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## 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
 $erfc(x) = 2π ∫x∞ e-t2 dt = 1-erf(x) .$
Unless stated otherwise in the Users' Note, s15adc calls the complementary error function supplied by the compiler used for your implementation; as such, details of the underlying algorithm should be obtained from the documentation supplied by the compiler vendor. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
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 $|x|\le \stackrel{^}{x}$ the value of $\mathrm{erfc}\left(x\right)$ is based on the following rational Chebyshev expansion for $\mathrm{erf}\left(x\right)$:
 $erf(x) ≈ xRℓ,m (x2) ,$
where ${R}_{\ell ,m}$ denotes a rational function of degree $\ell$ in the numerator and $m$ in the denominator.
For $|x|>\stackrel{^}{x}$ the value of $\mathrm{erfc}\left(x\right)$ is based on a rational Chebyshev expansion for $\mathrm{erfc}\left(x\right)$: for $\stackrel{^}{x}<|x|\le 4$ the value is based on the expansion
 $erfc(x) ≈ ex2 Rℓ,m (x) ;$
and for $|x|>4$ it is based on the expansion
 $erfc(x) ≈ ex2 x (1π+1x2Rℓ,m(1/x2)) .$
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 $|x|\ge {x}_{\mathrm{hi}}$ 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

Unless stated otherwise in the Users' Note, s15adc calls the complementary error function supplied by the compiler used for your implementation. The following discussion only applies if the Users' Note for your implementation indicates that the compiler's supplied function was not available.
If $\delta$ and $\epsilon$ are relative errors in the argument and result, respectively, then in principle
 $|ε|≃ | 2x e -x2 πerfc(x) δ| .$
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 $x$.

## 8Parallelism and Performance

### 9.1Internal Changes

Internal changes have been made to this function as follows:
• At Mark 27.1:
Modified to use a compiler-supplied erfc function, when available.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

## 10Example

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