Package org.hipparchus.util

Class 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 initialized on first use, so that the setup time does not penalize methods that don't need them.

    Note that FastMath is extensively used inside Hipparchus, so by calling some algorithms, the overhead when the the tables need to be initialized 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:

    The following methods are found in Math/StrictMath since 1.6 only, they are provided by FastMath even in 1.5 Java virtual machines

    • Field Summary

      Fields 
      Modifier and Type Field 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

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method 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 <T extends CalculusFieldElement<T>>
      T      		 abs​(T x)
      Absolute value.
      static int absExact​(int x)
      Absolute value.
      static long absExact​(long x)
      Absolute value.
      static double acos​(double x)
      Compute the arc cosine of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 acos​(T x)
      Compute the arc cosine of a number.
      static double acosh​(double a)
      Compute the inverse hyperbolic cosine of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 acosh​(T 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 <T extends CalculusFieldElement<T>>
      T      		 asin​(T x)
      Compute the arc sine of a number.
      static double asinh​(double a)
      Compute the inverse hyperbolic sine of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 asinh​(T a)
      Compute the inverse hyperbolic sine of a number.
      static double atan​(double x)
      Arctangent function
      static <T extends CalculusFieldElement<T>>
      T      		 atan​(T x)
      Arctangent function
      static double atan2​(double y, double x)
      Two arguments arctangent function
      static <T extends CalculusFieldElement<T>>
      T      		 atan2​(T y, T x)
      Two arguments arctangent function
      static double atanh​(double a)
      Compute the inverse hyperbolic tangent of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 atanh​(T a)
      Compute the inverse hyperbolic tangent of a number.
      static double cbrt​(double x)
      Compute the cubic root of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 cbrt​(T x)
      Compute the cubic root of a number.
      static double ceil​(double x)
      Get the smallest whole number larger than x.
      static <T extends CalculusFieldElement<T>>
      T      		 ceil​(T x)
      Get the smallest whole number larger than x.
      static int ceilDiv​(int a, int b)
      Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static long ceilDiv​(long a, int b)
      Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static long ceilDiv​(long a, long b)
      Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static int ceilDivExact​(int a, int b)
      Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static long ceilDivExact​(long a, long b)
      Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static int ceilMod​(int a, int b)
      Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static int ceilMod​(long a, int b)
      Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static long ceilMod​(long a, long b)
      Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.
      static double clamp​(double value, double inf, double sup)
      Clamp a value within an interval.
      static float clamp​(float value, float inf, float sup)
      Clamp a value within an interval.
      static int clamp​(int value, int inf, int sup)
      Clamp a value within an interval.
      static int clamp​(long value, int inf, int sup)
      Clamp a value within an interval.
      static long clamp​(long value, long inf, long sup)
      Clamp a value within an interval.
      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 <T extends CalculusFieldElement<T>>
      T      		 copySign​(T magnitude, double sign)
      Returns the first argument with the sign of the second argument.
      static <T extends CalculusFieldElement<T>>
      T      		 copySign​(T magnitude, T sign)
      Returns the first argument with the sign of the second argument.
      static double cos​(double x)
      Cosine function.
      static <T extends CalculusFieldElement<T>>
      T      		 cos​(T x)
      Cosine function.
      static double cosh​(double x)
      Compute the hyperbolic cosine of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 cosh​(T 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 int divideExact​(int x, int y)
      Divide two integers, checking for overflow.
      static long divideExact​(long x, long y)
      Divide two long integers, checking for overflow.
      static double exp​(double x)
      Exponential function.
      static <T extends CalculusFieldElement<T>>
      T      		 exp​(T x)
      Exponential function.
      static double expm1​(double x)
      Compute exp(x) - 1
      static <T extends CalculusFieldElement<T>>
      T      		 expm1​(T x)
      Compute exp(x) - 1
      static double floor​(double x)
      Get the largest whole number smaller than x.
      static <T extends CalculusFieldElement<T>>
