Erfc

Defined in header <cmath>.

Description

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 num  (since C++23).

​Additional Overloads are provided for all integer types, which are treated as double.

Declarations

C++23

// 1)
/* floating-point-type */ erfc( /* floating-point-type */ num );
// 2)
float erfcf( float num );
// 3)
long double erfcl( long double num );

Additional Overloads

// 4)
template< class Integer >
double erfc ( Integer num );

C++11

// 1)
float erfc ( float num );
// 2)
double erfc ( double num );
//3)
long double erfc ( long double num );
// 4)
float erfcf( float num );
// 5)
long double erfcl( long double num );

Additional Overloads

// 6)
template< class Integer >
double erfc ( Integer num );

Parameters

​num - floating-point or integer value

Return value

If no errors occur, value of the complementary error function of num, that is math here, 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 Additional Overloads. 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)).

Examples

#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';
}

Result

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