign*a > 0.0)
 402                 result = result + sign;
 403             return result;
 404         }
 405     }
 406 
 407     /**
 408      * Returns the {@code double} value that is closest in value
 409      * to the argument and is equal to a mathematical integer. If two
 410      * {@code double} values that are mathematical integers are
 411      * equally close to the value of the argument, the result is the
 412      * integer value that is even. Special cases:
 413      * <ul><li>If the argument value is already equal to a mathematical
 414      * integer, then the result is the same as the argument.
 415      * <li>If the argument is NaN or an infinity or positive zero or negative
 416      * zero, then the result is the same as the argument.</ul>
 417      *
 418      * @param   a   a value.
 419      * @return  the closest floating-point value to {@code a} that is
 420      *          equal to a mathematical integer.
 421      * @author Joseph D. Darcy
 422      */
 423     public static double rint(double a) {
 424         /*
 425          * If the absolute value of a is not less than 2^52, it
 426          * is either a finite integer (the double format does not have
 427          * enough significand bits for a number that large to have any
 428          * fractional portion), an infinity, or a NaN.  In any of
 429          * these cases, rint of the argument is the argument.
 430          *
 431          * Otherwise, the sum (twoToThe52 + a ) will properly round
 432          * away any fractional portion of a since ulp(twoToThe52) ==
 433          * 1.0; subtracting out twoToThe52 from this sum will then be
 434          * exact and leave the rounded integer portion of a.
 435          *
 436          * This method does *not* need to be declared strictfp to get
 437          * fully reproducible results.  Whether or not a method is
 438          * declared strictfp can only make a difference in the
 439          * returned result if some operation would overflow or
 440          * underflow with strictfp semantics.  The operation
 441          * (twoToThe52 + a ) cannot overflow since large values of a
 442          * are screened out; the add cannot underflow since twoToThe52
 443          * is too large.  The subtraction ((twoToThe52 + a ) -
 444          * twoToThe52) will be exact as discussed above and thus
 445          * cannot overflow or meaningfully underflow.  Finally, the
 446          * last multiply in the return statement is by plus or minus
 447          * 1.0, which is exact too.
 448          */
 449         double twoToThe52 = (double)(1L << 52); // 2^52
 450         double sign = Math.copySign(1.0, a); // preserve sign info
 451         a = Math.abs(a);
 452 
 453         if (a < twoToThe52) { // E_min <= ilogb(a) <= 51
 454             a = ((twoToThe52 + a ) - twoToThe52);
 455         }
 456 
 457         return sign * a; // restore original sign
 458     }
 459 
 460     /**
 461      * Returns the angle <i>theta</i> from the conversion of rectangular
 462      * coordinates ({@code x},&nbsp;{@code y}) to polar
 463      * coordinates (r,&nbsp;<i>theta</i>).
 464      * This method computes the phase <i>theta</i> by computing an arc tangent
 465      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
 466      * cases:
 467      * <ul><li>If either argument is NaN, then the result is NaN.
 468      * <li>If the first argument is positive zero and the second argument
 469      * is positive, or the first argument is positive and finite and the
 470      * second argument is positive infinity, then the result is positive
 471      * zero.
 472      * <li>If the first argument is negative zero and the second argument
 473      * is positive, or the first argument is negative and finite and the
 474      * second argument is positive infinity, then the result is negative zero.
 475      * <li>If the first argument is positive zero and the second argument
 476      * is negative, or the first argument is positive and finite and the
 477      * second argument is negative infinity, then the result is the
 478      * {@code double} value closest to <i>pi</i>.
 479      * <li>If the first argument is negative zero and the second argument
 480      * is negative, or the first argument is negative and finite and the
 481      * second argument is negative infinity, then the result is the
 482      * {@code double} value closest to -<i>pi</i>.
 483      * <li>If the first argument is positive and the second argument is
 484      * positive zero or negative zero, or the first argument is positive
 485      * infinity and the second argument is finite, then the result is the
 486      * {@code double} value closest to <i>pi</i>/2.
 487      * <li>If the first argument is negative and the second argument is
 488      * positive zero or negative zero, or the first argument is negative
 489      * infinity and the second argument is finite, then the result is the
 490      * {@code double} value closest to -<i>pi</i>/2.
 491      * <li>If both arguments are positive infinity, then the result is the
 492      * {@code double} value closest to <i>pi</i>/4.
 493      * <li>If the first argument is positive infinity and the second argument
 494      * is negative infinity, then the result is the {@code double}
 495      * value closest to 3*<i>pi</i>/4.
 496      * <li>If the first argument is negative infinity and the second argument
 497      * is positive infinity, then the result is the {@code double} value
 498      * closest to -<i>pi</i>/4.
 499      * <li>If both arguments are negative infinity, then the result is the
 500      * {@code double} value closest to -3*<i>pi</i>/4.</ul>
 501      *
 502      * @param   y   the ordinate coordinate
 503      * @param   x   the abscissa coordinate
 504      * @return  the <i>theta</i> component of the point
 505      *          (<i>r</i>,&nbsp;<i>theta</i>)
 506      *          in polar coordinates that corresponds to the point
 507      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
 508      */
 509     public static native double atan2(double y, double x);
 510 
 511 
 512     /**
 513      * Returns the value of the first argument raised to the power of the
 514      * second argument. Special cases:
 515      *
 516      * <ul><li>If the second argument is positive or negative zero, then the
 517      * result is 1.0.
 518      * <li>If the second argument is 1.0, then the result is the same as the
 519      * first argument.
 520      * <li>If the second argument is NaN, then the result is NaN.
 521      * <li>If the first argument is NaN and the second argument is nonzero,
 522      * then the result is NaN.
 523      *
 524      * <li>If
 525      * <ul>
 526      * <li>the absolute value of the first argument is greater than 1
 527      * and the second argument is positive infinity, or
 528      * <li>the absolute value of the first argument is less than 1 and
 529      * the second argument is negative infinity,
 530      * </ul>
 531      * then the result is positive infinity.