      T      		 floor​(T 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, 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 floorDivExact​(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 floorDivExact​(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 int floorMod​(long 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 double fma​(double a, double b, double c)
      Compute Fused-multiply-add operation a * b + c.
      static float fma​(float a, float b, float c)
      Compute Fused-multiply-add operation a * b + c.
      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 <T extends CalculusFieldElement<T>>
      T      		 hypot​(T x, T 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 <T extends CalculusFieldElement<T>>
      T      		 IEEEremainder​(T dividend, double divisor)
      Computes the remainder as prescribed by the IEEE 754 standard.
      static <T extends CalculusFieldElement<T>>
      T      		 IEEEremainder​(T dividend, T 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 <T extends CalculusFieldElement<T>>
      T      		 log​(T x)
      Natural logarithm.
      static double log10​(double x)
      Compute the base 10 logarithm.
      static <T extends CalculusFieldElement<T>>
      T      		 log10​(T x)
      Compute the base 10 logarithm.
      static double log1p​(double x)
      Computes log(1 + x).
      static <T extends CalculusFieldElement<T>>
      T      		 log1p​(T x)
      Computes log(1 + x).
      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 <T extends CalculusFieldElement<T>>
      T      		 max​(T a, double b)
      Compute the maximum of two values
      static <T extends CalculusFieldElement<T>>
      T      		 max​(T a, T 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 <T extends CalculusFieldElement<T>>
      T      		 min​(T a, double b)
      Compute the minimum of two values
      static <T extends CalculusFieldElement<T>>
      T      		 min​(T a, T b)
      Compute the minimum of two values
      static int multiplyExact​(int a, int b)
      Multiply two numbers, detecting overflows.
      static long multiplyExact​(long a, int b)
      Multiply two numbers, detecting overflows.
      static long multiplyExact​(long a, long b)
      Multiply two numbers, detecting overflows.
      static long multiplyFull​(int a, int b)
      Multiply two integers and give an exact result without overflow.
      static long multiplyHigh​(long a, long b)
      Multiply two long integers and give the 64 most significant bits of the result.
      static int negateExact​(int x)
      Negates the argument.
      static long negateExact​(long x)
      Negates the argument.
      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 <T extends CalculusFieldElement<T>>
      double      		 norm​(T x)
      Norm.
      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 <T extends CalculusFieldElement<T>>
      T      		 pow​(T x, double y)
      Power function.
      static <T extends CalculusFieldElement<T>>
      T      		 pow​(T d, int e)
      Raise a double to an int power.
      static <T extends CalculusFieldElement<T>>
      T      		 pow​(T x, T y)
      Power function.
      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 <T extends CalculusFieldElement<T>>
      T      		 rint​(T 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 <T extends CalculusFieldElement<T>>
      long      		 round​(T x)
      Get the closest long 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 <T extends CalculusFieldElement<T>>
      T      		 scalb​(T d, int n)
      Multiply a double number by a power of 2.
      static <T extends CalculusFieldElement<T>>
      T      		 sign​(T a)
      Compute the sign of a number.
      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 <T extends CalculusFieldElement<T>>
      T      		 sin​(T x)
      Sine function.
      static SinCos sinCos​(double x)
      Combined Sine and Cosine function.
      static <T extends CalculusFieldElement<T>>
      FieldSinCos<T>      		 sinCos​(T x)
      Combined Sine and Cosine function.
      static double sinh​(double x)
      Compute the hyperbolic sine of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 sinh​(T x)
      Compute the hyperbolic sine of a number.
      static SinhCosh sinhCosh​(double x)
      Combined hyperbolic sine and hyperbolic cosine function.
      static <T extends CalculusFieldElement<T>>
      FieldSinhCosh<T>      		 sinhCosh​(T x)
      Combined hyperbolic sine and hyperbolic cosine function.
      static double sqrt​(double a)
      Compute the square root of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 sqrt​(T 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 <T extends CalculusFieldElement<T>>
      T      		 tan​(T x)
      Tangent function.
      static double tanh​(double x)
      Compute the hyperbolic tangent of a number.
      static <T extends CalculusFieldElement<T>>
      T      		 tanh​(T 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 <T extends CalculusFieldElement<T>>
      T      		 toDegrees​(T 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 <T extends CalculusFieldElement<T>>
      T      		 toRadians​(T 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.
      static <T extends CalculusFieldElement<T>>
      T      		 ulp​(T x)
      Compute least significant bit (Unit in Last Position) for a number.
      static long unsignedMultiplyHigh​(long a, long b)
      Multiply two long unsigned integers and give the 64 most significant bits of the unsigned result.

