dependence analysis
Patch from Preston Briggs <preston.briggs@gmail.com>.
This is an updated version of the dependence-analysis patch, including an MIV
test based on Banerjee's inequalities.
It's a fairly complete implementation of the paper
Practical Dependence Testing
Gina Goff, Ken Kennedy, and Chau-Wen Tseng
PLDI 1991
It cannot yet propagate constraints between coupled RDIV subscripts (discussed
in Section 5.3.2 of the paper).
It's organized as a FunctionPass with a single entry point that supports testing
for dependence between two instructions in a function. If there's no dependence,
it returns null. If there's a dependence, it returns a pointer to a Dependence
which can be queried about details (what kind of dependence, is it loop
independent, direction and distance vector entries, etc). I haven't included
every imaginable feature, but there's a good selection that should be adequate
for supporting many loop transformations. Of course, it can be extended as
necessary.
Included in the patch file are many test cases, commented with C code showing
the loops and array references.
git-svn-id: https://llvm.org/svn/llvm-project/llvm/trunk@165708 91177308-0d34-0410-b5e6-96231b3b80d8
diff --git a/lib/Analysis/DependenceAnalysis.cpp b/lib/Analysis/DependenceAnalysis.cpp
new file mode 100644
index 0000000..c7bec43
--- /dev/null
+++ b/lib/Analysis/DependenceAnalysis.cpp
@@ -0,0 +1,3781 @@
+//===-- DependenceAnalysis.cpp - DA Implementation --------------*- C++ -*-===//
+//
+// The LLVM Compiler Infrastructure
+//
+// This file is distributed under the University of Illinois Open Source
+// License. See LICENSE.TXT for details.
+//
+//===----------------------------------------------------------------------===//
+//
+// DependenceAnalysis is an LLVM pass that analyses dependences between memory
+// accesses. Currently, it is an (incomplete) implementation of the approach
+// described in
+//
+// Practical Dependence Testing
+// Goff, Kennedy, Tseng
+// PLDI 1991
+//
+// There's a single entry point that analyzes the dependence between a pair
+// of memory references in a function, returning either NULL, for no dependence,
+// or a more-or-less detailed description of the dependence between them.
+//
+// Currently, the implementation cannot propagate constraints between
+// coupled RDIV subscripts and lacks a multi-subscript MIV test.
+// Both of these are conservative weaknesses;
+// that is, not a source of correctness problems.
+//
+// The implementation depends on the GEP instruction to
+// differentiate subscripts. Since Clang linearizes subscripts
+// for most arrays, we give up some precision (though the existing MIV tests
+// will help). We trust that the GEP instruction will eventually be extended.
+// In the meantime, we should explore Maslov's ideas about delinearization.
+//
+// We should pay some careful attention to the possibility of integer overflow
+// in the implementation of the various tests. This could happen with Add,
+// Subtract, or Multiply, with both APInt's and SCEV's.
+//
+// Some non-linear subscript pairs can be handled by the GCD test
+// (and perhaps other tests).
+// Should explore how often these things occur.
+//
+// Finally, it seems like certain test cases expose weaknesses in the SCEV
+// simplification, especially in the handling of sign and zero extensions.
+// It could be useful to spend time exploring these.
+//
+// Please note that this is work in progress and the interface is subject to
+// change.
+//
+//===----------------------------------------------------------------------===//
+// //
+// In memory of Ken Kennedy, 1945 - 2007 //
+// //
+//===----------------------------------------------------------------------===//
+
+#define DEBUG_TYPE "da"
+
+#include "llvm/Analysis/DependenceAnalysis.h"
+#include "llvm/ADT/Statistic.h"
+#include "llvm/Instructions.h"
+#include "llvm/Operator.h"
+#include "llvm/Analysis/ValueTracking.h"
+#include "llvm/Support/Debug.h"
+#include "llvm/Support/ErrorHandling.h"
+#include "llvm/Support/InstIterator.h"
+
+using namespace llvm;
+
+//===----------------------------------------------------------------------===//
+// statistics
+
+STATISTIC(TotalArrayPairs, "Array pairs tested");
+STATISTIC(SeparableSubscriptPairs, "Separable subscript pairs");
+STATISTIC(CoupledSubscriptPairs, "Coupled subscript pairs");
+STATISTIC(NonlinearSubscriptPairs, "Nonlinear subscript pairs");
+STATISTIC(ZIVapplications, "ZIV applications");
+STATISTIC(ZIVindependence, "ZIV independence");
+STATISTIC(StrongSIVapplications, "Strong SIV applications");
+STATISTIC(StrongSIVsuccesses, "Strong SIV successes");
+STATISTIC(StrongSIVindependence, "Strong SIV independence");
+STATISTIC(WeakCrossingSIVapplications, "Weak-Crossing SIV applications");
+STATISTIC(WeakCrossingSIVsuccesses, "Weak-Crossing SIV successes");
+STATISTIC(WeakCrossingSIVindependence, "Weak-Crossing SIV independence");
+STATISTIC(ExactSIVapplications, "Exact SIV applications");
+STATISTIC(ExactSIVsuccesses, "Exact SIV successes");
+STATISTIC(ExactSIVindependence, "Exact SIV independence");
+STATISTIC(WeakZeroSIVapplications, "Weak-Zero SIV applications");
+STATISTIC(WeakZeroSIVsuccesses, "Weak-Zero SIV successes");
+STATISTIC(WeakZeroSIVindependence, "Weak-Zero SIV independence");
+STATISTIC(ExactRDIVapplications, "Exact RDIV applications");
+STATISTIC(ExactRDIVindependence, "Exact RDIV independence");
+STATISTIC(SymbolicRDIVapplications, "Symbolic RDIV applications");
+STATISTIC(SymbolicRDIVindependence, "Symbolic RDIV independence");
+STATISTIC(DeltaApplications, "Delta applications");
+STATISTIC(DeltaSuccesses, "Delta successes");
+STATISTIC(DeltaIndependence, "Delta independence");
+STATISTIC(DeltaPropagations, "Delta propagations");
+STATISTIC(GCDapplications, "GCD applications");
+STATISTIC(GCDsuccesses, "GCD successes");
+STATISTIC(GCDindependence, "GCD independence");
+STATISTIC(BanerjeeApplications, "Banerjee applications");
+STATISTIC(BanerjeeIndependence, "Banerjee independence");
+STATISTIC(BanerjeeSuccesses, "Banerjee successes");
+
+//===----------------------------------------------------------------------===//
+// basics
+
+INITIALIZE_PASS_BEGIN(DependenceAnalysis, "da",
+ "Dependence Analysis", true, true)
+INITIALIZE_PASS_DEPENDENCY(LoopInfo)
+INITIALIZE_PASS_DEPENDENCY(ScalarEvolution)
+INITIALIZE_AG_DEPENDENCY(AliasAnalysis)
+INITIALIZE_PASS_END(DependenceAnalysis, "da",
+ "Dependence Analysis", true, true)
+
+char DependenceAnalysis::ID = 0;
+
+
+FunctionPass *llvm::createDependenceAnalysisPass() {
+ return new DependenceAnalysis();
+}
+
+
+bool DependenceAnalysis::runOnFunction(Function &F) {
+ this->F = &F;
+ AA = &getAnalysis<AliasAnalysis>();
+ SE = &getAnalysis<ScalarEvolution>();
+ LI = &getAnalysis<LoopInfo>();
+ return false;
+}
+
+
+void DependenceAnalysis::releaseMemory() {
+}
+
+
+void DependenceAnalysis::getAnalysisUsage(AnalysisUsage &AU) const {
+ AU.setPreservesAll();
+ AU.addRequiredTransitive<AliasAnalysis>();
+ AU.addRequiredTransitive<ScalarEvolution>();
+ AU.addRequiredTransitive<LoopInfo>();
+}
+
+
+// Used to test the dependence analyzer.
+// Looks through the function, noting the first store instruction
+// and the first load instruction
+// (which always follows the first load in our tests).
+// Calls depends() and prints out the result.
+// Ignores all other instructions.
+static
+void dumpExampleDependence(raw_ostream &OS, Function *F,
+ DependenceAnalysis *DA) {
+ for (inst_iterator SrcI = inst_begin(F), SrcE = inst_end(F);
+ SrcI != SrcE; ++SrcI) {
+ if (const StoreInst *Src = dyn_cast<StoreInst>(&*SrcI)) {
+ for (inst_iterator DstI = SrcI, DstE = inst_end(F);
+ DstI != DstE; ++DstI) {
+ if (const LoadInst *Dst = dyn_cast<LoadInst>(&*DstI)) {
+ OS << "da analyze - ";
+ if (Dependence *D = DA->depends(Src, Dst, true)) {
+ D->dump(OS);
+ for (unsigned Level = 1; Level <= D->getLevels(); Level++) {
+ if (D->isSplitable(Level)) {
+ OS << "da analyze - split level = " << Level;
+ OS << ", iteration = " << *DA->getSplitIteration(D, Level);
+ OS << "!\n";
+ }
+ }
+ delete D;
+ }
+ else
+ OS << "none!\n";
+ return;
+ }
+ }
+ }
+ }
+}
+
+
+void DependenceAnalysis::print(raw_ostream &OS, const Module*) const {
+ dumpExampleDependence(OS, F, const_cast<DependenceAnalysis *>(this));
+}
+
+//===----------------------------------------------------------------------===//
+// Dependence methods
+
+// Returns true if this is an input dependence.
+bool Dependence::isInput() const {
+ return Src->mayReadFromMemory() && Dst->mayReadFromMemory();
+}
+
+
+// Returns true if this is an output dependence.
+bool Dependence::isOutput() const {
+ return Src->mayWriteToMemory() && Dst->mayWriteToMemory();
+}
+
+
+// Returns true if this is an flow (aka true) dependence.
+bool Dependence::isFlow() const {
+ return Src->mayWriteToMemory() && Dst->mayReadFromMemory();
+}
+
+
+// Returns true if this is an anti dependence.
+bool Dependence::isAnti() const {
+ return Src->mayReadFromMemory() && Dst->mayWriteToMemory();
+}
+
+
+// Returns true if a particular level is scalar; that is,
+// if no subscript in the source or destination mention the induction
+// variable associated with the loop at this level.
+// Leave this out of line, so it will serve as a virtual method anchor
+bool Dependence::isScalar(unsigned level) const {
+ return false;
+}
+
+
+//===----------------------------------------------------------------------===//
+// FullDependence methods
+
+FullDependence::FullDependence(const Instruction *Source,
+ const Instruction *Destination,
+ bool PossiblyLoopIndependent,
+ unsigned CommonLevels) :
+ Dependence(Source, Destination),
+ Levels(CommonLevels),
+ LoopIndependent(PossiblyLoopIndependent) {
+ Consistent = true;
+ DV = CommonLevels ? new DVEntry[CommonLevels] : NULL;
+}
+
+// The rest are simple getters that hide the implementation.
+
+// getDirection - Returns the direction associated with a particular level.
+unsigned FullDependence::getDirection(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].Direction;
+}
+
+
+// Returns the distance (or NULL) associated with a particular level.
+const SCEV *FullDependence::getDistance(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].Distance;
+}
+
+
+// Returns true if a particular level is scalar; that is,
+// if no subscript in the source or destination mention the induction
+// variable associated with the loop at this level.
+bool FullDependence::isScalar(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].Scalar;
+}
+
+
+// Returns true if peeling the first iteration from this loop
+// will break this dependence.
+bool FullDependence::isPeelFirst(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].PeelFirst;
+}
+
+
+// Returns true if peeling the last iteration from this loop
+// will break this dependence.
+bool FullDependence::isPeelLast(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].PeelLast;
+}
+
+
+// Returns true if splitting this loop will break the dependence.
+bool FullDependence::isSplitable(unsigned Level) const {
+ assert(0 < Level && Level <= Levels && "Level out of range");
+ return DV[Level - 1].Splitable;
+}
+
+
+//===----------------------------------------------------------------------===//
+// DependenceAnalysis::Constraint methods
+
+// If constraint is a point <X, Y>, returns X.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getX() const {
+ assert(Kind == Point && "Kind should be Point");
+ return A;
+}
+
+
+// If constraint is a point <X, Y>, returns Y.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getY() const {
+ assert(Kind == Point && "Kind should be Point");
+ return B;
+}
+
+
+// If constraint is a line AX + BY = C, returns A.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getA() const {
+ assert((Kind == Line || Kind == Distance) &&
+ "Kind should be Line (or Distance)");
+ return A;
+}
+
+
+// If constraint is a line AX + BY = C, returns B.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getB() const {
+ assert((Kind == Line || Kind == Distance) &&
+ "Kind should be Line (or Distance)");
+ return B;
+}
+
+
+// If constraint is a line AX + BY = C, returns C.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getC() const {
+ assert((Kind == Line || Kind == Distance) &&
+ "Kind should be Line (or Distance)");
+ return C;
+}
+
+
+// If constraint is a distance, returns D.
+// Otherwise assert.
+const SCEV *DependenceAnalysis::Constraint::getD() const {
+ assert(Kind == Distance && "Kind should be Distance");
+ return SE->getNegativeSCEV(C);
+}
+
+
+// Returns the loop associated with this constraint.
+const Loop *DependenceAnalysis::Constraint::getAssociatedLoop() const {
+ assert((Kind == Distance || Kind == Line || Kind == Point) &&
+ "Kind should be Distance, Line, or Point");
+ return AssociatedLoop;
+}
+
+
+void DependenceAnalysis::Constraint::setPoint(const SCEV *X,
+ const SCEV *Y,
+ const Loop *CurLoop) {
+ Kind = Point;
+ A = X;
+ B = Y;
+ AssociatedLoop = CurLoop;
+}
+
+
+void DependenceAnalysis::Constraint::setLine(const SCEV *AA,
+ const SCEV *BB,
+ const SCEV *CC,
+ const Loop *CurLoop) {
+ Kind = Line;
+ A = AA;
+ B = BB;
+ C = CC;
+ AssociatedLoop = CurLoop;
+}
+
+
+void DependenceAnalysis::Constraint::setDistance(const SCEV *D,
+ const Loop *CurLoop) {
+ Kind = Distance;
+ A = SE->getConstant(D->getType(), 1);
+ B = SE->getNegativeSCEV(A);
+ C = SE->getNegativeSCEV(D);
+ AssociatedLoop = CurLoop;
+}
+
+
+void DependenceAnalysis::Constraint::setEmpty() {
+ Kind = Empty;
+}
+
+
+void DependenceAnalysis::Constraint::setAny(ScalarEvolution *NewSE) {
+ SE = NewSE;
+ Kind = Any;
+}
+
+
+// For debugging purposes. Dumps the constraint out to OS.
+void DependenceAnalysis::Constraint::dump(raw_ostream &OS) const {
+ if (isEmpty())
+ OS << " Empty\n";
+ else if (isAny())
+ OS << " Any\n";
+ else if (isPoint())
+ OS << " Point is <" << *getX() << ", " << *getY() << ">\n";
+ else if (isDistance())
+ OS << " Distance is " << *getD() <<
+ " (" << *getA() << "*X + " << *getB() << "*Y = " << *getC() << ")\n";
+ else if (isLine())
+ OS << " Line is " << *getA() << "*X + " <<
+ *getB() << "*Y = " << *getC() << "\n";
+ else
+ llvm_unreachable("unknown constraint type in Constraint::dump");
+}
+
+
+// Updates X with the intersection
+// of the Constraints X and Y. Returns true if X has changed.
+// Corresponds to Figure 4 from the paper
+//
+// Practical Dependence Testing
+// Goff, Kennedy, Tseng
+// PLDI 1991
+bool DependenceAnalysis::intersectConstraints(Constraint *X,
+ const Constraint *Y) {
+ ++DeltaApplications;
+ DEBUG(dbgs() << "\tintersect constraints\n");
+ DEBUG(dbgs() << "\t X ="; X->dump(dbgs()));
+ DEBUG(dbgs() << "\t Y ="; Y->dump(dbgs()));
+ assert(!Y->isPoint() && "Y must not be a Point");
+ if (X->isAny()) {
+ if (Y->isAny())
+ return false;
+ *X = *Y;
+ return true;
+ }
+ if (X->isEmpty())
+ return false;
+ if (Y->isEmpty()) {
+ X->setEmpty();
+ return true;
+ }
+
+ if (X->isDistance() && Y->isDistance()) {
+ DEBUG(dbgs() << "\t intersect 2 distances\n");
+ if (isKnownPredicate(CmpInst::ICMP_EQ, X->getD(), Y->getD()))
+ return false;
+ if (isKnownPredicate(CmpInst::ICMP_NE, X->getD(), Y->getD())) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ // Hmmm, interesting situation.
+ // I guess if either is constant, keep it and ignore the other.
+ if (isa<SCEVConstant>(Y->getD())) {
+ *X = *Y;
+ return true;
+ }
+ return false;
+ }
+
+ // At this point, the pseudo-code in Figure 4 of the paper
+ // checks if (X->isPoint() && Y->isPoint()).
+ // This case can't occur in our implementation,
+ // since a Point can only arise as the result of intersecting
+ // two Line constraints, and the right-hand value, Y, is never
+ // the result of an intersection.
+ assert(!(X->isPoint() && Y->isPoint()) &&
+ "We shouldn't ever see X->isPoint() && Y->isPoint()");
+
+ if (X->isLine() && Y->isLine()) {
+ DEBUG(dbgs() << "\t intersect 2 lines\n");
+ const SCEV *Prod1 = SE->getMulExpr(X->getA(), Y->getB());
+ const SCEV *Prod2 = SE->getMulExpr(X->getB(), Y->getA());
+ if (isKnownPredicate(CmpInst::ICMP_EQ, Prod1, Prod2)) {
+ // slopes are equal, so lines are parallel
+ DEBUG(dbgs() << "\t\tsame slope\n");
+ Prod1 = SE->getMulExpr(X->getC(), Y->getB());
+ Prod2 = SE->getMulExpr(X->getB(), Y->getC());
+ if (isKnownPredicate(CmpInst::ICMP_EQ, Prod1, Prod2))
+ return false;
+ if (isKnownPredicate(CmpInst::ICMP_NE, Prod1, Prod2)) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ return false;
+ }
+ if (isKnownPredicate(CmpInst::ICMP_NE, Prod1, Prod2)) {
+ // slopes differ, so lines intersect
+ DEBUG(dbgs() << "\t\tdifferent slopes\n");
+ const SCEV *C1B2 = SE->getMulExpr(X->getC(), Y->getB());
+ const SCEV *C1A2 = SE->getMulExpr(X->getC(), Y->getA());
+ const SCEV *C2B1 = SE->getMulExpr(Y->getC(), X->getB());
+ const SCEV *C2A1 = SE->getMulExpr(Y->getC(), X->getA());
+ const SCEV *A1B2 = SE->getMulExpr(X->getA(), Y->getB());
+ const SCEV *A2B1 = SE->getMulExpr(Y->getA(), X->getB());
+ const SCEVConstant *C1A2_C2A1 =
+ dyn_cast<SCEVConstant>(SE->getMinusSCEV(C1A2, C2A1));
+ const SCEVConstant *C1B2_C2B1 =
+ dyn_cast<SCEVConstant>(SE->getMinusSCEV(C1B2, C2B1));
+ const SCEVConstant *A1B2_A2B1 =
+ dyn_cast<SCEVConstant>(SE->getMinusSCEV(A1B2, A2B1));
+ const SCEVConstant *A2B1_A1B2 =
+ dyn_cast<SCEVConstant>(SE->getMinusSCEV(A2B1, A1B2));
+ if (!C1B2_C2B1 || !C1A2_C2A1 ||
+ !A1B2_A2B1 || !A2B1_A1B2)
+ return false;
+ APInt Xtop = C1B2_C2B1->getValue()->getValue();
+ APInt Xbot = A1B2_A2B1->getValue()->getValue();
+ APInt Ytop = C1A2_C2A1->getValue()->getValue();
+ APInt Ybot = A2B1_A1B2->getValue()->getValue();
+ DEBUG(dbgs() << "\t\tXtop = " << Xtop << "\n");
+ DEBUG(dbgs() << "\t\tXbot = " << Xbot << "\n");
+ DEBUG(dbgs() << "\t\tYtop = " << Ytop << "\n");
+ DEBUG(dbgs() << "\t\tYbot = " << Ybot << "\n");
+ APInt Xq = Xtop; // these need to be initialized, even
+ APInt Xr = Xtop; // though they're just going to be overwritten
+ APInt::sdivrem(Xtop, Xbot, Xq, Xr);
+ APInt Yq = Ytop;
+ APInt Yr = Ytop;;
+ APInt::sdivrem(Ytop, Ybot, Yq, Yr);
+ if (Xr != 0 || Yr != 0) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ DEBUG(dbgs() << "\t\tX = " << Xq << ", Y = " << Yq << "\n");
+ if (Xq.slt(0) || Yq.slt(0)) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ if (const SCEVConstant *CUB =
+ collectConstantUpperBound(X->getAssociatedLoop(), Prod1->getType())) {
+ APInt UpperBound = CUB->getValue()->getValue();
+ DEBUG(dbgs() << "\t\tupper bound = " << UpperBound << "\n");
+ if (Xq.sgt(UpperBound) || Yq.sgt(UpperBound)) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ }
+ X->setPoint(SE->getConstant(Xq),
+ SE->getConstant(Yq),
+ X->getAssociatedLoop());
+ ++DeltaSuccesses;
+ return true;
+ }
+ return false;
+ }
+
+ // if (X->isLine() && Y->isPoint()) This case can't occur.
