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/*
* Copyright (c) 1994, 2026, 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
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
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;
import static java.lang.Double.*;
/**
* 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 <dfn>{@index ulp}s</dfn>, {@index "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
* <dfn>correctly rounded</dfn>. A {@index "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 <dfn>{@index "semi-monotonic"}</dfn>: 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
* {@code int} and {@code long} primitive types. The developer should
* choose the primitive type to ensure that arithmetic operations
* consistently produce correct results, which in some cases means the
* operations will not overflow the range of values of the
* computation. The best practice is to choose the primitive type and
* algorithm to avoid overflow. In cases where the size is {@code int}
* or {@code long} and overflow errors need to be detected, the
* methods whose names end with {@code Exact} throw an {@code
* ArithmeticException} when the results overflow.
*
* <h2><a id=Ieee754RecommendedOps>IEEE 754 Recommended
* Operations</a></h2>
*
* The 2019 revision of the IEEE 754 floating-point standard includes
* a section of recommended operations and the semantics of those
* operations if they are included in a programming environment. The
* recommended operations present in this class include {@link sin
* sin}, {@link cos cos}, {@link tan tan}, {@link asin asin}, {@link
* acos acos}, {@link atan atan}, {@link exp exp}, {@link expm1
* expm1}, {@link log log}, {@link log10 log10}, {@link log1p log1p},
* {@link sinh sinh}, {@link cosh cosh}, {@link tanh tanh}, {@link asinh asinh},
* {@link acosh acosh}, {@link atanh atanh}, {@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.
*
* @spec https://standards.ieee.org/ieee/754/6210/
* IEEE Standard for Floating-Point Arithmetic
*
* @since 1.0
*/
public final class Math {
/**
* Don't let anyone instantiate this class.
*/
private Math() {}
/**
* The {@code double} value that is closer than any other to
* <i>e</i>, the base of the natural logarithms.
*/
public static final double E = 2.718281828459045;
/**
* The {@code double} value that is closer than any other to
* <i>pi</i> (π), the ratio of the circumference of a circle to
* its diameter.
*/
public static final double PI = 3.141592653589793;
/**
* The {@code double} value that is closer than any other to
* <i>tau</i> (τ), the ratio of the circumference of a circle
* to its radius.
*
* @apiNote
* The value of <i>pi</i> is one half that of <i>tau</i>; in other
* words, <i>tau</i> is double <i>pi</i> .
*
* @since 19
*/
public static final double TAU = 2.0 * PI;
/**
* Constant by which to multiply an angular value in degrees to obtain an
* angular value in radians.
*/
private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
/**
* Constant by which to multiply an angular value in radians to obtain an
* angular value in degrees.
*/
private static final double RADIANS_TO_DEGREES = 57.29577951308232;
/**
* Returns the trigonometric sine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a an angle, in radians.
* @return the sine of the argument.
*/
@IntrinsicCandidate
public static double sin(double a) {
return StrictMath.sin(a); // default impl. delegates to StrictMath
}
/**
* Returns the trigonometric cosine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is {@code 1.0}.
*</ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a an angle, in radians.
* @return the cosine of the argument.
*/
@IntrinsicCandidate
public static double cos(double a) {
return StrictMath.cos(a); // default impl. delegates to StrictMath
}
/**
* Returns the trigonometric tangent of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the result
* is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
*
* <p>The computed result must be within 1.25 ulps 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.
*
* @apiNote
* This method corresponds to the squareRoot operation defined in
* IEEE 754.
*
* @param a a value.
* @return the positive square root of {@code a}.
* If the argument is NaN or less than zero, the result is NaN.
*/
@IntrinsicCandidate
public static double sqrt(double a) {
return StrictMath.sqrt(a); // default impl. delegates to StrictMath
// Note that hardware sqrt instructions
// frequently can be directly used by JITs
// and should be much faster than doing
// Math.sqrt in software.
}
/**
* Returns the cube root of a {@code double} value. For
* positive finite {@code x}, {@code cbrt(-x) ==
* -cbrt(x)}; that is, the cube root of a negative value is
* the negative of the cube root of that value's magnitude.
*
* Special cases:
*
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <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
*/
@IntrinsicCandidate
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)}.
*
* @apiNote
* This method corresponds to the roundToIntegralTowardPositive
* operation defined in IEEE 754.
*
* @param a a value.
* @return the smallest (closest to negative infinity)
* floating-point value that is greater than or equal to
* the argument and is equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double ceil(double a) {
return StrictMath.ceil(a); // default impl. delegates to StrictMath
}
/**
* Returns the largest (closest to positive infinity)
* {@code double} value that is less than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* <ul><li>If the argument value is already equal to a
* mathematical integer, then the result is the same as the
* argument. <li>If the argument is NaN or an infinity or
* positive zero or negative zero, then the result is the same as
* the argument.</ul>
*
* @apiNote
* This method corresponds to the roundToIntegralTowardNegative
* operation defined in IEEE 754.
*
* @param a a value.
* @return the largest (closest to positive infinity)
* floating-point value that less than or equal to the argument
* and is equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double floor(double a) {
return StrictMath.floor(a); // default impl. delegates to StrictMath
}
/**
* Returns the {@code double} value that is closest in value
* to the argument and is equal to a mathematical integer. If two
* {@code double} values that are mathematical integers are
* equally close, the result is the integer value that is
* even. Special cases:
* <ul><li>If the argument value is already equal to a mathematical
* integer, then the result is the same as the argument.
* <li>If the argument is NaN or an infinity or positive zero or negative
* zero, then the result is the same as the argument.</ul>
*
* @apiNote
* This method corresponds to the roundToIntegralTiesToEven
* operation defined in IEEE 754.
