Julia at Full Tilt: Profiling and Optimizations

Paul Schrimpf

2024-04-30

Introduction

About Me

• UBC Economics, research in econometrics, industrial organization
• My Website
• my github
• Teach courses using Julia
  • ECON 622: Computational Economics with Data Science Applications
  • ECON 567 (empirical industrial organization)

Useful Resources

• Performance Tips in Julia Manual

• How to optimise Julia code: A practical guide Nissen (2022)

• Optimizing Serial Code Rackauckas (2019)

• Coding for performance (basis for this talk) Schrimpf (2019 (revised 2024))

Overview

Basics

Avoid Premature Optimization

• Complete, correct \(>>\) fast, incorrect, unfinished
• Clear, maintainable \(>\) fast, incomprensible (almost always)
• But some practices can both make code faster and clearer

Functions

• Julia functions are JIT compiled, global scripts are not
• Code needs to be in a function for full performance
• Organizing code into functions is also better for readability and maintenance

Type Stability

• To generate efficient code, the Julia compiler needs to know the types of all variables
• Given the types of the inputs of a function, the types of its intermediate variables and output should be deterministic

Type Stable: Example

using BenchmarkTools
function unstabletrick(x, t)
  sum(xi < t ? xi : t for xi in x)
end

t = 0.5
n = 10000
x = (rand(n).-0.5)*10

Type Stable: Example

Stable: Float to Float

@benchmark unstabletrick(x, 0.)

BenchmarkTools.Trial: 10000 samples with 3 evaluations.
 Range (min … max):  8.870 μs … 136.207 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     9.429 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   9.774 μs ±   2.080 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █   █ ▇  ▅ ▃  ▂                          ▁▁ ▁▁ ▁            ▂
  █▁▁▁████▇█▆██▆█▄▅▇▄▄▃▄▁▁▃▅▁▃▃▃▁▁▁▁▁▁▅▅▆▇▇████████▇▇▇▆▆▆▆▆▆▄ █
  8.87 μs      Histogram: log(frequency) by time      16.5 μs <
 Memory estimate: 16 bytes, allocs estimate: 1.

Stable: Int to Int

xint = round.(Int, x)
@benchmark unstabletrick(xint, 0)

BenchmarkTools.Trial: 10000 samples with 10 evaluations.
 Range (min … max):  1.753 μs …   7.859 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     1.783 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   1.896 μs ± 400.146 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▇▁▄▄▄▃▂                                                    ▂
  █████████▇█▆▆▃▄▅▅▆▆▅▄▄▁▄▅▅▅▄▅▆▅▅▆▅▆▇▇▇▅▅▄▄▄▃▅▆▆▇███▇█▇▇▇▅▅▄ █
  1.75 μs      Histogram: log(frequency) by time      3.64 μs <
 Memory estimate: 16 bytes, allocs estimate: 1.

Unstable: (Int & Float) to (Int | Float)

@benchmark unstabletrick(xint, 0.)

BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  48.619 μs … 98.266 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     49.680 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   50.257 μs ±  2.712 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▂▆█▇▆▆▆▆▄▂   ▁                                              ▂
  ██████████▆▇███▇▇█▇▇▇▇▅▆▆▃▄▄▆▅▄▁▄▃▃▁▅▆▄▃▃▁▁▁▃▁▃▁▁▃▁▁▄▄▃▄▅▄▅ █
  48.6 μs      Histogram: log(frequency) by time      66.1 μs <
 Memory estimate: 16 bytes, allocs estimate: 1.

• 6-32x slowdown!

Detecting Type Instability

@code_warntype unstabletrick(xint,0)

MethodInstance for unstabletrick(::Vector{Int64}, ::Int64)
  from unstabletrick(x, t) @ Main In[2]:2
Arguments
  #self#::Core.Const(unstabletrick)
  x::Vector{Int64}
  t::Int64
Locals
  #13::var"#13#14"{Int64}
Body::Int64
1 ─ %1 = Main.:(var"#13#14")::Core.Const(var"#13#14")
│   %2 = Core.typeof(t)::Core.Const(Int64)
│   %3 = Core.apply_type(%1, %2)::Core.Const(var"#13#14"{Int64})
│        (#13 = %new(%3, t))
│   %5 = #13::var"#13#14"{Int64}
│   %6 = Base.Generator(%5, x)::Base.Generator{Vector{Int64}, var"#13#14"{Int64}}
│   %7 = Main.sum(%6)::Int64
└──      return %7

@code_warntype unstabletrick(xint,0.)

MethodInstance for unstabletrick(::Vector{Int64}, ::Float64)
  from unstabletrick(x, t) @ Main In[2]:2
Arguments
  #self#::Core.Const(unstabletrick)
  x::Vector{Int64}
  t::Float64
Locals
  #13::var"#13#14"{Float64}
Body::Union{Float64, Int64}
1 ─ %1 = Main.:(var"#13#14")::Core.Const(var"#13#14")
│   %2 = Core.typeof(t)::Core.Const(Float64)
│   %3 = Core.apply_type(%1, %2)::Core.Const(var"#13#14"{Float64})
│        (#13 = %new(%3, t))
│   %5 = #13::var"#13#14"{Float64}
│   %6 = Base.Generator(%5, x)::Base.Generator{Vector{Int64}, var"#13#14"{Float64}}
│   %7 = Main.sum(%6)::Union{Float64, Int64}
└──      return %7

Extended Example

using ForwardDiff, LinearAlgebra, Distributions
function statparts(gi::Function)
  function P(A) # projection matrix
    A*pinv(A'*A)*A'
  end
  function(θ)
    giθ = gi(θ)
    p = length(θ)
    (n, k) = size(giθ)
    Ω = cov(giθ)
    gn=mean(gi(θ), dims=1)'
    Gi= ForwardDiff.jacobian(gi,θ)
    Gi = reshape(Gi, n , k, p)
    G = mean(Gi, dims=1)
    Γ = zeros(eltype(Gi),p,k,k)
    D = zeros(eltype(Gi),k, p)
    for j in 1:p
      for i in 1:n
        Γ[j,:,:] += (Gi[i,:,j] .- G[1,:,j]) * giθ[i,:]'
      end
      Γ[j,:,:] ./= n
      D[:,j] = G[1,:,j] - Γ[j,:,:]*inv(Ω)*gn
    end
    return(n,k,p,gn, Ω, D, P)
  end
end

function klm(gi::Function)
  SP = statparts(gi)
  function(θ)
    (n,k,p,gn, Ω, D, P) = SP(θ)
    return n*(gn'*Ω^(-1/2)*P(Ω^(-1/2)*D)*Ω^(-1/2)*gn)[1]
  end
end

import Random
function simulate_ivshare(n,β,γ,ρ)
  z = randn(n, size(γ)[1])
  endo = randn(n, length(β))
  x = z*γ .+ endo
  ξ = rand(Normal(0,sqrt((1.0-ρ^2))),n).+endo[:,1]*ρ
  y = cdf.(Logistic(), x*β .+ ξ)
  return((y=y,x=x,z=z))
end
n = 100
k = 2
iv = 3
β0 = ones(k)
π0 = vcat(5*I,ones(iv-k,k))
ρ = 0.5
Random.seed!(622)
(y,x,z) = simulate_ivshare(n,β0,π0,ρ)

function gi_ivshare(β,y,x,z)
  ξ = quantile.(Logistic(),y) .- x*β
  ξ.*z
end

gi = let y=y, x=x, z=z
  β->gi_ivshare(β,y,x,z)
end

#20 (generic function with 1 method)

@code_warntype statparts(gi)(β0)

MethodInstance for (::var"#15#17"{var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}, var"#P#16"})(::Vector{Float64})
  from (::var"#15#17")(θ) @ Main In[8]:6
Arguments
  #self#::var"#15#17"{var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}, var"#P#16"}
  θ::Vector{Float64}
Locals
  @_3::Union{Nothing, Tuple{Int64, Int64}}
  @_4::Int64
  D::Union{Array{Float64, 3}, Matrix}
  Γ::Union{Array{Float64, 4}, Array{_A, 3} where _A}
  G::Any
  Gi::Any
  gn::Adjoint{Float64, Matrix{Float64}}
  Ω::Matrix{Float64}
  k::Int64
  n::Int64
  p::Int64
  giθ::Matrix{Float64}
  @_15::Union{Nothing, Tuple{Int64, Int64}}
  j::Int64
  i::Int64
Body::Tuple{Int64, Int64, Int64, Adjoint{Float64, Matrix{Float64}}, Matrix{Float64}, Union{Array{Float64, 3}, Matrix}, var"#P#16"}
1 ─ %1  = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│         (giθ = (%1)(θ))
│         (p = Main.length(θ))
│   %4  = Main.size(giθ)::Tuple{Int64, Int64}
│   %5  = Base.indexed_iterate(%4, 1)::Core.PartialStruct(Tuple{Int64, Int64}, Any[Int64, Core.Const(2)])
│         (n = Core.getfield(%5, 1))
│         (@_4 = Core.getfield(%5, 2))
│   %8  = Base.indexed_iterate(%4, 2, @_4::Core.Const(2))::Core.PartialStruct(Tuple{Int64, Int64}, Any[Int64, Core.Const(3)])
│         (k = Core.getfield(%8, 1))
│         (Ω = Main.cov(giθ))
│   %11 = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│   %12 = (%11)(θ)::Matrix{Float64}
│   %13 = (:dims,)::Core.Const((:dims,))
│   %14 = Core.apply_type(Core.NamedTuple, %13)::Core.Const(NamedTuple{(:dims,)})
│   %15 = Core.tuple(1)::Core.Const((1,))
│   %16 = (%14)(%15)::Core.Const((dims = 1,))
│   %17 = Core.kwcall(%16, Main.mean, %12)::Matrix{Float64}
│         (gn = Main.:var"'"(%17))
│   %19 = ForwardDiff.jacobian::Core.Const(ForwardDiff.jacobian)
│   %20 = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│         (Gi = (%19)(%20, θ))
│         (Gi = Main.reshape(Gi, n, k, p))
│   %23 = (:dims,)::Core.Const((:dims,))
│   %24 = Core.apply_type(Core.NamedTuple, %23)::Core.Const(NamedTuple{(:dims,)})
│   %25 = Core.tuple(1)::Core.Const((1,))
│   %26 = (%24)(%25)::Core.Const((dims = 1,))
│         (G = Core.kwcall(%26, Main.mean, Gi))
│   %28 = Main.eltype(Gi)::Any
│   %29 = p::Int64
│   %30 = k::Int64
│         (Γ = Main.zeros(%28, %29, %30, k))
│   %32 = Main.eltype(Gi)::Any
│   %33 = k::Int64
│         (D = Main.zeros(%32, %33, p))
│   %35 = (1:p)::Core.PartialStruct(UnitRange{Int64}, Any[Core.Const(1), Int64])
│         (@_3 = Base.iterate(%35))
│   %37 = (@_3 === nothing)::Bool
│   %38 = Base.not_int(%37)::Bool
└──       goto #7 if not %38
2 ┄ %40 = @_3::Tuple{Int64, Int64}
│         (j = Core.getfield(%40, 1))
│   %42 = Core.getfield(%40, 2)::Int64
│   %43 = (1:n)::Core.PartialStruct(UnitRange{Int64}, Any[Core.Const(1), Int64])
│         (@_15 = Base.iterate(%43))
│   %45 = (@_15 === nothing)::Bool
│   %46 = Base.not_int(%45)::Bool
└──       goto #5 if not %46
3 ┄ %48 = @_15::Tuple{Int64, Int64}
│         (i = Core.getfield(%48, 1))
│   %50 = Core.getfield(%48, 2)::Int64
│   %51 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix
│   %52 = Main.:-::Core.Const(-)
│   %53 = Base.getindex(Gi, i, Main.:(:), j)::Any
│   %54 = Base.getindex(G, 1, Main.:(:), j)::Any
│   %55 = Base.broadcasted(%52, %53, %54)::Any
│   %56 = Base.materialize(%55)::Any
│   %57 = Base.getindex(giθ, i, Main.:(:))::Vector{Float64}
│   %58 = Main.:var"'"(%57)::Adjoint{Float64, Vector{Float64}}
│   %59 = (%56 * %58)::Any
│   %60 = (%51 + %59)::Any
│         Base.setindex!(Γ, %60, j, Main.:(:), Main.:(:))
│         (@_15 = Base.iterate(%43, %50))
│   %63 = (@_15 === nothing)::Bool
│   %64 = Base.not_int(%63)::Bool
└──       goto #5 if not %64
4 ─       goto #3
5 ┄ %67 = Base.dotview(Γ, j, Main.:(:), Main.:(:))::SubArray{_A, 2, P, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}, Base.Slice{Base.OneTo{Int64}}}, true} where {_A, P<:(Array{_A, 3} where _A)}
│   %68 = Main.:/::Core.Const(/)
│   %69 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix
│   %70 = Base.broadcasted(%68, %69, n)::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{2}, Nothing, typeof(/), <:Tuple{Matrix, Int64}}
│         Base.materialize!(%67, %70)
│   %72 = Base.getindex(G, 1, Main.:(:), j)::Any
│   %73 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix
│   %74 = Main.inv(Ω)::Matrix{Float64}
│   %75 = (%73 * %74 * gn)::Any
│   %76 = (%72 - %75)::Any
│         Base.setindex!(D, %76, Main.:(:), j)
│         (@_3 = Base.iterate(%35, %42))
│   %79 = (@_3 === nothing)::Bool
│   %80 = Base.not_int(%79)::Bool
└──       goto #7 if not %80
6 ─       goto #2
7 ┄ %83 = n::Int64
│   %84 = k::Int64
│   %85 = p::Int64
│   %86 = gn::Adjoint{Float64, Matrix{Float64}}
│   %87 = Ω::Matrix{Float64}
│   %88 = D::Union{Array{Float64, 3}, Matrix}
│   %89 = Core.getfield(#self#, :P)::Core.Const(var"#P#16"())
│   %90 = Core.tuple(%83, %84, %85, %86, %87, %88, %89)::Tuple{Int64, Int64, Int64, Adjoint{Float64, Matrix{Float64}}, Matrix{Float64}, Union{Array{Float64, 3}, Matrix}, var"#P#16"}
└──       return %90

@benchmark statparts(gi)(β0)

BenchmarkTools.Trial: 8695 samples with 1 evaluation.
 Range (min … max):  515.108 μs …   5.377 ms  ┊ GC (min … max): 0.00% … 86.93%
 Time  (median):     539.470 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   570.285 μs ± 288.814 μs  ┊ GC (mean ± σ):  3.46% ±  6.04%
        ▄▇█▄                                                     
  ▁▁▁▂▄██████▅▄▃▃▂▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▁▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  515 μs           Histogram: frequency by time          676 μs <
 Memory estimate: 221.16 KiB, allocs estimate: 3910.

