on floating-point numbers

[Chris Angelico]

On Sun, Sep 5, 2021 at 12:48 PM Hope Rouselle <hrouselle at jevedi.com> wrote:
>
> Chris Angelico <rosuav at gmail.com> writes:
>
> > On Fri, Sep 3, 2021 at 4:58 AM Hope Rouselle <hrouselle at jevedi.com> wrote:
> >>
> >> Hope Rouselle <hrouselle at jevedi.com> writes:
> >>
> >> > 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.)
> >>
> >> Suppose these numbers are prices in dollar, never going beyond cents.
> >> Would it be safe to multiply each one of them by 100 and therefore work
> >> with cents only?  For instance
> >
> > Yes and no. It absolutely *is* safe to always work with cents, but to
> > do that, you have to be consistent: ALWAYS work with cents, never with
> > floating point dollars.
> >
> > (Or whatever other unit you choose to use. Most currencies have a
> > smallest-normally-used-unit, with other currency units (where present)
> > being whole number multiples of that minimal unit. Only in forex do
> > you need to concern yourself with fractional cents or fractional yen.)
> >
> > But multiplying a set of floats by 100 won't necessarily solve your
> > problem; you may have already fallen victim to the flaw of assuming
> > that the numbers are represented accurately.
>
> Hang on a second.  I see it's always safe to work with cents, but I'm
> only confident to say that when one gives me cents to start with.  In
> other words, if one gives me integers from the start.  (Because then, of
> course, I don't even have floats to worry about.)  If I'm given 1.17,
> say, I am not confident that I could turn this number into 117 by
> multiplying it by 100.  And that was the question.  Can I always
> multiply such IEEE 754 dollar amounts by 100?
>
> Considering your last paragraph above, I should say: if one gives me an
> accurate floating-point representation, can I assume a multiplication of
> it by 100 remains accurately representable in IEEE 754?

Humans usually won't give you IEEE 754 floats. What they'll usually
give you is a text string. Let's say you ask someone to type in the
prices of various items, the quantities thereof, and the shipping. You
take strings like "1.17" (or "$1.17"), and you parse that into the
integer 117.

> Hm, I think I see what you're saying.  You're saying multiplication and
> division in IEEE 754 is perfectly safe --- so long as the numbers you
> start with are accurately representable in IEEE 754 and assuming no
> overflow or underflow would occur.  (Addition and subtraction are not
> safe.)
>

All operations are equally valid. Anything that causes rounding can
cause loss of data, and that can happen with multiplication/division
as well as addition/subtraction. But yes, with the caveats you give,
everything is safe.

ChrisA