java/8 : java/lang/Math.java

Math
https://openjdk.java.net/
GPLv2 + Classpath Exception
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package java.lang;
import java.util.Random;

import sun.misc.FloatConsts;
import sun.misc.DoubleConsts;

The class Math contains methods for performing basic numeric operations such as the elementary exponential, logarithm, square root, and trigonometric functions. Unlike some of the numeric methods of class StrictMath, all implementations of the equivalent functions of class Math are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required. 
By default many of the Math methods simply call the equivalent method in 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 Math methods. Such higher-performance implementations still must conform to the specification for Math. 
The quality of implementation specifications concern two properties, accuracy of the returned result and monotonicity of the method. Accuracy of the floating-point Math methods is measured in terms of ulps, units in the last place. For a given floating-point format, an ulp of a specific real number value is the distance between the two floating-point values bracketing that numerical value. When discussing the accuracy of a method as a whole rather than at a specific argument, the number of ulps cited is for the worst-case error at any argument. If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is correctly
rounded. 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 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 semi-monotonic: 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.
 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 int or long and overflow errors need to be detected, the methods addExact, subtractExact, multiplyExact, and toIntExact throw an ArithmeticException when the results overflow. For other arithmetic operations such as divide, absolute value, increment, decrement, and negation overflow occurs only with a specific minimum or maximum value and should be checked against the minimum or maximum as appropriate. 
Author:  unascribed,  Joseph D. Darcy
Since:   JDK1.0/**
 * The class {@code Math} contains methods for performing basic
 * numeric operations such as the elementary exponential, logarithm,
 * square root, and trigonometric functions.
 *
 * <p>Unlike some of the numeric methods of class
 * {@code StrictMath}, all implementations of the equivalent
 * functions of class {@code Math} are not defined to return the
 * bit-for-bit same results.  This relaxation permits
 * better-performing implementations where strict reproducibility is
 * not required.
 *
 * <p>By default many of the {@code Math} methods simply call
 * the equivalent method in {@code StrictMath} for their
 * implementation.  Code generators are encouraged to use
 * platform-specific native libraries or microprocessor instructions,
 * where available, to provide higher-performance implementations of
 * {@code Math} methods.  Such higher-performance
 * implementations still must conform to the specification for
 * {@code Math}.
 *
 * <p>The quality of implementation specifications concern two
 * properties, accuracy of the returned result and monotonicity of the
 * method.  Accuracy of the floating-point {@code Math} methods is
 * measured in terms of <i>ulps</i>, units in the last place.  For a
 * given floating-point format, an {@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 {@code addExact},
 * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
 * throw an {@code ArithmeticException} when the results overflow.
 * For other arithmetic operations such as divide, absolute value,
 * increment, decrement, and negation overflow occurs only with
 * a specific minimum or maximum value and should be checked against
 * the minimum or maximum as appropriate.
 *
 * @author  unascribed
 * @author  Joseph D. Darcy
 * @since   JDK1.0
 */

public final class Math {

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

    The double value that is closer than any other to e, the base of the natural logarithms.
/**
     * The {@code double} value that is closer than any other to
     * <i>e</i>, the base of the natural logarithms.
     */
    public static final double E = 2.7182818284590452354;

    The double value that is closer than any other to pi, the ratio of the circumference of a circle to its
diameter.
/**
     * The {@code double} value that is closer than any other to
     * <i>pi</i>, the ratio of the circumference of a circle to its
     * diameter.
     */
    public static final double PI = 3.14159265358979323846;

    Returns the trigonometric sine of an angle.  Special cases:
If the argument is NaN or an infinity, then the
result is NaN.
If the argument is zero, then the result is a zero with the
same sign as the argument.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   an angle, in radians.
Returns:  the sine of the argument./**
     * Returns the trigonometric sine of an angle.  Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the sine of the argument.
     */
    public static double sin(double a) {
        return StrictMath.sin(a); // default impl. delegates to StrictMath
    }

    Returns the trigonometric cosine of an angle. Special cases:
If the argument is NaN or an infinity, then the
result is NaN.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   an angle, in radians.
Returns:  the cosine of the argument./**
     * Returns the trigonometric cosine of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the cosine of the argument.
     */
    public static double cos(double a) {
        return StrictMath.cos(a); // default impl. delegates to StrictMath
    }

    Returns the trigonometric tangent of an angle.  Special cases:
If the argument is NaN or an infinity, then the result
is NaN.
If the argument is zero, then the result is a zero with the
same sign as the argument.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   an angle, in radians.
Returns:  the tangent of the argument./**
     * Returns the trigonometric tangent of an angle.  Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the result
     * is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the tangent of the argument.
     */
    public static double tan(double a) {
        return StrictMath.tan(a); // default impl. delegates to StrictMath
    }

    Returns the arc sine of a value; the returned angle is in the
range -pi/2 through pi/2.  Special cases:
If the argument is NaN or its absolute value is greater
than 1, then the result is NaN.
If the argument is zero, then the result is a zero with the
same sign as the argument.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   the value whose arc sine is to be returned.
Returns:  the arc sine of the argument./**
     * 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 pi.  Special case:
If the argument is NaN or its absolute value is greater
than 1, then the result is NaN.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   the value whose arc cosine is to be returned.
Returns:  the arc cosine of the argument./**
     * Returns the arc cosine of a value; the returned angle is in the
     * range 0.0 through <i>pi</i>.  Special case:
     * <ul><li>If the argument is NaN or its absolute value is greater
     * than 1, then the result is NaN.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the value whose arc cosine is to be returned.
     * @return  the arc cosine of the argument.
     */
    public static double acos(double a) {
        return StrictMath.acos(a); // default impl. delegates to StrictMath
    }

    Returns the arc tangent of a value; the returned angle is in the
range -pi/2 through pi/2.  Special cases:
If the argument is NaN, then the result is NaN.
If the argument is zero, then the result is a zero with the
same sign as the argument.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   the value whose arc tangent is to be returned.
Returns:  the arc tangent of the argument./**
     * Returns the arc tangent of a value; the returned angle is in the
     * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
     * <ul><li>If the argument is NaN, then the result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the value whose arc tangent is to be returned.
     * @return  the arc tangent of the argument.
     */
    public static double atan(double a) {
        return StrictMath.atan(a); // default impl. delegates to StrictMath
    }

    Converts an angle measured in degrees to an approximately
equivalent angle measured in radians.  The conversion from
degrees to radians is generally inexact.
Params: angdeg –   an angle, in degrees
Returns:  the measurement of the angle angdeg in radians.
Since:   1.2/**
     * Converts an angle measured in degrees to an approximately
     * equivalent angle measured in radians.  The conversion from
     * degrees to radians is generally inexact.
     *
     * @param   angdeg   an angle, in degrees
     * @return  the measurement of the angle {@code angdeg}
     *          in radians.
     * @since   1.2
     */
    public static double toRadians(double angdeg) {
        return angdeg / 180.0 * PI;
    }

    Converts an angle measured in radians to an approximately
equivalent angle measured in degrees.  The conversion from
radians to degrees is generally inexact; users should
not expect cos(toRadians(90.0)) to exactly equal 0.0. 
Params: angrad –   an angle, in radians
Returns:  the measurement of the angle angrad in degrees.
Since:   1.2/**
     * Converts an angle measured in radians to an approximately
     * equivalent angle measured in degrees.  The conversion from
     * radians to degrees is generally inexact; users should
     * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
     * equal {@code 0.0}.
     *
     * @param   angrad   an angle, in radians
     * @return  the measurement of the angle {@code angrad}
     *          in degrees.
     * @since   1.2
     */
    public static double toDegrees(double angrad) {
        return angrad * 180.0 / PI;
    }

    Returns Euler's number e raised to the power of a double value. Special cases: If the argument is NaN, the result is NaN.
If the argument is positive infinity, then the result is
positive infinity.
If the argument is negative infinity, then the result is
positive zero.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   the exponent to raise e to.
Returns:  the value e^a,
         where e is the base of the natural logarithms./**
     * Returns Euler's number <i>e</i> raised to the power of a
     * {@code double} value.  Special cases:
     * <ul><li>If the argument is NaN, the result is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is negative infinity, then the result is
     * positive zero.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the exponent to raise <i>e</i> to.
     * @return  the value <i>e</i><sup>{@code a}</sup>,
     *          where <i>e</i> is the base of the natural logarithms.
     */
    public static double exp(double a) {
        return StrictMath.exp(a); // default impl. delegates to StrictMath
    }

