real analysis - Look for a one-to-one function that maps a square to R

I am looking for a one-to-one function which maps (0,1)^2 to R. It is preferable that the function doesn't involve trig functions.
I have tried several mappings like $\ln(\frac{x_2}{1-x_1}),$ but they are not one-to-one. The challenge for me is the one-to-one requirement.

I have read Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$. I like the idea there, but I need to use this function to do further calculation, so it has to be in explicit form. Is it possible to find such a function?

I appreciate any ideas and comments.