FastMath (Apache Commons Math 3.6.1 API)

org.apache.commons.math3.util

## Class FastMath

- java.lang.Object
- - org.apache.commons.math3.util.FastMath

-     public class FastMath
      extends Object

  Faster, more accurate, portable alternative to `Math` and `StrictMath` for large scale computation.

  FastMath is a drop-in replacement for both Math and StrictMath. This means that for any method in Math (say `Math.sin(x)` or `Math.cbrt(y)`), user can directly change the class and use the methods as is (using `FastMath.sin(x)` or `FastMath.cbrt(y)` in the previous example).

  FastMath speed is achieved by relying heavily on optimizing compilers to native code present in many JVMs today and use of large tables. The larger tables are lazily initialised on first use, so that the setup time does not penalise methods that don't need them.

  Note that FastMath is extensively used inside Apache Commons Math, so by calling some algorithms, the overhead when the the tables need to be intialised will occur regardless of the end-user calling FastMath methods directly or not. Performance figures for a specific JVM and hardware can be evaluated by running the FastMathTestPerformance tests in the test directory of the source distribution.

  FastMath accuracy should be mostly independent of the JVM as it relies only on IEEE-754 basic operations and on embedded tables. Almost all operations are accurate to about 0.5 ulp throughout the domain range. This statement, of course is only a rough global observed behavior, it is not a guarantee for every double numbers input (see William Kahan's Table Maker's Dilemma).

  FastMath additionally implements the following methods not found in Math/StrictMath:

  - `asinh(double)`
  - `acosh(double)`
  - `atanh(double)`
  The following methods are found in Math/StrictMath since 1.6 only, they are provided by FastMath even in 1.5 Java virtual machines
  - `copySign(double, double)`
  - `getExponent(double)`
  - `nextAfter(double,double)`
  - `nextUp(double)`
  - `scalb(double, int)`
  - `copySign(float, float)`
  - `getExponent(float)`
  - `nextAfter(float,double)`
  - `nextUp(float)`
  - `scalb(float, int)`
  Since:
    2.2

