13. The Need for Speed

13.1. Overview

Computer scientists often classify programming languages according to the following two categories.

High level languages aim to maximize productivity by

  • being easy to read, write and debug

  • automating standard tasks (e.g., memory management)

  • being interactive, etc.

Low level languages aim for speed and control, which they achieve by

  • being closer to the metal (direct access to CPU, memory, etc.)

  • requiring a relatively large amount of information from the user (e.g., all data types must be specified)

Traditionally we understand this as a trade off

  • high productivity or high performance

  • optimized for humans or optimized for machines

One of the great strengths of Julia is that it pushes out the curve, achieving both high productivity and high performance with relatively little fuss.

The word “relatively” is important here, however…

In simple programs, excellent performance is often trivial to achieve.

For longer, more sophisticated programs, you need to be aware of potential stumbling blocks.

This lecture covers the key points.

13.1.1. Requirements

You should read our earlier lecture on types, methods and multiple dispatch before this one.

using LinearAlgebra, Statistics

13.2. Understanding Multiple Dispatch in Julia

This section provides more background on how methods, functions, and types are connected.

13.2.1. Methods and Functions

The precise data type is important, for reasons of both efficiency and mathematical correctness.

For example consider 1 + 1 vs. 1.0 + 1.0 or [1 0] + [0 1].

On a CPU, integer and floating point addition are different things, using a different set of instructions.

Julia handles this problem by storing multiple, specialized versions of functions like addition, one for each data type or set of data types.

These individual specialized versions are called methods.

When an operation like addition is requested, the Julia compiler inspects the type of data to be acted on and hands it out to the appropriate method.

This process is called multiple dispatch.

Like all “infix” operators, 1 + 1 has the alternative syntax +(1, 1)

+(1, 1)
2

This operator + is itself a function with multiple methods.

We can investigate them using the @which macro, which shows the method to which a given call is dispatched

x, y = 1.0, 1.0
@which +(x, y)
+(x::Float64, y::Float64) in Base at float.jl:326

We see that the operation is sent to the + method that specializes in adding floating point numbers.

Here’s the integer case

x, y = 1, 1
@which +(x, y)
+(x::T, y::T) where T<:Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8} in Base at int.jl:87

This output says that the call has been dispatched to the + method responsible for handling integer values.

(We’ll learn more about the details of this syntax below)

Here’s another example, with complex numbers

x, y = 1.0 + 1.0im, 1.0 + 1.0im
@which +(x, y)
+(z::Complex, w::Complex) in Base at complex.jl:275

Again, the call has been dispatched to a + method specifically designed for handling the given data type.

13.2.1.1. Adding Methods

It’s straightforward to add methods to existing functions.

For example, we can’t at present add an integer and a string in Julia (i.e. 100 + "100" is not valid syntax).

This is sensible behavior, but if you want to change it there’s nothing to stop you.

import Base: +  # enables adding methods to the + function
+(x::Integer, y::String) = x + parse(Int, y)
@show +(100, "100")
@show 100 + "100";  # equivalent
100 + "100" = 200
100 + "100" = 200

The above code is not executed to avoid any chance of a method invalidation, where are a source of compile-time latency.

13.2.2. Understanding the Compilation Process

We can now be a little bit clearer about what happens when you call a function on given types.

Suppose we execute the function call f(a, b) where a and b are of concrete types S and T respectively.

The Julia interpreter first queries the types of a and b to obtain the tuple (S, T).

It then parses the list of methods belonging to f, searching for a match.

If it finds a method matching (S, T) it calls that method.

If not, it looks to see whether the pair (S, T) matches any method defined for immediate parent types.

For example, if S is Float64 and T is ComplexF32 then the immediate parents are AbstractFloat and Number respectively

supertype(Float64)
AbstractFloat
supertype(ComplexF32)
Number

Hence the interpreter looks next for a method of the form f(x::AbstractFloat, y::Number).

If the interpreter can’t find a match in immediate parents (supertypes) it proceeds up the tree, looking at the parents of the last type it checked at each iteration.

