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      * Constant by which to multiply an angular value in degrees to obtain an
 102      * angular value in radians.
 103      */
 104     private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
 105 
 106     /**
 107      * Constant by which to multiply an angular value in radians to obtain an
 108      * angular value in degrees.
 109      */
 110 
 111     private static final double RADIANS_TO_DEGREES = 57.29577951308232;
 112 
 113     /**
 114      * Returns the trigonometric sine of an angle. Special cases:
 115      * <ul><li>If the argument is NaN or an infinity, then the
 116      * result is NaN.
 117      * <li>If the argument is zero, then the result is a zero with the
 118      * same sign as the argument.</ul>
 119      *
 120      * @param   a   an angle, in radians.
 121      * @return  the sine of the argument.
 122      */
 123     public static native double sin(double a);
 124 
 125     /**
 126      * Returns the trigonometric cosine of an angle. Special cases:
 127      * <ul><li>If the argument is NaN or an infinity, then the
 128      * result is NaN.</ul>
 129      *
 130      * @param   a   an angle, in radians.
 131      * @return  the cosine of the argument.
 132      */
 133     public static native double cos(double a);
 134 
 135     /**
 136      * Returns the trigonometric tangent of an angle. Special cases:
 137      * <ul><li>If the argument is NaN or an infinity, then the result
 138      * 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   an angle, in radians.
 143      * @return  the tangent of the argument.
 144      */
 145     public static native double tan(double a);
 146 
 147     /**
 148      * Returns the arc sine of a value; the returned angle is in the
 149      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 150      * <ul><li>If the argument is NaN or its absolute value is greater
 151      * than 1, then the result is NaN.
 152      * <li>If the argument is zero, then the result is a zero with the
 153      * same sign as the argument.</ul>
 154      *
 155      * @param   a   the value whose arc sine is to be returned.
 156      * @return  the arc sine of the argument.
 157      */
 158     public static native double asin(double a);
 159 
 160     /**
 161      * Returns the arc cosine of a value; the returned angle is in the
 162      * range 0.0 through <i>pi</i>.  Special case:
 163      * <ul><li>If the argument is NaN or its absolute value is greater
 164      * than 1, then the result is NaN.</ul>
 165      *
 166      * @param   a   the value whose arc cosine is to be returned.
 167      * @return  the arc cosine of the argument.
 168      */
 169     public static native double acos(double a);
 170 
 171     /**
 172      * Returns the arc tangent of a value; the returned angle is in the
 173      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
 174      * <ul><li>If the argument is NaN, then the result is NaN.
 175      * <li>If the argument is zero, then the result is a zero with the
 176      * same sign as the argument.</ul>
 177      *
 178      * @param   a   the value whose arc tangent is to be returned.
 179      * @return  the arc tangent of the argument.
 180      */
 181     public static native double atan(double a);
 182 
 183     /**
 184      * Converts an angle measured in degrees to an approximately
 185      * equivalent angle measured in radians.  The conversion from
 186      * degrees to radians is generally inexact.
 187      *
 188      * @param   angdeg   an angle, in degrees
 189      * @return  the measurement of the angle {@code angdeg}
 190      *          in radians.
 191      */
 192     public static strictfp double toRadians(double angdeg) {
 193         // Do not delegate to Math.toRadians(angdeg) because
 194         // this method has the strictfp modifier.
 195         return angdeg * DEGREES_TO_RADIANS;
 196     }
 197 
 198     /**
 199      * Converts an angle measured in radians to an approximately
 200      * equivalent angle measured in degrees.  The conversion from
 201      * radians to degrees is generally inexact; users should
 202      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
 203      * equal {@code 0.0}.
 204      *
 205      * @param   angrad   an angle, in radians
 206      * @return  the measurement of the angle {@code angrad}
 207      *          in degrees.
 208      */
 209     public static strictfp double toDegrees(double angrad) {
 210         // Do not delegate to Math.toDegrees(angrad) because
 211         // this method has the strictfp modifier.
 212         return angrad * RADIANS_TO_DEGREES;
 213     }
 214 
 215     /**
 216      * Returns Euler's number <i>e</i> raised to the power of a
 217      * {@code double} value. Special cases:
 218      * <ul><li>If the argument is NaN, the result is NaN.
 219      * <li>If the argument is positive infinity, then the result is
 220      * positive infinity.
 221      * <li>If the argument is negative infinity, then the result is
 222      * positive zero.</ul>
 223      *
 224      * @param   a   the exponent to raise <i>e</i> to.
 225      * @return  the value <i>e</i><sup>{@code a}</sup>,
 226      *          where <i>e</i> is the base of the natural logarithms.
 227      */
 228     public static native double exp(double a);
 229 
 230     /**
 231      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
 232      * value. Special cases:
 233      * <ul><li>If the argument is NaN or less than zero, then the result
 234      * is NaN.
 235      * <li>If the argument is positive infinity, then the result is
 236      * positive infinity.
 237      * <li>If the argument is positive zero or negative zero, then the
 238      * result is negative infinity.</ul>
 239      *
 240      * @param   a   a value
 241      * @return  the value ln&nbsp;{@code a}, the natural logarithm of
 242      *          {@code a}.
 243      */
 244     public static native double log(double a);
 245 
 246 
 247     /**
 248      * Returns the base 10 logarithm of a {@code double} value.
 249      * Special cases:
 250      *
 251      * <ul><li>If the argument is NaN or less than zero, then the result
 252      * is NaN.
 253      * <li>If the argument is positive infinity, then the result is
 254      * positive infinity.
 255      * <li>If the argument is positive zero or negative zero, then the
 256      * result is negative infinity.
 257      * <li> If the argument is equal to 10<sup><i>n</i></sup> for
 258      * integer <i>n</i>, then the result is <i>n</i>.
 259      * </ul>
 260      *
 261      * @param   a   a value
 262      * @return  the base 10 logarithm of  {@code a}.
 263      * @since 1.5
 264      */
 265     public static native double log10(double a);
 266 
 267     /**
 268      * Returns the correctly rounded positive square root of a
 269      * {@code double} value.
 270      * Special cases:
 271      * <ul><li>If the argument is NaN or less than zero, then the result
 272      * is NaN.
 273      * <li>If the argument is positive infinity, then the result is positive
 274      * infinity.
 275      * <li>If the argument is positive zero or negative zero, then the
 276      * result is the same as the argument.</ul>
 277      * Otherwise, the result is the {@code double} value closest to
 278      * the true mathematical square root of the argument value.
 279      *
 280      * @param   a   a value.
 281      * @return  the positive square root of {@code a}.
 282      */
 283     public static native double sqrt(double a);
 284 
 285     /**
 286      * Returns the cube root of a {@code double} value.  For
 287      * positive finite {@code x}, {@code cbrt(-x) ==
 288      * -cbrt(x)}; that is, the cube root of a negative value is
 289      * the negative of the cube root of that value's magnitude.
 290      * Special cases:
 291      *
 292      * <ul>
 293      *
 294      * <li>If the argument is NaN, then the result is NaN.
 295      *
 296      * <li>If the argument is infinite, then the result is an infinity
 297      * with the same sign as the argument.
 298      *
 299      * <li>If the argument is zero, then the result is a zero with the
 300      * same sign as the argument.
 301      *
 302      * </ul>
 303      *
 304      * @param   a   a value.
 305      * @return  the cube root of {@code a}.
