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  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.

  *

  * This code is free software; you can redistribute it and/or modify it

  * under the terms of the GNU General Public License version 2 only, as

  * published by the Free Software Foundation.  Oracle designates this

  * particular file as subject to the "Classpath" exception as provided

  * by Oracle in the LICENSE file that accompanied this code.

  *

  * This code is distributed in the hope that it will be useful, but WITHOUT

  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or

  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License

  * version 2 for more details (a copy is included in the LICENSE file that

  * accompanied this code).

  *

  * You should have received a copy of the GNU General Public License version

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 package java.lang;

 import java.util.Random;

 import sun.misc.FpUtils;

 import sun.misc.DoubleConsts;

 /**

  * The class {@code StrictMath} contains methods for performing basic

  * numeric operations such as the elementary exponential, logarithm,

  * square root, and trigonometric functions.

  *

  * <p>To help ensure portability of Java programs, the definitions of

  * some of the numeric functions in this package require that they

  * produce the same results as certain published algorithms. These

  * algorithms are available from the well-known network library

  * {@code netlib} as the package "Freely Distributable Math

  * Library," <a

  * href="ftp://ftp.netlib.org/fdlibm.tar">{@code fdlibm}</a>. These

  * algorithms, which are written in the C programming language, are

  * then to be understood as executed with all floating-point

  * operations following the rules of Java floating-point arithmetic.

  *

  * <p>The Java math library is defined with respect to

  * {@code fdlibm} version 5.3. Where {@code fdlibm} provides

  * more than one definition for a function (such as

  * {@code acos}), use the "IEEE 754 core function" version

  * (residing in a file whose name begins with the letter

  * {@code e}).  The methods which require {@code fdlibm}

  * semantics are {@code sin}, {@code cos}, {@code tan},

  * {@code asin}, {@code acos}, {@code atan},

  * {@code exp}, {@code log}, {@code log10},

  * {@code cbrt}, {@code atan2}, {@code pow},

  * {@code sinh}, {@code cosh}, {@code tanh},

  * {@code hypot}, {@code expm1}, and {@code log1p}.

  *

  * @author  unascribed

  * @author  Joseph D. Darcy

  * @since   1.3

  */

 public final class StrictMath {

     /**

      * Don't let anyone instantiate this class.

      */

     private StrictMath() {}

     /**

      * The {@code double} value that is closer than any other to

      * <i>e</i>, the base of the natural logarithms.

      */

     public static final double E = 2.7182818284590452354;

     /**

      * The {@code double} value that is closer than any other to

      * <i>pi</i>, the ratio of the circumference of a circle to its

      * diameter.

      */

     public static final double PI = 3.14159265358979323846;

     /**

      * Returns the trigonometric sine of an angle. Special cases:

      * <ul><li>If the argument is NaN or an infinity, then the

      * result is NaN.

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.</ul>

      *

      * @param   a   an angle, in radians.

      * @return  the sine of the argument.

      */

     public static native double sin(double a);

     /**

      * Returns the trigonometric cosine of an angle. Special cases:

      * <ul><li>If the argument is NaN or an infinity, then the

      * result is NaN.</ul>

      *

      * @param   a   an angle, in radians.

      * @return  the cosine of the argument.

      */

     public static native double cos(double a);

     /**

      * Returns the trigonometric tangent of an angle. Special cases:

      * <ul><li>If the argument is NaN or an infinity, then the result

      * is NaN.

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.</ul>

      *

      * @param   a   an angle, in radians.

      * @return  the tangent of the argument.

      */

     public static native double tan(double a);

     /**

      * Returns the arc sine of a value; the returned angle is in the

      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:

      * <ul><li>If the argument is NaN or its absolute value is greater

      * than 1, then the result is NaN.

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.</ul>

      *

      * @param   a   the value whose arc sine is to be returned.

      * @return  the arc sine of the argument.

      */

     public static native double asin(double a);

     /**

      * Returns the arc cosine of a value; the returned angle is in the

      * range 0.0 through <i>pi</i>.  Special case:

      * <ul><li>If the argument is NaN or its absolute value is greater

      * than 1, then the result is NaN.</ul>

      *

      * @param   a   the value whose arc cosine is to be returned.

      * @return  the arc cosine of the argument.

      */

     public static native double acos(double a);

     /**

      * Returns the arc tangent of a value; the returned angle is in the

      * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:

      * <ul><li>If the argument is NaN, then the result is NaN.

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.</ul>

      *

      * @param   a   the value whose arc tangent is to be returned.

      * @return  the arc tangent of the argument.

      */

     public static native double atan(double a);

     /**

      * Converts an angle measured in degrees to an approximately

      * equivalent angle measured in radians.  The conversion from

      * degrees to radians is generally inexact.

      *

      * @param   angdeg   an angle, in degrees

      * @return  the measurement of the angle {@code angdeg}

      *          in radians.

      */

     public static strictfp double toRadians(double angdeg) {

         return angdeg / 180.0 * PI;

     }

     /**

      * Converts an angle measured in radians to an approximately

      * equivalent angle measured in degrees.  The conversion from

      * radians to degrees is generally inexact; users should

      * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly

      * equal {@code 0.0}.

      *

      * @param   angrad   an angle, in radians

      * @return  the measurement of the angle {@code angrad}

      *          in degrees.

      */

     public static strictfp double toDegrees(double angrad) {

         return angrad * 180.0 / PI;

     }

     /**

      * Returns Euler's number <i>e</i> raised to the power of a

      * {@code double} value. Special cases:

      * <ul><li>If the argument is NaN, the result is NaN.

      * <li>If the argument is positive infinity, then the result is

      * positive infinity.

      * <li>If the argument is negative infinity, then the result is

      * positive zero.</ul>

      *

      * @param   a   the exponent to raise <i>e</i> to.

