xref: /external/llvm/lib/Analysis/ScalarEvolutionExpander.cpp

//===- ScalarEvolutionExpander.cpp - Scalar Evolution Analysis --*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// This file contains the implementation of the scalar evolution expander,
// which is used to generate the code corresponding to a given scalar evolution
// expression.
//
//===----------------------------------------------------------------------===//

#include "llvm/Analysis/ScalarEvolutionExpander.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/IntrinsicInst.h"
#include "llvm/LLVMContext.h"
#include "llvm/Target/TargetData.h"
#include "llvm/ADT/STLExtras.h"
using namespace llvm;

/// InsertNoopCastOfTo - Insert a cast of V to the specified type,
/// which must be possible with a noop cast, doing what we can to share
/// the casts.
Value *SCEVExpander::InsertNoopCastOfTo(Value *V, const Type *Ty) {
  Instruction::CastOps Op = CastInst::getCastOpcode(V, false, Ty, false);
  assert((Op == Instruction::BitCast ||
          Op == Instruction::PtrToInt ||
          Op == Instruction::IntToPtr) &&
         "InsertNoopCastOfTo cannot perform non-noop casts!");
  assert(SE.getTypeSizeInBits(V->getType()) == SE.getTypeSizeInBits(Ty) &&
         "InsertNoopCastOfTo cannot change sizes!");

  // Short-circuit unnecessary bitcasts.
  if (Op == Instruction::BitCast && V->getType() == Ty)
    return V;

  // Short-circuit unnecessary inttoptr<->ptrtoint casts.
  if ((Op == Instruction::PtrToInt || Op == Instruction::IntToPtr) &&
      SE.getTypeSizeInBits(Ty) == SE.getTypeSizeInBits(V->getType())) {
    if (CastInst *CI = dyn_cast<CastInst>(V))
      if ((CI->getOpcode() == Instruction::PtrToInt ||
           CI->getOpcode() == Instruction::IntToPtr) &&
          SE.getTypeSizeInBits(CI->getType()) ==
          SE.getTypeSizeInBits(CI->getOperand(0)->getType()))
        return CI->getOperand(0);
    if (ConstantExpr *CE = dyn_cast<ConstantExpr>(V))
      if ((CE->getOpcode() == Instruction::PtrToInt ||
           CE->getOpcode() == Instruction::IntToPtr) &&
          SE.getTypeSizeInBits(CE->getType()) ==
          SE.getTypeSizeInBits(CE->getOperand(0)->getType()))
        return CE->getOperand(0);
  }

  if (Constant *C = dyn_cast<Constant>(V))
    return ConstantExpr::getCast(Op, C, Ty);

  if (Argument *A = dyn_cast<Argument>(V)) {
    // Check to see if there is already a cast!
    for (Value::use_iterator UI = A->use_begin(), E = A->use_end();
         UI != E; ++UI)
      if ((*UI)->getType() == Ty)
        if (CastInst *CI = dyn_cast<CastInst>(cast<Instruction>(*UI)))
          if (CI->getOpcode() == Op) {
            // If the cast isn't the first instruction of the function, move it.
            if (BasicBlock::iterator(CI) !=
                A->getParent()->getEntryBlock().begin()) {
              // Recreate the cast at the beginning of the entry block.
              // The old cast is left in place in case it is being used
              // as an insert point.
              Instruction *NewCI =
                CastInst::Create(Op, V, Ty, "",
                                 A->getParent()->getEntryBlock().begin());
              NewCI->takeName(CI);
              CI->replaceAllUsesWith(NewCI);
              return NewCI;
            }
            return CI;
          }

    Instruction *I = CastInst::Create(Op, V, Ty, V->getName(),
                                      A->getParent()->getEntryBlock().begin());
    rememberInstruction(I);
    return I;
  }

  Instruction *I = cast<Instruction>(V);

  // Check to see if there is already a cast.  If there is, use it.
  for (Value::use_iterator UI = I->use_begin(), E = I->use_end();
       UI != E; ++UI) {
    if ((*UI)->getType() == Ty)
      if (CastInst *CI = dyn_cast<CastInst>(cast<Instruction>(*UI)))
        if (CI->getOpcode() == Op) {
          BasicBlock::iterator It = I; ++It;
          if (isa<InvokeInst>(I))
            It = cast<InvokeInst>(I)->getNormalDest()->begin();
          while (isa<PHINode>(It)) ++It;
          if (It != BasicBlock::iterator(CI)) {
            // Recreate the cast after the user.
            // The old cast is left in place in case it is being used
            // as an insert point.
            Instruction *NewCI = CastInst::Create(Op, V, Ty, "", It);
            NewCI->takeName(CI);
            CI->replaceAllUsesWith(NewCI);
            rememberInstruction(NewCI);
            return NewCI;
          }
          rememberInstruction(CI);
          return CI;
        }
  }
  BasicBlock::iterator IP = I; ++IP;
  if (InvokeInst *II = dyn_cast<InvokeInst>(I))
    IP = II->getNormalDest()->begin();
  while (isa<PHINode>(IP)) ++IP;
  Instruction *CI = CastInst::Create(Op, V, Ty, V->getName(), IP);
  rememberInstruction(CI);
  return CI;
}

/// InsertBinop - Insert the specified binary operator, doing a small amount
/// of work to avoid inserting an obviously redundant operation.
Value *SCEVExpander::InsertBinop(Instruction::BinaryOps Opcode,
                                 Value *LHS, Value *RHS) {
  // Fold a binop with constant operands.
  if (Constant *CLHS = dyn_cast<Constant>(LHS))
    if (Constant *CRHS = dyn_cast<Constant>(RHS))
      return ConstantExpr::get(Opcode, CLHS, CRHS);

  // Do a quick scan to see if we have this binop nearby.  If so, reuse it.
  unsigned ScanLimit = 6;
  BasicBlock::iterator BlockBegin = Builder.GetInsertBlock()->begin();
  // Scanning starts from the last instruction before the insertion point.
  BasicBlock::iterator IP = Builder.GetInsertPoint();
  if (IP != BlockBegin) {
    --IP;
    for (; ScanLimit; --IP, --ScanLimit) {
      // Don't count dbg.value against the ScanLimit, to avoid perturbing the
      // generated code.
      if (isa<DbgInfoIntrinsic>(IP))
        ScanLimit++;
      if (IP->getOpcode() == (unsigned)Opcode && IP->getOperand(0) == LHS &&
          IP->getOperand(1) == RHS)
        return IP;
      if (IP == BlockBegin) break;
    }
  }

  // Save the original insertion point so we can restore it when we're done.
  BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
  BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();

  // Move the insertion point out of as many loops as we can.
  while (const Loop *L = SE.LI->getLoopFor(Builder.GetInsertBlock())) {
    if (!L->isLoopInvariant(LHS) || !L->isLoopInvariant(RHS)) break;
    BasicBlock *Preheader = L->getLoopPreheader();
    if (!Preheader) break;

    // Ok, move up a level.
    Builder.SetInsertPoint(Preheader, Preheader->getTerminator());
  }

  // If we haven't found this binop, insert it.
  Value *BO = Builder.CreateBinOp(Opcode, LHS, RHS, "tmp");
  rememberInstruction(BO);

  // Restore the original insert point.
  if (SaveInsertBB)
    restoreInsertPoint(SaveInsertBB, SaveInsertPt);

  return BO;
}

/// FactorOutConstant - Test if S is divisible by Factor, using signed
/// division. If so, update S with Factor divided out and return true.
/// S need not be evenly divisible if a reasonable remainder can be
/// computed.
/// TODO: When ScalarEvolution gets a SCEVSDivExpr, this can be made
/// unnecessary; in its place, just signed-divide Ops[i] by the scale and
/// check to see if the divide was folded.
static bool FactorOutConstant(const SCEV *&S,
                              const SCEV *&Remainder,
                              const SCEV *Factor,
                              ScalarEvolution &SE,
                              const TargetData *TD) {
  // Everything is divisible by one.
  if (Factor->isOne())
    return true;

  // x/x == 1.
  if (S == Factor) {
    S = SE.getConstant(S->getType(), 1);
    return true;
  }

  // For a Constant, check for a multiple of the given factor.
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S)) {
    // 0/x == 0.
    if (C->isZero())
      return true;
    // Check for divisibility.
    if (const SCEVConstant *FC = dyn_cast<SCEVConstant>(Factor)) {
      ConstantInt *CI =
        ConstantInt::get(SE.getContext(),
                         C->getValue()->getValue().sdiv(
                                                   FC->getValue()->getValue()));
      // If the quotient is zero and the remainder is non-zero, reject
      // the value at this scale. It will be considered for subsequent
      // smaller scales.
      if (!CI->isZero()) {
        const SCEV *Div = SE.getConstant(CI);
        S = Div;
        Remainder =
          SE.getAddExpr(Remainder,
                        SE.getConstant(C->getValue()->getValue().srem(
                                                  FC->getValue()->getValue())));
        return true;
      }
    }
  }

  // In a Mul, check if there is a constant operand which is a multiple
  // of the given factor.
  if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(S)) {
    if (TD) {
      // With TargetData, the size is known. Check if there is a constant
      // operand which is a multiple of the given factor. If so, we can
      // factor it.
      const SCEVConstant *FC = cast<SCEVConstant>(Factor);
      if (const SCEVConstant *C = dyn_cast<SCEVConstant>(M->getOperand(0)))
        if (!C->getValue()->getValue().srem(FC->getValue()->getValue())) {
          SmallVector<const SCEV *, 4> NewMulOps(M->op_begin(), M->op_end());
          NewMulOps[0] =
            SE.getConstant(C->getValue()->getValue().sdiv(
                                                   FC->getValue()->getValue()));
          S = SE.getMulExpr(NewMulOps);
          return true;
        }
    } else {
      // Without TargetData, check if Factor can be factored out of any of the
      // Mul's operands. If so, we can just remove it.
      for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i) {
        const SCEV *SOp = M->getOperand(i);
        const SCEV *Remainder = SE.getConstant(SOp->getType(), 0);
        if (FactorOutConstant(SOp, Remainder, Factor, SE, TD) &&
            Remainder->isZero()) {
          SmallVector<const SCEV *, 4> NewMulOps(M->op_begin(), M->op_end());
          NewMulOps[i] = SOp;
          S = SE.getMulExpr(NewMulOps);
          return true;
        }
      }
    }
  }

