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Tullio is a very flexible einsum macro. It understands many array operations written in index notation -- not just matrix multiplication and permutations, but also convolutions, stencils, scatter/gather, and broadcasting. For example:

@tullio M[x,y,c] := N[x+i, y+j,c] * K[i,j]     # sum over i,j, and create M

@tullio S[x] = P[x,y] * log(Q[x,y] / R[y])     # sum over y, and write into S

@tullio A[i,j] += B[i,k,l] * C[l,j] * D[k,j]   # sum over k,l, and add to values in A

@tullio (*) Z[j] := X[ind[k],j] * exp(-Y[k])   # product over k

Used by itself the macro writes ordinary nested loops much like Einsum.@einsum. One difference is that it can parse more expressions, and infer ranges for their indices. Another is that it will use multi-threading (via Threads.@spawn) and recursive tiling, on large enough arrays. But it also co-operates with various other packages, provided they are loaded before the macro is called:

  • It uses LoopVectorization.@avx to speed many things up. (Disable with keyword avx=false.) On a good day this will match the speed of OpenBLAS for matrix multiplication.

  • It uses KernelAbstractions.@kernel (plus CUDAKernels) to make a GPU version. (Disable with cuda=false.) This is somewhat experimental, and may not be fast.

The macro also tries to provide a gradient for use with Tracker or (via ChainRules) for Zygote, Yota, etc. (Disable with grad=false, or nograd=A.) This is done in one of two ways:

  • By default it takes a symbolic derivative of the right hand side expression. This works for reductions over + or min/max. The functions as typed must be known, mostly from DiffRules.

  • The option grad=Dual uses instead ForwardDiff to differentiate the right hand side (only for reductions over +). This allows for more complicated expressions.

The entire right hand side is summed over the full possible range of any indices not appearing on the left. Pipe operators |> or <| indicate functions to be performed outside the sum, for example:

@tullio lse[j] := log <| exp(mat[i,j])   # vec(log.(sum(exp.(mat), dims=1)))

The option @tullio verbose=true will cause it to print index ranges, symbolic derivatives, and notices when it is unable to use the packages mentioned above. And verbose=2 will print everything.

If it's useful in academic work, you can cite it with this DOI: DOI


Index notation for some simple functions:

using Pkg; Pkg.add("Tullio")
using Tullio, Test
M = rand(1:20, 3, 7)

@tullio S[1,c] := M[r,c]  # sum over r ∈ 1:3, for each c ∈ 1:7
@test S == sum(M, dims=1) 

@tullio Q[ρ,c] := M[ρ,c] + sqrt(S[1,c])  # loop over ρ & c, no sum -- broadcasting
@test Q  M .+ sqrt.(S)

mult(M,Q) = @tullio P[x,y] := M[x,c] * Q[y,c]  # sum over c ∈ 1:7 -- matrix multiplication
@test mult(M,Q)  M * transpose(Q)

R = [rand(Int8, 3, 4) for δ in 1:5]

@tullio T[j,i,δ] := R[δ][i,j] + 10im  # three nested loops -- concatenation
@test T == permutedims(cat(R...; dims=3), (2,1,3)) .+ 10im

@tullio (max) X[i] := abs2(T[j,i,δ])  # reduce using max, over j and δ
@test X == dropdims(maximum(abs2, T, dims=(1,3)), dims=(1,3))

dbl!(M, S) = @tullio M[r,c] = 2 * S[1,c]  # write into existing matrix, M .= 2 .* S
dbl!(M, S)
@test all(M[r,c] == 2*S[1,c] for r  1:3, c  1:7)

More complicated examples:

using Tullio
A = [abs2(i - 11) for i in 1:21]

# Downsample -- range of i is that allowed by both terms:
@tullio B[i] := (A[2i] + A[2i+1])/2  # 1:10 == intersect(1:10, 0:10)

# Shifts -- range of i calculated in terms of that given for j:
@tullio M[i,j] := A[i+j-1]  (j in 1:15)  # i in 1:7
@tullio M[i+_,j] := A[i+j]  (j in 1:15)  # i in 0:6, automatic shift "i+_"

using OffsetArrays # Convolve a filter:
K = OffsetArray([1,-1,2,-1,1], -2:2)
@tullio C[i] := A[i+j] * K[j]    # j ∈ -2:2 implies i ∈ 3:19

