Potential-functional theory is an intriguing alternative to density-functional theory for solving electronicstructure problems. We derive and solve equations using interacting potential functionals. A semiclassical approximation to exchange in one dimension with hard-wall boundary conditions is found to be almost exact (compared to standard density-functional approximations). The variational stability of this approximation is tested, and its far greater accuracy relative to the local-density approximation demonstrated. Even a fully orbital-free potential-functional calculation yields little error relative to exact exchange, for more than one orbital.