  • If it eventually finds a matching method, it invokes that method.

  • If not, we get an error.

This is the process that leads to the following error (since we only added the + for adding Integer and String above)

@show (typeof(100.0) <: Integer) == false
100.0 + "100"
(typeof(100.0) <: Integer) == false = true
MethodError: no method matching +(::Float64, ::String)
Closest candidates are:
  +(::Any, ::Any, ::Any, ::Any...) at operators.jl:560
  +(::Union{Float16, Float32, Float64}, ::BigFloat) at mpfr.jl:392
  +(::AbstractFloat, ::Bool) at bool.jl:102
  ...

Stacktrace:
 [1] top-level scope
   @ In[9]:2
 [2] eval
   @ ./boot.jl:360 [inlined]
 [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
   @ Base ./loading.jl:1094

Because the dispatch procedure starts from concrete types and works upwards, dispatch always invokes the most specific method available.

For example, if you have methods for function f that handle

  1. (Float64, Int64) pairs

  2. (Number, Number) pairs

and you call f with f(0.5, 1) then the first method will be invoked.

This makes sense because (hopefully) the first method is optimized for exactly this kind of data.

The second method is probably more of a “catch all” method that handles other data in a less optimal way.

Here’s another simple example, involving a user-defined function

function q(x)  # or q(x::Any)
    println("Default (Any) method invoked")
end

function q(x::Number)
    println("Number method invoked")
end

function q(x::Integer)
    println("Integer method invoked")
end
q (generic function with 3 methods)

Let’s now run this and see how it relates to our discussion of method dispatch above

q(3)
Integer method invoked
q(3.0)
Number method invoked
q("foo")
Default (Any) method invoked

Since typeof(3) <: Int64 <: Integer <: Number, the call q(3) proceeds up the tree to Integer and invokes q(x::Integer).

On the other hand, 3.0 is a Float64, which is not a subtype of Integer.

Hence the call q(3.0) continues up to q(x::Number).

Finally, q("foo") is handled by the function operating on Any, since String is not a subtype of Number or Integer.

13.2.3. Analyzing Function Return Types

For the most part, time spent “optimizing” Julia code to run faster is about ensuring the compiler can correctly deduce types for all functions.

The macro @code_warntype gives us a hint

x = [1, 2, 3]
f(x) = 2x
@code_warntype f(x)
Variables
  #self#
::Core.Const(f)
  x::Vector{Int64}

Body::Vector{Int64}
1 ─ %1 = (
2 * x)
::Vector{Int64}
└──      return %1

The @code_warntype macro compiles f(x) using the type of x as an example – i.e., the [1, 2, 3] is used as a prototype for analyzing the compilation, rather than simply calculating the value.

Here, the Body::Array{Int64,1} tells us the type of the return value of the function, when called with types like [1, 2, 3], is always a vector of integers.

In contrast, consider a function potentially returning nothing, as in this lecture

f(x) = x > 0.0 ? x : nothing
@code_warntype f(1)
Variables
  #self#::Core.Const(f)
  x::Int64

Body
::Union{Nothing, Int64}
1 ─ %1 = (x > 0.0)::Bool
└──      goto #3 if not %1
2 ─      return x
3 ─      return Main.nothing

This states that the compiler determines the return type when called with an integer (like 1) could be one of two different types, Body::Union{Nothing, Int64}.

A final example is a variation on the above, which returns the maximum of x and 0.

f(x) = x > 0.0 ? x : 0.0
@code_warntype f(1)
Variables
  #self#::Core.Const(f)
  x::Int64

Body::Union{Float64, Int64}
1 ─ %1 = (x > 0.0)::Bool
└──      goto #3 if not %1
2 ─      return x
3 ─      return 0.0

Which shows that, when called with an integer, the type could be that integer or the floating point 0.0.

