  Our goal is to find accurate and efficient algorithms, when they exist, for
evaluating rational expressions containing floating point numbers, and for
computing matrix factorizations (like LU and the SVD) of matrices with rational
expressions as entries. More precisely, {\em accuracy} means the relative error
in the output must be less than one (no matter how tiny the output is), and
{\em efficiency} means that the algorithm runs in polynomial time. Our goal is
challenging because our accuracy demand is much stricter than usual.
