Module java.base
Package java.lang

# Class Math

java.lang.Object
java.lang.Math

public final class Math extends Object
The class `Math` contains methods for performing basic numeric operations such as the elementary exponential, logarithm, square root, and trigonometric functions.

Unlike some of the numeric methods of class `StrictMath`, all implementations of the equivalent functions of class `Math` are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required.

By default many of the `Math` methods simply call the equivalent method in `StrictMath` for their implementation. Code generators are encouraged to use platform-specific native libraries or microprocessor instructions, where available, to provide higher-performance implementations of `Math` methods. Such higher-performance implementations still must conform to the specification for `Math`.

The quality of implementation specifications concern two properties, accuracy of the returned result and monotonicity of the method. Accuracy of the floating-point `Math` methods is measured in terms of ulps, units in the last place. For a given floating-point format, an ulp of a specific real number value is the distance between the two floating-point values bracketing that numerical value. When discussing the accuracy of a method as a whole rather than at a specific argument, the number of ulps cited is for the worst-case error at any argument. If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is correctly rounded. A correctly rounded method is generally the best a floating-point approximation can be; however, it is impractical for many floating-point methods to be correctly rounded. Instead, for the `Math` class, a larger error bound of 1 or 2 ulps is allowed for certain methods. Informally, with a 1 ulp error bound, when the exact result is a representable number, the exact result should be returned as the computed result; otherwise, either of the two floating-point values which bracket the exact result may be returned. For exact results large in magnitude, one of the endpoints of the bracket may be infinite. Besides accuracy at individual arguments, maintaining proper relations between the method at different arguments is also important. Therefore, most methods with more than 0.5 ulp errors are required to be semi-monotonic: whenever the mathematical function is non-decreasing, so is the floating-point approximation, likewise, whenever the mathematical function is non-increasing, so is the floating-point approximation. Not all approximations that have 1 ulp accuracy will automatically meet the monotonicity requirements.

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`, `toIntExact`, `incrementExact`, `decrementExact` and `negateExact` throw an `ArithmeticException` when the results overflow. For the arithmetic operations divide and absolute value, overflow occurs only with a specific minimum or maximum value and should be checked against the minimum or maximum as appropriate.

## IEEE 754 Recommended Operations

The 2019 revision of the IEEE 754 floating-point standard includes a section of recommended operations and the semantics of those operations if they are included in a programming environment. The recommended operations present in this class include `sin`, `cos`, `tan`, `asin`, `acos`, `atan`, `exp`, `expm1`, `log`, `log10`, `log1p`, `sinh`, `cosh`, `tanh`, `hypot`, and `pow`. (The `sqrt` operation is a required part of IEEE 754 from a different section of the standard.) The special case behavior of the recommended operations generally follows the guidance of the IEEE 754 standard. However, the `pow` method defines different behavior for some arguments, as noted in its specification. The IEEE 754 standard defines its operations to be correctly rounded, which is a more stringent quality of implementation condition than required for most of the methods in question that are also included in this class.

Since:
  1.0

- ## Field Summary

  Fields
  Modifier and Type
  Field
  Description
  `static final double`
  `E`
  The `double` value that is closer than any other to e, the base of the natural logarithms.
  `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.

- ## Method Summary

  Modifier and Type
  Method
  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 int`
  `absExact(int a)`
  Returns the mathematical absolute value of an `int` value if it is exactly representable as an `int`, throwing `ArithmeticException` if the result overflows the positive `int` range.
  `static long`
  `absExact(long a)`
  Returns the mathematical absolute value of an `long` value if it is exactly representable as an `long`, throwing `ArithmeticException` if the result overflows the positive `long` range.
  `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 double`
  `copySign(double magnitude, double sign)`
  Returns the first floating-point argument with the sign of the second floating-point argument.
  `static float`
  `copySign(float magnitude, float 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 int`
  `decrementExact(int a)`
  Returns the argument decremented by one, throwing an exception if the result overflows an `int`.
  `static long`
  `decrementExact(long a)`
  Returns the argument decremented by one, throwing an exception if the result overflows a `long`.
  `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, int y)`
  Returns the largest (closest to positive infinity) `long` 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 int`
  `floorMod(int x, int y)`
  Returns the floor modulus of the `int` arguments.
  `static int`
  `floorMod(long x, int y)`
  Returns the floor modulus of the `long` and `int` arguments.
  `static long`
  `floorMod(long x, long y)`
  Returns the floor modulus of the `long` arguments.
  `static double`
  `fma(double a, double b, double c)`
  Returns the fused multiply add of the three arguments; that is, returns the exact product of the first two arguments summed with the third argument and then rounded once to the nearest `double`.
  `static float`
  `fma(float a, float b, float c)`
  Returns the fused multiply add of the three arguments; that is, returns the exact product of the first two arguments summed with the third argument and then rounded once to the nearest `float`.
  `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 double`
  `IEEEremainder(double f1, double f2)`
  Computes the remainder operation on two arguments as prescribed by the IEEE 754 standard.
  `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 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 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, int y)`
  Returns the product of the arguments, throwing an exception if the result overflows a `long`.
  `static long`
  `multiplyExact(long x, long y)`
  Returns the product of the arguments, throwing an exception if the result overflows a `long`.
  `static long`
  `multiplyFull(int x, int y)`
  Returns the exact mathematical product of the arguments.
  `static long`
  `multiplyHigh(long x, long y)`
  Returns as a `long` the most significant 64 bits of the 128-bit product of two 64-bit factors.
  `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 double`
  `nextUp(double d)`
  Returns the floating-point value adjacent to `d` in the direction of positive infinity.
  `static float`
  `nextUp(float f)`
  Returns the floating-point 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 to positive infinity.
  `static int`
  `round(float a)`
  Returns the closest `int` to the argument, with ties rounding to positive infinity.
  `static double`
  `scalb(double d, int scaleFactor)`
  Returns `d` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply.
  `static float`
  `scalb(float f, int scaleFactor)`
  Returns `f` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply.
  `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 int`
  `subtractExact(int x, int y)`
  Returns the difference of the arguments, throwing an exception if the result overflows an `int`.
  `static long`
  `subtractExact(long x, long y)`
  Returns the difference of the arguments, throwing an exception if the result overflows a `long`.
  `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.