On the other hand, if we use change the function to return 0 if x <= 0, it is type-unstable with floating point.

f(x) = x > 0.0 ? x : 0
@code_warntype f(1.0)
Variables
  #self#::Core.Const(f)
  x::Float64

Body::Union{Float64, Int64}
1 ─ %1 = (x > 0.0)::Bool
└──      goto #3 if not %1
2 ─      return x
3 ─      return 0

The solution is to use the zero(x) function which returns the additive identity element of type x.

On the other hand, if we change the function to return 0 if x <= 0, it is type-unstable with floating point.

@show zero(2.3)
@show zero(4)
@show zero(2.0 + 3im)

f(x) = x > 0.0 ? x : zero(x)
@code_warntype f(1.0)
zero(2.3) = 0.0
zero(4) = 0
zero(2.0 + 3im) = 0.0 + 0.0im
Variables
  #self#::Core.Const(f)
  x::Float64

Body::Float64
1 ─ %1 = (x > 0.0)::Bool
└──      goto #3 if not %1
2 ─      return x
3 ─ %4 = Main.zero(x)::Core.Const(0.0)
└──      return %4

13.3. Foundations

Let’s think about how quickly code runs, taking as given

  • hardware configuration

  • algorithm (i.e., set of instructions to be executed)

We’ll start by discussing the kinds of instructions that machines understand.

13.3.1. Machine Code

All instructions for computers end up as machine code.

Writing fast code — expressing a given algorithm so that it runs quickly — boils down to producing efficient machine code.

You can do this yourself, by hand, if you want to.

Typically this is done by writing assembly, which is a symbolic representation of machine code.

Here’s some assembly code implementing a function that takes arguments \(a, b\) and returns \(2a + 8b\)

    pushq   %rbp
    movq    %rsp, %rbp
    addq    %rdi, %rdi
    leaq    (%rdi,%rsi,8), %rax
    popq    %rbp
    retq
    nopl    (%rax)

Note that this code is specific to one particular piece of hardware that we use — different machines require different machine code.

If you ever feel tempted to start rewriting your economic model in assembly, please restrain yourself.

It’s far more sensible to give these instructions in a language like Julia, where they can be easily written and understood.

function f(a, b)
    y = 2a + 8b
    return y
end
f (generic function with 2 methods)

or Python

def f(a, b):
    y = 2 * a + 8 * b
    return y

or even C

int f(int a, int b) {
    int y = 2 * a + 8 * b;
    return y;
}

In any of these languages we end up with code that is much easier for humans to write, read, share and debug.

We leave it up to the machine itself to turn our code into machine code.

How exactly does this happen?

13.3.2. Generating Machine Code

The process for turning high level code into machine code differs across languages.

Let’s look at some of the options and how they differ from one another.

13.3.2.1. AOT Compiled Languages

Traditional compiled languages like Fortran, C and C++ are a reasonable option for writing fast code.

Indeed, the standard benchmark for performance is still well-written C or Fortran.

These languages compile down to efficient machine code because users are forced to provide a lot of detail on data types and how the code will execute.

The compiler therefore has ample information for building the corresponding machine code ahead of time (AOT) in a way that

  • organizes the data optimally in memory and

  • implements efficient operations as required for the task in hand

At the same time, the syntax and semantics of C and Fortran are verbose and unwieldy when compared to something like Julia.

Moreover, these low level languages lack the interactivity that’s so crucial for scientific work.

13.3.2.2. Interpreted Languages

Interpreted languages like Python generate machine code “on the fly”, during program execution.

This allows them to be flexible and interactive.

Moreover, programmers can leave many tedious details to the runtime environment, such as

  • specifying variable types

  • memory allocation/deallocation, etc.

But all this convenience and flexibility comes at a cost: it’s hard to turn instructions written in these languages into efficient machine code.

For example, consider what happens when Python adds a long list of numbers together.

Typically the runtime environment has to check the type of these objects one by one before it figures out how to add them.

This involves substantial overheads.

There are also significant overheads associated with accessing the data values themselves, which might not be stored contiguously in memory.

