IndVarSimplify.cpp

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//===- IndVarSimplify.cpp - Induction Variable Elimination ----------------===//
//
// 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 transformation analyzes and transforms the induction variables (and
// computations derived from them) into simpler forms suitable for subsequent
// analysis and transformation.
//
// If the trip count of a loop is computable, this pass also makes the following
// changes:
//   1. The exit condition for the loop is canonicalized to compare the
//      induction value against the exit value.  This turns loops like:
//        'for (i = 7; i*i < 1000; ++i)' into 'for (i = 0; i != 25; ++i)'
//   2. Any use outside of the loop of an expression derived from the indvar
//      is changed to compute the derived value outside of the loop, eliminating
//      the dependence on the exit value of the induction variable.  If the only
//      purpose of the loop is to compute the exit value of some derived
//      expression, this transformation will make the loop dead.
//
//===----------------------------------------------------------------------===//

#include "llvm/Transforms/Scalar/IndVarSimplify.h"
#include "llvm/ADT/APFloat.h"
#include "llvm/ADT/APInt.h"
#include "llvm/ADT/ArrayRef.h"
#include "llvm/ADT/DenseMap.h"
#include "llvm/ADT/None.h"
#include "llvm/ADT/Optional.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/ADT/SmallSet.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/ADT/iterator_range.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/LoopPass.h"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/Analysis/ScalarEvolutionExpander.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Analysis/TargetLibraryInfo.h"
#include "llvm/Analysis/TargetTransformInfo.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/Transforms/Utils/Local.h"
#include "llvm/IR/BasicBlock.h"
#include "llvm/IR/Constant.h"
#include "llvm/IR/ConstantRange.h"
#include "llvm/IR/Constants.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/DerivedTypes.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/Function.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/Operator.h"
#include "llvm/IR/PassManager.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/IR/Type.h"
#include "llvm/IR/Use.h"
#include "llvm/IR/User.h"
#include "llvm/IR/Value.h"
#include "llvm/IR/ValueHandle.h"
#include "llvm/Pass.h"
#include "llvm/Support/Casting.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/Compiler.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/ErrorHandling.h"
#include "llvm/Support/MathExtras.h"
#include "llvm/Support/raw_ostream.h"
#include "llvm/Transforms/Scalar.h"
#include "llvm/Transforms/Scalar/LoopPassManager.h"
#include "llvm/Transforms/Utils/BasicBlockUtils.h"
#include "llvm/Transforms/Utils/LoopUtils.h"
#include "llvm/Transforms/Utils/SimplifyIndVar.h"
#include <cassert>
#include <cstdint>
#include <utility>

using namespace llvm;

#define DEBUG_TYPE "indvars"

STATISTIC(NumWidened     , "Number of indvars widened");
STATISTIC(NumReplaced    , "Number of exit values replaced");
STATISTIC(NumLFTR        , "Number of loop exit tests replaced");
STATISTIC(NumElimExt     , "Number of IV sign/zero extends eliminated");
STATISTIC(NumElimIV      , "Number of congruent IVs eliminated");

// Trip count verification can be enabled by default under NDEBUG if we
// implement a strong expression equivalence checker in SCEV. Until then, we
// use the verify-indvars flag, which may assert in some cases.
static cl::opt<bool> VerifyIndvars(
  "verify-indvars", cl::Hidden,
  cl::desc("Verify the ScalarEvolution result after running indvars"));

enum ReplaceExitVal { NeverRepl, OnlyCheapRepl, NoHardUse, AlwaysRepl };

static cl::opt<ReplaceExitVal> ReplaceExitValue(
    "replexitval", cl::Hidden, cl::init(OnlyCheapRepl),
    cl::desc("Choose the strategy to replace exit value in IndVarSimplify"),
    cl::values(clEnumValN(NeverRepl, "never", "never replace exit value"),
               clEnumValN(OnlyCheapRepl, "cheap",
                          "only replace exit value when the cost is cheap"),
               clEnumValN(NoHardUse, "noharduse",
                          "only replace exit values when loop def likely dead"),
               clEnumValN(AlwaysRepl, "always",
                          "always replace exit value whenever possible")));

static cl::opt<bool> UsePostIncrementRanges(
  "indvars-post-increment-ranges", cl::Hidden,
  cl::desc("Use post increment control-dependent ranges in IndVarSimplify"),
  cl::init(true));

static cl::opt<bool>
DisableLFTR("disable-lftr", cl::Hidden, cl::init(false),
            cl::desc("Disable Linear Function Test Replace optimization"));

static cl::opt<bool>
LoopPredication("indvars-predicate-loops", cl::Hidden, cl::init(false),
                cl::desc("Predicate conditions in read only loops"));


namespace {

struct RewritePhi;

class IndVarSimplify {
  LoopInfo *LI;
  ScalarEvolution *SE;
  DominatorTree *DT;
  const DataLayout &DL;
  TargetLibraryInfo *TLI;
  const TargetTransformInfo *TTI;

  SmallVector<WeakTrackingVH, 16> DeadInsts;

  bool isValidRewrite(Value *FromVal, Value *ToVal);

  bool handleFloatingPointIV(Loop *L, PHINode *PH);
  bool rewriteNonIntegerIVs(Loop *L);

  bool simplifyAndExtend(Loop *L, SCEVExpander &Rewriter, LoopInfo *LI);
  /// Try to eliminate loop exits based on analyzeable exit counts
  bool optimizeLoopExits(Loop *L, SCEVExpander &Rewriter);
  /// Try to form loop invariant tests for loop exits by changing how many
  /// iterations of the loop run when that is unobservable.
  bool predicateLoopExits(Loop *L, SCEVExpander &Rewriter);

  bool canLoopBeDeleted(Loop *L, SmallVector<RewritePhi, 8> &RewritePhiSet);
  bool rewriteLoopExitValues(Loop *L, SCEVExpander &Rewriter);
  bool rewriteFirstIterationLoopExitValues(Loop *L);
  bool hasHardUserWithinLoop(const Loop *L, const Instruction *I) const;

  bool linearFunctionTestReplace(Loop *L, BasicBlock *ExitingBB,
                                 const SCEV *ExitCount,
                                 PHINode *IndVar, SCEVExpander &Rewriter);

  bool sinkUnusedInvariants(Loop *L);

public:
  IndVarSimplify(LoopInfo *LI, ScalarEvolution *SE, DominatorTree *DT,
                 const DataLayout &DL, TargetLibraryInfo *TLI,
                 TargetTransformInfo *TTI)
      : LI(LI), SE(SE), DT(DT), DL(DL), TLI(TLI), TTI(TTI) {}

  bool run(Loop *L);
};

} // end anonymous namespace

/// Return true if the SCEV expansion generated by the rewriter can replace the
/// original value. SCEV guarantees that it produces the same value, but the way
/// it is produced may be illegal IR.  Ideally, this function will only be
/// called for verification.
bool IndVarSimplify::isValidRewrite(Value *FromVal, Value *ToVal) {
  // If an SCEV expression subsumed multiple pointers, its expansion could
  // reassociate the GEP changing the base pointer. This is illegal because the
  // final address produced by a GEP chain must be inbounds relative to its
  // underlying object. Otherwise basic alias analysis, among other things,
  // could fail in a dangerous way. Ultimately, SCEV will be improved to avoid
  // producing an expression involving multiple pointers. Until then, we must
  // bail out here.
  //
  // Retrieve the pointer operand of the GEP. Don't use GetUnderlyingObject
  // because it understands lcssa phis while SCEV does not.
  Value *FromPtr = FromVal;
  Value *ToPtr = ToVal;
  if (auto *GEP = dyn_cast<GEPOperator>(FromVal)) {
    FromPtr = GEP->getPointerOperand();
  }
  if (auto *GEP = dyn_cast<GEPOperator>(ToVal)) {
    ToPtr = GEP->getPointerOperand();
  }
  if (FromPtr != FromVal || ToPtr != ToVal) {
    // Quickly check the common case
    if (FromPtr == ToPtr)
      return true;

    // SCEV may have rewritten an expression that produces the GEP's pointer
    // operand. That's ok as long as the pointer operand has the same base
    // pointer. Unlike GetUnderlyingObject(), getPointerBase() will find the
    // base of a recurrence. This handles the case in which SCEV expansion
    // converts a pointer type recurrence into a nonrecurrent pointer base
    // indexed by an integer recurrence.

    // If the GEP base pointer is a vector of pointers, abort.
    if (!FromPtr->getType()->isPointerTy() || !ToPtr->getType()->isPointerTy())
      return false;

    const SCEV *FromBase = SE->getPointerBase(SE->getSCEV(FromPtr));
    const SCEV *ToBase = SE->getPointerBase(SE->getSCEV(ToPtr));
    if (FromBase == ToBase)
      return true;

    LLVM_DEBUG(dbgs() << "INDVARS: GEP rewrite bail out " << *FromBase
                      << " != " << *ToBase << "\n");

    return false;
  }
  return true;
}

/// Determine the insertion point for this user. By default, insert immediately
/// before the user. SCEVExpander or LICM will hoist loop invariants out of the
/// loop. For PHI nodes, there may be multiple uses, so compute the nearest
/// common dominator for the incoming blocks. A nullptr can be returned if no
/// viable location is found: it may happen if User is a PHI and Def only comes
/// to this PHI from unreachable blocks.
static Instruction *getInsertPointForUses(Instruction *User, Value *Def,
                                          DominatorTree *DT, LoopInfo *LI) {
  PHINode *PHI = dyn_cast<PHINode>(User);
  if (!PHI)
    return User;

  Instruction *InsertPt = nullptr;
  for (unsigned i = 0, e = PHI->getNumIncomingValues(); i != e; ++i) {
    if (PHI->getIncomingValue(i) != Def)
      continue;

    BasicBlock *InsertBB = PHI->getIncomingBlock(i);

    if (!DT->isReachableFromEntry(InsertBB))
      continue;

    if (!InsertPt) {
      InsertPt = InsertBB->getTerminator();
      continue;
    }
    InsertBB = DT->findNearestCommonDominator(InsertPt->getParent(), InsertBB);
    InsertPt = InsertBB->getTerminator();
  }

  // If we have skipped all inputs, it means that Def only comes to Phi from
  // unreachable blocks.
  if (!InsertPt)
    return nullptr;

  auto *DefI = dyn_cast<Instruction>(Def);
  if (!DefI)
    return InsertPt;

  assert(DT->dominates(DefI, InsertPt) && "def does not dominate all uses");

  auto *L = LI->getLoopFor(DefI->getParent());
  assert(!L || L->contains(LI->getLoopFor(InsertPt->getParent())));

  for (auto *DTN = (*DT)[InsertPt->getParent()]; DTN; DTN = DTN->getIDom())
    if (LI->getLoopFor(DTN->getBlock()) == L)
      return DTN->getBlock()->getTerminator();

  llvm_unreachable("DefI dominates InsertPt!");
}

//===----------------------------------------------------------------------===//
// rewriteNonIntegerIVs and helpers. Prefer integer IVs.
//===----------------------------------------------------------------------===//

/// Convert APF to an integer, if possible.
static bool ConvertToSInt(const APFloat &APF, int64_t &IntVal) {
  bool isExact = false;
  // See if we can convert this to an int64_t
  uint64_t UIntVal;
  if (APF.convertToInteger(makeMutableArrayRef(UIntVal), 64, true,
                           APFloat::rmTowardZero, &isExact) != APFloat::opOK ||
      !isExact)
    return false;
  IntVal = UIntVal;
  return true;
}

/// If the loop has floating induction variable then insert corresponding
/// integer induction variable if possible.
/// For example,
/// for(double i = 0; i < 10000; ++i)
///   bar(i)
/// is converted into
/// for(int i = 0; i < 10000; ++i)
///   bar((double)i);
bool IndVarSimplify::handleFloatingPointIV(Loop *L, PHINode *PN) {
  unsigned IncomingEdge = L->contains(PN->getIncomingBlock(0));
  unsigned BackEdge     = IncomingEdge^1;

  // Check incoming value.
  auto *InitValueVal = dyn_cast<ConstantFP>(PN->getIncomingValue(IncomingEdge));

  int64_t InitValue;
  if (!InitValueVal || !ConvertToSInt(InitValueVal->getValueAPF(), InitValue))
    return false;

  // Check IV increment. Reject this PN if increment operation is not
  // an add or increment value can not be represented by an integer.
  auto *Incr = dyn_cast<BinaryOperator>(PN->getIncomingValue(BackEdge));
  if (Incr == nullptr || Incr->getOpcode() != Instruction::FAdd) return false;

  // If this is not an add of the PHI with a constantfp, or if the constant fp
  // is not an integer, bail out.
  ConstantFP *IncValueVal = dyn_cast<ConstantFP>(Incr->getOperand(1));
  int64_t IncValue;
  if (IncValueVal == nullptr || Incr->getOperand(0) != PN ||
      !ConvertToSInt(IncValueVal->getValueAPF(), IncValue))
    return false;

  // Check Incr uses. One user is PN and the other user is an exit condition
  // used by the conditional terminator.
  Value::user_iterator IncrUse = Incr->user_begin();
  Instruction *U1 = cast<Instruction>(*IncrUse++);
  if (IncrUse == Incr->user_end()) return false;
  Instruction *U2 = cast<Instruction>(*IncrUse++);
  if (IncrUse != Incr->user_end()) return false;

  // Find exit condition, which is an fcmp.  If it doesn't exist, or if it isn't
  // only used by a branch, we can't transform it.
  FCmpInst *Compare = dyn_cast<FCmpInst>(U1);
  if (!Compare)
    Compare = dyn_cast<FCmpInst>(U2);
  if (!Compare || !Compare->hasOneUse() ||
      !isa<BranchInst>(Compare->user_back()))
    return false;

  BranchInst *TheBr = cast<BranchInst>(Compare->user_back());

  // We need to verify that the branch actually controls the iteration count
  // of the loop.  If not, the new IV can overflow and no one will notice.
  // The branch block must be in the loop and one of the successors must be out
  // of the loop.
  assert(TheBr->isConditional() && "Can't use fcmp if not conditional");
  if (!L->contains(TheBr->getParent()) ||
      (L->contains(TheBr->getSuccessor(0)) &&
       L->contains(TheBr->getSuccessor(1))))
    return false;

  // If it isn't a comparison with an integer-as-fp (the exit value), we can't
  // transform it.
  ConstantFP *ExitValueVal = dyn_cast<ConstantFP>(Compare->getOperand(1));
  int64_t ExitValue;
  if (ExitValueVal == nullptr ||
      !ConvertToSInt(ExitValueVal->getValueAPF(), ExitValue))
    return false;

  // Find new predicate for integer comparison.
  CmpInst::Predicate NewPred = CmpInst::BAD_ICMP_PREDICATE;
  switch (Compare->getPredicate()) {
  default: return false;  // Unknown comparison.
  case CmpInst::FCMP_OEQ:
  case CmpInst::FCMP_UEQ: NewPred = CmpInst::ICMP_EQ; break;
  case CmpInst::FCMP_ONE:
  case CmpInst::FCMP_UNE: NewPred = CmpInst::ICMP_NE; break;
  case CmpInst::FCMP_OGT:
  case CmpInst::FCMP_UGT: NewPred = CmpInst::ICMP_SGT; break;
  case CmpInst::FCMP_OGE:
  case CmpInst::FCMP_UGE: NewPred = CmpInst::ICMP_SGE; break;
  case CmpInst::FCMP_OLT:
  case CmpInst::FCMP_ULT: NewPred = CmpInst::ICMP_SLT; break;
  case CmpInst::FCMP_OLE:
  case CmpInst::FCMP_ULE: NewPred = CmpInst::ICMP_SLE; break;
  }

  // We convert the floating point induction variable to a signed i32 value if
  // we can.  This is only safe if the comparison will not overflow in a way
  // that won't be trapped by the integer equivalent operations.  Check for this
  // now.
  // TODO: We could use i64 if it is native and the range requires it.