*
* @param a a {@code double} value.
* @return the closest floating-point value to {@code a} that is
* equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double rint(double a) {
return StrictMath.rint(a); // default impl. delegates to StrictMath
}
/**
* Returns the angle <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
*/
@IntrinsicCandidate
public static int round(float a) {
int intBits = Float.floatToRawIntBits(a);
int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
>> (FloatConsts.SIGNIFICAND_WIDTH - 1);
int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
+ FloatConsts.EXP_BIAS) - biasedExp;
if ((shift & -32) == 0) { // shift >= 0 && shift < 32
// a is a finite number such that pow(2,-32) <= ulp(a) < 1
int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
| (FloatConsts.SIGNIF_BIT_MASK + 1));
if (intBits < 0) {
r = -r;
}
// In the comments below each Java expression evaluates to the value
// the corresponding mathematical expression:
// (r) evaluates to a / ulp(a)
// (r >> shift) evaluates to floor(a * 2)
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
return ((r >> shift) + 1) >> 1;
} else {
// a is either
// - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
// - an infinity or NaN
return (int) a;
}
}
/**
* Returns the closest {@code long} to the argument, with ties
* rounding to positive infinity.
*
* <p>Special cases:
* <ul><li>If the argument is NaN, the result is 0.
* <li>If the argument is negative infinity or any value less than or
* equal to the value of {@code Long.MIN_VALUE}, the result is
* equal to the value of {@code Long.MIN_VALUE}.
* <li>If the argument is positive infinity or any value greater than or
* equal to the value of {@code Long.MAX_VALUE}, the result is
* equal to the value of {@code Long.MAX_VALUE}.</ul>
*
* @param a a floating-point value to be rounded to a
* {@code long}.
* @return the value of the argument rounded to the nearest
* {@code long} value.
* @see java.lang.Long#MAX_VALUE
* @see java.lang.Long#MIN_VALUE
*/
@IntrinsicCandidate
public static long round(double a) {
long longBits = Double.doubleToRawLongBits(a);
long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
>> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
+ DoubleConsts.EXP_BIAS) - biasedExp;
if ((shift & -64) == 0) { // shift >= 0 && shift < 64
// a is a finite number such that pow(2,-64) <= ulp(a) < 1
long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
| (DoubleConsts.SIGNIF_BIT_MASK + 1));
if (longBits < 0) {
r = -r;
}
// In the comments below each Java expression evaluates to the value
// the corresponding mathematical expression:
// (r) evaluates to a / ulp(a)
// (r >> shift) evaluates to floor(a * 2)
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
return ((r >> shift) + 1) >> 1;
} else {
// a is either
// - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
// - an infinity or NaN
return (long) a;
}
}
private static final class RandomNumberGeneratorHolder {
static final Random randomNumberGenerator = new Random();
}
/**
* Returns a {@code double} value with a positive sign, greater
* than or equal to {@code 0.0} and less than {@code 1.0}.
* Returned values are chosen pseudorandomly with (approximately)
* uniform distribution from that range.
*
* <p>When this method is first called, it creates a single new
* pseudorandom-number generator, exactly as if by the expression
*
* <blockquote>{@code new java.util.Random()}</blockquote>
*
* This new pseudorandom-number generator is used thereafter for
* all calls to this method and is used nowhere else.
*
* <p>This method is properly synchronized to allow correct use by
* more than one thread. However, if many threads need to generate
* pseudorandom numbers at a great rate, it may reduce contention
* for each thread to have its own pseudorandom-number generator.
*
* @apiNote
* As the largest {@code double} value less than {@code 1.0}
* is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
* {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
*
* <blockquote><pre>{@code
* double f = Math.random()/Math.nextDown(1.0);
* double x = x1*(1.0 - f) + x2*f;
* }</pre></blockquote>
*
* @return a pseudorandom {@code double} greater than or equal
* to {@code 0.0} and less than {@code 1.0}.
* @see #nextDown(double)
* @see Random#nextDouble()
*/
public static double random() {
return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
}
/**
* Returns the sum of its arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int addExact(int x, int y) {
int r = x + y;
// HD 2-12 Overflow iff both arguments have the opposite sign of the result
if (((x ^ r) & (y ^ r)) < 0) {
throw new ArithmeticException("integer overflow");
}
return r;
}
/**
* Returns the sum of its arguments,
* throwing an exception if the result overflows a {@code long}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long addExact(long x, long y) {
long r = x + y;
// HD 2-12 Overflow iff both arguments have the opposite sign of the result
if (((x ^ r) & (y ^ r)) < 0) {
throw new ArithmeticException("long overflow");
}
return r;
}
/**
* Returns the difference of the arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int subtractExact(int x, int y) {
int r = x - y;
// HD 2-12 Overflow iff the arguments have different signs and
// the sign of the result is different from the sign of x
if (((x ^ y) & (x ^ r)) < 0) {
throw new ArithmeticException("integer overflow");
}
return r;
}
/**
* Returns the difference of the arguments,
* throwing an exception if the result overflows a {@code long}.
*
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long subtractExact(long x, long y) {
long r = x - y;
// HD 2-12 Overflow iff the arguments have different signs and
// the sign of the result is different from the sign of x
if (((x ^ y) & (x ^ r)) < 0) {
throw new ArithmeticException("long overflow");
}
return r;
}
/**
* Returns the product of the arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int multiplyExact(int x, int y) {
long r = (long)x * (long)y;
if ((int)r != r) {
throw new ArithmeticException("integer overflow");
}
return (int)r;
}