# 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 {@linkplain #ulp(double) ulp} of a specific real number value is the distance between the two floating-point values bracketing that numerical value. When discussing the accuracy of a method as a whole rather than at a specific argument, the number of ulps cited is for the worst-case error at any argument. If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is 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.

The platform uses signed two's complement integer arithmetic with int and long primitive types. The developer should choose the primitive type to ensure that arithmetic operations consistently produce correct results, which in some cases means the operations will not overflow the range of values of the computation. The best practice is to choose the primitive type and algorithm to avoid overflow. In cases where the size is `int` or `long` and overflow errors need to be detected, the methods `addExact`, `subtractExact`, `multiplyExact`, and `toIntExact` throw an `ArithmeticException` when the results overflow. For other arithmetic operations such as divide, absolute value, increment, decrement, and negation overflow occurs only with a specific minimum or maximum value and should be checked against the minimum or maximum as appropriate.

### Constant Summary

 double E The `double` value that is closer than any other to e, the base of the natural logarithms. double PI The `double` value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.

### Public Method Summary

 static double IEEEremainder(double f1, double f2) Computes the remainder operation on two arguments as prescribed by the IEEE 754 standard. static long abs(long a) Returns the absolute value of a `long` value. static int abs(int a) Returns the absolute value of an `int` value. static float abs(float a) Returns the absolute value of a `float` value. static double abs(double a) Returns the absolute value of a `double` 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 int addExact(int x, int y) Returns the sum of its arguments, throwing an exception if the result overflows an `int`. static long addExact(long x, long y) Returns the sum of its arguments, throwing an exception if the result overflows a `long`. 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 float copySign(float magnitude, float sign) Returns the first floating-point argument with the sign of the second floating-point argument. static double copySign(double magnitude, double sign) Returns the first floating-point argument with the sign of the second floating-point 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 long decrementExact(long a) Returns the argument decremented by one, throwing an exception if the result overflows a `long`. static int decrementExact(int a) Returns the argument decremented by one, throwing an exception if the result overflows an `int`. static double exp(double a) Returns Euler's number e raised to the power of a `double` value. static double expm1(double x) Returns ex -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 floorDiv(int x, int y) Returns the largest (closest to positive infinity) `int` value that is less than or equal to the algebraic quotient. static long floorDiv(long x, long y) Returns the largest (closest to positive infinity) `long` value that is less than or equal to the algebraic quotient. static long floorMod(long x, long y) Returns the floor modulus of the `long` arguments. static int floorMod(int x, int y) Returns the floor modulus of the `int` arguments. 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(x2 +y2) without intermediate overflow or underflow. static int incrementExact(int a) Returns the argument incremented by one, throwing an exception if the result overflows an `int`. static long incrementExact(long a) Returns the argument incremented by one, throwing an exception if the result overflows a `long`. 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 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 float max(float a, float b) Returns the greater of two `float` values. static double max(double a, double b) Returns the greater of two `double` values. static float min(float a, float b) Returns the smaller of two `float` values. static double min(double a, double b) Returns the smaller of two `double` 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 int multiplyExact(int x, int y) Returns the product of the arguments, throwing an exception if the result overflows an `int`. static long multiplyExact(long x, long y) Returns the product of the arguments, throwing an exception if the result overflows a `long`. static int negateExact(int a) Returns the negation of the argument, throwing an exception if the result overflows an `int`. static long negateExact(long a) Returns the negation of the argument, throwing an exception if the result overflows a `long`. static double nextAfter(double start, double direction) Returns the floating-point number adjacent to the first argument in the direction of the second argument. static float nextAfter(float start, double direction) Returns the floating-point number adjacent to the first argument in the direction of the second argument. static double nextDown(double d) Returns the floating-point value adjacent to `d` in the direction of negative infinity. static float nextDown(float f) Returns the floating-point value adjacent to `f` in the direction of negative infinity. static float nextUp(float f) Returns the floating-point value adjacent to `f` in the direction of positive infinity. static double nextUp(double d) Returns the floating-point value adjacent to `d` 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 to positive infinity. static int round(float a) Returns the closest `int` to the argument, with ties rounding to positive infinity. static float scalb(float f, int scaleFactor) Returns `f` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply to a member of the float value set. static double scalb(double d, int scaleFactor) Returns `d` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply to a member of the double 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 long subtractExact(long x, long y) Returns the difference of the arguments, throwing an exception if the result overflows a `long`. static int subtractExact(int x, int y) Returns the difference of the arguments, throwing an exception if the result overflows an `int`. 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 int toIntExact(long value) Returns the value of the `long` argument; throwing an exception if the value overflows an `int`. 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.