  ### Methods declared in class java.lang.Object

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

- ## Field Details

  - ### E

    public static final double E
    The `double` value that is closer than any other to e, the base of the natural logarithms.
    See Also:
      - Constant Field Values

  - ### PI

    public static final double PI
    The `double` value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.
    See Also:
      - Constant Field Values

- ## Method Details

  - ### sin

    public static double sin(double a)
    Returns the trigonometric sine of an angle. Special cases:
    - If the argument is NaN or an infinity, then the result is NaN.
    - If the argument is zero, then the result is a zero with the same sign as the argument.

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

    Parameters:
      `a` - an angle, in radians.
    Returns:
      the sine of the argument.

  - ### cos

    public static double cos(double a)
    Returns the trigonometric cosine of an angle. Special cases:
    - If the argument is NaN or an infinity, then the result is NaN.
    - If the argument is zero, then the result is `1.0`.

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

    Parameters:
      `a` - an angle, in radians.
    Returns:
      the cosine of the argument.

  - ### tan

    public static double tan(double a)
    Returns the trigonometric tangent of an angle. Special cases:
    - If the argument is NaN or an infinity, then the result is NaN.
    - If the argument is zero, then the result is a zero with the same sign as the argument.

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

    Parameters:
      `a` - an angle, in radians.
    Returns:
      the tangent of the argument.

  - ### asin

    public static double asin(double a)
    Returns the arc sine of a value; the returned angle is in the range -pi/2 through pi/2. Special cases:
    - If the argument is NaN or its absolute value is greater than 1, then the result is NaN.
    - If the argument is zero, then the result is a zero with the same sign as the argument.

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

    Parameters:
      `a` - the value whose arc sine is to be returned.
    Returns:
      the arc sine of the argument.

  - ### acos

    public static double acos(double a)
    Returns the arc cosine of a value; the returned angle is in the range 0.0 through pi. Special case:
    - If the argument is NaN or its absolute value is greater than 1, then the result is NaN.
    - If the argument is `1.0`, 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 value whose arc cosine is to be returned.
    Returns:
      the arc cosine of the argument.

  - ### atan

    public static double atan(double a)
    Returns the arc tangent of a value; the returned angle is in the range -pi/2 through pi/2. Special cases:
    - If the argument is NaN, then the result is NaN.
    - If the argument is zero, then the result is a zero with the same sign as the argument.
    - If the argument is infinite, then the result is the closest value to pi/2 with the same sign as the input.

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

    Parameters:
      `a` - the value whose arc tangent is to be returned.
    Returns:
      the arc tangent of the argument.

  - ### toRadians

    public static double toRadians(double angdeg)
    Converts an angle measured in degrees to an approximately equivalent angle measured in radians. The conversion from degrees to radians is generally inexact.
    Parameters:
      `angdeg` - an angle, in degrees
    Returns:
      the measurement of the angle `angdeg` in radians.
    Since:
      1.2

  - ### toDegrees

    public static double toDegrees(double angrad)
    Converts an angle measured in radians to an approximately equivalent angle measured in degrees. The conversion from radians to degrees is generally inexact; users should not expect `cos(toRadians(90.0))` to exactly equal `0.0`.
    Parameters:
      `angrad` - an angle, in radians
    Returns:
      the measurement of the angle `angrad` in degrees.
    Since:
      1.2

  - ### exp

    public static double exp(double a)
    Returns Euler's number e raised to the power of a `double` value. Special cases:
    - If the argument is NaN, the result is NaN.
    - If the argument is positive infinity, then the result is positive infinity.
    - If the argument is negative infinity, then the result is positive zero.
    - If the argument is zero, then the result is `1.0`.

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

    Parameters:
      `a` - the exponent to raise e to.
    Returns:
      the value e`a`, where e is the base of the natural logarithms.

  - ### log

    public static double log(double a)
    Returns the natural logarithm (base e) of a `double` value. Special cases:
    - If the argument is NaN or less than zero, then the result is NaN.
    - If the argument is positive infinity, then the result is positive infinity.
    - If the argument is positive zero or negative zero, then the result is negative infinity.
    - If the argument is `1.0`, 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` - a value
    Returns:
      the value ln `a`, the natural logarithm of `a`.

