Module CSE

Common subexpression elimination over RTL. This optimization proceeds by value numbering over extended basic blocks.

Require Import Coqlib.
Require Import Maps.
Require Import Errors.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Op.
Require Import Registers.
Require Import RTL.
Require Import RTLtyping.
Require Import Kildall.
Require Import CombineOp.

Value numbering

The idea behind value numbering algorithms is to associate abstract identifiers (``value numbers'') to the contents of registers at various program points, and record equations between these identifiers. For instance, consider the instruction r1 = add(r2, r3) and assume that r2 and r3 are mapped to abstract identifiers x and y respectively at the program point just before this instruction. At the program point just after, we can add the equation z = add(x, y) and associate r1 with z, where z is a fresh abstract identifier. However, if we already knew an equation u = add(x, y), we can preferably add no equation and just associate r1 with u. If there exists a register r4 mapped with u at this point, we can then replace the instruction r1 = add(r2, r3) by a move instruction r1 = r4, therefore eliminating a common subexpression and reusing the result of an earlier addition. Abstract identifiers / value numbers are represented by positive integers. Equations are of the form valnum = rhs, where the right-hand sides rhs are either arithmetic operations or memory loads.

Definition eq_valnum: forall (x y: valnum), {x=y}+{x<>y} := peq.

Definition eq_list_valnum (x y: list valnum) : {x=y}+{x<>y}.
Proof.
decide equality. apply eq_valnum.
Qed.

Definition eq_rhs (x y: rhs) : {x=y}+{x<>y}.
Proof.
generalize Int.eq_dec; intro.
generalize Float.eq_dec; intro.
generalize eq_operation; intro.
assert (forall (x y: memory_chunk), {x=y}+{x<>y}). decide equality.
generalize eq_valnum; intro.
generalize eq_list_valnum; intro.
decide equality.
Qed.

A value numbering is a collection of equations between value numbers plus a partial map from registers to value numbers. Additionally, we maintain the next unused value number, so as to easily generate fresh value numbers.

Record numbering : Type := mknumbering {
num_next: valnum; (* first unused value number *)
num_eqs: list (valnum * rhs); (* valid equations *)
num_reg: PTree.t valnum; (* mapping register to valnum *)
num_val: PMap.t (list reg) (* reverse mapping valnum to regs containing it *)
}.

Definition empty_numbering :=
mknumbering 1%positive nil (PTree.empty valnum) (PMap.init nil).

valnum_reg n r returns the value number for the contents of register r. If none exists, a fresh value number is returned and associated with register r. The possibly updated numbering is also returned. valnum_regs is similar, but for a list of registers.

Definition valnum_reg (n: numbering) (r: reg) : numbering * valnum :=
match PTree.get r n.(num_reg) with
| Some v => (n, v)
| None =>
let v := n.(num_next) in
(mknumbering (Psucc v)
n.(num_eqs)
(PTree.set r v n.(num_reg))
(PMap.set v (r :: nil) n.(num_val)),
v)
end.

Fixpoint valnum_regs (n: numbering) (rl: list reg)
{struct rl} : numbering * list valnum :=
match rl with
| nil =>
(n, nil)
| r1 :: rs =>
let (n1, v1) := valnum_reg n r1 in
let (ns, vs) := valnum_regs n1 rs in
(ns, v1 :: vs)
end.

find_valnum_rhs rhs eqs searches the list of equations eqs for an equation of the form vn = rhs for some value number vn. If found, Some vn is returned, otherwise None is returned.

Fixpoint find_valnum_rhs (r: rhs) (eqs: list (valnum * rhs))
{struct eqs} : option valnum :=
match eqs with
| nil => None
| (v, r') :: eqs1 =>
if eq_rhs r r' then Some v else find_valnum_rhs r eqs1
end.

find_valnum_num vn eqs searches the list of equations eqs for an equation of the form vn = rhs for some equation rhs. If found, Some rhs is returned, otherwise None is returned.

Fixpoint find_valnum_num (v: valnum) (eqs: list (valnum * rhs))
{struct eqs} : option rhs :=
match eqs with
| nil => None
| (v', r') :: eqs1 =>
if peq v v' then Some r' else find_valnum_num v eqs1
end.