# Index by the values in K
@tullio D[i,j] := A[2K[j]+i] ÷ K[j] # extrema(K)==(-1,2) implies i ∈ 3:17

# Wrapped & padded:
@tullio M[i,j] := A[mod(i+j)]  (j in 1:15, i in 1:15)   # wraps around, mod(i+j, axes(A,1))
@tullio M[i,j] := A[clamp(i+j)]  (j in 1:15, i in 1:15) # instead repeats "100"
@tullio M[i+_,j] := A[pad(i+j, 3)]  (j in 1:15)         # fills with zeros

using FFTW # Functions of the indices are OK:
S = [0,1,0,0, 0,0,0,0]
fft(S)  @tullio F[k] := S[x] * exp(-im*pi/8 * (k-1) * x)  (k  axes(S,1))

# Finalisers <| or |> are applied after sum (the two are equivalent):
@tullio N2[j] := sqrt <| M[i,j]^2     # N2 ≈ map(norm, eachcol(M))
@tullio n3[_] := A[i]^3  |> (_)^(1/3) # n3[1] ≈ norm(A,3), with _ anon. func.

# Reduction over any function:
@tullio (*) P[i] := A[i+k]  (k in 0:2) # product
@tullio (max) X[i,_] := D[i,j]         # maximum(D, dims=2), almost

min1(x,y) = ifelse(first(x) < first(y), x, y); # findmin(D, dims=1), almost:
@tullio (min1) Ts[j+_] := (D[i,j], (i,j))  init=(typemax(Int), (0,0))

# Access to fields & arrays -- this uses j ∈ eachindex(first(N).c)
N = [(a=i, b=i^2, c=fill(i^3,3)) for i in 1:10]
@tullio T[i,j] := (N[i].a // 1, N[i].c[j])

# Functions which create arrays are evaluated once:
@tullio R[i,j] := abs.((rand(Int8, 5)[i], rand(Int8, 5)[j]))

using NamedDims, AxisKeys # Dimension names, plus pretty printing:
@tullio M[row=i, col=j, z=k] := A[i+j-1]  (j in 1:15, k in 1:2)
@tullio S[i] := M[col=j-i, z=k, row=i+1] # sum over j,k

Fast & Slow

When used with LoopVectorization, on straightforward matrix multiplication of real numbers, @tullio tends to be about as fast as OpenBLAS. Depending on the size, and on your computer. Here's a speed comparison on mine: v2.5.

This race is a useful diagnostic, but isn't really the goal. There is little point in avoiding using BLAS libraries, if you want precisely what they are optimised to give you. One of the things @tullio is often very fast at is weird tensor contractions, for which you would otherwise need permutedims:

using Tullio, LoopVectorization, NNlib, BenchmarkTools

# Batched matmul, with batch index first in B:
bmm_rev(A, B) = @tullio C[i,k,b] := A[i,j,b] * B[b,k,j]  # (sum over j)

A = randn(20,30,500); B = randn(500,40,30);
bmm_rev(A, B)  NNlib.batched_mul(A, permutedims(B, (3,2,1)))  # true

@btime bmm_rev($A, $B);  # 317.526 μs μs, same speed as un-permuted
@btime NNlib.batched_mul($A, permutedims($B, (3,2,1)));  # 1.478 ms, with MKL

Complex numbers aren't handled by LoopVectorization, so will be much slower.

Chained multiplication is also very slow, because it doesn't know there's a better algorithm. Here it just makes 4 loops, instead of multiplying sequentially, 30^4 instead of 2 * 30^3 operations:

M1, M2, M3 = randn(30,30), randn(30,30), randn(30,30);
@btime $M1 * $M2 * $M3;                                   #  3.525 μs
@btime @tullio M4[i,l] := $M1[i,j] * $M2[j,k] * $M3[k,l]; # 30.401 μs

Or slightly less obviously:

M, Σ = randn(100,100), randn(100,100);
@tullio R4[i, j] := (M[μ, i] - M[μ,j])' * Σ[μ,ν] * (M[ν, i] - M[ν, j]);
  S = M' * Σ * M  # two N^3 operations, instead of one N^4
  @tullio R3[i,j] := S[i,i] + S[j,j] - S[i,j] - S[j,i]
R3  R4

Another thing Tullio can be very fast at is broadcast reductions, where it can avoid large allocations. Here LoopVectorization is speeding up log, and Tullio is handling tiled memory access and multi-threading:

sum_opp(X, Y=X) = @tullio s := X[i,j] * log(Y[j,i])
sum_part(X, Y=X) = @tullio S[i] := X[i,j] * log(Y[j,i])

