jdk/src/share/classes/java/lang/Math.java

/*
 * Copyright 1994-2006 Sun Microsystems, Inc.  All Rights Reserved.
 * 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.  Sun designates this
 * particular file as subject to the "Classpath" exception as provided
 * by Sun 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
 * 2 along with this work; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
 *
 * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
 * CA 95054 USA or visit www.sun.com if you need additional information or
 * have any questions.
 */

package java.lang;
import java.util.Random;


/**
 * The class {@code Math} contains methods for performing basic
 * numeric operations such as the elementary exponential, logarithm,
 * square root, and trigonometric functions.
 *
 * <p>Unlike some of the numeric methods of class
 * {@code StrictMath}, all implementations of the equivalent
 * functions of class {@code Math} are not defined to return the
 * bit-for-bit same results.  This relaxation permits
 * better-performing implementations where strict reproducibility is
 * not required.
 *
 * <p>By default many of the {@code Math} methods simply call
 * the equivalent method in {@code StrictMath} for their
 * implementation.  Code generators are encouraged to use
 * platform-specific native libraries or microprocessor instructions,
 * where available, to provide higher-performance implementations of
 * {@code Math} methods.  Such higher-performance
 * implementations still must conform to the specification for
 * {@code Math}.
 *
 * <p>The quality of implementation specifications concern two
 * properties, accuracy of the returned result and monotonicity of the
 * method.  Accuracy of the floating-point {@code Math} methods
 * is measured in terms of <i>ulps</i>, units in the last place.  For
 * a given floating-point format, an ulp of a specific real number
 * value is the distance between the two floating-point values
 * bracketing that numerical value.  When discussing the accuracy of a
 * method as a whole rather than at a specific argument, the number of
 * ulps cited is for the worst-case error at any argument.  If a
 * method always has an error less than 0.5 ulps, the method always
 * returns the floating-point number nearest the exact result; such a
 * method is <i>correctly rounded</i>.  A correctly rounded method is
 * generally the best a floating-point approximation can be; however,
 * it is impractical for many floating-point methods to be correctly
 * rounded.  Instead, for the {@code Math} class, a larger error
 * bound of 1 or 2 ulps is allowed for certain methods.  Informally,
 * with a 1 ulp error bound, when the exact result is a representable
 * number, the exact result should be returned as the computed result;
 * otherwise, either of the two floating-point values which bracket
 * the exact result may be returned.  For exact results large in
 * magnitude, one of the endpoints of the bracket may be infinite.
 * Besides accuracy at individual arguments, maintaining proper
 * relations between the method at different arguments is also
 * important.  Therefore, most methods with more than 0.5 ulp errors
 * are required to be <i>semi-monotonic</i>: whenever the mathematical
 * function is non-decreasing, so is the floating-point approximation,
 * likewise, whenever the mathematical function is non-increasing, so
 * is the floating-point approximation.  Not all approximations that
 * have 1 ulp accuracy will automatically meet the monotonicity
 * requirements.
 *
 * @author  unascribed
 * @author  Joseph D. Darcy
 * @since   JDK1.0
 */

public final class Math {

    /**
     * Don't let anyone instantiate this class.
     */
    private Math() {}

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the sine of the argument.
     */
    public static double sin(double a) {
        return StrictMath.sin(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the cosine of the argument.
     */
    public static double cos(double a) {
        return StrictMath.cos(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the tangent of the argument.
     */
    public static double tan(double a) {
        return StrictMath.tan(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the value whose arc sine is to be returned.
     * @return  the arc sine of the argument.
     */
    public static double asin(double a) {
        return StrictMath.asin(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the value whose arc cosine is to be returned.
     * @return  the arc cosine of the argument.
     */
    public static double acos(double a) {
        return StrictMath.acos(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the value whose arc tangent is to be returned.
     * @return  the arc tangent of the argument.
     */
    public static double atan(double a) {
        return StrictMath.atan(a); // default impl. delegates to StrictMath
    }

    /**
     * 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.
     * @since   1.2
     */
    public static 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.
     * @since   1.2
     */
    public static 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @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 double exp(double a) {
        return StrictMath.exp(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   a value
     * @return  the value ln&nbsp;{@code a}, the natural logarithm of
     *          {@code a}.
     */
    public static double log(double a) {
        return StrictMath.log(a); // default impl. delegates to StrictMath
    }

    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   a value
     * @return  the base 10 logarithm of  {@code a}.
     * @since 1.5
     */
    public static double log10(double a) {
        return StrictMath.log10(a); // default impl. delegates to StrictMath
    }

    /**
     * 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}.
     *          If the argument is NaN or less than zero, the result is NaN.
     */
    public static double sqrt(double a) {
        return StrictMath.sqrt(a); // default impl. delegates to StrictMath
                                   // Note that hardware sqrt instructions
                                   // frequently can be directly used by JITs
                                   // and should be much faster than doing
                                   // Math.sqrt in software.
    }


    /**
     * 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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     *
     * @param   a   a value.
     * @return  the cube root of {@code a}.
     * @since 1.5
     */
    public static double cbrt(double a) {
        return StrictMath.cbrt(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 double IEEEremainder(double f1, double f2) {
        return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
    }

    /**
     * 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 Math.ceil(x)} is exactly the
     * value of {@code -Math.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 StrictMath.ceil(a); // default impl. delegates to StrictMath
    }

