# NAG FL Interfaces15acf (compcdf_​normal)

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## 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 <nag.h>
 double s15acf_ (const double *x, Integer *ifail)
The routine may be called by the names s15acf or nagf_specfun_compcdf_normal.

## 3Description

s15acf evaluates an approximate value for the complement of the cumulative Normal distribution function
 $Q(x) = 12π ∫x∞ e-u2/2 du .$
The routine is based on the fact that
 $Q(x) = 12 erfc(x2)$
and it calls s15adf to obtain the necessary value of $\mathit{erfc}$, the complementary error function.
NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$Integer Input/Output
There are no failure exits from this routine. The argument ifail is included for consistency with other routines in this chapter.

None.

## 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
 $|ε|≃ | x e -x2/2 2πQ(x) δ| .$
For $x$ negative or small positive this factor is always less than $1$ 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)