  // The start/stride/exit values must all fit in signed i32.
  if (!isInt<32>(InitValue) || !isInt<32>(IncValue) || !isInt<32>(ExitValue))
    return false;

  // If not actually striding (add x, 0.0), avoid touching the code.
  if (IncValue == 0)
    return false;

  // Positive and negative strides have different safety conditions.
  if (IncValue > 0) {
    // If we have a positive stride, we require the init to be less than the
    // exit value.
    if (InitValue >= ExitValue)
      return false;

    uint32_t Range = uint32_t(ExitValue-InitValue);
    // Check for infinite loop, either:
    // while (i <= Exit) or until (i > Exit)
    if (NewPred == CmpInst::ICMP_SLE || NewPred == CmpInst::ICMP_SGT) {
      if (++Range == 0) return false;  // Range overflows.
    }

    unsigned Leftover = Range % uint32_t(IncValue);

    // If this is an equality comparison, we require that the strided value
    // exactly land on the exit value, otherwise the IV condition will wrap
    // around and do things the fp IV wouldn't.
    if ((NewPred == CmpInst::ICMP_EQ || NewPred == CmpInst::ICMP_NE) &&
        Leftover != 0)
      return false;

    // If the stride would wrap around the i32 before exiting, we can't
    // transform the IV.
    if (Leftover != 0 && int32_t(ExitValue+IncValue) < ExitValue)
      return false;
  } else {
    // If we have a negative stride, we require the init to be greater than the
    // exit value.
    if (InitValue <= ExitValue)
      return false;

    uint32_t Range = uint32_t(InitValue-ExitValue);
    // Check for infinite loop, either:
    // while (i >= Exit) or until (i < Exit)
    if (NewPred == CmpInst::ICMP_SGE || NewPred == CmpInst::ICMP_SLT) {
      if (++Range == 0) return false;  // Range overflows.
    }

    unsigned Leftover = Range % uint32_t(-IncValue);

    // If this is an equality comparison, we require that the strided value
    // exactly land on the exit value, otherwise the IV condition will wrap
    // around and do things the fp IV wouldn't.
    if ((NewPred == CmpInst::ICMP_EQ || NewPred == CmpInst::ICMP_NE) &&
        Leftover != 0)
      return false;

    // If the stride would wrap around the i32 before exiting, we can't
    // transform the IV.
    if (Leftover != 0 && int32_t(ExitValue+IncValue) > ExitValue)
      return false;
  }

  IntegerType *Int32Ty = Type::getInt32Ty(PN->getContext());

  // Insert new integer induction variable.
  PHINode *NewPHI = PHINode::Create(Int32Ty, 2, PN->getName()+".int", PN);
  NewPHI->addIncoming(ConstantInt::get(Int32Ty, InitValue),
                      PN->getIncomingBlock(IncomingEdge));

  Value *NewAdd =
    BinaryOperator::CreateAdd(NewPHI, ConstantInt::get(Int32Ty, IncValue),
                              Incr->getName()+".int", Incr);
  NewPHI->addIncoming(NewAdd, PN->getIncomingBlock(BackEdge));

  ICmpInst *NewCompare = new ICmpInst(TheBr, NewPred, NewAdd,
                                      ConstantInt::get(Int32Ty, ExitValue),
                                      Compare->getName());

  // In the following deletions, PN may become dead and may be deleted.
  // Use a WeakTrackingVH to observe whether this happens.
  WeakTrackingVH WeakPH = PN;

  // Delete the old floating point exit comparison.  The branch starts using the
  // new comparison.
  NewCompare->takeName(Compare);
  Compare->replaceAllUsesWith(NewCompare);
  RecursivelyDeleteTriviallyDeadInstructions(Compare, TLI);

  // Delete the old floating point increment.
  Incr->replaceAllUsesWith(UndefValue::get(Incr->getType()));
  RecursivelyDeleteTriviallyDeadInstructions(Incr, TLI);

  // If the FP induction variable still has uses, this is because something else
  // in the loop uses its value.  In order to canonicalize the induction
  // variable, we chose to eliminate the IV and rewrite it in terms of an
  // int->fp cast.
  //
  // We give preference to sitofp over uitofp because it is faster on most
  // platforms.
  if (WeakPH) {
    Value *Conv = new SIToFPInst(NewPHI, PN->getType(), "indvar.conv",
                                 &*PN->getParent()->getFirstInsertionPt());
    PN->replaceAllUsesWith(Conv);
    RecursivelyDeleteTriviallyDeadInstructions(PN, TLI);
  }
  return true;
}

bool IndVarSimplify::rewriteNonIntegerIVs(Loop *L) {
  // First step.  Check to see if there are any floating-point recurrences.
  // If there are, change them into integer recurrences, permitting analysis by
  // the SCEV routines.
  BasicBlock *Header = L->getHeader();

  SmallVector<WeakTrackingVH, 8> PHIs;
  for (PHINode &PN : Header->phis())
    PHIs.push_back(&PN);

  bool Changed = false;
  for (unsigned i = 0, e = PHIs.size(); i != e; ++i)
    if (PHINode *PN = dyn_cast_or_null<PHINode>(&*PHIs[i]))
      Changed |= handleFloatingPointIV(L, PN);

  // If the loop previously had floating-point IV, ScalarEvolution
  // may not have been able to compute a trip count. Now that we've done some
  // re-writing, the trip count may be computable.
  if (Changed)
    SE->forgetLoop(L);
  return Changed;
}

namespace {

// Collect information about PHI nodes which can be transformed in
// rewriteLoopExitValues.
struct RewritePhi {
  PHINode *PN;

  // Ith incoming value.
  unsigned Ith;

  // Exit value after expansion.
  Value *Val;

  // High Cost when expansion.
  bool HighCost;

  RewritePhi(PHINode *P, unsigned I, Value *V, bool H)
      : PN(P), Ith(I), Val(V), HighCost(H) {}
};

} // end anonymous namespace

//===----------------------------------------------------------------------===//
// rewriteLoopExitValues - Optimize IV users outside the loop.
// As a side effect, reduces the amount of IV processing within the loop.
//===----------------------------------------------------------------------===//

bool IndVarSimplify::hasHardUserWithinLoop(const Loop *L, const Instruction *I) const {
  SmallPtrSet<const Instruction *, 8> Visited;
  SmallVector<const Instruction *, 8> WorkList;
  Visited.insert(I);
  WorkList.push_back(I);
  while (!WorkList.empty()) {
    const Instruction *Curr = WorkList.pop_back_val();
    // This use is outside the loop, nothing to do.
    if (!L->contains(Curr))
      continue;
    // Do we assume it is a "hard" use which will not be eliminated easily?
    if (Curr->mayHaveSideEffects())
      return true;
    // Otherwise, add all its users to worklist.
    for (auto U : Curr->users()) {
      auto *UI = cast<Instruction>(U);
      if (Visited.insert(UI).second)
        WorkList.push_back(UI);
    }
  }
  return false;
}

/// Check to see if this loop has a computable loop-invariant execution count.
/// If so, this means that we can compute the final value of any expressions
/// that are recurrent in the loop, and substitute the exit values from the loop
/// into any instructions outside of the loop that use the final values of the
/// current expressions.
///
/// This is mostly redundant with the regular IndVarSimplify activities that
/// happen later, except that it's more powerful in some cases, because it's
/// able to brute-force evaluate arbitrary instructions as long as they have
/// constant operands at the beginning of the loop.
bool IndVarSimplify::rewriteLoopExitValues(Loop *L, SCEVExpander &Rewriter) {
  // Check a pre-condition.
  assert(L->isRecursivelyLCSSAForm(*DT, *LI) &&
         "Indvars did not preserve LCSSA!");

  SmallVector<BasicBlock*, 8> ExitBlocks;
  L->getUniqueExitBlocks(ExitBlocks);

  SmallVector<RewritePhi, 8> RewritePhiSet;
  // Find all values that are computed inside the loop, but used outside of it.
  // Because of LCSSA, these values will only occur in LCSSA PHI Nodes.  Scan
  // the exit blocks of the loop to find them.
  for (BasicBlock *ExitBB : ExitBlocks) {
    // If there are no PHI nodes in this exit block, then no values defined
    // inside the loop are used on this path, skip it.
    PHINode *PN = dyn_cast<PHINode>(ExitBB->begin());
    if (!PN) continue;

    unsigned NumPreds = PN->getNumIncomingValues();

    // Iterate over all of the PHI nodes.
    BasicBlock::iterator BBI = ExitBB->begin();
    while ((PN = dyn_cast<PHINode>(BBI++))) {
      if (PN->use_empty())
        continue; // dead use, don't replace it

      if (!SE->isSCEVable(PN->getType()))
        continue;

      // It's necessary to tell ScalarEvolution about this explicitly so that
      // it can walk the def-use list and forget all SCEVs, as it may not be
      // watching the PHI itself. Once the new exit value is in place, there
      // may not be a def-use connection between the loop and every instruction
      // which got a SCEVAddRecExpr for that loop.
      SE->forgetValue(PN);

      // Iterate over all of the values in all the PHI nodes.
      for (unsigned i = 0; i != NumPreds; ++i) {
        // If the value being merged in is not integer or is not defined
        // in the loop, skip it.
        Value *InVal = PN->getIncomingValue(i);
        if (!isa<Instruction>(InVal))
          continue;

        // If this pred is for a subloop, not L itself, skip it.
        if (LI->getLoopFor(PN->getIncomingBlock(i)) != L)
          continue; // The Block is in a subloop, skip it.

        // Check that InVal is defined in the loop.
        Instruction *Inst = cast<Instruction>(InVal);
        if (!L->contains(Inst))
          continue;

        // Okay, this instruction has a user outside of the current loop
        // and varies predictably *inside* the loop.  Evaluate the value it
        // contains when the loop exits, if possible.  We prefer to start with
        // expressions which are true for all exits (so as to maximize
        // expression reuse by the SCEVExpander), but resort to per-exit
        // evaluation if that fails.  
        const SCEV *ExitValue = SE->getSCEVAtScope(Inst, L->getParentLoop());
        if (isa<SCEVCouldNotCompute>(ExitValue) ||
            !SE->isLoopInvariant(ExitValue, L) ||
            !isSafeToExpand(ExitValue, *SE)) {
          // TODO: This should probably be sunk into SCEV in some way; maybe a
          // getSCEVForExit(SCEV*, L, ExitingBB)?  It can be generalized for
          // most SCEV expressions and other recurrence types (e.g. shift
          // recurrences).  Is there existing code we can reuse?
          const SCEV *ExitCount = SE->getExitCount(L, PN->getIncomingBlock(i));
          if (isa<SCEVCouldNotCompute>(ExitCount))
            continue;
          if (auto *AddRec = dyn_cast<SCEVAddRecExpr>(SE->getSCEV(Inst)))
            if (AddRec->getLoop() == L)
              ExitValue = AddRec->evaluateAtIteration(ExitCount, *SE);
          if (isa<SCEVCouldNotCompute>(ExitValue) ||
              !SE->isLoopInvariant(ExitValue, L) ||
              !isSafeToExpand(ExitValue, *SE))
            continue;
        }
        
        // Computing the value outside of the loop brings no benefit if it is
        // definitely used inside the loop in a way which can not be optimized
        // away.  Avoid doing so unless we know we have a value which computes
        // the ExitValue already.  TODO: This should be merged into SCEV
        // expander to leverage its knowledge of existing expressions.
        if (ReplaceExitValue != AlwaysRepl &&
            !isa<SCEVConstant>(ExitValue) && !isa<SCEVUnknown>(ExitValue) &&
            hasHardUserWithinLoop(L, Inst))
          continue;

        bool HighCost = Rewriter.isHighCostExpansion(ExitValue, L, Inst);
        Value *ExitVal = Rewriter.expandCodeFor(ExitValue, PN->getType(), Inst);

        LLVM_DEBUG(dbgs() << "INDVARS: RLEV: AfterLoopVal = " << *ExitVal
                          << '\n'
                          << "  LoopVal = " << *Inst << "\n");

        if (!isValidRewrite(Inst, ExitVal)) {
          DeadInsts.push_back(ExitVal);
          continue;
        }

#ifndef NDEBUG
        // If we reuse an instruction from a loop which is neither L nor one of
        // its containing loops, we end up breaking LCSSA form for this loop by
        // creating a new use of its instruction.
        if (auto *ExitInsn = dyn_cast<Instruction>(ExitVal))
          if (auto *EVL = LI->getLoopFor(ExitInsn->getParent()))
            if (EVL != L)
              assert(EVL->contains(L) && "LCSSA breach detected!");
#endif

        // Collect all the candidate PHINodes to be rewritten.
        RewritePhiSet.emplace_back(PN, i, ExitVal, HighCost);
      }
    }
  }

  bool LoopCanBeDel = canLoopBeDeleted(L, RewritePhiSet);

  bool Changed = false;
  // Transformation.
  for (const RewritePhi &Phi : RewritePhiSet) {
    PHINode *PN = Phi.PN;
    Value *ExitVal = Phi.Val;

    // Only do the rewrite when the ExitValue can be expanded cheaply.
    // If LoopCanBeDel is true, rewrite exit value aggressively.
    if (ReplaceExitValue == OnlyCheapRepl && !LoopCanBeDel && Phi.HighCost) {
      DeadInsts.push_back(ExitVal);
      continue;
    }

    Changed = true;
    ++NumReplaced;
    Instruction *Inst = cast<Instruction>(PN->getIncomingValue(Phi.Ith));
    PN->setIncomingValue(Phi.Ith, ExitVal);

    // If this instruction is dead now, delete it. Don't do it now to avoid
    // invalidating iterators.
    if (isInstructionTriviallyDead(Inst, TLI))
      DeadInsts.push_back(Inst);

    // Replace PN with ExitVal if that is legal and does not break LCSSA.
    if (PN->getNumIncomingValues() == 1 &&
        LI->replacementPreservesLCSSAForm(PN, ExitVal)) {
      PN->replaceAllUsesWith(ExitVal);
      PN->eraseFromParent();
    }
  }

  // The insertion point instruction may have been deleted; clear it out
  // so that the rewriter doesn't trip over it later.
  Rewriter.clearInsertPoint();
  return Changed;
}

//===---------------------------------------------------------------------===//
// rewriteFirstIterationLoopExitValues: Rewrite loop exit values if we know
// they will exit at the first iteration.
//===---------------------------------------------------------------------===//

/// Check to see if this loop has loop invariant conditions which lead to loop
/// exits. If so, we know that if the exit path is taken, it is at the first
/// loop iteration. This lets us predict exit values of PHI nodes that live in
/// loop header.
bool IndVarSimplify::rewriteFirstIterationLoopExitValues(Loop *L) {
  // Verify the input to the pass is already in LCSSA form.
  assert(L->isLCSSAForm(*DT));

  SmallVector<BasicBlock *, 8> ExitBlocks;
  L->getUniqueExitBlocks(ExitBlocks);

  bool MadeAnyChanges = false;
  for (auto *ExitBB : ExitBlocks) {
    // If there are no more PHI nodes in this exit block, then no more
    // values defined inside the loop are used on this path.
    for (PHINode &PN : ExitBB->phis()) {
      for (unsigned IncomingValIdx = 0, E = PN.getNumIncomingValues();
           IncomingValIdx != E; ++IncomingValIdx) {
        auto *IncomingBB = PN.getIncomingBlock(IncomingValIdx);

        // Can we prove that the exit must run on the first iteration if it
        // runs at all?  (i.e. early exits are fine for our purposes, but
        // traces which lead to this exit being taken on the 2nd iteration
        // aren't.)  Note that this is about whether the exit branch is
        // executed, not about whether it is taken.
        if (!L->getLoopLatch() ||
            !DT->dominates(IncomingBB, L->getLoopLatch()))
          continue;

        // Get condition that leads to the exit path.
        auto *TermInst = IncomingBB->getTerminator();

        Value *Cond = nullptr;
        if (auto *BI = dyn_cast<BranchInst>(TermInst)) {
          // Must be a conditional branch, otherwise the block
          // should not be in the loop.
          Cond = BI->getCondition();
        } else if (auto *SI = dyn_cast<SwitchInst>(TermInst))
          Cond = SI->getCondition();
        else
          continue;

        if (!L->isLoopInvariant(Cond))
          continue;

        auto *ExitVal = dyn_cast<PHINode>(PN.getIncomingValue(IncomingValIdx));

        // Only deal with PHIs in the loop header.
        if (!ExitVal || ExitVal->getParent() != L->getHeader())
          continue;

        // If ExitVal is a PHI on the loop header, then we know its
        // value along this exit because the exit can only be taken
        // on the first iteration.
        auto *LoopPreheader = L->getLoopPreheader();
        assert(LoopPreheader && "Invalid loop");
        int PreheaderIdx = ExitVal->getBasicBlockIndex(LoopPreheader);
        if (PreheaderIdx != -1) {
          assert(ExitVal->getParent() == L->getHeader() &&
                 "ExitVal must be in loop header");
          MadeAnyChanges = true;
          PN.setIncomingValue(IncomingValIdx,
                              ExitVal->getIncomingValue(PreheaderIdx));
        }
      }
    }
  }
  return MadeAnyChanges;
}

/// Check whether it is possible to delete the loop after rewriting exit
/// value. If it is possible, ignore ReplaceExitValue and do rewriting
/// aggressively.
bool IndVarSimplify::canLoopBeDeleted(
    Loop *L, SmallVector<RewritePhi, 8> &RewritePhiSet) {
  BasicBlock *Preheader = L->getLoopPreheader();
  // If there is no preheader, the loop will not be deleted.
  if (!Preheader)
    return false;

