Long-range lattice summation techniques such as the particle-mesh Ewald (PME) algorithm for electrostatics have been revolutionary to the precision and accuracy of molecular simulations in general. Despite the performance penalty associated with lattice summation electrostatics, few biomolecular simulations today are performed without it. There are increasingly strong arguments for moving in the same direction for Lennard-Jones (LJ) interactions, and by using geometric approximations of the combination rules in reciprocal space, we have been able to make a very high-performance implementation available in GROMACS. Here, we present a new way to correct for these approximations to achieve exact treatment of Lorentz-Berthelot combination rules within the cutoff, and only a very small approximation error remains outside the cutoff (a part that would be completely ignored without LJ-PME). This not only improves accuracy by almost an order of magnitude but also achieves absolute biomolecular simulation performance that is an order of magnitude faster than any other available lattice summation technique for LJ interactions. The implementation includes both CPU and GPU acceleration, and its combination with improved scaling LJ-PME simulations now provides performance close to the truncated potential methods in GROMACS but with much higher accuracy.