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Begin debugging trilinear cases
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@mewilhel
mewilhel committed Nov 6, 2020
1 parent cbe7bc4 commit e15f0b8
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Showing 4 changed files with 297 additions and 52 deletions.
5 changes: 3 additions & 2 deletions src/McCormick.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -63,7 +63,7 @@ export MC, cc, cv, Intv, lo, hi, cc_grad, cv_grad, cnst, +, -, *, /, convert,
relu, param_relu, leaky_relu, maxsig, maxtanh, softplus, pentanh,
sigmoid, bisigmoid, softsign, gelu, elu, selu, swish1,
positive, negative, lower_bnd, upper_bnd, bnd, xlogx,
<, <=, ==, fma, cbrt, abs2, sinpi, cospi, arh, xexpax
<, <=, ==, fma, cbrt, abs2, sinpi, cospi, arh, xexpax, trilinear

# Export kernel operators
export plus_kernel, minus_kernel, mult_kernel, div_kernel, max_kernel,
Expand All @@ -83,7 +83,8 @@ export plus_kernel, minus_kernel, mult_kernel, div_kernel, max_kernel,
bisigmoid_kernel, softsign_kernel, gelu_kernel, elu_kernel, selu_kernel,
swish1_kernel, positive_kernel, negative_kernel, lower_bnd_kernel,
upper_bnd_kernel, bnd_kernel, xlogx_kernel, fma_kernel, cbrt_kernel,
abs2_kernel, sinpi_kernel, cospi_kernel, arh_kernel, xexpax_kernel
abs2_kernel, sinpi_kernel, cospi_kernel, arh_kernel, xexpax_kernel,
trilinear_kernel

export seed_gradient, RelaxTag, NS, MV, Diff

Expand Down
1 change: 1 addition & 0 deletions src/forward_operators/forward.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -11,3 +11,4 @@ include("other.jl")
include("activation_functions.jl")
include("set_bounds.jl")
include("comparison.jl")
include("trilinear.jl")
182 changes: 132 additions & 50 deletions src/forward_operators/trilinear.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -57,7 +57,7 @@ function coeff6(i::Int64, a::Float64, b::Float64, c::Float64, d::Float64, e::Flo
(i === 3) && (return c)
(i === 4) && (return d)
(i === 5) && (return e)
(return f
return f
end

macro unpack_trilinear_bnd()
Expand All @@ -72,7 +72,7 @@ macro unpack_trilinear_bnd()
yzLL = yL*zL; yzLU = yL*zU;
yzUL = yU*zL; yzUU = yU*zU
xyzLLL = xyLL*zL; xyzLLU = xyLL*zU
xyzLUL = xyLU*zL; xzyULL = xyUL*zL
xyzLUL = xyLU*zL; xyzULL = xyUL*zL
xyzLUU = xyLU*zU; xyzULU = xyUL*zU
xyzUUL = xyUU*zL; xyzUUU = xyUU*zU
end)
Expand All @@ -86,12 +86,12 @@ macro unpack_trilinear_end()
coeff6(cvind, cv_az1, cv_az2, cv_az3, cv_az4, cv_az5, cv_az6)*z.cv_grad

cc, ccind = min6(cc1, cc2, cc3, cc4, cc5, cc6)
cc_grad = coeff6(cvind, $(cc_ax[1]), $(cc_ax[2]), $(cc_ax[3]), $(cc_ax[4]), $(cc_ax[5]), $(cc_ax[6]))*x.cc_grad +
coeff6(cvind, $(cc_ay[1]), $(cc_ay[2]), $(cc_ay[3]), $(cc_ay[4]), $(cc_ay[5]), $(cc_ay[6]))*y.cc_grad +
coeff6(cvind, $(cc_az[1]), $(cc_az[2]), $(cc_az[3]), $(cc_az[4]), $(cc_az[5]), $(cc_az[6]))*z.cc_grad
cc_grad = coeff6(cvind, cc_ax1, cc_ax2, cc_ax3, cc_ax4, cc_ax5, cc_ax6)*x.cc_grad +
coeff6(cvind, cc_ay1, cc_ay2, cc_ay3, cc_ay4, cc_ay5, cc_ay6)*y.cc_grad +
coeff6(cvind, cc_az1, cc_az2, cc_az3, cc_az4, cc_az5, cc_az6)*z.cc_grad

MC{N,T}(cv, cc, z, cv_grad, cc_grad, x.cnst && y.cnst && z.cnst)
end
MC{N,T}(cv, cc, q, cv_grad, cc_grad, x.cnst && y.cnst && z.cnst)
end)
end

function is_tri_case_1(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}) where {N,T<:RelaxTag}
Expand Down Expand Up @@ -130,47 +130,49 @@ Case 3.1 + Case 4.1 of Meyer-Floudas 2004
function trilinear_case_1(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

delZ = zU - zL
θcv = xyzLUU - xyzUUL - xyzLLU + xyzULU
θcc = xyzULL - xyzUUU - xyzLLL + xyzLUL
@show "RAN 1"

delX = xU - xL
θcv1 = xyzUUL - xyzLUU - xyzULL + xyzULU
θcv2 = xyzLLU - xyzULL - xyzLUU + xyzLUL

# define cv and coefficients
cv_b1 = -2.0*xyzUUU
cv_b2 = -(xyzLUL + xyzLUU)
cv_b3 = -(xyzLUL + xyzLLL)
cv_b4 = -(xyzULU + xyzULL)
cv_b5 = -(xyzULL + xyzLLL)
cv_b6 = -θcv*zL/delZ - xyzLUU - xyzULL + xyzUUL
cv_b1 = -2.0*xyzLLL
cv_b2 = -2.0*xyzUUU
cv_b3 = -(xyzLLU + xyzULU)
cv_b4 = -(xyzUUL + xyzLUL)
cv_b5 = -θcv1*xL/delX - xyzUUL - xyzULU + xyzLUU
cv_b6 = θcv2*xU/delX - xyzLLU - xyzLUL + xyzULL

cv_ax1, cv_ax2, cv_ax3, cv_ax4, cv_ax5, cv_ax6 = yzUU, yzUL, yzUL, yzLU, yzLL, yzLU
cv_ay1, cv_ay2, cv_ay3, cv_ay4, cv_ay5, cv_ay6 = xzUU, xzLU, xzLL, xzUL, xzLL, xzLU
cv_az1, cv_az2, cv_az3, cv_az4, cv_az5, cv_az6 = xyUU, xyLU, xyLL, xyUL, xyLL, θcv/delZ
cv_ax1, cv_ax2, cv_ax3, cv_ax4, cv_ax5, cv_ax6 = yzLL, yzUU, yzLU, yzUL, θcv1/delX, -θcv2/delX
cv_ay1, cv_ay2, cv_ay3, cv_ay4, cv_ay5, cv_ay6 = xzLL, xzUU, xzLU, xzUL, xzUL, xzLU
cv_az1, cv_az2, cv_az3, cv_az4, cv_az5, cv_az6 = xyLL, xyUU, xyUL, xyLU, xyUL, xyLU

cv1 = cv_ax1*x.cv + cv_ay1*y.cv + cv_az1*z.cv + cv_b1
cv2 = cv_ax2*x.cv + cv_ay2*y.cv + cv_az2*z.cv + cv_b2
cv3 = cv_ax3*x.cv + cv_ay3*y.cv + cv_az3*z.cv + cv_b3
cv4 = cv_ax4*x.cv + cv_ay4*y.cv + cv_az4*z.cv + cv_b4
cv5 = cv_ax5*ifelse(cv_ax5 > 0.0, x.cv, -x.cc) + cv_ay5*y.cv + cv_az5*z.cv + cv_b5
cv6 = cv_ax6*ifelse(cv_ax6 > 0.0, x.cv, -x.cc) + cv_ay6*y.cv + cv_az6*z.cv + cv_b6