 306      * @since 1.5
 307      */
 308     public static native double cbrt(double a);
 309 
 310     /**
 311      * Computes the remainder operation on two arguments as prescribed
 312      * by the IEEE 754 standard.
 313      * The remainder value is mathematically equal to
 314      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
 315      * where <i>n</i> is the mathematical integer closest to the exact
 316      * mathematical value of the quotient {@code f1/f2}, and if two
 317      * mathematical integers are equally close to {@code f1/f2},
 318      * then <i>n</i> is the integer that is even. If the remainder is
 319      * zero, its sign is the same as the sign of the first argument.
 320      * Special cases:
 321      * <ul><li>If either argument is NaN, or the first argument is infinite,
 322      * or the second argument is positive zero or negative zero, then the
 323      * result is NaN.
 324      * <li>If the first argument is finite and the second argument is
 325      * infinite, then the result is the same as the first argument.</ul>
 326      *
 327      * @param   f1   the dividend.
 328      * @param   f2   the divisor.
 329      * @return  the remainder when {@code f1} is divided by
 330      *          {@code f2}.
 331      */
 332     public static native double IEEEremainder(double f1, double f2);
 333 
 334     /**
 335      * Returns the smallest (closest to negative infinity)
 336      * {@code double} value that is greater than or equal to the
 337      * argument and is equal to a mathematical integer. Special cases:
 338      * <ul><li>If the argument value is already equal to a
 339      * mathematical integer, then the result is the same as the
 340      * argument.  <li>If the argument is NaN or an infinity or
 341      * positive zero or negative zero, then the result is the same as
 342      * the argument.  <li>If the argument value is less than zero but
 343      * greater than -1.0, then the result is negative zero.</ul> Note
 344      * that the value of {@code StrictMath.ceil(x)} is exactly the
 345      * value of {@code -StrictMath.floor(-x)}.
 346      *
 347      * @param   a   a value.
 348      * @return  the smallest (closest to negative infinity)
 349      *          floating-point value that is greater than or equal to
 350      *          the argument and is equal to a mathematical integer.
 351      */
 352     public static double ceil(double a) {
 353         return floorOrCeil(a, -0.0, 1.0, 1.0);
 354     }
 355 
 356     /**
 357      * Returns the largest (closest to positive infinity)
 358      * {@code double} value that is less than or equal to the
 359      * argument and is equal to a mathematical integer. Special cases:
 360      * <ul><li>If the argument value is already equal to a
 361      * mathematical integer, then the result is the same as the
 362      * argument.  <li>If the argument is NaN or an infinity or
 363      * positive zero or negative zero, then the result is the same as
 364      * the argument.</ul>
 365      *
 366      * @param   a   a value.
 367      * @return  the largest (closest to positive infinity)
 368      *          floating-point value that less than or equal to the argument
 369      *          and is equal to a mathematical integer.
 370      */
 371     public static double floor(double a) {
 372         return floorOrCeil(a, -1.0, 0.0, -1.0);
 373     }
 374 
 375     /**
 376      * Internal method to share logic between floor and ceil.
 377      *
 378      * @param a the value to be floored or ceiled
 379      * @param negativeBoundary result for values in (-1, 0)
 380      * @param positiveBoundary result for values in (0, 1)
 381      * @param increment value to add when the argument is non-integral
 382      */
 383     private static double floorOrCeil(double a,
 384                                       double negativeBoundary,
 385                                       double positiveBoundary,
 386                                       double sign) {
 387         int exponent = Math.getExponent(a);
 388 
 389         if (exponent < 0) {
 390             /*
 391              * Absolute value of argument is less than 1.
 392              * floorOrceil(-0.0) => -0.0
 393              * floorOrceil(+0.0) => +0.0
 394              */
 395             return ((a == 0.0) ? a :
 396                     ( (a < 0.0) ?  negativeBoundary : positiveBoundary) );
 397         } else if (exponent >= 52) {
 398             /*
 399              * Infinity, NaN, or a value so large it must be integral.
 400              */
 401             return a;
 402         }
 403         // Else the argument is either an integral value already XOR it
 404         // has to be rounded to one.
 405         assert exponent >= 0 && exponent <= 51;
 406 
 407         long doppel = Double.doubleToRawLongBits(a);
 408         long mask   = DoubleConsts.SIGNIF_BIT_MASK >> exponent;
 409 
 410         if ( (mask & doppel) == 0L )
 411             return a; // integral value
 412         else {
 413             double result = Double.longBitsToDouble(doppel & (~mask));
 414             if (sign*a > 0.0)
 415                 result = result + sign;
 416             return result;
 417         }
 418     }
 419 
 420     /**
 421      * Returns the {@code double} value that is closest in value
 422      * to the argument and is equal to a mathematical integer. If two
 423      * {@code double} values that are mathematical integers are
 424      * equally close to the value of the argument, the result is the
 425      * integer value that is even. Special cases:
 426      * <ul><li>If the argument value is already equal to a mathematical
 427      * integer, then the result is the same as the argument.
 428      * <li>If the argument is NaN or an infinity or positive zero or negative
 429      * zero, then the result is the same as the argument.</ul>
 430      *
 431      * @param   a   a value.
 432      * @return  the closest floating-point value to {@code a} that is
 433      *          equal to a mathematical integer.
 434      * @author Joseph D. Darcy
 435      */
 436     public static double rint(double a) {
 437         /*
 438          * If the absolute value of a is not less than 2^52, it
 439          * is either a finite integer (the double format does not have
 440          * enough significand bits for a number that large to have any
 441          * fractional portion), an infinity, or a NaN.  In any of
 442          * these cases, rint of the argument is the argument.
 443          *
 444          * Otherwise, the sum (twoToThe52 + a ) will properly round
 445          * away any fractional portion of a since ulp(twoToThe52) ==
 446          * 1.0; subtracting out twoToThe52 from this sum will then be
 447          * exact and leave the rounded integer portion of a.
 448          *
 449          * This method does *not* need to be declared strictfp to get
 450          * fully reproducible results.  Whether or not a method is
 451          * declared strictfp can only make a difference in the
 452          * returned result if some operation would overflow or
 453          * underflow with strictfp semantics.  The operation
 454          * (twoToThe52 + a ) cannot overflow since large values of a
 455          * are screened out; the add cannot underflow since twoToThe52
 456          * is too large.  The subtraction ((twoToThe52 + a ) -
 457          * twoToThe52) will be exact as discussed above and thus
 458          * cannot overflow or meaningfully underflow.  Finally, the
 459          * last multiply in the return statement is by plus or minus
 460          * 1.0, which is exact too.
 461          */
 462         double twoToThe52 = (double)(1L << 52); // 2^52
 463         double sign = Math.copySign(1.0, a); // preserve sign info
 464         a = Math.abs(a);
 465 
 466         if (a < twoToThe52) { // E_min <= ilogb(a) <= 51
 467             a = ((twoToThe52 + a ) - twoToThe52);
 468         }
 469 
 470         return sign * a; // restore original sign
 471     }
 472 
 473     /**
 474      * Returns the angle <i>theta</i> from the conversion of rectangular
 475      * coordinates ({@code x},&nbsp;{@code y}) to polar
 476      * coordinates (r,&nbsp;<i>theta</i>).
 477      * This method computes the phase <i>theta</i> by computing an arc tangent
 478      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
 479      * cases:
 480      * <ul><li>If either argument is NaN, then the result is NaN.
 481      * <li>If the first argument is positive zero and the second argument
 482      * is positive, or the first argument is positive and finite and the
 483      * second argument is positive infinity, then the result is positive
 484      * zero.
