on floating-point numbers

[Peter J. Holzer]

On 2021-09-04 09:48:40 -0300, Hope Rouselle wrote:
> Christian Gollwitzer <auriocus at gmx.de> writes:
> > Am 02.09.21 um 15:51 schrieb Hope Rouselle:
> >>>>> ls = [7.23, 8.41, 6.15, 2.31, 7.73, 7.77]
> >>>>> sum(ls)
> >> 39.599999999999994
> >> 
> >>>>> ls = [8.41, 6.15, 2.31, 7.73, 7.77, 7.23]
> >>>>> sum(ls)
> >> 39.60000000000001
> >> All I did was to take the first number, 7.23, and move it to the
> >> last
> >> position in the list.  (So we have a violation of the commutativity of
> >> addition.)
> >
> > I believe it is not commutativity, but associativity, that is
> > violated. 

I agree.


> Shall we take this seriously?  (I will disagree, but that doesn't mean I
> am not grateful for your post.  Quite the contary.)  It in general
> violates associativity too, but the example above couldn't be referring
> to associativity because the second sum above could not be obtained from
> associativity alone.  Commutativity is required, applied to five pairs
> of numbers.  How can I go from
> 
>   7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
> 
> to 
> 
>   8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23?

Simple:

>>> 7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
39.599999999999994
>>> 7.23 + (8.41 + 6.15 + 2.31 + 7.73 + 7.77)
39.60000000000001

Due to commutativity, this is the same as

>>> (8.41 + 6.15 + 2.31 + 7.73 + 7.77) + 7.23
39.60000000000001

So commutativity is preserved but associativity is lost. (Of course a
single example doesn't prove that this is always the case, but it can be
seen from the guarantees that IEEE-754 arithmetic gives you that this is
actually the case).

        hp

-- 
   _  | Peter J. Holzer    | Story must make more sense than reality.
|_|_) |                    |
| |   | hjp at hjp.at         |    -- Charles Stross, "Creative writing
__/   | http://www.hjp.at/ |       challenge!"
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