Find X in a 4PL Curve Regression I'm working on a project where I need to replicate a calculation that is currently done by a legacy system, let's call this legacy system of 4P.
In the company nobody knows how the calculation is done, my goal is to find out how it works.
Let's start with the principle that we have the following points on a Cartesian plane.
Important, the points are not fixed, the user can change them at any time


*

*x: 0    y: 0.009

*x: 10   y: 0.242

*x: 20   y: 0.579

*x: 40   y: 1.445

*x: 80   y: 2.91

*x: 160  y: 3.5


The problem is apparently simple, given a value in Y, return the corresponding X value.
For example, based on the points reported, if the user informs Y = 0.161 the program should return the value of X = 9.4737.
My findings so far
Using the website https://www.aatbio.com/tools/four-parameter-logistic-4pl-curve-regression-online-calculator/ I discovered the following formula.
Formula
Using the site, when i input the value of Y = 0.161 the site returns X = 8.95, value close to that calculated by the 4P program.
Using the site and the program 4P I identified that four values ​​are used in the formula, however, the values ​​are different in the site and 4p.


*

*Intercept: website = 0.075 | 4PL = 0.0806

*Slope fator: website = 2.200 | 4PL = 2.3604

*Inflection point: website = 49.090 | 4PL = 47.0402

*Limit: website = 3.788 | 4PL = 3.6908


4PL Result values
Using the formula reported on the site with the values ​​generated in the 4P program you can find X = 9.4737
How can I solve this problem?
 A: One cannot give a definitive answer to your question because some pieces of information are missing. 
The website fits the data according to a supposed formula $$y=y_0+\frac{y_\infty-y_0}{1+\left(\frac{x_{I_p}}{x}\right)^{-S_f}}$$ 
This means that the parameters $y_0$ , $y_\infty$ , $S_f$ and $x_{I_p}$ are adjusted according to some criteria of proximity between $y_{given}$ and $y_{computed}$.
First question : Are you sure that 4PL use the same formula ? Probably yes.
Second question : are you sure that website and 4PL use the same criteria for fitting ? 
They are many criteria for example : Least mean square deviation, or least mean square relative deviation, or least mean absolute deviation, or etc. For each different criteria, the result is slightly different, depending on the scatter of the data. 
The results are exactly the same only if there is no scatter and if the supposed formula is the correct formula to model the experiment.
Since the results of website and 4PL are not exactly the same, probably they don't use the same criteria for fitting. If it is the case, it is not surprizing that the results be slightly different.

Example of different conclusions when different criteria of fitting are chosen :
Least mean square deviation = 
$\begin{cases}\text{Website : }0.00359\\
\text{4PL : }0.00434 \\ 
\text{The website is better.}\end{cases}$ 
Least mean absolute deviation =$ \begin{cases}\text{Website : }0.0583\\
\text{4PL : }0.0564 \\ 
\text{4PL is better.}\end{cases}$ 
It's up to you to chose which criteria is convenient for your use and then to find a convenient software in which the chosen criteria is implemented.