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `E` Napier's constant e, base of the natural logarithm.
    `static double` | `PI` Archimede's constant PI, ratio of circle circumference to diameter.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `static double` | `abs(double x)` Absolute value.
    `static float` | `abs(float x)` Absolute value.
    `static int` | `abs(int x)` Absolute value.
    `static long` | `abs(long x)` Absolute value.
    `static double` | `acos(double x)` Compute the arc cosine of a number.
    `static double` | `acosh(double a)` Compute the inverse hyperbolic cosine of a number.
    `static int` | `addExact(int a, int b)` Add two numbers, detecting overflows.
    `static long` | `addExact(long a, long b)` Add two numbers, detecting overflows.
    `static double` | `asin(double x)` Compute the arc sine of a number.
    `static double` | `asinh(double a)` Compute the inverse hyperbolic sine of a number.
    `static double` | `atan(double x)` Arctangent function
    `static double` | `atan2(double y, double x)` Two arguments arctangent function
    `static double` | `atanh(double a)` Compute the inverse hyperbolic tangent of a number.
    `static double` | `cbrt(double x)` Compute the cubic root of a number.
    `static double` | `ceil(double x)` Get the smallest whole number larger than x.
    `static double` | `copySign(double magnitude, double sign)` Returns the first argument with the sign of the second argument.
    `static float` | `copySign(float magnitude, float sign)` Returns the first argument with the sign of the second argument.
    `static double` | `cos(double x)` Cosine function.
    `static double` | `cosh(double x)` Compute the hyperbolic cosine of a number.
    `static int` | `decrementExact(int n)` Decrement a number, detecting overflows.
    `static long` | `decrementExact(long n)` Decrement a number, detecting overflows.
    `static double` | `exp(double x)` Exponential function.
    `static double` | `expm1(double x)` Compute exp(x) - 1
    `static double` | `floor(double x)` Get the largest whole number smaller than x.
    `static int` | `floorDiv(int a, int b)` Finds q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.
    `static long` | `floorDiv(long a, long b)` Finds q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.
    `static int` | `floorMod(int a, int b)` Finds r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.
    `static long` | `floorMod(long a, long b)` Finds r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.
    `static int` | `getExponent(double d)` Return the exponent of a double number, removing the bias.
    `static int` | `getExponent(float f)` Return the exponent of a float number, removing the bias.
    `static double` | `hypot(double x, double y)` Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2) avoiding intermediate overflow or underflow.
    `static double` | `IEEEremainder(double dividend, double divisor)` Computes the remainder as prescribed by the IEEE 754 standard.
    `static int` | `incrementExact(int n)` Increment a number, detecting overflows.
    `static long` | `incrementExact(long n)` Increment a number, detecting overflows.
    `static double` | `log(double x)` Natural logarithm.
    `static double` | `log(double base, double x)` Computes the logarithm in a given base.
    `static double` | `log10(double x)` Compute the base 10 logarithm.
    `static double` | `log1p(double x)` Computes log(1 + x).
    `static void` | `main(String[] a)` Print out contents of arrays, and check the length.
    `static double` | `max(double a, double b)` Compute the maximum of two values
    `static float` | `max(float a, float b)` Compute the maximum of two values
    `static int` | `max(int a, int b)` Compute the maximum of two values
    `static long` | `max(long a, long b)` Compute the maximum of two values
    `static double` | `min(double a, double b)` Compute the minimum of two values
    `static float` | `min(float a, float b)` Compute the minimum of two values
    `static int` | `min(int a, int b)` Compute the minimum of two values
    `static long` | `min(long a, long b)` Compute the minimum of two values
    `static int` | `multiplyExact(int a, int b)` Multiply two numbers, detecting overflows.
    `static long` | `multiplyExact(long a, long b)` Multiply two numbers, detecting overflows.
    `static double` | `nextAfter(double d, double direction)` Get the next machine representable number after a number, moving in the direction of another number.
    `static float` | `nextAfter(float f, double direction)` Get the next machine representable number after a number, moving in the direction of another number.
    `static double` | `nextDown(double a)` Compute next number towards negative infinity.
    `static float` | `nextDown(float a)` Compute next number towards negative infinity.
    `static double` | `nextUp(double a)` Compute next number towards positive infinity.
    `static float` | `nextUp(float a)` Compute next number towards positive infinity.
    `static double` | `pow(double x, double y)` Power function.
    `static double` | `pow(double d, int e)` Raise a double to an int power.
    `static double` | `pow(double d, long e)` Raise a double to a long power.
    `static double` | `random()` Returns a pseudo-random number between 0.0 and 1.0.
    `static double` | `rint(double x)` Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.
    `static long` | `round(double x)` Get the closest long to x.
    `static int` | `round(float x)` Get the closest int to x.
    `static double` | `scalb(double d, int n)` Multiply a double number by a power of 2.
    `static float` | `scalb(float f, int n)` Multiply a float number by a power of 2.
    `static double` | `signum(double a)` Compute the signum of a number.
    `static float` | `signum(float a)` Compute the signum of a number.
    `static double` | `sin(double x)` Sine function.
    `static double` | `sinh(double x)` Compute the hyperbolic sine of a number.
    `static double` | `sqrt(double a)` Compute the square root of a number.
    `static int` | `subtractExact(int a, int b)` Subtract two numbers, detecting overflows.
    `static long` | `subtractExact(long a, long b)` Subtract two numbers, detecting overflows.
    `static double` | `tan(double x)` Tangent function.
    `static double` | `tanh(double x)` Compute the hyperbolic tangent of a number.
    `static double` | `toDegrees(double x)` Convert radians to degrees, with error of less than 0.5 ULP
    `static int` | `toIntExact(long n)` Convert a long to interger, detecting overflows
    `static double` | `toRadians(double x)` Convert degrees to radians, with error of less than 0.5 ULP
    `static double` | `ulp(double x)` Compute least significant bit (Unit in Last Position) for a number.
    `static float` | `ulp(float x)` Compute least significant bit (Unit in Last Position) for a number.

