Transforms/Scalar/StraightLineStrengthReduce.cpp

//===-- StraightLineStrengthReduce.cpp - ------------------------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// This file implements straight-line strength reduction (SLSR). Unlike loop
// strength reduction, this algorithm is designed to reduce arithmetic
// redundancy in straight-line code instead of loops. It has proven to be
// effective in simplifying arithmetic statements derived from an unrolled loop.
// It can also simplify the logic of SeparateConstOffsetFromGEP.
//
// There are many optimizations we can perform in the domain of SLSR. This file
// for now contains only an initial step. Specifically, we look for strength
// reduction candidates in the following forms:
//
// Form 1: B + i * S
// Form 2: (B + i) * S
// Form 3: &B[i * S]
//
// where S is an integer variable, and i is a constant integer. If we found two
// candidates S1 and S2 in the same form and S1 dominates S2, we may rewrite S2
// in a simpler way with respect to S1. For example,
//
// S1: X = B + i * S
// S2: Y = B + i' * S   => X + (i' - i) * S
//
// S1: X = (B + i) * S
// S2: Y = (B + i') * S => X + (i' - i) * S
//
// S1: X = &B[i * S]
// S2: Y = &B[i' * S]   => &X[(i' - i) * S]
//
// Note: (i' - i) * S is folded to the extent possible.
//
// This rewriting is in general a good idea. The code patterns we focus on
// usually come from loop unrolling, so (i' - i) * S is likely the same
// across iterations and can be reused. When that happens, the optimized form
// takes only one add starting from the second iteration.
//
// When such rewriting is possible, we call S1 a "basis" of S2. When S2 has
// multiple bases, we choose to rewrite S2 with respect to its "immediate"
// basis, the basis that is the closest ancestor in the dominator tree.
//
// TODO:
//
// - Floating point arithmetics when fast math is enabled.
//
// - SLSR may decrease ILP at the architecture level. Targets that are very
//   sensitive to ILP may want to disable it. Having SLSR to consider ILP is
//   left as future work.
//
// - When (i' - i) is constant but i and i' are not, we could still perform
//   SLSR.
#include <vector>

#include "llvm/ADT/DenseSet.h"
#include "llvm/ADT/FoldingSet.h"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/Analysis/TargetTransformInfo.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/IRBuilder.h"
#include "llvm/IR/Module.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/Support/raw_ostream.h"
#include "llvm/Transforms/Scalar.h"

using namespace llvm;
using namespace PatternMatch;

namespace {

class StraightLineStrengthReduce : public FunctionPass {
public:
  // SLSR candidate. Such a candidate must be in one of the forms described in
  // the header comments.
  struct Candidate : public ilist_node<Candidate> {
    enum Kind {
      Invalid, // reserved for the default constructor
      Add,     // B + i * S
      Mul,     // (B + i) * S
      GEP,     // &B[..][i * S][..]
    };

    Candidate()
        : CandidateKind(Invalid), Base(nullptr), Index(nullptr),
          Stride(nullptr), Ins(nullptr), Basis(nullptr) {}
    Candidate(Kind CT, const SCEV *B, ConstantInt *Idx, Value *S,
              Instruction *I)
        : CandidateKind(CT), Base(B), Index(Idx), Stride(S), Ins(I),
          Basis(nullptr) {}
    Kind CandidateKind;
    const SCEV *Base;
    // Note that Index and Stride of a GEP candidate do not necessarily have the
    // same integer type. In that case, during rewriting, Stride will be
    // sign-extended or truncated to Index's type.
    ConstantInt *Index;
    Value *Stride;
    // The instruction this candidate corresponds to. It helps us to rewrite a
    // candidate with respect to its immediate basis. Note that one instruction
    // can correspond to multiple candidates depending on how you associate the
    // expression. For instance,
    //
    // (a + 1) * (b + 2)
    //
    // can be treated as
    //
    // <Base: a, Index: 1, Stride: b + 2>
    //
    // or
    //
    // <Base: b, Index: 2, Stride: a + 1>
    Instruction *Ins;
    // Points to the immediate basis of this candidate, or nullptr if we cannot
    // find any basis for this candidate.
    Candidate *Basis;
  };

  static char ID;

