Relationship between `PartitionArray` and `ProductRepr`

[mateuszbaran]

I've just noticed that RecursiveArrayTools.jl has something similar to ProductRepr from Manifolds.jl, and here it's called PartitionArray. It would be nice if both efforts could be unified to mutually gain from new developments. The main difference seems to be in the way they handle broadcasting. I don't know why they differ but the first difference I've noticed is that ProductRepr broadcasting is faster:

julia> a = ArrayPartition(SA[1, 2], SA[1 2; 3 4])
([1, 2], [1 2; 3 4])

julia> b = ArrayPartition(SA[1, 2], SA[1 2; 3 4])
([1, 2], [1 2; 3 4])

julia> c = ArrayPartition((@MVector [1, 2]), (@MMatrix [1 2; 3 4]))
([1, 2], [1 2; 3 4])

julia> ap = ProductRepr(SA[1, 2], SA[1 2; 3 4])
ProductRepr with 2 submanifold components:
 Component 1 =
  2-element SVector{2, Int64} with indices SOneTo(2):
   1
   2
 Component 2 =
  2×2 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
   1  2
   3  4

julia> bp = ProductRepr(SA[1, 2], SA[1 2; 3 4])
ProductRepr with 2 submanifold components:
 Component 1 =
  2-element SVector{2, Int64} with indices SOneTo(2):
   1
   2
 Component 2 =
  2×2 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
   1  2
   3  4

julia> cp = ProductRepr((@MVector [1, 2]), (@MMatrix [1 2; 3 4]))
ProductRepr with 2 submanifold components:
 Component 1 =
  2-element MVector{2, Int64} with indices SOneTo(2):
   1
   2
 Component 2 =
  2×2 MMatrix{2, 2, Int64, 4} with indices SOneTo(2)×SOneTo(2):
   1  2
   3  4

julia> f(a, b) = a .+ b
f (generic function with 1 method)

julia> f!(c, a, b) =  c.= a .+ b
f! (generic function with 1 method)

julia> @benchmark f($a, $b)
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  3.196 ns … 9.457 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     3.284 ns             ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.299 ns ± 0.146 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  3.2 ns         Histogram: frequency by time       3.52 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark f($ap, $bp)
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  0.023 ns … 4.012 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     0.025 ns             ┊ GC (median):    0.00%
 Time  (mean ± σ):   0.026 ns ± 0.040 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  023 ns         Histogram: frequency by time        028 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark f!($c, $a, $b)
BenchmarkTools.Trial: 10000 samples with 71 evaluations.
 Range (min … max):  836.789 ns … 83.129 μs  ┊ GC (min … max): 0.00% … 98.49%
 Time  (median):     878.359 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   975.382 ns ±  2.561 μs  ┊ GC (mean ± σ):  8.28% ±  3.12%

     ▁▄▆███▇▇▇▆▆▅▄▄▄▃▂▂▂▂▁ ▁     ▁▁▁ ▁▁▁▁▁                     ▃
  ▄▆▇█████████████████████████████████████████▇█▇█▇▇▆▆▆▇▇▆▇▆▆▆ █
  837 ns        Histogram: log(frequency) by time       1.1 μs <

 Memory estimate: 672 bytes, allocs estimate: 12.

julia> @benchmark f!($cp, $ap, $bp)
BenchmarkTools.Trial: 10000 samples with 755 evaluations.
 Range (min … max):  177.677 ns …   6.882 μs  ┊ GC (min … max):  0.00% … 97.06%
 Time  (median):     185.719 ns               ┊ GC (median):     0.00%
 Time  (mean ± σ):   227.402 ns ± 472.338 ns  ┊ GC (mean ± σ):  17.13% ±  7.94%

   ▂▆██▇▅▄▂▂▂▁▂▁▁▁▁▁                                            ▂
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  178 ns        Histogram: log(frequency) by time        290 ns <

 Memory estimate: 416 bytes, allocs estimate: 8.

Would someone here be willing to review some changes to broadcasting?

[ChrisRackauckas]

Yes, I'd review. It would be nice to make that faster.

[mateuszbaran]

Implemented in #163.