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	9.567 ns	×3.1
`A ⋅ a`	3.095 ns	62.063 ns	×20.1
`S ⋅ a`	3.095 ns	62.086 ns	×20.1
Double contraction
`A ⊡ A`	3.406 ns	11.111 ns	×3.3
`S ⊡ S`	3.406 ns	11.121 ns	×3.3
`B ⊡ A`	5.430 ns	132.229 ns	×24.4
`AA ⊡ A`	7.742 ns	145.648 ns	×18.8
`SS ⊡ S`	4.197 ns	145.752 ns	×34.7
Tensor product
`a ⊗ a`	3.406 ns	39.812 ns	×11.7
Cross product
`a × a`	3.406 ns	39.812 ns	×11.7
Determinant
`det(A)`	3.396 ns	182.065 ns	×53.6
`det(S)`	3.406 ns	183.010 ns	×53.7
Inverse
`inv(A)`	5.851 ns	497.513 ns	×85.0
`inv(S)`	4.879 ns	488.804 ns	×100.2
`inv(AA)`	912.927 ns	1.555 μs	×1.7
`inv(SS)`	321.294 ns	1.552 μs	×4.8

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

julia> versioninfo()
Julia Version 1.10.3
Commit 0b4590a5507 (2024-04-30 10:59 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)