Next, lets add support for

• Multiple datatypes (number and boolean)
• Calling external functions

In the process of doing so, we will learn about

• Tagged Representations
• Calling Conventions

Plan

Our plan will be to (start with boa) and add the following features:

1. Representing boolean values (and numbers)

2. Arithmetic Operations

3. Arithmetic Comparisons

4. Dynamic Checking (to ensure operators are well behaved)

1. Representation

Motivation: Why booleans?

In the year 2021, its a bit silly to use

• 0 for false and
• non-zero for true.

But really, boolean is a stepping stone to other data

• Pointers
• Tuples
• Structures
• Closures

The Key Issue

How to distinguish numbers from booleans?

• Need extra information to mark values as number or bool.

A Word

(A reminder for me, since I always get mixed up)

• A Bit is 1-bit
  
• A Byte is 8-bits
• A Word is 2-bytes = 16 bits
• A Double Word is 2-words = 4-bytes = 32 bits
• A Quad Word is 4-words = 8-bytes = 64 bits

We are working in x86_64 where the default size is a qword

• Registers are 64-bits
• Arithmetic is 64-bits
• Stack slots should be 64-bits
• etc.

Option 1: Use Two (Quad-)Words

How to distinguish numbers from booleans?

Need extra information to mark values as number or bool.

First word is 0 means bool, is 1 means number, 2 means pointer etc.

Pros

• Can have lots of different types, but

Cons

• Takes up double memory,

• Operators +, - require two memory reads.

In short, rather wasteful! We don’t need so many types.

Option 2: Use a Tag Bit

Can distinguish two types with a single bit.

Least Significant Bit (LSB) is

• 0 for number
• 1 for boolean

Question: why not 0 for boolean and 1 for number?

Tag Bit: Numbers

So number is the binary representation shifted left by 1 bit

• Lowest bit is always 0
• Remaining bits are number’s binary representation

For example, in binary:

Or in hexadecimal:

Tag Bit: Booleans

Most Significant Bit (MSB) is

• 1 for true
• 0 for false

For example

Or, in HEX

(eliding the 32/8 zeros in the “most-significant” DWORD)

Types

Lets extend our source types with boolean constants

data Expr a
  = ...
  | Boolean Bool a

Correspondingly, we extend our assembly Arg (values) with

data Arg
  = ...
  | HexConst  Int

So, our examples become:

Transforms

Next, lets update our implementation

The parse, anf and tag stages are straightforward.

Lets focus on the compile function.

A TypeClass for Representing Constants

Its convenient to introduce a type class describing Haskell types that can be represented as x86 arguments:

class Repr a where
  repr :: a -> Arg

We can now define instances for Int and Bool as:

instance Repr Int where
  repr n = Const (Data.Bits.shift n 1) -- left-shift `n` by 1

instance Repr Bool where
  repr False = HexConst 0x00000001
  repr True  = HexConst 0x80000001

Immediate Values to Arguments

Boolean b is an immediate value (like Number n).

Lets extend immArg that transforms an immediate expression to an x86 argument.

immArg :: Env -> ImmTag -> Arg
immArg (Var    x _)  = ...
immArg (Number n _)  = repr n
immArg (Boolean b _) = repr b

Compiling Constants

Finally, we can easily update the compile function as:

compileEnv :: Env -> AnfTagE -> Asm
compileEnv _ e@(Number _ _)  = [IMov (Reg RAX) (immArg env e)]
compileEnv _ e@(Boolean _ _) = [IMov (Reg RAX) (immArg env e)]

(The other cases remain unchanged.)

Lets run some tests to double check.

QUIZ

What is the result of:

ghci> exec "15"

A. Error

B. 0

C. 15

D. 30

Output Representation

Say what?! Need to update our run-time printer in main.c

void print(int val){
  if (val == CONST_TRUE)
    printf("true");
  else if (val == CONST_FALSE)
    printf("false");
  else // should be a number!
    printf("%d", d >> 1);  // shift right to remove tag bit.
}

and now we get:

ghci> exec "15"
15

Can you think of some other tests we should write?

