How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions.

I have the following data points:

x y z 1 0.5 0.01 1 1 0.01 1 2 0.01 1 10 0.01 1.3 0.5 0.015 1.3 1 0.0177 1.3 2 0.023 1.3 10 0.066 1.5 0.5 0.018 1.5 1 0.0223 1.5 2 0.031 1.5 10 0.1

Assume z is the output, and x and y is the input, and no variable can be 0.

1. Given these sample data points, how can I predict z given x and y?
2. Is there a mathematical relationship between the variables?
3. How can I find an equation that relates these variables?

A rough drawing of the points $(y,z)$ on a graph shows that $z(y)$ is quite linear for each one of the three values of $x$ : That is on the form $z\simeq Ay+B$

Again, a rough drawing of $(A,x)$ and $(B,x)$ on a graph shows that they are almost linear functions of $x$ : That is on the form $A\simeq a_1x+b_1$ and $B\simeq a_2x+b_2$

This draw us to consider the function $z \simeq (a_1x+b_1)y+(a_2x+b_2)$ which can be expressed as : $$z \simeq Axy+By+Cx+D$$

Then, we can proceed on a more accurate manner : A linear regression in order to evaluate the coefficients $A,B,C,D$ for the best fit according to the mean square deviation :

The mean absolute error ( https://en.wikipedia.org/wiki/Mean_absolute_error ) is : $$MAE=0.00032$$

In ADDITION :

The preliminary drawings made to observe the almost linear relationships in the given data :

This was a preliminary search which leads to the selected relationship $z \simeq Axy+By+Cx+D$. The numerical values appearing below are of no use for the computation of the coefficients $A,B,C,D$ as shown above.