    • 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

      • sinhCosh

        public static SinhCosh sinhCosh​(double x)
        Combined hyperbolic sine and hyperbolic cosine function.
        Parameters:
        x - Argument.
        Returns:
        [sinh(x), cosh(x)]

      • sinhCosh

        public static <T extends CalculusFieldElement<T>> FieldSinhCosh<T> sinhCosh​(T x)
        Combined hyperbolic sine and hyperbolic cosine function.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Argument.
        Returns:
        [sinh(x), cosh(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

      • 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

      • clamp

        public static int clamp​(int value,
                        int inf,
                        int sup)
        Clamp a value within an interval.
        Parameters:
        value - value to clamp
        inf - lower bound of the clamping interval
        sup - upper bound of the clamping interval
        Returns:
        value clamped within [inf; sup], or value if already within bounds.
        Since:
        3.0

      • clamp

        public static long clamp​(long value,
                         long inf,
                         long sup)
        Clamp a value within an interval.
        Parameters:
        value - value to clamp
        inf - lower bound of the clamping interval
        sup - upper bound of the clamping interval
        Returns:
        value clamped within [inf; sup], or value if already within bounds.
        Since:
        3.0

      • clamp

        public static int clamp​(long value,
                        int inf,
                        int sup)
        Clamp a value within an interval.
        Parameters:
        value - value to clamp
        inf - lower bound of the clamping interval
        sup - upper bound of the clamping interval
        Returns:
        value clamped within [inf; sup], or value if already within bounds.
        Since:
        3.0

      • clamp

        public static float clamp​(float value,
                          float inf,
                          float sup)
        Clamp a value within an interval.

        This method assumes -0.0 is below +0.0

        Parameters:
        value - value to clamp
        inf - lower bound of the clamping interval
        sup - upper bound of the clamping interval
        Returns:
        value clamped within [inf; sup], or value if already within bounds.
        Since:
        3.0

      • clamp

        public static double clamp​(double value,
                           double inf,
                           double sup)
        Clamp a value within an interval.

        This method assumes -0.0 is below +0.0

        Parameters:
        value - value to clamp
        inf - lower bound of the clamping interval
        sup - upper bound of the clamping interval
        Returns:
        value clamped within [inf; sup], or value if already within bounds.
        Since:
        3.0

      • 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.

      • 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

      • 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

      • 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)

      • sinCos

        public static SinCos sinCos​(double x)
        Combined Sine and Cosine function.
        Parameters:
        x - Argument.
        Returns:
        [sin(x), cos(x)]

      • sinCos

        public static <T extends CalculusFieldElement<T>> FieldSinCos<T> sinCos​(T x)
        Combined Sine and Cosine function.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Argument.
        Returns:
        [sin(x), cos(x)]
        Since:
        1.4

      • 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)

      • absExact

        public static int absExact​(int x)
        Absolute value.
        Parameters:
        x - number from which absolute value is requested
        Returns:
        abs(x), or throws an exception for Integer.MIN_VALUE
        Since:
        2.0

      • absExact

        public static long absExact​(long x)
        Absolute value.
        Parameters:
        x - number from which absolute value is requested
        Returns:
        abs(x), or throws an exception for Long.MIN_VALUE
        Since:
        2.0

      • abs

        public static float abs​(float x)
        Absolute value.
        Parameters:
        x - number from which absolute value is requested
        Returns:
        abs(x)
        Since:
        2.0

      • abs

        public static double abs​(double x)
        Absolute value.
        Parameters:
        x - number from which absolute value is requested
        Returns:
        abs(x)

      • negateExact

        public static int negateExact​(int x)
        Negates the argument.
        Parameters:
        x - number from which opposite value is requested
        Returns:
        -x, or throws an exception for Integer.MIN_VALUE
        Since:
        2.0

      • negateExact

        public static long negateExact​(long x)
        Negates the argument.
        Parameters:
        x - number from which opposite value is requested
        Returns:
        -x, or throws an exception for Long.MIN_VALUE
        Since:
        2.0

      • 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.
        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 MathRuntimeException
        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:
        MathRuntimeException - if n cannot fit into an int

