std:: erfc, std:: erfcf, std:: erfcl

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

double erfc ( double num ) ;

long double erfc ( long double num ) ;
(until C++23)
/* floating-point-type */
erfc ( /* floating-point-type */ num ) ;
(since C++23)
(constexpr since C++26)
float erfcf ( float num ) ;
(2) (since C++11)
(constexpr since C++26)
long double erfcl ( long double num ) ;
(3) (since C++11)
(constexpr since C++26)
Defined in header <cmath>
template < class Integer >
double erfc ( Integer num ) ;
(A) (constexpr since C++26)
1-3) Computes the complementary error function of num , that is 1.0 - std:: erf ( num ) , but without loss of precision for large num . The library provides overloads of std::erfc 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 complementary error function of num , that is
2
√ π
∫ ∞num e -t 2 d t or 1-erf(num) , is returned.

If a range error occurs due to underflow, the correct result (after rounding) 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 is returned.
  • If the argument is -∞, 2 is returned.
  • If the argument is NaN, NaN is returned.

Notes

For the IEEE-compatible type double , underflow is guaranteed if num > 26.55 .

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 :: erfc ( num ) has the same effect as std :: erfc ( static_cast < double > ( num ) ) .

Example

Run this code 
#include <cmath>
#include <iomanip>
#include <iostream>
 
double normalCDF(double x) // Phi(-∞, x) aka N(x)
{
    return std::erfc(-x / std::sqrt(2)) / 2;
}
 
int main()
{
    std::cout << "normal cumulative distribution function:\n"
              << std::fixed << std::setprecision(2);
    for (double n = 0; n < 1; n += 0.1)
        std::cout << "normalCDF(" << n << ") = " << 100 * normalCDF(n) << "%\n";
 
    std::cout << "special values:\n"
              << "erfc(-Inf) = " << std::erfc(-INFINITY) << '\n'
              << "erfc(Inf) = " << std::erfc(INFINITY) << '\n';
}

Output: 

normal cumulative distribution function:
normalCDF(0.00) = 50.00%
normalCDF(0.10) = 53.98%
normalCDF(0.20) = 57.93%
normalCDF(0.30) = 61.79%
normalCDF(0.40) = 65.54%
normalCDF(0.50) = 69.15%
normalCDF(0.60) = 72.57%
normalCDF(0.70) = 75.80%
normalCDF(0.80) = 78.81%
normalCDF(0.90) = 81.59%
normalCDF(1.00) = 84.13%
special values:
erfc(-Inf) = 2.00
erfc(Inf) = 0.00

See also

(C++11) (C++11) (C++11)
error function
(function)
C documentation for erfc

External links

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