  StraightLineStrengthReduce()
      : FunctionPass(ID), DL(nullptr), DT(nullptr), TTI(nullptr) {
    initializeStraightLineStrengthReducePass(*PassRegistry::getPassRegistry());
  }

  void getAnalysisUsage(AnalysisUsage &AU) const override {
    AU.addRequired<DominatorTreeWrapperPass>();
    AU.addRequired<ScalarEvolution>();
    AU.addRequired<TargetTransformInfoWrapperPass>();
    // We do not modify the shape of the CFG.
    AU.setPreservesCFG();
  }

  bool doInitialization(Module &M) override {
    DL = &M.getDataLayout();
    return false;
  }

  bool runOnFunction(Function &F) override;

private:
  // Returns true if Basis is a basis for C, i.e., Basis dominates C and they
  // share the same base and stride.
  bool isBasisFor(const Candidate &Basis, const Candidate &C);
  // Returns whether the candidate can be folded into an addressing mode.
  bool isFoldable(const Candidate &C, TargetTransformInfo *TTI,
                  const DataLayout *DL);
  // Returns true if C is already in a simplest form and not worth being
  // rewritten.
  bool isSimplestForm(const Candidate &C);
  // Checks whether I is in a candidate form. If so, adds all the matching forms
  // to Candidates, and tries to find the immediate basis for each of them.
  void allocateCandidatesAndFindBasis(Instruction *I);
  // Allocate candidates and find bases for Add instructions.
  void allocateCandidatesAndFindBasisForAdd(Instruction *I);
  // Given I = LHS + RHS, factors RHS into i * S and makes (LHS + i * S) a
  // candidate.
  void allocateCandidatesAndFindBasisForAdd(Value *LHS, Value *RHS,
                                            Instruction *I);
  // Allocate candidates and find bases for Mul instructions.
  void allocateCandidatesAndFindBasisForMul(Instruction *I);
  // Splits LHS into Base + Index and, if succeeds, calls
  // allocateCandidatesAndFindBasis.
  void allocateCandidatesAndFindBasisForMul(Value *LHS, Value *RHS,
                                            Instruction *I);
  // Allocate candidates and find bases for GetElementPtr instructions.
  void allocateCandidatesAndFindBasisForGEP(GetElementPtrInst *GEP);
  // A helper function that scales Idx with ElementSize before invoking
  // allocateCandidatesAndFindBasis.
  void allocateCandidatesAndFindBasisForGEP(const SCEV *B, ConstantInt *Idx,
                                            Value *S, uint64_t ElementSize,
                                            Instruction *I);
  // Adds the given form <CT, B, Idx, S> to Candidates, and finds its immediate
  // basis.
  void allocateCandidatesAndFindBasis(Candidate::Kind CT, const SCEV *B,
                                      ConstantInt *Idx, Value *S,
                                      Instruction *I);
  // Rewrites candidate C with respect to Basis.
  void rewriteCandidateWithBasis(const Candidate &C, const Candidate &Basis);
  // A helper function that factors ArrayIdx to a product of a stride and a
  // constant index, and invokes allocateCandidatesAndFindBasis with the
  // factorings.
  void factorArrayIndex(Value *ArrayIdx, const SCEV *Base, uint64_t ElementSize,
                        GetElementPtrInst *GEP);
  // Emit code that computes the "bump" from Basis to C. If the candidate is a
  // GEP and the bump is not divisible by the element size of the GEP, this
  // function sets the BumpWithUglyGEP flag to notify its caller to bump the
  // basis using an ugly GEP.
  static Value *emitBump(const Candidate &Basis, const Candidate &C,
                         IRBuilder<> &Builder, const DataLayout *DL,
                         bool &BumpWithUglyGEP);

  const DataLayout *DL;
  DominatorTree *DT;
  ScalarEvolution *SE;
  TargetTransformInfo *TTI;
  ilist<Candidate> Candidates;
  // Temporarily holds all instructions that are unlinked (but not deleted) by
  // rewriteCandidateWithBasis. These instructions will be actually removed
  // after all rewriting finishes.
  std::vector<Instruction *> UnlinkedInstructions;
};
}  // anonymous namespace

char StraightLineStrengthReduce::ID = 0;
INITIALIZE_PASS_BEGIN(StraightLineStrengthReduce, "slsr",
                      "Straight line strength reduction", false, false)
INITIALIZE_PASS_DEPENDENCY(DominatorTreeWrapperPass)
INITIALIZE_PASS_DEPENDENCY(ScalarEvolution)
INITIALIZE_PASS_DEPENDENCY(TargetTransformInfoWrapperPass)
INITIALIZE_PASS_END(StraightLineStrengthReduce, "slsr",
                    "Straight line strength reduction", false, false)