      • incrementExact

        public static int incrementExact​(int n)
                          throws MathRuntimeException
        Increment a number, detecting overflows.
        Parameters:
        n - number to increment
        Returns:
        n+1 if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • incrementExact

        public static long incrementExact​(long n)
                           throws MathRuntimeException
        Increment a number, detecting overflows.
        Parameters:
        n - number to increment
        Returns:
        n+1 if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • decrementExact

        public static int decrementExact​(int n)
                          throws MathRuntimeException
        Decrement a number, detecting overflows.
        Parameters:
        n - number to decrement
        Returns:
        n-1 if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • decrementExact

        public static long decrementExact​(long n)
                           throws MathRuntimeException
        Decrement a number, detecting overflows.
        Parameters:
        n - number to decrement
        Returns:
        n-1 if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • addExact

        public static int addExact​(int a,
                           int b)
                    throws MathRuntimeException
        Add two numbers, detecting overflows.
        Parameters:
        a - first number to add
        b - second number to add
        Returns:
        a+b if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • addExact

        public static long addExact​(long a,
                            long b)
                     throws MathRuntimeException
        Add two numbers, detecting overflows.
        Parameters:
        a - first number to add
        b - second number to add
        Returns:
        a+b if no overflows occur
        Throws:
        MathRuntimeException - if an overflow occurs

      • 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:
        MathRuntimeException - if an overflow occurs

      • 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:
        MathRuntimeException - if an overflow occurs

      • 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:
        MathRuntimeException - if an overflow occurs

      • multiplyExact

        public static long multiplyExact​(long 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:
        MathRuntimeException - if an overflow occurs
        Since:
        1.3

      • 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:
        MathRuntimeException - if an overflow occurs

      • multiplyFull

        public static long multiplyFull​(int a,
                                int b)
        Multiply two integers and give an exact result without overflow.
        Parameters:
        a - first factor
        b - second factor
        Returns:
        a * b exactly
        Since:
        1.3

      • multiplyHigh

        public static long multiplyHigh​(long a,
                                long b)
        Multiply two long integers and give the 64 most significant bits of the result.

        Beware that as Java primitive long are always considered to be signed, there are some intermediate values a and b for which a * b exceeds Long.MAX_VALUE but this method will still return 0l. This happens for example for a = 2³¹ and b = 2³² as a * b = 2⁶³ = Long.MAX_VALUE + 1, so it exceeds the max value for a long, but still fits in 64 bits, so this method correctly returns 0l in this case, but multiplication result would be considered negative (and in fact equal to Long.MIN_VALUE

        Parameters:
        a - first factor
        b - second factor
        Returns:
        a * b / 264
        Since:
        1.3

      • unsignedMultiplyHigh

        public static long unsignedMultiplyHigh​(long a,
                                        long b)
        Multiply two long unsigned integers and give the 64 most significant bits of the unsigned result.

        Beware that as Java primitive long are always considered to be signed, there are some intermediate values a and b for which a * b exceeds Long.MAX_VALUE but this method will still return 0l. This happens for example for a = 2³¹ and b = 2³² as a * b = 2⁶³ = Long.MAX_VALUE + 1, so it exceeds the max value for a long, but still fits in 64 bits, so this method correctly returns 0l in this case, but multiplication result would be considered negative (and in fact equal to Long.MIN_VALUE

        Parameters:
        a - first factor
        b - second factor
        Returns:
        a * b / 264
        Since:
        3.0

      • divideExact

        public static int divideExact​(int x,
                              int y)
        Divide two integers, checking for overflow.
        Parameters:
        x - dividend
        y - divisor
        Returns:
        x / y
        Throws:
        MathRuntimeException - if an overflow occurs
        Since:
        3.0

      • divideExact

        public static long divideExact​(long x,
                               long y)
        Divide two long integers, checking for overflow.
        Parameters:
        x - dividend
        y - divisor
        Returns:
        x / y
        Throws:
        MathRuntimeException - if an overflow occurs
        Since:
        3.0

      • ceilDiv

        public static int ceilDiv​(int a,
                          int b)
                   throws MathRuntimeException
        Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • ceilDivExact

        public static int ceilDivExact​(int a,
                               int b)
                        throws MathRuntimeException
        Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0 or if a == Integer.MIN_VALUE and b = -1
        Since:
        3.0