function statparts_stable(gi::F) where {F <: Function}
  function P(A) # projection matrix
    A*pinv(A'*A)*A'
  end
  function(θ)
    giθ = gi(θ)
    p = length(θ)
    (n, k) = size(giθ)
    Ω = cov(giθ)
    gn=mean(gi(θ), dims=1)'
    Gi = similar(gn,n,k,p)
    Gi= ForwardDiff.jacobian!(Gi,gi,θ)
    G = mean(Gi, dims=1)
    Γ = zeros(eltype(Gi),p,k,k)
    D = zeros(eltype(Gi),k, p)
    for j in 1:p
      for i in 1:n
        Γ[j,:,:] += (Gi[i,:,j] .- G[1,:,j]) * giθ[i,:]'
      end
      Γ[j,:,:] ./= n
      D[:,j] = G[1,:,j] - Γ[j,:,:]*inv(Ω)*gn
    end
    return(n,k,p,gn, Ω, D, P)
  end
end

function klm_stable(gi::F) where {F <: Function}
  SP = statparts_stable(gi)
  function(θ)
    (n,k,p,gn, Ω, D, P) = SP(θ)
    λ, v = eigen(Ω)
    irΩ = v*diagm(λ.^(-1/2))*v'
    return n*(gn'*irΩ*P(irΩ*D)*irΩ*gn)[1]
  end
end

klm_stable (generic function with 1 method)

@code_warntype statparts_stable(gi)(β0)

MethodInstance for (::var"#22#24"{var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}, var"#P#23"})(::Vector{Float64})
  from (::var"#22#24")(θ) @ Main In[11]:5
Arguments
  #self#::var"#22#24"{var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}, var"#P#23"}
  θ::Vector{Float64}
Locals
  @_3::Union{Nothing, Tuple{Int64, Int64}}
  @_4::Int64
  D::Matrix{Float64}
  Γ::Array{Float64, 3}
  G::Array{Float64, 3}
  Gi::Array{Float64, 3}
  gn::Adjoint{Float64, Matrix{Float64}}
  Ω::Matrix{Float64}
  k::Int64
  n::Int64
  p::Int64
  giθ::Matrix{Float64}
  @_15::Union{Nothing, Tuple{Int64, Int64}}
  j::Int64
  i::Int64
Body::Tuple{Int64, Int64, Int64, Adjoint{Float64, Matrix{Float64}}, Matrix{Float64}, Matrix{Float64}, var"#P#23"}
1 ─ %1  = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│         (giθ = (%1)(θ))
│         (p = Main.length(θ))
│   %4  = Main.size(giθ)::Tuple{Int64, Int64}
│   %5  = Base.indexed_iterate(%4, 1)::Core.PartialStruct(Tuple{Int64, Int64}, Any[Int64, Core.Const(2)])
│         (n = Core.getfield(%5, 1))
│         (@_4 = Core.getfield(%5, 2))
│   %8  = Base.indexed_iterate(%4, 2, @_4::Core.Const(2))::Core.PartialStruct(Tuple{Int64, Int64}, Any[Int64, Core.Const(3)])
│         (k = Core.getfield(%8, 1))
│         (Ω = Main.cov(giθ))
│   %11 = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│   %12 = (%11)(θ)::Matrix{Float64}
│   %13 = (:dims,)::Core.Const((:dims,))
│   %14 = Core.apply_type(Core.NamedTuple, %13)::Core.Const(NamedTuple{(:dims,)})
│   %15 = Core.tuple(1)::Core.Const((1,))
│   %16 = (%14)(%15)::Core.Const((dims = 1,))
│   %17 = Core.kwcall(%16, Main.mean, %12)::Matrix{Float64}
│         (gn = Main.:var"'"(%17))
│         (Gi = Main.similar(gn, n, k, p))
│   %20 = ForwardDiff.jacobian!::Core.Const(ForwardDiff.jacobian!)
│   %21 = Gi::Array{Float64, 3}
│   %22 = Core.getfield(#self#, :gi)::var"#20#21"{Vector{Float64}, Matrix{Float64}, Matrix{Float64}}
│         (Gi = (%20)(%21, %22, θ))
│   %24 = (:dims,)::Core.Const((:dims,))
│   %25 = Core.apply_type(Core.NamedTuple, %24)::Core.Const(NamedTuple{(:dims,)})
│   %26 = Core.tuple(1)::Core.Const((1,))
│   %27 = (%25)(%26)::Core.Const((dims = 1,))
│         (G = Core.kwcall(%27, Main.mean, Gi))
│   %29 = Main.eltype(Gi)::Core.Const(Float64)
│   %30 = p::Int64
│   %31 = k::Int64
│         (Γ = Main.zeros(%29, %30, %31, k))
│   %33 = Main.eltype(Gi)::Core.Const(Float64)
│   %34 = k::Int64
│         (D = Main.zeros(%33, %34, p))
│   %36 = (1:p)::Core.PartialStruct(UnitRange{Int64}, Any[Core.Const(1), Int64])
│         (@_3 = Base.iterate(%36))
│   %38 = (@_3 === nothing)::Bool
│   %39 = Base.not_int(%38)::Bool
└──       goto #7 if not %39
2 ┄ %41 = @_3::Tuple{Int64, Int64}
│         (j = Core.getfield(%41, 1))
│   %43 = Core.getfield(%41, 2)::Int64
│   %44 = (1:n)::Core.PartialStruct(UnitRange{Int64}, Any[Core.Const(1), Int64])
│         (@_15 = Base.iterate(%44))
│   %46 = (@_15 === nothing)::Bool
│   %47 = Base.not_int(%46)::Bool
└──       goto #5 if not %47
3 ┄ %49 = @_15::Tuple{Int64, Int64}
│         (i = Core.getfield(%49, 1))
│   %51 = Core.getfield(%49, 2)::Int64
│   %52 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix{Float64}
│   %53 = Main.:-::Core.Const(-)
│   %54 = Base.getindex(Gi, i, Main.:(:), j)::Vector{Float64}
│   %55 = Base.getindex(G, 1, Main.:(:), j)::Vector{Float64}
│   %56 = Base.broadcasted(%53, %54, %55)::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{1}, Nothing, typeof(-), Tuple{Vector{Float64}, Vector{Float64}}}
│   %57 = Base.materialize(%56)::Vector{Float64}
│   %58 = Base.getindex(giθ, i, Main.:(:))::Vector{Float64}
│   %59 = Main.:var"'"(%58)::Adjoint{Float64, Vector{Float64}}
│   %60 = (%57 * %59)::Matrix{Float64}
│   %61 = (%52 + %60)::Matrix{Float64}
│         Base.setindex!(Γ, %61, j, Main.:(:), Main.:(:))
│         (@_15 = Base.iterate(%44, %51))
│   %64 = (@_15 === nothing)::Bool
│   %65 = Base.not_int(%64)::Bool
└──       goto #5 if not %65
4 ─       goto #3
5 ┄ %68 = Base.dotview(Γ, j, Main.:(:), Main.:(:))::SubArray{Float64, 2, Array{Float64, 3}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}, Base.Slice{Base.OneTo{Int64}}}, true}
│   %69 = Main.:/::Core.Const(/)
│   %70 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix{Float64}
│   %71 = Base.broadcasted(%69, %70, n)::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{2}, Nothing, typeof(/), Tuple{Matrix{Float64}, Int64}}
│         Base.materialize!(%68, %71)
│   %73 = Base.getindex(G, 1, Main.:(:), j)::Vector{Float64}
│   %74 = Base.getindex(Γ, j, Main.:(:), Main.:(:))::Matrix{Float64}
│   %75 = Main.inv(Ω)::Matrix{Float64}
│   %76 = (%74 * %75 * gn)::Matrix{Float64}
│   %77 = (%73 - %76)::Matrix{Float64}
│         Base.setindex!(D, %77, Main.:(:), j)
│         (@_3 = Base.iterate(%36, %43))
│   %80 = (@_3 === nothing)::Bool
│   %81 = Base.not_int(%80)::Bool
└──       goto #7 if not %81
6 ─       goto #2
7 ┄ %84 = n::Int64
│   %85 = k::Int64
│   %86 = p::Int64
│   %87 = gn::Adjoint{Float64, Matrix{Float64}}
│   %88 = Ω::Matrix{Float64}
│   %89 = D::Matrix{Float64}
│   %90 = Core.getfield(#self#, :P)::Core.Const(var"#P#23"())
│   %91 = Core.tuple(%84, %85, %86, %87, %88, %89, %90)::Tuple{Int64, Int64, Int64, Adjoint{Float64, Matrix{Float64}}, Matrix{Float64}, Matrix{Float64}, var"#P#23"}
└──       return %91

@benchmark statparts_stable(gi)(β0)

BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):   95.746 μs …   4.246 ms  ┊ GC (min … max): 0.00% … 95.12%
 Time  (median):     118.570 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   141.684 μs ± 225.594 μs  ┊ GC (mean ± σ):  9.54% ±  5.84%
   ▃▅▇████▇▇▆▆▅▅▄▄▃▂▂▂▁▁ ▁▁                         ▁           ▃
  ▇███████████████████████████▇█▆▆▇▆▆▇▇▆▇▇▆▆▇▇▆▇█▇█▇██▇█▇▇▇▆▆▇▇ █
  95.7 μs       Histogram: log(frequency) by time        282 μs <
 Memory estimate: 173.17 KiB, allocs estimate: 1466.

using Test
@test klm(gi)(β0) ≈ klm_stable(gi)(β0)

Test Passed

• See https://schrimpf.github.io/ARGridBootstrap.jl/assignment.html#statparts for more information

Targeted Optimizations

Profiling

using Profile, ProfileCanvas
function profilehtmlstring()
  buf = IOBuffer()
  show(buf, MIME("text/html"), ProfileCanvas.view(Profile.fetch()))
  s=String(take!(buf))
  println("\n<br><br>\n"*s*"\n<br>\n")
end
function profileiframe(filename="proftmp.html")
  #buf = IOBuffer()
  #show(buf, MIME("text/html"), ProfileCanvas.view(Profile.fetch()))
  #s=String(take!(buf))
  #s=replace(s, "\"" => "&quot" )
  #  HTML("<iframe srcdata=\""*s*"\" width=\"1200\"  height=\"650\"></iframe>\n")
  #ProfileCanvas.html_file(filename)

  #HTML("<iframe src=\""*filename*"\" width=\"1200\"  height=\"650\"></iframe>\n")
  open(filename, "w") do io
    show(io, MIME("text/html"), ProfileCanvas.view(Profile.fetch()))
  end
  HTML("<iframe src=\""*filename*"\" width=\"1500\"  height=\"650\"></iframe>\n")
end

Profile.clear();
Profile.init(n=10^7,delay=0.00001);
klm_stable(gi)(β0)
foo(n) = sum(klm_stable(gi)(β0) for i in 1:n) # loop so it runs longer to get better profile
foo(1) # precompile outside of profiling
@profile foo(1_000)
profileiframe("klmprof.html") # only needed for quarto, just use `@profview foo(1_000)` elsewhere

Reducing Allocations

• Allocating memory is slow
• Reduce allocations by:
  • Using @views instead of slices
  • Pre-allocating and reusing arrays
  • Eliminate dynamic allocations with StaticArrays or similar compile time known size types

Reduced Allocations

function statparts_fast(gi::F) where {F <: Function}
  function P(A::AbstractMatrix) # projection matrix
    A*pinv(A'*A)*A'
  end
  let gi=gi
    function(θ)
      giθ = gi(θ)
      p = length(θ)
      (n, k) = size(giθ)
      Ω = Hermitian(cov(giθ))
      gn=mean(gi(θ), dims=1)'
      iΩgn = Ω \ gn
      Gi = similar(gn,n,k,p)
      ForwardDiff.jacobian!(Gi,gi,θ)
      G = mean(Gi, dims=1)
      Γ = zeros(eltype(Gi),p,k,k)
      D = zeros(eltype(Gi),k, p)
      @inbounds for j in 1:p
        @inbounds for i in 1:n
          @views Γ[j,:,:] .+= (Gi[i,:,j] .- G[1,:,j]) * giθ[i,:]'
        end
        Γ[j,:,:] ./= n
        @views D[:,j] .= G[1,:,j] .- Γ[j,:,:]*iΩgn
      end
      return(n,k,p,gn, Ω, D, P)
    end
  end
end

function klm_fast(gi::F) where {F <: Function}
  SP = statparts_fast(gi)
  function(θ)
    (n,k,p,gn, Ω, D, P) = SP(θ)
    λ, v = eigen(Ω)
    irΩ = v*diagm(λ.^(-1/2))*v'
    return n*(gn'*irΩ*P(irΩ*D)*irΩ*gn)[1]
  end
end

klm_fast (generic function with 1 method)

@benchmark klm_fast(gi)(β0)

BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  68.806 μs …   4.973 ms  ┊ GC (min … max):  0.00% … 89.78%
 Time  (median):     76.166 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   89.047 μs ± 220.860 μs  ┊ GC (mean ± σ):  11.87% ±  4.70%
         ▄▆█▇▆▃▁                                                
  ▁▁▁▂▃▄█████████▇▅▅▄▃▃▂▂▂▂▂▂▂▂▂▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  68.8 μs         Histogram: frequency by time          108 μs <
 Memory estimate: 118.53 KiB, allocs estimate: 501.

@test klm_fast(gi)(β0) ≈ klm(gi)(β0)

Test Passed

StaticArrays

struct Integrator{Tx, Tw}
    x::Tx
    w::Tw
end

function Integrator(dx::Distribution, n=100)
    x = [rand(dx) for _ in 1:n]
    w = Base.Iterators.Repeated(1/n)
    Integrator(x,w)
end

(∫::Integrator)(f) = sum((xw)->f(xw[1])*xw[2], zip(∫.x, ∫.w))

function share(δ, Σ, x, ∫)
  J,K = size(x)
  (length(δ) == J) || error("length(δ)=$(length(δ)) != size(x,1)=$J")
  (K,K) == size(Σ) || error("size(x,2)=$K != size(Σ)=$(size(Σ))")
  xΣ = x*Σ
  function shareν(ν)
    s = δ + xΣ*ν
    smax=max(0,maximum(s))
    s = s .- smax
    s = exp.(s)
    s *= 1/(sum(s) + exp(0-smax))
    return(s)
  end
  return(∫(shareν))
end

share (generic function with 1 method)

Heap allocated arrays

J = 10
K = 5
δ = rand(J)
X = randn(J,K)
Σ = I + zeros(K,K)
∫ = Integrator(MvNormal(zeros(K),I))

@benchmark share(δ,Σ,X,∫)

BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):  51.257 μs …   4.396 ms  ┊ GC (min … max): 0.00% … 96.60%
 Time  (median):     73.012 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   85.758 μs ± 182.146 μs  ┊ GC (mean ± σ):  9.59% ±  4.44%
             ▃▄▃▆▅▆█▇▄▂                                         
  ▁▁▂▂▂▂▃▃▄▅▇███████████▆▄▃▃▃▂▃▃▂▂▂▂▂▃▃▃▂▃▃▃▃▂▃▂▂▂▂▃▂▂▂▂▂▂▁▁▁▁ ▃
  51.3 μs         Histogram: frequency by time          125 μs <
 Memory estimate: 98.78 KiB, allocs estimate: 700.