+ assert(!(X->isLine() && Y->isPoint()) && "This case should never occur");
+
+ if (X->isPoint() && Y->isLine()) {
+ DEBUG(dbgs() << "\t intersect Point and Line\n");
+ const SCEV *A1X1 = SE->getMulExpr(Y->getA(), X->getX());
+ const SCEV *B1Y1 = SE->getMulExpr(Y->getB(), X->getY());
+ const SCEV *Sum = SE->getAddExpr(A1X1, B1Y1);
+ if (isKnownPredicate(CmpInst::ICMP_EQ, Sum, Y->getC()))
+ return false;
+ if (isKnownPredicate(CmpInst::ICMP_NE, Sum, Y->getC())) {
+ X->setEmpty();
+ ++DeltaSuccesses;
+ return true;
+ }
+ return false;
+ }
+
+ llvm_unreachable("shouldn't reach the end of Constraint intersection");
+ return false;
+}
+
+
+//===----------------------------------------------------------------------===//
+// DependenceAnalysis methods
+
+// For debugging purposes. Dumps a dependence to OS.
+void Dependence::dump(raw_ostream &OS) const {
+ bool Splitable = false;
+ if (isConfused())
+ OS << "confused";
+ else {
+ if (isConsistent())
+ OS << "consistent ";
+ if (isFlow())
+ OS << "flow";
+ else if (isOutput())
+ OS << "output";
+ else if (isAnti())
+ OS << "anti";
+ else if (isInput())
+ OS << "input";
+ unsigned Levels = getLevels();
+ if (Levels) {
+ OS << " [";
+ for (unsigned II = 1; II <= Levels; ++II) {
+ if (isSplitable(II))
+ Splitable = true;
+ if (isPeelFirst(II))
+ OS << 'p';
+ const SCEV *Distance = getDistance(II);
+ if (Distance)
+ OS << *Distance;
+ else if (isScalar(II))
+ OS << "S";
+ else {
+ unsigned Direction = getDirection(II);
+ if (Direction == DVEntry::ALL)
+ OS << "*";
+ else {
+ if (Direction & DVEntry::LT)
+ OS << "<";
+ if (Direction & DVEntry::EQ)
+ OS << "=";
+ if (Direction & DVEntry::GT)
+ OS << ">";
+ }
+ }
+ if (isPeelLast(II))
+ OS << 'p';
+ if (II < Levels)
+ OS << " ";
+ }
+ if (isLoopIndependent())
+ OS << "|<";
+ OS << "]";
+ if (Splitable)
+ OS << " splitable";
+ }
+ }
+ OS << "!\n";
+}
+
+
+
+static
+AliasAnalysis::AliasResult underlyingObjectsAlias(AliasAnalysis *AA,
+ const Value *A,
+ const Value *B) {
+ const Value *AObj = GetUnderlyingObject(A);
+ const Value *BObj = GetUnderlyingObject(B);
+ return AA->alias(AObj, AA->getTypeStoreSize(AObj->getType()),
+ BObj, AA->getTypeStoreSize(BObj->getType()));
+}
+
+
+// Returns true if the load or store can be analyzed. Atomic and volatile
+// operations have properties which this analysis does not understand.
+static
+bool isLoadOrStore(const Instruction *I) {
+ if (const LoadInst *LI = dyn_cast<LoadInst>(I))
+ return LI->isUnordered();
+ else if (const StoreInst *SI = dyn_cast<StoreInst>(I))
+ return SI->isUnordered();
+ return false;
+}
+
+
+static
+const Value *getPointerOperand(const Instruction *I) {
+ if (const LoadInst *LI = dyn_cast<LoadInst>(I))
+ return LI->getPointerOperand();
+ if (const StoreInst *SI = dyn_cast<StoreInst>(I))
+ return SI->getPointerOperand();
+ llvm_unreachable("Value is not load or store instruction");
+ return 0;
+}
+
+
+// Examines the loop nesting of the Src and Dst
+// instructions and establishes their shared loops. Sets the variables
+// CommonLevels, SrcLevels, and MaxLevels.
+// The source and destination instructions needn't be contained in the same
+// loop. The routine establishNestingLevels finds the level of most deeply
+// nested loop that contains them both, CommonLevels. An instruction that's
+// not contained in a loop is at level = 0. MaxLevels is equal to the level
+// of the source plus the level of the destination, minus CommonLevels.
+// This lets us allocate vectors MaxLevels in length, with room for every
+// distinct loop referenced in both the source and destination subscripts.
+// The variable SrcLevels is the nesting depth of the source instruction.
+// It's used to help calculate distinct loops referenced by the destination.
+// Here's the map from loops to levels:
+// 0 - unused
+// 1 - outermost common loop
+// ... - other common loops
+// CommonLevels - innermost common loop
+// ... - loops containing Src but not Dst
+// SrcLevels - innermost loop containing Src but not Dst
+// ... - loops containing Dst but not Src
+// MaxLevels - innermost loops containing Dst but not Src
+// Consider the follow code fragment:
+// for (a = ...) {
+// for (b = ...) {
+// for (c = ...) {
+// for (d = ...) {
+// A[] = ...;
+// }
+// }
+// for (e = ...) {
+// for (f = ...) {
+// for (g = ...) {
+// ... = A[];
+// }
+// }
+// }
+// }
+// }
+// If we're looking at the possibility of a dependence between the store
+// to A (the Src) and the load from A (the Dst), we'll note that they
+// have 2 loops in common, so CommonLevels will equal 2 and the direction
+// vector for Result will have 2 entries. SrcLevels = 4 and MaxLevels = 7.
+// A map from loop names to loop numbers would look like
+// a - 1
+// b - 2 = CommonLevels
+// c - 3
+// d - 4 = SrcLevels
+// e - 5
+// f - 6
+// g - 7 = MaxLevels
+void DependenceAnalysis::establishNestingLevels(const Instruction *Src,
+ const Instruction *Dst) {
+ const BasicBlock *SrcBlock = Src->getParent();
+ const BasicBlock *DstBlock = Dst->getParent();
+ unsigned SrcLevel = LI->getLoopDepth(SrcBlock);
+ unsigned DstLevel = LI->getLoopDepth(DstBlock);
+ const Loop *SrcLoop = LI->getLoopFor(SrcBlock);
+ const Loop *DstLoop = LI->getLoopFor(DstBlock);
+ SrcLevels = SrcLevel;
+ MaxLevels = SrcLevel + DstLevel;
+ while (SrcLevel > DstLevel) {
+ SrcLoop = SrcLoop->getParentLoop();
+ SrcLevel--;
+ }
+ while (DstLevel > SrcLevel) {
+ DstLoop = DstLoop->getParentLoop();
+ DstLevel--;
+ }
+ while (SrcLoop != DstLoop) {
+ SrcLoop = SrcLoop->getParentLoop();
+ DstLoop = DstLoop->getParentLoop();
+ SrcLevel--;
+ }
+ CommonLevels = SrcLevel;
+ MaxLevels -= CommonLevels;
+}
+
+
+// Given one of the loops containing the source, return
+// its level index in our numbering scheme.
+unsigned DependenceAnalysis::mapSrcLoop(const Loop *SrcLoop) const {
+ return SrcLoop->getLoopDepth();
+}
+
+
+// Given one of the loops containing the destination,
+// return its level index in our numbering scheme.
+unsigned DependenceAnalysis::mapDstLoop(const Loop *DstLoop) const {
+ unsigned D = DstLoop->getLoopDepth();
+ if (D > CommonLevels)
+ return D - CommonLevels + SrcLevels;
+ else
+ return D;
+}
+
+
+// Returns true if Expression is loop invariant in LoopNest.
+bool DependenceAnalysis::isLoopInvariant(const SCEV *Expression,
+ const Loop *LoopNest) const {
+ if (!LoopNest)
+ return true;
+ return SE->isLoopInvariant(Expression, LoopNest) &&
+ isLoopInvariant(Expression, LoopNest->getParentLoop());
+}
+
+
+
+// Finds the set of loops from the LoopNest that
+// have a level <= CommonLevels and are referred to by the SCEV Expression.
+void DependenceAnalysis::collectCommonLoops(const SCEV *Expression,
+ const Loop *LoopNest,
+ SmallBitVector &Loops) const {
+ while (LoopNest) {
+ unsigned Level = LoopNest->getLoopDepth();
+ if (Level <= CommonLevels && !SE->isLoopInvariant(Expression, LoopNest))
+ Loops.set(Level);
+ LoopNest = LoopNest->getParentLoop();
+ }
+}
+
+
+// removeMatchingExtensions - Examines a subscript pair.
+// If the source and destination are identically sign (or zero)
+// extended, it strips off the extension in an effect to simplify
+// the actual analysis.
+void DependenceAnalysis::removeMatchingExtensions(Subscript *Pair) {
+ const SCEV *Src = Pair->Src;
+ const SCEV *Dst = Pair->Dst;
+ if ((isa<SCEVZeroExtendExpr>(Src) && isa<SCEVZeroExtendExpr>(Dst)) ||
+ (isa<SCEVSignExtendExpr>(Src) && isa<SCEVSignExtendExpr>(Dst))) {
+ const SCEVCastExpr *SrcCast = cast<SCEVCastExpr>(Src);
+ const SCEVCastExpr *DstCast = cast<SCEVCastExpr>(Dst);
+ if (SrcCast->getType() == DstCast->getType()) {
+ Pair->Src = SrcCast->getOperand();
+ Pair->Dst = DstCast->getOperand();
+ }
+ }
+}
+
+
+// Examine the scev and return true iff it's linear.
+// Collect any loops mentioned in the set of "Loops".
+bool DependenceAnalysis::checkSrcSubscript(const SCEV *Src,
+ const Loop *LoopNest,
+ SmallBitVector &Loops) {
+ const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Src);
+ if (!AddRec)
+ return isLoopInvariant(Src, LoopNest);
+ const SCEV *Start = AddRec->getStart();
+ const SCEV *Step = AddRec->getStepRecurrence(*SE);
+ if (!isLoopInvariant(Step, LoopNest))
+ return false;
+ Loops.set(mapSrcLoop(AddRec->getLoop()));
+ return checkSrcSubscript(Start, LoopNest, Loops);
+}
+
+
+
+// Examine the scev and return true iff it's linear.
+// Collect any loops mentioned in the set of "Loops".
+bool DependenceAnalysis::checkDstSubscript(const SCEV *Dst,
+ const Loop *LoopNest,
+ SmallBitVector &Loops) {
+ const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Dst);
+ if (!AddRec)
+ return isLoopInvariant(Dst, LoopNest);
+ const SCEV *Start = AddRec->getStart();
+ const SCEV *Step = AddRec->getStepRecurrence(*SE);
+ if (!isLoopInvariant(Step, LoopNest))
+ return false;
+ Loops.set(mapDstLoop(AddRec->getLoop()));
+ return checkDstSubscript(Start, LoopNest, Loops);
+}
+
+
+// Examines the subscript pair (the Src and Dst SCEVs)
+// and classifies it as either ZIV, SIV, RDIV, MIV, or Nonlinear.
+// Collects the associated loops in a set.
+DependenceAnalysis::Subscript::ClassificationKind
+DependenceAnalysis::classifyPair(const SCEV *Src, const Loop *SrcLoopNest,
+ const SCEV *Dst, const Loop *DstLoopNest,
+ SmallBitVector &Loops) {
+ SmallBitVector SrcLoops(MaxLevels + 1);
+ SmallBitVector DstLoops(MaxLevels + 1);
+ if (!checkSrcSubscript(Src, SrcLoopNest, SrcLoops))
+ return Subscript::NonLinear;
+ if (!checkDstSubscript(Dst, DstLoopNest, DstLoops))
+ return Subscript::NonLinear;
+ Loops = SrcLoops;
+ Loops |= DstLoops;
+ unsigned N = Loops.count();
+ if (N == 0)
+ return Subscript::ZIV;
+ if (N == 1)
+ return Subscript::SIV;
+ if (N == 2 && (SrcLoops.count() == 0 ||
+ DstLoops.count() == 0 ||
+ (SrcLoops.count() == 1 && DstLoops.count() == 1)))
+ return Subscript::RDIV;
+ return Subscript::MIV;
+}
+
+
+// A wrapper around SCEV::isKnownPredicate.
+// Looks for cases where we're interested in comparing for equality.
+// If both X and Y have been identically sign or zero extended,
+// it strips off the (confusing) extensions before invoking
+// SCEV::isKnownPredicate. Perhaps, someday, the ScalarEvolution package
+// will be similarly updated.
+//
+// If SCEV::isKnownPredicate can't prove the predicate,
+// we try simple subtraction, which seems to help in some cases
+// involving symbolics.
+bool DependenceAnalysis::isKnownPredicate(ICmpInst::Predicate Pred,
+ const SCEV *X,
+ const SCEV *Y) const {
+ if (Pred == CmpInst::ICMP_EQ ||
+ Pred == CmpInst::ICMP_NE) {
+ if ((isa<SCEVSignExtendExpr>(X) &&
+ isa<SCEVSignExtendExpr>(Y)) ||
+ (isa<SCEVZeroExtendExpr>(X) &&
+ isa<SCEVZeroExtendExpr>(Y))) {
+ const SCEVCastExpr *CX = cast<SCEVCastExpr>(X);
+ const SCEVCastExpr *CY = cast<SCEVCastExpr>(Y);
+ const SCEV *Xop = CX->getOperand();
+ const SCEV *Yop = CY->getOperand();
+ if (Xop->getType() == Yop->getType()) {
+ X = Xop;
+ Y = Yop;
+ }
+ }
+ }
+ if (SE->isKnownPredicate(Pred, X, Y))
+ return true;
+ // If SE->isKnownPredicate can't prove the condition,
+ // we try the brute-force approach of subtracting
+ // and testing the difference.
+ // By testing with SE->isKnownPredicate first, we avoid
+ // the possibility of overflow when the arguments are constants.
+ const SCEV *Delta = SE->getMinusSCEV(X, Y);
+ switch (Pred) {
+ case CmpInst::ICMP_EQ:
+ return Delta->isZero();
+ case CmpInst::ICMP_NE:
+ return SE->isKnownNonZero(Delta);
+ case CmpInst::ICMP_SGE:
+ return SE->isKnownNonNegative(Delta);
+ case CmpInst::ICMP_SLE:
+ return SE->isKnownNonPositive(Delta);
+ case CmpInst::ICMP_SGT:
+ return SE->isKnownPositive(Delta);
+ case CmpInst::ICMP_SLT:
+ return SE->isKnownNegative(Delta);
+ default:
+ llvm_unreachable("unexpected predicate in isKnownPredicate");
+ }
+}
+
+
+// All subscripts are all the same type.
+// Loop bound may be smaller (e.g., a char).
+// Should zero extend loop bound, since it's always >= 0.
+// This routine collects upper bound and extends if needed.
+// Return null if no bound available.
+const SCEV *DependenceAnalysis::collectUpperBound(const Loop *L,
+ Type *T) const {
+ if (SE->hasLoopInvariantBackedgeTakenCount(L)) {
+ const SCEV *UB = SE->getBackedgeTakenCount(L);
+ return SE->getNoopOrZeroExtend(UB, T);
+ }
+ return NULL;
+}
+
+
+// Calls collectUpperBound(), then attempts to cast it to SCEVConstant.
+// If the cast fails, returns NULL.
+const SCEVConstant *DependenceAnalysis::collectConstantUpperBound(const Loop *L,
+ Type *T
+ ) const {
+ if (const SCEV *UB = collectUpperBound(L, T))
+ return dyn_cast<SCEVConstant>(UB);
+ return NULL;
+}
+
+
+// testZIV -
+// When we have a pair of subscripts of the form [c1] and [c2],
+// where c1 and c2 are both loop invariant, we attack it using
+// the ZIV test. Basically, we test by comparing the two values,
+// but there are actually three possible results:
+// 1) the values are equal, so there's a dependence
+// 2) the values are different, so there's no dependence
+// 3) the values might be equal, so we have to assume a dependence.
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::testZIV(const SCEV *Src,
+ const SCEV *Dst,
+ FullDependence &Result) const {
+ DEBUG(dbgs() << " src = " << *Src << "\n");
+ DEBUG(dbgs() << " dst = " << *Dst << "\n");
+ ++ZIVapplications;
+ if (isKnownPredicate(CmpInst::ICMP_EQ, Src, Dst)) {
+ DEBUG(dbgs() << " provably dependent\n");
+ return false; // provably dependent
+ }
+ if (isKnownPredicate(CmpInst::ICMP_NE, Src, Dst)) {
+ DEBUG(dbgs() << " provably independent\n");
+ ++ZIVindependence;
+ return true; // provably independent
+ }
+ DEBUG(dbgs() << " possibly dependent\n");
+ Result.Consistent = false;
+ return false; // possibly dependent
+}
+
+
+// strongSIVtest -
+// From the paper, Practical Dependence Testing, Section 4.2.1
+//
+// When we have a pair of subscripts of the form [c1 + a*i] and [c2 + a*i],
+// where i is an induction variable, c1 and c2 are loop invariant,
+// and a is a constant, we can solve it exactly using the Strong SIV test.
+//
+// Can prove independence. Failing that, can compute distance (and direction).
+// In the presence of symbolic terms, we can sometimes make progress.