QUIZ

What is the result of

ghci> exec "let x = 15 in x"

A. Error

B. 0

C. 15

D. 30

QUIZ

What is the result of

>>> exec "if 3: 12 else: 49"

>>> exec "if 0: 12 else: 49"

>>> exec "if true: 12 else: 49"

>>> exec "if false: 12 else: 49"

A. Error

B. 0

C. 12

D. 49

Lets go and fix the code so the above do the right thing!

2. Arithmetic Operations

Constants like 2, 29, false are only useful if we can perform computations with them.

First lets see what happens with our arithmetic operators.

QUIZ: Addition

What will be the result of:

ghci> exec "12 + 4"

A. Does not compile

B. Run-time error (e.g. segmentation fault)

C. 16

D. 32

E. 0

Shifted Representation and Addition

We are representing a number n by shifting it left by 1

n has the machine representation 2*n

Thus, our source values have the following _representations:

That is, addition (and similarly, subtraction) works as is with the shifted representation.

QUIZ: Multiplication

What will be the result (using our code so far) of:

ghci> exec "12 * 4"

A. Does not compile

B. Run-time error (e.g. segmentation fault)

C. 24

D. 48

E. 96

Shifted Representation and Multiplication

We are representing a number n by shifting it left by 1

n has the machine representation 2*n

Thus, our source values have the following _representations:

Thus, multiplication ends up accumulating the factor of 2.

• Result is two times the desired one.

Strategy

Thus, our strategy for compiling arithmetic operations is:

Addition and Subtraction “just work” - as shifting “cancels out”,

Multiplication result must be “adjusted” by dividing-by-two

• i.e. right shifting by 1

Types

The source language does not change at all, for the Asm lets add a “right shift” instruction (shr):

data Instruction
  = ...
  | IShr    Arg   Arg

Transforms

We need only modify compileEnv to account for the “fixing up”

compileEnv :: Env -> AnfTagE -> [Instruction]
compileEnv env (Prim2 o v1 v2 _) = compilePrim2 env o v1 v2

where the helper compilePrim2 works for Prim2 (binary) operators and immediate arguments:

compilePrim2 :: Env -> Prim2 -> ImmE -> ImmE -> [Instruction]
compilePrim2 env Plus v1 v2   = [ IMov (Reg RAX) (immArg env v1)
                                , IAdd (Reg RAX) (immArg env v2)
                                ]
compilePrim2 env Minus v1 v2  = [ IMov (Reg RAX) (immArg env v1)
                                , ISub (Reg RAX) (immArg env v2)
                                ]
compilePrim2 env Times v1 v2  = [ IMov (Reg RAX) (immArg env v1)
                                , IMul (Reg RAX) (immArg env v2)
                                , IShr (Reg RAX) (Const 1)
                                ]

Tests

Lets take it out for a drive.

ghci> exec' "2 * (0 - 1)"
4611686018427387902

Whoa?!

Well, its easy to figure out if you look at the generated assembly:

mov rax, 4
imul rax, -2
shr rax, 1
ret

Two’s Complement

The negative result is in twos-complement format.

When we shift that right-by-one, we get the odd value

• does not “divide by two”

Solution: Signed/Arithmetic Shift

The instruction sar shift arithmetic right does what we want, namely:

• preserves the sign-bit when shifting
• i.e. doesn’t introduce a 0 by default

Transforms Revisited

Lets add sar to our target:

data Instruction
  = ...
  | ISar Arg Arg

and use it to fix the post-multiplication adjustment

• i.e. use ISar instead of IShr

compilePrim2 env Times v1 v2  = [ IMov (Reg RAX) (immArg env v1)
                                , IMul (Reg RAX) (immArg env v2)
                                , ISar (Reg RAX) (Const 1)
                                ]

After which all is well:

ghci> exec' "2 * (-1)"
-2

3. Arithmetic Comparisons

Next, lets try to implement comparisons:

ghci> exec "1 < 2"
...
boa: lib/Language/Boa/Compiler.hs:(104,1)-(106,43): Non-exhaustive patterns in function compilePrim2

Oops. Need to implement it first!

