/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
 2 √π

arg
e-t2
dt 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

```#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)

### See also

 erferfferfl (C99)(C99)(C99) computes error function (function)
Weisstein, Eric W. "Erfc." From MathWorld--A Wolfram Web Resource.

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