# NAG Library Routine Document

## 1Purpose

s15acf returns the value of the complement of the cumulative Normal distribution function, $Q\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s15acf ( x,
 Real (Kind=nag_wp) :: s15acf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s15acf_ (const double *x, Integer *ifail)

## 3Description

s15acf evaluates an approximate value for the complement of the cumulative Normal distribution function
 $Qx=12π∫x∞e-u2/2du.$
The routine is based on the fact that
 $Qx=12erfcx2$
and it calls s15adf to obtain the necessary value of $\mathit{erfc}$, the complementary error function.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

There are no failure exits from this routine. The argument ifail is included for consistency with other routines in this chapter.

## 7Accuracy

Because of its close relationship with $\mathit{erfc}$ the accuracy of this routine is very similar to that in s15adf. If $\epsilon$ and $\delta$ are the relative errors in result and argument, respectively, then in principle they are related by
 $ε≃ x e -x2/2 2πQx δ .$
For $x$ negative or small positive this factor is always less than one and accuracy is mainly limited by machine precision. For large positive $x$ we find $\epsilon \sim {x}^{2}\delta$ and hence to a certain extent relative accuracy is unavoidably lost. However the absolute error in the result, $E$, is given by
 $E≃ x e -x2/2 2π δ$
and since this factor is always less than one absolute accuracy can be guaranteed for all $x$.

## 8Parallelism and Performance

s15acf is not threaded in any implementation.

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.

### 10.1Program Text

Program Text (s15acfe.f90)

### 10.2Program Data

Program Data (s15acfe.d)

### 10.3Program Results

Program Results (s15acfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017