[issue1580] Use shorter float repr when possible

[Tim Peters]

Tim Peters added the comment:

Guido, right, for that to work reliably, double->str and str->double
must both round correctly on the platform doing the repr(), and
str->double must round correctly on the platform reading the string.

It's quite easy to understand why at a high level:  a simple (but
tedious) pigeonhole argument shows that there are more representable
doubles than there are possible 16-digit decimal outputs.  Therefore
there must exist distinct doubles (say, x and y) that map to the same
16-digit decimal output (in fact, there are cases where 7 consecutive
doubles all map to the same correctly-rounded 16-digit decimal string).
 When reading that back in, there's no room for error in deciding which
of {x, y} was /intended/ -- the str->double conversion has to be
flawless in deciding which of {x, y} is closest to the 16-digit decimal
input, and in case x and y are equally close, has to apply 754's
round-to-nearest/even rule correctly.  This can require exact
multi-thousand bit arithmetic in some cases, which is why the 754
standard doesn't require it (thought to be impractically expensive at
the time).

> which would really turn the educational issue into an educational
> nightmare, trying to explain why two numbers both print the same
> but compare unequal

Well, newbies see that in, e.g., spreadsheets every day.  You generally
have to work to get a spreadsheet to show you "all the digits"; in some
spreadsheets it's not even possible.  "Not all digits are displayed"
should be within the grasp of must students without suffering nightmares ;-)

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<http://bugs.python.org/issue1580>
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