This project aims at approximating the mathematical error function with a simple sigmoid function:
erf(x) ~= 2 / (1+e^(-bx))-1
This approximation is strong around zero, with x>1, and x<-1. In this project I aim at finding the value b with the smalles average error by gradient decend. This project compares it's predictions to the known values provided by math.erf().
Even though less accurate, this approximation can be inverted to a inverse error function:
erfinv(y) = -ln(2/(1+y)-1)/b
The first run of the gradient-descend function had the following properties:
| Property | Value |
|---|---|
| Starting Point | 2.405 |
| Gamma | 0.001 |
| Precision | 1E-7 |
| Max. Iterations | 2E+6 |
| Iterations | 1727 |
In the numeric approximations of integral and derivative the following variables were used:
| Variable | Value |
|---|---|
| Integral Min. | -3 |
| Integral Max. | +3 |
| Δx for derivative | 1E-7 |
| Δx for integral | 5E-4 |
The function produced an approximation of b and an average error :
| Result | Value |
|---|---|
| b | 2.4086376806430434 |
| Average Error | 0.0105561117608605 |