The resulting machine code is often complex and slow.

13.3.2.3. Just-in-time compilation

Just-in-time (JIT) compilation is an alternative approach that marries some of the advantages of AOT compilation and interpreted languages.

The basic idea is that functions for specific tasks are compiled as requested.

As long as the compiler has enough information about what the function does, it can in principle generate efficient machine code.

In some instances, all the information is supplied by the programmer.

In other cases, the compiler will attempt to infer missing information on the fly based on usage.

Through this approach, computing environments built around JIT compilers aim to

  • provide all the benefits of high level languages discussed above and, at the same time,

  • produce efficient instruction sets when functions are compiled down to machine code

13.4. JIT Compilation in Julia

JIT compilation is the approach used by Julia.

In an ideal setting, all information necessary to generate efficient native machine code is supplied or inferred.

In such a setting, Julia will be on par with machine code from low level languages.

13.4.1. An Example

Consider the function

function f(a, b)
    y = (a + 8b)^2
    return 7y
end
f (generic function with 2 methods)

Suppose we call f with integer arguments (e.g., z = f(1, 2)).

The JIT compiler now knows the types of a and b.

Moreover, it can infer types for other variables inside the function

  • e.g., y will also be an integer

It then compiles a specialized version of the function to handle integers and stores it in memory.

We can view the corresponding machine code using the @code_native macro

@code_native f(1, 2)
	
.text
; ┌ @ In[20]:2 within `f'
; │┌ @ int.jl:87 within `+'
	leaq	(%rdi,%rsi,8), %rcx
; │└
; │┌ @ intfuncs.jl:312 within `literal_pow'
; ││┌ @ int.jl:88 within `*'
	imulq	%rcx, %rcx
; │└└
; │ @ In[20]:3 within `f'
; │┌ @ int.jl:88 within `*'
	leaq	(,%rcx,8), %rax
	subq	%rcx, %rax
; │└
	retq
	nopw	%cs:(%rax,%rax)
; └

If we now call f again, but this time with floating point arguments, the JIT compiler will once more infer types for the other variables inside the function.

  • e.g., y will also be a float

It then compiles a new version to handle this type of argument.

@code_native f(1.0, 2.0)
	.text
; ┌ @ In[20]:1 within `f'
	movabsq	$.rodata.cst8, %rax
; │ @ In[20]:2 within `f'
; │┌ @ promotion.jl:322 within `*' @ float.jl:332
	vmulsd	(%rax), %xmm1, %xmm1
; │└
; │┌ @ float.jl:326 within `+'
	vaddsd	%xmm0, %xmm1, %xmm0
; │└
; │┌ @ intfuncs.jl:312 within `literal_pow'
; ││┌ @ float.jl:332 within `*'
	vmulsd	%xmm0, %xmm0, %xmm0
	movabsq	$140435319105392, %rax          # imm = 0x7FB9A54F2770
; │└└
; │ @ In[20]:3 within `f'
; │┌ @ promotion.jl:322 within `*' @ float.jl:332
	vmulsd	(%rax), %xmm0, %xmm0
; │└
	retq
	nopw	%cs:(%rax,%rax)
; └

Subsequent calls using either floats or integers are now routed to the appropriate compiled code.

13.4.2. Potential Problems

In some senses, what we saw above was a best case scenario.

Sometimes the JIT compiler produces messy, slow machine code.

This happens when type inference fails or the compiler has insufficient information to optimize effectively.

The next section looks at situations where these problems arise and how to get around them.

13.5. Fast and Slow Julia Code

To summarize what we’ve learned so far, Julia provides a platform for generating highly efficient machine code with relatively little effort by combining

  1. JIT compilation

  2. Optional type declarations and type inference to pin down the types of variables and hence compile efficient code

  3. Multiple dispatch to facilitate specialization and optimization of compiled code for different data types

But the process is not flawless, and hiccups can occur.

The purpose of this section is to highlight potential issues and show you how to circumvent them.

