Is there any way to optimize array additions and multiplications with transposes?

Specific Domains Numerics
,
1

The question is related to python - Optimizing array additions and multiplications with transposes - Stack Overflow

Given A is an array, I want to evaluate sum += p(A) * factor(p)

where p is a permutation/transpose, factor(p) is a list of prefactors. For example, A is a 2-dimensional array, one target is (in terms of numpy as np)

0.1*A + 0.1*np.transpose(A,(1,0))

I would like to improve the efficiency of this type of operation. One answer I got is

One solution is to apply the transposition in an order that maximize cache locality from the previous transposition (fast but especially hard to implement). A simpler (but slower) solution is simply to manually work on the (i0, i1, i2, i4, i3) case at the same time than the (i0, i1, i2, i3, i4) case using the previously-provided optimized Numba transposition code.

Is there any solution in Julia that can optimize the cache locality or simply be significantly faster than Python?

2

For the uninitiated: is np.transpose() lazy?

It probably depends: do you have some kind of benchmark comparing NumPy and Julia?

3

Not really. I only compared a bit with Fortran and Numpy, not Julia. I am very new to Julia, just ask in advance.

4

what’s the question here? Julia’s transpose is lazy, when in doubt, use something like Tullio.jl or just write your for loop?

Have you tried it? Julia should be faster than Python when things get complicated

5

Cool, what did you see (would be interesting to know how far away from Fortran we are)?

6

The question is to optimize a series of sum += p(A) * factor(A), at least better than numby or numba hopefully. It was linked to StackOverflow question. So it may not look clear :frowning:

Thank you so much. I will have a look. I googled Julia with transpose, did not find a way similar to Python, which can specify like (1,0,2,3) to higher dimensional array. Maybe Tullio.ji is the right place.

Sorry, what does lazy mean in transpose

7

It doesn’t eagerly copy the matrix but simply switches indices when accessing the data.

OK, this

np.transpose does not physically transpose the array in memory .

answers my question about NumPy. I’m not sure if I’m willing to translate your benchmark to Julia without you trying it first.

8

Cool, what did you see (would be interesting to know how far away from Fortran we are)?

That can be tricky. Since there is some trick to optimize the performance of Fortran at least in transpose.
See my stupid question in Expected list of ‘lower-bound :’ or list of ‘lower-bound : upper-bound’ specifications at (1) in Fortran pointer - Stack Overflow
& comment linked to
Transposition of a matrix by multithread in Fortran - Stack Overflow

A naive transpose together with array addition in Fortran does not much better than numpy (since I noticed the comment from [Ian Bush], I did not do much benchmark on Fortran. And Ian Bush’s approach may be complicated to apply to high dimensional array, at least to me :frowning: )

1 Like
9

that’s OK. I can try myself and see how it works. If I met some difficulty, I will ask.

2 Likes
10

Good, at least (having learned from StackOverflow that your matrices are pretty big) we can give you some obligatory advice: your kernel probably should look something like

@btime (B .= 0.1.*(A .+ transpose(A))) setup=(A=rand(10_000, 10_000); B=similar(A))

yielding

  1.952 s (0 allocations: 0 bytes)

that way avoiding allocation of intermediate results (via broadcasting) and pre-allocating the result B. Recommended way to measure is using BenchmarkTools’s @btime or @benchmark (avoid using global non constants there). If you are seeing unreasonable amounts of allocations when testing, better call us;) Later we could try to parallelize this one (although I’d somehow suspect the task is memory bound).

Edit: stupid me: broadcasting is needed, not @views.
Another edit: and obviously I didn’t try LoopVectorization and Tullio yet.

11

Thanks a lot. I did a preliminary test with a rank-4 array. I apologize that I changed a bit from

For example, A is a 2-dimensional array, one target is (in terms of numpy as np) 0.1A + 0.1np.transpose(A,(1,0))

since ultimately I would like to calculate higher-rank arrays with different factors in multiplications, similar to the link in the original question. Not sure if this is the best approach, that I gradually expand my test from the simplest.

The results from Julia and Numpy are comparable :frowning:

Here is my Julia Code

using BenchmarkTools, Tullio 

n = 30
A = rand(n, n, n, n)
B = rand(n, n, n, n)

@btime (@tullio C[a,b,c,d] := A[b,a,c,d])
@tullio C[a,b,c,d] := A[b,a,c,d]
@btime 0.1*A + 0.2*C

My Python code

import numpy as np
import time

n0=n1=n2=n3=30
M = np.random.random((n0,n1,n2,n3))

n_loop = 100

start = time.time()
for i in range(n_loop):
    M2 = np.add( 0.1*M, 0.2*np.transpose(M,(1,0,2,3)) )
end = time.time()
print('time M2 transpose', (end - start)/n_loop)

The results from Julia is

  1.455 ms (3 allocations: 6.18 MiB)
  3.179 ms (9 allocations: 18.54 MiB)

from Python

time M2 transpose 0.00432762861251831

The sum of the two steps from Julia is similar to Python. Maybe there are tricks in Julia I simply don’t know :frowning:

1 Like
12

I checked the following Python script

import numpy as np
import time

n0=n1=n2=n3=30
A = np.random.random((n0,n1,n2,n3))

n_loop = 100

start = time.time()
for i in range(n_loop):
    B = np.add( 0.1*A, 0.2*np.transpose(A,(1,0,2,3)) )
end = time.time()
print((end - start)/n_loop * 1000, " ms")

against Julia’s

using BenchmarkTools, LoopVectorization, Tullio

function test1(A)
    B = similar(A)
    for i in 1:size(A,1), j in 1:size(A,2)
        # B[i,j,:,:] = 0.1*A[i,j,:,:] + 0.2*A[i,j,:,:]'
        B[:,:,i,j] = 0.1*A[:,:,i,j] + 0.2*A[:,:,i,j]'
    end
    B
end    

function test2(A)
    B = similar(A)
    for i in 1:size(A,1), j in 1:size(A,2)
        # @. @views B[i,j,:,:] = 0.1*A[i,j,:,:] + 0.2*A[i,j,:,:]'
        @. @views B[:,:,i,j] = 0.1*A[:,:,i,j] + 0.2*A[:,:,i,j]'
    end
    B
end    

function test3(A)
    # @tullio B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[i,j,l,k]
    @tullio B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]
    B
end    

@btime test1(A) setup=(n=30; A=rand(n,n,n,n))
@btime test2(A) setup=(n=30; A=rand(n,n,n,n))
@btime test3(A) setup=(n=30; A=rand(n,n,n,n))
nothing

Python yields

4.956309795379639  ms

Julia yields

  7.886 ms (5403 allocations: 37.81 MiB) # naive
  2.253 ms (3 allocations: 6.18 MiB) # naive, @views, broadcasting
  1.323 ms (44 allocations: 6.18 MiB) # @tullio

Observations:

  • for the naive versions ordering of memory access is relevant
  • I didn’t manage to test a version with LoopVectorization’s @turbo
13

I think nobody has given the most obvious translation, which uses permutedims (eager) or PermutedDimsArray (lazy) in place of np.transpose. For me this is faster than the other Base solutions.