    Returns the natural logarithm (base e) of a double value. Special cases: If the argument is NaN or less than zero, then the result
is NaN.
If the argument is positive infinity, then the result is
positive infinity.
If the argument is positive zero or negative zero, then the
result is negative infinity.
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   a value
Returns:  the value ln a, the natural logarithm of a./**
     * Returns the natural logarithm (base <i>e</i>) of a {@code double}
     * value.  Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is negative infinity.</ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   a value
     * @return  the value ln&nbsp;{@code a}, the natural logarithm of
     *          {@code a}.
     */
    public static double log(double a) {
        return StrictMath.log(a); // default impl. delegates to StrictMath
    }

    Returns the base 10 logarithm of a double value. Special cases: If the argument is NaN or less than zero, then the result
is NaN.
If the argument is positive infinity, then the result is
positive infinity.
If the argument is positive zero or negative zero, then the
result is negative infinity.
 If the argument is equal to 10ⁿ for
integer n, then the result is n.

The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   a value
Returns:  the base 10 logarithm of a.
Since: 1.5/**
     * Returns the base 10 logarithm of a {@code double} value.
     * Special cases:
     *
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is negative infinity.
     * <li> If the argument is equal to 10<sup><i>n</i></sup> for
     * integer <i>n</i>, then the result is <i>n</i>.
     * </ul>
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   a value
     * @return  the base 10 logarithm of  {@code a}.
     * @since 1.5
     */
    public static double log10(double a) {
        return StrictMath.log10(a); // default impl. delegates to StrictMath
    }

    Returns the correctly rounded positive square root of a double value. Special cases: If the argument is NaN or less than zero, then the result
is NaN.
If the argument is positive infinity, then the result is positive
infinity.
If the argument is positive zero or negative zero, then the
result is the same as the argument.
 Otherwise, the result is the double value closest to the true mathematical square root of the argument value. 
Params: a –   a value.
Returns:  the positive square root of a. If the argument is NaN or less than zero, the result is NaN./**
     * Returns the correctly rounded positive square root of a
     * {@code double} value.
     * Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is positive
     * infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is the same as the argument.</ul>
     * Otherwise, the result is the {@code double} value closest to
     * the true mathematical square root of the argument value.
     *
     * @param   a   a value.
     * @return  the positive square root of {@code a}.
     *          If the argument is NaN or less than zero, the result is NaN.
     */
    public static double sqrt(double a) {
        return StrictMath.sqrt(a); // default impl. delegates to StrictMath
                                   // Note that hardware sqrt instructions
                                   // frequently can be directly used by JITs
                                   // and should be much faster than doing
                                   // Math.sqrt in software.
    }


    Returns the cube root of a double value. For positive finite x, 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: 
If the argument is NaN, then the result is NaN.
If the argument is infinite, then the result is an infinity
with the same sign as the argument.
If the argument is zero, then the result is a zero with the
same sign as the argument.

The computed result must be within 1 ulp of the exact result.
Params: a –   a value.
Returns:  the cube root of a.
Since: 1.5/**
     * 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
f1 - f2 × n,
where n is the mathematical integer closest to the exact mathematical value of the quotient f1/f2, and if two mathematical integers are equally close to f1/f2, then n 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:
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.
If the first argument is finite and the second argument is
infinite, then the result is the same as the first argument.
Params: f1 –   the dividend.
f2 –   the divisor.
Returns:  the remainder when f1 is divided by f2./**
     * 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) double value that is greater than or equal to the argument and is equal to a mathematical integer. Special cases: If the argument value is already equal to a
mathematical integer, then the result is the same as the
argument.  
If the argument is NaN or an infinity or
positive zero or negative zero, then the result is the same as
the argument.  
If the argument value is less than zero but
greater than -1.0, then the result is negative zero.
 Note that the value of Math.ceil(x) is exactly the value of -Math.floor(-x). 
Params: a –   a value.
Returns:  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./**
     * Returns the smallest (closest to negative infinity)
     * {@code double} value that is greater than or equal to the
     * argument and is equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a
     * mathematical integer, then the result is the same as the
     * argument.  <li>If the argument is NaN or an infinity or
     * positive zero or negative zero, then the result is the same as
     * the argument.  <li>If the argument value is less than zero but
     * greater than -1.0, then the result is negative zero.</ul> Note
     * that the value of {@code Math.ceil(x)} is exactly the
     * value of {@code -Math.floor(-x)}.
     *
     *
     * @param   a   a value.
     * @return  the smallest (closest to negative infinity)
     *          floating-point value that is greater than or equal to
     *          the argument and is equal to a mathematical integer.
     */
    public static double ceil(double a) {
        return StrictMath.ceil(a); // default impl. delegates to StrictMath
    }

    Returns the largest (closest to positive infinity) double value that is less than or equal to the argument and is equal to a mathematical integer. Special cases: If the argument value is already equal to a
mathematical integer, then the result is the same as the
argument.  
If the argument is NaN or an infinity or
positive zero or negative zero, then the result is the same as
the argument.
Params: a –   a value.
Returns:  the largest (closest to positive infinity)
         floating-point value that less than or equal to the argument
         and is equal to a mathematical integer./**
     * Returns the largest (closest to positive infinity)
     * {@code double} value that is less than or equal to the
     * argument and is equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a
     * mathematical integer, then the result is the same as the
     * argument.  <li>If the argument is NaN or an infinity or
     * positive zero or negative zero, then the result is the same as
     * the argument.</ul>
     *
     * @param   a   a value.
     * @return  the largest (closest to positive infinity)
     *          floating-point value that less than or equal to the argument
     *          and is equal to a mathematical integer.
     */
    public static double floor(double a) {
        return StrictMath.floor(a); // default impl. delegates to StrictMath
    }

    Returns the double value that is closest in value to the argument and is equal to a mathematical integer. If two double values that are mathematical integers are equally close, the result is the integer value that is even. Special cases: If the argument value is already equal to a mathematical
integer, then the result is the same as the argument.
If the argument is NaN or an infinity or positive zero or negative
zero, then the result is the same as the argument.
Params: a –  a double value.
Returns:  the closest floating-point value to a that is equal to a mathematical integer./**
     * Returns the {@code double} value that is closest in value
     * to the argument and is equal to a mathematical integer. If two
     * {@code double} values that are mathematical integers are
     * equally close, the result is the integer value that is
     * even. Special cases:
     * <ul><li>If the argument value is already equal to a mathematical
     * integer, then the result is the same as the argument.
     * <li>If the argument is NaN or an infinity or positive zero or negative
     * zero, then the result is the same as the argument.</ul>
     *
     * @param   a   a {@code double} value.
     * @return  the closest floating-point value to {@code a} that is
     *          equal to a mathematical integer.
     */
    public static double rint(double a) {
        return StrictMath.rint(a); // default impl. delegates to StrictMath
    }

