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

Source abstract

In their book, Scientific Computing on the Itanium , Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form ab + 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 2 u +7 u 2 +6 u 3 . Furthermore, by building an example for which the relative error is asymptotically (as p → ∞ or, equivalently, as u → 0) equivalent to 2 u , we show that our error bound is asymptotically optimal.