How to scale an image RATIOnally?

[PixLab]

I just generated a grid for test purpose (Filters > Render > Patterns > Grid...)
Then
Filters > G'mic_Qt > Deformations (I'm almost sure that in the "Testing" part of G'MIC you'll find something, I did not scroll down to the Testing directory/part, it's just too huge inside)

Anyway I think there is things there to start with like...

Cartesian distortion (I tried your equation see below)
conformal maps
logarithmic distortion
pixel push
and few others which at first glance seems to not do the job, but after tweaking many sliders....