Type stability, deprecation



Below, I have an extract of code that does
a) automatic differentiation (I know, ForwardDiff.jl does it better, but I want to learn) and
b) some “tensor” operation.

Running the same test on plain old Float64 is faster by a factor of 100ish, should be 10ish.

Profiling reveals two things I do not understand:

  1. my elementary operations on my autodiff ∂ℝ type are type-unstable. Why on earth?
  2. in the function/operator ∘, the call to zeros wastes time in “deprecated”. Am I not following the book?

I am running Julia 0.6.0.

Thank you in advance!

using ForwardDiff

ι(a,d::Int)                            = 1:size(a,d)
ι(a::Vector)                           = 1:size(a,1)
ι(a::StepRangeLen)                     = 1:size(a,1)
using     StaticArrays

ℤ  = Int64
ℝ  = Float64
ℝ1 = Vector{ℝ}
ι(a,d::Int)                            = 1:size(a,d)
ι(a::Vector)                           = 1:size(a,1)
ι(a::StepRangeLen)                     = 1:size(a,1)

struct ∂ℝ{N} <:Real
    x  :: ℝ
    dx :: SVector{N,ℝ}

∂ℝ(x,dx::ℝ1)                          = ∂ℝ(x,SVector{size(dx,1),ℝ}(dx))
Base.convert(::Type{∂ℝ{N}},x::Union{ℤ,ℝ}) where{N}             = ∂ℝ{N}(convert(ℝ,x),zeros(ℝ,N))
Base.promote_rule(::Type{∂ℝ{N}},::Type{<:Union{ℤ,ℝ}}) where{N} = ∂ℝ{N}
Base.show(io::IO,a::∂ℝ)               = print(io,a.x," + ɛ⋅",a.dx,"\n")

variate(a::ℝ)                         =  ∂ℝ(a ,[1.])
variate(a::ℝ1)                        = [∂ℝ(a[i],[i==j?1.:0. for j=ι(a,1)]) for i=ι(a,1)]  

Base.:(+)(a::∂ℝ{N},b::∂ℝ{N}) where{N}         = ∂ℝ{N}(a.x+b.x,  a.dx.+b.dx)
Base.:(+)(a:: ℝ,b::∂ℝ{N})    where{N}         = ∂ℝ{N}(a  +b.x,        b.dx)
Base.:(+)(a::∂ℝ{N},b:: ℝ)    where{N}         = ∂ℝ{N}(a.x+b  ,  a.dx      )
Base.:(*)(a::∂ℝ{N},b::∂ℝ{N}) where{N}         = ∂ℝ{N}(a.x*b.x,  a.dx.*b.x.+b.dx.*a.x)
Base.:(*)(a:: ℝ,b::∂ℝ{N})    where{N}         = ∂ℝ{N}(a  *b.x,             b.dx.*a  )
Base.:(*)(a::∂ℝ{N},b:: ℝ)    where{N}         = ∂ℝ{N}(a.x*b  ,  a.dx.*b             )

function ∘(a::Array{Float64,3},b::Vector{T}) where {T} 
    c = zeros(T,size(a,1),size(a,2))
    for k=ι(a,3),j=ι(a,2),i=ι(a,1)
        @inbounds c[i,j] += a[i,j,k]*b[k]
    return c

function foo(a,b)
    c = a∘b
    for i = 1:10000
    c = a∘b
    return c

a = randn(3,4,5)
b = randn(5)
@time c = foo(a,b)                                       # plain old Float64
@time ForwardDiff.jacobian(b->foo(a,b),b) # ForwardDiff - fast!
b = variate(b)  
@time c = foo(a,b)                                       # my stuff, "wasting ti'ime"
if false
@profile c=foo(a,b)
using ProfileView


Your dR function cannot be type-stable because it depends on a global ® whose value can change at any time. Make them const, eg. const ℝ = Float64.


This accelerates my benchmark by a factor 100!!!
I spent a lot of time pondering this one, and I would not have found this answer without help.

Une fois de plus, merci!



On a different note, your conversion/promotion rules look really weird. If you define a proper zero-function then you don’t need them:

Base.zero(T::Type{Var_RR{N}}) where {N} = Var_RR(zero(RR), zero(SVector{N,RR}))


Hi “foobar”. Sorry I missed your post. I have added your “zero” snippet to my code with thanks.

I still seem to need the conversion/promotion, it looks like I add an integer to a “Var_RR” somewhere.

Good tip!