 485      * <li>If the first argument is negative zero and the second argument
 486      * is positive, or the first argument is negative and finite and the
 487      * second argument is positive infinity, then the result is negative zero.
 488      * <li>If the first argument is positive zero and the second argument
 489      * is negative, or the first argument is positive and finite and the
 490      * second argument is negative infinity, then the result is the
 491      * {@code double} value closest to <i>pi</i>.
 492      * <li>If the first argument is negative zero and the second argument
 493      * is negative, or the first argument is negative and finite and the
 494      * second argument is negative infinity, then the result is the
 495      * {@code double} value closest to -<i>pi</i>.
 496      * <li>If the first argument is positive and the second argument is
 497      * positive zero or negative zero, or the first argument is positive
 498      * infinity and the second argument is finite, then the result is the
 499      * {@code double} value closest to <i>pi</i>/2.
 500      * <li>If the first argument is negative and the second argument is
 501      * positive zero or negative zero, or the first argument is negative
 502      * infinity and the second argument is finite, then the result is the
 503      * {@code double} value closest to -<i>pi</i>/2.
 504      * <li>If both arguments are positive infinity, then the result is the
 505      * {@code double} value closest to <i>pi</i>/4.
 506      * <li>If the first argument is positive infinity and the second argument
 507      * is negative infinity, then the result is the {@code double}
 508      * value closest to 3*<i>pi</i>/4.
 509      * <li>If the first argument is negative infinity and the second argument
 510      * is positive infinity, then the result is the {@code double} value
 511      * closest to -<i>pi</i>/4.
 512      * <li>If both arguments are negative infinity, then the result is the
 513      * {@code double} value closest to -3*<i>pi</i>/4.</ul>
 514      *
 515      * @param   y   the ordinate coordinate
 516      * @param   x   the abscissa coordinate
 517      * @return  the <i>theta</i> component of the point
 518      *          (<i>r</i>,&nbsp;<i>theta</i>)
 519      *          in polar coordinates that corresponds to the point
 520      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
 521      */
 522     public static native double atan2(double y, double x);
 523 
 524 
 525     /**
 526      * Returns the value of the first argument raised to the power of the
 527      * second argument. Special cases:
 528      *
 529      * <ul><li>If the second argument is positive or negative zero, then the
 530      * result is 1.0.
 531      * <li>If the second argument is 1.0, then the result is the same as the
 532      * first argument.
 533      * <li>If the second argument is NaN, then the result is NaN.
 534      * <li>If the first argument is NaN and the second argument is nonzero,
 535      * then the result is NaN.
 536      *
 537      * <li>If
 538      * <ul>
 539      * <li>the absolute value of the first argument is greater than 1
 540      * and the second argument is positive infinity, or
 541      * <li>the absolute value of the first argument is less than 1 and
 542      * the second argument is negative infinity,
 543      * </ul>
 544      * then the result is positive infinity.
 545      *
 546      * <li>If
 547      * <ul>
 548      * <li>the absolute value of the first argument is greater than 1 and
 549      * the second argument is negative infinity, or
 550      * <li>the absolute value of the
 551      * first argument is less than 1 and the second argument is positive
 552      * infinity,
 553      * </ul>
 554      * then the result is positive zero.
 555      *
 556      * <li>If the absolute value of the first argument equals 1 and the
 557      * second argument is infinite, then the result is NaN.
 558      *
 559      * <li>If
 560      * <ul>
 561      * <li>the first argument is positive zero and the second argument
 562      * is greater than zero, or
 563      * <li>the first argument is positive infinity and the second
 564      * argument is less than zero,
 565      * </ul>
 566      * then the result is positive zero.
 567      *
 568      * <li>If
 569      * <ul>
 570      * <li>the first argument is positive zero and the second argument
 571      * is less than zero, or
 572      * <li>the first argument is positive infinity and the second
 573      * argument is greater than zero,
 574      * </ul>
 575      * then the result is positive infinity.
 576      *
 577      * <li>If
 578      * <ul>
 579      * <li>the first argument is negative zero and the second argument
 580      * is greater than zero but not a finite odd integer, or
 581      * <li>the first argument is negative infinity and the second
 582      * argument is less than zero but not a finite odd integer,
 583      * </ul>
 584      * then the result is positive zero.
 585      *
 586      * <li>If
 587      * <ul>
 588      * <li>the first argument is negative zero and the second argument
 589      * is a positive finite odd integer, or
 590      * <li>the first argument is negative infinity and the second
 591      * argument is a negative finite odd integer,
 592      * </ul>
 593      * then the result is negative zero.
 594      *
 595      * <li>If
 596      * <ul>
 597      * <li>the first argument is negative zero and the second argument
 598      * is less than zero but not a finite odd integer, or
 599      * <li>the first argument is negative infinity and the second
 600      * argument is greater than zero but not a finite odd integer,
 601      * </ul>
 602      * then the result is positive infinity.
 603      *
 604      * <li>If
 605      * <ul>
 606      * <li>the first argument is negative zero and the second argument
 607      * is a negative finite odd integer, or
 608      * <li>the first argument is negative infinity and the second
 609      * argument is a positive finite odd integer,
 610      * </ul>
 611      * then the result is negative infinity.
 612      *
 613      * <li>If the first argument is finite and less than zero
 614      * <ul>
 615      * <li> if the second argument is a finite even integer, the
 616      * result is equal to the result of raising the absolute value of
 617      * the first argument to the power of the second argument
 618      *
 619      * <li>if the second argument is a finite odd integer, the result
 620      * is equal to the negative of the result of raising the absolute
 621      * value of the first argument to the power of the second
 622      * argument
 623      *
 624      * <li>if the second argument is finite and not an integer, then
 625      * the result is NaN.
 626      * </ul>
 627      *
 628      * <li>If both arguments are integers, then the result is exactly equal
 629      * to the mathematical result of raising the first argument to the power
 630      * of the second argument if that result can in fact be represented
 631      * exactly as a {@code double} value.</ul>
 632      *
 633      * <p>(In the foregoing descriptions, a floating-point value is
 634      * considered to be an integer if and only if it is finite and a
 635      * fixed point of the method {@link #ceil ceil} or,
 636      * equivalently, a fixed point of the method {@link #floor
 637      * floor}. A value is a fixed point of a one-argument
 638      * method if and only if the result of applying the method to the
 639      * value is equal to the value.)
 640      *
 641      * @param   a   base.
 642      * @param   b   the exponent.
 643      * @return  the value {@code a}<sup>{@code b}</sup>.
 644      */
 645     public static native double pow(double a, double b);
 646 
 647     /**
 648      * Returns the closest {@code int} to the argument, with ties
 649      * rounding to positive infinity.
 650      *
 651      * <p>Special cases:
 652      * <ul><li>If the argument is NaN, the result is 0.
 653      * <li>If the argument is negative infinity or any value less than or
 654      * equal to the value of {@code Integer.MIN_VALUE}, the result is
 655      * equal to the value of {@code Integer.MIN_VALUE}.
 656      * <li>If the argument is positive infinity or any value greater than or
 657      * equal to the value of {@code Integer.MAX_VALUE}, the result is
 658      * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 659      *
 660      * @param   a   a floating-point value to be rounded to an integer.
 661      * @return  the value of the argument rounded to the nearest
 662      *          {@code int} value.
