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

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 * published by the Free Software Foundation.  Oracle designates this

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

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

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package java.lang;

import java.util.Random;

import sun.misc.FloatConsts;

import sun.misc.DoubleConsts;

import jdk.internal.HotSpotIntrinsicCandidate;

/**

 * 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   1.0

 */

public final class Math {

    /**

     * Don't let anyone instantiate this class.

     */

    private Math() {}

    /**

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

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

     */

    public static final double E = 2.7182818284590452354;

    /**

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

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

     * diameter.

     */

    public static final double PI = 3.14159265358979323846;

    /**

     * Constant by which to multiply an angular value in degrees to obtain an

     * angular value in radians.

     */

    private static final double DEGREES_TO_RADIANS = 0.017453292519943295;

    /**

     * Constant by which to multiply an angular value in radians to obtain an

     * angular value in degrees.

     */

    private static final double RADIANS_TO_DEGREES = 57.29577951308232;

    /**

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

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

     * result is NaN.

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

     * same sign as the argument.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

     * @param   a   an angle, in radians.

     * @return  the sine of the argument.

     */

    @HotSpotIntrinsicCandidate

    public static double sin(double a) {

        return StrictMath.sin(a); // default impl. delegates to StrictMath

    }

    /**

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

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

     * result is NaN.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

     * @param   a   an angle, in radians.

     * @return  the cosine of the argument.

     */

    @HotSpotIntrinsicCandidate

    public static double cos(double a) {

        return StrictMath.cos(a); // default impl. delegates to StrictMath

    }

    /**

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

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

     * is NaN.

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

     * same sign as the argument.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

     * @param   a   an angle, in radians.

     * @return  the tangent of the argument.

     */

    @HotSpotIntrinsicCandidate

    public static double tan(double a) {

        return StrictMath.tan(a); // default impl. delegates to StrictMath

    }

    /**

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

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

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

     * than 1, then the result is NaN.

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

     * same sign as the argument.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

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

     * @return  the arc sine of the argument.

     */

    public static double asin(double a) {

        return StrictMath.asin(a); // default impl. delegates to StrictMath

    }

    /**

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

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

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

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

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

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

     * @return  the arc cosine of the argument.

     */

    public static double acos(double a) {

        return StrictMath.acos(a); // default impl. delegates to StrictMath

    }

    /**

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

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

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

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

     * same sign as the argument.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

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

     * @return  the arc tangent of the argument.

     */

    public static double atan(double a) {

        return StrictMath.atan(a); // default impl. delegates to StrictMath

    }

    /**

     * Converts an angle measured in degrees to an approximately

     * equivalent angle measured in radians.  The conversion from

     * degrees to radians is generally inexact.

     *

     * @param   angdeg   an angle, in degrees

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

     *          in radians.

     * @since   1.2

     */

    public static double toRadians(double angdeg) {

        return angdeg * DEGREES_TO_RADIANS;

    }

    /**

     * Converts an angle measured in radians to an approximately

     * equivalent angle measured in degrees.  The conversion from

     * radians to degrees is generally inexact; users should

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

     * equal {@code 0.0}.

     *

     * @param   angrad   an angle, in radians

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

     *          in degrees.

     * @since   1.2

     */

    public static double toDegrees(double angrad) {

        return angrad * RADIANS_TO_DEGREES;

    }

    /**

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

     * {@code double} value.  Special cases:

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

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

     * positive infinity.

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

     * positive zero.</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.

     */

    @HotSpotIntrinsicCandidate

    public static double exp(double a) {

        return StrictMath.exp(a); // default impl. delegates to StrictMath

    }

    /**

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

     * value.  Special cases:

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

     * is NaN.

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

     * positive infinity.

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

     * result is negative infinity.</ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

     * @param   a   a value

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

     *          {@code a}.

     */

    @HotSpotIntrinsicCandidate

    public static double log(double a) {

        return StrictMath.log(a); // default impl. delegates to StrictMath

    }

    /**

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

     * Special cases:

     *

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

     * is NaN.

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

     * positive infinity.

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

     * result is negative infinity.

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

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

     * </ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     * Results must be semi-monotonic.

     *

     * @param   a   a value

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

     * @since 1.5

     */

    @HotSpotIntrinsicCandidate

    public static double log10(double a) {

        return StrictMath.log10(a); // default impl. delegates to StrictMath

    }

    /**

     * Returns the correctly rounded positive square root of a

     * {@code double} value.

     * Special cases:

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

     * is NaN.

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

     * infinity.

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

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

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

     * the true mathematical square root of the argument value.

     *

     * @param   a   a value.

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

     *          If the argument is NaN or less than zero, the result is NaN.

     */

    @HotSpotIntrinsicCandidate

    public static double sqrt(double a) {

        return StrictMath.sqrt(a); // default impl. delegates to StrictMath

                                   // Note that hardware sqrt instructions

                                   // frequently can be directly used by JITs

                                   // and should be much faster than doing

                                   // Math.sqrt in software.

    }

    /**

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

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

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

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

     *

     * Special cases:

     *

     * <ul>

     *

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

     *

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

     * with the same sign as the argument.

     *

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

     * same sign as the argument.

     *

     * </ul>

     *

     * <p>The computed result must be within 1 ulp of the exact result.

     *

     * @param   a   a value.

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

     * @since 1.5

     */

    public static double cbrt(double a) {

        return StrictMath.cbrt(a);

    }

    /**

     * Computes the remainder operation on two arguments as prescribed

     * by the IEEE 754 standard.

     * The remainder value is mathematically equal to

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

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

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

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

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

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

     * Special cases:

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

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

     * result is NaN.

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

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

     *

     * @param   f1   the dividend.

     * @param   f2   the divisor.

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

     *          {@code f2}.

     */

    public static double IEEEremainder(double f1, double f2) {

        return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath

    }

    /**

     * Returns the smallest (closest to negative infinity)

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

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

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

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

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

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

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

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

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

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

     *

     *

     * @param   a   a value.

     * @return  the smallest (closest to negative infinity)

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

     *          the argument and is equal to a mathematical integer.

     */

    public static double ceil(double a) {

        return StrictMath.ceil(a); // default impl. delegates to StrictMath

    }

    /**

     * Returns the largest (closest to positive infinity)

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

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

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

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

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

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

     * the argument.</ul>

     *

     * @param   a   a value.

     * @return  the largest (closest to positive infinity)

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

     *          and is equal to a mathematical integer.

     */

    public static double floor(double a) {

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

    }

    /**

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

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

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

     * equally close, the result is the integer value that is

     * even. Special cases:

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

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

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

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

     *

     * @param   a   a {@code double} value.

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

     *          equal to a mathematical integer.

     */

    public static double rint(double a) {

        return StrictMath.rint(a); // default impl. delegates to StrictMath

    }

    /**

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

     * coordinates ({@code x}, {@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