on floating-point numbers

[Peter J. Holzer]

On 2021-09-05 23:21:14 -0400, Richard Damon wrote:
> > On Sep 5, 2021, at 6:22 PM, Peter J. Holzer <hjp-python at hjp.at> wrote:
> > On 2021-09-04 10:01:23 -0400, Richard Damon wrote:
> >>> On 9/4/21 9:40 AM, Hope Rouselle wrote:
> >>> 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.)
> >>> 
> >> 
> >> Addition and Subtraction are just as safe, as long as you stay within
> >> the precision limits.
> > 
> > That depends a lot on what you call "safe", 
> > 
> > a * b / a will always be very close to b (unless there's an over- or
> > underflow), but a + b - a can be quite different from b.
> > 
> > In general when analyzing a numerical algorithm you have to pay a lot
> > more attention to addition and subtraction than to multiplication and
> > division.
> > 
> Yes, it depends on your definition of safe. If ‘close’ is good enough
> then multiplication is probably safer as the problems are in more
> extreme cases. If EXACT is the question, addition tends to be better.
> To have any chance, the numbers need to be somewhat low ‘precision’,
> which means the need to avoid arbitrary decimals.

If you have any "decimals" (i.e decimal digits to the right of your
decimal point) then the input values won't be exactly representable and
the nearest representation will use all available bits, thus losing some
precision with most additions.

> Once past that, as long as the numbers are of roughly the same
> magnitude, and are the sort of numbers you are apt to just write, you
> can tend to add a lot of them before you get enough bits to accumulate
> to have a problem.

But they won't be exact. You may not care about rounding errors in the
tenth digit after the point, but you are only close, not exact. So if
you are fine with a tiny rounding error here, why are you upset about
equally tiny rounding errors on multiplication?

> With multiplication, every multiply roughly adds the number of bits of
> precision, so you quickly run out, and one divide will have a chance
> to just end the process.

Nope. The relative error stays the same unlike for addition where is can
get very large very quickly.

        hp

-- 
   _  | Peter J. Holzer    | Story must make more sense than reality.
|_|_) |                    |
| |   | hjp at hjp.at         |    -- Charles Stross, "Creative writing
__/   | http://www.hjp.at/ |       challenge!"
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