      * @return  the value <i>e</i><sup>{@code a}</sup>,

      *          where <i>e</i> is the base of the natural logarithms.

      */

     public static native double exp(double a);

     /**

      * Returns the natural logarithm (base <i>e</i>) of a {@code double}

      * value. Special cases:

      * <ul><li>If the argument is NaN or less than zero, then the result

      * is NaN.

      * <li>If the argument is positive infinity, then the result is

      * positive infinity.

      * <li>If the argument is positive zero or negative zero, then the

      * result is negative infinity.</ul>

      *

      * @param   a   a value

      * @return  the value ln {@code a}, the natural logarithm of

      *          {@code a}.

      */

     public static native double log(double a);

     /**

      * Returns the base 10 logarithm of a {@code double} value.

      * Special cases:

      *

      * <ul><li>If the argument is NaN or less than zero, then the result

      * is NaN.

      * <li>If the argument is positive infinity, then the result is

      * positive infinity.

      * <li>If the argument is positive zero or negative zero, then the

      * result is negative infinity.

      * <li> If the argument is equal to 10<sup><i>n</i></sup> for

      * integer <i>n</i>, then the result is <i>n</i>.

      * </ul>

      *

      * @param   a   a value

      * @return  the base 10 logarithm of  {@code a}.

      * @since 1.5

      */

     public static native double log10(double a);

     /**

      * Returns the correctly rounded positive square root of a

      * {@code double} value.

      * Special cases:

      * <ul><li>If the argument is NaN or less than zero, then the result

      * is NaN.

      * <li>If the argument is positive infinity, then the result is positive

      * infinity.

      * <li>If the argument is positive zero or negative zero, then the

      * result is the same as the argument.</ul>

      * Otherwise, the result is the {@code double} value closest to

      * the true mathematical square root of the argument value.

      *

      * @param   a   a value.

      * @return  the positive square root of {@code a}.

      */

     public static native double sqrt(double a);

     /**

      * Returns the cube root of a {@code double} value.  For

      * positive finite {@code x}, {@code cbrt(-x) ==

      * -cbrt(x)}; that is, the cube root of a negative value is

      * the negative of the cube root of that value's magnitude.

      * Special cases:

      *

      * <ul>

      *

      * <li>If the argument is NaN, then the result is NaN.

      *

      * <li>If the argument is infinite, then the result is an infinity

      * with the same sign as the argument.

      *

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.

      *

      * </ul>

      *

      * @param   a   a value.

      * @return  the cube root of {@code a}.

      * @since 1.5

      */

     public static native double cbrt(double a);

     /**

      * Computes the remainder operation on two arguments as prescribed

      * by the IEEE 754 standard.

      * The remainder value is mathematically equal to

      * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,

      * where <i>n</i> is the mathematical integer closest to the exact

      * mathematical value of the quotient {@code f1/f2}, and if two

      * mathematical integers are equally close to {@code f1/f2},

      * then <i>n</i> is the integer that is even. If the remainder is

      * zero, its sign is the same as the sign of the first argument.

      * Special cases:

      * <ul><li>If either argument is NaN, or the first argument is infinite,

      * or the second argument is positive zero or negative zero, then the

      * result is NaN.

      * <li>If the first argument is finite and the second argument is

      * infinite, then the result is the same as the first argument.</ul>

      *

      * @param   f1   the dividend.

      * @param   f2   the divisor.

      * @return  the remainder when {@code f1} is divided by

      *          {@code f2}.

      */

     public static native double IEEEremainder(double f1, double f2);

     /**

      * Returns the smallest (closest to negative infinity)

      * {@code double} value that is greater than or equal to the

      * argument and is equal to a mathematical integer. Special cases:

      * <ul><li>If the argument value is already equal to a

      * mathematical integer, then the result is the same as the

      * argument.  <li>If the argument is NaN or an infinity or

      * positive zero or negative zero, then the result is the same as

      * the argument.  <li>If the argument value is less than zero but

      * greater than -1.0, then the result is negative zero.</ul> Note

      * that the value of {@code StrictMath.ceil(x)} is exactly the

      * value of {@code -StrictMath.floor(-x)}.

      *

      * @param   a   a value.

      * @return  the smallest (closest to negative infinity)

      *          floating-point value that is greater than or equal to

      *          the argument and is equal to a mathematical integer.

      */

     public static double ceil(double a) {

         return floorOrCeil(a, -0.0, 1.0, 1.0);

     }

     /**

      * Returns the largest (closest to positive infinity)

      * {@code double} value that is less than or equal to the

      * argument and is equal to a mathematical integer. Special cases:

      * <ul><li>If the argument value is already equal to a

      * mathematical integer, then the result is the same as the

      * argument.  <li>If the argument is NaN or an infinity or

      * positive zero or negative zero, then the result is the same as

      * the argument.</ul>

      *

      * @param   a   a value.

      * @return  the largest (closest to positive infinity)

      *          floating-point value that less than or equal to the argument

      *          and is equal to a mathematical integer.

      */

     public static double floor(double a) {

         return floorOrCeil(a, -1.0, 0.0, -1.0);

     }

     /**

      * Internal method to share logic between floor and ceil.

      *

      * @param a the value to be floored or ceiled

      * @param negativeBoundary result for values in (-1, 0)

      * @param positiveBoundary result for values in (0, 1)

      * @param increment value to add when the argument is non-integral

      */

     private static double floorOrCeil(double a,

                                       double negativeBoundary,

                                       double positiveBoundary,

                                       double sign) {

         int exponent = Math.getExponent(a);

         if (exponent < 0) {

             /*

              * Absolute value of argument is less than 1.

              * floorOrceil(-0.0) => -0.0

              * floorOrceil(+0.0) => +0.0

              */

             return ((a == 0.0) ? a :

                     ( (a < 0.0) ?  negativeBoundary : positiveBoundary) );

         } else if (exponent >= 52) {

             /*

              * Infinity, NaN, or a value so large it must be integral.

