Andy Messier wrote:
>> The reality is
>> that FP ops are defined only on a discrete set, and yield
>> values only on a discrete set. They are not defined on the
>> reals. The reals are a different set, and have different
>> operations on them. The fact that 0.7 is not representable
>> by a binary floating point number no more implies that
>> FP math is inexact, than does the exact same statement
>> that 7 / 10 in integers is 0. The result is not inexact,
>> nor incorrect. It is exact. It is correct. It just isn't the
>> same as what you get when you use reals instead of integers.
>
> The problem with this line of reasoning is that the floating point
> numbers are only useful insofar as they approximate real numbers. You
> wouldn't choose to use a floating-point representation on any
> mathematical basis for being a 'discrete set' with special properties.
ISTM that mprime uses FP numbers in a manner different from what you
say is their only useful application.
> So with regard to their applicability, they are only approximations
> of an ideal (which are real numbers). Hence the term 'inexact'. To
IMO, mprime is a perfect example of how FP numbers have applictions
other than you propose, and why thinking of FP numbers as other than a
discrete set is not the best way.
> say the result is what you want because it is a member of a 'discrete
> set' is misleading. It's more accurate to say you're accepting a
> certain amount of error because it's inconvenient to represent the
> desired result.
IEEE math was created dedicated to the proposition
that FP numbers should be thought of as a discrete set, and encourage
numerical analysts to use a system which had predictable results, as
opposed to the previous implementations, which had some of the
undesirable qualities of unpredictability (when using different
machines, you'd get different results), etc, which have come, in the
common lore, to be considered as inherent and inescapable attributes
of FP math. It's that idea that "well, it's all inexact, and we're
going to lose a few bits here and there, anyway" that has lead
to the deplorable condition of earlier FP implementations which
*were* inexact, unpredictable, etc.
Anyway, let's not argue. This isn't really an appropriate topic
for this venue, and if my suggestion doesn't appeal to the site
owner, then it doesn't, and that's it.
Mike
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