To elaborate on what @stevengj said: You’re pulling the most costly part of the computation out into the setup here, so you’re only measuring a small part of the actual cost of the operation. In a fair benchmark, Achol \ b wins handily. Here’s what I find:
julia> @benchmark $Achol \ $b
BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
Range (min … max): 248.917 μs … 388.625 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 261.416 μs ┊ GC (median): 0.00%
Time (mean ± σ): 263.493 μs ± 9.988 μs ┊ GC (mean ± σ): 0.00% ± 0.00%
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249 μs Histogram: frequency by time 289 μs <
Memory estimate: 8.00 KiB, allocs estimate: 1.
julia> @benchmark U \ (L \ $b) setup=(U = $Achol.U; L = $Achol.L) # Unfair
BenchmarkTools.Trial: 5858 samples with 1 evaluation per sample.
Range (min … max): 132.750 μs … 227.459 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 145.292 μs ┊ GC (median): 0.00%
Time (mean ± σ): 145.427 μs ± 5.723 μs ┊ GC (mean ± σ): 0.00% ± 0.00%
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133 μs Histogram: frequency by time 162 μs <
Memory estimate: 16.00 KiB, allocs estimate: 2.
julia> myldiv(chol, x) = chol.U \ (chol.L \ x)
myldiv (generic function with 1 method)
julia> @benchmark myldiv($Achol, $b) # Fair, but slower
BenchmarkTools.Trial: 5796 samples with 1 evaluation per sample.
Range (min … max): 659.125 μs … 2.103 ms ┊ GC (min … max): 0.00% … 57.96%
Time (median): 765.250 μs ┊ GC (median): 0.00%
Time (mean ± σ): 862.006 μs ± 209.153 μs ┊ GC (mean ± σ): 9.99% ± 14.68%
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659 μs Histogram: log(frequency) by time 1.5 ms <
Memory estimate: 7.65 MiB, allocs estimate: 4.