      • ceilDiv

        public static long ceilDiv​(long a,
                           long b)
                    throws MathRuntimeException
        Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • ceilDivExact

        public static long ceilDivExact​(long a,
                                long b)
                         throws MathRuntimeException
        Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0 or if a == Long.MIN_VALUE and b = -1
        Since:
        3.0

      • ceilDiv

        public static long ceilDiv​(long a,
                           int b)
                    throws MathRuntimeException
        Finds q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        q such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • ceilMod

        public static int ceilMod​(int a,
                          int b)
                   throws MathRuntimeException
        Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • ceilMod

        public static int ceilMod​(long a,
                          int b)
                   throws MathRuntimeException
        Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • ceilMod

        public static long ceilMod​(long a,
                           long b)
                    throws MathRuntimeException
        Finds r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0.

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

        Parameters:
        a - dividend
        b - divisor
        Returns:
        r such that a = q b + r with b < r <= 0 if b > 0 and 0 <= r < b if b < 0
        Throws:
        MathRuntimeException - if b == 0
        Since:
        3.0

      • floorDiv

        public static int floorDiv​(int a,
                           int b)
                    throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0
        See Also:
        floorMod(int, int)

      • floorDivExact

        public static int floorDivExact​(int a,
                                int b)
                         throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0 or if a == Integer.MIN_VALUE and b = -1
        Since:
        3.0
        See Also:
        floorMod(int, int)

      • floorDiv

        public static long floorDiv​(long a,
                            int b)
                     throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0
        Since:
        1.3
        See Also:
        floorMod(long, int)

      • floorDiv

        public static long floorDiv​(long a,
                            long b)
                     throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0
        See Also:
        floorMod(long, long)

      • floorDivExact

        public static long floorDivExact​(long a,
                                 long b)
                          throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0 or if a == Long.MIN_VALUE and b = -1
        Since:
        3.0
        See Also:
        floorMod(long, long)

      • floorMod

        public static int floorMod​(int a,
                           int b)
                    throws MathRuntimeException
        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:
        MathRuntimeException - if b == 0
        See Also:
        floorDiv(int, int)

      • floorMod

        public static int floorMod​(long 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.

        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:
        MathRuntimeException - if b == 0
        Since:
        1.3
        See Also:
        floorDiv(long, 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:
        MathRuntimeException - if b == 0
        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

      • sqrt

        public static <T extends CalculusFieldElement<T>> T sqrt​(T a)
        Compute the square root of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - number on which evaluation is done
        Returns:
        square root of a
        Since:
        1.3

      • cosh

        public static <T extends CalculusFieldElement<T>> T cosh​(T x)
        Compute the hyperbolic cosine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        hyperbolic cosine of x
        Since:
        1.3

      • sinh

        public static <T extends CalculusFieldElement<T>> T sinh​(T x)
        Compute the hyperbolic sine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        hyperbolic sine of x
        Since:
        1.3

      • tanh

        public static <T extends CalculusFieldElement<T>> T tanh​(T x)
        Compute the hyperbolic tangent of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        hyperbolic tangent of x
        Since:
        1.3

      • acosh

        public static <T extends CalculusFieldElement<T>> T acosh​(T a)
        Compute the inverse hyperbolic cosine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - number on which evaluation is done
        Returns:
        inverse hyperbolic cosine of a
        Since:
        1.3

      • asinh

        public static <T extends CalculusFieldElement<T>> T asinh​(T a)
        Compute the inverse hyperbolic sine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - number on which evaluation is done
        Returns:
        inverse hyperbolic sine of a
        Since:
        1.3

      • atanh

        public static <T extends CalculusFieldElement<T>> T atanh​(T a)
        Compute the inverse hyperbolic tangent of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - number on which evaluation is done
        Returns:
        inverse hyperbolic tangent of a
        Since:
        1.3

      • sign

        public static <T extends CalculusFieldElement<T>> T sign​(T a)
        Compute the sign of a number. The sign is -1 for negative numbers, +1 for positive numbers and 0 otherwise, for Complex number, it is extended on the unit circle (equivalent to z/|z|, with special handling for 0 and NaN)
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - number on which evaluation is done
        Returns:
        -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a
        Since:
        2.0

