precision problems in base conversion of rational numbers

[Raymond Hettinger]

[Terry Hancock]
> > Needless to say, the conventional floating point numbers in Python
> > are actually stored as *binary*, which is why there is a "decimal"
> > module (which is new).
> >
> > If you're going to be converting between bases anyway, it probably
> > makes little difference whether you are using the decimal module
> > or not, of course.  You'll have the same problems the conventional
> > float numbers do.

Not really.  floats won't do because they may not have sufficient
precision to differentiate rational values falling close the split
between representable values in a given base.  The decimal module
offers arbitrarily large precision for making sure the error-term is
small enough to not make a difference.



[Brian van den Broek]
> Thanks. mensanator provided the actual formula for my case. I had a
> "magic number" in my code by which I multiplied my desired level of
> "number of places in the representation" to obtain the value for
> decimal.getcontext.prec.
>
> mensanator wrote:
>
> > The value you want for x is log(17)/log(10) =
> > 1.2304489213782739285401698943283
>
> where x was my "magic number". I've not had a chance to think it
> through yet, but I feel confident that given the formula, I'll be able
> to work out *why* that is the formula I need.

That "formula" just gives a starting point estimate.  The required
decimal precision may be much higher.  If the rational falls very close
to the half-way point between two representable numbers, the
calculation needs to be retried with increased precision until the
split-point is definitive (when the error-term becomes less than the
distance to the next representable value).

For a simple example, convert both 10247448370872321 and
10247448370872319 from base ten to 4 digits of hex.  The calculations
need to be carried out to 15 places of hex (or 17 places of decimal)
just to determine whether the fourth hex digit is a 7 or 8:

    >>> hex(10247448370872321)
    '0x24680000000001L'
    >>> hex(10247448370872319)
    '0x2467ffffffffffL'

For an example of using decimal with iteratively increasing precision,
see the dsum() recipe at
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/393090 .


Raymond