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When I check "Reuse reference" and then use "Set main reference" the image moves a little. What could cause this?

I tried some more things. When I zoom in a few times from an original depth D (such as 9.61963041900E111) and then insert the depth D again in the location window, the image is not centered at where I zoomed in on as I would expect. No matter where I zoom to, when I set the depth to a specific value I get the same image, as if it's not influenced by the coordinates. Even when I change the coordinates radically (manual manipulation of the digits), what I see doesn't change much. When I don't set the depth and only zoom in and out and move the image around, the results do seem to be consistent.

So, you selected reuse reference, and while having that active, you selected set main reference and clicked somewhere else?
So the program things the position where you clicked is the position of the reference, but the reference was not recalculated.
So the reason is that the location of the reference is not where the program thinks it is :)

Then maybe I don't understand what reuse reference and set main reference do exactly. Some problems can be prevented by unchecking reuse reference, then using set main reference and then rechecking reuse reference. By doing so, the image doesn't move, however, zooming in and then setting a lower depth manually does still cause the image not to be centered around the coordinates. Does the program ignore the coordinates and use those of the reference or something?

When "Reuse reference" is active, the reference is simply never re-calculated.
If you zoom in at the side of the reference, and have this active, the view may look nice but series approximation will eventually distort the image since the distance from the reference get too far, since the reference is probably far out of screen.
As we saw on claude's and knighty's experiments with finding a theoretical way of using series approxiation, having too low threashold makes too many iterations to be skipped and the image is still looking nice but distorted.

Okay, now I know set main reference doesn't work with reuse ference enabled.

The distortion is not what I mean. It's that when reuse reference is enabled, changing the coordinates manually doesn't seem to affect the image much and I think that's weird. It can't be explained by distortion because dragging the image around works fine. I could set the coordinates to something completely different like (0, 0) and it still wouldn't affect the image.