TensorOperations.jl is often the fastest way to do permutedims on larger arrays. It has a smarter cache-friendly blocking algorithm than the one in Base. But how much this matters of course depends on size & permutation.

test4(A) = 0.1 .* A .+ 0.2 .* PermutedDimsArray(A, (2,1,3,4))

using Tullio, TensorOperations
test5(A) = @tullio B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]  avx=false  # no LV
test6(A) = @tensor B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]
3 Likes
14

Very instructive! Thanks, here are the numbers for

using BenchmarkTools, LoopVectorization, Tullio, TensorOperations

function test1(A)
    B = similar(A)
    for i in 1:size(A,1), j in 1:size(A,2)
        B[:,:,i,j] = 0.1*A[:,:,i,j] + 0.2*A[:,:,i,j]'
    end
    B
end    

function test2(A)
    B = similar(A)
    for i in 1:size(A,1), j in 1:size(A,2)
        @. @views B[:,:,i,j] = 0.1*A[:,:,i,j] + 0.2*A[:,:,i,j]'
    end
    B
end    

test3(A) = 0.1 .* A .+ 0.2 .* PermutedDimsArray(A, (2,1,3,4))
test4(A) = @tullio B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]
test5(A) = @tullio B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]  avx=false  # no LV
test6(A) = @tensor B[i,j,k,l] := 0.1*A[i,j,k,l] + 0.2*A[j,i,k,l]

@btime test1(A) setup=(n=30; A=rand(n,n,n,n))
@btime test2(A) setup=(n=30; A=rand(n,n,n,n))
@btime test3(A) setup=(n=30; A=rand(n,n,n,n))
@btime test4(A) setup=(n=30; A=rand(n,n,n,n))
@btime test5(A) setup=(n=30; A=rand(n,n,n,n))
@btime test6(A) setup=(n=30; A=rand(n,n,n,n))
nothing

from my system for comparison

  7.980 ms (5403 allocations: 37.81 MiB) # naive
  2.280 ms (3 allocations: 6.18 MiB) # naive, @views, broadcasting
  1.894 ms (12 allocations: 6.18 MiB) # permuted dims
  1.341 ms (44 allocations: 6.18 MiB) # @tullio
  1.365 ms (44 allocations: 6.18 MiB) # @tullio, no LV
  1.096 ms (131 allocations: 6.19 MiB) # @tensor
1 Like
15

In my system, it is

  8.389 ms (5403 allocations: 37.81 MiB)
  1.664 ms (3 allocations: 6.18 MiB)
  1.669 ms (12 allocations: 6.18 MiB)
  1.293 ms (3 allocations: 6.18 MiB)
  1.588 ms (3 allocations: 6.18 MiB)
  2.585 ms (21 allocations: 6.18 MiB)

approach 4 is the fastest. And faster than python by ~ a factor of 3.

I noticed I should use -O3 in Fotran test. The code and results are

    
  
  Program transpose
  
    integer, parameter :: dp = selected_real_kind(15, 307)

    real(dp), dimension(:, :, :, :), allocatable :: a, b
    Integer :: n1, n2, n3, n4, n, m_iter
    Integer :: l1, l2, l3, l4  
    Integer :: start, finish, rate
    real(dp) :: sum_time
    
    Write(*, *) 'n1, n2, n3, n4?'
    Read(*, *) n1, n2, n3, n4

    Allocate( a ( 1:n1, 1:n2, 1:n3, 1:n4 ) )
    Allocate( b ( 1:n1, 1:n2, 1:n3, 1:n4 ) )
    
    Call Random_number( a )
    
    m_iter = 100

    call system_clock( start, rate ) 
    do n = 1, m_iter  
      b = 0.1*a + 0.2*reshape(a, (/n1, n2, n3, n4/), order = (/2,1,3,4/) ) 
    end do
    call system_clock( finish, rate )
    sum_time = real( finish - start, dp ) / rate  

    write (*,*) 'reshape time', sum_time/m_iter 

    
    call system_clock( start, rate )
    do n = 1, m_iter  
      do l2 = 1, n2
        do l1 = 1, n1
          b(l1,l2,:,:) = 0.1*a(l1,l2,:,:) + 0.2*a(l2,l1,:,:)
        end do
      end do        
    end do
    call system_clock( finish, rate )
    sum_time =  real( finish - start, dp ) / rate  

    write (*,*) 'reduced loop time', sum_time/m_iter

    Call system_clock( start, rate )
    do n = 1, m_iter  
      do l4 = 1, n4
        do l3 = 1, n3     
          do l2 = 1, n2
            do l1 = 1, n1
              b(l1,l2,l3,l4) = 0.1*a(l1,l2,l3,l4) + 0.2*a(l2,l1,l3,l4)
            end do
          end do                    
        end do
      end do        
    end do
    Call system_clock( finish, rate )
    sum_time =  real( finish - start, dp ) / rate  

    write (*,*) 'all loop', sum_time/m_iter 

  End

by gfortran -O3 transpose.f90

 n1, n2, n3, n4?
30 30 30 30
 reshape time   3.3300000000000001E-003
 reduced loop time   1.0620000000000001E-002
 all loop   1.7000000000000001E-003

If I use loops explicily in Fortran without any fancy trick in transpose, Julia is a bit faster.

1 Like
16

Sorry there is one more question towards systematic test for all permutations.

Suppose I would like to compute
0.1 A[i,j,k,l] + 0.2 * A[i,j,l,k] + 0.3 * A[i,k,i,j] + 0.4 * A[i,k,j,i] + 0.5 * A[i,l,j,k]...
Is there any simple way to permutations, than specifying [i,k,j,i] etc (I can try to generate strings as the indices, but it may be a bit complicted)?
I used Get all Permutations of Array?

using BenchmarkTools, Combinatorics

function perms(a) 
    B = reverse(collect(permutations(a)))
    B
end

n=30
A=rand(Float64,(n,n,n,n))
@btime p = perms(A)

Got ERROR: LoadError: OverflowError: 531439031701620000 * 809997 overflowed for type Int64

And, if there is a prefactor equals zero, e.g., 0.0* A[i,l,k,j], can I skip the permutation to i,l,k,j? The permutations will generate all permutations as I understand it (In Python, I can use
np.transpose(A,(0,3,2,1)) to specify the given transpose.)

