On mantissa distributions in computing and Benford’s law. (English) Zbl 0662.65040

Schatte, Peter

J. Inf. Process. Cybern. 24, No. 9, 443-455 (1988).

In the course of a sufficiently long computation in floating-point arithmetic, the occurring mantissas have nearly logarithmic distribution. This phenomenon has important applications in computer design and in the probabilistic analysis of roundoff errors. In the paper, the mantissas of products and of sums are considered in case of the number of operands tending to infinity. Under general assumptions the mantissa distribution of a product exponentially converges to the logarithmic distribution, the mantissa distribution of a sum does not converge, and special methods must be used for limitation. A consequence of the logarithmic mantissa distribution in extensive computation is the logarithmic first digit distribution, i.e. the Benford law. The author explains the Benford law by limit theorems for random floating-point mantissas.

Reviewer: S.Grzegórski

MSC:

65G50

Roundoff error

Keywords:

probabilistic error analysis; floating-point arithmetic; logarithmic distribution; computer design; roundoff errors; Benford law; random floating-point mantissas