X = rand(1000,1000);
@btime sum_opp($X)                    #   499.814 μs (93 allocations: 3.97 KiB)
@btime sum($X .* log.(transpose($X))) # 8.759 ms (2 allocations: 7.63 MiB)

@btime sum_part($X)'                           #  1.599 ms (not the same computer!)
@btime sum($X .* log.(transpose($X)), dims=2)  # 13.292 ms

At present indices using pad, clamp or mod are also slow. These result in extra checks or operations at every iteration, not just around the edges:

conv1(x,k) = @tullio y[i+_, j+_] := x[i+a, j+b] * k[a,b]
conv2(x,k) = @tullio y[i+_, j+_] := x[2i-a, 2j-b] * k[a,b]
conv3(x,k) = @tullio y[i+_, j+_] := x[pad(i-a,3), pad(j-b,3)] * k[a,b]

x100 = rand(100,100); k7 = randn(7,7);
@btime conv1($x100, $k7); #  25.574 μs
@btime conv2($x100, $k7); #  44.590 μs
@btime conv3($x100, $k7); #  86.228 μs

using Flux
x104 = reshape(x100,(100,100,1,1)); k74 = reshape(k7,(7,7,1,1)); 
conv1(x100, k7)  @btime CrossCor($k74, false)($x104)       # 586.694 μs
conv2(x100, k7)  @btime Conv($k74, false, stride=2)($x104) # 901.573 μs
conv3(x100, k7)  @btime Conv($k74, false, pad=3)($x104)    # 932.658 μs

using DSP
@btime DSP.conv($x100, $k7); # 198.331 μs

Gradients & GPU

Derivatives & GPU
using Tullio
mul(A, B) = @tullio C[i,k] := A[i,j] * B[j,k]

A = rand(3,40); B = rand(40,500);
A * B  mul(A, B) # true

using Tracker # or Zygote
ΔA = Tracker.gradient((A,B) -> sum(mul(A, B)), A, B)[1]
ΔA  ones(3,500) * B' # true

using CUDA, CUDAKernels, KernelAbstractions # Now defined with a GPU version:
mul(A, B) = @tullio C[i,k] := A[i,j] * B[j,k]

cu(A * B)  mul(cu(A), cu(B)) # true

cu(ΔA)  Tracker.gradient((A,B) -> sum(mul(A, B)), cu(A), cu(B))[1] # true

# Reduction over min/max:
Tracker.gradient(x -> (@tullio (max) res := x[i]^3), [1,2,3,-2,-1,3])[1]

Some warnings are in order:

  • Complete reductions to a number will not work on the GPU at present. They were extremely slow, and a re-organisation of multi-threading for the CPU case killed them, sorry.
  • Gradients are not calculated for scalars, only arrays. Thus for example gradient(a -> (@tullio _ := $a * A[i]), 3.14) will be zero.
  • When using grad=Dual, the right hand side is evaluated a second time during the backward pass. This avoids needing memory to store partials, but if the function is expensive, it may be slow.

Larger Expressions

The expression need not be just one line, for example:

@tullio out[x, y] := @inbounds(begin  # sum over k
        a,b = off[k]
        mat[mod(x+a), mod(y+b)]
    end) (x in axes(mat,1), y in axes(mat,2)) grad=Dual nograd=off

Here the macro cannot infer the range of the output's indices x,y, so they must be provided explicitly. (If writing into an existing array, with out[x,y] = begin ... or +=, then ranges would be taken from there.) Because it sees assignment being made, it does not attempt to sum over a,b, and it assumes that indices could go out of bounds so does not add @inbounds for you. (Although in fact mod(x+a) == mod(x+a, axes(mat,1)) is safe.) It will also not be able to take a symbolic derivative, but dual numbers will work fine.

More examples:

using Tullio, OffsetArrays

# A convolution with cyclic indices
mat = zeros(10,10,1); mat[2,2] = 101; mat[10,10] = 1;
@tullio kern[i,j] := 1/(1+i^2+j^2)  (i in -3:3, j in -3:3)

@tullio out[x,y,c] := begin
    xi = mod(x+i, axes(mat,1)) # xi = ... means that it won't be summed,
    # yj = mod(y+j, axes(mat,2))
    @inbounds trunc(Int, mat[xi, mod(y+j), c] * kern[i,j]) # and disables automatic @inbounds,
end (x in 1:10, y in 1:10) # and prevents range of x from being inferred.

