on floating-point numbers

[Chris Angelico]

On Sun, Sep 5, 2021 at 12:50 PM Hope Rouselle <hrouselle at jevedi.com> wrote:
>
> 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.)
>
>      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.
>

Show me the pairs of numbers. You'll find that they are not the same
numbers. Commutativity is specifically that a+b == b+a and you won't
find any situation where that is violated.

As soon as you go to three or more numbers, what you're doing is
changing which numbers get added first, which is this:

a + (b + c) != (a + b) + c

and this can most certainly be violated due to intermediate rounding.

ChrisA