# define cc and coefficients
cc_b1 = -2.0*xyzUUL
cc_b2 = -(xyzULU + xyzULL)
cc_b3 = -(xyzLUU + xyzLLU)
cc_b1 = -(xyzUUL + xyzULL)
cc_b2 = -(xyzUUL + xyzLUL)
cc_b3 = -(xyzULU + xyzULL)
cc_b4 = -(xyzLUU + xyzLUL)
cc_b5 = -(xyzULU + xyzLLU)
cc_b6 = θcc*zU/delZ - xyzULL - xyzLUL + xyzUUU
cc_b6 = -(xyzLUU + xyzLLU)

cc_ax1, cc_ax2, cc_ax3, cc_ax4, cc_ax5, cc_ax6 = yzUL, yzLL, yzUU, yzUU, yzLU, yzLU
cc_ay1, cc_ay2, cc_ay3, cc_ay4, cc_ay5, cc_ay6 = xzUL, xzUU, xzLU, xzLL, xzUU, xzLU
cc_az1, cc_az2, cc_az3, cc_az4, cc_az5, cc_az6 = xyUU, xyUL, xyLL, xyLU, xyLL, θcc/delZ

cv1 = cv_ax1*x.cv + cv_ay1*y.cv + cv_az1*z.cv + cv_b1
cv2 = cv_ax2*x.cv + cv_ay2*y.cv + cv_az2*z.cc + cv_b2
cv3 = cv_ax3*x.cv + cv_ay3*y.cc + cv_az3*z.cc + cv_b3
cv4 = cv_ax4*x.cv + cv_ay4*y.cc + cv_az4*z.cv + cv_b4
cv5 = cv_ax5*x.cv + cv_ay5*y.cc + cv_az5*z.cc + cv_b5
cv6 = cv_ax6*x.cv + cv_ay6*y.cv + cv_az6*ifelse(cv_az6 > 0.0, z.cv, -z.cc) + cv_b6
cc_ax1, cc_ax2, cc_ax3, cc_ax4, cc_ax5, cc_ax6 = yzLL, yzUL, yzLL, yzUU, yzLU, yzUU
cc_ay1, cc_ay2, cc_ay3, cc_ay4, cc_ay5, cc_ay6 = xzUL, xzLL, xzUU, xzLL, xzUU, xzLU
cc_az1, cc_az2, cc_az3, cc_az4, cc_az5, cc_az6 = xyUU, xyUU, xyUL, xyLU, xyLL, xyLL

cc1 = cc_ax1*x.cc + cc_ay1*y.cc + cc_az1*z.cc + cc_b1
cc2 = cc_ax2*x.cc + cc_ay2*y.cc + cc_az2*z.cv + cc_b2
cc3 = cc_ax3*x.cc + cc_ay3*y.cc + cc_az3*z.cv + cc_b3
cc2 = cc_ax2*x.cc + cc_ay2*y.cc + cc_az2*z.cc + cc_b2
cc3 = cc_ax3*x.cc + cc_ay3*y.cc + cc_az3*z.cc + cc_b3
cc4 = cc_ax4*x.cc + cc_ay4*y.cc + cc_az4*z.cc + cc_b4
cc5 = cc_ax5*x.cc + cc_ay5*y.cc + cc_az5*z.cv + cc_b5
cc6 = cc_ax6*x.cc + cc_ay6*y.cc + cc_az6*ifelse(cc_az6 > 0.0, z.cc, -z.cv) + cc_b6
cc5 = cc_ax5*x.cc + cc_ay5*y.cc + cc_az5*z.cc + cc_b5
cc6 = cc_ax6*x.cc + cc_ay6*y.cc + cc_az6*z.cc + cc_b6

