Math.java

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package java.lang;

import java.math.BigDecimal;
import java.util.Random;
import jdk.internal.math.FloatConsts;
import jdk.internal.math.DoubleConsts;
import jdk.internal.vm.annotation.IntrinsicCandidate;

/**
 * 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
 * {@link java.lang.StrictMath 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 {@linkplain #ulp(double) 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.
 *
 * <p>
 * The platform uses signed two's complement integer arithmetic with
 * int and long primitive types.  The developer should choose
 * the primitive type to ensure that arithmetic operations consistently
 * produce correct results, which in some cases means the operations
 * will not overflow the range of values of the computation.
 * The best practice is to choose the primitive type and algorithm to avoid
 * overflow. In cases where the size is {@code int} or {@code long} and
 * overflow errors need to be detected, the methods whose names end with
 * {@code Exact} throw an {@code ArithmeticException} when the results overflow.
 *
 * <h2><a id=Ieee754RecommendedOps>IEEE 754 Recommended
 * Operations</a></h2>
 *
 * The 2019 revision of the IEEE 754 floating-point standard includes
 * a section of recommended operations and the semantics of those
 * operations if they are included in a programming environment. The
 * recommended operations present in this class include {@link sin
 * sin}, {@link cos cos}, {@link tan tan}, {@link asin asin}, {@link
 * acos acos}, {@link atan atan}, {@link exp exp}, {@link expm1
 * expm1}, {@link log log}, {@link log10 log10}, {@link log1p log1p},
 * {@link sinh sinh}, {@link cosh cosh}, {@link tanh tanh}, {@link
 * hypot hypot}, and {@link pow pow}.  (The {@link sqrt sqrt}
 * operation is a required part of IEEE 754 from a different section
 * of the standard.) The special case behavior of the recommended
 * operations generally follows the guidance of the IEEE 754
 * standard. However, the {@code pow} method defines different
 * behavior for some arguments, as noted in its {@linkplain pow
 * specification}. The IEEE 754 standard defines its operations to be
 * correctly rounded, which is a more stringent quality of
 * implementation condition than required for most of the methods in
 * question that are also included in this class.
 *
 * @see <a href="https://standards.ieee.org/ieee/754/6210/">
 *      <cite>IEEE Standard for Floating-Point Arithmetic</cite></a>
 *
 * @author  Joseph D. Darcy
 * @since   1.0
 */

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 = 2.718281828459045;

    /**
     * The {@code double} value that is closer than any other to
     * <i>pi</i> (&pi;), the ratio of the circumference of a circle to
     * its diameter.
     */
    public static final double PI = 3.141592653589793;

    /**
     * The {@code double} value that is closer than any other to
     * <i>tau</i> (&tau;), the ratio of the circumference of a circle
     * to its radius.
     *
     * @apiNote
     * The value of <i>pi</i> is one half that of <i>tau</i>; in other
     * words, <i>tau</i> is double <i>pi</i> .
     *
     * @since 19
     */
    public static final double TAU = 2.0 * PI;

    /**
     * Constant by which to multiply an angular value in degrees to obtain an
     * angular value in radians.
     */
    private static final double DEGREES_TO_RADIANS = 0.017453292519943295;

    /**
     * Constant by which to multiply an angular value in radians to obtain an
     * angular value in degrees.
     */
    private static final double RADIANS_TO_DEGREES = 57.29577951308232;

    /**
     * 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.
     */
    @IntrinsicCandidate
    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.
     * <li>If the argument is zero, then the result is {@code 1.0}.
     *</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.
     */
    @IntrinsicCandidate
    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.
     */
    @IntrinsicCandidate
    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.
     * <li>If the argument is {@code 1.0}, 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 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.
     * <li>If the argument is {@linkplain Double#isInfinite infinite},
     * then the result is the closest value to <i>pi</i>/2 with the
     * same sign as the input.
     * </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 * DEGREES_TO_RADIANS;
    }

    /**
     * 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 * RADIANS_TO_DEGREES;
    }

    /**
     * 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.
     * <li>If the argument is zero, then the result is {@code 1.0}.
     * </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.
     */
    @IntrinsicCandidate
    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.
     * <li>If the argument is {@code 1.0}, 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   a value
     * @return  the value ln&nbsp;{@code a}, the natural logarithm of
     *          {@code a}.
     */
    @IntrinsicCandidate
    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>. In particular,
     * if the argument is {@code 1.0} (10<sup>0</sup>), 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   a value
     * @return  the base 10 logarithm of  {@code a}.
     * @since 1.5
     */
    @IntrinsicCandidate
    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.
     */
    @IntrinsicCandidate
    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.
     */
    @IntrinsicCandidate
    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.
     */
    @IntrinsicCandidate
    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.
     */
    @IntrinsicCandidate
    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.
     *
     * @apiNote
     * For <i>y</i> with a positive sign and finite nonzero
     * <i>x</i>, the exact mathematical value of {@code atan2} is
     * equal to:
     * <ul>
     * <li>If <i>x</i> {@literal >} 0, atan(abs(<i>y</i>/<i>x</i>))
     * <li>If <i>x</i> {@literal <} 0, &pi; - atan(abs(<i>y</i>/<i>x</i>))
     * </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.
     */
    @IntrinsicCandidate
    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.
     *
     * @apiNote
     * The special cases definitions of this method differ from the
     * special case definitions of the IEEE 754 recommended {@code
     * pow} operation for &plusmn;{@code 1.0} raised to an infinite
     * power. This method treats such cases as indeterminate and
     * specifies a NaN is returned. The IEEE 754 specification treats
     * the infinite power as a large integer (large-magnitude
     * floating-point numbers are numerically integers, specifically
     * even integers) and therefore specifies {@code 1.0} be returned.
     *
     * @param   a   the base.
     * @param   b   the exponent.
     * @return  the value {@code a}<sup>{@code b}</sup>.
     */
    @IntrinsicCandidate
    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, with ties
     * rounding to positive infinity.
     *
     * <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
     */
    @IntrinsicCandidate
    public static int round(float a) {
        int intBits = Float.floatToRawIntBits(a);
        int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
                >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
        int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
                + FloatConsts.EXP_BIAS) - biasedExp;
        if ((shift & -32) == 0) { // shift >= 0 && shift < 32
            // a is a finite number such that pow(2,-32) <= ulp(a) < 1
            int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
                    | (FloatConsts.SIGNIF_BIT_MASK + 1));
            if (intBits < 0) {
                r = -r;
            }
            // In the comments below each Java expression evaluates to the value
            // the corresponding mathematical expression:
            // (r) evaluates to a / ulp(a)
            // (r >> shift) evaluates to floor(a * 2)
            // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
            // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
            return ((r >> shift) + 1) >> 1;
        } else {
            // a is either
            // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
            // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
            // - an infinity or NaN
            return (int) a;
        }
    }

