# Module ConstpropOp

Static analysis and strength reduction for operators and conditions. This is the machine-dependent part of Constprop.

Require Import Coqlib.
Require Import Ast.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Op.
Require Import Registers.

# Static analysis

To each pseudo-register at each program point, the static analysis associates a compile-time approximation taken from the following set.

Inductive approx : Type :=
| Novalue: approx
| Unknown: approx
| I: int -> approx
| F: float -> approx
| S: ident -> int -> approx.
The value is the address of the given global symbol plus the given integer offset.

We now define the abstract interpretations of conditions and operators over this set of approximations. For instance, the abstract interpretation of the operator Oaddf applied to two expressions a and b is F(Float.add f g) if a and b have static approximations Vfloat f and Vfloat g respectively, and Unknown otherwise. The static approximations are defined by large pattern-matchings over the approximations of the results. We write these matchings in the indirect style described in file Cmconstr to avoid excessive duplication of cases in proofs.

Inductive eval_static_condition_cases: forall (cond: condition) (vl: list approx), Type :=
| eval_static_condition_case1:
forall c n1 n2,
eval_static_condition_cases (Ccomp c) (I n1 :: I n2 :: nil)
| eval_static_condition_case2:
forall c n1 n2,
eval_static_condition_cases (Ccompu c) (I n1 :: I n2 :: nil)
| eval_static_condition_case3:
forall c n n1,
eval_static_condition_cases (Ccompimm c n) (I n1 :: nil)
| eval_static_condition_case4:
forall c n n1,
eval_static_condition_cases (Ccompuimm c n) (I n1 :: nil)
| eval_static_condition_case5:
forall c n1 n2,
eval_static_condition_cases (Ccompf c) (F n1 :: F n2 :: nil)
| eval_static_condition_case6:
forall c n1 n2,
eval_static_condition_cases (Cnotcompf c) (F n1 :: F n2 :: nil)
| eval_static_condition_case7:
forall n n1,
eval_static_condition_cases (Cmaskzero n) (I n1 :: nil)
| eval_static_condition_case8:
forall n n1,
eval_static_condition_cases (Cmasknotzero n) (I n1 :: nil)
| eval_static_condition_default:
forall (cond: condition) (vl: list approx),
eval_static_condition_cases cond vl.

Definition eval_static_condition_match (cond: condition) (vl: list approx) :=
match cond as z1, vl as z2 return eval_static_condition_cases z1 z2 with
| Ccomp c, I n1 :: I n2 :: nil =>
eval_static_condition_case1 c n1 n2
| Ccompu c, I n1 :: I n2 :: nil =>
eval_static_condition_case2 c n1 n2
| Ccompimm c n, I n1 :: nil =>
eval_static_condition_case3 c n n1
| Ccompuimm c n, I n1 :: nil =>
eval_static_condition_case4 c n n1
| Ccompf c, F n1 :: F n2 :: nil =>
eval_static_condition_case5 c n1 n2
| Cnotcompf c, F n1 :: F n2 :: nil =>
eval_static_condition_case6 c n1 n2
| Cmaskzero n, I n1 :: nil =>
eval_static_condition_case7 n n1
| Cmasknotzero n, I n1 :: nil =>
eval_static_condition_case8 n n1
| cond, vl =>
eval_static_condition_default cond vl
end.

Definition eval_static_condition (cond: condition) (vl: list approx) :=
match eval_static_condition_match cond vl with
| eval_static_condition_case1 c n1 n2 =>
Some(Int.cmp c n1 n2)
| eval_static_condition_case2 c n1 n2 =>
Some(Int.cmpu c n1 n2)
| eval_static_condition_case3 c n n1 =>
Some(Int.cmp c n1 n)
| eval_static_condition_case4 c n n1 =>
Some(Int.cmpu c n1 n)
| eval_static_condition_case5 c n1 n2 =>
Some(Float.cmp c n1 n2)
| eval_static_condition_case6 c n1 n2 =>
Some(negb(Float.cmp c n1 n2))
| eval_static_condition_case7 n n1 =>
Some(Int.eq (Int.and n1 n) Int.zero)
| eval_static_condition_case8 n n1 =>
Some(negb(Int.eq (Int.and n1 n) Int.zero))
| eval_static_condition_default cond vl =>
None
end.

