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products/Sources/formale Sprachen/JAVA/Openjdk/src/java.base/share/classes/java/lang/ (Sun/Oracle ^©) Datei vom 13.11.2022 mit Größe 127 kB

Quelle Math.java Sprache: JAVA

/*
* Copyright (c) 1994, 2022, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
* by Oracle in the LICENSE file that accompanied this code.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
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* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
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package java.lang;

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

/**
* The class {@code Math} contains methods for performing basic
* numeric operations such as the elementary exponential, logarithm,
* square root, and trigonometric functions.
*
* Unlike some of the numeric methods of class
* {@link java.lang.StrictMath StrictMath}, all implementations of the equivalent
* functions of class {@code Math} are not defined to return the
* bit-for-bit same results. This relaxation permits
* better-performing implementations where strict reproducibility is
* not required.
*
* 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}.
*
* 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 ulps, 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 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 {@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
* 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 {@code int} or {@code long} and
* overflow errors need to be detected, the methods whose names end with
* {@code Exact} throw an {@code ArithmeticException} when the results overflow.
*
* <h2><a id=Ieee754RecommendedOps>IEEE 754 Recommended
* Operations</a></h2>
*
* The 2019 revision of the IEEE 754 floating-point standard includes
* a section of recommended operations and the semantics of those
* operations if they are included in a programming environment. The
* recommended operations present in this class include {@link sin
* sin}, {@link cos cos}, {@link tan tan}, {@link asin asin}, {@link
* acos acos}, {@link atan atan}, {@link exp exp}, {@link expm1
* expm1}, {@link log log}, {@link log10 log10}, {@link log1p log1p},
* {@link sinh sinh}, {@link cosh cosh}, {@link tanh tanh}, {@link
* hypot hypot}, and {@link pow pow}. (The {@link sqrt sqrt}
* operation is a required part of IEEE 754 from a different section
* of the standard.) The special case behavior of the recommended
* operations generally follows the guidance of the IEEE 754
* standard. However, the {@code pow} method defines different
* behavior for some arguments, as noted in its {@linkplain pow
* specification}. The IEEE 754 standard defines its operations to be
* correctly rounded, which is a more stringent quality of
* implementation condition than required for most of the methods in
* question that are also included in this class.
*
* @see <a href="https://standards.ieee.org/ieee/754/6210/">
* <cite>IEEE Standard for Floating-Point Arithmetic</cite></a>
*
* @author Joseph D. Darcy
* @since 1.0
*/

public final class Math {

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

 /**
 * The {@code double} value that is closer than any other to
 * e, the base of the natural logarithms.
 */
 public static final double E = 2.718281828459045;

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

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

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

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

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

 /**
 * Returns the trigonometric cosine of an angle. Special cases:
 * <ul><li>If the argument is NaN or an infinity, then the
 * result is NaN.
 * <li>If the argument is zero, then the result is {@code 1.0}.
 *</ul>
 *
 * The computed result must be within 1 ulp of the exact result.
 * Results must be semi-monotonic.
 *
 * @param a an angle, in radians.
 * @return the cosine of the argument.
 */
 @IntrinsicCandidate
 public static double cos(double a) {
 return StrictMath.cos(a); // default impl. delegates to StrictMath
 }

