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erfc, erfcf, erfcl

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erfc
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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.

Contents

Parameters

arg - floating-point value

Return value

If no errors occur, value of the complementary error function of arg , that is
2
√ π
∫ ∞
arg e -t 2
d t 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 <math.h>
#include <stdio.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));
    printf("special values:\n"
           "erfc(-Inf) = %f\n"
           "erfc(Inf) = %f\n",
           erfc(-INFINITY),
           erfc(INFINITY));
}

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) = 2.000000
erfc(Inf) = 0.000000

References

  • C23 standard (ISO/IEC 9899:2024):
  • 7.12.8.2 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)
  • C17 standard (ISO/IEC 9899:2018):
  • 7.12.8.2 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)
  • C11 standard (ISO/IEC 9899:2011):
  • 7.12.8.2 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):
  • 7.12.8.2 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

(C99) (C99) (C99)
computes error function
(function)
C++ documentation for erfc

External links

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