text: //===- LoopReroll.cpp - Loop rerolling pass -------------------------------===//
  //
  // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
  // See https://llvm.org/LICENSE.txt for license information.
  // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
  //
  //===----------------------------------------------------------------------===//
  //
  // This pass implements a simple loop reroller.
  //
  //===----------------------------------------------------------------------===//
  
  #include "llvm/ADT/APInt.h"
  #include "llvm/ADT/BitVector.h"
  #include "llvm/ADT/DenseMap.h"
  #include "llvm/ADT/DenseSet.h"
  #include "llvm/ADT/MapVector.h"
  #include "llvm/ADT/STLExtras.h"
  #include "llvm/ADT/SmallPtrSet.h"
  #include "llvm/ADT/SmallVector.h"
  #include "llvm/ADT/Statistic.h"
  #include "llvm/Analysis/AliasAnalysis.h"
  #include "llvm/Analysis/AliasSetTracker.h"
  #include "llvm/Analysis/LoopInfo.h"
  #include "llvm/Analysis/LoopPass.h"
  #include "llvm/Analysis/ScalarEvolution.h"
  #include "llvm/Analysis/ScalarEvolutionExpressions.h"
  #include "llvm/Analysis/TargetLibraryInfo.h"
  #include "llvm/Analysis/ValueTracking.h"
  #include "llvm/IR/BasicBlock.h"
  #include "llvm/IR/Constants.h"
  #include "llvm/IR/DataLayout.h"
  #include "llvm/IR/DerivedTypes.h"
  #include "llvm/IR/Dominators.h"
  #include "llvm/IR/IRBuilder.h"
  #include "llvm/IR/InstrTypes.h"
  #include "llvm/IR/Instruction.h"
  #include "llvm/IR/Instructions.h"
  #include "llvm/IR/IntrinsicInst.h"
  #include "llvm/IR/Intrinsics.h"
  #include "llvm/IR/Module.h"
  #include "llvm/IR/Type.h"
  #include "llvm/IR/Use.h"
  #include "llvm/IR/User.h"
  #include "llvm/IR/Value.h"
  #include "llvm/InitializePasses.h"
  #include "llvm/Pass.h"
  #include "llvm/Support/Casting.h"
  #include "llvm/Support/CommandLine.h"
  #include "llvm/Support/Debug.h"
  #include "llvm/Support/raw_ostream.h"
  #include "llvm/Transforms/Scalar.h"
  #include "llvm/Transforms/Scalar/LoopReroll.h"
  #include "llvm/Transforms/Utils.h"
  #include "llvm/Transforms/Utils/BasicBlockUtils.h"
  #include "llvm/Transforms/Utils/Local.h"
  #include "llvm/Transforms/Utils/LoopUtils.h"
  #include "llvm/Transforms/Utils/ScalarEvolutionExpander.h"
  #include <cassert>
  #include <cstddef>
  #include <cstdint>
  #include <cstdlib>
  #include <iterator>
  #include <map>
  #include <utility>
  
  using namespace llvm;
  
  #define DEBUG_TYPE "loop-reroll"
  
  STATISTIC(NumRerolledLoops, "Number of rerolled loops");
  
  static cl::opt<unsigned>
  NumToleratedFailedMatches("reroll-num-tolerated-failed-matches", cl::init(400),
                            cl::Hidden,
                            cl::desc("The maximum number of failures to tolerate"
                                     " during fuzzy matching. (default: 400)"));
  
  // This loop re-rolling transformation aims to transform loops like this:
  //
  // int foo(int a);
  // void bar(int *x) {
  //   for (int i = 0; i < 500; i += 3) {
  //     foo(i);
  //     foo(i+1);
  //     foo(i+2);
  //   }
  // }
  //
  // into a loop like this:
  //
  // void bar(int *x) {
  //   for (int i = 0; i < 500; ++i)
  //     foo(i);
  // }
  //
  // It does this by looking for loops that, besides the latch code, are composed
  // of isomorphic DAGs of instructions, with each DAG rooted at some increment
  // to the induction variable, and where each DAG is isomorphic to the DAG
  // rooted at the induction variable (excepting the sub-DAGs which root the
  // other induction-variable increments). In other words, we're looking for loop
  // bodies of the form:
  //
  // %iv = phi [ (preheader, ...), (body, %iv.next) ]
  // f(%iv)
  // %iv.1 = add %iv, 1                <-- a root increment
  // f(%iv.1)
  // %iv.2 = add %iv, 2                <-- a root increment
  // f(%iv.2)
  // %iv.scale_m_1 = add %iv, scale-1  <-- a root increment
  // f(%iv.scale_m_1)
  // ...
  // %iv.next = add %iv, scale
  // %cmp = icmp(%iv, ...)
  // br %cmp, header, exit
  //
  // where each f(i) is a set of instructions that, collectively, are a function
  // only of i (and other loop-invariant values).
  //
  // As a special case, we can also reroll loops like this:
  //
  // int foo(int);
  // void bar(int *x) {
  //   for (int i = 0; i < 500; ++i) {
  //     x[3*i] = foo(0);
  //     x[3*i+1] = foo(0);
  //     x[3*i+2] = foo(0);
  //   }
  // }
  //
  // into this:
  //
  // void bar(int *x) {
  //   for (int i = 0; i < 1500; ++i)
  //     x[i] = foo(0);
  // }
  //
  // in which case, we're looking for inputs like this:
  //
  // %iv = phi [ (preheader, ...), (body, %iv.next) ]
  // %scaled.iv = mul %iv, scale
  // f(%scaled.iv)
  // %scaled.iv.1 = add %scaled.iv, 1
  // f(%scaled.iv.1)
  // %scaled.iv.2 = add %scaled.iv, 2
  // f(%scaled.iv.2)
  // %scaled.iv.scale_m_1 = add %scaled.iv, scale-1
  // f(%scaled.iv.scale_m_1)
  // ...
  // %iv.next = add %iv, 1
  // %cmp = icmp(%iv, ...)
  // br %cmp, header, exit
  
  namespace {
  
    enum IterationLimits {
      /// The maximum number of iterations that we'll try and reroll.
      IL_MaxRerollIterations = 32,
      /// The bitvector index used by loop induction variables and other
      /// instructions that belong to all iterations.
      IL_All,
      IL_End
    };
  
    class LoopRerollLegacyPass : public LoopPass {
    public:
      static char ID; // Pass ID, replacement for typeid
  
      LoopRerollLegacyPass() : LoopPass(ID) {
        initializeLoopRerollLegacyPassPass(*PassRegistry::getPassRegistry());
      }
  
      bool runOnLoop(Loop *L, LPPassManager &LPM) override;
  
      void getAnalysisUsage(AnalysisUsage &AU) const override {
        AU.addRequired<TargetLibraryInfoWrapperPass>();
        getLoopAnalysisUsage(AU);
      }
    };
  
    class LoopReroll {
    public:
      LoopReroll(AliasAnalysis *AA, LoopInfo *LI, ScalarEvolution *SE,
                 TargetLibraryInfo *TLI, DominatorTree *DT, bool PreserveLCSSA)
          : AA(AA), LI(LI), SE(SE), TLI(TLI), DT(DT),
            PreserveLCSSA(PreserveLCSSA) {}
      bool runOnLoop(Loop *L);
  
    protected:
      AliasAnalysis *AA;
      LoopInfo *LI;
      ScalarEvolution *SE;
      TargetLibraryInfo *TLI;
      DominatorTree *DT;
      bool PreserveLCSSA;
  
      using SmallInstructionVector = SmallVector<Instruction *, 16>;
      using SmallInstructionSet = SmallPtrSet<Instruction *, 16>;
  
      // Map between induction variable and its increment
      DenseMap<Instruction *, int64_t> IVToIncMap;
  
      // For loop with multiple induction variable, remember the one used only to
      // control the loop.
      Instruction *LoopControlIV;
  
      // A chain of isomorphic instructions, identified by a single-use PHI
      // representing a reduction. Only the last value may be used outside the
      // loop.
      struct SimpleLoopReduction {
        SimpleLoopReduction(Instruction *P, Loop *L) : Instructions(1, P) {
          assert(isa<PHINode>(P) && "First reduction instruction must be a PHI");
          add(L);
        }
  
        bool valid() const {
          return Valid;
        }
  
        Instruction *getPHI() const {
          assert(Valid && "Using invalid reduction");
          return Instructions.front();
        }
  
