
Java™ Platform Standard Ed. 6 

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java.lang.Object java.lang.Math
public final class Math
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
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.
Field Summary  

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  

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. 
static int 
round(float a)
Returns the closest int to the argument. 
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. 
Methods inherited from class java.lang.Object 

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

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.
Method Detail 

public 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 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 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 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 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 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 toRadians(double angdeg)
angdeg
 an angle, in degrees
angdeg
in radians.public static double toDegrees(double angrad)
cos(toRadians(90.0))
to exactly
equal 0.0
.
angrad
 an angle, in radians
angrad
in degrees.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 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 value
a
, 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 value
a
.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 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 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 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 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 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 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 coordinate
public 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 int round(float a)
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
.
a
 a floatingpoint value to be rounded to an integer.
int
value.Integer.MAX_VALUE
,
Integer.MIN_VALUE
public static long round(double a)
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
.
a
 a floatingpoint value to be rounded to a
long
.
long
value.Long.MAX_VALUE
,
Long.MIN_VALUE
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
This new pseudorandomnumber 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 pseudorandomnumber generator.
double
greater than or equal
to 0.0
and less than 1.0
.Random.nextDouble()
public 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 determined
Integer.MIN_VALUE
public 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 determined
Long.MIN_VALUE
public 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 determined
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 determined
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
.Long.MAX_VALUE
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
.Long.MAX_VALUE
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 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 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
.Long.MIN_VALUE
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
.Long.MIN_VALUE
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 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 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 returned
public 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
public static double signum(double d)
Special Cases:
d
 the floatingpoint value whose signum is to be returned
public static float signum(float f)
Special Cases:
f
 the floatingpoint value whose signum is to be returned
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 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 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 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 value
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 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 value
x
+ 1), the natural
log of x
+ 1public static double copySign(double magnitude, double sign)
StrictMath.copySign
method, this method does not require NaN sign
arguments to be treated as positive values; implementations are
permitted to treat some NaN arguments as positive and other NaN
arguments as negative to allow greater performance.
magnitude
 the parameter providing the magnitude of the resultsign
 the parameter providing the sign of the result
magnitude
and the sign of sign
.public static float copySign(float magnitude, float sign)
StrictMath.copySign
method, this method does not require NaN sign
arguments to be treated as positive values; implementations are
permitted to treat some NaN arguments as positive and other NaN
arguments as negative to allow greater performance.
magnitude
 the parameter providing the magnitude of the resultsign
 the parameter providing the sign of the result
magnitude
and the sign of sign
.public static int getExponent(float f)
float
. Special cases:
Float.MAX_EXPONENT
+ 1.
Float.MIN_EXPONENT
1.
f
 a float
value
public static int getExponent(double d)
double
. Special cases:
Double.MAX_EXPONENT
+ 1.
Double.MIN_EXPONENT
1.
d
 a double
value
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 returned
start
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 returned
start
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 value
public 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 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}

Java™ Platform Standard Ed. 6 

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