StochasticArithmetic

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.

More precisely, 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 $x\circ y$ is replaced with

\[\bigcirc_{\text{rnd}}(x,y) = \left|\begin{array}{ll} \bigtriangledown(x\circ y) & \text{with probability $\displaystyle\frac12$}, \\[1em] \bigtriangleup(x\circ y) & \text{with probability $\displaystyle\frac12$}, \end{array}\right.\]

where for any real value $a\in\mathbb{R}$, $\bigtriangleup(a)$ and $\bigtriangledown(a)$ respectively denote its upward- and downward-rounded floating-point values.

Operations in this package do not rely on changing the hardware rounding mode, since this can be vey inefficient. Instead, the implementation proposed here follows the same overall design as the Verrou tool: Error-Free Transformations are used to efficiently simulate directed rounding modes. New stochastic Floating-Point types rely on this to model regular FP numbers, on which arithmetic operations get randomly rounded upwards or downwards.