NAME¶

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

Synopsis¶

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

double erfc ( double num ); (until C++23)

long double erfc ( long double num );
/* floating-point-type */ (since C++23)
erfc ( /* floating-point-type */ num ); (constexpr since C++26)
float erfcf( float num ); (1) (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 > (A) (constexpr since C++26)
double erfc ( Integer num );

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 (since C++11)
treated as double.

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^-t2
dt 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¶

// 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¶

erf
erff
erfl error function
(C++11) (function)
(C++11)
(C++11)
C documentation for
erfc

External links¶

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