  A type of exponential correction to General Relativity gives viable modified
gravity model of dark energy. The model behaves as $R-2\Lambda$ at large
curvature where an effective cosmological constant appears, but it becomes zero
in flat space time. The cosmic evolution of the main density parameters is
consistent with current observations. The thin shell conditions for the Solar
system were analyzed. Apart from satisfying cosmological and local gravity
restrictions, the model may also show measurable differences with $\Lambda$CDM
at recent times. The current value of the deviation parameter $m$ for scales
relevant to the matter power spectrum can be larger than $10^{-6}$. The growth
index of matter density perturbations is clearly different from that of the
$\Lambda$CDM. The theoretical predictions of the model for the weighted growth
rate were analyzed in the light of the $f\sigma_8$-tension.