Update the num_val mapping prior to a redefinition of register r.

Definition forget_reg (n: numbering) (rd: reg) : PMap.t (list reg) :=
match PTree.get rd n.(num_reg) with
| None => n.(num_val)
| Some v => PMap.set v (List.remove peq rd (PMap.get v n.(num_val))) n.(num_val)
end.

Definition update_reg (n: numbering) (rd: reg) (vn: valnum) : PMap.t (list reg) :=
let nv := forget_reg n rd in PMap.set vn (rd :: PMap.get vn nv) nv.

add_rhs n rd rhs updates the value numbering n to reflect the computation of the operation or load represented by rhs and the storing of the result in register rd. If an equation vn = rhs is known, register rd is set to vn. Otherwise, a fresh value number vn is generated and associated with rd, and the equation vn = rhs is added.

Definition add_rhs (n: numbering) (rd: reg) (rh: rhs) : numbering :=
match find_valnum_rhs rh n.(num_eqs) with
| Some vres =>
mknumbering n.(num_next) n.(num_eqs)
(PTree.set rd vres n.(num_reg))
(update_reg n rd vres)
| None =>
mknumbering (Psucc n.(num_next))
((n.(num_next), rh) :: n.(num_eqs))
(PTree.set rd n.(num_next) n.(num_reg))
(update_reg n rd n.(num_next))
end.

add_op n rd op rs specializes add_rhs for the case of an arithmetic operation. The right-hand side corresponding to op and the value numbers for the argument registers rs is built and added to n as described in add_rhs. If op is a move instruction, we simply assign the value number of the source register to the destination register, since we know that the source and destination registers have exactly the same value. This enables more common subexpressions to be recognized. For instance:
`     z = add(x, y);  u = x; v = add(u, y);`
Since u and x have the same value number, the second add is recognized as computing the same result as the first add, and therefore u and z have the same value number.

Definition add_op (n: numbering) (rd: reg) (op: operation) (rs: list reg) :=
match is_move_operation op rs with
| Some r =>
let (n1, v) := valnum_reg n r in
mknumbering n1.(num_next) n1.(num_eqs)
(PTree.set rd v n1.(num_reg)) (update_reg n1 rd v)
| None =>
let (n1, vs) := valnum_regs n rs in
add_rhs n1 rd (Op op vs)
end.

(rs: list reg) :=
let (n1, vs) := valnum_regs n rs in

add_unknown n rd returns a numbering where rd is mapped to a fresh value number, and no equations are added. This is useful to model instructions with unpredictable results such as Ibuiltin.

Definition add_unknown (n: numbering) (rd: reg) :=
mknumbering (Psucc n.(num_next))
n.(num_eqs)
(PTree.set rd n.(num_next) n.(num_reg))
(forget_reg n rd).

kill_equations pred n remove all equations satisfying predicate pred.

Fixpoint kill_eqs (pred: rhs -> bool) (eqs: list (valnum * rhs)) : list (valnum * rhs) :=
match eqs with
| nil => nil
| eq :: rem => if pred (snd eq) then kill_eqs pred rem else eq :: kill_eqs pred rem
end.

Definition kill_equations (pred: rhs -> bool) (n: numbering) : numbering :=
mknumbering n.(num_next)
(kill_eqs pred n.(num_eqs))
n.(num_reg) n.(num_val).

kill_loads n removes all equations involving memory loads, as well as those involving memory-dependent operators. It is used to reflect the effect of a builtin operation, which can change memory in unpredictable ways and potentially invalidate all such equations.

Definition filter_loads (r: rhs) : bool :=
match r with
| Op op _ => op_depends_on_memory op
| Load _ _ _ => true
end.

Definition kill_loads (n: numbering) : numbering :=

add_store n chunk addr rargs rsrc updates the numbering n to reflect the effect of a store instruction Istore chunk addr rargs rsrc. Equations involving the memory state are removed from n, unless we can prove that the store does not invalidate them. Then, an equations rsrc = Load chunk addr rargs is added to reflect the known content of the stored memory area, but only if chunk is a "full-size" quantity (Mint32 or Mfloat64).