 532      *
 533      * <li>If
 534      * <ul>
 535      * <li>the absolute value of the first argument is greater than 1 and
 536      * the second argument is negative infinity, or
 537      * <li>the absolute value of the
 538      * first argument is less than 1 and the second argument is positive
 539      * infinity,
 540      * </ul>
 541      * then the result is positive zero.
 542      *
 543      * <li>If the absolute value of the first argument equals 1 and the
 544      * second argument is infinite, then the result is NaN.
 545      *
 546      * <li>If
 547      * <ul>
 548      * <li>the first argument is positive zero and the second argument
 549      * is greater than zero, or
 550      * <li>the first argument is positive infinity and the second
 551      * argument is less than zero,
 552      * </ul>
 553      * then the result is positive zero.
 554      *
 555      * <li>If
 556      * <ul>
 557      * <li>the first argument is positive zero and the second argument
 558      * is less than zero, or
 559      * <li>the first argument is positive infinity and the second
 560      * argument is greater than zero,
 561      * </ul>
 562      * then the result is positive infinity.
 563      *
 564      * <li>If
 565      * <ul>
 566      * <li>the first argument is negative zero and the second argument
 567      * is greater than zero but not a finite odd integer, or
 568      * <li>the first argument is negative infinity and the second
 569      * argument is less than zero but not a finite odd integer,
 570      * </ul>
 571      * then the result is positive zero.
 572      *
 573      * <li>If
 574      * <ul>
 575      * <li>the first argument is negative zero and the second argument
 576      * is a positive finite odd integer, or
 577      * <li>the first argument is negative infinity and the second
 578      * argument is a negative finite odd integer,
 579      * </ul>
 580      * then the result is negative zero.
 581      *
 582      * <li>If
 583      * <ul>
 584      * <li>the first argument is negative zero and the second argument
 585      * is less than zero but not a finite odd integer, or
 586      * <li>the first argument is negative infinity and the second
 587      * argument is greater than zero but not a finite odd integer,
 588      * </ul>
 589      * then the result is positive infinity.
 590      *
 591      * <li>If
 592      * <ul>
 593      * <li>the first argument is negative zero and the second argument
 594      * is a negative finite odd integer, or
 595      * <li>the first argument is negative infinity and the second
 596      * argument is a positive finite odd integer,
 597      * </ul>
 598      * then the result is negative infinity.
 599      *
 600      * <li>If the first argument is finite and less than zero
 601      * <ul>
 602      * <li> if the second argument is a finite even integer, the
 603      * result is equal to the result of raising the absolute value of
 604      * the first argument to the power of the second argument
 605      *
 606      * <li>if the second argument is a finite odd integer, the result
 607      * is equal to the negative of the result of raising the absolute
 608      * value of the first argument to the power of the second
 609      * argument
 610      *
 611      * <li>if the second argument is finite and not an integer, then
 612      * the result is NaN.
 613      * </ul>
 614      *
 615      * <li>If both arguments are integers, then the result is exactly equal
 616      * to the mathematical result of raising the first argument to the power
 617      * of the second argument if that result can in fact be represented
 618      * exactly as a {@code double} value.</ul>
 619      *
 620      * <p>(In the foregoing descriptions, a floating-point value is
 621      * considered to be an integer if and only if it is finite and a
 622      * fixed point of the method {@link #ceil ceil} or,
 623      * equivalently, a fixed point of the method {@link #floor
 624      * floor}. A value is a fixed point of a one-argument
 625      * method if and only if the result of applying the method to the
 626      * value is equal to the value.)
 627      *
 628      * @param   a   base.
 629      * @param   b   the exponent.
 630      * @return  the value {@code a}<sup>{@code b}</sup>.
 631      */
 632     public static native double pow(double a, double b);
 633 
 634     /**
 635      * Returns the closest {@code int} to the argument, with ties
 636      * rounding to positive infinity.
 637      *
 638      * <p>Special cases:
 639      * <ul><li>If the argument is NaN, the result is 0.
 640      * <li>If the argument is negative infinity or any value less than or
 641      * equal to the value of {@code Integer.MIN_VALUE}, the result is
 642      * equal to the value of {@code Integer.MIN_VALUE}.
 643      * <li>If the argument is positive infinity or any value greater than or
 644      * equal to the value of {@code Integer.MAX_VALUE}, the result is
 645      * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 646      *
 647      * @param   a   a floating-point value to be rounded to an integer.
 648      * @return  the value of the argument rounded to the nearest
 649      *          {@code int} value.
 650      * @see     java.lang.Integer#MAX_VALUE
 651      * @see     java.lang.Integer#MIN_VALUE
 652      */
 653     public static int round(float a) {
 654         return Math.round(a);
 655     }
 656 
 657     /**
 658      * Returns the closest {@code long} to the argument, with ties
 659      * rounding to positive infinity.
 660      *
 661      * <p>Special cases:
 662      * <ul><li>If the argument is NaN, the result is 0.
 663      * <li>If the argument is negative infinity or any value less than or
 664      * equal to the value of {@code Long.MIN_VALUE}, the result is
 665      * equal to the value of {@code Long.MIN_VALUE}.
 666      * <li>If the argument is positive infinity or any value greater than or
 667      * equal to the value of {@code Long.MAX_VALUE}, the result is
 668      * equal to the value of {@code Long.MAX_VALUE}.</ul>
 669      *
 670      * @param   a  a floating-point value to be rounded to a
 671      *          {@code long}.
 672      * @return  the value of the argument rounded to the nearest
 673      *          {@code long} value.
 674      * @see     java.lang.Long#MAX_VALUE
 675      * @see     java.lang.Long#MIN_VALUE
 676      */
 677     public static long round(double a) {
 678         return Math.round(a);
 679     }
 680 
 681     private static final class RandomNumberGeneratorHolder {
 682         static final Random randomNumberGenerator = new Random();
 683     }
 684 
 685     /**
 686      * Returns a {@code double} value with a positive sign, greater
 687      * than or equal to {@code 0.0} and less than {@code 1.0}.