  // In LoopDeletion pass Loop can be deleted when ExitingBlocks.size() > 1.
  // We obviate multiple ExitingBlocks case for simplicity.
  // TODO: If we see testcase with multiple ExitingBlocks can be deleted
  // after exit value rewriting, we can enhance the logic here.
  SmallVector<BasicBlock *, 4> ExitingBlocks;
  L->getExitingBlocks(ExitingBlocks);
  SmallVector<BasicBlock *, 8> ExitBlocks;
  L->getUniqueExitBlocks(ExitBlocks);
  if (ExitBlocks.size() != 1 || ExitingBlocks.size() != 1)
    return false;

  BasicBlock *ExitBlock = ExitBlocks[0];
  BasicBlock::iterator BI = ExitBlock->begin();
  while (PHINode *P = dyn_cast<PHINode>(BI)) {
    Value *Incoming = P->getIncomingValueForBlock(ExitingBlocks[0]);

    // If the Incoming value of P is found in RewritePhiSet, we know it
    // could be rewritten to use a loop invariant value in transformation
    // phase later. Skip it in the loop invariant check below.
    bool found = false;
    for (const RewritePhi &Phi : RewritePhiSet) {
      unsigned i = Phi.Ith;
      if (Phi.PN == P && (Phi.PN)->getIncomingValue(i) == Incoming) {
        found = true;
        break;
      }
    }

    Instruction *I;
    if (!found && (I = dyn_cast<Instruction>(Incoming)))
      if (!L->hasLoopInvariantOperands(I))
        return false;

    ++BI;
  }

  for (auto *BB : L->blocks())
    if (llvm::any_of(*BB, [](Instruction &I) {
          return I.mayHaveSideEffects();
        }))
      return false;

  return true;
}

//===----------------------------------------------------------------------===//
//  IV Widening - Extend the width of an IV to cover its widest uses.
//===----------------------------------------------------------------------===//

namespace {

// Collect information about induction variables that are used by sign/zero
// extend operations. This information is recorded by CollectExtend and provides
// the input to WidenIV.
struct WideIVInfo {
  PHINode *NarrowIV = nullptr;

  // Widest integer type created [sz]ext
  Type *WidestNativeType = nullptr;

  // Was a sext user seen before a zext?
  bool IsSigned = false;
};

} // end anonymous namespace

/// Update information about the induction variable that is extended by this
/// sign or zero extend operation. This is used to determine the final width of
/// the IV before actually widening it.
static void visitIVCast(CastInst *Cast, WideIVInfo &WI, ScalarEvolution *SE,
                        const TargetTransformInfo *TTI) {
  bool IsSigned = Cast->getOpcode() == Instruction::SExt;
  if (!IsSigned && Cast->getOpcode() != Instruction::ZExt)
    return;

  Type *Ty = Cast->getType();
  uint64_t Width = SE->getTypeSizeInBits(Ty);
  if (!Cast->getModule()->getDataLayout().isLegalInteger(Width))
    return;

  // Check that `Cast` actually extends the induction variable (we rely on this
  // later).  This takes care of cases where `Cast` is extending a truncation of
  // the narrow induction variable, and thus can end up being narrower than the
  // "narrow" induction variable.
  uint64_t NarrowIVWidth = SE->getTypeSizeInBits(WI.NarrowIV->getType());
  if (NarrowIVWidth >= Width)
    return;

  // Cast is either an sext or zext up to this point.
  // We should not widen an indvar if arithmetics on the wider indvar are more
  // expensive than those on the narrower indvar. We check only the cost of ADD
  // because at least an ADD is required to increment the induction variable. We
  // could compute more comprehensively the cost of all instructions on the
  // induction variable when necessary.
  if (TTI &&
      TTI->getArithmeticInstrCost(Instruction::Add, Ty) >
          TTI->getArithmeticInstrCost(Instruction::Add,
                                      Cast->getOperand(0)->getType())) {
    return;
  }

  if (!WI.WidestNativeType) {
    WI.WidestNativeType = SE->getEffectiveSCEVType(Ty);
    WI.IsSigned = IsSigned;
    return;
  }

  // We extend the IV to satisfy the sign of its first user, arbitrarily.
  if (WI.IsSigned != IsSigned)
    return;

  if (Width > SE->getTypeSizeInBits(WI.WidestNativeType))
    WI.WidestNativeType = SE->getEffectiveSCEVType(Ty);
}

namespace {

/// Record a link in the Narrow IV def-use chain along with the WideIV that
/// computes the same value as the Narrow IV def.  This avoids caching Use*
/// pointers.
struct NarrowIVDefUse {
  Instruction *NarrowDef = nullptr;
  Instruction *NarrowUse = nullptr;
  Instruction *WideDef = nullptr;

  // True if the narrow def is never negative.  Tracking this information lets
  // us use a sign extension instead of a zero extension or vice versa, when
  // profitable and legal.
  bool NeverNegative = false;

  NarrowIVDefUse(Instruction *ND, Instruction *NU, Instruction *WD,
                 bool NeverNegative)
      : NarrowDef(ND), NarrowUse(NU), WideDef(WD),
        NeverNegative(NeverNegative) {}
};

/// The goal of this transform is to remove sign and zero extends without
/// creating any new induction variables. To do this, it creates a new phi of
/// the wider type and redirects all users, either removing extends or inserting
/// truncs whenever we stop propagating the type.
class WidenIV {
  // Parameters
  PHINode *OrigPhi;
  Type *WideType;

  // Context
  LoopInfo        *LI;
  Loop            *L;
  ScalarEvolution *SE;
  DominatorTree   *DT;

  // Does the module have any calls to the llvm.experimental.guard intrinsic
  // at all? If not we can avoid scanning instructions looking for guards.
  bool HasGuards;

  // Result
  PHINode *WidePhi = nullptr;
  Instruction *WideInc = nullptr;
  const SCEV *WideIncExpr = nullptr;
  SmallVectorImpl<WeakTrackingVH> &DeadInsts;

  SmallPtrSet<Instruction *,16> Widened;
  SmallVector<NarrowIVDefUse, 8> NarrowIVUsers;

  enum ExtendKind { ZeroExtended, SignExtended, Unknown };

  // A map tracking the kind of extension used to widen each narrow IV
  // and narrow IV user.
  // Key: pointer to a narrow IV or IV user.
  // Value: the kind of extension used to widen this Instruction.
  DenseMap<AssertingVH<Instruction>, ExtendKind> ExtendKindMap;

  using DefUserPair = std::pair<AssertingVH<Value>, AssertingVH<Instruction>>;

  // A map with control-dependent ranges for post increment IV uses. The key is
  // a pair of IV def and a use of this def denoting the context. The value is
  // a ConstantRange representing possible values of the def at the given
  // context.
  DenseMap<DefUserPair, ConstantRange> PostIncRangeInfos;

  Optional<ConstantRange> getPostIncRangeInfo(Value *Def,
                                              Instruction *UseI) {
    DefUserPair Key(Def, UseI);
    auto It = PostIncRangeInfos.find(Key);
    return It == PostIncRangeInfos.end()
               ? Optional<ConstantRange>(None)
               : Optional<ConstantRange>(It->second);
  }

  void calculatePostIncRanges(PHINode *OrigPhi);
  void calculatePostIncRange(Instruction *NarrowDef, Instruction *NarrowUser);

  void updatePostIncRangeInfo(Value *Def, Instruction *UseI, ConstantRange R) {
    DefUserPair Key(Def, UseI);
    auto It = PostIncRangeInfos.find(Key);
    if (It == PostIncRangeInfos.end())
      PostIncRangeInfos.insert({Key, R});
    else
      It->second = R.intersectWith(It->second);
  }

public:
  WidenIV(const WideIVInfo &WI, LoopInfo *LInfo, ScalarEvolution *SEv,
          DominatorTree *DTree, SmallVectorImpl<WeakTrackingVH> &DI,
          bool HasGuards)
      : OrigPhi(WI.NarrowIV), WideType(WI.WidestNativeType), LI(LInfo),
        L(LI->getLoopFor(OrigPhi->getParent())), SE(SEv), DT(DTree),
        HasGuards(HasGuards), DeadInsts(DI) {
    assert(L->getHeader() == OrigPhi->getParent() && "Phi must be an IV");
    ExtendKindMap[OrigPhi] = WI.IsSigned ? SignExtended : ZeroExtended;
  }

  PHINode *createWideIV(SCEVExpander &Rewriter);

protected:
  Value *createExtendInst(Value *NarrowOper, Type *WideType, bool IsSigned,
                          Instruction *Use);

  Instruction *cloneIVUser(NarrowIVDefUse DU, const SCEVAddRecExpr *WideAR);
  Instruction *cloneArithmeticIVUser(NarrowIVDefUse DU,
                                     const SCEVAddRecExpr *WideAR);
  Instruction *cloneBitwiseIVUser(NarrowIVDefUse DU);

  ExtendKind getExtendKind(Instruction *I);

  using WidenedRecTy = std::pair<const SCEVAddRecExpr *, ExtendKind>;

  WidenedRecTy getWideRecurrence(NarrowIVDefUse DU);

  WidenedRecTy getExtendedOperandRecurrence(NarrowIVDefUse DU);

  const SCEV *getSCEVByOpCode(const SCEV *LHS, const SCEV *RHS,
                              unsigned OpCode) const;

  Instruction *widenIVUse(NarrowIVDefUse DU, SCEVExpander &Rewriter);

  bool widenLoopCompare(NarrowIVDefUse DU);
  bool widenWithVariantLoadUse(NarrowIVDefUse DU);
  void widenWithVariantLoadUseCodegen(NarrowIVDefUse DU);

  void pushNarrowIVUsers(Instruction *NarrowDef, Instruction *WideDef);
};

} // end anonymous namespace

Value *WidenIV::createExtendInst(Value *NarrowOper, Type *WideType,
                                 bool IsSigned, Instruction *Use) {
  // Set the debug location and conservative insertion point.
  IRBuilder<> Builder(Use);
  // Hoist the insertion point into loop preheaders as far as possible.
  for (const Loop *L = LI->getLoopFor(Use->getParent());
       L && L->getLoopPreheader() && L->isLoopInvariant(NarrowOper);
       L = L->getParentLoop())
    Builder.SetInsertPoint(L->getLoopPreheader()->getTerminator());

  return IsSigned ? Builder.CreateSExt(NarrowOper, WideType) :
                    Builder.CreateZExt(NarrowOper, WideType);
}

/// Instantiate a wide operation to replace a narrow operation. This only needs
/// to handle operations that can evaluation to SCEVAddRec. It can safely return
/// 0 for any operation we decide not to clone.
Instruction *WidenIV::cloneIVUser(NarrowIVDefUse DU,
                                  const SCEVAddRecExpr *WideAR) {
  unsigned Opcode = DU.NarrowUse->getOpcode();
  switch (Opcode) {
  default:
    return nullptr;
  case Instruction::Add:
  case Instruction::Mul:
  case Instruction::UDiv:
  case Instruction::Sub:
    return cloneArithmeticIVUser(DU, WideAR);

  case Instruction::And:
  case Instruction::Or:
  case Instruction::Xor:
  case Instruction::Shl:
  case Instruction::LShr:
  case Instruction::AShr:
    return cloneBitwiseIVUser(DU);
  }
}

Instruction *WidenIV::cloneBitwiseIVUser(NarrowIVDefUse DU) {
  Instruction *NarrowUse = DU.NarrowUse;
  Instruction *NarrowDef = DU.NarrowDef;
  Instruction *WideDef = DU.WideDef;

  LLVM_DEBUG(dbgs() << "Cloning bitwise IVUser: " << *NarrowUse << "\n");

  // Replace NarrowDef operands with WideDef. Otherwise, we don't know anything
  // about the narrow operand yet so must insert a [sz]ext. It is probably loop
  // invariant and will be folded or hoisted. If it actually comes from a
  // widened IV, it should be removed during a future call to widenIVUse.
  bool IsSigned = getExtendKind(NarrowDef) == SignExtended;
  Value *LHS = (NarrowUse->getOperand(0) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(0), WideType,
                                      IsSigned, NarrowUse);
  Value *RHS = (NarrowUse->getOperand(1) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(1), WideType,
                                      IsSigned, NarrowUse);

  auto *NarrowBO = cast<BinaryOperator>(NarrowUse);
  auto *WideBO = BinaryOperator::Create(NarrowBO->getOpcode(), LHS, RHS,
                                        NarrowBO->getName());
  IRBuilder<> Builder(NarrowUse);
  Builder.Insert(WideBO);
  WideBO->copyIRFlags(NarrowBO);
  return WideBO;
}

Instruction *WidenIV::cloneArithmeticIVUser(NarrowIVDefUse DU,
                                            const SCEVAddRecExpr *WideAR) {
  Instruction *NarrowUse = DU.NarrowUse;
  Instruction *NarrowDef = DU.NarrowDef;
  Instruction *WideDef = DU.WideDef;

  LLVM_DEBUG(dbgs() << "Cloning arithmetic IVUser: " << *NarrowUse << "\n");

  unsigned IVOpIdx = (NarrowUse->getOperand(0) == NarrowDef) ? 0 : 1;

  // We're trying to find X such that
  //
  //  Widen(NarrowDef `op` NonIVNarrowDef) == WideAR == WideDef `op.wide` X
  //
  // We guess two solutions to X, sext(NonIVNarrowDef) and zext(NonIVNarrowDef),
  // and check using SCEV if any of them are correct.

  // Returns true if extending NonIVNarrowDef according to `SignExt` is a
  // correct solution to X.
  auto GuessNonIVOperand = [&](bool SignExt) {
    const SCEV *WideLHS;
    const SCEV *WideRHS;

    auto GetExtend = [this, SignExt](const SCEV *S, Type *Ty) {
      if (SignExt)
        return SE->getSignExtendExpr(S, Ty);
      return SE->getZeroExtendExpr(S, Ty);
    };

    if (IVOpIdx == 0) {
      WideLHS = SE->getSCEV(WideDef);
      const SCEV *NarrowRHS = SE->getSCEV(NarrowUse->getOperand(1));
      WideRHS = GetExtend(NarrowRHS, WideType);
    } else {
      const SCEV *NarrowLHS = SE->getSCEV(NarrowUse->getOperand(0));
      WideLHS = GetExtend(NarrowLHS, WideType);
      WideRHS = SE->getSCEV(WideDef);
    }

    // WideUse is "WideDef `op.wide` X" as described in the comment.
    const SCEV *WideUse = nullptr;

    switch (NarrowUse->getOpcode()) {
    default:
      llvm_unreachable("No other possibility!");

    case Instruction::Add:
      WideUse = SE->getAddExpr(WideLHS, WideRHS);
      break;

    case Instruction::Mul:
      WideUse = SE->getMulExpr(WideLHS, WideRHS);
      break;

    case Instruction::UDiv:
      WideUse = SE->getUDivExpr(WideLHS, WideRHS);
      break;

    case Instruction::Sub:
      WideUse = SE->getMinusSCEV(WideLHS, WideRHS);
      break;
    }

    return WideUse == WideAR;
  };

  bool SignExtend = getExtendKind(NarrowDef) == SignExtended;
  if (!GuessNonIVOperand(SignExtend)) {
    SignExtend = !SignExtend;
    if (!GuessNonIVOperand(SignExtend))
      return nullptr;
  }

  Value *LHS = (NarrowUse->getOperand(0) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(0), WideType,
                                      SignExtend, NarrowUse);
  Value *RHS = (NarrowUse->getOperand(1) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(1), WideType,
                                      SignExtend, NarrowUse);

  auto *NarrowBO = cast<BinaryOperator>(NarrowUse);
  auto *WideBO = BinaryOperator::Create(NarrowBO->getOpcode(), LHS, RHS,
                                        NarrowBO->getName());

  IRBuilder<> Builder(NarrowUse);
  Builder.Insert(WideBO);
  WideBO->copyIRFlags(NarrowBO);
  return WideBO;
}

WidenIV::ExtendKind WidenIV::getExtendKind(Instruction *I) {
  auto It = ExtendKindMap.find(I);
  assert(It != ExtendKindMap.end() && "Instruction not yet extended!");
  return It->second;
}

const SCEV *WidenIV::getSCEVByOpCode(const SCEV *LHS, const SCEV *RHS,
                                     unsigned OpCode) const {
  if (OpCode == Instruction::Add)
    return SE->getAddExpr(LHS, RHS);
  if (OpCode == Instruction::Sub)
    return SE->getMinusSCEV(LHS, RHS);
  if (OpCode == Instruction::Mul)
    return SE->getMulExpr(LHS, RHS);

  llvm_unreachable("Unsupported opcode.");
}

/// No-wrap operations can transfer sign extension of their result to their
/// operands. Generate the SCEV value for the widened operation without
/// actually modifying the IR yet. If the expression after extending the
/// operands is an AddRec for this loop, return the AddRec and the kind of
/// extension used.
WidenIV::WidenedRecTy WidenIV::getExtendedOperandRecurrence(NarrowIVDefUse DU) {
  // Handle the common case of add<nsw/nuw>
  const unsigned OpCode = DU.NarrowUse->getOpcode();
  // Only Add/Sub/Mul instructions supported yet.
  if (OpCode != Instruction::Add && OpCode != Instruction::Sub &&
      OpCode != Instruction::Mul)
    return {nullptr, Unknown};

