Metaprogramming

The strongest legacy of Lisp in the Julia language is its metaprogramming support. Like Lisp, Julia represents its own code as a data structure of the language itself. Since code is represented by objects that can be created and manipulated from within the language, it is possible for a program to transform and generate its own code. This allows sophisticated code generation without extra build steps, and also allows true Lisp-style macros operating at the level of abstract syntax trees. In contrast, preprocessor “macro” systems, like that of C and C++, perform textual manipulation and substitution before any actual parsing or interpretation occurs. Because all data types and code in Julia are represented by Julia data structures, powerful reflection capabilities are available to explore the internals of a program and its types just like any other data.

Program representation

Every Julia program starts life as a string:

julia> prog = "1 + 1"
"1 + 1"

What happens next?

The next step is to parse each string into an object called an expression, represented by the Julia type Expr:

julia> ex1 = parse(prog)
:(1 + 1)

julia> typeof(ex1)
Expr

Expr objects contain three parts:

• a Symbol identifying the kind of expression. A symbol is an interned string identifier (more discussion below).

julia> ex1.head
:call

• the expression arguments, which may be symbols, other expressions, or literal values:

julia> ex1.args
3-element Array{Any,1}:
  :+
 1
 1

• finally, the expression result type, which may be annotated by the user or inferred by the compiler (and may be ignored completely for the purposes of this chapter):

julia> ex1.typ
Any

Expressions may also be constructed directly in prefix notation:

julia> ex2 = Expr(:call, :+, 1, 1)
:(1 + 1)

The two expressions constructed above – by parsing and by direct construction – are equivalent:

julia> ex1 == ex2
true

The key point here is that Julia code is internally represented as a data structure that is accessible from the language itself.

The dump() function provides indented and annotated display of Expr objects:

julia> dump(ex2)
Expr
  head: Symbol call
  args: Array{Any}((3,))
    1: Symbol +
    2: Int64 1
    3: Int64 1
  typ: Any

Expr objects may also be nested:

julia> ex3 = parse("(4 + 4) / 2")
:((4 + 4) / 2)

Another way to view expressions is with Meta.show_sexpr, which displays the S-expression form of a given Expr, which may look very familiar to users of Lisp. Here’s an example illustrating the display on a nested Expr:

julia> Meta.show_sexpr(ex3)
(:call, :/, (:call, :+, 4, 4), 2)

Symbols

The : character has two syntactic purposes in Julia. The first form creates a Symbol, an interned string used as one building-block of expressions:

julia> :foo
:foo

julia> typeof(ans)
Symbol

The Symbol constructor takes any number of arguments and creates a new symbol by concatenating their string representations together:

julia> :foo == Symbol("foo")
true

julia> Symbol("func",10)
:func10

julia> Symbol(:var,'_',"sym")
:var_sym

In the context of an expression, symbols are used to indicate access to variables; when an expression is evaluated, a symbol is replaced with the value bound to that symbol in the appropriate scope.

Sometimes extra parentheses around the argument to : are needed to avoid ambiguity in parsing.:

julia> :(:)
:(:)

julia> :(::)
:(::)

Expressions and evaluation

Quoting

The second syntactic purpose of the : character is to create expression objects without using the explicit Expr constructor. This is referred to as quoting. The : character, followed by paired parentheses around a single statement of Julia code, produces an Expr object based on the enclosed code. Here is example of the short form used to quote an arithmetic expression:

julia> ex = :(a+b*c+1)
:(a + b * c + 1)

julia> typeof(ex)
Expr

(to view the structure of this expression, try ex.head and ex.args, or use dump() as above)

Note that equivalent expressions may be constructed using parse() or the direct Expr form:

julia>      :(a + b*c + 1)  ==
       parse("a + b*c + 1") ==
       Expr(:call, :+, :a, Expr(:call, :*, :b, :c), 1)
true

Expressions provided by the parser generally only have symbols, other expressions, and literal values as their args, whereas expressions constructed by Julia code can have arbitrary run-time values without literal forms as args. In this specific example, + and a are symbols, *(b,c) is a subexpression, and 1 is a literal 64-bit signed integer.

