Performance Tips¶

In the following sections, we briefly go through a few techniques that can help make your Julia code run as fast as possible.

Avoid global variables¶

A global variable might have its value, and therefore its type, change at any point. This makes it difficult for the compiler to optimize code using global variables. Variables should be local, or passed as arguments to functions, whenever possible.

Any code that is performance critical or being benchmarked should be inside a function.

We find that global names are frequently constants, and declaring them as such greatly improves performance:

const DEFAULT_VAL = 0


Uses of non-constant globals can be optimized by annotating their types at the point of use:

global x
y = f(x::Int + 1)


Writing functions is better style. It leads to more reusable code and clarifies what steps are being done, and what their inputs and outputs are.

NOTE: All code in the REPL is evaluated in global scope, so a variable defined and assigned at toplevel will be a global variable.

In the following REPL session:

julia> x = 1.0


is equivalent to:

julia> global x = 1.0


so all the performance issues discussed previously apply.

Measure performance with @time and pay attention to memory allocation¶

The most useful tool for measuring performance is the @time macro. The following example illustrates good working style:

julia> function f(n)
s = 0
for i = 1:n
s += i/2
end
s
end
f (generic function with 1 method)

julia> @time f(1)
elapsed time: 0.004710563 seconds (93504 bytes allocated)
0.5

julia> @time f(10^6)
elapsed time: 0.04123202 seconds (32002136 bytes allocated)
2.5000025e11


On the first call (@time f(1)), f gets compiled. (If you’ve not yet used @time in this session, it will also compile functions needed for timing.) You should not take the results of this run seriously. For the second run, note that in addition to reporting the time, it also indicated that a large amount of memory was allocated. This is the single biggest advantage of @time vs. functions like tic() and toc(), which only report time.

Unexpected memory allocation is almost always a sign of some problem with your code, usually a problem with type-stability. Consequently, in addition to the allocation itself, it’s very likely that the code generated for your function is far from optimal. Take such indications seriously and follow the advice below.

As a teaser, note that an improved version of this function allocates no memory (except to pass back the result back to the REPL) and has an order of magnitude faster execution after the first call:

julia> @time f_improved(1)   # first call
elapsed time: 0.003702172 seconds (78944 bytes allocated)
0.5

julia> @time f_improved(10^6)
elapsed time: 0.004313644 seconds (112 bytes allocated)
2.5000025e11


Below you’ll learn how to spot the problem with f and how to fix it.

In some situations, your function may need to allocate memory as part of its operation, and this can complicate the simple picture above. In such cases, consider using one of the tools below to diagnose problems, or write a version of your function that separates allocation from its algorithmic aspects (see Pre-allocating outputs).

Tools¶

Julia and its package ecosystem includes tools that may help you diagnose problems and improve the performance of your code:

• Profiling allows you to measure the performance of your running code and identify lines that serve as bottlenecks. For complex projects, the ProfileView package can help you visualize your profiling results.
• Unexpectedly-large memory allocations—as reported by @time, @allocated, or the profiler (through calls to the garbage-collection routines)—hint that there might be issues with your code. If you don’t see another reason for the allocations, suspect a type problem. You can also start Julia with the --track-allocation=user option and examine the resulting *.mem files to see information about where those allocations occur. See Memory allocation analysis.
• @code_warntype generates a representation of your code that can be helpful in finding expressions that result in type uncertainty. See @code_warntype below.
• The Lint and TypeCheck packages can also warn you of certain types of programming errors.

Avoid containers with abstract type parameters¶

When working with parameterized types, including arrays, it is best to avoid parameterizing with abstract types where possible.

Consider the following:

a = Real[]    # typeof(a) = Array{Real,1}
if (f = rand()) < .8
push!(a, f)
end


Because a is a an array of abstract type Real, it must be able to hold any Real value. Since Real objects can be of arbitrary size and structure, a must be represented as an array of pointers to individually allocated Real objects. Because f will always be a Float64, we should instead, use:

a = Float64[] # typeof(a) = Array{Float64,1}


which will create a contiguous block of 64-bit floating-point values that can be manipulated efficiently.

See also the discussion under Parametric Types.

Type declarations¶

In many languages with optional type declarations, adding declarations is the principal way to make code run faster. This is not the case in Julia. In Julia, the compiler generally knows the types of all function arguments, local variables, and expressions. However, there are a few specific instances where declarations are helpful.