Edited: found PermutedDimsArray(A, (2,1,3,4)) in test3 is similar to Python transpose

1 Like
17

This has been implicitly addressed, but I’d like to point out that this allocates two intermediate arrays plus one output array. With broadcasting you can fuse the operations to avoid the intermediate arrays, like this

0.1 .*A .+ 0.2 .* C

Or avoid any allocation by reusing C:

C .= 0.1 .*A .+ 0.2 .* C

Also, when benchmarking, interpolate non-const globals:

@btime 0.1 .*$A .+ 0.2 .* $C
2 Likes
18

I extened my Julia code to include more permutations, closer to the objective in the original stackoverflow question.

Here is the Julia code

using BenchmarkTools, Tullio, TensorOperations, Combinatorics, LoopVectorization

function perms(a) 
    B = collect(permutations(a))
    B
end


function perm_add_4(n)
    sum_4 = zeros(n, n, n, n)
    for i = 1:24
        sum_4 = sum_4  .+  factor_p[i] .* PermutedDimsArray(A, P[i])
    end     
end    

function perm_add_5(n)
    sum_5 = zeros(n, n, n, n, n)
    for i = 1:120
        sum_5 = sum_5  .+  factor_p[i] .* PermutedDimsArray(A, P[i])
    end     
end    


P = perms([1,2,3,4])
factor_p = [0.1:0.1:2.4;]
n = 30
A = rand(n, n, n, n)

perm_add_4(n)
@btime perm_add_4(n)


P = perms([1,2,3,4,5])
factor_p = [0.1:0.1:12.0;]
n = 20
A = rand(n, n, n, n, n)
@btime perm_add_5(n)

But, this style is easy to adapt with test3(A) with PermutedDimsArray, harder for test4 ~ test6.

From TensorOperations.jl is often the fastest way to do permutedims on larger arrays. It has a smarter cache-friendly blocking algorithm than the one in Base. But how much this matters of course depends on size & permutation., is there any simple method to adapt with TensorOperations.jl?

1 Like
19

I tried to apply Tullio/tensor for more permutations similar to test4/6. I tested for 6 dimensional array. @tensor works. I did not figure out any elegant solution. I used Python to generate a long Julia code. But, @tullio leads to

ERROR: LoadError: BoundsError: attempt to access 721-element Vector{LoopVectorization.ArrayReferenceMeta} at index [0]

