In this paper, we present an improvement of a mathematically equivalent representation of the Colebrook-White (CW) equation to compute friction factors for turbulent flow in rough pipes. This new form is simple and very well-suited for accurately estimating the friction factor, because no iterative calculations are necessary. Specifically, the friction factor is expressed as the sum of known simple functions and an unknown correction term. This correction term satisfies an auxiliary equation that can be accurately and easily solved with predictable error bounds over the complete range of pipe roughness and Reynolds number values encountered in practice. The simplest case, with the unknown correction term set to zero, resulted in friction factor estimates with errors of < 1%. A simple linear approximation of the correction term resulted in a maximum error of 3.64 x 10(-4)%, whereas friction factor estimates from a continued-fractions-based approximation had a maximum error of 1.04 x 10(-10)%. These maximum errors are significantly lower than any of the explicit approximations that have been proposed to date for the CW equation. The equation presented in this study is entirely theoretical and eliminates the need for best-fit parameters or complicated initial guesses that are an integral component of the various other approximations proposed to date. The simplicity with which this new equation can be solved, coupled with its smooth and predictable error behavior, should make it the method of choice for estimating turbulent flow friction factor in rough pipes.