on floating-point numbers

[Christian Gollwitzer]

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.

	Christian