Optimizing `Fixed` --> `Float` conversions

[kimikage]

See: #129 (comment)

Although this has been postponed, this is a issue on conversion, not arithmetic. So, it might be better to include this in the next release (v0.8.0).

[kimikage]

(::Type{Tf})(x::Fixed{T,f}) where {Tf <: AbstractFloat, T, f} = Tf(Tf(x.i) * Tf(@exp2(-f)))
Base.Float16(x::Fixed{T,f}) where {T, f} = Float16(Float32(x))
Base.Float32(x::Fixed{T,f}) where {T, f} = Float32(x.i) * Float32(@exp2(-f))
Base.Float64(x::Fixed{T,f}) where {T, f} = Float64(x.i) * @exp2(-f)

[kimikage]

Benchmark

There seems to be no significant difference between Julia versions or between operating systems.
There is a slowdown in converting Vec3{Fixed{Int16}} arrays to Vec3{Float32} arrays, but this is mainly a problem with the LLVM backend.

Script

using BenchmarkTools
using FixedPointNumbers
struct Vec3{T <: Real}
    x::T; y::T; z::T
end
struct Vec4{T <: Real}
    x::T; y::T; z::T; w::T
end
Vec3{T}(v::Vec3{T}) where {T} = v
Vec3{T}(v::Vec3{U}) where {T, U} = Vec3{T}(v.x, v.y, v.z) 
Vec4{T}(v::Vec4{T}) where {T} = v
Vec4{T}(v::Vec4{U}) where {T, U} = Vec4{T}(v.x, v.y, v.z, v.w) 
Base.rand(::Type{Vec3{T}}) where {T} = Vec3{T}(rand(T), rand(T), rand(T))
Base.rand(::Type{Vec4{T}}) where {T} = Vec4{T}(rand(T), rand(T), rand(T), rand(T))
function Base.rand(::Type{T}, sz::Dims) where {T <: Union{Vec3, Vec4}}
    A = Array{T}(undef, sz)
    for i in eachindex(A); A[i] = rand(T); end
    return A
end
Ts = (Q0f7, Q4f3, Q0f15, Q12f3, Q0f31, Q28f3, Q0f63, Q60f3)

mat3s = [rand(Vec3{T}, 64, 64) for T in Ts];
mat4s = [rand(Vec4{T}, 64, 64) for T in Ts];

for mat in mat3s
    println(eltype(mat), "-> Float32")
    @btime Vec3{Float32}.(view($mat,:,:))
end

for mat in mat3s
    println(eltype(mat), "-> Float64")
    @btime Vec3{Float64}.(view($mat,:,:))
end

for mat in mat4s
    println(eltype(mat), "-> Float32")
    @btime Vec4{Float32}.(view($mat,:,:))
end

for mat in mat4s
    println(eltype(mat), "-> Float64")
    @btime Vec4{Float64}.(view($mat,:,:))
end

Julia v1.3.1 x86_64-w64-mingw32

julia> versioninfo()
Julia Version 1.3.1
Commit 2d5741174c (2019-12-30 21:36 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.1 (ORCJIT, skylake)

Matrix of Vec3 (unit: μs)

w64	Float32 master	Float32 optimized		Float64 master	Float64 optimized
Q0f7	3.943	3.350		10.700	5.575
Q4f3	3.886	3.362		10.900	5.420
Q0f15	4.043	4.850		11.100	5.480
Q12f3	4.057	4.850		11.001	5.840
Q0f31	5.960	5.000		9.199	5.520
Q28f3	6.000	5.000		9.199	5.640
Q0f63	30.600	5.183		48.299	6.150
Q60f3	25.799	5.617		47.401	6.150

Matrix of Vec4 (unit: μs)

w64	Float32 master	Float32 optimized		Float64 master	Float64 optimized
Q0f7	16.399	3.643		17.199	5.700
Q4f3	15.500	3.614		17.200	6.100
Q0f15	15.100	3.643		17.100	5.801
Q12f3	15.000	3.750		20.699	6.200
Q0f31	13.899	3.783		17.001	5.499
Q28f3	13.800	3.743		17.199	7.133
Q0f63	57.300	13.499		65.000	10.800
Q60f3	60.800	13.699		66.699	10.600

Julia v1.0.5 x86_64-pc-linux-gnu on WSL

julia> versioninfo()
Julia Version 1.0.5
Commit 3af96bcefc (2019-09-09 19:06 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.0 (ORCJIT, skylake)

Matrix of Vec3 (unit: μs)

linux	Float32 master	Float32 optimized		Float64 master	Float64 optimized
Q0f7	3.914	3.362		10.900	5.400
Q4f3	4.000	3.425		11.300	5.620
Q0f15	4.057	5.033		11.800	5.800
Q12f3	4.057	5.050		11.700	5.875
Q0f31	6.100	5.133		9.800	5.640
Q28f3	6.120	5.150		9.600	5.700
Q0f63	48.700	5.450		51.900	6.180
Q60f3	44.400	5.417		46.900	6.375

Matrix of Vec4 (unit: μs)

linux	Float32 master	Float32 optimized		Float64 master	Float64 optimized
Q0f7	14.200	3.471		16.500	6.467
Q4f3	14.700	3.414		16.700	6.633
Q0f15	14.000	3.614		16.100	6.667
Q12f3	13.500	3.557		16.300	5.967
Q0f31	12.300	3.486		14.400	7.000
Q28f3	12.200	3.543		14.300	7.300
Q0f63	62.900	14.300		64.400	11.400
Q60f3	56.500	14.200		56.500	11.600