  We study the computational complexity of a robust version of the problem of
testing two univariate C-finite functions for eventual inequality at large
times. Specifically, working in the bit-model of real computation, we consider
the eventual inequality testing problem for real functions that are specified
by homogeneous linear Cauchy problems with arbitrary real coefficients and
initial values. In order to assign to this problem a well-defined computational
complexity, we develop a natural notion of polynomial-time decidability of
subsets of computable metric spaces which extends our recently introduced
notion of maximal partial decidability. We show that eventual inequality of
C-finite functions is polynomial-time decidable in this sense.