Declare specific types for fields of composite types¶

Given a user-defined type like the following:

type Foo
field
end


the compiler will not generally know the type of foo.field, since it might be modified at any time to refer to a value of a different type. It will help to declare the most specific type possible, such as field::Float64 or field::Array{Int64,1}.

Annotate values taken from untyped locations¶

It is often convenient to work with data structures that may contain values of any type, such as the original Foo type above, or cell arrays (arrays of type Array{Any}). But, if you’re using one of these structures and happen to know the type of an element, it helps to share this knowledge with the compiler:

function foo(a::Array{Any,1})
x = a[1]::Int32
b = x+1
...
end


Here, we happened to know that the first element of a would be an Int32. Making an annotation like this has the added benefit that it will raise a run-time error if the value is not of the expected type, potentially catching certain bugs earlier.

Declare types of keyword arguments¶

Keyword arguments can have declared types:

function with_keyword(x; name::Int = 1)
...
end


Functions are specialized on the types of keyword arguments, so these declarations will not affect performance of code inside the function. However, they will reduce the overhead of calls to the function that include keyword arguments.

Functions with keyword arguments have near-zero overhead for call sites that pass only positional arguments.

Passing dynamic lists of keyword arguments, as in f(x; keywords...), can be slow and should be avoided in performance-sensitive code.

Break functions into multiple definitions¶

Writing a function as many small definitions allows the compiler to directly call the most applicable code, or even inline it.

Here is an example of a “compound function” that should really be written as multiple definitions:

function norm(A)
if isa(A, Vector)
return sqrt(real(dot(A,A)))
elseif isa(A, Matrix)
return max(svd(A)[2])
else
error("norm: invalid argument")
end
end


This can be written more concisely and efficiently as:

norm(x::Vector) = sqrt(real(dot(x,x)))
norm(A::Matrix) = max(svd(A)[2])


Write “type-stable” functions¶

When possible, it helps to ensure that a function always returns a value of the same type. Consider the following definition:

pos(x) = x < 0 ? 0 : x


Although this seems innocent enough, the problem is that 0 is an integer (of type Int) and x might be of any type. Thus, depending on the value of x, this function might return a value of either of two types. This behavior is allowed, and may be desirable in some cases. But it can easily be fixed as follows:

pos(x) = x < 0 ? zero(x) : x


There is also a one() function, and a more general oftype(x,y) function, which returns y converted to the type of x.

Avoid changing the type of a variable¶

An analogous “type-stability” problem exists for variables used repeatedly within a function:

function foo()
x = 1
for i = 1:10
x = x/bar()
end
return x
end


Local variable x starts as an integer, and after one loop iteration becomes a floating-point number (the result of / operator). This makes it more difficult for the compiler to optimize the body of the loop. There are several possible fixes:

• Initialize x with x = 1.0
• Declare the type of x: x::Float64 = 1
• Use an explicit conversion: x = one(T)

Separate kernel functions¶

Many functions follow a pattern of performing some set-up work, and then running many iterations to perform a core computation. Where possible, it is a good idea to put these core computations in separate functions. For example, the following contrived function returns an array of a randomly-chosen type:

function strange_twos(n)
a = Array(rand(Bool) ? Int64 : Float64, n)
for i = 1:n
a[i] = 2
end
return a
end


This should be written as:

function fill_twos!(a)
for i=1:length(a)
a[i] = 2
end
end

function strange_twos(n)
a = Array(rand(Bool) ? Int64 : Float64, n)
fill_twos!(a)
return a
end


Julia’s compiler specializes code for argument types at function boundaries, so in the original implementation it does not know the type of a during the loop (since it is chosen randomly). Therefore the second version is generally faster since the inner loop can be recompiled as part of fill_twos! for different types of a.

The second form is also often better style and can lead to more code reuse.

This pattern is used in several places in the standard library. For example, see hvcat_fill in abstractarray.jl, or the fill! function, which we could have used instead of writing our own fill_twos!.

Functions like strange_twos occur when dealing with data of uncertain type, for example data loaded from an input file that might contain either integers, floats, strings, or something else.

