/C

# 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.

### Parameters

 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
arg
e-t2
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

### Notes

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

### Example

#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));
}

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

• 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)

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