/C

# erf, erff, erfl

Defined in header <math.h>
float       erff( float arg );
(1) (since C99)
double      erf( double arg );
(2) (since C99)
long double erfl( long double arg );
(3) (since C99)
Defined in header <tgmath.h>
#define erf( arg )
(4) (since C99)
1-3) Computes the error function of arg.
4) Type-generic macro: If arg has type long double, erfl is called. Otherwise, if arg has integer type or the type double, erf is called. Otherwise, erff is called.

### Parameters

 arg - floating point value

### Return value

If no errors occur, value of the error function of arg, that is $$\frac{2}{\sqrt{\pi} }\int_{0}^{arg}{e^{-{t^2} }\mathsf{d}t}$$2/√πarg
0
e-t2
dt, is returned. If a range error occurs due to underflow, the correct result (after rounding), that is $$\frac{2\cdot arg}{\sqrt{\pi} }$$2*arg/√π, 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, ±0 is returned
• If the argument is ±∞, ±1 is returned
• If the argument is NaN, NaN is returned

### Notes

Underflow is guaranteed if |arg| < DBL_MIN*(sqrt(π)/2). $$\operatorname{erf}(\frac{x}{\sigma \sqrt{2} })$$erf(

x/σ√2) is the probability that a measurement whose errors are subject to a normal distribution with standard deviation $$\sigma$$σ is less than $$x$$x away from the mean value.

### Example

#include <stdio.h>
#include <math.h>
double phi(double x1, double x2)
{
return (erf(x2/sqrt(2)) - erf(x1/sqrt(2)))/2;
}
int main(void)
{
puts("normal variate probabilities:");
for(int n=-4; n<4; ++n)
printf("[%2d:%2d]: %5.2f%%\n", n, n+1, 100*phi(n, n+1));

puts("special values:");
printf("erf(-0) = %f\n", erf(-0.0));
printf("erf(Inf) = %f\n", erf(INFINITY));
}

Output:

normal variate probabilities:
[-4:-3]:  0.13%
[-3:-2]:  2.14%
[-2:-1]: 13.59%
[-1: 0]: 34.13%
[ 0: 1]: 34.13%
[ 1: 2]: 13.59%
[ 2: 3]:  2.14%
[ 3: 4]:  0.13%
special values:
erf(-0) = -0.000000
erf(Inf) = 1.000000

### References

• C11 standard (ISO/IEC 9899:2011):
• 7.12.8.1 The erf functions (p: 249)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• F.10.5.1 The erf functions (p: 525)
• C99 standard (ISO/IEC 9899:1999):
• 7.12.8.1 The erf functions (p: 230)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• F.9.5.1 The erf functions (p: 462)

 erfcerfcferfcl (C99)(C99)(C99) computes complementary error function (function) C++ documentation for erf