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

 /**
 * Returns the arc sine of a value; the returned angle is in the
 * range -pi/2 through pi/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>
 *
 * 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:
 * <ul><li>If the argument is NaN or its absolute value is greater
 * than 1, then the result is NaN.
 * <li>If the argument is {@code 1.0}, the result is positive zero.
 * </ul>
 *
 * 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:
 * <ul><li>If the argument is NaN, then the result is NaN.
 * <li>If the argument is zero, then the result is a zero with the
 * same sign as the argument.
 * <li>If the argument is {@linkplain Double#isInfinite infinite},
 * then the result is the closest value to pi/2 with the
 * same sign as the input.
 * </ul>
 *
 * 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
 * not 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 e raised to the power of a
 * {@code double} value. Special cases:
 * <ul><li>If the argument is NaN, the result is NaN.
 * <li>If the argument is positive infinity, then the result is
 * positive infinity.
 * <li>If the argument is negative infinity, then the result is
 * positive zero.
 * <li>If the argument is zero, then the result is {@code 1.0}.
 * </ul>
 *
 * The computed result must be within 1 ulp of the exact result.
 * Results must be semi-monotonic.
 *
 * @param a the exponent to raise e to.
 * @return the value e{@code a},
 * where e is the base of the natural logarithms.
 */
 @IntrinsicCandidate
 public static double exp(double a) {
 return StrictMath.exp(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the natural logarithm (base e) of a {@code double}
 * value. Special cases:
 * <ul><li>If the argument is NaN or less than zero, then the result
 * is NaN.
 * <li>If the argument is positive infinity, then the result is
 * positive infinity.
 * <li>If the argument is positive zero or negative zero, then the
 * result is negative infinity.
 * <li>If the argument is {@code 1.0}, then the result is positive
 * zero.
 * </ul>
 *
 * 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}.
 */
 @IntrinsicCandidate
 public static double log(double a) {
 return StrictMath.log(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the base 10 logarithm of a {@code double} value.
 * Special cases:
 *
 * <ul><li>If the argument is NaN or less than zero, then the result
 * is NaN.
 * <li>If the argument is positive infinity, then the result is
 * positive infinity.
 * <li>If the argument is positive zero or negative zero, then the
 * result is negative infinity.
 * <li>If the argument is equal to 10n for
 * integer n, then the result is n. In particular,
 * if the argument is {@code 1.0} (100), then the
 * result is positive zero.
 * </ul>
 *
 * The computed result must be within 1 ulp of the exact result.
 * Results must be semi-monotonic.
 *
 * @param a a value
 * @return the base 10 logarithm of {@code a}.
 * @since 1.5
 */
 @IntrinsicCandidate
 public static double log10(double a) {
 return StrictMath.log10(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the correctly rounded positive square root of a
 * {@code double} value.
 * Special cases:
 * <ul><li>If the argument is NaN or less than zero, then the result
 * is NaN.
 * <li>If the argument is positive infinity, then the result is positive
 * infinity.
 * <li>If the argument is positive zero or negative zero, then the
 * result is the same as the argument.</ul>
 * Otherwise, the result is the {@code double} value closest to
 * the true mathematical square root of the argument value.
 *
 * @apiNote
 * This method corresponds to the squareRoot operation defined in
 * IEEE 754.
 *
 * @param a a value.
 * @return the positive square root of {@code a}.
 * If the argument is NaN or less than zero, the result is NaN.
 */
 @IntrinsicCandidate
 public static double sqrt(double a) {
 return StrictMath.sqrt(a); // default impl. delegates to StrictMath
 // Note that hardware sqrt instructions
 // frequently can be directly used by JITs
 // and should be much faster than doing
 // Math.sqrt in software.
 }

