[Bug math/28472] pow(10, i) accuracy

["wdijkstr at arm dot com" <sourceware-bugzilla@sourceware.org>]

https://sourceware.org/bugzilla/show_bug.cgi?id=28472

--- Comment #17 from Wilco <wdijkstr at arm dot com> ---
(In reply to Vincent Lefèvre from comment #16)
> (In reply to Wilco from comment #15)
> > GLIBC double precision pow is the most accurate of all libraries tested at
> > 0.523 ULP [1].
> 
> What you forget is that this is the accuracy *tested* on arbitrary values.
> The actual accuracy may be worse. And this is the case here, with an
> accuracy larger than 1 ulp, according to the results in Comment #6!

Please see the implementation - it documents the accuracy across the full input
ranges. The worst-case reported by random testing is slightly lower due to not
being able to test all input values.

And comment #6 discusses exp10, which had a known ULP of 2.01 in previous
GLIBCs.

> > The new exp10 is also the most accurate of the 13 tested math libraries.
> > 
> > If you complain about inaccuracies in the most accurate library then maybe
> > your expectations are a little bit off...
> 
> In the present case, it may be far worse than the most accurate libraries
> (well, it is difficult to say, due to the random tests). The result returned
> by glibc is not even faithfully rounded. So the user is right to complain,
> even though there is no guarantee from the ISO C standard.

No, it's not difficult to say. We computed the accuracy and have *documented*
it in the source code. So it's not only not a "guess", it's actually impossible
to get cases that are worse. Ie. if we have an algorithm that does < 0.55ULP
before rounding, we can't ever get a 2 ULP error.

> > The fact is, binary floating point cannot represent all powers of 10. If you
> > don't like the rounding behaviour of floating point, don't use floating
> > point.
> 
> Don't blame floating point if this is a poor implementation.

Even the old exp10 wasn't disastrously bad like j0/j1/y0/y0/tgamma.

> > Note compilers optimize pow (C, x) into exp (x * log (C)) with -Ofast.
> > However if C is a multiple of 2 or 10, we could use exp2 or exp10 for
> > slightly better accuracy.
> 
> I suppose that you mean "if C is a *power* of 2 or 10".

Correct.

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