  // In an AddRec, check if both start and step are divisible.
  if (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(S)) {
    const SCEV *Step = A->getStepRecurrence(SE);
    const SCEV *StepRem = SE.getConstant(Step->getType(), 0);
    if (!FactorOutConstant(Step, StepRem, Factor, SE, TD))
      return false;
    if (!StepRem->isZero())
      return false;
    const SCEV *Start = A->getStart();
    if (!FactorOutConstant(Start, Remainder, Factor, SE, TD))
      return false;
    S = SE.getAddRecExpr(Start, Step, A->getLoop());
    return true;
  }

  return false;
}

/// SimplifyAddOperands - Sort and simplify a list of add operands. NumAddRecs
/// is the number of SCEVAddRecExprs present, which are kept at the end of
/// the list.
///
static void SimplifyAddOperands(SmallVectorImpl<const SCEV *> &Ops,
                                const Type *Ty,
                                ScalarEvolution &SE) {
  unsigned NumAddRecs = 0;
  for (unsigned i = Ops.size(); i > 0 && isa<SCEVAddRecExpr>(Ops[i-1]); --i)
    ++NumAddRecs;
  // Group Ops into non-addrecs and addrecs.
  SmallVector<const SCEV *, 8> NoAddRecs(Ops.begin(), Ops.end() - NumAddRecs);
  SmallVector<const SCEV *, 8> AddRecs(Ops.end() - NumAddRecs, Ops.end());
  // Let ScalarEvolution sort and simplify the non-addrecs list.
  const SCEV *Sum = NoAddRecs.empty() ?
                    SE.getConstant(Ty, 0) :
                    SE.getAddExpr(NoAddRecs);
  // If it returned an add, use the operands. Otherwise it simplified
  // the sum into a single value, so just use that.
  Ops.clear();
  if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Sum))
    Ops.insert(Ops.end(), Add->op_begin(), Add->op_end());
  else if (!Sum->isZero())
    Ops.push_back(Sum);
  // Then append the addrecs.
  Ops.insert(Ops.end(), AddRecs.begin(), AddRecs.end());
}

/// SplitAddRecs - Flatten a list of add operands, moving addrec start values
/// out to the top level. For example, convert {a + b,+,c} to a, b, {0,+,d}.
/// This helps expose more opportunities for folding parts of the expressions
/// into GEP indices.
///
static void SplitAddRecs(SmallVectorImpl<const SCEV *> &Ops,
                         const Type *Ty,
                         ScalarEvolution &SE) {
  // Find the addrecs.
  SmallVector<const SCEV *, 8> AddRecs;
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    while (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(Ops[i])) {
      const SCEV *Start = A->getStart();
      if (Start->isZero()) break;
      const SCEV *Zero = SE.getConstant(Ty, 0);
      AddRecs.push_back(SE.getAddRecExpr(Zero,
                                         A->getStepRecurrence(SE),
                                         A->getLoop()));
      if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Start)) {
        Ops[i] = Zero;
        Ops.insert(Ops.end(), Add->op_begin(), Add->op_end());
        e += Add->getNumOperands();
      } else {
        Ops[i] = Start;
      }
    }
  if (!AddRecs.empty()) {
    // Add the addrecs onto the end of the list.
    Ops.insert(Ops.end(), AddRecs.begin(), AddRecs.end());
    // Resort the operand list, moving any constants to the front.
    SimplifyAddOperands(Ops, Ty, SE);
  }
}

/// expandAddToGEP - Expand an addition expression with a pointer type into
/// a GEP instead of using ptrtoint+arithmetic+inttoptr. This helps
/// BasicAliasAnalysis and other passes analyze the result. See the rules
/// for getelementptr vs. inttoptr in
/// http://llvm.org/docs/LangRef.html#pointeraliasing
/// for details.
///
/// Design note: The correctness of using getelementptr here depends on
/// ScalarEvolution not recognizing inttoptr and ptrtoint operators, as
/// they may introduce pointer arithmetic which may not be safely converted
/// into getelementptr.
///
/// Design note: It might seem desirable for this function to be more
/// loop-aware. If some of the indices are loop-invariant while others
/// aren't, it might seem desirable to emit multiple GEPs, keeping the
/// loop-invariant portions of the overall computation outside the loop.
/// However, there are a few reasons this is not done here. Hoisting simple
/// arithmetic is a low-level optimization that often isn't very
/// important until late in the optimization process. In fact, passes
/// like InstructionCombining will combine GEPs, even if it means
/// pushing loop-invariant computation down into loops, so even if the
/// GEPs were split here, the work would quickly be undone. The
/// LoopStrengthReduction pass, which is usually run quite late (and
/// after the last InstructionCombining pass), takes care of hoisting
/// loop-invariant portions of expressions, after considering what
/// can be folded using target addressing modes.
///
Value *SCEVExpander::expandAddToGEP(const SCEV *const *op_begin,
                                    const SCEV *const *op_end,
                                    const PointerType *PTy,
                                    const Type *Ty,
                                    Value *V) {
  const Type *ElTy = PTy->getElementType();
  SmallVector<Value *, 4> GepIndices;
  SmallVector<const SCEV *, 8> Ops(op_begin, op_end);
  bool AnyNonZeroIndices = false;

  // Split AddRecs up into parts as either of the parts may be usable
  // without the other.
  SplitAddRecs(Ops, Ty, SE);

