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

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Defined in header <cmath>
(1)
float erf ( float num ) ;

double erf ( double num ) ;

long double erf ( long double num ) ;
(until C++23)
/* floating-point-type */
erf ( /* floating-point-type */ num ) ;
(since C++23)
(constexpr since C++26)
float erff ( float num ) ;
(2) (since C++11)
(constexpr since C++26)
long double erfl ( long double num ) ;
(3) (since C++11)
(constexpr since C++26)
Defined in header <cmath>
template < class Integer >
double erf ( Integer num ) ;
(A) (constexpr since C++26)
1-3) Computes the error function of num . The library provides overloads of std::erf for all cv-unqualified floating-point types as the type of the parameter. (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double .
(since C++11)

Parameters

num - floating-point or integer value

Return value

If no errors occur, value of the error function of num , that is
2
√ π
∫ num0 e -t 2 d t , is returned.
If a range error occurs due to underflow, the correct result (after rounding), that is
2*num
√ 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 | num | < DBL_MIN * ( std:: sqrt ( π ) / 2 ) .

erf(
x
σ √ 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.

The additional overloads are not required to be provided exactly as (A) . They only need to be sufficient to ensure that for their argument num of integer type, std :: erf ( num ) has the same effect as std :: erf ( static_cast < double > ( num ) ) .

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

(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.
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