      • exp

        public static <T extends CalculusFieldElement<T>> T exp​(T 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.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a double
        Returns:
        double ex
        Since:
        1.3

      • expm1

        public static <T extends CalculusFieldElement<T>> T expm1​(T x)
        Compute exp(x) - 1
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number to compute shifted exponential
        Returns:
        exp(x) - 1
        Since:
        1.3

      • log

        public static <T extends CalculusFieldElement<T>> T log​(T x)
        Natural logarithm.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a double
        Returns:
        log(x)
        Since:
        1.3

      • log1p

        public static <T extends CalculusFieldElement<T>> T log1p​(T x)
        Computes log(1 + x).
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Number.
        Returns:
        log(1 + x).
        Since:
        1.3

      • log10

        public static <T extends CalculusFieldElement<T>> T log10​(T x)
        Compute the base 10 logarithm.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a number
        Returns:
        log10(x)
        Since:
        1.3

      • pow

        public static <T extends CalculusFieldElement<T>> T pow​(T x,
                                                        T y)
        Power function. Compute xy.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a double
        y - a double
        Returns:
        xy
        Since:
        1.3

      • pow

        public static <T extends CalculusFieldElement<T>> T pow​(T x,
                                                        double y)
        Power function. Compute xy.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a double
        y - a double
        Returns:
        xy
        Since:
        1.7

      • pow

        public static <T extends CalculusFieldElement<T>> T pow​(T d,
                                                        int e)
        Raise a double to an int power.
        Type Parameters:
        T - the type of the field element
        Parameters:
        d - Number to raise.
        e - Exponent.
        Returns:
        de
        Since:
        1.3

      • sin

        public static <T extends CalculusFieldElement<T>> T sin​(T x)
        Sine function.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Argument.
        Returns:
        sin(x)
        Since:
        1.3

      • cos

        public static <T extends CalculusFieldElement<T>> T cos​(T x)
        Cosine function.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Argument.
        Returns:
        cos(x)
        Since:
        1.3

      • tan

        public static <T extends CalculusFieldElement<T>> T tan​(T x)
        Tangent function.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - Argument.
        Returns:
        tan(x)
        Since:
        1.3

      • atan

        public static <T extends CalculusFieldElement<T>> T atan​(T x)
        Arctangent function
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a number
        Returns:
        atan(x)
        Since:
        1.3

      • atan2

        public static <T extends CalculusFieldElement<T>> T atan2​(T y,
                                                          T x)
        Two arguments arctangent function
        Type Parameters:
        T - the type of the field element
        Parameters:
        y - ordinate
        x - abscissa
        Returns:
        phase angle of point (x,y) between -PI and PI
        Since:
        1.3

      • asin

        public static <T extends CalculusFieldElement<T>> T asin​(T x)
        Compute the arc sine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        arc sine of x
        Since:
        1.3

      • acos

        public static <T extends CalculusFieldElement<T>> T acos​(T x)
        Compute the arc cosine of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        arc cosine of x
        Since:
        1.3

      • cbrt

        public static <T extends CalculusFieldElement<T>> T cbrt​(T x)
        Compute the cubic root of a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number on which evaluation is done
        Returns:
        cubic root of x
        Since:
        1.3

      • norm

        public static <T extends CalculusFieldElement<T>> double norm​(T x)
        Norm.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which norm is requested
        Returns:
        norm(x)
        Since:
        2.0

      • abs

        public static <T extends CalculusFieldElement<T>> T abs​(T x)
        Absolute value.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which absolute value is requested
        Returns:
        abs(x)
        Since:
        2.0

      • toRadians

        public static <T extends CalculusFieldElement<T>> T toRadians​(T x)
        Convert degrees to radians, with error of less than 0.5 ULP
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - angle in degrees
        Returns:
        x converted into radians

      • toDegrees

        public static <T extends CalculusFieldElement<T>> T toDegrees​(T x)
        Convert radians to degrees, with error of less than 0.5 ULP
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - angle in radians
        Returns:
        x converted into degrees

