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

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)
Additional overloads (since C++11)
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)

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 \(\frac{2}{\sqrt{\pi} }\int_{num}^{\infty}{e^{-{t^2} }\mathsf{d}t}\)2/√π∫∞
nume^-t2dt or \({\small 1-\operatorname{erf}(num)}\)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

#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

External links

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

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https://en.cppreference.com/w/cpp/numeric/math/erfc

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