 688      * Returned values are chosen pseudorandomly with (approximately)
 689      * uniform distribution from that range.
 690      *
 691      * <p>When this method is first called, it creates a single new
 692      * pseudorandom-number generator, exactly as if by the expression
 693      *
 694      * <blockquote>{@code new java.util.Random()}</blockquote>
 695      *
 696      * This new pseudorandom-number generator is used thereafter for
 697      * all calls to this method and is used nowhere else.
 698      *
 699      * <p>This method is properly synchronized to allow correct use by
 700      * more than one thread. However, if many threads need to generate
 701      * pseudorandom numbers at a great rate, it may reduce contention
 702      * for each thread to have its own pseudorandom-number generator.
 703      *
 704      * @return  a pseudorandom {@code double} greater than or equal
 705      * to {@code 0.0} and less than {@code 1.0}.
 706      * @see Random#nextDouble()
 707      */
 708     public static double random() {
 709         return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 710     }
 711 
 712     /**
 713      * Returns the sum of its arguments,
 714      * throwing an exception if the result overflows an {@code int}.
 715      *
 716      * @param x the first value
 717      * @param y the second value
 718      * @return the result
 719      * @throws ArithmeticException if the result overflows an int
 720      * @see Math#addExact(int,int)
 721      * @since 1.8
 722      */
 723     public static int addExact(int x, int y) {
 724         return Math.addExact(x, y);
 725     }
 726 
 727     /**
 728      * Returns the sum of its arguments,
 729      * throwing an exception if the result overflows a {@code long}.
 730      *
 731      * @param x the first value
 732      * @param y the second value
 733      * @return the result
 734      * @throws ArithmeticException if the result overflows a long
 735      * @see Math#addExact(long,long)
 736      * @since 1.8
 737      */
 738     public static long addExact(long x, long y) {
 739         return Math.addExact(x, y);
 740     }
 741 
 742     /**
 743      * Returns the difference of the arguments,
 744      * throwing an exception if the result overflows an {@code int}.
 745      *
 746      * @param x the first value
 747      * @param y the second value to subtract from the first
 748      * @return the result
 749      * @throws ArithmeticException if the result overflows an int
 750      * @see Math#subtractExact(int,int)
 751      * @since 1.8
 752      */
 753     public static int subtractExact(int x, int y) {
 754         return Math.subtractExact(x, y);
 755     }
 756 
 757     /**
 758      * Returns the difference of the arguments,
 759      * throwing an exception if the result overflows a {@code long}.
 760      *
 761      * @param x the first value
 762      * @param y the second value to subtract from the first
 763      * @return the result
 764      * @throws ArithmeticException if the result overflows a long
 765      * @see Math#subtractExact(long,long)
 766      * @since 1.8
 767      */
 768     public static long subtractExact(long x, long y) {
 769         return Math.subtractExact(x, y);
 770     }
 771 
 772     /**
 773      * Returns the product of the arguments,
 774      * throwing an exception if the result overflows an {@code int}.
 775      *
 776      * @param x the first value
 777      * @param y the second value
 778      * @return the result
 779      * @throws ArithmeticException if the result overflows an int
 780      * @see Math#multiplyExact(int,int)
 781      * @since 1.8
 782      */
 783     public static int multiplyExact(int x, int y) {
 784         return Math.multiplyExact(x, y);
 785     }
 786 
 787     /**
 788      * Returns the product of the arguments,
 789      * throwing an exception if the result overflows a {@code long}.
 790      *
 791      * @param x the first value
 792      * @param y the second value
 793      * @return the result
 794      * @throws ArithmeticException if the result overflows a long
 795      * @see Math#multiplyExact(long,long)
 796      * @since 1.8
 797      */
 798     public static long multiplyExact(long x, long y) {
 799         return Math.multiplyExact(x, y);
 800     }
 801 
 802     /**
 803      * Returns the value of the {@code long} argument;
 804      * throwing an exception if the value overflows an {@code int}.
 805      *
 806      * @param value the long value
 807      * @return the argument as an int
 808      * @throws ArithmeticException if the {@code argument} overflows an int
 809      * @see Math#toIntExact(long)
 810      * @since 1.8
 811      */
 812     public static int toIntExact(long value) {
 813         return Math.toIntExact(value);
 814     }
 815 
 816     /**
 817      * Returns the largest (closest to positive infinity)
 818      * {@code int} value that is less than or equal to the algebraic quotient.
 819      * There is one special case, if the dividend is the
 820      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
 821      * then integer overflow occurs and
 822      * the result is equal to the {@code Integer.MIN_VALUE}.
 823      * <p>
 824      * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
 825      * a comparison to the integer division {@code /} operator.
 826      *
 827      * @param x the dividend
 828      * @param y the divisor
 829      * @return the largest (closest to positive infinity)
 830      * {@code int} value that is less than or equal to the algebraic quotient.
 831      * @throws ArithmeticException if the divisor {@code y} is zero
 832      * @see Math#floorDiv(int, int)
 833      * @see Math#floor(double)
 834      * @since 1.8
 835      */
 836     public static int floorDiv(int x, int y) {
 837         return Math.floorDiv(x, y);
 838     }
 839 
 840     /**
 841      * Returns the largest (closest to positive infinity)
 842      * {@code long} value that is less than or equal to the algebraic quotient.
 843      * There is one special case, if the dividend is the
 844      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
 845      * then integer overflow occurs and
 846      * the result is equal to the {@code Long.MIN_VALUE}.
 847      * <p>
 848      * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
 849      * a comparison to the integer division {@code /} operator.
 850      *
 851      * @param x the dividend
 852      * @param y the divisor
 853      * @return the largest (closest to positive infinity)
 854      * {@code long} value that is less than or equal to the algebraic quotient.