 663      * @see     java.lang.Integer#MAX_VALUE
 664      * @see     java.lang.Integer#MIN_VALUE
 665      */
 666     public static int round(float a) {
 667         return Math.round(a);
 668     }
 669 
 670     /**
 671      * Returns the closest {@code long} to the argument, with ties
 672      * rounding to positive infinity.
 673      *
 674      * <p>Special cases:
 675      * <ul><li>If the argument is NaN, the result is 0.
 676      * <li>If the argument is negative infinity or any value less than or
 677      * equal to the value of {@code Long.MIN_VALUE}, the result is
 678      * equal to the value of {@code Long.MIN_VALUE}.
 679      * <li>If the argument is positive infinity or any value greater than or
 680      * equal to the value of {@code Long.MAX_VALUE}, the result is
 681      * equal to the value of {@code Long.MAX_VALUE}.</ul>
 682      *
 683      * @param   a  a floating-point value to be rounded to a
 684      *          {@code long}.
 685      * @return  the value of the argument rounded to the nearest
 686      *          {@code long} value.
 687      * @see     java.lang.Long#MAX_VALUE
 688      * @see     java.lang.Long#MIN_VALUE
 689      */
 690     public static long round(double a) {
 691         return Math.round(a);
 692     }
 693 
 694     private static final class RandomNumberGeneratorHolder {
 695         static final Random randomNumberGenerator = new Random();
 696     }
 697 
 698     /**
 699      * Returns a {@code double} value with a positive sign, greater
 700      * than or equal to {@code 0.0} and less than {@code 1.0}.
 701      * Returned values are chosen pseudorandomly with (approximately)
 702      * uniform distribution from that range.
 703      *
 704      * <p>When this method is first called, it creates a single new
 705      * pseudorandom-number generator, exactly as if by the expression
 706      *
 707      * <blockquote>{@code new java.util.Random()}</blockquote>
 708      *
 709      * This new pseudorandom-number generator is used thereafter for
 710      * all calls to this method and is used nowhere else.
 711      *
 712      * <p>This method is properly synchronized to allow correct use by
 713      * more than one thread. However, if many threads need to generate
 714      * pseudorandom numbers at a great rate, it may reduce contention
 715      * for each thread to have its own pseudorandom-number generator.
 716      *
 717      * @return  a pseudorandom {@code double} greater than or equal
 718      * to {@code 0.0} and less than {@code 1.0}.
 719      * @see Random#nextDouble()
 720      */
 721     public static double random() {
 722         return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 723     }
 724 
 725     /**
 726      * Returns the sum of its arguments,
 727      * throwing an exception if the result overflows an {@code int}.
 728      *
 729      * @param x the first value
 730      * @param y the second value
 731      * @return the result
 732      * @throws ArithmeticException if the result overflows an int
 733      * @see Math#addExact(int,int)
 734      * @since 1.8
 735      */
 736     public static int addExact(int x, int y) {
 737         return Math.addExact(x, y);
 738     }
 739 
 740     /**
 741      * Returns the sum of its arguments,
 742      * throwing an exception if the result overflows a {@code long}.
 743      *
 744      * @param x the first value
 745      * @param y the second value
 746      * @return the result
 747      * @throws ArithmeticException if the result overflows a long
 748      * @see Math#addExact(long,long)
 749      * @since 1.8
 750      */
 751     public static long addExact(long x, long y) {
 752         return Math.addExact(x, y);
 753     }
 754 
 755     /**
 756      * Returns the difference of the arguments,
 757      * throwing an exception if the result overflows an {@code int}.
 758      *
 759      * @param x the first value
 760      * @param y the second value to subtract from the first
 761      * @return the result
 762      * @throws ArithmeticException if the result overflows an int
 763      * @see Math#subtractExact(int,int)
 764      * @since 1.8
 765      */
 766     public static int subtractExact(int x, int y) {
 767         return Math.subtractExact(x, y);
 768     }
 769 
 770     /**
 771      * Returns the difference of the arguments,
 772      * throwing an exception if the result overflows a {@code long}.
 773      *
 774      * @param x the first value
 775      * @param y the second value to subtract from the first
 776      * @return the result
 777      * @throws ArithmeticException if the result overflows a long
 778      * @see Math#subtractExact(long,long)
 779      * @since 1.8
 780      */
 781     public static long subtractExact(long x, long y) {
 782         return Math.subtractExact(x, y);
 783     }
 784 
 785     /**
 786      * Returns the product of the arguments,
 787      * throwing an exception if the result overflows an {@code int}.
 788      *
 789      * @param x the first value
 790      * @param y the second value
 791      * @return the result
 792      * @throws ArithmeticException if the result overflows an int
 793      * @see Math#multiplyExact(int,int)
 794      * @since 1.8
 795      */
 796     public static int multiplyExact(int x, int y) {
 797         return Math.multiplyExact(x, y);
 798     }
 799 
 800     /**
 801      * Returns the product of the arguments,
 802      * throwing an exception if the result overflows a {@code long}.
 803      *
 804      * @param x the first value
 805      * @param y the second value
 806      * @return the result
 807      * @throws ArithmeticException if the result overflows a long
 808      * @see Math#multiplyExact(long,long)
 809      * @since 1.8
 810      */
 811     public static long multiplyExact(long x, long y) {
 812         return Math.multiplyExact(x, y);
 813     }
 814 
 815     /**
 816      * Returns the value of the {@code long} argument;
 817      * throwing an exception if the value overflows an {@code int}.
 818      *
 819      * @param value the long value
 820      * @return the argument as an int
 821      * @throws ArithmeticException if the {@code argument} overflows an int
 822      * @see Math#toIntExact(long)
 823      * @since 1.8
 824      */
 825     public static int toIntExact(long value) {
 826         return Math.toIntExact(value);
 827     }
 828 
 829     /**
 830      * Returns the largest (closest to positive infinity)
 831      * {@code int} value that is less than or equal to the algebraic quotient.
 832      * There is one special case, if the dividend is the
 833      * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
 834      * then integer overflow occurs and
 835      * the result is equal to the {@code Integer.MIN_VALUE}.
 836      * <p>
 837      * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
 838      * a comparison to the integer division {@code /} operator.
 839      *
 840      * @param x the dividend
 841      * @param y the divisor
 842      * @return the largest (closest to positive infinity)
 843      * {@code int} value that is less than or equal to the algebraic quotient.
 844      * @throws ArithmeticException if the divisor {@code y} is zero
 845      * @see Math#floorDiv(int, int)
 846      * @see Math#floor(double)
 847      * @since 1.8
 848      */
 849     public static int floorDiv(int x, int y) {
 850         return Math.floorDiv(x, y);
 851     }
 852 
 853     /**
 854      * Returns the largest (closest to positive infinity)
 855      * {@code long} value that is less than or equal to the algebraic quotient.
 856      * There is one special case, if the dividend is the
 857      * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
 858      * then integer overflow occurs and
 859      * the result is equal to the {@code Long.MIN_VALUE}.
 860      * <p>
 861      * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
 862      * a comparison to the integer division {@code /} operator.
 863      *
 864      * @param x the dividend
 865      * @param y the divisor
 866      * @return the largest (closest to positive infinity)
 867      * {@code long} value that is less than or equal to the algebraic quotient.
 868      * @throws ArithmeticException if the divisor {@code y} is zero
 869      * @see Math#floorDiv(long, long)
 870      * @see Math#floor(double)
 871      * @since 1.8
 872      */
 873     public static long floorDiv(long x, long y) {
 874         return Math.floorDiv(x, y);
 875     }
 876 
 877     /**
 878      * Returns the floor modulus of the {@code int} arguments.