+//
+// If there's a dependence,
+//
+// c1 + a*i = c2 + a*i'
+//
+// The dependence distance is
+//
+// d = i' - i = (c1 - c2)/a
+//
+// A dependence only exists if d is an integer and abs(d) <= U, where U is the
+// loop's upper bound. If a dependence exists, the dependence direction is
+// defined as
+//
+// { < if d > 0
+// direction = { = if d = 0
+// { > if d < 0
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::strongSIVtest(const SCEV *Coeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *CurLoop,
+ unsigned Level,
+ FullDependence &Result,
+ Constraint &NewConstraint) const {
+ DEBUG(dbgs() << "\tStrong SIV test\n");
+ DEBUG(dbgs() << "\t Coeff = " << *Coeff);
+ DEBUG(dbgs() << ", " << *Coeff->getType() << "\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst);
+ DEBUG(dbgs() << ", " << *SrcConst->getType() << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst);
+ DEBUG(dbgs() << ", " << *DstConst->getType() << "\n");
+ ++StrongSIVapplications;
+ assert(0 < Level && Level <= CommonLevels && "level out of range");
+ Level--;
+
+ const SCEV *Delta = SE->getMinusSCEV(SrcConst, DstConst);
+ DEBUG(dbgs() << "\t Delta = " << *Delta);
+ DEBUG(dbgs() << ", " << *Delta->getType() << "\n");
+
+ // check that |Delta| < iteration count
+ if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
+ DEBUG(dbgs() << "\t UpperBound = " << *UpperBound);
+ DEBUG(dbgs() << ", " << *UpperBound->getType() << "\n");
+ const SCEV *AbsDelta =
+ SE->isKnownNonNegative(Delta) ? Delta : SE->getNegativeSCEV(Delta);
+ const SCEV *AbsCoeff =
+ SE->isKnownNonNegative(Coeff) ? Coeff : SE->getNegativeSCEV(Coeff);
+ const SCEV *Product = SE->getMulExpr(UpperBound, AbsCoeff);
+ if (isKnownPredicate(CmpInst::ICMP_SGT, AbsDelta, Product)) {
+ // Distance greater than trip count - no dependence
+ ++StrongSIVindependence;
+ ++StrongSIVsuccesses;
+ return true;
+ }
+ }
+
+ // Can we compute distance?
+ if (isa<SCEVConstant>(Delta) && isa<SCEVConstant>(Coeff)) {
+ APInt ConstDelta = cast<SCEVConstant>(Delta)->getValue()->getValue();
+ APInt ConstCoeff = cast<SCEVConstant>(Coeff)->getValue()->getValue();
+ APInt Distance = ConstDelta; // these need to be initialized
+ APInt Remainder = ConstDelta;
+ APInt::sdivrem(ConstDelta, ConstCoeff, Distance, Remainder);
+ DEBUG(dbgs() << "\t Distance = " << Distance << "\n");
+ DEBUG(dbgs() << "\t Remainder = " << Remainder << "\n");
+ // Make sure Coeff divides Delta exactly
+ if (Remainder != 0) {
+ // Coeff doesn't divide Distance, no dependence
+ ++StrongSIVindependence;
+ ++StrongSIVsuccesses;
+ return true;
+ }
+ Result.DV[Level].Distance = SE->getConstant(Distance);
+ NewConstraint.setDistance(SE->getConstant(Distance), CurLoop);
+ if (Distance.sgt(0))
+ Result.DV[Level].Direction &= Dependence::DVEntry::LT;
+ else if (Distance.slt(0))
+ Result.DV[Level].Direction &= Dependence::DVEntry::GT;
+ else
+ Result.DV[Level].Direction &= Dependence::DVEntry::EQ;
+ ++StrongSIVsuccesses;
+ }
+ else if (Delta->isZero()) {
+ // since 0/X == 0
+ Result.DV[Level].Distance = Delta;
+ NewConstraint.setDistance(Delta, CurLoop);
+ Result.DV[Level].Direction &= Dependence::DVEntry::EQ;
+ ++StrongSIVsuccesses;
+ }
+ else {
+ if (Coeff->isOne()) {
+ DEBUG(dbgs() << "\t Distance = " << *Delta << "\n");
+ Result.DV[Level].Distance = Delta; // since X/1 == X
+ NewConstraint.setDistance(Delta, CurLoop);
+ }
+ else {
+ Result.Consistent = false;
+ NewConstraint.setLine(Coeff,
+ SE->getNegativeSCEV(Coeff),
+ SE->getNegativeSCEV(Delta), CurLoop);
+ }
+
+ // maybe we can get a useful direction
+ bool DeltaMaybeZero = !SE->isKnownNonZero(Delta);
+ bool DeltaMaybePositive = !SE->isKnownNonPositive(Delta);
+ bool DeltaMaybeNegative = !SE->isKnownNonNegative(Delta);
+ bool CoeffMaybePositive = !SE->isKnownNonPositive(Coeff);
+ bool CoeffMaybeNegative = !SE->isKnownNonNegative(Coeff);
+ // The double negatives above are confusing.
+ // It helps to read !SE->isKnownNonZero(Delta)
+ // as "Delta might be Zero"
+ unsigned NewDirection = Dependence::DVEntry::NONE;
+ if ((DeltaMaybePositive && CoeffMaybePositive) ||
+ (DeltaMaybeNegative && CoeffMaybeNegative))
+ NewDirection = Dependence::DVEntry::LT;
+ if (DeltaMaybeZero)
+ NewDirection |= Dependence::DVEntry::EQ;
+ if ((DeltaMaybeNegative && CoeffMaybePositive) ||
+ (DeltaMaybePositive && CoeffMaybeNegative))
+ NewDirection |= Dependence::DVEntry::GT;
+ if (NewDirection < Result.DV[Level].Direction)
+ ++StrongSIVsuccesses;
+ Result.DV[Level].Direction &= NewDirection;
+ }
+ return false;
+}
+
+
+// weakCrossingSIVtest -
+// From the paper, Practical Dependence Testing, Section 4.2.2
+//
+// When we have a pair of subscripts of the form [c1 + a*i] and [c2 - a*i],
+// where i is an induction variable, c1 and c2 are loop invariant,
+// and a is a constant, we can solve it exactly using the
+// Weak-Crossing SIV test.
+//
+// Given c1 + a*i = c2 - a*i', we can look for the intersection of
+// the two lines, where i = i', yielding
+//
+// c1 + a*i = c2 - a*i
+// 2a*i = c2 - c1
+// i = (c2 - c1)/2a
+//
+// If i < 0, there is no dependence.
+// If i > upperbound, there is no dependence.
+// If i = 0 (i.e., if c1 = c2), there's a dependence with distance = 0.
+// If i = upperbound, there's a dependence with distance = 0.
+// If i is integral, there's a dependence (all directions).
+// If the non-integer part = 1/2, there's a dependence (<> directions).
+// Otherwise, there's no dependence.
+//
+// Can prove independence. Failing that,
+// can sometimes refine the directions.
+// Can determine iteration for splitting.
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::weakCrossingSIVtest(const SCEV *Coeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *CurLoop,
+ unsigned Level,
+ FullDependence &Result,
+ Constraint &NewConstraint,
+ const SCEV *&SplitIter) const {
+ DEBUG(dbgs() << "\tWeak-Crossing SIV test\n");
+ DEBUG(dbgs() << "\t Coeff = " << *Coeff << "\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst << "\n");
+ ++WeakCrossingSIVapplications;
+ assert(0 < Level && Level <= CommonLevels && "Level out of range");
+ Level--;
+ Result.Consistent = false;
+ const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ NewConstraint.setLine(Coeff, Coeff, Delta, CurLoop);
+ if (Delta->isZero()) {
+ Result.DV[Level].Direction &= ~Dependence::DVEntry::LT;
+ Result.DV[Level].Direction &= ~Dependence::DVEntry::GT;
+ ++WeakCrossingSIVsuccesses;
+ if (!Result.DV[Level].Direction) {
+ ++WeakCrossingSIVindependence;
+ return true;
+ }
+ Result.DV[Level].Distance = Delta; // = 0
+ return false;
+ }
+ const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(Coeff);
+ if (!ConstCoeff)
+ return false;
+
+ Result.DV[Level].Splitable = true;
+ if (SE->isKnownNegative(ConstCoeff)) {
+ ConstCoeff = dyn_cast<SCEVConstant>(SE->getNegativeSCEV(ConstCoeff));
+ assert(ConstCoeff &&
+ "dynamic cast of negative of ConstCoeff should yield constant");
+ Delta = SE->getNegativeSCEV(Delta);
+ }
+ assert(SE->isKnownPositive(ConstCoeff) && "ConstCoeff should be positive");
+
+ // compute SplitIter for use by DependenceAnalysis::getSplitIteration()
+ SplitIter =
+ SE->getUDivExpr(SE->getSMaxExpr(SE->getConstant(Delta->getType(), 0),
+ Delta),
+ SE->getMulExpr(SE->getConstant(Delta->getType(), 2),
+ ConstCoeff));
+ DEBUG(dbgs() << "\t Split iter = " << *SplitIter << "\n");
+
+ const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
+ if (!ConstDelta)
+ return false;
+
+ // We're certain that ConstCoeff > 0; therefore,
+ // if Delta < 0, then no dependence.
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ DEBUG(dbgs() << "\t ConstCoeff = " << *ConstCoeff << "\n");
+ if (SE->isKnownNegative(Delta)) {
+ // No dependence, Delta < 0
+ ++WeakCrossingSIVindependence;
+ ++WeakCrossingSIVsuccesses;
+ return true;
+ }
+
+ // We're certain that Delta > 0 and ConstCoeff > 0.
+ // Check Delta/(2*ConstCoeff) against upper loop bound
+ if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
+ DEBUG(dbgs() << "\t UpperBound = " << *UpperBound << "\n");
+ const SCEV *ConstantTwo = SE->getConstant(UpperBound->getType(), 2);
+ const SCEV *ML = SE->getMulExpr(SE->getMulExpr(ConstCoeff, UpperBound),
+ ConstantTwo);
+ DEBUG(dbgs() << "\t ML = " << *ML << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SGT, Delta, ML)) {
+ // Delta too big, no dependence
+ ++WeakCrossingSIVindependence;
+ ++WeakCrossingSIVsuccesses;
+ return true;
+ }
+ if (isKnownPredicate(CmpInst::ICMP_EQ, Delta, ML)) {
+ // i = i' = UB
+ Result.DV[Level].Direction &= ~Dependence::DVEntry::LT;
+ Result.DV[Level].Direction &= ~Dependence::DVEntry::GT;
+ ++WeakCrossingSIVsuccesses;
+ if (!Result.DV[Level].Direction) {
+ ++WeakCrossingSIVindependence;
+ return true;
+ }
+ Result.DV[Level].Splitable = false;
+ Result.DV[Level].Distance = SE->getConstant(Delta->getType(), 0);
+ return false;
+ }
+ }
+
+ // check that Coeff divides Delta
+ APInt APDelta = ConstDelta->getValue()->getValue();
+ APInt APCoeff = ConstCoeff->getValue()->getValue();
+ APInt Distance = APDelta; // these need to be initialzed
+ APInt Remainder = APDelta;
+ APInt::sdivrem(APDelta, APCoeff, Distance, Remainder);
+ DEBUG(dbgs() << "\t Remainder = " << Remainder << "\n");
+ if (Remainder != 0) {
+ // Coeff doesn't divide Delta, no dependence
+ ++WeakCrossingSIVindependence;
+ ++WeakCrossingSIVsuccesses;
+ return true;
+ }
+ DEBUG(dbgs() << "\t Distance = " << Distance << "\n");
+
+ // if 2*Coeff doesn't divide Delta, then the equal direction isn't possible
+ APInt Two = APInt(Distance.getBitWidth(), 2, true);
+ Remainder = Distance.srem(Two);
+ DEBUG(dbgs() << "\t Remainder = " << Remainder << "\n");
+ if (Remainder != 0) {
+ // Equal direction isn't possible
+ Result.DV[Level].Direction &= ~Dependence::DVEntry::EQ;
+ ++WeakCrossingSIVsuccesses;
+ }
+ return false;
+}
+
+
+// Kirch's algorithm, from
+//
+// Optimizing Supercompilers for Supercomputers
+// Michael Wolfe
+// MIT Press, 1989
+//
+// Program 2.1, page 29.
+// Computes the GCD of AM and BM.
+// Also finds a solution to the equation ax - by = gdc(a, b).
+// Returns true iff the gcd divides Delta.
+static
+bool findGCD(unsigned Bits, APInt AM, APInt BM, APInt Delta,
+ APInt &G, APInt &X, APInt &Y) {
+ APInt A0(Bits, 1, true), A1(Bits, 0, true);
+ APInt B0(Bits, 0, true), B1(Bits, 1, true);
+ APInt G0 = AM.abs();
+ APInt G1 = BM.abs();
+ APInt Q = G0; // these need to be initialized
+ APInt R = G0;
+ APInt::sdivrem(G0, G1, Q, R);
+ while (R != 0) {
+ APInt A2 = A0 - Q*A1; A0 = A1; A1 = A2;
+ APInt B2 = B0 - Q*B1; B0 = B1; B1 = B2;
+ G0 = G1; G1 = R;
+ APInt::sdivrem(G0, G1, Q, R);
+ }
+ G = G1;
+ DEBUG(dbgs() << "\t GCD = " << G << "\n");
+ X = AM.slt(0) ? -A1 : A1;
+ Y = BM.slt(0) ? B1 : -B1;
+
+ // make sure gcd divides Delta
+ R = Delta.srem(G);
+ if (R != 0)
+ return true; // gcd doesn't divide Delta, no dependence
+ Q = Delta.sdiv(G);
+ X *= Q;
+ Y *= Q;
+ return false;
+}
+
+
+static
+APInt floorOfQuotient(APInt A, APInt B) {
+ APInt Q = A; // these need to be initialized
+ APInt R = A;
+ APInt::sdivrem(A, B, Q, R);
+ if (R == 0)
+ return Q;
+ if ((A.sgt(0) && B.sgt(0)) ||
+ (A.slt(0) && B.slt(0)))
+ return Q;
+ else
+ return Q - 1;
+}
+
+
+static
+APInt ceilingOfQuotient(APInt A, APInt B) {
+ APInt Q = A; // these need to be initialized
+ APInt R = A;
+ APInt::sdivrem(A, B, Q, R);
+ if (R == 0)
+ return Q;
+ if ((A.sgt(0) && B.sgt(0)) ||
+ (A.slt(0) && B.slt(0)))
+ return Q + 1;
+ else
+ return Q;
+}
+
+
+static
+APInt maxAPInt(APInt A, APInt B) {
+ return A.sgt(B) ? A : B;
+}
+
+
+static
+APInt minAPInt(APInt A, APInt B) {
+ return A.slt(B) ? A : B;
+}
+
+
+// exactSIVtest -
+// When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*i],
+// where i is an induction variable, c1 and c2 are loop invariant, and a1
+// and a2 are constant, we can solve it exactly using an algorithm developed
+// by Banerjee and Wolfe. See Section 2.5.3 in
+//
+// Optimizing Supercompilers for Supercomputers
+// Michael Wolfe
+// MIT Press, 1989
+//
+// It's slower than the specialized tests (strong SIV, weak-zero SIV, etc),
+// so use them if possible. They're also a bit better with symbolics and,
+// in the case of the strong SIV test, can compute Distances.
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::exactSIVtest(const SCEV *SrcCoeff,
+ const SCEV *DstCoeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *CurLoop,
+ unsigned Level,
+ FullDependence &Result,
+ Constraint &NewConstraint) const {
+ DEBUG(dbgs() << "\tExact SIV test\n");
+ DEBUG(dbgs() << "\t SrcCoeff = " << *SrcCoeff << " = AM\n");
+ DEBUG(dbgs() << "\t DstCoeff = " << *DstCoeff << " = BM\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst << "\n");
+ ++ExactSIVapplications;
+ assert(0 < Level && Level <= CommonLevels && "Level out of range");
+ Level--;
+ Result.Consistent = false;
+ const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ NewConstraint.setLine(SrcCoeff, SE->getNegativeSCEV(DstCoeff),
+ Delta, CurLoop);
+ const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
+ const SCEVConstant *ConstSrcCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
+ const SCEVConstant *ConstDstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
+ if (!ConstDelta || !ConstSrcCoeff || !ConstDstCoeff)
+ return false;
+
+ // find gcd
+ APInt G, X, Y;
+ APInt AM = ConstSrcCoeff->getValue()->getValue();
+ APInt BM = ConstDstCoeff->getValue()->getValue();
+ unsigned Bits = AM.getBitWidth();
+ if (findGCD(Bits, AM, BM, ConstDelta->getValue()->getValue(), G, X, Y)) {
+ // gcd doesn't divide Delta, no dependence
+ ++ExactSIVindependence;
+ ++ExactSIVsuccesses;
+ return true;
+ }
+
+ DEBUG(dbgs() << "\t X = " << X << ", Y = " << Y << "\n");
+
+ // since SCEV construction normalizes, LM = 0
+ APInt UM(Bits, 1, true);
+ bool UMvalid = false;
+ // UM is perhaps unavailable, let's check
+ if (const SCEVConstant *CUB =
+ collectConstantUpperBound(CurLoop, Delta->getType())) {
+ UM = CUB->getValue()->getValue();
+ DEBUG(dbgs() << "\t UM = " << UM << "\n");
+ UMvalid = true;
+ }
+
+ APInt TU(APInt::getSignedMaxValue(Bits));
+ APInt TL(APInt::getSignedMinValue(Bits));
+
+ // test(BM/G, LM-X) and test(-BM/G, X-UM)
+ APInt TMUL = BM.sdiv(G);
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(-X, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ if (UMvalid) {
+ TU = minAPInt(TU, floorOfQuotient(UM - X, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ }
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(-X, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ if (UMvalid) {
+ TL = maxAPInt(TL, ceilingOfQuotient(UM - X, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ }
+ }
+
+ // test(AM/G, LM-Y) and test(-AM/G, Y-UM)
+ TMUL = AM.sdiv(G);
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(-Y, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ if (UMvalid) {
+ TU = minAPInt(TU, floorOfQuotient(UM - Y, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ }
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(-Y, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ if (UMvalid) {
+ TL = maxAPInt(TL, ceilingOfQuotient(UM - Y, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ }
+ }
+ if (TL.sgt(TU)) {
+ ++ExactSIVindependence;
+ ++ExactSIVsuccesses;
+ return true;
+ }
+
+ // explore directions
+ unsigned NewDirection = Dependence::DVEntry::NONE;
+
+ // less than
+ APInt SaveTU(TU); // save these
+ APInt SaveTL(TL);
+ DEBUG(dbgs() << "\t exploring LT direction\n");
+ TMUL = AM - BM;
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(X - Y + 1, TMUL));
+ DEBUG(dbgs() << "\t\t TL = " << TL << "\n");
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(X - Y + 1, TMUL));
+ DEBUG(dbgs() << "\t\t TU = " << TU << "\n");
+ }
+ if (TL.sle(TU)) {
+ NewDirection |= Dependence::DVEntry::LT;
+ ++ExactSIVsuccesses;
+ }
+
+ // equal
+ TU = SaveTU; // restore
+ TL = SaveTL;
+ DEBUG(dbgs() << "\t exploring EQ direction\n");
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(X - Y, TMUL));
+ DEBUG(dbgs() << "\t\t TL = " << TL << "\n");
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(X - Y, TMUL));
+ DEBUG(dbgs() << "\t\t TU = " << TU << "\n");
+ }
+ TMUL = BM - AM;
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(Y - X, TMUL));
+ DEBUG(dbgs() << "\t\t TL = " << TL << "\n");
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(Y - X, TMUL));
+ DEBUG(dbgs() << "\t\t TU = " << TU << "\n");
+ }
+ if (TL.sle(TU)) {
+ NewDirection |= Dependence::DVEntry::EQ;
+ ++ExactSIVsuccesses;
+ }
+
+ // greater than
+ TU = SaveTU; // restore
+ TL = SaveTL;
+ DEBUG(dbgs() << "\t exploring GT direction\n");
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(Y - X + 1, TMUL));
+ DEBUG(dbgs() << "\t\t TL = " << TL << "\n");
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(Y - X + 1, TMUL));
+ DEBUG(dbgs() << "\t\t TU = " << TU << "\n");
+ }
+ if (TL.sle(TU)) {
+ NewDirection |= Dependence::DVEntry::GT;
+ ++ExactSIVsuccesses;
+ }
+
+ // finished
+ Result.DV[Level].Direction &= NewDirection;
+ if (Result.DV[Level].Direction == Dependence::DVEntry::NONE)
+ ++ExactSIVindependence;
+ return Result.DV[Level].Direction == Dependence::DVEntry::NONE;
+}
+
+
+
+// Return true if the divisor evenly divides the dividend.
+static
+bool isRemainderZero(const SCEVConstant *Dividend,
+ const SCEVConstant *Divisor) {
+ APInt ConstDividend = Dividend->getValue()->getValue();
+ APInt ConstDivisor = Divisor->getValue()->getValue();
+ return ConstDividend.srem(ConstDivisor) == 0;
+}
+
+
+// weakZeroSrcSIVtest -
+// From the paper, Practical Dependence Testing, Section 4.2.2
+//
+// When we have a pair of subscripts of the form [c1] and [c2 + a*i],
+// where i is an induction variable, c1 and c2 are loop invariant,
+// and a is a constant, we can solve it exactly using the
+// Weak-Zero SIV test.
+//
+// Given
+//
+// c1 = c2 + a*i
+//
+// we get
+//
+// (c1 - c2)/a = i
+//
+// If i is not an integer, there's no dependence.
+// If i < 0 or > UB, there's no dependence.
+// If i = 0, the direction is <= and peeling the
+// 1st iteration will break the dependence.
+// If i = UB, the direction is >= and peeling the
+// last iteration will break the dependence.
+// Otherwise, the direction is *.
+//
+// Can prove independence. Failing that, we can sometimes refine
+// the directions. Can sometimes show that first or last
+// iteration carries all the dependences (so worth peeling).
+//
+// (see also weakZeroDstSIVtest)
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::weakZeroSrcSIVtest(const SCEV *DstCoeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *CurLoop,
+ unsigned Level,
+ FullDependence &Result,
+ Constraint &NewConstraint) const {
+ // For the WeakSIV test, it's possible the loop isn't common to
+ // the Src and Dst loops. If it isn't, then there's no need to
+ // record a direction.
+ DEBUG(dbgs() << "\tWeak-Zero (src) SIV test\n");
+ DEBUG(dbgs() << "\t DstCoeff = " << *DstCoeff << "\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst << "\n");
+ ++WeakZeroSIVapplications;
+ assert(0 < Level && Level <= MaxLevels && "Level out of range");
+ Level--;
+ Result.Consistent = false;
+ const SCEV *Delta = SE->getMinusSCEV(SrcConst, DstConst);
+ NewConstraint.setLine(SE->getConstant(Delta->getType(), 0),
+ DstCoeff, Delta, CurLoop);
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_EQ, SrcConst, DstConst)) {
+ if (Level < CommonLevels) {
+ Result.DV[Level].Direction &= Dependence::DVEntry::LE;
+ Result.DV[Level].PeelFirst = true;
+ ++WeakZeroSIVsuccesses;
+ }
+ return false; // dependences caused by first iteration
+ }
+ const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
+ if (!ConstCoeff)
+ return false;
+ const SCEV *AbsCoeff =
+ SE->isKnownNegative(ConstCoeff) ?
+ SE->getNegativeSCEV(ConstCoeff) : ConstCoeff;
+ const SCEV *NewDelta =
+ SE->isKnownNegative(ConstCoeff) ? SE->getNegativeSCEV(Delta) : Delta;
+
+ // check that Delta/SrcCoeff < iteration count
+ // really check NewDelta < count*AbsCoeff
+ if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
+ DEBUG(dbgs() << "\t UpperBound = " << *UpperBound << "\n");
+ const SCEV *Product = SE->getMulExpr(AbsCoeff, UpperBound);
+ if (isKnownPredicate(CmpInst::ICMP_SGT, NewDelta, Product)) {
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+ if (isKnownPredicate(CmpInst::ICMP_EQ, NewDelta, Product)) {
+ // dependences caused by last iteration
+ if (Level < CommonLevels) {
+ Result.DV[Level].Direction &= Dependence::DVEntry::GE;
+ Result.DV[Level].PeelLast = true;
+ ++WeakZeroSIVsuccesses;
+ }
+ return false;
+ }
+ }
+
+ // check that Delta/SrcCoeff >= 0
+ // really check that NewDelta >= 0
+ if (SE->isKnownNegative(NewDelta)) {
+ // No dependence, newDelta < 0
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+
+ // if SrcCoeff doesn't divide Delta, then no dependence
+ if (isa<SCEVConstant>(Delta) &&
+ !isRemainderZero(cast<SCEVConstant>(Delta), ConstCoeff)) {
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+ return false;
+}
+
+
+// weakZeroDstSIVtest -
+// From the paper, Practical Dependence Testing, Section 4.2.2
+//
+// When we have a pair of subscripts of the form [c1 + a*i] and [c2],
+// where i is an induction variable, c1 and c2 are loop invariant,
+// and a is a constant, we can solve it exactly using the
+// Weak-Zero SIV test.
+//
+// Given
+//
+// c1 + a*i = c2
+//
+// we get
+//
+// i = (c2 - c1)/a
+//
+// If i is not an integer, there's no dependence.
+// If i < 0 or > UB, there's no dependence.
+// If i = 0, the direction is <= and peeling the
+// 1st iteration will break the dependence.
+// If i = UB, the direction is >= and peeling the
+// last iteration will break the dependence.
+// Otherwise, the direction is *.
+//
+// Can prove independence. Failing that, we can sometimes refine
+// the directions. Can sometimes show that first or last
+// iteration carries all the dependences (so worth peeling).
+//
+// (see also weakZeroSrcSIVtest)
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::weakZeroDstSIVtest(const SCEV *SrcCoeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *CurLoop,
+ unsigned Level,
+ FullDependence &Result,
+ Constraint &NewConstraint) const {
+ // For the WeakSIV test, it's possible the loop isn't common to the
+ // Src and Dst loops. If it isn't, then there's no need to record a direction.
+ DEBUG(dbgs() << "\tWeak-Zero (dst) SIV test\n");
+ DEBUG(dbgs() << "\t SrcCoeff = " << *SrcCoeff << "\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst << "\n");
+ ++WeakZeroSIVapplications;
+ assert(0 < Level && Level <= SrcLevels && "Level out of range");
+ Level--;
+ Result.Consistent = false;
+ const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
+ NewConstraint.setLine(SrcCoeff, SE->getConstant(Delta->getType(), 0),
+ Delta, CurLoop);
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_EQ, DstConst, SrcConst)) {
+ if (Level < CommonLevels) {
+ Result.DV[Level].Direction &= Dependence::DVEntry::LE;
+ Result.DV[Level].PeelFirst = true;
+ ++WeakZeroSIVsuccesses;
+ }
+ return false; // dependences caused by first iteration
+ }
+ const SCEVConstant *ConstCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
+ if (!ConstCoeff)
+ return false;
+ const SCEV *AbsCoeff =
+ SE->isKnownNegative(ConstCoeff) ?
+ SE->getNegativeSCEV(ConstCoeff) : ConstCoeff;
+ const SCEV *NewDelta =
+ SE->isKnownNegative(ConstCoeff) ? SE->getNegativeSCEV(Delta) : Delta;
+
+ // check that Delta/SrcCoeff < iteration count
+ // really check NewDelta < count*AbsCoeff
+ if (const SCEV *UpperBound = collectUpperBound(CurLoop, Delta->getType())) {
+ DEBUG(dbgs() << "\t UpperBound = " << *UpperBound << "\n");
+ const SCEV *Product = SE->getMulExpr(AbsCoeff, UpperBound);
+ if (isKnownPredicate(CmpInst::ICMP_SGT, NewDelta, Product)) {
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+ if (isKnownPredicate(CmpInst::ICMP_EQ, NewDelta, Product)) {
+ // dependences caused by last iteration
+ if (Level < CommonLevels) {
+ Result.DV[Level].Direction &= Dependence::DVEntry::GE;
+ Result.DV[Level].PeelLast = true;
+ ++WeakZeroSIVsuccesses;
+ }
+ return false;
+ }
+ }
+
+ // check that Delta/SrcCoeff >= 0
+ // really check that NewDelta >= 0
+ if (SE->isKnownNegative(NewDelta)) {
+ // No dependence, newDelta < 0
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+
+ // if SrcCoeff doesn't divide Delta, then no dependence
+ if (isa<SCEVConstant>(Delta) &&
+ !isRemainderZero(cast<SCEVConstant>(Delta), ConstCoeff)) {
+ ++WeakZeroSIVindependence;
+ ++WeakZeroSIVsuccesses;
+ return true;
+ }
+ return false;
+}
+
+
+// exactRDIVtest - Tests the RDIV subscript pair for dependence.
+// Things of the form [c1 + a*i] and [c2 + b*j],
+// where i and j are induction variable, c1 and c2 are loop invariant,
+// and a and b are constants.
+// Returns true if any possible dependence is disproved.
+// Marks the result as inconsistant.
+// Works in some cases that symbolicRDIVtest doesn't, and vice versa.
+bool DependenceAnalysis::exactRDIVtest(const SCEV *SrcCoeff,
+ const SCEV *DstCoeff,
+ const SCEV *SrcConst,
+ const SCEV *DstConst,
+ const Loop *SrcLoop,
+ const Loop *DstLoop,
+ FullDependence &Result) const {
+ DEBUG(dbgs() << "\tExact RDIV test\n");
+ DEBUG(dbgs() << "\t SrcCoeff = " << *SrcCoeff << " = AM\n");
+ DEBUG(dbgs() << "\t DstCoeff = " << *DstCoeff << " = BM\n");
+ DEBUG(dbgs() << "\t SrcConst = " << *SrcConst << "\n");
+ DEBUG(dbgs() << "\t DstConst = " << *DstConst << "\n");
+ ++ExactRDIVapplications;
+ Result.Consistent = false;
+ const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
+ DEBUG(dbgs() << "\t Delta = " << *Delta << "\n");
+ const SCEVConstant *ConstDelta = dyn_cast<SCEVConstant>(Delta);
+ const SCEVConstant *ConstSrcCoeff = dyn_cast<SCEVConstant>(SrcCoeff);
+ const SCEVConstant *ConstDstCoeff = dyn_cast<SCEVConstant>(DstCoeff);
+ if (!ConstDelta || !ConstSrcCoeff || !ConstDstCoeff)
+ return false;
+
+ // find gcd
+ APInt G, X, Y;
+ APInt AM = ConstSrcCoeff->getValue()->getValue();
+ APInt BM = ConstDstCoeff->getValue()->getValue();
+ unsigned Bits = AM.getBitWidth();
+ if (findGCD(Bits, AM, BM, ConstDelta->getValue()->getValue(), G, X, Y)) {
+ // gcd doesn't divide Delta, no dependence
+ ++ExactRDIVindependence;
+ return true;
+ }
+
+ DEBUG(dbgs() << "\t X = " << X << ", Y = " << Y << "\n");
+
+ // since SCEV construction seems to normalize, LM = 0
+ APInt SrcUM(Bits, 1, true);
+ bool SrcUMvalid = false;
+ // SrcUM is perhaps unavailable, let's check
+ if (const SCEVConstant *UpperBound =
+ collectConstantUpperBound(SrcLoop, Delta->getType())) {
+ SrcUM = UpperBound->getValue()->getValue();
+ DEBUG(dbgs() << "\t SrcUM = " << SrcUM << "\n");
+ SrcUMvalid = true;
+ }
+
+ APInt DstUM(Bits, 1, true);
+ bool DstUMvalid = false;
+ // UM is perhaps unavailable, let's check
+ if (const SCEVConstant *UpperBound =
+ collectConstantUpperBound(DstLoop, Delta->getType())) {
+ DstUM = UpperBound->getValue()->getValue();
+ DEBUG(dbgs() << "\t DstUM = " << DstUM << "\n");
+ DstUMvalid = true;
+ }
+
+ APInt TU(APInt::getSignedMaxValue(Bits));
+ APInt TL(APInt::getSignedMinValue(Bits));
+
+ // test(BM/G, LM-X) and test(-BM/G, X-UM)
+ APInt TMUL = BM.sdiv(G);
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(-X, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ if (SrcUMvalid) {
+ TU = minAPInt(TU, floorOfQuotient(SrcUM - X, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ }
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(-X, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ if (SrcUMvalid) {
+ TL = maxAPInt(TL, ceilingOfQuotient(SrcUM - X, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ }
+ }
+
+ // test(AM/G, LM-Y) and test(-AM/G, Y-UM)
+ TMUL = AM.sdiv(G);
+ if (TMUL.sgt(0)) {
+ TL = maxAPInt(TL, ceilingOfQuotient(-Y, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ if (DstUMvalid) {
+ TU = minAPInt(TU, floorOfQuotient(DstUM - Y, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ }
+ }
+ else {
+ TU = minAPInt(TU, floorOfQuotient(-Y, TMUL));
+ DEBUG(dbgs() << "\t TU = " << TU << "\n");
+ if (DstUMvalid) {
+ TL = maxAPInt(TL, ceilingOfQuotient(DstUM - Y, TMUL));
+ DEBUG(dbgs() << "\t TL = " << TL << "\n");
+ }
+ }
+ if (TL.sgt(TU))
+ ++ExactRDIVindependence;
+ return TL.sgt(TU);
+}
+
+
+// symbolicRDIVtest -
+// In Section 4.5 of the Practical Dependence Testing paper,the authors
+// introduce a special case of Banerjee's Inequalities (also called the
+// Extreme-Value Test) that can handle some of the SIV and RDIV cases,
+// particularly cases with symbolics. Since it's only able to disprove
+// dependence (not compute distances or directions), we'll use it as a
+// fall back for the other tests.
+//
+// When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*j]
+// where i and j are induction variables and c1 and c2 are loop invariants,
+// we can use the symbolic tests to disprove some dependences, serving as a
+// backup for the RDIV test. Note that i and j can be the same variable,
+// letting this test serve as a backup for the various SIV tests.
+//
+// For a dependence to exist, c1 + a1*i must equal c2 + a2*j for some
+// 0 <= i <= N1 and some 0 <= j <= N2, where N1 and N2 are the (normalized)
+// loop bounds for the i and j loops, respectively. So, ...
+//
+// c1 + a1*i = c2 + a2*j
+// a1*i - a2*j = c2 - c1
+//
+// To test for a dependence, we compute c2 - c1 and make sure it's in the
+// range of the maximum and minimum possible values of a1*i - a2*j.
+// Considering the signs of a1 and a2, we have 4 possible cases:
+//
+// 1) If a1 >= 0 and a2 >= 0, then
+// a1*0 - a2*N2 <= c2 - c1 <= a1*N1 - a2*0
+// -a2*N2 <= c2 - c1 <= a1*N1
+//
+// 2) If a1 >= 0 and a2 <= 0, then
+// a1*0 - a2*0 <= c2 - c1 <= a1*N1 - a2*N2
+// 0 <= c2 - c1 <= a1*N1 - a2*N2
+//
+// 3) If a1 <= 0 and a2 >= 0, then
+// a1*N1 - a2*N2 <= c2 - c1 <= a1*0 - a2*0
+// a1*N1 - a2*N2 <= c2 - c1 <= 0
+//
+// 4) If a1 <= 0 and a2 <= 0, then
+// a1*N1 - a2*0 <= c2 - c1 <= a1*0 - a2*N2
+// a1*N1 <= c2 - c1 <= -a2*N2
+//
+// return true if dependence disproved
+bool DependenceAnalysis::symbolicRDIVtest(const SCEV *A1,
+ const SCEV *A2,
+ const SCEV *C1,
+ const SCEV *C2,
+ const Loop *Loop1,
+ const Loop *Loop2) const {
+ ++SymbolicRDIVapplications;
+ DEBUG(dbgs() << "\ttry symbolic RDIV test\n");
+ DEBUG(dbgs() << "\t A1 = " << *A1);
+ DEBUG(dbgs() << ", type = " << *A1->getType() << "\n");
+ DEBUG(dbgs() << "\t A2 = " << *A2 << "\n");
+ DEBUG(dbgs() << "\t C1 = " << *C1 << "\n");
+ DEBUG(dbgs() << "\t C2 = " << *C2 << "\n");
+ const SCEV *N1 = collectUpperBound(Loop1, A1->getType());
+ const SCEV *N2 = collectUpperBound(Loop2, A1->getType());
+ DEBUG(if (N1) dbgs() << "\t N1 = " << *N1 << "\n");
+ DEBUG(if (N2) dbgs() << "\t N2 = " << *N2 << "\n");
+ const SCEV *C2_C1 = SE->getMinusSCEV(C2, C1);
+ const SCEV *C1_C2 = SE->getMinusSCEV(C1, C2);
+ DEBUG(dbgs() << "\t C2 - C1 = " << *C2_C1 << "\n");
+ DEBUG(dbgs() << "\t C1 - C2 = " << *C1_C2 << "\n");
+ if (SE->isKnownNonNegative(A1)) {
+ if (SE->isKnownNonNegative(A2)) {
+ // A1 >= 0 && A2 >= 0
+ if (N1) {
+ // make sure that c2 - c1 <= a1*N1
+ const SCEV *A1N1 = SE->getMulExpr(A1, N1);
+ DEBUG(dbgs() << "\t A1*N1 = " << *A1N1 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SGT, C2_C1, A1N1)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ if (N2) {
+ // make sure that -a2*N2 <= c2 - c1, or a2*N2 >= c1 - c2
+ const SCEV *A2N2 = SE->getMulExpr(A2, N2);
+ DEBUG(dbgs() << "\t A2*N2 = " << *A2N2 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SLT, A2N2, C1_C2)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ }
+ else if (SE->isKnownNonPositive(A2)) {
+ // a1 >= 0 && a2 <= 0
+ if (N1 && N2) {
+ // make sure that c2 - c1 <= a1*N1 - a2*N2
+ const SCEV *A1N1 = SE->getMulExpr(A1, N1);
+ const SCEV *A2N2 = SE->getMulExpr(A2, N2);
+ const SCEV *A1N1_A2N2 = SE->getMinusSCEV(A1N1, A2N2);
+ DEBUG(dbgs() << "\t A1*N1 - A2*N2 = " << *A1N1_A2N2 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SGT, C2_C1, A1N1_A2N2)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ // make sure that 0 <= c2 - c1
+ if (SE->isKnownNegative(C2_C1)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ }
+ else if (SE->isKnownNonPositive(A1)) {
+ if (SE->isKnownNonNegative(A2)) {
+ // a1 <= 0 && a2 >= 0
+ if (N1 && N2) {
+ // make sure that a1*N1 - a2*N2 <= c2 - c1
+ const SCEV *A1N1 = SE->getMulExpr(A1, N1);
+ const SCEV *A2N2 = SE->getMulExpr(A2, N2);
+ const SCEV *A1N1_A2N2 = SE->getMinusSCEV(A1N1, A2N2);
+ DEBUG(dbgs() << "\t A1*N1 - A2*N2 = " << *A1N1_A2N2 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SGT, A1N1_A2N2, C2_C1)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ // make sure that c2 - c1 <= 0
+ if (SE->isKnownPositive(C2_C1)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ else if (SE->isKnownNonPositive(A2)) {
+ // a1 <= 0 && a2 <= 0
+ if (N1) {
+ // make sure that a1*N1 <= c2 - c1
+ const SCEV *A1N1 = SE->getMulExpr(A1, N1);
+ DEBUG(dbgs() << "\t A1*N1 = " << *A1N1 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SGT, A1N1, C2_C1)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ if (N2) {
+ // make sure that c2 - c1 <= -a2*N2, or c1 - c2 >= a2*N2
+ const SCEV *A2N2 = SE->getMulExpr(A2, N2);
+ DEBUG(dbgs() << "\t A2*N2 = " << *A2N2 << "\n");
+ if (isKnownPredicate(CmpInst::ICMP_SLT, C1_C2, A2N2)) {
+ ++SymbolicRDIVindependence;
+ return true;
+ }
+ }
+ }
+ }
+ return false;
+}
+
+
+// testSIV -
+// When we have a pair of subscripts of the form [c1 + a1*i] and [c2 - a2*i]
+// where i is an induction variable, c1 and c2 are loop invariant, and a1 and
+// a2 are constant, we attack it with an SIV test. While they can all be
+// solved with the Exact SIV test, it's worthwhile to use simpler tests when
+// they apply; they're cheaper and sometimes more precise.
