public final class Math extends Object
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 bitforbit same
results. This relaxation permits betterperforming 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
platformspecific native libraries or microprocessor instructions, where available, to provide
higherperformance implementations of Math
methods. Such higherperformance
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 floatingpoint Math
methods is
measured in terms of ulps, units in the last place. For a given floatingpoint format, an
ulp of a specific real number value is the distance between the two floatingpoint 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 worstcase error at any argument. If
a method always has an error less than 0.5 ulps, the method always returns the floatingpoint
number nearest the exact result; such a method is correctly rounded. A correctly rounded
method is generally the best a floatingpoint approximation can be; however, it is impractical
for many floatingpoint 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 floatingpoint 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 semimonotonic: whenever the mathematical function is
nondecreasing, so is the floatingpoint approximation, likewise, whenever the mathematical
function is nonincreasing, so is the floatingpoint approximation. Not all approximations that
have 1 ulp accuracy will automatically meet the monotonicity requirements.
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. 
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 
cbrt(double a)
Returns the cube root of a
double value. 
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 
copySign(double magnitude,
double sign)
Returns the first floatingpoint argument with the sign of the second floatingpoint argument.

static float 
copySign(float magnitude,
float sign)
Returns the first floatingpoint argument with the sign of the second floatingpoint argument.

static double 
cos(double a)
Returns the trigonometric cosine of an angle.

static double 
cosh(double x)
Returns the hyperbolic cosine of a
double value. 
static double 
exp(double a)
Returns Euler's number e raised to the power of a
double value. 
static double 
expm1(double x)
Returns e^{x} 1.

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 int 
getExponent(double d)
Returns the unbiased exponent used in the representation of a
double . 
static int 
getExponent(float f)
Returns the unbiased exponent used in the representation of a
float . 
static double 
hypot(double x,
double y)
Returns sqrt(x^{2} +y^{2}) without intermediate overflow or
underflow.

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 
log1p(double x)
Returns the natural logarithm of the sum of the argument and 1.

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 
nextAfter(double start,
double direction)
Returns the floatingpoint number adjacent to the first argument in the direction of the second
argument.

static float 
nextAfter(float start,
double direction)
Returns the floatingpoint number adjacent to the first argument in the direction of the second
argument.

static double 
nextUp(double d)
Returns the floatingpoint value adjacent to
d in the direction of positive infinity. 
static float 
nextUp(float f)
Returns the floatingpoint value adjacent to
f in the direction of positive infinity. 
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 
scalb(double d,
int scaleFactor)
Return
d × 2^{scaleFactor} rounded as if performed by a single
correctly rounded floatingpoint multiply to a member of the double value set. 
static float 
scalb(float f,
int scaleFactor)
Return
f × 2^{scaleFactor} rounded as if performed by a single
correctly rounded floatingpoint multiply to a member of the float value set. 
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 
sinh(double x)
Returns the hyperbolic sine of a
double value. 
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 
tanh(double x)
Returns the hyperbolic tangent of a
double value. 
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.

static double 
ulp(double d)
Returns the size of an ulp of the argument.

static float 
ulp(float f)
Returns the size of an ulp of the argument.

