path: root/backend/CSE.v

(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** 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 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 valnum := positive.

Inductive rhs : Type :=
  | Op: operation -> list valnum -> rhs
  | Load: memory_chunk -> addressing -> list valnum -> rhs.
*)

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. Defined.

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.
  generalize eq_addressing; intro.
  assert (forall (x y: memory_chunk), {x=y}+{x<>y}). decide equality.
  generalize eq_valnum; intro.
  generalize eq_list_valnum; intro.
  decide equality.
Defined.

(** 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;                  (**r first unused value number *)
  num_eqs: list (valnum * rhs);       (**r valid equations *)
  num_reg: PTree.t valnum;           (**r mapping register to valnum *)
  num_val: PMap.t (list reg)         (**r 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.

(** [add_load n rd chunk addr rs] specializes [add_rhs] for the case of a
  memory load.  The right-hand side corresponding to [chunk], [addr]
  and the value numbers for the argument registers [rs] is built
  and added to [n] as described in [add_rhs]. *)

Definition add_load (n: numbering) (rd: reg) 
                    (chunk: memory_chunk) (addr: addressing)
                    (rs: list reg) :=
  let (n1, vs) := valnum_regs n rs in
  add_rhs n1 rd (Load chunk addr vs).

(** [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 :=
  kill_equations filter_loads n.

(** [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] or [Mint64]). *)

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
  | Load chunk' addr' vl' =>
      negb(eq_list_valnum vl vl' && addressing_separated chunk addr chunk' addr')
  end.

Definition add_store (n: numbering) (chunk: memory_chunk) (addr: addressing)
                                   (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
  | Mint32 | Mint64 | Mfloat64 | Mfloat64al32 => add_rhs n2 rsrc (Load chunk addr vargs)
  | _ => 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)
  | Load chunk addr vl =>
      exists a,
      eval_addressing ge sp addr (List.map valuation vl) = Some 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, for instructions that modify neither registers nor
  the memory, we keep the numbering unchanged.

  For builtin invocations [Ibuiltin], we have three strategies:
- Forget all equations.  This is appropriate for builtins that can be
  turned into function calls ([EF_external], [EF_malloc], [EF_free]).
- Forget equations involving loads but keep equations over registers.
  This is appropriate for builtins that modify memory, e.g. [EF_memcpy].
- Keep all equations, taking advantage of the fact that neither memory
  nor registers are modified.  This is appropriate for annotations,
  for inlined builtin functions, and for volatile loads.
*)

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 =>
          add_op before res op args
      | Iload chunk addr args dst s =>
          add_load before dst chunk addr args
      | Istore chunk addr args src s =>
          add_store before chunk addr args src
      | Icall sig ros args res s =>
          empty_numbering
      | Itailcall sig ros args =>
          empty_numbering
      | Ibuiltin ef args res s =>
          match ef with
          | EF_external _ _ | EF_malloc | EF_free | EF_inline_asm _ =>
              empty_numbering
          | EF_vstore _ | EF_vstore_global _ _ _ | EF_memcpy _ _ =>
              add_unknown (kill_loads before) res
          | EF_builtin _ _ | EF_vload _ | EF_vload_global _ _ _
          | EF_annot _ _ | EF_annot_val _ _ =>
              add_unknown before res
          end
      | 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
  | Iload chunk addr args dst s =>
      let (n1, vl) := valnum_regs n args in
      match find_rhs n1 (Load chunk addr vl) with
      | Some r =>
          Iop Omove (r :: nil) dst s
      | None =>
          let (addr', args') := reduce _ combine_addr n1 addr args vl in
          Iload chunk addr' args' dst s
      end
  | Istore chunk addr args src s =>
      let (n1, vl) := valnum_regs n args in
      let (addr', args') := reduce _ combine_addr n1 addr args vl 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.