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NAG C Library Manual

# NAG Library Function Documentnag_cumul_normal (s15abc)

## 1  Purpose

nag_cumul_normal (s15abc) returns the value of the cumulative Normal distribution function $P\left(x\right)$.

## 2  Specification

 #include #include
 double nag_cumul_normal (double x)

## 3  Description

nag_cumul_normal (s15abc) evaluates the cumulative Normal distribution function
 $P x = 1 2π ∫ -∞ x e - u 2 / 2 du .$
The function is based on the fact that
 $P x = 1 2 erfc -x / 2 .$

## 4  References

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

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.

None.

## 7  Accuracy

If $\epsilon$ and $\delta$ are the relative errors in result and argument, respectively, they are in principle related by $\left|\epsilon \right|\simeq \left|\left({xe}^{-{x}^{2}/2}/\sqrt{2\pi }P\left(x\right)\right)\delta \right|$.
For $x$ small and for $x$ positive the multiplying factor is always less than one and accuracy is mainly limited by machine precision. For large negative $x$ we find $\epsilon \sim {x}^{2}$ and hence to a certain extent relative accuracy is unavoidably lost. However, the absolute error in the result, $E$, is given by $\left|E\right|\simeq \left|\left({xe}^{-{x}^{2}/2}/\sqrt{2\pi }\right)\right|>\delta$, and since this multiplying factor is always less than one, absolute accuracy can be guaranteed for all $x$.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s15abce.c)

### 9.2  Program Data

Program Data (s15abce.d)

### 9.3  Program Results

Program Results (s15abce.r)