[issue2819] Full precision summation

[Mark Dickinson]

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

[Tim]
> if you're concerned about speed, it would probably pay to eliminate all
> library calls except one to frexp().

Indeed it would!  Here's a revised patch that avoids use of fmod.  Speed is comparable with the current 
version.  Here are some timings for summing random values on OS X/Core 2 Duo, on a non-debug build of the 
trunk. lsum is the patched version, msum is the version in the current trunk; timings are in seconds.

                                                |  lsum    |  msum
------------------------------------------------+----------+---------
50000 random values in [0, 1)                   | 0.003091 | 0.002540
50000 random values in [0, 1), sorted           | 0.003202 | 0.003043
50000 random values in [0, 1), reverse order    | 0.003201 | 0.002587
50000 random values in [-1, 1)                  | 0.003229 | 0.002829
50000 random values in [-1, 1), sorted          | 0.003183 | 0.002629
50000 random values in [-1, 1), reverse order   | 0.003195 | 0.002731
50000 random exponential values                 | 0.003994 | 0.006178
50000 random exponential values, sorted         | 0.003713 | 0.007933
50000 random exponential values, reverse order  | 0.003714 | 0.002849

Note that lsum doesn't suffer from the 'fragmentation of partials' problem that slows down msum for sorted 
datasets.

I'll do some timings on Linux/x86 as well.

Added file: http://bugs.python.org/file11014/fsum10.patch

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