    - ### Methods inherited from class java.lang.Object

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

- - ### Field Detail

    - #### PI

          public static final double PI

      Archimede's constant PI, ratio of circle circumference to diameter.
      See Also:
        Constant Field Values

    - #### E

          public static final double E

      Napier's constant e, base of the natural logarithm.
      See Also:
        Constant Field Values

  - ### Method Detail

    - #### sqrt

          public static double sqrt(double a)

      Compute the square root of a number.

      Note: this implementation currently delegates to `Math.sqrt(double)`

      Parameters:
        `a` - number on which evaluation is done
      Returns:
        square root of a

    - #### cosh

          public static double cosh(double x)

      Compute the hyperbolic cosine of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        hyperbolic cosine of x

    - #### sinh

          public static double sinh(double x)

      Compute the hyperbolic sine of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        hyperbolic sine of x

    - #### tanh

          public static double tanh(double x)

      Compute the hyperbolic tangent of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        hyperbolic tangent of x

    - #### acosh

          public static double acosh(double a)

      Compute the inverse hyperbolic cosine of a number.
      Parameters:
        `a` - number on which evaluation is done
      Returns:
        inverse hyperbolic cosine of a

    - #### asinh

          public static double asinh(double a)

      Compute the inverse hyperbolic sine of a number.
      Parameters:
        `a` - number on which evaluation is done
      Returns:
        inverse hyperbolic sine of a

    - #### atanh

          public static double atanh(double a)

      Compute the inverse hyperbolic tangent of a number.
      Parameters:
        `a` - number on which evaluation is done
      Returns:
        inverse hyperbolic tangent of a

    - #### signum

          public static double signum(double a)

      Compute the signum of a number. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
      Parameters:
        `a` - number on which evaluation is done
      Returns:
        -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a

    - #### signum

          public static float signum(float a)

      Compute the signum of a number. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise
      Parameters:
        `a` - number on which evaluation is done
      Returns:
        -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a

    - #### nextUp

          public static double nextUp(double a)

      Compute next number towards positive infinity.
      Parameters:
        `a` - number to which neighbor should be computed
      Returns:
        neighbor of a towards positive infinity

    - #### nextUp

          public static float nextUp(float a)

      Compute next number towards positive infinity.
      Parameters:
        `a` - number to which neighbor should be computed
      Returns:
        neighbor of a towards positive infinity

    - #### nextDown

          public static double nextDown(double a)

      Compute next number towards negative infinity.
      Parameters:
        `a` - number to which neighbor should be computed
      Returns:
        neighbor of a towards negative infinity
      Since:
        3.4

    - #### nextDown

          public static float nextDown(float a)

      Compute next number towards negative infinity.
      Parameters:
        `a` - number to which neighbor should be computed
      Returns:
        neighbor of a towards negative infinity
      Since:
        3.4

    - #### random

          public static double random()

      Returns a pseudo-random number between 0.0 and 1.0.

      Note: this implementation currently delegates to `Math.random()`

      Returns:
        a random number between 0.0 and 1.0

    - #### exp

          public static double exp(double x)

      Exponential function. Computes exp(x), function result is nearly rounded. It will be correctly rounded to the theoretical value for 99.9% of input values, otherwise it will have a 1 ULP error. Method: Lookup intVal = exp(int(x)) Lookup fracVal = exp(int(x-int(x) / 1024.0) * 1024.0 ); Compute z as the exponential of the remaining bits by a polynomial minus one exp(x) = intVal * fracVal * (1 + z) Accuracy: Calculation is done with 63 bits of precision, so result should be correctly rounded for 99.9% of input values, with less than 1 ULP error otherwise.
      Parameters:
        `x` - a double
      Returns:
        double ex

    - #### expm1

          public static double expm1(double x)

      Compute exp(x) - 1
      Parameters:
        `x` - number to compute shifted exponential
      Returns:
        exp(x) - 1

    - #### log

          public static double log(double x)

      Natural logarithm.
      Parameters:
        `x` - a double
      Returns:
        log(x)

    - #### log1p

          public static double log1p(double x)

      Computes log(1 + x).
      Parameters:
        `x` - Number.
      Returns:
        `log(1 + x)`.

