  We present a detailed study of roundoff errors in probabilistic
floating-point computations. We derive closed-form expressions for the
distribution of roundoff errors associated with a random variable, and we prove
that roundoff errors are generally close to being uncorrelated with their
generating distribution. Based on these theoretical advances, we propose a
model of IEEE floating-point arithmetic for numerical expressions with
probabilistic inputs and an algorithm for evaluating this model. Our algorithm
provides rigorous bounds to the output and error distributions of arithmetic
expressions over random variables, evaluated in the presence of roundoff
errors. It keeps track of complex dependencies between random variables using
an SMT solver, and is capable of providing sound but tight probabilistic bounds
to roundoff errors using symbolic affine arithmetic. We implemented the
algorithm in the PAF tool, and evaluated it on FPBench, a standard benchmark
suite for the analysis of roundoff errors. Our evaluation shows that PAF
computes tighter bounds than current state-of-the-art on almost all benchmarks.