    Returns the angle theta from the conversion of rectangular coordinates (x, y) to polar coordinates (r, theta).
This method computes the phase theta by computing an arc tangent of y/x in the range of -pi to pi. Special
cases:
If either argument is NaN, then the result is NaN.
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.
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.
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 double value closest to pi.
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 double value closest to -pi.
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 double value closest to pi/2.
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 double value closest to -pi/2.
If both arguments are positive infinity, then the result is the double value closest to pi/4.
If the first argument is positive infinity and the second argument is negative infinity, then the result is the double value closest to 3*pi/4.
If the first argument is negative infinity and the second argument is positive infinity, then the result is the double value closest to -pi/4.
If both arguments are negative infinity, then the result is the double value closest to -3*pi/4.
The computed result must be within 2 ulps of the exact result.
Results must be semi-monotonic.
Params: y –   the ordinate coordinate
x –   the abscissa coordinate
Returns:  the theta component of the point
         (r, theta)
         in polar coordinates that corresponds to the point
         (x, y) in Cartesian coordinates./**
     * Returns the angle <i>theta</i> from the conversion of rectangular
     * coordinates ({@code x},&nbsp;{@code y}) to polar
     * coordinates (r,&nbsp;<i>theta</i>).
     * This method computes the phase <i>theta</i> by computing an arc tangent
     * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
     * cases:
     * <ul><li>If either argument is NaN, then the result is NaN.
     * <li>If the first argument is positive zero and the second argument
     * is positive, or the first argument is positive and finite and the
     * second argument is positive infinity, then the result is positive
     * zero.
     * <li>If the first argument is negative zero and the second argument
     * is positive, or the first argument is negative and finite and the
     * second argument is positive infinity, then the result is negative zero.
     * <li>If the first argument is positive zero and the second argument
     * is negative, or the first argument is positive and finite and the
     * second argument is negative infinity, then the result is the
     * {@code double} value closest to <i>pi</i>.
     * <li>If the first argument is negative zero and the second argument
     * is negative, or the first argument is negative and finite and the
     * second argument is negative infinity, then the result is the
     * {@code double} value closest to -<i>pi</i>.
     * <li>If the first argument is positive and the second argument is
     * positive zero or negative zero, or the first argument is positive
     * infinity and the second argument is finite, then the result is the
     * {@code double} value closest to <i>pi</i>/2.
     * <li>If the first argument is negative and the second argument is
     * positive zero or negative zero, or the first argument is negative
     * infinity and the second argument is finite, then the result is the
     * {@code double} value closest to -<i>pi</i>/2.
     * <li>If both arguments are positive infinity, then the result is the
     * {@code double} value closest to <i>pi</i>/4.
     * <li>If the first argument is positive infinity and the second argument
     * is negative infinity, then the result is the {@code double}
     * value closest to 3*<i>pi</i>/4.
     * <li>If the first argument is negative infinity and the second argument
     * is positive infinity, then the result is the {@code double} value
     * closest to -<i>pi</i>/4.
     * <li>If both arguments are negative infinity, then the result is the
     * {@code double} value closest to -3*<i>pi</i>/4.</ul>
     *
     * <p>The computed result must be within 2 ulps of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   y   the ordinate coordinate
     * @param   x   the abscissa coordinate
     * @return  the <i>theta</i> component of the point
     *          (<i>r</i>,&nbsp;<i>theta</i>)
     *          in polar coordinates that corresponds to the point
     *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
     */
    public static double atan2(double y, double x) {
        return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
    }

    Returns the value of the first argument raised to the power of the
second argument. Special cases:
If the second argument is positive or negative zero, then the
result is 1.0.
If the second argument is 1.0, then the result is the same as the
first argument.
If the second argument is NaN, then the result is NaN.
If the first argument is NaN and the second argument is nonzero,
then the result is NaN.
If

the absolute value of the first argument is greater than 1
and the second argument is positive infinity, or
the absolute value of the first argument is less than 1 and
the second argument is negative infinity,

then the result is positive infinity.
If

the absolute value of the first argument is greater than 1 and
the second argument is negative infinity, or
the absolute value of the
first argument is less than 1 and the second argument is positive
infinity,

then the result is positive zero.
If the absolute value of the first argument equals 1 and the
second argument is infinite, then the result is NaN.
If

the first argument is positive zero and the second argument
is greater than zero, or
the first argument is positive infinity and the second
argument is less than zero,

then the result is positive zero.
If

the first argument is positive zero and the second argument
is less than zero, or
the first argument is positive infinity and the second
argument is greater than zero,

then the result is positive infinity.
If

the first argument is negative zero and the second argument
is greater than zero but not a finite odd integer, or
the first argument is negative infinity and the second
argument is less than zero but not a finite odd integer,

then the result is positive zero.
If

the first argument is negative zero and the second argument
is a positive finite odd integer, or
the first argument is negative infinity and the second
argument is a negative finite odd integer,

then the result is negative zero.
If

the first argument is negative zero and the second argument
is less than zero but not a finite odd integer, or
the first argument is negative infinity and the second
argument is greater than zero but not a finite odd integer,

then the result is positive infinity.
If

the first argument is negative zero and the second argument
is a negative finite odd integer, or
the first argument is negative infinity and the second
argument is a positive finite odd integer,

then the result is negative infinity.
If the first argument is finite and less than zero

 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
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
if the second argument is finite and not an integer, then
the result is NaN.

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 double value.
(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 ceil or, equivalently, a fixed point of the method 
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.) 
The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: a –   the base.
b –   the exponent.
Returns:  the value a^b./**
     * Returns the value of the first argument raised to the power of the
     * second argument. Special cases:
     *
     * <ul><li>If the second argument is positive or negative zero, then the
     * result is 1.0.
     * <li>If the second argument is 1.0, then the result is the same as the
     * first argument.
     * <li>If the second argument is NaN, then the result is NaN.
     * <li>If the first argument is NaN and the second argument is nonzero,
     * then the result is NaN.
     *
     * <li>If
     * <ul>
     * <li>the absolute value of the first argument is greater than 1
     * and the second argument is positive infinity, or
     * <li>the absolute value of the first argument is less than 1 and
     * the second argument is negative infinity,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the absolute value of the first argument is greater than 1 and
     * the second argument is negative infinity, or
     * <li>the absolute value of the
     * first argument is less than 1 and the second argument is positive
     * infinity,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If the absolute value of the first argument equals 1 and the
     * second argument is infinite, then the result is NaN.
     *
     * <li>If
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is greater than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is less than zero,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is less than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is greater than zero,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is greater than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is less than zero but not a finite odd integer,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a positive finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a negative finite odd integer,
     * </ul>
     * then the result is negative zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is less than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is greater than zero but not a finite odd integer,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a negative finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a positive finite odd integer,
     * </ul>
     * then the result is negative infinity.
     *
     * <li>If the first argument is finite and less than zero
     * <ul>
     * <li> if the second argument is a finite even integer, the
     * result is equal to the result of raising the absolute value of
     * the first argument to the power of the second argument
     *
     * <li>if the second argument is a finite odd integer, the result
     * is equal to the negative of the result of raising the absolute
     * value of the first argument to the power of the second
     * argument
     *
     * <li>if the second argument is finite and not an integer, then
     * the result is NaN.
     * </ul>
     *
     * <li>If both arguments are integers, then the result is exactly equal
     * to the mathematical result of raising the first argument to the power
     * of the second argument if that result can in fact be represented
     * exactly as a {@code double} value.</ul>
     *
     * <p>(In the foregoing descriptions, a floating-point value is
     * considered to be an integer if and only if it is finite and a
     * fixed point of the method {@link #ceil ceil} or,
     * equivalently, a fixed point of the method {@link #floor
     * floor}. A value is a fixed point of a one-argument
     * method if and only if the result of applying the method to the
     * value is equal to the value.)
     *
     * <p>The computed result must be within 1 ulp of the exact result.
     * Results must be semi-monotonic.
     *
     * @param   a   the base.
     * @param   b   the exponent.
     * @return  the value {@code a}<sup>{@code b}</sup>.
     */
    public static double pow(double a, double b) {
        return StrictMath.pow(a, b); // default impl. delegates to StrictMath
    }

