# Seven Lines of Julia (examples sought)

For a fairer comparison:

• Python has mean, median, and geometric_mean functions that could be employed like the Julia library functions.
• There’s no arbitrary recursion depth here.
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Thanks.
It was a snip from the 1st comment I saw on Twitter.

Language syntax comparison suffer from the same type of problem as language performance comparison:
it depends on the ability of the person writing the code & on the actual language

The world would be better off if there was a “language elegance game” just like there is a Benchmarks Game.

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Wooaah! remarkable! Could you explain how that magic trick works?

``````            Fⁿ(x,n)=  ∘(fill(F,n)...)(x)
``````

Thanks!

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∘ is the function composition function. `∘(f,g,h)(x)` turns into `f(g(h(x)))`.
`fill(F,n)` populates an array with n times the function F. They are composed by ∘ to form the function

`x -> F(F(F(F...F(x)))...)` .

That’s is, you just call it with any argument you like.

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Here’s my seven lines (excluding imports). It calculates and plots pursuit curves using DifferentialEquations.jl. I’ve renamed the pursuer and pursued to `fox` and `rabbit` for clarity. It’s kind of fun to play around with different paths for the rabbit to take. I have a parameter, `k`, which alters the speed of the fox.

The fourth line beginning `Rx, Ry,...` is a little goofy to fit within the space constraint, but it works and makes what I’m plotting clear as day.

``````using Plots, DifferentialEquations, LinearAlgebra

rabbit(t) = [cos(t), sin(t)] # Rabbit Running in a Circle
fox(u, k, t) = k * (rabbit(t) - u) / norm(rabbit(t)-u) # Fox chases rabbit at speed k

prob = ODEProblem(fox, [2.0, 0.0], (0.0, 10), 0.8)
sol = solve(prob, saveat=0.1);
Rx, Ry, Fx, Fy = [getindex.(vcat.(rabbit.(sol.t), sol.u), i) for i=1:4]

plot(Rx, Ry, label="Rabbit", lw=sol.t)
plot!(Fx, Fy, label="Fox", lw=sol.t, aspect_ratio=:equal)
``````

Here’s a bonus animation:

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Don’t forget one of the most evocative tools in Julia-land, the splat, which is needed to take the array-typed output of `fill` and use the elements of that array to populate the individual arguments of `∘`.

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I had to submit another one. I’m still leaning about the ApproxFun.jl package, so I’m happy to hear your feedback. This my first real trial of this package other than running a few of the examples.

The following 7 lines (excluding imports) calculates deflection, angle, moment, and shear of a uniformly loaded cantilevered beam and plots the results. This uses ApproxFun.jl find to solution of an ODE. The solution to this ODE is a function which defines the vertical displacement of the beam (under certain assumptions). Figure of the setup given below.

The line beginning `B = ...` defines the boundary conditions. It’s a fourth order ODE so there are four boundary conditions. The vertical displacement and first derivative (ie, angle) are both zero at the fixed end of the beam (`leftendpoint`). The moment and shear in the beam (2nd and 3rd) derivatives are zero at the free end of the beam (`rightendpoint`).

The solution is generated on the next line, `v = ...`. The boundary conditions, `B`, are all set to 0 using `zeros(4)...`. The differential equation is defined `E*I*Derivative()^4` and gets set to a uniform load `one(z)` (shown as q in the image above).

The rest is just plotting. It’s ugly again to fit into seven lines.

``````using ApproxFun, Plots
L, E, I = 12.0, 1.0, 1.0
d = 0..L
z = Fun(identity, d)
B = [[Evaluation(d,leftendpoint,k) for k=0:1]... ; [Evaluation(d,rightendpoint,k) for k=2:3]... ;]
v = [B; E*I*Derivative()^4] \ [ zeros(4)..., one(z)]
func_name = zip([v, v', v'', v'''], ["Deflection", "Angle", "Momement", "Shear"])
plot([plot(z, f, title=n, label="") for (f,n) in func_name]..., lw=3)
``````

Result:

It’s easy enough to change the load function:

``````v = [B; E*I*Derivative()^4] \ [ zeros(4)..., sin(z)]
``````

I haven’t really messed with the units to ensure the solution is scaled correctly, but it does seem to match the analytical solution of the uniformly loaded beam quite well. The comparison:

``````plot(z, v, lw=3, label="ApproxFun", title="Deflection | Comparison", legend=:left)
plot!(z-> (z^2)*(z^2 - 4*L*z + 6L^2)/(24*E*I), 0, L, lw=3, ls=:dash, label="Analytical")
``````

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• eight or nine easily read lines are better than seven munched up lines
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Hey!
I love your code, but you should use

``````Fn(n) = reduce(\circ, fill(F, n))(x)
...
Fn(100)[1,1,2,3,5]
``````
1 Like

This is not my code, I saw it on Twitter. But yes it’s true that reducing is faster than splatting. But I think the point here was the clarity of the code not really the performance.