 /**
 * Returns the cube root of a {@code double} value. For
 * positive finite {@code x}, {@code cbrt(-x) ==
 * -cbrt(x)}; that is, the cube root of a negative value is
 * the negative of the cube root of that value's magnitude.
 *
 * Special cases:
 *
 * <ul>
 *
 * <li>If the argument is NaN, then the result is NaN.
 *
 * <li>If the argument is infinite, then the result is an infinity
 * with the same sign as the argument.
 *
 * <li>If the argument is zero, then the result is a zero with the
 * same sign as the argument.
 *
 * </ul>
 *
 * 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 - f2</code> × n,
 * where n 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 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:
 * <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)}.
 *
 * @apiNote
 * This method corresponds to the roundToIntegralTowardPositive
 * operation defined in IEEE 754.
 *
 * @param a a value.
 * @return the smallest (closest to negative infinity)
 * floating-point value that is greater than or equal to
 * the argument and is equal to a mathematical integer.
 */
 @IntrinsicCandidate
 public static double ceil(double a) {
 return StrictMath.ceil(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the largest (closest to positive infinity)
 * {@code double} value that is less than or equal to the
 * argument and is equal to a mathematical integer. Special cases:
 * <ul><li>If the argument value is already equal to a
 * mathematical integer, then the result is the same as the
 * argument. <li>If the argument is NaN or an infinity or
 * positive zero or negative zero, then the result is the same as
 * the argument.</ul>
 *
 * @apiNote
 * This method corresponds to the roundToIntegralTowardNegative
 * operation defined in IEEE 754.
 *
 * @param a a value.
 * @return the largest (closest to positive infinity)
 * floating-point value that less than or equal to the argument
 * and is equal to a mathematical integer.
 */
 @IntrinsicCandidate
 public static double floor(double a) {
 return StrictMath.floor(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the {@code double} value that is closest in value
 * to the argument and is equal to a mathematical integer. If two
 * {@code double} values that are mathematical integers are
 * equally close, the result is the integer value that is
 * even. Special cases:
 * <ul><li>If the argument value is already equal to a mathematical
 * integer, then the result is the same as the argument.
 * <li>If the argument is NaN or an infinity or positive zero or negative
 * zero, then the result is the same as the argument.</ul>
 *
 * @apiNote
 * This method corresponds to the roundToIntegralTiesToEven
 * operation defined in IEEE 754.
 *
 * @param a a {@code double} value.
 * @return the closest floating-point value to {@code a} that is
 * equal to a mathematical integer.
 */
 @IntrinsicCandidate
 public static double rint(double a) {
 return StrictMath.rint(a); // default impl. delegates to StrictMath
 }

 /**
 * Returns the angle theta from the conversion of rectangular
 * coordinates ({@code x}, {@code y}) to polar
 * coordinates (r, theta).
 * This method computes the phase theta by computing an arc tangent
 * of {@code y/x} in the range of -pi to pi. 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 pi.
 * <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 -pi.
 * <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 pi/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 -pi/2.
 * <li>If both arguments are positive infinity, then the result is the
 * {@code double} value closest to pi/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*pi/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 -pi/4.
 * <li>If both arguments are negative infinity, then the result is the
 * {@code double} value closest to -3*pi/4.</ul>
 *
 * The computed result must be within 2 ulps of the exact result.
 * Results must be semi-monotonic.
 *
 * @apiNote
 * For y with a positive sign and finite nonzero
 * x, the exact mathematical value of {@code atan2} is
 * equal to:
 * <ul>
 * <li>If x {@literal >} 0, atan(abs(y/x))
 * <li>If x {@literal <} 0, π - atan(abs(y/x))
 * </ul>
 *
 * @param y the ordinate coordinate
 * @param x the abscissa coordinate
 * @return the theta component of the point
 * (r, theta)
 * in polar coordinates that corresponds to the point
 * (x, y) in Cartesian coordinates.
 */
 @IntrinsicCandidate
 public static double atan2(double y, double x) {
 return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
 }

