Interval comparisons in mixed-format (and mixed-radix) environments
Dan Zuras' remark
> That is, subset with a smidge at both ends when those ends
> are finite.
reminded me of a point that occurred to me while reading the
various motions on comparisons:
Don't we also need "as equal as possible"?
It seems to me that some refinement algorithms would need this
for stopping conditions. The needed relation is that A is the
tightest hull of B, which differs from equality when B has more
precision than A, or is in a different radix.
The relation can be derived if each format defines a constant
interval SMIDGE that is [0, Dmin] for that format, i.e. the
smallest representable non-singleton positive interval.
Then my relation would be: A <= B <= (A+SMIDGE)
or: (A-SMIDGE) <= B <= A
(Note that SMIDGE itself will be blown up to match the exponent
of A, as needed to satisfy containment for a minimal increment.)
SMIDGE can also be used to widen an interval by the smallest
possible amount, one-sided (A+SMIDGE or A-SMIDGE) or two-sided
((A+SMIDGE)-SMIDGE). The latter is in fact inflate(A) as defined
in section 5.3 of the Vienna Proposal.
Michel.
---Sent: 2010-10-04 00:07:22 UTC