FunctionPass *llvm::createStraightLineStrengthReducePass() {
  return new StraightLineStrengthReduce();
}

bool StraightLineStrengthReduce::isBasisFor(const Candidate &Basis,
                                            const Candidate &C) {
  return (Basis.Ins != C.Ins && // skip the same instruction
          // Basis must dominate C in order to rewrite C with respect to Basis.
          DT->dominates(Basis.Ins->getParent(), C.Ins->getParent()) &&
          // They share the same base, stride, and candidate kind.
          Basis.Base == C.Base &&
          Basis.Stride == C.Stride &&
          Basis.CandidateKind == C.CandidateKind);
}

static bool isGEPFoldable(GetElementPtrInst *GEP,
                          const TargetTransformInfo *TTI,
                          const DataLayout *DL) {
  GlobalVariable *BaseGV = nullptr;
  int64_t BaseOffset = 0;
  bool HasBaseReg = false;
  int64_t Scale = 0;

  if (GlobalVariable *GV = dyn_cast<GlobalVariable>(GEP->getPointerOperand()))
    BaseGV = GV;
  else
    HasBaseReg = true;

  gep_type_iterator GTI = gep_type_begin(GEP);
  for (auto I = GEP->idx_begin(); I != GEP->idx_end(); ++I, ++GTI) {
    if (isa<SequentialType>(*GTI)) {
      int64_t ElementSize = DL->getTypeAllocSize(GTI.getIndexedType());
      if (ConstantInt *ConstIdx = dyn_cast<ConstantInt>(*I)) {
        BaseOffset += ConstIdx->getSExtValue() * ElementSize;
      } else {
        // Needs scale register.
        if (Scale != 0) {
          // No addressing mode takes two scale registers.
          return false;
        }
        Scale = ElementSize;
      }
    } else {
      StructType *STy = cast<StructType>(*GTI);
      uint64_t Field = cast<ConstantInt>(*I)->getZExtValue();
      BaseOffset += DL->getStructLayout(STy)->getElementOffset(Field);
    }
  }
  return TTI->isLegalAddressingMode(GEP->getType()->getElementType(), BaseGV,
                                    BaseOffset, HasBaseReg, Scale);
}

// Returns whether (Base + Index * Stride) can be folded to an addressing mode.
static bool isAddFoldable(const SCEV *Base, ConstantInt *Index, Value *Stride,
                          TargetTransformInfo *TTI) {
  return TTI->isLegalAddressingMode(Base->getType(), nullptr, 0, true,
                                    Index->getSExtValue());
}

bool StraightLineStrengthReduce::isFoldable(const Candidate &C,
                                            TargetTransformInfo *TTI,
                                            const DataLayout *DL) {
  if (C.CandidateKind == Candidate::Add)
    return isAddFoldable(C.Base, C.Index, C.Stride, TTI);
  if (C.CandidateKind == Candidate::GEP)
    return isGEPFoldable(cast<GetElementPtrInst>(C.Ins), TTI, DL);
  return false;
}

// Returns true if GEP has zero or one non-zero index.
static bool hasOnlyOneNonZeroIndex(GetElementPtrInst *GEP) {
  unsigned NumNonZeroIndices = 0;
  for (auto I = GEP->idx_begin(); I != GEP->idx_end(); ++I) {
    ConstantInt *ConstIdx = dyn_cast<ConstantInt>(*I);
    if (ConstIdx == nullptr || !ConstIdx->isZero())
      ++NumNonZeroIndices;
  }
  return NumNonZeroIndices <= 1;
}

bool StraightLineStrengthReduce::isSimplestForm(const Candidate &C) {
  if (C.CandidateKind == Candidate::Add) {
    // B + 1 * S or B + (-1) * S
    return C.Index->isOne() || C.Index->isMinusOne();
  }
  if (C.CandidateKind == Candidate::Mul) {
    // (B + 0) * S
    return C.Index->isZero();
  }
  if (C.CandidateKind == Candidate::GEP) {
    // (char*)B + S or (char*)B - S
    return ((C.Index->isOne() || C.Index->isMinusOne()) &&
            hasOnlyOneNonZeroIndex(cast<GetElementPtrInst>(C.Ins)));
  }
  return false;
}

// TODO: We currently implement an algorithm whose time complexity is linear in
// the number of existing candidates. However, we could do better by using
// ScopedHashTable. Specifically, while traversing the dominator tree, we could
// maintain all the candidates that dominate the basic block being traversed in
// a ScopedHashTable. This hash table is indexed by the base and the stride of
// a candidate. Therefore, finding the immediate basis of a candidate boils down
// to one hash-table look up.
void StraightLineStrengthReduce::allocateCandidatesAndFindBasis(
    Candidate::Kind CT, const SCEV *B, ConstantInt *Idx, Value *S,
    Instruction *I) {
  Candidate C(CT, B, Idx, S, I);
  // SLSR can complicate an instruction in two cases:
  //
  // 1. If we can fold I into an addressing mode, computing I is likely free or
  // takes only one instruction.
  //
  // 2. I is already in a simplest form. For example, when
  //      X = B + 8 * S
  //      Y = B + S,
  //    rewriting Y to X - 7 * S is probably a bad idea.
  //
  // In the above cases, we still add I to the candidate list so that I can be
  // the basis of other candidates, but we leave I's basis blank so that I
  // won't be rewritten.
  if (!isFoldable(C, TTI, DL) && !isSimplestForm(C)) {
    // Try to compute the immediate basis of C.
    unsigned NumIterations = 0;
    // Limit the scan radius to avoid running in quadratice time.
    static const unsigned MaxNumIterations = 50;
    for (auto Basis = Candidates.rbegin();
         Basis != Candidates.rend() && NumIterations < MaxNumIterations;
         ++Basis, ++NumIterations) {
      if (isBasisFor(*Basis, C)) {
        C.Basis = &(*Basis);
        break;
      }
    }
  }
  // Regardless of whether we find a basis for C, we need to push C to the
  // candidate list so that it can be the basis of other candidates.
  Candidates.push_back(C);
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasis(
    Instruction *I) {
  switch (I->getOpcode()) {
  case Instruction::Add:
    allocateCandidatesAndFindBasisForAdd(I);
    break;
  case Instruction::Mul:
    allocateCandidatesAndFindBasisForMul(I);
    break;
  case Instruction::GetElementPtr:
    allocateCandidatesAndFindBasisForGEP(cast<GetElementPtrInst>(I));
    break;
  }
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForAdd(
    Instruction *I) {
  // Try matching B + i * S.
  if (!isa<IntegerType>(I->getType()))
    return;

  assert(I->getNumOperands() == 2 && "isn't I an add?");
  Value *LHS = I->getOperand(0), *RHS = I->getOperand(1);
  allocateCandidatesAndFindBasisForAdd(LHS, RHS, I);
  if (LHS != RHS)
    allocateCandidatesAndFindBasisForAdd(RHS, LHS, I);
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForAdd(
    Value *LHS, Value *RHS, Instruction *I) {
  Value *S = nullptr;
  ConstantInt *Idx = nullptr;
  if (match(RHS, m_Mul(m_Value(S), m_ConstantInt(Idx)))) {
    // I = LHS + RHS = LHS + Idx * S
    allocateCandidatesAndFindBasis(Candidate::Add, SE->getSCEV(LHS), Idx, S, I);
  } else if (match(RHS, m_Shl(m_Value(S), m_ConstantInt(Idx)))) {
    // I = LHS + RHS = LHS + (S << Idx) = LHS + S * (1 << Idx)
    APInt One(Idx->getBitWidth(), 1);
    Idx = ConstantInt::get(Idx->getContext(), One << Idx->getValue());
    allocateCandidatesAndFindBasis(Candidate::Add, SE->getSCEV(LHS), Idx, S, I);
  } else {
    // At least, I = LHS + 1 * RHS
    ConstantInt *One = ConstantInt::get(cast<IntegerType>(I->getType()), 1);
    allocateCandidatesAndFindBasis(Candidate::Add, SE->getSCEV(LHS), One, RHS,
                                   I);
  }
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForMul(
    Value *LHS, Value *RHS, Instruction *I) {
  Value *B = nullptr;
  ConstantInt *Idx = nullptr;
  // Only handle the canonical operand ordering.
  if (match(LHS, m_Add(m_Value(B), m_ConstantInt(Idx)))) {
    // If LHS is in the form of "Base + Index", then I is in the form of
    // "(Base + Index) * RHS".
    allocateCandidatesAndFindBasis(Candidate::Mul, SE->getSCEV(B), Idx, RHS, I);
  } else {
    // Otherwise, at least try the form (LHS + 0) * RHS.
    ConstantInt *Zero = ConstantInt::get(cast<IntegerType>(I->getType()), 0);
    allocateCandidatesAndFindBasis(Candidate::Mul, SE->getSCEV(LHS), Zero, RHS,
                                  I);
  }
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForMul(
    Instruction *I) {
  // Try matching (B + i) * S.
  // TODO: we could extend SLSR to float and vector types.
  if (!isa<IntegerType>(I->getType()))
    return;

