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

 

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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 \(\frac{2}{\sqrt{\pi} }\int_{0}^{arg}{e^{-{t^2} }\mathsf{d}t}\)

2

√π

∫arg0e^-t2dt, is returned. If a range error occurs due to underflow, the correct result (after rounding), that is \(\frac{2\cdot arg}{\sqrt{\pi} }\)

2*arg

√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)

\(\operatorname{erf}(\frac{x}{\sigma \sqrt{2} })\)erf(

x

σ√2

) is the probability that a measurement whose errors are subject to a normal distribution with standard deviation \(\sigma\)σ is less than \(x\)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 <cmath>
#include <iomanip>
#include <iostream>
 
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

See also

erfcerfcferfcl (C++11)(C++11)(C++11)	complementary error function (function)
C documentation for erf

External links

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