[Cfp-interest 2123] Re: AI about updating definition of normalized

[Vincent Lefevre]

On 2021-08-21 20:36:04 -0700, Jim Thomas wrote:
> The model defines floating-point numbers as having a fixed precision p and requires the type to include all normalized model numbers with precision p. For double-double I think that means the precision is p = 106 (it could be taken to be less than 106, but to what advantage?), and numbers represented with extra precision (and some other representations e.g. (DBL_MAX, DBL_MAX)) are not  floating-point numbers. 

I would say that (DBL_MAX, DBL_MAX) and, more generally, all pairs (x,y)
such that |y| > ulp(x)/2 are trap representations: the algorithms will
probably fail on such values, unless they first do a normalization step
with TwoSum (or equivalent). I suppose that it is less costly to do
normalization at the end of each operation, when needed: a FastTwoSum
should be sufficient in general, and the number of outputs is less than
the number of inputs (since an output can be reused as several inputs).

In any case, the normalization of (DBL_MAX, DBL_MAX) would overflow at
DBL_MAX + DBL_MAX.

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