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	/*
	* Copyright (c) 1994, 2021, 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).
	*
	* You should have received a copy of the GNU General Public License version
	* 2 along with this work; if not, write to the Free Software Foundation,
	* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
	*
	* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 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.
	*
	* <p>Unlike some of the numeric methods of class
	* {@link java.lang.StrictMath StrictMath}, all implementations of the equivalent
	* functions of class {@code Math} are not defined to return the
	* bit-for-bit same results. This relaxation permits
	* better-performing implementations where strict reproducibility is
	* not required.
	*
	* <p>By default many of the {@code Math} methods simply call
	* the equivalent method in {@code StrictMath} for their
	* implementation. Code generators are encouraged to use
	* platform-specific native libraries or microprocessor instructions,
	* where available, to provide higher-performance implementations of
	* {@code Math} methods. Such higher-performance
	* implementations still must conform to the specification for
	* {@code Math}.
	*
	* <p>The quality of implementation specifications concern two
	* properties, accuracy of the returned result and monotonicity of the
	* method. Accuracy of the floating-point {@code Math} methods is
	* measured in terms of <i>ulps</i>, units in the last place. For a
	* given floating-point format, an {@linkplain #ulp(double) ulp} of a
	* specific real number value is the distance between the two
	* floating-point values bracketing that numerical value. When
	* discussing the accuracy of a method as a whole rather than at a
	* specific argument, the number of ulps cited is for the worst-case
	* error at any argument. If a method always has an error less than
	* 0.5 ulps, the method always returns the floating-point number
	* nearest the exact result; such a method is <i>correctly
	* rounded</i>. A correctly rounded method is generally the best a
	* floating-point approximation can be; however, it is impractical for
	* many floating-point methods to be correctly rounded. Instead, for
	* the {@code Math} class, a larger error bound of 1 or 2 ulps is
	* allowed for certain methods. Informally, with a 1 ulp error bound,
	* when the exact result is a representable number, the exact result
	* should be returned as the computed result; otherwise, either of the
	* two floating-point values which bracket the exact result may be
	* returned. For exact results large in magnitude, one of the
	* endpoints of the bracket may be infinite. Besides accuracy at
	* individual arguments, maintaining proper relations between the
	* method at different arguments is also important. Therefore, most
	* methods with more than 0.5 ulp errors are required to be
	* <i>semi-monotonic</i>: whenever the mathematical function is
	* non-decreasing, so is the floating-point approximation, likewise,
	* whenever the mathematical function is non-increasing, so is the
	* floating-point approximation. Not all approximations that have 1
	* ulp accuracy will automatically meet the monotonicity requirements.
	*
	* <p>
	* The platform uses signed two's complement integer arithmetic with
	* int and long primitive types. The developer should choose
	* the primitive type to ensure that arithmetic operations consistently
	* produce correct results, which in some cases means the operations
	* will not overflow the range of values of the computation.
	* The best practice is to choose the primitive type and algorithm to avoid
	* overflow. In cases where the size is {@code int} or {@code long} and
	* overflow errors need to be detected, the methods {@code addExact},
	* {@code subtractExact}, {@code multiplyExact}, {@code toIntExact},
	* {@code incrementExact}, {@code decrementExact} and {@code negateExact}
	* throw an {@code ArithmeticException} when the results overflow.
	* For the arithmetic operations divide and absolute value, overflow
	* occurs only with a specific minimum or maximum value and
	* should be checked against the minimum or maximum as appropriate.
	*
	* <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.
	*
	* @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.
	*/
	@IntrinsicCandidate
	public static double sin(double a) {
	return StrictMath.sin(a); // default impl. delegates to StrictMath
	}

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

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

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

	/**
	* Returns the arc cosine of a value; the returned angle is in the
	* range 0.0 through <i>pi</i>. Special case:
	* <ul><li>If the argument is NaN or its absolute value is greater
	* than 1, then the result is NaN.
	* <li>If the argument is {@code 1.0}, the result is positive zero.
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	* Results must be semi-monotonic.
	*
	* @param a the value whose arc cosine is to be returned.
	* @return the arc cosine of the argument.
	*/
	public static double acos(double a) {
	return StrictMath.acos(a); // default impl. delegates to StrictMath
	}