public static final double E
double
value that is closer than any other to e, the base of the natural
logarithms.public static final double PI
double
value that is closer than any other to pi, the ratio of the
circumference of a circle to its diameter.public static double abs(double 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)
a
 the argument whose absolute value is to be determinedpublic static float abs(float 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))
a
 the argument whose absolute value is to be determinedpublic static int abs(int a)
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.
a
 the argument whose absolute value is to be determinedpublic static long abs(long 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.
a
 the argument whose absolute value is to be determinedpublic static double acos(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 the value whose arc cosine is to be returned.public static double asin(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 the value whose arc sine is to be returned.public static double atan(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 the value whose arc tangent is to be returned.public static double atan2(double y, double x)
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:
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 semimonotonic.
y
 the ordinate coordinatex
 the abscissa coordinatepublic static double cbrt(double 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.
a
 a value.a
.public static double ceil(double a)
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)
.a
 a value.public static double copySign(double magnitude, double sign)
magnitude
 the parameter providing the magnitude of the resultsign
 the parameter providing the sign of the resultmagnitude
and the sign of sign
.public static float copySign(float magnitude, float sign)
magnitude
 the parameter providing the magnitude of the resultsign
 the parameter providing the sign of the resultmagnitude
and the sign of sign
.public static double cos(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 an angle, in radians.public static double cosh(double x)
double
value. The hyperbolic cosine of x is
defined to be (e^{x} + e^{x})/2 where e is
Euler's number.
Special cases:
1.0
.
The computed result must be within 2.5 ulps of the exact result.
x
 The number whose hyperbolic cosine is to be returned.x
.public static double exp(double a)
double
value. Special cases:
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 the exponent to raise e to.public static double expm1(double x)
expm1(x)
+ 1 is much closer to the true result of e^{x} than
exp(x)
.
Special cases:
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic. The
result of expm1
for any finite input must be greater than or equal to 1.0
. Note
that once the exact result of e^{x}  1 is within 1/2 ulp of the
limit value 1, 1.0
should be returned.
x
 the exponent to raise e to in the computation of e^{x
} 1.public static double floor(double a)
double
value that is less than or
equal to the argument and is equal to a mathematical integer. Special cases:
a
 a value.public static int getExponent(double d)
double
. Special cases:
Double.MAX_EXPONENT
+ 1.
Double.MIN_EXPONENT
1.
d
 a double
valuepublic static int getExponent(float f)
float
. Special cases:
Float.MAX_EXPONENT
+ 1.
Float.MIN_EXPONENT
1.
f
 a float
valuepublic static double hypot(double x, double y)
Special cases:
The computed result must be within 1 ulp of the exact result. If one parameter is held constant, the results must be semimonotonic in the other parameter.
x
 a valuey
 a valuepublic static double IEEEremainder(double f1, double f2)
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
 the dividend.f2
 the divisor.f1
is divided by f2
.public static double log(double a)
double
value. Special cases:
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 a valuea
, the natural logarithm of a
.public static double log10(double a)
double
value. Special cases:
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 a valuea
.public static double log1p(double x)
x
, the result of log1p(x)
is much closer to the true result of ln(1 + x
)
than the floatingpoint evaluation of log(1.0+x)
.
Special cases:
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
x
 a valuex
+ 1), the natural log of x
+ 1public static double max(double a, double b)
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.a
 an argument.b
 another argument.a
and b
.public static float max(float a, float b)
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.a
 an argument.b
 another argument.a
and b
.public static int max(int a, int b)
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.a
 an argument.b
 another argument.a
and b
.public static long max(long a, long b)
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.a
 an argument.b
 another argument.a
and b
.public static double min(double a, double b)
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.a
 an argument.b
 another argument.a
and b
.public static float min(float a, float b)
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.a
 an argument.b
 another argument.a
and b
.public static int min(int a, int b)
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.a
 an argument.b
 another argument.a
and b
.public static long min(long a, long b)
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.a
 an argument.b
 another argument.a
and b
.public static double nextAfter(double start, double direction)
Special cases:
direction
is returned unchanged (as implied by
the requirement of returning the second argument if the arguments compare as equal).
start
is ±Double.MIN_VALUE
and direction
has a value such
that the result should have a smaller magnitude, then a zero with the same sign as start
is returned.
start
is infinite and direction
has a value such that the result should
have a smaller magnitude, Double.MAX_VALUE
with the same sign as start
is
returned.
start
is equal to ± Double.MAX_VALUE
and direction
has a
value such that the result should have a larger magnitude, an infinity with same sign as
start
is returned.
start
 starting floatingpoint valuedirection
 value indicating which of start
's neighbors or start
should be returnedstart
in the direction of
direction