  // One operand (NarrowDef) has already been extended to WideDef. Now determine
  // if extending the other will lead to a recurrence.
  const unsigned ExtendOperIdx =
      DU.NarrowUse->getOperand(0) == DU.NarrowDef ? 1 : 0;
  assert(DU.NarrowUse->getOperand(1-ExtendOperIdx) == DU.NarrowDef && "bad DU");

  const SCEV *ExtendOperExpr = nullptr;
  const OverflowingBinaryOperator *OBO =
    cast<OverflowingBinaryOperator>(DU.NarrowUse);
  ExtendKind ExtKind = getExtendKind(DU.NarrowDef);
  if (ExtKind == SignExtended && OBO->hasNoSignedWrap())
    ExtendOperExpr = SE->getSignExtendExpr(
      SE->getSCEV(DU.NarrowUse->getOperand(ExtendOperIdx)), WideType);
  else if(ExtKind == ZeroExtended && OBO->hasNoUnsignedWrap())
    ExtendOperExpr = SE->getZeroExtendExpr(
      SE->getSCEV(DU.NarrowUse->getOperand(ExtendOperIdx)), WideType);
  else
    return {nullptr, Unknown};

  // When creating this SCEV expr, don't apply the current operations NSW or NUW
  // flags. This instruction may be guarded by control flow that the no-wrap
  // behavior depends on. Non-control-equivalent instructions can be mapped to
  // the same SCEV expression, and it would be incorrect to transfer NSW/NUW
  // semantics to those operations.
  const SCEV *lhs = SE->getSCEV(DU.WideDef);
  const SCEV *rhs = ExtendOperExpr;

  // Let's swap operands to the initial order for the case of non-commutative
  // operations, like SUB. See PR21014.
  if (ExtendOperIdx == 0)
    std::swap(lhs, rhs);
  const SCEVAddRecExpr *AddRec =
      dyn_cast<SCEVAddRecExpr>(getSCEVByOpCode(lhs, rhs, OpCode));

  if (!AddRec || AddRec->getLoop() != L)
    return {nullptr, Unknown};

  return {AddRec, ExtKind};
}

/// Is this instruction potentially interesting for further simplification after
/// widening it's type? In other words, can the extend be safely hoisted out of
/// the loop with SCEV reducing the value to a recurrence on the same loop. If
/// so, return the extended recurrence and the kind of extension used. Otherwise
/// return {nullptr, Unknown}.
WidenIV::WidenedRecTy WidenIV::getWideRecurrence(NarrowIVDefUse DU) {
  if (!SE->isSCEVable(DU.NarrowUse->getType()))
    return {nullptr, Unknown};

  const SCEV *NarrowExpr = SE->getSCEV(DU.NarrowUse);
  if (SE->getTypeSizeInBits(NarrowExpr->getType()) >=
      SE->getTypeSizeInBits(WideType)) {
    // NarrowUse implicitly widens its operand. e.g. a gep with a narrow
    // index. So don't follow this use.
    return {nullptr, Unknown};
  }

  const SCEV *WideExpr;
  ExtendKind ExtKind;
  if (DU.NeverNegative) {
    WideExpr = SE->getSignExtendExpr(NarrowExpr, WideType);
    if (isa<SCEVAddRecExpr>(WideExpr))
      ExtKind = SignExtended;
    else {
      WideExpr = SE->getZeroExtendExpr(NarrowExpr, WideType);
      ExtKind = ZeroExtended;
    }
  } else if (getExtendKind(DU.NarrowDef) == SignExtended) {
    WideExpr = SE->getSignExtendExpr(NarrowExpr, WideType);
    ExtKind = SignExtended;
  } else {
    WideExpr = SE->getZeroExtendExpr(NarrowExpr, WideType);
    ExtKind = ZeroExtended;
  }
  const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(WideExpr);
  if (!AddRec || AddRec->getLoop() != L)
    return {nullptr, Unknown};
  return {AddRec, ExtKind};
}

/// This IV user cannot be widened. Replace this use of the original narrow IV
/// with a truncation of the new wide IV to isolate and eliminate the narrow IV.
static void truncateIVUse(NarrowIVDefUse DU, DominatorTree *DT, LoopInfo *LI) {
  auto *InsertPt = getInsertPointForUses(DU.NarrowUse, DU.NarrowDef, DT, LI);
  if (!InsertPt)
    return;
  LLVM_DEBUG(dbgs() << "INDVARS: Truncate IV " << *DU.WideDef << " for user "
                    << *DU.NarrowUse << "\n");
  IRBuilder<> Builder(InsertPt);
  Value *Trunc = Builder.CreateTrunc(DU.WideDef, DU.NarrowDef->getType());
  DU.NarrowUse->replaceUsesOfWith(DU.NarrowDef, Trunc);
}

/// If the narrow use is a compare instruction, then widen the compare
//  (and possibly the other operand).  The extend operation is hoisted into the
// loop preheader as far as possible.
bool WidenIV::widenLoopCompare(NarrowIVDefUse DU) {
  ICmpInst *Cmp = dyn_cast<ICmpInst>(DU.NarrowUse);
  if (!Cmp)
    return false;

  // We can legally widen the comparison in the following two cases:
  //
  //  - The signedness of the IV extension and comparison match
  //
  //  - The narrow IV is always positive (and thus its sign extension is equal
  //    to its zero extension).  For instance, let's say we're zero extending
  //    %narrow for the following use
  //
  //      icmp slt i32 %narrow, %val   ... (A)
  //
  //    and %narrow is always positive.  Then
  //
  //      (A) == icmp slt i32 sext(%narrow), sext(%val)
  //          == icmp slt i32 zext(%narrow), sext(%val)
  bool IsSigned = getExtendKind(DU.NarrowDef) == SignExtended;
  if (!(DU.NeverNegative || IsSigned == Cmp->isSigned()))
    return false;

  Value *Op = Cmp->getOperand(Cmp->getOperand(0) == DU.NarrowDef ? 1 : 0);
  unsigned CastWidth = SE->getTypeSizeInBits(Op->getType());
  unsigned IVWidth = SE->getTypeSizeInBits(WideType);
  assert(CastWidth <= IVWidth && "Unexpected width while widening compare.");

  // Widen the compare instruction.
  auto *InsertPt = getInsertPointForUses(DU.NarrowUse, DU.NarrowDef, DT, LI);
  if (!InsertPt)
    return false;
  IRBuilder<> Builder(InsertPt);
  DU.NarrowUse->replaceUsesOfWith(DU.NarrowDef, DU.WideDef);

  // Widen the other operand of the compare, if necessary.
  if (CastWidth < IVWidth) {
    Value *ExtOp = createExtendInst(Op, WideType, Cmp->isSigned(), Cmp);
    DU.NarrowUse->replaceUsesOfWith(Op, ExtOp);
  }
  return true;
}

/// If the narrow use is an instruction whose two operands are the defining
/// instruction of DU and a load instruction, then we have the following:
/// if the load is hoisted outside the loop, then we do not reach this function
/// as scalar evolution analysis works fine in widenIVUse with variables
/// hoisted outside the loop and efficient code is subsequently generated by
/// not emitting truncate instructions. But when the load is not hoisted
/// (whether due to limitation in alias analysis or due to a true legality),
/// then scalar evolution can not proceed with loop variant values and
/// inefficient code is generated. This function handles the non-hoisted load
/// special case by making the optimization generate the same type of code for
/// hoisted and non-hoisted load (widen use and eliminate sign extend
/// instruction). This special case is important especially when the induction
/// variables are affecting addressing mode in code generation.
bool WidenIV::widenWithVariantLoadUse(NarrowIVDefUse DU) {
  Instruction *NarrowUse = DU.NarrowUse;
  Instruction *NarrowDef = DU.NarrowDef;
  Instruction *WideDef = DU.WideDef;

  // Handle the common case of add<nsw/nuw>
  const unsigned OpCode = NarrowUse->getOpcode();
  // Only Add/Sub/Mul instructions are supported.
  if (OpCode != Instruction::Add && OpCode != Instruction::Sub &&
      OpCode != Instruction::Mul)
    return false;

  // The operand that is not defined by NarrowDef of DU. Let's call it the
  // other operand.
  unsigned ExtendOperIdx = DU.NarrowUse->getOperand(0) == NarrowDef ? 1 : 0;
  assert(DU.NarrowUse->getOperand(1 - ExtendOperIdx) == DU.NarrowDef &&
         "bad DU");

  const SCEV *ExtendOperExpr = nullptr;
  const OverflowingBinaryOperator *OBO =
    cast<OverflowingBinaryOperator>(NarrowUse);
  ExtendKind ExtKind = getExtendKind(NarrowDef);
  if (ExtKind == SignExtended && OBO->hasNoSignedWrap())
    ExtendOperExpr = SE->getSignExtendExpr(
      SE->getSCEV(NarrowUse->getOperand(ExtendOperIdx)), WideType);
  else if (ExtKind == ZeroExtended && OBO->hasNoUnsignedWrap())
    ExtendOperExpr = SE->getZeroExtendExpr(
      SE->getSCEV(NarrowUse->getOperand(ExtendOperIdx)), WideType);
  else
    return false;

  // We are interested in the other operand being a load instruction.
  // But, we should look into relaxing this restriction later on.
  auto *I = dyn_cast<Instruction>(NarrowUse->getOperand(ExtendOperIdx));
  if (I && I->getOpcode() != Instruction::Load)
    return false;

  // Verifying that Defining operand is an AddRec
  const SCEV *Op1 = SE->getSCEV(WideDef);
  const SCEVAddRecExpr *AddRecOp1 = dyn_cast<SCEVAddRecExpr>(Op1);
  if (!AddRecOp1 || AddRecOp1->getLoop() != L)
    return false;
  // Verifying that other operand is an Extend.
  if (ExtKind == SignExtended) {
    if (!isa<SCEVSignExtendExpr>(ExtendOperExpr))
      return false;
  } else {
    if (!isa<SCEVZeroExtendExpr>(ExtendOperExpr))
      return false;
  }

  if (ExtKind == SignExtended) {
    for (Use &U : NarrowUse->uses()) {
      SExtInst *User = dyn_cast<SExtInst>(U.getUser());
      if (!User || User->getType() != WideType)
        return false;
    }
  } else { // ExtKind == ZeroExtended
    for (Use &U : NarrowUse->uses()) {
      ZExtInst *User = dyn_cast<ZExtInst>(U.getUser());
      if (!User || User->getType() != WideType)
        return false;
    }
  }

  return true;
}

/// Special Case for widening with variant Loads (see
/// WidenIV::widenWithVariantLoadUse). This is the code generation part.
void WidenIV::widenWithVariantLoadUseCodegen(NarrowIVDefUse DU) {
  Instruction *NarrowUse = DU.NarrowUse;
  Instruction *NarrowDef = DU.NarrowDef;
  Instruction *WideDef = DU.WideDef;

  ExtendKind ExtKind = getExtendKind(NarrowDef);

  LLVM_DEBUG(dbgs() << "Cloning arithmetic IVUser: " << *NarrowUse << "\n");

  // Generating a widening use instruction.
  Value *LHS = (NarrowUse->getOperand(0) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(0), WideType,
                                      ExtKind, NarrowUse);
  Value *RHS = (NarrowUse->getOperand(1) == NarrowDef)
                   ? WideDef
                   : createExtendInst(NarrowUse->getOperand(1), WideType,
                                      ExtKind, NarrowUse);

  auto *NarrowBO = cast<BinaryOperator>(NarrowUse);
  auto *WideBO = BinaryOperator::Create(NarrowBO->getOpcode(), LHS, RHS,
                                        NarrowBO->getName());
  IRBuilder<> Builder(NarrowUse);
  Builder.Insert(WideBO);
  WideBO->copyIRFlags(NarrowBO);

  if (ExtKind == SignExtended)
    ExtendKindMap[NarrowUse] = SignExtended;
  else
    ExtendKindMap[NarrowUse] = ZeroExtended;

  // Update the Use.
  if (ExtKind == SignExtended) {
    for (Use &U : NarrowUse->uses()) {
      SExtInst *User = dyn_cast<SExtInst>(U.getUser());
      if (User && User->getType() == WideType) {
        LLVM_DEBUG(dbgs() << "INDVARS: eliminating " << *User << " replaced by "
                          << *WideBO << "\n");
        ++NumElimExt;
        User->replaceAllUsesWith(WideBO);
        DeadInsts.emplace_back(User);
      }
    }
  } else { // ExtKind == ZeroExtended
    for (Use &U : NarrowUse->uses()) {
      ZExtInst *User = dyn_cast<ZExtInst>(U.getUser());
      if (User && User->getType() == WideType) {
        LLVM_DEBUG(dbgs() << "INDVARS: eliminating " << *User << " replaced by "
                          << *WideBO << "\n");
        ++NumElimExt;
        User->replaceAllUsesWith(WideBO);
        DeadInsts.emplace_back(User);
      }
    }
  }
}

/// Determine whether an individual user of the narrow IV can be widened. If so,
/// return the wide clone of the user.
Instruction *WidenIV::widenIVUse(NarrowIVDefUse DU, SCEVExpander &Rewriter) {
  assert(ExtendKindMap.count(DU.NarrowDef) &&
         "Should already know the kind of extension used to widen NarrowDef");

  // Stop traversing the def-use chain at inner-loop phis or post-loop phis.
  if (PHINode *UsePhi = dyn_cast<PHINode>(DU.NarrowUse)) {
    if (LI->getLoopFor(UsePhi->getParent()) != L) {
      // For LCSSA phis, sink the truncate outside the loop.
      // After SimplifyCFG most loop exit targets have a single predecessor.
      // Otherwise fall back to a truncate within the loop.
      if (UsePhi->getNumOperands() != 1)
        truncateIVUse(DU, DT, LI);
      else {
        // Widening the PHI requires us to insert a trunc.  The logical place
        // for this trunc is in the same BB as the PHI.  This is not possible if
        // the BB is terminated by a catchswitch.
        if (isa<CatchSwitchInst>(UsePhi->getParent()->getTerminator()))
          return nullptr;

        PHINode *WidePhi =
          PHINode::Create(DU.WideDef->getType(), 1, UsePhi->getName() + ".wide",
                          UsePhi);
        WidePhi->addIncoming(DU.WideDef, UsePhi->getIncomingBlock(0));
        IRBuilder<> Builder(&*WidePhi->getParent()->getFirstInsertionPt());
        Value *Trunc = Builder.CreateTrunc(WidePhi, DU.NarrowDef->getType());
        UsePhi->replaceAllUsesWith(Trunc);
        DeadInsts.emplace_back(UsePhi);
        LLVM_DEBUG(dbgs() << "INDVARS: Widen lcssa phi " << *UsePhi << " to "
                          << *WidePhi << "\n");
      }
      return nullptr;
    }
  }

  // This narrow use can be widened by a sext if it's non-negative or its narrow
  // def was widended by a sext. Same for zext.
  auto canWidenBySExt = [&]() {
    return DU.NeverNegative || getExtendKind(DU.NarrowDef) == SignExtended;
  };
  auto canWidenByZExt = [&]() {
    return DU.NeverNegative || getExtendKind(DU.NarrowDef) == ZeroExtended;
  };

  // Our raison d'etre! Eliminate sign and zero extension.
  if ((isa<SExtInst>(DU.NarrowUse) && canWidenBySExt()) ||
      (isa<ZExtInst>(DU.NarrowUse) && canWidenByZExt())) {
    Value *NewDef = DU.WideDef;
    if (DU.NarrowUse->getType() != WideType) {
      unsigned CastWidth = SE->getTypeSizeInBits(DU.NarrowUse->getType());
      unsigned IVWidth = SE->getTypeSizeInBits(WideType);
      if (CastWidth < IVWidth) {
        // The cast isn't as wide as the IV, so insert a Trunc.
        IRBuilder<> Builder(DU.NarrowUse);
        NewDef = Builder.CreateTrunc(DU.WideDef, DU.NarrowUse->getType());
      }
      else {
        // A wider extend was hidden behind a narrower one. This may induce
        // another round of IV widening in which the intermediate IV becomes
        // dead. It should be very rare.
        LLVM_DEBUG(dbgs() << "INDVARS: New IV " << *WidePhi
                          << " not wide enough to subsume " << *DU.NarrowUse
                          << "\n");
        DU.NarrowUse->replaceUsesOfWith(DU.NarrowDef, DU.WideDef);
        NewDef = DU.NarrowUse;
      }
    }
    if (NewDef != DU.NarrowUse) {
      LLVM_DEBUG(dbgs() << "INDVARS: eliminating " << *DU.NarrowUse
                        << " replaced by " << *DU.WideDef << "\n");
      ++NumElimExt;
      DU.NarrowUse->replaceAllUsesWith(NewDef);
      DeadInsts.emplace_back(DU.NarrowUse);
    }
    // Now that the extend is gone, we want to expose it's uses for potential
    // further simplification. We don't need to directly inform SimplifyIVUsers
    // of the new users, because their parent IV will be processed later as a
    // new loop phi. If we preserved IVUsers analysis, we would also want to
    // push the uses of WideDef here.

