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

1 files changed, 3781 insertions, 0 deletions

diff --git a/lib/Analysis/DependenceAnalysis.cpp b/lib/Analysis/DependenceAnalysis.cpp
new file mode 100644
index 0000000000..c7bec4323c
--- /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;
+}