 855      * @throws ArithmeticException if the divisor {@code y} is zero
 856      * @see Math#floorDiv(long, long)
 857      * @see Math#floor(double)
 858      * @since 1.8
 859      */
 860     public static long floorDiv(long x, long y) {
 861         return Math.floorDiv(x, y);
 862     }
 863 
 864     /**
 865      * Returns the floor modulus of the {@code int} arguments.
 866      * <p>
 867      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
 868      * has the same sign as the divisor {@code y}, and
 869      * is in the range of {@code -abs(y) < r < +abs(y)}.
 870      * <p>
 871      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
 872      * <ul>
 873      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
 874      * </ul>
 875      * <p>
 876      * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
 877      * a comparison to the {@code %} operator.
 878      *
 879      * @param x the dividend
 880      * @param y the divisor
 881      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
 882      * @throws ArithmeticException if the divisor {@code y} is zero
 883      * @see Math#floorMod(int, int)
 884      * @see StrictMath#floorDiv(int, int)
 885      * @since 1.8
 886      */
 887     public static int floorMod(int x, int y) {
 888         return Math.floorMod(x , y);
 889     }
 890     /**
 891      * Returns the floor modulus of the {@code long} arguments.
 892      * <p>
 893      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
 894      * has the same sign as the divisor {@code y}, and
 895      * is in the range of {@code -abs(y) < r < +abs(y)}.
 896      * <p>
 897      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
 898      * <ul>
 899      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
 900      * </ul>
 901      * <p>
 902      * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
 903      * a comparison to the {@code %} operator.
 904      *
 905      * @param x the dividend
 906      * @param y the divisor
 907      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
 908      * @throws ArithmeticException if the divisor {@code y} is zero
 909      * @see Math#floorMod(long, long)
 910      * @see StrictMath#floorDiv(long, long)
 911      * @since 1.8
 912      */
 913     public static long floorMod(long x, long y) {
 914         return Math.floorMod(x, y);
 915     }
 916 
 917     /**
 918      * Returns the absolute value of an {@code int} value.
 919      * If the argument is not negative, the argument is returned.
 920      * If the argument is negative, the negation of the argument is returned.
 921      *
 922      * <p>Note that if the argument is equal to the value of
 923      * {@link Integer#MIN_VALUE}, the most negative representable
 924      * {@code int} value, the result is that same value, which is
 925      * negative.
 926      *
 927      * @param   a   the  argument whose absolute value is to be determined.
 928      * @return  the absolute value of the argument.
 929      */
 930     public static int abs(int a) {
 931         return Math.abs(a);
 932     }
 933 
 934     /**
 935      * Returns the absolute value of a {@code long} value.
 936      * If the argument is not negative, the argument is returned.
 937      * If the argument is negative, the negation of the argument is returned.
 938      *
 939      * <p>Note that if the argument is equal to the value of
 940      * {@link Long#MIN_VALUE}, the most negative representable
 941      * {@code long} value, the result is that same value, which
 942      * is negative.
 943      *
 944      * @param   a   the  argument whose absolute value is to be determined.
 945      * @return  the absolute value of the argument.
 946      */
 947     public static long abs(long a) {
 948         return Math.abs(a);
 949     }
 950 
 951     /**
 952      * Returns the absolute value of a {@code float} value.
 953      * If the argument is not negative, the argument is returned.
 954      * If the argument is negative, the negation of the argument is returned.
 955      * Special cases:
 956      * <ul><li>If the argument is positive zero or negative zero, the
 957      * result is positive zero.
 958      * <li>If the argument is infinite, the result is positive infinity.
 959      * <li>If the argument is NaN, the result is NaN.</ul>
 960      * In other words, the result is the same as the value of the expression:
 961      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
 962      *
 963      * @param   a   the argument whose absolute value is to be determined
 964      * @return  the absolute value of the argument.
 965      */
 966     public static float abs(float a) {
 967         return Math.abs(a);
 968     }
 969 
 970     /**
 971      * Returns the absolute value of a {@code double} value.
 972      * If the argument is not negative, the argument is returned.
 973      * If the argument is negative, the negation of the argument is returned.
 974      * Special cases:
 975      * <ul><li>If the argument is positive zero or negative zero, the result
 976      * is positive zero.
 977      * <li>If the argument is infinite, the result is positive infinity.
 978      * <li>If the argument is NaN, the result is NaN.</ul>
 979      * In other words, the result is the same as the value of the expression:
 980      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
 981      *
 982      * @param   a   the argument whose absolute value is to be determined
 983      * @return  the absolute value of the argument.
 984      */
 985     public static double abs(double a) {
 986         return Math.abs(a);
 987     }
 988 
 989     /**
 990      * Returns the greater of two {@code int} values. That is, the
 991      * result is the argument closer to the value of
 992      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
 993      * the result is that same value.
 994      *
 995      * @param   a   an argument.
 996      * @param   b   another argument.
 997      * @return  the larger of {@code a} and {@code b}.
 998      */
 999     public static int max(int a, int b) {
1000         return Math.max(a, b);
1001     }
1002 
1003     /**
1004      * Returns the greater of two {@code long} values. That is, the
1005      * result is the argument closer to the value of
1006      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1007      * the result is that same value.
1008      *
1009      * @param   a   an argument.
1010      * @param   b   another argument.
1011      * @return  the larger of {@code a} and {@code b}.
1012         */
1013     public static long max(long a, long b) {
1014         return Math.max(a, b);
1015     }
1016 
1017     /**
1018      * Returns the greater of two {@code float} values.  That is,
1019      * the result is the argument closer to positive infinity. If the
1020      * arguments have the same value, the result is that same
1021      * value. If either value is NaN, then the result is NaN.  Unlike
1022      * the numerical comparison operators, this method considers
1023      * negative zero to be strictly smaller than positive zero. If one
1024      * argument is positive zero and the other negative zero, the
1025      * result is positive zero.
