/*
* @(#)Math.java 1.69 04/06/14
*
* Copyright 2004 Sun Microsystems, Inc. All rights reserved.
* SUN PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
*/
package java.lang;
import java.util.Random;
/**
* The class Math
contains methods for performing basic
* numeric operations such as the elementary exponential, logarithm,
* square root, and trigonometric functions.
*
*
Unlike some of the numeric methods of class
* StrictMath
, all implementations of the equivalent
* functions of class Math
are not defined to return the
* bit-for-bit same results. This relaxation permits
* better-performing implementations where strict reproducibility is
* not required.
*
*
By default many of the Math
methods simply call
* the equivalent method in StrictMath
for their
* implementation. Code generators are encouraged to use
* platform-specific native libraries or microprocessor instructions,
* where available, to provide higher-performance implementations of
* Math
methods. Such higher-performance
* implementations still must conform to the specification for
* Math
.
*
*
The quality of implementation specifications concern two
* properties, accuracy of the returned result and monotonicity of the
* method. Accuracy of the floating-point Math
methods
* is measured in terms of ulps, units in the last place. For
* a given floating-point format, an ulp of a specific real number
* value is the distance between the two floating-point values
* bracketing that numerical value. When discussing the accuracy of a
* method as a whole rather than at a specific argument, the number of
* ulps cited is for the worst-case error at any argument. If a
* method always has an error less than 0.5 ulps, the method always
* returns the floating-point number nearest the exact result; such a
* method is correctly rounded. A correctly rounded method is
* generally the best a floating-point approximation can be; however,
* it is impractical for many floating-point methods to be correctly
* rounded. Instead, for the Math
class, a larger error
* bound of 1 or 2 ulps is allowed for certain methods. Informally,
* with a 1 ulp error bound, when the exact result is a representable
* number, the exact result should be returned as the computed result;
* otherwise, either of the two floating-point values which bracket
* the exact result may be returned. For exact results large in
* magnitude, one of the endpoints of the bracket may be infinite.
* Besides accuracy at individual arguments, maintaining proper
* relations between the method at different arguments is also
* important. Therefore, most methods with more than 0.5 ulp errors
* are required to be semi-monotonic: whenever the mathematical
* function is non-decreasing, so is the floating-point approximation,
* likewise, whenever the mathematical function is non-increasing, so
* is the floating-point approximation. Not all approximations that
* have 1 ulp accuracy will automatically meet the monotonicity
* requirements.
*
* @author unascribed
* @author Joseph D. Darcy
* @version 1.69, 06/14/04
* @since JDK1.0
*/
public final class Math {
/**
* Don't let anyone instantiate this class.
*/
private Math() {}
/**
* The double
value that is closer than any other to
* e, the base of the natural logarithms.
*/
public static final double E = 2.7182818284590452354;
/**
* The double
value that is closer than any other to
* pi, the ratio of the circumference of a circle to its
* diameter.
*/
public static final double PI = 3.14159265358979323846;
/**
* Returns the trigonometric sine of an angle. Special cases:
*
The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the sine of the argument. */ public static double sin(double a) { return StrictMath.sin(a); // default impl. delegates to StrictMath } /** * Returns the trigonometric cosine of an angle. Special cases: *
The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the cosine of the argument. */ public static double cos(double a) { return StrictMath.cos(a); // default impl. delegates to StrictMath } /** * Returns the trigonometric tangent of an angle. Special cases: *
The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the tangent of the argument. */ public static double tan(double a) { return StrictMath.tan(a); // default impl. delegates to StrictMath } /** * Returns the arc sine of an angle, in the range of -pi/2 through * pi/2. Special cases: *
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 an angle, in the range of 0.0 through * pi. Special case: *
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 an angle, in the range of -pi/2 * through pi/2. Special cases: *
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 angdeg
* in radians.
* @since 1.2
*/
public static double toRadians(double angdeg) {
return angdeg / 180.0 * PI;
}
/**
* Converts an angle measured in radians to an approximately
* equivalent angle measured in degrees. The conversion from
* radians to degrees is generally inexact; users should
* not expect cos(toRadians(90.0))
to exactly
* equal 0.0
.
*
* @param angrad an angle, in radians
* @return the measurement of the angle angrad
* in degrees.
* @since 1.2
*/
public static double toDegrees(double angrad) {
return angrad * 180.0 / PI;
}
/**
* Returns Euler's number e raised to the power of a
* double
value. Special cases:
*
The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the exponent to raise e to.
* @return the value ea
,
* where e is the base of the natural logarithms.
*/
public static double exp(double a) {
return StrictMath.exp(a); // default impl. delegates to StrictMath
}
/**
* Returns the natural logarithm (base e) of a double
* value. Special cases:
*
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 a
, the natural logarithm of
* a
.