Stack allocated arrays

using StaticArrays
sδ = SVector{J}(δ)
sΣ = SMatrix{K,K}(Σ)
sX = SMatrix{J,K}(X)
nd = length(∫.x)
iw = SVector{nd}(fill(1/nd,nd))
ix = [SVector{K}(x) for x ∈ ∫.x]
s∫ = Integrator(ix,iw)

@benchmark share(sδ,sΣ,sX,s∫)

BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (min … max):   9.665 μs … 34.069 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):      9.816 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   10.148 μs ±  1.266 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▂▇█▇█▆▃▂▂▁▁        ▁▅▆▅▅▃▁   ▁▄▄▄▄▂       ▁          ▁▁▁    ▃
  ██████████████▆▅▅▄▄█████████▅████████▇▆▅▆███▇▆▃▃▃▁▁▁▅████▆▆ █
  9.66 μs      Histogram: log(frequency) by time      11.6 μs <
 Memory estimate: 96 bytes, allocs estimate: 1.

Memory Considerations

• Computations are faster when operating accessing contiguous chunks of memory
  • Access arrays by columns
• Data moves RAM ⇒ Cache ⇒ CPU Registers
  • RAM ⇒ Cache much slower than Cache ⇒ CPU Registers
  • Can see big benefits from small code that fits on cache
  • CPU prefetches data from RAM ⇒ Cache, by predicting what will be needed
  • Needed data not on Cache when needed is a “cache miss”, these are costly
  • Predictable code without branches and accessing contiguous memory helps avoid caches misses

Single Instruction, Multiple Data

• CPUs can perform the same operation on multiple numbers at the same time
  • “Vectorized instructions”
  • Current generation x86 CPUs have 512 bit registers, can operate on 8 Float64 values at once
• Compiler tries to use vectorized instructions when possible
  • Loop of fixed length (no break or continue)
  • No branching
  • Re-ordering allowed (indicate with @simd)

SISD

function slowdot(a,b)
  out = one(promote_type(eltype(a),eltype(b)))
  for i in eachindex(a)
    out += a[i]*b[i]
  end
  out
end

n = 1_000
a = rand(n)
b = rand(n)
@code_llvm slowdot(a,b)

;  @ In[24]:1 within `slowdot`
define double @julia_slowdot_4835({}* noundef nonnull align 16 dereferenceable(40) %0, {}* noundef nonnull align 16 dereferenceable(40) %1) #0 {
top:
;  @ In[24]:3 within `slowdot`
; ┌ @ abstractarray.jl:321 within `eachindex`
; │┌ @ abstractarray.jl:137 within `axes1`
; ││┌ @ abstractarray.jl:98 within `axes`
; │││┌ @ array.jl:191 within `size`
      %2 = bitcast {}* %0 to { i8*, i64, i16, i16, i32 }*
      %arraylen_ptr = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %2, i64 0, i32 1
      %arraylen = load i64, i64* %arraylen_ptr, align 8
; └└└└
; ┌ @ range.jl:897 within `iterate`
; │┌ @ range.jl:672 within `isempty`
; ││┌ @ operators.jl:378 within `>`
; │││┌ @ int.jl:83 within `<`
      %.not.not = icmp eq i64 %arraylen, 0
; └└└└
  br i1 %.not.not, label %L31, label %L13.preheader

L13.preheader:                                    ; preds = %top
  %3 = bitcast {}* %1 to { i8*, i64, i16, i16, i32 }*
  %arraylen_ptr8 = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %3, i64 0, i32 1
  %arraylen9 = load i64, i64* %arraylen_ptr8, align 8
  %4 = bitcast {}* %0 to double**
  %arrayptr23 = load double*, double** %4, align 8
  %5 = bitcast {}* %1 to double**
  %arrayptr1524 = load double*, double** %5, align 8
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %smin = call i64 @llvm.smin.i64(i64 %arraylen9, i64 0)
   %6 = sub i64 %arraylen9, %smin
   %smax = call i64 @llvm.smax.i64(i64 %smin, i64 -1)
   %7 = add nsw i64 %smax, 1
   %8 = mul nuw nsw i64 %6, %7
   %exit.mainloop.at = call i64 @llvm.umin.i64(i64 %arraylen, i64 %8)
   %.not = icmp eq i64 %exit.mainloop.at, 0
   br i1 %.not, label %main.pseudo.exit, label %idxend13.preheader

idxend13.preheader:                               ; preds = %L13.preheader
; └
;  @ In[24]:5 within `slowdot`
  %9 = add nsw i64 %exit.mainloop.at, -1
  %xtraiter = and i64 %exit.mainloop.at, 7
  %10 = icmp ult i64 %9, 7
  br i1 %10, label %main.exit.selector.unr-lcssa, label %idxend13.preheader.new

idxend13.preheader.new:                           ; preds = %idxend13.preheader
  %unroll_iter = and i64 %exit.mainloop.at, 9223372036854775800
  br label %idxend13

L31:                                              ; preds = %idxend13.postloop, %main.exit.selector, %top
  %value_phi20 = phi double [ 1.000000e+00, %top ], [ %.lcssa50, %main.exit.selector ], [ %63, %idxend13.postloop ]
  ret double %value_phi20

oob:                                              ; preds = %L13.postloop
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %errorbox = alloca i64, align 8
   store i64 %value_phi3.postloop, i64* %errorbox, align 8
   call void @ijl_bounds_error_ints({}* %0, i64* nonnull %errorbox, i64 1)
   unreachable

oob11:                                            ; preds = %idxend.postloop
   %errorbox12 = alloca i64, align 8
   store i64 %value_phi3.postloop, i64* %errorbox12, align 8
   call void @ijl_bounds_error_ints({}* %1, i64* nonnull %errorbox12, i64 1)
   unreachable

idxend13:                                         ; preds = %idxend13, %idxend13.preheader.new
   %value_phi3 = phi i64 [ 1, %idxend13.preheader.new ], [ %50, %idxend13 ]
   %value_phi5 = phi double [ 1.000000e+00, %idxend13.preheader.new ], [ %49, %idxend13 ]
   %niter = phi i64 [ 0, %idxend13.preheader.new ], [ %niter.next.7, %idxend13 ]
   %11 = add nsw i64 %value_phi3, -1
   %12 = getelementptr inbounds double, double* %arrayptr23, i64 %11
   %arrayref = load double, double* %12, align 8
   %13 = getelementptr inbounds double, double* %arrayptr1524, i64 %11
   %arrayref16 = load double, double* %13, align 8
; └
; ┌ @ float.jl:411 within `*`
   %14 = fmul double %arrayref, %arrayref16
; └
; ┌ @ float.jl:409 within `+`
   %15 = fadd double %value_phi5, %14
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %16 = add nuw nsw i64 %value_phi3, 1
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %17 = getelementptr inbounds double, double* %arrayptr23, i64 %value_phi3
   %arrayref.1 = load double, double* %17, align 8
   %18 = getelementptr inbounds double, double* %arrayptr1524, i64 %value_phi3
   %arrayref16.1 = load double, double* %18, align 8
; └
; ┌ @ float.jl:411 within `*`
   %19 = fmul double %arrayref.1, %arrayref16.1
; └
; ┌ @ float.jl:409 within `+`
   %20 = fadd double %15, %19
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %21 = add nuw nsw i64 %value_phi3, 2
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %22 = getelementptr inbounds double, double* %arrayptr23, i64 %16
   %arrayref.2 = load double, double* %22, align 8
   %23 = getelementptr inbounds double, double* %arrayptr1524, i64 %16
   %arrayref16.2 = load double, double* %23, align 8
; └
; ┌ @ float.jl:411 within `*`
   %24 = fmul double %arrayref.2, %arrayref16.2
; └
; ┌ @ float.jl:409 within `+`
   %25 = fadd double %20, %24
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %26 = add nuw nsw i64 %value_phi3, 3
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %27 = getelementptr inbounds double, double* %arrayptr23, i64 %21
   %arrayref.3 = load double, double* %27, align 8
   %28 = getelementptr inbounds double, double* %arrayptr1524, i64 %21
   %arrayref16.3 = load double, double* %28, align 8
; └
; ┌ @ float.jl:411 within `*`
   %29 = fmul double %arrayref.3, %arrayref16.3
; └
; ┌ @ float.jl:409 within `+`
   %30 = fadd double %25, %29
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %31 = add nuw nsw i64 %value_phi3, 4
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %32 = getelementptr inbounds double, double* %arrayptr23, i64 %26
   %arrayref.4 = load double, double* %32, align 8
   %33 = getelementptr inbounds double, double* %arrayptr1524, i64 %26
   %arrayref16.4 = load double, double* %33, align 8
; └
; ┌ @ float.jl:411 within `*`
   %34 = fmul double %arrayref.4, %arrayref16.4
; └
; ┌ @ float.jl:409 within `+`
   %35 = fadd double %30, %34
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %36 = add nuw nsw i64 %value_phi3, 5
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %37 = getelementptr inbounds double, double* %arrayptr23, i64 %31
   %arrayref.5 = load double, double* %37, align 8
   %38 = getelementptr inbounds double, double* %arrayptr1524, i64 %31
   %arrayref16.5 = load double, double* %38, align 8
; └
; ┌ @ float.jl:411 within `*`
   %39 = fmul double %arrayref.5, %arrayref16.5
; └
; ┌ @ float.jl:409 within `+`
   %40 = fadd double %35, %39
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %41 = add nuw nsw i64 %value_phi3, 6
; └
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %42 = getelementptr inbounds double, double* %arrayptr23, i64 %36
   %arrayref.6 = load double, double* %42, align 8
   %43 = getelementptr inbounds double, double* %arrayptr1524, i64 %36
   %arrayref16.6 = load double, double* %43, align 8
; └
; ┌ @ float.jl:411 within `*`
   %44 = fmul double %arrayref.6, %arrayref16.6
; └
; ┌ @ float.jl:409 within `+`
   %45 = fadd double %40, %44
; └
; ┌ @ essentials.jl:13 within `getindex`
   %46 = getelementptr inbounds double, double* %arrayptr23, i64 %41
   %arrayref.7 = load double, double* %46, align 8
   %47 = getelementptr inbounds double, double* %arrayptr1524, i64 %41
   %arrayref16.7 = load double, double* %47, align 8
; └
; ┌ @ float.jl:411 within `*`
   %48 = fmul double %arrayref.7, %arrayref16.7
; └
; ┌ @ float.jl:409 within `+`
   %49 = fadd double %45, %48
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %50 = add nuw nsw i64 %value_phi3, 8
; └
  %niter.next.7 = add nuw i64 %niter, 8
  %niter.ncmp.7.not = icmp eq i64 %niter.next.7, %unroll_iter
  br i1 %niter.ncmp.7.not, label %main.exit.selector.unr-lcssa.loopexit, label %idxend13

main.exit.selector.unr-lcssa.loopexit:            ; preds = %idxend13
; ┌ @ range.jl:901 within `iterate`
   %51 = add nuw nsw i64 %value_phi3, 7
; └
  br label %main.exit.selector.unr-lcssa

main.exit.selector.unr-lcssa:                     ; preds = %main.exit.selector.unr-lcssa.loopexit, %idxend13.preheader
  %value_phi3.lcssa.ph = phi i64 [ undef, %idxend13.preheader ], [ %51, %main.exit.selector.unr-lcssa.loopexit ]
  %.lcssa50.ph = phi double [ undef, %idxend13.preheader ], [ %49, %main.exit.selector.unr-lcssa.loopexit ]
  %.lcssa49.ph = phi i64 [ undef, %idxend13.preheader ], [ %50, %main.exit.selector.unr-lcssa.loopexit ]
  %value_phi3.unr = phi i64 [ 1, %idxend13.preheader ], [ %50, %main.exit.selector.unr-lcssa.loopexit ]
  %value_phi5.unr = phi double [ 1.000000e+00, %idxend13.preheader ], [ %49, %main.exit.selector.unr-lcssa.loopexit ]
  %lcmp.mod.not = icmp eq i64 %xtraiter, 0
  br i1 %lcmp.mod.not, label %main.exit.selector, label %idxend13.epil

idxend13.epil:                                    ; preds = %idxend13.epil, %main.exit.selector.unr-lcssa
  %value_phi3.epil = phi i64 [ %57, %idxend13.epil ], [ %value_phi3.unr, %main.exit.selector.unr-lcssa ]
  %value_phi5.epil = phi double [ %56, %idxend13.epil ], [ %value_phi5.unr, %main.exit.selector.unr-lcssa ]
  %epil.iter = phi i64 [ %epil.iter.next, %idxend13.epil ], [ 0, %main.exit.selector.unr-lcssa ]
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %52 = add nsw i64 %value_phi3.epil, -1
   %53 = getelementptr inbounds double, double* %arrayptr23, i64 %52
   %arrayref.epil = load double, double* %53, align 8
   %54 = getelementptr inbounds double, double* %arrayptr1524, i64 %52
   %arrayref16.epil = load double, double* %54, align 8
; └
; ┌ @ float.jl:411 within `*`
   %55 = fmul double %arrayref.epil, %arrayref16.epil
; └
; ┌ @ float.jl:409 within `+`
   %56 = fadd double %value_phi5.epil, %55
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %57 = add nuw nsw i64 %value_phi3.epil, 1
; └
  %epil.iter.next = add i64 %epil.iter, 1
  %epil.iter.cmp.not = icmp eq i64 %epil.iter.next, %xtraiter
  br i1 %epil.iter.cmp.not, label %main.exit.selector, label %idxend13.epil

main.exit.selector:                               ; preds = %idxend13.epil, %main.exit.selector.unr-lcssa
  %value_phi3.lcssa = phi i64 [ %value_phi3.lcssa.ph, %main.exit.selector.unr-lcssa ], [ %value_phi3.epil, %idxend13.epil ]
;  @ In[24]:4 within `slowdot`
; ┌ @ float.jl:409 within `+`
   %.lcssa50 = phi double [ %.lcssa50.ph, %main.exit.selector.unr-lcssa ], [ %56, %idxend13.epil ]
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
   %.lcssa49 = phi i64 [ %.lcssa49.ph, %main.exit.selector.unr-lcssa ], [ %57, %idxend13.epil ]
; └
  %58 = icmp ult i64 %value_phi3.lcssa, %arraylen
  br i1 %58, label %main.pseudo.exit, label %L31

main.pseudo.exit:                                 ; preds = %main.exit.selector, %L13.preheader
  %value_phi3.copy = phi i64 [ 1, %L13.preheader ], [ %.lcssa49, %main.exit.selector ]
  %value_phi5.copy = phi double [ 1.000000e+00, %L13.preheader ], [ %.lcssa50, %main.exit.selector ]
  br label %L13.postloop

