Monte Carlo Arithmetic: a framework for the statistical analysis of roundo error

Monte Carlo Arithmetic (MCA) is an extension of standard oating-point arithmetic that exploits randomness in basic oating-point operations. MCA uses randomization to implement random rounding | which forces roundoo errors to be randomly distributed | and random unrounding | which randomly extends the inputs of arithmetic operations to arbitrary precision. Random rounding can be used to produce roundoo errors that are truly random and uncorrelated, and that have zero expected bias. Random unrounding detects catastrophic cancellation , which is the primary way that signiicant digits are lost in numerical computation. Randomization also can be used to vary precision dynamically, and to implement inexact values (values known to only a few signiicant digits). Randomization has both theoretical and practical beneets. It has the eeect of transforming any oating-point computation into a Monte Carlo computation, and roundoo analysis into statistical analysis. By running a program multiple times, one directly measures the sensitivity of particular outputs to random perturbations of particular inputs. Also, MCA gives a way to avoid some anomalies of oating-point arithmetic. For example, while oating-point summation is not associative, Monte Carlo summation turns out to bèstatistically associative' up to the standard error of the sum. Generally, it gives a diierent perspective on the study of error.