1026      *
1027      * @param   a   an argument.
1028      * @param   b   another argument.
1029      * @return  the larger of {@code a} and {@code b}.
1030      */
1031     public static float max(float a, float b) {
1032         return Math.max(a, b);
1033     }
1034 
1035     /**
1036      * Returns the greater of two {@code double} values.  That
1037      * is, the result is the argument closer to positive infinity. If
1038      * the arguments have the same value, the result is that same
1039      * value. If either value is NaN, then the result is NaN.  Unlike
1040      * the numerical comparison operators, this method considers
1041      * negative zero to be strictly smaller than positive zero. If one
1042      * argument is positive zero and the other negative zero, the
1043      * result is positive zero.
1044      *
1045      * @param   a   an argument.
1046      * @param   b   another argument.
1047      * @return  the larger of {@code a} and {@code b}.
1048      */
1049     public static double max(double a, double b) {
1050         return Math.max(a, b);
1051     }
1052 
1053     /**
1054      * Returns the smaller of two {@code int} values. That is,
1055      * the result the argument closer to the value of
1056      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1057      * value, the result is that same value.
1058      *
1059      * @param   a   an argument.
1060      * @param   b   another argument.
1061      * @return  the smaller of {@code a} and {@code b}.
1062      */
1063     public static int min(int a, int b) {
1064         return Math.min(a, b);
1065     }
1066 
1067     /**
1068      * Returns the smaller of two {@code long} values. That is,
1069      * the result is the argument closer to the value of
1070      * {@link Long#MIN_VALUE}. If the arguments have the same
1071      * value, the result is that same value.
1072      *
1073      * @param   a   an argument.
1074      * @param   b   another argument.
1075      * @return  the smaller of {@code a} and {@code b}.
1076      */
1077     public static long min(long a, long b) {
1078         return Math.min(a, b);
1079     }
1080 
1081     /**
1082      * Returns the smaller of two {@code float} values.  That is,
1083      * the result is the value closer to negative infinity. If the
1084      * arguments have the same value, the result is that same
1085      * value. If either value is NaN, then the result is NaN.  Unlike
1086      * the numerical comparison operators, this method considers
1087      * negative zero to be strictly smaller than positive zero.  If
1088      * one argument is positive zero and the other is negative zero,
1089      * the result is negative zero.
1090      *
1091      * @param   a   an argument.
1092      * @param   b   another argument.
1093      * @return  the smaller of {@code a} and {@code b.}
1094      */
1095     public static float min(float a, float b) {
1096         return Math.min(a, b);
1097     }
1098 
1099     /**
1100      * Returns the smaller of two {@code double} values.  That
1101      * is, the result is the value closer to negative infinity. If the
1102      * arguments have the same value, the result is that same
1103      * value. If either value is NaN, then the result is NaN.  Unlike
1104      * the numerical comparison operators, this method considers
1105      * negative zero to be strictly smaller than positive zero. If one
1106      * argument is positive zero and the other is negative zero, the
1107      * result is negative zero.
1108      *
1109      * @param   a   an argument.
1110      * @param   b   another argument.
1111      * @return  the smaller of {@code a} and {@code b}.
1112      */
1113     public static double min(double a, double b) {
1114         return Math.min(a, b);
1115     }
1116 
1117     /**
1118      * Returns the size of an ulp of the argument.  An ulp, unit in
1119      * the last place, of a {@code double} value is the positive
1120      * distance between this floating-point value and the {@code
1121      * double} value next larger in magnitude.  Note that for non-NaN
1122      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1123      *
1124      * <p>Special Cases:
1125      * <ul>
1126      * <li> If the argument is NaN, then the result is NaN.
1127      * <li> If the argument is positive or negative infinity, then the
1128      * result is positive infinity.
1129      * <li> If the argument is positive or negative zero, then the result is
1130      * {@code Double.MIN_VALUE}.
1131      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1132      * the result is equal to 2<sup>971</sup>.
1133      * </ul>
1134      *
1135      * @param d the floating-point value whose ulp is to be returned
1136      * @return the size of an ulp of the argument
1137      * @author Joseph D. Darcy
1138      * @since 1.5
1139      */
1140     public static double ulp(double d) {
1141         return Math.ulp(d);
1142     }
1143 
1144     /**
1145      * Returns the size of an ulp of the argument.  An ulp, unit in
1146      * the last place, of a {@code float} value is the positive
1147      * distance between this floating-point value and the {@code
1148      * float} value next larger in magnitude.  Note that for non-NaN
1149      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1150      *
1151      * <p>Special Cases:
1152      * <ul>
1153      * <li> If the argument is NaN, then the result is NaN.
1154      * <li> If the argument is positive or negative infinity, then the
1155      * result is positive infinity.
1156      * <li> If the argument is positive or negative zero, then the result is
1157      * {@code Float.MIN_VALUE}.
1158      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1159      * the result is equal to 2<sup>104</sup>.
1160      * </ul>
1161      *
1162      * @param f the floating-point value whose ulp is to be returned
1163      * @return the size of an ulp of the argument
1164      * @author Joseph D. Darcy
1165      * @since 1.5
1166      */
1167     public static float ulp(float f) {
1168         return Math.ulp(f);
1169     }
1170 
1171     /**
1172      * Returns the signum function of the argument; zero if the argument
1173      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1174      * argument is less than zero.
1175      *
1176      * <p>Special Cases:
1177      * <ul>
1178      * <li> If the argument is NaN, then the result is NaN.
1179      * <li> If the argument is positive zero or negative zero, then the
1180      *      result is the same as the argument.
1181      * </ul>
1182      *
1183      * @param d the floating-point value whose signum is to be returned
1184      * @return the signum function of the argument
1185      * @author Joseph D. Darcy
1186      * @since 1.5
1187      */
1188     public static double signum(double d) {
1189         return Math.signum(d);
1190     }
1191 
1192     /**
1193      * Returns the signum function of the argument; zero if the argument
1194      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1195      * argument is less than zero.
