java.lang

Class Math

java.lang.Object

java.lang.Math

public final class Math
extends Object

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.

Since:

JDK1.0

Author:

unascribed, Joseph D. Darcy

Field Summary

Fields

Modifier and Type	Field and Description
static double	E The double value that is closer than any other to e, the base of the natural logarithms.
static double	PI The double value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.

Method Summary

Methods

Modifier and Type	Method and Description
static double	abs(double a) Returns the absolute value of a double value.
static float	abs(float a) Returns the absolute value of a float value.
static int	abs(int a) Returns the absolute value of an int value.
static long	abs(long a) Returns the absolute value of a long value.
static double	acos(double a) Returns the arc cosine of a value; the returned angle is in the range 0.0 through pi.
static double	asin(double a) Returns the arc sine of a value; the returned angle is in the range -pi/2 through pi/2.
static double	atan(double a) Returns the arc tangent of a value; the returned angle is in the range -pi/2 through pi/2.
static double	atan2(double y, double x) Returns the angle theta from the conversion of rectangular coordinates (x, y) to polar coordinates (r, theta).
static double	ceil(double a) 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.
static double	cos(double a) Returns the trigonometric cosine of an angle.
static double	exp(double a) Returns Euler's number e raised to the power of a double value.
static double	floor(double a) 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.
static double	IEEEremainder(double f1, double f2) Computes the remainder operation on two arguments as prescribed by the IEEE 754 standard.
static double	log(double a) Returns the natural logarithm (base e) of a double value.
static double	log10(double a) Returns the base 10 logarithm of a double value.
static double	max(double a, double b) Returns the greater of two double values.
static float	max(float a, float b) Returns the greater of two float values.
static int	max(int a, int b) Returns the greater of two int values.
static long	max(long a, long b) Returns the greater of two long values.
static double	min(double a, double b) Returns the smaller of two double values.
static float	min(float a, float b) Returns the smaller of two float values.
static int	min(int a, int b) Returns the smaller of two int values.
static long	min(long a, long b) Returns the smaller of two long values.
static double	pow(double a, double b) Returns the value of the first argument raised to the power of the second argument.
static double	random() Returns a double value with a positive sign, greater than or equal to 0.0 and less than 1.0.
static double	rint(double a) Returns the double value that is closest in value to the argument and is equal to a mathematical integer.
static long	round(double a) Returns the closest long to the argument, with ties rounding up.
static int	round(float a) Returns the closest int to the argument, with ties rounding up.
static double	signum(double d) 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.
static float	signum(float f) 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.
static double	sin(double a) Returns the trigonometric sine of an angle.
static double	sqrt(double a) Returns the correctly rounded positive square root of a double value.
static double	tan(double a) Returns the trigonometric tangent of an angle.
static double	toDegrees(double angrad) Converts an angle measured in radians to an approximately equivalent angle measured in degrees.
static double	toRadians(double angdeg) Converts an angle measured in degrees to an approximately equivalent angle measured in radians.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

E

public static final double E

The double value that is closer than any other to e, the base of the natural logarithms.

See Also:

Constant Field Values

PI

public static final double PI

The double value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.

See Also:

Constant Field Values

Method Detail

sin

public static double sin(double a)

Returns the trigonometric sine of an angle. Special cases:

• If the argument is NaN or an infinity, then the result is NaN.
• If the argument is zero, then the result is a zero with the same sign as the argument.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - an angle, in radians.

Returns:

the sine of the argument.

cos

public static double cos(double a)

Returns the trigonometric cosine of an angle. Special cases:

• If the argument is NaN or an infinity, then the result is NaN.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - an angle, in radians.

Returns:

the cosine of the argument.

tan

public static double tan(double a)

Returns the trigonometric tangent of an angle. Special cases:

• If the argument is NaN or an infinity, then the result is NaN.
• If the argument is zero, then the result is a zero with the same sign as the argument.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - an angle, in radians.

Returns:

the tangent of the argument.

asin

public static double asin(double a)

Returns the arc sine of a value; the returned angle is in the range -pi/2 through pi/2. Special cases:

• If the argument is NaN or its absolute value is greater than 1, then the result is NaN.
• If the argument is zero, then the result is a zero with the same sign as the argument.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - the value whose arc sine is to be returned.

Returns:

the arc sine of the argument.

acos

public static double acos(double a)

Returns the arc cosine of a value; the returned angle is in the range 0.0 through pi. Special case:

• If the argument is NaN or its absolute value is greater than 1, then the result is NaN.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - the value whose arc cosine is to be returned.

Returns:

the arc cosine of the argument.

atan

public static double atan(double a)

Returns the arc tangent of a value; the returned angle is in the range -pi/2 through pi/2. Special cases:

• If the argument is NaN, then the result is NaN.
• If the argument is zero, then the result is a zero with the same sign as the argument.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - the value whose arc tangent is to be returned.

Returns:

the arc tangent of the argument.

toRadians

public static double toRadians(double angdeg)

Converts an angle measured in degrees to an approximately equivalent angle measured in radians. The conversion from degrees to radians is generally inexact.

