[r6rs-discuss] [Formal] Formal Comment: NaN should be considered a number, not a real

From: Thomas Lord <lord>
Date: Fri, 22 Jun 2007 16:59:56 -0700

Joe Marshall wrote:
> I disagree.
>
>> Floating point numbers may be interpreted as certain
>> Rationals, to be sure. (I'm capitalizing set names
>> when I mean the mathematical sets as opposed to
>> Scheme's type categories.)
>>
>> However, the semantics of those floating point numbers
>> comes from regarding each of those rationals
>> as the name for an interval of the Reals.
>
> This is incorrect, and again, this is a popular misconception.
> Interval arithmetic doesn't really work as a model for floating-point.
> The problem is that intervals have width, and that is not modeled
> in the floating point representation.


Intervals are modeled in the representation plus the
rounding rules plus the rules about infinity (and so,
by implication, the rules about NaN).

If floating point were "simply" dyadic rationals then
there would be no rounding rules (for example).



>
> The IEEE standard requires that floating point numbers are treated
> as *exact* numbers. Operations are required to return the *exact*
> answer if that answer can be represented, and the closest representable
> answer otherwise.

In any "algebra" of a finite set of intervals, where each interval
has a characteristic/canonical member, we would be surprised
by anything else.

Our two descriptions of the situation, yours and mine, are essentially
duals. To talk about the specific intervals, in my view, you have to
name the finite set of characteristic rationals and define operations
in terms of those. To talk about the definitions of the operators,
in your view (e.g., rounding rules and inf rules), you need to point
out the intervals. It's the same thing and we're just talking about
which description is less confusing (which means, in some sense,
that a faux argument about it is a good way to rehearse the cannon
within which these dual descriptions occur).


I don't see how you can teach people how to figure out how to
figure out error terms unless you emphasize the interval view.
I also don't see any simpler way to explain +/- inf.




>
>> The rational
>> number stored in the bits of a floating point number
>> are the "canonical members" of those intervals. (The
>> inf intervals are exceptional: they have no characteristic
>> member and the semantics of floating point math reflects
>> this.)
>
> This, too, is incorrect and a different, but common, misinterpretation.
> Floating point numbers are not `stand-ins' for numbers within
> intervals.


They make no sense otherwise. You're just wrong on that
point.




>
>> When we write floating point numbers like "0.1", we
>> are essentially writing "There exists a number X in the
>> interval [0.1-epsilon_a .. 0.1+epsilon_b]" and we go
>> from there.
>
> No. A floating point number like 0.1 is interpreted by
> the computer as the exact *representable* number closest
> to 0.1 If the computer uses decimal floats, it is exactly
> 1/10, if double-precision binary it is *exactly*
> 3602879701896397/36028797018963968
>
> There is no epsilon anywhere in there.

The epsilon rules appear just as soon as you try to add
the floating point number denoted by "0.1" to some
other floating point number. The epsilon appears
in both rounding rules and in the (lemma from the definitions
of operators) error term rules.



>
>> The "terms" in floating point equations
>> are all existentially qualified statements like that. The
>> "operators" in floating point equations reliably draw
>> certain inferences from one or two of those terms, generating
>> a new term of the same form (or NaN): "there exists
>> A near 0.25 and there exists B near 0.5 so there
>> exists C near 0.75"
>
> This is wrong. The floating point operations are defined on
> exact inputs, in your example, A would be exactly
> 4503599627370496/18014398509481984,
> B would be exactly
> 4503599627370496/9007199254740992
> and the answer C would be exactly
> 6755399441055744/9007199254740992

It is an artifact of the way that IEEE floats are
defined that you can use them for a certain
weird class of computations over a finite set
of dyadic rationals but that isn't the main gist
of what floats are about.




>
> Note that no rounding takes place at all anywhere
> in this operation.


Yes it does. The base terms in the math come
with an error term that is implicit in the representations.
The result comes with an error term. The rounding rules
relate those error terms, even though when interpreted
as exact dyadic rationals you get an exact dyadic rational
result (I assume, I didn't look at your specific numbers).



>
>> The particular inferences an FPU will draw are, in part, defined with
>> reference to the characteristic dyadic rationals of the
>> finite intervals in the system -- but the semantics are
>> really "about" those intervals (the ones implied by those
>> existentially qualified statements). You can also see that the
>> semantics are really "about" the intervals, not the rationals
>> per se, from the treatment of inf or the treatment of rounding.
>
> As I mentioned before, the floating-point unit has no notion
> whatsoever of widths, which is of primary importance in
> interval arithmetic. The semantics are defined in terms of
> exact rationals. In fact, the interval model will fail on the
> edge cases.


Show me what you mean by an "edge case" because I think
you are confused about what I am saying.



>
>> The interval view, as opposed to seeing floating points
>> as Rationals, is also a good way to make sense of figuring
>> out error terms for results from an FPU under various
>> conditions and assumptions.
>
> I disagree, and I'm in good company. Kahan, for example,
> explicitly notes that floating point numbers are *not* intervals.


Let's get down to it then. I care not about you claims of
authority -- please do prove me wrong on some point.
(I suspect the main confusion in your reference to Kahan is
over the detailed definition of "are intervals" but, we'll see.)


-t
Received on Fri Jun 22 2007 - 19:59:56 UTC