    // No further widening is needed. The deceased [sz]ext had done it for us.
    return nullptr;
  }

  // Does this user itself evaluate to a recurrence after widening?
  WidenedRecTy WideAddRec = getExtendedOperandRecurrence(DU);
  if (!WideAddRec.first)
    WideAddRec = getWideRecurrence(DU);

  assert((WideAddRec.first == nullptr) == (WideAddRec.second == Unknown));
  if (!WideAddRec.first) {
    // If use is a loop condition, try to promote the condition instead of
    // truncating the IV first.
    if (widenLoopCompare(DU))
      return nullptr;

    // We are here about to generate a truncate instruction that may hurt
    // performance because the scalar evolution expression computed earlier
    // in WideAddRec.first does not indicate a polynomial induction expression.
    // In that case, look at the operands of the use instruction to determine
    // if we can still widen the use instead of truncating its operand.
    if (widenWithVariantLoadUse(DU)) {
      widenWithVariantLoadUseCodegen(DU);
      return nullptr;
    }

    // This user does not evaluate to a recurrence after widening, so don't
    // follow it. Instead insert a Trunc to kill off the original use,
    // eventually isolating the original narrow IV so it can be removed.
    truncateIVUse(DU, DT, LI);
    return nullptr;
  }
  // Assume block terminators cannot evaluate to a recurrence. We can't to
  // insert a Trunc after a terminator if there happens to be a critical edge.
  assert(DU.NarrowUse != DU.NarrowUse->getParent()->getTerminator() &&
         "SCEV is not expected to evaluate a block terminator");

  // Reuse the IV increment that SCEVExpander created as long as it dominates
  // NarrowUse.
  Instruction *WideUse = nullptr;
  if (WideAddRec.first == WideIncExpr &&
      Rewriter.hoistIVInc(WideInc, DU.NarrowUse))
    WideUse = WideInc;
  else {
    WideUse = cloneIVUser(DU, WideAddRec.first);
    if (!WideUse)
      return nullptr;
  }
  // Evaluation of WideAddRec ensured that the narrow expression could be
  // extended outside the loop without overflow. This suggests that the wide use
  // evaluates to the same expression as the extended narrow use, but doesn't
  // absolutely guarantee it. Hence the following failsafe check. In rare cases
  // where it fails, we simply throw away the newly created wide use.
  if (WideAddRec.first != SE->getSCEV(WideUse)) {
    LLVM_DEBUG(dbgs() << "Wide use expression mismatch: " << *WideUse << ": "
                      << *SE->getSCEV(WideUse) << " != " << *WideAddRec.first
                      << "\n");
    DeadInsts.emplace_back(WideUse);
    return nullptr;
  }

  // if we reached this point then we are going to replace
  // DU.NarrowUse with WideUse. Reattach DbgValue then.
  replaceAllDbgUsesWith(*DU.NarrowUse, *WideUse, *WideUse, *DT);

  ExtendKindMap[DU.NarrowUse] = WideAddRec.second;
  // Returning WideUse pushes it on the worklist.
  return WideUse;
}

/// Add eligible users of NarrowDef to NarrowIVUsers.
void WidenIV::pushNarrowIVUsers(Instruction *NarrowDef, Instruction *WideDef) {
  const SCEV *NarrowSCEV = SE->getSCEV(NarrowDef);
  bool NonNegativeDef =
      SE->isKnownPredicate(ICmpInst::ICMP_SGE, NarrowSCEV,
                           SE->getConstant(NarrowSCEV->getType(), 0));
  for (User *U : NarrowDef->users()) {
    Instruction *NarrowUser = cast<Instruction>(U);

    // Handle data flow merges and bizarre phi cycles.
    if (!Widened.insert(NarrowUser).second)
      continue;

    bool NonNegativeUse = false;
    if (!NonNegativeDef) {
      // We might have a control-dependent range information for this context.
      if (auto RangeInfo = getPostIncRangeInfo(NarrowDef, NarrowUser))
        NonNegativeUse = RangeInfo->getSignedMin().isNonNegative();
    }

    NarrowIVUsers.emplace_back(NarrowDef, NarrowUser, WideDef,
                               NonNegativeDef || NonNegativeUse);
  }
}

/// Process a single induction variable. First use the SCEVExpander to create a
/// wide induction variable that evaluates to the same recurrence as the
/// original narrow IV. Then use a worklist to forward traverse the narrow IV's
/// def-use chain. After widenIVUse has processed all interesting IV users, the
/// narrow IV will be isolated for removal by DeleteDeadPHIs.
///
/// It would be simpler to delete uses as they are processed, but we must avoid
/// invalidating SCEV expressions.
PHINode *WidenIV::createWideIV(SCEVExpander &Rewriter) {
  // Is this phi an induction variable?
  const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(SE->getSCEV(OrigPhi));
  if (!AddRec)
    return nullptr;

  // Widen the induction variable expression.
  const SCEV *WideIVExpr = getExtendKind(OrigPhi) == SignExtended
                               ? SE->getSignExtendExpr(AddRec, WideType)
                               : SE->getZeroExtendExpr(AddRec, WideType);

  assert(SE->getEffectiveSCEVType(WideIVExpr->getType()) == WideType &&
         "Expect the new IV expression to preserve its type");

  // Can the IV be extended outside the loop without overflow?
  AddRec = dyn_cast<SCEVAddRecExpr>(WideIVExpr);
  if (!AddRec || AddRec->getLoop() != L)
    return nullptr;

  // An AddRec must have loop-invariant operands. Since this AddRec is
  // materialized by a loop header phi, the expression cannot have any post-loop
  // operands, so they must dominate the loop header.
  assert(
      SE->properlyDominates(AddRec->getStart(), L->getHeader()) &&
      SE->properlyDominates(AddRec->getStepRecurrence(*SE), L->getHeader()) &&
      "Loop header phi recurrence inputs do not dominate the loop");

  // Iterate over IV uses (including transitive ones) looking for IV increments
  // of the form 'add nsw %iv, <const>'. For each increment and each use of
  // the increment calculate control-dependent range information basing on
  // dominating conditions inside of the loop (e.g. a range check inside of the
  // loop). Calculated ranges are stored in PostIncRangeInfos map.
  //
  // Control-dependent range information is later used to prove that a narrow
  // definition is not negative (see pushNarrowIVUsers). It's difficult to do
  // this on demand because when pushNarrowIVUsers needs this information some
  // of the dominating conditions might be already widened.
  if (UsePostIncrementRanges)
    calculatePostIncRanges(OrigPhi);

  // The rewriter provides a value for the desired IV expression. This may
  // either find an existing phi or materialize a new one. Either way, we
  // expect a well-formed cyclic phi-with-increments. i.e. any operand not part
  // of the phi-SCC dominates the loop entry.
  Instruction *InsertPt = &L->getHeader()->front();
  WidePhi = cast<PHINode>(Rewriter.expandCodeFor(AddRec, WideType, InsertPt));

  // Remembering the WideIV increment generated by SCEVExpander allows
  // widenIVUse to reuse it when widening the narrow IV's increment. We don't
  // employ a general reuse mechanism because the call above is the only call to
  // SCEVExpander. Henceforth, we produce 1-to-1 narrow to wide uses.
  if (BasicBlock *LatchBlock = L->getLoopLatch()) {
    WideInc =
      cast<Instruction>(WidePhi->getIncomingValueForBlock(LatchBlock));
    WideIncExpr = SE->getSCEV(WideInc);
    // Propagate the debug location associated with the original loop increment
    // to the new (widened) increment.
    auto *OrigInc =
      cast<Instruction>(OrigPhi->getIncomingValueForBlock(LatchBlock));
    WideInc->setDebugLoc(OrigInc->getDebugLoc());
  }

  LLVM_DEBUG(dbgs() << "Wide IV: " << *WidePhi << "\n");
  ++NumWidened;

  // Traverse the def-use chain using a worklist starting at the original IV.
  assert(Widened.empty() && NarrowIVUsers.empty() && "expect initial state" );

  Widened.insert(OrigPhi);
  pushNarrowIVUsers(OrigPhi, WidePhi);

  while (!NarrowIVUsers.empty()) {
    NarrowIVDefUse DU = NarrowIVUsers.pop_back_val();

    // Process a def-use edge. This may replace the use, so don't hold a
    // use_iterator across it.
    Instruction *WideUse = widenIVUse(DU, Rewriter);

    // Follow all def-use edges from the previous narrow use.
    if (WideUse)
      pushNarrowIVUsers(DU.NarrowUse, WideUse);

    // widenIVUse may have removed the def-use edge.
    if (DU.NarrowDef->use_empty())
      DeadInsts.emplace_back(DU.NarrowDef);
  }

  // Attach any debug information to the new PHI.
  replaceAllDbgUsesWith(*OrigPhi, *WidePhi, *WidePhi, *DT);

  return WidePhi;
}

/// Calculates control-dependent range for the given def at the given context
/// by looking at dominating conditions inside of the loop
void WidenIV::calculatePostIncRange(Instruction *NarrowDef,
                                    Instruction *NarrowUser) {
  using namespace llvm::PatternMatch;

  Value *NarrowDefLHS;
  const APInt *NarrowDefRHS;
  if (!match(NarrowDef, m_NSWAdd(m_Value(NarrowDefLHS),
                                 m_APInt(NarrowDefRHS))) ||
      !NarrowDefRHS->isNonNegative())
    return;

  auto UpdateRangeFromCondition = [&] (Value *Condition,
                                       bool TrueDest) {
    CmpInst::Predicate Pred;
    Value *CmpRHS;
    if (!match(Condition, m_ICmp(Pred, m_Specific(NarrowDefLHS),
                                 m_Value(CmpRHS))))
      return;

    CmpInst::Predicate P =
            TrueDest ? Pred : CmpInst::getInversePredicate(Pred);

    auto CmpRHSRange = SE->getSignedRange(SE->getSCEV(CmpRHS));
    auto CmpConstrainedLHSRange =
            ConstantRange::makeAllowedICmpRegion(P, CmpRHSRange);
    auto NarrowDefRange = CmpConstrainedLHSRange.addWithNoWrap(
        *NarrowDefRHS, OverflowingBinaryOperator::NoSignedWrap);

    updatePostIncRangeInfo(NarrowDef, NarrowUser, NarrowDefRange);
  };

  auto UpdateRangeFromGuards = [&](Instruction *Ctx) {
    if (!HasGuards)
      return;

    for (Instruction &I : make_range(Ctx->getIterator().getReverse(),
                                     Ctx->getParent()->rend())) {
      Value *C = nullptr;
      if (match(&I, m_Intrinsic<Intrinsic::experimental_guard>(m_Value(C))))
        UpdateRangeFromCondition(C, /*TrueDest=*/true);
    }
  };

  UpdateRangeFromGuards(NarrowUser);

  BasicBlock *NarrowUserBB = NarrowUser->getParent();
  // If NarrowUserBB is statically unreachable asking dominator queries may
  // yield surprising results. (e.g. the block may not have a dom tree node)
  if (!DT->isReachableFromEntry(NarrowUserBB))
    return;

  for (auto *DTB = (*DT)[NarrowUserBB]->getIDom();
       L->contains(DTB->getBlock());
       DTB = DTB->getIDom()) {
    auto *BB = DTB->getBlock();
    auto *TI = BB->getTerminator();
    UpdateRangeFromGuards(TI);

    auto *BI = dyn_cast<BranchInst>(TI);
    if (!BI || !BI->isConditional())
      continue;

    auto *TrueSuccessor = BI->getSuccessor(0);
    auto *FalseSuccessor = BI->getSuccessor(1);

    auto DominatesNarrowUser = [this, NarrowUser] (BasicBlockEdge BBE) {
      return BBE.isSingleEdge() &&
             DT->dominates(BBE, NarrowUser->getParent());
    };

    if (DominatesNarrowUser(BasicBlockEdge(BB, TrueSuccessor)))
      UpdateRangeFromCondition(BI->getCondition(), /*TrueDest=*/true);

    if (DominatesNarrowUser(BasicBlockEdge(BB, FalseSuccessor)))
      UpdateRangeFromCondition(BI->getCondition(), /*TrueDest=*/false);
  }
}

/// Calculates PostIncRangeInfos map for the given IV
void WidenIV::calculatePostIncRanges(PHINode *OrigPhi) {
  SmallPtrSet<Instruction *, 16> Visited;
  SmallVector<Instruction *, 6> Worklist;
  Worklist.push_back(OrigPhi);
  Visited.insert(OrigPhi);

  while (!Worklist.empty()) {
    Instruction *NarrowDef = Worklist.pop_back_val();

    for (Use &U : NarrowDef->uses()) {
      auto *NarrowUser = cast<Instruction>(U.getUser());

      // Don't go looking outside the current loop.
      auto *NarrowUserLoop = (*LI)[NarrowUser->getParent()];
      if (!NarrowUserLoop || !L->contains(NarrowUserLoop))
        continue;

      if (!Visited.insert(NarrowUser).second)
        continue;

      Worklist.push_back(NarrowUser);

      calculatePostIncRange(NarrowDef, NarrowUser);
    }
  }
}

//===----------------------------------------------------------------------===//
//  Live IV Reduction - Minimize IVs live across the loop.
//===----------------------------------------------------------------------===//

//===----------------------------------------------------------------------===//
//  Simplification of IV users based on SCEV evaluation.
//===----------------------------------------------------------------------===//

namespace {

class IndVarSimplifyVisitor : public IVVisitor {
  ScalarEvolution *SE;
  const TargetTransformInfo *TTI;
  PHINode *IVPhi;

public:
  WideIVInfo WI;

  IndVarSimplifyVisitor(PHINode *IV, ScalarEvolution *SCEV,
                        const TargetTransformInfo *TTI,
                        const DominatorTree *DTree)
    : SE(SCEV), TTI(TTI), IVPhi(IV) {
    DT = DTree;
    WI.NarrowIV = IVPhi;
  }

  // Implement the interface used by simplifyUsersOfIV.
  void visitCast(CastInst *Cast) override { visitIVCast(Cast, WI, SE, TTI); }
};

} // end anonymous namespace

/// Iteratively perform simplification on a worklist of IV users. Each
/// successive simplification may push more users which may themselves be
/// candidates for simplification.
///
/// Sign/Zero extend elimination is interleaved with IV simplification.
bool IndVarSimplify::simplifyAndExtend(Loop *L,
                                       SCEVExpander &Rewriter,
                                       LoopInfo *LI) {
  SmallVector<WideIVInfo, 8> WideIVs;

  auto *GuardDecl = L->getBlocks()[0]->getModule()->getFunction(
          Intrinsic::getName(Intrinsic::experimental_guard));
  bool HasGuards = GuardDecl && !GuardDecl->use_empty();

  SmallVector<PHINode*, 8> LoopPhis;
  for (BasicBlock::iterator I = L->getHeader()->begin(); isa<PHINode>(I); ++I) {
    LoopPhis.push_back(cast<PHINode>(I));
  }
  // Each round of simplification iterates through the SimplifyIVUsers worklist
  // for all current phis, then determines whether any IVs can be
  // widened. Widening adds new phis to LoopPhis, inducing another round of
  // simplification on the wide IVs.
  bool Changed = false;
  while (!LoopPhis.empty()) {
    // Evaluate as many IV expressions as possible before widening any IVs. This
    // forces SCEV to set no-wrap flags before evaluating sign/zero
    // extension. The first time SCEV attempts to normalize sign/zero extension,
    // the result becomes final. So for the most predictable results, we delay
    // evaluation of sign/zero extend evaluation until needed, and avoid running
    // other SCEV based analysis prior to simplifyAndExtend.
    do {
      PHINode *CurrIV = LoopPhis.pop_back_val();

      // Information about sign/zero extensions of CurrIV.
      IndVarSimplifyVisitor Visitor(CurrIV, SE, TTI, DT);

      Changed |=
          simplifyUsersOfIV(CurrIV, SE, DT, LI, DeadInsts, Rewriter, &Visitor);

      if (Visitor.WI.WidestNativeType) {
        WideIVs.push_back(Visitor.WI);
      }
    } while(!LoopPhis.empty());

    for (; !WideIVs.empty(); WideIVs.pop_back()) {
      WidenIV Widener(WideIVs.back(), LI, SE, DT, DeadInsts, HasGuards);
      if (PHINode *WidePhi = Widener.createWideIV(Rewriter)) {
        Changed = true;
        LoopPhis.push_back(WidePhi);
      }
    }
  }
  return Changed;
}

