Improve performance of derangement/subfactorial with iterative implementation

[FedericoStra]

This PR changes the implementation of derangement, hence also subfactorial,
to use the recursive formula !n = (n-1) * (!(n-1) + !(n-2)) presented here.

For values such as subfactorial(100) I get a huge speed-up, like ~10x.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 96.93%. Comparing base (403dcb4) to head (e2b762b).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #146      +/-   ##
==========================================
+ Coverage   96.89%   96.93%   +0.03%     
==========================================
  Files           8        8              
  Lines         805      815      +10     
==========================================
+ Hits          780      790      +10     
  Misses         25       25              

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[FedericoStra]

With this change I pass from

julia> @benchmark subfactorial(100)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  185.527 μs … 47.124 ms  ┊ GC (min … max):  0.00% … 60.54%
 Time  (median):     189.635 μs              ┊ GC (median):     0.00%
 Time  (mean ± σ):   267.191 μs ±  1.711 ms  ┊ GC (mean ± σ):  14.89% ±  2.31%

  ▆█▇▆▆▅▄▃▂▃▃▃▃▂▁▁▁▂▁▁▁                                        ▂
  █████████████████████▇▇▇▅▆▇▅▄▆▆█▇█▇▇▇▇▇▆▆▅▅▃▅▄▄▃▄▄▃▁▄▁▁▄▃▃▁▃ █
  186 μs        Histogram: log(frequency) by time       273 μs <

 Memory estimate: 81.06 KiB, allocs estimate: 2931.

to

julia> @benchmark subfactorial(100)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  12.113 μs …  49.734 ms  ┊ GC (min … max):  0.00% … 60.39%
 Time  (median):     15.437 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   29.177 μs ± 762.966 μs  ┊ GC (mean ± σ):  25.19% ±  0.96%

          ▂▅▆█▂  ▂▄     ▄▅▁
  ▁▁▁▁▁▁▂▄█████▆▅███▅▅▃▅███▆▅▅▄▃▃▂▂▁▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▃
  12.1 μs         Histogram: frequency by time         24.2 μs <

 Memory estimate: 14.09 KiB, allocs estimate: 499.

[FedericoStra]

Changes

• I replaced the recursive formula with the simpler !n = n * !(n-1) + (-1)^n.
• I replaced all operations on BigInts with in-place functions from Base.GMP.MPZ to reduce allocations.

Consequences
The performance improved by another factor ~10x:

julia> @benchmark subfactorial(100)
BenchmarkTools.Trial: 10000 samples with 10 evaluations.
 Range (min … max):  1.427 μs … 137.132 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     1.480 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   1.538 μs ±   1.549 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  ▃█▆▆▄▂                                                      ▂
  ██████▆▃▃▃▅██▇▆▆▅▃▄▄▄▁▃▄▄▃▁▄▃▃▁▄▄▁▁▄▁▁▃▁▁▃▄▁▃▁▁▁▁▃▃▁▁▃▁▁▁▃▅ █
  1.43 μs      Histogram: log(frequency) by time      3.05 μs <

 Memory estimate: 112 bytes, allocs estimate: 11.

[inkydragon]

Need a rebase to master.
If CI looks good, I'll merge this pr.

[FedericoStra]

There seems to be a small regression in codecov/project stemming from "indirect changes" in src/combinations.jl. I'm not familiar with codevov so I don't know what this means exactly and how this PR affects code coverage in src/combinations.jl.

[FedericoStra]

There seems to be a small regression in codecov/project stemming from "indirect changes" in src/combinations.jl. I'm not familiar with codevov so I don't know what this means exactly and how this PR affects code coverage in src/combinations.jl.

Ok, rebasing resolved the issue.