  Classical probabilistic rounding error analysis is particularly well suited
to stochastic rounding (SR), and it yields strong results when dealing with
floating-point algorithms that rely heavily on summation. For many numerical
linear algebra algorithms, one can prove probabilistic error bounds that grow
as O(nu), where n is the problem size and u is the unit roundoff. These
probabilistic bounds are asymptotically tighter than the worst-case ones, which
grow as O(nu). For certain classes of algorithms, SR has been shown to be
unbiased. However, all these results were derived under the assumption that SR
is implemented exactly, which typically requires a number of random bits that
is too large to be suitable for practical implementations. We investigate the
effect of the number of random bits on the probabilistic rounding error
analysis of SR. To this end, we introduce a new rounding mode,
limited-precision SR. By taking into account the number r of random bits used,
this new rounding mode matches hardware implementations accurately, unlike the
ideal SR operator generally used in the literature. We show that this new
rounding mode is biased and that the bias is a function of r. As r approaches
infinity, however, the bias disappears, and limited-precision SR converges to
the ideal, unbiased SR operator. We develop a novel model for probabilistic
error analysis of algorithms employing SR. Several numerical examples
corroborate our theoretical findings.