    Returns the closest int to the argument, with ties rounding to positive infinity. 
Special cases:
If the argument is NaN, the result is 0.
If the argument is negative infinity or any value less than or equal to the value of Integer.MIN_VALUE, the result is equal to the value of Integer.MIN_VALUE. 
If the argument is positive infinity or any value greater than or equal to the value of Integer.MAX_VALUE, the result is equal to the value of Integer.MAX_VALUE.
Params: a –   a floating-point value to be rounded to an integer.
See Also: Integer.MAX_VALUE
Integer.MIN_VALUE
Returns:  the value of the argument rounded to the nearest int value./**
     * 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
     */
    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 long to the argument, with ties rounding to positive infinity. Special cases:
If the argument is NaN, the result is 0.
If the argument is negative infinity or any value less than or equal to the value of Long.MIN_VALUE, the result is equal to the value of Long.MIN_VALUE. 
If the argument is positive infinity or any value greater than or equal to the value of Long.MAX_VALUE, the result is equal to the value of Long.MAX_VALUE.
Params: a –  a floating-point value to be rounded to a long.
See Also: Long.MAX_VALUE
Long.MIN_VALUE
Returns:  the value of the argument rounded to the nearest long value./**
     * 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
     */
    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 double value with a positive sign, greater than or equal to 0.0 and less than 1.0. Returned values are chosen pseudorandomly with (approximately) uniform distribution from that range. When this method is first called, it creates a single new
pseudorandom-number generator, exactly as if by the expression
new java.util.Random()
This new pseudorandom-number generator is used thereafter for
all calls to this method and is used nowhere else.
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.
See Also: Random.nextDouble()
Returns:  a pseudorandom double greater than or equal to 0.0 and less than 1.0./**
     * Returns a {@code double} value with a positive sign, greater
     * than or equal to {@code 0.0} and less than {@code 1.0}.
     * Returned values are chosen pseudorandomly with (approximately)
     * uniform distribution from that range.
     *
     * <p>When this method is first called, it creates a single new
     * pseudorandom-number generator, exactly as if by the expression
     *
     * <blockquote>{@code new java.util.Random()}</blockquote>
     *
     * This new pseudorandom-number generator is used thereafter for
     * all calls to this method and is used nowhere else.
     *
     * <p>This method is properly synchronized to allow correct use by
     * more than one thread. However, if many threads need to generate
     * pseudorandom numbers at a great rate, it may reduce contention
     * for each thread to have its own pseudorandom-number generator.
     *
     * @return  a pseudorandom {@code double} greater than or equal
     * to {@code 0.0} and less than {@code 1.0}.
     * @see Random#nextDouble()
     */
    public static double random() {
        return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
    }

    Returns the sum of its arguments, throwing an exception if the result overflows an int. 
Params: x – the first value
y – the second value
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * 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
     */
    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 long. 
Params: x – the first value
y – the second value
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * 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
     */
    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 int. 
Params: x – the first value
y – the second value to subtract from the first
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * 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
     */
    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 than 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 long. 
Params: x – the first value
y – the second value to subtract from the first
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * 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
     */
    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 than 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 int. 
Params: x – the first value
y – the second value
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * 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
     */
    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 long. 
Params: x – the first value
y – the second value
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * 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
     */
    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 argument incremented by one, throwing an exception if the result overflows an int. 
Params: a – the value to increment
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * Returns the argument incremented by one, throwing an exception if the
     * result overflows an {@code int}.
     *
     * @param a the value to increment
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    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 long. 
Params: a – the value to increment
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * Returns the argument incremented by one, throwing an exception if the
     * result overflows a {@code long}.
     *
     * @param a the value to increment
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    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 int. 
Params: a – the value to decrement
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * Returns the argument decremented by one, throwing an exception if the
     * result overflows an {@code int}.
     *
     * @param a the value to decrement
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    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 long. 
Params: a – the value to decrement
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * Returns the argument decremented by one, throwing an exception if the
     * result overflows a {@code long}.
     *
     * @param a the value to decrement
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    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 int. 
Params: a – the value to negate
Throws: ArithmeticException – if the result overflows an int
Returns: the result
Since: 1.8/**
     * Returns the negation of the argument, throwing an exception if the
     * result overflows an {@code int}.
     *
     * @param a the value to negate
     * @return the result
     * @throws ArithmeticException if the result overflows an int
     * @since 1.8
     */
    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 long. 
Params: a – the value to negate
Throws: ArithmeticException – if the result overflows a long
Returns: the result
Since: 1.8/**
     * Returns the negation of the argument, throwing an exception if the
     * result overflows a {@code long}.
     *
     * @param a the value to negate
     * @return the result
     * @throws ArithmeticException if the result overflows a long
     * @since 1.8
     */
    public static long negateExact(long a) {
        if (a == Long.MIN_VALUE) {
            throw new ArithmeticException("long overflow");
        }

        return -a;
    }

    Returns the value of the long argument; throwing an exception if the value overflows an int. 
Params: value – the long value
Throws: ArithmeticException – if the argument overflows an int
Returns: the argument as an int
Since: 1.8/**
     * 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 largest (closest to positive infinity) int value that is less than or equal to the algebraic quotient. There is one special case, if the dividend is the Integer.MIN_VALUE and the divisor is -1, then integer overflow occurs and the result is equal to the Integer.MIN_VALUE. 
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 than truncation
when the exact result is negative.

  If the signs of the arguments are the same, the results of floorDiv and the / operator are the same. 
 For example, floorDiv(4, 3) == 1 and (4 / 3) == 1.
  If the signs of the arguments are different, the quotient is negative and floorDiv returns the integer less than or equal to the quotient and the / operator returns the integer closest to zero.
 For example, floorDiv(-4, 3) == -2, whereas (-4 / 3) == -1. 


Params: x – the dividend
y – the divisor
Throws: ArithmeticException – if the divisor y is zero
See Also: floorMod(int, int)
floor(double)
Returns: the largest (closest to positive infinity) int value that is less than or equal to the algebraic quotient.
Since: 1.8/**
     * 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 the
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to the {@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 than truncation
     * when the exact result 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,  the quotient is negative and
     *       {@code floorDiv} returns the integer less than or equal to the quotient
     *       and the {@code /} operator returns the integer closest to zero.<br>
     *       For example, {@code floorDiv(-4, 3) == -2},
     *       whereas {@code (-4 / 3) == -1}.
     *   </li>
     * </ul>
     * <p>
     *
     * @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) {
        int r = x / y;
        // if the signs are different and modulo not zero, round down
        if ((x ^ y) < 0 && (r * y != x)) {
            r--;
        }
        return r;
    }

    Returns the largest (closest to positive infinity) long value that is less than or equal to the algebraic quotient. There is one special case, if the dividend is the Long.MIN_VALUE and the divisor is -1, then integer overflow occurs and the result is equal to the Long.MIN_VALUE. 
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 than truncation
when the exact result is negative.
 For examples, see floorDiv(int, int). 
Params: x – the dividend
y – the divisor
Throws: ArithmeticException – if the divisor y is zero
See Also: floorMod(long, long)
floor(double)
Returns: the largest (closest to positive infinity) long value that is less than or equal to the algebraic quotient.
Since: 1.8/**
     * 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 the
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to the {@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 than truncation
     * when the exact result 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) {
        long r = x / y;
        // if the signs are different and modulo not zero, round down
        if ((x ^ y) < 0 && (r * y != x)) {
            r--;
        }
        return r;
    }

    Returns the floor modulus of the int arguments.  The floor modulus is x - (floorDiv(x, y) * y), has the same sign as the divisor y, and is in the range of -abs(y) < r < +abs(y). 
 The relationship between floorDiv and floorMod is such that: 

  floorDiv(x, y) * y + floorMod(x, y) == x 
 The difference in values between floorMod and the % operator is due to the difference between floorDiv that returns the integer less than or equal to the quotient and the / operator that returns the integer closest to zero. 