Access arrays in memory order, along columns¶

Multidimensional arrays in Julia are stored in column-major order. This means that arrays are stacked one column at a time. This can be verified using the vec function or the syntax [:] as shown below (notice that the array is ordered [1 3 2 4], not [1 2 3 4]):

julia> x = [1 2; 3 4]
2x2 Array{Int64,2}:
1  2
3  4

julia> x[:]
4-element Array{Int64,1}:
1
3
2
4


This convention for ordering arrays is common in many languages like Fortran, Matlab, and R (to name a few). The alternative to column-major ordering is row-major ordering, which is the convention adopted by C and Python (numpy) among other languages. Remembering the ordering of arrays can have significant performance effects when looping over arrays. A rule of thumb to keep in mind is that with column-major arrays, the first index changes most rapidly. Essentially this means that looping will be faster if the inner-most loop index is the first to appear in a slice expression.

Consider the following contrived example. Imagine we wanted to write a function that accepts a Vector and returns a square Matrix with either the rows or the columns filled with copies of the input vector. Assume that it is not important whether rows or columns are filled with these copies (perhaps the rest of the code can be easily adapted accordingly). We could conceivably do this in at least four ways (in addition to the recommended call to the built-in repmat()):

function copy_cols{T}(x::Vector{T})
n = size(x, 1)
out = Array(eltype(x), n, n)
for i=1:n
out[:, i] = x
end
out
end

function copy_rows{T}(x::Vector{T})
n = size(x, 1)
out = Array(eltype(x), n, n)
for i=1:n
out[i, :] = x
end
out
end

function copy_col_row{T}(x::Vector{T})
n = size(x, 1)
out = Array(T, n, n)
for col=1:n, row=1:n
out[row, col] = x[row]
end
out
end

function copy_row_col{T}(x::Vector{T})
n = size(x, 1)
out = Array(T, n, n)
for row=1:n, col=1:n
out[row, col] = x[col]
end
out
end


Now we will time each of these functions using the same random 10000 by 1 input vector:

julia> x = randn(10000);

julia> fmt(f) = println(rpad(string(f)*": ", 14, ' '), @elapsed f(x))

julia> map(fmt, Any[copy_cols, copy_rows, copy_col_row, copy_row_col]);
copy_cols:    0.331706323
copy_rows:    1.799009911
copy_col_row: 0.415630047
copy_row_col: 1.721531501


Notice that copy_cols is much faster than copy_rows. This is expected because copy_cols respects the column-based memory layout of the Matrix and fills it one column at a time. Additionally, copy_col_row is much faster than copy_row_col because it follows our rule of thumb that the first element to appear in a slice expression should be coupled with the inner-most loop.

Pre-allocating outputs¶

If your function returns an Array or some other complex type, it may have to allocate memory. Unfortunately, oftentimes allocation and its converse, garbage collection, are substantial bottlenecks.

Sometimes you can circumvent the need to allocate memory on each function call by preallocating the output. As a trivial example, compare

function xinc(x)
return [x, x+1, x+2]
end

function loopinc()
y = 0
for i = 1:10^7
ret = xinc(i)
y += ret[2]
end
y
end


with

function xinc!{T}(ret::AbstractVector{T}, x::T)
ret[1] = x
ret[2] = x+1
ret[3] = x+2
nothing
end

function loopinc_prealloc()
ret = Array(Int, 3)
y = 0
for i = 1:10^7
xinc!(ret, i)
y += ret[2]
end
y
end


Timing results:

julia> @time loopinc()
elapsed time: 1.955026528 seconds (1279975584 bytes allocated)
50000015000000

julia> @time loopinc_prealloc()
elapsed time: 0.078639163 seconds (144 bytes allocated)
50000015000000


Preallocation has other advantages, for example by allowing the caller to control the “output” type from an algorithm. In the example above, we could have passed a SubArray rather than an Array, had we so desired.

Taken to its extreme, pre-allocation can make your code uglier, so performance measurements and some judgment may be required.

Avoid string interpolation for I/O¶

When writing data to a file (or other I/O device), forming extra intermediate strings is a source of overhead. Instead of:

println(file, "$a$b")


use:

println(file, a, " ", b)


The first version of the code forms a string, then writes it to the file, while the second version writes values directly to the file. Also notice that in some cases string interpolation can be harder to read. Consider:

println(file, "$(f(a))$(f(b))")


versus:

println(file, f(a), f(b))


Optimize network I/O during parallel execution¶

When executing a remote function in parallel:

responses = cell(nworkers())
@sync begin
for (idx, pid) in enumerate(workers())
@async responses[idx] = remotecall_fetch(pid, foo, args...)
end
end


is faster than:

refs = cell(nworkers())
for (idx, pid) in enumerate(workers())
refs[idx] = @spawnat pid foo(args...)
end
responses = [fetch(r) for r in refs]


The former results in a single network round-trip to every worker, while the latter results in two network calls - first by the @spawnat and the second due to the fetch (or even a wait). The fetch/wait is also being executed serially resulting in an overall poorer performance.