              */

             return a;

         }

         // Else the argument is either an integral value already XOR it

         // has to be rounded to one.

         assert exponent >= 0 && exponent <= 51;

         long doppel = Double.doubleToRawLongBits(a);

         long mask   = DoubleConsts.SIGNIF_BIT_MASK >> exponent;

         if ( (mask & doppel) == 0L )

             return a; // integral value

         else {

             double result = Double.longBitsToDouble(doppel & (~mask));

             if (sign*a > 0.0)

                 result = result + sign;

             return result;

         }

     }

     /**

      * Returns the {@code double} value that is closest in value

      * to the argument and is equal to a mathematical integer. If two

      * {@code double} values that are mathematical integers are

      * equally close to the value of the argument, the result is the

      * integer value that is even. Special cases:

      * <ul><li>If the argument value is already equal to a mathematical

      * integer, then the result is the same as the argument.

      * <li>If the argument is NaN or an infinity or positive zero or negative

      * zero, then the result is the same as the argument.</ul>

      *

      * @param   a   a value.

      * @return  the closest floating-point value to {@code a} that is

      *          equal to a mathematical integer.

      * @author Joseph D. Darcy

      */

     public static double rint(double a) {

         /*

          * If the absolute value of a is not less than 2^52, it

          * is either a finite integer (the double format does not have

          * enough significand bits for a number that large to have any

          * fractional portion), an infinity, or a NaN.  In any of

          * these cases, rint of the argument is the argument.

          *

          * Otherwise, the sum (twoToThe52 + a ) will properly round

          * away any fractional portion of a since ulp(twoToThe52) ==

          * 1.0; subtracting out twoToThe52 from this sum will then be

          * exact and leave the rounded integer portion of a.

          *

          * This method does *not* need to be declared strictfp to get

          * fully reproducible results.  Whether or not a method is

          * declared strictfp can only make a difference in the

          * returned result if some operation would overflow or

          * underflow with strictfp semantics.  The operation

          * (twoToThe52 + a ) cannot overflow since large values of a

          * are screened out; the add cannot underflow since twoToThe52

          * is too large.  The subtraction ((twoToThe52 + a ) -

          * twoToThe52) will be exact as discussed above and thus

          * cannot overflow or meaningfully underflow.  Finally, the

          * last multiply in the return statement is by plus or minus

          * 1.0, which is exact too.

          */

         double twoToThe52 = (double)(1L << 52); // 2^52

         double sign = FpUtils.rawCopySign(1.0, a); // preserve sign info

         a = Math.abs(a);

         if (a < twoToThe52) { // E_min <= ilogb(a) <= 51

             a = ((twoToThe52 + a ) - twoToThe52);

         }

         return sign * a; // restore original sign

     }

     /**

      * Returns the angle <i>theta</i> from the conversion of rectangular

      * coordinates ({@code x}, {@code y}) to polar

      * coordinates (r,&nbsp;<i>theta</i>).

      * This method computes the phase <i>theta</i> by computing an arc tangent

      * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special

      * cases:

      * <ul><li>If either argument is NaN, then the result is NaN.

      * <li>If the first argument is positive zero and the second argument

      * is positive, or the first argument is positive and finite and the

      * second argument is positive infinity, then the result is positive

      * zero.

      * <li>If the first argument is negative zero and the second argument

      * is positive, or the first argument is negative and finite and the

      * second argument is positive infinity, then the result is negative zero.

      * <li>If the first argument is positive zero and the second argument

      * is negative, or the first argument is positive and finite and the

      * second argument is negative infinity, then the result is the

      * {@code double} value closest to <i>pi</i>.

      * <li>If the first argument is negative zero and the second argument

      * is negative, or the first argument is negative and finite and the

      * second argument is negative infinity, then the result is the

      * {@code double} value closest to -<i>pi</i>.

      * <li>If the first argument is positive and the second argument is

      * positive zero or negative zero, or the first argument is positive

      * infinity and the second argument is finite, then the result is the

      * {@code double} value closest to <i>pi</i>/2.

      * <li>If the first argument is negative and the second argument is

      * positive zero or negative zero, or the first argument is negative

      * infinity and the second argument is finite, then the result is the

      * {@code double} value closest to -<i>pi</i>/2.

      * <li>If both arguments are positive infinity, then the result is the

      * {@code double} value closest to <i>pi</i>/4.

      * <li>If the first argument is positive infinity and the second argument

      * is negative infinity, then the result is the {@code double}

      * value closest to 3*<i>pi</i>/4.

      * <li>If the first argument is negative infinity and the second argument

      * is positive infinity, then the result is the {@code double} value

      * closest to -<i>pi</i>/4.

      * <li>If both arguments are negative infinity, then the result is the

      * {@code double} value closest to -3*<i>pi</i>/4.</ul>

      *

      * @param   y   the ordinate coordinate

      * @param   x   the abscissa coordinate

      * @return  the <i>theta</i> component of the point

      *          (<i>r</i>,&nbsp;<i>theta</i>)

      *          in polar coordinates that corresponds to the point

      *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.

      */

     public static native double atan2(double y, double x);

     /**

      * Returns the value of the first argument raised to the power of the

      * second argument. Special cases:

      *

      * <ul><li>If the second argument is positive or negative zero, then the

      * result is 1.0.

      * <li>If the second argument is 1.0, then the result is the same as the

      * first argument.

      * <li>If the second argument is NaN, then the result is NaN.

      * <li>If the first argument is NaN and the second argument is nonzero,

      * then the result is NaN.

      *

      * <li>If

      * <ul>

      * <li>the absolute value of the first argument is greater than 1

      * and the second argument is positive infinity, or

      * <li>the absolute value of the first argument is less than 1 and

      * the second argument is negative infinity,

      * </ul>

      * then the result is positive infinity.