## Constants

#### public static final double E

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

Constant Value: 2.718281828459045

#### 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.

Constant Value: 3.141592653589793

## Public Methods

#### 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. the divisor.
##### Returns
• the remainder when `f1` is divided by `f2`.

#### 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### public static int addExact(int x, int y)

Returns the sum of its arguments, throwing an exception if the result overflows an `int`.

##### Parameters
x the first value the second value
• the result
##### Throws
ArithmeticException if the result overflows an int

#### public static long addExact(long x, long y)

Returns the sum of its arguments, throwing an exception if the result overflows a `long`.

##### Parameters
x the first value the second value
• the result
##### Throws
ArithmeticException if the result overflows a long

#### 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.

#### 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.

#### 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 the abscissa coordinate
##### Returns
• the theta component of the point (rtheta) in polar coordinates that corresponds to the point (xy) in Cartesian coordinates.

#### public static double cbrt(double a)

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:

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

##### Parameters
a a value.
##### Returns
• the cube root of `a`.

#### 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.

#### public static float copySign(float magnitude, float sign)

Returns the first floating-point argument with the sign of the second floating-point argument. Note that unlike the `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.

##### Parameters
magnitude the parameter providing the magnitude of the result the parameter providing the sign of the result
##### Returns
• a value with the magnitude of `magnitude` and the sign of `sign`.

#### public static double copySign(double magnitude, double sign)

Returns the first floating-point argument with the sign of the second floating-point argument. Note that unlike the `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.

##### Parameters
magnitude the parameter providing the magnitude of the result the parameter providing the sign of the result
##### Returns
• a value with the magnitude of `magnitude` and the sign of `sign`.

#### 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.

##### Returns
• the cosine of the argument.

#### public static double cosh(double 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:

• If the argument is NaN, then the result is NaN.
• If the argument is infinite, then the result is positive infinity.
• If the argument is zero, then the result is `1.0`.

The computed result must be within 2.5 ulps of the exact result.

##### Parameters
x The number whose hyperbolic cosine is to be returned.
##### Returns
• The hyperbolic cosine of `x`.

#### public static long decrementExact(long a)

Returns the argument decremented by one, throwing an exception if the result overflows a `long`.

##### Parameters
a the value to decrement
• the result
##### Throws
ArithmeticException if the result overflows a long

#### public static int decrementExact(int a)

Returns the argument decremented by one, throwing an exception if the result overflows an `int`.

##### Parameters
a the value to decrement
• the result
##### Throws
ArithmeticException if the result overflows an int

#### 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.

#### public static double expm1(double x)

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:

• 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 -1.0.
• 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. 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.

##### Parameters
x the exponent to raise e to in the computation of e`x` -1.
##### Returns
• the value e`x` - 1.

#### 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.

#### public static int floorDiv(int x, int y)

Returns the largest (closest to positive infinity) `int` value that is less than or equal to the algebraic quotient. There is one special case, if the dividend is the {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is `-1`, then integer overflow occurs and the result is equal to the `Integer.MIN_VALUE`.

Normal integer division operates under the round to zero rounding mode (truncation). This operation instead acts under the round toward negative infinity (floor) rounding mode. The floor rounding mode gives different results than truncation when the exact result is negative.

• If the signs of the arguments are the same, the results of `floorDiv` and the `/` operator are the same.
For example, `floorDiv(4, 3) == 1` and `(4 / 3) == 1`.
• If the signs of the arguments are different, the quotient is negative and `floorDiv` returns the integer less than or equal to the quotient and the `/` operator returns the integer closest to zero.
For example, `floorDiv(-4, 3) == -2`, whereas `(-4 / 3) == -1`.