Inductive eval_static_addressing_cases: forall (addr: addressing) (vl: list approx), Type :=
forall n n1,
eval_static_addressing_cases (Aindexed n) (I n1::nil)
forall n id ofs,
eval_static_addressing_cases (Aindexed n) (S id ofs::nil)
forall n n1 n2,
eval_static_addressing_cases (Aindexed2 n) (I n1::I n2::nil)
forall n id ofs n2,
eval_static_addressing_cases (Aindexed2 n) (S id ofs::I n2::nil)
forall n n1 id ofs,
eval_static_addressing_cases (Aindexed2 n) (I n1::S id ofs::nil)
forall sc n n1,
eval_static_addressing_cases (Ascaled sc n) (I n1::nil)
forall sc n n1 n2,
eval_static_addressing_cases (Aindexed2scaled sc n) (I n1::I n2::nil)
forall sc n id ofs n2,
eval_static_addressing_cases (Aindexed2scaled sc n) (S id ofs::I n2::nil)
forall id ofs,
eval_static_addressing_cases (Aglobal id ofs) (nil)
forall id ofs n1,
eval_static_addressing_cases (Abased id ofs) (I n1::nil)
forall sc id ofs n1,
eval_static_addressing_cases (Abasedscaled sc id ofs) (I n1::nil)

match addr as z1, vl as z2 return eval_static_addressing_cases z1 z2 with
| Aindexed n, I n1::nil =>
| Aindexed n, S id ofs::nil =>
eval_static_addressing_case2 n id ofs
| Aindexed2 n, I n1::I n2::nil =>
eval_static_addressing_case3 n n1 n2
| Aindexed2 n, S id ofs::I n2::nil =>
eval_static_addressing_case4 n id ofs n2
| Aindexed2 n, I n1::S id ofs::nil =>
eval_static_addressing_case5 n n1 id ofs
| Ascaled sc n, I n1::nil =>
eval_static_addressing_case6 sc n n1
| Aindexed2scaled sc n, I n1::I n2::nil =>
eval_static_addressing_case7 sc n n1 n2
| Aindexed2scaled sc n, S id ofs::I n2::nil =>
eval_static_addressing_case8 sc n id ofs n2
| Aglobal id ofs, nil =>
| Abased id ofs, I n1::nil =>
eval_static_addressing_case10 id ofs n1
| Abasedscaled sc id ofs, I n1::nil =>
eval_static_addressing_case11 sc id ofs n1
| addr, vl =>
end.

| eval_static_addressing_case1 n n1 =>
I (Int.add n1 n)
| eval_static_addressing_case2 n id ofs =>
S id (Int.add ofs n)
| eval_static_addressing_case3 n n1 n2 =>
| eval_static_addressing_case4 n id ofs n2 =>
S id (Int.add (Int.add ofs n2) n)
| eval_static_addressing_case5 n n1 id ofs =>
S id (Int.add (Int.add ofs n1) n)
| eval_static_addressing_case6 sc n n1 =>
I (Int.add (Int.mul n1 sc) n)
| eval_static_addressing_case7 sc n n1 n2 =>
I (Int.add n1 (Int.add (Int.mul n2 sc) n))
| eval_static_addressing_case8 sc n id ofs n2 =>
S id (Int.add ofs (Int.add (Int.mul n2 sc) n))
| eval_static_addressing_case9 id ofs =>
S id ofs
| eval_static_addressing_case10 id ofs n1 =>
S id (Int.add ofs n1)
| eval_static_addressing_case11 sc id ofs n1 =>
S id (Int.add ofs (Int.mul sc n1))
Unknown
end.