# A stencil?
offsets = [(a,b) for a in -2:2 for b in -2:2 if a>=b] # vector of tuples

@tullio out[x,y,1] = begin
        a,b = offsets[k]
        i = clamp(x+a, extrema(axes(mat,1))...)
        # j = clamp(y+b, extrema(axes(mat,2))...) # can be written clamp(y+b)
        @inbounds mat[i, clamp(y+b), 1] * 10
    end # ranges of x,y read from out[x,y,1]

# Applying a vector of functions
fs = [sin, cos, tan]
xs = randn(3,100)
@tullio ys[r,c] := (fs[r])(xs[r,c])

using Zygote, ForwardDiff
rowmap(fs, xs) = @tullio ys[r,c] := (fs[r])(xs[r,c]) grad=Dual nograd=fs
Zygote.gradient(sumrowmap, fs, ones(3,2))
[f'(1) for f in fs] # agrees

Keyword Options

The default setting is: @tullio threads=true fastmath=true avx=true tensor=true cuda=256 grad=Base verbose=false A[i,j] := ...

  • threads=false turns off threading, while threads=64^3 sets a threshold size at which to divide the work (replacing the macro's best guess).
  • avx=false turns off the use of LoopVectorization, while avx=4 inserts @avx unroll=4 for i in ....
  • grad=false turns off gradient calculation, and grad=Dual switches it to use ForwardDiff (which must be loaded).
  • nograd=A turns of the gradient calculation just for A, and nograd=(A,B,C) does this for several arrays.
  • tensor=false turns off the use of TensorOperations.
  • Assignment xi = ... removes xi from the list of indices: its range is note calculated, and it will not be summed over. It also disables @inbounds since this is now up to you.
  • verbose=true prints things like the index ranges inferred, and gradient calculations. verbose=2 prints absolutely everything.
  • A[i,j] := ... makes a new array, while A[i,j] = ... and A[i,j] += ... write into an existing one. A[row=i, col=j] := ... makes a new NamedDimsArray.
  • @tullio (*) A[i,j] := ... is a product, as is @tullio A[i,j] *= .... For other reductions, @tullio (f) A[i,j] ^= ... is an in-place update.
  • init=0.0 gives the initial value for reductions. For +, *, min, min, &, | it has sensible defaults, for other reductions uses zero.


  • Indices without shifts must have the same range everywhere they appear, but those with shifts (even A[i+0]) run over the intersection of possible ranges.
  • Shifted output indices must start at 1, unless OffsetArrays is visible in the calling module.
  • The use of @avx, and the calculation of gradients, are switched off by sufficiently complex syntax (such as arrays of arrays).
  • Gradient hooks are attached for any or all of ReverseDiff, Tracker & Zygote. These packages need not be loaded when the macro is run.
  • Gradients are only defined for reductions over (+) (default) and min, max.
  • GPU kernels are only constructed when both KernelAbstractions and CUDA are visible. The default cuda=256 is passed to kernel(CUDA(), 256).
  • The CPU kernels from KernelAbstractions are called only when threads=false; they are not at present very fast, but perhaps useful for testing.


  • A[i] := i^2 (i in 1:10) is how you specify a range for indices when this can't be inferred.
  • A[i] := B[i, $col] - C[i, 2] is how you fix one index to a constant (to prevent col being summed over).
  • A[i] := $d * B[i] is the preferred way to include other constants. Note that no gradient is calculated for d.
  • Within indexing, A[mod(i), clamp(j)] both maps i & j to lie within axes(A), and disables inference of their ranges from A.
  • Similarly, A[pad(i,3)] extends the range of i, inserting zeros outside of A. Instead of zero, pad=NaN uses this value as padding. The implementation of this (and mod, clamp) is not very fast at present.
  • On the left, when making a new array, an underscore like A[i+_] := inserts whatever shift is needed to make A one-based.
  • Tullio.@printgrad (x+y)*log(x/z) x y z prints out how symbolic derivatives will be done.


  • Tullio.@tensor is a macro which uses TensorOperations to evaluate expressions, but provides gradient definitions. (Previously this was automatic behaviour, when TensorOperations.jl was loaded & the expression was suitable.)
  • Tullio.@einsum is a variant with a few changes, to allow the running of Einsum.jl's tests.