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::testSIV(const SCEV *Src,
+ const SCEV *Dst,
+ unsigned &Level,
+ FullDependence &Result,
+ Constraint &NewConstraint,
+ const SCEV *&SplitIter) const {
+ DEBUG(dbgs() << " src = " << *Src << "\n");
+ DEBUG(dbgs() << " dst = " << *Dst << "\n");
+ const SCEVAddRecExpr *SrcAddRec = dyn_cast<SCEVAddRecExpr>(Src);
+ const SCEVAddRecExpr *DstAddRec = dyn_cast<SCEVAddRecExpr>(Dst);
+ if (SrcAddRec && DstAddRec) {
+ const SCEV *SrcConst = SrcAddRec->getStart();
+ const SCEV *DstConst = DstAddRec->getStart();
+ const SCEV *SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
+ const SCEV *DstCoeff = DstAddRec->getStepRecurrence(*SE);
+ const Loop *CurLoop = SrcAddRec->getLoop();
+ assert(CurLoop == DstAddRec->getLoop() &&
+ "both loops in SIV should be same");
+ Level = mapSrcLoop(CurLoop);
+ bool disproven;
+ if (SrcCoeff == DstCoeff)
+ disproven = strongSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
+ Level, Result, NewConstraint);
+ else if (SrcCoeff == SE->getNegativeSCEV(DstCoeff))
+ disproven = weakCrossingSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
+ Level, Result, NewConstraint, SplitIter);
+ else
+ disproven = exactSIVtest(SrcCoeff, DstCoeff, SrcConst, DstConst, CurLoop,
+ Level, Result, NewConstraint);
+ return disproven ||
+ gcdMIVtest(Src, Dst, Result) ||
+ symbolicRDIVtest(SrcCoeff, DstCoeff, SrcConst, DstConst, CurLoop, CurLoop);
+ }
+ if (SrcAddRec) {
+ const SCEV *SrcConst = SrcAddRec->getStart();
+ const SCEV *SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
+ const SCEV *DstConst = Dst;
+ const Loop *CurLoop = SrcAddRec->getLoop();
+ Level = mapSrcLoop(CurLoop);
+ return weakZeroDstSIVtest(SrcCoeff, SrcConst, DstConst, CurLoop,
+ Level, Result, NewConstraint) ||
+ gcdMIVtest(Src, Dst, Result);
+ }
+ if (DstAddRec) {
+ const SCEV *DstConst = DstAddRec->getStart();
+ const SCEV *DstCoeff = DstAddRec->getStepRecurrence(*SE);
+ const SCEV *SrcConst = Src;
+ const Loop *CurLoop = DstAddRec->getLoop();
+ Level = mapDstLoop(CurLoop);
+ return weakZeroSrcSIVtest(DstCoeff, SrcConst, DstConst,
+ CurLoop, Level, Result, NewConstraint) ||
+ gcdMIVtest(Src, Dst, Result);
+ }
+ llvm_unreachable("SIV test expected at least one AddRec");
+ return false;
+}
+
+
+// testRDIV -
+// When we have a pair of subscripts of the form [c1 + a1*i] and [c2 + a2*j]
+// where i and j are induction variables, c1 and c2 are loop invariant,
+// and a1 and a2 are constant, we can solve it exactly with an easy adaptation
+// of the Exact SIV test, the Restricted Double Index Variable (RDIV) test.
+// It doesn't make sense to talk about distance or direction in this case,
+// so there's no point in making special versions of the Strong SIV test or
+// the Weak-crossing SIV test.
+//
+// With minor algebra, this test can also be used for things like
+// [c1 + a1*i + a2*j][c2].
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::testRDIV(const SCEV *Src,
+ const SCEV *Dst,
+ FullDependence &Result) const {
+ // we have 3 possible situations here:
+ // 1) [a*i + b] and [c*j + d]
+ // 2) [a*i + c*j + b] and [d]
+ // 3) [b] and [a*i + c*j + d]
+ // We need to find what we've got and get organized
+
+ const SCEV *SrcConst, *DstConst;
+ const SCEV *SrcCoeff, *DstCoeff;
+ const Loop *SrcLoop, *DstLoop;
+
+ DEBUG(dbgs() << " src = " << *Src << "\n");
+ DEBUG(dbgs() << " dst = " << *Dst << "\n");
+ const SCEVAddRecExpr *SrcAddRec = dyn_cast<SCEVAddRecExpr>(Src);
+ const SCEVAddRecExpr *DstAddRec = dyn_cast<SCEVAddRecExpr>(Dst);
+ if (SrcAddRec && DstAddRec) {
+ SrcConst = SrcAddRec->getStart();
+ SrcCoeff = SrcAddRec->getStepRecurrence(*SE);
+ SrcLoop = SrcAddRec->getLoop();
+ DstConst = DstAddRec->getStart();
+ DstCoeff = DstAddRec->getStepRecurrence(*SE);
+ DstLoop = DstAddRec->getLoop();
+ }
+ else if (SrcAddRec) {
+ if (const SCEVAddRecExpr *tmpAddRec =
+ dyn_cast<SCEVAddRecExpr>(SrcAddRec->getStart())) {
+ SrcConst = tmpAddRec->getStart();
+ SrcCoeff = tmpAddRec->getStepRecurrence(*SE);
+ SrcLoop = tmpAddRec->getLoop();
+ DstConst = Dst;
+ DstCoeff = SE->getNegativeSCEV(SrcAddRec->getStepRecurrence(*SE));
+ DstLoop = SrcAddRec->getLoop();
+ }
+ else
+ llvm_unreachable("RDIV reached by surprising SCEVs");
+ }
+ else if (DstAddRec) {
+ if (const SCEVAddRecExpr *tmpAddRec =
+ dyn_cast<SCEVAddRecExpr>(DstAddRec->getStart())) {
+ DstConst = tmpAddRec->getStart();
+ DstCoeff = tmpAddRec->getStepRecurrence(*SE);
+ DstLoop = tmpAddRec->getLoop();
+ SrcConst = Src;
+ SrcCoeff = SE->getNegativeSCEV(DstAddRec->getStepRecurrence(*SE));
+ SrcLoop = DstAddRec->getLoop();
+ }
+ else
+ llvm_unreachable("RDIV reached by surprising SCEVs");
+ }
+ else
+ llvm_unreachable("RDIV expected at least one AddRec");
+ return exactRDIVtest(SrcCoeff, DstCoeff,
+ SrcConst, DstConst,
+ SrcLoop, DstLoop,
+ Result) ||
+ gcdMIVtest(Src, Dst, Result) ||
+ symbolicRDIVtest(SrcCoeff, DstCoeff,
+ SrcConst, DstConst,
+ SrcLoop, DstLoop);
+}
+
+
+// Tests the single-subscript MIV pair (Src and Dst) for dependence.
+// Return true if dependence disproved.
+// Can sometimes refine direction vectors.
+bool DependenceAnalysis::testMIV(const SCEV *Src,
+ const SCEV *Dst,
+ const SmallBitVector &Loops,
+ FullDependence &Result) const {
+ DEBUG(dbgs() << " src = " << *Src << "\n");
+ DEBUG(dbgs() << " dst = " << *Dst << "\n");
+ Result.Consistent = false;
+ return gcdMIVtest(Src, Dst, Result) ||
+ banerjeeMIVtest(Src, Dst, Loops, Result);
+}
+
+
+// Given a product, e.g., 10*X*Y, returns the first constant operand,
+// in this case 10. If there is no constant part, returns NULL.
+static
+const SCEVConstant *getConstantPart(const SCEVMulExpr *Product) {
+ for (unsigned Op = 0, Ops = Product->getNumOperands(); Op < Ops; Op++) {
+ if (const SCEVConstant *Constant = dyn_cast<SCEVConstant>(Product->getOperand(Op)))
+ return Constant;
+ }
+ return NULL;
+}
+
+
+//===----------------------------------------------------------------------===//
+// gcdMIVtest -
+// Tests an MIV subscript pair for dependence.
+// Returns true if any possible dependence is disproved.
+// Marks the result as inconsistant.
+// Can sometimes disprove the equal direction for 1 or more loops,
+// as discussed in Michael Wolfe's book,
+// High Performance Compilers for Parallel Computing, page 235.
+//
+// We spend some effort (code!) to handle cases like
+// [10*i + 5*N*j + 15*M + 6], where i and j are induction variables,
+// but M and N are just loop-invariant variables.
+// This should help us handle linearized subscripts;
+// also makes this test a useful backup to the various SIV tests.
+//
+// It occurs to me that the presence of loop-invariant variables
+// changes the nature of the test from "greatest common divisor"
+// to "a common divisor!"
+bool DependenceAnalysis::gcdMIVtest(const SCEV *Src,
+ const SCEV *Dst,
+ FullDependence &Result) const {
+ DEBUG(dbgs() << "starting gcd\n");
+ ++GCDapplications;
+ unsigned BitWidth = Src->getType()->getIntegerBitWidth();
+ APInt RunningGCD = APInt::getNullValue(BitWidth);
+
+ // Examine Src coefficients.
+ // Compute running GCD and record source constant.
+ // Because we're looking for the constant at the end of the chain,
+ // we can't quit the loop just because the GCD == 1.
+ const SCEV *Coefficients = Src;
+ while (const SCEVAddRecExpr *AddRec =
+ dyn_cast<SCEVAddRecExpr>(Coefficients)) {
+ const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
+ const SCEVConstant *Constant = dyn_cast<SCEVConstant>(Coeff);
+ if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Coeff))
+ // If the coefficient is the product of a constant and other stuff,
+ // we can use the constant in the GCD computation.
+ Constant = getConstantPart(Product);
+ if (!Constant)
+ return false;
+ APInt ConstCoeff = Constant->getValue()->getValue();
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
+ Coefficients = AddRec->getStart();
+ }
+ const SCEV *SrcConst = Coefficients;
+
+ // Examine Dst coefficients.
+ // Compute running GCD and record destination constant.
+ // Because we're looking for the constant at the end of the chain,
+ // we can't quit the loop just because the GCD == 1.
+ Coefficients = Dst;
+ while (const SCEVAddRecExpr *AddRec =
+ dyn_cast<SCEVAddRecExpr>(Coefficients)) {
+ const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
+ const SCEVConstant *Constant = dyn_cast<SCEVConstant>(Coeff);
+ if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Coeff))
+ // If the coefficient is the product of a constant and other stuff,
+ // we can use the constant in the GCD computation.
+ Constant = getConstantPart(Product);
+ if (!Constant)
+ return false;
+ APInt ConstCoeff = Constant->getValue()->getValue();
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
+ Coefficients = AddRec->getStart();
+ }
+ const SCEV *DstConst = Coefficients;
+
+ APInt ExtraGCD = APInt::getNullValue(BitWidth);
+ const SCEV *Delta = SE->getMinusSCEV(DstConst, SrcConst);
+ DEBUG(dbgs() << " Delta = " << *Delta << "\n");
+ const SCEVConstant *Constant = dyn_cast<SCEVConstant>(Delta);
+ if (const SCEVAddExpr *Sum = dyn_cast<SCEVAddExpr>(Delta)) {
+ // If Delta is a sum of products, we may be able to make further progress.
+ for (unsigned Op = 0, Ops = Sum->getNumOperands(); Op < Ops; Op++) {
+ const SCEV *Operand = Sum->getOperand(Op);
+ if (isa<SCEVConstant>(Operand)) {
+ assert(!Constant && "Surprised to find multiple constants");
+ Constant = cast<SCEVConstant>(Operand);
+ }
+ else if (isa<SCEVMulExpr>(Operand)) {
+ // Search for constant operand to participate in GCD;
+ // If none found; return false.
+ const SCEVConstant *ConstOp =
+ getConstantPart(cast<SCEVMulExpr>(Operand));
+ APInt ConstOpValue = ConstOp->getValue()->getValue();
+ ExtraGCD = APIntOps::GreatestCommonDivisor(ExtraGCD,
+ ConstOpValue.abs());
+ }
+ else
+ return false;
+ }
+ }
+ if (!Constant)
+ return false;
+ APInt ConstDelta = cast<SCEVConstant>(Constant)->getValue()->getValue();
+ DEBUG(dbgs() << " ConstDelta = " << ConstDelta << "\n");
+ if (ConstDelta == 0)
+ return false;
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ExtraGCD);
+ DEBUG(dbgs() << " RunningGCD = " << RunningGCD << "\n");
+ APInt Remainder = ConstDelta.srem(RunningGCD);
+ if (Remainder != 0) {
+ ++GCDindependence;
+ return true;
+ }
+
+ // Try to disprove equal directions.
+ // For example, given a subscript pair [3*i + 2*j] and [i' + 2*j' - 1],
+ // the code above can't disprove the dependence because the GCD = 1.
+ // So we consider what happen if i = i' and what happens if j = j'.
+ // If i = i', we can simplify the subscript to [2*i + 2*j] and [2*j' - 1],
+ // which is infeasible, so we can disallow the = direction for the i level.
+ // Setting j = j' doesn't help matters, so we end up with a direction vector
+ // of [<>, *]
+ //
+ // Given A[5*i + 10*j*M + 9*M*N] and A[15*i + 20*j*M - 21*N*M + 5],
+ // we need to remember that the constant part is 5 and the RunningGCD should
+ // be initialized to ExtraGCD = 30.
+ DEBUG(dbgs() << " ExtraGCD = " << ExtraGCD << '\n');
+
+ bool Improved = false;
+ Coefficients = Src;
+ while (const SCEVAddRecExpr *AddRec =
+ dyn_cast<SCEVAddRecExpr>(Coefficients)) {
+ Coefficients = AddRec->getStart();
+ const Loop *CurLoop = AddRec->getLoop();
+ RunningGCD = ExtraGCD;
+ const SCEV *SrcCoeff = AddRec->getStepRecurrence(*SE);
+ const SCEV *DstCoeff = SE->getMinusSCEV(SrcCoeff, SrcCoeff);
+ const SCEV *Inner = Src;
+ while (RunningGCD != 1 && isa<SCEVAddRecExpr>(Inner)) {
+ AddRec = cast<SCEVAddRecExpr>(Inner);
+ const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
+ if (CurLoop == AddRec->getLoop())
+ ; // SrcCoeff == Coeff
+ else {
+ if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Coeff))
+ // If the coefficient is the product of a constant and other stuff,
+ // we can use the constant in the GCD computation.
+ Constant = getConstantPart(Product);
+ else
+ Constant = cast<SCEVConstant>(Coeff);
+ APInt ConstCoeff = Constant->getValue()->getValue();
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
+ }
+ Inner = AddRec->getStart();
+ }
+ Inner = Dst;
+ while (RunningGCD != 1 && isa<SCEVAddRecExpr>(Inner)) {
+ AddRec = cast<SCEVAddRecExpr>(Inner);
+ const SCEV *Coeff = AddRec->getStepRecurrence(*SE);
+ if (CurLoop == AddRec->getLoop())
+ DstCoeff = Coeff;
+ else {
+ if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Coeff))
+ // If the coefficient is the product of a constant and other stuff,
+ // we can use the constant in the GCD computation.
+ Constant = getConstantPart(Product);
+ else
+ Constant = cast<SCEVConstant>(Coeff);
+ APInt ConstCoeff = Constant->getValue()->getValue();
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
+ }
+ Inner = AddRec->getStart();
+ }
+ Delta = SE->getMinusSCEV(SrcCoeff, DstCoeff);
+ if (const SCEVMulExpr *Product = dyn_cast<SCEVMulExpr>(Delta))
+ // If the coefficient is the product of a constant and other stuff,
+ // we can use the constant in the GCD computation.
+ Constant = getConstantPart(Product);
+ else if (isa<SCEVConstant>(Delta))
+ Constant = cast<SCEVConstant>(Delta);
+ else {
+ // The difference of the two coefficients might not be a product
+ // or constant, in which case we give up on this direction.
+ continue;
+ }
+ APInt ConstCoeff = Constant->getValue()->getValue();
+ RunningGCD = APIntOps::GreatestCommonDivisor(RunningGCD, ConstCoeff.abs());
+ DEBUG(dbgs() << "\tRunningGCD = " << RunningGCD << "\n");
+ if (RunningGCD != 0) {
+ Remainder = ConstDelta.srem(RunningGCD);
+ DEBUG(dbgs() << "\tRemainder = " << Remainder << "\n");
+ if (Remainder != 0) {
+ unsigned Level = mapSrcLoop(CurLoop);
+ Result.DV[Level - 1].Direction &= ~Dependence::DVEntry::EQ;
+ Improved = true;
+ }
+ }
+ }
+ if (Improved)
+ ++GCDsuccesses;
+ DEBUG(dbgs() << "all done\n");
+ return false;
+}
+
+
+//===----------------------------------------------------------------------===//
+// banerjeeMIVtest -
+// Use Banerjee's Inequalities to test an MIV subscript pair.
+// (Wolfe, in the race-car book, calls this the Extreme Value Test.)
+// Generally follows the discussion in Section 2.5.2 of
+//
+// Optimizing Supercompilers for Supercomputers
+// Michael Wolfe
+//
+// The inequalities given on page 25 are simplified in that loops are
+// normalized so that the lower bound is always 0 and the stride is always 1.
+// For example, Wolfe gives
+//
+// LB^<_k = (A^-_k - B_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
+//
+// where A_k is the coefficient of the kth index in the source subscript,
+// B_k is the coefficient of the kth index in the destination subscript,
+// U_k is the upper bound of the kth index, L_k is the lower bound of the Kth
+// index, and N_k is the stride of the kth index. Since all loops are normalized
+// by the SCEV package, N_k = 1 and L_k = 0, allowing us to simplify the
+// equation to
+//
+// LB^<_k = (A^-_k - B_k)^- (U_k - 0 - 1) + (A_k - B_k)0 - B_k 1
+// = (A^-_k - B_k)^- (U_k - 1) - B_k
+//
+// Similar simplifications are possible for the other equations.
+//
+// When we can't determine the number of iterations for a loop,
+// we use NULL as an indicator for the worst case, infinity.
+// When computing the upper bound, NULL denotes +inf;
+// for the lower bound, NULL denotes -inf.
+//
+// Return true if dependence disproved.
+bool DependenceAnalysis::banerjeeMIVtest(const SCEV *Src,
+ const SCEV *Dst,
+ const SmallBitVector &Loops,
+ FullDependence &Result) const {
+ DEBUG(dbgs() << "starting Banerjee\n");
+ ++BanerjeeApplications;
+ DEBUG(dbgs() << " Src = " << *Src << '\n');
+ const SCEV *A0;
+ CoefficientInfo *A = collectCoeffInfo(Src, true, A0);
+ DEBUG(dbgs() << " Dst = " << *Dst << '\n');
+ const SCEV *B0;
+ CoefficientInfo *B = collectCoeffInfo(Dst, false, B0);
+ BoundInfo *Bound = new BoundInfo[MaxLevels + 1];
+ const SCEV *Delta = SE->getMinusSCEV(B0, A0);
+ DEBUG(dbgs() << "\tDelta = " << *Delta << '\n');
+
+ // Compute bounds for all the * directions.
+ DEBUG(dbgs() << "\tBounds[*]\n");
+ for (unsigned K = 1; K <= MaxLevels; ++K) {
+ Bound[K].Iterations = A[K].Iterations ? A[K].Iterations : B[K].Iterations;
+ Bound[K].Direction = Dependence::DVEntry::ALL;
+ Bound[K].DirSet = Dependence::DVEntry::NONE;
+ findBoundsALL(A, B, Bound, K);
+#ifndef NDEBUG
+ DEBUG(dbgs() << "\t " << K << '\t');
+ if (Bound[K].Lower[Dependence::DVEntry::ALL])
+ DEBUG(dbgs() << *Bound[K].Lower[Dependence::DVEntry::ALL] << '\t');
+ else
+ DEBUG(dbgs() << "-inf\t");
+ if (Bound[K].Upper[Dependence::DVEntry::ALL])
+ DEBUG(dbgs() << *Bound[K].Upper[Dependence::DVEntry::ALL] << '\n');
+ else
+ DEBUG(dbgs() << "+inf\n");
+#endif
+ }
+
+ // Test the *, *, *, ... case.
