J.-M. Chesneaux

Modélisation théorique et conditions de validité de la méthode CESTAC

C.R.A.S., Paris, Série 1, tome 307, 1988, pp.417-422.

The probabilistic CESTAC method of LA PORTE and VIGNES consists in randomly perturbing the last bit of the mantissa of each intermediate result. Then a statistical estimate (of beta %) of the accuracy of the final result is obtained with Student's test. We are faced with two problems : Is Student's test good for estimating the average of the informatical results and what is the relation between the exact result and this average ?

It has been shown that the mantissa was logarithmically distributed. It also has been proved that under this assumption round-off errors are uniformally distributed on for chopping arithmetic and on [-1/2,+1/2] for rounding arithmetic. A new modelization of an informatical result after rounding and perturbation is established under the following hypothesis : i) the probabilistic assumptions for the rounding arithmetic are still true for random perturbed arithmetic, ii) neither intermediate result is an informatical zero in the mean of VIGNES iii) the first order in respect to 2^(-p) is valid. Under theses assumptions, it has been shown that the mathematical result in the expectation of the informatical result. Then, the validity of the Student's test on the modelization (and so the validity of the CESTAC method) is proved by showing that the corrections on 1-beta is at most O.OO76 for chopping and 0.048 for rounding.