    /**
     * Returns the closest {@code long} to the argument, with ties
     * rounding to positive infinity.
     *
     * <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
     */
    @IntrinsicCandidate
    public static long round(double a) {
        long longBits = Double.doubleToRawLongBits(a);
        long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
                >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
        long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
                + DoubleConsts.EXP_BIAS) - biasedExp;
        if ((shift & -64) == 0) { // shift >= 0 && shift < 64
            // a is a finite number such that pow(2,-64) <= ulp(a) < 1
            long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
                    | (DoubleConsts.SIGNIF_BIT_MASK + 1));
            if (longBits < 0) {
                r = -r;
            }
            // In the comments below each Java expression evaluates to the value
            // the corresponding mathematical expression:
            // (r) evaluates to a / ulp(a)
            // (r >> shift) evaluates to floor(a * 2)
            // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
            // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
            return ((r >> shift) + 1) >> 1;
        } else {
            // a is either
            // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
            // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
            // - an infinity or NaN
            return (long) a;
        }
    }

    private static final class RandomNumberGeneratorHolder {
        static final Random 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.
     *
     * @apiNote
     * As the largest {@code double} value less than {@code 1.0}
     * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
     * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
     *
     * <blockquote><pre>{@code
     * double f = Math.random()/Math.nextDown(1.0);
     * double x = x1*(1.0 - f) + x2*f;
     * }</pre></blockquote>
     *
     * @return  a pseudorandom {@code double} greater than or equal
     * to {@code 0.0} and less than {@code 1.0}.
     * @see #nextDown(double)
     * @see Random#nextDouble()
     */
    public static double random() {
        return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
    }

    /**
     * Returns the sum of its arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int addExact(int x, int y) {
        int r = x + y;
        // HD 2-12 Overflow iff both arguments have the opposite sign of the result
        if (((x ^ r) & (y ^ r)) < 0) {
            throw new ArithmeticException("integer overflow");
        }
        return r;
    }

    /**
     * Returns the sum of its arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long addExact(long x, long y) {
        long r = x + y;
        // HD 2-12 Overflow iff both arguments have the opposite sign of the result
        if (((x ^ r) & (y ^ r)) < 0) {
            throw new ArithmeticException("long overflow");
        }
        return r;
    }

    /**
     * Returns the difference of the arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value to subtract from the first
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int subtractExact(int x, int y) {
        int r = x - y;
        // HD 2-12 Overflow iff the arguments have different signs and
        // the sign of the result is different from the sign of x
        if (((x ^ y) & (x ^ r)) < 0) {
            throw new ArithmeticException("integer overflow");
        }
        return r;
    }

    /**
     * Returns the difference of the arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value to subtract from the first
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long subtractExact(long x, long y) {
        long r = x - y;
        // HD 2-12 Overflow iff the arguments have different signs and
        // the sign of the result is different from the sign of x
        if (((x ^ y) & (x ^ r)) < 0) {
            throw new ArithmeticException("long overflow");
        }
        return r;
    }

    /**
     * Returns the product of the arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int multiplyExact(int x, int y) {
        long r = (long)x * (long)y;
        if ((int)r != r) {
            throw new ArithmeticException("integer overflow");
        }
        return (int)r;
    }

    /**
     * Returns the product of the arguments, throwing an exception if the result
     * overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 9
     */
    public static long multiplyExact(long x, int y) {
        return multiplyExact(x, (long)y);
    }

    /**
     * Returns the product of the arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long multiplyExact(long x, long y) {
        long r = x * y;
        long ax = Math.abs(x);
        long ay = Math.abs(y);
        if (((ax | ay) >>> 31 != 0)) {
            // Some bits greater than 2^31 that might cause overflow
            // Check the result using the divide operator
            // and check for the special case of Long.MIN_VALUE * -1
           if (((y != 0) && (r / y != x)) ||
               (x == Long.MIN_VALUE && y == -1)) {
                throw new ArithmeticException("long overflow");
            }
        }
        return r;
    }

    /**
     * Returns the quotient of the arguments, throwing an exception if the
     * result overflows an {@code int}.  Such overflow occurs in this method if
     * {@code x} is {@link Integer#MIN_VALUE} and {@code y} is {@code -1}.
     * In contrast, if {@code Integer.MIN_VALUE / -1} were evaluated directly,
     * the result would be {@code Integer.MIN_VALUE} and no exception would be
     * thrown.
     * <p>
     * If {@code y} is zero, an {@code ArithmeticException} is thrown
     * (JLS {@jls 15.17.2}).
     * <p>
     * The built-in remainder operator "{@code %}" is a suitable counterpart
     * both for this method and for the built-in division operator "{@code /}".
     *
     * @param x the dividend
     * @param y the divisor
     * @return the quotient {@code x / y}
     * @throws ArithmeticException if {@code y} is zero or the quotient
     * overflows an int
     * @jls 15.17.2 Division Operator /
     * @since 18
     */
    public static int divideExact(int x, int y) {
        int q = x / y;
        if ((x & y & q) >= 0) {
            return q;
        }
        throw new ArithmeticException("integer overflow");
    }

    /**
     * Returns the quotient of the arguments, throwing an exception if the
     * result overflows a {@code long}.  Such overflow occurs in this method if
     * {@code x} is {@link Long#MIN_VALUE} and {@code y} is {@code -1}.
     * In contrast, if {@code Long.MIN_VALUE / -1} were evaluated directly,
     * the result would be {@code Long.MIN_VALUE} and no exception would be
     * thrown.
     * <p>
     * If {@code y} is zero, an {@code ArithmeticException} is thrown
     * (JLS {@jls 15.17.2}).
     * <p>
     * The built-in remainder operator "{@code %}" is a suitable counterpart
     * both for this method and for the built-in division operator "{@code /}".
     *
     * @param x the dividend
     * @param y the divisor
     * @return the quotient {@code x / y}
     * @throws ArithmeticException if {@code y} is zero or the quotient
     * overflows a long
     * @jls 15.17.2 Division Operator /
     * @since 18
     */
    public static long divideExact(long x, long y) {
        long q = x / y;
        if ((x & y & q) >= 0) {
            return q;
        }
        throw new ArithmeticException("long overflow");
    }