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

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

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

	/**
	* Returns Euler's number <i>e</i> raised to the power of a
	* {@code double} value. Special cases:
	* <ul><li>If the argument is NaN, the result is NaN.
	* <li>If the argument is positive infinity, then the result is
	* positive infinity.
	* <li>If the argument is negative infinity, then the result is
	* positive zero.
	* <li>If the argument is zero, then the result is {@code 1.0}.
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	* Results must be semi-monotonic.
	*
	* @param a the exponent to raise <i>e</i> to.
	* @return the value <i>e</i><sup>{@code a}</sup>,
	* where <i>e</i> is the base of the natural logarithms.
	*/
	@IntrinsicCandidate
	public static double exp(double a) {
	return StrictMath.exp(a); // default impl. delegates to StrictMath
	}

	/**
	* Returns the natural logarithm (base <i>e</i>) of a {@code double}
	* value. Special cases:
	* <ul><li>If the argument is NaN or less than zero, then the result
	* is NaN.
	* <li>If the argument is positive infinity, then the result is
	* positive infinity.
	* <li>If the argument is positive zero or negative zero, then the
	* result is negative infinity.
	* <li>If the argument is {@code 1.0}, then the result is positive
	* zero.
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	* Results must be semi-monotonic.
	*
	* @param a a value
	* @return the value ln {@code a}, the natural logarithm of
	* {@code a}.
	*/
	@IntrinsicCandidate
	public static double log(double a) {
	return StrictMath.log(a); // default impl. delegates to StrictMath
	}

	/**
	* Returns the base 10 logarithm of a {@code double} value.
	* Special cases:
	*
	* <ul><li>If the argument is NaN or less than zero, then the result
	* is NaN.
	* <li>If the argument is positive infinity, then the result is
	* positive infinity.
	* <li>If the argument is positive zero or negative zero, then the
	* result is negative infinity.
	* <li>If the argument is equal to 10<sup><i>n</i></sup> for
	* integer <i>n</i>, then the result is <i>n</i>. In particular,
	* if the argument is {@code 1.0} (10<sup>0</sup>), then the
	* result is positive zero.
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	* Results must be semi-monotonic.
	*
	* @param a a value
	* @return the base 10 logarithm of {@code a}.
	* @since 1.5
	*/
	@IntrinsicCandidate
	public static double log10(double a) {
	return StrictMath.log10(a); // default impl. delegates to StrictMath
	}

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


	/**
	* Returns the cube root of a {@code double} value. For
	* positive finite {@code x}, {@code cbrt(-x) ==
	* -cbrt(x)}; that is, the cube root of a negative value is
	* the negative of the cube root of that value's magnitude.
	*
	* Special cases:
	*
	* <ul>
	*
	* <li>If the argument is NaN, then the result is NaN.
	*
	* <li>If the argument is infinite, then the result is an infinity
	* with the same sign as the argument.
	*
	* <li>If the argument is zero, then the result is a zero with the
	* same sign as the argument.
	*
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	*
	* @param a a value.
	* @return the cube root of {@code a}.
	* @since 1.5
	*/
	public static double cbrt(double a) {
	return StrictMath.cbrt(a);
	}

	/**
	* Computes the remainder operation on two arguments as prescribed
	* by the IEEE 754 standard.
	* The remainder value is mathematically equal to
	* <code>f1 - f2</code> × <i>n</i>,
	* where <i>n</i> is the mathematical integer closest to the exact
	* mathematical value of the quotient {@code f1/f2}, and if two
	* mathematical integers are equally close to {@code f1/f2},
	* then <i>n</i> is the integer that is even. If the remainder is
	* zero, its sign is the same as the sign of the first argument.
	* Special cases:
	* <ul><li>If either argument is NaN, or the first argument is infinite,
	* or the second argument is positive zero or negative zero, then the
	* result is NaN.
	* <li>If the first argument is finite and the second argument is
	* infinite, then the result is the same as the first argument.</ul>
	*
	* @param f1 the dividend.
	* @param f2 the divisor.
	* @return the remainder when {@code f1} is divided by
	* {@code f2}.
	*/
	public static double IEEEremainder(double f1, double f2) {
	return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
	}

