on floating-point numbers

[Hope Rouselle]

Christian Gollwitzer <auriocus at gmx.de> writes:

> Am 04.09.21 um 14:48 schrieb Hope Rouselle:
>> Christian Gollwitzer <auriocus at gmx.de> writes:
>> 
>>> Am 02.09.21 um 15:51 schrieb Hope Rouselle:
>>>> Just sharing a case of floating-point numbers.  Nothing needed to be
>>>> solved or to be figured out.  Just bringing up conversation.
>>>> (*) An introduction to me
>>>> I don't understand floating-point numbers from the inside out, but I
>>>> do
>>>> know how to work with base 2 and scientific notation.  So the idea of
>>>> expressing a number as
>>>>     mantissa * base^{power}
>>>> is not foreign to me. (If that helps you to perhaps instruct me on
>>>> what's going on here.)
>>>> (*) A presentation of the behavior
>>>>
>>>>>>> import sys
>>>>>>> sys.version
>>>> '3.8.10 (tags/v3.8.10:3d8993a, May 3 2021, 11:48:03) [MSC v.1928 64
>>>> bit (AMD64)]'
>>>>
>>>>>>> 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.
>> 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.)
>
> With the parens it will become more obvious.
>
>>       7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
>>     = 8.41 + 7.23 + 6.15 + 2.31 + 7.73 + 7.77
>
> The sum is evaluated as
>
> 	(((7.23 + 8.41) + 6.15 + ...)
>
> For the first shift, you are correct that commutativity will result in
>
> 	(((8.41 + 7.23) + 6.15 + ...)
>
> But you can't go in one step to
>
> 	(((8.41 + 6.15) + 7.23 + ...)
>
> with  the commutativity law alone. Instead, a sequence of
> associativity and commutativity is required to move the 7.23 out of
> the first pair of parentheses.
>
> And what I was trying to say, the commutative steps *are* equal in
> floating point arithmetics, whereas the associative steps are not.

Oh, I see it.  Very good point!  Lesson learned.