erf, erff, erfl
float erff( float arg );
| (1) | (since C99) |
double erf( double arg );
| (2) | (since C99) |
long double erfl( long double arg );
| (3) | (since C99) |
#define erf( arg )
| (4) | (since C99) |
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-t2
d
t, 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)
See also
(C99)(C99)(C99) | computes complementary error function (function) |
C++ documentation for erf |
External links
Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource.