[issue2819] Full precision summation

[Mark Dickinson]

Mark Dickinson <dickinsm at gmail.com> added the comment:

Here's a patch, in final form, that replaces fsum with an lsum-based
implementation.  In brief, the accumulated sum-so-far is represented in
the form

  huge_integer * 2**(smallest_possible_exponent)

and the huge_integer is stored in base 2**30, with a signed-digit representation (digits 
in the range [-2**29, 2**29).

What are the chances of getting this in before next week's beta?

I did toy with a base 2**52 version, with digits stored as doubles.  It's attractive for 
a couple of reasons: (1) each 53-bit double straddles exactly two digits, which makes 
the inner loop more predictable and removes some branches, and (2) one can make some 
optimizations (e.g. being sloppy about transferring single-bit carries to the next digit 
up) based on the assumption that the input is unlikely to have more than 2**51 summands.  The result was slightly faster on OS X, and slower on Linux;  the final rounding code 
also became a little more complicated (as a result of not being able to do bit 
operations on a double easily), and making sure that things work for non IEEE doubles is 
a bit of a pain.  So in the end I abandoned this approach.

Added file: http://bugs.python.org/file11108/fsum11.patch

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