# Analytic Error Function and Numeric Inverse Obtained by Geometric Means

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of Craig’s Integral Representation

## 3. Power Series Expansion

## 4. Approximations for the Inverse Error Function

## 5. Discussion

`Slackware 15.0`(

`Linux 5.15.19`) on an ordinary HP laptop with an Intel® Core™2 Duo CPU P8600 @ 2.4GHz with 3MB memory used. The dependence of the CPU time for the calculation was estimated by calculating the value ${10}^{6}$ times in sequence. The speed of the calculation did not depend on the value for E, as the precision was not optimized. This would be required for practical application. Using an arbitrary starting value $E=0.8$, we performed this test, and the results are shown in Table 1. An analysis of this table showed that a further step in the degree p doubled the runtime while the dynamics for increasing n added a constant value of approximately $0.06$ seconds to the result. Though the increase in the dynamics required the solution of a linear system of equations and the coding of the results, this endeavor was justified, as by using the dynamics, we could increase the precision of the results without sacrificing the computational speed.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Runtime experiment for our algorithm under C for $E=0.8$ and different values of n and p (CPU time in seconds). As indicated, the errors are in the last displayed digit, i.e., $\pm 0.01$ s.

$\mathit{n}=0$ | $\mathit{n}=1$ | $\mathit{n}=2$ | $\mathit{n}=3$ | $\mathit{n}=4$ | $\mathit{n}=5$ | |
---|---|---|---|---|---|---|

$p=0$ | $0.07\left(1\right)$ | $0.13\left(1\right)$ | $0.17\left(1\right)$ | $0.21\left(1\right)$ | $0.31\left(1\right)$ | $0.56\left(1\right)$ |

$p=1$ | $0.14\left(1\right)$ | $0.20\left(1\right)$ | $0.24\left(1\right)$ | $0.29\left(1\right)$ | $0.39\left(1\right)$ | $0.63\left(1\right)$ |

$p=2$ | $0.25\left(1\right)$ | $0.32\left(1\right)$ | $0.35\left(1\right)$ | $0.40\left(1\right)$ | $0.50\left(1\right)$ | $0.75\left(1\right)$ |

$\mathit{E}=$ | $\mathit{n}=0$ | $\mathit{n}=1$ | $\mathit{n}=2$ | $\mathit{n}=3$ | $\mathit{n}=4$ | $\mathit{n}=5$ | [12] |
---|---|---|---|---|---|---|---|

$0.7$ | $0.732995$ | $0.732868$ | $0.732869$ | $0.732869$ | $0.732869$ | $0.732869$ | $0.17$ |

$0.8$ | $0.906326$ | $0.906193$ | $0.906194$ | $0.906194$ | $0.906194$ | $0.906194$ | $0.19$ |

$0.9$ | $1.163247$ | $1.163085$ | $1.163087$ | $1.163087$ | $1.163087$ | $1.163087$ | $0.35$ |

$0.99$ | $1.821691$ | $1.821376$ | $1.821387$ | $1.821386$ | $1.821386$ | $1.821386$ | $1.95$ |

$0.999$ | $2.326608$ | $2.326762$ | $2.326752$ | $2.326754$ | $2.326754$ | $2.326754$ | $14.62$ |

$0.9999$ | $2.749217$ | $2.751197$ | $2.751034$ | $2.751076$ | $2.751056$ | $2.751971$ | $128.30$ |

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**MDPI and ACS Style**

Martila, D.; Groote, S.
Analytic Error Function and Numeric Inverse Obtained by Geometric Means. *Stats* **2023**, *6*, 431-437.
https://doi.org/10.3390/stats6010026

**AMA Style**

Martila D, Groote S.
Analytic Error Function and Numeric Inverse Obtained by Geometric Means. *Stats*. 2023; 6(1):431-437.
https://doi.org/10.3390/stats6010026

**Chicago/Turabian Style**

Martila, Dmitri, and Stefan Groote.
2023. "Analytic Error Function and Numeric Inverse Obtained by Geometric Means" *Stats* 6, no. 1: 431-437.
https://doi.org/10.3390/stats6010026