+ bool Disproved = false;
+ if (testBounds(Dependence::DVEntry::ALL, 0, Bound, Delta)) {
+ // Explore the direction vector hierarchy.
+ unsigned DepthExpanded = 0;
+ unsigned NewDeps = exploreDirections(1, A, B, Bound,
+ Loops, DepthExpanded, Delta);
+ if (NewDeps > 0) {
+ bool Improved = false;
+ for (unsigned K = 1; K <= CommonLevels; ++K) {
+ if (Loops[K]) {
+ unsigned Old = Result.DV[K - 1].Direction;
+ Result.DV[K - 1].Direction = Old & Bound[K].DirSet;
+ Improved |= Old != Result.DV[K - 1].Direction;
+ if (!Result.DV[K - 1].Direction) {
+ Improved = false;
+ Disproved = true;
+ break;
+ }
+ }
+ }
+ if (Improved)
+ ++BanerjeeSuccesses;
+ }
+ else {
+ ++BanerjeeIndependence;
+ Disproved = true;
+ }
+ }
+ else {
+ ++BanerjeeIndependence;
+ Disproved = true;
+ }
+ delete [] Bound;
+ delete [] A;
+ delete [] B;
+ return Disproved;
+}
+
+
+// Hierarchically expands the direction vector
+// search space, combining the directions of discovered dependences
+// in the DirSet field of Bound. Returns the number of distinct
+// dependences discovered. If the dependence is disproved,
+// it will return 0.
+unsigned DependenceAnalysis::exploreDirections(unsigned Level,
+ CoefficientInfo *A,
+ CoefficientInfo *B,
+ BoundInfo *Bound,
+ const SmallBitVector &Loops,
+ unsigned &DepthExpanded,
+ const SCEV *Delta) const {
+ if (Level > CommonLevels) {
+ // record result
+ DEBUG(dbgs() << "\t[");
+ for (unsigned K = 1; K <= CommonLevels; ++K) {
+ if (Loops[K]) {
+ Bound[K].DirSet |= Bound[K].Direction;
+#ifndef NDEBUG
+ switch (Bound[K].Direction) {
+ case Dependence::DVEntry::LT:
+ DEBUG(dbgs() << " <");
+ break;
+ case Dependence::DVEntry::EQ:
+ DEBUG(dbgs() << " =");
+ break;
+ case Dependence::DVEntry::GT:
+ DEBUG(dbgs() << " >");
+ break;
+ case Dependence::DVEntry::ALL:
+ DEBUG(dbgs() << " *");
+ break;
+ default:
+ llvm_unreachable("unexpected Bound[K].Direction");
+ }
+#endif
+ }
+ }
+ DEBUG(dbgs() << " ]\n");
+ return 1;
+ }
+ if (Loops[Level]) {
+ if (Level > DepthExpanded) {
+ DepthExpanded = Level;
+ // compute bounds for <, =, > at current level
+ findBoundsLT(A, B, Bound, Level);
+ findBoundsGT(A, B, Bound, Level);
+ findBoundsEQ(A, B, Bound, Level);
+#ifndef NDEBUG
+ DEBUG(dbgs() << "\tBound for level = " << Level << '\n');
+ DEBUG(dbgs() << "\t <\t");
+ if (Bound[Level].Lower[Dependence::DVEntry::LT])
+ DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::LT] << '\t');
+ else
+ DEBUG(dbgs() << "-inf\t");
+ if (Bound[Level].Upper[Dependence::DVEntry::LT])
+ DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::LT] << '\n');
+ else
+ DEBUG(dbgs() << "+inf\n");
+ DEBUG(dbgs() << "\t =\t");
+ if (Bound[Level].Lower[Dependence::DVEntry::EQ])
+ DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::EQ] << '\t');
+ else
+ DEBUG(dbgs() << "-inf\t");
+ if (Bound[Level].Upper[Dependence::DVEntry::EQ])
+ DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::EQ] << '\n');
+ else
+ DEBUG(dbgs() << "+inf\n");
+ DEBUG(dbgs() << "\t >\t");
+ if (Bound[Level].Lower[Dependence::DVEntry::GT])
+ DEBUG(dbgs() << *Bound[Level].Lower[Dependence::DVEntry::GT] << '\t');
+ else
+ DEBUG(dbgs() << "-inf\t");
+ if (Bound[Level].Upper[Dependence::DVEntry::GT])
+ DEBUG(dbgs() << *Bound[Level].Upper[Dependence::DVEntry::GT] << '\n');
+ else
+ DEBUG(dbgs() << "+inf\n");
+#endif
+ }
+
+ unsigned NewDeps = 0;
+
+ // test bounds for <, *, *, ...
+ if (testBounds(Dependence::DVEntry::LT, Level, Bound, Delta))
+ NewDeps += exploreDirections(Level + 1, A, B, Bound,
+ Loops, DepthExpanded, Delta);
+
+ // Test bounds for =, *, *, ...
+ if (testBounds(Dependence::DVEntry::EQ, Level, Bound, Delta))
+ NewDeps += exploreDirections(Level + 1, A, B, Bound,
+ Loops, DepthExpanded, Delta);
+
+ // test bounds for >, *, *, ...
+ if (testBounds(Dependence::DVEntry::GT, Level, Bound, Delta))
+ NewDeps += exploreDirections(Level + 1, A, B, Bound,
+ Loops, DepthExpanded, Delta);
+
+ Bound[Level].Direction = Dependence::DVEntry::ALL;
+ return NewDeps;
+ }
+ else
+ return exploreDirections(Level + 1, A, B, Bound, Loops, DepthExpanded, Delta);
+}
+
+
+// Returns true iff the current bounds are plausible.
+bool DependenceAnalysis::testBounds(unsigned char DirKind,
+ unsigned Level,
+ BoundInfo *Bound,
+ const SCEV *Delta) const {
+ Bound[Level].Direction = DirKind;
+ if (const SCEV *LowerBound = getLowerBound(Bound))
+ if (isKnownPredicate(CmpInst::ICMP_SGT, LowerBound, Delta))
+ return false;
+ if (const SCEV *UpperBound = getUpperBound(Bound))
+ if (isKnownPredicate(CmpInst::ICMP_SGT, Delta, UpperBound))
+ return false;
+ return true;
+}
+
+
+// Computes the upper and lower bounds for level K
+// using the * direction. Records them in Bound.
+// Wolfe gives the equations
+//
+// LB^*_k = (A^-_k - B^+_k)(U_k - L_k) + (A_k - B_k)L_k
+// UB^*_k = (A^+_k - B^-_k)(U_k - L_k) + (A_k - B_k)L_k
+//
+// Since we normalize loops, we can simplify these equations to
+//
+// LB^*_k = (A^-_k - B^+_k)U_k
+// UB^*_k = (A^+_k - B^-_k)U_k
+//
+// We must be careful to handle the case where the upper bound is unknown.
+// Note that the lower bound is always <= 0
+// and the upper bound is always >= 0.
+void DependenceAnalysis::findBoundsALL(CoefficientInfo *A,
+ CoefficientInfo *B,
+ BoundInfo *Bound,
+ unsigned K) const {
+ Bound[K].Lower[Dependence::DVEntry::ALL] = NULL; // Default value = -infinity.
+ Bound[K].Upper[Dependence::DVEntry::ALL] = NULL; // Default value = +infinity.
+ if (Bound[K].Iterations) {
+ Bound[K].Lower[Dependence::DVEntry::ALL] =
+ SE->getMulExpr(SE->getMinusSCEV(A[K].NegPart, B[K].PosPart),
+ Bound[K].Iterations);
+ Bound[K].Upper[Dependence::DVEntry::ALL] =
+ SE->getMulExpr(SE->getMinusSCEV(A[K].PosPart, B[K].NegPart),
+ Bound[K].Iterations);
+ }
+ else {
+ // If the difference is 0, we won't need to know the number of iterations.
+ if (isKnownPredicate(CmpInst::ICMP_EQ, A[K].NegPart, B[K].PosPart))
+ Bound[K].Lower[Dependence::DVEntry::ALL] =
+ SE->getConstant(A[K].Coeff->getType(), 0);
+ if (isKnownPredicate(CmpInst::ICMP_EQ, A[K].PosPart, B[K].NegPart))
+ Bound[K].Upper[Dependence::DVEntry::ALL] =
+ SE->getConstant(A[K].Coeff->getType(), 0);
+ }
+}
+
+
+// Computes the upper and lower bounds for level K
+// using the = direction. Records them in Bound.
+// Wolfe gives the equations
+//
+// LB^=_k = (A_k - B_k)^- (U_k - L_k) + (A_k - B_k)L_k
+// UB^=_k = (A_k - B_k)^+ (U_k - L_k) + (A_k - B_k)L_k
+//
+// Since we normalize loops, we can simplify these equations to
+//
+// LB^=_k = (A_k - B_k)^- U_k
+// UB^=_k = (A_k - B_k)^+ U_k
+//
+// We must be careful to handle the case where the upper bound is unknown.
+// Note that the lower bound is always <= 0
+// and the upper bound is always >= 0.
+void DependenceAnalysis::findBoundsEQ(CoefficientInfo *A,
+ CoefficientInfo *B,
+ BoundInfo *Bound,
+ unsigned K) const {
+ Bound[K].Lower[Dependence::DVEntry::EQ] = NULL; // Default value = -infinity.
+ Bound[K].Upper[Dependence::DVEntry::EQ] = NULL; // Default value = +infinity.
+ if (Bound[K].Iterations) {
+ const SCEV *Delta = SE->getMinusSCEV(A[K].Coeff, B[K].Coeff);
+ const SCEV *NegativePart = getNegativePart(Delta);
+ Bound[K].Lower[Dependence::DVEntry::EQ] =
+ SE->getMulExpr(NegativePart, Bound[K].Iterations);
+ const SCEV *PositivePart = getPositivePart(Delta);
+ Bound[K].Upper[Dependence::DVEntry::EQ] =
+ SE->getMulExpr(PositivePart, Bound[K].Iterations);
+ }
+ else {
+ // If the positive/negative part of the difference is 0,
+ // we won't need to know the number of iterations.
+ const SCEV *Delta = SE->getMinusSCEV(A[K].Coeff, B[K].Coeff);
+ const SCEV *NegativePart = getNegativePart(Delta);
+ if (NegativePart->isZero())
+ Bound[K].Lower[Dependence::DVEntry::EQ] = NegativePart; // Zero
+ const SCEV *PositivePart = getPositivePart(Delta);
+ if (PositivePart->isZero())
+ Bound[K].Upper[Dependence::DVEntry::EQ] = PositivePart; // Zero
+ }
+}
+
+
+// Computes the upper and lower bounds for level K
+// using the < direction. Records them in Bound.
+// Wolfe gives the equations
+//
+// LB^<_k = (A^-_k - B_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
+// UB^<_k = (A^+_k - B_k)^+ (U_k - L_k - N_k) + (A_k - B_k)L_k - B_k N_k
+//
+// Since we normalize loops, we can simplify these equations to
+//
+// LB^<_k = (A^-_k - B_k)^- (U_k - 1) - B_k
+// UB^<_k = (A^+_k - B_k)^+ (U_k - 1) - B_k
+//
+// We must be careful to handle the case where the upper bound is unknown.
+void DependenceAnalysis::findBoundsLT(CoefficientInfo *A,
+ CoefficientInfo *B,
+ BoundInfo *Bound,
+ unsigned K) const {
+ Bound[K].Lower[Dependence::DVEntry::LT] = NULL; // Default value = -infinity.
+ Bound[K].Upper[Dependence::DVEntry::LT] = NULL; // Default value = +infinity.
+ if (Bound[K].Iterations) {
+ const SCEV *Iter_1 =
+ SE->getMinusSCEV(Bound[K].Iterations,
+ SE->getConstant(Bound[K].Iterations->getType(), 1));
+ const SCEV *NegPart =
+ getNegativePart(SE->getMinusSCEV(A[K].NegPart, B[K].Coeff));
+ Bound[K].Lower[Dependence::DVEntry::LT] =
+ SE->getMinusSCEV(SE->getMulExpr(NegPart, Iter_1), B[K].Coeff);
+ const SCEV *PosPart =
+ getPositivePart(SE->getMinusSCEV(A[K].PosPart, B[K].Coeff));
+ Bound[K].Upper[Dependence::DVEntry::LT] =
+ SE->getMinusSCEV(SE->getMulExpr(PosPart, Iter_1), B[K].Coeff);
+ }
+ else {
+ // If the positive/negative part of the difference is 0,
+ // we won't need to know the number of iterations.
+ const SCEV *NegPart =
+ getNegativePart(SE->getMinusSCEV(A[K].NegPart, B[K].Coeff));
+ if (NegPart->isZero())
+ Bound[K].Lower[Dependence::DVEntry::LT] = SE->getNegativeSCEV(B[K].Coeff);
+ const SCEV *PosPart =
+ getPositivePart(SE->getMinusSCEV(A[K].PosPart, B[K].Coeff));
+ if (PosPart->isZero())
+ Bound[K].Upper[Dependence::DVEntry::LT] = SE->getNegativeSCEV(B[K].Coeff);
+ }
+}
+
+
+// Computes the upper and lower bounds for level K
+// using the > direction. Records them in Bound.
+// Wolfe gives the equations
+//
+// LB^>_k = (A_k - B^+_k)^- (U_k - L_k - N_k) + (A_k - B_k)L_k + A_k N_k
+// UB^>_k = (A_k - B^-_k)^+ (U_k - L_k - N_k) + (A_k - B_k)L_k + A_k N_k
+//
+// Since we normalize loops, we can simplify these equations to
+//
+// LB^>_k = (A_k - B^+_k)^- (U_k - 1) + A_k
+// UB^>_k = (A_k - B^-_k)^+ (U_k - 1) + A_k
+//
+// We must be careful to handle the case where the upper bound is unknown.
+void DependenceAnalysis::findBoundsGT(CoefficientInfo *A,
+ CoefficientInfo *B,
+ BoundInfo *Bound,
+ unsigned K) const {
+ Bound[K].Lower[Dependence::DVEntry::GT] = NULL; // Default value = -infinity.
+ Bound[K].Upper[Dependence::DVEntry::GT] = NULL; // Default value = +infinity.
+ if (Bound[K].Iterations) {
+ const SCEV *Iter_1 =
+ SE->getMinusSCEV(Bound[K].Iterations,
+ SE->getConstant(Bound[K].Iterations->getType(), 1));
+ const SCEV *NegPart =
+ getNegativePart(SE->getMinusSCEV(A[K].Coeff, B[K].PosPart));
+ Bound[K].Lower[Dependence::DVEntry::GT] =
+ SE->getAddExpr(SE->getMulExpr(NegPart, Iter_1), A[K].Coeff);
+ const SCEV *PosPart =
+ getPositivePart(SE->getMinusSCEV(A[K].Coeff, B[K].NegPart));
+ Bound[K].Upper[Dependence::DVEntry::GT] =
+ SE->getAddExpr(SE->getMulExpr(PosPart, Iter_1), A[K].Coeff);
+ }
+ else {
+ // If the positive/negative part of the difference is 0,
+ // we won't need to know the number of iterations.
+ const SCEV *NegPart = getNegativePart(SE->getMinusSCEV(A[K].Coeff, B[K].PosPart));
+ if (NegPart->isZero())
+ Bound[K].Lower[Dependence::DVEntry::GT] = A[K].Coeff;
+ const SCEV *PosPart = getPositivePart(SE->getMinusSCEV(A[K].Coeff, B[K].NegPart));
+ if (PosPart->isZero())
+ Bound[K].Upper[Dependence::DVEntry::GT] = A[K].Coeff;
+ }
+}
+
+
+// X^+ = max(X, 0)
+const SCEV *DependenceAnalysis::getPositivePart(const SCEV *X) const {
+ return SE->getSMaxExpr(X, SE->getConstant(X->getType(), 0));
+}
+
+
+// X^- = min(X, 0)
+const SCEV *DependenceAnalysis::getNegativePart(const SCEV *X) const {
+ return SE->getSMinExpr(X, SE->getConstant(X->getType(), 0));
+}
+
+
+// Walks through the subscript,
+// collecting each coefficient, the associated loop bounds,
+// and recording its positive and negative parts for later use.
+DependenceAnalysis::CoefficientInfo *
+DependenceAnalysis::collectCoeffInfo(const SCEV *Subscript,
+ bool SrcFlag,
+ const SCEV *&Constant) const {
+ const SCEV *Zero = SE->getConstant(Subscript->getType(), 0);
+ CoefficientInfo *CI = new CoefficientInfo[MaxLevels + 1];
+ for (unsigned K = 1; K <= MaxLevels; ++K) {
+ CI[K].Coeff = Zero;
+ CI[K].PosPart = Zero;
+ CI[K].NegPart = Zero;
+ CI[K].Iterations = NULL;
+ }
+ while (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Subscript)) {
+ const Loop *L = AddRec->getLoop();
+ unsigned K = SrcFlag ? mapSrcLoop(L) : mapDstLoop(L);
+ CI[K].Coeff = AddRec->getStepRecurrence(*SE);
+ CI[K].PosPart = getPositivePart(CI[K].Coeff);
+ CI[K].NegPart = getNegativePart(CI[K].Coeff);
+ CI[K].Iterations = collectUpperBound(L, Subscript->getType());
+ Subscript = AddRec->getStart();
+ }
+ Constant = Subscript;
+#ifndef NDEBUG
+ DEBUG(dbgs() << "\tCoefficient Info\n");
+ for (unsigned K = 1; K <= MaxLevels; ++K) {
+ DEBUG(dbgs() << "\t " << K << "\t" << *CI[K].Coeff);
+ DEBUG(dbgs() << "\tPos Part = ");
+ DEBUG(dbgs() << *CI[K].PosPart);
+ DEBUG(dbgs() << "\tNeg Part = ");
+ DEBUG(dbgs() << *CI[K].NegPart);
+ DEBUG(dbgs() << "\tUpper Bound = ");
+ if (CI[K].Iterations)
+ DEBUG(dbgs() << *CI[K].Iterations);
+ else
+ DEBUG(dbgs() << "+inf");
+ DEBUG(dbgs() << '\n');
+ }
+ DEBUG(dbgs() << "\t Constant = " << *Subscript << '\n');
+#endif
+ return CI;
+}
+
+
+// Looks through all the bounds info and
+// computes the lower bound given the current direction settings
+// at each level. If the lower bound for any level is -inf,
+// the result is -inf.
+const SCEV *DependenceAnalysis::getLowerBound(BoundInfo *Bound) const {
+ const SCEV *Sum = Bound[1].Lower[Bound[1].Direction];
+ for (unsigned K = 2; Sum && K <= MaxLevels; ++K) {
+ if (Bound[K].Lower[Bound[K].Direction])
+ Sum = SE->getAddExpr(Sum, Bound[K].Lower[Bound[K].Direction]);
+ else
+ Sum = NULL;
+ }
+ return Sum;
+}
+
+
+// Looks through all the bounds info and
+// computes the upper bound given the current direction settings
+// at each level. If the upper bound at any level is +inf,
+// the result is +inf.
+const SCEV *DependenceAnalysis::getUpperBound(BoundInfo *Bound) const {
+ const SCEV *Sum = Bound[1].Upper[Bound[1].Direction];
+ for (unsigned K = 2; Sum && K <= MaxLevels; ++K) {
+ if (Bound[K].Upper[Bound[K].Direction])
+ Sum = SE->getAddExpr(Sum, Bound[K].Upper[Bound[K].Direction]);
+ else
+ Sum = NULL;
+ }
+ return Sum;
+}
+
+
+//===----------------------------------------------------------------------===//
+// Constraint manipulation for Delta test.
