On relative errors of floating-point operations: Optimal bounds and applications
Claude-Pierre Jeannerod, Siegfried M. Rump · Mathematics of Computation · 2016
Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function f l \mathrm {fl} and barring underflow and overflow, such models bound the relative errors E 1 ( t ) = | t − f l ( t ) | / | t | E_1(t) = |t-\mathrm {fl}(t)|/|t| and E 2 ( t ) = | t − f l ( t ) | / | f l ( t ) | E_2(t) = |t-\mathrm {fl}(t)|/|\mathrm {fl}(t)| by the unit roundoff u u . This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t t and in the case where t t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 ( t ) E_1(t) and E 2 ( t ) E_2(t)