Clustering when data is represented by multiple functional forms, all at once

I wish to cluster similar data where I have a collection of many $y$ vs $x$ data. A third variable $z$ also exists, but z doesn't always affect $y$ whereas $x$ always affects $z$.

In terms of $x$ and $z$, y has five possible functional forms:

1. $y = \frac{f(x,A)}{g(x,B)}$
2. $y = \frac{f(x,A)}{g(x,B\times h(z,C))}$
3. $y = \frac{f(x,A)}{g(x\times h(z,D),B)}$
4. $y = \frac{f(x,A)}{g(x,B)\times j(z,E)}$
5. $y = \frac{f(x,A)}{g(x\times h(z,D),B\times h(z,C))}$

One of those forms best describes results from each experimental run i.e. each $y$ vs $x$ data I have. My goal is to cluster data obtained from many experiments (multiple $y$ vs $x$ data) where we have a mix of those forms.

How does one go about doing this?

Points to note:

• We don't know what the $x$ vs $z$ relation is.
• The graph of each of those functional forms looks similar. The form that best fits data from an experimental run is chosen for that particular run.