There is a second syntactic form of quoting for multiple expressions: blocks of code enclosed in quote ... end. Note that this form introduces QuoteNode elements to the expression tree, which must be considered when directly manipulating an expression tree generated from quote blocks. For other purposes, :( ... ) and quote .. end blocks are treated identically.

julia> ex = quote
           x = 1
           y = 2
           x + y
       end
quote  # none, line 2:
    x = 1 # none, line 3:
    y = 2 # none, line 4:
    x + y
end

julia> typeof(ex)
Expr

Interpolation

Direct construction of Expr objects with value arguments is powerful, but Expr constructors can be tedious compared to “normal” Julia syntax. As an alternative, Julia allows “splicing” or interpolation of literals or expressions into quoted expressions. Interpolation is indicated by the $ prefix.

In this example, the literal value of a is interpolated:

julia> a = 1;

julia> ex = :($a + b)
:(1 + b)

Interpolating into an unquoted expression is not supported and will cause a compile-time error:

julia> $a + b
ERROR: unsupported or misplaced expression $
 ...

In this example, the tuple (1,2,3) is interpolated as an expression into a conditional test:

julia> ex = :(a in $:((1,2,3)) )
:(a in (1,2,3))

Interpolating symbols into a nested expression requires enclosing each symbol in an enclosing quote block:

julia> :( :a in $( :(:a + :b) ) )
                   ^^^^^^^^^^
                   quoted inner expression

The use of $ for expression interpolation is intentionally reminiscent of string interpolation and command interpolation. Expression interpolation allows convenient, readable programmatic construction of complex Julia expressions.

eval() and effects

Given an expression object, one can cause Julia to evaluate (execute) it at global scope using eval():

julia> :(1 + 2)
:(1 + 2)

julia> eval(ans)
3

julia> ex = :(a + b)
:(a + b)

julia> eval(ex)
ERROR: UndefVarError: b not defined
 ...

julia> a = 1; b = 2;

julia> eval(ex)
3

Every module has its own eval() function that evaluates expressions in its global scope. Expressions passed to eval() are not limited to returning values — they can also have side-effects that alter the state of the enclosing module’s environment:

julia> ex = :(x = 1)
:(x = 1)

julia> x
ERROR: UndefVarError: x not defined
 ...

julia> eval(ex)
1

julia> x
1

Here, the evaluation of an expression object causes a value to be assigned to the global variable x.

Since expressions are just Expr objects which can be constructed programmatically and then evaluated, it is possible to dynamically generate arbitrary code which can then be run using eval(). Here is a simple example:

julia> a = 1;

julia> ex = Expr(:call, :+, a, :b)
:(1 + b)

julia> a = 0; b = 2;

julia> eval(ex)
3

The value of a is used to construct the expression ex which applies the + function to the value 1 and the variable b. Note the important distinction between the way a and b are used:

• The value of the variable a at expression construction time is used as an immediate value in the expression. Thus, the value of a when the expression is evaluated no longer matters: the value in the expression is already 1, independent of whatever the value of a might be.
• On the other hand, the symbol :b is used in the expression construction, so the value of the variable b at that time is irrelevant — :b is just a symbol and the variable b need not even be defined. At expression evaluation time, however, the value of the symbol :b is resolved by looking up the value of the variable b.

Functions on Expressions

As hinted above, one extremely useful feature of Julia is the capability to generate and manipulate Julia code within Julia itself. We have already seen one example of a function returning Expr objects: the parse() function, which takes a string of Julia code and returns the corresponding Expr. A function can also take one or more Expr objects as arguments, and return another Expr. Here is a simple, motivating example:

julia> function math_expr(op, op1, op2)
         expr = Expr(:call, op, op1, op2)
         return expr
       end

 julia>  ex = math_expr(:+, 1, Expr(:call, :*, 4, 5))
 :(1 + 4*5)

 julia> eval(ex)
 21

As another example, here is a function that doubles any numeric argument, but leaves expressions alone:

julia> function make_expr2(op, opr1, opr2)
         opr1f, opr2f = map(x -> isa(x, Number) ? 2*x : x, (opr1, opr2))
         retexpr = Expr(:call, op, opr1f, opr2f)

         return retexpr
   end
make_expr2 (generic function with 1 method)

julia> make_expr2(:+, 1, 2)
:(2 + 4)

julia> ex = make_expr2(:+, 1, Expr(:call, :*, 5, 8))
:(2 + 5 * 8)

julia> eval(ex)
42

Macros

Macros provide a method to include generated code in the final body of a program. A macro maps a tuple of arguments to a returned expression, and the resulting expression is compiled directly rather than requiring a runtime eval() call. Macro arguments may include expressions, literal values, and symbols.

Basics

Here is an extraordinarily simple macro:

macro sayhello()
    return :( println("Hello, world!") )
end

Macros have a dedicated character in Julia’s syntax: the @ (at-sign), followed by the unique name declared in a macro NAME ... end block. In this example, the compiler will replace all instances of @sayhello with:

:( println("Hello, world!") )

When @sayhello is given at the REPL, the expression executes immediately, thus we only see the evaluation result:

julia> @sayhello()
"Hello, world!"

Now, consider a slightly more complex macro:

julia> macro sayhello(name)
           return :( println("Hello, ", $name) )
       end

This macro takes one argument: name. When @sayhello is encountered, the quoted expression is expanded to interpolate the value of the argument into the final expression:

julia> @sayhello("human")
Hello, human

We can view the quoted return expression using the function macroexpand() (important note: this is an extremely useful tool for debugging macros):

julia> ex = macroexpand( :(@sayhello("human")) )
:(println("Hello, ","human"))
                    ^^^^^^^
                    interpolated: now a literal string

julia> typeof(ex)
Expr

Hold up: why macros?

We have already seen a function f(::Expr...) -> Expr in a previous section. In fact, macroexpand() is also such a function. So, why do macros exist?

Macros are necessary because they execute when code is parsed, therefore, macros allow the programmer to generate and include fragments of customized code before the full program is run. To illustrate the difference, consider the following example:

julia> macro twostep(arg)
           println("I execute at parse time. The argument is: ", arg)

           return :(println("I execute at runtime. The argument is: ", $arg))
       end

julia> ex = macroexpand( :(@twostep :(1, 2, 3)) );
I execute at parse time. The argument is: :((1,2,3))

The first call to println() is executed when macroexpand() is called. The resulting expression contains only the second println:

julia> typeof(ex)
Expr

julia> ex
:(println("I execute at runtime. The argument is: ",$(Expr(:copyast, :(:((1,2,3)))))))

julia> eval(ex)
I execute at runtime. The argument is: (1,2,3)

Macro invocation

Macros are invoked with the following general syntax:

@name expr1 expr2 ...
@name(expr1, expr2, ...)

Note the distinguishing @ before the macro name and the lack of commas between the argument expressions in the first form, and the lack of whitespace after @name in the second form. The two styles should not be mixed. For example, the following syntax is different from the examples above; it passes the tuple (expr1, expr2, ...) as one argument to the macro:

@name (expr1, expr2, ...)

It is important to emphasize that macros receive their arguments as expressions, literals, or symbols. One way to explore macro arguments is to call the show() function within the macro body:

julia> macro showarg(x)
   show(x)
   # ... remainder of macro, returning an expression
end


julia> @showarg(a)
(:a,)

julia> @showarg(1+1)
:(1 + 1)

julia> @showarg(println("Yo!"))
:(println("Yo!"))

Building an advanced macro

Here is a simplified definition of Julia’s @assert macro:

macro assert(ex)
    return :( $ex ? nothing : throw(AssertionError($(string(ex)))) )
end

This macro can be used like this:

julia> @assert 1==1.0

julia> @assert 1==0
ERROR: AssertionError: 1 == 0
 ...

