Add more analytical unit tests

[mikeingold]

More progress toward #67

• Ellipsoid
• Hexahedron
• Tetrahedron
• Torus
• Triangle

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 100.00%. Comparing base (7818474) to head (32ddbcb).
Report is 3 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff            @@
##              main      #166   +/-   ##
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  Coverage   100.00%   100.00%           
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  Files           18        18           
  Lines          181       181           
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  Hits           181       181           

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Benchmark Results

	main	32ddbcb...	main/32ddbcb330eb19...
Differentials/Differential	0.202 ± 0.0011 μs	0.205 ± 0.002 μs	0.985
Differentials/Jacobian	0.167 ± 0.001 μs	0.167 ± 0.001 μs	1
Integrals/Segment/Scalar GaussKronrod	0.826 ± 0.0068 μs	0.841 ± 0.0091 μs	0.982
Integrals/Segment/Scalar GaussLegendre	1.64 ± 0.007 μs	1.62 ± 0.009 μs	1.01
Integrals/Segment/Scalar HAdaptiveCubature	1.16 ± 0.13 μs	1.16 ± 0.12 μs	1
Integrals/Segment/Vector GaussKronrod	3.12 ± 0.083 μs	3.03 ± 0.068 μs	1.03
Integrals/Segment/Vector GaussLegendre	19.5 ± 0.73 μs	17.9 ± 0.58 μs	1.09
Integrals/Segment/Vector HAdaptiveCubature	3.9 ± 0.1 μs	3.89 ± 0.1 μs	1
Integrals/Sphere/Scalar GaussKronrod	0.0728 ± 0.0018 ms	0.0728 ± 0.0019 ms	1
Integrals/Sphere/Scalar GaussLegendre	1.82 ± 0.0025 ms	1.82 ± 0.0031 ms	0.997
Integrals/Sphere/Scalar HAdaptiveCubature	0.047 ± 0.00011 ms	0.0471 ± 0.00014 ms	0.997
Integrals/Sphere/Vector GaussKronrod	0.107 ± 0.0016 ms	0.108 ± 0.0016 ms	0.987
Integrals/Sphere/Vector GaussLegendre	3.43 ± 0.074 ms	3.62 ± 0.094 ms	0.948
Integrals/Sphere/Vector HAdaptiveCubature	0.0995 ± 0.0016 ms	0.0982 ± 0.0013 ms	1.01
Rules/GaussLegendre	21.8 ± 0.56 μs	21.7 ± 0.77 μs	1.01
Specializations/Scalar GaussLegendre/BezierCurve	0.25 ± 0.0071 ms	0.251 ± 0.0022 ms	0.999
Specializations/Scalar GaussLegendre/Line	6.95 ± 0.056 μs	7.11 ± 0.064 μs	0.977
Specializations/Scalar GaussLegendre/Plane	0.747 ± 0.0016 ms	0.745 ± 0.002 ms	1
Specializations/Scalar GaussLegendre/Ray	5.97 ± 0.052 μs	6.08 ± 0.055 μs	0.981
Specializations/Scalar GaussLegendre/Rope	0.0508 ± 0.00021 ms	0.0516 ± 0.0003 ms	0.986
Specializations/Scalar GaussLegendre/Tetrahedron	0.158 ± 0.0017 s	0.157 ± 0.0021 s	1
Specializations/Scalar GaussLegendre/Triangle	0.604 ± 0.0075 ms	0.606 ± 0.0074 ms	0.996
time_to_load	1.53 ± 0.0053 s	1.51 ± 0.018 s	1.01

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[mikeingold]

Not sure why Downgrade fails...

[JoshuaLampert]

Not sure why Downgrade fails...

That's why: https://github.com/JuliaGeometry/MeshIntegrals.jl/actions/runs/12840564494/job/35809354687?pr=166#step:8:477. The necessary diff is also given there. Aqua.jl wants the entries to be ordered alphabetically. This wasn't caught in the previous PR already because we skip CI for changes under docs/.

[JoshuaLampert]

LGTM.
Just out of interest: How did you come up with the functions and their solutions? Did you compute the solutions by hand, did you use some symbolic solver or similar, or did you find them in some reference, or something else?

[mikeingold]

Not sure why Downgrade fails...

That's why: https://github.com/JuliaGeometry/MeshIntegrals.jl/actions/runs/12840564494/job/35809354687?pr=166#step:8:477. The necessary diff is also given there. Aqua.jl wants the entries to be ordered alphabetically. This wasn't caught in the previous PR already because we skip CI for changes under docs/.

Oooohhhhh… I guess they’re using the sorting convention that all upper-case letters come before lower-case letters.

[mikeingold]

Just out of interest: How did you come up with the functions and their solutions? Did you compute the solutions by hand, did you use some symbolic solver or similar, or did you find them in some reference, or something else?

Usually I hand-derive integral problems and just choose an integrand function that produces a non-zero solution but is also still pretty reasonable to derive by hand.

The Hexahedron and Ellipsoid solutions here came from the observation that both have degenerate cases, i.e. they can just be turned into a cube and a sphere.

The others in this PR actually came from experimenting with an LLM (GPT 4o) which was surprisingly competent at suggesting test cases with step-by-step derived solutions. I just guided the model a bit and then validated the provided solution by hand.

[JoshuaLampert]

Just out of interest: How did you come up with the functions and their solutions? Did you compute the solutions by hand, did you use some symbolic solver or similar, or did you find them in some reference, or something else?

Usually I hand-derive integral problems and just choose an integrand function that produces a non-zero solution but is also still pretty reasonable to derive by hand.

The Hexahedron and Ellipsoid solutions here came from the observation that both have degenerate cases, i.e. they can just be turned into a cube and a sphere.

The others in this PR actually came from experimenting with an LLM (GPT 4o) which was surprisingly competent at suggesting test cases with step-by-step derived solutions. I just guided the model a bit and then validated the provided solution by hand.

Sounds good.

[mikeingold]

Thanks!