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In their book, Scientific Computing on the Itanium, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form ab&plus;cd in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if p is the precision of the floating-point format and if u = 2-p, the relative error of the result is of order u. We improve their proof to show that the relative error is bounded by 2u&plus;7u2&plus;6u3. Furthermore, by building an example for which the relative error is asymptotically (as p → ∞ or, equivalently, as u → 0) equivalent to 2u, we show that our error bound is asymptotically optimal.

On the Error of Computing ab + cd using Cornea, Harrison and Tang's Method