## 1Purpose

s15adc returns the value of the complementary error function, $\mathrm{erfc}\left(x\right)$.

## 2Specification

 #include
The function may be called by the names: s15adc, nag_specfun_erfc_real or nag_erfc.

## 3Description

s15adc calculates an approximate value for the complement of the error function
 $erfcx = 2π ∫x∞ e-t2 dt = 1-erfx .$
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 $\mathrm{erfc}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erfx ≈ xRℓ,m x2 ,$
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 $\mathrm{erfc}\left(x\right)$ 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
 $erfcx ≈ ex2 Rℓ,m x ;$
and for $\left|x\right|>4$ it is based on the expansion
 $erfcx ≈ ex2 x 1π + 1x2 Rℓ,m 1/x2 .$
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 X02BEC).
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 $x\ge {x}_{\mathrm{hi}}$, s15adc returns $\mathrm{erfc}\left(x\right)=0$; for $x\le -{x}_{\mathrm{hi}}$ it returns $\mathrm{erfc}\left(x\right)=2$.

## 4References

NIST Digital Library of Mathematical Functions
Cody W J (1969) Rational Chebyshev approximations for the error function Math.Comp. 23 631–637

## 5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.

None.

## 7Accuracy

If $\delta$ and $\epsilon$ are relative errors in the argument and result, respectively, then in principle
 $ε≃ 2x e -x2 πerfcx δ .$
That is, the relative error in the argument, $x$, is amplified by a factor $\frac{2x{e}^{-{x}^{2}}}{\sqrt{\pi }\mathrm{erfc}\left(x\right)}$ in the result.
The behaviour of this factor is shown in Figure 1.
It should be noted that near $x=0$ this factor behaves as $\frac{2x}{\sqrt{\pi }}$ and hence the accuracy is largely determined by the machine precision. Also, for large negative $x$, where the factor is $\text{}\sim \frac{x{e}^{-{x}^{2}}}{\sqrt{\pi }}$, accuracy is mainly limited by machine precision. However, for large positive $x$, the factor becomes $\text{}\sim 2{x}^{2}$ and to an extent relative accuracy is necessarily lost. The absolute accuracy $E$ is given by
 $E ≃ 2xe-x2π δ$
so absolute accuracy is guaranteed for all $x$.

None.

## 10Example

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