  assert(I->getNumOperands() == 2 && "isn't I a mul?");
  Value *LHS = I->getOperand(0), *RHS = I->getOperand(1);
  allocateCandidatesAndFindBasisForMul(LHS, RHS, I);
  if (LHS != RHS) {
    // Symmetrically, try to split RHS to Base + Index.
    allocateCandidatesAndFindBasisForMul(RHS, LHS, I);
  }
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForGEP(
    const SCEV *B, ConstantInt *Idx, Value *S, uint64_t ElementSize,
    Instruction *I) {
  // I = B + sext(Idx *nsw S) * ElementSize
  //   = B + (sext(Idx) * sext(S)) * ElementSize
  //   = B + (sext(Idx) * ElementSize) * sext(S)
  // Casting to IntegerType is safe because we skipped vector GEPs.
  IntegerType *IntPtrTy = cast<IntegerType>(DL->getIntPtrType(I->getType()));
  ConstantInt *ScaledIdx = ConstantInt::get(
      IntPtrTy, Idx->getSExtValue() * (int64_t)ElementSize, true);
  allocateCandidatesAndFindBasis(Candidate::GEP, B, ScaledIdx, S, I);
}

void StraightLineStrengthReduce::factorArrayIndex(Value *ArrayIdx,
                                                  const SCEV *Base,
                                                  uint64_t ElementSize,
                                                  GetElementPtrInst *GEP) {
  // At least, ArrayIdx = ArrayIdx *nsw 1.
  allocateCandidatesAndFindBasisForGEP(
      Base, ConstantInt::get(cast<IntegerType>(ArrayIdx->getType()), 1),
      ArrayIdx, ElementSize, GEP);
  Value *LHS = nullptr;
  ConstantInt *RHS = nullptr;
  // One alternative is matching the SCEV of ArrayIdx instead of ArrayIdx
  // itself. This would allow us to handle the shl case for free. However,
  // matching SCEVs has two issues:
  //
  // 1. this would complicate rewriting because the rewriting procedure
  // would have to translate SCEVs back to IR instructions. This translation
  // is difficult when LHS is further evaluated to a composite SCEV.
  //
  // 2. ScalarEvolution is designed to be control-flow oblivious. It tends
  // to strip nsw/nuw flags which are critical for SLSR to trace into
  // sext'ed multiplication.
  if (match(ArrayIdx, m_NSWMul(m_Value(LHS), m_ConstantInt(RHS)))) {
    // SLSR is currently unsafe if i * S may overflow.
    // GEP = Base + sext(LHS *nsw RHS) * ElementSize
    allocateCandidatesAndFindBasisForGEP(Base, RHS, LHS, ElementSize, GEP);
  } else if (match(ArrayIdx, m_NSWShl(m_Value(LHS), m_ConstantInt(RHS)))) {
    // GEP = Base + sext(LHS <<nsw RHS) * ElementSize
    //     = Base + sext(LHS *nsw (1 << RHS)) * ElementSize
    APInt One(RHS->getBitWidth(), 1);
    ConstantInt *PowerOf2 =
        ConstantInt::get(RHS->getContext(), One << RHS->getValue());
    allocateCandidatesAndFindBasisForGEP(Base, PowerOf2, LHS, ElementSize, GEP);
  }
}

void StraightLineStrengthReduce::allocateCandidatesAndFindBasisForGEP(
    GetElementPtrInst *GEP) {
  // TODO: handle vector GEPs
  if (GEP->getType()->isVectorTy())
    return;

  const SCEV *GEPExpr = SE->getSCEV(GEP);
  Type *IntPtrTy = DL->getIntPtrType(GEP->getType());

