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).

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 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

All 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 <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)
static long addExact(long a, long b)
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 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 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 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.
• ### 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.
Constant Field Values
• #### E

public static final double E
Napier's constant e, base of the natural logarithm.
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
• #### 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

public static double toRadians(double x)
Convert degrees to radians, with error of less than 0.5 ULP
Parameters:
x - angle in degrees
Returns:
• #### 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
• #### 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

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

public static long addExact(long a,
long b)
throws MathRuntimeException
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
• #### 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
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
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
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
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
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
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

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:
• #### 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