.public static float nextAfter(float start, double direction)
Special cases:
direction
is returned.
start
is ±Float.MIN_VALUE
and direction
has a value such
that the result should have a smaller magnitude, then a zero with the same sign as start
is returned.
start
is infinite and direction
has a value such that the result should
have a smaller magnitude, Float.MAX_VALUE
with the same sign as start
is
returned.
start
is equal to ± Float.MAX_VALUE
and direction
has a
value such that the result should have a larger magnitude, an infinity with same sign as
start
is returned.
start
 starting floatingpoint valuedirection
 value indicating which of start
's neighbors or start
should be returnedstart
in the direction of
direction
.public static double nextUp(double d)
d
in the direction of positive infinity.
This method is semantically equivalent to nextAfter(d,
Double.POSITIVE_INFINITY)
; however, a nextUp
implementation may run faster than its
equivalent nextAfter
call.
Special Cases:
Double.MIN_VALUE
d
 starting floatingpoint valuepublic static float nextUp(float f)
f
in the direction of positive infinity.
This method is semantically equivalent to nextAfter(f,
Float.POSITIVE_INFINITY)
; however, a nextUp
implementation may run faster than its
equivalent nextAfter
call.
Special Cases:
Float.MIN_VALUE
f
 starting floatingpoint valuepublic static double pow(double a, double b)
double
value.
(In the foregoing descriptions, a floatingpoint 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 oneargument 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 semimonotonic.
a
 the base.b
 the exponent.a
^{b}.public static double random()
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 pseudorandomnumber generator, exactly as if by the expression
new java.util.Random()
This new pseudorandomnumber 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 pseudorandomnumber generator.
double
greater than or equal to 0.0
and less than
1.0
.Random.nextDouble()
public static double rint(double a)
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:
a
 a double
value.a
that is equal to a mathematical integer.public static long round(double a)
long
to the argument, with ties rounding up.
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
.
a
 a floatingpoint value to be rounded to a long
.long
value.Long.MAX_VALUE
,
Long.MIN_VALUE
public static int round(float a)
int
to the argument, with ties rounding up.
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
.
a
 a floatingpoint value to be rounded to an integer.int
value.Integer.MAX_VALUE
,
Integer.MIN_VALUE
public static double scalb(double d, int scaleFactor)
d
× 2^{scaleFactor} rounded as if performed by a single
correctly rounded floatingpoint multiply to a member of the double value set. See the Java
Language Specification for a discussion of floatingpoint value sets. If the exponent of the
result is between Double.MIN_EXPONENT
and Double.MAX_EXPONENT
, the answer is
calculated exactly. If the exponent of the result would be larger than
Double.MAX_EXPONENT
, an infinity is returned. Note that if the result is subnormal,
precision may be lost; that is, when scalb(x, n)
is subnormal,
scalb(scalb(x, n), n)
may not equal x. When the result is nonNaN, the result has
the same sign as d
.
Special cases:
d
 number to be scaled by a power of two.scaleFactor
 power of 2 used to scale d
d
× 2^{scaleFactor}public static float scalb(float f, int scaleFactor)
f
× 2^{scaleFactor} rounded as if performed by a single
correctly rounded floatingpoint multiply to a member of the float value set. See the Java
Language Specification for a discussion of floatingpoint value sets. If the exponent of the
result is between Float.MIN_EXPONENT
and Float.MAX_EXPONENT
, the answer is
calculated exactly. If the exponent of the result would be larger than
Float.MAX_EXPONENT
, an infinity is returned. Note that if the result is subnormal,
precision may be lost; that is, when scalb(x, n)
is subnormal,
scalb(scalb(x, n), n)
may not equal x. When the result is nonNaN, the result has
the same sign as f
.
Special cases:
f
 number to be scaled by a power of two.scaleFactor
 power of 2 used to scale f
f
× 2^{scaleFactor}public static double signum(double d)
Special Cases:
d
 the floatingpoint value whose signum is to be returnedpublic static float signum(float f)
Special Cases:
f
 the floatingpoint value whose signum is to be returnedpublic static double sin(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 an angle, in radians.public static double sinh(double x)
double
value. The hyperbolic sine of x is defined
to be (e^{x}  e^{x})/2 where e is Euler's number.
Special cases:
The computed result must be within 2.5 ulps of the exact result.
x
 The number whose hyperbolic sine is to be returned.x
.public static double sqrt(double a)
double
value. Special cases:
double
value closest to the true mathematical square root of
the argument value.a
 a value.a
. If the argument is NaN or less than zero, the
result is NaN.public static double tan(double a)
The computed result must be within 1 ulp of the exact result. Results must be semimonotonic.
a
 an angle, in radians.public static double tanh(double x)
double
value. The hyperbolic tangent of x is
defined to be (e^{x}  e^{x})/(e^{x} +
e^{x}), in other words, sinh(x)/
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.
x
 The number whose hyperbolic tangent is to be returned.x
.public static double toDegrees(double angrad)
cos(toRadians(90.0))
to exactly equal 0.0
.angrad
 an angle, in radiansangrad
in degrees.public static double toRadians(double angdeg)
angdeg
 an angle, in degreesangdeg
in radians.public static double ulp(double d)
double
value is the positive
distance between this floatingpoint value and the double
value next larger in magnitude.
Note that for nonNaN x, ulp(x) == ulp(x)
.
Special Cases:
Double.MIN_VALUE
.
Double.MAX_VALUE
, then the result is equal to
2^{971}.
d
 the floatingpoint value whose ulp is to be returnedpublic static float ulp(float f)
float
value is the positive
distance between this floatingpoint value and the float
value next larger in magnitude.
Note that for nonNaN x, ulp(x) == ulp(x)
.
Special Cases:
Float.MIN_VALUE
.
Float.MAX_VALUE
, then the result is equal to
2^{104}.
f
 the floatingpoint value whose ulp is to be returned