Parameters:

angdeg - an angle, in degrees

Returns:

the measurement of the angle angdeg in radians.

Since:

1.2

toDegrees

public static double toDegrees(double angrad)

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.

Parameters:

angrad - an angle, in radians

Returns:

the measurement of the angle angrad in degrees.

Since:

1.2

exp

public static double exp(double a)

Returns Euler's number e raised to the power of a double value. Special cases:

• If the argument is NaN, the result is NaN.
• If the argument is positive infinity, then the result is positive infinity.
• If the argument is negative infinity, then the result is positive zero.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - the exponent to raise e to.

Returns:

the value e^a, where e is the base of the natural logarithms.

log

public static double log(double a)

Returns the natural logarithm (base e) of a double value. Special cases:

• If the argument is NaN or less than zero, then the result is NaN.
• If the argument is positive infinity, then the result is positive infinity.
• If the argument is positive zero or negative zero, then the result is negative infinity.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - a value

Returns:

the value ln a, the natural logarithm of a.

log10

public static double log10(double a)

Returns the base 10 logarithm of a double value. Special cases:

• If the argument is NaN or less than zero, then the result is NaN.
• If the argument is positive infinity, then the result is positive infinity.
• If the argument is positive zero or negative zero, then the result is negative infinity.
• If the argument is equal to 10ⁿ for integer n, then the result is n.

The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

Parameters:

a - a value

Returns:

the base 10 logarithm of a.

Since:

1.5

sqrt

public static double sqrt(double a)

Returns the correctly rounded positive square root of a double value. Special cases:

• If the argument is NaN or less than zero, then the result is NaN.
• If the argument is positive infinity, then the result is positive infinity.
• If the argument is positive zero or negative zero, then the result is the same as the argument.

Otherwise, the result is the double value closest to the true mathematical square root of the argument value.

Parameters:

a - a value.

Returns:

the positive square root of a. If the argument is NaN or less than zero, the result is NaN.

ceil

public static double ceil(double a)

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:

• If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
• If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.
• If the argument value is less than zero but greater than -1.0, then the result is negative zero.

Note that the value of Math.ceil(x) is exactly the value of -Math.floor(-x).

Parameters:

a - a value.

Returns:

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.

floor

public static double floor(double a)

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:

• If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
• If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.

Parameters:

a - a value.

Returns:

the largest (closest to positive infinity) floating-point value that less than or equal to the argument and is equal to a mathematical integer.

IEEEremainder

public static double IEEEremainder(double f1,
                   double f2)

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:

• 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.
• If the first argument is finite and the second argument is infinite, then the result is the same as the first argument.

Parameters:

f1 - the dividend.

f2 - the divisor.

Returns:

the remainder when f1 is divided by f2.

rint

public static double rint(double a)

Returns the 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:

• If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
• If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.

Parameters:

a - a double value.

Returns:

the closest floating-point value to a that is equal to a mathematical integer.

atan2

public static double atan2(double y,
           double x)

Returns the angle theta from the conversion of rectangular coordinates (x, y) to polar coordinates (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:

• If either argument is NaN, then the result is NaN.
• 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.
• 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.
• 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 double value closest to pi.
• 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 double value closest to -pi.
• 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 double value closest to pi/2.
• 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 double value closest to -pi/2.
• If both arguments are positive infinity, then the result is the double value closest to pi/4.
• If the first argument is positive infinity and the second argument is negative infinity, then the result is the double value closest to 3*pi/4.
• If the first argument is negative infinity and the second argument is positive infinity, then the result is the double value closest to -pi/4.
• If both arguments are negative infinity, then the result is the double value closest to -3*pi/4.

The computed result must be within 2 ulps of the exact result. Results must be semi-monotonic.

Parameters:

y - the ordinate coordinate

x - the abscissa coordinate

Returns:

the theta component of the point (r, theta) in polar coordinates that corresponds to the point (x, y) in Cartesian coordinates.

pow

public static double pow(double a,
         double b)

Returns the value of the first argument raised to the power of the second argument. Special cases:

