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

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< cpp‎ | numeric‎ | math

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)
SIMD overload (since C++26) |  |
Defined in header `<simd>` |  |
template< /*math-floating-point*/ V > constexpr /*deduced-simd-t*/<V> erfc ( const V& v_num ); | (S) | (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)

S) The SIMD overload performs an element-wise `std::erfc` on v_num.

    (See `math-floating-point` and `deduced-simd-t` for their definitions.)

(since C++26)

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 \(\frac{2}{\sqrt{\pi} }\int_{num}^{\infty}{e^{-{t^2} }\mathsf{d}t}\)

2
√π

∫∞
num e-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

erferfferfl (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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