  // Descend down the pointer's type and attempt to convert the other
  // operands into GEP indices, at each level. The first index in a GEP
  // indexes into the array implied by the pointer operand; the rest of
  // the indices index into the element or field type selected by the
  // preceding index.
  for (;;) {
    // If the scale size is not 0, attempt to factor out a scale for
    // array indexing.
    SmallVector<const SCEV *, 8> ScaledOps;
    if (ElTy->isSized()) {
      const SCEV *ElSize = SE.getSizeOfExpr(ElTy);
      if (!ElSize->isZero()) {
        SmallVector<const SCEV *, 8> NewOps;
        for (unsigned i = 0, e = Ops.size(); i != e; ++i) {
          const SCEV *Op = Ops[i];
          const SCEV *Remainder = SE.getConstant(Ty, 0);
          if (FactorOutConstant(Op, Remainder, ElSize, SE, SE.TD)) {
            // Op now has ElSize factored out.
            ScaledOps.push_back(Op);
            if (!Remainder->isZero())
              NewOps.push_back(Remainder);
            AnyNonZeroIndices = true;
          } else {
            // The operand was not divisible, so add it to the list of operands
            // we'll scan next iteration.
            NewOps.push_back(Ops[i]);
          }
        }
        // If we made any changes, update Ops.
        if (!ScaledOps.empty()) {
          Ops = NewOps;
          SimplifyAddOperands(Ops, Ty, SE);
        }
      }
    }

    // Record the scaled array index for this level of the type. If
    // we didn't find any operands that could be factored, tentatively
    // assume that element zero was selected (since the zero offset
    // would obviously be folded away).
    Value *Scaled = ScaledOps.empty() ?
                    Constant::getNullValue(Ty) :
                    expandCodeFor(SE.getAddExpr(ScaledOps), Ty);
    GepIndices.push_back(Scaled);

    // Collect struct field index operands.
    while (const StructType *STy = dyn_cast<StructType>(ElTy)) {
      bool FoundFieldNo = false;
      // An empty struct has no fields.
      if (STy->getNumElements() == 0) break;
      if (SE.TD) {
        // With TargetData, field offsets are known. See if a constant offset
        // falls within any of the struct fields.
        if (Ops.empty()) break;
        if (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[0]))
          if (SE.getTypeSizeInBits(C->getType()) <= 64) {
            const StructLayout &SL = *SE.TD->getStructLayout(STy);
            uint64_t FullOffset = C->getValue()->getZExtValue();
            if (FullOffset < SL.getSizeInBytes()) {
              unsigned ElIdx = SL.getElementContainingOffset(FullOffset);
              GepIndices.push_back(
                  ConstantInt::get(Type::getInt32Ty(Ty->getContext()), ElIdx));
              ElTy = STy->getTypeAtIndex(ElIdx);
              Ops[0] =
                SE.getConstant(Ty, FullOffset - SL.getElementOffset(ElIdx));
              AnyNonZeroIndices = true;
              FoundFieldNo = true;
            }
          }
      } else {
        // Without TargetData, just check for an offsetof expression of the
        // appropriate struct type.
        for (unsigned i = 0, e = Ops.size(); i != e; ++i)
          if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(Ops[i])) {
            const Type *CTy;
            Constant *FieldNo;
            if (U->isOffsetOf(CTy, FieldNo) && CTy == STy) {
              GepIndices.push_back(FieldNo);
              ElTy =
                STy->getTypeAtIndex(cast<ConstantInt>(FieldNo)->getZExtValue());
              Ops[i] = SE.getConstant(Ty, 0);
              AnyNonZeroIndices = true;
              FoundFieldNo = true;
              break;
            }
          }
      }
      // If no struct field offsets were found, tentatively assume that
      // field zero was selected (since the zero offset would obviously
      // be folded away).
      if (!FoundFieldNo) {
        ElTy = STy->getTypeAtIndex(0u);
        GepIndices.push_back(
          Constant::getNullValue(Type::getInt32Ty(Ty->getContext())));
      }
    }

    if (const ArrayType *ATy = dyn_cast<ArrayType>(ElTy))
      ElTy = ATy->getElementType();
    else
      break;
  }

  // If none of the operands were convertible to proper GEP indices, cast
  // the base to i8* and do an ugly getelementptr with that. It's still
  // better than ptrtoint+arithmetic+inttoptr at least.
  if (!AnyNonZeroIndices) {
    // Cast the base to i8*.
    V = InsertNoopCastOfTo(V,
       Type::getInt8PtrTy(Ty->getContext(), PTy->getAddressSpace()));

    // Expand the operands for a plain byte offset.
    Value *Idx = expandCodeFor(SE.getAddExpr(Ops), Ty);

    // Fold a GEP with constant operands.
    if (Constant *CLHS = dyn_cast<Constant>(V))
      if (Constant *CRHS = dyn_cast<Constant>(Idx))
        return ConstantExpr::getGetElementPtr(CLHS, &CRHS, 1);

    // Do a quick scan to see if we have this GEP nearby.  If so, reuse it.
    unsigned ScanLimit = 6;
    BasicBlock::iterator BlockBegin = Builder.GetInsertBlock()->begin();
    // Scanning starts from the last instruction before the insertion point.
    BasicBlock::iterator IP = Builder.GetInsertPoint();
    if (IP != BlockBegin) {
      --IP;
      for (; ScanLimit; --IP, --ScanLimit) {
        // Don't count dbg.value against the ScanLimit, to avoid perturbing the
        // generated code.
        if (isa<DbgInfoIntrinsic>(IP))
          ScanLimit++;
        if (IP->getOpcode() == Instruction::GetElementPtr &&
            IP->getOperand(0) == V && IP->getOperand(1) == Idx)
          return IP;
        if (IP == BlockBegin) break;
      }
    }

