# Module Constprop

Constant propagation over RTL. This is one of the optimizations performed at RTL level. It proceeds by a standard dataflow analysis and the corresponding code rewriting.

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Globalenvs.
Require Import Op.
Require Import Registers.
Require Import RTL.
Require Import Lattice.
Require Import Kildall.
Require Import ConstpropOp.

# Static analysis

The type approx of compile-time approximations of values is defined in the machine-dependent part ConstpropOp.

We equip this type of approximations with a semi-lattice structure. The ordering is inclusion between the sets of values denoted by the approximations.

Module Approx <: SEMILATTICE_WITH_TOP.
Definition t := approx.
Definition eq (x y: t) := (x = y).
Definition eq_refl: forall x, eq x x := (@refl_equal t).
Definition eq_sym: forall x y, eq x y -> eq y x := (@sym_equal t).
Definition eq_trans: forall x y z, eq x y -> eq y z -> eq x z := (@trans_equal t).
Lemma eq_dec: forall (x y: t), {x=y} + {x<>y}.
Proof.
decide equality.
apply Int.eq_dec.
apply Float.eq_dec.
apply Int.eq_dec.
apply ident_eq.
apply Int.eq_dec.
Qed.
Definition beq (x y: t) := if eq_dec x y then true else false.
Lemma beq_correct: forall x y, beq x y = true -> x = y.
Proof.
unfold beq; intros. destruct (eq_dec x y). auto. congruence.
Qed.

Definition ge (x y: t) : Prop := x = Unknown \/ y = Novalue \/ x = y.

Lemma ge_refl: forall x y, eq x y -> ge x y.
Proof.
unfold eq, ge; tauto.
Qed.
Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Proof.
unfold ge; intuition congruence.
Qed.
Lemma ge_compat: forall x x' y y', eq x x' -> eq y y' -> ge x y -> ge x' y'.
Proof.
unfold eq, ge; intros; congruence.
Qed.
Definition bot := Novalue.
Definition top := Unknown.
Lemma ge_bot: forall x, ge x bot.
Proof.
unfold ge, bot; tauto.
Qed.
Lemma ge_top: forall x, ge top x.
Proof.
unfold ge, bot; tauto.
Qed.
Definition lub (x y: t) : t :=
if eq_dec x y then x else
match x, y with
| Novalue, _ => y
| _, Novalue => x
| _, _ => Unknown
end.
Lemma ge_lub_left: forall x y, ge (lub x y) x.
Proof.
unfold lub; intros.
case (eq_dec x y); intro.
apply ge_refl. apply eq_refl.
destruct x; destruct y; unfold ge; tauto.
Qed.
Lemma ge_lub_right: forall x y, ge (lub x y) y.
Proof.
unfold lub; intros.
case (eq_dec x y); intro.
apply ge_refl. subst. apply eq_refl.
destruct x; destruct y; unfold ge; tauto.
Qed.
End Approx.

Module D := LPMap Approx.

We keep track of read-only global variables (i.e. "const" global variables in C) as a map from their names to their initialization data.

Definition global_approx : Type := PTree.t (list init_data).

Given some initialization data and a byte offset, compute a static approximation of the result of a memory load from a memory block initialized with this data.

Fixpoint eval_load_init (chunk: memory_chunk) (pos: Z) (il: list init_data): approx :=
match il with
| nil => Unknown
| Init_int8 n :: il' =>
if zeq pos 0 then
match chunk with
| Mint8unsigned => I (Int.zero_ext 8 n)
| Mint8signed => I (Int.sign_ext 8 n)
| _ => Unknown
end
else eval_load_init chunk (pos - 1) il'
| Init_int16 n :: il' =>
if zeq pos 0 then
match chunk with
| Mint16unsigned => I (Int.zero_ext 16 n)
| Mint16signed => I (Int.sign_ext 16 n)
| _ => Unknown
end
else eval_load_init chunk (pos - 2) il'
| Init_int32 n :: il' =>
if zeq pos 0
then match chunk with Mint32 => I n | _ => Unknown end
else eval_load_init chunk (pos - 4) il'
| Init_float32 n :: il' =>
if zeq pos 0
then match chunk with
| Mfloat32 => if propagate_float_constants tt then F (Float.singleoffloat n) else Unknown
| _ => Unknown
end
else eval_load_init chunk (pos - 4) il'
| Init_float64 n :: il' =>
if zeq pos 0
then match chunk with
| Mfloat64 => if propagate_float_constants tt then F n else Unknown
| _ => Unknown
end
else eval_load_init chunk (pos - 8) il'
| Init_addrof symb ofs :: il' =>
if zeq pos 0
then match chunk with Mint32 => G symb ofs | _ => Unknown end
else eval_load_init chunk (pos - 4) il'
| Init_space n :: il' =>
eval_load_init chunk (pos - Zmax n 0) il'
end.

Compute a static approximation for the result of a load at an address whose approximation is known. If the approximation points to a global variable, and this global variable is read-only, we use its initialization data to determine a static approximation. Otherwise, Unknown is returned.

| G symb ofs =>
match gapp!symb with
| None => Unknown
| Some il => eval_load_init chunk (Int.unsigned ofs) il
end
| _ => Unknown
end.