Fix deprecation warnings¶

A deprecated function internally performs a lookup in order to print a relevant warning only once. This extra lookup can cause a significant slowdown, so all uses of deprecated functions should be modified as suggested by the warnings.

Tweaks¶

These are some minor points that might help in tight inner loops.

• Avoid unnecessary arrays. For example, instead of sum([x,y,z]) use x+y+z.
• Use abs2(z) instead of abs(z)^2 for complex z. In general, try to rewrite code to use abs2() instead of abs() for complex arguments.
• Use div(x,y) for truncating division of integers instead of trunc(x/y), fld(x,y) instead of floor(x/y), and cld(x,y) instead of ceil(x/y).

Performance Annotations¶

Sometimes you can enable better optimization by promising certain program properties.

• Use @inbounds to eliminate array bounds checking within expressions. Be certain before doing this. If the subscripts are ever out of bounds, you may suffer crashes or silent corruption.
• Use @fastmath to allow floating point optimizations that are correct for real numbers, but lead to differences for IEEE numbers. Be careful when doing this, as this may change numerical results. This corresponds to the -ffast-math option of clang.
• Write @simd in front of for loops that are amenable to vectorization. This feature is experimental and could change or disappear in future versions of Julia.

Note: While @simd needs to be placed directly in front of a loop, both @inbounds and @fastmath can be applied to several statements at once, e.g. using begin ... end, or even to a whole function.

Here is an example with both @inbounds and @simd markup:

function inner( x, y )
s = zero(eltype(x))
for i=1:length(x)
@inbounds s += x[i]*y[i]
end
s
end

function innersimd( x, y )
s = zero(eltype(x))
@simd for i=1:length(x)
@inbounds s += x[i]*y[i]
end
s
end

function timeit( n, reps )
x = rand(Float32,n)
y = rand(Float32,n)
s = zero(Float64)
time = @elapsed for j in 1:reps
s+=inner(x,y)
end
println("GFlop        = ",2.0*n*reps/time*1E-9)
time = @elapsed for j in 1:reps
s+=innersimd(x,y)
end
println("GFlop (SIMD) = ",2.0*n*reps/time*1E-9)
end

timeit(1000,1000)


On a computer with a 2.4GHz Intel Core i5 processor, this produces:

GFlop        = 1.9467069505224963
GFlop (SIMD) = 17.578554163920018


The range for a @simd for loop should be a one-dimensional range. A variable used for accumulating, such as s in the example, is called a reduction variable. By using @simd, you are asserting several properties of the loop:

• It is safe to execute iterations in arbitrary or overlapping order, with special consideration for reduction variables.
• Floating-point operations on reduction variables can be reordered, possibly causing different results than without @simd.
• No iteration ever waits on another iteration to make forward progress.

A loop containing break, continue, or @goto will cause a compile-time error.

Using @simd merely gives the compiler license to vectorize. Whether it actually does so depends on the compiler. To actually benefit from the current implementation, your loop should have the following additional properties:

• The loop must be an innermost loop.
• The loop body must be straight-line code. This is why @inbounds is currently needed for all array accesses. The compiler can sometimes turn short &&, ||, and ?: expressions into straight-line code, if it is safe to evaluate all operands unconditionally. Consider using ifelse() instead of ?: in the loop if it is safe to do so.
• Accesses must have a stride pattern and cannot be “gathers” (random-index reads) or “scatters” (random-index writes).
• The stride should be unit stride.
• In some simple cases, for example with 2-3 arrays accessed in a loop, the LLVM auto-vectorization may kick in automatically, leading to no further speedup with @simd.