  - ### log10

    public static double log10(double a)
    Returns the base 10 logarithm of a `double` value. Special cases:
    - If the argument is NaN or less than zero, then the result is NaN.
    - If the argument is positive infinity, then the result is positive infinity.
    - If the argument is positive zero or negative zero, then the result is negative infinity.
    - If the argument is equal to 10n for integer n, then the result is n. In particular, if the argument is `1.0` (100), 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` - a value
    Returns:
      the base 10 logarithm of `a`.
    Since:
      1.5

  - ### sqrt

    public static double sqrt(double a)
    Returns the correctly rounded positive square root of a `double` value. Special cases:
    - If the argument is NaN or less than zero, then the result is NaN.
    - If the argument is positive infinity, then the result is positive infinity.
    - If the argument is positive zero or negative zero, then the result is the same as the argument.
    Otherwise, the result is the `double` value closest to the true mathematical square root of the argument value.
    Parameters:
      `a` - a value.
    Returns:
      the positive square root of `a`. If the argument is NaN or less than zero, the result is NaN.

  - ### cbrt

    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`.
    Since:
      1.5

  - ### IEEEremainder

    public static double IEEEremainder(double f1, double f2)
    Computes the remainder operation on two arguments as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to `f1 - f2` × n, where n is the mathematical integer closest to the exact mathematical value of the quotient `f1/f2`, and if two mathematical integers are equally close to `f1/f2`, then n is the integer that is even. If the remainder is zero, its sign is the same as the sign of the first argument. Special cases:
    - If either argument is NaN, or the first argument is infinite, or the second argument is positive zero or negative zero, then the result is NaN.
    - If the first argument is finite and the second argument is infinite, then the result is the same as the first argument.
    Parameters:
      `f1` - the dividend.
      `f2` - the divisor.
    Returns:
      the remainder when `f1` is divided by `f2`.

  - ### ceil

    public static double ceil(double a)
    Returns the smallest (closest to negative infinity) `double` value that is greater than or equal to the argument and is equal to a mathematical integer. Special cases:
    - If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
    - If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.
    - If the argument value is less than zero but greater than -1.0, then the result is negative zero.
    Note that the value of `Math.ceil(x)` is exactly the value of `-Math.floor(-x)`.
    Parameters:
      `a` - a value.
    Returns:
      the smallest (closest to negative infinity) floating-point value that is greater than or equal to the argument and is equal to a mathematical integer.

  - ### floor

    public static double floor(double a)
    Returns the largest (closest to positive infinity) `double` value that is less than or equal to the argument and is equal to a mathematical integer. Special cases:
    - If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
    - If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.
    Parameters:
      `a` - a value.
    Returns:
      the largest (closest to positive infinity) floating-point value that less than or equal to the argument and is equal to a mathematical integer.

  - ### rint

    public static double rint(double a)
    Returns the `double` value that is closest in value to the argument and is equal to a mathematical integer. If two `double` values that are mathematical integers are equally close, the result is the integer value that is even. Special cases:
    - If the argument value is already equal to a mathematical integer, then the result is the same as the argument.
    - If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.
    Parameters:
      `a` - a `double` value.
    Returns:
      the closest floating-point value to `a` that is equal to a mathematical integer.

  - ### atan2

    public static double atan2(double y, double x)
    Returns the angle theta from the conversion of rectangular coordinates (`x`, `y`) to polar coordinates (r, theta). This method computes the phase theta by computing an arc tangent of `y/x` in the range of -pi to pi. Special cases:
    - If either argument is NaN, then the result is NaN.
    - If the first argument is positive zero and the second argument is positive, or the first argument is positive and finite and the second argument is positive infinity, then the result is positive zero.
    - If the first argument is negative zero and the second argument is positive, or the first argument is negative and finite and the second argument is positive infinity, then the result is negative zero.
    - If the first argument is positive zero and the second argument is negative, or the first argument is positive and finite and the second argument is negative infinity, then the result is the `double` value closest to pi.
    - If the first argument is negative zero and the second argument is negative, or the first argument is negative and finite and the second argument is negative infinity, then the result is the `double` value closest to -pi.
    - If the first argument is positive and the second argument is positive zero or negative zero, or the first argument is positive infinity and the second argument is finite, then the result is the `double` value closest to pi/2.
    - If the first argument is negative and the second argument is positive zero or negative zero, or the first argument is negative infinity and the second argument is finite, then the result is the `double` value closest to -pi/2.
    - If both arguments are positive infinity, then the result is the `double` value closest to pi/4.
    - If the first argument is positive infinity and the second argument is negative infinity, then the result is the `double` value closest to 3*pi/4.
    - If the first argument is negative infinity and the second argument is positive infinity, then the result is the `double` value closest to -pi/4.
    - If both arguments are negative infinity, then the result is the `double` value closest to -3*pi/4.