• If the second argument is positive or negative zero, then the result is 1.0.
• If the second argument is 1.0, then the result is the same as the first argument.
• If the second argument is NaN, then the result is NaN.
• If the first argument is NaN and the second argument is nonzero, then the result is NaN.
• If
  • the absolute value of the first argument is greater than 1 and the second argument is positive infinity, or
  • the absolute value of the first argument is less than 1 and the second argument is negative infinity,
  then the result is positive infinity.
• If
  • the absolute value of the first argument is greater than 1 and the second argument is negative infinity, or
  • the absolute value of the first argument is less than 1 and the second argument is positive infinity,
  then the result is positive zero.
• If the absolute value of the first argument equals 1 and the second argument is infinite, then the result is NaN.
• If
  • the first argument is positive zero and the second argument is greater than zero, or
  • the first argument is positive infinity and the second argument is less than zero,
  then the result is positive zero.
• If
  • the first argument is positive zero and the second argument is less than zero, or
  • the first argument is positive infinity and the second argument is greater than zero,
  then the result is positive infinity.
• If
  • the first argument is negative zero and the second argument is greater than zero but not a finite odd integer, or
  • the first argument is negative infinity and the second argument is less than zero but not a finite odd integer,
  then the result is positive zero.
• If
  • the first argument is negative zero and the second argument is a positive finite odd integer, or
  • the first argument is negative infinity and the second argument is a negative finite odd integer,
  then the result is negative zero.
• If
  • the first argument is negative zero and the second argument is less than zero but not a finite odd integer, or
  • the first argument is negative infinity and the second argument is greater than zero but not a finite odd integer,
  then the result is positive infinity.
• If
  • the first argument is negative zero and the second argument is a negative finite odd integer, or
  • the first argument is negative infinity and the second argument is a positive finite odd integer,
  then the result is negative infinity.
• If the first argument is finite and less than zero
  • 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
  • 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
  • if the second argument is finite and not an integer, then the result is NaN.
• 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 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 ceil or, equivalently, a fixed point of the method 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.

Parameters:

a - the base.

b - the exponent.

Returns:

the value ab.

round

public static int round(float a)

Returns the closest int to the argument, with ties rounding up.

Special cases:

• If the argument is NaN, the result is 0.
• If the argument is negative infinity or any value less than or equal to the value of Integer.MIN_VALUE, the result is equal to the value of Integer.MIN_VALUE.
• If the argument is positive infinity or any value greater than or equal to the value of Integer.MAX_VALUE, the result is equal to the value of Integer.MAX_VALUE.

Parameters:

a - a floating-point value to be rounded to an integer.

Returns:

the value of the argument rounded to the nearest int value.

See Also:

Integer.MAX_VALUE, Integer.MIN_VALUE

round

public static long round(double a)

Returns the closest long to the argument, with ties rounding up.

Special cases:

• If the argument is NaN, the result is 0.
• If the argument is negative infinity or any value less than or equal to the value of Long.MIN_VALUE, the result is equal to the value of Long.MIN_VALUE.
• If the argument is positive infinity or any value greater than or equal to the value of Long.MAX_VALUE, the result is equal to the value of Long.MAX_VALUE.

Parameters:

a - a floating-point value to be rounded to a long.

Returns:

the value of the argument rounded to the nearest long value.

See Also:

Long.MAX_VALUE, Long.MIN_VALUE

random

public static double 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

new java.util.Random()

This new pseudorandom-number generator is used thereafter for all calls to this method and is used nowhere else.

This method is properly synchronized to allow correct use by more than one thread. However, if many threads need to generate pseudorandom numbers at a great rate, it may reduce contention for each thread to have its own pseudorandom-number generator.

Returns:

a pseudorandom double greater than or equal to 0.0 and less than 1.0.

See Also:

Random#nextDouble()

abs

public static int abs(int a)

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.

Parameters:

a - the argument whose absolute value is to be determined

Returns:

the absolute value of the argument.

abs

public static long abs(long 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.

Parameters:

a - the argument whose absolute value is to be determined

Returns:

the absolute value of the argument.

abs

public static float abs(float 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:

• If the argument is positive zero or negative zero, the result is positive zero.
• If the argument is infinite, the result is positive infinity.
• If the argument is NaN, the result is NaN.

In other words, the result is the same as the value of the expression:

Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))

Parameters:

a - the argument whose absolute value is to be determined

Returns:

the absolute value of the argument.

abs

public static double abs(double 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:

• If the argument is positive zero or negative zero, the result is positive zero.
• If the argument is infinite, the result is positive infinity.
• If the argument is NaN, the result is NaN.

In other words, the result is the same as the value of the expression:

Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)

Parameters:

a - the argument whose absolute value is to be determined

Returns:

the absolute value of the argument.

max

public static int max(int a,
      int b)

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.

Parameters:

a - an argument.

b - another argument.

Returns:

the larger of a and b.

max

public static long max(long a,
       long 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the larger of a and b.

max

public static float max(float a,
        float b)

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.

Parameters:

a - an argument.

b - another argument.

Returns:

the larger of a and b.

max

public static double max(double a,
         double 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the larger of a and b.

min

public static int min(int a,
      int 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the smaller of a and b.

min

public static long min(long a,
       long 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the smaller of a and b.

min

public static float min(float a,
        float 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the smaller of a and b.

min

public static double min(double a,
         double 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.

Parameters:

a - an argument.

b - another argument.

Returns:

the smaller of a and b.

signum

public static double signum(double d)

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.

Special Cases:

• If the argument is NaN, then the result is NaN.
• If the argument is positive zero or negative zero, then the result is the same as the argument.

Parameters:

d - the floating-point value whose signum is to be returned

Returns:

the signum function of the argument

Since:

1.5

signum

public static float signum(float f)

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.

Special Cases:

• If the argument is NaN, then the result is NaN.
• If the argument is positive zero or negative zero, then the result is the same as the argument.

Parameters:

f - the floating-point value whose signum is to be returned

Returns:

the signum function of the argument

Since:

1.5