on floating-point numbers

[Chris Angelico]

On Fri, Sep 3, 2021 at 10:42 PM jak <nospam at please.ty> wrote:
>
> Il 03/09/2021 09:07, Julio Di Egidio ha scritto:
> > On Friday, 3 September 2021 at 01:22:28 UTC+2, Chris Angelico wrote:
> >> On Fri, Sep 3, 2021 at 8:15 AM Dennis Lee Bieber <wlf... at ix.netcom.com> wrote:
> >>> On Fri, 3 Sep 2021 04:43:02 +1000, Chris Angelico <ros... at gmail.com>
> >>> declaimed the following:
> >>>
> >>>> The naive summation algorithm used by sum() is compatible with a
> >>>> variety of different data types - even lists, although it's documented
> >>>> as being intended for numbers - but if you know for sure that you're
> >>>> working with floats, there's a more accurate algorithm available to
> >>>> you.
> >>>>
> >>>>>>> math.fsum([7.23, 8.41, 6.15, 2.31, 7.73, 7.77])
> >>>> 39.6
> >>>>>>> math.fsum([8.41, 6.15, 2.31, 7.73, 7.77, 7.23])
> >>>> 39.6
> >>>>
> >>>> It seeks to minimize loss to repeated rounding and is, I believe,
> >>>> independent of data order.
> >>>
> >>> Most likely it sorts the data so the smallest values get summed first,
> >>> and works its way up to the larger values. That way it minimizes the losses
> >>> that occur when denormalizing a value (to set the exponent equal to that of
> >>> the next larger value).
> >>>
> >> I'm not sure, but that sounds familiar. It doesn't really matter
> >> though - the docs just say that it is an "accurate floating point
> >> sum", so the precise algorithm is an implementation detail.
> >
> > The docs are quite misleading there, it is not accurate without further qualifications.
> >
> > <https://docs.python.org/3.8/library/math.html#math.fsum>
> > <https://code.activestate.com/recipes/393090/>
> >
>
> https://en.wikipedia.org/wiki/IEEE_754

I believe the definition of "accurate" here is that, if you take all
of the real numbers represented by those floats, add them all together
with mathematical accuracy, and then take the nearest representable
float, that will be the exact value that fsum will return. In other
words, its accuracy is exactly as good as the final result can be.

ChrisA