    /**
     * 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 StrictMath.floor(a); // default impl. delegates to StrictMath
    }

    /**
     * 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, 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 {@code double} value.
     * @return  the closest floating-point value to {@code a} that is
     *          equal to a mathematical integer.
     */
    public static double rint(double a) {
        return StrictMath.rint(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the angle <i>theta</i> from the conversion of rectangular
     * coordinates ({@code x},&nbsp;{@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>
     *
     * <p>The computed result must be within 2 ulps of the exact result.
     * Results must be semi-monotonic.
     *
     * @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 double atan2(double y, double x) {
        return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
    }

    /**
     * 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.)
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the base.
     * @param   b   the exponent.
     * @return  the value {@code a}<sup>{@code b}</sup>.
     */
    public static double pow(double a, double b) {
        return StrictMath.pow(a, b); // default impl. delegates to StrictMath
    }

    /**
     * Returns the closest {@code int} to the argument. The
     * result is rounded to an integer by adding 1/2, taking the
     * floor of the result, and casting the result to type {@code int}.
     * In other words, the result is equal to the value of the expression:
     * <p>{@code (int)Math.floor(a + 0.5f)}
     * <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 (int)floor(a + 0.5f);
    }

    /**
     * Returns the closest {@code long} to the argument. The result
     * is rounded to an integer by adding 1/2, taking the floor of the
     * result, and casting the result to type {@code long}. In other
     * words, the result is equal to the value of the expression:
     * <p>{@code (long)Math.floor(a + 0.5d)}
     * <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 (long)floor(a + 0.5d);
    }

    private static Random randomNumberGenerator;

    private static synchronized void initRNG() {
        if (randomNumberGenerator == null)
            randomNumberGenerator = new Random();
    }

    /**
     * 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     java.util.Random#nextDouble()
     */
    public static double random() {
        if (randomNumberGenerator == null) initRNG();
        return randomNumberGenerator.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;
    }

    private static long negativeZeroFloatBits = Float.floatToIntBits(-0.0f);
    private static long negativeZeroDoubleBits = Double.doubleToLongBits(-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.floatToIntBits(a) == negativeZeroFloatBits)) {
            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.doubleToLongBits(a) == negativeZeroDoubleBits)) {
            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.floatToIntBits(b) == negativeZeroFloatBits)) {
            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.doubleToLongBits(b) == negativeZeroDoubleBits)) {
            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>
     *
     * <p>The computed result must be within 2.5 ulps of the exact result.
     *
     * @param   x The number whose hyperbolic sine is to be returned.
     * @return  The hyperbolic sine of {@code x}.
     * @since 1.5
     */
    public static double sinh(double x) {
        return StrictMath.sinh(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>
     *
     * <p>The computed result must be within 2.5 ulps of the exact result.
     *
     * @param   x The number whose hyperbolic cosine is to be returned.
     * @return  The hyperbolic cosine of {@code x}.
     * @since 1.5
     */
    public static double cosh(double x) {
        return StrictMath.cosh(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>
     *
     * <p>The computed result must be within 2.5 ulps of the exact result.
     * The result of {@code tanh} for any finite input must have
     * an absolute value less than or equal to 1.  Note that once the
     * exact result of tanh is within 1/2 of an ulp of the limit value
     * of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
     * be returned.
     *
     * @param   x The number whose hyperbolic tangent is to be returned.
     * @return  The hyperbolic tangent of {@code x}.
     * @since 1.5
     */
    public static double tanh(double x) {
        return StrictMath.tanh(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>
     *
     * <p>The computed result must be within 1 ulp of the exact
     * result.  If one parameter is held constant, the results must be
     * semi-monotonic in the other parameter.
     *
     * @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 double hypot(double x, double y) {
        return StrictMath.hypot(x, 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)}&nbsp;+&nbsp;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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.  The result of
     * {@code expm1} for any finite input must be greater than or
     * equal to {@code -1.0}.  Note that once the exact result of
     * <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
     * ulp of the limit value -1, {@code -1.0} should be
     * returned.
     *
     * @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 double expm1(double x) {
        return StrictMath.expm1(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>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   x   a value
     * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
     * log of {@code x}&nbsp;+&nbsp;1
     * @since 1.5
     */
    public static double log1p(double x) {
        return StrictMath.log1p(x);
    }

    /**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  Note that unlike the {@link
     * StrictMath#copySign(double, double) StrictMath.copySign}
     * method, this method does not require NaN {@code sign}
     * arguments to be treated as positive values; implementations are
     * permitted to treat some NaN arguments as positive and other NaN
     * arguments as negative to allow greater performance.
     *
     * @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.rawCopySign(magnitude, sign);
    }

    /**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  Note that unlike the {@link
     * StrictMath#copySign(float, float) StrictMath.copySign}
     * method, this method does not require NaN {@code sign}
     * arguments to be treated as positive values; implementations are
     * permitted to treat some NaN arguments as positive and other NaN
     * arguments as negative to allow greater performance.
     *
     * @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.rawCopySign(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
     * @return the unbiased exponent of the argument
     * @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
     * @return the unbiased exponent of the argument
     * @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
     * &plusmn;{@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
     * &plusmn;{@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} &times;
     * 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} &times;
     * 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);
    }
}