      • scalb

        public static <T extends CalculusFieldElement<T>> T scalb​(T d,
                                                          int n)
        Multiply a double number by a power of 2.
        Type Parameters:
        T - the type of the field element
        Parameters:
        d - number to multiply
        n - power of 2
        Returns:
        d × 2n
        Since:
        1.3

      • ulp

        public static <T extends CalculusFieldElement<T>> T ulp​(T x)
        Compute least significant bit (Unit in Last Position) for a number.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which ulp is requested
        Returns:
        ulp(x)
        Since:
        2.0

      • floor

        public static <T extends CalculusFieldElement<T>> T floor​(T x)
        Get the largest whole number smaller than x.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which floor is requested
        Returns:
        a double number f such that f is an integer f <= x < f + 1.0
        Since:
        1.3

      • ceil

        public static <T extends CalculusFieldElement<T>> T ceil​(T x)
        Get the smallest whole number larger than x.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which ceil is requested
        Returns:
        a double number c such that c is an integer c - 1.0 < x <= c
        Since:
        1.3

      • rint

        public static <T extends CalculusFieldElement<T>> T rint​(T x)
        Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.
        Type Parameters:
        T - the type of the field element
        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
        Since:
        1.3

      • round

        public static <T extends CalculusFieldElement<T>> long round​(T x)
        Get the closest long to x.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - number from which closest long is requested
        Returns:
        closest long to x
        Since:
        1.3

      • min

        public static <T extends CalculusFieldElement<T>> T min​(T a,
                                                        T b)
        Compute the minimum of two values
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - first value
        b - second value
        Returns:
        a if a is lesser or equal to b, b otherwise
        Since:
        1.3

      • min

        public static <T extends CalculusFieldElement<T>> T min​(T a,
                                                        double b)
        Compute the minimum of two values
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - first value
        b - second value
        Returns:
        a if a is lesser or equal to b, b otherwise
        Since:
        1.3

      • max

        public static <T extends CalculusFieldElement<T>> T max​(T a,
                                                        T b)
        Compute the maximum of two values
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - first value
        b - second value
        Returns:
        b if a is lesser or equal to b, a otherwise
        Since:
        1.3

      • max

        public static <T extends CalculusFieldElement<T>> T max​(T a,
                                                        double b)
        Compute the maximum of two values
        Type Parameters:
        T - the type of the field element
        Parameters:
        a - first value
        b - second value
        Returns:
        b if a is lesser or equal to b, a otherwise
        Since:
        1.3

      • hypot

        public static <T extends CalculusFieldElement<T>> T hypot​(T x,
                                                          T 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.
        Type Parameters:
        T - the type of the field element
        Parameters:
        x - a value
        y - a value
        Returns:
        sqrt(x2 +y2)
        Since:
        1.3

      • IEEEremainder

        public static <T extends CalculusFieldElement<T>> T IEEEremainder​(T 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.
        Type Parameters:
        T - the type of the field element
        Parameters:
        dividend - the number to be divided
        divisor - the number by which to divide
        Returns:
        the remainder, rounded
        Since:
        1.3

      • IEEEremainder

        public static <T extends CalculusFieldElement<T>> T IEEEremainder​(T dividend,
                                                                  T 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.
        Type Parameters:
        T - the type of the field element
        Parameters:
        dividend - the number to be divided
        divisor - the number by which to divide
        Returns:
        the remainder, rounded
        Since:
        1.3

      • copySign

        public static <T extends CalculusFieldElement<T>> T copySign​(T magnitude,
                                                             T sign)
        Returns the first argument with the sign of the second argument. A NaN sign argument is treated as positive.
        Type Parameters:
        T - the type of the field element
        Parameters:
        magnitude - the value to return
        sign - the sign for the returned value
        Returns:
        the magnitude with the same sign as the sign argument
        Since:
        1.3

      • copySign

        public static <T extends CalculusFieldElement<T>> T copySign​(T magnitude,
                                                             double sign)
        Returns the first argument with the sign of the second argument. A NaN sign argument is treated as positive.
        Type Parameters:
        T - the type of the field element
        Parameters:
        magnitude - the value to return
        sign - the sign for the returned value
        Returns:
        the magnitude with the same sign as the sign argument
        Since:
        1.3