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

    API Note:
      For y with a positive sign and finite nonzero x, the exact mathematical value of `atan2` is equal to:
      - If x > 0, atan(abs(y/x))
      - If x < 0, π - atan(abs(y/x))
    Parameters:
      `y` - the ordinate coordinate
      `x` - the abscissa coordinate
    Returns:
      the theta component of the point (r, theta) in polar coordinates that corresponds to the point (x, y) in Cartesian coordinates.

  - ### pow

    public static double pow(double a, double b)
    Returns the value of the first argument raised to the power of the second argument. Special cases:
    - If the second argument is positive or negative zero, then the result is 1.0.
    - If the second argument is 1.0, then the result is the same as the first argument.
    - If the second argument is NaN, then the result is NaN.
    - If the first argument is NaN and the second argument is nonzero, then the result is NaN.
    - If
      - the absolute value of the first argument is greater than 1 and the second argument is positive infinity, or
      - the absolute value of the first argument is less than 1 and the second argument is negative infinity,
      then the result is positive infinity.
    - If
      - the absolute value of the first argument is greater than 1 and the second argument is negative infinity, or
      - the absolute value of the first argument is less than 1 and the second argument is positive infinity,
      then the result is positive zero.
    - If the absolute value of the first argument equals 1 and the second argument is infinite, then the result is NaN.
    - If
      - the first argument is positive zero and the second argument is greater than zero, or
      - the first argument is positive infinity and the second argument is less than zero,
      then the result is positive zero.
    - If
      - the first argument is positive zero and the second argument is less than zero, or
      - the first argument is positive infinity and the second argument is greater than zero,
      then the result is positive infinity.
    - If
      - the first argument is negative zero and the second argument is greater than zero but not a finite odd integer, or
      - the first argument is negative infinity and the second argument is less than zero but not a finite odd integer,
      then the result is positive zero.
    - If
      - the first argument is negative zero and the second argument is a positive finite odd integer, or
      - the first argument is negative infinity and the second argument is a negative finite odd integer,
      then the result is negative zero.
    - If
      - the first argument is negative zero and the second argument is less than zero but not a finite odd integer, or
      - the first argument is negative infinity and the second argument is greater than zero but not a finite odd integer,
      then the result is positive infinity.
    - If
      - the first argument is negative zero and the second argument is a negative finite odd integer, or
      - the first argument is negative infinity and the second argument is a positive finite odd integer,
      then the result is negative infinity.
    - If the first argument is finite and less than zero
      - if the second argument is a finite even integer, the result is equal to the result of raising the absolute value of the first argument to the power of the second argument
      - if the second argument is a finite odd integer, the result is equal to the negative of the result of raising the absolute value of the first argument to the power of the second argument
      - if the second argument is finite and not an integer, then the result is NaN.
    - If both arguments are integers, then the result is exactly equal to the mathematical result of raising the first argument to the power of the second argument if that result can in fact be represented exactly as a `double` value.

    (In the foregoing descriptions, a floating-point value is considered to be an integer if and only if it is finite and a fixed point of the method `ceil` or, equivalently, a fixed point of the method `floor`. A value is a fixed point of a one-argument method if and only if the result of applying the method to the value is equal to the value.)

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

    API Note:
      The special cases definitions of this method differ from the special case definitions of the IEEE 754 recommended `pow` operation for ±`1.0` raised to an infinite power. This method treats such cases as indeterminate and specifies a NaN is returned. The IEEE 754 specification treats the infinite power as a large integer (large-magnitude floating-point numbers are numerically integers, specifically even integers) and therefore specifies `1.0` be returned.
    Parameters:
      `a` - the base.
      `b` - the exponent.
    Returns:
      the value `a``b`.

  - ### round

    public static int round(float a)
    Returns the closest `int` to the argument, with ties rounding to positive infinity.

    Special cases:

    - If the argument is NaN, the result is 0.
    - If the argument is negative infinity or any value less than or equal to the value of `Integer.MIN_VALUE`, the result is equal to the value of `Integer.MIN_VALUE`.
    - If the argument is positive infinity or any value greater than or equal to the value of `Integer.MAX_VALUE`, the result is equal to the value of `Integer.MAX_VALUE`.
    Parameters:
      `a` - a floating-point value to be rounded to an integer.
    Returns:
      the value of the argument rounded to the nearest `int` value.
    See Also:
      - `Integer.MAX_VALUE`
      - `Integer.MIN_VALUE`

  - ### round

    public static long round(double a)
    Returns the closest `long` to the argument, with ties rounding to positive infinity.

    Special cases:

    - If the argument is NaN, the result is 0.
    - If the argument is negative infinity or any value less than or equal to the value of `Long.MIN_VALUE`, the result is equal to the value of `Long.MIN_VALUE`.
    - If the argument is positive infinity or any value greater than or equal to the value of `Long.MAX_VALUE`, the result is equal to the value of `Long.MAX_VALUE`.
    Parameters:
      `a` - a floating-point value to be rounded to a `long`.
    Returns:
      the value of the argument rounded to the nearest `long` value.
    See Also:
      - `Long.MAX_VALUE`
      - `Long.MIN_VALUE`

  - ### random

    public static double random()
    Returns a `double` value with a positive sign, greater than or equal to `0.0` and less than `1.0`. Returned values are chosen pseudorandomly with (approximately) uniform distribution from that range.

