Use Float32 to calculate DE_2000 for low precision colors

[kimikage]

This speeds up the calculation of DE_2000 while ensuring the accuracy of 5e-5.
This changes not only the type of intermediate variables, but also the return type. (cf. #477)
This also fixes typos in the DE_2000 test data. (separated to #499)

julia> a, b = Lab(50, -1.3802, -84.2814), Lab(50, 0.0000, -82.7485);

julia> colordiff(Lab{BigFloat}(a), Lab{BigFloat}(b)) |> Float64
0.9999988647524654

julia> colordiff(Lab{Float64}(a), Lab{Float64}(b))
0.9999988647524661

julia> colordiff(Lab{Float32}(a), Lab{Float32}(b))
1.0000001f0

julia> colordiff(Lab{BigFloat}.(Lab{Float32}.((a, b)))...) |> Float32
0.99999994f0

Benchmark

julia> a = Lab{Float32}.(rand(RGB{Float32}, 1000, 1000));

julia> b = Lab{Float32}.(rand(RGB{Float32}, 1000, 1000));

julia> @btime colordiff.($a, $b);
  174.563 ms (2 allocations: 7.63 MiB) # v0.12.8 (`Float64`)
   69.342 ms (2 allocations: 3.81 MiB) # this PR (`Float32`)

julia> a64, b64 = Lab{Float64}.(a), Lab{Float64}.(b);

julia> @btime colordiff.($a64, $b64);
  107.792 ms (2 allocations: 7.63 MiB) # this PR

julia> versioninfo()
Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, tigerlake)

[codecov]

Codecov Report

Merging #494 (ba054ea) into master (2a0b92d) will increase coverage by 0.15%.
The diff coverage is 98.14%.

@@            Coverage Diff             @@
##           master     #494      +/-   ##
==========================================
+ Coverage   93.64%   93.80%   +0.15%     
==========================================
  Files           9        9              
  Lines        1118     1146      +28     
==========================================
+ Hits         1047     1075      +28     
  Misses         71       71              

Impacted Files	Coverage Δ
src/utilities.jl	98.68% <87.50%> (ø)
src/differences.jl	99.25% <100.00%> (+0.19%)	⬆️

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[kimikage]

At the moment, this uses Float32 entirely, in order to identify the effect of precompilation. However, in practice, using Float32 instead of Float64 outside of bottlenecks should not result in significant speed improvements.

Probably, atands and cosds are the bottlenecks.
Edit: atand has been improved in PR #495. cosd was replaced by cos in PR #498, and _de2000_t(::Float32) will be specialized.
Edit2: _de2000_t(::Float32) and _de2000_rot(::Float32) were specialized.

[kimikage]

I would like to determine if it is worth calculating delta_h in DE_2000 with Float32, after improving the accuracy of delta_h.
cf. #477 (comment) (Edit: cf. PR #501)

[johnnychen94]

FWIW, I doubt that anyone is really using it for DE_2000, but CUDA devices are partialcularily optimized for Float32 operations and not for Float64.

[kimikage]

Probably the last bottleneck is mean_hue. As we learned in high school, a vector bisecting the angle of two direction vectors can be represented by the composition of their unit vectors. This method reduces the costly atan to a single call, but causes a loss of significance when the mean hue is near 0°, 90°, 180°, or 270°. Perhaps rotating the entire field will work around the problem.
Edit: By rotating each hue by 90° inward when the hue difference is greater than 90°, the updated hue difference will be less than 90°, which will avoid the loss of digits in one of the a* or b*.
Edit2: cf. PR #505

I've spent too much time improving delta_h, so I'd like to merge this PR first, with mean_hue being a future task.

[kimikage]

Since this change is related to issue #496 as well as speed, I would like to merge this.