    - #### log10

          public static double log10(double x)

      Compute the base 10 logarithm.
      Parameters:
        `x` - a number
      Returns:
        log10(x)

    - #### log

          public static double log(double base,
                   double x)

      Computes the logarithm in a given base. Returns `NaN` if either argument is negative. If `base` is 0 and `x` is positive, 0 is returned. If `base` is positive and `x` is 0, `Double.NEGATIVE_INFINITY` is returned. If both arguments are 0, the result is `NaN`.
      Parameters:
        `base` - Base of the logarithm, must be greater than 0.
        `x` - Argument, must be greater than 0.
      Returns:
        the value of the logarithm, i.e. the number `y` such that `basey = x`.
      Since:
        1.2 (previously in `MathUtils`, moved as of version 3.0)

    - #### pow

          public static double pow(double x,
                   double y)

      Power function. Compute x^y.
      Parameters:
        `x` - a double
        `y` - a double
      Returns:
        double

    - #### pow

          public static double pow(double d,
                   int e)

      Raise a double to an int power.
      Parameters:
        `d` - Number to raise.
        `e` - Exponent.
      Returns:
        de
      Since:
        3.1

    - #### pow

          public static double pow(double d,
                   long e)

      Raise a double to a long power.
      Parameters:
        `d` - Number to raise.
        `e` - Exponent.
      Returns:
        de
      Since:
        3.6

    - #### sin

          public static double sin(double x)

      Sine function.
      Parameters:
        `x` - Argument.
      Returns:
        sin(x)

    - #### cos

          public static double cos(double x)

      Cosine function.
      Parameters:
        `x` - Argument.
      Returns:
        cos(x)

    - #### tan

          public static double tan(double x)

      Tangent function.
      Parameters:
        `x` - Argument.
      Returns:
        tan(x)

    - #### atan

          public static double atan(double x)

      Arctangent function
      Parameters:
        `x` - a number
      Returns:
        atan(x)

    - #### atan2

          public static double atan2(double y,
                     double x)

      Two arguments arctangent function
      Parameters:
        `y` - ordinate
        `x` - abscissa
      Returns:
        phase angle of point (x,y) between `-PI` and `PI`

    - #### asin

          public static double asin(double x)

      Compute the arc sine of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        arc sine of x

    - #### acos

          public static double acos(double x)

      Compute the arc cosine of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        arc cosine of x

    - #### cbrt

          public static double cbrt(double x)

      Compute the cubic root of a number.
      Parameters:
        `x` - number on which evaluation is done
      Returns:
        cubic root of x

    - #### toRadians

          public static double toRadians(double x)

      Convert degrees to radians, with error of less than 0.5 ULP
      Parameters:
        `x` - angle in degrees
      Returns:
        x converted into radians

    - #### toDegrees

          public static double toDegrees(double x)

      Convert radians to degrees, with error of less than 0.5 ULP
      Parameters:
        `x` - angle in radians
      Returns:
        x converted into degrees

    - #### abs

          public static int abs(int x)

      Absolute value.
      Parameters:
        `x` - number from which absolute value is requested
      Returns:
        abs(x)

    - #### abs

          public static long abs(long x)

      Absolute value.
      Parameters:
        `x` - number from which absolute value is requested
      Returns:
        abs(x)

    - #### abs

          public static float abs(float x)

      Absolute value.
      Parameters:
        `x` - number from which absolute value is requested
      Returns:
        abs(x)

    - #### abs

          public static double abs(double x)