Inductive eval_static_operation_cases: forall (op: operation) (vl: list approx), Type :=
| eval_static_operation_case1:
forall v1,
eval_static_operation_cases (Omove) (v1::nil)
| eval_static_operation_case2:
forall n,
eval_static_operation_cases (Ointconst n) (nil)
| eval_static_operation_case3:
forall n,
eval_static_operation_cases (Ofloatconst n) (nil)
| eval_static_operation_case4:
forall n1,
eval_static_operation_cases (Ocast8signed) (I n1 :: nil)
| eval_static_operation_case5:
forall n1,
eval_static_operation_cases (Ocast8unsigned) (I n1 :: nil)
| eval_static_operation_case6:
forall n1,
eval_static_operation_cases (Ocast16signed) (I n1 :: nil)
| eval_static_operation_case7:
forall n1,
eval_static_operation_cases (Ocast16unsigned) (I n1 :: nil)
| eval_static_operation_case8:
forall n1,
eval_static_operation_cases (Oneg) (I n1 :: nil)
| eval_static_operation_case9:
forall n1 n2,
eval_static_operation_cases (Osub) (I n1 :: I n2 :: nil)
| eval_static_operation_case10:
forall s1 n1 n2,
eval_static_operation_cases (Osub) (S s1 n1 :: I n2 :: nil)
| eval_static_operation_case11:
forall n1 n2,
eval_static_operation_cases (Omul) (I n1 :: I n2 :: nil)
| eval_static_operation_case12:
forall n n1,
eval_static_operation_cases (Omulimm n) (I n1 :: nil)
| eval_static_operation_case13:
forall n1 n2,
eval_static_operation_cases (Odiv) (I n1 :: I n2 :: nil)
| eval_static_operation_case14:
forall n1 n2,
eval_static_operation_cases (Odivu) (I n1 :: I n2 :: nil)
| eval_static_operation_case15:
forall n1 n2,
eval_static_operation_cases (Omod) (I n1 :: I n2 :: nil)
| eval_static_operation_case16:
forall n1 n2,
eval_static_operation_cases (Omodu) (I n1 :: I n2 :: nil)
| eval_static_operation_case17:
forall n1 n2,
eval_static_operation_cases (Oand) (I n1 :: I n2 :: nil)
| eval_static_operation_case18:
forall n n1,
eval_static_operation_cases (Oandimm n) (I n1 :: nil)
| eval_static_operation_case19:
forall n1 n2,
eval_static_operation_cases (Oor) (I n1 :: I n2 :: nil)
| eval_static_operation_case20:
forall n n1,
eval_static_operation_cases (Oorimm n) (I n1 :: nil)
| eval_static_operation_case21:
forall n1 n2,
eval_static_operation_cases (Oxor) (I n1 :: I n2 :: nil)
| eval_static_operation_case22:
forall n n1,
eval_static_operation_cases (Oxorimm n) (I n1 :: nil)
| eval_static_operation_case23:
forall n1 n2,
eval_static_operation_cases (Oshl) (I n1 :: I n2 :: nil)
| eval_static_operation_case24:
forall n n1,
eval_static_operation_cases (Oshlimm n) (I n1 :: nil)
| eval_static_operation_case25:
forall n1 n2,
eval_static_operation_cases (Oshr) (I n1 :: I n2 :: nil)
| eval_static_operation_case26:
forall n n1,
eval_static_operation_cases (Oshrimm n) (I n1 :: nil)
| eval_static_operation_case27:
forall n n1,
eval_static_operation_cases (Oshrximm n) (I n1 :: nil)
| eval_static_operation_case28:
forall n1 n2,
eval_static_operation_cases (Oshru) (I n1 :: I n2 :: nil)
| eval_static_operation_case29:
forall n n1,
eval_static_operation_cases (Oshruimm n) (I n1 :: nil)
| eval_static_operation_case30:
forall n n1,
eval_static_operation_cases (Ororimm n) (I n1 :: nil)
| eval_static_operation_case31:
forall mode vl,
eval_static_operation_cases (Olea mode) (vl)
| eval_static_operation_case32:
forall n1,
eval_static_operation_cases (Onegf) (F n1 :: nil)
| eval_static_operation_case34:
forall n1 n2,
eval_static_operation_cases (Oaddf) (F n1 :: F n2 :: nil)
| eval_static_operation_case35:
forall n1 n2,
eval_static_operation_cases (Osubf) (F n1 :: F n2 :: nil)
| eval_static_operation_case36:
forall n1 n2,
eval_static_operation_cases (Omulf) (F n1 :: F n2 :: nil)
| eval_static_operation_case37:
forall n1 n2,
eval_static_operation_cases (Odivf) (F n1 :: F n2 :: nil)
| eval_static_operation_case38:
forall n1,
eval_static_operation_cases (Osingleoffloat) (F n1 :: nil)
| eval_static_operation_case39:
forall n1,
eval_static_operation_cases (Ointoffloat) (F n1 :: nil)
| eval_static_operation_case41:
forall n1,
eval_static_operation_cases (Ofloatofint) (I n1 :: nil)
| eval_static_operation_case43:
forall c vl,
eval_static_operation_cases (Ocmp c) vl
| eval_static_operation_default:
forall (op: operation) (vl: list approx),
eval_static_operation_cases op vl.

