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(* *********************************************************************)
(* *)
(* 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. *)
(* *)
(* *********************************************************************)
(** 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 (** No value possible, code is unreachable. *)
| Unknown: approx (** All values are possible,
no compile-time information is available. *)
| I: int -> approx (** A known integer value. *)
| F: float -> approx (** A known floating-point value. *)
| G: ident -> int -> approx
(** The value is the address of the given global
symbol plus the given integer offset. *)
| S: int -> approx. (** The value is the stack pointer plus the 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 [SelectOp] to avoid excessive
duplication of cases in proofs. *)
Nondetfunction eval_static_condition (cond: condition) (vl: list approx) :=
match cond, vl with
| Ccomp c, I n1 :: I n2 :: nil => Some(Int.cmp c n1 n2)
| Ccompu c, I n1 :: I n2 :: nil => Some(Int.cmpu c n1 n2)
| Ccompimm c n, I n1 :: nil => Some(Int.cmp c n1 n)
| Ccompuimm c n, I n1 :: nil => Some(Int.cmpu c n1 n)
| Ccompf c, F n1 :: F n2 :: nil => Some(Float.cmp c n1 n2)
| Cnotcompf c, F n1 :: F n2 :: nil => Some(negb(Float.cmp c n1 n2))
| Cmaskzero n, I n1 :: nil => Some(Int.eq (Int.and n1 n) Int.zero)
| Cmasknotzero n, I n1::nil => Some(negb(Int.eq (Int.and n1 n) Int.zero))
| _, _ => None
end.
Definition eval_static_condition_val (cond: condition) (vl: list approx) :=
match eval_static_condition cond vl with
| None => Unknown
| Some b => I(if b then Int.one else Int.zero)
end.
Definition eval_static_intoffloat (f: float) :=
match Float.intoffloat f with Some x => I x | None => Unknown end.
Nondetfunction eval_static_addressing (addr: addressing) (vl: list approx) :=
match addr, vl with
| Aindexed n, I n1::nil => I (Int.add n1 n)
| Aindexed n, G id ofs::nil => G id (Int.add ofs n)
| Aindexed n, S ofs::nil => S (Int.add ofs n)
| Aindexed2 n, I n1::I n2::nil => I (Int.add (Int.add n1 n2) n)
| Aindexed2 n, G id ofs::I n2::nil => G id (Int.add (Int.add ofs n2) n)
| Aindexed2 n, I n1::G id ofs::nil => G id (Int.add (Int.add ofs n1) n)
| Aindexed2 n, S ofs::I n2::nil => S (Int.add (Int.add ofs n2) n)
| Aindexed2 n, I n1::S ofs::nil => S (Int.add (Int.add ofs n1) n)
| Ascaled sc n, I n1::nil => I (Int.add (Int.mul n1 sc) n)
| Aindexed2scaled sc n, I n1::I n2::nil => I (Int.add n1 (Int.add (Int.mul n2 sc) n))
| Aindexed2scaled sc n, G id ofs::I n2::nil => G id (Int.add ofs (Int.add (Int.mul n2 sc) n))
| Aindexed2scaled sc n, S ofs::I n2::nil => S (Int.add ofs (Int.add (Int.mul n2 sc) n))
| Aglobal id ofs, nil => G id ofs
| Abased id ofs, I n1::nil => G id (Int.add ofs n1)
| Abasedscaled sc id ofs, I n1::nil => G id (Int.add ofs (Int.mul sc n1))
| Ainstack ofs, nil => S ofs
| _, _ => Unknown
end.
Nondetfunction eval_static_operation (op: operation) (vl: list approx) :=
match op, vl with
| Omove, v1::nil => v1
| Ointconst n, nil => I n
| Ofloatconst n, nil => F n
| Ocast8signed, I n1 :: nil => I(Int.sign_ext 8 n1)
| Ocast8unsigned, I n1 :: nil => I(Int.zero_ext 8 n1)
| Ocast16signed, I n1 :: nil => I(Int.sign_ext 16 n1)
| Ocast16unsigned, I n1 :: nil => I(Int.zero_ext 16 n1)
| Oneg, I n1 :: nil => I(Int.neg n1)
| Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2)
| Osub, G s1 n1 :: I n2 :: nil => G s1 (Int.sub n1 n2)
| Omul, I n1 :: I n2 :: nil => I(Int.mul n1 n2)
| Omulimm n, I n1 :: nil => I(Int.mul n1 n)
| Odiv, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
| Odivu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
| Omod, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.mods n1 n2)
| Omodu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.modu n1 n2)
| Oand, I n1 :: I n2 :: nil => I(Int.and n1 n2)
| Oandimm n, I n1 :: nil => I(Int.and n1 n)
| Oor, I n1 :: I n2 :: nil => I(Int.or n1 n2)
| Oorimm n, I n1 :: nil => I(Int.or n1 n)
| Oxor, I n1 :: I n2 :: nil => I(Int.xor n1 n2)
| Oxorimm n, I n1 :: nil => I(Int.xor n1 n)
| Oshl, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown
| Oshlimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shl n1 n) else Unknown
| Oshr, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown
| Oshrimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown
| Oshrximm n, I n1 :: nil => if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown
| Oshru, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown
| Oshruimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shru n1 n) else Unknown
| Ororimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.ror n1 n) else Unknown
| Olea mode, vl => eval_static_addressing mode vl
| Onegf, F n1 :: nil => F(Float.neg n1)
| Oabsf, F n1 :: nil => F(Float.abs n1)
| Oaddf, F n1 :: F n2 :: nil => F(Float.add n1 n2)
| Osubf, F n1 :: F n2 :: nil => F(Float.sub n1 n2)
| Omulf, F n1 :: F n2 :: nil => F(Float.mul n1 n2)
| Odivf, F n1 :: F n2 :: nil => F(Float.div n1 n2)
| Osingleoffloat, F n1 :: nil => F(Float.singleoffloat n1)
| Ointoffloat, F n1 :: nil => eval_static_intoffloat n1
| Ofloatofint, I n1 :: nil => F(Float.floatofint n1)
| Ocmp c, vl => eval_static_condition_val c 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.