	/**
	* Returns the smallest (closest to negative infinity)
	* {@code double} value that is greater than or equal to the
	* argument and is equal to a mathematical integer. Special cases:
	* <ul><li>If the argument value is already equal to a
	* mathematical integer, then the result is the same as the
	* argument. <li>If the argument is NaN or an infinity or
	* positive zero or negative zero, then the result is the same as
	* the argument. <li>If the argument value is less than zero but
	* greater than -1.0, then the result is negative zero.</ul> Note
	* that the value of {@code Math.ceil(x)} is exactly the
	* value of {@code -Math.floor(-x)}.
	*
	*
	* @param a a value.
	* @return the smallest (closest to negative infinity)
	* floating-point value that is greater than or equal to
	* the argument and is equal to a mathematical integer.
	*/
	@IntrinsicCandidate
	public static double ceil(double a) {
	return StrictMath.ceil(a); // default impl. delegates to StrictMath
	}

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

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

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

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

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

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

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

	/**
	* Returns a {@code double} value with a positive sign, greater
	* than or equal to {@code 0.0} and less than {@code 1.0}.
	* Returned values are chosen pseudorandomly with (approximately)
	* uniform distribution from that range.
	*
	* <p>When this method is first called, it creates a single new
	* pseudorandom-number generator, exactly as if by the expression
	*
	* <blockquote>{@code new java.util.Random()}</blockquote>
	*
	* This new pseudorandom-number generator is used thereafter for
	* all calls to this method and is used nowhere else.
	*
	* <p>This method is properly synchronized to allow correct use by
	* more than one thread. However, if many threads need to generate
	* pseudorandom numbers at a great rate, it may reduce contention
	* for each thread to have its own pseudorandom-number generator.
	*
	* @apiNote
	* As the largest {@code double} value less than {@code 1.0}
	* is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
	* {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
	*
	* <blockquote><pre>{@code
	* double f = Math.random()/Math.nextDown(1.0);
	* double x = x1(1.0 - f) + x2f;
	* }</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 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
	* @since 9
	*/
	@IntrinsicCandidate
	public static long multiplyHigh(long x, long y) {
	if (x < 0 \|\| y < 0) {
	// 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);
	} else {
	// Use Karatsuba technique with two base 2^32 digits.
	long x1 = x >>> 32;
	long y1 = y >>> 32;
	long x2 = x & 0xFFFFFFFFL;
	long y2 = y & 0xFFFFFFFFL;
	long A = x1 * y1;
	long B = x2 * y2;
	long C = (x1 + x2) * (y1 + y2);
	long K = C - A - B;
	return (((B >>> 32) + K) >>> 32) + A;
	}
	}

	/**
	* 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 {@code Integer.MIN_VALUE}.
	* <p>
	* Normal integer division operates under the round to zero rounding mode
	* (truncation). This operation instead acts under the round toward
	* negative infinity (floor) rounding mode.
	* The floor rounding mode gives different results from truncation
	* when the exact 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>
	*
	* @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)
	* {@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 {@code Long.MIN_VALUE}.
	* <p>
	* Normal integer division operates under the round to zero rounding mode
	* (truncation). This operation instead acts under the round toward
	* negative infinity (floor) rounding mode.
	* The floor rounding mode gives different results from truncation
	* when the exact result is 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 int} 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 the
	* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
	* then integer overflow occurs and
	* the result is equal to {@code Long.MIN_VALUE}.
	* <p>
	* Normal integer division operates under the round to zero rounding mode
	* (truncation). This operation instead acts under the round toward
	* negative infinity (floor) rounding mode.
	* The floor rounding mode gives different results from truncation
	* when the exact result is 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 {@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};   and {@code (+4 % +3) == +1}</li>
	* <li>{@code floorMod(-4, -3) == -1};   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};   and {@code (+4 % -3) == +1}</li>
	* <li>{@code floorMod(-4, +3) == +2};   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 mod = x % y;
	// if the signs are different and modulo not zero, adjust result
	if ((mod ^ y) < 0 && mod != 0) {
	mod += y;
	}
	return mod;
	}

	/**
	* Returns the floor modulus of the {@code long} and {@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>
	* For examples, see {@link #floorMod(int, int)}.
	*
	* @param x the dividend
	* @param y the divisor
	* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
	* @throws ArithmeticException if the divisor {@code y} is zero
	* @see #floorDiv(long, int)
	* @since 9
	*/
	public static int floorMod(long x, int y) {
	// Result cannot overflow the range of int.
	return (int)floorMod(x, (long)y);
	}