Definition eval_static_operation_match (op: operation) (vl: list approx) :=
match op as z1, vl as z2 return eval_static_operation_cases z1 z2 with
| Omove, v1::nil =>
eval_static_operation_case1 v1
| Ointconst n, nil =>
eval_static_operation_case2 n
| Ofloatconst n, nil =>
eval_static_operation_case3 n
| Ocast8signed, I n1 :: nil =>
eval_static_operation_case4 n1
| Ocast8unsigned, I n1 :: nil =>
eval_static_operation_case5 n1
| Ocast16signed, I n1 :: nil =>
eval_static_operation_case6 n1
| Ocast16unsigned, I n1 :: nil =>
eval_static_operation_case7 n1
| Oneg, I n1 :: nil =>
eval_static_operation_case8 n1
| Osub, I n1 :: I n2 :: nil =>
eval_static_operation_case9 n1 n2
| Osub, S s1 n1 :: I n2 :: nil =>
eval_static_operation_case10 s1 n1 n2
| Omul, I n1 :: I n2 :: nil =>
eval_static_operation_case11 n1 n2
| Omulimm n, I n1 :: nil =>
eval_static_operation_case12 n n1
| Odiv, I n1 :: I n2 :: nil =>
eval_static_operation_case13 n1 n2
| Odivu, I n1 :: I n2 :: nil =>
eval_static_operation_case14 n1 n2
| Omod, I n1 :: I n2 :: nil =>
eval_static_operation_case15 n1 n2
| Omodu, I n1 :: I n2 :: nil =>
eval_static_operation_case16 n1 n2
| Oand, I n1 :: I n2 :: nil =>
eval_static_operation_case17 n1 n2
| Oandimm n, I n1 :: nil =>
eval_static_operation_case18 n n1
| Oor, I n1 :: I n2 :: nil =>
eval_static_operation_case19 n1 n2
| Oorimm n, I n1 :: nil =>
eval_static_operation_case20 n n1
| Oxor, I n1 :: I n2 :: nil =>
eval_static_operation_case21 n1 n2
| Oxorimm n, I n1 :: nil =>
eval_static_operation_case22 n n1
| Oshl, I n1 :: I n2 :: nil =>
eval_static_operation_case23 n1 n2
| Oshlimm n, I n1 :: nil =>
eval_static_operation_case24 n n1
| Oshr, I n1 :: I n2 :: nil =>
eval_static_operation_case25 n1 n2
| Oshrimm n, I n1 :: nil =>
eval_static_operation_case26 n n1
| Oshrximm n, I n1 :: nil =>
eval_static_operation_case27 n n1
| Oshru, I n1 :: I n2 :: nil =>
eval_static_operation_case28 n1 n2
| Oshruimm n, I n1 :: nil =>
eval_static_operation_case29 n n1
| Ororimm n, I n1 :: nil =>
eval_static_operation_case30 n n1
| Olea mode, vl =>
eval_static_operation_case31 mode vl
| Onegf, F n1 :: nil =>
eval_static_operation_case32 n1
| Oaddf, F n1 :: F n2 :: nil =>
eval_static_operation_case34 n1 n2
| Osubf, F n1 :: F n2 :: nil =>
eval_static_operation_case35 n1 n2
| Omulf, F n1 :: F n2 :: nil =>
eval_static_operation_case36 n1 n2
| Odivf, F n1 :: F n2 :: nil =>
eval_static_operation_case37 n1 n2
| Osingleoffloat, F n1 :: nil =>
eval_static_operation_case38 n1
| Ointoffloat, F n1 :: nil =>
eval_static_operation_case39 n1
| Ofloatofint, I n1 :: nil =>
eval_static_operation_case41 n1
| Ocmp c, vl =>
eval_static_operation_case43 c vl
| op, vl =>
eval_static_operation_default op vl
end.