        Instruction *getReducedValue() const {
          assert(Valid && "Using invalid reduction");
          return Instructions.back();
        }
  
        Instruction *get(size_t i) const {
          assert(Valid && "Using invalid reduction");
          return Instructions[i+1];
        }
  
        Instruction *operator [] (size_t i) const { return get(i); }
  
        // The size, ignoring the initial PHI.
        size_t size() const {
          assert(Valid && "Using invalid reduction");
          return Instructions.size()-1;
        }
  
        using iterator = SmallInstructionVector::iterator;
        using const_iterator = SmallInstructionVector::const_iterator;
  
        iterator begin() {
          assert(Valid && "Using invalid reduction");
          return std::next(Instructions.begin());
        }
  
        const_iterator begin() const {
          assert(Valid && "Using invalid reduction");
          return std::next(Instructions.begin());
        }
  
        iterator end() { return Instructions.end(); }
        const_iterator end() const { return Instructions.end(); }
  
      protected:
        bool Valid = false;
        SmallInstructionVector Instructions;
  
        void add(Loop *L);
      };
  
      // The set of all reductions, and state tracking of possible reductions
      // during loop instruction processing.
      struct ReductionTracker {
        using SmallReductionVector = SmallVector<SimpleLoopReduction, 16>;
  
        // Add a new possible reduction.
        void addSLR(SimpleLoopReduction &SLR) { PossibleReds.push_back(SLR); }
  
        // Setup to track possible reductions corresponding to the provided
        // rerolling scale. Only reductions with a number of non-PHI instructions
        // that is divisible by the scale are considered. Three instructions sets
        // are filled in:
        //   - A set of all possible instructions in eligible reductions.
        //   - A set of all PHIs in eligible reductions
        //   - A set of all reduced values (last instructions) in eligible
        //     reductions.
        void restrictToScale(uint64_t Scale,
                             SmallInstructionSet &PossibleRedSet,
                             SmallInstructionSet &PossibleRedPHISet,
                             SmallInstructionSet &PossibleRedLastSet) {
          PossibleRedIdx.clear();
          PossibleRedIter.clear();
          Reds.clear();
  
          for (unsigned i = 0, e = PossibleReds.size(); i != e; ++i)
            if (PossibleReds[i].size() % Scale == 0) {
              PossibleRedLastSet.insert(PossibleReds[i].getReducedValue());
              PossibleRedPHISet.insert(PossibleReds[i].getPHI());
  
              PossibleRedSet.insert(PossibleReds[i].getPHI());
              PossibleRedIdx[PossibleReds[i].getPHI()] = i;
              for (Instruction *J : PossibleReds[i]) {
                PossibleRedSet.insert(J);
                PossibleRedIdx[J] = i;
              }
            }
        }
  
        // The functions below are used while processing the loop instructions.
  
        // Are the two instructions both from reductions, and furthermore, from
        // the same reduction?
        bool isPairInSame(Instruction *J1, Instruction *J2) {
          DenseMap<Instruction *, int>::iterator J1I = PossibleRedIdx.find(J1);
          if (J1I != PossibleRedIdx.end()) {
            DenseMap<Instruction *, int>::iterator J2I = PossibleRedIdx.find(J2);
            if (J2I != PossibleRedIdx.end() && J1I->second == J2I->second)
              return true;
          }
  
          return false;
        }
  
        // The two provided instructions, the first from the base iteration, and
        // the second from iteration i, form a matched pair. If these are part of
        // a reduction, record that fact.
        void recordPair(Instruction *J1, Instruction *J2, unsigned i) {
          if (PossibleRedIdx.count(J1)) {
            assert(PossibleRedIdx.count(J2) &&
                   "Recording reduction vs. non-reduction instruction?");
  
            PossibleRedIter[J1] = 0;
            PossibleRedIter[J2] = i;
  
            int Idx = PossibleRedIdx[J1];
            assert(Idx == PossibleRedIdx[J2] &&
                   "Recording pair from different reductions?");
            Reds.insert(Idx);
          }
        }
  
        // The functions below can be called after we've finished processing all
        // instructions in the loop, and we know which reductions were selected.
  
        bool validateSelected();
        void replaceSelected();
  
      protected:
        // The vector of all possible reductions (for any scale).
        SmallReductionVector PossibleReds;
  
        DenseMap<Instruction *, int> PossibleRedIdx;
        DenseMap<Instruction *, int> PossibleRedIter;
        DenseSet<int> Reds;
      };
  
      // A DAGRootSet models an induction variable being used in a rerollable
      // loop. For example,
      //
      //   x[i*3+0] = y1
      //   x[i*3+1] = y2
      //   x[i*3+2] = y3
      //
      //   Base instruction -> i*3
      //                    +---+----+
      //                   /    |     \
      //               ST[y1]  +1     +2  <-- Roots
      //                        |      |
      //                      ST[y2] ST[y3]
      //
      // There may be multiple DAGRoots, for example:
      //
      //   x[i*2+0] = ...   (1)
      //   x[i*2+1] = ...   (1)
      //   x[i*2+4] = ...   (2)
      //   x[i*2+5] = ...   (2)
      //   x[(i+1234)*2+5678] = ... (3)
      //   x[(i+1234)*2+5679] = ... (3)
      //
      // The loop will be rerolled by adding a new loop induction variable,
      // one for the Base instruction in each DAGRootSet.
      //
      struct DAGRootSet {
        Instruction *BaseInst;
        SmallInstructionVector Roots;
  
        // The instructions between IV and BaseInst (but not including BaseInst).
        SmallInstructionSet SubsumedInsts;
      };
  
      // The set of all DAG roots, and state tracking of all roots
      // for a particular induction variable.
      struct DAGRootTracker {
        DAGRootTracker(LoopReroll *Parent, Loop *L, Instruction *IV,
                       ScalarEvolution *SE, AliasAnalysis *AA,
                       TargetLibraryInfo *TLI, DominatorTree *DT, LoopInfo *LI,
                       bool PreserveLCSSA,
                       DenseMap<Instruction *, int64_t> &IncrMap,
                       Instruction *LoopCtrlIV)
            : Parent(Parent), L(L), SE(SE), AA(AA), TLI(TLI), DT(DT), LI(LI),
              PreserveLCSSA(PreserveLCSSA), IV(IV), IVToIncMap(IncrMap),
              LoopControlIV(LoopCtrlIV) {}
  
        /// Stage 1: Find all the DAG roots for the induction variable.
        bool findRoots();
  
        /// Stage 2: Validate if the found roots are valid.
        bool validate(ReductionTracker &Reductions);
  
        /// Stage 3: Assuming validate() returned true, perform the
        /// replacement.
        /// @param BackedgeTakenCount The backedge-taken count of L.
        void replace(const SCEV *BackedgeTakenCount);
  
      protected:
        using UsesTy = MapVector<Instruction *, BitVector>;
  
        void findRootsRecursive(Instruction *IVU,
                                SmallInstructionSet SubsumedInsts);
        bool findRootsBase(Instruction *IVU, SmallInstructionSet SubsumedInsts);
        bool collectPossibleRoots(Instruction *Base,
                                  std::map<int64_t,Instruction*> &Roots);
        bool validateRootSet(DAGRootSet &DRS);
  
        bool collectUsedInstructions(SmallInstructionSet &PossibleRedSet);
        void collectInLoopUserSet(const SmallInstructionVector &Roots,
                                  const SmallInstructionSet &Exclude,
                                  const SmallInstructionSet &Final,
                                  DenseSet<Instruction *> &Users);
        void collectInLoopUserSet(Instruction *Root,
                                  const SmallInstructionSet &Exclude,
                                  const SmallInstructionSet &Final,
                                  DenseSet<Instruction *> &Users);
  
        UsesTy::iterator nextInstr(int Val, UsesTy &In,
                                   const SmallInstructionSet &Exclude,
                                   UsesTy::iterator *StartI=nullptr);
        bool isBaseInst(Instruction *I);
        bool isRootInst(Instruction *I);
        bool instrDependsOn(Instruction *I,
                            UsesTy::iterator Start,
                            UsesTy::iterator End);
        void replaceIV(DAGRootSet &DRS, const SCEV *Start, const SCEV *IncrExpr);
  
        LoopReroll *Parent;
  