	/**
	* Returns the floor modulus of the {@code long} arguments.
	* <p>
	* The floor modulus is {@code 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) {
	long mod = x % y;
	// if the signs are different and modulo not zero, adjust result
	if ((x ^ y) < 0 && mod != 0) {
	mod += y;
	}
	return mod;
	}

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

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

	/**
	* Returns the absolute value of a {@code long} value.
	* If the argument is not negative, the argument is returned.
	* If the argument is negative, the negation of the argument is returned.
	*
	* <p>Note that if the argument is equal to the value of {@link
	* Long#MIN_VALUE}, the most negative representable {@code long}
	* value, the result is that same value, which is negative. In
	* contrast, the {@link Math#absExact(long)} method throws an
	* {@code ArithmeticException} for this value.
	*
	* @param a the argument whose absolute value is to be determined
	* @return the absolute value of the argument.
	* @see Math#absExact(long)
	*/
	@IntrinsicCandidate
	public static long abs(long a) {
	return (a < 0) ? -a : a;
	}

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

	/**
	* Returns the absolute value of a {@code float} value.
	* If the argument is not negative, the argument is returned.
	* If the argument is negative, the negation of the argument is returned.
	* Special cases:
	* <ul><li>If the argument is positive zero or negative zero, the
	* result is positive zero.
	* <li>If the argument is infinite, the result is positive infinity.
	* <li>If the argument is NaN, the result is NaN.</ul>
	*
	* @apiNote As implied by the above, one valid implementation of
	* this method is given by the expression below which computes a
	* {@code float} with the same exponent and significand as the
	* argument but with a guaranteed zero sign bit indicating a
	* positive value:<br>
	* {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
	*
	* @param a the argument whose absolute value is to be determined
	* @return the absolute value of the argument.
	*/
	@IntrinsicCandidate
	public static float abs(float a) {
	return (a <= 0.0F) ? 0.0F - a : a;
	}

	/**
	* Returns the absolute value of a {@code double} value.
	* If the argument is not negative, the argument is returned.
	* If the argument is negative, the negation of the argument is returned.
	* Special cases:
	* <ul><li>If the argument is positive zero or negative zero, the result
	* is positive zero.
	* <li>If the argument is infinite, the result is positive infinity.
	* <li>If the argument is NaN, the result is NaN.</ul>
	*
	* @apiNote As implied by the above, one valid implementation of
	* this method is given by the expression below which computes a
	* {@code double} with the same exponent and significand as the
	* argument but with a guaranteed zero sign bit indicating a
	* positive value:<br>
	* {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
	*
	* @param a the argument whose absolute value is to be determined
	* @return the absolute value of the argument.
	*/
	@IntrinsicCandidate
	public static double abs(double a) {
	return (a <= 0.0D) ? 0.0D - a : a;
	}

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

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

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

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

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

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

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

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

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

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

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

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

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

	/**
	* Returns the size of an ulp of the argument. An ulp, unit in
	* the last place, of a {@code double} value is the positive
	* distance between this floating-point value and the {@code
	* double} value next larger in magnitude. Note that for non-NaN
	* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
	*
	* <p>Special Cases:
	* <ul>
	* <li> If the argument is NaN, then the result is NaN.
	* <li> If the argument is positive or negative infinity, then the
	* result is positive infinity.
	* <li> If the argument is positive or negative zero, then the result is
	* {@code Double.MIN_VALUE}.
	* <li> If the argument is ±{@code Double.MAX_VALUE}, then
	* the result is equal to 2<sup>971</sup>.
	* </ul>
	*
	* @param d the floating-point value whose ulp is to be returned
	* @return the size of an ulp of the argument
	* @author Joseph D. Darcy
	* @since 1.5
	*/
	public static double ulp(double d) {
	int exp = getExponent(d);