Examples:

  If the signs of the arguments are the same, the results of floorMod and the % operator are the same. 

      
      floorMod(4, 3) == 1; and (4 % 3) == 1
      
  
If the signs of the arguments are different, the results differ from the % operator.

     
     floorMod(+4, -3) == -2; and (+4 % -3) == +1 
     floorMod(-4, +3) == +2; and (-4 % +3) == -1 
     floorMod(-4, -3) == -1; and (-4 % -3) == -1  
     
  

 If the signs of arguments are unknown and a positive modulus is needed it can be computed as (floorMod(x, y) + abs(y)) % abs(y). 
Params: x – the dividend
y – the divisor
Throws: ArithmeticException – if the divisor y is zero
See Also: floorDiv(int, int)
Returns: the floor modulus x - (floorDiv(x, y) * y)
Since: 1.8/**
     * Returns the floor modulus of the {@code int} arguments.
     * <p>
     * The floor modulus is {@code x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y}, 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}
     * </ul>
     * <p>
     * The difference in values between {@code floorMod} and
     * the {@code %} operator is due to the difference between
     * {@code floorDiv} that returns the integer less than or equal to the quotient
     * and the {@code /} operator that returns the integer closest to zero.
     * <p>
     * Examples:
     * <ul>
     *   <li>If the signs of the arguments are the same, the results
     *       of {@code floorMod} and the {@code %} operator are the same.  <br>
     *       <ul>
     *       <li>{@code floorMod(4, 3) == 1}; &nbsp; and {@code (4 % 3) == 1}</li>
     *       </ul>
     *   <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br>
     *      <ul>
     *      <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>
     *      <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1 } </li>
     *      </ul>
     *   </li>
     * </ul>
     * <p>
     * If the signs of arguments are unknown and a positive modulus
     * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}.
     *
     * @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) {
        int r = x - floorDiv(x, y) * y;
        return r;
    }

    Returns the floor modulus of the long arguments.  The floor modulus is x - (floorDiv(x, y) * y), has the same sign as the divisor y, and is in the range of -abs(y) < r < +abs(y). 
 The relationship between floorDiv and floorMod is such that: 

  floorDiv(x, y) * y + floorMod(x, y) == x 
 For examples, see floorMod(int, int). 
Params: x – the dividend
y – the divisor
Throws: ArithmeticException – if the divisor y is zero
See Also: floorDiv(long, long)
Returns: the floor modulus x - (floorDiv(x, y) * y)
Since: 1.8/**
     * Returns the floor modulus of the {@code long} arguments.
     * <p>
     * The floor modulus is {@code x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y}, 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}
     * </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) {
        return x - floorDiv(x, y) * y;
    }

    Returns the absolute value of an int value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned. Note that if the argument is equal to the value of Integer.MIN_VALUE, the most negative representable int value, the result is that same value, which is negative. 
Params: a –   the argument whose absolute value is to be determined
Returns:  the absolute value of the argument./**
     * Returns the absolute value of an {@code int} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     *
     * <p>Note that if the argument is equal to the value of
     * {@link Integer#MIN_VALUE}, the most negative representable
     * {@code int} value, the result is that same value, which is
     * negative.
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    public static int abs(int a) {
        return (a < 0) ? -a : a;
    }

    Returns the absolute value of a long value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned. Note that if the argument is equal to the value of Long.MIN_VALUE, the most negative representable long value, the result is that same value, which is negative. 
Params: a –   the argument whose absolute value is to be determined
Returns:  the absolute value of the argument./**
     * Returns the absolute value of a {@code long} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     *
     * <p>Note that if the argument is equal to the value of
     * {@link Long#MIN_VALUE}, the most negative representable
     * {@code long} value, the result is that same value, which
     * is negative.
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    public static long abs(long a) {
        return (a < 0) ? -a : a;
    }

    Returns the absolute value of a 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: If the argument is positive zero or negative zero, the
result is positive zero.
If the argument is infinite, the result is positive infinity.
If the argument is NaN, the result is NaN.
In other words, the result is the same as the value of the expression:
Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a)) 
Params: a –   the argument whose absolute value is to be determined
Returns:  the absolute value of the argument./**
     * Returns the absolute value of a {@code float} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the
     * result is positive zero.
     * <li>If the argument is infinite, the result is positive infinity.
     * <li>If the argument is NaN, the result is NaN.</ul>
     * In other words, the result is the same as the value of the expression:
     * <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))}
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    public static float abs(float a) {
        return (a <= 0.0F) ? 0.0F - a : a;
    }

    Returns the absolute value of a 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: If the argument is positive zero or negative zero, the result
is positive zero.
If the argument is infinite, the result is positive infinity.
If the argument is NaN, the result is NaN.
In other words, the result is the same as the value of the expression:
Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1) 
Params: a –   the argument whose absolute value is to be determined
Returns:  the absolute value of the argument./**
     * Returns the absolute value of a {@code double} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the result
     * is positive zero.
     * <li>If the argument is infinite, the result is positive infinity.
     * <li>If the argument is NaN, the result is NaN.</ul>
     * In other words, the result is the same as the value of the expression:
     * <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    public static double abs(double a) {
        return (a <= 0.0D) ? 0.0D - a : a;
    }

    Returns the greater of two int values. That is, the result is the argument closer to the value of Integer.MAX_VALUE. If the arguments have the same value, the result is that same value. 
Params: a –   an argument.
b –   another argument.
Returns:  the larger of a and b./**
     * Returns the greater of two {@code int} values. That is, the
     * result is the argument closer to the value of
     * {@link Integer#MAX_VALUE}. If the arguments have the same value,
     * the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of {@code a} and {@code b}.
     */
    public static int max(int a, int b) {
        return (a >= b) ? a : b;
    }

    Returns the greater of two long values. That is, the result is the argument closer to the value of Long.MAX_VALUE. If the arguments have the same value, the result is that same value. 
Params: a –   an argument.
b –   another argument.
Returns:  the larger of a and b./**
     * Returns the greater of two {@code long} values. That is, the
     * result is the argument closer to the value of
     * {@link Long#MAX_VALUE}. If the arguments have the same value,
     * the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of {@code a} and {@code b}.
     */
    public static long max(long a, long b) {
        return (a >= b) ? a : b;
    }

    // Use raw bit-wise conversions on guaranteed non-NaN arguments.
    private static long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
    private static long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);

    Returns the greater of two 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. 
Params: a –   an argument.
b –   another argument.
Returns:  the larger of a and b./**
     * Returns the greater of two {@code float} values.  That is,
     * the result is the argument closer to positive infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of {@code a} and {@code b}.
     */
    public static float max(float a, float b) {
        if (a != a)
            return a;   // a is NaN
        if ((a == 0.0f) &&
            (b == 0.0f) &&
            (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
            // Raw conversion ok since NaN can't map to -0.0.
            return b;
        }
        return (a >= b) ? a : b;
    }

    Returns the greater of two 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. 
Params: a –   an argument.
b –   another argument.
Returns:  the larger of a and b./**
     * Returns the greater of two {@code double} values.  That
     * is, the result is the argument closer to positive infinity. If
     * the arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of {@code a} and {@code b}.
     */
    public static double max(double a, double b) {
        if (a != a)
            return a;   // a is NaN
        if ((a == 0.0d) &&
            (b == 0.0d) &&
            (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
            // Raw conversion ok since NaN can't map to -0.0.
            return b;
        }
        return (a >= b) ? a : b;
    }

    Returns the smaller of two int values. That is, the result the argument closer to the value of Integer.MIN_VALUE. If the arguments have the same value, the result is that same value. 
Params: a –   an argument.
b –   another argument.
Returns:  the smaller of a and b./**
     * Returns the smaller of two {@code int} values. That is,
     * the result the argument closer to the value of
     * {@link Integer#MIN_VALUE}.  If the arguments have the same
     * value, the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of {@code a} and {@code b}.
     */
    public static int min(int a, int b) {
        return (a <= b) ? a : b;
    }

    Returns the smaller of two long values. That is, the result is the argument closer to the value of Long.MIN_VALUE. If the arguments have the same value, the result is that same value. 
Params: a –   an argument.
b –   another argument.
Returns:  the smaller of a and 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 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. 
Params: a –   an argument.
b –   another argument.
Returns:  the smaller of a and b./**
     * Returns the smaller of two {@code float} values.  That is,
     * the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero.  If
     * one argument is positive zero and the other is negative zero,
     * the result is negative zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of {@code a} and {@code b}.
     */
    public static float min(float a, float b) {
        if (a != a)
            return a;   // a is NaN
        if ((a == 0.0f) &&
            (b == 0.0f) &&
            (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
            // Raw conversion ok since NaN can't map to -0.0.
            return b;
        }
        return (a <= b) ? a : b;
    }

    Returns the smaller of two 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. 
Params: a –   an argument.
b –   another argument.
Returns:  the smaller of a and b./**
     * Returns the smaller of two {@code double} values.  That
     * is, the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other is negative zero, the
     * result is negative zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of {@code a} and {@code b}.
     */
    public static double min(double a, double b) {
        if (a != a)
            return a;   // a is NaN
        if ((a == 0.0d) &&
            (b == 0.0d) &&
            (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
            // Raw conversion ok since NaN can't map to -0.0.
            return b;
        }
        return (a <= b) ? a : b;
    }

    Returns the size of an ulp of the argument. An ulp, unit in the last place, of a double value is the positive distance between this floating-point value and the 
double value next larger in magnitude. Note that for non-NaN x, ulp(-x) == ulp(x).
Special Cases:

 If the argument is NaN, then the result is NaN.
 If the argument is positive or negative infinity, then the
result is positive infinity.
 If the argument is positive or negative zero, then the result is Double.MIN_VALUE. 
 If the argument is ±Double.MAX_VALUE, then the result is equal to 2⁹⁷¹.