1196      *
1197      * <p>Special Cases:
1198      * <ul>
1199      * <li> If the argument is NaN, then the result is NaN.
1200      * <li> If the argument is positive zero or negative zero, then the
1201      *      result is the same as the argument.
1202      * </ul>
1203      *
1204      * @param f the floating-point value whose signum is to be returned
1205      * @return the signum function of the argument
1206      * @author Joseph D. Darcy
1207      * @since 1.5
1208      */
1209     public static float signum(float f) {
1210         return Math.signum(f);
1211     }
1212 
1213     /**
1214      * Returns the hyperbolic sine of a {@code double} value.
1215      * The hyperbolic sine of <i>x</i> is defined to be
1216      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1217      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1218      *
1219      * <p>Special cases:
1220      * <ul>
1221      *
1222      * <li>If the argument is NaN, then the result is NaN.
1223      *
1224      * <li>If the argument is infinite, then the result is an infinity
1225      * with the same sign as the argument.
1226      *
1227      * <li>If the argument is zero, then the result is a zero with the
1228      * same sign as the argument.
1229      *
1230      * </ul>
1231      *
1232      * @param   x The number whose hyperbolic sine is to be returned.
1233      * @return  The hyperbolic sine of {@code x}.
1234      * @since 1.5
1235      */
1236     public static native double sinh(double x);
1237 
1238     /**
1239      * Returns the hyperbolic cosine of a {@code double} value.
1240      * The hyperbolic cosine of <i>x</i> is defined to be
1241      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1242      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1243      *
1244      * <p>Special cases:
1245      * <ul>
1246      *
1247      * <li>If the argument is NaN, then the result is NaN.
1248      *
1249      * <li>If the argument is infinite, then the result is positive
1250      * infinity.
1251      *
1252      * <li>If the argument is zero, then the result is {@code 1.0}.
1253      *
1254      * </ul>
1255      *
1256      * @param   x The number whose hyperbolic cosine is to be returned.
1257      * @return  The hyperbolic cosine of {@code x}.
1258      * @since 1.5
1259      */
1260     public static native double cosh(double x);
1261 
1262     /**
1263      * Returns the hyperbolic tangent of a {@code double} value.
1264      * The hyperbolic tangent of <i>x</i> is defined to be
1265      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1266      * in other words, {@linkplain Math#sinh
1267      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1268      * that the absolute value of the exact tanh is always less than
1269      * 1.
1270      *
1271      * <p>Special cases:
1272      * <ul>
1273      *
1274      * <li>If the argument is NaN, then the result is NaN.
1275      *
1276      * <li>If the argument is zero, then the result is a zero with the
1277      * same sign as the argument.
1278      *
1279      * <li>If the argument is positive infinity, then the result is
1280      * {@code +1.0}.
1281      *
1282      * <li>If the argument is negative infinity, then the result is
1283      * {@code -1.0}.
1284      *
1285      * </ul>
1286      *
1287      * @param   x The number whose hyperbolic tangent is to be returned.
1288      * @return  The hyperbolic tangent of {@code x}.
1289      * @since 1.5
1290      */
1291     public static native double tanh(double x);
1292 
1293     /**
1294      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1295      * without intermediate overflow or underflow.
1296      *
1297      * <p>Special cases:
1298      * <ul>
1299      *
1300      * <li> If either argument is infinite, then the result
1301      * is positive infinity.
1302      *
1303      * <li> If either argument is NaN and neither argument is infinite,
1304      * then the result is NaN.
1305      *
1306      * </ul>
1307      *
1308      * @param x a value
1309      * @param y a value
1310      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1311      * without intermediate overflow or underflow
1312      * @since 1.5
1313      */
1314     public static native double hypot(double x, double y);
1315 
1316     /**
1317      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
1318      * <i>x</i> near 0, the exact sum of
1319      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
1320      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1321      *
1322      * <p>Special cases:
1323      * <ul>
1324      * <li>If the argument is NaN, the result is NaN.
1325      *
1326      * <li>If the argument is positive infinity, then the result is
1327      * positive infinity.
1328      *
1329      * <li>If the argument is negative infinity, then the result is
1330      * -1.0.
1331      *
1332      * <li>If the argument is zero, then the result is a zero with the
1333      * same sign as the argument.
1334      *
1335      * </ul>
1336      *
1337      * @param   x   the exponent to raise <i>e</i> to in the computation of
1338      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
1339      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
1340      * @since 1.5
1341      */
1342     public static native double expm1(double x);
1343 
1344     /**
1345      * Returns the natural logarithm of the sum of the argument and 1.
1346      * Note that for small values {@code x}, the result of
1347      * {@code log1p(x)} is much closer to the true result of ln(1
1348      * + {@code x}) than the floating-point evaluation of
1349      * {@code log(1.0+x)}.
1350      *
1351      * <p>Special cases:
1352      * <ul>
1353      *
1354      * <li>If the argument is NaN or less than -1, then the result is
1355      * NaN.
1356      *
1357      * <li>If the argument is positive infinity, then the result is
1358      * positive infinity.
1359      *
1360      * <li>If the argument is negative one, then the result is
1361      * negative infinity.
1362      *
1363      * <li>If the argument is zero, then the result is a zero with the
1364      * same sign as the argument.
1365      *
1366      * </ul>
1367      *
1368      * @param   x   a value
1369      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
1370      * log of {@code x}&nbsp;+&nbsp;1
1371      * @since 1.5
1372      */
1373     public static native double log1p(double x);
1374 
1375     /**
1376      * Returns the first floating-point argument with the sign of the
1377      * second floating-point argument.  For this method, a NaN
1378      * {@code sign} argument is always treated as if it were
1379      * positive.