@unpack_trilinear_end()
end
Expand All @@ -183,6 +185,8 @@ Case 3.2 + Case 4.2 of Meyer-Floudas 2004
function trilinear_case_2(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 2"

delZ = zU - zL
θcv = xyzLUU - xyzUUL - xyzLLU + xyzULU
θcc = xyzULL - xyzUUU - xyzLLL + xyzLUL
Expand Down Expand Up @@ -236,6 +240,8 @@ Case 3.3 + Case 4.3 of Meyer-Floudas 2004
function trilinear_case_3(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 3"

delY = yU - yL
delZ = zU - zL
θcv1 = xyzLUU - xyzUUL - xyzLLU + xyzULU
Expand Down Expand Up @@ -320,6 +326,8 @@ Case 3.4 + Case 4.4 of Meyer-Floudas 2004
function trilinear_case_4(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 4"

delX = xU - xL
delY = yU - yL
delZ = zU - zL
Expand All @@ -340,11 +348,11 @@ function trilinear_case_4(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float6
cv_b3 = -2.0*xyzUUU
cv_b4 = -2.0*xyzULL
cv_b5 = θcv4
#cv_b6 = TODO? eq 6 not in paper
cv_b6 = Inf #TODO? eq 6 not in paper

cv_ax1, cv_ax2, cv_ax3, cv_ax4, cv_ax5 = yzUL, yzLU, yzUU, yzLL, θcv1 # TODO? eq 6 not in paper
cv_ay1, cv_ay2, cv_ay3, cv_ay4, cv_ay5 = xzLL, xzLU, xzUU, xzUL, θcv2 # TODO? eq 6 not in paper
cv_az1, cv_az2, cv_az3, cv_az4, cv_az5 = xyLU, xyLL, xyUU, xyUL, θcv3 # TODO? eq 6 not in paper
cv_ax1, cv_ax2, cv_ax3, cv_ax4, cv_ax5, cv_ax6 = yzUL, yzLU, yzUU, yzLL, θcv1, Inf # TODO? eq 6 not in paper
cv_ay1, cv_ay2, cv_ay3, cv_ay4, cv_ay5, cv_ay6 = xzLL, xzLU, xzUU, xzUL, θcv2, Inf # TODO? eq 6 not in paper
cv_az1, cv_az2, cv_az3, cv_az4, cv_az5, cv_az6 = xyLU, xyLL, xyUU, xyUL, θcv3, Inf # TODO? eq 6 not in paper

cv1 = cv_ax1*x.cc + cv_ay1*y.cv + cv_az1*z.cc + cv_b1
cv2 = cv_ax2*x.cc + cv_ay2*y.cc + cv_az2*z.cv + cv_b2
Expand Down Expand Up @@ -420,11 +428,11 @@ function trilinear_case_4(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float6
cc_b3 = -2.0*xyzULU
cc_b4 = -2.0*xyzLUU
cc_b5 = θcc4
#cc_b6 = TODO? eq 6 not in paper
cc_b6 = Inf #TODO? eq 6 not in paper

cc_ax1, cc_ax2, cc_ax3, cc_ax4, cc_ax5 = yzLL, yzUL, yzLU, yzUU, θcc1
cc_ay1, cc_ay2, cc_ay3, cc_ay4, cc_ay5 = xzLL, xzUL, xzUU, xzLU, θcc2
cc_az1, cc_az2, cc_az3, cc_az4, cc_az5 = xyLL, xyUU, xyUL, xyLU, θcc3
cc_ax1, cc_ax2, cc_ax3, cc_ax4, cc_ax5, cc_ax6 = yzLL, yzUL, yzLU, yzUU, θcc1, Inf
cc_ay1, cc_ay2, cc_ay3, cc_ay4, cc_ay5, cc_ay6 = xzLL, xzUL, xzUU, xzLU, θcc2, Inf
cc_az1, cc_az2, cc_az3, cc_az4, cc_az5, cc_az6 = xyLL, xyUU, xyUL, xyLU, θcc3, Inf

cc1 = cc_ax1*x.cc + cc_ay1*y.cc + cc_az1*z.cc + cc_b1
cc2 = cc_ax2*x.cv + cc_ay2*y.cv + cc_az2*z.cc + cc_b2
Expand Down Expand Up @@ -496,6 +504,8 @@ Case 3.5 + Case 4.5 of Meyer-Floudas 2004
function trilinear_case_5(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 5"

