std::erfc

 

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>
float       erfc( float arg );		(since C++11)
double      erfc( double arg );		(since C++11)
long double erfc( long double arg );		(since C++11)
double      erfc( Integral arg );		(since C++11)

1-3) Computes the complementary error function of arg, that is 1.0-erf(arg), but without loss of precision for large arg

4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to 2) (the argument is cast to double).

Parameters

arg

-

value of a floating-point or Integral type

Return value

If no errors occur, value of the complementary error function of arg, that is

2

√π

∫∞arge^-t2dt or 1-erf(arg), 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 arg > 26.55.

Example

Run this code

#include <iostream>
#include <cmath>
#include <iomanip>
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) 57.93%
normalCDF(0.20) 65.54%
normalCDF(0.30) 72.57%
normalCDF(0.40) 78.81%
normalCDF(0.50) 84.13%
normalCDF(0.60) 88.49%
normalCDF(0.70) 91.92%
normalCDF(0.80) 94.52%
normalCDF(0.90) 96.41%
normalCDF(1.00) 97.72%
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.