  We reformulate the algorithm of Gr{\o}nbech-Jensen and Farago (GJF) for
Langevin dynamics simulations at constant temperature. The GJF algorithm has
become increasingly popular in molecular dynamics simulations because it
provides robust (i.e., insensitive to variations in the time step) and accurate
configurational sampling of the phase space with larger time steps than other
Langevin thermostats. In the original derivation [Mol. Phys. {\bf 111}, 983
(2013)], the algorithm was formulated as a velocity-Verlet type integrator with
an in-site velocity variable. Here, we reformulate it as a leap frog scheme
with a half-step velocity variable. In contrast to the original form, the
reforumlated one also provides robust and accurate estimations of kinetic
measures such as the average kinetic energy. We analytically prove that the
newly presented algorithm gives the exact configurational and kinetic
temperatures of a harmonic oscillator for any time step smaller than the Verlet
stability limit, and use computer simulations to demonstrate the
configurational and kinetic robustness of the algorithm in strongly non-linear
systems. This property of the new formulation of the GJF thermostat makes it
very attractive for implementation in computer simulations.