      Absolute value.
      Parameters:
        `x` - number from which absolute value is requested
      Returns:
        abs(x)

    - #### ulp

          public static double ulp(double x)

      Compute least significant bit (Unit in Last Position) for a number.
      Parameters:
        `x` - number from which ulp is requested
      Returns:
        ulp(x)

    - #### ulp

          public static float ulp(float x)

      Compute least significant bit (Unit in Last Position) for a number.
      Parameters:
        `x` - number from which ulp is requested
      Returns:
        ulp(x)

    - #### scalb

          public static double scalb(double d,
                     int n)

      Multiply a double number by a power of 2.
      Parameters:
        `d` - number to multiply
        `n` - power of 2
      Returns:
        d × 2n

    - #### scalb

          public static float scalb(float f,
                    int n)

      Multiply a float number by a power of 2.
      Parameters:
        `f` - number to multiply
        `n` - power of 2
      Returns:
        f × 2n

    - #### nextAfter

          public static double nextAfter(double d,
                         double direction)

      Get the next machine representable number after a number, moving in the direction of another number.

      The ordering is as follows (increasing):

      - -INFINITY
      - -MAX_VALUE
      - -MIN_VALUE
      - -0.0
      - +0.0
      - +MIN_VALUE
      - +MAX_VALUE
      - +INFINITY

      If arguments compare equal, then the second argument is returned.

      If `direction` is greater than `d`, the smallest machine representable number strictly greater than `d` is returned; if less, then the largest representable number strictly less than `d` is returned.

      If `d` is infinite and direction does not bring it back to finite numbers, it is returned unchanged.

      Parameters:
        `d` - base number
        `direction` - (the only important thing is whether `direction` is greater or smaller than `d`)
      Returns:
        the next machine representable number in the specified direction

    - #### nextAfter

          public static float nextAfter(float f,
                        double direction)

      Get the next machine representable number after a number, moving in the direction of another number.

      The ordering is as follows (increasing):

      - -INFINITY
      - -MAX_VALUE
      - -MIN_VALUE
      - -0.0
      - +0.0
      - +MIN_VALUE
      - +MAX_VALUE
      - +INFINITY

      If arguments compare equal, then the second argument is returned.

      If `direction` is greater than `f`, the smallest machine representable number strictly greater than `f` is returned; if less, then the largest representable number strictly less than `f` is returned.

      If `f` is infinite and direction does not bring it back to finite numbers, it is returned unchanged.

      Parameters:
        `f` - base number
        `direction` - (the only important thing is whether `direction` is greater or smaller than `f`)
      Returns:
        the next machine representable number in the specified direction

    - #### floor

          public static double floor(double x)

      Get the largest whole number smaller than x.
      Parameters:
        `x` - number from which floor is requested
      Returns:
        a double number f such that f is an integer f <= x < f + 1.0

    - #### ceil

          public static double ceil(double x)

      Get the smallest whole number larger than x.
      Parameters:
        `x` - number from which ceil is requested
      Returns:
        a double number c such that c is an integer c - 1.0 < x <= c

    - #### rint

          public static double rint(double x)

      Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.
      Parameters:
        `x` - number from which nearest whole number is requested
      Returns:
        a double number r such that r is an integer r - 0.5 <= x <= r + 0.5

    - #### round

          public static long round(double x)

      Get the closest long to x.
      Parameters:
        `x` - number from which closest long is requested
      Returns:
        closest long to x

    - #### round

          public static int round(float x)

      Get the closest int to x.
      Parameters:
        `x` - number from which closest int is requested
      Returns:
        closest int to x

    - #### min

          public static int min(int a,
                int b)

      Compute the minimum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        a if a is lesser or equal to b, b otherwise

    - #### min

          public static long min(long a,
                 long b)

      Compute the minimum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        a if a is lesser or equal to b, b otherwise

    - #### min

          public static float min(float a,
                  float b)

      Compute the minimum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        a if a is lesser or equal to b, b otherwise

