  The role of an exponential function of the scalar curvature in the modified
gravity is analyzed. Two models are proposed. A toy model that complies with
local and cosmological constraints and gives appropriate qualitative
description of the cosmic evolution. This model contains a saddle
matter-dominant critical point that can evolve towards a late time de Sitter
attractor. Initial conditions have been proposed, showing that this toy model
has an acceptable matter era and gives an approximate qualitative behavior of
cosmic evolution. A second viable model, behaves very close to $\Lambda$CDM at
early times and can satisfy local and cosmological constraints. It behaves as
$R-2\Lambda$ at $R\rightarrow \infty$ and tends to zero at $R\rightarrow 0$,
containing flat spacetime solution. The model gives viable cosmological
trajectories that, as the first model, connect the matter dominated point with
a late time de Sitter attractor. The cosmic evolution of the main density
parameters in this model is consistent with current observations with an
equation of state very close to $-1$.
