Query: Neumaier's mode 'accurate'
Dear Arnold and P1788 members
I have a query about mode 'accurate' described in 3.1 - 3.3 of your proposal text.
Consider a delivered result, say zz = xx op yy in your notation, and the exact mathematical result
zz0 = { x op y | x in xx, y in yy } .
For the other two modes, zz can simply be described in terms of zz0 and the finite-precision hull operation:
zz = hull_F(zz0) in mode 'tightest'
zz contains zz0 in mode 'valid'
where F is the set of finite-precision numbers in the given format, and hull_F(S) is the smallest interval containing S
and whose endpoints are members of F. (This assumes a lower-upper-bound representation, not midpoint-
radius.)
For mode 'accurate', this is not so. It can only be defined in terms of the operation, op, and its inputs xx and yy. Am
I right that it is a kind of backward-error definition of tightness? This is not a criticism, but I would like to know
more background. Has someone written up the consequences and use of this concept?
John Pryce
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