1380      *
1381      * @param magnitude  the parameter providing the magnitude of the result
1382      * @param sign   the parameter providing the sign of the result
1383      * @return a value with the magnitude of {@code magnitude}
1384      * and the sign of {@code sign}.
1385      * @since 1.6
1386      */
1387     public static double copySign(double magnitude, double sign) {
1388         return Math.copySign(magnitude, (Double.isNaN(sign)?1.0d:sign));
1389     }
1390 
1391     /**
1392      * Returns the first floating-point argument with the sign of the
1393      * second floating-point argument.  For this method, a NaN
1394      * {@code sign} argument is always treated as if it were
1395      * positive.
1396      *
1397      * @param magnitude  the parameter providing the magnitude of the result
1398      * @param sign   the parameter providing the sign of the result
1399      * @return a value with the magnitude of {@code magnitude}
1400      * and the sign of {@code sign}.
1401      * @since 1.6
1402      */
1403     public static float copySign(float magnitude, float sign) {
1404         return Math.copySign(magnitude, (Float.isNaN(sign)?1.0f:sign));
1405     }
1406     /**
1407      * Returns the unbiased exponent used in the representation of a
1408      * {@code float}.  Special cases:
1409      *
1410      * <ul>
1411      * <li>If the argument is NaN or infinite, then the result is
1412      * {@link Float#MAX_EXPONENT} + 1.
1413      * <li>If the argument is zero or subnormal, then the result is
1414      * {@link Float#MIN_EXPONENT} -1.
1415      * </ul>
1416      * @param f a {@code float} value
1417      * @return the unbiased exponent of the argument
1418      * @since 1.6
1419      */
1420     public static int getExponent(float f) {
1421         return Math.getExponent(f);
1422     }
1423 
1424     /**
1425      * Returns the unbiased exponent used in the representation of a
1426      * {@code double}.  Special cases:
1427      *
1428      * <ul>
1429      * <li>If the argument is NaN or infinite, then the result is
1430      * {@link Double#MAX_EXPONENT} + 1.
1431      * <li>If the argument is zero or subnormal, then the result is
1432      * {@link Double#MIN_EXPONENT} -1.
1433      * </ul>
1434      * @param d a {@code double} value
1435      * @return the unbiased exponent of the argument
1436      * @since 1.6
1437      */
1438     public static int getExponent(double d) {
1439         return Math.getExponent(d);
1440     }
1441 
1442     /**
1443      * Returns the floating-point number adjacent to the first
1444      * argument in the direction of the second argument.  If both
1445      * arguments compare as equal the second argument is returned.
1446      *
1447      * <p>Special cases:
1448      * <ul>
1449      * <li> If either argument is a NaN, then NaN is returned.
1450      *
1451      * <li> If both arguments are signed zeros, {@code direction}
1452      * is returned unchanged (as implied by the requirement of
1453      * returning the second argument if the arguments compare as
1454      * equal).
1455      *
1456      * <li> If {@code start} is
1457      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
1458      * has a value such that the result should have a smaller
1459      * magnitude, then a zero with the same sign as {@code start}
1460      * is returned.
1461      *
1462      * <li> If {@code start} is infinite and
1463      * {@code direction} has a value such that the result should
1464      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1465      * same sign as {@code start} is returned.
1466      *
1467      * <li> If {@code start} is equal to &plusmn;
1468      * {@link Double#MAX_VALUE} and {@code direction} has a
1469      * value such that the result should have a larger magnitude, an
1470      * infinity with same sign as {@code start} is returned.
1471      * </ul>
1472      *
1473      * @param start  starting floating-point value
1474      * @param direction value indicating which of
1475      * {@code start}'s neighbors or {@code start} should
1476      * be returned
1477      * @return The floating-point number adjacent to {@code start} in the
1478      * direction of {@code direction}.
1479      * @since 1.6
1480      */
1481     public static double nextAfter(double start, double direction) {
1482         return Math.nextAfter(start, direction);
1483     }
1484 
1485     /**
1486      * Returns the floating-point number adjacent to the first
1487      * argument in the direction of the second argument.  If both
1488      * arguments compare as equal a value equivalent to the second argument
1489      * is returned.
1490      *
1491      * <p>Special cases:
1492      * <ul>
1493      * <li> If either argument is a NaN, then NaN is returned.
1494      *
1495      * <li> If both arguments are signed zeros, a value equivalent
1496      * to {@code direction} is returned.
1497      *
1498      * <li> If {@code start} is
1499      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
1500      * has a value such that the result should have a smaller
1501      * magnitude, then a zero with the same sign as {@code start}
1502      * is returned.
1503      *
1504      * <li> If {@code start} is infinite and
1505      * {@code direction} has a value such that the result should
1506      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1507      * same sign as {@code start} is returned.
1508      *
1509      * <li> If {@code start} is equal to &plusmn;
1510      * {@link Float#MAX_VALUE} and {@code direction} has a
1511      * value such that the result should have a larger magnitude, an
1512      * infinity with same sign as {@code start} is returned.
1513      * </ul>
1514      *
1515      * @param start  starting floating-point value
1516      * @param direction value indicating which of
1517      * {@code start}'s neighbors or {@code start} should
1518      * be returned
1519      * @return The floating-point number adjacent to {@code start} in the
1520      * direction of {@code direction}.
1521      * @since 1.6
1522      */
1523     public static float nextAfter(float start, double direction) {
1524         return Math.nextAfter(start, direction);
1525     }
1526 
1527     /**
1528      * Returns the floating-point value adjacent to {@code d} in
1529      * the direction of positive infinity.  This method is
1530      * semantically equivalent to {@code nextAfter(d,
1531      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
1532      * implementation may run faster than its equivalent
1533      * {@code nextAfter} call.
1534      *
1535      * <p>Special Cases:
1536      * <ul>
1537      * <li> If the argument is NaN, the result is NaN.
1538      *
1539      * <li> If the argument is positive infinity, the result is
1540      * positive infinity.