        // Members of Parent, replicated here for brevity.
        Loop *L;
        ScalarEvolution *SE;
        AliasAnalysis *AA;
        TargetLibraryInfo *TLI;
        DominatorTree *DT;
        LoopInfo *LI;
        bool PreserveLCSSA;
  
        // The loop induction variable.
        Instruction *IV;
  
        // Loop step amount.
        int64_t Inc;
  
        // Loop reroll count; if Inc == 1, this records the scaling applied
        // to the indvar: a[i*2+0] = ...; a[i*2+1] = ... ;
        // If Inc is not 1, Scale = Inc.
        uint64_t Scale;
  
        // The roots themselves.
        SmallVector<DAGRootSet,16> RootSets;
  
        // All increment instructions for IV.
        SmallInstructionVector LoopIncs;
  
        // Map of all instructions in the loop (in order) to the iterations
        // they are used in (or specially, IL_All for instructions
        // used in the loop increment mechanism).
        UsesTy Uses;
  
        // Map between induction variable and its increment
        DenseMap<Instruction *, int64_t> &IVToIncMap;
  
        Instruction *LoopControlIV;
      };
  
      // Check if it is a compare-like instruction whose user is a branch
      bool isCompareUsedByBranch(Instruction *I) {
        auto *TI = I->getParent()->getTerminator();
        if (!isa<BranchInst>(TI) || !isa<CmpInst>(I))
          return false;
        return I->hasOneUse() && TI->getOperand(0) == I;
      };
  
      bool isLoopControlIV(Loop *L, Instruction *IV);
      void collectPossibleIVs(Loop *L, SmallInstructionVector &PossibleIVs);
      void collectPossibleReductions(Loop *L,
             ReductionTracker &Reductions);
      bool reroll(Instruction *IV, Loop *L, BasicBlock *Header,
                  const SCEV *BackedgeTakenCount, ReductionTracker &Reductions);
    };
  
  } // end anonymous namespace
  
  char LoopRerollLegacyPass::ID = 0;
  
  INITIALIZE_PASS_BEGIN(LoopRerollLegacyPass, "loop-reroll", "Reroll loops",
                        false, false)
  INITIALIZE_PASS_DEPENDENCY(LoopPass)
  INITIALIZE_PASS_DEPENDENCY(TargetLibraryInfoWrapperPass)
  INITIALIZE_PASS_END(LoopRerollLegacyPass, "loop-reroll", "Reroll loops", false,
                      false)
  
  Pass *llvm::createLoopRerollPass() { return new LoopRerollLegacyPass; }
  
  // Returns true if the provided instruction is used outside the given loop.
  // This operates like Instruction::isUsedOutsideOfBlock, but considers PHIs in
  // non-loop blocks to be outside the loop.
  static bool hasUsesOutsideLoop(Instruction *I, Loop *L) {
    for (User *U : I->users()) {
      if (!L->contains(cast<Instruction>(U)))
        return true;
    }
    return false;
  }
  
  // Check if an IV is only used to control the loop. There are two cases:
  // 1. It only has one use which is loop increment, and the increment is only
  // used by comparison and the PHI (could has sext with nsw in between), and the
  // comparison is only used by branch.
  // 2. It is used by loop increment and the comparison, the loop increment is
  // only used by the PHI, and the comparison is used only by the branch.
  bool LoopReroll::isLoopControlIV(Loop *L, Instruction *IV) {
    unsigned IVUses = IV->getNumUses();
    if (IVUses != 2 && IVUses != 1)
      return false;
  
    for (auto *User : IV->users()) {
      int32_t IncOrCmpUses = User->getNumUses();
      bool IsCompInst = isCompareUsedByBranch(cast<Instruction>(User));
  
      // User can only have one or two uses.
      if (IncOrCmpUses != 2 && IncOrCmpUses != 1)
        return false;
  
      // Case 1
      if (IVUses == 1) {
        // The only user must be the loop increment.
        // The loop increment must have two uses.
        if (IsCompInst || IncOrCmpUses != 2)
          return false;
      }
  
      // Case 2
      if (IVUses == 2 && IncOrCmpUses != 1)
        return false;
  
      // The users of the IV must be a binary operation or a comparison
      if (auto *BO = dyn_cast<BinaryOperator>(User)) {
        if (BO->getOpcode() == Instruction::Add) {
          // Loop Increment
          // User of Loop Increment should be either PHI or CMP
          for (auto *UU : User->users()) {
            if (PHINode *PN = dyn_cast<PHINode>(UU)) {
              if (PN != IV)
                return false;
            }
            // Must be a CMP or an ext (of a value with nsw) then CMP
            else {
              Instruction *UUser = dyn_cast<Instruction>(UU);
              // Skip SExt if we are extending an nsw value
              // TODO: Allow ZExt too
              if (BO->hasNoSignedWrap() && UUser && UUser->hasOneUse() &&
                  isa<SExtInst>(UUser))
                UUser = dyn_cast<Instruction>(*(UUser->user_begin()));
              if (!isCompareUsedByBranch(UUser))
                return false;
            }
          }
        } else
          return false;
        // Compare : can only have one use, and must be branch
      } else if (!IsCompInst)
        return false;
    }
    return true;
  }
  
  // Collect the list of loop induction variables with respect to which it might
  // be possible to reroll the loop.
  void LoopReroll::collectPossibleIVs(Loop *L,
                                      SmallInstructionVector &PossibleIVs) {
    BasicBlock *Header = L->getHeader();
    for (BasicBlock::iterator I = Header->begin(),
         IE = Header->getFirstInsertionPt(); I != IE; ++I) {
      if (!isa<PHINode>(I))
        continue;
      if (!I->getType()->isIntegerTy() && !I->getType()->isPointerTy())
        continue;
  
      if (const SCEVAddRecExpr *PHISCEV =
              dyn_cast<SCEVAddRecExpr>(SE->getSCEV(&*I))) {
        if (PHISCEV->getLoop() != L)
          continue;
        if (!PHISCEV->isAffine())
          continue;
        auto IncSCEV = dyn_cast<SCEVConstant>(PHISCEV->getStepRecurrence(*SE));
        if (IncSCEV) {
          IVToIncMap[&*I] = IncSCEV->getValue()->getSExtValue();
          LLVM_DEBUG(dbgs() << "LRR: Possible IV: " << *I << " = " << *PHISCEV
                            << "\n");
  
          if (isLoopControlIV(L, &*I)) {
            assert(!LoopControlIV && "Found two loop control only IV");
            LoopControlIV = &(*I);
            LLVM_DEBUG(dbgs() << "LRR: Possible loop control only IV: " << *I
                              << " = " << *PHISCEV << "\n");
          } else
            PossibleIVs.push_back(&*I);
        }
      }
    }
  }
  
  // Add the remainder of the reduction-variable chain to the instruction vector
  // (the initial PHINode has already been added). If successful, the object is
  // marked as valid.
  void LoopReroll::SimpleLoopReduction::add(Loop *L) {
    assert(!Valid && "Cannot add to an already-valid chain");
  
    // The reduction variable must be a chain of single-use instructions
    // (including the PHI), except for the last value (which is used by the PHI
    // and also outside the loop).
    Instruction *C = Instructions.front();
    if (C->user_empty())
      return;
  
    do {
      C = cast<Instruction>(*C->user_begin());
      if (C->hasOneUse()) {
        if (!C->isBinaryOp())
          return;
  
        if (!(isa<PHINode>(Instructions.back()) ||
              C->isSameOperationAs(Instructions.back())))
          return;
  
        Instructions.push_back(C);
      }
    } while (C->hasOneUse());
  
    if (Instructions.size() < 2 ||
        !C->isSameOperationAs(Instructions.back()) ||
        C->use_empty())
      return;
  
    // C is now the (potential) last instruction in the reduction chain.
    for (User *U : C->users()) {
      // The only in-loop user can be the initial PHI.
      if (L->contains(cast<Instruction>(U)))
        if (cast<Instruction>(U) != Instructions.front())
          return;
    }
  
    Instructions.push_back(C);
    Valid = true;
  }
  
  // Collect the vector of possible reduction variables.
  void LoopReroll::collectPossibleReductions(Loop *L,
    ReductionTracker &Reductions) {
    BasicBlock *Header = L->getHeader();
    for (BasicBlock::iterator I = Header->begin(),
         IE = Header->getFirstInsertionPt(); I != IE; ++I) {
      if (!isa<PHINode>(I))
        continue;
      if (!I->getType()->isSingleValueType())
        continue;
  
