  In the last few decades, extensions of General Relativity have reached always
more attention especially in view of possible breakdowns of the standard
$\Lambda$CDM paradigm at intermediate and high redshift regimes. If General
Relativity would not be the ultimate theory of gravity, modifying Einstein's
gravity in the homogeneous and isotropic universe may likely represent a viable
path toward the description of current universe acceleration. We here focus our
attention on two classes of extended theories, i.e. the $f(R)$ and
$f(R,G)$-gravity. We parameterize the so-obtained Hubble function by means of
effective barotropic fluids, by calibrating the shapes of our curves through
some of the most suitable dark energy parameterizations, XCDM, CPL, WP.
Afterwards, by virtue of the correspondence between the Ricci scalar and the
Gauss-Bonnet topological invariant with the redshift $z$, we rewrite $f(R,G)$
in terms of corresponding $f(z)$ auxiliary functions. This scheme enables one
to get numerical shapes for $f(R,G)$ and $f(R)$ models, through a
coarse-grained inverse scattering procedure. Although our procedure agrees with
the simplest extensions of general relativity, it leaves open the possibility
that the most suitable forms of $f(R)$ and $f(R,G)$ are rational Pad\'e
polynomials of first orders. These approximations seem to be compatible with
numerical reconstructions within intermediate redshift domains and match fairly
well small redshift tests.