 879      * <p>
 880      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
 881      * has the same sign as the divisor {@code y}, and
 882      * is in the range of {@code -abs(y) < r < +abs(y)}.
 883      * <p>
 884      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
 885      * <ul>
 886      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
 887      * </ul>
 888      * <p>
 889      * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
 890      * a comparison to the {@code %} operator.
 891      *
 892      * @param x the dividend
 893      * @param y the divisor
 894      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
 895      * @throws ArithmeticException if the divisor {@code y} is zero
 896      * @see Math#floorMod(int, int)
 897      * @see StrictMath#floorDiv(int, int)
 898      * @since 1.8
 899      */
 900     public static int floorMod(int x, int y) {
 901         return Math.floorMod(x , y);
 902     }
 903     /**
 904      * Returns the floor modulus of the {@code long} arguments.
 905      * <p>
 906      * The floor modulus is {@code x - (floorDiv(x, y) * y)},
 907      * has the same sign as the divisor {@code y}, and
 908      * is in the range of {@code -abs(y) < r < +abs(y)}.
 909      * <p>
 910      * The relationship between {@code floorDiv} and {@code floorMod} is such that:
 911      * <ul>
 912      *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
 913      * </ul>
 914      * <p>
 915      * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
 916      * a comparison to the {@code %} operator.
 917      *
 918      * @param x the dividend
 919      * @param y the divisor
 920      * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
 921      * @throws ArithmeticException if the divisor {@code y} is zero
 922      * @see Math#floorMod(long, long)
 923      * @see StrictMath#floorDiv(long, long)
 924      * @since 1.8
 925      */
 926     public static long floorMod(long x, long y) {
 927         return Math.floorMod(x, y);
 928     }
 929 
 930     /**
 931      * Returns the absolute value of an {@code int} value.
 932      * If the argument is not negative, the argument is returned.
 933      * If the argument is negative, the negation of the argument is returned.
 934      *
 935      * <p>Note that if the argument is equal to the value of
 936      * {@link Integer#MIN_VALUE}, the most negative representable
 937      * {@code int} value, the result is that same value, which is
 938      * negative.
 939      *
 940      * @param   a   the  argument whose absolute value is to be determined.
 941      * @return  the absolute value of the argument.
 942      */
 943     public static int abs(int a) {
 944         return Math.abs(a);
 945     }
 946 
 947     /**
 948      * Returns the absolute value of a {@code long} value.
 949      * If the argument is not negative, the argument is returned.
 950      * If the argument is negative, the negation of the argument is returned.
 951      *
 952      * <p>Note that if the argument is equal to the value of
 953      * {@link Long#MIN_VALUE}, the most negative representable
 954      * {@code long} value, the result is that same value, which
 955      * is negative.
 956      *
 957      * @param   a   the  argument whose absolute value is to be determined.
 958      * @return  the absolute value of the argument.
 959      */
 960     public static long abs(long a) {
 961         return Math.abs(a);
 962     }
 963 
 964     /**
 965      * Returns the absolute value of a {@code float} value.
 966      * If the argument is not negative, the argument is returned.
 967      * If the argument is negative, the negation of the argument is returned.
 968      * Special cases:
 969      * <ul><li>If the argument is positive zero or negative zero, the
 970      * result is positive zero.
 971      * <li>If the argument is infinite, the result is positive infinity.
 972      * <li>If the argument is NaN, the result is NaN.</ul>
 973      * In other words, the result is the same as the value of the expression:
 974      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
 975      *
 976      * @param   a   the argument whose absolute value is to be determined
 977      * @return  the absolute value of the argument.
 978      */
 979     public static float abs(float a) {
 980         return Math.abs(a);
 981     }
 982 
 983     /**
 984      * Returns the absolute value of a {@code double} value.
 985      * If the argument is not negative, the argument is returned.
 986      * If the argument is negative, the negation of the argument is returned.
 987      * Special cases:
 988      * <ul><li>If the argument is positive zero or negative zero, the result
 989      * is positive zero.
 990      * <li>If the argument is infinite, the result is positive infinity.
 991      * <li>If the argument is NaN, the result is NaN.</ul>
 992      * In other words, the result is the same as the value of the expression:
 993      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
 994      *
 995      * @param   a   the argument whose absolute value is to be determined
 996      * @return  the absolute value of the argument.
 997      */
 998     public static double abs(double a) {
 999         return Math.abs(a);
1000     }
1001 
1002     /**
1003      * Returns the greater of two {@code int} values. That is, the
1004      * result is the argument closer to the value of
1005      * {@link Integer#MAX_VALUE}. If the arguments have the same value,
1006      * the result is that same value.
1007      *
1008      * @param   a   an argument.
1009      * @param   b   another argument.
1010      * @return  the larger of {@code a} and {@code b}.
1011      */
1012     public static int max(int a, int b) {
1013         return Math.max(a, b);
1014     }
1015 
1016     /**
1017      * Returns the greater of two {@code long} values. That is, the
1018      * result is the argument closer to the value of
1019      * {@link Long#MAX_VALUE}. If the arguments have the same value,
1020      * the result is that same value.
1021      *
1022      * @param   a   an argument.
1023      * @param   b   another argument.
1024      * @return  the larger of {@code a} and {@code b}.
1025         */
1026     public static long max(long a, long b) {
1027         return Math.max(a, b);
1028     }
1029 
1030     /**
1031      * Returns the greater of two {@code float} values.  That is,
1032      * the result is the argument closer to positive infinity. If the
1033      * arguments have the same value, the result is that same
1034      * value. If either value is NaN, then the result is NaN.  Unlike
1035      * the numerical comparison operators, this method considers
1036      * negative zero to be strictly smaller than positive zero. If one
1037      * argument is positive zero and the other negative zero, the
1038      * result is positive zero.
1039      *
1040      * @param   a   an argument.
1041      * @param   b   another argument.
1042      * @return  the larger of {@code a} and {@code b}.
1043      */
1044     public static float max(float a, float b) {
1045         return Math.max(a, b);
1046     }
1047 
1048     /**
1049      * Returns the greater of two {@code double} values.  That
1050      * is, the result is the argument closer to positive infinity. If
1051      * the arguments have the same value, the result is that same
1052      * value. If either value is NaN, then the result is NaN.  Unlike
1053      * the numerical comparison operators, this method considers
1054      * negative zero to be strictly smaller than positive zero. If one
1055      * argument is positive zero and the other negative zero, the
1056      * result is positive zero.
1057      *
1058      * @param   a   an argument.
1059      * @param   b   another argument.
1060      * @return  the larger of {@code a} and {@code b}.
1061      */
1062     public static double max(double a, double b) {
1063         return Math.max(a, b);
1064     }
1065 
1066     /**
1067      * Returns the smaller of two {@code int} values. That is,
1068      * the result the argument closer to the value of
1069      * {@link Integer#MIN_VALUE}.  If the arguments have the same
1070      * value, the result is that same value.
1071      *
1072      * @param   a   an argument.
1073      * @param   b   another argument.
1074      * @return  the smaller of {@code a} and {@code b}.
1075      */
1076     public static int min(int a, int b) {
1077         return Math.min(a, b);
1078     }
1079 
1080     /**
1081      * Returns the smaller of two {@code long} values. That is,
1082      * the result is the argument closer to the value of
1083      * {@link Long#MIN_VALUE}. If the arguments have the same
1084      * value, the result is that same value.