    When this method is first called, it creates a single new pseudorandom-number generator, exactly as if by the expression

    > `new java.util.Random()`

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

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

    API Note:
      As the largest `double` value less than `1.0` is `Math.nextDown(1.0)`, a value `x` in the closed range `[x1,x2]` where `x1<=x2` may be defined by the statements

      >      double f = Math.random()/Math.nextDown(1.0);
      >      double x = x1*(1.0 - f) + x2*f;

    Returns:
      a pseudorandom `double` greater than or equal to `0.0` and less than `1.0`.
    See Also:
      - `nextDown(double)`
      - `Random.nextDouble()`

  - ### addExact

    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
      `y` - the second value
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### addExact

    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
      `y` - the second value
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### subtractExact

    public static int subtractExact(int x, int y)
    Returns the difference of the arguments, throwing an exception if the result overflows an `int`.
    Parameters:
      `x` - the first value
      `y` - the second value to subtract from the first
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### subtractExact

    public static long subtractExact(long x, long y)
    Returns the difference of the arguments, throwing an exception if the result overflows a `long`.
    Parameters:
      `x` - the first value
      `y` - the second value to subtract from the first
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### multiplyExact

    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
      `y` - the second value
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### multiplyExact

    public static long multiplyExact(long x, int y)
    Returns the product of the arguments, throwing an exception if the result overflows a `long`.
    Parameters:
      `x` - the first value
      `y` - the second value
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      9

  - ### multiplyExact

    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
      `y` - the second value
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### incrementExact

    public static int incrementExact(int a)
    Returns the argument incremented by one, throwing an exception if the result overflows an `int`. The overflow only occurs for the maximum value.
    Parameters:
      `a` - the value to increment
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### incrementExact

    public static long incrementExact(long a)
    Returns the argument incremented by one, throwing an exception if the result overflows a `long`. The overflow only occurs for the maximum value.
    Parameters:
      `a` - the value to increment
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### decrementExact

    public static int decrementExact(int a)
    Returns the argument decremented by one, throwing an exception if the result overflows an `int`. The overflow only occurs for the minimum value.
    Parameters:
      `a` - the value to decrement
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### decrementExact

    public static long decrementExact(long a)
    Returns the argument decremented by one, throwing an exception if the result overflows a `long`. The overflow only occurs for the minimum value.
    Parameters:
      `a` - the value to decrement
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### negateExact

    public static int negateExact(int a)
    Returns the negation of the argument, throwing an exception if the result overflows an `int`. The overflow only occurs for the minimum value.
    Parameters:
      `a` - the value to negate
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows an int
    Since:
      1.8

  - ### negateExact

    public static long negateExact(long a)
    Returns the negation of the argument, throwing an exception if the result overflows a `long`. The overflow only occurs for the minimum value.
    Parameters:
      `a` - the value to negate
    Returns:
      the result
    Throws:
      `ArithmeticException` - if the result overflows a long
    Since:
      1.8

  - ### toIntExact

    public static int toIntExact(long value)
    Returns the value of the `long` argument, throwing an exception if the value overflows an `int`.
    Parameters:
      `value` - the long value
    Returns:
      the argument as an int
    Throws:
      `ArithmeticException` - if the `argument` overflows an int
    Since:
      1.8

  - ### multiplyFull

    public static long multiplyFull(int x, int y)
    Returns the exact mathematical product of the arguments.
    Parameters:
      `x` - the first value
      `y` - the second value
    Returns:
      the result
    Since:
      9

  - ### multiplyHigh

    public static long multiplyHigh(long x, long y)
    Returns as a `long` the most significant 64 bits of the 128-bit product of two 64-bit factors.
    Parameters:
      `x` - the first value
      `y` - the second value
    Returns:
      the result
    Since:
      9

  - ### floorDiv

    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 Integer.MIN_VALUE and the divisor is `-1`, then integer overflow occurs and the result is equal to `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 from 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
      `y` - 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
    Since:
      1.8
    See Also:
      - `floorMod(int, int)`
      - `floor(double)`

  - ### floorDiv

    public static long floorDiv(long x, int 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 Long.MIN_VALUE and the divisor is `-1`, then integer overflow occurs and the result is equal to `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 from truncation when the exact result is negative.

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

    Parameters:
      `x` - the dividend
      `y` - 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
    Since:
      9
    See Also:
      - `floorMod(long, int)`
      - `floor(double)`

  - ### floorDiv

    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 Long.MIN_VALUE and the divisor is `-1`, then integer overflow occurs and the result is equal to `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 from truncation when the exact result is negative.