# define cv and coefficients
cv_b1 = -(xyzLUL + xyzLLL)
cv_b2 = -(xyzLUL + xyzLUU)
Expand Down Expand Up @@ -524,7 +534,7 @@ function trilinear_case_5(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float6
θcc2 = xyzULU - xyzLLL - xyzUUU + xyzLUU

cc_b1 = -2.0*xyzLLU
Cc_b2 = -2.0*xyzUUL
cc_b2 = -2.0*xyzUUL
cc_b3 = -(xyzULU + xyzULL)
cc_b4 = -(xyzLUU + xyzLUL)
cc_b5 = θcc1*zU/delZ - xyzULL - xyzLUL + xyzUUU
Expand Down Expand Up @@ -578,6 +588,8 @@ Case 3.6 + Case 4.6 of Meyer-Floudas 2004
function trilinear_case_6(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 6"

delY = yU - yL
θcv = xyzULU - xyzUUL - xyzLLU + xyzLLL
θcc = xyzLUL - xyzULL - xyzLUU + xyzUUU
Expand Down Expand Up @@ -631,6 +643,8 @@ Case 3.7 + Case 4.7 of Meyer-Floudas 2004
function trilinear_case_7(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 7"

delX = xU - xL
delY = yU - yL

Expand Down Expand Up @@ -716,6 +730,8 @@ Case 3.8 + Case 4.8 of Meyer-Floudas 2004
function trilinear_case_8(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 8"

# define cv and coefficients
if (xyzLLL + xyzUUU <= xyzLLL + xyzUUU) &&
(xyzLLL + xyzUUU <= xyzLUL + xyzULU)
Expand Down Expand Up @@ -798,17 +814,19 @@ Case 3.9 + Case 4.9 of Meyer-Floudas 2004
function trilinear_case_9(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 9"

delX = xU - xL
θcv = xyzUUL - xyzLLL - xyzUUU + xyzULU
θcc = xyzLUL - xyzULL - xyzLUU + xyzLLU

# define cv and coefficients
cv_b1 = -2.0*xyzULL
cv_b2 = -(xyzLUU + xyzLUL)
cv_b3 = -(xyzLUL + xyzLLI)
cv_b3 = -(xyzLUU + xyzLLU)
cv_b4 = -(xyzULU + xyzLLU)
cv_b5 = -(xyzLUL + xyzUUL)
cv_b6 = -θcv*xL/delX - xyzUUL - xyzULU + xyzLLL
cv_b6 = -θcv*xL/delX - xyzLLL - xyzUUU + xyzULU

cv_ax1, cv_ax2, cv_ax3, cv_ax4, cv_ax5, cv_ax6 = yzLL, yzUU, yzUU, yzLU, yzUL, θcv/delX
cv_ay1, cv_ay2, cv_ay3, cv_ay4, cv_ay5, cv_ay6 = xzUL, xzLL, xzLU, xzUU, xzLL, xzUU
Expand Down Expand Up @@ -851,6 +869,10 @@ Case 3.10 + Case 4.10 of Meyer-Floudas 2004
function trilinear_case_10(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float64}) where {N,T<:RelaxTag}
@unpack_trilinear_bnd()

@show "RAN 10"

delX = xU - xL

θcc1 = xyzLUL - xyzULL - xyzLUU + xyzLLU
θcc2 = xyzUUL - xyzLUU - xyzULL + xyzULU

Expand All @@ -867,7 +889,7 @@ function trilinear_case_10(x::MC{N,T}, y::MC{N,T}, z::MC{N,T}, q::Interval{Float
cv_az1, cv_az2, cv_az3, cv_az4, cv_az5, cv_az6 = xyUU, xyLU, xyUU, xyLL, xyLL, xyUL