Is there any solution/more elegant approach? Thank you very much



test10(A) = @tullio  B[i,j,k,l,m,n] := 0.1* A[i, j, k, l, m, n] +
0.2* A[i, j, k, l, n, m] +
0.30000000000000004* A[i, j, k, m, l, n] +
0.4* A[i, j, k, n, l, m] +
0.5* A[i, j, k, m, n, l] +
0.6000000000000001* A[i, j, k, n, m, l] +
0.7000000000000001* A[i, j, l, k, m, n] +
0.8* A[i, j, l, k, n, m] +
0.9* A[i, j, m, k, l, n] +
1.0* A[i, j, n, k, l, m] +
1.1* A[i, j, m, k, n, l] +
1.2000000000000002* A[i, j, n, k, m, l] +
1.3* A[i, j, l, m, k, n] +
1.4000000000000001* A[i, j, l, n, k, m] +
1.5* A[i, j, m, l, k, n] +
1.6* A[i, j, n, l, k, m] +
1.7000000000000002* A[i, j, m, n, k, l] +
1.8* A[i, j, n, m, k, l] +
1.9000000000000001* A[i, j, l, m, n, k] +
2.0* A[i, j, l, n, m, k] +
2.1* A[i, j, m, l, n, k] +
2.2* A[i, j, n, l, m, k] +
2.3000000000000003* A[i, j, m, n, l, k] +
2.4000000000000004* A[i, j, n, m, l, k] +
2.5* A[i, k, j, l, m, n] +
2.6* A[i, k, j, l, n, m] +
2.7* A[i, k, j, m, l, n] +
2.8000000000000003* A[i, k, j, n, l, m] +
2.9000000000000004* A[i, k, j, m, n, l] +
3.0* A[i, k, j, n, m, l] +
3.1* A[i, l, j, k, m, n] +
3.2* A[i, l, j, k, n, m] +
3.3000000000000003* A[i, m, j, k, l, n] +
3.4000000000000004* A[i, n, j, k, l, m] +
3.5* A[i, m, j, k, n, l] +
3.6* A[i, n, j, k, m, l] +
3.7* A[i, l, j, m, k, n] +
3.8000000000000003* A[i, l, j, n, k, m] +
3.9000000000000004* A[i, m, j, l, k, n] +
4.0* A[i, n, j, l, k, m] +
4.1000000000000005* A[i, m, j, n, k, l] +
4.2* A[i, n, j, m, k, l] +
4.3* A[i, l, j, m, n, k] +
4.4* A[i, l, j, n, m, k] +
4.5* A[i, m, j, l, n, k] +
4.6000000000000005* A[i, n, j, l, m, k] +
4.7* A[i, m, j, n, l, k] +
4.800000000000001* A[i, n, j, m, l, k] +
4.9* A[i, k, l, j, m, n] +
5.0* A[i, k, l, j, n, m] +
5.1000000000000005* A[i, k, m, j, l, n] +
5.2* A[i, k, n, j, l, m] +
5.300000000000001* A[i, k, m, j, n, l] +
5.4* A[i, k, n, j, m, l] +
5.5* A[i, l, k, j, m, n] +
5.6000000000000005* A[i, l, k, j, n, m] +
5.7* A[i, m, k, j, l, n] +
5.800000000000001* A[i, n, k, j, l, m] +
5.9* A[i, m, k, j, n, l] +
6.0* A[i, n, k, j, m, l] +
6.1000000000000005* A[i, l, m, j, k, n] +
6.2* A[i, l, n, j, k, m] +
6.300000000000001* A[i, m, l, j, k, n] +
6.4* A[i, n, l, j, k, m] +
6.5* A[i, m, n, j, k, l] +
6.6000000000000005* A[i, n, m, j, k, l] +
6.7* A[i, l, m, j, n, k] +
6.800000000000001* A[i, l, n, j, m, k] +
6.9* A[i, m, l, j, n, k] +
7.0* A[i, n, l, j, m, k] +
7.1000000000000005* A[i, m, n, j, l, k] +
7.2* A[i, n, m, j, l, k] +
7.300000000000001* A[i, k, l, m, j, n] +
7.4* A[i, k, l, n, j, m] +
7.5* A[i, k, m, l, j, n] +
7.6000000000000005* A[i, k, n, l, j, m] +
7.7* A[i, k, m, n, j, l] +
7.800000000000001* A[i, k, n, m, j, l] +
7.9* A[i, l, k, m, j, n] +
8.0* A[i, l, k, n, j, m] +
8.1* A[i, m, k, l, j, n] +
8.200000000000001* A[i, n, k, l, j, m] +
8.3* A[i, m, k, n, j, l] +
8.4* A[i, n, k, m, j, l] +
8.5* A[i, l, m, k, j, n] +
8.6* A[i, l, n, k, j, m] +
8.700000000000001* A[i, m, l, k, j, n] +
8.8* A[i, n, l, k, j, m] +
8.9* A[i, m, n, k, j, l] +
9.0* A[i, n, m, k, j, l] +
9.1* A[i, l, m, n, j, k] +
9.200000000000001* A[i, l, n, m, j, k] +
9.3* A[i, m, l, n, j, k] +
9.4* A[i, n, l, m, j, k] +
9.5* A[i, m, n, l, j, k] +
9.600000000000001* A[i, n, m, l, j, k] +
9.700000000000001* A[i, k, l, m, n, j] +
9.8* A[i, k, l, n, m, j] +
9.9* A[i, k, m, l, n, j] +
10.0* A[i, k, n, l, m, j] +
10.100000000000001* A[i, k, m, n, l, j] +
10.200000000000001* A[i, k, n, m, l, j] +
10.3* A[i, l, k, m, n, j] +
10.4* A[i, l, k, n, m, j] +
10.5* A[i, m, k, l, n, j] +
10.600000000000001* A[i, n, k, l, m, j] +
10.700000000000001* A[i, m, k, n, l, j] +
10.8* A[i, n, k, m, l, j] +
10.9* A[i, l, m, k, n, j] +
11.0* A[i, l, n, k, m, j] +
11.100000000000001* A[i, m, l, k, n, j] +
11.200000000000001* A[i, n, l, k, m, j] +
11.3* A[i, m, n, k, l, j] +
11.4* A[i, n, m, k, l, j] +
11.5* A[i, l, m, n, k, j] +
11.600000000000001* A[i, l, n, m, k, j] +
11.700000000000001* A[i, m, l, n, k, j] +
11.8* A[i, n, l, m, k, j] +
11.9* A[i, m, n, l, k, j] +
12.0* A[i, n, m, l, k, j] +
12.100000000000001* A[j, i, k, l, m, n] +
12.200000000000001* A[j, i, k, l, n, m] +
12.3* A[j, i, k, m, l, n] +
12.4* A[j, i, k, n, l, m] +
12.5* A[j, i, k, m, n, l] +
12.600000000000001* A[j, i, k, n, m, l] +
12.700000000000001* A[j, i, l, k, m, n] +
12.8* A[j, i, l, k, n, m] +
12.9* A[j, i, m, k, l, n] +
13.0* A[j, i, n, k, l, m] +
13.100000000000001* A[j, i, m, k, n, l] +
13.200000000000001* A[j, i, n, k, m, l] +
13.3* A[j, i, l, m, k, n] +
13.4* A[j, i, l, n, k, m] +
13.5* A[j, i, m, l, k, n] +
13.600000000000001* A[j, i, n, l, k, m] +
13.700000000000001* A[j, i, m, n, k, l] +
13.8* A[j, i, n, m, k, l] +
13.9* A[j, i, l, m, n, k] +
14.0* A[j, i, l, n, m, k] +
14.100000000000001* A[j, i, m, l, n, k] +
14.200000000000001* A[j, i, n, l, m, k] +
14.3* A[j, i, m, n, l, k] +
14.4* A[j, i, n, m, l, k] +
