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`	2.100 ns	12.212 ns	×5.8
`A ⋅ a`	3.100 ns	77.838 ns	×25.1
`S ⋅ a`	3.100 ns	77.550 ns	×25.0
Double contraction
`A ⊡ A`	5.700 ns	16.333 ns	×2.9
`S ⊡ S`	3.300 ns	15.516 ns	×4.7
`B ⊡ A`	8.208 ns	235.068 ns	×28.6
`AA ⊡ A`	9.209 ns	261.733 ns	×28.4
`SS ⊡ S`	5.700 ns	261.988 ns	×46.0
Tensor product
`a ⊗ a`	4.800 ns	53.966 ns	×11.2
Cross product
`a × a`	4.800 ns	53.966 ns	×11.2
Determinant
`det(A)`	2.800 ns	277.866 ns	×99.2
`det(S)`	3.000 ns	272.076 ns	×90.7
Inverse
`inv(A)`	24.198 ns	738.632 ns	×30.5
`inv(S)`	14.529 ns	664.179 ns	×45.7
`inv(AA)`	1.460 μs	2.311 μs	×1.6
`inv(SS)`	657.525 ns	2.311 μs	×3.5

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

julia> versioninfo()
Julia Version 1.5.4
Commit 69fcb5745b (2021-03-11 19:13 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Xeon(R) CPU E5-2673 v3 @ 2.40GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-9.0.1 (ORCJIT, haswell)