W. Kahan, Homepage, How Futile are Mindless Assessments of Roundoff in Floating-Point Computation? here. Treasure
Numerical data piles up and numerical programs grow ever more ambitious and complicated while their users become, on average, far less knowlegeable about numerical error-analysis, though no less clever than their predecessors about subjects they care to learn. Consequently numerical anomalies go mostly unobserved or, if observed, routinely misdiagnosed. Fortunately most of them don’t matter. Most computations don’t matter.
Of all the many ways in which floating-point computation can go astray, only roundoff, which should rarely have to be taken seriously, is considered seriously in what follows. Over/underflow and other diversions will have to be discussed some other time. Neither shall we consider here computations so numerically unstable that their faults become obvious after any testing adequate to satisfy requirements of Due Diligence. Our main concerns hereunder are errors hard to find. Some errors are designed not to be found; these will be considered next in §2 and §3.