L13.postloop:                                     ; preds = %idxend13.postloop, %main.pseudo.exit
  %value_phi3.postloop = phi i64 [ %64, %idxend13.postloop ], [ %value_phi3.copy, %main.pseudo.exit ]
  %value_phi5.postloop = phi double [ %63, %idxend13.postloop ], [ %value_phi5.copy, %main.pseudo.exit ]
;  @ In[24]:4 within `slowdot`
; ┌ @ essentials.jl:13 within `getindex`
   %59 = add i64 %value_phi3.postloop, -1
   %inbounds.postloop = icmp ult i64 %59, %arraylen
   br i1 %inbounds.postloop, label %idxend.postloop, label %oob

idxend.postloop:                                  ; preds = %L13.postloop
   %inbounds10.postloop = icmp ult i64 %59, %arraylen9
   br i1 %inbounds10.postloop, label %idxend13.postloop, label %oob11

idxend13.postloop:                                ; preds = %idxend.postloop
   %60 = getelementptr inbounds double, double* %arrayptr23, i64 %59
   %arrayref.postloop = load double, double* %60, align 8
   %61 = getelementptr inbounds double, double* %arrayptr1524, i64 %59
   %arrayref16.postloop = load double, double* %61, align 8
; └
; ┌ @ float.jl:411 within `*`
   %62 = fmul double %arrayref.postloop, %arrayref16.postloop
; └
; ┌ @ float.jl:409 within `+`
   %63 = fadd double %value_phi5.postloop, %62
; └
;  @ In[24]:5 within `slowdot`
; ┌ @ range.jl:901 within `iterate`
; │┌ @ promotion.jl:521 within `==`
    %.not.not25.postloop = icmp eq i64 %value_phi3.postloop, %arraylen
; │└
   %64 = add nuw nsw i64 %value_phi3.postloop, 1
; └
  br i1 %.not.not25.postloop, label %L31, label %L13.postloop
}

SIMD

function fastdot(a,b)
  out = one(promote_type(eltype(a),eltype(b)))
  @simd for i in eachindex(a)
    out += a[i]*b[i]
  end
  out
end

@code_llvm fastdot(a,b)

;  @ In[25]:1 within `fastdot`
define double @julia_fastdot_4859({}* noundef nonnull align 16 dereferenceable(40) %0, {}* noundef nonnull align 16 dereferenceable(40) %1) #0 {
top:
;  @ In[25]:3 within `fastdot`
; ┌ @ simdloop.jl:69 within `macro expansion`
; │┌ @ abstractarray.jl:321 within `eachindex`
; ││┌ @ abstractarray.jl:137 within `axes1`
; │││┌ @ abstractarray.jl:98 within `axes`
; ││││┌ @ array.jl:191 within `size`
       %2 = bitcast {}* %0 to { i8*, i64, i16, i16, i32 }*
       %arraylen_ptr = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %2, i64 0, i32 1
       %arraylen = load i64, i64* %arraylen_ptr, align 8
; │└└└└
; │ @ simdloop.jl:72 within `macro expansion`
; │┌ @ int.jl:83 within `<`
    %.not = icmp eq i64 %arraylen, 0
; │└
   br i1 %.not, label %L32, label %L13.lr.ph

L13.lr.ph:                                        ; preds = %top
   %3 = bitcast {}* %1 to { i8*, i64, i16, i16, i32 }*
   %arraylen_ptr4 = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %3, i64 0, i32 1
   %arraylen5 = load i64, i64* %arraylen_ptr4, align 8
   %4 = bitcast {}* %0 to double**
   %arrayptr17 = load double*, double** %4, align 8
   %5 = bitcast {}* %1 to double**
   %arrayptr1118 = load double*, double** %5, align 8
; │ @ simdloop.jl:75 within `macro expansion`
   %6 = add i64 %arraylen5, -9223372036854775807
   %smax = call i64 @llvm.smax.i64(i64 %6, i64 0)
   %7 = sub i64 %arraylen5, %smax
   %arraylen5.lobit = ashr i64 %arraylen5, 63
   %8 = add nsw i64 %arraylen5.lobit, 1
   %9 = mul nuw nsw i64 %7, %8
   %smin27 = call i64 @llvm.smin.i64(i64 %arraylen, i64 %9)
   %exit.mainloop.at = call i64 @llvm.smax.i64(i64 %smin27, i64 0)
   %.not39 = icmp slt i64 %smin27, 1
   br i1 %.not39, label %main.pseudo.exit, label %idxend9.preheader

idxend9.preheader:                                ; preds = %L13.lr.ph
   %min.iters.check = icmp ult i64 %exit.mainloop.at, 16
   br i1 %min.iters.check, label %scalar.ph, label %vector.ph

vector.ph:                                        ; preds = %idxend9.preheader
   %n.vec = and i64 %exit.mainloop.at, 9223372036854775792
   %10 = add nsw i64 %n.vec, -16
   %11 = lshr exact i64 %10, 4
   %12 = add nuw nsw i64 %11, 1
   %xtraiter = and i64 %12, 1
   %13 = icmp eq i64 %10, 0
   br i1 %13, label %middle.block.unr-lcssa, label %vector.ph.new

vector.ph.new:                                    ; preds = %vector.ph
   %unroll_iter = and i64 %12, 2305843009213693950
   br label %vector.body

vector.body:                                      ; preds = %vector.body, %vector.ph.new
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ int.jl:87 within `+`
     %index = phi i64 [ 0, %vector.ph.new ], [ %index.next.1, %vector.body ]
     %vec.phi = phi <4 x double> [ <double 1.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph.new ], [ %58, %vector.body ]
     %vec.phi44 = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph.new ], [ %59, %vector.body ]
     %vec.phi45 = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph.new ], [ %60, %vector.body ]
     %vec.phi46 = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph.new ], [ %61, %vector.body ]
     %niter = phi i64 [ 0, %vector.ph.new ], [ %niter.next.1, %vector.body ]
; │└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %14 = getelementptr inbounds double, double* %arrayptr17, i64 %index
    %15 = bitcast double* %14 to <4 x double>*
    %wide.load = load <4 x double>, <4 x double>* %15, align 8
    %16 = getelementptr inbounds double, double* %14, i64 4
    %17 = bitcast double* %16 to <4 x double>*
    %wide.load47 = load <4 x double>, <4 x double>* %17, align 8
    %18 = getelementptr inbounds double, double* %14, i64 8
    %19 = bitcast double* %18 to <4 x double>*
    %wide.load48 = load <4 x double>, <4 x double>* %19, align 8
    %20 = getelementptr inbounds double, double* %14, i64 12
    %21 = bitcast double* %20 to <4 x double>*
    %wide.load49 = load <4 x double>, <4 x double>* %21, align 8
    %22 = getelementptr inbounds double, double* %arrayptr1118, i64 %index
    %23 = bitcast double* %22 to <4 x double>*
    %wide.load50 = load <4 x double>, <4 x double>* %23, align 8
    %24 = getelementptr inbounds double, double* %22, i64 4
    %25 = bitcast double* %24 to <4 x double>*
    %wide.load51 = load <4 x double>, <4 x double>* %25, align 8
    %26 = getelementptr inbounds double, double* %22, i64 8
    %27 = bitcast double* %26 to <4 x double>*
    %wide.load52 = load <4 x double>, <4 x double>* %27, align 8
    %28 = getelementptr inbounds double, double* %22, i64 12
    %29 = bitcast double* %28 to <4 x double>*
    %wide.load53 = load <4 x double>, <4 x double>* %29, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %30 = fmul contract <4 x double> %wide.load, %wide.load50
    %31 = fmul contract <4 x double> %wide.load47, %wide.load51
    %32 = fmul contract <4 x double> %wide.load48, %wide.load52
    %33 = fmul contract <4 x double> %wide.load49, %wide.load53
; │└
; │┌ @ float.jl:409 within `+`
    %34 = fadd reassoc contract <4 x double> %vec.phi, %30
    %35 = fadd reassoc contract <4 x double> %vec.phi44, %31
    %36 = fadd reassoc contract <4 x double> %vec.phi45, %32
    %37 = fadd reassoc contract <4 x double> %vec.phi46, %33
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ int.jl:87 within `+`
     %index.next = or i64 %index, 16
; │└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %38 = getelementptr inbounds double, double* %arrayptr17, i64 %index.next
    %39 = bitcast double* %38 to <4 x double>*
    %wide.load.1 = load <4 x double>, <4 x double>* %39, align 8
    %40 = getelementptr inbounds double, double* %38, i64 4
    %41 = bitcast double* %40 to <4 x double>*
    %wide.load47.1 = load <4 x double>, <4 x double>* %41, align 8
    %42 = getelementptr inbounds double, double* %38, i64 8
    %43 = bitcast double* %42 to <4 x double>*
    %wide.load48.1 = load <4 x double>, <4 x double>* %43, align 8
    %44 = getelementptr inbounds double, double* %38, i64 12
    %45 = bitcast double* %44 to <4 x double>*
    %wide.load49.1 = load <4 x double>, <4 x double>* %45, align 8
    %46 = getelementptr inbounds double, double* %arrayptr1118, i64 %index.next
    %47 = bitcast double* %46 to <4 x double>*
    %wide.load50.1 = load <4 x double>, <4 x double>* %47, align 8
    %48 = getelementptr inbounds double, double* %46, i64 4
    %49 = bitcast double* %48 to <4 x double>*
    %wide.load51.1 = load <4 x double>, <4 x double>* %49, align 8
    %50 = getelementptr inbounds double, double* %46, i64 8
    %51 = bitcast double* %50 to <4 x double>*
    %wide.load52.1 = load <4 x double>, <4 x double>* %51, align 8
    %52 = getelementptr inbounds double, double* %46, i64 12
    %53 = bitcast double* %52 to <4 x double>*
    %wide.load53.1 = load <4 x double>, <4 x double>* %53, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %54 = fmul contract <4 x double> %wide.load.1, %wide.load50.1
    %55 = fmul contract <4 x double> %wide.load47.1, %wide.load51.1
    %56 = fmul contract <4 x double> %wide.load48.1, %wide.load52.1
    %57 = fmul contract <4 x double> %wide.load49.1, %wide.load53.1
; │└
; │┌ @ float.jl:409 within `+`
    %58 = fadd reassoc contract <4 x double> %34, %54
    %59 = fadd reassoc contract <4 x double> %35, %55
    %60 = fadd reassoc contract <4 x double> %36, %56
    %61 = fadd reassoc contract <4 x double> %37, %57
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ int.jl:87 within `+`
     %index.next.1 = add nuw i64 %index, 32
     %niter.next.1 = add i64 %niter, 2
     %niter.ncmp.1 = icmp eq i64 %niter.next.1, %unroll_iter
     br i1 %niter.ncmp.1, label %middle.block.unr-lcssa, label %vector.body

middle.block.unr-lcssa:                           ; preds = %vector.body, %vector.ph
     %.lcssa63.ph = phi <4 x double> [ undef, %vector.ph ], [ %58, %vector.body ]
     %.lcssa62.ph = phi <4 x double> [ undef, %vector.ph ], [ %59, %vector.body ]
     %.lcssa61.ph = phi <4 x double> [ undef, %vector.ph ], [ %60, %vector.body ]
     %.lcssa60.ph = phi <4 x double> [ undef, %vector.ph ], [ %61, %vector.body ]
     %index.unr = phi i64 [ 0, %vector.ph ], [ %index.next.1, %vector.body ]
     %vec.phi.unr = phi <4 x double> [ <double 1.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph ], [ %58, %vector.body ]
     %vec.phi44.unr = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph ], [ %59, %vector.body ]
     %vec.phi45.unr = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph ], [ %60, %vector.body ]
     %vec.phi46.unr = phi <4 x double> [ <double -0.000000e+00, double -0.000000e+00, double -0.000000e+00, double -0.000000e+00>, %vector.ph ], [ %61, %vector.body ]
     %lcmp.mod.not = icmp eq i64 %xtraiter, 0
     br i1 %lcmp.mod.not, label %middle.block, label %vector.body.epil.preheader

vector.body.epil.preheader:                       ; preds = %middle.block.unr-lcssa
; │└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %62 = getelementptr inbounds double, double* %arrayptr17, i64 %index.unr
    %63 = bitcast double* %62 to <4 x double>*
    %wide.load.epil = load <4 x double>, <4 x double>* %63, align 8
    %64 = getelementptr inbounds double, double* %62, i64 4
    %65 = bitcast double* %64 to <4 x double>*
    %wide.load47.epil = load <4 x double>, <4 x double>* %65, align 8
    %66 = getelementptr inbounds double, double* %62, i64 8
    %67 = bitcast double* %66 to <4 x double>*
    %wide.load48.epil = load <4 x double>, <4 x double>* %67, align 8
    %68 = getelementptr inbounds double, double* %62, i64 12
    %69 = bitcast double* %68 to <4 x double>*
    %wide.load49.epil = load <4 x double>, <4 x double>* %69, align 8
    %70 = getelementptr inbounds double, double* %arrayptr1118, i64 %index.unr
    %71 = bitcast double* %70 to <4 x double>*
    %wide.load50.epil = load <4 x double>, <4 x double>* %71, align 8
    %72 = getelementptr inbounds double, double* %70, i64 4
    %73 = bitcast double* %72 to <4 x double>*
    %wide.load51.epil = load <4 x double>, <4 x double>* %73, align 8
    %74 = getelementptr inbounds double, double* %70, i64 8
    %75 = bitcast double* %74 to <4 x double>*
    %wide.load52.epil = load <4 x double>, <4 x double>* %75, align 8
    %76 = getelementptr inbounds double, double* %70, i64 12
    %77 = bitcast double* %76 to <4 x double>*
    %wide.load53.epil = load <4 x double>, <4 x double>* %77, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %78 = fmul contract <4 x double> %wide.load.epil, %wide.load50.epil
    %79 = fmul contract <4 x double> %wide.load47.epil, %wide.load51.epil
    %80 = fmul contract <4 x double> %wide.load48.epil, %wide.load52.epil
    %81 = fmul contract <4 x double> %wide.load49.epil, %wide.load53.epil
; │└
; │┌ @ float.jl:409 within `+`
    %82 = fadd reassoc contract <4 x double> %vec.phi.unr, %78
    %83 = fadd reassoc contract <4 x double> %vec.phi44.unr, %79
    %84 = fadd reassoc contract <4 x double> %vec.phi45.unr, %80
    %85 = fadd reassoc contract <4 x double> %vec.phi46.unr, %81
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   br label %middle.block

middle.block:                                     ; preds = %vector.body.epil.preheader, %middle.block.unr-lcssa
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ float.jl:409 within `+`
    %.lcssa63 = phi <4 x double> [ %.lcssa63.ph, %middle.block.unr-lcssa ], [ %82, %vector.body.epil.preheader ]
    %.lcssa62 = phi <4 x double> [ %.lcssa62.ph, %middle.block.unr-lcssa ], [ %83, %vector.body.epil.preheader ]
    %.lcssa61 = phi <4 x double> [ %.lcssa61.ph, %middle.block.unr-lcssa ], [ %84, %vector.body.epil.preheader ]
    %.lcssa60 = phi <4 x double> [ %.lcssa60.ph, %middle.block.unr-lcssa ], [ %85, %vector.body.epil.preheader ]
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %bin.rdx = fadd reassoc contract <4 x double> %.lcssa62, %.lcssa63
   %bin.rdx54 = fadd reassoc contract <4 x double> %.lcssa61, %bin.rdx
   %bin.rdx55 = fadd reassoc contract <4 x double> %.lcssa60, %bin.rdx54
   %86 = call reassoc contract double @llvm.vector.reduce.fadd.v4f64(double -0.000000e+00, <4 x double> %bin.rdx55)
   %cmp.n = icmp eq i64 %exit.mainloop.at, %n.vec
   br i1 %cmp.n, label %main.exit.selector, label %scalar.ph

scalar.ph:                                        ; preds = %middle.block, %idxend9.preheader
   %bc.resume.val = phi i64 [ %n.vec, %middle.block ], [ 0, %idxend9.preheader ]
   %bc.merge.rdx = phi double [ %86, %middle.block ], [ 1.000000e+00, %idxend9.preheader ]
   br label %idxend9

