# Symbolics.jl: Speed of derivative computations vs. SymEngine.jl

**URL:** <https://discourse.julialang.org/t/symbolics-jl-speed-of-derivative-computations-vs-symengine-jl/68803>\
**Category:** Performance\
**Tags:** question, symbolics\
**Created:** [September 27, 2021, 10:13am UTC](https://discourse.julialang.org/t/symbolics-jl-speed-of-derivative-computations-vs-symengine-jl/68803 "2021-09-27T10:13:50Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![ranocha](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ranocha/32/35588_2.png) [@ranocha](https://discourse.julialang.org/u/ranocha)\
**Post date:** [September 27, 2021, 10:13am UTC](https://discourse.julialang.org/t/symbolics-jl-speed-of-derivative-computations-vs-symengine-jl/68803/1 "2021-09-27T10:13:50Z")

</div>

I benchmarked some derivative computations using Symbolics.jl, SymEngine.jl, and SymPy.jl. As expected, SymPy.jl was the slowest solution in my case. However, SymEngine.jl was significantly faster than Symbolics.jl. Here is an example:

```julia
julia> using Pkg; Pkg.activate(temp=true); Pkg.add(["SymEngine", "Symbolics", "BenchmarkTools"])

julia> using SymEngine, Symbolics, BenchmarkTools

julia> f(u) = u[2] / (u[1]^2 + u[2]^2)
f (generic function with 1 method)

julia> u_symbolics = @variables u1 u2

julia> expand_derivatives(( Differential(u_symbolics[1])(f(u_symbolics)) ))
-2u1*(u2 / ((u1^2 + u2^2)^2))

julia> @benchmark expand_derivatives($( Differential(u_symbolics[1])(f(u_symbolics)) ))
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max): 19.477 μs … 4.431 ms ┊ GC (min … max): 0.00% … 99.14%
 Time (median): 21.305 μs ┊ GC (median): 0.00%
 Time (mean ± σ): 22.915 μs ± 61.455 μs ┊ GC (mean ± σ): 3.78% ± 1.40%

     ▃█▇▄                                                      
  ▂▃▅█████▆▆▅▅▄▄▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▁▂▂▁▁▁▁▂▁▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂ ▃
  19.5 μs Histogram: frequency by time 37.5 μs <

 Memory estimate: 10.97 KiB, allocs estimate: 233.

julia> u_symengine = @vars u1 u2
(u1, u2)

julia> diff((f(u_symengine)), (u_symengine[1]))
-2*u2*u1/(u1^2 + u2^2)^2

julia> @benchmark diff($(f(u_symengine)), $(u_symengine[1]))
BenchmarkTools.Trial: 10000 samples with 7 evaluations.
 Range (min … max): 4.198 μs … 35.084 μs ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 4.268 μs ┊ GC (median): 0.00%
 Time (mean ± σ): 4.320 μs ± 510.445 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

  ▄█▇▆▅▂▁▁▁▁ ▂
  ████████████▆▆▅▃▅▃▃▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▃▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆▅ █
  4.2 μs Histogram: log(frequency) by time 6.1 μs <

 Memory estimate: 336 bytes, allocs estimate: 20.

```

1. Am I doing something wrong with Symbolics.jl?
2. Is there a way for me to get better performance of such derivative calculations using Symbolics.jl?

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [September 27, 2021, 3:55pm UTC](https://discourse.julialang.org/t/symbolics-jl-speed-of-derivative-computations-vs-symengine-jl/68803/2 "2021-09-27T15:55:40Z")

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Open an issue. Worth profiling.

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<div class="post-metadata">

**Author:** ![ranocha](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ranocha/32/35588_2.png) [@ranocha](https://discourse.julialang.org/u/ranocha)\
**Post date:** [September 27, 2021, 6:24pm UTC](https://discourse.julialang.org/t/symbolics-jl-speed-of-derivative-computations-vs-symengine-jl/68803/3 "2021-09-27T18:24:24Z")

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[Done](https://github.com/JuliaSymbolics/Symbolics.jl/issues/395)