Re: Midpoint paper (2012-02-08 version)
On 2012-02-11 11:18:40 -0800, Dan Zuras Intervals wrote:
> OK, I'm going to resist your suggestion that we
> include formatOf varients of midpoint in our
> proposal on the grounds that I don't understand
> the implications.
>
> Indeed, let me mention that this entire discussion
> is outside the scope of the proposal that Nate & I
> are making.
>
> But, that having been said, you raise valid issues
> that apply to ALL formatOf functions. John, I
> think some caveats are going to be needed to cover
> the sorts of problems that can happen to any
> formatOf function that is strictly narrower in
> range & precision than its inputs.
I don't think this is a problem. But it seems that you forgot that
there are two very different cases for formatOf inf-sup midpoint:
1. The output format is smaller than the input format. Indeed,
there are "problems", but this is the well-known problem of
floating-point: a limited range and precision. If the user
wants to change the format to a smaller one, there may be a
good reason. In IEEE 754, formatOf operations can be useful
as a final operation to implement correct rounding (without
double rounding). IIRC, there's an example in our article
@Article{KLLLM2009a,
AUTHOR = {P. Kornerup and Ch. Lauter and V. Lefèvre and N. Louvet and J.-M. Muller},
TITLE = {Computing Correctly Rounded Integer Powers in Floating-Point Arithmetic},
JOURNAL = {{ACM} Transactions on Mathematical Software},
VOLUME = {37},
NUMBER = {1},
MONTH = jan,
YEAR = {2010},
URL = {http://doi.acm.org/10.1145/1644001.1644005},
DOI = {10.1145/1644001.1644005}
}
2. The output format is larger than the input format. In this case,
there are no caveats. On the contrary, it can even allow to get
the exact midpoint (no rounding).
For implicit interval types, a midpoint may necessarily be some form
of a formatOf operation.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)