# erf, erff, erfl

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< c‎ | numeric‎ | math

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 {{enwiki}Error function|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
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

Run this code

    #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

- 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

erfc erfcf erfcl (C99)(C99)(C99) | computes complementary error function (function)

### External links

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

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