##### Parameters
x the dividend the divisor
##### Returns
• the largest (closest to positive infinity) `int` value that is less than or equal to the algebraic quotient.
##### Throws
ArithmeticException if the divisor `y` is zero
• `floorMod(int, int)`
• `floor(double)`

#### public static long floorDiv(long x, long y)

Returns the largest (closest to positive infinity) `long` value that is less than or equal to the algebraic quotient. There is one special case, if the dividend is the {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is `-1`, then integer overflow occurs and the result is equal to the `Long.MIN_VALUE`.

Normal integer division operates under the round to zero rounding mode (truncation). This operation instead acts under the round toward negative infinity (floor) rounding mode. The floor rounding mode gives different results than truncation when the exact result is negative.

For examples, see `floorDiv(int, int)`.

##### Parameters
x the dividend the divisor
##### Returns
• the largest (closest to positive infinity) `long` value that is less than or equal to the algebraic quotient.
##### Throws
ArithmeticException if the divisor `y` is zero
• `floorMod(long, long)`
• `floor(double)`

#### public static long floorMod(long x, long y)

Returns the floor modulus of the `long` arguments.

The floor modulus is `x - (floorDiv(x, y) * y)`, has the same sign as the divisor `y`, and is in the range of `-abs(y) < r < +abs(y)`.

The relationship between `floorDiv` and `floorMod` is such that:

• `floorDiv(x, y) * y + floorMod(x, y) == x`

For examples, see `floorMod(int, int)`.

##### Parameters
x the dividend the divisor
##### Returns
• the floor modulus `x - (floorDiv(x, y) * y)`
##### Throws
ArithmeticException if the divisor `y` is zero
• `floorDiv(long, long)`

#### public static int floorMod(int x, int y)

Returns the floor modulus of the `int` arguments.

The floor modulus is `x - (floorDiv(x, y) * y)`, has the same sign as the divisor `y`, and is in the range of `-abs(y) < r < +abs(y)`.

The relationship between `floorDiv` and `floorMod` is such that:

• `floorDiv(x, y) * y + floorMod(x, y) == x`

The difference in values between `floorMod` and the `%` operator is due to the difference between `floorDiv` that returns the integer less than or equal to the quotient and the `/` operator that returns the integer closest to zero.

Examples:

• If the signs of the arguments are the same, the results of `floorMod` and the `%` operator are the same.
• `floorMod(4, 3) == 1`;   and `(4 % 3) == 1`
• If the signs of the arguments are different, the results differ from the `%` operator.
• `floorMod(+4, -3) == -2`;   and `(+4 % -3) == +1`
• `floorMod(-4, +3) == +2`;   and `(-4 % +3) == -1`
• `floorMod(-4, -3) == -1`;   and `(-4 % -3) == -1 `

If the signs of arguments are unknown and a positive modulus is needed it can be computed as `(floorMod(x, y) + abs(y)) % abs(y)`.

##### Parameters
x the dividend the divisor
##### Returns
• the floor modulus `x - (floorDiv(x, y) * y)`
##### Throws
ArithmeticException if the divisor `y` is zero
• `floorDiv(int, int)`

#### public static int getExponent(double d)

Returns the unbiased exponent used in the representation of a `double`. Special cases:

• If the argument is NaN or infinite, then the result is `Double.MAX_EXPONENT` + 1.
• If the argument is zero or subnormal, then the result is `Double.MIN_EXPONENT` -1.

##### Parameters
d a `double` value
##### Returns
• the unbiased exponent of the argument

#### public static int getExponent(float f)

Returns the unbiased exponent used in the representation of a `float`. Special cases:

• If the argument is NaN or infinite, then the result is `Float.MAX_EXPONENT` + 1.
• If the argument is zero or subnormal, then the result is `Float.MIN_EXPONENT` -1.

##### Parameters
f a `float` value
##### Returns
• the unbiased exponent of the argument

#### public static double hypot(double x, double y)

Returns sqrt(x2 +y2) without intermediate overflow or underflow.

Special cases:

• If either argument is infinite, then the result is positive infinity.
• If either argument is NaN and neither argument is infinite, then the result is NaN.