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

    Parameters:
      `x` - the dividend
      `y` - 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
    Since:
      1.8
    See Also:
      - `floorMod(long, long)`
      - `floor(double)`

  - ### floorMod

    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`
      - `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`

    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
      `y` - the divisor
    Returns:
      the floor modulus `x - (floorDiv(x, y) * y)`
    Throws:
      `ArithmeticException` - if the divisor `y` is zero
    Since:
      1.8
    See Also:
      - `floorDiv(int, int)`

  - ### floorMod

    public static int floorMod(long x, int y)
    Returns the floor modulus of the `long` and `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`

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

    Parameters:
      `x` - the dividend
      `y` - the divisor
    Returns:
      the floor modulus `x - (floorDiv(x, y) * y)`
    Throws:
      `ArithmeticException` - if the divisor `y` is zero
    Since:
      9
    See Also:
      - `floorDiv(long, int)`

  - ### floorMod

    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
      `y` - the divisor
    Returns:
      the floor modulus `x - (floorDiv(x, y) * y)`
    Throws:
      `ArithmeticException` - if the divisor `y` is zero
    Since:
      1.8
    See Also:
      - `floorDiv(long, long)`

  - ### abs

    public static int abs(int a)
    Returns the absolute value of an `int` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.

    Note that if the argument is equal to the value of `Integer.MIN_VALUE`, the most negative representable `int` value, the result is that same value, which is negative. In contrast, the `absExact(int)` method throws an `ArithmeticException` for this value.

    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument.
    See Also:
      - `absExact(int)`

  - ### absExact

    public static int absExact(int a)
    Returns the mathematical absolute value of an `int` value if it is exactly representable as an `int`, throwing `ArithmeticException` if the result overflows the positive `int` range.

    Since the range of two's complement integers is asymmetric with one additional negative value (JLS 4.2.1), the mathematical absolute value of `Integer.MIN_VALUE` overflows the positive `int` range, so an exception is thrown for that argument.

    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument, unless overflow occurs
    Throws:
      `ArithmeticException` - if the argument is `Integer.MIN_VALUE`
    Since:
      15
    See Also:
      - `abs(int)`

  - ### abs

    public static long abs(long a)
    Returns the absolute value of a `long` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.

    Note that if the argument is equal to the value of `Long.MIN_VALUE`, the most negative representable `long` value, the result is that same value, which is negative. In contrast, the `absExact(long)` method throws an `ArithmeticException` for this value.

    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument.
    See Also:
      - `absExact(long)`

  - ### absExact

    public static long absExact(long a)
    Returns the mathematical absolute value of an `long` value if it is exactly representable as an `long`, throwing `ArithmeticException` if the result overflows the positive `long` range.

    Since the range of two's complement integers is asymmetric with one additional negative value (JLS 4.2.1), the mathematical absolute value of `Long.MIN_VALUE` overflows the positive `long` range, so an exception is thrown for that argument.

    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument, unless overflow occurs
    Throws:
      `ArithmeticException` - if the argument is `Long.MIN_VALUE`
    Since:
      15
    See Also:
      - `abs(long)`

  - ### abs

    public static float abs(float a)
    Returns the absolute value of a `float` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned. Special cases:
    - If the argument is positive zero or negative zero, the result is positive zero.
    - If the argument is infinite, the result is positive infinity.
    - If the argument is NaN, the result is NaN.
    API Note:
      As implied by the above, one valid implementation of this method is given by the expression below which computes a `float` with the same exponent and significand as the argument but with a guaranteed zero sign bit indicating a positive value:
      `Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))`
    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument.

  - ### abs

    public static double abs(double a)
    Returns the absolute value of a `double` value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned. Special cases:
    - If the argument is positive zero or negative zero, the result is positive zero.
    - If the argument is infinite, the result is positive infinity.
    - If the argument is NaN, the result is NaN.
    API Note:
      As implied by the above, one valid implementation of this method is given by the expression below which computes a `double` with the same exponent and significand as the argument but with a guaranteed zero sign bit indicating a positive value:
      `Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)`
    Parameters:
      `a` - the argument whose absolute value is to be determined
    Returns:
      the absolute value of the argument.

  - ### max

    public static int max(int a, int b)
    Returns the greater of two `int` values. That is, the result is the argument closer to the value of `Integer.MAX_VALUE`. If the arguments have the same value, the result is that same value.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the larger of `a` and `b`.

  - ### max

    public static long max(long a, long b)
    Returns the greater of two `long` values. That is, the result is the argument closer to the value of `Long.MAX_VALUE`. If the arguments have the same value, the result is that same value.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the larger of `a` and `b`.

  - ### max

    public static float max(float a, float b)
    Returns the greater of two `float` values. That is, the result is the argument closer to positive infinity. If the arguments have the same value, the result is that same value. If either value is NaN, then the result is NaN. Unlike the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero. If one argument is positive zero and the other negative zero, the result is positive zero.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the larger of `a` and `b`.

  - ### max

    public static double max(double a, double b)
    Returns the greater of two `double` values. That is, the result is the argument closer to positive infinity. If the arguments have the same value, the result is that same value. If either value is NaN, then the result is NaN. Unlike the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero. If one argument is positive zero and the other negative zero, the result is positive zero.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the larger of `a` and `b`.

  - ### min

    public static int min(int a, int b)
    Returns the smaller of two `int` values. That is, the result the argument closer to the value of `Integer.MIN_VALUE`. If the arguments have the same value, the result is that same value.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the smaller of `a` and `b`.

  - ### min

    public static long min(long a, long b)
    Returns the smaller of two `long` values. That is, the result is the argument closer to the value of `Long.MIN_VALUE`. If the arguments have the same value, the result is that same value.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the smaller of `a` and `b`.