Author: Joseph D. Darcy
Params: d – the floating-point value whose ulp is to be returned
Returns: the size of an ulp of the argument
Since: 1.5/**
     * 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);

        switch(exp) {
        case DoubleConsts.MAX_EXPONENT+1:       // NaN or infinity
            return Math.abs(d);

        case DoubleConsts.MIN_EXPONENT-1:       // zero or subnormal
            return Double.MIN_VALUE;

        default:
            assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;

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

    Returns the size of an ulp of the argument. An ulp, unit in the last place, of a float value is the positive distance between this floating-point value and the 
float value next larger in magnitude. Note that for non-NaN x, ulp(-x) == ulp(x).
Special Cases:

 If the argument is NaN, then the result is NaN.
 If the argument is positive or negative infinity, then the
result is positive infinity.
 If the argument is positive or negative zero, then the result is Float.MIN_VALUE. 
 If the argument is ±Float.MAX_VALUE, then the result is equal to 2¹⁰⁴.

Author: Joseph D. Darcy
Params: f – the floating-point value whose ulp is to be returned
Returns: the size of an ulp of the argument
Since: 1.5/**
     * 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);

        switch(exp) {
        case FloatConsts.MAX_EXPONENT+1:        // NaN or infinity
            return Math.abs(f);

        case FloatConsts.MIN_EXPONENT-1:        // zero or subnormal
            return FloatConsts.MIN_VALUE;

        default:
            assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;

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

 If the argument is NaN, then the result is NaN.
 If the argument is positive zero or negative zero, then the
     result is the same as the argument.

Author: Joseph D. Darcy
Params: d – the floating-point value whose signum is to be returned
Returns: the signum function of the argument
Since: 1.5/**
     * Returns the signum function of the argument; zero if the argument
     * is zero, 1.0 if the argument is greater than zero, -1.0 if the
     * argument is less than zero.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive zero or negative zero, then the
     *      result is the same as the argument.
     * </ul>
     *
     * @param d the floating-point value whose signum is to be returned
     * @return the signum function of the argument
     * @author Joseph D. Darcy
     * @since 1.5
     */
    public static double signum(double d) {
        return (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.
Special Cases:

 If the argument is NaN, then the result is NaN.
 If the argument is positive zero or negative zero, then the
     result is the same as the argument.

Author: Joseph D. Darcy
Params: f – the floating-point value whose signum is to be returned
Returns: the signum function of the argument
Since: 1.5/**
     * Returns the signum function of the argument; zero if the argument
     * is zero, 1.0f if the argument is greater than zero, -1.0f if the
     * argument is less than zero.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive zero or negative zero, then the
     *      result is the same as the argument.
     * </ul>
     *
     * @param f the floating-point value whose signum is to be returned
     * @return the signum function of the argument
     * @author Joseph D. Darcy
     * @since 1.5
     */
    public static float signum(float f) {
        return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
    }

    Returns the hyperbolic sine of a double value. The hyperbolic sine of x is defined to be
(e^x - e^-x)/2
where e is Euler's number. Special cases:

If the argument is NaN, then the result is NaN.
If the argument is infinite, then the result is an infinity
with the same sign as the argument.
If the argument is zero, then the result is a zero with the
same sign as the argument.

The computed result must be within 2.5 ulps of the exact result.
Params: x – The number whose hyperbolic sine is to be returned.
Returns:  The hyperbolic sine of x.
Since: 1.5/**
     * 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 double value. The hyperbolic cosine of x is defined to be
(e^x + e^-x)/2
where e is Euler's number. Special cases:

If the argument is NaN, then the result is NaN.
If the argument is infinite, then the result is positive
infinity.
If the argument is zero, then the result is 1.0. 
The computed result must be within 2.5 ulps of the exact result.
Params: x – The number whose hyperbolic cosine is to be returned.
Returns:  The hyperbolic cosine of x.
Since: 1.5/**
     * 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 double value. The hyperbolic tangent of x is defined to be
(e^x - e^-x)/(e^x + e^-x), in other words, 
sinh(x)/cosh(x). Note that the absolute value of the exact tanh is always less than 1. Special cases:

If the argument is NaN, then the result is NaN.
If the argument is zero, then the result is a zero with the
same sign as the argument.
If the argument is positive infinity, then the result is +1.0. 
If the argument is negative infinity, then the result is -1.0. 
The computed result must be within 2.5 ulps of the exact result. The result of 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 ±1, correctly signed ±1.0 should be returned. 
Params: x – The number whose hyperbolic tangent is to be returned.
Returns:  The hyperbolic tangent of x.
Since: 1.5/**
     * 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(x² +y²)
without intermediate overflow or underflow.
Special cases:

 If either argument is infinite, then the result
is positive infinity.
 If either argument is NaN and neither argument is infinite,
then the result is NaN.

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.
Params: x – a value
y – a value
Returns: sqrt(x² +y²)
without intermediate overflow or underflow
Since: 1.5/**
     * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     * without intermediate overflow or underflow.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li> If either argument is infinite, then the result
     * is positive infinity.
     *
     * <li> If either argument is NaN and neither argument is infinite,
     * then the result is NaN.
     *
     * </ul>
     *
     * <p>The computed result must be within 1 ulp of the exact
     * result.  If one parameter is held constant, the results must be
     * semi-monotonic in the other parameter.
     *
     * @param x a value
     * @param y a value
     * @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     * without intermediate overflow or underflow
     * @since 1.5
     */
    public static double hypot(double x, double y) {
        return StrictMath.hypot(x, y);
    }

    Returns e^x -1.  Note that for values of
x near 0, the exact sum of expm1(x) + 1 is much closer to the true result of e^x than exp(x). Special cases:

If the argument is NaN, the result is NaN.
If the argument is positive infinity, then the result is
positive infinity.
If the argument is negative infinity, then the result is
-1.0.
If the argument is zero, then the result is a zero with the
same sign as the argument.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic. The result of expm1 for any finite input must be greater than or equal to -1.0. Note that once the exact result of e^x - 1 is within 1/2 ulp of the limit value -1, -1.0 should be returned. 
Params: x –   the exponent to raise e to in the computation of
             e^x -1.
Returns:  the value e^x - 1.
Since: 1.5/**
     * 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 x, the result of log1p(x) is much closer to the true result of ln(1 + x) than the floating-point evaluation of log(1.0+x). Special cases:

If the argument is NaN or less than -1, then the result is
NaN.
If the argument is positive infinity, then the result is
positive infinity.
If the argument is negative one, then the result is
negative infinity.
If the argument is zero, then the result is a zero with the
same sign as the argument.