Definition filter_after_store (chunk: memory_chunk) (addr: addressing) (vl: list valnum) (r: rhs) : bool :=
match r with
| Op op vl' => op_depends_on_memory op
end.

(rargs: list reg) (rsrc: reg) : numbering :=
let (n1, vargs) := valnum_regs n rargs in
let n2 := kill_equations (filter_after_store chunk addr vargs) n1 in
match chunk with
| _ => n2
end.

reg_valnum n vn returns a register that is mapped to value number vn, or None if no such register exists.

Definition reg_valnum (n: numbering) (vn: valnum) : option reg :=
match PMap.get vn n.(num_val) with
| nil => None
| r :: rs => Some r
end.

Fixpoint regs_valnums (n: numbering) (vl: list valnum) : option (list reg) :=
match vl with
| nil => Some nil
| v1 :: vs =>
match reg_valnum n v1, regs_valnums n vs with
| Some r1, Some rs => Some (r1 :: rs)
| _, _ => None
end
end.

find_rhs return a register that already holds the result of the given arithmetic operation or memory load, according to the given numbering. None is returned if no such register exists.

Definition find_rhs (n: numbering) (rh: rhs) : option reg :=
match find_valnum_rhs rh n.(num_eqs) with
| None => None
| Some vres => reg_valnum n vres
end.

Experimental: take advantage of known equations to select more efficient forms of operations, addressing modes, and conditions.

Section REDUCE.

Variable A: Type.
Variable f: (valnum -> option rhs) -> A -> list valnum -> option (A * list valnum).
Variable n: numbering.

Fixpoint reduce_rec (niter: nat) (op: A) (args: list valnum) : option(A * list reg) :=
match niter with
| O => None
| S niter' =>
match f (fun v => find_valnum_num v n.(num_eqs)) op args with
| None => None
| Some(op', args') =>
match reduce_rec niter' op' args' with
| None =>
match regs_valnums n args' with Some rl => Some(op', rl) | None => None end
| Some _ as res =>
res
end
end
end.

Definition reduce (op: A) (rl: list reg) (vl: list valnum) : A * list reg :=
match reduce_rec 4%nat op vl with
| None => (op, rl)
| Some res => res
end.

End REDUCE.

The static analysis

We now define a notion of satisfiability of a numbering. This semantic notion plays a central role in the correctness proof (see CSEproof), but is defined here because we need it to define the ordering over numberings used in the static analysis. A numbering is satisfiable in a given register environment and memory state if there exists a valuation, mapping value numbers to actual values, that validates both the equations and the association of registers to value numbers.

Definition equation_holds
(valuation: valnum -> val)
(ge: genv) (sp: val) (m: mem)
(vres: valnum) (rh: rhs) : Prop :=
match rh with
| Op op vl =>
eval_operation ge sp op (List.map valuation vl) m =
Some (valuation vres)
exists a,
Mem.loadv chunk m a = Some (valuation vres)
end.

Definition numbering_holds
(valuation: valnum -> val)
(ge: genv) (sp: val) (rs: regset) (m: mem) (n: numbering) : Prop :=
(forall vn rh,
In (vn, rh) n.(num_eqs) ->
equation_holds valuation ge sp m vn rh)
/\ (forall r vn,
PTree.get r n.(num_reg) = Some vn -> rs#r = valuation vn).

Definition numbering_satisfiable
(ge: genv) (sp: val) (rs: regset) (m: mem) (n: numbering) : Prop :=
exists valuation, numbering_holds valuation ge sp rs m n.

Lemma empty_numbering_satisfiable:
forall ge sp rs m, numbering_satisfiable ge sp rs m empty_numbering.
Proof.
intros; red.
exists (fun (vn: valnum) => Vundef). split; simpl; intros.
elim H.
rewrite PTree.gempty in H. discriminate.
Qed.

We now equip the type numbering with a partial order and a greatest element. The partial order is based on entailment: n1 is greater than n2 if n1 is satisfiable whenever n2 is. The greatest element is, of course, the empty numbering (no equations).