L32:                                              ; preds = %idxend9.postloop, %main.exit.selector, %top
   %value_phi13 = phi double [ 1.000000e+00, %top ], [ %.lcssa43, %main.exit.selector ], [ %97, %idxend9.postloop ]
; │ @ simdloop.jl:86 within `macro expansion`
   ret double %value_phi13

oob:                                              ; preds = %L13.postloop
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %errorbox = alloca i64, align 8
    store i64 %93, i64* %errorbox, align 8
    call void @ijl_bounds_error_ints({}* %0, i64* nonnull %errorbox, i64 1)
    unreachable

oob7:                                             ; preds = %idxend.postloop
    %errorbox8 = alloca i64, align 8
    store i64 %93, i64* %errorbox8, align 8
    call void @ijl_bounds_error_ints({}* %1, i64* nonnull %errorbox8, i64 1)
    unreachable

idxend9:                                          ; preds = %idxend9, %scalar.ph
    %value_phi124 = phi i64 [ %87, %idxend9 ], [ %bc.resume.val, %scalar.ph ]
    %value_phi23 = phi double [ %91, %idxend9 ], [ %bc.merge.rdx, %scalar.ph ]
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ int.jl:87 within `+`
     %87 = add nuw nsw i64 %value_phi124, 1
; │└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %88 = getelementptr inbounds double, double* %arrayptr17, i64 %value_phi124
    %arrayref = load double, double* %88, align 8
    %89 = getelementptr inbounds double, double* %arrayptr1118, i64 %value_phi124
    %arrayref12 = load double, double* %89, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %90 = fmul contract double %arrayref, %arrayref12
; │└
; │┌ @ float.jl:409 within `+`
    %91 = fadd reassoc contract double %value_phi23, %90
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %exitcond.not = icmp eq i64 %87, %exit.mainloop.at
   br i1 %exitcond.not, label %main.exit.selector, label %idxend9

main.exit.selector:                               ; preds = %idxend9, %middle.block
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ float.jl:409 within `+`
    %.lcssa43 = phi double [ %86, %middle.block ], [ %91, %idxend9 ]
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %92 = icmp ult i64 %exit.mainloop.at, %arraylen
   br i1 %92, label %main.pseudo.exit, label %L32

main.pseudo.exit:                                 ; preds = %main.exit.selector, %L13.lr.ph
   %value_phi23.copy = phi double [ 1.000000e+00, %L13.lr.ph ], [ %.lcssa43, %main.exit.selector ]
   br label %L13.postloop

L13.postloop:                                     ; preds = %idxend9.postloop, %main.pseudo.exit
   %value_phi124.postloop = phi i64 [ %exit.mainloop.at, %main.pseudo.exit ], [ %93, %idxend9.postloop ]
   %value_phi23.postloop = phi double [ %value_phi23.copy, %main.pseudo.exit ], [ %97, %idxend9.postloop ]
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ int.jl:87 within `+`
     %93 = add nuw nsw i64 %value_phi124.postloop, 1
; │└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[25]:4
; │┌ @ essentials.jl:13 within `getindex`
    %inbounds.postloop = icmp ult i64 %value_phi124.postloop, %arraylen
    br i1 %inbounds.postloop, label %idxend.postloop, label %oob

idxend.postloop:                                  ; preds = %L13.postloop
    %inbounds6.postloop = icmp ult i64 %value_phi124.postloop, %arraylen5
    br i1 %inbounds6.postloop, label %idxend9.postloop, label %oob7

idxend9.postloop:                                 ; preds = %idxend.postloop
    %94 = getelementptr inbounds double, double* %arrayptr17, i64 %value_phi124.postloop
    %arrayref.postloop = load double, double* %94, align 8
    %95 = getelementptr inbounds double, double* %arrayptr1118, i64 %value_phi124.postloop
    %arrayref12.postloop = load double, double* %95, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %96 = fmul contract double %arrayref.postloop, %arrayref12.postloop
; │└
; │┌ @ float.jl:409 within `+`
    %97 = fadd reassoc contract double %value_phi23.postloop, %96
; │└
; │ @ simdloop.jl:75 within `macro expansion`
; │┌ @ int.jl:83 within `<`
    %.not16.postloop = icmp ult i64 %93, %arraylen
; │└
   br i1 %.not16.postloop, label %L13.postloop, label %L32
; └
}

Multiple faster than Single

@benchmark slowdot($a,$b)

BenchmarkTools.Trial: 10000 samples with 53 evaluations.
 Range (min … max):  878.717 ns …  1.934 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     879.415 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   925.088 ns ± 97.759 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █      ▆   ▅   ▂    ▂                                        ▁
  █▁▁▁▁▁▅█▄▁▄█▄▃▄█▃▁▁▁█▁▄▁▁▆▁▁▁▁▁▁▁▁▃▆▇▇▇█▇▇▇█████▇▇▇▇▆▆▆▅▆▅▆▄ █
  879 ns        Histogram: log(frequency) by time      1.33 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

@benchmark fastdot($a,$b)

BenchmarkTools.Trial: 10000 samples with 930 evaluations.
 Range (min … max):  108.141 ns … 175.028 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     108.315 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   110.789 ns ±   5.424 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▆     ▁        ▄▄▄       ▃▂        ▁                         ▁
  ██▁▁▁▁▇█▆▆▆▅▆▆▆▅███▁▃▁▁▁▃▃██▄▅▆▅▆▇▇▇█▇▅▆▆▆▇▆▆▇▇▆▆▆▇▅▆▇▆▅▅▄▅▆▅ █
  108 ns        Histogram: log(frequency) by time        133 ns <
 Memory estimate: 0 bytes, allocs estimate: 0.

Beyond @simd

• @simd is somewhat conservative in its vectorization
• The LoopVectorization provides a more aggresive auto-vectorization macro, @turbo
  • Unless new maintainer steps forward, will not work with Julia 1.11 and newer
  • Does not work for all loops
• Can manually write SIMD code
  • Using SIMD.jl
  • Using llvmcall (not recommended)

SIMD.jl Example

• Based on https://schrimpf.github.io/ARGridBootstrap.jl/argridboot.html#simd

using LoopVectorization, SIMD

T = 1000
e = zeros(T)
y = randn(T)
θ = ones(3);

function resids!(e, yin, θ)
  T = length(yin)
  @inbounds @simd for t in 2:T
    e[t-1] = yin[t] - θ[1] - θ[2]*t - θ[3]*yin[t-1]
  end
  nothing
end

@code_llvm resids!(e,y, θ)

;  @ In[29]:1 within `resids!`
define nonnull {}* @"japi1_resids!_4960"({}* %function, {}** noalias nocapture noundef readonly %args, i32 %nargs) #0 {
top:
  %stackargs = alloca {}**, align 8
  store volatile {}** %args, {}*** %stackargs, align 8
  %0 = getelementptr inbounds {}*, {}** %args, i64 1
  %1 = load {}*, {}** %0, align 8
;  @ In[29]:2 within `resids!`
; ┌ @ essentials.jl:10 within `length`
   %2 = bitcast {}* %1 to { i8*, i64, i16, i16, i32 }*
   %arraylen_ptr = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %2, i64 0, i32 1
   %arraylen = load i64, i64* %arraylen_ptr, align 8
; └
;  @ In[29]:3 within `resids!`
; ┌ @ simdloop.jl within `macro expansion`
   %3 = call i64 @llvm.umax.i64(i64 %arraylen, i64 1)
; │ @ simdloop.jl:71 within `macro expansion`
; │┌ @ simdloop.jl:51 within `simd_inner_length`
; ││┌ @ range.jl:761 within `length`
; │││┌ @ int.jl:87 within `+`
      %4 = add nsw i64 %3, -1
; │└└└
; │ @ simdloop.jl:72 within `macro expansion`
; │┌ @ int.jl:83 within `<`
    %5 = icmp ult i64 %arraylen, 2
    %.not23.not = icmp eq i64 %4, 0
; │└
   %or.cond = select i1 %5, i1 true, i1 %.not23.not
   br i1 %or.cond, label %L56, label %L19.lr.ph

L19.lr.ph:                                        ; preds = %top
   %6 = getelementptr inbounds {}*, {}** %args, i64 2
   %7 = bitcast {}** %6 to double***
   %8 = load double**, double*** %7, align 8
   %9 = bitcast {}** %args to double***
   %10 = load double**, double*** %9, align 8
   %11 = bitcast {}* %1 to double**
   %arrayptr17 = load double*, double** %11, align 8
   %arrayptr418 = load double*, double** %8, align 8
   %12 = getelementptr inbounds double, double* %arrayptr418, i64 1
   %13 = getelementptr inbounds double, double* %arrayptr418, i64 2
   %arrayptr1622 = load double*, double** %10, align 8
; │ @ simdloop.jl:75 within `macro expansion`
   %min.iters.check = icmp ult i64 %4, 8
   br i1 %min.iters.check, label %scalar.ph, label %vector.memcheck

vector.memcheck:                                  ; preds = %L19.lr.ph
   %scevgep = getelementptr double, double* %arrayptr1622, i64 %4
   %scevgep28 = getelementptr double, double* %arrayptr17, i64 %3
   %scevgep31 = getelementptr double, double* %arrayptr418, i64 3
   %bound0 = icmp ult double* %arrayptr1622, %scevgep28
   %bound1 = icmp ult double* %arrayptr17, %scevgep
   %found.conflict = and i1 %bound0, %bound1
   %bound033 = icmp ult double* %arrayptr1622, %scevgep31
   %bound134 = icmp ult double* %arrayptr418, %scevgep
   %found.conflict35 = and i1 %bound033, %bound134
   %conflict.rdx = or i1 %found.conflict, %found.conflict35
   br i1 %conflict.rdx, label %scalar.ph, label %vector.ph

vector.ph:                                        ; preds = %vector.memcheck
   %n.vec = and i64 %4, -4
   %14 = add nsw i64 %n.vec, -4
   %15 = lshr exact i64 %14, 2
   %16 = add nuw nsw i64 %15, 1
   %xtraiter = and i64 %16, 1
   %17 = icmp eq i64 %14, 0
   br i1 %17, label %middle.block.unr-lcssa, label %vector.ph.new

vector.ph.new:                                    ; preds = %vector.ph
   %unroll_iter = and i64 %16, 9223372036854775806
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %.pre = load double, double* %arrayptr418, align 8
    %.pre43 = load double, double* %12, align 8
    %.pre44 = load double, double* %13, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   br label %vector.body

vector.body:                                      ; preds = %vector.body, %vector.ph.new
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %index = phi i64 [ 0, %vector.ph.new ], [ %index.next.1, %vector.body ]
    %vec.ind = phi <4 x i64> [ <i64 0, i64 1, i64 2, i64 3>, %vector.ph.new ], [ %vec.ind.next.1, %vector.body ]
    %niter = phi i64 [ 0, %vector.ph.new ], [ %niter.next.1, %vector.body ]
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:929 within `getindex`
; │││┌ @ int.jl:87 within `+`
      %18 = add nuw nsw <4 x i64> %vec.ind, <i64 2, i64 2, i64 2, i64 2>
; │└└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %19 = or i64 %index, 1
    %20 = getelementptr inbounds double, double* %arrayptr17, i64 %19
    %21 = bitcast double* %20 to <4 x double>*
    %wide.load = load <4 x double>, <4 x double>* %21, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %broadcast.splatinsert = insertelement <4 x double> poison, double %.pre, i64 0
   %broadcast.splat = shufflevector <4 x double> %broadcast.splatinsert, <4 x double> poison, <4 x i32> zeroinitializer
   %22 = fsub <4 x double> %wide.load, %broadcast.splat
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %broadcast.splatinsert36 = insertelement <4 x double> poison, double %.pre43, i64 0
        %broadcast.splat37 = shufflevector <4 x double> %broadcast.splatinsert36, <4 x double> poison, <4 x i32> zeroinitializer
        %23 = sitofp <4 x i64> %18 to <4 x double>
; │└└└└└
; │ @ simdloop.jl:75 within `macro expansion`
   %24 = fmul <4 x double> %broadcast.splat37, %23
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %25 = fsub <4 x double> %22, %24
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %broadcast.splatinsert39 = insertelement <4 x double> poison, double %.pre44, i64 0
    %broadcast.splat40 = shufflevector <4 x double> %broadcast.splatinsert39, <4 x double> poison, <4 x i32> zeroinitializer
    %26 = getelementptr inbounds double, double* %arrayptr17, i64 %index
    %27 = bitcast double* %26 to <4 x double>*
    %wide.load38 = load <4 x double>, <4 x double>* %27, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %28 = fmul <4 x double> %broadcast.splat40, %wide.load38
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %29 = fsub <4 x double> %25, %28
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %30 = getelementptr inbounds double, double* %arrayptr1622, i64 %index
    %31 = bitcast double* %30 to <4 x double>*
    store <4 x double> %29, <4 x double>* %31, align 8
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %index.next = or i64 %index, 4
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:929 within `getindex`
; │││┌ @ int.jl:87 within `+`
      %32 = add <4 x i64> %vec.ind, <i64 6, i64 6, i64 6, i64 6>
; │└└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %33 = or i64 %index, 5
    %34 = getelementptr inbounds double, double* %arrayptr17, i64 %33
    %35 = bitcast double* %34 to <4 x double>*
    %wide.load.1 = load <4 x double>, <4 x double>* %35, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %36 = fsub <4 x double> %wide.load.1, %broadcast.splat
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %37 = sitofp <4 x i64> %32 to <4 x double>
; │└└└└└
; │ @ simdloop.jl:75 within `macro expansion`
   %38 = fmul <4 x double> %broadcast.splat37, %37
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %39 = fsub <4 x double> %36, %38
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %40 = getelementptr inbounds double, double* %arrayptr17, i64 %index.next
    %41 = bitcast double* %40 to <4 x double>*
    %wide.load38.1 = load <4 x double>, <4 x double>* %41, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %42 = fmul <4 x double> %broadcast.splat40, %wide.load38.1
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %43 = fsub <4 x double> %39, %42
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %44 = getelementptr inbounds double, double* %arrayptr1622, i64 %index.next
    %45 = bitcast double* %44 to <4 x double>*
    store <4 x double> %43, <4 x double>* %45, align 8
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %index.next.1 = add nuw i64 %index, 8
    %vec.ind.next.1 = add <4 x i64> %vec.ind, <i64 8, i64 8, i64 8, i64 8>
    %niter.next.1 = add i64 %niter, 2
    %niter.ncmp.1 = icmp eq i64 %niter.next.1, %unroll_iter
    br i1 %niter.ncmp.1, label %middle.block.unr-lcssa.loopexit, label %vector.body

