# Analytic error function and numeric inverse obtained by geometric means

## TL;DR

This paper presents a geometric approach to deriving the error function's integral representation and its inverse, offering new formulas and insights useful for high-speed Monte Carlo simulations.

## Contribution

It introduces a geometric method to derive the error function's integral form and inverse approximation formulas, enhancing computational efficiency.

## Key findings

- Derived the Craig formula using geometric considerations
- Proved convergence of the power series expansion
- Developed systematic formulas for approximating the inverse error function

## Abstract

Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same geometric means finally help to systematically derive handy formulas that approximate the inverse error function. Our approach can be used for applications in e.g.\ high-speed Monte Carlo simulations where this function is used extensively.