How to compare samples with different distributions?

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I have different implementations of functions in a computer program.
Now, I want to find out if one of the implementations shows an increase in performance.

To do that, I implemented a slow function and 2 faster versions and measured the run-time for each. The problem is, that the recorded samples have different distributions.

For example:

The sample of the slow function $X$ is normal distributed, whereas the sample of a faster version $Y$ is not (Gamma- or F like distribution, sometimes equal distributed ).

I want to show that: $\bar{Y} < \bar{X}$

By calculating the mean run-times, I can see that the faster version $Y$ provides an improvement. But how can I use statistical tests to show or prove this?

I only know the t-test, z-test but they assume t- or normal distributed samples which is not the case for my data.

Thank you for your help :)