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
2
√π
∫arg
0e-t2
dt, is returned. If a range error occurs due to underflow, the correct result (after rounding), that is
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). erf(

x
σ√2
) is the probability that a measurement whose errors are subject to a normal distribution with standard deviation σ is less than 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)

See also

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

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