//===----------------------------------------------------------------------===//
//  linearFunctionTestReplace and its kin. Rewrite the loop exit condition.
//===----------------------------------------------------------------------===//

/// Given an Value which is hoped to be part of an add recurance in the given
/// loop, return the associated Phi node if so.  Otherwise, return null.  Note
/// that this is less general than SCEVs AddRec checking.  
static PHINode *getLoopPhiForCounter(Value *IncV, Loop *L) {
  Instruction *IncI = dyn_cast<Instruction>(IncV);
  if (!IncI)
    return nullptr;

  switch (IncI->getOpcode()) {
  case Instruction::Add:
  case Instruction::Sub:
    break;
  case Instruction::GetElementPtr:
    // An IV counter must preserve its type.
    if (IncI->getNumOperands() == 2)
      break;
    LLVM_FALLTHROUGH;
  default:
    return nullptr;
  }

  PHINode *Phi = dyn_cast<PHINode>(IncI->getOperand(0));
  if (Phi && Phi->getParent() == L->getHeader()) {
    if (L->isLoopInvariant(IncI->getOperand(1)))
      return Phi;
    return nullptr;
  }
  if (IncI->getOpcode() == Instruction::GetElementPtr)
    return nullptr;

  // Allow add/sub to be commuted.
  Phi = dyn_cast<PHINode>(IncI->getOperand(1));
  if (Phi && Phi->getParent() == L->getHeader()) {
    if (L->isLoopInvariant(IncI->getOperand(0)))
      return Phi;
  }
  return nullptr;
}

/// Whether the current loop exit test is based on this value.  Currently this
/// is limited to a direct use in the loop condition.
static bool isLoopExitTestBasedOn(Value *V, BasicBlock *ExitingBB) {
  BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
  ICmpInst *ICmp = dyn_cast<ICmpInst>(BI->getCondition());
  // TODO: Allow non-icmp loop test.
  if (!ICmp)
    return false;

  // TODO: Allow indirect use.
  return ICmp->getOperand(0) == V || ICmp->getOperand(1) == V;
}

/// linearFunctionTestReplace policy. Return true unless we can show that the
/// current exit test is already sufficiently canonical.
static bool needsLFTR(Loop *L, BasicBlock *ExitingBB) {
  assert(L->getLoopLatch() && "Must be in simplified form");

  // Avoid converting a constant or loop invariant test back to a runtime
  // test.  This is critical for when SCEV's cached ExitCount is less precise
  // than the current IR (such as after we've proven a particular exit is
  // actually dead and thus the BE count never reaches our ExitCount.)
  BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
  if (L->isLoopInvariant(BI->getCondition()))
    return false;
  
  // Do LFTR to simplify the exit condition to an ICMP.
  ICmpInst *Cond = dyn_cast<ICmpInst>(BI->getCondition());
  if (!Cond)
    return true;

  // Do LFTR to simplify the exit ICMP to EQ/NE
  ICmpInst::Predicate Pred = Cond->getPredicate();
  if (Pred != ICmpInst::ICMP_NE && Pred != ICmpInst::ICMP_EQ)
    return true;

  // Look for a loop invariant RHS
  Value *LHS = Cond->getOperand(0);
  Value *RHS = Cond->getOperand(1);
  if (!L->isLoopInvariant(RHS)) {
    if (!L->isLoopInvariant(LHS))
      return true;
    std::swap(LHS, RHS);
  }
  // Look for a simple IV counter LHS
  PHINode *Phi = dyn_cast<PHINode>(LHS);
  if (!Phi)
    Phi = getLoopPhiForCounter(LHS, L);

  if (!Phi)
    return true;

  // Do LFTR if PHI node is defined in the loop, but is *not* a counter.
  int Idx = Phi->getBasicBlockIndex(L->getLoopLatch());
  if (Idx < 0)
    return true;

  // Do LFTR if the exit condition's IV is *not* a simple counter.
  Value *IncV = Phi->getIncomingValue(Idx);
  return Phi != getLoopPhiForCounter(IncV, L);
}

/// Return true if undefined behavior would provable be executed on the path to
/// OnPathTo if Root produced a posion result.  Note that this doesn't say
/// anything about whether OnPathTo is actually executed or whether Root is
/// actually poison.  This can be used to assess whether a new use of Root can
/// be added at a location which is control equivalent with OnPathTo (such as
/// immediately before it) without introducing UB which didn't previously
/// exist.  Note that a false result conveys no information.  
static bool mustExecuteUBIfPoisonOnPathTo(Instruction *Root,
                                          Instruction *OnPathTo, 
                                          DominatorTree *DT) {
  // Basic approach is to assume Root is poison, propagate poison forward
  // through all users we can easily track, and then check whether any of those
  // users are provable UB and must execute before out exiting block might
  // exit.

  // The set of all recursive users we've visited (which are assumed to all be
  // poison because of said visit)
  SmallSet<const Value *, 16> KnownPoison;
  SmallVector<const Instruction*, 16> Worklist;
  Worklist.push_back(Root);
  while (!Worklist.empty()) {
    const Instruction *I = Worklist.pop_back_val();

    // If we know this must trigger UB on a path leading our target.
    if (mustTriggerUB(I, KnownPoison) && DT->dominates(I, OnPathTo))
      return true;
    
    // If we can't analyze propagation through this instruction, just skip it
    // and transitive users.  Safe as false is a conservative result.
    if (!propagatesFullPoison(I) && I != Root)
      continue;

    if (KnownPoison.insert(I).second)
      for (const User *User : I->users())
        Worklist.push_back(cast<Instruction>(User));
  }

  // Might be non-UB, or might have a path we couldn't prove must execute on
  // way to exiting bb. 
  return false;
}

/// Recursive helper for hasConcreteDef(). Unfortunately, this currently boils
/// down to checking that all operands are constant and listing instructions
/// that may hide undef.
static bool hasConcreteDefImpl(Value *V, SmallPtrSetImpl<Value*> &Visited,
                               unsigned Depth) {
  if (isa<Constant>(V))
    return !isa<UndefValue>(V);

  if (Depth >= 6)
    return false;

  // Conservatively handle non-constant non-instructions. For example, Arguments
  // may be undef.
  Instruction *I = dyn_cast<Instruction>(V);
  if (!I)
    return false;

  // Load and return values may be undef.
  if(I->mayReadFromMemory() || isa<CallInst>(I) || isa<InvokeInst>(I))
    return false;

  // Optimistically handle other instructions.
  for (Value *Op : I->operands()) {
    if (!Visited.insert(Op).second)
      continue;
    if (!hasConcreteDefImpl(Op, Visited, Depth+1))
      return false;
  }
  return true;
}

/// Return true if the given value is concrete. We must prove that undef can
/// never reach it.
///
/// TODO: If we decide that this is a good approach to checking for undef, we
/// may factor it into a common location.
static bool hasConcreteDef(Value *V) {
  SmallPtrSet<Value*, 8> Visited;
  Visited.insert(V);
  return hasConcreteDefImpl(V, Visited, 0);
}

/// Return true if this IV has any uses other than the (soon to be rewritten)
/// loop exit test.
static bool AlmostDeadIV(PHINode *Phi, BasicBlock *LatchBlock, Value *Cond) {
  int LatchIdx = Phi->getBasicBlockIndex(LatchBlock);
  Value *IncV = Phi->getIncomingValue(LatchIdx);

  for (User *U : Phi->users())
    if (U != Cond && U != IncV) return false;

  for (User *U : IncV->users())
    if (U != Cond && U != Phi) return false;
  return true;
}

/// Return true if the given phi is a "counter" in L.  A counter is an
/// add recurance (of integer or pointer type) with an arbitrary start, and a
/// step of 1.  Note that L must have exactly one latch.
static bool isLoopCounter(PHINode* Phi, Loop *L,
                          ScalarEvolution *SE) {
  assert(Phi->getParent() == L->getHeader());
  assert(L->getLoopLatch());
  
  if (!SE->isSCEVable(Phi->getType()))
    return false;

  const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(SE->getSCEV(Phi));
  if (!AR || AR->getLoop() != L || !AR->isAffine())
    return false;

  const SCEV *Step = dyn_cast<SCEVConstant>(AR->getStepRecurrence(*SE));
  if (!Step || !Step->isOne())
    return false;

  int LatchIdx = Phi->getBasicBlockIndex(L->getLoopLatch());
  Value *IncV = Phi->getIncomingValue(LatchIdx);
  return (getLoopPhiForCounter(IncV, L) == Phi);
}

/// Search the loop header for a loop counter (anadd rec w/step of one)
/// suitable for use by LFTR.  If multiple counters are available, select the
/// "best" one based profitable heuristics.
///
/// BECount may be an i8* pointer type. The pointer difference is already
/// valid count without scaling the address stride, so it remains a pointer
/// expression as far as SCEV is concerned.
static PHINode *FindLoopCounter(Loop *L, BasicBlock *ExitingBB,
                                const SCEV *BECount,
                                ScalarEvolution *SE, DominatorTree *DT) {
  uint64_t BCWidth = SE->getTypeSizeInBits(BECount->getType());

  Value *Cond = cast<BranchInst>(ExitingBB->getTerminator())->getCondition();

  // Loop over all of the PHI nodes, looking for a simple counter.
  PHINode *BestPhi = nullptr;
  const SCEV *BestInit = nullptr;
  BasicBlock *LatchBlock = L->getLoopLatch();
  assert(LatchBlock && "Must be in simplified form");
  const DataLayout &DL = L->getHeader()->getModule()->getDataLayout();

  for (BasicBlock::iterator I = L->getHeader()->begin(); isa<PHINode>(I); ++I) {
    PHINode *Phi = cast<PHINode>(I);
    if (!isLoopCounter(Phi, L, SE))
      continue;

    // Avoid comparing an integer IV against a pointer Limit.
    if (BECount->getType()->isPointerTy() && !Phi->getType()->isPointerTy())
      continue;

    const auto *AR = cast<SCEVAddRecExpr>(SE->getSCEV(Phi));

    // AR may be a pointer type, while BECount is an integer type.
    // AR may be wider than BECount. With eq/ne tests overflow is immaterial.
    // AR may not be a narrower type, or we may never exit.
    uint64_t PhiWidth = SE->getTypeSizeInBits(AR->getType());
    if (PhiWidth < BCWidth || !DL.isLegalInteger(PhiWidth))
      continue;

    // Avoid reusing a potentially undef value to compute other values that may
    // have originally had a concrete definition.
    if (!hasConcreteDef(Phi)) {
      // We explicitly allow unknown phis as long as they are already used by
      // the loop exit test.  This is legal since performing LFTR could not
      // increase the number of undef users. 
      Value *IncPhi = Phi->getIncomingValueForBlock(LatchBlock);
      if (!isLoopExitTestBasedOn(Phi, ExitingBB) &&
          !isLoopExitTestBasedOn(IncPhi, ExitingBB))
        continue;
    }

    // Avoid introducing undefined behavior due to poison which didn't exist in
    // the original program.  (Annoyingly, the rules for poison and undef
    // propagation are distinct, so this does NOT cover the undef case above.)
    // We have to ensure that we don't introduce UB by introducing a use on an
    // iteration where said IV produces poison.  Our strategy here differs for
    // pointers and integer IVs.  For integers, we strip and reinfer as needed,
    // see code in linearFunctionTestReplace.  For pointers, we restrict
    // transforms as there is no good way to reinfer inbounds once lost.
    if (!Phi->getType()->isIntegerTy() &&
        !mustExecuteUBIfPoisonOnPathTo(Phi, ExitingBB->getTerminator(), DT))
      continue;
    
    const SCEV *Init = AR->getStart();

    if (BestPhi && !AlmostDeadIV(BestPhi, LatchBlock, Cond)) {
      // Don't force a live loop counter if another IV can be used.
      if (AlmostDeadIV(Phi, LatchBlock, Cond))
        continue;

      // Prefer to count-from-zero. This is a more "canonical" counter form. It
      // also prefers integer to pointer IVs.
      if (BestInit->isZero() != Init->isZero()) {
        if (BestInit->isZero())
          continue;
      }
      // If two IVs both count from zero or both count from nonzero then the
      // narrower is likely a dead phi that has been widened. Use the wider phi
      // to allow the other to be eliminated.
      else if (PhiWidth <= SE->getTypeSizeInBits(BestPhi->getType()))
        continue;
    }
    BestPhi = Phi;
    BestInit = Init;
  }
  return BestPhi;
}

/// Insert an IR expression which computes the value held by the IV IndVar
/// (which must be an loop counter w/unit stride) after the backedge of loop L
/// is taken ExitCount times.
static Value *genLoopLimit(PHINode *IndVar, BasicBlock *ExitingBB,
                           const SCEV *ExitCount, bool UsePostInc, Loop *L,
                           SCEVExpander &Rewriter, ScalarEvolution *SE) {
  assert(isLoopCounter(IndVar, L, SE));
  const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(SE->getSCEV(IndVar));
  const SCEV *IVInit = AR->getStart();

  // IVInit may be a pointer while ExitCount is an integer when FindLoopCounter
  // finds a valid pointer IV. Sign extend ExitCount in order to materialize a
  // GEP. Avoid running SCEVExpander on a new pointer value, instead reusing
  // the existing GEPs whenever possible.
  if (IndVar->getType()->isPointerTy() &&
      !ExitCount->getType()->isPointerTy()) {
    // IVOffset will be the new GEP offset that is interpreted by GEP as a
    // signed value. ExitCount on the other hand represents the loop trip count,
    // which is an unsigned value. FindLoopCounter only allows induction
    // variables that have a positive unit stride of one. This means we don't
    // have to handle the case of negative offsets (yet) and just need to zero
    // extend ExitCount.
    Type *OfsTy = SE->getEffectiveSCEVType(IVInit->getType());
    const SCEV *IVOffset = SE->getTruncateOrZeroExtend(ExitCount, OfsTy);
    if (UsePostInc)
      IVOffset = SE->getAddExpr(IVOffset, SE->getOne(OfsTy));

    // Expand the code for the iteration count.
    assert(SE->isLoopInvariant(IVOffset, L) &&
           "Computed iteration count is not loop invariant!");

    // We could handle pointer IVs other than i8*, but we need to compensate for
    // gep index scaling.
    assert(SE->getSizeOfExpr(IntegerType::getInt64Ty(IndVar->getContext()),
                             cast<PointerType>(IndVar->getType())
                                 ->getElementType())->isOne() &&
           "unit stride pointer IV must be i8*");

    const SCEV *IVLimit = SE->getAddExpr(IVInit, IVOffset);
    BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
    return Rewriter.expandCodeFor(IVLimit, IndVar->getType(), BI);
  } else {
    // In any other case, convert both IVInit and ExitCount to integers before
    // comparing. This may result in SCEV expansion of pointers, but in practice
    // SCEV will fold the pointer arithmetic away as such:
    // BECount = (IVEnd - IVInit - 1) => IVLimit = IVInit (postinc).
    //
    // Valid Cases: (1) both integers is most common; (2) both may be pointers
    // for simple memset-style loops.
    //
    // IVInit integer and ExitCount pointer would only occur if a canonical IV
    // were generated on top of case #2, which is not expected.

    assert(AR->getStepRecurrence(*SE)->isOne() && "only handles unit stride");
    // For unit stride, IVCount = Start + ExitCount with 2's complement
    // overflow.

