@Article{Muller:2014:OEC,
  author =       "Jean-Michel Muller",
  title =        "On the error of computing $ab + cd$ using {Cornea},
                  {Harrison} and {Tang}’s method",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume  =      41,
  number  =      2,
  accepted =     "28 February 2014",
  upcoming =     "true", 
  abstract =     "
                  In their book \textit{Scientific Computing on The Itanium}
                  [Cornea et al. 2002], Cornea, Harrison and Tang 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 $2u + 7u^2 + 6u^3$. Furthermore, by building
                  an example for which the relative error is asymptotically
                  (as $p \rightarrow \Inf$ or, equivalently, as $u \rightarrow
                  0$) equivalent to $2u$, we show that our error bound is 
                  asymptotically optimal.",
}