  We investigate an ability of the exponential power-law inflation to be
phenomenologically correct model of the early universe. GR scalar cosmology
equations we study in Ivanov-Salopek-Bond (or Hamilton-Jacobi like)
representation where the Hubble parameter $H$ is the function of a scalar field
$\phi$. Such approach admits calculation of the potential for given $H(\phi)$
and consequently reconstruction of $f(R)$ gravity in parametric form. By this
manner the Starobinsky potential and non-minimal Higgs potential (and
consequently the corresponding $f(R)$ gravity) were reconstructed using
constraints on model's parameters. Also comparison to observation (PLANCK 2018)
data shows that both models give correct values for scalar spectral index and
tensor-to-scalar ratio under wide range of exponential-power-law model's
parameters.
