Precision of float and bfloat



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.