14.5* A[k, i, j, l, m, n] +
14.600000000000001* A[k, i, j, l, n, m] +
14.700000000000001* A[k, i, j, m, l, n] +
14.8* A[k, i, j, n, l, m] +
14.9* A[k, i, j, m, n, l] +
15.0* A[k, i, j, n, m, l] +
15.100000000000001* A[l, i, j, k, m, n] +
15.200000000000001* A[l, i, j, k, n, m] +
15.3* A[m, i, j, k, l, n] +
15.4* A[n, i, j, k, l, m] +
15.5* A[m, i, j, k, n, l] +
15.600000000000001* A[n, i, j, k, m, l] +
15.700000000000001* A[l, i, j, m, k, n] +
15.8* A[l, i, j, n, k, m] +
15.9* A[m, i, j, l, k, n] +
16.0* A[n, i, j, l, k, m] +
16.1* A[m, i, j, n, k, l] +
16.2* A[n, i, j, m, k, l] +
16.3* A[l, i, j, m, n, k] +
16.400000000000002* A[l, i, j, n, m, k] +
16.5* A[m, i, j, l, n, k] +
16.6* A[n, i, j, l, m, k] +
16.7* A[m, i, j, n, l, k] +
16.8* A[n, i, j, m, l, k] +
16.900000000000002* A[k, i, l, j, m, n] +
17.0* A[k, i, l, j, n, m] +
17.1* A[k, i, m, j, l, n] +
17.2* A[k, i, n, j, l, m] +
17.3* A[k, i, m, j, n, l] +
17.400000000000002* A[k, i, n, j, m, l] +
17.5* A[l, i, k, j, m, n] +
17.6* A[l, i, k, j, n, m] +
17.7* A[m, i, k, j, l, n] +
17.8* A[n, i, k, j, l, m] +
17.900000000000002* A[m, i, k, j, n, l] +
18.0* A[n, i, k, j, m, l] +
18.1* A[l, i, m, j, k, n] +
18.2* A[l, i, n, j, k, m] +
18.3* A[m, i, l, j, k, n] +
18.400000000000002* A[n, i, l, j, k, m] +
18.5* A[m, i, n, j, k, l] +
18.6* A[n, i, m, j, k, l] +
18.7* A[l, i, m, j, n, k] +
18.8* A[l, i, n, j, m, k] +
18.900000000000002* A[m, i, l, j, n, k] +
19.0* A[n, i, l, j, m, k] +
19.1* A[m, i, n, j, l, k] +
19.200000000000003* A[n, i, m, j, l, k] +
19.3* A[k, i, l, m, j, n] +
19.400000000000002* A[k, i, l, n, j, m] +
19.5* A[k, i, m, l, j, n] +
19.6* A[k, i, n, l, j, m] +
19.700000000000003* A[k, i, m, n, j, l] +
19.8* A[k, i, n, m, j, l] +
19.900000000000002* A[l, i, k, m, j, n] +
20.0* A[l, i, k, n, j, m] +
20.1* A[m, i, k, l, j, n] +
20.200000000000003* A[n, i, k, l, j, m] +
20.3* A[m, i, k, n, j, l] +
20.400000000000002* A[n, i, k, m, j, l] +
20.5* A[l, i, m, k, j, n] +
20.6* A[l, i, n, k, j, m] +
20.700000000000003* A[m, i, l, k, j, n] +
20.8* A[n, i, l, k, j, m] +
20.900000000000002* A[m, i, n, k, j, l] +
21.0* A[n, i, m, k, j, l] +
21.1* A[l, i, m, n, j, k] +
21.200000000000003* A[l, i, n, m, j, k] +
21.3* A[m, i, l, n, j, k] +
21.400000000000002* A[n, i, l, m, j, k] +
21.5* A[m, i, n, l, j, k] +
21.6* A[n, i, m, l, j, k] +
21.700000000000003* A[k, i, l, m, n, j] +
21.8* A[k, i, l, n, m, j] +
21.900000000000002* A[k, i, m, l, n, j] +
22.0* A[k, i, n, l, m, j] +
22.1* A[k, i, m, n, l, j] +
22.200000000000003* A[k, i, n, m, l, j] +
22.3* A[l, i, k, m, n, j] +
22.400000000000002* A[l, i, k, n, m, j] +
22.5* A[m, i, k, l, n, j] +
22.6* A[n, i, k, l, m, j] +
22.700000000000003* A[m, i, k, n, l, j] +
22.8* A[n, i, k, m, l, j] +
22.900000000000002* A[l, i, m, k, n, j] +
23.0* A[l, i, n, k, m, j] +
23.1* A[m, i, l, k, n, j] +
23.200000000000003* A[n, i, l, k, m, j] +
23.3* A[m, i, n, k, l, j] +
23.400000000000002* A[n, i, m, k, l, j] +
23.5* A[l, i, m, n, k, j] +
23.6* A[l, i, n, m, k, j] +
23.700000000000003* A[m, i, l, n, k, j] +
23.8* A[n, i, l, m, k, j] +
23.900000000000002* A[m, i, n, l, k, j] +
24.0* A[n, i, m, l, k, j] +
24.1* A[j, k, i, l, m, n] +
24.200000000000003* A[j, k, i, l, n, m] +
24.3* A[j, k, i, m, l, n] +
24.400000000000002* A[j, k, i, n, l, m] +
24.5* A[j, k, i, m, n, l] +
24.6* A[j, k, i, n, m, l] +
24.700000000000003* A[j, l, i, k, m, n] +
24.8* A[j, l, i, k, n, m] +
24.900000000000002* A[j, m, i, k, l, n] +
25.0* A[j, n, i, k, l, m] +
25.1* A[j, m, i, k, n, l] +
25.200000000000003* A[j, n, i, k, m, l] +
25.3* A[j, l, i, m, k, n] +
25.400000000000002* A[j, l, i, n, k, m] +
25.5* A[j, m, i, l, k, n] +
25.6* A[j, n, i, l, k, m] +
25.700000000000003* A[j, m, i, n, k, l] +
25.8* A[j, n, i, m, k, l] +
25.900000000000002* A[j, l, i, m, n, k] +
26.0* A[j, l, i, n, m, k] +
26.1* A[j, m, i, l, n, k] +
26.200000000000003* A[j, n, i, l, m, k] +
26.3* A[j, m, i, n, l, k] +
26.400000000000002* A[j, n, i, m, l, k] +
26.5* A[k, j, i, l, m, n] +
26.6* A[k, j, i, l, n, m] +
26.700000000000003* A[k, j, i, m, l, n] +
26.8* A[k, j, i, n, l, m] +
26.900000000000002* A[k, j, i, m, n, l] +
27.0* A[k, j, i, n, m, l] +
27.1* A[l, j, i, k, m, n] +
27.200000000000003* A[l, j, i, k, n, m] +
27.3* A[m, j, i, k, l, n] +
27.400000000000002* A[n, j, i, k, l, m] +
27.5* A[m, j, i, k, n, l] +
27.6* A[n, j, i, k, m, l] +
27.700000000000003* A[l, j, i, m, k, n] +
27.8* A[l, j, i, n, k, m] +
27.900000000000002* A[m, j, i, l, k, n] +
28.0* A[n, j, i, l, k, m] +
28.1* A[m, j, i, n, k, l] +
28.200000000000003* A[n, j, i, m, k, l] +
28.3* A[l, j, i, m, n, k] +
28.400000000000002* A[l, j, i, n, m, k] +
28.5* A[m, j, i, l, n, k] +
28.6* A[n, j, i, l, m, k] +
28.700000000000003* A[m, j, i, n, l, k] +
28.8* A[n, j, i, m, l, k] +
28.900000000000002* A[k, l, i, j, m, n] +
29.0* A[k, l, i, j, n, m] +
29.1* A[k, m, i, j, l, n] +
29.200000000000003* A[k, n, i, j, l, m] +
29.3* A[k, m, i, j, n, l] +
29.400000000000002* A[k, n, i, j, m, l] +
29.5* A[l, k, i, j, m, n] +
29.6* A[l, k, i, j, n, m] +
29.700000000000003* A[m, k, i, j, l, n] +
29.8* A[n, k, i, j, l, m] +
29.900000000000002* A[m, k, i, j, n, l] +
30.0* A[n, k, i, j, m, l] +
30.1* A[l, m, i, j, k, n] +
30.200000000000003* A[l, n, i, j, k, m] +
30.3* A[m, l, i, j, k, n] +
30.400000000000002* A[n, l, i, j, k, m] +
30.5* A[m, n, i, j, k, l] +
30.6* A[n, m, i, j, k, l] +
30.700000000000003* A[l, m, i, j, n, k] +
30.8* A[l, n, i, j, m, k] +
30.900000000000002* A[m, l, i, j, n, k] +
31.0* A[n, l, i, j, m, k] +
31.1* A[m, n, i, j, l, k] +
31.200000000000003* A[n, m, i, j, l, k] +
31.3* A[k, l, i, m, j, n] +
31.400000000000002* A[k, l, i, n, j, m] +
31.5* A[k, m, i, l, j, n] +
31.6* A[k, n, i, l, j, m] +
31.700000000000003* A[k, m, i, n, j, l] +
31.8* A[k, n, i, m, j, l] +
31.900000000000002* A[l, k, i, m, j, n] +
32.0* A[l, k, i, n, j, m] +
32.1* A[m, k, i, l, j, n] +
32.2* A[n, k, i, l, j, m] +
32.300000000000004* A[m, k, i, n, j, l] +
32.4* A[n, k, i, m, j, l] +
32.5* A[l, m, i, k, j, n] +
32.6* A[l, n, i, k, j, m] +
32.7* A[m, l, i, k, j, n] +
32.800000000000004* A[n, l, i, k, j, m] +
32.9* A[m, n, i, k, j, l] +
33.0* A[n, m, i, k, j, l] +
33.1* A[l, m, i, n, j, k] +
33.2* A[l, n, i, m, j, k] +
33.300000000000004* A[m, l, i, n, j, k] +
33.4* A[n, l, i, m, j, k] +
33.5* A[m, n, i, l, j, k] +
33.6* A[n, m, i, l, j, k] +
33.7* A[k, l, i, m, n, j] +
33.800000000000004* A[k, l, i, n, m, j] +
33.9* A[k, m, i, l, n, j] +
34.0* A[k, n, i, l, m, j] +
34.1* A[k, m, i, n, l, j] +
34.2* A[k, n, i, m, l, j] +
34.300000000000004* A[l, k, i, m, n, j] +
34.4* A[l, k, i, n, m, j] +
34.5* A[m, k, i, l, n, j] +
34.6* A[n, k, i, l, m, j] +
34.7* A[m, k, i, n, l, j] +
34.800000000000004* A[n, k, i, m, l, j] +
34.9* A[l, m, i, k, n, j] +
35.0* A[l, n, i, k, m, j] +
35.1* A[m, l, i, k, n, j] +
35.2* A[n, l, i, k, m, j] +
35.300000000000004* A[m, n, i, k, l, j] +
35.4* A[n, m, i, k, l, j] +
35.5* A[l, m, i, n, k, j] +
35.6* A[l, n, i, m, k, j] +
35.7* A[m, l, i, n, k, j] +
35.800000000000004* A[n, l, i, m, k, j] +
35.9* A[m, n, i, l, k, j] +
36.0* A[n, m, i, l, k, j] +
36.1* A[j, k, l, i, m, n] +
36.2* A[j, k, l, i, n, m] +
36.300000000000004* A[j, k, m, i, l, n] +
36.4* A[j, k, n, i, l, m] +
36.5* A[j, k, m, i, n, l] +
36.6* A[j, k, n, i, m, l] +
36.7* A[j, l, k, i, m, n] +
36.800000000000004* A[j, l, k, i, n, m] +
36.9* A[j, m, k, i, l, n] +
37.0* A[j, n, k, i, l, m] +
37.1* A[j, m, k, i, n, l] +
37.2* A[j, n, k, i, m, l] +
37.300000000000004* A[j, l, m, i, k, n] +
37.4* A[j, l, n, i, k, m] +
37.5* A[j, m, l, i, k, n] +
37.6* A[j, n, l, i, k, m] +
37.7* A[j, m, n, i, k, l] +
37.800000000000004* A[j, n, m, i, k, l] +
37.9* A[j, l, m, i, n, k] +
38.0* A[j, l, n, i, m, k] +
38.1* A[j, m, l, i, n, k] +
38.2* A[j, n, l, i, m, k] +
38.300000000000004* A[j, m, n, i, l, k] +
38.400000000000006* A[j, n, m, i, l, k] +
38.5* A[k, j, l, i, m, n] +
38.6* A[k, j, l, i, n, m] +
38.7* A[k, j, m, i, l, n] +
38.800000000000004* A[k, j, n, i, l, m] +
38.900000000000006* A[k, j, m, i, n, l] +
39.0* A[k, j, n, i, m, l] +
39.1* A[l, j, k, i, m, n] +
39.2* A[l, j, k, i, n, m] +
39.300000000000004* A[m, j, k, i, l, n] +
39.400000000000006* A[n, j, k, i, l, m] +
39.5* A[m, j, k, i, n, l] +
39.6* A[n, j, k, i, m, l] +
39.7* A[l, j, m, i, k, n] +
39.800000000000004* A[l, j, n, i, k, m] +
39.900000000000006* A[m, j, l, i, k, n] +
40.0* A[n, j, l, i, k, m] +
40.1* A[m, j, n, i, k, l] +
40.2* A[n, j, m, i, k, l] +
40.300000000000004* A[l, j, m, i, n, k] +
40.400000000000006* A[l, j, n, i, m, k] +
40.5* A[m, j, l, i, n, k] +
40.6* A[n, j, l, i, m, k] +
40.7* A[m, j, n, i, l, k] +
40.800000000000004* A[n, j, m, i, l, k] +
40.900000000000006* A[k, l, j, i, m, n] +
41.0* A[k, l, j, i, n, m] +
41.1* A[k, m, j, i, l, n] +
41.2* A[k, n, j, i, l, m] +
41.300000000000004* A[k, m, j, i, n, l] +
41.400000000000006* A[k, n, j, i, m, l] +
41.5* A[l, k, j, i, m, n] +
41.6* A[l, k, j, i, n, m] +
41.7* A[m, k, j, i, l, n] +
41.800000000000004* A[n, k, j, i, l, m] +
41.900000000000006* A[m, k, j, i, n, l] +
42.0* A[n, k, j, i, m, l] +
42.1* A[l, m, j, i, k, n] +
42.2* A[l, n, j, i, k, m] +
42.300000000000004* A[m, l, j, i, k, n] +
42.400000000000006* A[n, l, j, i, k, m] +
42.5* A[m, n, j, i, k, l] +
42.6* A[n, m, j, i, k, l] +
42.7* A[l, m, j, i, n, k] +
42.800000000000004* A[l, n, j, i, m, k] +
42.900000000000006* A[m, l, j, i, n, k] +
43.0* A[n, l, j, i, m, k] +
43.1* A[m, n, j, i, l, k] +
43.2* A[n, m, j, i, l, k] +
43.300000000000004* A[k, l, m, i, j, n] +
43.400000000000006* A[k, l, n, i, j, m] +
43.5* A[k, m, l, i, j, n] +
43.6* A[k, n, l, i, j, m] +
43.7* A[k, m, n, i, j, l] +
43.800000000000004* A[k, n, m, i, j, l] +
43.900000000000006* A[l, k, m, i, j, n] +
44.0* A[l, k, n, i, j, m] +
44.1* A[m, k, l, i, j, n] +
44.2* A[n, k, l, i, j, m] +
44.300000000000004* A[m, k, n, i, j, l] +
44.400000000000006* A[n, k, m, i, j, l] +
44.5* A[l, m, k, i, j, n] +
44.6* A[l, n, k, i, j, m] +
44.7* A[m, l, k, i, j, n] +
44.800000000000004* A[n, l, k, i, j, m] +
44.900000000000006* A[m, n, k, i, j, l] +
45.0* A[n, m, k, i, j, l] +
45.1* A[l, m, n, i, j, k] +
45.2* A[l, n, m, i, j, k] +
45.300000000000004* A[m, l, n, i, j, k] +
45.400000000000006* A[n, l, m, i, j, k] +
45.5* A[m, n, l, i, j, k] +
45.6* A[n, m, l, i, j, k] +
45.7* A[k, l, m, i, n, j] +
45.800000000000004* A[k, l, n, i, m, j] +
45.900000000000006* A[k, m, l, i, n, j] +
46.0* A[k, n, l, i, m, j] +
46.1* A[k, m, n, i, l, j] +
46.2* A[k, n, m, i, l, j] +
46.300000000000004* A[l, k, m, i, n, j] +
46.400000000000006* A[l, k, n, i, m, j] +
46.5* A[m, k, l, i, n, j] +
46.6* A[n, k, l, i, m, j] +
46.7* A[m, k, n, i, l, j] +
46.800000000000004* A[n, k, m, i, l, j] +
46.900000000000006* A[l, m, k, i, n, j] +
47.0* A[l, n, k, i, m, j] +
47.1* A[m, l, k, i, n, j] +
47.2* A[n, l, k, i, m, j] +
47.300000000000004* A[m, n, k, i, l, j] +
47.400000000000006* A[n, m, k, i, l, j] +
47.5* A[l, m, n, i, k, j] +
47.6* A[l, n, m, i, k, j] +
47.7* A[m, l, n, i, k, j] +
47.800000000000004* A[n, l, m, i, k, j] +
47.900000000000006* A[m, n, l, i, k, j] +
48.0* A[n, m, l, i, k, j] +
48.1* A[j, k, l, m, i, n] +
48.2* A[j, k, l, n, i, m] +
48.300000000000004* A[j, k, m, l, i, n] +
48.400000000000006* A[j, k, n, l, i, m] +
48.5* A[j, k, m, n, i, l] +
48.6* A[j, k, n, m, i, l] +
48.7* A[j, l, k, m, i, n] +
48.800000000000004* A[j, l, k, n, i, m] +
48.900000000000006* A[j, m, k, l, i, n] +
49.0* A[j, n, k, l, i, m] +
49.1* A[j, m, k, n, i, l] +
49.2* A[j, n, k, m, i, l] +
49.300000000000004* A[j, l, m, k, i, n] +
49.400000000000006* A[j, l, n, k, i, m] +
49.5* A[j, m, l, k, i, n] +
49.6* A[j, n, l, k, i, m] +
49.7* A[j, m, n, k, i, l] +
49.800000000000004* A[j, n, m, k, i, l] +
49.900000000000006* A[j, l, m, n, i, k] +
50.0* A[j, l, n, m, i, k] +
50.1* A[j, m, l, n, i, k] +
50.2* A[j, n, l, m, i, k] +
50.300000000000004* A[j, m, n, l, i, k] +
50.400000000000006* A[j, n, m, l, i, k] +
50.5* A[k, j, l, m, i, n] +
50.6* A[k, j, l, n, i, m] +
50.7* A[k, j, m, l, i, n] +
50.800000000000004* A[k, j, n, l, i, m] +
50.900000000000006* A[k, j, m, n, i, l] +
51.0* A[k, j, n, m, i, l] +
51.1* A[l, j, k, m, i, n] +
51.2* A[l, j, k, n, i, m] +
51.300000000000004* A[m, j, k, l, i, n] +
51.400000000000006* A[n, j, k, l, i, m] +
51.5* A[m, j, k, n, i, l] +
51.6* A[n, j, k, m, i, l] +
51.7* A[l, j, m, k, i, n] +
51.800000000000004* A[l, j, n, k, i, m] +
51.900000000000006* A[m, j, l, k, i, n] +
52.0* A[n, j, l, k, i, m] +
52.1* A[m, j, n, k, i, l] +
52.2* A[n, j, m, k, i, l] +
52.300000000000004* A[l, j, m, n, i, k] +
52.400000000000006* A[l, j, n, m, i, k] +
52.5* A[m, j, l, n, i, k] +
52.6* A[n, j, l, m, i, k] +
52.7* A[m, j, n, l, i, k] +
52.800000000000004* A[n, j, m, l, i, k] +
52.900000000000006* A[k, l, j, m, i, n] +
53.0* A[k, l, j, n, i, m] +
53.1* A[k, m, j, l, i, n] +
53.2* A[k, n, j, l, i, m] +
53.300000000000004* A[k, m, j, n, i, l] +
53.400000000000006* A[k, n, j, m, i, l] +
53.5* A[l, k, j, m, i, n] +
53.6* A[l, k, j, n, i, m] +
53.7* A[m, k, j, l, i, n] +
53.800000000000004* A[n, k, j, l, i, m] +
53.900000000000006* A[m, k, j, n, i, l] +
54.0* A[n, k, j, m, i, l] +
54.1* A[l, m, j, k, i, n] +
54.2* A[l, n, j, k, i, m] +
54.300000000000004* A[m, l, j, k, i, n] +
54.400000000000006* A[n, l, j, k, i, m] +
54.5* A[m, n, j, k, i, l] +
54.6* A[n, m, j, k, i, l] +
54.7* A[l, m, j, n, i, k] +
54.800000000000004* A[l, n, j, m, i, k] +
54.900000000000006* A[m, l, j, n, i, k] +
55.0* A[n, l, j, m, i, k] +
55.1* A[m, n, j, l, i, k] +
55.2* A[n, m, j, l, i, k] +
55.300000000000004* A[k, l, m, j, i, n] +
55.400000000000006* A[k, l, n, j, i, m] +
55.5* A[k, m, l, j, i, n] +
55.6* A[k, n, l, j, i, m] +
55.7* A[k, m, n, j, i, l] +
55.800000000000004* A[k, n, m, j, i, l] +
55.900000000000006* A[l, k, m, j, i, n] +
56.0* A[l, k, n, j, i, m] +
56.1* A[m, k, l, j, i, n] +
56.2* A[n, k, l, j, i, m] +
56.300000000000004* A[m, k, n, j, i, l] +
56.400000000000006* A[n, k, m, j, i, l] +
56.5* A[l, m, k, j, i, n] +
56.6* A[l, n, k, j, i, m] +
56.7* A[m, l, k, j, i, n] +
56.800000000000004* A[n, l, k, j, i, m] +
56.900000000000006* A[m, n, k, j, i, l] +
57.0* A[n, m, k, j, i, l] +
57.1* A[l, m, n, j, i, k] +
57.2* A[l, n, m, j, i, k] +
57.300000000000004* A[m, l, n, j, i, k] +
57.400000000000006* A[n, l, m, j, i, k] +
57.5* A[m, n, l, j, i, k] +
57.6* A[n, m, l, j, i, k] +
57.7* A[k, l, m, n, i, j] +
57.800000000000004* A[k, l, n, m, i, j] +
57.900000000000006* A[k, m, l, n, i, j] +
58.0* A[k, n, l, m, i, j] +
58.1* A[k, m, n, l, i, j] +