1085      *
1086      * @param   a   an argument.
1087      * @param   b   another argument.
1088      * @return  the smaller of {@code a} and {@code b}.
1089      */
1090     public static long min(long a, long b) {
1091         return Math.min(a, b);
1092     }
1093 
1094     /**
1095      * Returns the smaller of two {@code float} values.  That is,
1096      * the result is the value closer to negative infinity. If the
1097      * arguments have the same value, the result is that same
1098      * value. If either value is NaN, then the result is NaN.  Unlike
1099      * the numerical comparison operators, this method considers
1100      * negative zero to be strictly smaller than positive zero.  If
1101      * one argument is positive zero and the other is negative zero,
1102      * the result is negative zero.
1103      *
1104      * @param   a   an argument.
1105      * @param   b   another argument.
1106      * @return  the smaller of {@code a} and {@code b.}
1107      */
1108     public static float min(float a, float b) {
1109         return Math.min(a, b);
1110     }
1111 
1112     /**
1113      * Returns the smaller of two {@code double} values.  That
1114      * is, the result is the value closer to negative infinity. If the
1115      * arguments have the same value, the result is that same
1116      * value. If either value is NaN, then the result is NaN.  Unlike
1117      * the numerical comparison operators, this method considers
1118      * negative zero to be strictly smaller than positive zero. If one
1119      * argument is positive zero and the other is negative zero, the
1120      * result is negative zero.
1121      *
1122      * @param   a   an argument.
1123      * @param   b   another argument.
1124      * @return  the smaller of {@code a} and {@code b}.
1125      */
1126     public static double min(double a, double b) {
1127         return Math.min(a, b);
1128     }
1129 
1130     /**
1131      * Returns the size of an ulp of the argument.  An ulp, unit in
1132      * the last place, of a {@code double} value is the positive
1133      * distance between this floating-point value and the {@code
1134      * double} value next larger in magnitude.  Note that for non-NaN
1135      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1136      *
1137      * <p>Special Cases:
1138      * <ul>
1139      * <li> If the argument is NaN, then the result is NaN.
1140      * <li> If the argument is positive or negative infinity, then the
1141      * result is positive infinity.
1142      * <li> If the argument is positive or negative zero, then the result is
1143      * {@code Double.MIN_VALUE}.
1144      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
1145      * the result is equal to 2<sup>971</sup>.
1146      * </ul>
1147      *
1148      * @param d the floating-point value whose ulp is to be returned
1149      * @return the size of an ulp of the argument
1150      * @author Joseph D. Darcy
1151      * @since 1.5
1152      */
1153     public static double ulp(double d) {
1154         return Math.ulp(d);
1155     }
1156 
1157     /**
1158      * Returns the size of an ulp of the argument.  An ulp, unit in
1159      * the last place, of a {@code float} value is the positive
1160      * distance between this floating-point value and the {@code
1161      * float} value next larger in magnitude.  Note that for non-NaN
1162      * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
1163      *
1164      * <p>Special Cases:
1165      * <ul>
1166      * <li> If the argument is NaN, then the result is NaN.
1167      * <li> If the argument is positive or negative infinity, then the
1168      * result is positive infinity.
1169      * <li> If the argument is positive or negative zero, then the result is
1170      * {@code Float.MIN_VALUE}.
1171      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
1172      * the result is equal to 2<sup>104</sup>.
1173      * </ul>
1174      *
1175      * @param f the floating-point value whose ulp is to be returned
1176      * @return the size of an ulp of the argument
1177      * @author Joseph D. Darcy
1178      * @since 1.5
1179      */
1180     public static float ulp(float f) {
1181         return Math.ulp(f);
1182     }
1183 
1184     /**
1185      * Returns the signum function of the argument; zero if the argument
1186      * is zero, 1.0 if the argument is greater than zero, -1.0 if the
1187      * argument is less than zero.
1188      *
1189      * <p>Special Cases:
1190      * <ul>
1191      * <li> If the argument is NaN, then the result is NaN.
1192      * <li> If the argument is positive zero or negative zero, then the
1193      *      result is the same as the argument.
1194      * </ul>
1195      *
1196      * @param d the floating-point value whose signum is to be returned
1197      * @return the signum function of the argument
1198      * @author Joseph D. Darcy
1199      * @since 1.5
1200      */
1201     public static double signum(double d) {
1202         return Math.signum(d);
1203     }
1204 
1205     /**
1206      * Returns the signum function of the argument; zero if the argument
1207      * is zero, 1.0f if the argument is greater than zero, -1.0f if the
1208      * argument is less than zero.
1209      *
1210      * <p>Special Cases:
1211      * <ul>
1212      * <li> If the argument is NaN, then the result is NaN.
1213      * <li> If the argument is positive zero or negative zero, then the
1214      *      result is the same as the argument.
1215      * </ul>
1216      *
1217      * @param f the floating-point value whose signum is to be returned
1218      * @return the signum function of the argument
1219      * @author Joseph D. Darcy
1220      * @since 1.5
1221      */
1222     public static float signum(float f) {
1223         return Math.signum(f);
1224     }
1225 
1226     /**
1227      * Returns the hyperbolic sine of a {@code double} value.
1228      * The hyperbolic sine of <i>x</i> is defined to be
1229      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
1230      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1231      *
1232      * <p>Special cases:
1233      * <ul>
1234      *
1235      * <li>If the argument is NaN, then the result is NaN.
1236      *
1237      * <li>If the argument is infinite, then the result is an infinity
1238      * with the same sign as the argument.
1239      *
1240      * <li>If the argument is zero, then the result is a zero with the
1241      * same sign as the argument.
1242      *
1243      * </ul>
1244      *
1245      * @param   x The number whose hyperbolic sine is to be returned.
1246      * @return  The hyperbolic sine of {@code x}.
1247      * @since 1.5
1248      */
1249     public static native double sinh(double x);
1250 
1251     /**
1252      * Returns the hyperbolic cosine of a {@code double} value.
1253      * The hyperbolic cosine of <i>x</i> is defined to be
1254      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
1255      * where <i>e</i> is {@linkplain Math#E Euler's number}.
1256      *
1257      * <p>Special cases:
1258      * <ul>
1259      *
1260      * <li>If the argument is NaN, then the result is NaN.
1261      *
1262      * <li>If the argument is infinite, then the result is positive
1263      * infinity.
1264      *
1265      * <li>If the argument is zero, then the result is {@code 1.0}.
1266      *
1267      * </ul>
1268      *
1269      * @param   x The number whose hyperbolic cosine is to be returned.
1270      * @return  The hyperbolic cosine of {@code x}.
1271      * @since 1.5
1272      */
1273     public static native double cosh(double x);
1274 
1275     /**
1276      * Returns the hyperbolic tangent of a {@code double} value.
1277      * The hyperbolic tangent of <i>x</i> is defined to be
1278      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
1279      * in other words, {@linkplain Math#sinh
1280      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
1281      * that the absolute value of the exact tanh is always less than
1282      * 1.
1283      *
1284      * <p>Special cases:
1285      * <ul>
1286      *
1287      * <li>If the argument is NaN, then the result is NaN.
1288      *
1289      * <li>If the argument is zero, then the result is a zero with the
1290      * same sign as the argument.
1291      *
1292      * <li>If the argument is positive infinity, then the result is
1293      * {@code +1.0}.
1294      *
1295      * <li>If the argument is negative infinity, then the result is
1296      * {@code -1.0}.