    /**
     * Returns the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     * This method is identical to {@link #floorDiv(int,int)} except that it
     * throws an {@code ArithmeticException} when the dividend is
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is
     * {@code -1} instead of ignoring the integer overflow and returning
     * {@code Integer.MIN_VALUE}.
     * <p>
     * The floor modulus method {@link #floorMod(int,int)} is a suitable
     * counterpart both for this method and for the {@link #floorDiv(int,int)}
     * method.
     * <p>
     * For examples, see {@link #floorDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero, or the
     * dividend {@code x} is {@code Integer.MIN_VALUE} and the divisor {@code y}
     * is {@code -1}.
     * @see #floorDiv(int, int)
     * @since 18
     */
    public static int floorDivExact(int x, int y) {
        final int q = x / y;
        if ((x & y & q) >= 0) {
            // if the signs are different and modulo not zero, round down
            if ((x ^ y) < 0 && (q * y != x)) {
                return q - 1;
            }
            return q;
        }
        throw new ArithmeticException("integer overflow");
    }

    /**
     * Returns the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * This method is identical to {@link #floorDiv(long,long)} except that it
     * throws an {@code ArithmeticException} when the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is
     * {@code -1} instead of ignoring the integer overflow and returning
     * {@code Long.MIN_VALUE}.
     * <p>
     * The floor modulus method {@link #floorMod(long,long)} is a suitable
     * counterpart both for this method and for the {@link #floorDiv(long,long)}
     * method.
     * <p>
     * For examples, see {@link #floorDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero, or the
     * dividend {@code x} is {@code Long.MIN_VALUE} and the divisor {@code y}
     * is {@code -1}.
     * @see #floorDiv(long,long)
     * @since 18
     */
    public static long floorDivExact(long x, long y) {
        final long q = x / y;
        if ((x & y & q) >= 0) {
            // if the signs are different and modulo not zero, round down
            if ((x ^ y) < 0 && (q * y != x)) {
                return q - 1;
            }
            return q;
        }
        throw new ArithmeticException("long overflow");
    }

    /**
     * Returns the smallest (closest to negative infinity)
     * {@code int} value that is greater than or equal to the algebraic quotient.
     * This method is identical to {@link #ceilDiv(int,int)} except that it
     * throws an {@code ArithmeticException} when the dividend is
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is
     * {@code -1} instead of ignoring the integer overflow and returning
     * {@code Integer.MIN_VALUE}.
     * <p>
     * The ceil modulus method {@link #ceilMod(int,int)} is a suitable
     * counterpart both for this method and for the {@link #ceilDiv(int,int)}
     * method.
     * <p>
     * For examples, see {@link #ceilDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the smallest (closest to negative infinity)
     * {@code int} value that is greater than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero, or the
     * dividend {@code x} is {@code Integer.MIN_VALUE} and the divisor {@code y}
     * is {@code -1}.
     * @see #ceilDiv(int, int)
     * @since 18
     */
    public static int ceilDivExact(int x, int y) {
        final int q = x / y;
        if ((x & y & q) >= 0) {
            // if the signs are the same and modulo not zero, round up
            if ((x ^ y) >= 0 && (q * y != x)) {
                return q + 1;
            }
            return q;
        }
        throw new ArithmeticException("integer overflow");
    }

    /**
     * Returns the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * This method is identical to {@link #ceilDiv(long,long)} except that it
     * throws an {@code ArithmeticException} when the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is
     * {@code -1} instead of ignoring the integer overflow and returning
     * {@code Long.MIN_VALUE}.
     * <p>
     * The ceil modulus method {@link #ceilMod(long,long)} is a suitable
     * counterpart both for this method and for the {@link #ceilDiv(long,long)}
     * method.
     * <p>
     * For examples, see {@link #ceilDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero, or the
     * dividend {@code x} is {@code Long.MIN_VALUE} and the divisor {@code y}
     * is {@code -1}.
     * @see #ceilDiv(long,long)
     * @since 18
     */
    public static long ceilDivExact(long x, long y) {
        final long q = x / y;
        if ((x & y & q) >= 0) {
            // if the signs are the same and modulo not zero, round up
            if ((x ^ y) >= 0 && (q * y != x)) {
                return q + 1;
            }
            return q;
        }
        throw new ArithmeticException("long overflow");
    }

    /**
     * Returns the argument incremented by one, throwing an exception if the
     * result overflows an {@code int}.
     * The overflow only occurs for {@linkplain Integer#MAX_VALUE the maximum value}.
     *
     * @param a the value to increment
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int incrementExact(int a) {
        if (a == Integer.MAX_VALUE) {
            throw new ArithmeticException("integer overflow");
        }

        return a + 1;
    }

    /**
     * Returns the argument incremented by one, throwing an exception if the
     * result overflows a {@code long}.
     * The overflow only occurs for {@linkplain Long#MAX_VALUE the maximum value}.
     *
     * @param a the value to increment
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long incrementExact(long a) {
        if (a == Long.MAX_VALUE) {
            throw new ArithmeticException("long overflow");
        }

        return a + 1L;
    }

    /**
     * Returns the argument decremented by one, throwing an exception if the
     * result overflows an {@code int}.
     * The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
     *
     * @param a the value to decrement
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int decrementExact(int a) {
        if (a == Integer.MIN_VALUE) {
            throw new ArithmeticException("integer overflow");
        }

        return a - 1;
    }

    /**
     * Returns the argument decremented by one, throwing an exception if the
     * result overflows a {@code long}.
     * The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
     *
     * @param a the value to decrement
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long decrementExact(long a) {
        if (a == Long.MIN_VALUE) {
            throw new ArithmeticException("long overflow");
        }

        return a - 1L;
    }

    /**
     * Returns the negation of the argument, throwing an exception if the
     * result overflows an {@code int}.
     * The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
     *
     * @param a the value to negate
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    @IntrinsicCandidate
    public static int negateExact(int a) {
        if (a == Integer.MIN_VALUE) {
            throw new ArithmeticException("integer overflow");
        }

        return -a;
    }

    /**
     * Returns the negation of the argument, throwing an exception if the
     * result overflows a {@code long}.
     * The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
     *
     * @param a the value to negate
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    @IntrinsicCandidate
    public static long negateExact(long a) {
        if (a == Long.MIN_VALUE) {
            throw new ArithmeticException("long overflow");
        }

        return -a;
    }

    /**
     * Returns the value of the {@code long} argument,
     * throwing an exception if the value overflows an {@code int}.
     *
     * @param value the long value
     * @return the argument as an int
     * @throws ArithmeticException if the {@code argument} overflows an int
     * @since 1.8
     */
    public static int toIntExact(long value) {
        if ((int)value != value) {
            throw new ArithmeticException("integer overflow");
        }
        return (int)value;
    }