      SimpleLoopReduction SLR(&*I, L);
      if (!SLR.valid())
        continue;
  
      LLVM_DEBUG(dbgs() << "LRR: Possible reduction: " << *I << " (with "
                        << SLR.size() << " chained instructions)\n");
      Reductions.addSLR(SLR);
    }
  }
  
  // Collect the set of all users of the provided root instruction. This set of
  // users contains not only the direct users of the root instruction, but also
  // all users of those users, and so on. There are two exceptions:
  //
  //   1. Instructions in the set of excluded instructions are never added to the
  //   use set (even if they are users). This is used, for example, to exclude
  //   including root increments in the use set of the primary IV.
  //
  //   2. Instructions in the set of final instructions are added to the use set
  //   if they are users, but their users are not added. This is used, for
  //   example, to prevent a reduction update from forcing all later reduction
  //   updates into the use set.
  void LoopReroll::DAGRootTracker::collectInLoopUserSet(
    Instruction *Root, const SmallInstructionSet &Exclude,
    const SmallInstructionSet &Final,
    DenseSet<Instruction *> &Users) {
    SmallInstructionVector Queue(1, Root);
    while (!Queue.empty()) {
      Instruction *I = Queue.pop_back_val();
      if (!Users.insert(I).second)
        continue;
  
      if (!Final.count(I))
        for (Use &U : I->uses()) {
          Instruction *User = cast<Instruction>(U.getUser());
          if (PHINode *PN = dyn_cast<PHINode>(User)) {
            // Ignore "wrap-around" uses to PHIs of this loop's header.
            if (PN->getIncomingBlock(U) == L->getHeader())
              continue;
          }
  
          if (L->contains(User) && !Exclude.count(User)) {
            Queue.push_back(User);
          }
        }
  
      // We also want to collect single-user "feeder" values.
      for (Use &U : I->operands()) {
        if (Instruction *Op = dyn_cast<Instruction>(U))
          if (Op->hasOneUse() && L->contains(Op) && !Exclude.count(Op) &&
              !Final.count(Op))
            Queue.push_back(Op);
      }
    }
  }
  
  // Collect all of the users of all of the provided root instructions (combined
  // into a single set).
  void LoopReroll::DAGRootTracker::collectInLoopUserSet(
    const SmallInstructionVector &Roots,
    const SmallInstructionSet &Exclude,
    const SmallInstructionSet &Final,
    DenseSet<Instruction *> &Users) {
    for (Instruction *Root : Roots)
      collectInLoopUserSet(Root, Exclude, Final, Users);
  }
  
  static bool isUnorderedLoadStore(Instruction *I) {
    if (LoadInst *LI = dyn_cast<LoadInst>(I))
      return LI->isUnordered();
    if (StoreInst *SI = dyn_cast<StoreInst>(I))
      return SI->isUnordered();
    if (MemIntrinsic *MI = dyn_cast<MemIntrinsic>(I))
      return !MI->isVolatile();
    return false;
  }
  
  /// Return true if IVU is a "simple" arithmetic operation.
  /// This is used for narrowing the search space for DAGRoots; only arithmetic
  /// and GEPs can be part of a DAGRoot.
  static bool isSimpleArithmeticOp(User *IVU) {
    if (Instruction *I = dyn_cast<Instruction>(IVU)) {
      switch (I->getOpcode()) {
      default: return false;
      case Instruction::Add:
      case Instruction::Sub:
      case Instruction::Mul:
      case Instruction::Shl:
      case Instruction::AShr:
      case Instruction::LShr:
      case Instruction::GetElementPtr:
      case Instruction::Trunc:
      case Instruction::ZExt:
      case Instruction::SExt:
        return true;
      }
    }
    return false;
  }
  
  static bool isLoopIncrement(User *U, Instruction *IV) {
    BinaryOperator *BO = dyn_cast<BinaryOperator>(U);
  
    if ((BO && BO->getOpcode() != Instruction::Add) ||
        (!BO && !isa<GetElementPtrInst>(U)))
      return false;
  
    for (auto *UU : U->users()) {
      PHINode *PN = dyn_cast<PHINode>(UU);
      if (PN && PN == IV)
        return true;
    }
    return false;
  }
  
  bool LoopReroll::DAGRootTracker::
  collectPossibleRoots(Instruction *Base, std::map<int64_t,Instruction*> &Roots) {
    SmallInstructionVector BaseUsers;
  
    for (auto *I : Base->users()) {
      ConstantInt *CI = nullptr;
  
      if (isLoopIncrement(I, IV)) {
        LoopIncs.push_back(cast<Instruction>(I));
        continue;
      }
  
      // The root nodes must be either GEPs, ORs or ADDs.
      if (auto *BO = dyn_cast<BinaryOperator>(I)) {
        if (BO->getOpcode() == Instruction::Add ||
            BO->getOpcode() == Instruction::Or)
          CI = dyn_cast<ConstantInt>(BO->getOperand(1));
      } else if (auto *GEP = dyn_cast<GetElementPtrInst>(I)) {
        Value *LastOperand = GEP->getOperand(GEP->getNumOperands()-1);
        CI = dyn_cast<ConstantInt>(LastOperand);
      }
  
      if (!CI) {
        if (Instruction *II = dyn_cast<Instruction>(I)) {
          BaseUsers.push_back(II);
          continue;
        } else {
          LLVM_DEBUG(dbgs() << "LRR: Aborting due to non-instruction: " << *I
                            << "\n");
          return false;
        }
      }
  
      int64_t V = std::abs(CI->getValue().getSExtValue());
      if (Roots.find(V) != Roots.end())
        // No duplicates, please.
        return false;
  
      Roots[V] = cast<Instruction>(I);
    }
  
    // Make sure we have at least two roots.
    if (Roots.empty() || (Roots.size() == 1 && BaseUsers.empty()))
      return false;
  
    // If we found non-loop-inc, non-root users of Base, assume they are
    // for the zeroth root index. This is because "add %a, 0" gets optimized
    // away.
    if (BaseUsers.size()) {
      if (Roots.find(0) != Roots.end()) {
        LLVM_DEBUG(dbgs() << "LRR: Multiple roots found for base - aborting!\n");
        return false;
      }
      Roots[0] = Base;
    }
  
    // Calculate the number of users of the base, or lowest indexed, iteration.
    unsigned NumBaseUses = BaseUsers.size();
    if (NumBaseUses == 0)
      NumBaseUses = Roots.begin()->second->getNumUses();
  
    // Check that every node has the same number of users.
    for (auto &KV : Roots) {
      if (KV.first == 0)
        continue;
      if (!KV.second->hasNUses(NumBaseUses)) {
        LLVM_DEBUG(dbgs() << "LRR: Aborting - Root and Base #users not the same: "
                          << "#Base=" << NumBaseUses
                          << ", #Root=" << KV.second->getNumUses() << "\n");
        return false;
      }
    }
  
    return true;
  }
  
  void LoopReroll::DAGRootTracker::
  findRootsRecursive(Instruction *I, SmallInstructionSet SubsumedInsts) {
    // Does the user look like it could be part of a root set?
    // All its users must be simple arithmetic ops.
    if (I->hasNUsesOrMore(IL_MaxRerollIterations + 1))
      return;
  
    if (I != IV && findRootsBase(I, SubsumedInsts))
      return;
  
    SubsumedInsts.insert(I);
  
    for (User *V : I->users()) {
      Instruction *I = cast<Instruction>(V);
      if (is_contained(LoopIncs, I))
        continue;
  
      if (!isSimpleArithmeticOp(I))
        continue;
  
      // The recursive call makes a copy of SubsumedInsts.
      findRootsRecursive(I, SubsumedInsts);
    }
  }
  
  bool LoopReroll::DAGRootTracker::validateRootSet(DAGRootSet &DRS) {
    if (DRS.Roots.empty())
      return false;
  
    // If the value of the base instruction is used outside the loop, we cannot
    // reroll the loop. Check for other root instructions is unnecessary because
    // they don't match any base instructions if their values are used outside.
    if (hasUsesOutsideLoop(DRS.BaseInst, L))
      return false;
  
    // Consider a DAGRootSet with N-1 roots (so N different values including
    //   BaseInst).
    // Define d = Roots[0] - BaseInst, which should be the same as
    //   Roots[I] - Roots[I-1] for all I in [1..N).
    // Define D = BaseInst@J - BaseInst@J-1, where "@J" means the value at the
    //   loop iteration J.
    //
    // Now, For the loop iterations to be consecutive:
    //   D = d * N
    const auto *ADR = dyn_cast<SCEVAddRecExpr>(SE->getSCEV(DRS.BaseInst));
    if (!ADR)
      return false;
  
    // Check that the first root is evenly spaced.
    unsigned N = DRS.Roots.size() + 1;
    const SCEV *StepSCEV = SE->getMinusSCEV(SE->getSCEV(DRS.Roots[0]), ADR);
    const SCEV *ScaleSCEV = SE->getConstant(StepSCEV->getType(), N);
    if (ADR->getStepRecurrence(*SE) != SE->getMulExpr(StepSCEV, ScaleSCEV))
      return false;
  
    // Check that the remainling roots are evenly spaced.
    for (unsigned i = 1; i < N - 1; ++i) {
      const SCEV *NewStepSCEV = SE->getMinusSCEV(SE->getSCEV(DRS.Roots[i]),
                                                 SE->getSCEV(DRS.Roots[i-1]));
      if (NewStepSCEV != StepSCEV)
        return false;
    }
  
    return true;
  }
  
  bool LoopReroll::DAGRootTracker::
  findRootsBase(Instruction *IVU, SmallInstructionSet SubsumedInsts) {
    // The base of a RootSet must be an AddRec, so it can be erased.
    const auto *IVU_ADR = dyn_cast<SCEVAddRecExpr>(SE->getSCEV(IVU));
    if (!IVU_ADR || IVU_ADR->getLoop() != L)
      return false;
  
    std::map<int64_t, Instruction*> V;
    if (!collectPossibleRoots(IVU, V))
      return false;
  
    // If we didn't get a root for index zero, then IVU must be
    // subsumed.
    if (V.find(0) == V.end())
      SubsumedInsts.insert(IVU);
  
    // Partition the vector into monotonically increasing indexes.
    DAGRootSet DRS;
    DRS.BaseInst = nullptr;
  
    SmallVector<DAGRootSet, 16> PotentialRootSets;
  
    for (auto &KV : V) {
      if (!DRS.BaseInst) {
        DRS.BaseInst = KV.second;
        DRS.SubsumedInsts = SubsumedInsts;
      } else if (DRS.Roots.empty()) {
        DRS.Roots.push_back(KV.second);
      } else if (V.find(KV.first - 1) != V.end()) {
        DRS.Roots.push_back(KV.second);
      } else {
        // Linear sequence terminated.
        if (!validateRootSet(DRS))
          return false;
  
        // Construct a new DAGRootSet with the next sequence.
        PotentialRootSets.push_back(DRS);
        DRS.BaseInst = KV.second;
        DRS.Roots.clear();
      }
    }
  
    if (!validateRootSet(DRS))
      return false;
  
    PotentialRootSets.push_back(DRS);
  
    RootSets.append(PotentialRootSets.begin(), PotentialRootSets.end());
  
    return true;
  }
  
  bool LoopReroll::DAGRootTracker::findRoots() {
    Inc = IVToIncMap[IV];
  
    assert(RootSets.empty() && "Unclean state!");
    if (std::abs(Inc) == 1) {
      for (auto *IVU : IV->users()) {
        if (isLoopIncrement(IVU, IV))
          LoopIncs.push_back(cast<Instruction>(IVU));
      }
      findRootsRecursive(IV, SmallInstructionSet());
      LoopIncs.push_back(IV);
    } else {
      if (!findRootsBase(IV, SmallInstructionSet()))
        return false;
    }
  
    // Ensure all sets have the same size.
    if (RootSets.empty()) {
      LLVM_DEBUG(dbgs() << "LRR: Aborting because no root sets found!\n");
      return false;
    }
    for (auto &V : RootSets) {
      if (V.Roots.empty() || V.Roots.size() != RootSets[0].Roots.size()) {
        LLVM_DEBUG(
            dbgs()
            << "LRR: Aborting because not all root sets have the same size\n");
        return false;
      }
    }
  
    Scale = RootSets[0].Roots.size() + 1;
  
    if (Scale > IL_MaxRerollIterations) {
      LLVM_DEBUG(dbgs() << "LRR: Aborting - too many iterations found. "
                        << "#Found=" << Scale
                        << ", #Max=" << IL_MaxRerollIterations << "\n");
      return false;
    }
  
    LLVM_DEBUG(dbgs() << "LRR: Successfully found roots: Scale=" << Scale
                      << "\n");
  
    return true;
  }
  
  bool LoopReroll::DAGRootTracker::collectUsedInstructions(SmallInstructionSet &PossibleRedSet) {
    // Populate the MapVector with all instructions in the block, in order first,
    // so we can iterate over the contents later in perfect order.
    for (auto &I : *L->getHeader()) {
      Uses[&I].resize(IL_End);
    }
  
    SmallInstructionSet Exclude;
    for (auto &DRS : RootSets) {
      Exclude.insert(DRS.Roots.begin(), DRS.Roots.end());
      Exclude.insert(DRS.SubsumedInsts.begin(), DRS.SubsumedInsts.end());
      Exclude.insert(DRS.BaseInst);
    }
    Exclude.insert(LoopIncs.begin(), LoopIncs.end());
  
    for (auto &DRS : RootSets) {
      DenseSet<Instruction*> VBase;
      collectInLoopUserSet(DRS.BaseInst, Exclude, PossibleRedSet, VBase);
      for (auto *I : VBase) {
        Uses[I].set(0);
      }
  
      unsigned Idx = 1;
      for (auto *Root : DRS.Roots) {
        DenseSet<Instruction*> V;
        collectInLoopUserSet(Root, Exclude, PossibleRedSet, V);
  
        // While we're here, check the use sets are the same size.
        if (V.size() != VBase.size()) {
          LLVM_DEBUG(dbgs() << "LRR: Aborting - use sets are different sizes\n");
          return false;
        }
  
        for (auto *I : V) {
          Uses[I].set(Idx);
        }
        ++Idx;
      }
  
      // Make sure our subsumed instructions are remembered too.
      for (auto *I : DRS.SubsumedInsts) {
        Uses[I].set(IL_All);
      }
    }
  
    // Make sure the loop increments are also accounted for.
  
    Exclude.clear();
    for (auto &DRS : RootSets) {
      Exclude.insert(DRS.Roots.begin(), DRS.Roots.end());
      Exclude.insert(DRS.SubsumedInsts.begin(), DRS.SubsumedInsts.end());
      Exclude.insert(DRS.BaseInst);
    }
  
    DenseSet<Instruction*> V;
    collectInLoopUserSet(LoopIncs, Exclude, PossibleRedSet, V);
    for (auto *I : V) {
      if (I->mayHaveSideEffects()) {
        LLVM_DEBUG(dbgs() << "LRR: Aborting - "
                          << "An instruction which does not belong to any root "
                          << "sets must not have side effects: " << *I);
        return false;
      }
      Uses[I].set(IL_All);
    }
  
    return true;
  }
  
  /// Get the next instruction in "In" that is a member of set Val.
  /// Start searching from StartI, and do not return anything in Exclude.
  /// If StartI is not given, start from In.begin().
  LoopReroll::DAGRootTracker::UsesTy::iterator
  LoopReroll::DAGRootTracker::nextInstr(int Val, UsesTy &In,
                                        const SmallInstructionSet &Exclude,
                                        UsesTy::iterator *StartI) {
    UsesTy::iterator I = StartI ? *StartI : In.begin();
    while (I != In.end() && (I->second.test(Val) == 0 ||
                             Exclude.contains(I->first)))
      ++I;
    return I;
  }
  
  bool LoopReroll::DAGRootTracker::isBaseInst(Instruction *I) {
    for (auto &DRS : RootSets) {
      if (DRS.BaseInst == I)
        return true;
    }
    return false;
  }
  
  bool LoopReroll::DAGRootTracker::isRootInst(Instruction *I) {
    for (auto &DRS : RootSets) {
      if (is_contained(DRS.Roots, I))
        return true;
    }
    return false;
  }
  
