On 2/14/2013 6:17 AM, Vincent Lefevre wrote:
On 2013-02-13 10:22:02 -0600, Ralph Baker Kearfott wrote:
(We have defined
``flavors'' and will specify common properties of all
flavors and different interval formats. However, the
core interval data type we have mostly nailed down
is an inf-sup, 754-conforming data type. Data types
involving exact representations would fall under flavors
that have not yet been defined or non-754 data types we
haven't extensively considered to date. Exact representations
might be useful in achieving standard compliance with
underlying computations in implementations, but it probably
is not our business specifying details of implementations,
beyond checking feasibility or practicality of implementation.)
Exact representations would not be provided by a different flavor
(an exact representation makes sense in every flavor), but would
be some specific parameterized implicit interval type (this is
already covered by "Multi-precision interval types", but P1788
gives little specification).
There is a distinction between a multi-precision type and an exact type. For
example, finding the tightest enclosure for pi in 1000-bit binary floats is
possible, but finding the tightest rational enclosure for pi makes no sense.
It is quite possible to implement intervals of rational numbers for rational arithmetic
starting with rational numbers as input, and it is in some respects a vast
simplification, since the rational arithmetic does not underflow or overflow.
I was surprised to see in 11.11.7 that the standard seems to be
anticipating implementations in which two numbers (scalars) x,y can be produced
such that the implementation cannot tell whether x<=y.
Or this computation "can be hard". This seems so fundamentally a
part of interval arithmetic implementation that it must be
a solved problem if the implementation is to work.
I mention it in this thread since it is easily solved with
exact results, regardless of differing precisions, different radix
representations, etc.
Of course comparisons of scalars can typically be made without resorting to arbitrary
rational representation, e.g. as is done with range-reduction for trig functions
where a high-precision version of pi is used.
The version of 11.11.7 I am looking at seems to have a lot of fluff. If I read the
third paragraph, last sentence correctly, neither false positives nor false negatives
are acceptable, and so the predicate (L<=U) must be computed correctly.
In other words,
If L>U the interval is Empty. That's all, for the first 5 paragraphs.