    // For integer IVs, truncate the IV before computing IVInit + BECount,
    // unless we know apriori that the limit must be a constant when evaluated
    // in the bitwidth of the IV.  We prefer (potentially) keeping a truncate
    // of the IV in the loop over a (potentially) expensive expansion of the
    // widened exit count add(zext(add)) expression.
    if (SE->getTypeSizeInBits(IVInit->getType())
        > SE->getTypeSizeInBits(ExitCount->getType())) {
      if (isa<SCEVConstant>(IVInit) && isa<SCEVConstant>(ExitCount))
        ExitCount = SE->getZeroExtendExpr(ExitCount, IVInit->getType());
      else
        IVInit = SE->getTruncateExpr(IVInit, ExitCount->getType());
    }

    const SCEV *IVLimit = SE->getAddExpr(IVInit, ExitCount);

    if (UsePostInc)
      IVLimit = SE->getAddExpr(IVLimit, SE->getOne(IVLimit->getType()));

    // Expand the code for the iteration count.
    assert(SE->isLoopInvariant(IVLimit, L) &&
           "Computed iteration count is not loop invariant!");
    // Ensure that we generate the same type as IndVar, or a smaller integer
    // type. In the presence of null pointer values, we have an integer type
    // SCEV expression (IVInit) for a pointer type IV value (IndVar).
    Type *LimitTy = ExitCount->getType()->isPointerTy() ?
      IndVar->getType() : ExitCount->getType();
    BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
    return Rewriter.expandCodeFor(IVLimit, LimitTy, BI);
  }
}

/// This method rewrites the exit condition of the loop to be a canonical !=
/// comparison against the incremented loop induction variable.  This pass is
/// able to rewrite the exit tests of any loop where the SCEV analysis can
/// determine a loop-invariant trip count of the loop, which is actually a much
/// broader range than just linear tests.
bool IndVarSimplify::
linearFunctionTestReplace(Loop *L, BasicBlock *ExitingBB,
                          const SCEV *ExitCount,
                          PHINode *IndVar, SCEVExpander &Rewriter) {
  assert(L->getLoopLatch() && "Loop no longer in simplified form?");
  assert(isLoopCounter(IndVar, L, SE));
  Instruction * const IncVar =
    cast<Instruction>(IndVar->getIncomingValueForBlock(L->getLoopLatch()));

  // Initialize CmpIndVar to the preincremented IV.
  Value *CmpIndVar = IndVar;
  bool UsePostInc = false;

  // If the exiting block is the same as the backedge block, we prefer to
  // compare against the post-incremented value, otherwise we must compare
  // against the preincremented value.
  if (ExitingBB == L->getLoopLatch()) {
    // For pointer IVs, we chose to not strip inbounds which requires us not
    // to add a potentially UB introducing use.  We need to either a) show
    // the loop test we're modifying is already in post-inc form, or b) show
    // that adding a use must not introduce UB.
    bool SafeToPostInc =
        IndVar->getType()->isIntegerTy() ||
        isLoopExitTestBasedOn(IncVar, ExitingBB) ||
        mustExecuteUBIfPoisonOnPathTo(IncVar, ExitingBB->getTerminator(), DT);
    if (SafeToPostInc) {
      UsePostInc = true;
      CmpIndVar = IncVar;
    }
  }

  // It may be necessary to drop nowrap flags on the incrementing instruction
  // if either LFTR moves from a pre-inc check to a post-inc check (in which
  // case the increment might have previously been poison on the last iteration
  // only) or if LFTR switches to a different IV that was previously dynamically
  // dead (and as such may be arbitrarily poison). We remove any nowrap flags
  // that SCEV didn't infer for the post-inc addrec (even if we use a pre-inc
  // check), because the pre-inc addrec flags may be adopted from the original
  // instruction, while SCEV has to explicitly prove the post-inc nowrap flags.
  // TODO: This handling is inaccurate for one case: If we switch to a
  // dynamically dead IV that wraps on the first loop iteration only, which is
  // not covered by the post-inc addrec. (If the new IV was not dynamically
  // dead, it could not be poison on the first iteration in the first place.)
  if (auto *BO = dyn_cast<BinaryOperator>(IncVar)) {
    const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(SE->getSCEV(IncVar));
    if (BO->hasNoUnsignedWrap())
      BO->setHasNoUnsignedWrap(AR->hasNoUnsignedWrap());
    if (BO->hasNoSignedWrap())
      BO->setHasNoSignedWrap(AR->hasNoSignedWrap());
  }

  Value *ExitCnt = genLoopLimit(
      IndVar, ExitingBB, ExitCount, UsePostInc, L, Rewriter, SE);
  assert(ExitCnt->getType()->isPointerTy() ==
             IndVar->getType()->isPointerTy() &&
         "genLoopLimit missed a cast");

  // Insert a new icmp_ne or icmp_eq instruction before the branch.
  BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
  ICmpInst::Predicate P;
  if (L->contains(BI->getSuccessor(0)))
    P = ICmpInst::ICMP_NE;
  else
    P = ICmpInst::ICMP_EQ;

  IRBuilder<> Builder(BI);

  // The new loop exit condition should reuse the debug location of the
  // original loop exit condition.
  if (auto *Cond = dyn_cast<Instruction>(BI->getCondition()))
    Builder.SetCurrentDebugLocation(Cond->getDebugLoc());

  // For integer IVs, if we evaluated the limit in the narrower bitwidth to
  // avoid the expensive expansion of the limit expression in the wider type,
  // emit a truncate to narrow the IV to the ExitCount type.  This is safe
  // since we know (from the exit count bitwidth), that we can't self-wrap in
  // the narrower type.
  unsigned CmpIndVarSize = SE->getTypeSizeInBits(CmpIndVar->getType());
  unsigned ExitCntSize = SE->getTypeSizeInBits(ExitCnt->getType());
  if (CmpIndVarSize > ExitCntSize) {
    assert(!CmpIndVar->getType()->isPointerTy() &&
           !ExitCnt->getType()->isPointerTy());

    // Before resorting to actually inserting the truncate, use the same
    // reasoning as from SimplifyIndvar::eliminateTrunc to see if we can extend
    // the other side of the comparison instead.  We still evaluate the limit
    // in the narrower bitwidth, we just prefer a zext/sext outside the loop to
    // a truncate within in.  
    bool Extended = false;
    const SCEV *IV = SE->getSCEV(CmpIndVar);
    const SCEV *TruncatedIV = SE->getTruncateExpr(SE->getSCEV(CmpIndVar),
                                                  ExitCnt->getType());
    const SCEV *ZExtTrunc =
      SE->getZeroExtendExpr(TruncatedIV, CmpIndVar->getType());
    
    if (ZExtTrunc == IV) {
      Extended = true;
      ExitCnt = Builder.CreateZExt(ExitCnt, IndVar->getType(),
                                   "wide.trip.count");
    } else {
      const SCEV *SExtTrunc =
        SE->getSignExtendExpr(TruncatedIV, CmpIndVar->getType());
      if (SExtTrunc == IV) {
        Extended = true;
        ExitCnt = Builder.CreateSExt(ExitCnt, IndVar->getType(),
                                     "wide.trip.count");
      }
    }

    if (Extended) {
      bool Discard;
      L->makeLoopInvariant(ExitCnt, Discard);
    } else 
      CmpIndVar = Builder.CreateTrunc(CmpIndVar, ExitCnt->getType(),
                                      "lftr.wideiv");
  }
  LLVM_DEBUG(dbgs() << "INDVARS: Rewriting loop exit condition to:\n"
                    << "      LHS:" << *CmpIndVar << '\n'
                    << "       op:\t" << (P == ICmpInst::ICMP_NE ? "!=" : "==")
                    << "\n"
                    << "      RHS:\t" << *ExitCnt << "\n"
                    << "ExitCount:\t" << *ExitCount << "\n"
                    << "  was: " << *BI->getCondition() << "\n");

  Value *Cond = Builder.CreateICmp(P, CmpIndVar, ExitCnt, "exitcond");
  Value *OrigCond = BI->getCondition();
  // It's tempting to use replaceAllUsesWith here to fully replace the old
  // comparison, but that's not immediately safe, since users of the old
  // comparison may not be dominated by the new comparison. Instead, just
  // update the branch to use the new comparison; in the common case this
  // will make old comparison dead.
  BI->setCondition(Cond);
  DeadInsts.push_back(OrigCond);

  ++NumLFTR;
  return true;
}

//===----------------------------------------------------------------------===//
//  sinkUnusedInvariants. A late subpass to cleanup loop preheaders.
//===----------------------------------------------------------------------===//

/// If there's a single exit block, sink any loop-invariant values that
/// were defined in the preheader but not used inside the loop into the
/// exit block to reduce register pressure in the loop.
bool IndVarSimplify::sinkUnusedInvariants(Loop *L) {
  BasicBlock *ExitBlock = L->getExitBlock();
  if (!ExitBlock) return false;

  BasicBlock *Preheader = L->getLoopPreheader();
  if (!Preheader) return false;

  bool MadeAnyChanges = false;
  BasicBlock::iterator InsertPt = ExitBlock->getFirstInsertionPt();
  BasicBlock::iterator I(Preheader->getTerminator());
  while (I != Preheader->begin()) {
    --I;
    // New instructions were inserted at the end of the preheader.
    if (isa<PHINode>(I))
      break;

    // Don't move instructions which might have side effects, since the side
    // effects need to complete before instructions inside the loop.  Also don't
    // move instructions which might read memory, since the loop may modify
    // memory. Note that it's okay if the instruction might have undefined
    // behavior: LoopSimplify guarantees that the preheader dominates the exit
    // block.
    if (I->mayHaveSideEffects() || I->mayReadFromMemory())
      continue;

    // Skip debug info intrinsics.
    if (isa<DbgInfoIntrinsic>(I))
      continue;

    // Skip eh pad instructions.
    if (I->isEHPad())
      continue;

    // Don't sink alloca: we never want to sink static alloca's out of the
    // entry block, and correctly sinking dynamic alloca's requires
    // checks for stacksave/stackrestore intrinsics.
    // FIXME: Refactor this check somehow?
    if (isa<AllocaInst>(I))
      continue;

    // Determine if there is a use in or before the loop (direct or
    // otherwise).
    bool UsedInLoop = false;
    for (Use &U : I->uses()) {
      Instruction *User = cast<Instruction>(U.getUser());
      BasicBlock *UseBB = User->getParent();
      if (PHINode *P = dyn_cast<PHINode>(User)) {
        unsigned i =
          PHINode::getIncomingValueNumForOperand(U.getOperandNo());
        UseBB = P->getIncomingBlock(i);
      }
      if (UseBB == Preheader || L->contains(UseBB)) {
        UsedInLoop = true;
        break;
      }
    }

    // If there is, the def must remain in the preheader.
    if (UsedInLoop)
      continue;

    // Otherwise, sink it to the exit block.
    Instruction *ToMove = &*I;
    bool Done = false;

    if (I != Preheader->begin()) {
      // Skip debug info intrinsics.
      do {
        --I;
      } while (isa<DbgInfoIntrinsic>(I) && I != Preheader->begin());

      if (isa<DbgInfoIntrinsic>(I) && I == Preheader->begin())
        Done = true;
    } else {
      Done = true;
    }

    MadeAnyChanges = true;
    ToMove->moveBefore(*ExitBlock, InsertPt);
    if (Done) break;
    InsertPt = ToMove->getIterator();
  }

  return MadeAnyChanges;
}

/// Return a symbolic upper bound for the backedge taken count of the loop.
/// This is more general than getConstantMaxBackedgeTakenCount as it returns
/// an arbitrary expression as opposed to only constants.
/// TODO: Move into the ScalarEvolution class.
static const SCEV* getMaxBackedgeTakenCount(ScalarEvolution &SE,
                                            DominatorTree &DT, Loop *L) {
  SmallVector<BasicBlock*, 16> ExitingBlocks;
  L->getExitingBlocks(ExitingBlocks);

  // Form an expression for the maximum exit count possible for this loop. We
  // merge the max and exact information to approximate a version of
  // getConstantMaxBackedgeTakenCount which isn't restricted to just constants.
  SmallVector<const SCEV*, 4> ExitCounts;
  const SCEV *MaxConstEC = SE.getConstantMaxBackedgeTakenCount(L);
  if (!isa<SCEVCouldNotCompute>(MaxConstEC))
    ExitCounts.push_back(MaxConstEC);
  for (BasicBlock *ExitingBB : ExitingBlocks) {
    const SCEV *ExitCount = SE.getExitCount(L, ExitingBB);
    if (!isa<SCEVCouldNotCompute>(ExitCount)) {
      assert(DT.dominates(ExitingBB, L->getLoopLatch()) &&
             "We should only have known counts for exiting blocks that "
             "dominate latch!");
      ExitCounts.push_back(ExitCount);
    }
  }
  if (ExitCounts.empty())
    return SE.getCouldNotCompute();
  return SE.getUMinFromMismatchedTypes(ExitCounts);
}

bool IndVarSimplify::optimizeLoopExits(Loop *L, SCEVExpander &Rewriter) {
  SmallVector<BasicBlock*, 16> ExitingBlocks;
  L->getExitingBlocks(ExitingBlocks);

  // Remove all exits which aren't both rewriteable and analyzeable.
  auto NewEnd = llvm::remove_if(ExitingBlocks,
                                [&](BasicBlock *ExitingBB) {
    // If our exitting block exits multiple loops, we can only rewrite the
    // innermost one.  Otherwise, we're changing how many times the innermost
    // loop runs before it exits. 
    if (LI->getLoopFor(ExitingBB) != L)
      return true;

    // Can't rewrite non-branch yet.
    BranchInst *BI = dyn_cast<BranchInst>(ExitingBB->getTerminator());
    if (!BI)
      return true;

    // If already constant, nothing to do.
    if (isa<Constant>(BI->getCondition()))
      return true;
    
    const SCEV *ExitCount = SE->getExitCount(L, ExitingBB);
    if (isa<SCEVCouldNotCompute>(ExitCount))
      return true;
    return false;
   });
  ExitingBlocks.erase(NewEnd, ExitingBlocks.end());

  if (ExitingBlocks.empty())
    return false;
  
  // Get a symbolic upper bound on the loop backedge taken count.  
  const SCEV *MaxExitCount = getMaxBackedgeTakenCount(*SE, *DT, L);
  if (isa<SCEVCouldNotCompute>(MaxExitCount))
    return false;

  // Visit our exit blocks in order of dominance.  We know from the fact that
  // all exits (left) are analyzeable that the must be a total dominance order
  // between them as each must dominate the latch.  The visit order only
  // matters for the provably equal case.  
  llvm::sort(ExitingBlocks,
             [&](BasicBlock *A, BasicBlock *B) {
               // std::sort sorts in ascending order, so we want the inverse of
               // the normal dominance relation.
               if (DT->properlyDominates(A, B)) return true;
               if (DT->properlyDominates(B, A)) return false;
               llvm_unreachable("expected total dominance order!");
             });
#ifdef ASSERT
  for (unsigned i = 1; i < ExitingBlocks.size(); i++) {
    assert(DT->dominates(ExitingBlocks[i-1], ExitingBlocks[i]));
  }
#endif
  
  auto FoldExit = [&](BasicBlock *ExitingBB, bool IsTaken) {
    BranchInst *BI = cast<BranchInst>(ExitingBB->getTerminator());
    bool ExitIfTrue = !L->contains(*succ_begin(ExitingBB));
    auto *OldCond = BI->getCondition();
    auto *NewCond = ConstantInt::get(OldCond->getType(),
                                     IsTaken ? ExitIfTrue : !ExitIfTrue);
    BI->setCondition(NewCond);
    if (OldCond->use_empty())
      DeadInsts.push_back(OldCond);
  };

  bool Changed = false;
  SmallSet<const SCEV*, 8> DominatingExitCounts;
  for (BasicBlock *ExitingBB : ExitingBlocks) {
    const SCEV *ExitCount = SE->getExitCount(L, ExitingBB);
    assert(!isa<SCEVCouldNotCompute>(ExitCount) && "checked above");
    
    // If we know we'd exit on the first iteration, rewrite the exit to
    // reflect this.  This does not imply the loop must exit through this
    // exit; there may be an earlier one taken on the first iteration.
    // TODO: Given we know the backedge can't be taken, we should go ahead
    // and break it.  Or at least, kill all the header phis and simplify.
    if (ExitCount->isZero()) {
      FoldExit(ExitingBB, true);
      Changed = true;
      continue;
    }

    // If we end up with a pointer exit count, bail.  Note that we can end up
    // with a pointer exit count for one exiting block, and not for another in
    // the same loop.
    if (!ExitCount->getType()->isIntegerTy() ||
        !MaxExitCount->getType()->isIntegerTy())
      continue;
    
    Type *WiderType =
      SE->getWiderType(MaxExitCount->getType(), ExitCount->getType());
    ExitCount = SE->getNoopOrZeroExtend(ExitCount, WiderType);
    MaxExitCount = SE->getNoopOrZeroExtend(MaxExitCount, WiderType);
    assert(MaxExitCount->getType() == ExitCount->getType());
    
    // Can we prove that some other exit must be taken strictly before this
    // one?
    if (SE->isLoopEntryGuardedByCond(L, CmpInst::ICMP_ULT,
                                     MaxExitCount, ExitCount)) {
      FoldExit(ExitingBB, false);
      Changed = true;
      continue;
    }

    // As we run, keep track of which exit counts we've encountered.  If we
    // find a duplicate, we've found an exit which would have exited on the
    // exiting iteration, but (from the visit order) strictly follows another
    // which does the same and is thus dead. 
    if (!DominatingExitCounts.insert(ExitCount).second) {
      FoldExit(ExitingBB, false);
      Changed = true;
      continue;
    }

    // TODO: There might be another oppurtunity to leverage SCEV's reasoning
    // here.  If we kept track of the min of dominanting exits so far, we could
    // discharge exits with EC >= MDEC. This is less powerful than the existing
    // transform (since later exits aren't considered), but potentially more
    // powerful for any case where SCEV can prove a >=u b, but neither a == b
    // or a >u b.  Such a case is not currently known.
  }
  return Changed;
}

bool IndVarSimplify::predicateLoopExits(Loop *L, SCEVExpander &Rewriter) {
  SmallVector<BasicBlock*, 16> ExitingBlocks;
  L->getExitingBlocks(ExitingBlocks);

  bool Changed = false;