      *

      * <li>If

      * <ul>

      * <li>the absolute value of the first argument is greater than 1 and

      * the second argument is negative infinity, or

      * <li>the absolute value of the

      * first argument is less than 1 and the second argument is positive

      * infinity,

      * </ul>

      * then the result is positive zero.

      *

      * <li>If the absolute value of the first argument equals 1 and the

      * second argument is infinite, then the result is NaN.

      *

      * <li>If

      * <ul>

      * <li>the first argument is positive zero and the second argument

      * is greater than zero, or

      * <li>the first argument is positive infinity and the second

      * argument is less than zero,

      * </ul>

      * then the result is positive zero.

      *

      * <li>If

      * <ul>

      * <li>the first argument is positive zero and the second argument

      * is less than zero, or

      * <li>the first argument is positive infinity and the second

      * argument is greater than zero,

      * </ul>

      * then the result is positive infinity.

      *

      * <li>If

      * <ul>

      * <li>the first argument is negative zero and the second argument

      * is greater than zero but not a finite odd integer, or

      * <li>the first argument is negative infinity and the second

      * argument is less than zero but not a finite odd integer,

      * </ul>

      * then the result is positive zero.

      *

      * <li>If

      * <ul>

      * <li>the first argument is negative zero and the second argument

      * is a positive finite odd integer, or

      * <li>the first argument is negative infinity and the second

      * argument is a negative finite odd integer,

      * </ul>

      * then the result is negative zero.

      *

      * <li>If

      * <ul>

      * <li>the first argument is negative zero and the second argument

      * is less than zero but not a finite odd integer, or

      * <li>the first argument is negative infinity and the second

      * argument is greater than zero but not a finite odd integer,

      * </ul>

      * then the result is positive infinity.

      *

      * <li>If

      * <ul>

      * <li>the first argument is negative zero and the second argument

      * is a negative finite odd integer, or

      * <li>the first argument is negative infinity and the second

      * argument is a positive finite odd integer,

      * </ul>

      * then the result is negative infinity.

      *

      * <li>If the first argument is finite and less than zero

      * <ul>

      * <li> if the second argument is a finite even integer, the

      * result is equal to the result of raising the absolute value of

      * the first argument to the power of the second argument

      *

      * <li>if the second argument is a finite odd integer, the result

      * is equal to the negative of the result of raising the absolute

      * value of the first argument to the power of the second

      * argument

      *

      * <li>if the second argument is finite and not an integer, then

      * the result is NaN.

      * </ul>

      *

      * <li>If both arguments are integers, then the result is exactly equal

      * to the mathematical result of raising the first argument to the power

      * of the second argument if that result can in fact be represented

      * exactly as a {@code double} value.</ul>

      *

      * <p>(In the foregoing descriptions, a floating-point value is

      * considered to be an integer if and only if it is finite and a

      * fixed point of the method {@link #ceil ceil} or,

      * equivalently, a fixed point of the method {@link #floor

      * floor}. A value is a fixed point of a one-argument

      * method if and only if the result of applying the method to the

      * value is equal to the value.)

      *

      * @param   a   base.

      * @param   b   the exponent.

      * @return  the value {@code a}<sup>{@code b}</sup>.

      */

     public static native double pow(double a, double b);

     /**

      * Returns the closest {@code int} to the argument, with ties

      * rounding up.

      *

      * <p>Special cases:

      * <ul><li>If the argument is NaN, the result is 0.

      * <li>If the argument is negative infinity or any value less than or

      * equal to the value of {@code Integer.MIN_VALUE}, the result is

      * equal to the value of {@code Integer.MIN_VALUE}.

      * <li>If the argument is positive infinity or any value greater than or

      * equal to the value of {@code Integer.MAX_VALUE}, the result is

      * equal to the value of {@code Integer.MAX_VALUE}.</ul>

      *

      * @param   a   a floating-point value to be rounded to an integer.

      * @return  the value of the argument rounded to the nearest

      *          {@code int} value.

      * @see     java.lang.Integer#MAX_VALUE

      * @see     java.lang.Integer#MIN_VALUE

      */

     public static int round(float a) {

         return Math.round(a);

     }

     /**

      * Returns the closest {@code long} to the argument, with ties

      * rounding up.

      *

      * <p>Special cases:

      * <ul><li>If the argument is NaN, the result is 0.

      * <li>If the argument is negative infinity or any value less than or

      * equal to the value of {@code Long.MIN_VALUE}, the result is

      * equal to the value of {@code Long.MIN_VALUE}.

      * <li>If the argument is positive infinity or any value greater than or

      * equal to the value of {@code Long.MAX_VALUE}, the result is

      * equal to the value of {@code Long.MAX_VALUE}.</ul>

      *

      * @param   a  a floating-point value to be rounded to a

      *          {@code long}.

      * @return  the value of the argument rounded to the nearest

      *          {@code long} value.

      * @see     java.lang.Long#MAX_VALUE

      * @see     java.lang.Long#MIN_VALUE

      */

     public static long round(double a) {

         return Math.round(a);

     }

     private static Random randomNumberGenerator;

     private static synchronized Random initRNG() {

         Random rnd = randomNumberGenerator;

         return (rnd == null) ? (randomNumberGenerator = new Random()) : rnd;

     }

     /**

      * Returns a {@code double} value with a positive sign, greater

      * than or equal to {@code 0.0} and less than {@code 1.0}.

      * Returned values are chosen pseudorandomly with (approximately)

      * uniform distribution from that range.

      *

      * <p>When this method is first called, it creates a single new

      * pseudorandom-number generator, exactly as if by the expression

      *

      * <blockquote>{@code new java.util.Random()}</blockquote>

      *

      * This new pseudorandom-number generator is used thereafter for

      * all calls to this method and is used nowhere else.

