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/√π∫arg0e-t2dt, 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),

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 <math.h>
#include <stdio.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

See also

erfcerfcferfcl
(C99)(C99)(C99)
 computes complementary error function
(function)
C++ documentation for erf
Weisstein, Eric W. "Erf." From MathWorld — A Wolfram Web Resource.

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