*/
public static double log(double a) {
return StrictMath.log(a); // default impl. delegates to StrictMath
}
/**
* Returns the base 10 logarithm of a double
value.
* Special cases:
*
*
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 a
.
* @since 1.5
*/
public static double log10(double a) {
return StrictMath.log10(a); // default impl. delegates to StrictMath
}
/**
* Returns the correctly rounded positive square root of a
* double
value.
* Special cases:
*
double
value closest to
* the true mathematical square root of the argument value.
*
* @param a a value.
* @return the positive square root of a
.
* If the argument is NaN or less than zero, the result is NaN.
*/
public static double sqrt(double a) {
return StrictMath.sqrt(a); // default impl. delegates to StrictMath
// Note that hardware sqrt instructions
// frequently can be directly used by JITs
// and should be much faster than doing
// Math.sqrt in software.
}
/**
* Returns the cube root of a double
value. For
* positive finite x
, cbrt(-x) ==
* -cbrt(x)
; that is, the cube root of a negative value is
* the negative of the cube root of that value's magnitude.
*
* Special cases:
*
* The computed result must be within 1 ulp of the exact result.
*
* @param a a value.
* @return the cube root of a
.
* @since 1.5
*/
public static double cbrt(double a) {
return StrictMath.cbrt(a);
}
/**
* Computes the remainder operation on two arguments as prescribed
* by the IEEE 754 standard.
* The remainder value is mathematically equal to
* f1 - f2
× n,
* where n is the mathematical integer closest to the exact
* mathematical value of the quotient f1/f2
, and if two
* mathematical integers are equally close to f1/f2
,
* then n is the integer that is even. If the remainder is
* zero, its sign is the same as the sign of the first argument.
* Special cases:
*
f1
is divided by
* f2
.
*/
public static double IEEEremainder(double f1, double f2) {
return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
}
/**
* Returns the smallest (closest to negative infinity)
* double
value that is greater than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* Math.ceil(x)
is exactly the
* value of -Math.floor(-x)
.
*
*
* @param a a value.
* @return the smallest (closest to negative infinity)
* floating-point value that is greater than or equal to
* the argument and is equal to a mathematical integer.
*/
public static double ceil(double a) {
return StrictMath.ceil(a); // default impl. delegates to StrictMath
}
/**
* Returns the largest (closest to positive infinity)
* double
value that is less than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* double
value that is closest in value
* to the argument and is equal to a mathematical integer. If two
* double
values that are mathematical integers are
* equally close, the result is the integer value that is
* even. Special cases:
* double
value.
* @return the closest floating-point value to a
that is
* equal to a mathematical integer.
*/
public static double rint(double a) {
return StrictMath.rint(a); // default impl. delegates to StrictMath
}
/**
* Converts rectangular coordinates (x
, y
)
* to polar (r, theta).
* This method computes the phase theta by computing an arc tangent
* of y/x
in the range of -pi to pi. Special
* cases:
* double
value closest to pi.
* double
value closest to -pi.
* double
value closest to pi/2.
* double
value closest to -pi/2.
* double
value closest to pi/4.
* double
* value closest to 3*pi/4.
* double
value
* closest to -pi/4.
* double
value closest to -3*pi/4.The computed result must be within 2 ulps of the exact result. * Results must be semi-monotonic. * * @param y the ordinate coordinate * @param x the abscissa coordinate * @return the theta component of the point * (r, theta) * in polar coordinates that corresponds to the point * (x, y) in Cartesian coordinates. */ public static double atan2(double y, double x) { return StrictMath.atan2(y, x); // default impl. delegates to StrictMath } /** * Returns the value of the first argument raised to the power of the * second argument. Special cases: * *
double
value.(In the foregoing descriptions, a floating-point value is * considered to be an integer if and only if it is finite and a * fixed point of the method {@link #ceil ceil} or, * equivalently, a fixed point of the method {@link #floor * floor}. A value is a fixed point of a one-argument * method if and only if the result of applying the method to the * value is equal to the value.) * *
The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the base.
* @param b the exponent.
* @return the value ab
.
*/
public static double pow(double a, double b) {
return StrictMath.pow(a, b); // default impl. delegates to StrictMath
}
/**
* Returns the closest int
to the argument. The
* result is rounded to an integer by adding 1/2, taking the
* floor of the result, and casting the result to type int
.
* In other words, the result is equal to the value of the expression:
*
(int)Math.floor(a + 0.5f)*
* Special cases: *
Integer.MIN_VALUE
, the result is
* equal to the value of Integer.MIN_VALUE
.
* Integer.MAX_VALUE
, the result is
* equal to the value of Integer.MAX_VALUE
.int
value.