  - ### min

    public static float min(float a, float b)
    Returns the smaller of two `float` values. That is, the result is the value closer to negative infinity. If the arguments have the same value, the result is that same value. If either value is NaN, then the result is NaN. Unlike the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero. If one argument is positive zero and the other is negative zero, the result is negative zero.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the smaller of `a` and `b`.

  - ### min

    public static double min(double a, double b)
    Returns the smaller of two `double` values. That is, the result is the value closer to negative infinity. If the arguments have the same value, the result is that same value. If either value is NaN, then the result is NaN. Unlike the numerical comparison operators, this method considers negative zero to be strictly smaller than positive zero. If one argument is positive zero and the other is negative zero, the result is negative zero.
    Parameters:
      `a` - an argument.
      `b` - another argument.
    Returns:
      the smaller of `a` and `b`.

  - ### fma

    public static double fma(double a, double b, double c)
    Returns the fused multiply add of the three arguments; that is, returns the exact product of the first two arguments summed with the third argument and then rounded once to the nearest `double`. The rounding is done using the round to nearest even rounding mode. In contrast, if `a * b + c` is evaluated as a regular floating-point expression, two rounding errors are involved, the first for the multiply operation, the second for the addition operation.

    Special cases:

    - If any argument is NaN, the result is NaN.
    - If one of the first two arguments is infinite and the other is zero, the result is NaN.
    - If the exact product of the first two arguments is infinite (in other words, at least one of the arguments is infinite and the other is neither zero nor NaN) and the third argument is an infinity of the opposite sign, the result is NaN.

    Note that `fma(a, 1.0, c)` returns the same result as (`a + c`). However, `fma(a, b, +0.0)` does not always return the same result as (`a * b`) since `fma(-0.0, +0.0, +0.0)` is `+0.0` while (`-0.0 * +0.0`) is `-0.0`; `fma(a, b, -0.0)` is equivalent to (`a * b`) however.

    API Note:
      This method corresponds to the fusedMultiplyAdd operation defined in IEEE 754-2008.
    Parameters:
      `a` - a value
      `b` - a value
      `c` - a value
    Returns:
      (a × b + c) computed, as if with unlimited range and precision, and rounded once to the nearest `double` value
    Since:
      9

  - ### fma

    public static float fma(float a, float b, float c)
    Returns the fused multiply add of the three arguments; that is, returns the exact product of the first two arguments summed with the third argument and then rounded once to the nearest `float`. The rounding is done using the round to nearest even rounding mode. In contrast, if `a * b + c` is evaluated as a regular floating-point expression, two rounding errors are involved, the first for the multiply operation, the second for the addition operation.

    Special cases:

    - If any argument is NaN, the result is NaN.
    - If one of the first two arguments is infinite and the other is zero, the result is NaN.
    - If the exact product of the first two arguments is infinite (in other words, at least one of the arguments is infinite and the other is neither zero nor NaN) and the third argument is an infinity of the opposite sign, the result is NaN.

    Note that `fma(a, 1.0f, c)` returns the same result as (`a + c`). However, `fma(a, b, +0.0f)` does not always return the same result as (`a * b`) since `fma(-0.0f, +0.0f, +0.0f)` is `+0.0f` while (`-0.0f * +0.0f`) is `-0.0f`; `fma(a, b, -0.0f)` is equivalent to (`a * b`) however.

    API Note:
      This method corresponds to the fusedMultiplyAdd operation defined in IEEE 754-2008.
    Parameters:
      `a` - a value
      `b` - a value
      `c` - a value
    Returns:
      (a × b + c) computed, as if with unlimited range and precision, and rounded once to the nearest `float` value
    Since:
      9

  - ### ulp

    public static double ulp(double d)
    Returns the size of an ulp of the argument. An ulp, unit in the last place, of a `double` value is the positive distance between this floating-point value and the `double` value next larger in magnitude. Note that for non-NaN x, `ulp(-x) == ulp(x)`.

    Special Cases:

    - If the argument is NaN, then the result is NaN.
    - If the argument is positive or negative infinity, then the result is positive infinity.
    - If the argument is positive or negative zero, then the result is `Double.MIN_VALUE`.
    - If the argument is ±`Double.MAX_VALUE`, then the result is equal to 2971.
    Parameters:
      `d` - the floating-point value whose ulp is to be returned
    Returns:
      the size of an ulp of the argument
    Since:
      1.5

  - ### ulp

    public static float ulp(float f)
    Returns the size of an ulp of the argument. An ulp, unit in the last place, of a `float` value is the positive distance between this floating-point value and the `float` value next larger in magnitude. Note that for non-NaN x, `ulp(-x) == ulp(x)`.

    Special Cases:

    - If the argument is NaN, then the result is NaN.
    - If the argument is positive or negative infinity, then the result is positive infinity.
    - If the argument is positive or negative zero, then the result is `Float.MIN_VALUE`.
    - If the argument is ±`Float.MAX_VALUE`, then the result is equal to 2104.
    Parameters:
      `f` - the floating-point value whose ulp is to be returned
    Returns:
      the size of an ulp of the argument
    Since:
      1.5

  - ### signum

    public static double signum(double d)
    Returns the signum function of the argument; zero if the argument is zero, 1.0 if the argument is greater than zero, -1.0 if the argument is less than zero.