13.5.1. BenchmarkTools

The main Julia package for benchmarking is BenchmarkTools.jl.

Below, we’ll use the @btime macro it exports to evaluate the performance of Julia code.

As mentioned in an earlier lecture, we can also save benchmark results to a file and guard against performance regressions in code.

For more, see the package docs.

13.5.2. Global Variables

Global variables are names assigned to values outside of any function or type definition.

The are convenient and novice programmers typically use them with abandon.

But global variables are also dangerous, especially in medium to large size programs, since

  • they can affect what happens in any part of your program

  • they can be changed by any function

This makes it much harder to be certain about what some small part of a given piece of code actually commands.

Here’s a useful discussion on the topic.

When it comes to JIT compilation, global variables create further problems.

The reason is that the compiler can never be sure of the type of the global variable, or even that the type will stay constant while a given function runs.

To illustrate, consider this code, where b is global

b = 1.0
function g(a)
    global b
    tmp = a
    for i in 1:1_000_000
        tmp = tmp + a + b
    end
    return tmp
end
g (generic function with 1 method)

The code executes relatively slowly and uses a huge amount of memory.

using BenchmarkTools

@btime g(1.0)
  43.998 ms (3000001 allocations: 45.78 MiB)
2.000001e6

If you look at the corresponding machine code you will see that it’s a mess.

@code_native g(1.0)
	.text
; ┌ @ In[23]:2 within `g'
	pushq	%rbp
	movq	%rsp, %rbp
	pushq	%r15
	pushq	%r14
	pushq	%r13
	pushq	%r12
	pushq	%rbx
	andq	$-32, %rsp
	subq	$96, %rsp
	vmovsd	%xmm0, 8(%rsp)
	movabsq	$jl_system_image_data, %r14
	vxorps	%xmm0, %xmm0, %xmm0
	vmovaps	%ymm0, 32(%rsp)
	movq	$0, 64(%rsp)
	movq	%fs:0, %rax
	leaq	-32768(%rax), %rdi
	movq	$12, 32(%rsp)
	movq	-32768(%rax), %rcx
	movq	%rcx, 40(%rsp)
	leaq	32(%rsp), %rcx
	movq	%rcx, -32768(%rax)
	movabsq	$140435686286265, %rax          # imm = 0x7FB9BB31E3B9
	movq	%rdi, (%rsp)
	movl	$1400, %esi                     # imm = 0x578
	movl	$16, %edx
	vzeroupper
	callq	*%rax
	movq	%rax, %rbx
	movq	%r14, -8(%rax)
	vmovsd	8(%rsp), %xmm0                  # xmm0 = mem[0],zero
	vmovsd	%xmm0, (%rax)
	movl	$1000000, %r13d                 # imm = 0xF4240
	leaq	16(%rsp), %r12
	nopw	%cs:(%rax,%rax)
; │ @ In[23] within `g'
L176:
	movabsq	$jl_system_image_data, %r15
; │ @ In[23]:6 within `g'
	movq	-83094568(%r15), %r14
	movq	%r14, 64(%rsp)
	movq	%rbx, 56(%rsp)
	movq	(%rsp), %rdi
; │┌ @ operators.jl:560 within `+'
	movl	$1400, %esi                     # imm = 0x578
	movl	$16, %edx
	movabsq	$140435686286265, %rax          # imm = 0x7FB9BB31E3B9
	callq	*%rax
	movq	%r15, -8(%rax)
	vmovsd	8(%rsp), %xmm0                  # xmm0 = mem[0],zero
	vmovsd	%xmm0, (%rax)
	movq	%rax, 48(%rsp)
	movq	%rbx, 16(%rsp)
	movq	%rax, 24(%rsp)
	movabsq	$jl_system_image_data, %r15
	movq	%r15, %rdi
	movq	%r12, %rsi
	movl	$2, %edx
	movabsq	$jl_apply_generic, %rbx
	callq	*%rbx
	movq	%rax, 48(%rsp)
	movq	%rax, 16(%rsp)
	movq	%r14, 24(%rsp)
	movq	%r15, %rdi
	movq	%r12, %rsi
	movl	$2, %edx
	callq	*%rbx
	movq	%rax, %rbx
; │└
; │┌ @ range.jl:674 within `iterate'
; ││┌ @ promotion.jl:410 within `=='
	decq	%r13
; │└└
	jne	L176
; │ @ In[23] within `g'
	movq	40(%rsp), %rax
	movq	(%rsp), %rcx
	movq	%rax, (%rcx)
; │ @ In[23]:8 within `g'
	movq	%rbx, %rax
	leaq	-40(%rbp), %rsp
	popq	%rbx
	popq	%r12
	popq	%r13
	popq	%r14
	popq	%r15
	popq	%rbp
	retq
	nopl	(%rax)
; └

