Re: True and false infinities and zeros (was: Table 4 proposal...)
Lee Winter wrote:
> 754 does not have infinite arguments. It has overflows.
Clause 3.2 of 754-2008 clearly states that the mathematical model underlying
the standard, i.e. the domain of Level 1, is the "extended reals" with two
signed infinities, which is a layman'ss description of a well-known and
well-defined mathematical structure, namely the two-point compactification
of the real line. In this structure (which is NOT a field) there are two
elements which are not real numbers, but for which the field operations are
defined (some definitions are partial, in the technical sense, just like
division by zero in a field). Where my recollection fails me is whether
the two signed zeros are part of the mathematical structure or not; I'm
more familiar with the one-point compactification which does not suffer
from this issue. Perhaps we should look at Prof. Kahan's early papers
on this subject as FP standardization took its course from 1979 to 1985.
The real issue is that floating-point arguments to operations are treated
as exact, and the rest follows. It would have been difficult to justify
the tie breaking rules is this assumption was not made. So even if an
infinity or a zero is the result of over/underflow, that is forgotten when
it is used as an argument to another operation. The only thing that is
remembered (in the sticky flags) is that the computation may have been
compromised by this faulty but necessary assumption.
I agree that in 754 an infinity does not represent "the rest of the number
line" -- but it does in 1788. Also note that 1788 is NOT based on the
mathematical Extended Reals, but on the regular finite Reals, and the use
of 754 infinities is restricted to denoting unboundedness. Any other symbol
could have been used, e.g. a NaN with a particular payload -- but the rules
of 754 arithmetic with infinities are ALMOST just right for most operations
so that implementations can avoid excessive case-by-case analysis. So in
the end it is only by convention that 1788 Infinity denotes "the rest of the
number line".
Michel.
---Sent: 2012-03-23 03:42:10 UTC