    - #### min

          public static double min(double a,
                   double b)

      Compute the minimum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        a if a is lesser or equal to b, b otherwise

    - #### max

          public static int max(int a,
                int b)

      Compute the maximum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        b if a is lesser or equal to b, a otherwise

    - #### max

          public static long max(long a,
                 long b)

      Compute the maximum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        b if a is lesser or equal to b, a otherwise

    - #### max

          public static float max(float a,
                  float b)

      Compute the maximum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        b if a is lesser or equal to b, a otherwise

    - #### max

          public static double max(double a,
                   double b)

      Compute the maximum of two values
      Parameters:
        `a` - first value
        `b` - second value
      Returns:
        b if a is lesser or equal to b, a otherwise

    - #### hypot

          public static double hypot(double x,
                     double y)

      Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2)
      avoiding intermediate overflow or underflow.
      - If either argument is infinite, then the result is positive infinity.
      - else, if either argument is NaN then the result is NaN.
      Parameters:
        `x` - a value
        `y` - a value
      Returns:
        sqrt(x2 +y2)

    - #### IEEEremainder

          public static double IEEEremainder(double dividend,
                             double divisor)

      Computes the remainder as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to `x - y*n` where `n` is the mathematical integer closest to the exact mathematical value of the quotient `x/y`. If two mathematical integers are equally close to `x/y` then `n` is the integer that is even.

      - If either operand is NaN, the result is NaN.
      - If the result is not NaN, the sign of the result equals the sign of the dividend.
      - If the dividend is an infinity, or the divisor is a zero, or both, the result is NaN.
      - If the dividend is finite and the divisor is an infinity, the result equals the dividend.
      - If the dividend is a zero and the divisor is finite, the result equals the dividend.

      Note: this implementation currently delegates to `StrictMath.IEEEremainder(double, double)`

      Parameters:
        `dividend` - the number to be divided
        `divisor` - the number by which to divide
      Returns:
        the remainder, rounded

    - #### toIntExact

          public static int toIntExact(long n)
                                throws MathArithmeticException

      Convert a long to interger, detecting overflows
      Parameters:
        `n` - number to convert to int
      Returns:
        integer with same valie as n if no overflows occur
      Throws:
        `MathArithmeticException` - if n cannot fit into an int
      Since:
        3.4

    - #### incrementExact

          public static int incrementExact(int n)
                                    throws MathArithmeticException

      Increment a number, detecting overflows.
      Parameters:
        `n` - number to increment
      Returns:
        n+1 if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### incrementExact

          public static long incrementExact(long n)
                                     throws MathArithmeticException

      Increment a number, detecting overflows.
      Parameters:
        `n` - number to increment
      Returns:
        n+1 if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### decrementExact

          public static int decrementExact(int n)
                                    throws MathArithmeticException

      Decrement a number, detecting overflows.
      Parameters:
        `n` - number to decrement
      Returns:
        n-1 if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### decrementExact

          public static long decrementExact(long n)
                                     throws MathArithmeticException

      Decrement a number, detecting overflows.
      Parameters:
        `n` - number to decrement
      Returns:
        n-1 if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### addExact

          public static int addExact(int a,
                     int b)
                              throws MathArithmeticException

      Add two numbers, detecting overflows.
      Parameters:
        `a` - first number to add
        `b` - second number to add
      Returns:
        a+b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### addExact

          public static long addExact(long a,
                      long b)
                               throws MathArithmeticException

      Add two numbers, detecting overflows.
      Parameters:
        `a` - first number to add
        `b` - second number to add
      Returns:
        a+b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### subtractExact

          public static int subtractExact(int a,
                          int b)

      Subtract two numbers, detecting overflows.
      Parameters:
        `a` - first number
        `b` - second number to subtract from a
      Returns:
        a-b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### subtractExact

          public static long subtractExact(long a,
                           long b)