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

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

	/**
	* Returns the size of an ulp of the argument. An ulp, unit in
	* the last place, of a {@code float} value is the positive
	* distance between this floating-point value and the {@code
	* float} value next larger in magnitude. Note that for non-NaN
	* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
	*
	* <p>Special Cases:
	* <ul>
	* <li> If the argument is NaN, then the result is NaN.
	* <li> If the argument is positive or negative infinity, then the
	* result is positive infinity.
	* <li> If the argument is positive or negative zero, then the result is
	* {@code Float.MIN_VALUE}.
	* <li> If the argument is ±{@code Float.MAX_VALUE}, then
	* the result is equal to 2<sup>104</sup>.
	* </ul>
	*
	* @param f the floating-point value whose ulp is to be returned
	* @return the size of an ulp of the argument
	* @author Joseph D. Darcy
	* @since 1.5
	*/
	public static float ulp(float f) {
	int exp = getExponent(f);

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

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

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

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

	/**
	* Returns the hyperbolic sine of a {@code double} value.
	* The hyperbolic sine of <i>x</i> is defined to be
	* (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
	* where <i>e</i> is {@linkplain Math#E Euler's number}.
	*
	* <p>Special cases:
	* <ul>
	*
	* <li>If the argument is NaN, then the result is NaN.
	*
	* <li>If the argument is infinite, then the result is an infinity
	* with the same sign as the argument.
	*
	* <li>If the argument is zero, then the result is a zero with the
	* same sign as the argument.
	*
	* </ul>
	*
	* <p>The computed result must be within 2.5 ulps of the exact result.
	*
	* @param x The number whose hyperbolic sine is to be returned.
	* @return The hyperbolic sine of {@code x}.
	* @since 1.5
	*/
	public static double sinh(double x) {
	return StrictMath.sinh(x);
	}

	/**
	* Returns the hyperbolic cosine of a {@code double} value.
	* The hyperbolic cosine of <i>x</i> is defined to be
	* (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
	* where <i>e</i> is {@linkplain Math#E Euler's number}.
	*
	* <p>Special cases:
	* <ul>
	*
	* <li>If the argument is NaN, then the result is NaN.
	*
	* <li>If the argument is infinite, then the result is positive
	* infinity.
	*
	* <li>If the argument is zero, then the result is {@code 1.0}.
	*
	* </ul>
	*
	* <p>The computed result must be within 2.5 ulps of the exact result.
	*
	* @param x The number whose hyperbolic cosine is to be returned.
	* @return The hyperbolic cosine of {@code x}.
	* @since 1.5
	*/
	public static double cosh(double x) {
	return StrictMath.cosh(x);
	}

	/**
	* Returns the hyperbolic tangent of a {@code double} value.
	* The hyperbolic tangent of <i>x</i> is defined to be
	* (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + 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 ±1, correctly signed ±{@code 1.0} should
	* be returned.
	*
	* @param x The number whose hyperbolic tangent is to be returned.
	* @return The hyperbolic tangent of {@code x}.
	* @since 1.5
	*/
	public static double tanh(double x) {
	return StrictMath.tanh(x);
	}

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

	/**
	* Returns <i>e</i><sup>x</sup> -1. Note that for values of
	* <i>x</i> near 0, the exact sum of
	* {@code expm1(x)} + 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> - 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> -1.
	* @return the value <i>e</i><sup>{@code x}</sup> - 1.
	* @since 1.5
	*/
	public static double expm1(double x) {
	return StrictMath.expm1(x);
	}

	/**
	* Returns the natural logarithm of the sum of the argument and 1.
	* Note that for small values {@code x}, the result of
	* {@code log1p(x)} is much closer to the true result of ln(1
	* + {@code x}) than the floating-point evaluation of
	* {@code log(1.0+x)}.
	*
	* <p>Special cases:
	*
	* <ul>
	*
	* <li>If the argument is NaN or less than -1, then the result is
	* NaN.
	*
	* <li>If the argument is positive infinity, then the result is
	* positive infinity.
	*
	* <li>If the argument is negative one, then the result is
	* negative infinity.
	*
	* <li>If the argument is zero, then the result is a zero with the
	* same sign as the argument.
	*
	* </ul>
	*
	* <p>The computed result must be within 1 ulp of the exact result.
	* Results must be semi-monotonic.
	*
	* @param x a value
	* @return the value ln({@code x} + 1), the natural
	* log of {@code x} + 1
	* @since 1.5
	*/
	public static double log1p(double x) {
	return StrictMath.log1p(x);
	}