# define cc and coefficients
cc_b1 = -2.0*xyLLL
cc_b1 = -2.0*xyzLLL
cc_b2 = -2.0*xyzUUU
cc_b3 = -(xyzLLU + xyzULU)
cc_b4 = -(xyzLUL + xyzUUL)
Expand Down Expand Up @@ -919,7 +941,8 @@ function mult_kernel(x1::MC{N,T}, x2::MC{N,T}, x3::MC{N,T}, z::Interval{Float64}
return trilinear_case_10(x1, x2, x3, z)
end

@inline function *(x1::MC{N,T}, x2::MC{N,T}, x3::MC{N,T}) where {N, T<:Union{NS,MV}}
@inline function trilinear(x1::MC{N,T}, x2::MC{N,T}, x3::MC{N,T}) where {N, T<:Union{NS,MV}}
println("WARNING: VALID SUBGRADIENTS HAVE NOT YET BEEN DEFINED FOR trilinear()")
if x1 == x2
if x1 == x3
z = x1.Intv^3
Expand All @@ -933,3 +956,62 @@ end
end
return mult_kernel(x1, x2, x3, z)
end

#=
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}) where {N, T<:Union{NS,MV}} = *(*(a,b,c),d)
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::Float64) where {N, T<:Union{NS,MV}} = *(*(a,b,c),d)
*(a::MC{N,T}, b::MC{N,T}, c::Float64, d::MC{N,T}) where {N, T<:Union{NS,MV}} = *(*(a,b,d),c)
*(a::MC{N,T}, b::Float64, c::MC{N,T}, d::MC{N,T}) where {N, T<:Union{NS,MV}} = *(*(a,c,d),b)
*(a::Float64, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}) where {N, T<:Union{NS,MV}} = *(*(b,c,d),a)
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(*(a,b,c),d,e)
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}, e::Float64) where {N, T<:Union{NS,MV}} = *(a,b,c,d)*e
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::Float64, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,b,c,e)*d
*(a::MC{N,T}, b::MC{N,T}, c::Float64, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,b,d,e)*c
*(a::MC{N,T}, b::Float64, c::MC{N,T}, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,c,d,e)*b
*(a::Float64, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(b,c,d,e)*a
*(a::Float64, b::Float64, c::MC{N,T}, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(c,d,e)*a*b
*(a::Float64, b::MC{N,T}, c::Float64, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(b,d,e)*a*c
*(a::Float64, b::MC{N,T}, c::MC{N,T}, d::Float64, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(b,c,e)*a*d
*(a::Float64, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}, e::Float64) where {N, T<:Union{NS,MV}} = *(b,c,d)*a*e
*(a::MC{N,T}, b::Float64, c::Float64, d::MC{N,T}, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,d,e)*b*c
*(a::MC{N,T}, b::Float64, c::MC{N,T}, d::Float64, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,c,e)*b*d
*(a::MC{N,T}, b::Float64, c::MC{N,T}, d::MC{N,T}, e::Float64) where {N, T<:Union{NS,MV}} = *(a,c,d)*b*e
*(a::MC{N,T}, b::MC{N,T}, c::Float64, d::Float64, e::MC{N,T}) where {N, T<:Union{NS,MV}} = *(a,b,e)*c*d
*(a::MC{N,T}, b::MC{N,T}, c::Float64, d::MC{N,T}, e::Float64) where {N, T<:Union{NS,MV}} = *(a,b,d)*c*e
*(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::Float64, e::Float64) where {N, T<:Union{NS,MV}} = *(a,b,c)*d*e
function *(a::MC{N,T}, b::MC{N,T}, c::MC{N,T}, d::MC{N,T}, e::MC{N,T}, f::MC{N,T}) where {N, T<:Union{NS,MV}}
*(*(a,b,c),*(d,e,f))
end
=#

#=
function *(a,b,c,d,e,f,gs...)
y = *(a,b,c,d,e,f)
n = length(js)
t1 = a
t2 = b
t3 = c
for k = 1:n
y = @inbounds x[k]
!is_last
if mod(k)
end
end
for x in js;
y = op(y,x);
end
y
end
=#

#=
function prod(A::Array{MC{N,T},Q}; dims) where {N, Q, T<:Union{NS,MV}}
end
=#
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