In place of the written syntax, the macro call is expanded at parse time to its returned result. This is equivalent to writing:

1==1.0 ? nothing : throw(AssertionError("1==1.0"))
1==0 ? nothing : throw(AssertionError("1==0"))

That is, in the first call, the expression :(1==1.0) is spliced into the test condition slot, while the value of string(:(1==1.0)) is spliced into the assertion message slot. The entire expression, thus constructed, is placed into the syntax tree where the @assert macro call occurs. Then at execution time, if the test expression evaluates to true, then nothing is returned, whereas if the test is false, an error is raised indicating the asserted expression that was false. Notice that it would not be possible to write this as a function, since only the value of the condition is available and it would be impossible to display the expression that computed it in the error message.

The actual definition of @assert in the standard library is more complicated. It allows the user to optionally specify their own error message, instead of just printing the failed expression. Just like in functions with a variable number of arguments, this is specified with an ellipses following the last argument:

macro assert(ex, msgs...)
    msg_body = isempty(msgs) ? ex : msgs[1]
    msg = string(msg_body)
    return :($ex ? nothing : throw(AssertionError($msg)))
end

Now @assert has two modes of operation, depending upon the number of arguments it receives! If there’s only one argument, the tuple of expressions captured by msgs will be empty and it will behave the same as the simpler definition above. But now if the user specifies a second argument, it is printed in the message body instead of the failing expression. You can inspect the result of a macro expansion with the aptly named macroexpand() function:

julia> macroexpand(:(@assert a==b))
:(if a == b
        nothing
    else
        (Base.throw)(Base.Main.Base.AssertionError("a == b"))
    end)

julia> macroexpand(:(@assert a==b "a should equal b!"))
:(if a == b
        nothing
    else
        (Base.throw)(Base.Main.Base.AssertionError("a should equal b!"))
    end)

There is yet another case that the actual @assert macro handles: what if, in addition to printing “a should equal b,” we wanted to print their values? One might naively try to use string interpolation in the custom message, e.g., @assert a==b "a ($a) should equal b ($b)!", but this won’t work as expected with the above macro. Can you see why? Recall from string interpolation that an interpolated string is rewritten to a call to string(). Compare:

julia> typeof(:("a should equal b"))
String

julia> typeof(:("a ($a) should equal b ($b)!"))
Expr

julia> dump(:("a ($a) should equal b ($b)!"))
Expr
  head: Symbol string
  args: Array{Any}((5,))
    1: String "a ("
    2: Symbol a
    3: String ") should equal b ("
    4: Symbol b
    5: String ")!"
  typ: Any

So now instead of getting a plain string in msg_body, the macro is receiving a full expression that will need to be evaluated in order to display as expected. This can be spliced directly into the returned expression as an argument to the string() call; see error.jl for the complete implementation.

The @assert macro makes great use of splicing into quoted expressions to simplify the manipulation of expressions inside the macro body.

Hygiene

An issue that arises in more complex macros is that of hygiene. In short, macros must ensure that the variables they introduce in their returned expressions do not accidentally clash with existing variables in the surrounding code they expand into. Conversely, the expressions that are passed into a macro as arguments are often expected to evaluate in the context of the surrounding code, interacting with and modifying the existing variables. Another concern arises from the fact that a macro may be called in a different module from where it was defined. In this case we need to ensure that all global variables are resolved to the correct module. Julia already has a major advantage over languages with textual macro expansion (like C) in that it only needs to consider the returned expression. All the other variables (such as msg in @assert above) follow the normal scoping block behavior.

To demonstrate these issues, let us consider writing a @time macro that takes an expression as its argument, records the time, evaluates the expression, records the time again, prints the difference between the before and after times, and then has the value of the expression as its final value. The macro might look like this:

macro time(ex)
  return quote
    local t0 = time()
    local val = $ex
    local t1 = time()
    println("elapsed time: ", t1-t0, " seconds")
    val
  end
end

Here, we want t0, t1, and val to be private temporary variables, and we want time to refer to the time() function in the standard library, not to any time variable the user might have (the same applies to println). Imagine the problems that could occur if the user expression ex also contained assignments to a variable called t0, or defined its own time variable. We might get errors, or mysteriously incorrect behavior.