* @see java.lang.Integer#MAX_VALUE
* @see java.lang.Integer#MIN_VALUE
*/
public static int round(float a) {
return (int)floor(a + 0.5f);
}
/**
* Returns the closest long
to the argument. The result
* is rounded to an integer by adding 1/2, taking the floor of the
* result, and casting the result to type long
. In other
* words, the result is equal to the value of the expression:
* (long)Math.floor(a + 0.5d)*
* Special cases: *
Long.MIN_VALUE
, the result is
* equal to the value of Long.MIN_VALUE
.
* Long.MAX_VALUE
, the result is
* equal to the value of Long.MAX_VALUE
.long
.
* @return the value of the argument rounded to the nearest
* long
value.
* @see java.lang.Long#MAX_VALUE
* @see java.lang.Long#MIN_VALUE
*/
public static long round(double a) {
return (long)floor(a + 0.5d);
}
private static Random randomNumberGenerator;
private static synchronized void initRNG() {
if (randomNumberGenerator == null)
randomNumberGenerator = new Random();
}
/**
* Returns a double
value with a positive sign, greater
* than or equal to 0.0
and less than 1.0
.
* Returned values are chosen pseudorandomly with (approximately)
* uniform distribution from that range.
*
* When this method is first called, it creates a single new * pseudorandom-number generator, exactly as if by the expression *
This * new pseudorandom-number generator is used thereafter for all * calls to this method and is used nowhere else. * *new java.util.Random
This method is properly synchronized to allow correct use by
* more than one thread. However, if many threads need to generate
* pseudorandom numbers at a great rate, it may reduce contention
* for each thread to have its own pseudorandom-number generator.
*
* @return a pseudorandom double
greater than or equal
* to 0.0
and less than 1.0
.
* @see java.util.Random#nextDouble()
*/
public static double random() {
if (randomNumberGenerator == null) initRNG();
return randomNumberGenerator.nextDouble();
}
/**
* Returns the absolute value of an int
value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
*
*
Note that if the argument is equal to the value of
* Integer.MIN_VALUE
, the most negative representable
* int
value, the result is that same value, which is
* negative.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
* @see java.lang.Integer#MIN_VALUE
*/
public static int abs(int a) {
return (a < 0) ? -a : a;
}
/**
* Returns the absolute value of a long
value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
*
*
Note that if the argument is equal to the value of
* Long.MIN_VALUE
, the most negative representable
* long
value, the result is that same value, which
* is negative.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
* @see java.lang.Long#MIN_VALUE
*/
public static long abs(long a) {
return (a < 0) ? -a : a;
}
/**
* Returns the absolute value of a float
value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
*
Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))* * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static float abs(float a) { return (a <= 0.0F) ? 0.0F - a : a; } /** * Returns the absolute value of a
double
value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
* Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
*/
public static double abs(double a) {
return (a <= 0.0D) ? 0.0D - a : a;
}
/**
* Returns the greater of two int
values. That is, the
* result is the argument closer to the value of
* Integer.MAX_VALUE
. If the arguments have the same value,
* the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the larger of a
and b
.
* @see java.lang.Long#MAX_VALUE
*/
public static int max(int a, int b) {
return (a >= b) ? a : b;
}
/**
* Returns the greater of two long
values. That is, the
* result is the argument closer to the value of
* Long.MAX_VALUE
. If the arguments have the same value,
* the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the larger of a
and b
.
* @see java.lang.Long#MAX_VALUE
*/
public static long max(long a, long b) {
return (a >= b) ? a : b;
}
private static long negativeZeroFloatBits = Float.floatToIntBits(-0.0f);
private static long negativeZeroDoubleBits = Double.doubleToLongBits(-0.0d);
/**
* Returns the greater of two float
values. That is,
* the result is the argument closer to positive infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
*
* @param a an argument.
* @param b another argument.
* @return the larger of a
and b
.
*/
public static float max(float a, float b) {
if (a != a) return a; // a is NaN
if ((a == 0.0f) && (b == 0.0f)
&& (Float.floatToIntBits(a) == negativeZeroFloatBits)) {
return b;
}
return (a >= b) ? a : b;
}
/**
* Returns the greater of two double
values. That
* is, the result is the argument closer to positive infinity. If
* the arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
*
* @param a an argument.
* @param b another argument.
* @return the larger of a
and b
.
*/
public static double max(double a, double b) {
if (a != a) return a; // a is NaN
if ((a == 0.0d) && (b == 0.0d)
&& (Double.doubleToLongBits(a) == negativeZeroDoubleBits)) {
return b;
}
return (a >= b) ? a : b;
}
/**
* Returns the smaller of two int
values. That is,
* the result the argument closer to the value of
* Integer.MIN_VALUE
. If the arguments have the same
* value, the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of a
and b
.
