Stochastic Arithmetic in Multiprecision

Abstract

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates round-off error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of real-life applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic systems modeled by the logistic map.

rpe v5: an emulator for reduced floating-point precision in large numerical simulations

“…A variety of tools also exist for automatically tuning floating-point precision, and producing mixed-precision versions of double-precision programs. Tools such as SAM (Graillat et al, 2011), Precimonious (Rubio-González et al, 2013), CRAFT (Lam et al, 2013), and PROMISE (Graillat et al, 2016) all provide automated methods for producing a correct mixed-precision implementation. The automated tuning approach usually requires the specification of an appropriate error threshold used to determine the suitability of a particular automatically generated implementation of a program.…”

Auto-tuning for floating-point precision with Discrete Stochastic Arithmetic

“…Indeed we can mention as an example a numerical expression proposed by Rump [23] that provides the same first digits in single, double and extended precision. But in fact, that result, detected by DSA as numerical noise [9], is totally different from the correct evaluation. Therefore PROMISE uses DSA to control the numerical quality of the computed result.…”

“…9 } in our example. Therefore variables v 2 and v 6 should be declared in double precision in the program returned by the PROMISE tool.…”

Numerical validation in quadruple precision using stochastic arithmetic

“…The SAM library [8] enables one to estimate with DSA rounding errors in arbitrary precision programs. SAM is based on the MPFR [6] arbitrary precision library and can be used in programs written in C/C++.…”

“…CADNA has been successfully used for the numerical validation of academic and industrial simulation codes in various domains. Another library called SAM 2 [8] uses DSA to control the numerical stability of arbitrary precision programs. SAM is based on the MPFR 3 [6] arbitrary precision library.…”