      *

      * <p>This method is properly synchronized to allow correct use by

      * more than one thread. However, if many threads need to generate

      * pseudorandom numbers at a great rate, it may reduce contention

      * for each thread to have its own pseudorandom number generator.

      *

      * @return  a pseudorandom {@code double} greater than or equal

      * to {@code 0.0} and less than {@code 1.0}.

      * @see Random#nextDouble()

      */

     public static double random() {

         Random rnd = randomNumberGenerator;

         if (rnd == null) rnd = initRNG();

         return rnd.nextDouble();

     }

     /**

      * Returns the absolute value of an {@code int} value..

      * If the argument is not negative, the argument is returned.

      * If the argument is negative, the negation of the argument is returned.

      *

      * <p>Note that if the argument is equal to the value of

      * {@link Integer#MIN_VALUE}, the most negative representable

      * {@code int} value, the result is that same value, which is

      * negative.

      *

      * @param   a   the  argument whose absolute value is to be determined.

      * @return  the absolute value of the argument.

      */

     public static int abs(int a) {

         return (a < 0) ? -a : a;

     }

     /**

      * Returns the absolute value of a {@code long} value.

      * If the argument is not negative, the argument is returned.

      * If the argument is negative, the negation of the argument is returned.

      *

      * <p>Note that if the argument is equal to the value of

      * {@link Long#MIN_VALUE}, the most negative representable

      * {@code long} value, the result is that same value, which

      * is negative.

      *

      * @param   a   the  argument whose absolute value is to be determined.

      * @return  the absolute value of the argument.

      */

     public static long abs(long a) {

         return (a < 0) ? -a : a;

     }

     /**

      * Returns the absolute value of a {@code float} value.

      * If the argument is not negative, the argument is returned.

      * If the argument is negative, the negation of the argument is returned.

      * Special cases:

      * <ul><li>If the argument is positive zero or negative zero, the

      * result is positive zero.

      * <li>If the argument is infinite, the result is positive infinity.

      * <li>If the argument is NaN, the result is NaN.</ul>

      * In other words, the result is the same as the value of the expression:

      * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}

      *

      * @param   a   the argument whose absolute value is to be determined

      * @return  the absolute value of the argument.

      */

     public static float abs(float a) {

         return (a <= 0.0F) ? 0.0F - a : a;

     }

     /**

      * Returns the absolute value of a {@code double} value.

      * If the argument is not negative, the argument is returned.

      * If the argument is negative, the negation of the argument is returned.

      * Special cases:

      * <ul><li>If the argument is positive zero or negative zero, the result

      * is positive zero.

      * <li>If the argument is infinite, the result is positive infinity.

      * <li>If the argument is NaN, the result is NaN.</ul>

      * In other words, the result is the same as the value of the expression:

      * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}

      *

      * @param   a   the argument whose absolute value is to be determined

      * @return  the absolute value of the argument.

      */

     public static double abs(double a) {

         return (a <= 0.0D) ? 0.0D - a : a;

     }

     /**

      * Returns the greater of two {@code int} values. That is, the

      * result is the argument closer to the value of

      * {@link Integer#MAX_VALUE}. If the arguments have the same value,

      * the result is that same value.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the larger of {@code a} and {@code b}.

      */

     public static int max(int a, int b) {

         return (a >= b) ? a : b;

     }

     /**

      * Returns the greater of two {@code long} values. That is, the

      * result is the argument closer to the value of

      * {@link Long#MAX_VALUE}. If the arguments have the same value,

      * the result is that same value.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the larger of {@code a} and {@code b}.

         */

     public static long max(long a, long b) {

         return (a >= b) ? a : b;

     }

     // Use raw bit-wise conversions on guaranteed non-NaN arguments.

     private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);

     private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);

     /**

      * Returns the greater of two {@code float} values.  That is,

      * the result is the argument closer to positive infinity. If the

      * arguments have the same value, the result is that same

      * value. If either value is NaN, then the result is NaN.  Unlike

      * the numerical comparison operators, this method considers

      * negative zero to be strictly smaller than positive zero. If one

      * argument is positive zero and the other negative zero, the

      * result is positive zero.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the larger of {@code a} and {@code b}.

      */

     public static float max(float a, float b) {

         if (a != a)

             return a;   // a is NaN

         if ((a == 0.0f) &&

             (b == 0.0f) &&

             (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {

             // Raw conversion ok since NaN can't map to -0.0.

             return b;

         }

         return (a >= b) ? a : b;

     }

     /**

      * Returns the greater of two {@code double} values.  That

      * is, the result is the argument closer to positive infinity. If

      * the arguments have the same value, the result is that same

      * value. If either value is NaN, then the result is NaN.  Unlike

      * the numerical comparison operators, this method considers

      * negative zero to be strictly smaller than positive zero. If one

      * argument is positive zero and the other negative zero, the

      * result is positive zero.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the larger of {@code a} and {@code b}.

      */

     public static double max(double a, double b) {

         if (a != a)

             return a;   // a is NaN

         if ((a == 0.0d) &&

             (b == 0.0d) &&

             (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {

             // Raw conversion ok since NaN can't map to -0.0.

             return b;

         }

         return (a >= b) ? a : b;

     }

     /**

      * Returns the smaller of two {@code int} values. That is,

      * the result the argument closer to the value of

      * {@link Integer#MIN_VALUE}.  If the arguments have the same

      * value, the result is that same value.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the smaller of {@code a} and {@code b}.

      */

     public static int min(int a, int b) {

         return (a <= b) ? a : b;

     }

     /**

      * Returns the smaller of two {@code long} values. That is,

      * the result is the argument closer to the value of

      * {@link Long#MIN_VALUE}. If the arguments have the same

      * value, the result is that same value.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the smaller of {@code a} and {@code b}.

