reduce compile times by unrolling fastbroadcast on array

[ChrisRackauckas]

With SciML/DiffEqBase.jl#722 as well.

using OrdinaryDiffEq, SnoopCompile, ForwardDiff

lorenz = (du,u,p,t) -> begin
        du[1] = 10.0(u[2]-u[1])
        du[2] = u[1]*(28.0-u[3]) - u[2]
        du[3] = u[1]*u[2] - (8/3)*u[3]
end

u0 = [1.0;0.0;0.0]; tspan = (0.0,100.0);
prob = ODEProblem(lorenz,u0,tspan); alg = Rodas5();
tinf = @snoopi_deep ForwardDiff.gradient(u0 -> sum(solve(ODEProblem(lorenz,u0,tspan),alg)), u0)
tinf = @snoopi_deep ForwardDiff.gradient(u0 -> sum(solve(ODEProblem(lorenz,u0,tspan),alg)), u0)

Before:

#First
InferenceTimingNode: 1.849625/14.538148 on Core.Compiler.Timings.ROOT() with 32 direct children

#Second
InferenceTimingNode: 1.531660/4.170409 on Core.Compiler.Timings.ROOT() with 12 direct children

After:

#First
InferenceTimingNode: 1.181086/3.320321 on Core.Compiler.Timings.ROOT() with 32 direct children

#Second
InferenceTimingNode: 0.998814/1.650488 on Core.Compiler.Timings.ROOT() with 11 direct children

[chriselrod]

Perhaps inv(dt) and then multiply by that?

In theory, @fastmath would do that for you, if it weren't for Julia's ordering the LLVM passes in a way that moves the hoisted division back into the loop.