It is relatively well known in the scientific computing community that the use of finite-precision floating-point (FP) computations (as opposed to computations with real, infinite-precision numbers) can be the source of quality losses in the computed results.
Stochastic Arithmetic is one of many ways which can be used to diagnose FP-related problems. In standard IEEE-754-compliant Floating-Point Arithmetic, the result of each floating-point operation is rounded to the nearest representable floating-point value. When some floating-point operations produce results which are not representable as floating-point values (as is the case, for example of 1/3 in base 10, or 1/10 in binary representations), some information is lost during this rounding operation. Stochastic Arithmetic models this loss of accuracy using random variables.
This package implements the CESTAC arithmetic [Jean Vignes and Michel La Porte. Error Analysis in Computing, 1974] in which the result of each FP operation is randomly rounded upwards or downwards. When a complete computation is run with Stochastic Arithmetic, its results become realizations of a random variable. Studying the distribution of this random variable helps understanding the global impact of round-off errors on the computation.
Taking the example of an ill-conditioned dot product:
julia> x = ...
100-element Array{Float64,1}:
-47.03899089072116
-2302.2081944322417
-156472.61080419843
15263.008992086134
-213.23078574079136
[...]
julia> y = ...
100-element Array{Float64,1}:
-1.2528717788873576
1078.9625210090078
-31748.947458491493
-0.015303734559162641
4.784478272815557
[...]
julia> using LinearAlgebra
julia> dot(x, y)
0.8098193831511069
julia> @reliable_digits dot(SFloat64.(x), SFloat64.(y))
(0.8097595996923976, 3.798286681716415)
In the last instruction above, vectors were converted to a stochastic type
(SFloat64), effectively transforming the result of the dot product into a
random variable. Macro call @reliable_digits performs the computation a few
times and computes statistics estimating that:
- the averaged result is approximately 0.810,
- only 3 to 4 (decimal) digits should be relied upon in this result; the other are likely numerical noise.
See the full documentation for more details.
CESTAC and similar stochastic methods are currently implemented in various tools, such as:
-
CADNA: a library to be used by Fortran and C/C++ programs;
-
Verificarlo: an LLVM-based tool instrumenting programs at compile-time;
-
Verrou: a Valgrind-based tool which dynamically instruments binary executables. The techniques used in StochasticArithmetic.jl closely follow those implemented in Verrou.