# NAG Library Routine Document

## 1Purpose

s15abf returns the value of the cumulative Normal distribution function, $P\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s15abf ( x,
 Real (Kind=nag_wp) :: s15abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s15abf_ (const double *x, Integer *ifail)

## 3Description

s15abf evaluates an approximate value for the cumulative Normal distribution function
 $Px = 12π ∫-∞x e-u2/2 du .$
The routine is based on the fact that
 $Px = 12 erfc-x2$
and it calls s15adf to obtain a value of $\mathit{erfc}$ for the appropriate argument.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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, they are in principle related by
 $ε≃ x e -12 x2 2πPx δ$
so that the relative error in the argument, $x$, is amplified by a factor, $\frac{x{e}^{-\frac{1}{2}{x}^{2}}}{\sqrt{2\pi }P\left(x\right)}$, in the result.
For $x$ small and for $x$ positive this factor is always less than $1$ and accuracy is mainly limited by machine precision.
For large negative $x$ the factor behaves like $\text{}\sim {x}^{2}$ and hence to a certain extent relative accuracy is unavoidably lost.
However, the absolute error in the result, $E$, is given by
 $E≃ x e -12 x2 2π δ$
so absolute accuracy can be guaranteed for all $x$.

## 8Parallelism and Performance

s15abf 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 (s15abfe.f90)

### 10.2Program Data

Program Data (s15abfe.d)

### 10.3Program Results

Program Results (s15abfe.r)