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

 

C++

Compiler support

Freestanding and hosted

Language

Standard library

Standard library headers

Named requirements

Feature test macros (C++20)

Language support library

Concepts library (C++20)

Metaprogramming library (C++11)

Diagnostics library

General utilities library

Strings library

Containers library

Iterators library

Ranges library (C++20)

Algorithms library

Numerics library

Localizations library

Input/output library

Filesystem library (C++17)

Regular expressions library (C++11)

Concurrency support library (C++11)

Execution support library (C++26)

Technical specifications

Symbols index

External libraries

 

Numerics library

Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array (valarray)
Pseudo-random number generation
Factor operations
gcd (C++17)	lcm (C++17)
Interpolations
midpoint (C++20)	lerp (C++20)
Saturation arithmetic
add_sat (C++26) sub_sat (C++26) saturate_cast (C++26)	mul_sat (C++26) div_sat (C++26)
Generic numeric operations
iota (C++11) ranges::iota (C++23) accumulate inner_product adjacent_difference partial_sum	reduce (C++17) transform_reduce (C++17) inclusive_scan (C++17) exclusive_scan (C++17) transform_inclusive_scan (C++17) transform_exclusive_scan (C++17)
Bit operations
has_single_bit (C++20)     bit_cast (C++20) bit_ceil (C++20) bit_floor (C++20) bit_width (C++20) rotl (C++20) rotr (C++20)	popcount (C++20) countl_zero (C++20) countl_one (C++20) countr_zero (C++20) countr_one (C++20) byteswap (C++23) endian (C++20)

 

Common mathematical functions

Functions
Basic operations
abs(int)labsllabsimaxabs (C++11)   abs(float)fabs divldivlldivimaxdiv (C++11)	fmod remainder (C++11)   remquo (C++11) fma (C++11) fmax (C++11) fmin (C++11) fdim (C++11) nannanfnanl (C++11)(C++11)(C++11)
Exponential functions
exp exp2 (C++11) expm1 (C++11)	log log10 log1p (C++11) log2 (C++11)
Power functions
sqrt cbrt (C++11)	hypot (C++11) pow
Trigonometric and hyperbolic functions
sin cos tan asin acos atan atan2	sinh cosh tanh asinh (C++11) acosh (C++11) atanh (C++11)
Error and gamma functions
erf (C++11) erfc (C++11)	lgamma (C++11) tgamma (C++11)

Nearest integer floating point operations
ceil floor roundlroundllround (C++11)(C++11)(C++11)	trunc (C++11) nearbyint (C++11) rintlrintllrint (C++11)(C++11)(C++11)
Floating point manipulation functions
ldexp scalbnscalbln (C++11)(C++11) ilogb (C++11) logb (C++11)	frexp modf nextafternexttoward (C++11)(C++11) copysign (C++11)
Classification and comparison
fpclassify (C++11) isfinite (C++11) isinf (C++11) isnan (C++11) isnormal (C++11) signbit (C++11)	isgreater (C++11) isgreaterequal (C++11) isless (C++11) islessequal (C++11) islessgreater (C++11) isunordered (C++11)
Types
div_t ldiv_t lldiv_t (C++11)	imaxdiv_t (C++11) float_t (C++11) double_t (C++11)
Macro constants
HUGE_VALFHUGE_VALHUGE_VALL (C++11)(C++11) math_errhandlingMATH_ERRNOMATH_ERREXCEPT (C++11)   INFINITY (C++11) NAN (C++11)	Classification FP_NORMALFP_SUBNORMALFP_ZEROFP_INFINITEFP_NAN (C++11)(C++11)(C++11)(C++11)(C++11)

 

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)

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

∫∞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

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.