[WIP] Rework broadcast - #41

Took awhile to understand but it looks good. I'm going to have to use this other places :). What's the reason for this being labelled WIP?

For benchmarks it would be good to retest SciML/OrdinaryDiffEq.jl#122 (comment)

What's the reason for this being labelled WIP?

I was waiting on you to review and I would have this see some more testing. Developing this was helpful since it pointed out issues in DistributedArrays implementation that I am working on fixing, and the example we discussed doesn't work at all for DArrays yet.
So I want to make sure that works before we merge this.

Update of SciML/OrdinaryDiffEq.jl#122 :

using DiffEqBase, OrdinaryDiffEq, RecursiveArrayTools, DiffEqDevTools
using BenchmarkTools

u0 = zeros(100)
v0 = ones(100)
f1 = function (du,v,u,p,t)
  du .= v
end
f2 = function (dv,v,u,p,t)
  dv .= -u
end
function (::typeof(f2))(::Type{Val{:analytic}}, x, y0)
  u0, v0 = y0
  ArrayPartition(u0*cos(x) + v0*sin(x), -u0*sin(x) + v0*cos(x))
end

prob = DynamicalODEProblem(f1,f2,u0,v0,(0.0,5.0))

@benchmark solve(prob,RK4(),dt=1/2,adaptive=false,save_everystep=false)

function f(du,u,p,t)
    uu = @view u[1:100]
    v = @view u[101:end]
    du[1:100] .= uu
    du[101:200] .= v
end
u02 = [zeros(100);ones(100)]
tspan=(0.0,5.0)
prob = ODEProblem(f,u02,tspan)
@benchmark sol = solve(prob,RK4(),dt=1/2,save_everystep=false)

BenchmarkTools.Trial: 
  memory estimate:  70.98 KiB
  allocs estimate:  505
  --------------
  minimum time:     47.131 μs (0.00% GC)
  median time:      53.279 μs (0.00% GC)
  mean time:        69.464 μs (8.20% GC)
  maximum time:     2.615 ms (93.94% GC)
  --------------
  samples:          10000
  evals/sample:     1

BenchmarkTools.Trial: 
  memory estimate:  149.80 KiB
  allocs estimate:  513
  --------------
  minimum time:     67.330 μs (0.00% GC)
  median time:      75.821 μs (0.00% GC)
  mean time:        97.848 μs (10.04% GC)
  maximum time:     2.486 ms (94.94% GC)
  --------------
  samples:          10000
  evals/sample:     1

Wow. It's looking pretty fantastic now performance-wise especially considering the fact that there's still a broadcast performance bug for Array{Float64} which might be involved.

Tests on OrdinaryDiffEq.jl seem to pass. I say seem to because there's something that I think is unrelated that I'm tracking down. Those tests put everything through the gauntlet so it's pretty safe if they pass. I'll make sure OrdinaryDiffEqExtendedTests.jl gets ran as well.

Yeah, everything passes and the benchmarks look good.