  /// Return true if instruction I depends on any instruction between
  /// Start and End.
  bool LoopReroll::DAGRootTracker::instrDependsOn(Instruction *I,
                                                  UsesTy::iterator Start,
                                                  UsesTy::iterator End) {
    for (auto *U : I->users()) {
      for (auto It = Start; It != End; ++It)
        if (U == It->first)
          return true;
    }
    return false;
  }
  
  static bool isIgnorableInst(const Instruction *I) {
    if (isa<DbgInfoIntrinsic>(I))
      return true;
    const IntrinsicInst* II = dyn_cast<IntrinsicInst>(I);
    if (!II)
      return false;
    switch (II->getIntrinsicID()) {
      default:
        return false;
      case Intrinsic::annotation:
      case Intrinsic::ptr_annotation:
      case Intrinsic::var_annotation:
      // TODO: the following intrinsics may also be allowed:
      //   lifetime_start, lifetime_end, invariant_start, invariant_end
        return true;
    }
    return false;
  }
  
  bool LoopReroll::DAGRootTracker::validate(ReductionTracker &Reductions) {
    // We now need to check for equivalence of the use graph of each root with
    // that of the primary induction variable (excluding the roots). Our goal
    // here is not to solve the full graph isomorphism problem, but rather to
    // catch common cases without a lot of work. As a result, we will assume
    // that the relative order of the instructions in each unrolled iteration
    // is the same (although we will not make an assumption about how the
    // different iterations are intermixed). Note that while the order must be
    // the same, the instructions may not be in the same basic block.
  
    // An array of just the possible reductions for this scale factor. When we
    // collect the set of all users of some root instructions, these reduction
    // instructions are treated as 'final' (their uses are not considered).
    // This is important because we don't want the root use set to search down
    // the reduction chain.
    SmallInstructionSet PossibleRedSet;
    SmallInstructionSet PossibleRedLastSet;
    SmallInstructionSet PossibleRedPHISet;
    Reductions.restrictToScale(Scale, PossibleRedSet,
                               PossibleRedPHISet, PossibleRedLastSet);
  
    // Populate "Uses" with where each instruction is used.
    if (!collectUsedInstructions(PossibleRedSet))
      return false;
  
    // Make sure we mark the reduction PHIs as used in all iterations.
    for (auto *I : PossibleRedPHISet) {
      Uses[I].set(IL_All);
    }
  
    // Make sure we mark loop-control-only PHIs as used in all iterations. See
    // comment above LoopReroll::isLoopControlIV for more information.
    BasicBlock *Header = L->getHeader();
    if (LoopControlIV && LoopControlIV != IV) {
      for (auto *U : LoopControlIV->users()) {
        Instruction *IVUser = dyn_cast<Instruction>(U);
        // IVUser could be loop increment or compare
        Uses[IVUser].set(IL_All);
        for (auto *UU : IVUser->users()) {
          Instruction *UUser = dyn_cast<Instruction>(UU);
          // UUser could be compare, PHI or branch
          Uses[UUser].set(IL_All);
          // Skip SExt
          if (isa<SExtInst>(UUser)) {
            UUser = dyn_cast<Instruction>(*(UUser->user_begin()));
            Uses[UUser].set(IL_All);
          }
          // Is UUser a compare instruction?
          if (UU->hasOneUse()) {
            Instruction *BI = dyn_cast<BranchInst>(*UUser->user_begin());
            if (BI == cast<BranchInst>(Header->getTerminator()))
              Uses[BI].set(IL_All);
          }
        }
      }
    }
  
    // Make sure all instructions in the loop are in one and only one
    // set.
    for (auto &KV : Uses) {
      if (KV.second.count() != 1 && !isIgnorableInst(KV.first)) {
        LLVM_DEBUG(
            dbgs() << "LRR: Aborting - instruction is not used in 1 iteration: "
                   << *KV.first << " (#uses=" << KV.second.count() << ")\n");
        return false;
      }
    }
  
    LLVM_DEBUG(for (auto &KV
                    : Uses) {
      dbgs() << "LRR: " << KV.second.find_first() << "\t" << *KV.first << "\n";
    });
  
    for (unsigned Iter = 1; Iter < Scale; ++Iter) {
      // In addition to regular aliasing information, we need to look for
      // instructions from later (future) iterations that have side effects
      // preventing us from reordering them past other instructions with side
      // effects.
      bool FutureSideEffects = false;
      AliasSetTracker AST(*AA);
      // The map between instructions in f(%iv.(i+1)) and f(%iv).
      DenseMap<Value *, Value *> BaseMap;
  
      // Compare iteration Iter to the base.
      SmallInstructionSet Visited;
      auto BaseIt = nextInstr(0, Uses, Visited);
      auto RootIt = nextInstr(Iter, Uses, Visited);
      auto LastRootIt = Uses.begin();
  
      while (BaseIt != Uses.end() && RootIt != Uses.end()) {
        Instruction *BaseInst = BaseIt->first;
        Instruction *RootInst = RootIt->first;
  
        // Skip over the IV or root instructions; only match their users.
        bool Continue = false;
        if (isBaseInst(BaseInst)) {
          Visited.insert(BaseInst);
          BaseIt = nextInstr(0, Uses, Visited);
          Continue = true;
        }
        if (isRootInst(RootInst)) {
          LastRootIt = RootIt;
          Visited.insert(RootInst);
          RootIt = nextInstr(Iter, Uses, Visited);
          Continue = true;
        }
        if (Continue) continue;
  
        if (!BaseInst->isSameOperationAs(RootInst)) {
          // Last chance saloon. We don't try and solve the full isomorphism
          // problem, but try and at least catch the case where two instructions
          // *of different types* are round the wrong way. We won't be able to
          // efficiently tell, given two ADD instructions, which way around we
          // should match them, but given an ADD and a SUB, we can at least infer
          // which one is which.
          //
          // This should allow us to deal with a greater subset of the isomorphism
          // problem. It does however change a linear algorithm into a quadratic
          // one, so limit the number of probes we do.
          auto TryIt = RootIt;
          unsigned N = NumToleratedFailedMatches;
          while (TryIt != Uses.end() &&
                 !BaseInst->isSameOperationAs(TryIt->first) &&
                 N--) {
            ++TryIt;
            TryIt = nextInstr(Iter, Uses, Visited, &TryIt);
          }
  
          if (TryIt == Uses.end() || TryIt == RootIt ||
              instrDependsOn(TryIt->first, RootIt, TryIt)) {
            LLVM_DEBUG(dbgs() << "LRR: iteration root match failed at "
                              << *BaseInst << " vs. " << *RootInst << "\n");
            return false;
          }
  
          RootIt = TryIt;
          RootInst = TryIt->first;
        }
  
        // All instructions between the last root and this root
        // may belong to some other iteration. If they belong to a
        // future iteration, then they're dangerous to alias with.
        //
        // Note that because we allow a limited amount of flexibility in the order
        // that we visit nodes, LastRootIt might be *before* RootIt, in which
        // case we've already checked this set of instructions so we shouldn't
        // do anything.
        for (; LastRootIt < RootIt; ++LastRootIt) {
          Instruction *I = LastRootIt->first;
          if (LastRootIt->second.find_first() < (int)Iter)
            continue;
          if (I->mayWriteToMemory())
            AST.add(I);
          // Note: This is specifically guarded by a check on isa<PHINode>,
          // which while a valid (somewhat arbitrary) micro-optimization, is
          // needed because otherwise isSafeToSpeculativelyExecute returns
          // false on PHI nodes.
          if (!isa<PHINode>(I) && !isUnorderedLoadStore(I) &&
              !isSafeToSpeculativelyExecute(I))
            // Intervening instructions cause side effects.
            FutureSideEffects = true;
        }
  
        // Make sure that this instruction, which is in the use set of this
        // root instruction, does not also belong to the base set or the set of
        // some other root instruction.
        if (RootIt->second.count() > 1) {
          LLVM_DEBUG(dbgs() << "LRR: iteration root match failed at " << *BaseInst
                            << " vs. " << *RootInst << " (prev. case overlap)\n");
          return false;
        }
  
        // Make sure that we don't alias with any instruction in the alias set
        // tracker. If we do, then we depend on a future iteration, and we
        // can't reroll.
        if (RootInst->mayReadFromMemory())
          for (auto &K : AST) {
            if (K.aliasesUnknownInst(RootInst, *AA)) {
              LLVM_DEBUG(dbgs() << "LRR: iteration root match failed at "
                                << *BaseInst << " vs. " << *RootInst
                                << " (depends on future store)\n");
              return false;
            }
          }
  