Definition eval_static_operation (op: operation) (vl: list approx) :=
match eval_static_operation_match op vl with
| eval_static_operation_case1 v1 =>
v1
| eval_static_operation_case2 n =>
I n
| eval_static_operation_case3 n =>
F n
| eval_static_operation_case4 n1 =>
I(Int.sign_ext 8 n1)
| eval_static_operation_case5 n1 =>
I(Int.zero_ext 8 n1)
| eval_static_operation_case6 n1 =>
I(Int.sign_ext 16 n1)
| eval_static_operation_case7 n1 =>
I(Int.zero_ext 16 n1)
| eval_static_operation_case8 n1 =>
I(Int.neg n1)
| eval_static_operation_case9 n1 n2 =>
I(Int.sub n1 n2)
| eval_static_operation_case10 s1 n1 n2 =>
S s1 (Int.sub n1 n2)
| eval_static_operation_case11 n1 n2 =>
I(Int.mul n1 n2)
| eval_static_operation_case12 n n1 =>
I(Int.mul n1 n)
| eval_static_operation_case13 n1 n2 =>
if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
| eval_static_operation_case14 n1 n2 =>
if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
| eval_static_operation_case15 n1 n2 =>
if Int.eq n2 Int.zero then Unknown else I(Int.mods n1 n2)
| eval_static_operation_case16 n1 n2 =>
if Int.eq n2 Int.zero then Unknown else I(Int.modu n1 n2)
| eval_static_operation_case17 n1 n2 =>
I(Int.and n1 n2)
| eval_static_operation_case18 n n1 =>
I(Int.and n1 n)
| eval_static_operation_case19 n1 n2 =>
I(Int.or n1 n2)
| eval_static_operation_case20 n n1 =>
I(Int.or n1 n)
| eval_static_operation_case21 n1 n2 =>
I(Int.xor n1 n2)
| eval_static_operation_case22 n n1 =>
I(Int.xor n1 n)
| eval_static_operation_case23 n1 n2 =>
if Int.ltu n2 (Int.repr 32) then I(Int.shl n1 n2) else Unknown
| eval_static_operation_case24 n n1 =>
if Int.ltu n (Int.repr 32) then I(Int.shl n1 n) else Unknown
| eval_static_operation_case25 n1 n2 =>
if Int.ltu n2 (Int.repr 32) then I(Int.shr n1 n2) else Unknown
| eval_static_operation_case26 n n1 =>
if Int.ltu n (Int.repr 32) then I(Int.shr n1 n) else Unknown
| eval_static_operation_case27 n n1 =>
if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown
| eval_static_operation_case28 n1 n2 =>
if Int.ltu n2 (Int.repr 32) then I(Int.shru n1 n2) else Unknown
| eval_static_operation_case29 n n1 =>
if Int.ltu n (Int.repr 32) then I(Int.shru n1 n) else Unknown
| eval_static_operation_case30 n n1 =>
if Int.ltu n (Int.repr 32) then I(Int.ror n1 n) else Unknown
| eval_static_operation_case31 mode vl =>
| eval_static_operation_case32 n1 =>
F(Float.neg n1)
| eval_static_operation_case34 n1 n2 =>
| eval_static_operation_case35 n1 n2 =>
F(Float.sub n1 n2)
| eval_static_operation_case36 n1 n2 =>
F(Float.mul n1 n2)
| eval_static_operation_case37 n1 n2 =>
F(Float.div n1 n2)
| eval_static_operation_case38 n1 =>
F(Float.singleoffloat n1)
| eval_static_operation_case39 n1 =>
I(Float.intoffloat n1)
| eval_static_operation_case41 n1 =>
F(Float.floatofint n1)
| eval_static_operation_case43 c vl =>
match eval_static_condition c vl with
| None => Unknown
| Some b => I(if b then Int.one else Int.zero)
end
| eval_static_operation_default op vl =>
Unknown
end.