    Special Cases:

    - If the argument is NaN, then the result is NaN.
    - If the argument is positive zero or negative zero, then the result is the same as the argument.
    Parameters:
      `d` - the floating-point value whose signum is to be returned
    Returns:
      the signum function of the argument
    Since:
      1.5

  - ### signum

    public static float signum(float f)
    Returns the signum function of the argument; zero if the argument is zero, 1.0f if the argument is greater than zero, -1.0f if the argument is less than zero.

    Special Cases:

    - If the argument is NaN, then the result is NaN.
    - If the argument is positive zero or negative zero, then the result is the same as the argument.
    Parameters:
      `f` - the floating-point value whose signum is to be returned
    Returns:
      the signum function of the argument
    Since:
      1.5

  - ### sinh

    public static double sinh(double x)
    Returns the hyperbolic sine of a `double` value. The hyperbolic sine of x is defined to be (ex - e-x)/2 where e is Euler's number.

    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 2.5 ulps of the exact result.

    Parameters:
      `x` - The number whose hyperbolic sine is to be returned.
    Returns:
      The hyperbolic sine of `x`.
    Since:
      1.5

  - ### cosh

    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 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`.
    Since:
      1.5

  - ### tanh

    public static double tanh(double x)
    Returns the hyperbolic tangent of a `double` value. The hyperbolic tangent of x is defined to be (ex - e-x)/(ex + e-x), in other words, sinh(x)/cosh(x). Note that the absolute value of the exact tanh is always less than 1.

    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.
    - If the argument is positive infinity, then the result is `+1.0`.
    - If the argument is negative infinity, then the result is `-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.

    Parameters:
      `x` - The number whose hyperbolic tangent is to be returned.
    Returns:
      The hyperbolic tangent of `x`.
    Since:
      1.5

  - ### hypot

    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.
    - If both arguments are zero, the result is positive zero.

    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
      `y` - a value
    Returns:
      sqrt(x2 +y2) without intermediate overflow or underflow
    Since:
      1.5

  - ### expm1

    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.
    Since:
      1.5

  - ### log1p

    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
    Since:
      1.5

  - ### copySign

    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
      `sign` - the parameter providing the sign of the result
    Returns:
      a value with the magnitude of `magnitude` and the sign of `sign`.
    Since:
      1.6

  - ### copySign

    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
      `sign` - the parameter providing the sign of the result
    Returns:
      a value with the magnitude of `magnitude` and the sign of `sign`.
    Since:
      1.6

  - ### getExponent

    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
    Since:
      1.6

  - ### getExponent

    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
    Since:
      1.6

  - ### nextAfter

    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
      `direction` - 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`.
    Since:
      1.6

  - ### nextAfter

    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
      `direction` - 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`.
    Since:
      1.6

  - ### nextUp

    public static double nextUp(double d)
    Returns the floating-point value adjacent to `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:

    - If the argument is NaN, the result is NaN.
    - If the argument is positive infinity, the result is positive 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 positive infinity.
    Since:
      1.6

  - ### nextUp

    public static float nextUp(float f)
    Returns the floating-point value adjacent to `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:

    - If the argument is NaN, the result is NaN.
    - If the argument is positive infinity, the result is positive infinity.
    - If the argument is zero, the result is `Float.MIN_VALUE`
    Parameters:
      `f` - starting floating-point value
    Returns:
      The adjacent floating-point value closer to positive infinity.
    Since:
      1.6

  - ### nextDown

    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.
    Since:
      1.8

  - ### nextDown

    public static float nextDown(float f)
    Returns the floating-point value adjacent to `f` in the direction of negative infinity. This method is semantically equivalent to `nextAfter(f, Float.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 `-Float.MIN_VALUE`
    Parameters:
      `f` - starting floating-point value
    Returns:
      The adjacent floating-point value closer to negative infinity.
    Since:
      1.8

  - ### scalb

    public static double scalb(double d, int scaleFactor)
    Returns `d` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply. 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 non-NaN, the result has the same sign as `d`.

    Special cases:

    - If the first argument is NaN, NaN is returned.
    - If the first argument is infinite, then an infinity of the same sign is returned.
    - If the first argument is zero, then a zero of the same sign is returned.
    Parameters:
      `d` - number to be scaled by a power of two.
      `scaleFactor` - power of 2 used to scale `d`
    Returns:
      `d` × 2`scaleFactor`
    Since:
      1.6

  - ### scalb

    public static float scalb(float f, int scaleFactor)
    Returns `f` × 2`scaleFactor` rounded as if performed by a single correctly rounded floating-point multiply. 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 non-NaN, the result has the same sign as `f`.

    Special cases:

    - If the first argument is NaN, NaN is returned.
    - If the first argument is infinite, then an infinity of the same sign is returned.
    - If the first argument is zero, then a zero of the same sign is returned.
    Parameters:
      `f` - number to be scaled by a power of two.
      `scaleFactor` - power of 2 used to scale `f`
    Returns:
      `f` × 2`scaleFactor`
    Since:
      1.6