Benchmarks

a = rand(Vec{3})
A = rand(SecondOrderTensor{3})
S = rand(SymmetricSecondOrderTensor{3})
B = rand(Tensor{Tuple{3,3,3}})
AA = rand(FourthOrderTensor{3})
SS = rand(SymmetricFourthOrderTensor{3})

Operation	`Tensor`	`Array`	speed-up
Single contraction
`a ⋅ a`	2.785 ns	9.567 ns	×3.4
`A ⋅ a`	3.396 ns	65.700 ns	×19.3
`S ⋅ a`	3.706 ns	65.802 ns	×17.8
Double contraction
`A ⊡ A`	3.406 ns	11.111 ns	×3.3
`S ⊡ S`	3.396 ns	11.112 ns	×3.3
`B ⊡ A`	5.430 ns	130.664 ns	×24.1
`AA ⊡ A`	7.722 ns	155.259 ns	×20.1
`SS ⊡ S`	4.197 ns	146.812 ns	×35.0
Tensor product
`a ⊗ a`	3.406 ns	39.893 ns	×11.7
Cross product
`a × a`	3.406 ns	39.893 ns	×11.7
Determinant
`det(A)`	3.396 ns	190.460 ns	×56.1
`det(S)`	3.095 ns	188.991 ns	×61.1
Inverse
`inv(A)`	5.811 ns	495.482 ns	×85.3
`inv(S)`	4.929 ns	494.113 ns	×100.2
`inv(AA)`	928.091 ns	1.549 μs	×1.7
`inv(SS)`	357.704 ns	1.557 μs	×4.4

The benchmarks are generated by runbenchmarks.jl on the following system:

julia> versioninfo()
Julia Version 1.10.5
Commit 6f3fdf7b362 (2024-08-27 14:19 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)