58.2* A[k, n, m, l, i, j] +
58.300000000000004* A[l, k, m, n, i, j] +
58.400000000000006* A[l, k, n, m, i, j] +
58.5* A[m, k, l, n, i, j] +
58.6* A[n, k, l, m, i, j] +
58.7* A[m, k, n, l, i, j] +
58.800000000000004* A[n, k, m, l, i, j] +
58.900000000000006* A[l, m, k, n, i, j] +
59.0* A[l, n, k, m, i, j] +
59.1* A[m, l, k, n, i, j] +
59.2* A[n, l, k, m, i, j] +
59.300000000000004* A[m, n, k, l, i, j] +
59.400000000000006* A[n, m, k, l, i, j] +
59.5* A[l, m, n, k, i, j] +
59.6* A[l, n, m, k, i, j] +
59.7* A[m, l, n, k, i, j] +
59.800000000000004* A[n, l, m, k, i, j] +
59.900000000000006* A[m, n, l, k, i, j] +
60.0* A[n, m, l, k, i, j] +
60.1* A[j, k, l, m, n, i] +
60.2* A[j, k, l, n, m, i] +
60.300000000000004* A[j, k, m, l, n, i] +
60.400000000000006* A[j, k, n, l, m, i] +
60.5* A[j, k, m, n, l, i] +
60.6* A[j, k, n, m, l, i] +
60.7* A[j, l, k, m, n, i] +
60.800000000000004* A[j, l, k, n, m, i] +
60.900000000000006* A[j, m, k, l, n, i] +
61.0* A[j, n, k, l, m, i] +
61.1* A[j, m, k, n, l, i] +
61.2* A[j, n, k, m, l, i] +
61.300000000000004* A[j, l, m, k, n, i] +
61.400000000000006* A[j, l, n, k, m, i] +
61.5* A[j, m, l, k, n, i] +
61.6* A[j, n, l, k, m, i] +
61.7* A[j, m, n, k, l, i] +
61.800000000000004* A[j, n, m, k, l, i] +
61.900000000000006* A[j, l, m, n, k, i] +
62.0* A[j, l, n, m, k, i] +
62.1* A[j, m, l, n, k, i] +
62.2* A[j, n, l, m, k, i] +
62.300000000000004* A[j, m, n, l, k, i] +
62.400000000000006* A[j, n, m, l, k, i] +
62.5* A[k, j, l, m, n, i] +
62.6* A[k, j, l, n, m, i] +
62.7* A[k, j, m, l, n, i] +
62.800000000000004* A[k, j, n, l, m, i] +
62.900000000000006* A[k, j, m, n, l, i] +
63.0* A[k, j, n, m, l, i] +
63.1* A[l, j, k, m, n, i] +
63.2* A[l, j, k, n, m, i] +
63.300000000000004* A[m, j, k, l, n, i] +
63.400000000000006* A[n, j, k, l, m, i] +
63.5* A[m, j, k, n, l, i] +
63.6* A[n, j, k, m, l, i] +
63.7* A[l, j, m, k, n, i] +
63.800000000000004* A[l, j, n, k, m, i] +
63.900000000000006* A[m, j, l, k, n, i] +
64.0* A[n, j, l, k, m, i] +
64.10000000000001* A[m, j, n, k, l, i] +
64.2* A[n, j, m, k, l, i] +
64.3* A[l, j, m, n, k, i] +
64.4* A[l, j, n, m, k, i] +
64.5* A[m, j, l, n, k, i] +
64.60000000000001* A[n, j, l, m, k, i] +
64.7* A[m, j, n, l, k, i] +
64.8* A[n, j, m, l, k, i] +
64.9* A[k, l, j, m, n, i] +
65.0* A[k, l, j, n, m, i] +
65.10000000000001* A[k, m, j, l, n, i] +
65.2* A[k, n, j, l, m, i] +
65.3* A[k, m, j, n, l, i] +
65.4* A[k, n, j, m, l, i] +
65.5* A[l, k, j, m, n, i] +
65.60000000000001* A[l, k, j, n, m, i] +
65.7* A[m, k, j, l, n, i] +
65.8* A[n, k, j, l, m, i] +
65.9* A[m, k, j, n, l, i] +
66.0* A[n, k, j, m, l, i] +
66.10000000000001* A[l, m, j, k, n, i] +
66.2* A[l, n, j, k, m, i] +
66.3* A[m, l, j, k, n, i] +
66.4* A[n, l, j, k, m, i] +
66.5* A[m, n, j, k, l, i] +
66.60000000000001* A[n, m, j, k, l, i] +
66.7* A[l, m, j, n, k, i] +
66.8* A[l, n, j, m, k, i] +
66.9* A[m, l, j, n, k, i] +
67.0* A[n, l, j, m, k, i] +
67.10000000000001* A[m, n, j, l, k, i] +
67.2* A[n, m, j, l, k, i] +
67.3* A[k, l, m, j, n, i] +
67.4* A[k, l, n, j, m, i] +
67.5* A[k, m, l, j, n, i] +
67.60000000000001* A[k, n, l, j, m, i] +
67.7* A[k, m, n, j, l, i] +
67.8* A[k, n, m, j, l, i] +
67.9* A[l, k, m, j, n, i] +
68.0* A[l, k, n, j, m, i] +
68.10000000000001* A[m, k, l, j, n, i] +
68.2* A[n, k, l, j, m, i] +
68.3* A[m, k, n, j, l, i] +
68.4* A[n, k, m, j, l, i] +
68.5* A[l, m, k, j, n, i] +
68.60000000000001* A[l, n, k, j, m, i] +
68.7* A[m, l, k, j, n, i] +
68.8* A[n, l, k, j, m, i] +
68.9* A[m, n, k, j, l, i] +
69.0* A[n, m, k, j, l, i] +
69.10000000000001* A[l, m, n, j, k, i] +
69.2* A[l, n, m, j, k, i] +
69.3* A[m, l, n, j, k, i] +
69.4* A[n, l, m, j, k, i] +
69.5* A[m, n, l, j, k, i] +
69.60000000000001* A[n, m, l, j, k, i] +
69.7* A[k, l, m, n, j, i] +
69.8* A[k, l, n, m, j, i] +
69.9* A[k, m, l, n, j, i] +
70.0* A[k, n, l, m, j, i] +
70.10000000000001* A[k, m, n, l, j, i] +
70.2* A[k, n, m, l, j, i] +
70.3* A[l, k, m, n, j, i] +
70.4* A[l, k, n, m, j, i] +
70.5* A[m, k, l, n, j, i] +
70.60000000000001* A[n, k, l, m, j, i] +
70.7* A[m, k, n, l, j, i] +
70.8* A[n, k, m, l, j, i] +
70.9* A[l, m, k, n, j, i] +
71.0* A[l, n, k, m, j, i] +
71.10000000000001* A[m, l, k, n, j, i] +
71.2* A[n, l, k, m, j, i] +
71.3* A[m, n, k, l, j, i] +
71.4* A[n, m, k, l, j, i] +
71.5* A[l, m, n, k, j, i] +
71.60000000000001* A[l, n, m, k, j, i] +
71.7* A[m, l, n, k, j, i] +
71.8* A[n, l, m, k, j, i] +
71.9* A[m, n, l, k, j, i] +
72.0* A[n, m, l, k, j, i]

@btime test10(A) setup=(n=10; A=rand(n,n,n,n,n,n))
#println( B[1,1,1,1,1,1]/A[1,1,1,1,1,1])
20

700 terms is a lot. Maybe you want something more like this:

julia> begin
         B = zero.(A)
         for (i,p) in enumerate(permutations(1:ndims(A)))
           B .+= i/10 .* PermutedDimsArray(A, Tuple(p))
         end
         B
       end
2×2×2×2 Array{Float64, 4}:
1 Like