    /**
     * Returns the exact mathematical product of the arguments.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @since 9
     */
    public static long multiplyFull(int x, int y) {
        return (long)x * (long)y;
    }

    /**
     * Returns as a {@code long} the most significant 64 bits of the 128-bit
     * product of two 64-bit factors.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @see #unsignedMultiplyHigh
     * @since 9
     */
    @IntrinsicCandidate
    public static long multiplyHigh(long x, long y) {
        // Use technique from section 8-2 of Henry S. Warren, Jr.,
        // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174.
        long x1 = x >> 32;
        long x2 = x & 0xFFFFFFFFL;
        long y1 = y >> 32;
        long y2 = y & 0xFFFFFFFFL;

        long z2 = x2 * y2;
        long t = x1 * y2 + (z2 >>> 32);
        long z1 = t & 0xFFFFFFFFL;
        long z0 = t >> 32;
        z1 += x2 * y1;

        return x1 * y1 + z0 + (z1 >> 32);
    }

    /**
     * Returns as a {@code long} the most significant 64 bits of the unsigned
     * 128-bit product of two unsigned 64-bit factors.
     *
     * @param x the first value
     * @param y the second value
     * @return the result
     * @see #multiplyHigh
     * @since 18
     */
    @IntrinsicCandidate
    public static long unsignedMultiplyHigh(long x, long y) {
        // Compute via multiplyHigh() to leverage the intrinsic
        long result = Math.multiplyHigh(x, y);
        result += (y & (x >> 63)); // equivalent to `if (x < 0) result += y;`
        result += (x & (y >> 63)); // equivalent to `if (y < 0) result += x;`
        return result;
    }

    /**
     * Returns the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Integer.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * negative infinity (floor) rounding mode.
     * The floor rounding mode gives different results from truncation
     * when the exact quotient is not an integer and is negative.
     * <ul>
     *   <li>If the signs of the arguments are the same, the results of
     *       {@code floorDiv} and the {@code /} operator are the same.  <br>
     *       For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
     *   <li>If the signs of the arguments are different, {@code floorDiv}
     *       returns the largest integer less than or equal to the quotient
     *       while the {@code /} operator returns the smallest integer greater
     *       than or equal to the quotient.
     *       They differ if and only if the quotient is not an integer.<br>
     *       For example, {@code floorDiv(-4, 3) == -2},
     *       whereas {@code (-4 / 3) == -1}.
     *   </li>
     * </ul>
     *
     * @param x the dividend
     * @param y the divisor
     * @return the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorMod(int, int)
     * @see #floor(double)
     * @since 1.8
     */
    public static int floorDiv(int x, int y) {
        final int q = x / y;
        // if the signs are different and modulo not zero, round down
        if ((x ^ y) < 0 && (q * y != x)) {
            return q - 1;
        }
        return q;
    }

    /**
     * Returns the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Long.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * negative infinity (floor) rounding mode.
     * The floor rounding mode gives different results from truncation
     * when the exact result is not an integer and is negative.
     * <p>
     * For examples, see {@link #floorDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorMod(long, int)
     * @see #floor(double)
     * @since 9
     */
    public static long floorDiv(long x, int y) {
        return floorDiv(x, (long)y);
    }

    /**
     * Returns the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Long.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * negative infinity (floor) rounding mode.
     * The floor rounding mode gives different results from truncation
     * when the exact result is not an integer and is negative.
     * <p>
     * For examples, see {@link #floorDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorMod(long, long)
     * @see #floor(double)
     * @since 1.8
     */
    public static long floorDiv(long x, long y) {
        final long q = x / y;
        // if the signs are different and modulo not zero, round down
        if ((x ^ y) < 0 && (q * y != x)) {
            return q - 1;
        }
        return q;
    }

    /**
     * Returns the floor modulus of the {@code int} arguments.
     * <p>
     * The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
     * </ul>
     * <p>
     * The difference in values between {@code floorMod} and the {@code %} operator
     * is due to the difference between {@code floorDiv} and the {@code /}
     * operator, as detailed in {@linkplain #floorDiv(int, int)}.
     * <p>
     * Examples:
     * <ul>
     *   <li>Regardless of the signs of the arguments, {@code floorMod}(x, y)
     *       is zero exactly when {@code x % y} is zero as well.</li>
     *   <li>If neither {@code floorMod}(x, y) nor {@code x % y} is zero,
     *       they differ exactly when the signs of the arguments differ.<br>
     *       <ul>
     *       <li>{@code floorMod(+4, +3) == +1}; &nbsp; and {@code (+4 % +3) == +1}</li>
     *       <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1}</li>
     *       <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1}</li>
     *       <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1}</li>
     *       </ul>
     *   </li>
     * </ul>
     *
     * @param x the dividend
     * @param y the divisor
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorDiv(int, int)
     * @since 1.8
     */
    public static int floorMod(int x, int y) {
        final int r = x % y;
        // if the signs are different and modulo not zero, adjust result
        if ((x ^ y) < 0 && r != 0) {
            return r + y;
        }
        return r;
    }

    /**
     * Returns the floor modulus of the {@code long} and {@code int} arguments.
     * <p>
     * The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
     * </ul>
     * <p>
     * For examples, see {@link #floorMod(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorDiv(long, int)
     * @since 9
     */
    public static int floorMod(long x, int y) {
        // Result cannot overflow the range of int.
        return (int)floorMod(x, (long)y);
    }

    /**
     * Returns the floor modulus of the {@code long} arguments.
     * <p>
     * The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     *   <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
     * </ul>
     * <p>
     * For examples, see {@link #floorMod(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #floorDiv(long, long)
     * @since 1.8
     */
    public static long floorMod(long x, long y) {
        final long r = x % y;
        // if the signs are different and modulo not zero, adjust result
        if ((x ^ y) < 0 && r != 0) {
            return r + y;
        }
        return r;
    }

    /**
     * Returns the smallest (closest to negative infinity)
     * {@code int} value that is greater than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Integer.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * positive infinity (ceiling) rounding mode.
     * The ceiling rounding mode gives different results from truncation
     * when the exact quotient is not an integer and is positive.
     * <ul>
     *   <li>If the signs of the arguments are different, the results of
     *       {@code ceilDiv} and the {@code /} operator are the same.  <br>
     *       For example, {@code ceilDiv(-4, 3) == -1} and {@code (-4 / 3) == -1}.</li>
     *   <li>If the signs of the arguments are the same, {@code ceilDiv}
     *       returns the smallest integer greater than or equal to the quotient
     *       while the {@code /} operator returns the largest integer less
     *       than or equal to the quotient.
     *       They differ if and only if the quotient is not an integer.<br>
     *       For example, {@code ceilDiv(4, 3) == 2},
     *       whereas {@code (4 / 3) == 1}.
     *   </li>
     * </ul>
     *
     * @param x the dividend
     * @param y the divisor
     * @return the smallest (closest to negative infinity)
     * {@code int} value that is greater than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilMod(int, int)
     * @see #ceil(double)
     * @since 18
     */
    public static int ceilDiv(int x, int y) {
        final int q = x / y;
        // if the signs are the same and modulo not zero, round up
        if ((x ^ y) >= 0 && (q * y != x)) {
            return q + 1;
        }
        return q;
    }

    /**
     * Returns the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Long.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * positive infinity (ceiling) rounding mode.
     * The ceiling rounding mode gives different results from truncation
     * when the exact result is not an integer and is positive.
     * <p>
     * For examples, see {@link #ceilDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilMod(int, int)
     * @see #ceil(double)
     * @since 18
     */
    public static long ceilDiv(long x, int y) {
        return ceilDiv(x, (long)y);
    }

    /**
     * Returns the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * There is one special case: if the dividend is
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Long.MIN_VALUE}.
     * <p>
     * Normal integer division operates under the round to zero rounding mode
     * (truncation).  This operation instead acts under the round toward
     * positive infinity (ceiling) rounding mode.
     * The ceiling rounding mode gives different results from truncation
     * when the exact result is not an integer and is positive.
     * <p>
     * For examples, see {@link #ceilDiv(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the smallest (closest to negative infinity)
     * {@code long} value that is greater than or equal to the algebraic quotient.
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilMod(int, int)
     * @see #ceil(double)
     * @since 18
     */
    public static long ceilDiv(long x, long y) {
        final long q = x / y;
        // if the signs are the same and modulo not zero, round up
        if ((x ^ y) >= 0 && (q * y != x)) {
            return q + 1;
        }
        return q;
    }

    /**
     * Returns the ceiling modulus of the {@code int} arguments.
     * <p>
     * The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
     * has the opposite sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
     * <ul>
     *   <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
     * </ul>
     * <p>
     * The difference in values between {@code ceilMod} and the {@code %} operator
     * is due to the difference between {@code ceilDiv} and the {@code /}
     * operator, as detailed in {@linkplain #ceilDiv(int, int)}.
     * <p>
     * Examples:
     * <ul>
     *   <li>Regardless of the signs of the arguments, {@code ceilMod}(x, y)
     *       is zero exactly when {@code x % y} is zero as well.</li>
     *   <li>If neither {@code ceilMod}(x, y) nor {@code x % y} is zero,
     *       they differ exactly when the signs of the arguments are the same.<br>
     *       <ul>
     *       <li>{@code ceilMod(+4, +3) == -2}; &nbsp; and {@code (+4 % +3) == +1}</li>
     *       <li>{@code ceilMod(-4, -3) == +2}; &nbsp; and {@code (-4 % -3) == -1}</li>
     *       <li>{@code ceilMod(+4, -3) == +1}; &nbsp; and {@code (+4 % -3) == +1}</li>
     *       <li>{@code ceilMod(-4, +3) == -1}; &nbsp; and {@code (-4 % +3) == -1}</li>
     *       </ul>
     *   </li>
     * </ul>
     *
     * @param x the dividend
     * @param y the divisor
     * @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilDiv(int, int)
     * @since 18
     */
    public static int ceilMod(int x, int y) {
        final int r = x % y;
        // if the signs are the same and modulo not zero, adjust result
        if ((x ^ y) >= 0 && r != 0) {
            return r - y;
        }
        return r;
    }

    /**
     * Returns the ceiling modulus of the {@code long} and {@code int} arguments.
     * <p>
     * The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
     * has the opposite sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
     * <ul>
     *   <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
     * </ul>
     * <p>
     * For examples, see {@link #ceilMod(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilDiv(long, int)
     * @since 18
     */
    public static int ceilMod(long x, int y) {
        // Result cannot overflow the range of int.
        return (int)ceilMod(x, (long)y);
    }

    /**
     * Returns the ceiling modulus of the {@code long} arguments.
     * <p>
     * The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
     * has the opposite sign as the divisor {@code y} or is zero, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
     * <ul>
     *   <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
     * </ul>
     * <p>
     * For examples, see {@link #ceilMod(int, int)}.
     *
     * @param x the dividend
     * @param y the divisor
     * @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see #ceilDiv(long, long)
     * @since 18
     */
    public static long ceilMod(long x, long y) {
        final long r = x % y;
        // if the signs are the same and modulo not zero, adjust result
        if ((x ^ y) >= 0 && r != 0) {
            return r - y;
        }
        return r;
    }

    /**
     * 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. In
     * contrast, the {@link Math#absExact(int)} method throws an
     * {@code ArithmeticException} for this value.
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     * @see Math#absExact(int)
     */
    @IntrinsicCandidate
    public static int abs(int a) {
        return (a < 0) ? -a : a;
    }

    /**
     * Returns the mathematical absolute value of an {@code int} value
     * if it is exactly representable as an {@code int}, throwing
     * {@code ArithmeticException} if the result overflows the
     * positive {@code int} range.
     *
     * <p>Since the range of two's complement integers is asymmetric
     * with one additional negative value (JLS {@jls 4.2.1}), the
     * mathematical absolute value of {@link Integer#MIN_VALUE}
     * overflows the positive {@code int} range, so an exception is
     * thrown for that argument.
     *
     * @param  a  the argument whose absolute value is to be determined
     * @return the absolute value of the argument, unless overflow occurs
     * @throws ArithmeticException if the argument is {@link Integer#MIN_VALUE}
     * @see Math#abs(int)
     * @since 15
     */
    public static int absExact(int a) {
        if (a == Integer.MIN_VALUE)
            throw new ArithmeticException(
                "Overflow to represent absolute value of Integer.MIN_VALUE");
        else
            return abs(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. In
     * contrast, the {@link Math#absExact(long)} method throws an
     * {@code ArithmeticException} for this value.
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     * @see Math#absExact(long)
     */
    @IntrinsicCandidate
    public static long abs(long a) {
        return (a < 0) ? -a : a;
    }

    /**
     * Returns the mathematical absolute value of an {@code long} value
     * if it is exactly representable as an {@code long}, throwing
     * {@code ArithmeticException} if the result overflows the
     * positive {@code long} range.
     *
     * <p>Since the range of two's complement integers is asymmetric
     * with one additional negative value (JLS {@jls 4.2.1}), the
     * mathematical absolute value of {@link Long#MIN_VALUE} overflows
     * the positive {@code long} range, so an exception is thrown for
     * that argument.
     *
     * @param  a  the argument whose absolute value is to be determined
     * @return the absolute value of the argument, unless overflow occurs
     * @throws ArithmeticException if the argument is {@link Long#MIN_VALUE}
     * @see Math#abs(long)
     * @since 15
     */
    public static long absExact(long a) {
        if (a == Long.MIN_VALUE)
            throw new ArithmeticException(
                "Overflow to represent absolute value of Long.MIN_VALUE");
        else
            return abs(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>
     *
     * @apiNote As implied by the above, one valid implementation of
     * this method is given by the expression below which computes a
     * {@code float} with the same exponent and significand as the
     * argument but with a guaranteed zero sign bit indicating a
     * positive value:<br>
     * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    @IntrinsicCandidate
    public static float abs(float a) {
        // Convert to bit field form, zero the sign bit, and convert back
        return Float.intBitsToFloat(Float.floatToRawIntBits(a) & FloatConsts.MAG_BIT_MASK);
    }

    /**
     * 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>
     *
     * @apiNote As implied by the above, one valid implementation of
     * this method is given by the expression below which computes a
     * {@code double} with the same exponent and significand as the
     * argument but with a guaranteed zero sign bit indicating a
     * positive value:<br>
     * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    @IntrinsicCandidate
    public static double abs(double a) {
        // Convert to bit field form, zero the sign bit, and convert back
        return Double.longBitsToDouble(Double.doubleToRawLongBits(a) & DoubleConsts.MAG_BIT_MASK);

    }

    /**
     * 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}.
     */
    @IntrinsicCandidate
    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 final long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
    private static final 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}.
     */
    @IntrinsicCandidate
    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}.
     */
    @IntrinsicCandidate
    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}.
     */
    @IntrinsicCandidate
    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}.
     */
    @IntrinsicCandidate
    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}.
     */
    @IntrinsicCandidate
    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 fused multiply add of the three arguments; that is,
     * returns the exact product of the first two arguments summed
     * with the third argument and then rounded once to the nearest
     * {@code double}.
     *
     * The rounding is done using the {@linkplain
     * java.math.RoundingMode#HALF_EVEN round to nearest even
     * rounding mode}.
     *
     * In contrast, if {@code a * b + c} is evaluated as a regular
     * floating-point expression, two rounding errors are involved,
     * the first for the multiply operation, the second for the
     * addition operation.
     *
     * <p>Special cases:
     * <ul>
     * <li> If any argument is NaN, the result is NaN.
     *
     * <li> If one of the first two arguments is infinite and the
     * other is zero, the result is NaN.
     *
     * <li> If the exact product of the first two arguments is infinite
     * (in other words, at least one of the arguments is infinite and
     * the other is neither zero nor NaN) and the third argument is an
     * infinity of the opposite sign, the result is NaN.
     *
     * </ul>
     *
     * <p>Note that {@code fma(a, 1.0, c)} returns the same
     * result as ({@code a + c}).  However,
     * {@code fma(a, b, +0.0)} does <em>not</em> always return the
     * same result as ({@code a * b}) since
     * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while
     * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is
     * equivalent to ({@code a * b}) however.
     *
     * @apiNote This method corresponds to the fusedMultiplyAdd
     * operation defined in IEEE 754-2008.
     *
     * @param a a value
     * @param b a value
     * @param c a value
     *
     * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
     * computed, as if with unlimited range and precision, and rounded
     * once to the nearest {@code double} value
     *
     * @since 9
     */
    @IntrinsicCandidate
    public static double fma(double a, double b, double c) {
        /*
         * Infinity and NaN arithmetic is not quite the same with two
         * roundings as opposed to just one so the simple expression
         * "a * b + c" cannot always be used to compute the correct
         * result.  With two roundings, the product can overflow and
         * if the addend is infinite, a spurious NaN can be produced
         * if the infinity from the overflow and the infinite addend
         * have opposite signs.
         */

        // First, screen for and handle non-finite input values whose
        // arithmetic is not supported by BigDecimal.
        if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
            return Double.NaN;
        } else { // All inputs non-NaN
            boolean infiniteA = Double.isInfinite(a);
            boolean infiniteB = Double.isInfinite(b);
            boolean infiniteC = Double.isInfinite(c);
            double result;

            if (infiniteA || infiniteB || infiniteC) {
                if (infiniteA && b == 0.0 ||
                    infiniteB && a == 0.0 ) {
                    return Double.NaN;
                }
                double product = a * b;
                if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
                    // Intermediate overflow; might cause a
                    // spurious NaN if added to infinite c.
                    assert Double.isInfinite(c);
                    return c;
                } else {
                    result = product + c;
                    assert !Double.isFinite(result);
                    return result;
                }
            } else { // All inputs finite
                BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
                if (c == 0.0) { // Positive or negative zero
                    // If the product is an exact zero, use a
                    // floating-point expression to compute the sign
                    // of the zero final result. The product is an
                    // exact zero if and only if at least one of a and
                    // b is zero.
                    if (a == 0.0 || b == 0.0) {
                        return a * b + c;
                    } else {
                        // The sign of a zero addend doesn't matter if
                        // the product is nonzero. The sign of a zero
                        // addend is not factored in the result if the
                        // exact product is nonzero but underflows to
                        // zero; see IEEE-754 2008 section 6.3 "The
                        // sign bit".
                        return product.doubleValue();
                    }
                } else {
                    return product.add(new BigDecimal(c)).doubleValue();
                }
            }
        }
    }

    /**
     * Returns the fused multiply add of the three arguments; that is,
     * returns the exact product of the first two arguments summed
     * with the third argument and then rounded once to the nearest
     * {@code float}.
     *
     * The rounding is done using the {@linkplain
     * java.math.RoundingMode#HALF_EVEN round to nearest even
     * rounding mode}.
     *
     * In contrast, if {@code a * b + c} is evaluated as a regular
     * floating-point expression, two rounding errors are involved,
     * the first for the multiply operation, the second for the
     * addition operation.
     *
     * <p>Special cases:
     * <ul>
     * <li> If any argument is NaN, the result is NaN.
     *
     * <li> If one of the first two arguments is infinite and the
     * other is zero, the result is NaN.
     *
     * <li> If the exact product of the first two arguments is infinite
     * (in other words, at least one of the arguments is infinite and
     * the other is neither zero nor NaN) and the third argument is an
     * infinity of the opposite sign, the result is NaN.
     *
     * </ul>
     *
     * <p>Note that {@code fma(a, 1.0f, c)} returns the same
     * result as ({@code a + c}).  However,
     * {@code fma(a, b, +0.0f)} does <em>not</em> always return the
     * same result as ({@code a * b}) since
     * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
     * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
     * equivalent to ({@code a * b}) however.
     *
     * @apiNote This method corresponds to the fusedMultiplyAdd
     * operation defined in IEEE 754-2008.
     *
     * @param a a value
     * @param b a value
     * @param c a value
     *
     * @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
     * computed, as if with unlimited range and precision, and rounded
     * once to the nearest {@code float} value
     *
     * @since 9
     */
    @IntrinsicCandidate
    public static float fma(float a, float b, float c) {
        if (Float.isFinite(a) && Float.isFinite(b) && Float.isFinite(c)) {
            if (a == 0.0 || b == 0.0) {
                return a * b + c; // Handled signed zero cases
            } else {
                return (new BigDecimal((double)a * (double)b) // Exact multiply
                        .add(new BigDecimal((double)c)))      // Exact sum
                    .floatValue();                            // One rounding
                                                              // to a float value
            }
        } else {
            // At least one of a,b, and c is non-finite. The result
            // will be non-finite as well and will be the same
            // non-finite value under double as float arithmetic.
            return (float)fma((double)a, (double)b, (double)c);
        }
    }

    /**
     * Returns the size of an ulp of the argument.  An ulp, unit in
     * the last place, 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) {
        int exp = getExponent(d);

        return switch(exp) {
            case Double.MAX_EXPONENT + 1 -> Math.abs(d);      // NaN or infinity
            case Double.MIN_EXPONENT - 1 -> Double.MIN_VALUE; // zero or subnormal
            default -> {
                assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;

                // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
                exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH - 1);
                if (exp >= Double.MIN_EXPONENT) {
                    yield powerOfTwoD(exp);
                } else {
                    // return a subnormal result; left shift integer
                    // representation of Double.MIN_VALUE appropriate
                    // number of positions
                    yield Double.longBitsToDouble(1L <<
                            (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH - 1))));
                }
            }
        };
    }

    /**
     * Returns the size of an ulp of the argument.  An ulp, unit in
     * the last place, 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) {
        int exp = getExponent(f);

        return switch(exp) {
            case Float.MAX_EXPONENT + 1 -> Math.abs(f);     // NaN or infinity
            case Float.MIN_EXPONENT - 1 -> Float.MIN_VALUE; // zero or subnormal
            default -> {
                assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;

                // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
                exp = exp - (FloatConsts.SIGNIFICAND_WIDTH - 1);
                if (exp >= Float.MIN_EXPONENT) {
                    yield powerOfTwoF(exp);
                } else {
                    // return a subnormal result; left shift integer
                    // representation of FloatConsts.MIN_VALUE appropriate
                    // number of positions
                    yield Float.intBitsToFloat(1 <<
                            (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH - 1))));
                }
            }
        };
    }

    /**
     * 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
     */
    @IntrinsicCandidate
    public static double signum(double d) {
        return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, 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
     */
    @IntrinsicCandidate
    public static float signum(float f) {
        return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, 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.
     *
     * <li> If both arguments are zero, the result is positive zero.
     * </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
     */
    @IntrinsicCandidate
    public static double copySign(double magnitude, double sign) {
        return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
                                        (DoubleConsts.SIGN_BIT_MASK)) |
                                       (Double.doubleToRawLongBits(magnitude) &
                                        (DoubleConsts.EXP_BIT_MASK |
                                         DoubleConsts.SIGNIF_BIT_MASK)));
    }

    /**
     * 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
     */
    @IntrinsicCandidate
    public static float copySign(float magnitude, float sign) {
        return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
                                     (FloatConsts.SIGN_BIT_MASK)) |
                                    (Float.floatToRawIntBits(magnitude) &
                                     (FloatConsts.EXP_BIT_MASK |
                                      FloatConsts.SIGNIF_BIT_MASK)));
    }

    /**
     * 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) {
        /*
         * Bitwise convert f to integer, mask out exponent bits, shift
         * to the right and then subtract out float's bias adjust to
         * get true exponent value
         */
        return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
                (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
    }

    /**
     * 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) {
        /*
         * Bitwise convert d to long, mask out exponent bits, shift
         * to the right and then subtract out double's bias adjust to
         * get true exponent value.
         */
        return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
                      (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
    }

    /**
     * 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) {
        /*
         * The cases:
         *
         * nextAfter(+infinity, 0)  == MAX_VALUE
         * nextAfter(+infinity, +infinity)  == +infinity
         * nextAfter(-infinity, 0)  == -MAX_VALUE
         * nextAfter(-infinity, -infinity)  == -infinity
         *
         * are naturally handled without any additional testing
         */

        /*
         * IEEE 754 floating-point numbers are lexicographically
         * ordered if treated as signed-magnitude integers.
         * Since Java's integers are two's complement,
         * incrementing the two's complement representation of a
         * logically negative floating-point value *decrements*
         * the signed-magnitude representation. Therefore, when
         * the integer representation of a floating-point value
         * is negative, the adjustment to the representation is in
         * the opposite direction from what would initially be expected.
         */

        // Branch to descending case first as it is more costly than ascending
        // case due to start != 0.0d conditional.
        if (start > direction) { // descending
            if (start != 0.0d) {
                final long transducer = Double.doubleToRawLongBits(start);
                return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
            } else { // start == 0.0d && direction < 0.0d
                return -Double.MIN_VALUE;
            }
        } else if (start < direction) { // ascending
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
            // then bitwise convert start to integer.
            final long transducer = Double.doubleToRawLongBits(start + 0.0d);
            return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
        } else if (start == direction) {
            return direction;
        } else { // isNaN(start) || isNaN(direction)
            return 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) {
        /*
         * The cases:
         *
         * nextAfter(+infinity, 0)  == MAX_VALUE
         * nextAfter(+infinity, +infinity)  == +infinity
         * nextAfter(-infinity, 0)  == -MAX_VALUE
         * nextAfter(-infinity, -infinity)  == -infinity
         *
         * are naturally handled without any additional testing
         */

        /*
         * IEEE 754 floating-point numbers are lexicographically
         * ordered if treated as signed-magnitude integers.
         * Since Java's integers are two's complement,
         * incrementing the two's complement representation of a
         * logically negative floating-point value *decrements*
         * the signed-magnitude representation. Therefore, when
         * the integer representation of a floating-point value
         * is negative, the adjustment to the representation is in
         * the opposite direction from what would initially be expected.
         */

        // Branch to descending case first as it is more costly than ascending
        // case due to start != 0.0f conditional.
        if (start > direction) { // descending
            if (start != 0.0f) {
                final int transducer = Float.floatToRawIntBits(start);
                return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
            } else { // start == 0.0f && direction < 0.0f
                return -Float.MIN_VALUE;
            }
        } else if (start < direction) { // ascending
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
            // then bitwise convert start to integer.
            final int transducer = Float.floatToRawIntBits(start + 0.0f);
            return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
        } else if (start == direction) {
            return (float)direction;
        } else { // isNaN(start) || isNaN(direction)
            return start + (float)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) {
        // Use a single conditional and handle the likely cases first.
        if (d < Double.POSITIVE_INFINITY) {
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
            final long transducer = Double.doubleToRawLongBits(d + 0.0D);
            return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
        } else { // d is NaN or +Infinity
            return 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) {
        // Use a single conditional and handle the likely cases first.
        if (f < Float.POSITIVE_INFINITY) {
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
            final int transducer = Float.floatToRawIntBits(f + 0.0F);
            return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
        } else { // f is NaN or +Infinity
            return f;
        }
    }

    /**
     * Returns the floating-point value adjacent to {@code d} in
     * the direction of negative infinity.  This method is
     * semantically equivalent to {@code nextAfter(d,
     * Double.NEGATIVE_INFINITY)}; however, a
     * {@code nextDown} 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 negative infinity, the result is
     * negative infinity.
     *
     * <li> If the argument is zero, the result is
     * {@code -Double.MIN_VALUE}
     *
     * </ul>
     *
     * @param d  starting floating-point value
     * @return The adjacent floating-point value closer to negative
     * infinity.
     * @since 1.8
     */
    public static double nextDown(double d) {
        if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
            return d;
        else {
            if (d == 0.0)
                return -Double.MIN_VALUE;
            else
                return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
                                               ((d > 0.0d)?-1L:+1L));
        }
    }

    /**
     * Returns the floating-point value adjacent to {@code f} in
     * the direction of negative infinity.  This method is
     * semantically equivalent to {@code nextAfter(f,
     * Float.NEGATIVE_INFINITY)}; however, a
     * {@code nextDown} 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 negative infinity, the result is
     * negative infinity.
     *
     * <li> If the argument is zero, the result is
     * {@code -Float.MIN_VALUE}
     *
     * </ul>
     *
     * @param f  starting floating-point value
     * @return The adjacent floating-point value closer to negative
     * infinity.
     * @since 1.8
     */
    public static float nextDown(float f) {
        if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
            return f;
        else {
            if (f == 0.0f)
                return -Float.MIN_VALUE;
            else
                return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
                                            ((f > 0.0f)?-1:+1));
        }
    }

    /**
     * Returns {@code d} &times; 2<sup>{@code scaleFactor}</sup>
     * rounded as if performed by a single correctly rounded
     * floating-point multiply.  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) {
        /*
         * When scaling up, it does not matter what order the
         * multiply-store operations are done; the result will be
         * finite or overflow regardless of the operation ordering.
         * However, to get the correct result when scaling down, a
         * particular ordering must be used.
         *
         * When scaling down, the multiply-store operations are
         * sequenced so that it is not possible for two consecutive
         * multiply-stores to return subnormal results.  If one
         * multiply-store result is subnormal, the next multiply will
         * round it away to zero.  This is done by first multiplying
         * by 2 ^ (scaleFactor % n) and then multiplying several
         * times by 2^n as needed where n is the exponent of number
         * that is a convenient power of two.  In this way, at most one
         * real rounding error occurs.
         */

        // magnitude of a power of two so large that scaling a finite
        // nonzero value by it would be guaranteed to over or
        // underflow; due to rounding, scaling down takes an
        // additional power of two which is reflected here
        final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT +
                              DoubleConsts.SIGNIFICAND_WIDTH + 1;
        int exp_adjust = 0;
        int scale_increment = 0;
        double exp_delta = Double.NaN;

        // Make sure scaling factor is in a reasonable range

        if(scaleFactor < 0) {
            scaleFactor = Math.max(scaleFactor, -MAX_SCALE);
            scale_increment = -512;
            exp_delta = twoToTheDoubleScaleDown;
        }
        else {
            scaleFactor = Math.min(scaleFactor, MAX_SCALE);
            scale_increment = 512;
            exp_delta = twoToTheDoubleScaleUp;
        }

        // Calculate (scaleFactor % +/-512), 512 = 2^9, using
        // technique from "Hacker's Delight" section 10-2.
        int t = (scaleFactor >> 9-1) >>> 32 - 9;
        exp_adjust = ((scaleFactor + t) & (512 -1)) - t;

        d *= powerOfTwoD(exp_adjust);
        scaleFactor -= exp_adjust;

        while(scaleFactor != 0) {
            d *= exp_delta;
            scaleFactor -= scale_increment;
        }
        return d;
    }

    /**
     * Returns {@code f} &times; 2<sup>{@code scaleFactor}</sup>
     * rounded as if performed by a single correctly rounded
     * floating-point multiply.  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) {
        // magnitude of a power of two so large that scaling a finite
        // nonzero value by it would be guaranteed to over or
        // underflow; due to rounding, scaling down takes an
        // additional power of two which is reflected here
        final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
                              FloatConsts.SIGNIFICAND_WIDTH + 1;

        // Make sure scaling factor is in a reasonable range
        scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);

        /*
         * Since + MAX_SCALE for float fits well within the double
         * exponent range and + float -> double conversion is exact
         * the multiplication below will be exact. Therefore, the
         * rounding that occurs when the double product is cast to
         * float will be the correctly rounded float result.
         */
        return (float)((double)f*powerOfTwoD(scaleFactor));
    }

    // Constants used in scalb
    static double twoToTheDoubleScaleUp = powerOfTwoD(512);
    static double twoToTheDoubleScaleDown = powerOfTwoD(-512);

    /**
     * Returns a floating-point power of two in the normal range.
     */
    static double powerOfTwoD(int n) {
        assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
        return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) <<
                                        (DoubleConsts.SIGNIFICAND_WIDTH-1))
                                       & DoubleConsts.EXP_BIT_MASK);
    }

    /**
     * Returns a floating-point power of two in the normal range.
     */
    static float powerOfTwoF(int n) {
        assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
        return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
                                     (FloatConsts.SIGNIFICAND_WIDTH-1))
                                    & FloatConsts.EXP_BIT_MASK);
    }
}