1297      *
1298      * </ul>
1299      *
1300      * @param   x The number whose hyperbolic tangent is to be returned.
1301      * @return  The hyperbolic tangent of {@code x}.
1302      * @since 1.5
1303      */
1304     public static native double tanh(double x);
1305 
1306     /**
1307      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1308      * without intermediate overflow or underflow.
1309      *
1310      * <p>Special cases:
1311      * <ul>
1312      *
1313      * <li> If either argument is infinite, then the result
1314      * is positive infinity.
1315      *
1316      * <li> If either argument is NaN and neither argument is infinite,
1317      * then the result is NaN.
1318      *
1319      * </ul>
1320      *
1321      * @param x a value
1322      * @param y a value
1323      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
1324      * without intermediate overflow or underflow
1325      * @since 1.5
1326      */
1327     public static native double hypot(double x, double y);
1328 
1329     /**
1330      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
1331      * <i>x</i> near 0, the exact sum of
1332      * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
1333      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
1334      *
1335      * <p>Special cases:
1336      * <ul>
1337      * <li>If the argument is NaN, the result is NaN.
1338      *
1339      * <li>If the argument is positive infinity, then the result is
1340      * positive infinity.
1341      *
1342      * <li>If the argument is negative infinity, then the result is
1343      * -1.0.
1344      *
1345      * <li>If the argument is zero, then the result is a zero with the
1346      * same sign as the argument.
1347      *
1348      * </ul>
1349      *
1350      * @param   x   the exponent to raise <i>e</i> to in the computation of
1351      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.
1352      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
1353      * @since 1.5
1354      */
1355     public static native double expm1(double x);
1356 
1357     /**
1358      * Returns the natural logarithm of the sum of the argument and 1.
1359      * Note that for small values {@code x}, the result of
1360      * {@code log1p(x)} is much closer to the true result of ln(1
1361      * + {@code x}) than the floating-point evaluation of
1362      * {@code log(1.0+x)}.
1363      *
1364      * <p>Special cases:
1365      * <ul>
1366      *
1367      * <li>If the argument is NaN or less than -1, then the result is
1368      * NaN.
1369      *
1370      * <li>If the argument is positive infinity, then the result is
1371      * positive infinity.
1372      *
1373      * <li>If the argument is negative one, then the result is
1374      * negative infinity.
1375      *
1376      * <li>If the argument is zero, then the result is a zero with the
1377      * same sign as the argument.
1378      *
1379      * </ul>
1380      *
1381      * @param   x   a value
1382      * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
1383      * log of {@code x}&nbsp;+&nbsp;1
1384      * @since 1.5
1385      */
1386     public static native double log1p(double x);
1387 
1388     /**
1389      * Returns the first floating-point argument with the sign of the
1390      * second floating-point argument.  For this method, a NaN
1391      * {@code sign} argument is always treated as if it were
1392      * positive.
1393      *
1394      * @param magnitude  the parameter providing the magnitude of the result
1395      * @param sign   the parameter providing the sign of the result
1396      * @return a value with the magnitude of {@code magnitude}
1397      * and the sign of {@code sign}.
1398      * @since 1.6
1399      */
1400     public static double copySign(double magnitude, double sign) {
1401         return Math.copySign(magnitude, (Double.isNaN(sign)?1.0d:sign));
1402     }
1403 
1404     /**
1405      * Returns the first floating-point argument with the sign of the
1406      * second floating-point argument.  For this method, a NaN
1407      * {@code sign} argument is always treated as if it were
1408      * positive.
1409      *
1410      * @param magnitude  the parameter providing the magnitude of the result
1411      * @param sign   the parameter providing the sign of the result
1412      * @return a value with the magnitude of {@code magnitude}
1413      * and the sign of {@code sign}.
1414      * @since 1.6
1415      */
1416     public static float copySign(float magnitude, float sign) {
1417         return Math.copySign(magnitude, (Float.isNaN(sign)?1.0f:sign));
1418     }
1419     /**
1420      * Returns the unbiased exponent used in the representation of a
1421      * {@code float}.  Special cases:
1422      *
1423      * <ul>
1424      * <li>If the argument is NaN or infinite, then the result is
1425      * {@link Float#MAX_EXPONENT} + 1.
1426      * <li>If the argument is zero or subnormal, then the result is
1427      * {@link Float#MIN_EXPONENT} -1.
1428      * </ul>
1429      * @param f a {@code float} value
1430      * @return the unbiased exponent of the argument
1431      * @since 1.6
1432      */
1433     public static int getExponent(float f) {
1434         return Math.getExponent(f);
1435     }
1436 
1437     /**
1438      * Returns the unbiased exponent used in the representation of a
1439      * {@code double}.  Special cases:
1440      *
1441      * <ul>
1442      * <li>If the argument is NaN or infinite, then the result is
1443      * {@link Double#MAX_EXPONENT} + 1.
1444      * <li>If the argument is zero or subnormal, then the result is
1445      * {@link Double#MIN_EXPONENT} -1.
1446      * </ul>
1447      * @param d a {@code double} value
1448      * @return the unbiased exponent of the argument
1449      * @since 1.6
1450      */
1451     public static int getExponent(double d) {
1452         return Math.getExponent(d);
1453     }
1454 
1455     /**
1456      * Returns the floating-point number adjacent to the first
1457      * argument in the direction of the second argument.  If both
1458      * arguments compare as equal the second argument is returned.
1459      *
1460      * <p>Special cases:
1461      * <ul>
1462      * <li> If either argument is a NaN, then NaN is returned.
1463      *
1464      * <li> If both arguments are signed zeros, {@code direction}
1465      * is returned unchanged (as implied by the requirement of
1466      * returning the second argument if the arguments compare as
1467      * equal).
1468      *
1469      * <li> If {@code start} is
1470      * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
1471      * has a value such that the result should have a smaller
1472      * magnitude, then a zero with the same sign as {@code start}
1473      * is returned.
1474      *
1475      * <li> If {@code start} is infinite and
1476      * {@code direction} has a value such that the result should
1477      * have a smaller magnitude, {@link Double#MAX_VALUE} with the
1478      * same sign as {@code start} is returned.
1479      *
1480      * <li> If {@code start} is equal to &plusmn;
1481      * {@link Double#MAX_VALUE} and {@code direction} has a
1482      * value such that the result should have a larger magnitude, an
1483      * infinity with same sign as {@code start} is returned.
1484      * </ul>
1485      *
1486      * @param start  starting floating-point value
1487      * @param direction value indicating which of
1488      * {@code start}'s neighbors or {@code start} should
1489      * be returned
1490      * @return The floating-point number adjacent to {@code start} in the
1491      * direction of {@code direction}.
1492      * @since 1.6
1493      */
1494     public static double nextAfter(double start, double direction) {
1495         return Math.nextAfter(start, direction);
1496     }
1497 
1498     /**
1499      * Returns the floating-point number adjacent to the first
1500      * argument in the direction of the second argument.  If both
1501      * arguments compare as equal a value equivalent to the second argument
1502      * is returned.
1503      *
1504      * <p>Special cases:
1505      * <ul>
1506      * <li> If either argument is a NaN, then NaN is returned.
1507      *
1508      * <li> If both arguments are signed zeros, a value equivalent
1509      * to {@code direction} is returned.
1510      *
1511      * <li> If {@code start} is
1512      * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
1513      * has a value such that the result should have a smaller
1514      * magnitude, then a zero with the same sign as {@code start}
1515      * is returned.