middle.block.unr-lcssa.loopexit:                  ; preds = %vector.body
    %phi.bo = add <4 x i64> %vec.ind, <i64 10, i64 10, i64 10, i64 10>
    %phi.cast = sitofp <4 x i64> %phi.bo to <4 x double>
    br label %middle.block.unr-lcssa

middle.block.unr-lcssa:                           ; preds = %middle.block.unr-lcssa.loopexit, %vector.ph
    %index.unr = phi i64 [ 0, %vector.ph ], [ %index.next.1, %middle.block.unr-lcssa.loopexit ]
    %vec.ind.unr = phi <4 x double> [ <double 2.000000e+00, double 3.000000e+00, double 4.000000e+00, double 5.000000e+00>, %vector.ph ], [ %phi.cast, %middle.block.unr-lcssa.loopexit ]
    %lcmp.mod.not = icmp eq i64 %xtraiter, 0
    br i1 %lcmp.mod.not, label %middle.block, label %vector.body.epil.preheader

vector.body.epil.preheader:                       ; preds = %middle.block.unr-lcssa
    %46 = or i64 %index.unr, 1
    %47 = getelementptr inbounds double, double* %arrayptr17, i64 %46
    %48 = bitcast double* %47 to <4 x double>*
    %wide.load.epil = load <4 x double>, <4 x double>* %48, align 8
    %49 = load double, double* %arrayptr418, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %broadcast.splatinsert.epil = insertelement <4 x double> poison, double %49, i64 0
   %broadcast.splat.epil = shufflevector <4 x double> %broadcast.splatinsert.epil, <4 x double> poison, <4 x i32> zeroinitializer
   %50 = fsub <4 x double> %wide.load.epil, %broadcast.splat.epil
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %51 = load double, double* %12, align 8
; │└
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %broadcast.splatinsert36.epil = insertelement <4 x double> poison, double %51, i64 0
        %broadcast.splat37.epil = shufflevector <4 x double> %broadcast.splatinsert36.epil, <4 x double> poison, <4 x i32> zeroinitializer
; │└└└└└
; │ @ simdloop.jl:75 within `macro expansion`
   %52 = fmul <4 x double> %broadcast.splat37.epil, %vec.ind.unr
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %53 = fsub <4 x double> %50, %52
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %54 = load double, double* %13, align 8
    %broadcast.splatinsert39.epil = insertelement <4 x double> poison, double %54, i64 0
    %broadcast.splat40.epil = shufflevector <4 x double> %broadcast.splatinsert39.epil, <4 x double> poison, <4 x i32> zeroinitializer
    %55 = getelementptr inbounds double, double* %arrayptr17, i64 %index.unr
    %56 = bitcast double* %55 to <4 x double>*
    %wide.load38.epil = load <4 x double>, <4 x double>* %56, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   %57 = fmul <4 x double> %broadcast.splat40.epil, %wide.load38.epil
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ float.jl:410 within `-`
    %58 = fsub <4 x double> %53, %57
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %59 = getelementptr inbounds double, double* %arrayptr1622, i64 %index.unr
    %60 = bitcast double* %59 to <4 x double>*
    store <4 x double> %58, <4 x double>* %60, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   br label %middle.block

middle.block:                                     ; preds = %vector.body.epil.preheader, %middle.block.unr-lcssa
   %cmp.n = icmp eq i64 %4, %n.vec
   br i1 %cmp.n, label %L56, label %scalar.ph

scalar.ph:                                        ; preds = %middle.block, %vector.memcheck, %L19.lr.ph
   %bc.resume.val = phi i64 [ %n.vec, %middle.block ], [ 0, %L19.lr.ph ], [ 0, %vector.memcheck ]
   %61 = add nsw i64 %3, -2
   %xtraiter41 = and i64 %4, 1
   %lcmp.mod42.not = icmp eq i64 %xtraiter41, 0
   br i1 %lcmp.mod42.not, label %L19.prol.loopexit, label %L19.prol.preheader

L19.prol.preheader:                               ; preds = %scalar.ph
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:929 within `getindex`
; │││┌ @ int.jl:87 within `+`
      %62 = or i64 %bc.resume.val, 2
; │└└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %63 = or i64 %bc.resume.val, 1
    %64 = getelementptr inbounds double, double* %arrayptr17, i64 %63
    %arrayref.prol = load double, double* %64, align 8
    %arrayref5.prol = load double, double* %arrayptr418, align 8
; │└
; │┌ @ float.jl:410 within `-`
    %65 = fsub double %arrayref.prol, %arrayref5.prol
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref8.prol = load double, double* %12, align 8
; │└
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %66 = sitofp i64 %62 to double
; ││└└└└
; ││ @ promotion.jl:423 within `*` @ float.jl:411
    %67 = fmul double %arrayref8.prol, %66
; │└
; │┌ @ float.jl:410 within `-`
    %68 = fsub double %65, %67
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref11.prol = load double, double* %13, align 8
    %69 = getelementptr inbounds double, double* %arrayptr17, i64 %bc.resume.val
    %arrayref14.prol = load double, double* %69, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %70 = fmul double %arrayref11.prol, %arrayref14.prol
; │└
; │┌ @ float.jl:410 within `-`
    %71 = fsub double %68, %70
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %72 = getelementptr inbounds double, double* %arrayptr1622, i64 %bc.resume.val
    store double %71, double* %72, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
   br label %L19.prol.loopexit

L19.prol.loopexit:                                ; preds = %L19.prol.preheader, %scalar.ph
   %value_phi124.unr = phi i64 [ %bc.resume.val, %scalar.ph ], [ %63, %L19.prol.preheader ]
   %73 = icmp eq i64 %61, %bc.resume.val
   br i1 %73, label %L56, label %L19

L19:                                              ; preds = %L19, %L19.prol.loopexit
   %value_phi124 = phi i64 [ %74, %L19 ], [ %value_phi124.unr, %L19.prol.loopexit ]
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:929 within `getindex`
; │││┌ @ int.jl:87 within `+`
      %74 = add nuw nsw i64 %value_phi124, 2
; │└└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %75 = add nuw nsw i64 %value_phi124, 1
    %76 = getelementptr inbounds double, double* %arrayptr17, i64 %75
    %arrayref = load double, double* %76, align 8
    %arrayref5 = load double, double* %arrayptr418, align 8
; │└
; │┌ @ float.jl:410 within `-`
    %77 = fsub double %arrayref, %arrayref5
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref8 = load double, double* %12, align 8
; │└
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %78 = sitofp i64 %74 to double
; ││└└└└
; ││ @ promotion.jl:423 within `*` @ float.jl:411
    %79 = fmul double %arrayref8, %78
; │└
; │┌ @ float.jl:410 within `-`
    %80 = fsub double %77, %79
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref11 = load double, double* %13, align 8
    %81 = getelementptr inbounds double, double* %arrayptr17, i64 %value_phi124
    %arrayref14 = load double, double* %81, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %82 = fmul double %arrayref11, %arrayref14
; │└
; │┌ @ float.jl:410 within `-`
    %83 = fsub double %80, %82
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %84 = getelementptr inbounds double, double* %arrayptr1622, i64 %value_phi124
    store double %83, double* %84, align 8
; │└
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:929 within `getindex`
; │││┌ @ int.jl:87 within `+`
      %85 = add nuw nsw i64 %value_phi124, 3
; │└└└
; │ @ simdloop.jl:77 within `macro expansion` @ In[29]:4
; │┌ @ essentials.jl:13 within `getindex`
    %86 = getelementptr inbounds double, double* %arrayptr17, i64 %74
    %arrayref.1 = load double, double* %86, align 8
    %arrayref5.1 = load double, double* %arrayptr418, align 8
; │└
; │┌ @ float.jl:410 within `-`
    %87 = fsub double %arrayref.1, %arrayref5.1
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref8.1 = load double, double* %12, align 8
; │└
; │┌ @ promotion.jl:423 within `*`
; ││┌ @ promotion.jl:393 within `promote`
; │││┌ @ promotion.jl:370 within `_promote`
; ││││┌ @ number.jl:7 within `convert`
; │││││┌ @ float.jl:159 within `Float64`
        %88 = sitofp i64 %85 to double
; ││└└└└
; ││ @ promotion.jl:423 within `*` @ float.jl:411
    %89 = fmul double %arrayref8.1, %88
; │└
; │┌ @ float.jl:410 within `-`
    %90 = fsub double %87, %89
; │└
; │┌ @ essentials.jl:13 within `getindex`
    %arrayref11.1 = load double, double* %13, align 8
    %arrayref14.1 = load double, double* %76, align 8
; │└
; │┌ @ float.jl:411 within `*`
    %91 = fmul double %arrayref11.1, %arrayref14.1
; │└
; │┌ @ float.jl:410 within `-`
    %92 = fsub double %90, %91
; │└
; │┌ @ array.jl:1021 within `setindex!`
    %93 = getelementptr inbounds double, double* %arrayptr1622, i64 %75
    store double %92, double* %93, align 8
; │└
; │ @ simdloop.jl:75 within `macro expansion`
; │┌ @ int.jl:83 within `<`
    %exitcond.not.1 = icmp eq i64 %74, %4
; │└
   br i1 %exitcond.not.1, label %L56, label %L19

L56:                                              ; preds = %L19, %L19.prol.loopexit, %middle.block, %top
; │ @ simdloop.jl:76 within `macro expansion`
; │┌ @ simdloop.jl:54 within `simd_index`
; ││┌ @ range.jl:930 within `getindex`
     ret {}* inttoptr (i64 129421973389320 to {}*)
; └└└
}

@benchmark resids!($e,$y,$θ)

BenchmarkTools.Trial: 10000 samples with 57 evaluations.
 Range (min … max):  873.105 ns …   2.463 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     876.860 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   943.595 ns ± 120.792 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █     ▆  ▅▂  ▄   ▃   ▃        ▁                               ▁
  █▁▃▁▁▆█▅▆██▅▃█▅▅▅█▅▃▅█▆▇▇██▇▇▆█▇▇▇▇█▇▇▇██▇▇█▇▇▇▇▇▇▇▇▆▆▆▅▆▆▆▆▅ █
  873 ns        Histogram: log(frequency) by time       1.41 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

function resids_turbo!(e, yin, θ)
  T = length(yin)
  @turbo for t in 2:T
    e[t-1] = yin[t] - θ[1] - θ[2]*t - θ[3]*yin[t-1]
  end
  nothing
end

@code_llvm resids_turbo!(e,y, θ)

;  @ In[31]:1 within `resids_turbo!`
define nonnull {}* @"japi1_resids_turbo!_5519"({}* %function, {}** noalias nocapture noundef readonly %args, i32 %nargs) #0 {
top:
  %stackargs = alloca {}**, align 8
  store volatile {}** %args, {}*** %stackargs, align 8
  %0 = bitcast {}** %args to double***
  %1 = load double**, double*** %0, align 8
  %2 = getelementptr inbounds {}*, {}** %args, i64 1
  %3 = load {}*, {}** %2, align 8
  %4 = getelementptr inbounds {}*, {}** %args, i64 2
  %5 = bitcast {}** %4 to double***
  %6 = load double**, double*** %5, align 8
;  @ In[31]:2 within `resids_turbo!`
; ┌ @ essentials.jl:10 within `length`
   %7 = bitcast {}* %3 to { i8*, i64, i16, i16, i32 }*
   %arraylen_ptr = getelementptr inbounds { i8*, i64, i16, i16, i32 }, { i8*, i64, i16, i16, i32 }* %7, i64 0, i32 1
   %arraylen = load i64, i64* %arraylen_ptr, align 8
; └
;  @ In[31]:3 within `resids_turbo!`
; ┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/condense_loopset.jl:1179 within `macro expansion`
; │┌ @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:100 within `stridedpointer_preserve`
; ││┌ @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:18 within `memory_reference` @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:21
; │││┌ @ abstractarray.jl:1237 within `pointer`
; ││││┌ @ pointer.jl:65 within `unsafe_convert`
       %arrayptr225 = load double*, double** %1, align 8
; │└└└└
; │┌ @ abstractarray.jl:1291 within `getindex`
; ││┌ @ abstractarray.jl:1319 within `_getindex`
; │││┌ @ essentials.jl:13 within `getindex`
      %arrayptr25221 = load double*, double** %6, align 8
      %8 = getelementptr inbounds double, double* %arrayptr25221, i64 1
      %arrayref = load double, double* %8, align 8
      %9 = getelementptr inbounds double, double* %arrayptr25221, i64 2
      %arrayref28 = load double, double* %9, align 8
; │└└└
; │┌ @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:100 within `stridedpointer_preserve`
; ││┌ @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:18 within `memory_reference` @ /home/paul/.julia/packages/LayoutPointers/v9n88/src/stridedpointers.jl:21
; │││┌ @ abstractarray.jl:1237 within `pointer`
; ││││┌ @ pointer.jl:65 within `unsafe_convert`
       %10 = bitcast {}* %3 to double**
       %arrayptr30224 = load double*, double** %10, align 8
; │└└└└
; │┌ @ abstractarray.jl:1291 within `getindex`
; ││┌ @ abstractarray.jl:1319 within `_getindex`
; │││┌ @ essentials.jl:13 within `getindex`
      %arrayref33 = load double, double* %arrayptr25221, align 8
; │└└└
; │┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/condense_loopset.jl:390 within `gespf1`
; ││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:479 within `gesp`
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:442 within `increment_ptr`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:407 within `_gep`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:407 within `macro expansion`
        %ptr.1.i = getelementptr inbounds double, double* %arrayptr30224, i64 -2
        %ptr.1.i37 = getelementptr inbounds double, double* %arrayptr225, i64 -2
; │└└└└└
; │┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/reconstruct_loopset.jl:1107 within `_turbo_!`
; ││┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/reconstruct_loopset.jl:1107 within `macro expansion`
; │││┌ @ operators.jl:425 within `>=`
; ││││┌ @ operators.jl:401 within `<=`
; │││││┌ @ bool.jl:39 within `|`
        %11 = icmp ugt i64 %arraylen, 1
; │││└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:47 within `assume`
      call void @llvm.assume(i1 %11)
; │││└
; │││┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/modeling/graphs.jl:236 within `vcmpend`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `vsub_nsw`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
        %res.i = add nsw i64 %arraylen, -11
; ││││└└
; ││││┌ @ int.jl:514 within `<=`
       %.not232 = icmp ult i64 %arraylen, 13
; │││└└
     br i1 %.not232, label %L134, label %L97.lr.ph