Julia’s macro expander solves these problems in the following way. First, variables within a macro result are classified as either local or global. A variable is considered local if it is assigned to (and not declared global), declared local, or used as a function argument name. Otherwise, it is considered global. Local variables are then renamed to be unique (using the gensym() function, which generates new symbols), and global variables are resolved within the macro definition environment. Therefore both of the above concerns are handled; the macro’s locals will not conflict with any user variables, and time and println will refer to the standard library definitions.

One problem remains however. Consider the following use of this macro:

module MyModule
import Base.@time

time() = ... # compute something

@time time()
end

Here the user expression ex is a call to time, but not the same time function that the macro uses. It clearly refers to MyModule.time. Therefore we must arrange for the code in ex to be resolved in the macro call environment. This is done by “escaping” the expression with esc():

macro time(ex)
    ...
    local val = $(esc(ex))
    ...
end

An expression wrapped in this manner is left alone by the macro expander and simply pasted into the output verbatim. Therefore it will be resolved in the macro call environment.

This escaping mechanism can be used to “violate” hygiene when necessary, in order to introduce or manipulate user variables. For example, the following macro sets x to zero in the call environment:

macro zerox()
  return esc(:(x = 0))
end

function foo()
  x = 1
  @zerox
  x  # is zero
end

This kind of manipulation of variables should be used judiciously, but is occasionally quite handy.

Code Generation

When a significant amount of repetitive boilerplate code is required, it is common to generate it programmatically to avoid redundancy. In most languages, this requires an extra build step, and a separate program to generate the repetitive code. In Julia, expression interpolation and eval() allow such code generation to take place in the normal course of program execution. For example, the following code defines a series of operators on three arguments in terms of their 2-argument forms:

for op = (:+, :*, :&, :|, :$)
  eval(quote
    ($op)(a,b,c) = ($op)(($op)(a,b),c)
  end)
end

In this manner, Julia acts as its own preprocessor, and allows code generation from inside the language. The above code could be written slightly more tersely using the : prefix quoting form:

for op = (:+, :*, :&, :|, :$)
  eval(:(($op)(a,b,c) = ($op)(($op)(a,b),c)))
end

This sort of in-language code generation, however, using the eval(quote(...)) pattern, is common enough that Julia comes with a macro to abbreviate this pattern:

for op = (:+, :*, :&, :|, :$)
  @eval ($op)(a,b,c) = ($op)(($op)(a,b),c)
end

The @eval macro rewrites this call to be precisely equivalent to the above longer versions. For longer blocks of generated code, the expression argument given to @eval can be a block:

@eval begin
  # multiple lines
end

Non-Standard String Literals

Recall from Strings that string literals prefixed by an identifier are called non-standard string literals, and can have different semantics than un-prefixed string literals. For example:

• r"^\s*(?:#|$)" produces a regular expression object rather than a string
• b"DATA\xff\u2200" is a byte array literal for [68,65,84,65,255,226,136,128].

Perhaps surprisingly, these behaviors are not hard-coded into the Julia parser or compiler. Instead, they are custom behaviors provided by a general mechanism that anyone can use: prefixed string literals are parsed as calls to specially-named macros. For example, the regular expression macro is just the following:

macro r_str(p)
  Regex(p)
end

That’s all. This macro says that the literal contents of the string literal r"^\s*(?:#|$)" should be passed to the @r_str macro and the result of that expansion should be placed in the syntax tree where the string literal occurs. In other words, the expression r"^\s*(?:#|$)" is equivalent to placing the following object directly into the syntax tree:

Regex("^\\s*(?:#|\$)")

Not only is the string literal form shorter and far more convenient, but it is also more efficient: since the regular expression is compiled and the Regex object is actually created when the code is compiled, the compilation occurs only once, rather than every time the code is executed. Consider if the regular expression occurs in a loop:

for line = lines
  m = match(r"^\s*(?:#|$)", line)
  if m === nothing
    # non-comment
  else
    # comment
  end
end

Since the regular expression r"^\s*(?:#|$)" is compiled and inserted into the syntax tree when this code is parsed, the expression is only compiled once instead of each time the loop is executed. In order to accomplish this without macros, one would have to write this loop like this:

re = Regex("^\\s*(?:#|\$)")
for line = lines
  m = match(re, line)
  if m === nothing
    # non-comment
  else
    # comment
  end
end

