New evaluation routine

[saschatimme]

This new evaluation algorithm increases the memory footprint of a FixedPolynomial but for this we get performance increases with a factor of up to 7 (and probably even more for polynomials with higher degrees).

Here are the detailed benchmark results from the committed benchmark/evaluation.jl:
master

x + y + 1
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     84.496 ns (0.00% GC)
  median time:      86.308 ns (0.00% GC)
  mean time:        91.803 ns (0.00% GC)
  maximum time:     244.586 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     962

x^4 + 4x^3 + 6x^2 + 4x + y + 1
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     174.762 ns (0.00% GC)
  median time:      181.137 ns (0.00% GC)
  mean time:        191.230 ns (0.00% GC)
  maximum time:     472.054 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     723

50x^3 + 83x^2y + 24xy^2 + y^3 + 392x^2 + 414xy + 50y^2 - 28x + 59y - 100
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     350.547 ns (0.00% GC)
  median time:      363.752 ns (0.00% GC)
  mean time:        386.236 ns (0.00% GC)
  maximum time:     11.714 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     214

x^4y^2 + 10x^3y^2 + 3x^3y + 15x^2y^2 + xy^3 + 10x^2y + 10xy^2 + x^2 + 10xy + x + y
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     474.281 ns (0.00% GC)
  median time:      491.551 ns (0.00% GC)
  mean time:        530.601 ns (0.00% GC)
  maximum time:     1.745 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     196

this branch

  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     30.199 ns (0.00% GC)
  median time:      30.317 ns (0.00% GC)
  mean time:        33.423 ns (0.00% GC)
  maximum time:     822.494 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     994

x^4 + 4x^3 + 6x^2 + 4x + y + 1
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     41.132 ns (0.00% GC)
  median time:      44.613 ns (0.00% GC)
  mean time:        47.831 ns (0.00% GC)
  maximum time:     580.027 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     990

50x^3 + 83x^2y + 24xy^2 + y^3 + 392x^2 + 414xy + 50y^2 - 28x + 59y - 100
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     57.427 ns (0.00% GC)
  median time:      57.682 ns (0.00% GC)
  mean time:        63.204 ns (0.00% GC)
  maximum time:     348.563 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     983

x^4y^2 + 10x^3y^2 + 3x^3y + 15x^2y^2 + xy^3 + 10x^2y + 10xy^2 + x^2 + 10xy + x + y
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     61.178 ns (0.00% GC)
  median time:      65.876 ns (0.00% GC)
  mean time:        83.605 ns (0.00% GC)
  maximum time:     10.336 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     981

[codecov]

Codecov Report

Merging #1 into master will increase coverage by 0.36%.
The diff coverage is 100%.

@@            Coverage Diff             @@
##           master       #1      +/-   ##
==========================================
+ Coverage   94.17%   94.53%   +0.36%     
==========================================
  Files           3        3              
  Lines         223      238      +15     
==========================================
+ Hits          210      225      +15     
  Misses         13       13

Impacted Files	Coverage Δ
src/poly.jl	96.79% <100%> (+0.34%)	⬆️

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[blegat]

Do you think that the implementation of the evaluation in DynamicPolynomials could be improved ?

[saschatimme]

I don't think so (at least I don't see an obvious way to improve things).

This new algorithm relies heavily on these precomputations which I would have to be recomputed for each new polynomial. So I think we cannot have the cake and eat it too :/