Floating point arithmetic allows us to use a finite machine, the digital
computer, to reach conclusions about models based on continuous mathematics. In
this article we work in the other direction, that is, we present examples in
which continuous mathematics leads to sharp, simple and new results about the
evaluation of sums, square roots and dot products in floating point arithmetic.Comment: We changed a couple of paragraphs in order to give proper credit to
  previous, and present, work by S. Rump et al, and to point out that one our
  lemmas has been known for a long tim