      */

     public static long min(long a, long b) {

         return (a <= b) ? a : b;

     }

     /**

      * Returns the smaller of two {@code float} values.  That is,

      * the result is the value closer to negative infinity. If the

      * arguments have the same value, the result is that same

      * value. If either value is NaN, then the result is NaN.  Unlike

      * the numerical comparison operators, this method considers

      * negative zero to be strictly smaller than positive zero.  If

      * one argument is positive zero and the other is negative zero,

      * the result is negative zero.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the smaller of {@code a} and {@code b.}

      */

     public static float min(float a, float b) {

         if (a != a)

             return a;   // a is NaN

         if ((a == 0.0f) &&

             (b == 0.0f) &&

             (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {

             // Raw conversion ok since NaN can't map to -0.0.

             return b;

         }

         return (a <= b) ? a : b;

     }

     /**

      * Returns the smaller of two {@code double} values.  That

      * is, the result is the value closer to negative infinity. If the

      * arguments have the same value, the result is that same

      * value. If either value is NaN, then the result is NaN.  Unlike

      * the numerical comparison operators, this method considers

      * negative zero to be strictly smaller than positive zero. If one

      * argument is positive zero and the other is negative zero, the

      * result is negative zero.

      *

      * @param   a   an argument.

      * @param   b   another argument.

      * @return  the smaller of {@code a} and {@code b}.

      */

     public static double min(double a, double b) {

         if (a != a)

             return a;   // a is NaN

         if ((a == 0.0d) &&

             (b == 0.0d) &&

             (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {

             // Raw conversion ok since NaN can't map to -0.0.

             return b;

         }

         return (a <= b) ? a : b;

     }

     /**

      * Returns the size of an ulp of the argument.  An ulp of a

      * {@code double} value is the positive distance between this

      * floating-point value and the {@code double} value next

      * larger in magnitude.  Note that for non-NaN <i>x</i>,

      * <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, then the result is NaN.

      * <li> If the argument is positive or negative infinity, then the

      * result is positive infinity.

      * <li> If the argument is positive or negative zero, then the result is

      * {@code Double.MIN_VALUE}.

      * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then

      * the result is equal to 2<sup>971</sup>.

      * </ul>

      *

      * @param d the floating-point value whose ulp is to be returned

      * @return the size of an ulp of the argument

      * @author Joseph D. Darcy

      * @since 1.5

      */

     public static double ulp(double d) {

         return sun.misc.FpUtils.ulp(d);

     }

     /**

      * Returns the size of an ulp of the argument.  An ulp of a

      * {@code float} value is the positive distance between this

      * floating-point value and the {@code float} value next

      * larger in magnitude.  Note that for non-NaN <i>x</i>,

      * <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, then the result is NaN.

      * <li> If the argument is positive or negative infinity, then the

      * result is positive infinity.

      * <li> If the argument is positive or negative zero, then the result is

      * {@code Float.MIN_VALUE}.

      * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then

      * the result is equal to 2<sup>104</sup>.

      * </ul>

      *

      * @param f the floating-point value whose ulp is to be returned

      * @return the size of an ulp of the argument

      * @author Joseph D. Darcy

      * @since 1.5

      */

     public static float ulp(float f) {

         return sun.misc.FpUtils.ulp(f);

     }

     /**

      * Returns the signum function of the argument; zero if the argument

      * is zero, 1.0 if the argument is greater than zero, -1.0 if the

      * argument is less than zero.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, then the result is NaN.

      * <li> If the argument is positive zero or negative zero, then the

      *      result is the same as the argument.

      * </ul>

      *

      * @param d the floating-point value whose signum is to be returned

      * @return the signum function of the argument

      * @author Joseph D. Darcy

      * @since 1.5

      */

     public static double signum(double d) {

         return sun.misc.FpUtils.signum(d);

     }

     /**

      * Returns the signum function of the argument; zero if the argument

      * is zero, 1.0f if the argument is greater than zero, -1.0f if the

      * argument is less than zero.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, then the result is NaN.

      * <li> If the argument is positive zero or negative zero, then the

      *      result is the same as the argument.

      * </ul>

      *

      * @param f the floating-point value whose signum is to be returned

      * @return the signum function of the argument

      * @author Joseph D. Darcy

      * @since 1.5

      */

     public static float signum(float f) {

         return sun.misc.FpUtils.signum(f);

     }

     /**

      * Returns the hyperbolic sine of a {@code double} value.

      * The hyperbolic sine of <i>x</i> is defined to be

      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2

      * where <i>e</i> is {@linkplain Math#E Euler's number}.

      *

      * <p>Special cases:

      * <ul>

      *

      * <li>If the argument is NaN, then the result is NaN.

      *

      * <li>If the argument is infinite, then the result is an infinity

      * with the same sign as the argument.

      *

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.

      *

      * </ul>

      *

      * @param   x The number whose hyperbolic sine is to be returned.

      * @return  The hyperbolic sine of {@code x}.

      * @since 1.5

      */

     public static native double sinh(double x);

     /**

      * Returns the hyperbolic cosine of a {@code double} value.

      * The hyperbolic cosine of <i>x</i> is defined to be

      * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2

      * where <i>e</i> is {@linkplain Math#E Euler's number}.

      *

      * <p>Special cases:

      * <ul>

      *

      * <li>If the argument is NaN, then the result is NaN.

      *

      * <li>If the argument is infinite, then the result is positive

      * infinity.

      *

      * <li>If the argument is zero, then the result is {@code 1.0}.

      *

      * </ul>

      *

      * @param   x The number whose hyperbolic cosine is to be returned.

      * @return  The hyperbolic cosine of {@code x}.

      * @since 1.5

      */

     public static native double cosh(double x);

     /**

      * Returns the hyperbolic tangent of a {@code double} value.

      * The hyperbolic tangent of <i>x</i> is defined to be

      * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),

      * in other words, {@linkplain Math#sinh

      * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note

      * that the absolute value of the exact tanh is always less than

      * 1.