        // If we've past an instruction from a future iteration that may have
        // side effects, and this instruction might also, then we can't reorder
        // them, and this matching fails. As an exception, we allow the alias
        // set tracker to handle regular (unordered) load/store dependencies.
        if (FutureSideEffects && ((!isUnorderedLoadStore(BaseInst) &&
                                   !isSafeToSpeculativelyExecute(BaseInst)) ||
                                  (!isUnorderedLoadStore(RootInst) &&
                                   !isSafeToSpeculativelyExecute(RootInst)))) {
          LLVM_DEBUG(dbgs() << "LRR: iteration root match failed at " << *BaseInst
                            << " vs. " << *RootInst
                            << " (side effects prevent reordering)\n");
          return false;
        }
  
        // For instructions that are part of a reduction, if the operation is
        // associative, then don't bother matching the operands (because we
        // already know that the instructions are isomorphic, and the order
        // within the iteration does not matter). For non-associative reductions,
        // we do need to match the operands, because we need to reject
        // out-of-order instructions within an iteration!
        // For example (assume floating-point addition), we need to reject this:
        //   x += a[i]; x += b[i];
        //   x += a[i+1]; x += b[i+1];
        //   x += b[i+2]; x += a[i+2];
        bool InReduction = Reductions.isPairInSame(BaseInst, RootInst);
  
        if (!(InReduction && BaseInst->isAssociative())) {
          bool Swapped = false, SomeOpMatched = false;
          for (unsigned j = 0; j < BaseInst->getNumOperands(); ++j) {
            Value *Op2 = RootInst->getOperand(j);
  
            // If this is part of a reduction (and the operation is not
            // associatve), then we match all operands, but not those that are
            // part of the reduction.
            if (InReduction)
              if (Instruction *Op2I = dyn_cast<Instruction>(Op2))
                if (Reductions.isPairInSame(RootInst, Op2I))
                  continue;
  
            DenseMap<Value *, Value *>::iterator BMI = BaseMap.find(Op2);
            if (BMI != BaseMap.end()) {
              Op2 = BMI->second;
            } else {
              for (auto &DRS : RootSets) {
                if (DRS.Roots[Iter-1] == (Instruction*) Op2) {
                  Op2 = DRS.BaseInst;
                  break;
                }
              }
            }
  
            if (BaseInst->getOperand(Swapped ? unsigned(!j) : j) != Op2) {
              // If we've not already decided to swap the matched operands, and
              // we've not already matched our first operand (note that we could
              // have skipped matching the first operand because it is part of a
              // reduction above), and the instruction is commutative, then try
              // the swapped match.
              if (!Swapped && BaseInst->isCommutative() && !SomeOpMatched &&
                  BaseInst->getOperand(!j) == Op2) {
                Swapped = true;
              } else {
                LLVM_DEBUG(dbgs()
                           << "LRR: iteration root match failed at " << *BaseInst
                           << " vs. " << *RootInst << " (operand " << j << ")\n");
                return false;
              }
            }
  
            SomeOpMatched = true;
          }
        }
  
        if ((!PossibleRedLastSet.count(BaseInst) &&
             hasUsesOutsideLoop(BaseInst, L)) ||
            (!PossibleRedLastSet.count(RootInst) &&
             hasUsesOutsideLoop(RootInst, L))) {
          LLVM_DEBUG(dbgs() << "LRR: iteration root match failed at " << *BaseInst
                            << " vs. " << *RootInst << " (uses outside loop)\n");
          return false;
        }
  
        Reductions.recordPair(BaseInst, RootInst, Iter);
        BaseMap.insert(std::make_pair(RootInst, BaseInst));
  
        LastRootIt = RootIt;
        Visited.insert(BaseInst);
        Visited.insert(RootInst);
        BaseIt = nextInstr(0, Uses, Visited);
        RootIt = nextInstr(Iter, Uses, Visited);
      }
      assert(BaseIt == Uses.end() && RootIt == Uses.end() &&
             "Mismatched set sizes!");
    }
  
    LLVM_DEBUG(dbgs() << "LRR: Matched all iteration increments for " << *IV
                      << "\n");
  
    return true;
  }
  
  void LoopReroll::DAGRootTracker::replace(const SCEV *BackedgeTakenCount) {
    BasicBlock *Header = L->getHeader();
  
    // Compute the start and increment for each BaseInst before we start erasing
    // instructions.
    SmallVector<const SCEV *, 8> StartExprs;
    SmallVector<const SCEV *, 8> IncrExprs;
    for (auto &DRS : RootSets) {
      const SCEVAddRecExpr *IVSCEV =
          cast<SCEVAddRecExpr>(SE->getSCEV(DRS.BaseInst));
      StartExprs.push_back(IVSCEV->getStart());
      IncrExprs.push_back(SE->getMinusSCEV(SE->getSCEV(DRS.Roots[0]), IVSCEV));
    }
  
    // Remove instructions associated with non-base iterations.
    for (BasicBlock::reverse_iterator J = Header->rbegin(), JE = Header->rend();
         J != JE;) {
      unsigned I = Uses[&*J].find_first();
      if (I > 0 && I < IL_All) {
        LLVM_DEBUG(dbgs() << "LRR: removing: " << *J << "\n");
        J++->eraseFromParent();
        continue;
      }
  
      ++J;
    }
  
    // Rewrite each BaseInst using SCEV.
    for (size_t i = 0, e = RootSets.size(); i != e; ++i)
      // Insert the new induction variable.
      replaceIV(RootSets[i], StartExprs[i], IncrExprs[i]);
  
    { // Limit the lifetime of SCEVExpander.
      BranchInst *BI = cast<BranchInst>(Header->getTerminator());
      const DataLayout &DL = Header->getModule()->getDataLayout();
      SCEVExpander Expander(*SE, DL, "reroll");
      auto Zero = SE->getZero(BackedgeTakenCount->getType());
      auto One = SE->getOne(BackedgeTakenCount->getType());
      auto NewIVSCEV = SE->getAddRecExpr(Zero, One, L, SCEV::FlagAnyWrap);
      Value *NewIV =
          Expander.expandCodeFor(NewIVSCEV, BackedgeTakenCount->getType(),
                                 Header->getFirstNonPHIOrDbg());
      // FIXME: This arithmetic can overflow.
      auto TripCount = SE->getAddExpr(BackedgeTakenCount, One);
      auto ScaledTripCount = SE->getMulExpr(
          TripCount, SE->getConstant(BackedgeTakenCount->getType(), Scale));
      auto ScaledBECount = SE->getMinusSCEV(ScaledTripCount, One);
      Value *TakenCount =
          Expander.expandCodeFor(ScaledBECount, BackedgeTakenCount->getType(),
                                 Header->getFirstNonPHIOrDbg());
      Value *Cond =
          new ICmpInst(BI, CmpInst::ICMP_EQ, NewIV, TakenCount, "exitcond");
      BI->setCondition(Cond);
  
      if (BI->getSuccessor(1) != Header)
        BI->swapSuccessors();
    }
  
    SimplifyInstructionsInBlock(Header, TLI);
    DeleteDeadPHIs(Header, TLI);
  }
  
  void LoopReroll::DAGRootTracker::replaceIV(DAGRootSet &DRS,
                                             const SCEV *Start,
                                             const SCEV *IncrExpr) {
    BasicBlock *Header = L->getHeader();
    Instruction *Inst = DRS.BaseInst;
  
    const SCEV *NewIVSCEV =
        SE->getAddRecExpr(Start, IncrExpr, L, SCEV::FlagAnyWrap);
  
    { // Limit the lifetime of SCEVExpander.
      const DataLayout &DL = Header->getModule()->getDataLayout();
      SCEVExpander Expander(*SE, DL, "reroll");
      Value *NewIV = Expander.expandCodeFor(NewIVSCEV, Inst->getType(),
                                            Header->getFirstNonPHIOrDbg());
  
      for (auto &KV : Uses)
        if (KV.second.find_first() == 0)
          KV.first->replaceUsesOfWith(Inst, NewIV);
    }
  }
  