The transfer function for the dataflow analysis is straightforward. For Iop instructions, we set the approximation of the destination register to the result of executing abstractly the operation. For Iload instructions, we set the approximation of the destination register to the result of eval_static_load. For Icall and Ibuiltin, the destination register becomes Unknown. Other instructions keep the approximations unchanged, as they preserve the values of all registers.

Definition approx_reg (app: D.t) (r: reg) :=
D.get r app.

Definition approx_regs (app: D.t) (rl: list reg):=
List.map (approx_reg app) rl.

Definition transfer (gapp: global_approx) (f: function) (pc: node) (before: D.t) :=
match f.(fn_code)!pc with
| None => before
| Some i =>
match i with
| Iop op args res s =>
let a := eval_static_operation op (approx_regs before args) in
D.set res a before
let a := eval_static_load gapp chunk
D.set dst a before
| Icall sig ros args res s =>
D.set res Unknown before
| Ibuiltin ef args res s =>
D.set res Unknown before
| _ =>
before
end
end.

The static analysis itself is then an instantiation of Kildall's generic solver for forward dataflow inequations. analyze f returns a mapping from program points to mappings of pseudo-registers to approximations. It can fail to reach a fixpoint in a reasonable number of iterations, in which case we use the trivial mapping (program point -> D.top) instead.

Module DS := Dataflow_Solver(D)(NodeSetForward).

Definition analyze (gapp: global_approx) (f: RTL.function): PMap.t D.t :=
match DS.fixpoint (successors f) (transfer gapp f)
((f.(fn_entrypoint), D.top) :: nil) with
| None => PMap.init D.top
| Some res => res
end.

# Code transformation

The code transformation proceeds instruction by instruction. Operators whose arguments are all statically known are turned into ``load integer constant'', ``load float constant'' or ``load symbol address'' operations. Likewise for loads whose result can be statically predicted. Operators for which some but not all arguments are known are subject to strength reduction, and similarly for the addressing modes of load and store instructions. Conditional branches and multi-way branches are statically resolved into Inop instructions if possible. Other instructions are unchanged.

Definition transf_ros (app: D.t) (ros: reg + ident) : reg + ident :=
match ros with
| inl r =>
match D.get r app with
| G symb ofs => if Int.eq ofs Int.zero then inr _ symb else ros
| _ => ros
end
| inr s => ros
end.

Parameter generate_float_constants : unit -> bool.

Definition const_for_result (a: approx) : option operation :=
match a with
| I n => Some(Ointconst n)
| F n => if generate_float_constants tt then Some(Ofloatconst n) else None
| G symb ofs => Some(Oaddrsymbol symb ofs)
| S ofs => Some(Oaddrstack ofs)
| _ => None
end.

Definition transf_instr (gapp: global_approx) (app: D.t) (instr: instruction) :=
match instr with
| Iop op args res s =>
let a := eval_static_operation op (approx_regs app args) in
match const_for_result a with
| Some cop =>
Iop cop nil res s
| None =>
let (op', args') := op_strength_reduction op args (approx_regs app args) in
Iop op' args' res s
end
let a := eval_static_load gapp chunk
match const_for_result a with
| Some cop =>
Iop cop nil dst s
| None =>
end
| Istore chunk addr args src s =>
Istore chunk addr' args' src s
| Icall sig ros args res s =>
Icall sig (transf_ros app ros) args res s
| Itailcall sig ros args =>
Itailcall sig (transf_ros app ros) args
| Ibuiltin ef args res s =>
let (ef', args') := builtin_strength_reduction ef args (approx_regs app args) in
Ibuiltin ef' args' res s
| Icond cond args s1 s2 =>
match eval_static_condition cond (approx_regs app args) with
| Some b =>
if b then Inop s1 else Inop s2
| None =>
let (cond', args') := cond_strength_reduction cond args (approx_regs app args) in
Icond cond' args' s1 s2
end
| Ijumptable arg tbl =>
match approx_reg app arg with
| I n =>
match list_nth_z tbl (Int.unsigned n) with
| Some s => Inop s
| None => instr
end
| _ => instr
end
| _ =>
instr
end.

Definition transf_code (gapp: global_approx) (app: PMap.t D.t) (instrs: code) : code :=
PTree.map (fun pc instr => transf_instr gapp app!!pc instr) instrs.

Definition transf_function (gapp: global_approx) (f: function) : function :=
let approxs := analyze gapp f in
mkfunction
f.(fn_sig)
f.(fn_params)
f.(fn_stacksize)
(transf_code gapp approxs f.(fn_code))
f.(fn_entrypoint).

Definition transf_fundef (gapp: global_approx) (fd: fundef) : fundef :=
AST.transf_fundef (transf_function gapp) fd.

Fixpoint make_global_approx (gapp: global_approx) (vars: list (ident * globvar unit)) : global_approx :=
match vars with
| nil => gapp
| (id, gv) :: vars' =>
let gapp1 :=