      *

      * <p>Special cases:

      * <ul>

      *

      * <li>If the argument is NaN, then the result is NaN.

      *

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.

      *

      * <li>If the argument is positive infinity, then the result is

      * {@code +1.0}.

      *

      * <li>If the argument is negative infinity, then the result is

      * {@code -1.0}.

      *

      * </ul>

      *

      * @param   x The number whose hyperbolic tangent is to be returned.

      * @return  The hyperbolic tangent of {@code x}.

      * @since 1.5

      */

     public static native double tanh(double x);

     /**

      * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)

      * without intermediate overflow or underflow.

      *

      * <p>Special cases:

      * <ul>

      *

      * <li> If either argument is infinite, then the result

      * is positive infinity.

      *

      * <li> If either argument is NaN and neither argument is infinite,

      * then the result is NaN.

      *

      * </ul>

      *

      * @param x a value

      * @param y a value

      * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)

      * without intermediate overflow or underflow

      * @since 1.5

      */

     public static native double hypot(double x, double y);

     /**

      * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of

      * <i>x</i> near 0, the exact sum of

      * {@code expm1(x)} + 1 is much closer to the true

      * result of <i>e</i><sup>x</sup> than {@code exp(x)}.

      *

      * <p>Special cases:

      * <ul>

      * <li>If the argument is NaN, the result is NaN.

      *

      * <li>If the argument is positive infinity, then the result is

      * positive infinity.

      *

      * <li>If the argument is negative infinity, then the result is

      * -1.0.

      *

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.

      *

      * </ul>

      *

      * @param   x   the exponent to raise <i>e</i> to in the computation of

      *              <i>e</i><sup>{@code x}</sup>&nbsp;-1.

      * @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.

      * @since 1.5

      */

     public static native double expm1(double x);

     /**

      * Returns the natural logarithm of the sum of the argument and 1.

      * Note that for small values {@code x}, the result of

      * {@code log1p(x)} is much closer to the true result of ln(1

      * + {@code x}) than the floating-point evaluation of

      * {@code log(1.0+x)}.

      *

      * <p>Special cases:

      * <ul>

      *

      * <li>If the argument is NaN or less than -1, then the result is

      * NaN.

      *

      * <li>If the argument is positive infinity, then the result is

      * positive infinity.

      *

      * <li>If the argument is negative one, then the result is

      * negative infinity.

      *

      * <li>If the argument is zero, then the result is a zero with the

      * same sign as the argument.

      *

      * </ul>

      *

      * @param   x   a value

      * @return the value ln({@code x} + 1), the natural

      * log of {@code x} + 1

      * @since 1.5

      */

     public static native double log1p(double x);

     /**

      * Returns the first floating-point argument with the sign of the

      * second floating-point argument.  For this method, a NaN

      * {@code sign} argument is always treated as if it were

      * positive.

      *

      * @param magnitude  the parameter providing the magnitude of the result

      * @param sign   the parameter providing the sign of the result

      * @return a value with the magnitude of {@code magnitude}

      * and the sign of {@code sign}.

      * @since 1.6

      */

     public static double copySign(double magnitude, double sign) {

         return sun.misc.FpUtils.copySign(magnitude, sign);

     }

     /**

      * Returns the first floating-point argument with the sign of the

      * second floating-point argument.  For this method, a NaN

      * {@code sign} argument is always treated as if it were

      * positive.

      *

      * @param magnitude  the parameter providing the magnitude of the result

      * @param sign   the parameter providing the sign of the result

      * @return a value with the magnitude of {@code magnitude}

      * and the sign of {@code sign}.

      * @since 1.6

      */

     public static float copySign(float magnitude, float sign) {

         return sun.misc.FpUtils.copySign(magnitude, sign);

     }

     /**

      * Returns the unbiased exponent used in the representation of a

      * {@code float}.  Special cases:

      *

      * <ul>

      * <li>If the argument is NaN or infinite, then the result is

      * {@link Float#MAX_EXPONENT} + 1.

      * <li>If the argument is zero or subnormal, then the result is

      * {@link Float#MIN_EXPONENT} -1.

      * </ul>

      * @param f a {@code float} value

      * @since 1.6

      */

     public static int getExponent(float f) {

         return sun.misc.FpUtils.getExponent(f);

     }

     /**

      * Returns the unbiased exponent used in the representation of a

      * {@code double}.  Special cases:

      *

      * <ul>

      * <li>If the argument is NaN or infinite, then the result is

      * {@link Double#MAX_EXPONENT} + 1.

      * <li>If the argument is zero or subnormal, then the result is

      * {@link Double#MIN_EXPONENT} -1.

      * </ul>

      * @param d a {@code double} value

      * @since 1.6

      */

     public static int getExponent(double d) {

         return sun.misc.FpUtils.getExponent(d);

     }

     /**

      * Returns the floating-point number adjacent to the first

      * argument in the direction of the second argument.  If both

      * arguments compare as equal the second argument is returned.

      *

      * <p>Special cases:

      * <ul>

      * <li> If either argument is a NaN, then NaN is returned.

      *

      * <li> If both arguments are signed zeros, {@code direction}

      * is returned unchanged (as implied by the requirement of

      * returning the second argument if the arguments compare as

      * equal).

      *

      * <li> If {@code start} is

      * ±{@link Double#MIN_VALUE} and {@code direction}

      * has a value such that the result should have a smaller

      * magnitude, then a zero with the same sign as {@code start}

      * is returned.

      *

      * <li> If {@code start} is infinite and

      * {@code direction} has a value such that the result should

      * have a smaller magnitude, {@link Double#MAX_VALUE} with the

      * same sign as {@code start} is returned.

      *

      * <li> If {@code start} is equal to &plusmn;

      * {@link Double#MAX_VALUE} and {@code direction} has a

      * value such that the result should have a larger magnitude, an

      * infinity with same sign as {@code start} is returned.

