erfc, erfcf, erfcl

Defined in header <math.h>
float       erfcf( float arg );
(1) (since C99)
double      erfc( double arg );
(2) (since C99)
long double erfcl( long double arg );
(3) (since C99)
Defined in header <tgmath.h>
#define erfc( arg )
(4) (since C99)
1-3) Computes the complementary error function of arg, that is 1.0-erf(arg), but without loss of precision for large arg.
4) Type-generic macro: If arg has type long double, erfcl is called. Otherwise, if arg has integer type or the type double, erfc is called. Otherwise, erfcf is called.


arg - floating point value

Return value

If no errors occur, value of the complementary error function of arg, that is \(\frac{2}{\sqrt{\pi} }\int_{arg}^{\infty}{e^{-{t^2} }\mathsf{d}t}\)2/√π
dt or \({\small 1-\operatorname{erf}(arg)}\)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


For the IEEE-compatible type double, underflow is guaranteed if arg > 26.55.


#include <stdio.h>
#include <math.h>
double normalCDF(double x) // Phi(-∞, x) aka N(x)
    return erfc(-x/sqrt(2))/2;
int main(void)
    puts("normal cumulative distribution function:");
    for(double n=0; n<1; n+=0.1)
        printf("normalCDF(%.2f) %5.2f%%\n", n, 100*normalCDF(n));
    puts("special values:");
    printf("erfc(-Inf) = %f\n", erfc(-INFINITY));
    printf("erfc(Inf) = %f\n", erfc(INFINITY));


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) = 2.000000
erfc(Inf) = 0.000000


  • C11 standard (ISO/IEC 9899:2011):
    • The erfc functions (p: 249-250)
    • 7.25 Type-generic math <tgmath.h> (p: 373-375)
    • F.10.5.2 The erfc functions (p: 525)
  • C99 standard (ISO/IEC 9899:1999):
    • The erfc functions (p: 230)
    • 7.22 Type-generic math <tgmath.h> (p: 335-337)
    • F.9.5.2 The erfc functions (p: 462)

See also

computes error function
C++ documentation for erfc
Weisstein, Eric W. "Erfc." From MathWorld--A Wolfram Web Resource.

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