on floating-point numbers

[Joe Pfeiffer]

Hope Rouselle <hrouselle at jevedi.com> writes:
> Christian Gollwitzer <auriocus at gmx.de> writes:
>>
>> I believe it is not commutativity, but associativity, that is
>> violated. 
>
> 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?
>
> Perhaps only through various application of commutativity, namely the
> ones below. (I omit the parentheses for less typing.  I suppose that
> does not create much trouble.  There is no use of associativity below,
> except for the intented omission of parentheses.)
>
>      7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
>    = 8.41 + 7.23 + 6.15 + 2.31 + 7.73 + 7.77
>    = 8.41 + 6.15 + 7.23 + 2.31 + 7.73 + 7.77
>    = 8.41 + 6.15 + 2.31 + 7.23 + 7.73 + 7.77
>    = 8.41 + 6.15 + 2.31 + 7.73 + 7.23 + 7.77
>    = 8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23.

But these transformations depend on both commutativity and
associativity, precisely due to those omitted parentheses.  When you
transform

    7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77

into

    8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23.

it isn't just assuming commutativity, it's also assuming associativity
since it is changing 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.
    
If I use parentheses to modify the order of operations of the first line
to match that of the last, I get
    7.23 + (8.41 + 6.15 + 2.31 + 7.73 + 7.77)

Now, I get 39.60000000000001 evaluating either of them.