      * </ul>

      *

      * @param start  starting floating-point value

      * @param direction value indicating which of

      * {@code start}'s neighbors or {@code start} should

      * be returned

      * @return The floating-point number adjacent to {@code start} in the

      * direction of {@code direction}.

      * @since 1.6

      */

     public static double nextAfter(double start, double direction) {

         return sun.misc.FpUtils.nextAfter(start, direction);

     }

     /**

      * Returns the floating-point number adjacent to the first

      * argument in the direction of the second argument.  If both

      * arguments compare as equal a value equivalent to the second argument

      * is returned.

      *

      * <p>Special cases:

      * <ul>

      * <li> If either argument is a NaN, then NaN is returned.

      *

      * <li> If both arguments are signed zeros, a value equivalent

      * to {@code direction} is returned.

      *

      * <li> If {@code start} is

      * ±{@link Float#MIN_VALUE} and {@code direction}

      * has a value such that the result should have a smaller

      * magnitude, then a zero with the same sign as {@code start}

      * is returned.

      *

      * <li> If {@code start} is infinite and

      * {@code direction} has a value such that the result should

      * have a smaller magnitude, {@link Float#MAX_VALUE} with the

      * same sign as {@code start} is returned.

      *

      * <li> If {@code start} is equal to &plusmn;

      * {@link Float#MAX_VALUE} and {@code direction} has a

      * value such that the result should have a larger magnitude, an

      * infinity with same sign as {@code start} is returned.

      * </ul>

      *

      * @param start  starting floating-point value

      * @param direction value indicating which of

      * {@code start}'s neighbors or {@code start} should

      * be returned

      * @return The floating-point number adjacent to {@code start} in the

      * direction of {@code direction}.

      * @since 1.6

      */

     public static float nextAfter(float start, double direction) {

         return sun.misc.FpUtils.nextAfter(start, direction);

     }

     /**

      * Returns the floating-point value adjacent to {@code d} in

      * the direction of positive infinity.  This method is

      * semantically equivalent to {@code nextAfter(d,

      * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}

      * implementation may run faster than its equivalent

      * {@code nextAfter} call.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, the result is NaN.

      *

      * <li> If the argument is positive infinity, the result is

      * positive infinity.

      *

      * <li> If the argument is zero, the result is

      * {@link Double#MIN_VALUE}

      *

      * </ul>

      *

      * @param d starting floating-point value

      * @return The adjacent floating-point value closer to positive

      * infinity.

      * @since 1.6

      */

     public static double nextUp(double d) {

         return sun.misc.FpUtils.nextUp(d);

     }

     /**

      * Returns the floating-point value adjacent to {@code f} in

      * the direction of positive infinity.  This method is

      * semantically equivalent to {@code nextAfter(f,

      * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}

      * implementation may run faster than its equivalent

      * {@code nextAfter} call.

      *

      * <p>Special Cases:

      * <ul>

      * <li> If the argument is NaN, the result is NaN.

      *

      * <li> If the argument is positive infinity, the result is

      * positive infinity.

      *

      * <li> If the argument is zero, the result is

      * {@link Float#MIN_VALUE}

      *

      * </ul>

      *

      * @param f starting floating-point value

      * @return The adjacent floating-point value closer to positive

      * infinity.

      * @since 1.6

      */

     public static float nextUp(float f) {

         return sun.misc.FpUtils.nextUp(f);

     }

     /**

      * Return {@code d} ×

      * 2<sup>{@code scaleFactor}</sup> rounded as if performed

      * by a single correctly rounded floating-point multiply to a

      * member of the double value set.  See the Java

      * Language Specification for a discussion of floating-point

      * value sets.  If the exponent of the result is between {@link

      * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the

      * answer is calculated exactly.  If the exponent of the result

      * would be larger than {@code Double.MAX_EXPONENT}, an

      * infinity is returned.  Note that if the result is subnormal,

      * precision may be lost; that is, when {@code scalb(x, n)}

      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal

      * <i>x</i>.  When the result is non-NaN, the result has the same

      * sign as {@code d}.

      *

      * <p>Special cases:

      * <ul>

      * <li> If the first argument is NaN, NaN is returned.

      * <li> If the first argument is infinite, then an infinity of the

      * same sign is returned.

      * <li> If the first argument is zero, then a zero of the same

      * sign is returned.

      * </ul>

      *

      * @param d number to be scaled by a power of two.

      * @param scaleFactor power of 2 used to scale {@code d}

      * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>

      * @since 1.6

      */

     public static double scalb(double d, int scaleFactor) {

         return sun.misc.FpUtils.scalb(d, scaleFactor);

     }

     /**

      * Return {@code f} ×

      * 2<sup>{@code scaleFactor}</sup> rounded as if performed

      * by a single correctly rounded floating-point multiply to a

      * member of the float value set.  See the Java

      * Language Specification for a discussion of floating-point

      * value sets.  If the exponent of the result is between {@link

      * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the

      * answer is calculated exactly.  If the exponent of the result

      * would be larger than {@code Float.MAX_EXPONENT}, an

      * infinity is returned.  Note that if the result is subnormal,

      * precision may be lost; that is, when {@code scalb(x, n)}

      * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal

      * <i>x</i>.  When the result is non-NaN, the result has the same

      * sign as {@code f}.

      *

      * <p>Special cases:

      * <ul>

      * <li> If the first argument is NaN, NaN is returned.

      * <li> If the first argument is infinite, then an infinity of the

      * same sign is returned.

      * <li> If the first argument is zero, then a zero of the same

      * sign is returned.

      * </ul>

      *

      * @param f number to be scaled by a power of two.

      * @param scaleFactor power of 2 used to scale {@code f}

      * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>

      * @since 1.6

      */

     public static float scalb(float f, int scaleFactor) {

         return sun.misc.FpUtils.scalb(f, scaleFactor);

     }

 }