Tail calls
Here’s a little program to sum the first N natural numbers:
(define (sum n) (if (zero? n) n (+ n (sum (sub1 n))))) (print (sum (read-num)))
What will happen if we call (sum 1000000)
? We overflow the
stack.
In our interpreter, we can fix this:
(define (sum n total) (if (zero? n) total (sum (sub1 n) (+ n total)))) (print (sum (read-num) 0))
Our compiler, however, still segfaults. What’s going on here? The
question isn’t why our compiler segfaults–why
wouldn’t it? It has a finite amount of stack space, after all, and
we’re doing a lot of function calls here. Each call to
sum
adds a few values to the stack. So why isn’t the
interpreter overflowing the stack?
Let’s postpone the answer to that question. First, let’s
look at the assembly instructions we produce for our little
sum
program:
global _entry extern _error extern _read_num extern _print_value extern _print_newline _entry: mov [rsp + -24], rdi add rsp, -24 call _read_num sub rsp, -24 mov rdi, [rsp + -24] mov [rsp + -24], rax mov rax, 0 mov [rsp + -32], rax add rsp, -8 call _function_sum_957043065 sub rsp, -8 mov [rsp + -8], rdi mov rdi, rax add rsp, -8 call _print_value sub rsp, -8 mov rdi, [rsp + -8] mov rax, 159 ret _function_sum_957043065: mov rax, [rsp + -8] cmp rax, 0 mov rax, 0 setz al shl rax, 7 or rax, 31 cmp rax, 31 jz _else__2 mov rax, [rsp + -16] jmp _continue__3 _else__2: mov rax, [rsp + -8] sub rax, 4 mov [rsp + -40], rax mov rax, [rsp + -8] mov [rsp + -48], rax mov rax, [rsp + -16] mov r8, [rsp + -48] add rax, r8 mov [rsp + -48], rax add rsp, -24 call _function_sum_957043065 sub rsp, -24 _continue__3: ret
Look at the end of the program, there. In the else
case of
our conditional expression, we’re pushing a bunch of arguments to
the stack and then calling our function. After we come back from that
call we’ll execute two instructions: we’ll restore the stack
pointer and then return again. In other words, we’re really not
doing any work once our function returns! We have this stack frame
storing our local variables and function parameters, but we’re not
accessing any of them! It seems like once we’re doing that
function call, we shouldn’t really need an extra stack frame;
instead, we should be able to just re-use the one we already
have–essentially, replacing the call
with a
jmp
.
When can we do this? Well, we should be able to do this whenever we don’t have any work to do after a function is called. If we’re just going to return whatever another function returns, without modification, we should be able to re-use our stack frame: we’re guaranteed not to need any of the things we’ve stored there.
Function calls in this position are called tail calls. Take a look at this little program:
(define (f x) (+ 3 x)) (define (sum-f n total) (if (zero? n) total (sum-f (sub1 n) (+ (f n) total)))) (print (sum-f (read-num) 0))
Is the call to f
in tail position? No–after
f
returns, we have to do more work.
How about this program?
(define (even n) (if (zero? n) true (odd (sub1 n)))) (define (odd n) (if (zero? n) false (not (even n)))) (print (even (read-num)))
Is even
’s call to odd
a tail call? Yes.
Is odd
’s call to even
a tail call?