Moreover, if the compiler could not determine that the regex object was constant over all loops, certain optimizations might not be possible, making this version still less efficient than the more convenient literal form above. Of course, there are still situations where the non-literal form is more convenient: if one needs to interpolate a variable into the regular expression, one must take this more verbose approach; in cases where the regular expression pattern itself is dynamic, potentially changing upon each loop iteration, a new regular expression object must be constructed on each iteration. In the vast majority of use cases, however, regular expressions are not constructed based on run-time data. In this majority of cases, the ability to write regular expressions as compile-time values is invaluable.

The mechanism for user-defined string literals is deeply, profoundly powerful. Not only are Julia’s non-standard literals implemented using it, but also the command literal syntax (`echo "Hello, $person"`) is implemented with the following innocuous-looking macro:

macro cmd(str)
  :(cmd_gen($shell_parse(str)))
end

Of course, a large amount of complexity is hidden in the functions used in this macro definition, but they are just functions, written entirely in Julia. You can read their source and see precisely what they do — and all they do is construct expression objects to be inserted into your program’s syntax tree.

Generated functions

A very special macro is @generated, which allows you to define so-called generated functions. These have the capability to generate specialized code depending on the types of their arguments with more flexibility and/or less code than what can be achieved with multiple dispatch. While macros work with expressions at parsing-time and cannot access the types of their inputs, a generated function gets expanded at a time when the types of the arguments are known, but the function is not yet compiled.

Instead of performing some calculation or action, a generated function declaration returns a quoted expression which then forms the body for the method corresponding to the types of the arguments. When called, the body expression is compiled (or fetched from a cache, on subsequent calls) and only the returned expression - not the code that generated it - is evaluated. Thus, generated functions provide a flexible framework to move work from run-time to compile-time.

When defining generated functions, there are three main differences to ordinary functions:

1. You annotate the function declaration with the @generated macro. This adds some information to the AST that lets the compiler know that this is a generated function.
2. In the body of the generated function you only have access to the types of the arguments, not their values.
3. Instead of calculating something or performing some action, you return a quoted expression which, when evaluated, does what you want.

It’s easiest to illustrate this with an example. We can declare a generated function foo as

julia> @generated function foo(x)
           println(x)
           return :(x*x)
       end
foo (generic function with 1 method)

Note that the body returns a quoted expression, namely :(x*x), rather than just the value of x*x.

From the caller’s perspective, they are very similar to regular functions; in fact, you don’t have to know if you’re calling a regular or generated function - the syntax and result of the call is just the same. Let’s see how foo behaves:

# note: output is from println() statement in the body
julia> x = foo(2);
Int64

julia> x           # now we print x
4

julia> y = foo("bar");
String

julia> y
"barbar"

So, we see that in the body of the generated function, x is the type of the passed argument, and the value returned by the generated function, is the result of evaluating the quoted expression we returned from the definition, now with the value of x.

What happens if we evaluate foo again with a type that we have already used?

julia> foo(4)
16

Note that there is no printout of Int64. The body of the generated function is only executed once (not entirely true, see note below) when the method for that specific set of argument types is compiled. After that, the expression returned from the generated function on the first invocation is re-used as the method body.

The reason for the disclaimer above is that the number of times a generated function is generated is really an implementation detail; it might be only once, but it might also be more often. As a consequence, you should never write a generated function with side effects - when, and how often, the side effects occur is undefined. (This is true for macros too - and just like for macros, the use of eval() in a generated function is a sign that you’re doing something the wrong way.)

The example generated function foo above did not do anything a normal function foo(x)=x*x could not do, except printing the type on the first invocation (and incurring a higher compile-time cost). However, the power of a generated function lies in its ability to compute different quoted expression depending on the types passed to it:

julia> @generated function bar(x)
           if x <: Integer
               return :(x^2)
           else
               return :(x)
           end
       end
bar (generic function with 1 method)

julia> bar(4)
16
julia> bar("baz")
"baz"

(although of course this contrived example is easily implemented using multiple dispatch...)