# Operator strength reduction

We now define auxiliary functions for strength reduction of operators and addressing modes: replacing an operator with a cheaper one if some of its arguments are statically known. These are again large pattern-matchings expressed in indirect style.

Section STRENGTH_REDUCTION.

Variable app: reg -> approx.

Definition intval (r: reg) : option int :=
match app r with I n => Some n | _ => None end.

Inductive cond_strength_reduction_cases: condition -> list reg -> Type :=
| csr_case1:
forall c r1 r2,
cond_strength_reduction_cases (Ccomp c) (r1 :: r2 :: nil)
| csr_case2:
forall c r1 r2,
cond_strength_reduction_cases (Ccompu c) (r1 :: r2 :: nil)
| csr_default:
forall c rl,
cond_strength_reduction_cases c rl.

Definition cond_strength_reduction_match (cond: condition) (rl: list reg) :=
match cond as x, rl as y return cond_strength_reduction_cases x y with
| Ccomp c, r1 :: r2 :: nil =>
csr_case1 c r1 r2
| Ccompu c, r1 :: r2 :: nil =>
csr_case2 c r1 r2
| cond, rl =>
csr_default cond rl
end.

Definition cond_strength_reduction
(cond: condition) (args: list reg) : condition * list reg :=
match cond_strength_reduction_match cond args with
| csr_case1 c r1 r2 =>
match intval r1, intval r2 with
| Some n, _ =>
(Ccompimm (swap_comparison c) n, r2 :: nil)
| _, Some n =>
(Ccompimm c n, r1 :: nil)
| _, _ =>
(cond, args)
end
| csr_case2 c r1 r2 =>
match intval r1, intval r2 with
| Some n, _ =>
(Ccompuimm (swap_comparison c) n, r2 :: nil)
| _, Some n =>
(Ccompuimm c n, r1 :: nil)
| _, _ =>
(cond, args)
end
| csr_default cond args =>
(cond, args)
end.

Inductive addr_strength_reduction_cases: forall (addr: addressing) (args: list reg), Type :=
forall ofs r1,
addr_strength_reduction_cases (Aindexed ofs) (r1 :: nil)
forall ofs r1 r2,
addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil)
forall sc ofs r1 r2,
addr_strength_reduction_cases (Aindexed2scaled sc ofs) (r1 :: r2 :: nil)
forall id ofs r1,
addr_strength_reduction_cases (Abased id ofs) (r1 :: nil)
forall sc id ofs r1,
addr_strength_reduction_cases (Abasedscaled sc id ofs) (r1 :: nil)

match addr as z1, args as z2 return addr_strength_reduction_cases z1 z2 with
| Aindexed ofs, r1 :: nil =>
| Aindexed2 ofs, r1 :: r2 :: nil =>
addr_strength_reduction_case2 ofs r1 r2
| Aindexed2scaled sc ofs, r1 :: r2 :: nil =>
addr_strength_reduction_case3 sc ofs r1 r2
| Abased id ofs, r1 :: nil =>
addr_strength_reduction_case4 id ofs r1
| Abasedscaled sc id ofs, r1 :: nil =>
addr_strength_reduction_case5 sc id ofs r1
| addr, args =>
end.