      Subtract two numbers, detecting overflows.
      Parameters:
        `a` - first number
        `b` - second number to subtract from a
      Returns:
        a-b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### multiplyExact

          public static int multiplyExact(int a,
                          int b)

      Multiply two numbers, detecting overflows.
      Parameters:
        `a` - first number to multiply
        `b` - second number to multiply
      Returns:
        a*b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### multiplyExact

          public static long multiplyExact(long a,
                           long b)

      Multiply two numbers, detecting overflows.
      Parameters:
        `a` - first number to multiply
        `b` - second number to multiply
      Returns:
        a*b if no overflows occur
      Throws:
        `MathArithmeticException` - if an overflow occurs
      Since:
        3.4

    - #### floorDiv

          public static int floorDiv(int a,
                     int b)
                              throws MathArithmeticException

      Finds q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.

      This methods returns the same value as integer division when a and b are same signs, but returns a different value when they are opposite (i.e. q is negative).

      Parameters:
        `a` - dividend
        `b` - divisor
      Returns:
        q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0
      Throws:
        `MathArithmeticException` - if b == 0
      Since:
        3.4
      See Also:
        `floorMod(int, int)`

    - #### floorDiv

          public static long floorDiv(long a,
                      long b)
                               throws MathArithmeticException

      Finds q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.

      This methods returns the same value as integer division when a and b are same signs, but returns a different value when they are opposite (i.e. q is negative).

      Parameters:
        `a` - dividend
        `b` - divisor
      Returns:
        q such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0
      Throws:
        `MathArithmeticException` - if b == 0
      Since:
        3.4
      See Also:
        `floorMod(long, long)`

    - #### floorMod

          public static int floorMod(int a,
                     int b)
                              throws MathArithmeticException

      Finds r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.

      This methods returns the same value as integer modulo when a and b are same signs, but returns a different value when they are opposite (i.e. q is negative).

      Parameters:
        `a` - dividend
        `b` - divisor
      Returns:
        r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0
      Throws:
        `MathArithmeticException` - if b == 0
      Since:
        3.4
      See Also:
        `floorDiv(int, int)`

    - #### floorMod

          public static long floorMod(long a,
                      long b)

      Finds r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0.

      This methods returns the same value as integer modulo when a and b are same signs, but returns a different value when they are opposite (i.e. q is negative).

      Parameters:
        `a` - dividend
        `b` - divisor
      Returns:
        r such that a = q b + r with 0 <= r < b if b > 0 and b < r <= 0 if b < 0
      Throws:
        `MathArithmeticException` - if b == 0
      Since:
        3.4
      See Also:
        `floorDiv(long, long)`

    - #### copySign

          public static double copySign(double magnitude,
                        double sign)

      Returns the first argument with the sign of the second argument. A NaN `sign` argument is treated as positive.
      Parameters:
        `magnitude` - the value to return
        `sign` - the sign for the returned value
      Returns:
        the magnitude with the same sign as the `sign` argument

    - #### copySign

          public static float copySign(float magnitude,
                       float sign)

      Returns the first argument with the sign of the second argument. A NaN `sign` argument is treated as positive.
      Parameters:
        `magnitude` - the value to return
        `sign` - the sign for the returned value
      Returns:
        the magnitude with the same sign as the `sign` argument

    - #### getExponent

          public static int getExponent(double d)

      Return the exponent of a double number, removing the bias.

      For double numbers of the form 2x, the unbiased exponent is exactly x.

      Parameters:
        `d` - number from which exponent is requested
      Returns:
        exponent for d in IEEE754 representation, without bias

    - #### getExponent

          public static int getExponent(float f)

      Return the exponent of a float number, removing the bias.

      For float numbers of the form 2x, the unbiased exponent is exactly x.

      Parameters:
        `f` - number from which exponent is requested
      Returns:
        exponent for d in IEEE754 representation, without bias

    - #### main

          public static void main(String[] a)

      Print out contents of arrays, and check the length.

      used to generate the preset arrays originally.

      Parameters:
        `a` - unused

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