Nondetfunction cond_strength_reduction
(cond: condition) (args: list reg) (vl: list approx) :=
match cond, args, vl with
| Ccomp c, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
(Ccompimm (swap_comparison c) n1, r2 :: nil)
| Ccomp c, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
(Ccompimm c n2, r1 :: nil)
| Ccompu c, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
(Ccompuimm (swap_comparison c) n1, r2 :: nil)
| Ccompu c, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
(Ccompuimm c n2, r1 :: nil)
| _, _, _ =>
(cond, args)
end.
Nondetfunction addr_strength_reduction
(addr: addressing) (args: list reg) (vl: list approx) :=
match addr, args, vl with
| Aindexed ofs, r1 :: nil, G symb n :: nil =>
(Aglobal symb (Int.add n ofs), nil)
| Aindexed ofs, r1 :: nil, S n :: nil =>
(Ainstack (Int.add n ofs), nil)
| Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil =>
(Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
| Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil =>
(Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
| Aindexed2 ofs, r1 :: r2 :: nil, S n1 :: I n2 :: nil =>
(Ainstack (Int.add (Int.add n1 n2) ofs), nil)
| Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: S n2 :: nil =>
(Ainstack (Int.add (Int.add n1 n2) ofs), nil)
| Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
(Abased symb (Int.add n1 ofs), r2 :: nil)
| Aindexed2 ofs, r1 :: r2 :: nil, v1 :: G symb n2 :: nil =>
(Abased symb (Int.add n2 ofs), r1 :: nil)
| Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
(Aindexed (Int.add n1 ofs), r2 :: nil)
| Aindexed2 ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
(Aindexed (Int.add n2 ofs), r1 :: nil)
| Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil =>
(Aglobal symb (Int.add (Int.add n1 (Int.mul n2 sc)) ofs), nil)
| Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
(Abasedscaled sc symb (Int.add n1 ofs), r2 :: nil)
| Aindexed2scaled sc ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
(Aindexed (Int.add (Int.mul n2 sc) ofs), r1 :: nil)
| Abased id ofs, r1 :: nil, I n1 :: nil =>
(Aglobal id (Int.add ofs n1), nil)
| Abasedscaled sc id ofs, r1 :: nil, I n1 :: nil =>
(Aglobal id (Int.add ofs (Int.mul sc n1)), nil)
| _, _ =>
(addr, args)
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).
Definition make_divimm n (r1 r2: reg) :=
match Int.is_power2 n with
| Some l => if Int.ltu l (Int.repr 31)
then (Oshrximm l, r1 :: nil)
else (Odiv, r1 :: r2 :: nil)
| None => (Odiv, r1 :: r2 :: nil)
end.
Definition make_divuimm n (r1 r2: reg) :=
match Int.is_power2 n with
| Some l => make_shruimm l r1
| None => (Odivu, r1 :: r2 :: nil)
end.
Definition make_moduimm n (r1 r2: reg) :=
match Int.is_power2 n with
| Some l => (Oandimm (Int.sub n Int.one), r1 :: nil)
| None => (Omodu, r1 :: r2 :: nil)
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. *)
Nondetfunction op_strength_reduction
(op: operation) (args: list reg) (vl: list approx) :=
match op, args, vl with
| Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm (Int.neg n2) r1
| Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_mulimm n2 r1
| Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divimm n2 r1 r2
| Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divuimm n2 r1 r2
| Omodu, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_moduimm n2 r1 r2
| Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_andimm n2 r1
| Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_orimm n2 r1
| Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_xorimm n2 r1
| Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shlimm n2 r1
| Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shrimm n2 r1
| Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shruimm n2 r1
| Olea addr, args, vl =>
let (addr', args') := addr_strength_reduction addr args vl in
(Olea addr', args')
| Ocmp c, args, vl =>
let (c', args') := cond_strength_reduction c args vl in
(Ocmp c', args')
| _, _, _ => (op, args)
end.
Nondetfunction builtin_strength_reduction
(ef: external_function) (args: list reg) (vl: list approx) :=
match ef, args, vl with
| EF_vload chunk, r1 :: nil, G symb n1 :: nil =>
(EF_vload_global chunk symb n1, nil)
| EF_vstore chunk, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
(EF_vstore_global chunk symb n1, r2 :: nil)
| _, _, _ =>
(ef, args)
end.
End STRENGTH_REDUCTION.
|