How it Works

The following three macros all end up calling the same functions as does C = A * B:

@tensor C[i,j] := A[i,k] * B[k,j]         # TensorOperations.jl
@ein C[i,j] := A[i,k] * B[k,j]            # OMEinsum.jl
@matmul C[i,j] := sum(k) A[i,k] * B[k,j]  # TensorCast.jl

But this one writes its own for-loops:

@einsum C[i,j] := A[i,k] * B[k,j]         # Einsum.jl

expanding out to roughly this:

T = promote_type(eltype(A), eltype(B))
C = Array{T}(undef, size(A,1), size(B,2))
@inbounds for j in 1:size(B,2)
    for i in 1:size(A,1)
        acc = zero(T)
        for k in 1:size(A,2)
            acc += A[i,k] * B[k,j]
        C[i,j] = acc

Tullio does something similar, but working through a few functions. Taking a slightly more complicated example, this:

@tullio C[i,j] := tanh <| A[i,k] * B[k,j]

expands to roughly this:

function act!(::Type, C::AbstractArray{T}, A, B, ax_i, ax_j, ax_k, keep=nothing, final=true) where T
    @inbounds @fastmath for i in ax_i
        for j in ax_j
            acc = isnothing(keep) ? zero(T) : C[i,j]
            for k in ax_k
                acc += A[i,k] * B[k,j]
            C[i,j] = isnothing(final) ? acc : tanh(acc)

function make(A, B)
    ax_i = axes(A,1)
    ax_j = axes(B,2)
    ax_k = axes(A,2) # and check this is == axes(B,1)
    rhs(A,B,i,j,k) = tanh(A[i,k] * B[k,j])
    T = Core.Compiler.return_type(rhs, eltype.((A,B,1,1,1))) # plus a fallback
    C = similar(A, T, (ax_i, ax_j))
    Tullio.threader(act!, Array{T}, C, (A,B), (ax_i,ax_j), (ax_k,), +, 64^3)
    return C

C = Tullio.Eval(make, ∇make)(A, B)

This division allows it to dispatch to other methods of act!: one generated with @avx if LoopVectorization is loaded, and one for ::CuArray if KernelAbstractions is loaded.

It also allows threader to divide the work, calling act! many times, from different threads, on small tiles made by dividing the longest axis (say ax_i) in half, repeatedly. If it divides up ax_k, these are done sequentially, with keep=true on all ranges except the first, and final=nothing on all except the last. But ax_i and ax_j are safe to do in parallel.

Finally, Eval exists to give Zygote and friends somewhere to attach themselves. The gradient calculation looks roughly like this:

@adjoint function (e::Eval)(AB...)
    C = e.fwd(AB...)
    C, ΔC -> e.rev(ΔC, C, AB...)

function ∇act!(::Type, ΔC, ΔA, ΔB, C, A, B, ax_i, ax_j, ax_k, keep)
    for k in ax_k, i in ax_i, j in ax_j
        ex = ΔC[i,j] * (1-C[i,j])^2
        ΔA[i,k] += ex * B[k,j]
        ΔB[k,j] += A[i,k] * ex

function ∇make(ΔC, C, A, B)
    ΔA = similar(A) .= 0
    ΔB = similar(B) .= 0
    ax_i, ax_k = axes(A); ax_j = axes(B,2)
    Tullio.∇threader(∇act!, Array{T}, (ax_k,), (ax_i, ax_j), nothing)
    return (ΔA, ΔB)

In this case, it is the loop over k which can be safely broken among different threads, since both ΔA and ΔB have this index. Both ΔA and ΔB are filled in at once.

Notice that the derivative of y = tanh(z) has been written in terms of y (the final result of the forward pass) but free of z (the result of the sum, which was not saved). If this is not possible, it will fail.

If using grad=Dual, the gradient kernel looks different. This method cannot handle finalisers like tanh above, but for plain @tullio C[i,j] := A[i,k] * B[k,j] it would read:

function ∇act!(::Type, ΔC, ΔA, ΔB, C, A, B, ax_i, ax_j, ax_k, keep)
    eps1 = ForwardDiff.Dual(0, (1,0))
    eps2 = ForwardDiff.Dual(0, (0,1))
    for k in ax_k, i in ax_i, j in ax_j
        res = (A[i,k] + eps1) * (B[k,j] + eps2)
        ΔA[i,k] += ForwardDiff.partials(res, 1) * ΔC[i,j]
        ΔB[k,j] += ForwardDiff.partials(res, 2) * ΔC[i,j]

Writing @tullio verbose=2 will print all of these functions out.

Scalar reductions, such as @tullio s := A[i,j] * log(B[j,i]), are slightly different in that the act! function simply returns the sum, i.e. the variable acc above.


Back-end friends & relatives:

Front-end near-lookalikes:

Things you can't run: