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 multithreading (via Threads.@spawn
) and recursive tiling, on large enough arrays.
But it also cooperates with various other packages, provided they are loaded before the macro is called:

It uses
LoopVectorization.@avx
to speed many things up. (Disable with keywordavx=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 withcuda=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
+
ormin
/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:
Notation
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+j1] (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 * (k1) * 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+j1] (j in 1:15, k in 1:2)
@tullio S[i] := M[col=ji, 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, $B); # 317.526 μs μs, same speed as unpermuted
@btime NNlib.batched_mul($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] * $M3[k,l]; # 30.401 μs
Or slightly less obviously:
M, Σ = randn(100,100), randn(100,100);
@tullio R4[i, j] := (M[μ, i]  M[μ,j])' * Σ[μ,ν] * (M[ν, i]  M[ν, j]);
begin
S = M' * Σ * M # two N^3 operations, instead of one N^4
@tullio R3[i,j] := S[i,i] + S[j,j]  S[i,j]  S[j,i]
end;
R3 ≈ R4
Another thing Tullio can be very fast at is broadcast reductions, where it can avoid large allocations. Here LoopVectorization is speeding up log
, and Tullio is handling tiled memory access and multithreading:
sum_opp(X, Y=X) = @tullio s := X[i,j] * log(Y[j,i])
sum_part(X, Y=X) = @tullio S[i] := X[i,j] * log(Y[j,i])
X = rand(1000,1000);
@btime sum_opp($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($X)), dims=2) # 13.292 ms
At present indices using pad
, clamp
or mod
are also slow. These result in extra
checks or operations at every iteration, not just around the edges:
conv1(x,k) = @tullio y[i+_, j+_] := x[i+a, j+b] * k[a,b]
conv2(x,k) = @tullio y[i+_, j+_] := x[2ia, 2jb] * k[a,b]
conv3(x,k) = @tullio y[i+_, j+_] := x[pad(ia,3), pad(jb,3)] * k[a,b]
x100 = rand(100,100); k7 = randn(7,7);
@btime conv1($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 reorganisation of multithreading 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(sum∘rowmap, 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, whilethreads=64^3
sets a threshold size at which to divide the work (replacing the macro's best guess).avx=false
turns off the use ofLoopVectorization
, whileavx=4
inserts@avx unroll=4 for i in ...
.grad=false
turns off gradient calculation, andgrad=Dual
switches it to useForwardDiff
(which must be loaded).nograd=A
turns of the gradient calculation just forA
, andnograd=(A,B,C)
does this for several arrays.tensor=false
turns off the use ofTensorOperations
. Assignment
xi = ...
removesxi
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, whileA[i,j] = ...
andA[i,j] += ...
write into an existing one.A[row=i, col=j] := ...
makes a newNamedDimsArray
.@tullio (*) A[i,j] := ...
is a product, as is@tullio A[i,j] *= ...
. For other reductions,@tullio (f) A[i,j] ^= ...
is an inplace update.init=0.0
gives the initial value for reductions. For+
,*
,min
,min
,&
,
it has sensible defaults, for other reductions uses zero.
Implicit:
 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) andmin
,max
.  GPU kernels are only constructed when both
KernelAbstractions
andCUDA
are visible. The defaultcuda=256
is passed tokernel(CUDA(), 256)
.  The CPU kernels from
KernelAbstractions
are called only whenthreads=false
; they are not at present very fast, but perhaps useful for testing.
Extras:
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 preventcol
being summed over).A[i] := $d * B[i]
is the preferred way to include other constants. Note that no gradient is calculated ford
. Within indexing,
A[mod(i), clamp(j)]
both mapsi
&j
to lie withinaxes(A)
, and disables inference of their ranges fromA
.  Similarly,
A[pad(i,3)]
extends the range ofi
, inserting zeros outside ofA
. Instead of zero,pad=NaN
uses this value as padding. The implementation of this (andmod
,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 makeA
onebased. Tullio.@printgrad (x+y)*log(x/z) x y z
prints out how symbolic derivatives will be done.
Macros:
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 forloops:
@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]
end
C[i,j] = acc
end
end
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]
end
C[i,j] = isnothing(final) ? acc : tanh(acc)
end
end
end
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
end
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...)
end
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] * (1C[i,j])^2
ΔA[i,k] += ex * B[k,j]
ΔB[k,j] += A[i,k] * ex
end
end
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)
end
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]
end
end
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.
Elsewhere
Backend friends & relatives:

LoopVectorization.jl is used here, if available.

Gaius.jl and PaddedMatrices.jl build on that.

GPUifyLoops.jl and KernelAbstractions.jl generate GPUcompatible kernels.

ThreadsX.jl does threaded reductions, and much else.

Strided.jl does multithreaded broadcasting.
Frontend nearlookalikes:

Einsum.jl makes simple loops. See tests/einsum.jl where
using Tullio: @einsum
is an almostseamless replacement. 
TensorOperations.jl and OMEinsum.jl identify patterns on which they can call various basic operations. TensorRules.jl makes
@tensor
differentiable; see also TensorGrad.jl and TensorTrack.jl for earlier attempts. 
TensorCast.jl expresses everything as Julia array operations, broadcasting and reduction. (OMEinsum.jl also treats some cases as a special lazy broadcastreduction.)
Things you can't run:

Tortilla.jl seems to exist, publicly, only in this very nice talk.

ArrayMeta.jl was a Julia 0.5 take on some of this.

Tokamak.jl was another, see readme here.