The computed result must be within 1 ulp of the exact result.
Results must be semi-monotonic.
Params: x –   a value
Returns: the value ln(x + 1), the natural log of x + 1
Since: 1.5/**
     * 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 StrictMath.copySign method, this method does not require NaN 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. 
Params: magnitude –  the parameter providing the magnitude of the result
sign –   the parameter providing the sign of the result
Returns: a value with the magnitude of magnitude and the sign of sign.
Since: 1.6/**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  Note that unlike the {@link
     * StrictMath#copySign(double, double) StrictMath.copySign}
     * method, this method does not require NaN {@code sign}
     * arguments to be treated as positive values; implementations are
     * permitted to treat some NaN arguments as positive and other NaN
     * arguments as negative to allow greater performance.
     *
     * @param magnitude  the parameter providing the magnitude of the result
     * @param sign   the parameter providing the sign of the result
     * @return a value with the magnitude of {@code magnitude}
     * and the sign of {@code sign}.
     * @since 1.6
     */
    public static double copySign(double magnitude, double sign) {
        return 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 StrictMath.copySign method, this method does not require NaN 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. 
Params: magnitude –  the parameter providing the magnitude of the result
sign –   the parameter providing the sign of the result
Returns: a value with the magnitude of magnitude and the sign of sign.
Since: 1.6/**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  Note that unlike the {@link
     * StrictMath#copySign(float, float) StrictMath.copySign}
     * method, this method does not require NaN {@code sign}
     * arguments to be treated as positive values; implementations are
     * permitted to treat some NaN arguments as positive and other NaN
     * arguments as negative to allow greater performance.
     *
     * @param magnitude  the parameter providing the magnitude of the result
     * @param sign   the parameter providing the sign of the result
     * @return a value with the magnitude of {@code magnitude}
     * and the sign of {@code sign}.
     * @since 1.6
     */
    public static float copySign(float magnitude, float sign) {
        return 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 float. Special cases: 
If the argument is NaN or infinite, then the result is Float.MAX_EXPONENT + 1. 
If the argument is zero or subnormal, then the result is Float.MIN_EXPONENT -1. 
Params: f – a float value
Returns: the unbiased exponent of the argument
Since: 1.6/**
     * 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 double. Special cases: 
If the argument is NaN or infinite, then the result is Double.MAX_EXPONENT + 1. 
If the argument is zero or subnormal, then the result is Double.MIN_EXPONENT -1. 
Params: d – a double value
Returns: the unbiased exponent of the argument
Since: 1.6/**
     * 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.

Special cases:

 If either argument is a NaN, then NaN is returned.
 If both arguments are signed zeros, direction is returned unchanged (as implied by the requirement of returning the second argument if the arguments compare as equal). 
 If start is ±Double.MIN_VALUE and direction has a value such that the result should have a smaller magnitude, then a zero with the same sign as start is returned. 
 If start is infinite and direction has a value such that the result should have a smaller magnitude, Double.MAX_VALUE with the same sign as start is returned. 
 If start is equal to ± Double.MAX_VALUE and direction has a value such that the result should have a larger magnitude, an infinity with same sign as start is returned. 
Params: start –  starting floating-point value
direction – value indicating which of start's neighbors or start should be returned
Returns: The floating-point number adjacent to start in the direction of direction.
Since: 1.6/**
     * 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
         */

        // First check for NaN values
        if (Double.isNaN(start) || Double.isNaN(direction)) {
            // return a NaN derived from the input NaN(s)
            return start + direction;
        } else if (start == direction) {
            return direction;
        } else {        // start > direction or start < direction
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
            // then bitwise convert start to integer.
            long transducer = Double.doubleToRawLongBits(start + 0.0d);

            /*
             * 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 values
             * is less than zero, the adjustment to the representation
             * is in the opposite direction than would be expected at
             * first .
             */
            if (direction > start) { // Calculate next greater value
                transducer = transducer + (transducer >= 0L ? 1L:-1L);
            } else  { // Calculate next lesser value
                assert direction < start;
                if (transducer > 0L)
                    --transducer;
                else
                    if (transducer < 0L )
                        ++transducer;
                    /*
                     * transducer==0, the result is -MIN_VALUE
                     *
                     * The transition from zero (implicitly
                     * positive) to the smallest negative
                     * signed magnitude value must be done
                     * explicitly.
                     */
                    else
                        transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
            }

            return Double.longBitsToDouble(transducer);
        }
    }

    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.

Special cases:

 If either argument is a NaN, then NaN is returned.
 If both arguments are signed zeros, a value equivalent to direction is returned. 
 If start is ±Float.MIN_VALUE and direction has a value such that the result should have a smaller magnitude, then a zero with the same sign as start is returned. 
 If start is infinite and direction has a value such that the result should have a smaller magnitude, Float.MAX_VALUE with the same sign as start is returned. 
 If start is equal to ± Float.MAX_VALUE and direction has a value such that the result should have a larger magnitude, an infinity with same sign as start is returned. 
Params: start –  starting floating-point value
direction – value indicating which of start's neighbors or start should be returned
Returns: The floating-point number adjacent to start in the direction of direction.
Since: 1.6/**
     * 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
         */

        // First check for NaN values
        if (Float.isNaN(start) || Double.isNaN(direction)) {
            // return a NaN derived from the input NaN(s)
            return start + (float)direction;
        } else if (start == direction) {
            return (float)direction;
        } else {        // start > direction or start < direction
            // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
            // then bitwise convert start to integer.
            int transducer = Float.floatToRawIntBits(start + 0.0f);

            /*
             * 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 values
             * is less than zero, the adjustment to the representation
             * is in the opposite direction than would be expected at
             * first.
             */
            if (direction > start) {// Calculate next greater value
                transducer = transducer + (transducer >= 0 ? 1:-1);
            } else  { // Calculate next lesser value
                assert direction < start;
                if (transducer > 0)
                    --transducer;
                else
                    if (transducer < 0 )
                        ++transducer;
                    /*
                     * transducer==0, the result is -MIN_VALUE
                     *
                     * The transition from zero (implicitly
                     * positive) to the smallest negative
                     * signed magnitude value must be done
                     * explicitly.
                     */
                    else
                        transducer = FloatConsts.SIGN_BIT_MASK | 1;
            }

            return Float.intBitsToFloat(transducer);
        }
    }

    Returns the floating-point value adjacent to d in the direction of positive infinity. This method is semantically equivalent to nextAfter(d,
Double.POSITIVE_INFINITY); however, a nextUp implementation may run faster than its equivalent nextAfter call. Special Cases:

 If the argument is NaN, the result is NaN.
 If the argument is positive infinity, the result is
positive infinity.
 If the argument is zero, the result is Double.MIN_VALUE 
Params: d – starting floating-point value
Returns: The adjacent floating-point value closer to positive
infinity.
Since: 1.6/**
     * 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) {
        if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
            return d;
        else {
            d += 0.0d;
            return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
                                           ((d >= 0.0d)?+1L:-1L));
        }
    }

    Returns the floating-point value adjacent to f in the direction of positive infinity. This method is semantically equivalent to nextAfter(f,
Float.POSITIVE_INFINITY); however, a nextUp implementation may run faster than its equivalent nextAfter call. Special Cases:

 If the argument is NaN, the result is NaN.
 If the argument is positive infinity, the result is
positive infinity.
 If the argument is zero, the result is Float.MIN_VALUE 
Params: f – starting floating-point value
Returns: The adjacent floating-point value closer to positive
infinity.
Since: 1.6/**
     * 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) {
        if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
            return f;
        else {
            f += 0.0f;
            return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
                                        ((f >= 0.0f)?+1:-1));
        }
    }

    Returns the floating-point value adjacent to d in the direction of negative infinity. This method is semantically equivalent to nextAfter(d,
Double.NEGATIVE_INFINITY); however, a nextDown implementation may run faster than its equivalent nextAfter call. Special Cases:

 If the argument is NaN, the result is NaN.
 If the argument is negative infinity, the result is
negative infinity.
 If the argument is zero, the result is -Double.MIN_VALUE 
Params: d –  starting floating-point value
Returns: The adjacent floating-point value closer to negative
infinity.
Since: 1.8/**
     * 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 f in the direction of negative infinity. This method is semantically equivalent to nextAfter(f,
Float.NEGATIVE_INFINITY); however, a nextDown implementation may run faster than its equivalent nextAfter call. Special Cases:

 If the argument is NaN, the result is NaN.
 If the argument is negative infinity, the result is
negative infinity.
 If the argument is zero, the result is -Float.MIN_VALUE 
Params: f –  starting floating-point value
Returns: The adjacent floating-point value closer to negative
infinity.
Since: 1.8/**
     * 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 d × 2^scaleFactor rounded as if performed by a single correctly rounded floating-point multiply to a member of the double value set. See the Java Language Specification for a discussion of floating-point value sets. If the exponent of the result is between Double.MIN_EXPONENT and Double.MAX_EXPONENT, the answer is calculated exactly. If the exponent of the result would be larger than Double.MAX_EXPONENT, an infinity is returned. Note that if the result is subnormal, precision may be lost; that is, when scalb(x, n) is subnormal, scalb(scalb(x, n), -n) may not equal x. When the result is non-NaN, the result has the same sign as d. Special cases:

 If the first argument is NaN, NaN is returned.
 If the first argument is infinite, then an infinity of the
same sign is returned.
 If the first argument is zero, then a zero of the same
sign is returned.