* @see java.lang.Long#MIN_VALUE
*/
public static int min(int a, int b) {
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two long
values. That is,
* the result is the argument closer to the value of
* Long.MIN_VALUE
. If the arguments have the same
* value, the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of a
and b
.
* @see java.lang.Long#MIN_VALUE
*/
public static long min(long a, long b) {
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two float
values. That is,
* the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If
* one argument is positive zero and the other is negative zero,
* the result is negative zero.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of a
and b.
*/
public static float min(float a, float b) {
if (a != a) return a; // a is NaN
if ((a == 0.0f) && (b == 0.0f)
&& (Float.floatToIntBits(b) == negativeZeroFloatBits)) {
return b;
}
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two double
values. That
* is, the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other is negative zero, the
* result is negative zero.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of a
and b
.
*/
public static double min(double a, double b) {
if (a != a) return a; // a is NaN
if ((a == 0.0d) && (b == 0.0d)
&& (Double.doubleToLongBits(b) == negativeZeroDoubleBits)) {
return b;
}
return (a <= b) ? a : b;
}
/**
* Returns the size of an ulp of the argument. An ulp of a
* double
value is the positive distance between this
* floating-point value and the double
value next
* larger in magnitude. Note that for non-NaN x,
* ulp(-x) == ulp(x)
.
*
*
Special Cases: *
Double.MIN_VALUE
.
* Double.MAX_VALUE
, then
* the result is equal to 2971.
* float
value is the positive distance between this
* floating-point value and the float
value next
* larger in magnitude. Note that for non-NaN x,
* ulp(-x) == ulp(x)
.
*
* Special Cases: *
Float.MIN_VALUE
.
* Float.MAX_VALUE
, then
* the result is equal to 2104.
* Special Cases: *
Special Cases: *
double
value.
* The hyperbolic sine of x is defined to be
* (ex - e-x)/2
* where e is {@linkplain Math#E Euler's number}.
*
* Special cases: *
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 x
.
* @since 1.5
*/
public static double sinh(double x) {
return StrictMath.sinh(x);
}
/**
* Returns the hyperbolic cosine of a double
value.
* The hyperbolic cosine of x is defined to be
* (ex + e-x)/2
* where e is {@linkplain Math#E Euler's number}.
*
*
Special cases: *
1.0
.
*
* 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 x
.
* @since 1.5
*/
public static double cosh(double x) {
return StrictMath.cosh(x);
}
/**
* Returns the hyperbolic tangent of a double
value.
* The hyperbolic tangent of x is defined to be
* (ex - e-x)/(ex + e-x),
* in other words, {@linkplain Math#sinh
* sinh(x)}/{@linkplain Math#cosh cosh(x)}. Note
* that the absolute value of the exact tanh is always less than
* 1.
*
*
Special cases: *
+1.0
.
*
* -1.0
.
*
* The computed result must be within 2.5 ulps of the exact result.
* The result of tanh
for any finite input must have
* an absolute value less than or equal to 1. Note that once the
* exact result of tanh is within 1/2 of an ulp of the limit value
* of ±1, correctly signed ±1.0
should
* be returned.
*
* @param x The number whose hyperbolic tangent is to be returned.
* @return The hyperbolic tangent of x
.
* @since 1.5
*/
public static double tanh(double x) {
return StrictMath.tanh(x);
}
/**
* Returns sqrt(x2 +y2)
* without intermediate overflow or underflow.
*
*
Special cases: *
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(x2 +y2)
* without intermediate overflow or underflow
* @since 1.5
*/
public static double hypot(double x, double y) {
return StrictMath.hypot(x, y);
}
/**
* Returns ex -1. Note that for values of
* x near 0, the exact sum of
* expm1(x)
+ 1 is much closer to the true
* result of ex than exp(x)
.
*
*
Special cases: *
The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic. The result of
* expm1
for any finite input must be greater than or
* equal to -1.0
. Note that once the exact result of
* ex
- 1 is within 1/2
* ulp of the limit value -1, -1.0
should be
* returned.
*
* @param x the exponent to raise e to in the computation of
* ex
-1.
* @return the value ex
- 1.
*/
public static double expm1(double x) {
return StrictMath.expm1(x);
}
/**
* Returns the natural logarithm of the sum of the argument and 1.
* Note that for small values x
, the result of
* log1p(x)
is much closer to the true result of ln(1
* + x
) than the floating-point evaluation of
* log(1.0+x)
.
*
*
Special cases: * *
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(x
+ 1), the natural
* log of x
+ 1
*/
public static double log1p(double x) {
return StrictMath.log1p(x);
}
}