We can, of course, abuse this to produce some interesting behavior:

julia> @generated function baz(x)
           if rand() < .9
               return :(x^2)
           else
               return :("boo!")
           end
       end
baz (generic function with 1 method)

Since the body of the generated function is non-deterministic, its behavior is undefined; the expression returned on the first invocation will be used for all subsequent invocations with the same type (again, with the exception covered by the disclaimer above). When we call the generated function with x of a new type, rand() will be called again to see which method body to use for the new type. In this case, for one type out of ten, baz(x) will return the string "boo!".

Don’t copy these examples!

These examples are hopefully helpful to illustrate how generated functions work, both in the definition end and at the call site; however, don’t copy them, for the following reasons:

• the foo function has side-effects, and it is undefined exactly when, how often or how many times these side-effects will occur
• the bar function solves a problem that is better solved with multiple dispatch - defining bar(x) = x and bar(x::Integer) = x^2 will do the same thing, but it is both simpler and faster.
• the baz function is pathologically insane

Instead, now that we have a better understanding for how generated functions work, let’s use them to build some more advanced functionality...

An advanced example

Julia’s base library has a sub2ind() function to calculate a linear index into an n-dimensional array, based on a set of n multilinear indices - in other words, to calculate the index i that can be used to index into an array A using A[i], instead of A[x,y,z,...]. One possible implementation is the following:

function sub2ind_loop{N}(dims::NTuple{N}, I::Integer...)
    ind = I[N] - 1
    for i = N-1:-1:1
        ind = I[i]-1 + dims[i]*ind
    end
    return ind + 1
end

The same thing can be done using recursion:

sub2ind_rec(dims::Tuple{}) = 1
sub2ind_rec(dims::Tuple{},i1::Integer, I::Integer...) =
    i1==1 ? sub2ind_rec(dims,I...) : throw(BoundsError())
sub2ind_rec(dims::Tuple{Integer,Vararg{Integer}}, i1::Integer) = i1
sub2ind_rec(dims::Tuple{Integer,Vararg{Integer}}, i1::Integer, I::Integer...) =
    i1 + dims[1]*(sub2ind_rec(tail(dims),I...)-1)

Both these implementations, although different, do essentially the same thing: a runtime loop over the dimensions of the array, collecting the offset in each dimension into the final index.

However, all the information we need for the loop is embedded in the type information of the arguments. Thus, we can utilize generated functions to move the iteration to compile-time; in compiler parlance, we use generated functions to manually unroll the loop. The body becomes almost identical, but instead of calculating the linear index, we build up an expression that calculates the index:

@generated function sub2ind_gen{N}(dims::NTuple{N}, I::Integer...)
    ex = :(I[$N] - 1)
    for i = N-1:-1:1
        ex = :(I[$i] - 1 + dims[$i]*$ex)
    end
    return :($ex + 1)
end

What code will this generate?

An easy way to find out, is to extract the body into another (regular) function:

julia> @generated function sub2ind_gen{N}(dims::NTuple{N}, I::Integer...)
           sub2ind_gen_impl(dims, I...)
       end
sub2ind_gen (generic function with 1 method)

julia> function sub2ind_gen_impl{N}(dims::Type{NTuple{N}}, I...)
           length(I) == N || return :(error("partial indexing is unsupported"))
           ex = :(I[$N] - 1)
           for i = N-1:-1:1
               ex = :(I[$i] - 1 + dims[$i]*$ex)
           end
           return :($ex + 1)
       end
sub2ind_gen_impl (generic function with 1 method)

We can now execute sub2ind_gen_impl and examine the expression it returns:

julia> sub2ind_gen_impl(Tuple{Int,Int}, Int, Int)
:(((I[1] - 1) + dims[1] * (I[2] - 1)) + 1)

So, the method body that will be used here doesn’t include a loop at all - just indexing into the two tuples, multiplication and addition/subtraction. All the looping is performed compile-time, and we avoid looping during execution entirely. Thus, we only loop once per type, in this case once per N (except in edge cases where the function is generated more than once - see disclaimer above).