Here is an example with all three kinds of markup. This program first calculates the finite difference of a one-dimensional array, and then evaluates the L2-norm of the result:

function init!(u)
n = length(u)
dx = 1.0 / (n-1)
@fastmath @inbounds @simd for i in 1:n
u[i] = sin(2pi*dx*i)
end
end

function deriv!(u, du)
n = length(u)
dx = 1.0 / (n-1)
@fastmath @inbounds du[1] = (u[2] - u[1]) / dx
@fastmath @inbounds @simd for i in 2:n-1
du[i] = (u[i+1] - u[i-1]) / (2*dx)
end
@fastmath @inbounds du[n] = (u[n] - u[n-1]) / dx
end

function norm(u)
n = length(u)
T = eltype(u)
s = zero(T)
@fastmath @inbounds @simd for i in 1:n
s += u[i]^2
end
@fastmath @inbounds return sqrt(s/n)
end

function main()
n = 2000
u = Array(Float64, n)
init!(u)
du = similar(u)

deriv!(u, du)
nu = norm(du)

@time for i in 1:10^6
deriv!(u, du)
nu = norm(du)
end

println(nu)
end

main()


On a computer with a 2.7 GHz Intel Core i7 processor, this produces:

$julia wave.jl elapsed time: 1.207814709 seconds (0 bytes allocated) 4.443986180758243$ julia --math-mode=ieee wave.jl
elapsed time: 4.487083643 seconds (0 bytes allocated)
4.443986180758243


Here, the option --math-mode=ieee disables the @fastmath macro, so that we can compare results.

In this case, the speedup due to @fastmath is a factor of about 3.7. This is unusually large – in general, the speedup will be smaller. (In this particular example, the working set of the benchmark is small enough to fit into the L1 cache of the processor, so that memory access latency does not play a role, and computing time is dominated by CPU usage. In many real world programs this is not the case.) Also, in this case this optimization does not change the result – in general, the result will be slightly different. In some cases, especially for numerically unstable algorithms, the result can be very different.

The annotation @fastmath re-arranges floating point expressions, e.g. changing the order of evaluation, or assuming that certain special cases (inf, nan) cannot occur. In this case (and on this particular computer), the main difference is that the expression 1 / (2*dx) in the function deriv is hoisted out of the loop (i.e. calculated outside the loop), as if one had written idx = 1 / (2*dx). In the loop, the expression ... / (2*dx) then becomes ... * idx, which is much faster to evaluate. Of course, both the actual optimization that is applied by the compiler as well as the resulting speedup depend very much on the hardware. You can examine the change in generated code by using Julia’s code_native() function.

Treat Subnormal Numbers as Zeros¶

Subnormal numbers, formerly called denormal numbers, are useful in many contexts, but incur a performance penalty on some hardware. A call set_zero_subnormals(true) grants permission for floating-point operations to treat subnormal inputs or outputs as zeros, which may improve performance on some hardware. A call set_zero_subnormals(false) enforces strict IEEE behavior for subnormal numbers.

Below is an example where subnormals noticeably impact performance on some hardware:

function timestep{T}( b::Vector{T}, a::Vector{T}, Δt::T )
@assert length(a)==length(b)
n = length(b)
b[1] = 1                            # Boundary condition
for i=2:n-1
b[i] = a[i] + (a[i-1] - T(2)*a[i] + a[i+1]) * Δt
end
b[n] = 0                            # Boundary condition
end

function heatflow{T}( a::Vector{T}, nstep::Integer )
b = similar(a)
for t=1:div(nstep,2)                # Assume nstep is even
timestep(b,a,T(0.1))
timestep(a,b,T(0.1))
end
end

heatflow(zeros(Float32,10),2)           # Force compilation
for trial=1:6
a = zeros(Float32,1000)
set_zero_subnormals(iseven(trial))  # Odd trials use strict IEEE arithmetic
@time heatflow(a,1000)
end


This example generates many subnormal numbers because the values in a become an exponentially decreasing curve, which slowly flattens out over time.

Treating subnormals as zeros should be used with caution, because doing so breaks some identities, such as x-y==0 implies x==y:

julia> x=3f-38; y=2f-38;

julia> set_zero_subnormals(false); (x-y,x==y)
(1.0000001f-38,false)

julia> set_zero_subnormals(true); (x-y,x==y)
(0.0f0,false)