If we eliminate the global variable like so

function g(a, b)
    tmp = a
    for i in 1:1_000_000
        tmp = tmp + a + b
    end
    return tmp
end
g (generic function with 2 methods)

then execution speed improves dramatically. Furthermore, the number of allocations has dropped to zero.

@btime g(1.0, 1.0)
  2.673 ms (0 allocations: 0 bytes)
2.000001e6

Note that if you called @time instead, the first call would be slower as it would need to compile the function. Conversely, @btime discards the first call and runs it multiple times.

More information is available with @benchmark instead,

@benchmark g(1.0, 1.0)
BenchmarkTools.Trial: 1865 samples with 1 evaluation.
 Range (minmax):  2.673 ms2.707 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.679 ms             ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.679 ms ± 4.264 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █                                                          
  █▆▄▃▄▅▄▂▂▂▂▁▂▄▅▄▄▄▅▄▄▅▅▆▅▄▅▄▄▄▃▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂ ▃
  2.67 ms        Histogram: frequency by time       2.69 ms <

 Memory estimate: 0 bytes, allocs estimate: 0.

Also, the machine code is simple and clean

@code_native g(1.0, 1.0)
	.text
; ┌ @ In[26]:1 within `g'
	movl	$1000000, %eax                  # imm = 0xF4240
	vmovapd	%xmm0, %xmm2
	nopl	(%rax)
; │ @ In[26]:4 within `g'
; │┌ @ operators.jl:560 within `+' @ float.jl:326
L16:
	vaddsd	%xmm0, %xmm2, %xmm2
	vaddsd	%xmm1, %xmm2, %xmm2
; │└
; │┌ @ range.jl:674 within `iterate'
; ││┌ @ promotion.jl:410 within `=='
	decq	%rax
; │└└
	jne	L16
; │ @ In[26]:6 within `g'
	vmovapd	%xmm2, %xmm0
	retq
	nopw	%cs:(%rax,%rax)
; └

Now the compiler is certain of types throughout execution of the function and hence can optimize accordingly.

If global variations are strictly needed (and they almost never are) then you can declare them with a const to declare to Julia that the type never changes (the value can). For example,

const b_const = 1.0
function g_const(a)
    global b_const
    tmp = a
    for i in 1:1_000_000
        tmp = tmp + a + b_const
    end
    return tmp
end
@btime g_const(1)
  4.014 ms (0 allocations: 0 bytes)
2.000001e6

Now the compiler can again generate efficient machine code.

However, global variables within a function is almost always a bad idea. Instead, the b_const should be passed as a parameter to the function.

13.5.3. Composite Types with Abstract Field Types

Another scenario that trips up the JIT compiler is when composite types have fields with abstract types.

We met this issue earlier, when we discussed AR(1) models.

Let’s experiment, using, respectively,

  • an untyped field

  • a field with abstract type, and

  • parametric typing

As we’ll see, the last of these options gives us the best performance, while still maintaining significant flexibility.