  // Validate the selected reductions. All iterations must have an isomorphic
  // part of the reduction chain and, for non-associative reductions, the chain
  // entries must appear in order.
  bool LoopReroll::ReductionTracker::validateSelected() {
    // For a non-associative reduction, the chain entries must appear in order.
    for (int i : Reds) {
      int PrevIter = 0, BaseCount = 0, Count = 0;
      for (Instruction *J : PossibleReds[i]) {
        // Note that all instructions in the chain must have been found because
        // all instructions in the function must have been assigned to some
        // iteration.
        int Iter = PossibleRedIter[J];
        if (Iter != PrevIter && Iter != PrevIter + 1 &&
            !PossibleReds[i].getReducedValue()->isAssociative()) {
          LLVM_DEBUG(dbgs() << "LRR: Out-of-order non-associative reduction: "
                            << J << "\n");
          return false;
        }
  
        if (Iter != PrevIter) {
          if (Count != BaseCount) {
            LLVM_DEBUG(dbgs()
                       << "LRR: Iteration " << PrevIter << " reduction use count "
                       << Count << " is not equal to the base use count "
                       << BaseCount << "\n");
            return false;
          }
  
          Count = 0;
        }
  
        ++Count;
        if (Iter == 0)
          ++BaseCount;
  
        PrevIter = Iter;
      }
    }
  
    return true;
  }
  
  // For all selected reductions, remove all parts except those in the first
  // iteration (and the PHI). Replace outside uses of the reduced value with uses
  // of the first-iteration reduced value (in other words, reroll the selected
  // reductions).
  void LoopReroll::ReductionTracker::replaceSelected() {
    // Fixup reductions to refer to the last instruction associated with the
    // first iteration (not the last).
    for (int i : Reds) {
      int j = 0;
      for (int e = PossibleReds[i].size(); j != e; ++j)
        if (PossibleRedIter[PossibleReds[i][j]] != 0) {
          --j;
          break;
        }
  
      // Replace users with the new end-of-chain value.
      SmallInstructionVector Users;
      for (User *U : PossibleReds[i].getReducedValue()->users()) {
        Users.push_back(cast<Instruction>(U));
      }
  
      for (Instruction *User : Users)
        User->replaceUsesOfWith(PossibleReds[i].getReducedValue(),
                                PossibleReds[i][j]);
    }
  }
  
  // Reroll the provided loop with respect to the provided induction variable.
  // Generally, we're looking for a loop like this:
  //
  // %iv = phi [ (preheader, ...), (body, %iv.next) ]
  // f(%iv)
  // %iv.1 = add %iv, 1                <-- a root increment
  // f(%iv.1)
  // %iv.2 = add %iv, 2                <-- a root increment
  // f(%iv.2)
  // %iv.scale_m_1 = add %iv, scale-1  <-- a root increment
  // f(%iv.scale_m_1)
  // ...
  // %iv.next = add %iv, scale
  // %cmp = icmp(%iv, ...)
  // br %cmp, header, exit
  //
  // Notably, we do not require that f(%iv), f(%iv.1), etc. be isolated groups of
  // instructions. In other words, the instructions in f(%iv), f(%iv.1), etc. can
  // be intermixed with eachother. The restriction imposed by this algorithm is
  // that the relative order of the isomorphic instructions in f(%iv), f(%iv.1),
  // etc. be the same.
  //
  // First, we collect the use set of %iv, excluding the other increment roots.
  // This gives us f(%iv). Then we iterate over the loop instructions (scale-1)
  // times, having collected the use set of f(%iv.(i+1)), during which we:
  //   - Ensure that the next unmatched instruction in f(%iv) is isomorphic to
  //     the next unmatched instruction in f(%iv.(i+1)).
  //   - Ensure that both matched instructions don't have any external users
  //     (with the exception of last-in-chain reduction instructions).
  //   - Track the (aliasing) write set, and other side effects, of all
  //     instructions that belong to future iterations that come before the matched
  //     instructions. If the matched instructions read from that write set, then
  //     f(%iv) or f(%iv.(i+1)) has some dependency on instructions in
  //     f(%iv.(j+1)) for some j > i, and we cannot reroll the loop. Similarly,
  //     if any of these future instructions had side effects (could not be
  //     speculatively executed), and so do the matched instructions, when we
  //     cannot reorder those side-effect-producing instructions, and rerolling
  //     fails.
  //
  // Finally, we make sure that all loop instructions are either loop increment
  // roots, belong to simple latch code, parts of validated reductions, part of
  // f(%iv) or part of some f(%iv.i). If all of that is true (and all reductions
  // have been validated), then we reroll the loop.
  bool LoopReroll::reroll(Instruction *IV, Loop *L, BasicBlock *Header,
                          const SCEV *BackedgeTakenCount,
                          ReductionTracker &Reductions) {
    DAGRootTracker DAGRoots(this, L, IV, SE, AA, TLI, DT, LI, PreserveLCSSA,
                            IVToIncMap, LoopControlIV);
  
    if (!DAGRoots.findRoots())
      return false;
    LLVM_DEBUG(dbgs() << "LRR: Found all root induction increments for: " << *IV
                      << "\n");
  
    if (!DAGRoots.validate(Reductions))
      return false;
    if (!Reductions.validateSelected())
      return false;
    // At this point, we've validated the rerolling, and we're committed to
    // making changes!
  
    Reductions.replaceSelected();
    DAGRoots.replace(BackedgeTakenCount);
  
    ++NumRerolledLoops;
    return true;
  }
  
  bool LoopReroll::runOnLoop(Loop *L) {
    BasicBlock *Header = L->getHeader();
    LLVM_DEBUG(dbgs() << "LRR: F[" << Header->getParent()->getName() << "] Loop %"
                      << Header->getName() << " (" << L->getNumBlocks()
                      << " block(s))\n");
  
    // For now, we'll handle only single BB loops.
    if (L->getNumBlocks() > 1)
      return false;
  
    if (!SE->hasLoopInvariantBackedgeTakenCount(L))
      return false;
  
    const SCEV *BackedgeTakenCount = SE->getBackedgeTakenCount(L);
    LLVM_DEBUG(dbgs() << "\n Before Reroll:\n" << *(L->getHeader()) << "\n");
    LLVM_DEBUG(dbgs() << "LRR: backedge-taken count = " << *BackedgeTakenCount
                 << "\n");
  
    // First, we need to find the induction variable with respect to which we can
    // reroll (there may be several possible options).
    SmallInstructionVector PossibleIVs;
    IVToIncMap.clear();
    LoopControlIV = nullptr;
    collectPossibleIVs(L, PossibleIVs);
  
    if (PossibleIVs.empty()) {
      LLVM_DEBUG(dbgs() << "LRR: No possible IVs found\n");
      return false;
    }
  
    ReductionTracker Reductions;
    collectPossibleReductions(L, Reductions);
    bool Changed = false;
  
    // For each possible IV, collect the associated possible set of 'root' nodes
    // (i+1, i+2, etc.).
    for (Instruction *PossibleIV : PossibleIVs)
      if (reroll(PossibleIV, L, Header, BackedgeTakenCount, Reductions)) {
        Changed = true;
        break;
      }
    LLVM_DEBUG(dbgs() << "\n After Reroll:\n" << *(L->getHeader()) << "\n");
  
    // Trip count of L has changed so SE must be re-evaluated.
    if (Changed)
      SE->forgetLoop(L);
  
    return Changed;
  }
  
  bool LoopRerollLegacyPass::runOnLoop(Loop *L, LPPassManager &LPM) {
    if (skipLoop(L))
      return false;
  
    auto *AA = &getAnalysis<AAResultsWrapperPass>().getAAResults();
    auto *LI = &getAnalysis<LoopInfoWrapperPass>().getLoopInfo();
    auto *SE = &getAnalysis<ScalarEvolutionWrapperPass>().getSE();
    auto *TLI = &getAnalysis<TargetLibraryInfoWrapperPass>().getTLI(
        *L->getHeader()->getParent());
    auto *DT = &getAnalysis<DominatorTreeWrapperPass>().getDomTree();
    bool PreserveLCSSA = mustPreserveAnalysisID(LCSSAID);
  
    return LoopReroll(AA, LI, SE, TLI, DT, PreserveLCSSA).runOnLoop(L);
  }
  
  PreservedAnalyses LoopRerollPass::run(Loop &L, LoopAnalysisManager &AM,
                                        LoopStandardAnalysisResults &AR,
                                        LPMUpdater &U) {
    return LoopReroll(&AR.AA, &AR.LI, &AR.SE, &AR.TLI, &AR.DT, true).runOnLoop(&L)
               ? getLoopPassPreservedAnalyses()
               : PreservedAnalyses::all();
  }
  