accuracy of intrinsic SUM function

[N.M. Maclaren]

On Aug 24 2012, Steve Kargl wrote:
>
>The above algorithm probably represents a simple
>change to gfortran current in-lining of the naive
>algorithm.  Any other algorithm would involve 
>a much more complicated change, and would probably
>lead to a function call to a library routine.

My brief test indicates that it's a complete waste of effort.
I have coded up the algorithms, with pairwise needing 8.log_2(N)
words of scratch space, and here are some results on random
data (in C++, but so what?):

Please input a size:
100000000
Methods                Value                     Error     Time
Canonical              5.363276169432254392e+03
Kahan                  5.363276169432254392e+03  3.16e-17  0.804
Extended               5.363276169432254392e+03  1.29e-17  0.213
Pairwise               5.363276169432252573e+03  3.08e-16  0.555
Forwards               5.363276169433530413e+03  2.38e-13  0.241
numeric::accumulate    5.363276169433530413e+03  2.38e-13  0.241

This is on a single AMD system, but the fastest and most accurate
method is simply to use Intel native 80-bit arithmetic for a
sequential accumulation.  That makes fancy methods futile except
on non-Intel systems.  However, it might well be worth doing if
it doesn't already (I haven't checked).

I may try with some malice-aforethought data distributions, to see
if I can make pairwise less accurate on a distribution that is likely
to occur in practice, but not tonight.

You are welcome to my code, incidentally.


Regards,
Nick Maclaren.