| addr_strength_reduction_case1 ofs r1 =>
match app r1 with
| S symb n => (Aglobal symb (Int.add ofs n), nil)
| _ => (addr, args)
end
| addr_strength_reduction_case2 ofs r1 r2 =>
match app r1, app r2 with
| S symb n1, I n2 => (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
| I n1, S symb n2 => (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
| S symb n1, _ => (Abased symb (Int.add n1 ofs), r2 :: nil)
| _, S symb n2 => (Abased symb (Int.add n2 ofs), r1 :: nil)
| I n1, _ => (Aindexed (Int.add n1 ofs), r2 :: nil)
| _, I n2 => (Aindexed (Int.add n2 ofs), r1 :: nil)
| _, _ => (addr, args)
end
| addr_strength_reduction_case3 sc ofs r1 r2 =>
match app r1, app r2 with
| S symb n1, I n2 => (Aglobal symb (Int.add (Int.add n1 (Int.mul n2 sc)) ofs), nil)
| S symb n1, _ => (Abasedscaled sc symb (Int.add n1 ofs), r2 :: nil)
| _, I n2 => (Aindexed (Int.add (Int.mul n2 sc) ofs), r1 :: nil)
| _, _ => (addr, args)
end
| addr_strength_reduction_case4 id ofs r1 =>
match app r1 with
| I n1 => (Aglobal id (Int.add ofs n1), nil)
| _ => (addr, args)
end
| addr_strength_reduction_case5 sc id ofs r1 =>
match app r1 with
| I n1 => (Aglobal id (Int.add ofs (Int.mul sc n1)), nil)
| _ => (addr, args)
end
end.

Definition make_addimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Omove, r :: nil)
else (Oaddimm n, r :: nil).

Definition make_shlimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Omove, r :: nil)
else (Oshlimm n, r :: nil).

Definition make_shrimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Omove, r :: nil)
else (Oshrimm n, r :: nil).

Definition make_shruimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Omove, r :: nil)
else (Oshruimm n, r :: nil).

Definition make_mulimm (n: int) (r: reg) :=
if Int.eq n Int.zero then
(Ointconst Int.zero, nil)
else if Int.eq n Int.one then
(Omove, r :: nil)
else
match Int.is_power2 n with
| Some l => make_shlimm l r
| None => (Omulimm n, r :: nil)
end.

Definition make_andimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Ointconst Int.zero, nil)
else if Int.eq n Int.mone then (Omove, r :: nil)
else (Oandimm n, r :: nil).

Definition make_orimm (n: int) (r: reg) :=
if Int.eq n Int.zero then (Omove, r :: nil)
else if Int.eq n Int.mone then (Ointconst Int.mone, nil)
else (Oorimm n, r :: nil).

Definition make_xorimm (n: int) (r: reg) :=
if Int.eq n Int.zero
then (Omove, r :: nil)
else (Oxorimm n, r :: nil).

Inductive op_strength_reduction_cases: forall (op: operation) (args: list reg), Type :=
| op_strength_reduction_case2:
forall r1 r2,
op_strength_reduction_cases (Osub) (r1 :: r2 :: nil)
| op_strength_reduction_case3:
forall r1 r2,
op_strength_reduction_cases (Omul) (r1 :: r2 :: nil)
| op_strength_reduction_case4:
forall r1 r2,
op_strength_reduction_cases (Odiv) (r1 :: r2 :: nil)
| op_strength_reduction_case5:
forall r1 r2,
op_strength_reduction_cases (Odivu) (r1 :: r2 :: nil)
| op_strength_reduction_case7:
forall r1 r2,
op_strength_reduction_cases (Omodu) (r1 :: r2 :: nil)
| op_strength_reduction_case8:
forall r1 r2,
op_strength_reduction_cases (Oand) (r1 :: r2 :: nil)
| op_strength_reduction_case9:
forall r1 r2,
op_strength_reduction_cases (Oor) (r1 :: r2 :: nil)
| op_strength_reduction_case10:
forall r1 r2,
op_strength_reduction_cases (Oxor) (r1 :: r2 :: nil)
| op_strength_reduction_case11:
forall r1 r2,
op_strength_reduction_cases (Oshl) (r1 :: r2 :: nil)
| op_strength_reduction_case12:
forall r1 r2,
op_strength_reduction_cases (Oshr) (r1 :: r2 :: nil)
| op_strength_reduction_case13:
forall r1 r2,
op_strength_reduction_cases (Oshru) (r1 :: r2 :: nil)
| op_strength_reduction_case14:
op_strength_reduction_cases (Olea addr) (args)
| op_strength_reduction_case15:
forall c args,
op_strength_reduction_cases (Ocmp c) (args)
| op_strength_reduction_default:
forall (op: operation) (args: list reg),
op_strength_reduction_cases op args.

