Python -- floating point arithmetic

[Stefan Krah]

Adam Skutt <askutt at gmail.com> wrote:
> On Jul 8, 7:23 am, Mark Dickinson <dicki... at gmail.com> wrote:
> > On Jul 8, 11:58 am, Adam Skutt <ask... at gmail.com> wrote:
> >
> > > accurately.  Moreover, in general, it's impossible to even round
> > > operations involving transcendental functions to an arbitrary fixed-
> > > precision, you may need effectively infinite precision in order to the
> > > computation.
> >
> > Impossible?  Can you explain what you mean by this?  Doesn't the
> > decimal module do exactly that, giving correctly-rounded exp() and
> > log() results to arbitrary precision?
>
> You run into the table-maker's dilemma: there's no way to know in
> advance how many digits you need in order to have n bits of precision
> in the result.  For some computations, the number of bits required to
> get the desired precision can quickly overwhelm the finite limitations
> of your machine (e.g., you run out of RAM first or the time to compute
> the answer is simply unacceptable).

Yes, this appears to be unsolved yet, see also:

http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT

"Is it time to quit yet?  That's the  Table-Maker's Dilemma.  No general
 way exists to predict how many extra digits will have to be carried to
 compute a transcendental expression and round it  _correctly_  to some
 preassigned number of digits.  Even the fact  (if true)  that a finite
 number of extra digits will ultimately suffice may be a deep theorem."


However, in practice, mpfr rounds correctly and seems to be doing fine.

In addition to this, I've been running at least 6 months of continuous
tests comparing cdecimal and decimal, and neither log() nor exp() poses
a problem.

pow() is trickier. Exact results have to be weeded out before
attempting the correction loop for correct rounding, and this is
complicated.


For example, in decimal this expression takes a long time (in cdecimal
the power function is not correctly rounded):


Decimal('100.0') ** Decimal('-557.71e-742888888')



Stefan Krah