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.

##### Parameters
x a value a value
##### Returns
• sqrt(x2 +y2) without intermediate overflow or underflow

#### public static int incrementExact(int a)

Returns the argument incremented by one, throwing an exception if the result overflows an `int`.

##### Parameters
a the value to increment
• the result
##### Throws
ArithmeticException if the result overflows an int

#### public static long incrementExact(long a)

Returns the argument incremented by one, throwing an exception if the result overflows a `long`.

##### Parameters
a the value to increment
• the result
##### Throws
ArithmeticException if the result overflows a long

#### 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`.

#### 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`.

#### public static double log1p(double 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:

• If the argument is NaN or less than -1, then the result is NaN.
• If the argument is positive infinity, then the result is positive infinity.
• If the argument is negative one, then the result is negative infinity.
• 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
x a value
##### Returns
• the value ln(`x` + 1), the natural log of `x` + 1

#### 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. another argument.
##### Returns
• the larger of `a` and `b`.

#### 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. another argument.
##### Returns
• the larger of `a` and `b`.

#### 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. another argument.
##### Returns
• the larger of `a` and `b`.

#### 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. another argument.
##### Returns
• the larger of `a` and `b`.

#### 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. another argument.
##### Returns
• the smaller of `a` and `b`.

#### 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. another argument.
##### Returns
• the smaller of `a` and `b`.

#### 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. another argument.
##### Returns
• the smaller of `a` and `b`.

#### 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. another argument.
##### Returns
• the smaller of `a` and `b`.

#### public static int multiplyExact(int x, int y)

Returns the product of the arguments, throwing an exception if the result overflows an `int`.

##### Parameters
x the first value the second value
• the result
##### Throws
ArithmeticException if the result overflows an int

#### public static long multiplyExact(long x, long y)

Returns the product of the arguments, throwing an exception if the result overflows a `long`.

##### Parameters
x the first value the second value
• the result
##### Throws
ArithmeticException if the result overflows a long

#### public static int negateExact(int a)

Returns the negation of the argument, throwing an exception if the result overflows an `int`.

##### Parameters
a the value to negate
• the result
##### Throws
ArithmeticException if the result overflows an int

#### public static long negateExact(long a)

Returns the negation of the argument, throwing an exception if the result overflows a `long`.

##### Parameters
a the value to negate
• the result
##### Throws
ArithmeticException if the result overflows a long

#### public static double nextAfter(double start, double direction)

Returns the floating-point number adjacent to the first argument in the direction of the second argument. If both arguments compare as equal the second argument is returned.

Special cases:

• If either argument is a NaN, then NaN is returned.
• If both arguments are signed zeros, `direction` is returned unchanged (as implied by the requirement of returning the second argument if the arguments compare as equal).
• If `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.
• If `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.
• If `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.

##### Parameters
start starting floating-point value value indicating which of `start`'s neighbors or `start` should be returned
##### Returns
• The floating-point number adjacent to `start` in the direction of `direction`.

#### public static float nextAfter(float start, double direction)

Returns the floating-point number adjacent to the first argument in the direction of the second argument. If both arguments compare as equal a value equivalent to the second argument is returned.

Special cases:

• If either argument is a NaN, then NaN is returned.
• If both arguments are signed zeros, a value equivalent to `direction` is returned.
• If `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.
• If `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.
• If `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.

##### Parameters
start starting floating-point value value indicating which of `start`'s neighbors or `start` should be returned
##### Returns
• The floating-point number adjacent to `start` in the direction of `direction`.

#### public static double nextDown(double d)

Returns the floating-point value adjacent to `d` in the direction of negative infinity. This method is semantically equivalent to ```nextAfter(d, Double.NEGATIVE_INFINITY)```; however, a `nextDown` implementation may run faster than its equivalent `nextAfter` call.

Special Cases:

• If the argument is NaN, the result is NaN.
• If the argument is negative infinity, the result is negative infinity.
• If the argument is zero, the result is `-Double.MIN_VALUE`

##### Parameters
d starting floating-point value
##### Returns
• The adjacent floating-point value closer to negative infinity.