Here’s the untyped case

struct Foo_any
    a
end

Here’s the case of an abstract type on the field a

struct Foo_abstract
    a::Real
end

Finally, here’s the parametrically typed case (where the {T <: Real} is not necessary for performance, and could simply be {T}

struct Foo_concrete{T <: Real}
    a::T
end

Now we generate instances

fg = Foo_any(1.0)
fa = Foo_abstract(1.0)
fc = Foo_concrete(1.0)
Foo_concrete{Float64}(1.0)

In the last case, concrete type information for the fields is embedded in the object

typeof(fc)
Foo_concrete{Float64}

This is significant because such information is detected by the compiler.

13.5.3.1. Timing

Here’s a function that uses the field a of our objects

function f(foo)
    tmp = foo.a
    for i in 1:1_000_000
        tmp = i + foo.a
    end
    return tmp
end
f (generic function with 2 methods)

Let’s try timing our code, starting with the case without any constraints:

@btime f($fg)
  26.143 ms (1999489 allocations: 30.51 MiB)
1.000001e6

The timing is not very impressive.

Here’s the nasty looking machine code

@code_native f(fg)
	.text
; ┌ @ In[36]:1 within `f'
	pushq	%rbp
	pushq	%r15
	pushq	%r14
	pushq	%r13
	pushq	%r12
	pushq	%rbx
	subq	$56, %rsp
	vxorps	%xmm0, %xmm0, %xmm0
	vmovaps	%xmm0, (%rsp)
	movq	$0, 16(%rsp)
	movq	%fs:0, %rcx
; │ @ In[36]:4 within `f'
; │┌ @ Base.jl within `getproperty'
	movq	$4, (%rsp)
	movq	-32768(%rcx), %rax
	movq	%rax, 8(%rsp)
	movq	%rsp, %rax
	movq	%rcx, 32(%rsp)
	movq	%rax, -32768(%rcx)
	movq	(%rdi), %r13
	movl	$1, %ebx
	movabsq	$jl_box_int64, %rbp
	movabsq	$jl_apply_generic, %r12
	movabsq	$jl_system_image_data, %r14
	leaq	40(%rsp), %r15
	nopw	(%rax,%rax)
; │└
L128:
	movq	%rbx, %rdi
	callq	*%rbp
	movq	%rax, 16(%rsp)
	movq	%rax, 40(%rsp)
	movq	%r13, 48(%rsp)
	movq	%r14, %rdi
	movq	%r15, %rsi
	movl	$2, %edx
	callq	*%r12
; │┌ @ range.jl:674 within `iterate'
	incq	%rbx
; ││┌ @ promotion.jl:410 within `=='
	cmpq	$1000001, %rbx                  # imm = 0xF4241
; │└└
	jne	L128
; │ @ In[36] within `f'
	movq	8(%rsp), %rcx
	movq	32(%rsp), %rdx
	movq	%rcx, -32768(%rdx)
; │ @ In[36]:6 within `f'
	addq	$56, %rsp
	popq	%rbx
	popq	%r12
	popq	%r13
	popq	%r14
	popq	%r15
	popq	%rbp
	retq
	nop
; └

The abstract case is almost identical,

@btime f($fa)
  24.758 ms (1999489 allocations: 30.51 MiB)
1.000001e6

Note the large memory footprint.

The machine code is also long and complex, although we omit details.

Finally, let’s look at the parametrically typed version

@btime f($fc)
  1.700 ns (0 allocations: 0 bytes)
1.000001e6

Which is improbably small - since a runtime of 1-2 nanoseconds without any allocations suggests no computations really took place.

A hint is in the simplicity of the corresponding machine code

@code_native f(fc)
	.text
; ┌ @ In[36]:4 within `f'
; │┌ @ promotion.jl:321 within `+' @ float.jl:326
	vmovsd	(%rdi), %xmm0                   # xmm0 = mem[0],zero
	movabsq	$.rodata.cst8, %rax
	vaddsd	(%rax), %xmm0, %xmm0
; │└
; │ @ In[36]:6 within `f'
	retq
	nopw	%cs:(%rax,%rax)
; └

This machine code has none of the hallmark assembly instructions associated with a loop, in particular loops (e.g. for in julia) end up as jumps in the machine code (e.g. jne).