  gep_type_iterator GTI = gep_type_begin(GEP);
  for (auto I = GEP->idx_begin(); I != GEP->idx_end(); ++I) {
    if (!isa<SequentialType>(*GTI++))
      continue;
    Value *ArrayIdx = *I;
    // Compute the byte offset of this index.
    uint64_t ElementSize = DL->getTypeAllocSize(*GTI);
    const SCEV *ElementSizeExpr = SE->getSizeOfExpr(IntPtrTy, *GTI);
    const SCEV *ArrayIdxExpr = SE->getSCEV(ArrayIdx);
    ArrayIdxExpr = SE->getTruncateOrSignExtend(ArrayIdxExpr, IntPtrTy);
    const SCEV *LocalOffset =
        SE->getMulExpr(ArrayIdxExpr, ElementSizeExpr, SCEV::FlagNSW);
    // The base of this candidate equals GEPExpr less the byte offset of this
    // index.
    const SCEV *Base = SE->getMinusSCEV(GEPExpr, LocalOffset);
    factorArrayIndex(ArrayIdx, Base, ElementSize, GEP);
    // When ArrayIdx is the sext of a value, we try to factor that value as
    // well.  Handling this case is important because array indices are
    // typically sign-extended to the pointer size.
    Value *TruncatedArrayIdx = nullptr;
    if (match(ArrayIdx, m_SExt(m_Value(TruncatedArrayIdx))))
      factorArrayIndex(TruncatedArrayIdx, Base, ElementSize, GEP);
  }
}

// A helper function that unifies the bitwidth of A and B.
static void unifyBitWidth(APInt &A, APInt &B) {
  if (A.getBitWidth() < B.getBitWidth())
    A = A.sext(B.getBitWidth());
  else if (A.getBitWidth() > B.getBitWidth())
    B = B.sext(A.getBitWidth());
}

Value *StraightLineStrengthReduce::emitBump(const Candidate &Basis,
                                            const Candidate &C,
                                            IRBuilder<> &Builder,
                                            const DataLayout *DL,
                                            bool &BumpWithUglyGEP) {
  APInt Idx = C.Index->getValue(), BasisIdx = Basis.Index->getValue();
  unifyBitWidth(Idx, BasisIdx);
  APInt IndexOffset = Idx - BasisIdx;

  BumpWithUglyGEP = false;
  if (Basis.CandidateKind == Candidate::GEP) {
    APInt ElementSize(
        IndexOffset.getBitWidth(),
        DL->getTypeAllocSize(
            cast<GetElementPtrInst>(Basis.Ins)->getType()->getElementType()));
    APInt Q, R;
    APInt::sdivrem(IndexOffset, ElementSize, Q, R);
    if (R.getSExtValue() == 0)
      IndexOffset = Q;
    else
      BumpWithUglyGEP = true;
  }

  // Compute Bump = C - Basis = (i' - i) * S.
  // Common case 1: if (i' - i) is 1, Bump = S.
  if (IndexOffset.getSExtValue() == 1)
    return C.Stride;
  // Common case 2: if (i' - i) is -1, Bump = -S.
  if (IndexOffset.getSExtValue() == -1)
    return Builder.CreateNeg(C.Stride);

  // Otherwise, Bump = (i' - i) * sext/trunc(S). Note that (i' - i) and S may
  // have different bit widths.
  IntegerType *DeltaType =
      IntegerType::get(Basis.Ins->getContext(), IndexOffset.getBitWidth());
  Value *ExtendedStride = Builder.CreateSExtOrTrunc(C.Stride, DeltaType);
  if (IndexOffset.isPowerOf2()) {
    // If (i' - i) is a power of 2, Bump = sext/trunc(S) << log(i' - i).
    ConstantInt *Exponent = ConstantInt::get(DeltaType, IndexOffset.logBase2());
    return Builder.CreateShl(ExtendedStride, Exponent);
  }
  if ((-IndexOffset).isPowerOf2()) {
    // If (i - i') is a power of 2, Bump = -sext/trunc(S) << log(i' - i).
    ConstantInt *Exponent =
        ConstantInt::get(DeltaType, (-IndexOffset).logBase2());
    return Builder.CreateNeg(Builder.CreateShl(ExtendedStride, Exponent));
  }
  Constant *Delta = ConstantInt::get(DeltaType, IndexOffset);
  return Builder.CreateMul(ExtendedStride, Delta);
}

void StraightLineStrengthReduce::rewriteCandidateWithBasis(
    const Candidate &C, const Candidate &Basis) {
  assert(C.CandidateKind == Basis.CandidateKind && C.Base == Basis.Base &&
         C.Stride == Basis.Stride);
  // We run rewriteCandidateWithBasis on all candidates in a post-order, so the
  // basis of a candidate cannot be unlinked before the candidate.
  assert(Basis.Ins->getParent() != nullptr && "the basis is unlinked");