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Your code still wastes memory to create the useless vector `fill(F, n)`. Also, it doesn’t support the case `n=0`. Just do this instead

``````Fn(x, n) = foldl((y,_) -> F(y), 1:n; init=x)
``````

Edit: of course one can use a `StaticArrays.SVector{3}` to sqeeze out the last bit of performance.

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Just slightly over, but here’s a user-defined linear regression in Soss.jl:

``````julia> using Soss, MeasureTheory, SampleChainsDynamicHMC

julia> a = -0.2; b=2.0;

julia> x = randn(100); y = a .+ b .* x;

julia> m = @model x begin
a ~ Normal()
b ~ Normal()
y ~ For(x) do xj Normal(a + b * xj, 1) end
end;

julia> sample(DynamicHMCChain, m(x=x) | (;y=y))
4000-element MultiChain with 4 chains and schema (b = Float64, a = Float64)
(b = 1.985±0.097, a = -0.207±0.11)
``````
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It’s also standard notation for function iteration in the Dynamical Systems literature.
I agree it’s super neat!

Can you help me understand, what does “unsafe” & “type piracy” mean (in general & in this context)?

Btw, for composition I define a `Nest(f,x0,n)` function like Mathematica:

``````Nest(f,x0,n) = (n == 0) ? x0 : (f∘Nest)(f,x0,n-1)
import Base.^;
h^(n::Integer) = x0 -> foldl( (x, _) -> h(x), 1:n; init = x0)
``````
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Type piracy is when a module defines a method but neither the function nor any of the method’s parameter types are created by that module.

1 Like

Thanks, but what does that mean in this context?

``````Base.:^(f::Function, n) = n==1 ? f : f ∘ f^(n-1)
``````

That snippet uses `^` and `Function`, but both of those come from another module, Base. It would not be type piracy to define

``````struct CustomCounter end

Base.:^(f::Function, n::CustomCounter) = n==1 ? f : f ∘ f^(n-1)
``````

because `CustomCounter` is defined in this module.

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It’s also super slow and wasteful

``````julia> @time @btime (F^(10^3))([1,1,2,3,5])
660.172 μs (2001 allocations: 218.89 KiB)
10.475864 seconds (29.82 M allocations: 3.087 GiB, 4.50% gc time, 0.24% compilation time)
3-element Vector{Float64}:
2.089057949736859
2.089057949736859
2.089057949736859
``````

and doomed to fail miserably

``````julia> (F^(10^5))([1,1,2,3,5])
...
``````

Bonus: if one is really concerned about the type piracy, then

``````(f::Function)↑(n::Integer) = x0 -> foldl((x, _) -> f(x), 1:n; init = x0)
``````

is a suggestive alternative (`↑` is `\uparrow`).

Edit: in the first code snippet, `@time @btime` is a typo. The intended code was just `@time`, to take into consideration also the compilation time, instead of `@btime`.

4 Likes

Try my version w/ ‘foldl’

Yes your version with `foldl` is perfectly fine!

1 Like

I like Pink noise. It’s is Gaussian noise with power spectral density 1/fᵅ. One can sample it with the inverse Fourier transform

``````
# Make pink noise
using FFTW, Statistics
function pinknoise(m, n)
A = real(ifft((Complex{Float64}[i+j==2 ? 0. : (randn()+randn()*im)/sqrt((sin((i-1)*pi/m)^2 + sin((j-1)* pi/n)^2)) for i in 1:m, j in 1:n])))
real(A)/std(A)
end
A = pinknoise(500, 500)
``````

It most spectacular property is that it is invariant under conformal mappings, functions that locally preserves angles, but not necessarily lengths, for example the map `x + y*im -> ((x + y*im)/33)^2`.

``````# Make conformal map
getindexfc(A, i, j) = A[mod1(ceil(Int, i), size(A,1)), mod1(ceil(Int, j), size(A,2))]
finv(x) = reim(sqrt(x[1] + x[2]*im)*33) # use inverse for look-up
cfmap(A) = map(CartesianIndices(size(A))) do I
getindexfc(A, finv(I)...)
end
``````

This is how it acts on a grid matrix `G`:

``````[G  cfmap(G)]
``````

And here with pink noise

``````[A  cfmap(A)]
``````

PS: I took a bit extra effort to make it tiling, try `[A A A; A A A; A A A]`

27 Likes