	/**
	* Returns the first floating-point argument with the sign of the
	* second floating-point argument. Note that unlike the {@link
	* StrictMath#copySign(double, double) StrictMath.copySign}
	* method, this method does not require NaN {@code sign}
	* arguments to be treated as positive values; implementations are
	* permitted to treat some NaN arguments as positive and other NaN
	* arguments as negative to allow greater performance.
	*
	* @param magnitude the parameter providing the magnitude of the result
	* @param sign the parameter providing the sign of the result
	* @return a value with the magnitude of {@code magnitude}
	* and the sign of {@code sign}.
	* @since 1.6
	*/
	@IntrinsicCandidate
	public static double copySign(double magnitude, double sign) {
	return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
	(DoubleConsts.SIGN_BIT_MASK)) \|
	(Double.doubleToRawLongBits(magnitude) &
	(DoubleConsts.EXP_BIT_MASK \|
	DoubleConsts.SIGNIF_BIT_MASK)));
	}

	/**
	* Returns the first floating-point argument with the sign of the
	* second floating-point argument. Note that unlike the {@link
	* StrictMath#copySign(float, float) StrictMath.copySign}
	* method, this method does not require NaN {@code sign}
	* arguments to be treated as positive values; implementations are
	* permitted to treat some NaN arguments as positive and other NaN
	* arguments as negative to allow greater performance.
	*
	* @param magnitude the parameter providing the magnitude of the result
	* @param sign the parameter providing the sign of the result
	* @return a value with the magnitude of {@code magnitude}
	* and the sign of {@code sign}.
	* @since 1.6
	*/
	@IntrinsicCandidate
	public static float copySign(float magnitude, float sign) {
	return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
	(FloatConsts.SIGN_BIT_MASK)) \|
	(Float.floatToRawIntBits(magnitude) &
	(FloatConsts.EXP_BIT_MASK \|
	FloatConsts.SIGNIF_BIT_MASK)));
	}

	/**
	* Returns the unbiased exponent used in the representation of a
	* {@code float}. Special cases:
	*
	* <ul>
	* <li>If the argument is NaN or infinite, then the result is
	* {@link Float#MAX_EXPONENT} + 1.
	* <li>If the argument is zero or subnormal, then the result is
	* {@link Float#MIN_EXPONENT} -1.
	* </ul>
	* @param f a {@code float} value
	* @return the unbiased exponent of the argument
	* @since 1.6
	*/
	public static int getExponent(float f) {
	/*
	* Bitwise convert f to integer, mask out exponent bits, shift
	* to the right and then subtract out float's bias adjust to
	* get true exponent value
	*/
	return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
	(FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
	}

	/**
	* Returns the unbiased exponent used in the representation of a
	* {@code double}. Special cases:
	*
	* <ul>
	* <li>If the argument is NaN or infinite, then the result is
	* {@link Double#MAX_EXPONENT} + 1.
	* <li>If the argument is zero or subnormal, then the result is
	* {@link Double#MIN_EXPONENT} -1.
	* </ul>
	* @param d a {@code double} value
	* @return the unbiased exponent of the argument
	* @since 1.6
	*/
	public static int getExponent(double d) {
	/*
	* Bitwise convert d to long, mask out exponent bits, shift
	* to the right and then subtract out double's bias adjust to
	* get true exponent value.
	*/
	return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
	(DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
	}

	/**
	* Returns the floating-point number adjacent to the first
	* argument in the direction of the second argument. If both
	* arguments compare as equal the second argument is returned.
	*
	* <p>
	* Special cases:
	* <ul>
	* <li> If either argument is a NaN, then NaN is returned.
	*
	* <li> If both arguments are signed zeros, {@code direction}
	* is returned unchanged (as implied by the requirement of
	* returning the second argument if the arguments compare as
	* equal).
	*
	* <li> If {@code start} is
	* ±{@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 ±
	* {@link Double#MAX_VALUE} and {@code direction} has a
	* value such that the result should have a larger magnitude, an
	* infinity with same sign as {@code start} is returned.
	* </ul>
	*
	* @param start starting floating-point value
	* @param direction value indicating which of
	* {@code start}'s neighbors or {@code start} should
	* be returned
	* @return The floating-point number adjacent to {@code start} in the
	* direction of {@code direction}.
	* @since 1.6
	*/
	public static double nextAfter(double start, double direction) {
	/*
	* The cases:
	*
	* nextAfter(+infinity, 0) == MAX_VALUE
	* nextAfter(+infinity, +infinity) == +infinity
	* nextAfter(-infinity, 0) == -MAX_VALUE
	* nextAfter(-infinity, -infinity) == -infinity
	*
	* are naturally handled without any additional testing
	*/