Params: d – number to be scaled by a power of two.
scaleFactor – power of 2 used to scale d
Returns: d × 2^scaleFactor
Since: 1.6/**
     * Returns {@code d} &times;
     * 2<sup>{@code scaleFactor}</sup> rounded as if performed
     * by a single correctly rounded floating-point multiply to a
     * member of the double value set.  See the Java
     * Language Specification for a discussion of floating-point
     * value sets.  If the exponent of the result is between {@link
     * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
     * answer is calculated exactly.  If the exponent of the result
     * would be larger than {@code Double.MAX_EXPONENT}, an
     * infinity is returned.  Note that if the result is subnormal,
     * precision may be lost; that is, when {@code scalb(x, n)}
     * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
     * <i>x</i>.  When the result is non-NaN, the result has the same
     * sign as {@code d}.
     *
     * <p>Special cases:
     * <ul>
     * <li> If the first argument is NaN, NaN is returned.
     * <li> If the first argument is infinite, then an infinity of the
     * same sign is returned.
     * <li> If the first argument is zero, then a zero of the same
     * sign is returned.
     * </ul>
     *
     * @param d number to be scaled by a power of two.
     * @param scaleFactor power of 2 used to scale {@code d}
     * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
     * @since 1.6
     */
    public static double scalb(double d, int scaleFactor) {
        /*
         * This method does not need to be declared strictfp to
         * compute the same correct result on all platforms.  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 by 2^n as needed where n is the exponent of number
         * that is a covenient power of two.  In this way, at most one
         * real rounding error occurs.  If the double value set is
         * being used exclusively, the rounding will occur on a
         * multiply.  If the double-extended-exponent value set is
         * being used, the products will (perhaps) be exact but the
         * stores to d are guaranteed to round to the double value
         * set.
         *
         * It is _not_ a valid implementation to first multiply d by
         * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor %
         * MIN_EXPONENT) since even in a strictfp program double
         * rounding on underflow could occur; e.g. if the scaleFactor
         * argument was (MIN_EXPONENT - n) and the exponent of d was a
         * little less than -(MIN_EXPONENT - n), meaning the final
         * result would be subnormal.
         *
         * Since exact reproducibility of this method can be achieved
         * without any undue performance burden, there is no
         * compelling reason to allow double rounding on underflow in
         * scalb.
         */

        // 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 takes an
        // additional power of two which is reflected here
        final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.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 f × 2^scaleFactor rounded as if performed by a single correctly rounded floating-point multiply to a member of the float value set. See the Java Language Specification for a discussion of floating-point value sets. If the exponent of the result is between Float.MIN_EXPONENT and Float.MAX_EXPONENT, the answer is calculated exactly. If the exponent of the result would be larger than Float.MAX_EXPONENT, an infinity is returned. Note that if the result is subnormal, precision may be lost; that is, when scalb(x, n) is subnormal, scalb(scalb(x, n), -n) may not equal x. When the result is non-NaN, the result has the same sign as f. Special cases:

 If the first argument is NaN, NaN is returned.
 If the first argument is infinite, then an infinity of the
same sign is returned.
 If the first argument is zero, then a zero of the same
sign is returned.

Params: f – number to be scaled by a power of two.
scaleFactor – power of 2 used to scale f
Returns: f × 2^scaleFactor
Since: 1.6/**
     * Returns {@code f} &times;
     * 2<sup>{@code scaleFactor}</sup> rounded as if performed
     * by a single correctly rounded floating-point multiply to a
     * member of the float value set.  See the Java
     * Language Specification for a discussion of floating-point
     * value sets.  If the exponent of the result is between {@link
     * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
     * answer is calculated exactly.  If the exponent of the result
     * would be larger than {@code Float.MAX_EXPONENT}, an
     * infinity is returned.  Note that if the result is subnormal,
     * precision may be lost; that is, when {@code scalb(x, n)}
     * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
     * <i>x</i>.  When the result is non-NaN, the result has the same
     * sign as {@code f}.
     *
     * <p>Special cases:
     * <ul>
     * <li> If the first argument is NaN, NaN is returned.
     * <li> If the first argument is infinite, then an infinity of the
     * same sign is returned.
     * <li> If the first argument is zero, then a zero of the same
     * sign is returned.
     * </ul>
     *
     * @param f number to be scaled by a power of two.
     * @param scaleFactor power of 2 used to scale {@code f}
     * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
     * @since 1.6
     */
    public static float scalb(float f, int scaleFactor) {
        // 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 takes an
        // additional power of two which is reflected here
        final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.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.  Since
         * all operations other than the final multiply will be exact,
         * it is not necessary to declare this method strictfp.
         */
        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.
/**
     * Returns a floating-point power of two in the normal range.
     */
    static double powerOfTwoD(int n) {
        assert(n >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.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.
/**
     * Returns a floating-point power of two in the normal range.
     */
    static float powerOfTwoF(int n) {
        assert(n >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
        return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
                                     (FloatConsts.SIGNIFICAND_WIDTH-1))
                                    & FloatConsts.EXP_BIT_MASK);
    }
}
Params:	a – an angle, in radians.
Returns:	the sine of the argument.
Params:	a – the value whose arc sine is to be returned.
Returns:	the arc sine of the argument.
Params:	a – the value whose arc cosine is to be returned.
Returns:	the arc cosine of the argument.
Params:	a – the value whose arc tangent is to be returned.
Returns:	the arc tangent of the argument.
Params:	angdeg – an angle, in degrees
Returns:	the measurement of the angle `angdeg` in radians.
Since:	1.2
Params:	angrad – an angle, in radians
Returns:	the measurement of the angle `angrad` in degrees.
Since:	1.2
Params:	a – the exponent to raise e to.
Returns:	the value e^a, where e is the base of the natural logarithms.
Params:	a – a value
Returns:	the value ln `a`, the natural logarithm of `a`.
Params:	a – a value.
Returns:	the positive square root of `a`. If the argument is NaN or less than zero, the result is NaN.
Params:	f1 – the dividend. f2 – the divisor.
Returns:	the remainder when `f1` is divided by `f2`.
Params:	a – a `double` value.
Returns:	the closest floating-point value to `a` that is equal to a mathematical integer.
Params:	y – the ordinate coordinate x – the abscissa coordinate
Returns:	the theta component of the point (r, theta) in polar coordinates that corresponds to the point (x, y) in Cartesian coordinates.
Params:	a – the base. b – the exponent.
Returns:	the value `a`^b.
Params:	a – a floating-point value to be rounded to an integer.
See Also:	Integer.MAX_VALUE Integer.MIN_VALUE
Returns:	the value of the argument rounded to the nearest `int` value.
See Also:	Random.nextDouble()
Returns:	a pseudorandom `double` greater than or equal to `0.0` and less than `1.0`.
Params:	x – the first value y – the second value
Throws:	ArithmeticException – if the result overflows an int
Returns:	the result
Since:	1.8
Params:	a – the value to increment
Throws:	ArithmeticException – if the result overflows an int
Returns:	the result
Since:	1.8
Params:	a – the value to decrement
Throws:	ArithmeticException – if the result overflows an int
Returns:	the result
Since:	1.8
Params:	a – the value to negate
Throws:	ArithmeticException – if the result overflows an int
Returns:	the result
Since:	1.8
Params:	value – the long value
Throws:	ArithmeticException – if the `argument` overflows an int
Returns:	the argument as an int
Since:	1.8
/

java/ 8/ java/lang/Math.java
