See
http://portal.acm.org/citation.cfm?id=361581&dl=GUIDE&coll=GUIDE&CFID=56807517&CFTOKEN=25657548#
@article{361581,
author = {Richman, Paul L.},
title = {Automatic error analysis for determining precision},
journal = {Commun. ACM},
volume = {15},
number = {9},
year = {1972},
issn = {0001-0782},
pages = {813--817},
doi = {http://doi.acm.org/10.1145/361573.361581},
publisher = {ACM},
address = {New York, NY, USA},
}
Abstract
ABSTRACT
The problem considered is that of evaluating a rational expression to
within any desired tolerance on a computer which performs
variable-precision floating-point arithmetic operations. For example,
the expression might be &pgr;/(&pgr; + 1/2 - e) ?2), which is rational
in the data &pgr;, e, ?2. An automatic error analysis technique is given
for determining, directly from the results of a trial low-precision
interval arithmetic calculation, just how much precision and data
accuracy are required to achieve a desired final accuracy. The
techniques given generalize easily to the evaluation of many nonrational
expressions.
.........
For a more general class of expressions there is no complete algorithm
because it requires the solution of the zero-equivalence
problem, so there has to be some way to say that you can't do it.