L97.lr.ph:                                        ; preds = %top
     %ie.i = insertelement <4 x double> undef, double %arrayref33, i64 0
     %v.i = shufflevector <4 x double> %ie.i, <4 x double> undef, <4 x i32> zeroinitializer
     %12 = fneg fast double %arrayref28
     %ie.i63 = insertelement <4 x double> undef, double %12, i64 0
     %v.i64 = shufflevector <4 x double> %ie.i63, <4 x double> undef, <4 x i32> zeroinitializer
     %13 = fneg fast double %arrayref
     %ie.i70 = insertelement <4 x double> undef, double %13, i64 0
     %v.i71 = shufflevector <4 x double> %ie.i70, <4 x double> undef, <4 x i32> zeroinitializer
     br label %L97

L97:                                              ; preds = %L97, %L97.lr.ph
     %value_phi35233 = phi i64 [ 2, %L97.lr.ph ], [ %res.i91, %L97 ]
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:771 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:535 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i39 = getelementptr inbounds double, double* %ptr.1.i, i64 %value_phi35233
; ││││││└└└
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %ptr.1.i41 = bitcast double* %ptr.1.i39 to <4 x double>*
           %res.i42 = load <4 x double>, <4 x double>* %ptr.1.i41, align 8
           %ptr.1.i43 = getelementptr inbounds double, double* %ptr.1.i39, i64 4
           %ptr.2.i44 = bitcast double* %ptr.1.i43 to <4 x double>*
           %res.i45 = load <4 x double>, <4 x double>* %ptr.2.i44, align 8
           %ptr.1.i46 = getelementptr inbounds double, double* %ptr.1.i39, i64 8
           %ptr.2.i47 = bitcast double* %ptr.1.i46 to <4 x double>*
           %res.i48 = load <4 x double>, <4 x double>* %ptr.2.i47, align 8
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:55 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i49 = or i64 %value_phi35233, 1
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:771 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:535 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i50 = getelementptr inbounds double, double* %ptr.1.i, i64 %res.i49
; ││││││└└└
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %ptr.1.i52 = bitcast double* %ptr.1.i50 to <4 x double>*
           %res.i53 = load <4 x double>, <4 x double>* %ptr.1.i52, align 8
           %ptr.1.i54 = getelementptr inbounds double, double* %ptr.1.i50, i64 4
           %ptr.2.i55 = bitcast double* %ptr.1.i54 to <4 x double>*
           %res.i56 = load <4 x double>, <4 x double>* %ptr.2.i55, align 8
           %ptr.1.i57 = getelementptr inbounds double, double* %ptr.1.i50, i64 8
           %ptr.2.i58 = bitcast double* %ptr.1.i57 to <4 x double>*
           %res.i59 = load <4 x double>, <4 x double>* %ptr.2.i58, align 8
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:99 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:111 within `vsub_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i60 = fsub reassoc nsz arcp contract afn <4 x double> %res.i53, %v.i
; ││││││└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i61 = fsub reassoc nsz arcp contract afn <4 x double> %res.i56, %v.i
; ││││││└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:7
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i62 = fsub reassoc nsz arcp contract afn <4 x double> %res.i59, %v.i
; │││└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i65 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i64, <4 x double> %res.i42, <4 x double> %res.i60)
            %res.i66 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i64, <4 x double> %res.i45, <4 x double> %res.i61)
            %res.i67 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i64, <4 x double> %res.i48, <4 x double> %res.i62)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vector_width.jl:45 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i68 = add nuw nsw i64 %value_phi35233, 4
       %res.i69 = add nuw nsw i64 %value_phi35233, 8
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:203 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:254 within `vconvert`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %14 = sitofp i64 %value_phi35233 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i72 = insertelement <4 x double> undef, double %14, i64 0
               %v.i73 = shufflevector <4 x double> %ie.i72, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i74 = fadd fast <4 x double> %v.i73, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││││││└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %15 = sitofp i64 %res.i68 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i75 = insertelement <4 x double> undef, double %15, i64 0
               %v.i76 = shufflevector <4 x double> %ie.i75, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i77 = fadd fast <4 x double> %v.i76, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││││││└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:10
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %16 = sitofp i64 %res.i69 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i78 = insertelement <4 x double> undef, double %16, i64 0
               %v.i79 = shufflevector <4 x double> %ie.i78, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i80 = fadd fast <4 x double> %v.i79, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││└└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i81 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i71, <4 x double> %res.i74, <4 x double> %res.i65)
            %res.i82 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i71, <4 x double> %res.i77, <4 x double> %res.i66)
            %res.i83 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i71, <4 x double> %res.i80, <4 x double> %res.i67)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1803 within `_vstore!`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1174 within `_vstore_unroll!`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:870 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i84 = getelementptr inbounds double, double* %ptr.1.i37, i64 %value_phi35233
; ││││││└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:872 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:199 within `_vstore!`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `__vstore!`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `macro expansion`
           %ptr.1.i86 = bitcast double* %ptr.1.i84 to <4 x double>*
           store <4 x double> %res.i81, <4 x double>* %ptr.1.i86, align 8
           %ptr.1.i87 = getelementptr inbounds double, double* %ptr.1.i84, i64 4
           %ptr.2.i88 = bitcast double* %ptr.1.i87 to <4 x double>*
           store <4 x double> %res.i82, <4 x double>* %ptr.2.i88, align 8
           %ptr.1.i89 = getelementptr inbounds double, double* %ptr.1.i84, i64 8
           %ptr.2.i90 = bitcast double* %ptr.1.i89 to <4 x double>*
           store <4 x double> %res.i83, <4 x double>* %ptr.2.i90, align 8
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i91 = add nuw nsw i64 %value_phi35233, 12
; │││└└
; │││┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/modeling/graphs.jl:236 within `vcmpend`
; ││││┌ @ int.jl:514 within `<=`
       %.not = icmp sgt i64 %res.i91, %res.i
; │││└└
     br i1 %.not, label %L134, label %L97

L134:                                             ; preds = %L97, %top
     %value_phi35.lcssa = phi i64 [ 2, %top ], [ %res.i91, %L97 ]
; │││┌ @ /home/paul/.julia/packages/LoopVectorization/QgYWB/src/modeling/graphs.jl:229 within `cmpend`
; ││││┌ @ int.jl:514 within `<=`
       %.not226 = icmp ugt i64 %value_phi35.lcssa, %arraylen
; │││└└
     br i1 %.not226, label %L233, label %L136

L136:                                             ; preds = %L134
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:553 within `mask`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:520 within `_mask`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:520 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:503 within `_mask_cmp`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:499 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:53 within `vsub_nw`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:53 within `macro expansion`
            %res.i92 = add nuw i64 %arraylen, 2
; ││││││││└└
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:386 within `valrem`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:386 within `macro expansion`
; ││││││││││┌ @ int.jl:347 within `&`
             %17 = and i64 %res.i92, 3
; ││││││││└└└
; ││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:500 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:93 within `>=`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:810 within `vge`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/promotion.jl:141 within `itosize`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:209 within `rem`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:461 within `vrem`
; │││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:195 within `vconvert`
; │││││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; ││││││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
                   %ie.i93 = insertelement <4 x i64> undef, i64 %17, i64 0
                   %v.i94 = shufflevector <4 x i64> %ie.i93, <4 x i64> undef, <4 x i32> zeroinitializer
; ││││││││││└└└└└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:810 within `vge` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:730
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/masks.jl:690 within `macro expansion`
             %m.i = icmp uge <4 x i64> %v.i94, <i64 0, i64 1, i64 2, i64 3>
; │││└└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:55 within `vsub_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i96 = add nsw i64 %arraylen, -4
; │││└└
; │││┌ @ operators.jl:378 within `>`
; ││││┌ @ int.jl:83 within `<`
       %.not227 = icmp slt i64 %res.i96, %value_phi35.lcssa
; │││└└
     br i1 %.not227, label %L146, label %L165

L146:                                             ; preds = %L136
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
        %ptr.1.i99228 = getelementptr inbounds double, double* %ptr.1.i, i64 %value_phi35.lcssa
        %ptr.2.i100 = bitcast double* %ptr.1.i99228 to <4 x double>*
        %res.i101 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i100, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; ││││└└
; ││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:104 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:45 within `linear_index`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/cartesian_indexing.jl:5 within `tdot` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/cartesian_indexing.jl:9
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/lazymul.jl:61 within `lazymul`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 within `vmul_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
           %res.i102 = shl i64 %value_phi35.lcssa, 3
           %res.i103 = add nuw nsw i64 %res.i102, 8
; ││││└└└└└
; ││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
        %ptr.0.i104 = bitcast double* %ptr.1.i to i8*
        %ptr.1.i105 = getelementptr inbounds i8, i8* %ptr.0.i104, i64 %res.i103
        %ptr.2.i106 = bitcast i8* %ptr.1.i105 to <4 x double>*
        %res.i107 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i106, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; │││└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:94 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:96 within `promote`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191 within `vconvert`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
            %ie.i108 = insertelement <4 x double> undef, double %arrayref33, i64 0
            %v.i109 = shufflevector <4 x double> %ie.i108, <4 x double> undef, <4 x i32> zeroinitializer
; ││││└└└└└└
; ││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:95 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
        %res.i110 = fsub reassoc nsz arcp contract afn <4 x double> %res.i107, %v.i109
; │││└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %18 = fneg fast double %arrayref28
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191 within `vconvert`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i111 = insertelement <4 x double> undef, double %18, i64 0
              %v.i112 = shufflevector <4 x double> %ie.i111, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
         %res.i113 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i112, <4 x double> %res.i101, <4 x double> %res.i110)
; ││││└└└
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %19 = fneg fast double %arrayref
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191 within `vconvert`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i114 = insertelement <4 x double> undef, double %19, i64 0
              %v.i115 = shufflevector <4 x double> %ie.i114, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││││└└└└
; ││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:203 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││┌ @ number.jl:7 within `convert`
; ││││││││││┌ @ float.jl:159 within `Float64`
             %20 = sitofp i64 %value_phi35.lcssa to double
; │││││││││└└
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
             %ie.i116 = insertelement <4 x double> undef, double %20, i64 0
             %v.i117 = shufflevector <4 x double> %ie.i116, <4 x double> undef, <4 x i32> zeroinitializer
             %res.i118 = fadd fast <4 x double> %v.i117, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
         %res.i119 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i115, <4 x double> %res.i118, <4 x double> %res.i113)
; │││└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:212 within `_vstore!`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `__vstore!`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `macro expansion`
        %ptr.1.i122229 = getelementptr inbounds double, double* %ptr.1.i37, i64 %value_phi35.lcssa
        %ptr.2.i123 = bitcast double* %ptr.1.i122229 to <4 x double>*
        call void @llvm.masked.store.v4f64.p0v4f64(<4 x double> %res.i119, <4 x double>* nonnull %ptr.2.i123, i32 8, <4 x i1> %m.i)
; │││└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       br label %L233

L165:                                             ; preds = %L136
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:55 within `vsub_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i125 = add nsw i64 %arraylen, -8
; │││└└
; │││┌ @ operators.jl:378 within `>`
; ││││┌ @ int.jl:83 within `<`
       %.not230 = icmp slt i64 %res.i125, %value_phi35.lcssa
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:788 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:659 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i126 = getelementptr inbounds double, double* %ptr.1.i, i64 %value_phi35.lcssa
; ││││││└└└
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %ptr.1.i128 = bitcast double* %ptr.1.i126 to <4 x double>*
           %res.i129 = load <4 x double>, <4 x double>* %ptr.1.i128, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl within `_vload`
         %ptr.1.i130 = getelementptr inbounds double, double* %ptr.1.i126, i64 4
         %ptr.2.i131 = bitcast double* %ptr.1.i130 to <4 x double>*
; │││└└└└
     br i1 %.not230, label %L168, label %L196

L168:                                             ; preds = %L165
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:788 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:659 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
           %res.i132 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i131, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:55 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i133 = or i64 %value_phi35.lcssa, 1
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:788 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:659 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i134 = getelementptr inbounds double, double* %ptr.1.i, i64 %res.i133
; ││││││└└└
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %ptr.1.i136 = bitcast double* %ptr.1.i134 to <4 x double>*
           %res.i137 = load <4 x double>, <4 x double>* %ptr.1.i136, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
           %ptr.1.i138 = getelementptr inbounds double, double* %ptr.1.i134, i64 4
           %ptr.2.i139 = bitcast double* %ptr.1.i138 to <4 x double>*
           %res.i140 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i139, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:98 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:96 within `promote`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
            %ie.i141 = insertelement <4 x double> undef, double %arrayref33, i64 0
            %v.i142 = shufflevector <4 x double> %ie.i141, <4 x double> undef, <4 x i32> zeroinitializer
; ││││└└└└└└
; ││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:99 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:111 within `vsub_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i143 = fsub reassoc nsz arcp contract afn <4 x double> %res.i137, %v.i142
; ││││││└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:7
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i144 = fsub reassoc nsz arcp contract afn <4 x double> %res.i140, %v.i142
; │││└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %21 = fneg fast double %arrayref28
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i145 = insertelement <4 x double> undef, double %21, i64 0
              %v.i146 = shufflevector <4 x double> %ie.i145, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i147 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i146, <4 x double> %res.i129, <4 x double> %res.i143)
            %res.i148 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i146, <4 x double> %res.i132, <4 x double> %res.i144)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vector_width.jl:45 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i149 = add nuw nsw i64 %value_phi35.lcssa, 4
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %22 = fneg fast double %arrayref
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i150 = insertelement <4 x double> undef, double %22, i64 0
              %v.i151 = shufflevector <4 x double> %ie.i150, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││││└└└└
; ││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:203 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:254 within `vconvert`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %23 = sitofp i64 %value_phi35.lcssa to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i152 = insertelement <4 x double> undef, double %23, i64 0
               %v.i153 = shufflevector <4 x double> %ie.i152, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i154 = fadd fast <4 x double> %v.i153, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││││││└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:10
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %24 = sitofp i64 %res.i149 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i155 = insertelement <4 x double> undef, double %24, i64 0
               %v.i156 = shufflevector <4 x double> %ie.i155, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i157 = fadd fast <4 x double> %v.i156, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││└└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i158 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i151, <4 x double> %res.i154, <4 x double> %res.i147)
            %res.i159 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i151, <4 x double> %res.i157, <4 x double> %res.i148)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1826 within `_vstore!`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1473 within `_vstore_unroll!`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:870 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i160 = getelementptr inbounds double, double* %ptr.1.i37, i64 %value_phi35.lcssa
; ││││││└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:872 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:199 within `_vstore!`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `__vstore!`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `macro expansion`
           %ptr.1.i162 = bitcast double* %ptr.1.i160 to <4 x double>*
           store <4 x double> %res.i158, <4 x double>* %ptr.1.i162, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:212 within `_vstore!`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `__vstore!`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `macro expansion`
           %ptr.1.i163 = getelementptr inbounds double, double* %ptr.1.i160, i64 4
           %ptr.2.i164 = bitcast double* %ptr.1.i163 to <4 x double>*
           call void @llvm.masked.store.v4f64.p0v4f64(<4 x double> %res.i159, <4 x double>* nonnull %ptr.2.i164, i32 8, <4 x i1> %m.i)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       br label %L233