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

	// Branch to descending case first as it is more costly than ascending
	// case due to start != 0.0d conditional.
	if (start > direction) { // descending
	if (start != 0.0d) {
	final long transducer = Double.doubleToRawLongBits(start);
	return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
	} else { // start == 0.0d && direction < 0.0d
	return -Double.MIN_VALUE;
	}
	} else if (start < direction) { // ascending
	// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
	// then bitwise convert start to integer.
	final long transducer = Double.doubleToRawLongBits(start + 0.0d);
	return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
	} else if (start == direction) {
	return direction;
	} else { // isNaN(start) \|\| isNaN(direction)
	return start + direction;
	}
	}

	/**
	* Returns the floating-point number adjacent to the first
	* argument in the direction of the second argument. If both
	* arguments compare as equal a value equivalent to the second argument
	* is returned.
	*
	* <p>
	* Special cases:
	* <ul>
	* <li> If either argument is a NaN, then NaN is returned.
	*
	* <li> If both arguments are signed zeros, a value equivalent
	* to {@code direction} is returned.
	*
	* <li> If {@code start} is
	* ±{@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 ±
	* {@link Float#MAX_VALUE} and {@code direction} has a
	* value such that the result should have a larger magnitude, an
	* infinity with same sign as {@code start} is returned.
	* </ul>
	*
	* @param start starting floating-point value
	* @param direction value indicating which of
	* {@code start}'s neighbors or {@code start} should
	* be returned
	* @return The floating-point number adjacent to {@code start} in the
	* direction of {@code direction}.
	* @since 1.6
	*/
	public static float nextAfter(float start, double direction) {
	/*
	* The cases:
	*
	* nextAfter(+infinity, 0) == MAX_VALUE
	* nextAfter(+infinity, +infinity) == +infinity
	* nextAfter(-infinity, 0) == -MAX_VALUE
	* nextAfter(-infinity, -infinity) == -infinity
	*
	* are naturally handled without any additional testing
	*/

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

	// Branch to descending case first as it is more costly than ascending
	// case due to start != 0.0f conditional.
	if (start > direction) { // descending
	if (start != 0.0f) {
	final int transducer = Float.floatToRawIntBits(start);
	return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
	} else { // start == 0.0f && direction < 0.0f
	return -Float.MIN_VALUE;
	}
	} else if (start < direction) { // ascending
	// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
	// then bitwise convert start to integer.
	final int transducer = Float.floatToRawIntBits(start + 0.0f);
	return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
	} else if (start == direction) {
	return (float)direction;
	} else { // isNaN(start) \|\| isNaN(direction)
	return start + (float)direction;
	}
	}

	/**
	* Returns the floating-point value adjacent to {@code d} in
	* the direction of positive infinity. This method is
	* semantically equivalent to {@code nextAfter(d,
	* Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
	* implementation may run faster than its equivalent
	* {@code nextAfter} call.
	*
	* <p>Special Cases:
	* <ul>
	* <li> If the argument is NaN, the result is NaN.
	*
	* <li> If the argument is positive infinity, the result is
	* positive infinity.
	*
	* <li> If the argument is zero, the result is
	* {@link Double#MIN_VALUE}
	*
	* </ul>
	*
	* @param d starting floating-point value
	* @return The adjacent floating-point value closer to positive
	* infinity.
	* @since 1.6
	*/
	public static double nextUp(double d) {
	// Use a single conditional and handle the likely cases first.
	if (d < Double.POSITIVE_INFINITY) {
	// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
	final long transducer = Double.doubleToRawLongBits(d + 0.0D);
	return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
	} else { // d is NaN or +Infinity
	return d;
	}
	}