 /**
 * Returns the value of the first argument raised to the power of the
 * second argument. Special cases:
 *
 * <ul><li>If the second argument is positive or negative zero, then the
 * result is 1.0.
 * <li>If the second argument is 1.0, then the result is the same as the
 * first argument.
 * <li>If the second argument is NaN, then the result is NaN.
 * <li>If the first argument is NaN and the second argument is nonzero,
 * then the result is NaN.
 *
 * <li>If
 * <ul>
 * <li>the absolute value of the first argument is greater than 1
 * and the second argument is positive infinity, or
 * <li>the absolute value of the first argument is less than 1 and
 * the second argument is negative infinity,
 * </ul>
 * then the result is positive infinity.
 *
 * <li>If
 * <ul>
 * <li>the absolute value of the first argument is greater than 1 and
 * the second argument is negative infinity, or
 * <li>the absolute value of the
 * first argument is less than 1 and the second argument is positive
 * infinity,
 * </ul>
 * then the result is positive zero.
 *
 * <li>If the absolute value of the first argument equals 1 and the
 * second argument is infinite, then the result is NaN.
 *
 * <li>If
 * <ul>
 * <li>the first argument is positive zero and the second argument
 * is greater than zero, or
 * <li>the first argument is positive infinity and the second
 * argument is less than zero,
 * </ul>
 * then the result is positive zero.
 *
 * <li>If
 * <ul>
 * <li>the first argument is positive zero and the second argument
 * is less than zero, or
 * <li>the first argument is positive infinity and the second
 * argument is greater than zero,
 * </ul>
 * then the result is positive infinity.
 *
 * <li>If
 * <ul>
 * <li>the first argument is negative zero and the second argument
 * is greater than zero but not a finite odd integer, or
 * <li>the first argument is negative infinity and the second
 * argument is less than zero but not a finite odd integer,
 * </ul>
 * then the result is positive zero.
 *
 * <li>If
 * <ul>
 * <li>the first argument is negative zero and the second argument
 * is a positive finite odd integer, or
 * <li>the first argument is negative infinity and the second
 * argument is a negative finite odd integer,
 * </ul>
 * then the result is negative zero.
 *
 * <li>If
 * <ul>
 * <li>the first argument is negative zero and the second argument
 * is less than zero but not a finite odd integer, or
 * <li>the first argument is negative infinity and the second
 * argument is greater than zero but not a finite odd integer,
 * </ul>
 * then the result is positive infinity.
 *
 * <li>If
 * <ul>
 * <li>the first argument is negative zero and the second argument
 * is a negative finite odd integer, or
 * <li>the first argument is negative infinity and the second
 * argument is a positive finite odd integer,
 * </ul>
 * then the result is negative infinity.
 *
 * <li>If the first argument is finite and less than zero
 * <ul>
 * <li> if the second argument is a finite even integer, the
 * result is equal to the result of raising the absolute value of
 * the first argument to the power of the second argument
 *
 * <li>if the second argument is a finite odd integer, the result
 * is equal to the negative of the result of raising the absolute
 * value of the first argument to the power of the second
 * argument
 *
 * <li>if the second argument is finite and not an integer, then
 * the result is NaN.
 * </ul>
 *
 * <li>If both arguments are integers, then the result is exactly equal
 * to the mathematical result of raising the first argument to the power
 * of the second argument if that result can in fact be represented
 * exactly as a {@code double} value.</ul>
 *
 * (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.)
 *
 * The computed result must be within 1 ulp of the exact result.
 * Results must be semi-monotonic.
 *
 * @apiNote
 * The special cases definitions of this method differ from the
 * special case definitions of the IEEE 754 recommended {@code
 * pow} operation for ±{@code 1.0} raised to an infinite
 * power. This method treats such cases as indeterminate and
 * specifies a NaN is returned. The IEEE 754 specification treats
 * the infinite power as a large integer (large-magnitude
 * floating-point numbers are numerically integers, specifically
 * even integers) and therefore specifies {@code 1.0} be returned.
 *
 * @param a the base.
 * @param b the exponent.
 * @return the value {@code a}{@code b}.
 */
 @IntrinsicCandidate
 public static double pow(double a, double b) {
 return StrictMath.pow(a, b); // default impl. delegates to StrictMath
 }

 /**
 * Returns the closest {@code int} to the argument, with ties
 * rounding to positive infinity.
 *
 * 
 * Special cases:
 * <ul><li>If the argument is NaN, the result is 0.
 * <li>If the argument is negative infinity or any value less than or
 * equal to the value of {@code Integer.MIN_VALUE}, the result is
 * equal to the value of {@code Integer.MIN_VALUE}.
 * <li>If the argument is positive infinity or any value greater than or
 * equal to the value of {@code Integer.MAX_VALUE}, the result is
 * equal to the value of {@code Integer.MAX_VALUE}.</ul>
 *
 * @param a a floating-point value to be rounded to an integer.
 * @return the value of the argument rounded to the nearest
 * {@code int} value.
 * @see java.lang.Integer#MAX_VALUE
 * @see java.lang.Integer#MIN_VALUE
 */
 @IntrinsicCandidate
 public static int round(float a) {
 int intBits = Float.floatToRawIntBits(a);
 int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
 >> (FloatConsts.SIGNIFICAND_WIDTH - 1);
 int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
 + FloatConsts.EXP_BIAS) - biasedExp;
 if ((shift & -32) == 0) { // shift >= 0 && shift < 32
 // a is a finite number such that pow(2,-32) <= ulp(a) < 1
 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
 | (FloatConsts.SIGNIF_BIT_MASK + 1));
 if (intBits < 0) {
 r = -r;
 }
 // In the comments below each Java expression evaluates to the value
 // the corresponding mathematical expression:
 // (r) evaluates to a / ulp(a)
 // (r >> shift) evaluates to floor(a * 2)
 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 return ((r >> shift) + 1) >> 1;
 } else {
 // a is either
 // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 // - an infinity or NaN
 return (int) a;
 }
 }