    // Save the original insertion point so we can restore it when we're done.
    BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
    BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();

    // Move the insertion point out of as many loops as we can.
    while (const Loop *L = SE.LI->getLoopFor(Builder.GetInsertBlock())) {
      if (!L->isLoopInvariant(V) || !L->isLoopInvariant(Idx)) break;
      BasicBlock *Preheader = L->getLoopPreheader();
      if (!Preheader) break;

      // Ok, move up a level.
      Builder.SetInsertPoint(Preheader, Preheader->getTerminator());
    }

    // Emit a GEP.
    Value *GEP = Builder.CreateGEP(V, Idx, "uglygep");
    rememberInstruction(GEP);

    // Restore the original insert point.
    if (SaveInsertBB)
      restoreInsertPoint(SaveInsertBB, SaveInsertPt);

    return GEP;
  }

  // Save the original insertion point so we can restore it when we're done.
  BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
  BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();

  // Move the insertion point out of as many loops as we can.
  while (const Loop *L = SE.LI->getLoopFor(Builder.GetInsertBlock())) {
    if (!L->isLoopInvariant(V)) break;

    bool AnyIndexNotLoopInvariant = false;
    for (SmallVectorImpl<Value *>::const_iterator I = GepIndices.begin(),
         E = GepIndices.end(); I != E; ++I)
      if (!L->isLoopInvariant(*I)) {
        AnyIndexNotLoopInvariant = true;
        break;
      }
    if (AnyIndexNotLoopInvariant)
      break;

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

    // Ok, move up a level.
    Builder.SetInsertPoint(Preheader, Preheader->getTerminator());
  }

  // Insert a pretty getelementptr. Note that this GEP is not marked inbounds,
  // because ScalarEvolution may have changed the address arithmetic to
  // compute a value which is beyond the end of the allocated object.
  Value *Casted = V;
  if (V->getType() != PTy)
    Casted = InsertNoopCastOfTo(Casted, PTy);
  Value *GEP = Builder.CreateGEP(Casted,
                                 GepIndices.begin(),
                                 GepIndices.end(),
                                 "scevgep");
  Ops.push_back(SE.getUnknown(GEP));
  rememberInstruction(GEP);

  // Restore the original insert point.
  if (SaveInsertBB)
    restoreInsertPoint(SaveInsertBB, SaveInsertPt);

  return expand(SE.getAddExpr(Ops));
}

/// isNonConstantNegative - Return true if the specified scev is negated, but
/// not a constant.
static bool isNonConstantNegative(const SCEV *F) {
  const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(F);
  if (!Mul) return false;

  // If there is a constant factor, it will be first.
  const SCEVConstant *SC = dyn_cast<SCEVConstant>(Mul->getOperand(0));
  if (!SC) return false;

  // Return true if the value is negative, this matches things like (-42 * V).
  return SC->getValue()->getValue().isNegative();
}

/// PickMostRelevantLoop - Given two loops pick the one that's most relevant for
/// SCEV expansion. If they are nested, this is the most nested. If they are
/// neighboring, pick the later.
static const Loop *PickMostRelevantLoop(const Loop *A, const Loop *B,
                                        DominatorTree &DT) {
  if (!A) return B;
  if (!B) return A;
  if (A->contains(B)) return B;
  if (B->contains(A)) return A;
  if (DT.dominates(A->getHeader(), B->getHeader())) return B;
  if (DT.dominates(B->getHeader(), A->getHeader())) return A;
  return A; // Arbitrarily break the tie.
}

/// GetRelevantLoop - Get the most relevant loop associated with the given
/// expression, according to PickMostRelevantLoop.
static const Loop *GetRelevantLoop(const SCEV *S, LoopInfo &LI,
                                   DominatorTree &DT) {
  if (isa<SCEVConstant>(S))
    return 0;
  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    if (const Instruction *I = dyn_cast<Instruction>(U->getValue()))
      return LI.getLoopFor(I->getParent());
    return 0;
  }
  if (const SCEVNAryExpr *N = dyn_cast<SCEVNAryExpr>(S)) {
    const Loop *L = 0;
    if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(S))
      L = AR->getLoop();
    for (SCEVNAryExpr::op_iterator I = N->op_begin(), E = N->op_end();
         I != E; ++I)
      L = PickMostRelevantLoop(L, GetRelevantLoop(*I, LI, DT), DT);
    return L;
  }
  if (const SCEVCastExpr *C = dyn_cast<SCEVCastExpr>(S))
    return GetRelevantLoop(C->getOperand(), LI, DT);
  if (const SCEVUDivExpr *D = dyn_cast<SCEVUDivExpr>(S))
    return PickMostRelevantLoop(GetRelevantLoop(D->getLHS(), LI, DT),
                                GetRelevantLoop(D->getRHS(), LI, DT),
                                DT);
  llvm_unreachable("Unexpected SCEV type!");
}

namespace {

/// LoopCompare - Compare loops by PickMostRelevantLoop.
class LoopCompare {
  DominatorTree &DT;
public:
  explicit LoopCompare(DominatorTree &dt) : DT(dt) {}

  bool operator()(std::pair<const Loop *, const SCEV *> LHS,
                  std::pair<const Loop *, const SCEV *> RHS) const {
    // Compare loops with PickMostRelevantLoop.
    if (LHS.first != RHS.first)
      return PickMostRelevantLoop(LHS.first, RHS.first, DT) != LHS.first;