+
+// Given a linear SCEV,
+// return the coefficient (the step)
+// corresponding to the specified loop.
+// If there isn't one, return 0.
+// For example, given a*i + b*j + c*k, zeroing the coefficient
+// corresponding to the j loop would yield b.
+const SCEV *DependenceAnalysis::findCoefficient(const SCEV *Expr,
+ const Loop *TargetLoop) const {
+ const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
+ if (!AddRec)
+ return SE->getConstant(Expr->getType(), 0);
+ if (AddRec->getLoop() == TargetLoop)
+ return AddRec->getStepRecurrence(*SE);
+ return findCoefficient(AddRec->getStart(), TargetLoop);
+}
+
+
+// Given a linear SCEV,
+// return the SCEV given by zeroing out the coefficient
+// corresponding to the specified loop.
+// For example, given a*i + b*j + c*k, zeroing the coefficient
+// corresponding to the j loop would yield a*i + c*k.
+const SCEV *DependenceAnalysis::zeroCoefficient(const SCEV *Expr,
+ const Loop *TargetLoop) const {
+ const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
+ if (!AddRec)
+ return Expr; // ignore
+ if (AddRec->getLoop() == TargetLoop)
+ return AddRec->getStart();
+ return SE->getAddRecExpr(zeroCoefficient(AddRec->getStart(), TargetLoop),
+ AddRec->getStepRecurrence(*SE),
+ AddRec->getLoop(),
+ AddRec->getNoWrapFlags());
+}
+
+
+// Given a linear SCEV Expr,
+// return the SCEV given by adding some Value to the
+// coefficient corresponding to the specified TargetLoop.
+// For example, given a*i + b*j + c*k, adding 1 to the coefficient
+// corresponding to the j loop would yield a*i + (b+1)*j + c*k.
+const SCEV *DependenceAnalysis::addToCoefficient(const SCEV *Expr,
+ const Loop *TargetLoop,
+ const SCEV *Value) const {
+ const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Expr);
+ if (!AddRec) // create a new addRec
+ return SE->getAddRecExpr(Expr,
+ Value,
+ TargetLoop,
+ SCEV::FlagAnyWrap); // Worst case, with no info.
+ if (AddRec->getLoop() == TargetLoop) {
+ const SCEV *Sum = SE->getAddExpr(AddRec->getStepRecurrence(*SE), Value);
+ if (Sum->isZero())
+ return AddRec->getStart();
+ return SE->getAddRecExpr(AddRec->getStart(),
+ Sum,
+ AddRec->getLoop(),
+ AddRec->getNoWrapFlags());
+ }
+ return SE->getAddRecExpr(addToCoefficient(AddRec->getStart(),
+ TargetLoop, Value),
+ AddRec->getStepRecurrence(*SE),
+ AddRec->getLoop(),
+ AddRec->getNoWrapFlags());
+}
+
+
+// Review the constraints, looking for opportunities
+// to simplify a subscript pair (Src and Dst).
+// Return true if some simplification occurs.
+// If the simplification isn't exact (that is, if it is conservative
+// in terms of dependence), set consistent to false.
+// Corresponds to Figure 5 from the paper
+//
+// Practical Dependence Testing
+// Goff, Kennedy, Tseng
+// PLDI 1991
+bool DependenceAnalysis::propagate(const SCEV *&Src,
+ const SCEV *&Dst,
+ SmallBitVector &Loops,
+ SmallVector<Constraint, 4> &Constraints,
+ bool &Consistent) {
+ bool Result = false;
+ for (int LI = Loops.find_first(); LI >= 0; LI = Loops.find_next(LI)) {
+ DEBUG(dbgs() << "\t Constraint[" << LI << "] is");
+ DEBUG(Constraints[LI].dump(dbgs()));
+ if (Constraints[LI].isDistance())
+ Result |= propagateDistance(Src, Dst, Constraints[LI], Consistent);
+ else if (Constraints[LI].isLine())
+ Result |= propagateLine(Src, Dst, Constraints[LI], Consistent);
+ else if (Constraints[LI].isPoint())
+ Result |= propagatePoint(Src, Dst, Constraints[LI]);
+ }
+ return Result;
+}
+
+
+// Attempt to propagate a distance
+// constraint into a subscript pair (Src and Dst).
+// Return true if some simplification occurs.
+// If the simplification isn't exact (that is, if it is conservative
+// in terms of dependence), set consistent to false.
+bool DependenceAnalysis::propagateDistance(const SCEV *&Src,
+ const SCEV *&Dst,
+ Constraint &CurConstraint,
+ bool &Consistent) {
+ const Loop *CurLoop = CurConstraint.getAssociatedLoop();
+ DEBUG(dbgs() << "\t\tSrc is " << *Src << "\n");
+ const SCEV *A_K = findCoefficient(Src, CurLoop);
+ if (A_K->isZero())
+ return false;
+ const SCEV *DA_K = SE->getMulExpr(A_K, CurConstraint.getD());
+ Src = SE->getMinusSCEV(Src, DA_K);
+ Src = zeroCoefficient(Src, CurLoop);
+ DEBUG(dbgs() << "\t\tnew Src is " << *Src << "\n");
+ DEBUG(dbgs() << "\t\tDst is " << *Dst << "\n");
+ Dst = addToCoefficient(Dst, CurLoop, SE->getNegativeSCEV(A_K));
+ DEBUG(dbgs() << "\t\tnew Dst is " << *Dst << "\n");
+ if (!findCoefficient(Dst, CurLoop)->isZero())
+ Consistent = false;
+ return true;
+}
+
+
+// Attempt to propagate a line
+// constraint into a subscript pair (Src and Dst).
+// Return true if some simplification occurs.
+// If the simplification isn't exact (that is, if it is conservative
+// in terms of dependence), set consistent to false.
+bool DependenceAnalysis::propagateLine(const SCEV *&Src,
+ const SCEV *&Dst,
+ Constraint &CurConstraint,
+ bool &Consistent) {
+ const Loop *CurLoop = CurConstraint.getAssociatedLoop();
+ const SCEV *A = CurConstraint.getA();
+ const SCEV *B = CurConstraint.getB();
+ const SCEV *C = CurConstraint.getC();
+ DEBUG(dbgs() << "\t\tA = " << *A << ", B = " << *B << ", C = " << *C << "\n");
+ DEBUG(dbgs() << "\t\tSrc = " << *Src << "\n");
+ DEBUG(dbgs() << "\t\tDst = " << *Dst << "\n");
+ if (A->isZero()) {
+ const SCEVConstant *Bconst = dyn_cast<SCEVConstant>(B);
+ const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
+ if (!Bconst || !Cconst) return false;
+ APInt Beta = Bconst->getValue()->getValue();
+ APInt Charlie = Cconst->getValue()->getValue();
+ APInt CdivB = Charlie.sdiv(Beta);
+ assert(Charlie.srem(Beta) == 0 && "C should be evenly divisible by B");
+ const SCEV *AP_K = findCoefficient(Dst, CurLoop);
+ // Src = SE->getAddExpr(Src, SE->getMulExpr(AP_K, SE->getConstant(CdivB)));
+ Src = SE->getMinusSCEV(Src, SE->getMulExpr(AP_K, SE->getConstant(CdivB)));
+ Dst = zeroCoefficient(Dst, CurLoop);
+ if (!findCoefficient(Src, CurLoop)->isZero())
+ Consistent = false;
+ }
+ else if (B->isZero()) {
+ const SCEVConstant *Aconst = dyn_cast<SCEVConstant>(A);
+ const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
+ if (!Aconst || !Cconst) return false;
+ APInt Alpha = Aconst->getValue()->getValue();
+ APInt Charlie = Cconst->getValue()->getValue();
+ APInt CdivA = Charlie.sdiv(Alpha);
+ assert(Charlie.srem(Alpha) == 0 && "C should be evenly divisible by A");
+ const SCEV *A_K = findCoefficient(Src, CurLoop);
+ Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, SE->getConstant(CdivA)));
+ Src = zeroCoefficient(Src, CurLoop);
+ if (!findCoefficient(Dst, CurLoop)->isZero())
+ Consistent = false;
+ }
+ else if (isKnownPredicate(CmpInst::ICMP_EQ, A, B)) {
+ const SCEVConstant *Aconst = dyn_cast<SCEVConstant>(A);
+ const SCEVConstant *Cconst = dyn_cast<SCEVConstant>(C);
+ if (!Aconst || !Cconst) return false;
+ APInt Alpha = Aconst->getValue()->getValue();
+ APInt Charlie = Cconst->getValue()->getValue();
+ APInt CdivA = Charlie.sdiv(Alpha);
+ assert(Charlie.srem(Alpha) == 0 && "C should be evenly divisible by A");
+ const SCEV *A_K = findCoefficient(Src, CurLoop);
+ Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, SE->getConstant(CdivA)));
+ Src = zeroCoefficient(Src, CurLoop);
+ Dst = addToCoefficient(Dst, CurLoop, A_K);
+ if (!findCoefficient(Dst, CurLoop)->isZero())
+ Consistent = false;
+ }
+ else {
+ // paper is incorrect here, or perhaps just misleading
+ const SCEV *A_K = findCoefficient(Src, CurLoop);
+ Src = SE->getMulExpr(Src, A);
+ Dst = SE->getMulExpr(Dst, A);
+ Src = SE->getAddExpr(Src, SE->getMulExpr(A_K, C));
+ Src = zeroCoefficient(Src, CurLoop);
+ Dst = addToCoefficient(Dst, CurLoop, SE->getMulExpr(A_K, B));
+ if (!findCoefficient(Dst, CurLoop)->isZero())
+ Consistent = false;
+ }
+ DEBUG(dbgs() << "\t\tnew Src = " << *Src << "\n");
+ DEBUG(dbgs() << "\t\tnew Dst = " << *Dst << "\n");
+ return true;
+}
+
+
+// Attempt to propagate a point
+// constraint into a subscript pair (Src and Dst).
+// Return true if some simplification occurs.
+bool DependenceAnalysis::propagatePoint(const SCEV *&Src,
+ const SCEV *&Dst,
+ Constraint &CurConstraint) {
+ const Loop *CurLoop = CurConstraint.getAssociatedLoop();
+ const SCEV *A_K = findCoefficient(Src, CurLoop);
+ const SCEV *AP_K = findCoefficient(Dst, CurLoop);
+ const SCEV *XA_K = SE->getMulExpr(A_K, CurConstraint.getX());
+ const SCEV *YAP_K = SE->getMulExpr(AP_K, CurConstraint.getY());
+ DEBUG(dbgs() << "\t\tSrc is " << *Src << "\n");
+ Src = SE->getAddExpr(Src, SE->getMinusSCEV(XA_K, YAP_K));
+ Src = zeroCoefficient(Src, CurLoop);
+ DEBUG(dbgs() << "\t\tnew Src is " << *Src << "\n");
+ DEBUG(dbgs() << "\t\tDst is " << *Dst << "\n");
+ Dst = zeroCoefficient(Dst, CurLoop);
+ DEBUG(dbgs() << "\t\tnew Dst is " << *Dst << "\n");
+ return true;
+}
+
+
+// Update direction vector entry based on the current constraint.
+void DependenceAnalysis::updateDirection(Dependence::DVEntry &Level,
+ const Constraint &CurConstraint
+ ) const {
+ DEBUG(dbgs() << "\tUpdate direction, constraint =");
+ DEBUG(CurConstraint.dump(dbgs()));
+ if (CurConstraint.isAny())
+ ; // use defaults
+ else if (CurConstraint.isDistance()) {
+ // this one is consistent, the others aren't
+ Level.Scalar = false;
+ Level.Distance = CurConstraint.getD();
+ unsigned NewDirection = Dependence::DVEntry::NONE;
+ if (!SE->isKnownNonZero(Level.Distance)) // if may be zero
+ NewDirection = Dependence::DVEntry::EQ;
+ if (!SE->isKnownNonPositive(Level.Distance)) // if may be positive
+ NewDirection |= Dependence::DVEntry::LT;
+ if (!SE->isKnownNonNegative(Level.Distance)) // if may be negative
+ NewDirection |= Dependence::DVEntry::GT;
+ Level.Direction &= NewDirection;
+ }
+ else if (CurConstraint.isLine()) {
+ Level.Scalar = false;
+ Level.Distance = NULL;
+ // direction should be accurate
+ }
+ else if (CurConstraint.isPoint()) {
+ Level.Scalar = false;
+ Level.Distance = NULL;
+ unsigned NewDirection = Dependence::DVEntry::NONE;
+ if (!isKnownPredicate(CmpInst::ICMP_NE,
+ CurConstraint.getY(),
+ CurConstraint.getX()))
+ // if X may be = Y
+ NewDirection |= Dependence::DVEntry::EQ;
+ if (!isKnownPredicate(CmpInst::ICMP_SLE,
+ CurConstraint.getY(),
+ CurConstraint.getX()))
+ // if Y may be > X
+ NewDirection |= Dependence::DVEntry::LT;
+ if (!isKnownPredicate(CmpInst::ICMP_SGE,
+ CurConstraint.getY(),
+ CurConstraint.getX()))
+ // if Y may be < X
+ NewDirection |= Dependence::DVEntry::GT;
+ Level.Direction &= NewDirection;
+ }
+ else
+ llvm_unreachable("constraint has unexpected kind");
+}
+
+
+//===----------------------------------------------------------------------===//
+
+#ifndef NDEBUG
+// For debugging purposes, dump a small bit vector to dbgs().
+static void dumpSmallBitVector(SmallBitVector &BV) {
+ dbgs() << "{";
+ for (int VI = BV.find_first(); VI >= 0; VI = BV.find_next(VI)) {
+ dbgs() << VI;
+ if (BV.find_next(VI) >= 0)
+ dbgs() << ' ';
+ }
+ dbgs() << "}\n";
+}
+#endif
+
+
+// depends -
+// Returns NULL if there is no dependence.
+// Otherwise, return a Dependence with as many details as possible.
+// Corresponds to Section 3.1 in the paper
+//
+// Practical Dependence Testing
+// Goff, Kennedy, Tseng
+// PLDI 1991
+//
+// Care is required to keep the code below up to date w.r.t. this routine.
+Dependence *DependenceAnalysis::depends(const Instruction *Src,
+ const Instruction *Dst,
+ bool PossiblyLoopIndependent) {
+ if ((!Src->mayReadFromMemory() && !Src->mayWriteToMemory()) ||
+ (!Dst->mayReadFromMemory() && !Dst->mayWriteToMemory()))
+ // if both instructions don't reference memory, there's no dependence
+ return NULL;
+
+ if (!isLoadOrStore(Src) || !isLoadOrStore(Dst))
+ // can only analyze simple loads and stores, i.e., no calls, invokes, etc.
+ return new Dependence(Src, Dst);
+
+ const Value *SrcPtr = getPointerOperand(Src);
+ const Value *DstPtr = getPointerOperand(Dst);
+
+ switch (underlyingObjectsAlias(AA, DstPtr, SrcPtr)) {
+ case AliasAnalysis::MayAlias:
+ case AliasAnalysis::PartialAlias:
+ // cannot analyse objects if we don't understand their aliasing.
+ return new Dependence(Src, Dst);
+ case AliasAnalysis::NoAlias:
+ // If the objects noalias, they are distinct, accesses are independent.
+ return NULL;
+ case AliasAnalysis::MustAlias:
+ break; // The underlying objects alias; test accesses for dependence.
+ }
+
+ const GEPOperator *SrcGEP = dyn_cast<GEPOperator>(SrcPtr);
+ const GEPOperator *DstGEP = dyn_cast<GEPOperator>(DstPtr);
+ if (!SrcGEP || !DstGEP)
+ return new Dependence(Src, Dst); // missing GEP, assume dependence
+
+ if (SrcGEP->getPointerOperandType() != DstGEP->getPointerOperandType())
+ return new Dependence(Src, Dst); // different types, assume dependence
+
+ // establish loop nesting levels
+ establishNestingLevels(Src, Dst);
+ DEBUG(dbgs() << " common nesting levels = " << CommonLevels << "\n");
+ DEBUG(dbgs() << " maximum nesting levels = " << MaxLevels << "\n");
+
+ FullDependence Result(Src, Dst, PossiblyLoopIndependent, CommonLevels);
+ ++TotalArrayPairs;
+
+ // classify subscript pairs
+ unsigned Pairs = SrcGEP->idx_end() - SrcGEP->idx_begin();
+ SmallVector<Subscript, 4> Pair(Pairs);
+ for (unsigned SI = 0; SI < Pairs; ++SI) {
+ Pair[SI].Loops.resize(MaxLevels + 1);
+ Pair[SI].GroupLoops.resize(MaxLevels + 1);
+ Pair[SI].Group.resize(Pairs);
+ }
+ Pairs = 0;
+ for (GEPOperator::const_op_iterator SrcIdx = SrcGEP->idx_begin(),
+ SrcEnd = SrcGEP->idx_end(),
+ DstIdx = DstGEP->idx_begin(),
+ DstEnd = DstGEP->idx_end();
+ SrcIdx != SrcEnd && DstIdx != DstEnd;
+ ++SrcIdx, ++DstIdx, ++Pairs) {
+ Pair[Pairs].Src = SE->getSCEV(*SrcIdx);
+ Pair[Pairs].Dst = SE->getSCEV(*DstIdx);
+ removeMatchingExtensions(&Pair[Pairs]);
+ Pair[Pairs].Classification =
+ classifyPair(Pair[Pairs].Src, LI->getLoopFor(Src->getParent()),
+ Pair[Pairs].Dst, LI->getLoopFor(Dst->getParent()),
+ Pair[Pairs].Loops);
+ Pair[Pairs].GroupLoops = Pair[Pairs].Loops;
+ Pair[Pairs].Group.set(Pairs);
+ DEBUG(dbgs() << " subscript " << Pairs << "\n");
+ DEBUG(dbgs() << "\tsrc = " << *Pair[Pairs].Src << "\n");
+ DEBUG(dbgs() << "\tdst = " << *Pair[Pairs].Dst << "\n");
+ DEBUG(dbgs() << "\tclass = " << Pair[Pairs].Classification << "\n");
+ DEBUG(dbgs() << "\tloops = ");
+ DEBUG(dumpSmallBitVector(Pair[Pairs].Loops));
+ }
+
+ SmallBitVector Separable(Pairs);
+ SmallBitVector Coupled(Pairs);
+
+ // Partition subscripts into separable and minimally-coupled groups
+ // Algorithm in paper is algorithmically better;
+ // this may be faster in practice. Check someday.
+ //
+ // Here's an example of how it works. Consider this code:
+ //
+ // for (i = ...) {
+ // for (j = ...) {
+ // for (k = ...) {
+ // for (l = ...) {
+ // for (m = ...) {
+ // A[i][j][k][m] = ...;
+ // ... = A[0][j][l][i + j];
+ // }
+ // }
+ // }
+ // }
+ // }
+ //
+ // There are 4 subscripts here:
+ // 0 [i] and [0]
+ // 1 [j] and [j]
+ // 2 [k] and [l]
+ // 3 [m] and [i + j]
+ //
+ // We've already classified each subscript pair as ZIV, SIV, etc.,
+ // and collected all the loops mentioned by pair P in Pair[P].Loops.
+ // In addition, we've initialized Pair[P].GroupLoops to Pair[P].Loops
+ // and set Pair[P].Group = {P}.
+ //
+ // Src Dst Classification Loops GroupLoops Group
+ // 0 [i] [0] SIV {1} {1} {0}
+ // 1 [j] [j] SIV {2} {2} {1}
+ // 2 [k] [l] RDIV {3,4} {3,4} {2}
+ // 3 [m] [i + j] MIV {1,2,5} {1,2,5} {3}
+ //
+ // For each subscript SI 0 .. 3, we consider each remaining subscript, SJ.
+ // So, 0 is compared against 1, 2, and 3; 1 is compared against 2 and 3, etc.
