  In this paper we study the cubic fractional nonlinear Schrodinger equation
(NLS) on the torus and on the real line. Combining the normal form and the
restricted norm methods we prove that the nonlinear part of the solution is
smoother than the initial data. Our method applies to both focusing and
defocusing nonlinearities. In the case of full dispersion (NLS) and on the
torus, the gain is a full derivative, while on the real line we get a
derivative smoothing with an $\epsilon$ loss. Our result lowers the regularity
requirement of a recent theorem of Kappeler et al. on the periodic defocusing
cubic NLS, and extends it to the focusing case and to the real line. We also
obtain estimates on the higher order Sobolev norms of the global smooth
solutions in the defocusing case.
