For single input single output systems, we give a refinement d(c,r) of the generalized chordal metric d(c) introduced in [A. J. Sasane, Complex Anal. Oper. Theory, 7 (2013), pp. 1345-1356]. Our metric d(c,r) is given in terms of coprime factorizations, but it coincides with the extension of Vinnicombe's nu-metric given in Ball and Sasane [Complex Anal. Oper. Theory, 6 (2012), pp. 65-89] if the coprime factorizations happen to be normalized. The advantage of the metric d(c,r) introduced in this article is its easy computability (since it relies only on coprime factorizations and does not require normalized coprime factorizations). We also give concrete formulations of our abstract metric for standard classes of stable transfer functions.