No–after even
returns, there’s more work to do
(specifically, negating the value). We’ve decided that we want to
compile function calls differently when they are in tail position. So,
let’s add an argument to the compiler that will be
true
if the expression being compiled is in tail position
and false
otherwise. We’ll need to add it to every
call to compile_exp
; here are some of the more interesting
ones:
let rec compile_exp (defns : defn list) (tab : int symtab) (stack_index : int) (exp : s_exp) (is_tail : bool) : directive list = match exp with (* ... *) | Lst [Sym "print"; e] -> compile_exp defns tab stack_index e false @ [ Mov (stack_address stack_index, Reg Rdi) ; Mov (Reg Rdi, Reg Rax) ; Add (Reg Rsp, Imm (align_stack_index stack_index)) ; Call "print_value" ; Sub (Reg Rsp, Imm (align_stack_index stack_index)) ; Mov (Reg Rdi, stack_address stack_index) ; Mov (Reg Rax, operand_of_bool true) ] | Lst (Sym "do" :: exps) when List.length exps > 0 -> List.mapi (fun i exp -> compile_exp defns tab stack_index exp (if i = List.length exps - 1 then is_tail else false)) exps |> List.concat | Lst [Sym "if"; test_exp; then_exp; else_exp] -> let else_label = Util.gensym "else" in let continue_label = Util.gensym "continue" in compile_exp defns tab stack_index test_exp false @ [Cmp (Reg Rax, operand_of_bool false); Jz else_label] @ compile_exp defns tab stack_index then_exp is_tail @ [Jmp continue_label] @ [Label else_label] @ compile_exp defns tab stack_index else_exp is_tail @ [Label continue_label] | Lst [Sym "+"; e1; e2] -> compile_exp defns tab stack_index e1 false @ [Mov (stack_address stack_index, Reg Rax)] @ compile_exp defns tab (stack_index - 8) e2 false @ [Mov (Reg R8, stack_address stack_index)] @ [Add (Reg Rax, Reg R8)]
In our recursive calls, we have essentially two cases:
-
We’re compiling a subexpression and then doing something else.
When we compile the subexpression
is_tail
should befalse
. This is the case, for instance, with both operands to+
and every expression in the body of ado
except the last one. -
We’re compiling a subexpression and then not doing anything
else. If we’re already in tail position, this subexpression is
also in tail position. Otherwise, it’s not. This is the case for
both the
then
andelse
cases of anif
, and the last expression in ado
.
When we call compile_exp
, either to compile the body of a
function or our main program body, is_tail
will start out
as true
.
let compile_defn defns defn = let ftab = defn.args |> List.mapi (fun i arg -> (arg, -8 * (i + 1))) |> Symtab.of_list in [Label (defn_label defn.name)] @ compile_exp defns ftab (-8 * (List.length defn.args + 1)) defn.body true @ [Ret] let compile (program : s_exp list) : string = let defns, body = defns_and_body program in [ Global "entry" ; Extern "error" ; Extern "read_num" ; Extern "print_value" ; Extern "print_newline" ; Label "entry" ] @ compile_exp defns Symtab.empty (-8) body true @ [Ret] @ (List.map (compile_defn defns) defns |> List.concat) |> List.map string_of_directive |> String.concat "\n"
Now we need to use is_tail
to reuse the current stack frame
if a function call is in tail position. We can add another case like
this:
let rec compile_exp (defns : defn list) (tab : int symtab) (stack_index : int) (exp : s_exp) (is_tail : bool) : directive list = match exp with (* ... *) | Lst (Sym f :: args) when is_defn defns f && is_tail -> let defn = get_defn defns f in if List.length args = List.length defn.args then let compiled_args = args |> List.mapi (fun i arg -> compile_exp defns tab (stack_index - (8 * i)) arg false @ [Mov (stack_address (stack_index - (8 * i)), Reg Rax)]) |> List.concat in let moved_args = args |> List.mapi (fun i _ -> [ Mov (Reg R8, stack_address (stack_index - (8 * i))) ; Mov (stack_address ((i + 1) * -8), Reg R8) ]) |> List.concat in compiled_args @ moved_args @ [Jmp (defn_label f)] else raise (BadExpression exp)
We’re first compiling each argument and storing it in the next
available stack index. We then move all of these arguments to the base
of the stack (right after
rsp
), since that’s where function arguments go! Then,
we can just jump to the right label. Notice that we’re
not changing rsp
when we call a function in tail
position: this is exactly what it means for us to reuse a stack frame.
Tail calls in the interpreter
As we saw, our interpreter already seems to be doing this–it didn’t overflow the stack when we interpreted a tail-recursive program. Why? Well, the interpreter is written in OCaml, which properly implements tail calls by reusing stack frames. So as long as the interpreter’s calls to itself are in tail position, everything will work out! See the lecture capture for a little more about this.