Your model in the main post is the same as the pseudocode. But you might find that the estimated parameter theta is not correct, that’s because the volume change in the forward and backward transformation is not accounted for.
You can have a look at an example here: https://github.com/junpenglao/Planet_Sakaar_Data_Science/blob/master/Miscellaneous/Regression%20with%20a%20twist.ipynb
where f(y, theta) is a linear function. In the notebook, model2_ and model2 is quite similar, but only the output from model2 that accounted for the volume change is correct.
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