Here, the compiler was smart enough to realize that only the final step in the loop matters, i.e. it could generate the equivalent to f(a) = 1_000_000 + foo.a and then it can return that directly and skip the loop. These sorts of code optimizations are only possible if the compiler is able to use a great deal of information about the types.

Finally, note that if we compile a slightly different version of the function, which doesn’t actually return the value

function f_no_return(foo)
    for i in 1:1_000_000
        tmp = i + foo.a
    end
end
@code_native f_no_return(fc)
	.text
; ┌ @ In[42]:3 within `f_no_return'
	retq
	nopw	%cs:(%rax,%rax)
; └

We see that the code is even simpler. In effect, if figured out that because tmp wasn’t returned, and the foo.a could have no side effects (since it knows the type of a), that it doesn’t even need to execute any of the code in the function.

13.5.4. Type Inference

Consider the following function, which essentially does the same job as Julia’s sum() function but acts only on floating point data

function sum_float_array(x::AbstractVector{<:Number})
    sum = 0.0
    for i in eachindex(x)
        sum += x[i]
    end
    return sum
end
sum_float_array (generic function with 1 method)

Calls to this function run very quickly

x_range = range(0,  1, length = 100_000)
x = collect(x_range)
typeof(x)
Vector{Float64} (alias for Array{Float64, 1})
@btime sum_float_array($x)
  134.603 μs (0 allocations: 0 bytes)
50000.0

When Julia compiles this function, it knows that the data passed in as x will be an array of 64 bit floats. Hence it’s known to the compiler that the relevant method for + is always addition of floating point numbers.

But consider a version without that type annotation

function sum_array(x)
    sum = 0.0
    for i in eachindex(x)
        sum += x[i]
    end
    return sum
end
@btime sum_array($x)
  136.502 μs (0 allocations: 0 bytes)
50000.0

Note that this has the same running time as the one with the explicit types. In julia, there is (almost never) performance gain from declaring types, and if anything they can make things worse by limited the potential for specialized algorithms. See generic programming for more.

As an example within Julia code, look at the built-in sum for array

@btime sum($x)
  16.000 μs (0 allocations: 0 bytes)
50000.00000000001

Versus the underlying range

@btime sum($x_range)
  199.178 ns (0 allocations: 0 bytes)
50000.0

Note that the difference in speed is enormous—suggesting it is better to keep things in their more structured forms as long as possible. You can check the underlying source used for this with @which sum(x_range) to see the specialized algorithm used.

13.5.4.1. Type Inferences

Here’s the same function minus the type annotation in the function signature

function sum_array(x)
    sum = 0.0
    for i in eachindex(x)
        sum += x[i]
    end
    return sum
end
sum_array (generic function with 1 method)

When we run it with the same array of floating point numbers it executes at a similar speed as the function with type information.

@btime sum_array($x)
  135.902 μs (0 allocations: 0 bytes)
50000.0

The reason is that when sum_array() is first called on a vector of a given data type, a newly compiled version of the function is produced to handle that type.

In this case, since we’re calling the function on a vector of floats, we get a compiled version of the function with essentially the same internal representation as sum_float_array().

13.5.4.2. An Abstract Container

Things get tougher for the interpreter when the data type within the array is imprecise.

For example, the following snippet creates an array where the element type is Any

x = Any[ 1/i for i in 1:1e6 ];
eltype(x)
Any

Now summation is much slower and memory management is less efficient.

@btime sum_array($x)
  19.287 ms (1000000 allocations: 15.26 MiB)
14.392726722864989

13.6. Further Comments

Here are some final comments on performance.

13.6.1. Summary and Tips

Use functions to segregate operations into logically distinct blocks.

Data types will be determined at function boundaries.

If types are not supplied then they will be inferred.

If types are stable and can be inferred effectively your functions will run fast.

13.6.2. Further Reading

A good next stop for further reading is the relevant part of the Julia documentation.