Ensure type stability for Julia benchmarks

[GiggleLiu]

The julia code in this repo is not type stable, which causes more than 10x slow down.

Whenever the type is not stable, Julia can not compile efficient code for you.
For example, using global variables like this would cause type instability.

ADBench/tools/Julia/gmm_F.jl

Line 58 in 6c25250

d = size(means,1)

To avoid type instability, one should

• avoid using global variables, if it is nessesary, add const keyword.
• do not change the type of a variable inside a loop,
• try not using closure if not nessesary,

Type instability can be seen by typing @code_warntype f(x).
For more information, see performance tips
https://docs.julialang.org/en/v1/manual/performance-tips/

Also, please take a look at this package for benchmark.
https://github.com/JuliaCI/BenchmarkTools.jl

Fixing this issue will definitely make the benchmark in your paper more reliable.

[GiggleLiu]

I have setup a new repo to show how to fix the gaussian mixture Julia code here:
https://github.com/JuliaReverse/NiGaussianMixture.jl

ForwardDiff shows at least 1 order speed up.
The forwarddiff code is here
https://github.com/JuliaReverse/NiGaussianMixture.jl/blob/master/src/forwarddiff.jl

The benchmark script is here
https://github.com/JuliaReverse/NiGaussianMixture.jl/blob/master/benchmarks/forwarddiff.jl

It will definitely make the benchmark results more reliable.

If you are interested in benchmarking reversible programming AD framework NiLang. I will be happy to contribute to this part.

[GiggleLiu]

The code for the second benchmark, BA is here:

https://github.com/JuliaReverse/NiBundleAdjustment.jl/blob/master/README.md

ForwardDiff should still have some room for improvements judging from the allocation.
NiLang also shows a comparable performance with Tapenade.