L196:                                             ; preds = %L165
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:788 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:659 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %res.i172 = load <4 x double>, <4 x double>* %ptr.2.i131, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
           %ptr.1.i173 = getelementptr inbounds double, double* %ptr.1.i126, i64 8
           %ptr.2.i174 = bitcast double* %ptr.1.i173 to <4 x double>*
           %res.i175 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i174, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:55 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i176 = or i64 %value_phi35.lcssa, 1
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:788 within `_vload`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:659 within `_vload_unroll`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:60 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i177 = getelementptr inbounds double, double* %ptr.1.i, i64 %res.i176
; ││││││└└└
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:95 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:987 within `macro expansion`
           %ptr.1.i179 = bitcast double* %ptr.1.i177 to <4 x double>*
           %res.i180 = load <4 x double>, <4 x double>* %ptr.1.i179, align 8
           %ptr.1.i181 = getelementptr inbounds double, double* %ptr.1.i177, i64 4
           %ptr.2.i182 = bitcast double* %ptr.1.i181 to <4 x double>*
           %res.i183 = load <4 x double>, <4 x double>* %ptr.2.i182, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:105 within `_vload`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `__vload`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:997 within `macro expansion`
           %ptr.1.i184 = getelementptr inbounds double, double* %ptr.1.i177, i64 8
           %ptr.2.i185 = bitcast double* %ptr.1.i184 to <4 x double>*
           %res.i186 = call <4 x double> @llvm.masked.load.v4f64.p0v4f64(<4 x double>* nonnull %ptr.2.i185, i32 8, <4 x i1> %m.i, <4 x double> zeroinitializer)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:98 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:96 within `promote`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
            %ie.i187 = insertelement <4 x double> undef, double %arrayref33, i64 0
            %v.i188 = shufflevector <4 x double> %ie.i187, <4 x double> undef, <4 x i32> zeroinitializer
; ││││└└└└└└
; ││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:99 within `sub_fast`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:111 within `vsub_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i189 = fsub reassoc nsz arcp contract afn <4 x double> %res.i180, %v.i188
; ││││││└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i190 = fsub reassoc nsz arcp contract afn <4 x double> %res.i183, %v.i188
; ││││││└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:11 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:7
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `vsub_fast`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:116 within `macro expansion`
          %res.i191 = fsub reassoc nsz arcp contract afn <4 x double> %res.i186, %v.i188
; │││└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %25 = fneg fast double %arrayref28
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i192 = insertelement <4 x double> undef, double %25, i64 0
              %v.i193 = shufflevector <4 x double> %ie.i192, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i194 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i193, <4 x double> %res.i129, <4 x double> %res.i189)
            %res.i195 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i193, <4 x double> %res.i172, <4 x double> %res.i190)
            %res.i196 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i193, <4 x double> %res.i175, <4 x double> %res.i191)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vector_width.jl:45 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       %res.i197 = add nuw nsw i64 %value_phi35.lcssa, 4
       %res.i198 = add nuw nsw i64 %value_phi35.lcssa, 8
; │││└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:498 within `vfnmadd_fast`
; ││││┌ @ fastmath.jl:161 within `sub_fast`
       %26 = fneg fast double %arrayref
; ││││└
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:493 within `vfmadd_fast`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:392 within `vmuladd_fast`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/VectorizationBase.jl:100 within `promote`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:199 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:242 within `vconvert` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:191
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:151 within `vbroadcast`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:122 within `_vbroadcast`
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/vbroadcast.jl:95 within `macro expansion`
              %ie.i199 = insertelement <4 x double> undef, double %26, i64 0
              %v.i200 = shufflevector <4 x double> %ie.i199, <4 x double> undef, <4 x i32> zeroinitializer
; ││││││││└└└└
; ││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:203 within `convert`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/conversion.jl:254 within `vconvert`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap`
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %27 = sitofp i64 %value_phi35.lcssa to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i201 = insertelement <4 x double> undef, double %27, i64 0
               %v.i202 = shufflevector <4 x double> %ie.i201, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i203 = fadd fast <4 x double> %v.i202, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││││││└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %28 = sitofp i64 %res.i197 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i204 = insertelement <4 x double> undef, double %28, i64 0
               %v.i205 = shufflevector <4 x double> %ie.i204, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i206 = fadd fast <4 x double> %v.i205, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││││││└└└
; ││││││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 within `fmap` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:18 @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:10
; ││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:140 within `vconvert`
; │││││││││││┌ @ number.jl:7 within `convert`
; ││││││││││││┌ @ float.jl:159 within `Float64`
               %29 = sitofp i64 %res.i198 to double
; │││││││││││└└
; │││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:72 within `vrangeincr`
; ││││││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/ranges.jl:94 within `macro expansion`
               %ie.i207 = insertelement <4 x double> undef, double %29, i64 0
               %v.i208 = shufflevector <4 x double> %ie.i207, <4 x double> undef, <4 x i32> zeroinitializer
               %res.i209 = fadd fast <4 x double> %v.i208, <double 0.000000e+00, double 1.000000e+00, double 2.000000e+00, double 3.000000e+00>
; ││││││└└└└└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/base_defs.jl:393 within `vmuladd_fast` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:233
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `fmap`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/fmap.jl:25 within `macro expansion`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `vmuladd_fast`
; │││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/intrin_funcs.jl:438 within `macro expansion`
            %res.i210 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i200, <4 x double> %res.i203, <4 x double> %res.i194)
            %res.i211 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i200, <4 x double> %res.i206, <4 x double> %res.i195)
            %res.i212 = call reassoc nsz arcp contract afn <4 x double> @llvm.fmuladd.v4f64(<4 x double> %v.i200, <4 x double> %res.i209, <4 x double> %res.i196)
; │││└└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1826 within `_vstore!`
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:1473 within `_vstore_unroll!`
; │││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:870 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:436 within `gep`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `_gep`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:414 within `macro expansion`
           %ptr.1.i213 = getelementptr inbounds double, double* %ptr.1.i37, i64 %value_phi35.lcssa
; ││││││└└└
; ││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/vecunroll/memory.jl:872 within `macro expansion`
; ││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:199 within `_vstore!`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `__vstore!`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1482 within `macro expansion`
           %ptr.1.i215 = bitcast double* %ptr.1.i213 to <4 x double>*
           store <4 x double> %res.i210, <4 x double>* %ptr.1.i215, align 8
           %ptr.1.i216 = getelementptr inbounds double, double* %ptr.1.i213, i64 4
           %ptr.2.i217 = bitcast double* %ptr.1.i216 to <4 x double>*
           store <4 x double> %res.i211, <4 x double>* %ptr.2.i217, align 8
; │││││││└└
; │││││││ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/strided_pointers/stridedpointers.jl:212 within `_vstore!`
; │││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `__vstore!`
; ││││││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/memory_addr.jl:1667 within `macro expansion`
           %ptr.1.i218 = getelementptr inbounds double, double* %ptr.1.i213, i64 8
           %ptr.2.i219 = bitcast double* %ptr.1.i218 to <4 x double>*
           call void @llvm.masked.store.v4f64.p0v4f64(<4 x double> %res.i212, <4 x double>* nonnull %ptr.2.i219, i32 8, <4 x i1> %m.i)
; │││└└└└└└
; │││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/static.jl:53 within `vadd_nsw` @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49
; ││││┌ @ /home/paul/.julia/packages/VectorizationBase/6AO0m/src/llvm_intrin/binary_ops.jl:49 within `macro expansion`
       br label %L233

L233:                                             ; preds = %L196, %L168, %L146, %L134
; │└└└└
   ret {}* inttoptr (i64 129421973389320 to {}*)
; └
}

@benchmark resids_turbo!($e,$y,$θ)

BenchmarkTools.Trial: 10000 samples with 246 evaluations.
 Range (min … max):  307.780 ns … 599.557 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     309.004 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   316.058 ns ±  19.110 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▄          ▁▅▁     ▃       ▁                                 ▁
  ██▁▁▁▁▅▇▅▄▅▅███▆▆▄▄▇█▇▁▃▄▃▄██▃▅▆▆▆▅▇▇▆▆▆▅▆▆▆▆▆▆▅▆▅▅▆▆▅▅▆▆▅▅▆▅ █
  308 ns        Histogram: log(frequency) by time        400 ns <
 Memory estimate: 0 bytes, allocs estimate: 0.

Example: Manual SIMD

oneto(::Val{1}) = (1,)
oneto(::Val{N}) where N = (oneto(Val(N-1))..., N)

function resids_simd!(e,yin, θ, width::Val{N}=Val(8)) where N
  lane = VecRange{N}(0)
  tv=Vec{N,Float64}(oneto(Val(N)))
  θ1=-Vec{N,Float64}(θ[1])
  θ2=-Vec{N,Float64}(θ[2])
  θ3=-Vec{N,Float64}(θ[3])
  remainder = length(e) % N
  @inbounds for t ∈ 1:N:(length(e)-remainder) #eachindex(e)
    @fastmath e[t+lane] = muladd(θ2,tv,yin[t+1+lane])+muladd(θ3,yin[t+lane],θ1)
    @fastmath tv+=N
  end
  @inbounds for t ∈ (length(e)-remainder+1):length(e)
    @fastmath e[t] = muladd(-θ[2],t+1,yin[t+1])-muladd(θ[3],yin[t],θ[1])
  end
  nothing
end

@benchmark resids_simd!($e,$y,$θ)

BenchmarkTools.Trial: 10000 samples with 269 evaluations.
 Range (min … max):  292.390 ns …  23.285 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     294.654 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   309.271 ns ± 230.661 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ██    ▁    ▃█▅    ▄▅     ▄▃                                   ▂
  ██▁▁▁▄█▇▄▄▆███▅▄▅▄██▄▁▆▅▇██▅▇▆▅▆▇▇▅▆▆▆▇▇██▇▇▇▆▆▆▆▆▆▆▆▆▅▄▆▅▆▇▇ █
  292 ns        Histogram: log(frequency) by time        389 ns <
 Memory estimate: 0 bytes, allocs estimate: 0.

Example: Necessary Manual SIMD

function xx_xy!(xx,xy,yin)
  T = length(yin)
  xx .= zero(eltype(xx))
  xy .= zero(eltype(xy))
  @inbounds @fastmath @simd for t in 2:T # @turbo errors
    xx[1,3] += yin[t-1]
    xx[2,3] += t*yin[t-1]
    xx[3,3] += yin[t-1]^2
    xy[1] += yin[t]
    xy[2] += t*yin[t]
    xy[3] += yin[t-1]*yin[t]
  end
  xx[1,1] = T-1 # = 1'*1
  xx[1,2] = xx[2,1] = (T+1)*T/2 - 1 # sum(p+1:T)
  xx[2,2] = (2*(T)+1)*(T)*(T+1)/6 - 1 # sum((p+1:T).^2)
  xx[3,1] = xx[1,3]
  xx[3,2] = xx[2,3]
  nothing
end

xx = @MMatrix zeros(3,3)
xy = @MVector zeros(3)
@benchmark xx_xy!($xx,$xy,$y)

BenchmarkTools.Trial: 10000 samples with 9 evaluations.
 Range (min … max):  2.853 μs …  10.052 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     3.035 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.162 μs ± 502.763 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▆   █ ▇  ▅ ▂▂                                  ▁            ▂
  █▁▁▃█▃█▆▃█▁██▄▇▄▅▅▅▃▄▄▄▃▃▄▃▄▃▃▄▄▃▁▄▄▁▃▆▅▆▇▇███████▆▆▇▆▆▅▄▄▅ █
  2.85 μs      Histogram: log(frequency) by time      5.29 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

function xx_xy_simd!(xx,xy,yin, v::Val{N}=Val(32)) where {N}
  T = length(yin)
  remainder=(T-1) % N
  xx .= zero(eltype(xx))
  xy .= zero(eltype(xy))
  tv = Vec{N,eltype(yin)}(oneto(Val(N)))+1
  lane = VecRange{N}(0)
  @inbounds for t in 2:N:(T-remainder)
    xx[1,3] += sum(yin[t-1+lane])
    xx[2,3] += sum(yin[t-1+lane]*tv)
    xx[3,3] += sum(yin[t-1+lane]^2)
    xy[1] += sum(yin[t+lane])
    xy[2] += sum(tv*yin[t+lane])
    xy[3] += sum(yin[t-1+lane]*yin[t+lane])
    tv += N
  end
  @inbounds for t in (T-remainder+1):T
    xx[1,3] += yin[t-1]
    xx[2,3] += yin[t-1]*t
    xx[3,3] += yin[t-1]^2
    xy[1] += yin[t]
    xy[2] += t*yin[t]
    xy[3] += yin[t-1]*yin[t]
  end
  xx[1,1] = T-1 # = 1'*1
  xx[1,2] = xx[2,1] = (T+1)*T/2 - 1 # sum(2:T)
  xx[2,2] = (2*(T)+1)*(T)*(T+1)/6 - 1 # sum((2:T).^2)
  xx[3,1] = xx[1,3]
  xx[3,2] = xx[2,3]
  nothing
end

@benchmark xx_xy_simd!($xx,$xy,$y)

BenchmarkTools.Trial: 10000 samples with 66 evaluations.
 Range (min … max):  850.409 ns …  1.794 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     865.409 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   890.421 ns ± 65.304 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▆▇█▇     ▂▃▃▅▂ ▁▃▄▃  ▁ ▂                                     ▂
  ████▂▂▄▆▇██████████▄▇████▃▄▆▅▆▅▄▄▄▅▆▅▄▄▅▅▆▆▆▆▆▆▄▃▆▄▆▅▅▅▅▅▄▄▄ █
  850 ns        Histogram: log(frequency) by time      1.18 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

References

Nissen, Jakob Nybo. 2022. “How to Optimise Julia Code: A Practical Guide.” https://viralinstruction.com/posts/optimise/#how_to_optimise_julia_code_a_practical_guide.

Rackauckas, Chris. 2019. “Optimizing Serial Code.” https://book.sciml.ai/notes/02-Optimizing_Serial_Code/.

Schrimpf, Paul. 2019 (revised 2024). “Coding for Performance.” https://schrimpf.github.io/ARGridBootstrap.jl/argridboot.html.