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`	3.095 ns	8.825 ns	×2.9
`A ⋅ a`	2.855 ns	57.533 ns	×20.2
`S ⋅ a`	3.406 ns	57.604 ns	×16.9
Double contraction
`A ⊡ A`	3.706 ns	10.249 ns	×2.8
`S ⊡ S`	3.406 ns	10.249 ns	×3.0
`B ⊡ A`	5.439 ns	119.386 ns	×22.0
`AA ⊡ A`	7.732 ns	134.577 ns	×17.4
`SS ⊡ S`	3.857 ns	134.330 ns	×34.8
Tensor product
`a ⊗ a`	3.406 ns	36.641 ns	×10.8
Cross product
`a × a`	3.406 ns	36.641 ns	×10.8
Determinant
`det(A)`	3.095 ns	168.128 ns	×54.3
`det(S)`	3.095 ns	177.522 ns	×57.4
Inverse
`inv(A)`	5.350 ns	459.736 ns	×85.9
`inv(S)`	4.508 ns	465.612 ns	×103.3
`inv(AA)`	836.014 ns	1.442 μs	×1.7
`inv(SS)`	299.078 ns	1.438 μs	×4.8

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

julia> versioninfo()
Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 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)