    // If one operand is a non-constant negative and the other is not,
    // put the non-constant negative on the right so that a sub can
    // be used instead of a negate and add.
    if (isNonConstantNegative(LHS.second)) {
      if (!isNonConstantNegative(RHS.second))
        return false;
    } else if (isNonConstantNegative(RHS.second))
      return true;

    // Otherwise they are equivalent according to this comparison.
    return false;
  }
};

}

Value *SCEVExpander::visitAddExpr(const SCEVAddExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());

  // Collect all the add operands in a loop, along with their associated loops.
  // Iterate in reverse so that constants are emitted last, all else equal, and
  // so that pointer operands are inserted first, which the code below relies on
  // to form more involved GEPs.
  SmallVector<std::pair<const Loop *, const SCEV *>, 8> OpsAndLoops;
  for (std::reverse_iterator<SCEVAddExpr::op_iterator> I(S->op_end()),
       E(S->op_begin()); I != E; ++I)
    OpsAndLoops.push_back(std::make_pair(GetRelevantLoop(*I, *SE.LI, *SE.DT),
                                         *I));

  // Sort by loop. Use a stable sort so that constants follow non-constants and
  // pointer operands precede non-pointer operands.
  std::stable_sort(OpsAndLoops.begin(), OpsAndLoops.end(), LoopCompare(*SE.DT));

  // Emit instructions to add all the operands. Hoist as much as possible
  // out of loops, and form meaningful getelementptrs where possible.
  Value *Sum = 0;
  for (SmallVectorImpl<std::pair<const Loop *, const SCEV *> >::iterator
       I = OpsAndLoops.begin(), E = OpsAndLoops.end(); I != E; ) {
    const Loop *CurLoop = I->first;
    const SCEV *Op = I->second;
    if (!Sum) {
      // This is the first operand. Just expand it.
      Sum = expand(Op);
      ++I;
    } else if (const PointerType *PTy = dyn_cast<PointerType>(Sum->getType())) {
      // The running sum expression is a pointer. Try to form a getelementptr
      // at this level with that as the base.
      SmallVector<const SCEV *, 4> NewOps;
      for (; I != E && I->first == CurLoop; ++I)
        NewOps.push_back(I->second);
      Sum = expandAddToGEP(NewOps.begin(), NewOps.end(), PTy, Ty, Sum);
    } else if (const PointerType *PTy = dyn_cast<PointerType>(Op->getType())) {
      // The running sum is an integer, and there's a pointer at this level.
      // Try to form a getelementptr. If the running sum is instructions,
      // use a SCEVUnknown to avoid re-analyzing them.
      SmallVector<const SCEV *, 4> NewOps;
      NewOps.push_back(isa<Instruction>(Sum) ? SE.getUnknown(Sum) :
                                               SE.getSCEV(Sum));
      for (++I; I != E && I->first == CurLoop; ++I)
        NewOps.push_back(I->second);
      Sum = expandAddToGEP(NewOps.begin(), NewOps.end(), PTy, Ty, expand(Op));
    } else if (isNonConstantNegative(Op)) {
      // Instead of doing a negate and add, just do a subtract.
      Value *W = expandCodeFor(SE.getNegativeSCEV(Op), Ty);
      Sum = InsertNoopCastOfTo(Sum, Ty);
      Sum = InsertBinop(Instruction::Sub, Sum, W);
      ++I;
    } else {
      // A simple add.
      Value *W = expandCodeFor(Op, Ty);
      Sum = InsertNoopCastOfTo(Sum, Ty);
      // Canonicalize a constant to the RHS.
      if (isa<Constant>(Sum)) std::swap(Sum, W);
      Sum = InsertBinop(Instruction::Add, Sum, W);
      ++I;
    }
  }

  return Sum;
}

Value *SCEVExpander::visitMulExpr(const SCEVMulExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());

  // Collect all the mul operands in a loop, along with their associated loops.
  // Iterate in reverse so that constants are emitted last, all else equal.
  SmallVector<std::pair<const Loop *, const SCEV *>, 8> OpsAndLoops;
  for (std::reverse_iterator<SCEVMulExpr::op_iterator> I(S->op_end()),
       E(S->op_begin()); I != E; ++I)
    OpsAndLoops.push_back(std::make_pair(GetRelevantLoop(*I, *SE.LI, *SE.DT),
                                         *I));

  // Sort by loop. Use a stable sort so that constants follow non-constants.
  std::stable_sort(OpsAndLoops.begin(), OpsAndLoops.end(), LoopCompare(*SE.DT));

  // Emit instructions to mul all the operands. Hoist as much as possible
  // out of loops.
  Value *Prod = 0;
  for (SmallVectorImpl<std::pair<const Loop *, const SCEV *> >::iterator
       I = OpsAndLoops.begin(), E = OpsAndLoops.end(); I != E; ) {
    const SCEV *Op = I->second;
    if (!Prod) {
      // This is the first operand. Just expand it.
      Prod = expand(Op);
      ++I;
    } else if (Op->isAllOnesValue()) {
      // Instead of doing a multiply by negative one, just do a negate.
      Prod = InsertNoopCastOfTo(Prod, Ty);
      Prod = InsertBinop(Instruction::Sub, Constant::getNullValue(Ty), Prod);
      ++I;
    } else {
      // A simple mul.
      Value *W = expandCodeFor(Op, Ty);
      Prod = InsertNoopCastOfTo(Prod, Ty);
      // Canonicalize a constant to the RHS.
      if (isa<Constant>(Prod)) std::swap(Prod, W);
      Prod = InsertBinop(Instruction::Mul, Prod, W);
      ++I;
    }
  }

  return Prod;
}

Value *SCEVExpander::visitUDivExpr(const SCEVUDivExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());

