s15ab returns the value of the cumulative Normal distribution function, $P\left(x\right)$.

# Syntax

C# |
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public static double s15ab( double x ) |

Visual Basic |
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Public Shared Function s15ab ( _ x As Double _ ) As Double |

Visual C++ |
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public: static double s15ab( double x ) |

F# |
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static member s15ab : x : float -> float |

#### Parameters

- x
- Type: System..::..Double
*On entry*: the argument $x$ of the function.

#### Return Value

s15ab returns the value of the cumulative Normal distribution function, $P\left(x\right)$.

# Description

s15ab evaluates an approximate value for the cumulative Normal distribution function

The method is based on the fact that

and it calls s15ad to obtain a value of $\mathit{erfc}$ for the appropriate argument.

$$P\left(x\right)=\frac{1}{\sqrt{2\pi}}\underset{-\infty}{\overset{x}{\int}}{e}^{-{u}^{2}/2}du\text{.}$$ |

$$P\left(x\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{-x}{\sqrt{2}}\right)$$ |

# References

Abramowitz M and Stegun I A (1972)

*Handbook of Mathematical Functions*(3rd Edition) Dover Publications# Error Indicators and Warnings

None.

# Accuracy

Because of its close relationship with $\mathit{erfc}$, the accuracy of this method is very similar to that in s15ad. If $\epsilon $ and $\delta $ are the relative errors in result and argument, respectively, they are in principle related by

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.

$$\left|\epsilon \right|\simeq \left|\frac{x{e}^{-\frac{1}{2}{x}^{2}}}{\sqrt{2\pi}P\left(x\right)}\delta \right|$$ |

For $x$ small and for $x$ positive this factor is always less than one 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

so absolute accuracy can be guaranteed for all $x$.

$$\left|E\right|\simeq \left|\frac{x{e}^{-\frac{1}{2}{x}^{2}}}{\sqrt{2\pi}}\delta \right|$$ |

# Parallelism and Performance

None.

# Further Comments

None.

# Example

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

Example program (C#): s15abe.cs