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See the documentation of reduce: it modifies the element instead of returning a new copy. So:

sage: f = (x0^2-x1^2)/(x0-x1)
sage: f.reduce()
sage: f
x0 + x1

See the documentation of reduce: it modifies the element instead of returning a new copy. So:

sage: f = (x0^2-x1^2)/(x0-x1)
sage: f.reduce()
sage: f
x0 + x1

Also read in the documentation:

Automatically called for exact rings, but because it may be numerically unstable for inexact rings it must be called manually in that case.

So there is no point in calling it in your particular case.