Branch tunneling (optimization of branches to branches).

Require Import FunInd.

Require Import Coqlib Maps UnionFind.

Require Import AST.

Require Import LTL.

Branch tunneling shortens sequences of branches (with no intervening
computations) by rewriting the branch and conditional branch instructions
so that they jump directly to the end of the branch sequence.
For example:

L1: branch L2; L1: branch L3; L2; branch L3; becomes L2: branch L3; L3: instr; L3: instr; L4: if (cond) goto L1; L4: if (cond) goto L3;This optimization can be applied to several of our intermediate languages. We choose to perform it on the LTL language, after register allocation but before code linearization. Register allocation can delete instructions (such as dead computations or useless moves), therefore there are more opportunities for tunneling after allocation than before. Symmetrically, prior tunneling helps linearization to produce better code, e.g. by revealing that some branch instructions are dead code (as the "branch L3" in the example above).

The naive implementation of branch tunneling would replace
any branch to a node pc by a branch to the node
branch_target f pc, defined as follows:

branch_target f pc = branch_target f pc' if f(pc) = branch pc' = pc otherwiseHowever, this definition can fail to terminate if the program can contain loops consisting only of branches, as in

L1: branch L1;or

L1: branch L2; L2: branch L1;Coq warns us of this fact by not accepting the definition of branch_target above. To handle this problem, we proceed in two passes:

- The first pass populates a union-find data structure, adding equalities between PCs of blocks that are connected by branches and no other computation.

- The second pass rewrites the code, replacing every branch to a node pc by a branch to the canonical representative of the equivalence class of pc.

Module U := UnionFind.UF(PTree).

We start populating the union-find data structure by adding
equalities pc = pc' for every block pc: branch pc' in the function.

Definition record_branch (uf: U.t) (pc: node) (b: bblock) : U.t :=

match b with

| Lbranch s :: _ => U.union uf pc s

| _ => uf

end.

Definition record_branches (f: LTL.function) : U.t :=

PTree.fold record_branch f.(fn_code) U.empty.

An additional optimization opportunity comes from conditional branches.
Consider a block pc: cond ifso ifnot. If the ifso case
and the ifnot case jump to the same block pc'
(modulo intermediate branches), the block can be simplified into
pc: branch pc', and the equality pc = pc' can be added to the
union-find data structure.

In rare cases, the extra equation pc = pc' introduced by the
simplification of a conditional branch can trigger further simplifications
of other conditional branches. We therefore iterate the analysis
until no optimizable conditional branch remains.

The code c (first component of the st triple) starts identical
to the code fn.(fn_code) of the current function, but each time
conditional branch at pc is optimized, we remove the block at
pc from the code c. This guarantees termination of the
iteration.

Definition record_cond (st: code * U.t * bool) (pc: node) (b: bblock) : code * U.t * bool :=

match b with

| Lcond cond args s1 s2 :: _ =>

let '(c, u, _) := st in

if peq (U.repr u s1) (U.repr u s2)

then (PTree.remove pc c, U.union u pc s1, true)

else st

| _ =>

st

end.

Definition record_conds_1 (cu: code * U.t) : code * U.t * bool :=

let (c, u) := cu in PTree.fold record_cond c (c, u, false).

Definition measure_state (cu: code * U.t) : nat :=

PTree_Properties.cardinal (fst cu).

Function record_conds (cu: code * U.t) {measure measure_state cu} : U.t :=

let (cu', changed) := record_conds_1 cu in

if changed then record_conds cu' else snd cu.

Proof.

intros [c0 u0] [c1 u1].

set (P := fun (c: code) (s: code * U.t * bool) =>

(forall pc, c!pc = None -> (fst (fst s))!pc = c0!pc) /\

(PTree_Properties.cardinal (fst (fst s))

+ (if snd s then 1 else 0)

<= PTree_Properties.cardinal c0)%nat).

assert (A: P c0 (PTree.fold record_cond c0 (c0, u0, false))).

{ apply PTree_Properties.fold_rec; unfold P.

