Trivial simplification not happening.

[KronosTheLate]

I differentiated an expression. The result had a couple of glaring simplifications that Symbolics.jl does not seem to be able to see:

julia> using Symbolics

julia> @variables θ, n, x̄
3-element Vector{Num}:
  θ
  n
 x̄

julia> expr = n*log(1/(2*θ^3)) - 1/θ * n * x̄
n*log(1 / (2(θ^3))) + (-n*x̄) / θ

julia> Symbolics.derivative(expr, θ)
(n*x̄) / (θ^2) - 6n*(θ^2)*(1 / (4(θ^6)))*((1 / (2(θ^3)))^-1)

julia> simplify(ans)
(n*x̄) / (θ^2) - 6n*(θ^2)*(1 / (4(θ^6)))*((1 / (2(θ^3)))^-1)

In this final expression, there are 5 powers of theta that can be cancelled. This is something I expect a simplify command to do.
But the worst offender has to be ((1 / (2(θ^3)))^-1), which is equal to 2(θ^3).

I have no idea how to fix it, but as I consider it an issue, I though I should report it.

[ForceBru]

Same thing here. SymPy produces a very short expression, but Symbolics doesn't do obvious simplifications, especially with fractions.

Code

let K = 3
    @variables h[1:K] c[1:K] eps rho
    
    q(h) = (h + eps) / (1 + eps * K)
    
    ELBO = (
        sum(q(h[k]) * c[k] for k in 1:K)
        - sum(q(h[k]) * log(q(h[k])) for k in 1:K)
        - sum(log(h[k]) for k in 1:K) / rho
    )
    
    hess = Symbolics.hessian(ELBO, collect(h), simplify=true)
end

Output in Jupyter Lab:

What's up with -1 / ((eps + h) / (1 + 3eps)) / (1 + 3eps) in the numerator? That's just -1 / (h + eps). Why write -1 / ((eps + h) / (1 + 3eps)) instead of -(1+3eps) / (eps + h)? Why write h^{-2} / rho instead of 1 / (rho * h^2)? This is complexification, not simplification.

SymPy simplifies this nicely:

Versions

• Symbolics v4.2.3
• Julia 1.8

[ChrisRackauckas]

Did you try calling simplify_fractions?

[ForceBru]

@ChrisRackauckas, for my case, the Hessian becomes even more complicated. Symbolics.simplify_fractions.(hess)[1,1] returns this:

[ChrisRackauckas]

That's fun.

[KronosTheLate]

It seems like simplify_fractions has other issues as well: #487
Now I have no idea how these things are implemented, but I feel that two such glaring problems might point to some deeper problem with the function.

[KronosTheLate]

How the original example now works:

julia> using Symbolics

julia> @variables θ, n, x̄;

julia> expr = n*log(1/(2*θ^3)) - 1/θ * n * x̄;

julia> Symbolics.derivative(expr, θ)
(n*x̄) / (θ^2) - 6n*(θ^2)*(1 / (4(θ^6)))*((1 / (2(θ^3)))^-1)

julia> simplify(ans)
(n*x̄ - 3.0n*θ) / (θ^2)

Thanks a lot for the quick and effective solution!