  // An instruction can correspond to multiple candidates. Therefore, instead of
  // simply deleting an instruction when we rewrite it, we mark its parent as
  // nullptr (i.e. unlink it) so that we can skip the candidates whose
  // instruction is already rewritten.
  if (!C.Ins->getParent())
    return;

  IRBuilder<> Builder(C.Ins);
  bool BumpWithUglyGEP;
  Value *Bump = emitBump(Basis, C, Builder, DL, BumpWithUglyGEP);
  Value *Reduced = nullptr; // equivalent to but weaker than C.Ins
  switch (C.CandidateKind) {
  case Candidate::Add:
  case Candidate::Mul:
    if (BinaryOperator::isNeg(Bump)) {
      Reduced =
          Builder.CreateSub(Basis.Ins, BinaryOperator::getNegArgument(Bump));
    } else {
      Reduced = Builder.CreateAdd(Basis.Ins, Bump);
    }
    break;
  case Candidate::GEP:
    {
      Type *IntPtrTy = DL->getIntPtrType(C.Ins->getType());
      bool InBounds = cast<GetElementPtrInst>(C.Ins)->isInBounds();
      if (BumpWithUglyGEP) {
        // C = (char *)Basis + Bump
        unsigned AS = Basis.Ins->getType()->getPointerAddressSpace();
        Type *CharTy = Type::getInt8PtrTy(Basis.Ins->getContext(), AS);
        Reduced = Builder.CreateBitCast(Basis.Ins, CharTy);
        if (InBounds)
          Reduced =
              Builder.CreateInBoundsGEP(Builder.getInt8Ty(), Reduced, Bump);
        else
          Reduced = Builder.CreateGEP(Builder.getInt8Ty(), Reduced, Bump);
        Reduced = Builder.CreateBitCast(Reduced, C.Ins->getType());
      } else {
        // C = gep Basis, Bump
        // Canonicalize bump to pointer size.
        Bump = Builder.CreateSExtOrTrunc(Bump, IntPtrTy);
        if (InBounds)
          Reduced = Builder.CreateInBoundsGEP(nullptr, Basis.Ins, Bump);
        else
          Reduced = Builder.CreateGEP(nullptr, Basis.Ins, Bump);
      }
    }
    break;
  default:
    llvm_unreachable("C.CandidateKind is invalid");
  };
  Reduced->takeName(C.Ins);
  C.Ins->replaceAllUsesWith(Reduced);
  C.Ins->dropAllReferences();
  // Unlink C.Ins so that we can skip other candidates also corresponding to
  // C.Ins. The actual deletion is postponed to the end of runOnFunction.
  C.Ins->removeFromParent();
  UnlinkedInstructions.push_back(C.Ins);
}

bool StraightLineStrengthReduce::runOnFunction(Function &F) {
  if (skipOptnoneFunction(F))
    return false;

  TTI = &getAnalysis<TargetTransformInfoWrapperPass>().getTTI(F);
  DT = &getAnalysis<DominatorTreeWrapperPass>().getDomTree();
  SE = &getAnalysis<ScalarEvolution>();
  // Traverse the dominator tree in the depth-first order. This order makes sure
  // all bases of a candidate are in Candidates when we process it.
  for (auto node = GraphTraits<DominatorTree *>::nodes_begin(DT);
       node != GraphTraits<DominatorTree *>::nodes_end(DT); ++node) {
    for (auto &I : *node->getBlock())
      allocateCandidatesAndFindBasis(&I);
  }

  // Rewrite candidates in the reverse depth-first order. This order makes sure
  // a candidate being rewritten is not a basis for any other candidate.
  while (!Candidates.empty()) {
    const Candidate &C = Candidates.back();
    if (C.Basis != nullptr) {
      rewriteCandidateWithBasis(C, *C.Basis);
    }
    Candidates.pop_back();
  }

  // Delete all unlink instructions.
  for (auto I : UnlinkedInstructions) {
    delete I;
  }
  bool Ret = !UnlinkedInstructions.empty();
  UnlinkedInstructions.clear();
  return Ret;
}