- intros. destruct H0; split; auto. intros. rewrite <- H in H2. auto.

- simpl; split; intros. auto. simpl; lia.

- intros cd [[c u] changed] pc b NONE SOME [HR1 HR2]. simpl. split.

+ intros p EQ. rewrite PTree.gsspec in EQ. destruct (peq p pc); try discriminate.

unfold record_cond. destruct b as [ | [] b ]; auto.

destruct (peq (U.repr u s1) (U.repr u s2)); auto.

simpl. rewrite PTree.gro by auto. auto.

+ unfold record_cond. destruct b as [ | [] b ]; auto.

destruct (peq (U.repr u s1) (U.repr u s2)); auto.

simpl in *.

assert (SOME': c!pc = Some (Lcond cond args s1 s2 :: b)).

{ rewrite HR1 by auto. auto. }

generalize (PTree_Properties.cardinal_remove SOME').

destruct changed; lia.

}

unfold record_conds_1, measure_state; intros.

destruct A as [_ A]. rewrite teq in A. simpl in *.

lia.

Qed.

set (P := fun (c: code) (s: code * U.t * bool) =>

(forall pc, c!pc = None -> (fst (fst s))!pc = c0!pc) /\

(PTree_Properties.cardinal (fst (fst s))

+ (if snd s then 1 else 0)

<= PTree_Properties.cardinal c0)%nat).

assert (A: P c0 (PTree.fold record_cond c0 (c0, u0, false))).

{ apply PTree_Properties.fold_rec; unfold P.

- intros. destruct H0; split; auto. intros. rewrite <- H in H2. auto.

- simpl; split; intros. auto. simpl; lia.

- intros cd [[c u] changed] pc b NONE SOME [HR1 HR2]. simpl. split.

+ intros p EQ. rewrite PTree.gsspec in EQ. destruct (peq p pc); try discriminate.

unfold record_cond. destruct b as [ | [] b ]; auto.

destruct (peq (U.repr u s1) (U.repr u s2)); auto.

simpl. rewrite PTree.gro by auto. auto.

+ unfold record_cond. destruct b as [ | [] b ]; auto.

destruct (peq (U.repr u s1) (U.repr u s2)); auto.

simpl in *.

assert (SOME': c!pc = Some (Lcond cond args s1 s2 :: b)).

{ rewrite HR1 by auto. auto. }

generalize (PTree_Properties.cardinal_remove SOME').

destruct changed; lia.

}

unfold record_conds_1, measure_state; intros.

destruct A as [_ A]. rewrite teq in A. simpl in *.

lia.

Qed.

Definition record_gotos (f: LTL.function) : U.t :=

record_conds (f.(fn_code), record_branches f).

The code transformation rewrites all LTL instruction, replacing every
successor s of every instruction by the canonical representative
of its equivalence class in the union-find data structure.
Additionally, Lcond conditional branches are turned into Lbranch
unconditional branches whenever possible.

Definition tunnel_instr (u: U.t) (i: instruction) : instruction :=

match i with

| Lbranch s => Lbranch (U.repr u s)

| Lcond cond args s1 s2 =>

let s1' := U.repr u s1 in let s2' := U.repr u s2 in

if peq s1' s2'

then Lbranch s1'

else Lcond cond args s1' s2'

| Ljumptable arg tbl => Ljumptable arg (List.map (U.repr u) tbl)

| _ => i

end.

Definition tunnel_block (u: U.t) (b: bblock) : bblock :=

List.map (tunnel_instr u) b.

Definition tunnel_function (f: LTL.function) : LTL.function :=

let u := record_gotos f in

mkfunction

(fn_sig f)

(fn_stacksize f)

(PTree.map1 (tunnel_block u) (fn_code f))

(U.repr u (fn_entrypoint f)).

Definition tunnel_fundef (f: LTL.fundef) : LTL.fundef :=

transf_fundef tunnel_function f.

Definition tunnel_program (p: LTL.program) : LTL.program :=

transform_program tunnel_fundef p.