+ //
+ // We begin by comparing 0 and 1. The intersection of the GroupLoops is empty.
+ // Next, 0 and 2. Again, the intersection of their GroupLoops is empty.
+ // Next 0 and 3. The intersection of their GroupLoop = {1}, not empty,
+ // so Pair[3].Group = {0,3} and Done = false (that is, 0 will not be added
+ // to either Separable or Coupled).
+ //
+ // Next, we consider 1 and 2. The intersection of the GroupLoops is empty.
+ // Next, 1 and 3. The intersectionof their GroupLoops = {2}, not empty,
+ // so Pair[3].Group = {0, 1, 3} and Done = false.
+ //
+ // Next, we compare 2 against 3. The intersection of the GroupLoops is empty.
+ // Since Done remains true, we add 2 to the set of Separable pairs.
+ //
+ // Finally, we consider 3. There's nothing to compare it with,
+ // so Done remains true and we add it to the Coupled set.
+ // Pair[3].Group = {0, 1, 3} and GroupLoops = {1, 2, 5}.
+ //
+ // In the end, we've got 1 separable subscript and 1 coupled group.
+ for (unsigned SI = 0; SI < Pairs; ++SI) {
+ if (Pair[SI].Classification == Subscript::NonLinear) {
+ // ignore these, but collect loops for later
+ ++NonlinearSubscriptPairs;
+ collectCommonLoops(Pair[SI].Src,
+ LI->getLoopFor(Src->getParent()),
+ Pair[SI].Loops);
+ collectCommonLoops(Pair[SI].Dst,
+ LI->getLoopFor(Dst->getParent()),
+ Pair[SI].Loops);
+ Result.Consistent = false;
+ }
+ else if (Pair[SI].Classification == Subscript::ZIV) {
+ // always separable
+ Separable.set(SI);
+ }
+ else {
+ // SIV, RDIV, or MIV, so check for coupled group
+ bool Done = true;
+ for (unsigned SJ = SI + 1; SJ < Pairs; ++SJ) {
+ SmallBitVector Intersection = Pair[SI].GroupLoops;
+ Intersection &= Pair[SJ].GroupLoops;
+ if (Intersection.any()) {
+ // accumulate set of all the loops in group
+ Pair[SJ].GroupLoops |= Pair[SI].GroupLoops;
+ // accumulate set of all subscripts in group
+ Pair[SJ].Group |= Pair[SI].Group;
+ Done = false;
+ }
+ }
+ if (Done) {
+ if (Pair[SI].Group.count() == 1) {
+ Separable.set(SI);
+ ++SeparableSubscriptPairs;
+ }
+ else {
+ Coupled.set(SI);
+ ++CoupledSubscriptPairs;
+ }
+ }
+ }
+ }
+
+ DEBUG(dbgs() << " Separable = ");
+ DEBUG(dumpSmallBitVector(Separable));
+ DEBUG(dbgs() << " Coupled = ");
+ DEBUG(dumpSmallBitVector(Coupled));
+
+ Constraint NewConstraint;
+ NewConstraint.setAny(SE);
+
+ // test separable subscripts
+ for (int SI = Separable.find_first(); SI >= 0; SI = Separable.find_next(SI)) {
+ DEBUG(dbgs() << "testing subscript " << SI);
+ switch (Pair[SI].Classification) {
+ case Subscript::ZIV:
+ DEBUG(dbgs() << ", ZIV\n");
+ if (testZIV(Pair[SI].Src, Pair[SI].Dst, Result))
+ return NULL;
+ break;
+ case Subscript::SIV: {
+ DEBUG(dbgs() << ", SIV\n");
+ unsigned Level;
+ const SCEV *SplitIter = NULL;
+ if (testSIV(Pair[SI].Src, Pair[SI].Dst, Level,
+ Result, NewConstraint, SplitIter))
+ return NULL;
+ break;
+ }
+ case Subscript::RDIV:
+ DEBUG(dbgs() << ", RDIV\n");
+ if (testRDIV(Pair[SI].Src, Pair[SI].Dst, Result))
+ return NULL;
+ break;
+ case Subscript::MIV:
+ DEBUG(dbgs() << ", MIV\n");
+ if (testMIV(Pair[SI].Src, Pair[SI].Dst, Pair[SI].Loops, Result))
+ return NULL;
+ break;
+ default:
+ llvm_unreachable("subscript has unexpected classification");
+ }
+ }
+
+ if (Coupled.count()) {
+ // test coupled subscript groups
+ DEBUG(dbgs() << "starting on coupled subscripts\n");
+ DEBUG(dbgs() << "MaxLevels + 1 = " << MaxLevels + 1 << "\n");
+ SmallVector<Constraint, 4> Constraints(MaxLevels + 1);
+ for (unsigned II = 0; II <= MaxLevels; ++II)
+ Constraints[II].setAny(SE);
+ for (int SI = Coupled.find_first(); SI >= 0; SI = Coupled.find_next(SI)) {
+ DEBUG(dbgs() << "testing subscript group " << SI << " { ");
+ SmallBitVector Group(Pair[SI].Group);
+ SmallBitVector Sivs(Pairs);
+ SmallBitVector Mivs(Pairs);
+ SmallBitVector ConstrainedLevels(MaxLevels + 1);
+ for (int SJ = Group.find_first(); SJ >= 0; SJ = Group.find_next(SJ)) {
+ DEBUG(dbgs() << SJ << " ");
+ if (Pair[SJ].Classification == Subscript::SIV)
+ Sivs.set(SJ);
+ else
+ Mivs.set(SJ);
+ }
+ DEBUG(dbgs() << "}\n");
+ while (Sivs.any()) {
+ bool Changed = false;
+ for (int SJ = Sivs.find_first(); SJ >= 0; SJ = Sivs.find_next(SJ)) {
+ DEBUG(dbgs() << "testing subscript " << SJ << ", SIV\n");
+ // SJ is an SIV subscript that's part of the current coupled group
+ unsigned Level;
+ const SCEV *SplitIter = NULL;
+ DEBUG(dbgs() << "SIV\n");
+ if (testSIV(Pair[SJ].Src, Pair[SJ].Dst, Level,
+ Result, NewConstraint, SplitIter))
+ return NULL;
+ ConstrainedLevels.set(Level);
+ if (intersectConstraints(&Constraints[Level], &NewConstraint)) {
+ if (Constraints[Level].isEmpty()) {
+ ++DeltaIndependence;
+ return NULL;
+ }
+ Changed = true;
+ }
+ Sivs.reset(SJ);
+ }
+ if (Changed) {
+ // propagate, possibly creating new SIVs and ZIVs
+ DEBUG(dbgs() << " propagating\n");
+ DEBUG(dbgs() << "\tMivs = ");
+ DEBUG(dumpSmallBitVector(Mivs));
+ for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
+ // SJ is an MIV subscript that's part of the current coupled group
+ DEBUG(dbgs() << "\tSJ = " << SJ << "\n");
+ if (propagate(Pair[SJ].Src, Pair[SJ].Dst, Pair[SJ].Loops,
+ Constraints, Result.Consistent)) {
+ DEBUG(dbgs() << "\t Changed\n");
+ ++DeltaPropagations;
+ Pair[SJ].Classification =
+ classifyPair(Pair[SJ].Src, LI->getLoopFor(Src->getParent()),
+ Pair[SJ].Dst, LI->getLoopFor(Dst->getParent()),
+ Pair[SJ].Loops);
+ switch (Pair[SJ].Classification) {
+ case Subscript::ZIV:
+ DEBUG(dbgs() << "ZIV\n");
+ if (testZIV(Pair[SJ].Src, Pair[SJ].Dst, Result))
+ return NULL;
+ Mivs.reset(SJ);
+ break;
+ case Subscript::SIV:
+ Sivs.set(SJ);
+ Mivs.reset(SJ);
+ break;
+ case Subscript::RDIV:
+ case Subscript::MIV:
+ break;
+ default:
+ llvm_unreachable("bad subscript classification");
+ }
+ }
+ }
+ }
+ }
+
+ // test & propagate remaining RDIVs
+ for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
+ if (Pair[SJ].Classification == Subscript::RDIV) {
+ DEBUG(dbgs() << "RDIV test\n");
+ if (testRDIV(Pair[SJ].Src, Pair[SJ].Dst, Result))
+ return NULL;
+ // I don't yet understand how to propagate RDIV results
+ Mivs.reset(SJ);
+ }
+ }
+
+ // test remaining MIVs
+ // This code is temporary.
+ // Better to somehow test all remaining subscripts simultaneously.
+ for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
+ if (Pair[SJ].Classification == Subscript::MIV) {
+ DEBUG(dbgs() << "MIV test\n");
+ if (testMIV(Pair[SJ].Src, Pair[SJ].Dst, Pair[SJ].Loops, Result))
+ return NULL;
+ }
+ else
+ llvm_unreachable("expected only MIV subscripts at this point");
+ }
+
+ // update Result.DV from constraint vector
+ DEBUG(dbgs() << " updating\n");
+ for (int SJ = ConstrainedLevels.find_first();
+ SJ >= 0; SJ = ConstrainedLevels.find_next(SJ)) {
+ updateDirection(Result.DV[SJ - 1], Constraints[SJ]);
+ if (Result.DV[SJ - 1].Direction == Dependence::DVEntry::NONE)
+ return NULL;
+ }
+ }
+ }
+
+ // make sure Scalar flags are set correctly
+ SmallBitVector CompleteLoops(MaxLevels + 1);
+ for (unsigned SI = 0; SI < Pairs; ++SI)
+ CompleteLoops |= Pair[SI].Loops;
+ for (unsigned II = 1; II <= CommonLevels; ++II)
+ if (CompleteLoops[II])
+ Result.DV[II - 1].Scalar = false;
+
+ // make sure loopIndepent flag is set correctly
+ if (PossiblyLoopIndependent) {
+ for (unsigned II = 1; II <= CommonLevels; ++II) {
+ if (!(Result.getDirection(II) & Dependence::DVEntry::EQ)) {
+ Result.LoopIndependent = false;
+ break;
+ }
+ }
+ }
+
+ FullDependence *Final = new FullDependence(Result);
+ Result.DV = NULL;
+ return Final;
+}
+
+
+
+//===----------------------------------------------------------------------===//
+// getSplitIteration -
+// Rather than spend rarely-used space recording the splitting iteration
+// during the Weak-Crossing SIV test, we re-compute it on demand.
+// The re-computation is basically a repeat of the entire dependence test,
+// though simplified since we know that the dependence exists.
+// It's tedious, since we must go through all propagations, etc.
+//
+// Care is required to keep this code up to date w.r.t. the code above.
+//
+// Generally, the dependence analyzer will be used to build
+// a dependence graph for a function (basically a map from instructions
+// to dependences). Looking for cycles in the graph shows us loops
+// that cannot be trivially vectorized/parallelized.
+//
+// We can try to improve the situation by examining all the dependences
+// that make up the cycle, looking for ones we can break.
+// Sometimes, peeling the first or last iteration of a loop will break
+// dependences, and we've got flags for those possibilities.
+// Sometimes, splitting a loop at some other iteration will do the trick,
+// and we've got a flag for that case. Rather than waste the space to
+// record the exact iteration (since we rarely know), we provide
+// a method that calculates the iteration. It's a drag that it must work
+// from scratch, but wonderful in that it's possible.
+//
+// Here's an example:
+//
+// for (i = 0; i < 10; i++)
+// A[i] = ...
+// ... = A[11 - i]
+//
+// There's a loop-carried flow dependence from the store to the load,
+// found by the weak-crossing SIV test. The dependence will have a flag,
+// indicating that the dependence can be broken by splitting the loop.
+// Calling getSplitIteration will return 5.
+// Splitting the loop breaks the dependence, like so:
+//
+// for (i = 0; i <= 5; i++)
+// A[i] = ...
+// ... = A[11 - i]
+// for (i = 6; i < 10; i++)
+// A[i] = ...
+// ... = A[11 - i]
+//
+// breaks the dependence and allows us to vectorize/parallelize
+// both loops.
+const SCEV *DependenceAnalysis::getSplitIteration(const Dependence *Dep,
+ unsigned SplitLevel) {
+ assert(Dep && "expected a pointer to a Dependence");
+ assert(Dep->isSplitable(SplitLevel) &&
+ "Dep should be splitable at SplitLevel");
+ const Instruction *Src = Dep->getSrc();
+ const Instruction *Dst = Dep->getDst();
+ assert(Src->mayReadFromMemory() || Src->mayWriteToMemory());
+ assert(Dst->mayReadFromMemory() || Dst->mayWriteToMemory());
+ assert(isLoadOrStore(Src));
+ assert(isLoadOrStore(Dst));
+ const Value *SrcPtr = getPointerOperand(Src);
+ const Value *DstPtr = getPointerOperand(Dst);
+ assert(underlyingObjectsAlias(AA, DstPtr, SrcPtr) ==
+ AliasAnalysis::MustAlias);
+ const GEPOperator *SrcGEP = dyn_cast<GEPOperator>(SrcPtr);
+ const GEPOperator *DstGEP = dyn_cast<GEPOperator>(DstPtr);
+ assert(SrcGEP);
+ assert(DstGEP);
+ assert(SrcGEP->getPointerOperandType() == DstGEP->getPointerOperandType());
+
+ // establish loop nesting levels
+ establishNestingLevels(Src, Dst);
+
+ FullDependence Result(Src, Dst, false, CommonLevels);
+
+ // classify subscript pairs
+ unsigned Pairs = SrcGEP->idx_end() - SrcGEP->idx_begin();
+ SmallVector<Subscript, 4> Pair(Pairs);
+ for (unsigned SI = 0; SI < Pairs; ++SI) {
+ Pair[SI].Loops.resize(MaxLevels + 1);
+ Pair[SI].GroupLoops.resize(MaxLevels + 1);
+ Pair[SI].Group.resize(Pairs);
+ }
+ Pairs = 0;
+ for (GEPOperator::const_op_iterator SrcIdx = SrcGEP->idx_begin(),
+ SrcEnd = SrcGEP->idx_end(),
+ DstIdx = DstGEP->idx_begin(),
+ DstEnd = DstGEP->idx_end();
+ SrcIdx != SrcEnd && DstIdx != DstEnd;
+ ++SrcIdx, ++DstIdx, ++Pairs) {
+ Pair[Pairs].Src = SE->getSCEV(*SrcIdx);
+ Pair[Pairs].Dst = SE->getSCEV(*DstIdx);
+ Pair[Pairs].Classification =
+ classifyPair(Pair[Pairs].Src, LI->getLoopFor(Src->getParent()),
+ Pair[Pairs].Dst, LI->getLoopFor(Dst->getParent()),
+ Pair[Pairs].Loops);
+ Pair[Pairs].GroupLoops = Pair[Pairs].Loops;
+ Pair[Pairs].Group.set(Pairs);
+ }
+
+ SmallBitVector Separable(Pairs);
+ SmallBitVector Coupled(Pairs);
+
+ // partition subscripts into separable and minimally-coupled groups
+ for (unsigned SI = 0; SI < Pairs; ++SI) {
+ if (Pair[SI].Classification == Subscript::NonLinear) {
+ // ignore these, but collect loops for later
+ collectCommonLoops(Pair[SI].Src,
+ LI->getLoopFor(Src->getParent()),
+ Pair[SI].Loops);
+ collectCommonLoops(Pair[SI].Dst,
+ LI->getLoopFor(Dst->getParent()),
+ Pair[SI].Loops);
+ Result.Consistent = false;
+ }
+ else if (Pair[SI].Classification == Subscript::ZIV)
+ Separable.set(SI);
+ else {
+ // SIV, RDIV, or MIV, so check for coupled group
+ bool Done = true;
+ for (unsigned SJ = SI + 1; SJ < Pairs; ++SJ) {
+ SmallBitVector Intersection = Pair[SI].GroupLoops;
+ Intersection &= Pair[SJ].GroupLoops;
+ if (Intersection.any()) {
+ // accumulate set of all the loops in group
+ Pair[SJ].GroupLoops |= Pair[SI].GroupLoops;
+ // accumulate set of all subscripts in group
+ Pair[SJ].Group |= Pair[SI].Group;
+ Done = false;
+ }
+ }
+ if (Done) {
+ if (Pair[SI].Group.count() == 1)
+ Separable.set(SI);
+ else
+ Coupled.set(SI);
+ }
+ }
+ }
+
+ Constraint NewConstraint;
+ NewConstraint.setAny(SE);
+
+ // test separable subscripts
+ for (int SI = Separable.find_first(); SI >= 0; SI = Separable.find_next(SI)) {
+ switch (Pair[SI].Classification) {
+ case Subscript::SIV: {
+ unsigned Level;
+ const SCEV *SplitIter = NULL;
+ (void) testSIV(Pair[SI].Src, Pair[SI].Dst, Level,
+ Result, NewConstraint, SplitIter);
+ if (Level == SplitLevel) {
+ assert(SplitIter != NULL);
+ return SplitIter;
+ }
+ break;
+ }
+ case Subscript::ZIV:
+ case Subscript::RDIV:
+ case Subscript::MIV:
+ break;
+ default:
+ llvm_unreachable("subscript has unexpected classification");
+ }
+ }
+
+ if (Coupled.count()) {
+ // test coupled subscript groups
+ SmallVector<Constraint, 4> Constraints(MaxLevels + 1);
+ for (unsigned II = 0; II <= MaxLevels; ++II)
+ Constraints[II].setAny(SE);
+ for (int SI = Coupled.find_first(); SI >= 0; SI = Coupled.find_next(SI)) {
+ SmallBitVector Group(Pair[SI].Group);
+ SmallBitVector Sivs(Pairs);
+ SmallBitVector Mivs(Pairs);
+ SmallBitVector ConstrainedLevels(MaxLevels + 1);
+ for (int SJ = Group.find_first(); SJ >= 0; SJ = Group.find_next(SJ)) {
+ if (Pair[SJ].Classification == Subscript::SIV)
+ Sivs.set(SJ);
+ else
+ Mivs.set(SJ);
+ }
+ while (Sivs.any()) {
+ bool Changed = false;
+ for (int SJ = Sivs.find_first(); SJ >= 0; SJ = Sivs.find_next(SJ)) {
+ // SJ is an SIV subscript that's part of the current coupled group
+ unsigned Level;
+ const SCEV *SplitIter = NULL;
+ (void) testSIV(Pair[SJ].Src, Pair[SJ].Dst, Level,
+ Result, NewConstraint, SplitIter);
+ if (Level == SplitLevel && SplitIter)
+ return SplitIter;
+ ConstrainedLevels.set(Level);
+ if (intersectConstraints(&Constraints[Level], &NewConstraint))
+ Changed = true;
+ Sivs.reset(SJ);
+ }
+ if (Changed) {
+ // propagate, possibly creating new SIVs and ZIVs
+ for (int SJ = Mivs.find_first(); SJ >= 0; SJ = Mivs.find_next(SJ)) {
+ // SJ is an MIV subscript that's part of the current coupled group
+ if (propagate(Pair[SJ].Src, Pair[SJ].Dst,
+ Pair[SJ].Loops, Constraints, Result.Consistent)) {
+ Pair[SJ].Classification =
+ classifyPair(Pair[SJ].Src, LI->getLoopFor(Src->getParent()),
+ Pair[SJ].Dst, LI->getLoopFor(Dst->getParent()),
+ Pair[SJ].Loops);
+ switch (Pair[SJ].Classification) {
+ case Subscript::ZIV:
+ Mivs.reset(SJ);
+ break;
+ case Subscript::SIV:
+ Sivs.set(SJ);
+ Mivs.reset(SJ);
+ break;
+ case Subscript::RDIV:
+ case Subscript::MIV:
+ break;
+ default:
+ llvm_unreachable("bad subscript classification");
+ }
+ }
+ }
+ }
+ }
+ }
+ }
+ llvm_unreachable("somehow reached end of routine");
+ return NULL;
+}