1516      *
1517      * <li> If {@code start} is infinite and
1518      * {@code direction} has a value such that the result should
1519      * have a smaller magnitude, {@link Float#MAX_VALUE} with the
1520      * same sign as {@code start} is returned.
1521      *
1522      * <li> If {@code start} is equal to &plusmn;
1523      * {@link Float#MAX_VALUE} and {@code direction} has a
1524      * value such that the result should have a larger magnitude, an
1525      * infinity with same sign as {@code start} is returned.
1526      * </ul>
1527      *
1528      * @param start  starting floating-point value
1529      * @param direction value indicating which of
1530      * {@code start}'s neighbors or {@code start} should
1531      * be returned
1532      * @return The floating-point number adjacent to {@code start} in the
1533      * direction of {@code direction}.
1534      * @since 1.6
1535      */
1536     public static float nextAfter(float start, double direction) {
1537         return Math.nextAfter(start, direction);
1538     }
1539 
1540     /**
1541      * Returns the floating-point value adjacent to {@code d} in
1542      * the direction of positive infinity.  This method is
1543      * semantically equivalent to {@code nextAfter(d,
1544      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
1545      * implementation may run faster than its equivalent
1546      * {@code nextAfter} call.
1547      *
1548      * <p>Special Cases:
1549      * <ul>
1550      * <li> If the argument is NaN, the result is NaN.
1551      *
1552      * <li> If the argument is positive infinity, the result is
1553      * positive infinity.
1554      *
1555      * <li> If the argument is zero, the result is
1556      * {@link Double#MIN_VALUE}
1557      *
1558      * </ul>
1559      *
1560      * @param d starting floating-point value
1561      * @return The adjacent floating-point value closer to positive
1562      * infinity.
1563      * @since 1.6
1564      */
1565     public static double nextUp(double d) {
1566         return Math.nextUp(d);
1567     }
1568 
1569     /**
1570      * Returns the floating-point value adjacent to {@code f} in
1571      * the direction of positive infinity.  This method is
1572      * semantically equivalent to {@code nextAfter(f,
1573      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
1574      * implementation may run faster than its equivalent
1575      * {@code nextAfter} call.
1576      *
1577      * <p>Special Cases:
1578      * <ul>
1579      * <li> If the argument is NaN, the result is NaN.
1580      *
1581      * <li> If the argument is positive infinity, the result is
1582      * positive infinity.
1583      *
1584      * <li> If the argument is zero, the result is
1585      * {@link Float#MIN_VALUE}
1586      *
1587      * </ul>
1588      *
1589      * @param f starting floating-point value
1590      * @return The adjacent floating-point value closer to positive
1591      * infinity.
1592      * @since 1.6
1593      */
1594     public static float nextUp(float f) {
1595         return Math.nextUp(f);
1596     }
1597 
1598     /**
1599      * Returns the floating-point value adjacent to {@code d} in
1600      * the direction of negative infinity.  This method is
1601      * semantically equivalent to {@code nextAfter(d,
1602      * Double.NEGATIVE_INFINITY)}; however, a
1603      * {@code nextDown} implementation may run faster than its
1604      * equivalent {@code nextAfter} call.
1605      *
1606      * <p>Special Cases:
1607      * <ul>
1608      * <li> If the argument is NaN, the result is NaN.
1609      *
1610      * <li> If the argument is negative infinity, the result is
1611      * negative infinity.
1612      *
1613      * <li> If the argument is zero, the result is
1614      * {@code -Double.MIN_VALUE}
1615      *
1616      * </ul>
1617      *
1618      * @param d  starting floating-point value
1619      * @return The adjacent floating-point value closer to negative
1620      * infinity.
1621      * @since 1.8
1622      */
1623     public static double nextDown(double d) {
1624         return Math.nextDown(d);
1625     }
1626 
1627     /**
1628      * Returns the floating-point value adjacent to {@code f} in
1629      * the direction of negative infinity.  This method is
1630      * semantically equivalent to {@code nextAfter(f,
1631      * Float.NEGATIVE_INFINITY)}; however, a
1632      * {@code nextDown} implementation may run faster than its
1633      * equivalent {@code nextAfter} call.
1634      *
1635      * <p>Special Cases:
1636      * <ul>
1637      * <li> If the argument is NaN, the result is NaN.
1638      *
1639      * <li> If the argument is negative infinity, the result is
1640      * negative infinity.
1641      *
1642      * <li> If the argument is zero, the result is
1643      * {@code -Float.MIN_VALUE}
1644      *
1645      * </ul>
1646      *
1647      * @param f  starting floating-point value
1648      * @return The adjacent floating-point value closer to negative
1649      * infinity.
1650      * @since 1.8
1651      */
1652     public static float nextDown(float f) {
1653         return Math.nextDown(f);
1654     }
1655 
1656     /**
1657      * Returns {@code d} &times;
1658      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1659      * by a single correctly rounded floating-point multiply to a
1660      * member of the double value set.  See the Java
1661      * Language Specification for a discussion of floating-point
1662      * value sets.  If the exponent of the result is between {@link
1663      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
1664      * answer is calculated exactly.  If the exponent of the result
1665      * would be larger than {@code Double.MAX_EXPONENT}, an
1666      * infinity is returned.  Note that if the result is subnormal,
1667      * precision may be lost; that is, when {@code scalb(x, n)}
1668      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1669      * <i>x</i>.  When the result is non-NaN, the result has the same
1670      * sign as {@code d}.
1671      *
1672      * <p>Special cases:
1673      * <ul>
1674      * <li> If the first argument is NaN, NaN is returned.
1675      * <li> If the first argument is infinite, then an infinity of the
1676      * same sign is returned.
1677      * <li> If the first argument is zero, then a zero of the same
1678      * sign is returned.
1679      * </ul>
1680      *
1681      * @param d number to be scaled by a power of two.
1682      * @param scaleFactor power of 2 used to scale {@code d}
1683      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
1684      * @since 1.6
1685      */
1686     public static double scalb(double d, int scaleFactor) {
1687         return Math.scalb(d, scaleFactor);
1688     }
1689 
1690     /**
1691      * Returns {@code f} &times;
1692      * 2<sup>{@code scaleFactor}</sup> rounded as if performed
1693      * by a single correctly rounded floating-point multiply to a
1694      * member of the float value set.  See the Java
1695      * Language Specification for a discussion of floating-point
1696      * value sets.  If the exponent of the result is between {@link
1697      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
1698      * answer is calculated exactly.  If the exponent of the result
1699      * would be larger than {@code Float.MAX_EXPONENT}, an
1700      * infinity is returned.  Note that if the result is subnormal,
1701      * precision may be lost; that is, when {@code scalb(x, n)}
1702      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
1703      * <i>x</i>.  When the result is non-NaN, the result has the same
1704      * sign as {@code f}.
1705      *
1706      * <p>Special cases:
1707      * <ul>
1708      * <li> If the first argument is NaN, NaN is returned.
1709      * <li> If the first argument is infinite, then an infinity of the
1710      * same sign is returned.
1711      * <li> If the first argument is zero, then a zero of the same
1712      * sign is returned.
1713      * </ul>
1714      *
1715      * @param f number to be scaled by a power of two.
1716      * @param scaleFactor power of 2 used to scale {@code f}
1717      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
1718      * @since 1.6
1719      */
1720     public static float scalb(float f, int scaleFactor) {
1721         return Math.scalb(f, scaleFactor);
1722     }
1723 }