PolynomialEvaluationArrays

[saschatimme]

This introduces a new type PolynomialEvaluationArray. The idea is to extend the previous performance improvements to arrays of polynomials.
The implementation is quite straightforward:

• Find all occurring exponents
• Build up lookup tables for each polynomial
• Use a Horner schema to compute all values x[i]^k <-- here are the improvements
• Use the compute values to evaluate the polynomials

To achieve this I also modified the previous implementation for a single polynomial and improved the performance even more:

x + y + 1
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     19.861 ns (0.00% GC)
  median time:      20.354 ns (0.00% GC)
  mean time:        25.981 ns (0.00% GC)
  maximum time:     26.923 μs (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     995

x^4 + 4x^3 + 6x^2 + 4x + y + 1
BenchmarkTools.Trial: 
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     29.354 ns (0.00% GC)
  median time:      29.726 ns (0.00% GC)
  mean time:        32.476 ns (0.00% GC)
  maximum time:     213.355 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     994

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:     35.406 ns (0.00% GC)
  median time:      38.362 ns (0.00% GC)
  mean time:        41.891 ns (0.00% GC)
  maximum time:     665.141 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     992

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:     39.339 ns (0.00% GC)
  median time:      41.795 ns (0.00% GC)
  mean time:        43.575 ns (0.00% GC)
  maximum time:     175.543 ns (0.00% GC)
  --------------
  samples:          10000

I think the main reason that the performance is improved is that branch prediction is more successful than previously.

[codecov]

Codecov Report

Merging #4 into master will decrease coverage by 6.83%.
The diff coverage is 71.95%.

@@            Coverage Diff             @@
##           master       #4      +/-   ##
==========================================
- Coverage   94.62%   87.79%   -6.84%     
==========================================
  Files           3        5       +2     
  Lines         242      295      +53     
==========================================
+ Hits          229      259      +30     
- Misses         13       36      +23

Impacted Files	Coverage Δ
src/polynomialarray.jl	0% <0%> (ø)
src/poly.jl	96.37% <100%> (-0.5%)	⬇️
src/precomputation.jl	98.11% <98.11%> (ø)

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[codecov]

Codecov Report

Merging #4 into master will increase coverage by 0.98%.
The diff coverage is 100%.

@@            Coverage Diff            @@
##           master      #4      +/-   ##
=========================================
+ Coverage   94.62%   95.6%   +0.98%     
=========================================
  Files           3       5       +2     
  Lines         242     296      +54     
=========================================
+ Hits          229     283      +54     
  Misses         13      13

Impacted Files	Coverage Δ
src/precomputation.jl	100% <100%> (ø)
src/poly.jl	96.37% <100%> (-0.5%)	⬇️
src/polynomialarray.jl	100% <100%> (ø)

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