# NAG FL Interfaces15abf (cdf_​normal)

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## 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 <nag.h>
 double s15abf_ (const double *x, Integer *ifail)
The routine may be called by the names s15abf or nagf_specfun_cdf_normal.

## 3Description

s15abf evaluates an approximate value for the cumulative Normal distribution function
 $P(x) = 12π ∫-∞x e-u2/2 du .$
The routine is based on the fact that
 $P(x) = 12 erfc(-x2)$
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: $\mathbf{ifail}$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, they are in principle related by
 $|ε|≃ | x e -12 x2 2πP(x) δ|$
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)