Comment by semi-extrinsic

It's also what scientists have done to visualize solutions of PDEs since the 1960s. Author should download Paraview and give it a twirl, to get this perspective.

First create a mesh (Sources -> Plane for 2D, or Sources -> Box if you want to do it in 3D). Set reasonably high values for Resolution on this source. Then use a filter to apply your function, either Filters -> Alphabetical -> Calculator for easy stuff, or Filters -> Alphabetical -> Python Calculator if you want complicated stuff. The "coordsX" etc. are your spatial coordinates on the mesh. Pick whatever color map you want (diverging types are good for this), change the limits on coloring, use a log scale, whatever.

If you do this in 3D on a box, you can then use a slice to scrub through the result on an arbitrarily oriented plane. You could visualize translucent isosurfaces of constant "error" and raytrace them. Or you could take the gradient of your "error" and plot as a vector field. With a bit of leg work you can add a fourth coordinate (time) and make animations. And you can combine all of these. Sky is the limit.