The cross-stream migration of a single neutrally buoyant rigid sphere in tube flow is simulated by two packages, one (ALE) based on a moving and adaptive grid and another (DLM) using distributed Lagrange multipliers on a fixed grid. The two packages give results in good agreement with each other and with experiments. A lift law $L \,{=}\, CU_s (\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}})$ analogous to $L \,{=}\, \rho U\Gamma$ which was proposed and validated in two dimensions is validated in three dimensions here; $C$ is a constant depending on material and geometric parameters, $U_s$ is the slip velocity and it is positive, $\Omega_s$ is the slip angular velocity and $\Omega_{\hbox{\scriptsize{\it se}}}$ is the slip angular velocity when the sphere is in equilibrium at the Segré–Silberberg radius. The slip angular velocity discrepancy $\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}acute;$ is the circulation for the free particle and it changes sign with the lift. A method of constrained simulation is used to generate data which is processed for correlation formulas for the lift force, slip velocity, and equilibrium position. Our formulae predict the change of sign of the lift force which is necessary in the Segré–Silberberg effect. Our correlation formula is compared with analytical lift formulae in the literature and with the results of two-dimensional simulations. Our work establishes a general procedure for obtaining correlation formulae from numerical experiments. This procedure forms a link between numerical simulation and engineering practice.