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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∫arg
0e^-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),

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)

External links

Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource.

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https://en.cppreference.com/w/c/numeric/math/erf

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