 /**
 * Returns the closest {@code long} to the argument, with ties
 * rounding to positive infinity.
 *
 * Special cases:
 * <ul><li>If the argument is NaN, the result is 0.
 * <li>If the argument is negative infinity or any value less than or
 * equal to the value of {@code Long.MIN_VALUE}, the result is
 * equal to the value of {@code Long.MIN_VALUE}.
 * <li>If the argument is positive infinity or any value greater than or
 * equal to the value of {@code Long.MAX_VALUE}, the result is
 * equal to the value of {@code Long.MAX_VALUE}.</ul>
 *
 * @param a a floating-point value to be rounded to a
 * {@code long}.
 * @return the value of the argument rounded to the nearest
 * {@code long} value.
 * @see java.lang.Long#MAX_VALUE
 * @see java.lang.Long#MIN_VALUE
 */
 @IntrinsicCandidate
 public static long round(double a) {
 long longBits = Double.doubleToRawLongBits(a);
 long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
 >> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
 long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
 + DoubleConsts.EXP_BIAS) - biasedExp;
 if ((shift & -64) == 0) { // shift >= 0 && shift < 64
 // a is a finite number such that pow(2,-64) <= ulp(a) < 1
 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
 | (DoubleConsts.SIGNIF_BIT_MASK + 1));
 if (longBits < 0) {
 r = -r;
 }
 // In the comments below each Java expression evaluates to the value
 // the corresponding mathematical expression:
 // (r) evaluates to a / ulp(a)
 // (r >> shift) evaluates to floor(a * 2)
 // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
 // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
 return ((r >> shift) + 1) >> 1;
 } else {
 // a is either
 // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
 // - an infinity or NaN
 return (long) a;
 }
 }

 private static final class RandomNumberGeneratorHolder {
 static final Random randomNumberGenerator = new Random();
 }

 /**
 * Returns a {@code double} value with a positive sign, greater
 * than or equal to {@code 0.0} and less than {@code 1.0}.
 * Returned values are chosen pseudorandomly with (approximately)
 * uniform distribution from that range.
 *
 * 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.
 *
 * This method is properly synchronized to allow correct use by
 * more than one thread. However, if many threads need to generate
 * pseudorandom numbers at a great rate, it may reduce contention
 * for each thread to have its own pseudorandom-number generator.
 *
 * @apiNote
 * As the largest {@code double} value less than {@code 1.0}
 * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
 * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
 *
 * <blockquote><pre>{@code
 * double f = Math.random()/Math.nextDown(1.0);
 * double x = x1*(1.0 - f) + x2*f;
 * }</pre></blockquote>
 *
 * @return a pseudorandom {@code double} greater than or equal
 * to {@code 0.0} and less than {@code 1.0}.
 * @see #nextDown(double)
 * @see Random#nextDouble()
 */
 public static double random() {
 return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
 }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 return a + 1;
 }

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

 return a + 1L;
 }

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

 return a - 1;
 }

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

 return a - 1L;
 }

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

 return -a;
 }

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

 return -a;
 }

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

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

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

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

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

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

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

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

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

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

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

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

 /**
 * Returns the smallest (closest to negative infinity)
 * {@code int} value that is greater than or equal to the algebraic quotient.
 * There is one special case: if the dividend is
 * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
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