The analytic equation of state of nonideal Coulomb plasmas consisting of pointlike ions immersed in a polarizable electron background [ G. Chabrier and A. Y. Potekhin , Phys .
Rev .
E 58 , 4941 ( 1998 ) ] is improved , and its applicability range is considerably extended .
First , the fit of the electron screening contribution in the free energy of the Coulomb liquid is refined at high densities where the electrons are relativistic .
Second , we calculate the screening contribution for the Coulomb solid ( bcc and fcc ) and derive an analytic fitting expression .
Third , we propose a simple approximation to the internal and free energy of the liquid one-component plasma of ions , accurate within the numerical errors of the most recent Monte Carlo simulations .
We obtain an updated value of the coupling parameter at the solid-liquid phase transition for the one-component plasma : \Gamma _ { m } = 175.0 \pm 0.4 ( 1 \sigma ) .