In some applications, an alternative to zeroing subnormal numbers is to inject a tiny bit of noise. For example, instead of initializing a with zeros, initialize it with:

a = rand(Float32,1000) * 1.f-9


@code_warntype¶

The macro @code_warntype (or its function variant code_warntype()) can sometimes be helpful in diagnosing type-related problems. Here’s an example:

pos(x) = x < 0 ? 0 : x

function f(x)
y = pos(x)
sin(y*x+1)
end

julia> @code_warntype f(3.2)
Variables:
x::Float64
y::UNION(INT64,FLOAT64)
_var0::Float64
_var3::Tuple{Int64}
_var4::UNION(INT64,FLOAT64)
_var1::Float64
_var2::Float64

Body:
begin  # none, line 2:
_var0 = (top(box))(Float64,(top(sitofp))(Float64,0))
unless (top(box))(Bool,(top(or_int))((top(lt_float))(x::Float64,_var0::Float64)::Bool,(top(box))(Bool,(top(and_int))((top(box))(Bool,(top(and_int))((top(eq_float))(x::Float64,_var0::Float64)::Bool,(top(lt_float))(_var0::Float64,9.223372036854776e18)::Bool)),(top(slt_int))((top(box))(Int64,(top(fptosi))(Int64,_var0::Float64)),0)::Bool)))) goto 1
_var4 = 0
goto 2
1:
_var4 = x::Float64
2:
y = _var4::UNION(INT64,FLOAT64) # line 3:
_var1 = y::UNION(INT64,FLOAT64) * x::Float64::Float64
return (GlobalRef(Base.Math,:nan_dom_err))((top(ccall))($(Expr(:call1, :(top(tuple)), "sin", GlobalRef(Base.Math,:libm))),Float64,$(Expr(:call1, :(top(tuple)), :Float64)),_var2::Float64,0)::Float64,_var2::Float64)::Float64
end::Float64


Interpreting the output of @code_warntype, like that of its cousins @code_lowered, @code_typed, @code_llvm, and @code_native, takes a little practice. Your code is being presented in form that has been partially digested on its way to generating compiled machine code. Most of the expressions are annotated by a type, indicated by the ::T (where T might be Float64, for example). The most important characteristic of @code_warntype is that non-concrete types are displayed in red; in the above example, such output is shown in all-caps.

The top part of the output summarizes the type information for the different variables internal to the function. You can see that y, one of the variables you created, is a Union{Int64,Float64}, due to the type-instability of pos. There is another variable, _var4, which you can see also has the same type.

The next lines represent the body of f. The lines starting with a number followed by a colon (1:, 2:) are labels, and represent targets for jumps (via goto) in your code. Looking at the body, you can see that pos has been inlined into f—everything before 2: comes from code defined in pos.

Starting at 2:, the variable y is defined, and again annotated as a Union type. Next, we see that the compiler created the temporary variable _var1 to hold the result of y*x. Because a Float64 times either an Int64 or Float64 yields a Float64, all type-instability ends here. The net result is that f(x::Float64) will not be type-unstable in its output, even if some of the intermediate computations are type-unstable.

How you use this information is up to you. Obviously, it would be far and away best to fix pos to be type-stable: if you did so, all of the variables in f would be concrete, and its performance would be optimal. However, there are circumstances where this kind of ephemeral type instability might not matter too much: for example, if pos is never used in isolation, the fact that f‘s output is type-stable (for Float64 inputs) will shield later code from the propagating effects of type instability. This is particularly relevant in cases where fixing the type instability is difficult or impossible: for example, currently it’s not possible to infer the return type of an anonymous function. In such cases, the tips above (e.g., adding type annotations and/or breaking up functions) are your best tools to contain the “damage” from type instability.

The following examples may help you interpret expressions marked as containing non-leaf types:

• Function body ending in end::Union{T1,T2})
• Interpretation: function with unstable return type
• Suggestion: make the return value type-stable, even if you have to annotate it
• f(x::T)::Union{T1,T2}
• Interpretation: call to a type-unstable function
• Suggestion: fix the function, or if necessary annotate the return value
• (top(arrayref))(A::Array{Any,1},1)::Any
• Interpretation: accessing elements of poorly-typed arrays
• Suggestion: use arrays with better-defined types, or if necessary annotate the type of individual element accesses
• (top(getfield))(A::ArrayContainer{Float64},:data)::Array{Float64,N}
• Interpretation: getting a field that is of non-leaf type. In this case, ArrayContainer had a field data::Array{T}. But Array needs the dimension N, too, to be a concrete type.
• Suggestion: use concrete types like Array{T,3} or Array{T,N}, where N is now a parameter of ArrayContainer