  Value *LHS = expandCodeFor(S->getLHS(), Ty);
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(S->getRHS())) {
    const APInt &RHS = SC->getValue()->getValue();
    if (RHS.isPowerOf2())
      return InsertBinop(Instruction::LShr, LHS,
                         ConstantInt::get(Ty, RHS.logBase2()));
  }

  Value *RHS = expandCodeFor(S->getRHS(), Ty);
  return InsertBinop(Instruction::UDiv, LHS, RHS);
}

/// Move parts of Base into Rest to leave Base with the minimal
/// expression that provides a pointer operand suitable for a
/// GEP expansion.
static void ExposePointerBase(const SCEV *&Base, const SCEV *&Rest,
                              ScalarEvolution &SE) {
  while (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(Base)) {
    Base = A->getStart();
    Rest = SE.getAddExpr(Rest,
                         SE.getAddRecExpr(SE.getConstant(A->getType(), 0),
                                          A->getStepRecurrence(SE),
                                          A->getLoop()));
  }
  if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(Base)) {
    Base = A->getOperand(A->getNumOperands()-1);
    SmallVector<const SCEV *, 8> NewAddOps(A->op_begin(), A->op_end());
    NewAddOps.back() = Rest;
    Rest = SE.getAddExpr(NewAddOps);
    ExposePointerBase(Base, Rest, SE);
  }
}

/// getAddRecExprPHILiterally - Helper for expandAddRecExprLiterally. Expand
/// the base addrec, which is the addrec without any non-loop-dominating
/// values, and return the PHI.
PHINode *
SCEVExpander::getAddRecExprPHILiterally(const SCEVAddRecExpr *Normalized,
                                        const Loop *L,
                                        const Type *ExpandTy,
                                        const Type *IntTy) {
  // Reuse a previously-inserted PHI, if present.
  for (BasicBlock::iterator I = L->getHeader()->begin();
       PHINode *PN = dyn_cast<PHINode>(I); ++I)
    if (SE.isSCEVable(PN->getType()) &&
        (SE.getEffectiveSCEVType(PN->getType()) ==
         SE.getEffectiveSCEVType(Normalized->getType())) &&
        SE.getSCEV(PN) == Normalized)
      if (BasicBlock *LatchBlock = L->getLoopLatch()) {
        Instruction *IncV =
          cast<Instruction>(PN->getIncomingValueForBlock(LatchBlock));

        // Determine if this is a well-behaved chain of instructions leading
        // back to the PHI. It probably will be, if we're scanning an inner
        // loop already visited by LSR for example, but it wouldn't have
        // to be.
        do {
          if (IncV->getNumOperands() == 0 || isa<PHINode>(IncV)) {
            IncV = 0;
            break;
          }
          // If any of the operands don't dominate the insert position, bail.
          // Addrec operands are always loop-invariant, so this can only happen
          // if there are instructions which haven't been hoisted.
          for (User::op_iterator OI = IncV->op_begin()+1,
               OE = IncV->op_end(); OI != OE; ++OI)
            if (Instruction *OInst = dyn_cast<Instruction>(OI))
              if (!SE.DT->dominates(OInst, IVIncInsertPos)) {
                IncV = 0;
                break;
              }
          if (!IncV)
            break;
          // Advance to the next instruction.
          IncV = dyn_cast<Instruction>(IncV->getOperand(0));
          if (!IncV)
            break;
          if (IncV->mayHaveSideEffects()) {
            IncV = 0;
            break;
          }
        } while (IncV != PN);

        if (IncV) {
          // Ok, the add recurrence looks usable.
          // Remember this PHI, even in post-inc mode.
          InsertedValues.insert(PN);
          // Remember the increment.
          IncV = cast<Instruction>(PN->getIncomingValueForBlock(LatchBlock));
          rememberInstruction(IncV);
          if (L == IVIncInsertLoop)
            do {
              if (SE.DT->dominates(IncV, IVIncInsertPos))
                break;
              // Make sure the increment is where we want it. But don't move it
              // down past a potential existing post-inc user.
              IncV->moveBefore(IVIncInsertPos);
              IVIncInsertPos = IncV;
              IncV = cast<Instruction>(IncV->getOperand(0));
            } while (IncV != PN);
          return PN;
        }
      }

  // Save the original insertion point so we can restore it when we're done.
  BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
  BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();

  // Expand code for the start value.
  Value *StartV = expandCodeFor(Normalized->getStart(), ExpandTy,
                                L->getHeader()->begin());

  // Expand code for the step value. Insert instructions right before the
  // terminator corresponding to the back-edge. Do this before creating the PHI
  // so that PHI reuse code doesn't see an incomplete PHI. If the stride is
  // negative, insert a sub instead of an add for the increment (unless it's a
  // constant, because subtracts of constants are canonicalized to adds).
  const SCEV *Step = Normalized->getStepRecurrence(SE);
  bool isPointer = ExpandTy->isPointerTy();
  bool isNegative = !isPointer && isNonConstantNegative(Step);
  if (isNegative)
    Step = SE.getNegativeSCEV(Step);
  Value *StepV = expandCodeFor(Step, IntTy, L->getHeader()->begin());