How to implement comparisons?

Many ways to do this:

1. branches jne, jl, jg or

2. bit-twiddling.

Option 1: Comparisons via Branches

Key Idea:

Use the machine comparisons and branch

To implement arg1 < arg2

IF 
  arg1 < arg2

THEN 
  rax := <true>

ELSE 
  rax := <false>

mov rax, <arg1> 
cmp rax, <arg2>       # flags are set with comparison
jg false_label        # if cmp-greater then false else true
  mov rax, <true>     # assign to RAX := true
jmp exit_label
false_label: 
  mov rax, <false>    # assign to RAX := false
exit_label:

Option 2: Comparisons via Bit-Twiddling

Key idea:

A negative number’s most significant bit is 1

To implement arg1 < arg2, compute arg1 - arg2

• When result is negative, MSB is 1, ensure rax set to 0x80000001
• When result is non-negative, MSB is 0, ensure rax set to 0x00000001

1. Can extract msb by bitwise and with 0x8000000000000000.
2. Can shift msb to 32-position with shr
3. Can set tag bit by bitwise or with 0x00000001

So compilation strategy is:

mov rax, arg1
sub rax, arg2
and rax, 0x8000000000000000   ; mask out "sign" bit (msb)
shr rax, 32                   ; shift "sign" bit (msb) by 32
or  rax, 0x00000001           ; set tag bit to bool

Comparisons: Implementation

Lets go and extend:

1. The Instruction type

data Instruction
  = ...
  | IAnd    Arg   Arg
  | IOr     Arg   Arg

2. The instrAsm converter

instrAsm :: Instruction -> Text
instrAsm (IAnd a1 a2) = ...
instrAsm (IOr  a1 a2) = ...

3. The actual compilePrim2 function

Do in class

Exercise: Comparisons via Bit-Twiddling

• Can compute arg1 > arg2 by computing arg2 < arg1.
• Can compute arg1 != arg2 by computing arg1 < arg2 || arg2 < arg1
• Can compute arg1 = arg2 by computing ! (arg1 != arg2)

For the above, can you figure out how to implement:

1. Boolean ! ?
2. Boolean || ?
3. Boolean && ?

You may find these instructions useful

4. Dynamic Checking

We’ve added support for Number and Boolean but we have no way to ensure that we don’t write gibberish programs like:

2 + true

or

7 < false

In fact, lets try to see what happens with our code on the above:

ghci> exec "2 + true"

Oops.

Static vs. Dynamic Type Checking

Later we will add a static type system

• that rejects meaningless programs at compile time.

Now lets add a dynamic system

• that aborts execution with wrong operands at run time.

Checking Tags at Run-Time

Here are the allowed types of operands for each primitive operation.

Strategy: Asserting a Type

To check if arg is a number

• Suffices to check that the LSB is 0

• If not, jump to special error_non_int label

For example

mov rax, arg
mov rbx, rax              ; copy into rbx register
and rbx, 0x00000001       ; extract lsb
cmp rbx, 0                ; check if lsb equals 0
jne error_non_number      
...

at error_non_number we can call into a C function:

error_non_number:
  mov rdi, 0              ; pass error code
  mov rsi, rax            ; pass erroneous value
  call error              ; call run-time "error" function

Finally, the error function is part of the run-time and looks like:

void error(long code, long v){
   if (code == 0) {
     fprintf(stderr, "Error: expected a number but got %#010x\n", v);
   }
   else if (code == 1) {
     // print out message for errorcode 1 ...
   }
   else if (code == 2) {
     // print out message for errorcode 2 ...
   } ...
   exit(1);
 }

Strategy By Example

Lets implement the above in a simple file tests/output/int-check.s

section .text
extern error
extern print
global our_code_starts_here
our_code_starts_here:
  mov rax, 1                ; not a valid number
  mov rbx, rax              ; copy into rbx register
  and rbx, 0x00000001       ; extract lsb
  cmp rbx, 0                ; check if lsb equals 0
  jne error_non_number      
error_non_number:
  mov rdi, 0
  mov rsi, rax
  call error

Alas

make tests/output/int-check.result
... segmentation fault ...

