s15ac returns the value of the complement of the cumulative Normal distribution function, $Q\left(x\right)$.

# Syntax

C# |
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public static double s15ac( double x ) |

Visual Basic |
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Public Shared Function s15ac ( _ x As Double _ ) As Double |

Visual C++ |
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public: static double s15ac( double x ) |

F# |
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static member s15ac : x : float -> float |

#### Parameters

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

#### Return Value

s15ac returns the value of the complement of the cumulative Normal distribution function, $Q\left(x\right)$.

# Description

s15ac evaluates an approximate value for the complement of the cumulative Normal distribution function

The method is based on the fact that

and it calls s15ad to obtain the necessary value of $\mathit{erfc}$, the complementary error function.

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

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

# 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, then in principle they are related by

$$\left|\epsilon \right|\simeq \left|\frac{x{e}^{-{x}^{2}/2}}{\sqrt{2\pi}Q\left(x\right)}\delta \right|\text{.}$$ |

For $x$ negative or small positive this factor is always less than one 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

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

# 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#): s15ace.cs