﻿ s15ae Method
s15ae returns the value of the error function $\mathrm{erf}\left(x\right)$.

# Syntax

C#
```public static double s15ae(
double x
)```
Visual Basic
```Public Shared Function s15ae ( _
x As Double _
) As Double```
Visual C++
```public:
static double s15ae(
double x
)```
F#
```static member s15ae :
x : float -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.

#### Return Value

s15ae returns the value of the error function $\mathrm{erf}\left(x\right)$.

# Description

s15ae calculates an approximate value for the error function
 $erfx=2π∫0xe-t2dt=1-erfcx.$
Let $\stackrel{^}{x}$ be the root of the equation $\mathrm{erfc}\left(x\right)-\mathrm{erf}\left(x\right)=0$ (then $\stackrel{^}{x}\approx 0.46875$). For $\left|x\right|\le \stackrel{^}{x}$ the value of $\mathrm{erf}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erfx≈xRℓ,mx2,$
where ${R}_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.
For $\left|x\right|>\stackrel{^}{x}$ the value of $\mathrm{erf}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\stackrel{^}{x}<\left|x\right|\le 4$ the value is based on the expansion
 $erfcx≈ex2Rℓ,mx;$
and for $\left|x\right|>4$ it is based on the expansion
 $erfcx≈ex2x1π+1x2Rℓ,m1/x2.$
For each expansion, the specific values of $\ell$ and $m$ are selected to be minimal such that the maximum relative error in the expansion is of the order ${10}^{-d}$, where $d$ is the maximum number of decimal digits that can be accurately represented for the particular implementation (see x02be).
For $\left|x\right|\ge {x}_{\mathrm{hi}}$ there is a danger of setting underflow in $\mathrm{erfc}\left(x\right)$ (the value of ${x}_{\mathrm{hi}}$ is given in the Users' Note for your implementation). For $x\ge {x}_{\mathrm{hi}}$, s15ae returns $\mathrm{erf}\left(x\right)=1$; for $x\le -{x}_{\mathrm{hi}}$ it returns $\mathrm{erf}\left(x\right)=-1$.

# References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

# Error Indicators and Warnings

There are no failure exits from s15ae. The parameter _ifail has been included for consistency with other methods in this chapter.

None.

None.

# Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

Example program (C#): s15aee.cs

Example program data: s15aee.d

Example program results: s15aee.r