Module Numbering.
Definition t := numbering.
Definition ge (n1 n2: numbering) : Prop :=
forall ge sp rs m,
numbering_satisfiable ge sp rs m n2 ->
numbering_satisfiable ge sp rs m n1.
Definition top := empty_numbering.
Lemma top_ge: forall x, ge top x.
Proof.
intros; red; intros. unfold top. apply empty_numbering_satisfiable.
Qed.
Lemma refl_ge: forall x, ge x x.
Proof.
intros; red; auto.
Qed.
End Numbering.

We reuse the solver for forward dataflow inequations based on propagation over extended basic blocks defined in library Kildall.

Module Solver := BBlock_solver(Numbering).

The transfer function for the dataflow analysis returns the numbering ``after'' execution of the instruction at pc, as a function of the numbering ``before''. For Iop and Iload instructions, we add equations or reuse existing value numbers as described for add_op and add_load. For Istore instructions, we forget all equations involving memory loads. For Icall instructions, we could simply associate a fresh, unconstrained by equations value number to the result register. However, it is often undesirable to eliminate common subexpressions across a function call (there is a risk of increasing too much the register pressure across the call), so we just forget all equations and start afresh with an empty numbering. Finally, the remaining instructions modify neither registers nor the memory, so we keep the numbering unchanged.

Definition transfer (f: function) (pc: node) (before: numbering) :=
match f.(fn_code)!pc with
| None => before
| Some i =>
match i with
| Inop s =>
before
| Iop op args res s =>
| Istore chunk addr args src s =>
| Icall sig ros args res s =>
empty_numbering
| Itailcall sig ros args =>
empty_numbering
| Ibuiltin ef args res s =>
| Icond cond args ifso ifnot =>
before
| Ijumptable arg tbl =>
before
| Ireturn optarg =>
before
end
end.

The static analysis solves the dataflow inequations implied by the transfer function using the ``extended basic block'' solver, which produces sub-optimal solutions quickly. The result is a mapping from program points to numberings.

Definition analyze (f: RTL.function): option (PMap.t numbering) :=
Solver.fixpoint (successors f) (transfer f) f.(fn_entrypoint).

Code transformation

The code transformation is performed instruction by instruction. Iload instructions and non-trivial Iop instructions are turned into move instructions if their result is already available in a register, as indicated by the numbering inferred at that program point. Some operations are so cheap to compute that it is generally not worth reusing their results. These operations are detected by the function is_trivial_op in module Op.

Definition transf_instr (n: numbering) (instr: instruction) :=
match instr with
| Iop op args res s =>
if is_trivial_op op then instr else
let (n1, vl) := valnum_regs n args in
match find_rhs n1 (Op op vl) with
| Some r =>
Iop Omove (r :: nil) res s
| None =>
let (op', args') := reduce _ combine_op n1 op args vl in
Iop op' args' res s
end
let (n1, vl) := valnum_regs n args in
| Some r =>
Iop Omove (r :: nil) dst s
| None =>
end
| Istore chunk addr args src s =>
let (n1, vl) := valnum_regs n args in
Istore chunk addr' args' src s
| Icond cond args s1 s2 =>
let (n1, vl) := valnum_regs n args in
let (cond', args') := reduce _ combine_cond n1 cond args vl in
Icond cond' args' s1 s2
| _ =>
instr
end.

Definition transf_code (approxs: PMap.t numbering) (instrs: code) : code :=
PTree.map (fun pc instr => transf_instr approxs!!pc instr) instrs.

Definition transf_function (f: function) : res function :=
match type_function f with
| Error msg => Error msg
| OK tyenv =>
match analyze f with
| None => Error (msg "CSE failure")
| Some approxs =>
OK(mkfunction
f.(fn_sig)
f.(fn_params)
f.(fn_stacksize)
(transf_code approxs f.(fn_code))
f.(fn_entrypoint))
end
end.

Definition transf_fundef (f: fundef) : res fundef :=
AST.transf_partial_fundef transf_function f.

Definition transf_program (p: program) : res program :=
transform_partial_program transf_fundef p.