std::erf, std::erff, std::erfl

Defined in header `<cmath>`
float erf ( float arg ); float erff( float arg );	(1)	(since C++11)
double erf ( double arg );	(2)	(since C++11)
long double erf ( long double arg ); long double erfl( long double arg );	(3)	(since C++11)
double erf ( IntegralType arg );	(4)	(since C++11)

1-3) Computes the error function of arg.

4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (2) (the argument is cast to double).

Parameters

arg

-

value of a floating-point or integral type

Return value

If no errors occur, value of the error function of arg, that is

2 \sqrt π \int arg 0 e -t 2 d t

, is returned.

If a range error occurs due to underflow, the correct result (after rounding), that is

2*arg \sqrt π

is returned

Error handling

Errors are reported as specified in math_errhandling.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

If the argument is ±0, ±0 is returned
If the argument is ±∞, ±1 is returned
If the argument is NaN, NaN is returned

Notes

Underflow is guaranteed if |arg| < DBL_MIN*(sqrt(π)/2)

erf(x σ \sqrt 2)

is the probability that a measurement whose errors are subject to a normal distribution with standard deviation

σ

is less than

x

away from the mean value.

Example

The following example calculates the probability that a normal variate is on the interval (x1, x2)

Run this code

#include <iostream>
#include <cmath>
#include <iomanip>
double phi(double x1, double x2)
{
    return (std::erf(x2/std::sqrt(2)) - std::erf(x1/std::sqrt(2)))/2;
}
int main()
{
    std::cout << "normal variate probabilities:\n"
              << std::fixed << std::setprecision(2);
    for(int n=-4; n<4; ++n)
        std::cout << "[" << std::setw(2) << n << ":" << std::setw(2) << n+1 << "]: "
                  << std::setw(5) << 100*phi(n, n+1) << "%\n";
 
    std::cout << "special values:\n"
              << "erf(-0) = " << std::erf(-0.0) << '\n'
              << "erf(Inf) = " << std::erf(INFINITY) << '\n';
}

Output:

normal variate probabilities:
[-4:-3]:  0.13%
[-3:-2]:  2.14%
[-2:-1]: 13.59%
[-1: 0]: 34.13%
[ 0: 1]: 34.13%
[ 1: 2]: 13.59%
[ 2: 3]:  2.14%
[ 3: 4]:  0.13%
special values:
erf(-0) = -0.00
erf(Inf) = 1.00

External links

Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource.

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