  // Finally, see if we can rewrite our exit conditions into a loop invariant
  // form.  If we have a read-only loop, and we can tell that we must exit down
  // a path which does not need any of the values computed within the loop, we
  // can rewrite the loop to exit on the first iteration.  Note that this
  // doesn't either a) tell us the loop exits on the first iteration (unless
  // *all* exits are predicateable) or b) tell us *which* exit might be taken.
  // This transformation looks a lot like a restricted form of dead loop
  // elimination, but restricted to read-only loops and without neccesssarily
  // needing to kill the loop entirely. 
  if (!LoopPredication)
    return Changed;

  if (!SE->hasLoopInvariantBackedgeTakenCount(L))
    return Changed;

  // Note: ExactBTC is the exact backedge taken count *iff* the loop exits
  // through *explicit* control flow.  We have to eliminate the possibility of
  // implicit exits (see below) before we know it's truly exact.
  const SCEV *ExactBTC = SE->getBackedgeTakenCount(L);
  if (isa<SCEVCouldNotCompute>(ExactBTC) ||
      !SE->isLoopInvariant(ExactBTC, L) ||
      !isSafeToExpand(ExactBTC, *SE))
    return Changed;

  auto BadExit = [&](BasicBlock *ExitingBB) {
    // If our exiting block exits multiple loops, we can only rewrite the
    // innermost one.  Otherwise, we're changing how many times the innermost
    // loop runs before it exits. 
    if (LI->getLoopFor(ExitingBB) != L)
      return true;

    // Can't rewrite non-branch yet.
    BranchInst *BI = dyn_cast<BranchInst>(ExitingBB->getTerminator());
    if (!BI)
      return true;

    // If already constant, nothing to do.
    if (isa<Constant>(BI->getCondition()))
      return true;

    // If the exit block has phis, we need to be able to compute the values
    // within the loop which contains them.  This assumes trivially lcssa phis
    // have already been removed; TODO: generalize
    BasicBlock *ExitBlock =
    BI->getSuccessor(L->contains(BI->getSuccessor(0)) ? 1 : 0);
    if (!ExitBlock->phis().empty())
      return true;

    const SCEV *ExitCount = SE->getExitCount(L, ExitingBB);
    assert(!isa<SCEVCouldNotCompute>(ExactBTC) && "implied by having exact trip count");
    if (!SE->isLoopInvariant(ExitCount, L) ||
        !isSafeToExpand(ExitCount, *SE))
      return true;

    return false;
  };

  // If we have any exits which can't be predicated themselves, than we can't
  // predicate any exit which isn't guaranteed to execute before it.  Consider
  // two exits (a) and (b) which would both exit on the same iteration.  If we
  // can predicate (b), but not (a), and (a) preceeds (b) along some path, then
  // we could convert a loop from exiting through (a) to one exiting through
  // (b).  Note that this problem exists only for exits with the same exit
  // count, and we could be more aggressive when exit counts are known inequal.
  llvm::sort(ExitingBlocks,
            [&](BasicBlock *A, BasicBlock *B) {
              // std::sort sorts in ascending order, so we want the inverse of
              // the normal dominance relation, plus a tie breaker for blocks
              // unordered by dominance.
              if (DT->properlyDominates(A, B)) return true;
              if (DT->properlyDominates(B, A)) return false;
              return A->getName() < B->getName();
            });
  // Check to see if our exit blocks are a total order (i.e. a linear chain of
  // exits before the backedge).  If they aren't, reasoning about reachability
  // is complicated and we choose not to for now.
  for (unsigned i = 1; i < ExitingBlocks.size(); i++)
    if (!DT->dominates(ExitingBlocks[i-1], ExitingBlocks[i]))
      return Changed;

  // Given our sorted total order, we know that exit[j] must be evaluated
  // after all exit[i] such j > i.
  for (unsigned i = 0, e = ExitingBlocks.size(); i < e; i++)
    if (BadExit(ExitingBlocks[i])) {
      ExitingBlocks.resize(i);  
      break;
    }

  if (ExitingBlocks.empty())
    return Changed;

  // We rely on not being able to reach an exiting block on a later iteration
  // then it's statically compute exit count.  The implementaton of
  // getExitCount currently has this invariant, but assert it here so that
  // breakage is obvious if this ever changes..
  assert(llvm::all_of(ExitingBlocks, [&](BasicBlock *ExitingBB) {
        return DT->dominates(ExitingBB, L->getLoopLatch());
      }));

  // At this point, ExitingBlocks consists of only those blocks which are
  // predicatable.  Given that, we know we have at least one exit we can
  // predicate if the loop is doesn't have side effects and doesn't have any
  // implicit exits (because then our exact BTC isn't actually exact).
  // @Reviewers - As structured, this is O(I^2) for loop nests.  Any
  // suggestions on how to improve this?  I can obviously bail out for outer
  // loops, but that seems less than ideal.  MemorySSA can find memory writes,
  // is that enough for *all* side effects?
  for (BasicBlock *BB : L->blocks())
    for (auto &I : *BB)
      // TODO:isGuaranteedToTransfer
      if (I.mayHaveSideEffects() || I.mayThrow())
        return Changed;

  // Finally, do the actual predication for all predicatable blocks.  A couple
  // of notes here:
  // 1) We don't bother to constant fold dominated exits with identical exit
  //    counts; that's simply a form of CSE/equality propagation and we leave
  //    it for dedicated passes.
  // 2) We insert the comparison at the branch.  Hoisting introduces additional
  //    legality constraints and we leave that to dedicated logic.  We want to
  //    predicate even if we can't insert a loop invariant expression as
  //    peeling or unrolling will likely reduce the cost of the otherwise loop
  //    varying check.
  Rewriter.setInsertPoint(L->getLoopPreheader()->getTerminator());
  IRBuilder<> B(L->getLoopPreheader()->getTerminator());
  Value *ExactBTCV = nullptr; //lazy generated if needed
  for (BasicBlock *ExitingBB : ExitingBlocks) {
    const SCEV *ExitCount = SE->getExitCount(L, ExitingBB);

    auto *BI = cast<BranchInst>(ExitingBB->getTerminator());
    Value *NewCond;
    if (ExitCount == ExactBTC) {
      NewCond = L->contains(BI->getSuccessor(0)) ?
        B.getFalse() : B.getTrue();
    } else {
      Value *ECV = Rewriter.expandCodeFor(ExitCount);
      if (!ExactBTCV)
        ExactBTCV = Rewriter.expandCodeFor(ExactBTC);
      Value *RHS = ExactBTCV;
      if (ECV->getType() != RHS->getType()) {
        Type *WiderTy = SE->getWiderType(ECV->getType(), RHS->getType());
        ECV = B.CreateZExt(ECV, WiderTy);
        RHS = B.CreateZExt(RHS, WiderTy);
      }
      auto Pred = L->contains(BI->getSuccessor(0)) ?
        ICmpInst::ICMP_NE : ICmpInst::ICMP_EQ;
      NewCond = B.CreateICmp(Pred, ECV, RHS);
    }
    Value *OldCond = BI->getCondition();
    BI->setCondition(NewCond);
    if (OldCond->use_empty())
      DeadInsts.push_back(OldCond);
    Changed = true;
  }

  return Changed;
}

//===----------------------------------------------------------------------===//
//  IndVarSimplify driver. Manage several subpasses of IV simplification.
//===----------------------------------------------------------------------===//

bool IndVarSimplify::run(Loop *L) {
  // We need (and expect!) the incoming loop to be in LCSSA.
  assert(L->isRecursivelyLCSSAForm(*DT, *LI) &&
         "LCSSA required to run indvars!");
  bool Changed = false;

  // If LoopSimplify form is not available, stay out of trouble. Some notes:
  //  - LSR currently only supports LoopSimplify-form loops. Indvars'
  //    canonicalization can be a pessimization without LSR to "clean up"
  //    afterwards.
  //  - We depend on having a preheader; in particular,
  //    Loop::getCanonicalInductionVariable only supports loops with preheaders,
  //    and we're in trouble if we can't find the induction variable even when
  //    we've manually inserted one.
  //  - LFTR relies on having a single backedge.
  if (!L->isLoopSimplifyForm())
    return false;

  // If there are any floating-point recurrences, attempt to
  // transform them to use integer recurrences.
  Changed |= rewriteNonIntegerIVs(L);

#ifndef NDEBUG
  // Used below for a consistency check only
  const SCEV *BackedgeTakenCount = SE->getBackedgeTakenCount(L);
#endif

  // Create a rewriter object which we'll use to transform the code with.
  SCEVExpander Rewriter(*SE, DL, "indvars");
#ifndef NDEBUG
  Rewriter.setDebugType(DEBUG_TYPE);
#endif

  // Eliminate redundant IV users.
  //
  // Simplification works best when run before other consumers of SCEV. We
  // attempt to avoid evaluating SCEVs for sign/zero extend operations until
  // other expressions involving loop IVs have been evaluated. This helps SCEV
  // set no-wrap flags before normalizing sign/zero extension.
  Rewriter.disableCanonicalMode();
  Changed |= simplifyAndExtend(L, Rewriter, LI);

  // Check to see if we can compute the final value of any expressions
  // that are recurrent in the loop, and substitute the exit values from the
  // loop into any instructions outside of the loop that use the final values
  // of the current expressions.
  if (ReplaceExitValue != NeverRepl)
    Changed |= rewriteLoopExitValues(L, Rewriter);

  // Eliminate redundant IV cycles.
  NumElimIV += Rewriter.replaceCongruentIVs(L, DT, DeadInsts);

  // Try to eliminate loop exits based on analyzeable exit counts
  Changed |= optimizeLoopExits(L, Rewriter);
  
  // Try to form loop invariant tests for loop exits by changing how many
  // iterations of the loop run when that is unobservable.
  Changed |= predicateLoopExits(L, Rewriter);

  // If we have a trip count expression, rewrite the loop's exit condition
  // using it.  
  if (!DisableLFTR) {
    SmallVector<BasicBlock*, 16> ExitingBlocks;
    L->getExitingBlocks(ExitingBlocks);
    for (BasicBlock *ExitingBB : ExitingBlocks) {
      // Can't rewrite non-branch yet.
      if (!isa<BranchInst>(ExitingBB->getTerminator()))
        continue;

      // If our exitting block exits multiple loops, we can only rewrite the
      // innermost one.  Otherwise, we're changing how many times the innermost
      // loop runs before it exits. 
      if (LI->getLoopFor(ExitingBB) != L)
        continue;
      
      if (!needsLFTR(L, ExitingBB))
        continue;

      const SCEV *ExitCount = SE->getExitCount(L, ExitingBB);
      if (isa<SCEVCouldNotCompute>(ExitCount))
        continue;

      // This was handled above, but as we form SCEVs, we can sometimes refine
      // existing ones; this allows exit counts to be folded to zero which
      // weren't when optimizeLoopExits saw them.  Arguably, we should iterate
      // until stable to handle cases like this better.
      if (ExitCount->isZero())
        continue;
      
      PHINode *IndVar = FindLoopCounter(L, ExitingBB, ExitCount, SE, DT);
      if (!IndVar)
        continue;
      
      // Avoid high cost expansions.  Note: This heuristic is questionable in
      // that our definition of "high cost" is not exactly principled.  
      if (Rewriter.isHighCostExpansion(ExitCount, L))
        continue;

      // Check preconditions for proper SCEVExpander operation. SCEV does not
      // express SCEVExpander's dependencies, such as LoopSimplify. Instead
      // any pass that uses the SCEVExpander must do it. This does not work
      // well for loop passes because SCEVExpander makes assumptions about
      // all loops, while LoopPassManager only forces the current loop to be
      // simplified. 
      //
      // FIXME: SCEV expansion has no way to bail out, so the caller must
      // explicitly check any assumptions made by SCEV. Brittle.
      const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(ExitCount);
      if (!AR || AR->getLoop()->getLoopPreheader())
        Changed |= linearFunctionTestReplace(L, ExitingBB,
                                             ExitCount, IndVar,
                                             Rewriter);
    }
  }
  // Clear the rewriter cache, because values that are in the rewriter's cache
  // can be deleted in the loop below, causing the AssertingVH in the cache to
  // trigger.
  Rewriter.clear();

  // Now that we're done iterating through lists, clean up any instructions
  // which are now dead.
  while (!DeadInsts.empty())
    if (Instruction *Inst =
            dyn_cast_or_null<Instruction>(DeadInsts.pop_back_val()))
      Changed |= RecursivelyDeleteTriviallyDeadInstructions(Inst, TLI);

  // The Rewriter may not be used from this point on.

  // Loop-invariant instructions in the preheader that aren't used in the
  // loop may be sunk below the loop to reduce register pressure.
  Changed |= sinkUnusedInvariants(L);

  // rewriteFirstIterationLoopExitValues does not rely on the computation of
  // trip count and therefore can further simplify exit values in addition to
  // rewriteLoopExitValues.
  Changed |= rewriteFirstIterationLoopExitValues(L);

  // Clean up dead instructions.
  Changed |= DeleteDeadPHIs(L->getHeader(), TLI);

  // Check a post-condition.
  assert(L->isRecursivelyLCSSAForm(*DT, *LI) &&
         "Indvars did not preserve LCSSA!");

  // Verify that LFTR, and any other change have not interfered with SCEV's
  // ability to compute trip count.
#ifndef NDEBUG
  if (VerifyIndvars && !isa<SCEVCouldNotCompute>(BackedgeTakenCount)) {
    SE->forgetLoop(L);
    const SCEV *NewBECount = SE->getBackedgeTakenCount(L);
    if (SE->getTypeSizeInBits(BackedgeTakenCount->getType()) <
        SE->getTypeSizeInBits(NewBECount->getType()))
      NewBECount = SE->getTruncateOrNoop(NewBECount,
                                         BackedgeTakenCount->getType());
    else
      BackedgeTakenCount = SE->getTruncateOrNoop(BackedgeTakenCount,
                                                 NewBECount->getType());
    assert(BackedgeTakenCount == NewBECount && "indvars must preserve SCEV");
  }
#endif

  return Changed;
}

PreservedAnalyses IndVarSimplifyPass::run(Loop &L, LoopAnalysisManager &AM,
                                          LoopStandardAnalysisResults &AR,
                                          LPMUpdater &) {
  Function *F = L.getHeader()->getParent();
  const DataLayout &DL = F->getParent()->getDataLayout();

  IndVarSimplify IVS(&AR.LI, &AR.SE, &AR.DT, DL, &AR.TLI, &AR.TTI);
  if (!IVS.run(&L))
    return PreservedAnalyses::all();

  auto PA = getLoopPassPreservedAnalyses();
  PA.preserveSet<CFGAnalyses>();
  return PA;
}

namespace {

struct IndVarSimplifyLegacyPass : public LoopPass {
  static char ID; // Pass identification, replacement for typeid

  IndVarSimplifyLegacyPass() : LoopPass(ID) {
    initializeIndVarSimplifyLegacyPassPass(*PassRegistry::getPassRegistry());
  }

  bool runOnLoop(Loop *L, LPPassManager &LPM) override {
    if (skipLoop(L))
      return false;

    auto *LI = &getAnalysis<LoopInfoWrapperPass>().getLoopInfo();
    auto *SE = &getAnalysis<ScalarEvolutionWrapperPass>().getSE();
    auto *DT = &getAnalysis<DominatorTreeWrapperPass>().getDomTree();
    auto *TLIP = getAnalysisIfAvailable<TargetLibraryInfoWrapperPass>();
    auto *TLI = TLIP ? &TLIP->getTLI(*L->getHeader()->getParent()) : nullptr;
    auto *TTIP = getAnalysisIfAvailable<TargetTransformInfoWrapperPass>();
    auto *TTI = TTIP ? &TTIP->getTTI(*L->getHeader()->getParent()) : nullptr;
    const DataLayout &DL = L->getHeader()->getModule()->getDataLayout();

    IndVarSimplify IVS(LI, SE, DT, DL, TLI, TTI);
    return IVS.run(L);
  }

  void getAnalysisUsage(AnalysisUsage &AU) const override {
    AU.setPreservesCFG();
    getLoopAnalysisUsage(AU);
  }
};

} // end anonymous namespace

char IndVarSimplifyLegacyPass::ID = 0;

INITIALIZE_PASS_BEGIN(IndVarSimplifyLegacyPass, "indvars",
                      "Induction Variable Simplification", false, false)
INITIALIZE_PASS_DEPENDENCY(LoopPass)
INITIALIZE_PASS_END(IndVarSimplifyLegacyPass, "indvars",
                    "Induction Variable Simplification", false, false)

Pass *llvm::createIndVarSimplifyPass() {
  return new IndVarSimplifyLegacyPass();
}