What happened ?

Managing the Call Stack

To properly call into C functions (like error), we must play by the rules of the C calling convention

1. The local variables of an (executing) function are saved in its stack frame.
2. The start of the stack frame is saved in register rbp,
3. The start of the next frame is saved in register rsp.

Calling Convention

We must preserve the above invariant as follows:

In the Callee

At the start of the function

push rbp          ; SAVE (previous) caller's base-pointer on stack
mov rbp, rsp      ; set our base-pointer using the current stack-pointer
sub rsp, 8*N      ; ALLOCATE space for N local variables

At the end of the function

add rsp, 8*N0     ; FREE space for N local variables
pop rbp           ; RESTORE caller's base-pointer from stack
ret               ; return to caller

Fixed Strategy By Example

Lets implement the above in a simple file tests/output/int-check.s

section .text
extern error
extern print
global our_code_starts_here
our_code_starts_here:
  push rbp                  ; save caller's base-pointer
  mov rbp, rsp              ; set our base-pointer
  sub rsp, 1600             ; alloc '100' vars

  mov rax, 1                ; not a valid number
  mov rbx, rax              ; copy into rbx register
  and rbx, 0x00000001       ; extract lsb
  cmp rbx, 0                ; check if lsb equals 0
  jne error_non_number      

  add rsp, 1600             ; de-alloc '100' vars
  pop rbp                   ; restore caller's base-pointer
  ret
error_non_number:
  mov rdi, 0
  mov rsi, rax
  call error

Aha, now the above works!

make tests/output/int-check.result
... expected number but got ...

Q: What NEW thing does our compiler need to compute?

Hint: Why do we sub esp, 1600 above?

Types

Lets implement the above strategy.

To do so, we need a new data type for run-time types:

data Ty = TNumber | TBoolean

a new Label for the error

data Label
  = ...
  | TypeError Ty        -- Type Error Labels
  | Builtin   Text      -- Functions implemented in C

and thats it.

Transforms

The compiler must generate code to:

1. Perform dynamic type checks,
2. Exit by calling error if a failure occurs,
3. Manage the stack per the convention above.

1. Type Assertions

The key step in the implementation is to write a function

assertType :: Env -> IExp -> Ty -> [Instruction]
assertType env v ty
  = [ IMov (Reg RAX) (immArg env v)
    , IMov (Reg RBX) (Reg RAX)
    , IAnd (Reg RBX) (HexConst 0x00000001)
    , ICmp (Reg RBX) (typeTag  ty)
    , IJne (TypeError ty)
    ]

where typeTag is:

typeTag :: Ty -> Arg
typeTag TNumber  = HexConst 0x00000000
typeTag TBoolean = HexConst 0x00000001  

You can now splice assertType prior to doing the actual computations, e.g.

compilePrim2 :: Env -> Prim2 -> ImmE -> ImmE -> [Instruction]
compilePrim2 env Plus v1 v2   = assertType env v1 TNumber
                             ++ assertType env v2 TNumber  
                             ++ [ IMov (Reg RAX) (immArg env v1)
                                , IAdd (Reg RAX) (immArg env v2)
                                ]

2. Errors

We must also add code at the TypeError TNumber and TypeError TBoolean labels.

errorHandler :: Ty -> Asm
errorHandler t =
  [ ILabel   (TypeError t)        -- the expected-number error
  ,   IMov   (Reg RDI) (ecode t)  -- set the first  "code" param,
  ,   IMov   (Reg RSI) (Reg RAX)  -- set the second "value" param first,
  ,   ICall  (Builtin "error")    -- call the run-time's "error" function.  
  ]

ecode :: Ty -> Arg   
ecode TNumber  = Const 0
ecode TBoolean = Const 1

3. Stack Management

Maintaining rsp and rbp

We need to make sure that all our code respects the C calling convention..

To do so, just wrap the generated code, with instructions to save and restore rbp and rsp

compileBody :: AnfTagE -> Asm
compileBody e = entryCode e
             ++ compileEnv emptyEnv e
             ++ exitCode e

entryCode :: AnfTagE -> Asm
entryCode e = [ IPush (Reg RBP)                       -- SAVE caller's RBP
              , IMov  (Reg RBP) (Reg RSP)             -- SET our RBP
              , ISub  (Reg RSP) (Const (argBytes n))  -- ALLOC n local-vars
              ]
  where
    n       = countVars e

exitCode :: AnfTagE -> Asm
exitCode e = [ IAdd (Reg RSP) (Const (argBytes n))      -- FREE n local-vars
             , IPop (Reg RBP)                           -- RESTORE caller's RBP
             , IRet                                     -- RETURN to caller
             ]
  where
    n       = countVars e

the rsp needs to be a multiple of 16 so:

argBytes :: Int -> Int
argBytes n = 8 * n' 
  where 
    n' = if even n then n else n + 1

Q: But how shall we compute countVars?

Here’s a shady kludge:

countVars :: AnfTagE -> Int
countVars = 100

Obviously a sleazy hack (why?), but lets use it to test everything else; then we can fix it.

4. Computing the Size of the Stack

Ok, now that everything (else) seems to work, lets work out:

countVars :: AnfTagE -> Int

Finding the exact answer is undecidable in general (CSE 105), i.e. is impossible to compute.

However, it is easy to find an overapproximate heuristic, i.e.

• a value guaranteed to be larger than the than the max size,

• and which is reasonable in practice.

As usual, lets see if we can work out a heuristic by example.

QUIZ

How many stack slots/vars are needed for the following program?

1 + 2

A. 0

B. 1

C. 2

QUIZ

How many stack slots/vars are needed for the following program?

let x = 1
  , y = 2
  , z = 3
in
  x + y + z

A. 0

B. 1

C. 2

D. 3

E. 4

QUIZ

How many stack slots/vars are needed for the following program?

if true:
  let x = 1
    , y = 2
    , z = 3
  in
    x + y + z
else:
  0

A. 0

B. 1

C. 2

D. 3

E. 4

QUIZ

How many stack slots/vars are needed for the following program?

let x =
  let y =
    let z = 3  
    in z + 1
  in y + 1
in x + 1

A. 0

B. 1

C. 2

D. 3

E. 4

Strategy

Let countVars e be:

• The maximum number of let-binds in scope at any point inside e, i.e.

• The maximum size of the Env when compiling e

Lets work it out on a case-by-case basis:

• Immediate values like Number or Var
  • are compiled without pushing anything onto the Env
  • i.e. countVars = 0
• Binary Operations like Prim2 o v1 v2 take immediate values,
  • are compiled without pushing anything onto the Env
  • i.e. countVars = 0
• Branches like If v e1 e2 can go either way
  • can’t tell at compile-time
  • i.e. worst-case is larger of countVars e1 and countVars e2
• Let-bindings like Let x e1 e2 require
  • evaluating e1 and
  • pushing the result onto the stack and then evaluating e2
  • i.e. larger of countVars e1 and 1 + countVars e2

Implementation

We can implement the above a simple recursive function:

countVars :: AnfTagE -> Int  
countVars (If v e1 e2)  = max (countVars e1) (countVars e2)
countVars (Let x e1 e2) = max (countVars e1) (1 + countVars e2)
countVars _             = 0

Naive Heuristic is Naive

The above method is quite simplistic. For example, consider the expression:

let x = 1
  , y = 2
  , z = 3
in
    0

countVars would tell us that we need to allocate 3 stack spaces but clearly none of the variables are actually used.

Will revisit this problem later, when looking at optimizations.

Recap

We just saw how to add support for

• Multiple datatypes (number and boolean)
• Calling external functions

and in doing so, learned about

• Tagged Representations
• Calling Conventions

To get some practice, in your assignment, you will add:

1. Dynamic Checks for Arithmetic Overflows (see the jo and jno operations)
2. A Primitive print operation implemented by a function in the c run-time.

And next, we’ll see how to add user-defined functions.