sum(x, 2) is too slow

[lindahua]

I tried to give another example at GIST: https://gist.github.com/lindahua/4967432#file-ju_reduc-jl

Here is the benchmark result on my Mac:

sum(x, 1)      :  elapsed = 0.04232287406921387
fast_sum(x, 1) :  elapsed = 0.03795504570007324
sum(x, 2)      :  elapsed = 0.3422269821166992
fast_sum(x, 2) :  elapsed = 0.06840705871582031

So, sum(x, 1) is comparable to my implementation, but sum(x, 2) is 5x slower. The key problem is that it uses a perhaps easier approach -- simply doing reduction for each slide, instead of the cache-friendily approach that organizes the computation according to the memory layout.

Many reductions can be considered as recursively application of a binary operation can be implemented in such cache friendly way (such as, max, min, prod). There are also other functions can benefit from this (e.g. mean, var, and std).

[ViralBShah]

If you have a cache friendly code, it would be great to include it in base.

[ViralBShah]

I see the gist.

[lindahua]

The gist code is only for the purpose of illustration. Actual implementation has to take care of more things (e.g. generic arrays with dim > 2, etc).

I will consider more on how this can be implemented efficiently for general cases, (e.g. something as the follows)

X = rand(3, 4, 5, 6)
sum(X, 2)

[lindahua]

Closed by #5294.