Booleans II
Recall that we’re adding support for booleans to our interpreter and compiler. Booleans I covered adding them to our interpreter. Now we’ll turn to the compiler. Specifically, we’re going to add support for these expressions:
true
andfalse
, the two boolean values(not e)
, a unary operation which evaluates totrue
on the boolean valuefalse
andfalse
otherwise(num? e)
, a unary opertion which evaluates totrue
ife
is a number andfalse
otherwise(zero? e)
, a unary opertion which evaluates totrue
ife
is the number0
andfalse
otherwise
Types in the compiler
Now that we have an interpreter to test against, we can extend our compiler to support our new operations!
When our interpreter is executing a program, values of expressions are instances
of the value
datatype we just defined. We won’t be able to do that in the
compiler–we can’t define new datatypes in x86-64! Remember that when our
program is executing, its values live in registers (actually, just
rax
). Registers store 64-bit integers. Right now the values in our program are
all integers, so this works fine. But how will we add booleans? Take a second
and think about how you might implement this.
Well, we know that all of our values need to be represented, at runtime, as 64-bit integers. So instead of representing integers as themselves:
0 -> 0b00 1 -> 0b01 2 -> 0b10 3 -> 0b11 ...
We’re going to represent the integer x
as x << 2
(shifted left by two bits):
0 -> 0b0000 1 -> 0b0100 2 -> 0b1000 3 -> 0b1100
This is exactly equivalent to representing each integer x
as x * 4
.
This means our integers have to fit in 62 bits instead of 64. So our minimum
integer is now -(2**61)
and our maximum integer is (2**61) - 1
.
This also means there are a bunch of 64-bit integers (how many?) that are no
longer being used to represent values! All of our integer values now end with
00
. So anything that ends with a different pair of bits won’t be used to
represent a number. This means we can use some of them to represent booleans,
and other types!
First, though, let’s update our compiler to use this new representation for
integers. Integer constants will be easy–we’ll just shift them left. How will
we handle add1
and sub1
? Well, remember that our runtime representations are
the values multiplied by 4. Since multiplication distributes over addition (and
subtraction), we can just add (or subtract) 4 instead of 1! So:
let num_shift = 2 let num_mask = 0b11 let num_tag = 0b00 let rec compile_exp (exp : s_exp) : directive list = match exp with | Num n -> [Mov (Reg Rax, Imm (n lsl num_shift))] | Lst [Sym "add1"; arg] -> compile_exp arg @ [Add (Reg Rax, Imm (1 lsl num_shift))] | Lst [Sym "sub1"; arg] -> compile_exp arg @ [Sub (Reg Rax, Imm (1 lsl num_shift))] | e -> raise (BadExpression e)
(lsl
is “logical shift left.” We could also just multiply by 4, but it’s
clearer this way.)
What happens if we run a program now?
>>> compile_and_run "(add1 4)" 20
This makes sense–we’re printing out the runtime representation! We’ll need to fix that. We’ll edit our C runtime:
#include <stdio.h> #include <inttypes.h> #define num_shift 2 #define num_mask 0b11 #define num_tag 0b00 extern uint64_t entry(); void print_value(uint64_t value) { if ((value & num_mask) == num_tag) { int64_t ivalue = (int64_t)value; printf("%" PRIi64, ivalue >> num_shift); } else { printf("BAD VALUE %" PRIu64, value); } } int main(int argc, char **argv) { print_value(entry()); return 0; }
Boolean support in the runtime
While we’re editing the runtime, let’s also add support for booleans.
#include <stdio.h> #include <inttypes.h> #define num_shift 2 #define num_mask 0b11 #define num_tag 0b00 #define bool_shift 7 #define bool_mask 0b1111111 #define bool_tag 0b0011111 extern uint64_t entry(); void print_value(uint64_t value) { if ((value & num_mask) == num_tag) { int64_t ivalue = (int64_t)value; printf("%" PRIi64, ivalue >> num_shift); } else if ((value & bool_mask) == bool_tag) { if (value >> bool_shift) { printf("true"); } else { printf("false"); } } else { printf("BAD VALUE %" PRIu64, value); } } int main(int argc, char **argv) { print_value(entry()); return 0; }
We’ll need to recompile the runtime:
$ gcc -c runtime.c -o runtime.o
Boolean support in the compiler
We can now add support for true
and false
pretty easily:
let bool_shift = 7 let bool_mask = 0b1111111 let bool_tag = 0b0011111 let rec compile_exp (exp : s_exp) : directive list = match exp with (* some cases elided ... *) | Sym "true" -> [Mov (Reg Rax, Imm ((1 lsl bool_shift) lor bool_tag))] | Sym "false" -> [Mov (Reg Rax, Imm ((0 lsl bool_shift) lor bool_tag))]
Handling our other operations will be a little trickier. Let’s start with
not
. As a reminder, not
should evaluate to true
(i.e., should put the
runtime representation of true
into rax
!) when its argument is false
;
otherwise, it should evaluate to false
.
