  Free energy differences are a central quantity of interest in physics,
chemistry, and biology. We develop design principles that improve the precision
and accuracy of free energy estimators, which has potential applications to
screening for targeted drug discovery. Specifically, by exploiting the
connection between the work statistics of time-reversed protocol pairs, we
develop near-equilibrium approximations for moments of the excess work and
analyze the dominant contributions to the precision and accuracy of standard
nonequilibrium free-energy estimators. Within linear response,
minimum-dissipation protocols follow geodesics of the Riemannian metric induced
by the Stokes' friction tensor. We find the next-order contribution arises from
the rank-3 supra-Stokes' tensor that skews the geometric structure such that
minimum-dissipation protocols follow geodesics of a generalized cubic Finsler
metric. Thus, near equilibrium the supra-Stokes' tensor determines the
leading-order contribution to the bias of bidirectional free-energy estimators.