1541      *
1542      * <li> If the argument is zero, the result is
1543      * {@link Double#MIN_VALUE}
1544      *
1545      * </ul>
1546      *
1547      * @param d starting floating-point value
1548      * @return The adjacent floating-point value closer to positive
1549      * infinity.
1550      * @since 1.6
1551      */
1552     public static double nextUp(double d) {
1553         return Math.nextUp(d);
1554     }
1555 
1556     /**
1557      * Returns the floating-point value adjacent to {@code f} in
1558      * the direction of positive infinity.  This method is
1559      * semantically equivalent to {@code nextAfter(f,
1560      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
1561      * implementation may run faster than its equivalent
1562      * {@code nextAfter} call.
1563      *
1564      * <p>Special Cases:
1565      * <ul>
1566      * <li> If the argument is NaN, the result is NaN.
1567      *
1568      * <li> If the argument is positive infinity, the result is
1569      * positive infinity.
1570      *
1571      * <li> If the argument is zero, the result is
1572      * {@link Float#MIN_VALUE}
1573      *
1574      * </ul>
1575      *
1576      * @param f starting floating-point value
1577      * @return The adjacent floating-point value closer to positive
1578      * infinity.
1579      * @since 1.6
1580      */
1581     public static float nextUp(float f) {
1582         return Math.nextUp(f);
1583     }
1584 
1585     /**
1586      * Returns the floating-point value adjacent to {@code d} in
1587      * the direction of negative infinity.  This method is
1588      * semantically equivalent to {@code nextAfter(d,
1589      * Double.NEGATIVE_INFINITY)}; however, a
1590      * {@code nextDown} implementation may run faster than its
1591      * equivalent {@code nextAfter} call.
1592      *
1593      * <p>Special Cases:
1594      * <ul>
1595      * <li> If the argument is NaN, the result is NaN.
1596      *
1597      * <li> If the argument is negative infinity, the result is
1598      * negative infinity.
1599      *
1600      * <li> If the argument is zero, the result is
1601      * {@code -Double.MIN_VALUE}
1602      *
1603      * </ul>
1604      *
1605      * @param d  starting floating-point value
1606      * @return The adjacent floating-point value closer to negative
1607      * infinity.
1608      * @since 1.8
1609      */
1610     public static double nextDown(double d) {
1611         return Math.nextDown(d);
1612     }
1613 
1614     /**
1615      * Returns the floating-point value adjacent to {@code f} in
1616      * the direction of negative infinity.  This method is
1617      * semantically equivalent to {@code nextAfter(f,
1618      * Float.NEGATIVE_INFINITY)}; however, a
1619      * {@code nextDown} implementation may run faster than its
1620      * equivalent {@code nextAfter} call.
1621      *
1622      * <p>Special Cases:
1623      * <ul>
1624      * <li> If the argument is NaN, the result is NaN.
1625      *
1626      * <li> If the argument is negative infinity, the result is
1627      * negative infinity.
1628      *
1629      * <li> If the argument is zero, the result is
1630      * {@code -Float.MIN_VALUE}
1631      *
1632      * </ul>
1633      *
1634      * @param f  starting floating-point value
1635      * @return The adjacent floating-point value closer to negative
1636      * infinity.
1637      * @since 1.8
1638      */
1639     public static float nextDown(float f) {
1640         return Math.nextDown(f);
1641     }
1642 
1643     /**
1644      * Returns {@code d} &times;
1645      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1646      * by a single correctly rounded floating-point multiply to a
1647      * member of the double value set.  See the Java
1648      * Language Specification for a discussion of floating-point
1649      * value sets.  If the exponent of the result is between {@link
1650      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
1651      * answer is calculated exactly.  If the exponent of the result
1652      * would be larger than {@code Double.MAX_EXPONENT}, an
1653      * infinity is returned.  Note that if the result is subnormal,
1654      * precision may be lost; that is, when {@code scalb(x, n)}
1655      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1656      * <i>x</i>.  When the result is non-NaN, the result has the same
1657      * sign as {@code d}.
1658      *
1659      * <p>Special cases:
1660      * <ul>
1661      * <li> If the first argument is NaN, NaN is returned.
1662      * <li> If the first argument is infinite, then an infinity of the
1663      * same sign is returned.
1664      * <li> If the first argument is zero, then a zero of the same
1665      * sign is returned.
1666      * </ul>
1667      *
1668      * @param d number to be scaled by a power of two.
1669      * @param scaleFactor power of 2 used to scale {@code d}
1670      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
1671      * @since 1.6
1672      */
1673     public static double scalb(double d, int scaleFactor) {
1674         return Math.scalb(d, scaleFactor);
1675     }
1676 
1677     /**
1678      * Returns {@code f} &times;
1679      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1680      * by a single correctly rounded floating-point multiply to a
1681      * member of the float value set.  See the Java
1682      * Language Specification for a discussion of floating-point
1683      * value sets.  If the exponent of the result is between {@link
1684      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
1685      * answer is calculated exactly.  If the exponent of the result
1686      * would be larger than {@code Float.MAX_EXPONENT}, an
1687      * infinity is returned.  Note that if the result is subnormal,
1688      * precision may be lost; that is, when {@code scalb(x, n)}
1689      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1690      * <i>x</i>.  When the result is non-NaN, the result has the same
1691      * sign as {@code f}.
1692      *
1693      * <p>Special cases:
1694      * <ul>
1695      * <li> If the first argument is NaN, NaN is returned.
1696      * <li> If the first argument is infinite, then an infinity of the
1697      * same sign is returned.
1698      * <li> If the first argument is zero, then a zero of the same
1699      * sign is returned.
1700      * </ul>
1701      *
1702      * @param f number to be scaled by a power of two.
1703      * @param scaleFactor power of 2 used to scale {@code f}
1704      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
1705      * @since 1.6
1706      */
1707     public static float scalb(float f, int scaleFactor) {
1708         return Math.scalb(f, scaleFactor);
1709     }
1710 }