It seems like we need a way to compare the runtime representations of
values. For this, we’ll use the x86-64 instruction cmp
. cmp X,Y
compares X
to Y
. It then sets processor flags based on the result. There are a bunch of
flags, and we’ll talk about more of them later in the class; for now, we just
need to know that cmp
sets the flag ZF
to 1 if its arguments are the same
and 0
otherwise.
Flags aren’t like registers–we don’t access them directly in assembly
code1. These flags then modify the behavior of subsequent
instructions. We’ll see more examples of this next lecture when we talk about
conditionals. For now, we’re going to use another instruction, setz
, in order
to access ZF
. setz
takes a register2 and sets the last byte of that
register to 1 (i.e., 0b00000001
) if ZF
is set and 0 if ZF
is not set.
In pseudo-assembly, how we’re going to implement the not
operator:
not: cmp rax, 0b00011111 mov rax, 0 setz rax shl rax, 7 or rax, 0b0011111
So, now we can implement not
:
let bool_shift = 7 let bool_mask = 0b1111111 let bool_tag = 0b0011111 let rec compile_exp (exp : s_exp) : directive list = match exp with (* some cases elided ... *) | Sym "true" -> [Mov (Reg Rax, Imm ((1 lsl bool_shift) lor bool_tag))] | Sym "false" -> [Mov (Reg Rax, Imm ((0 lsl bool_shift) lor bool_tag))] | Lst [Sym "not"; arg] -> compile_exp arg @ [ Cmp (Reg Rax, Imm ((0 lsl bool_shift) lor bool_tag)) (* compare rax to false *) ; Mov (Reg Rax, Imm 0) (* zero out rax *) ; Setz (Reg Rax) (* 1 if ZF is set (meaning rax contained false), 0 otherwise *) ; Shl (Reg Rax, Imm bool_shift) (* rax << bool_shift *) ; Or (Reg Rax, Imm bool_tag) (* tag rax as a boolean: rax = rax | bool_tag *) ]
There’s some duplicate logic here. We’re going to make a helper function called
operand_of_bool
, which makes an instruction operand from a boolean using shift
and or:
let operand_of_bool (b : bool) : operand = Imm (((if b then 1 else 0) lsl bool_shift) lor bool_tag)
We can do the same thing for numbers:
let operand_of_num (x : int) : operand = Imm ((x lsl num_shift) lor num_tag)
(We include lor num_tag
here to be symmetric with operand_to_bool
, but
everything would work if we left it off–why?)
Lastly, we’re going to re-use the code to convert ZF
to a boolean:
let zf_to_bool: directive list = [Mov (Reg Rax, Imm 0) (* zero out rax *) ; Setz (Reg Rax) (* 1 if ZF is set, 0 otherwise *) ; Shl (Reg Rax, Imm bool_shift) (* rax << bool_shift *) ; Or (Reg Rax, Imm bool_tag) (* tag rax as a boolean: rax = rax | bool_tag *) ]
zf_to_bool
is a list, not a function. How is that possible? Won’t it depend on
the value we’re trying to convert? It does not! This is a list of instructions
that set rax
to the runtime representation of true
if ZF
is set and to the
runtime representation of false
otherwise.
Now we can implement zero?
easily:
let rec compile_exp (exp : s_exp) : directive list = match exp with (* some cases elided ... *) | Sym "true" -> [Mov (Reg Rax, operand_of_bool true)] | Sym "false" -> [Mov (Reg Rax, operand_of_bool false)] | Lst [Sym "not"; arg] -> compile_exp arg @ [ Cmp (Reg Rax, operand_of_bool false) ] @ zf_to_bool | Lst [Sym "zero?"; arg] -> compile_exp arg @ [ Cmp (Reg Rax, operand_of_num 0) ] @ zf_to_bool
Lastly, we can implement num?
. We can detect if a value is a number by looking
at the last two bits and seeing if they are both zero. We can do that like this:
let rec compile_exp (exp : s_exp) : directive list = match exp with (* some cases elided ... *) | Lst [Sym "num?"; arg] -> compile_exp arg @ [ And (Reg Rax, Imm num_mask); Cmp (Reg Rax, Imm num_tag) ] @ zf_to_bool
Footnotes:
Actually, all of the flags are packed together in the same special RFLAGS register
It actually just takes the lower
byte of a register, which are notated differently in assembly–for instance, the
lower byte of rax
is written al
. Our assembly library takes care of this, so
we won’t talk about it too much in class.