Definition op_strength_reduction_match (op: operation) (args: list reg) :=
match op as z1, args as z2 return op_strength_reduction_cases z1 z2 with
| Osub, r1 :: r2 :: nil =>
op_strength_reduction_case2 r1 r2
| Omul, r1 :: r2 :: nil =>
op_strength_reduction_case3 r1 r2
| Odiv, r1 :: r2 :: nil =>
op_strength_reduction_case4 r1 r2
| Odivu, r1 :: r2 :: nil =>
op_strength_reduction_case5 r1 r2
| Omodu, r1 :: r2 :: nil =>
op_strength_reduction_case7 r1 r2
| Oand, r1 :: r2 :: nil =>
op_strength_reduction_case8 r1 r2
| Oor, r1 :: r2 :: nil =>
op_strength_reduction_case9 r1 r2
| Oxor, r1 :: r2 :: nil =>
op_strength_reduction_case10 r1 r2
| Oshl, r1 :: r2 :: nil =>
op_strength_reduction_case11 r1 r2
| Oshr, r1 :: r2 :: nil =>
op_strength_reduction_case12 r1 r2
| Oshru, r1 :: r2 :: nil =>
op_strength_reduction_case13 r1 r2
| Olea addr, args =>
| Ocmp c, args =>
op_strength_reduction_case15 c args
| op, args =>
op_strength_reduction_default op args
end.

We must be careful to preserve 2-address constraints over the RTL code, which means that commutative operations cannot be specialized if their first argument is a constant.

Definition op_strength_reduction (op: operation) (args: list reg) :=
match op_strength_reduction_match op args with
| op_strength_reduction_case2 r1 r2 =>
match intval r2 with
| Some n => make_addimm (Int.neg n) r1
| _ => (op, args)
end
| op_strength_reduction_case3 r1 r2 =>
match intval r2 with
| Some n => make_mulimm n r1
| _ => (op, args)
end
| op_strength_reduction_case4 r1 r2 =>
match intval r2 with
| Some n =>
match Int.is_power2 n with
| Some l => if Int.ltu l (Int.repr 31)
then (Oshrximm l, r1 :: nil)
else (op, args)
| None => (op, args)
end
| None =>
(op, args)
end
| op_strength_reduction_case5 r1 r2 =>
match intval r2 with
| Some n =>
match Int.is_power2 n with
| Some l => make_shruimm l r1
| None => (op, args)
end
| None =>
(op, args)
end
| op_strength_reduction_case7 r1 r2 =>
match intval r2 with
| Some n =>
match Int.is_power2 n with
| Some l => (Oandimm (Int.sub n Int.one), r1 :: nil)
| None => (op, args)
end
| None =>
(op, args)
end
| op_strength_reduction_case8 r1 r2 =>
match intval r2 with
| Some n => make_andimm n r1
| _ => (op, args)
end
| op_strength_reduction_case9 r1 r2 =>
match intval r2 with
| Some n => make_orimm n r1
| _ => (op, args)
end
| op_strength_reduction_case10 r1 r2 =>
match intval r2 with
| Some n => make_xorimm n r1
| _ => (op, args)
end
| op_strength_reduction_case11 r1 r2 =>
match intval r2 with
| Some n =>
if Int.ltu n (Int.repr 32)
then make_shlimm n r1
else (op, args)
| _ => (op, args)
end
| op_strength_reduction_case12 r1 r2 =>
match intval r2 with
| Some n =>
if Int.ltu n (Int.repr 32)
then make_shrimm n r1
else (op, args)
| _ => (op, args)
end
| op_strength_reduction_case13 r1 r2 =>
match intval r2 with
| Some n =>
if Int.ltu n (Int.repr 32)
then make_shruimm n r1
else (op, args)
| _ => (op, args)
end
| op_strength_reduction_case14 addr args =>