	/**
	* Returns the floating-point value adjacent to {@code f} in
	* the direction of positive infinity. This method is
	* semantically equivalent to {@code nextAfter(f,
	* Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
	* implementation may run faster than its equivalent
	* {@code nextAfter} call.
	*
	* <p>Special Cases:
	* <ul>
	* <li> If the argument is NaN, the result is NaN.
	*
	* <li> If the argument is positive infinity, the result is
	* positive infinity.
	*
	* <li> If the argument is zero, the result is
	* {@link Float#MIN_VALUE}
	*
	* </ul>
	*
	* @param f starting floating-point value
	* @return The adjacent floating-point value closer to positive
	* infinity.
	* @since 1.6
	*/
	public static float nextUp(float f) {
	// Use a single conditional and handle the likely cases first.
	if (f < Float.POSITIVE_INFINITY) {
	// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
	final int transducer = Float.floatToRawIntBits(f + 0.0F);
	return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
	} else { // f is NaN or +Infinity
	return f;
	}
	}

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

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

	/**
	* Returns {@code d} × 2<sup>{@code scaleFactor}</sup>
	* rounded as if performed by a single correctly rounded
	* floating-point multiply. If the exponent of the result is
	* between {@link Double#MIN_EXPONENT} and {@link
	* Double#MAX_EXPONENT}, the answer is calculated exactly. If the
	* exponent of the result would be larger than {@code
	* Double.MAX_EXPONENT}, an infinity is returned. Note that if
	* the result is subnormal, precision may be lost; that is, when
	* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
	* -n)} may not equal <i>x</i>. When the result is non-NaN, the
	* result has the same sign as {@code d}.
	*
	* <p>Special cases:
	* <ul>
	* <li> If the first argument is NaN, NaN is returned.
	* <li> If the first argument is infinite, then an infinity of the
	* same sign is returned.
	* <li> If the first argument is zero, then a zero of the same
	* sign is returned.
	* </ul>
	*
	* @param d number to be scaled by a power of two.
	* @param scaleFactor power of 2 used to scale {@code d}
	* @return {@code d} × 2<sup>{@code scaleFactor}</sup>
	* @since 1.6
	*/
	public static double scalb(double d, int scaleFactor) {
	/*
	* When scaling up, it does not matter what order the
	* multiply-store operations are done; the result will be
	* finite or overflow regardless of the operation ordering.
	* However, to get the correct result when scaling down, a
	* particular ordering must be used.
	*
	* When scaling down, the multiply-store operations are
	* sequenced so that it is not possible for two consecutive
	* multiply-stores to return subnormal results. If one
	* multiply-store result is subnormal, the next multiply will
	* round it away to zero. This is done by first multiplying
	* by 2 ^ (scaleFactor % n) and then multiplying several
	* times by 2^n as needed where n is the exponent of number
	* that is a convenient power of two. In this way, at most one
	* real rounding error occurs.
	*/

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

	// Make sure scaling factor is in a reasonable range

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

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

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

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

	/**
	* Returns {@code f} × 2<sup>{@code scaleFactor}</sup>
	* rounded as if performed by a single correctly rounded
	* floating-point multiply. If the exponent of the result is
	* between {@link Float#MIN_EXPONENT} and {@link
	* Float#MAX_EXPONENT}, the answer is calculated exactly. If the
	* exponent of the result would be larger than {@code
	* Float.MAX_EXPONENT}, an infinity is returned. Note that if the
	* result is subnormal, precision may be lost; that is, when
	* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
	* -n)} may not equal <i>x</i>. When the result is non-NaN, the
	* result has the same sign as {@code f}.
	*
	* <p>Special cases:
	* <ul>
	* <li> If the first argument is NaN, NaN is returned.
	* <li> If the first argument is infinite, then an infinity of the
	* same sign is returned.
	* <li> If the first argument is zero, then a zero of the same
	* sign is returned.
	* </ul>
	*
	* @param f number to be scaled by a power of two.
	* @param scaleFactor power of 2 used to scale {@code f}
	* @return {@code f} × 2<sup>{@code scaleFactor}</sup>
	* @since 1.6
	*/
	public static float scalb(float f, int scaleFactor) {
	// magnitude of a power of two so large that scaling a finite
	// nonzero value by it would be guaranteed to over or
	// underflow; due to rounding, scaling down takes an
	// additional power of two which is reflected here
	final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
	FloatConsts.SIGNIFICAND_WIDTH + 1;

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

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

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

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

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

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