  // Create the PHI.
  Builder.SetInsertPoint(L->getHeader(), L->getHeader()->begin());
  PHINode *PN = Builder.CreatePHI(ExpandTy, "lsr.iv");
  rememberInstruction(PN);

  // Create the step instructions and populate the PHI.
  BasicBlock *Header = L->getHeader();
  for (pred_iterator HPI = pred_begin(Header), HPE = pred_end(Header);
       HPI != HPE; ++HPI) {
    BasicBlock *Pred = *HPI;

    // Add a start value.
    if (!L->contains(Pred)) {
      PN->addIncoming(StartV, Pred);
      continue;
    }

    // Create a step value and add it to the PHI. If IVIncInsertLoop is
    // non-null and equal to the addrec's loop, insert the instructions
    // at IVIncInsertPos.
    Instruction *InsertPos = L == IVIncInsertLoop ?
      IVIncInsertPos : Pred->getTerminator();
    Builder.SetInsertPoint(InsertPos->getParent(), InsertPos);
    Value *IncV;
    // If the PHI is a pointer, use a GEP, otherwise use an add or sub.
    if (isPointer) {
      const PointerType *GEPPtrTy = cast<PointerType>(ExpandTy);
      // If the step isn't constant, don't use an implicitly scaled GEP, because
      // that would require a multiply inside the loop.
      if (!isa<ConstantInt>(StepV))
        GEPPtrTy = PointerType::get(Type::getInt1Ty(SE.getContext()),
                                    GEPPtrTy->getAddressSpace());
      const SCEV *const StepArray[1] = { SE.getSCEV(StepV) };
      IncV = expandAddToGEP(StepArray, StepArray+1, GEPPtrTy, IntTy, PN);
      if (IncV->getType() != PN->getType()) {
        IncV = Builder.CreateBitCast(IncV, PN->getType(), "tmp");
        rememberInstruction(IncV);
      }
    } else {
      IncV = isNegative ?
        Builder.CreateSub(PN, StepV, "lsr.iv.next") :
        Builder.CreateAdd(PN, StepV, "lsr.iv.next");
      rememberInstruction(IncV);
    }
    PN->addIncoming(IncV, Pred);
  }

  // Restore the original insert point.
  if (SaveInsertBB)
    restoreInsertPoint(SaveInsertBB, SaveInsertPt);

  // Remember this PHI, even in post-inc mode.
  InsertedValues.insert(PN);

  return PN;
}

Value *SCEVExpander::expandAddRecExprLiterally(const SCEVAddRecExpr *S) {
  const Type *STy = S->getType();
  const Type *IntTy = SE.getEffectiveSCEVType(STy);
  const Loop *L = S->getLoop();

  // Determine a normalized form of this expression, which is the expression
  // before any post-inc adjustment is made.
  const SCEVAddRecExpr *Normalized = S;
  if (PostIncLoops.count(L)) {
    PostIncLoopSet Loops;
    Loops.insert(L);
    Normalized =
      cast<SCEVAddRecExpr>(TransformForPostIncUse(Normalize, S, 0, 0,
                                                  Loops, SE, *SE.DT));
  }

  // Strip off any non-loop-dominating component from the addrec start.
  const SCEV *Start = Normalized->getStart();
  const SCEV *PostLoopOffset = 0;
  if (!Start->properlyDominates(L->getHeader(), SE.DT)) {
    PostLoopOffset = Start;
    Start = SE.getConstant(Normalized->getType(), 0);
    Normalized =
      cast<SCEVAddRecExpr>(SE.getAddRecExpr(Start,
                                            Normalized->getStepRecurrence(SE),
                                            Normalized->getLoop()));
  }

  // Strip off any non-loop-dominating component from the addrec step.
  const SCEV *Step = Normalized->getStepRecurrence(SE);
  const SCEV *PostLoopScale = 0;
  if (!Step->dominates(L->getHeader(), SE.DT)) {
    PostLoopScale = Step;
    Step = SE.getConstant(Normalized->getType(), 1);
    Normalized =
      cast<SCEVAddRecExpr>(SE.getAddRecExpr(Start, Step,
                                            Normalized->getLoop()));
  }

  // Expand the core addrec. If we need post-loop scaling, force it to
  // expand to an integer type to avoid the need for additional casting.
  const Type *ExpandTy = PostLoopScale ? IntTy : STy;
  PHINode *PN = getAddRecExprPHILiterally(Normalized, L, ExpandTy, IntTy);

  // Accommodate post-inc mode, if necessary.
  Value *Result;
  if (!PostIncLoops.count(L))
    Result = PN;
  else {
    // In PostInc mode, use the post-incremented value.
    BasicBlock *LatchBlock = L->getLoopLatch();
    assert(LatchBlock && "PostInc mode requires a unique loop latch!");
    Result = PN->getIncomingValueForBlock(LatchBlock);
  }

  // Re-apply any non-loop-dominating scale.
  if (PostLoopScale) {
    Result = InsertNoopCastOfTo(Result, IntTy);
    Result = Builder.CreateMul(Result,
                               expandCodeFor(PostLoopScale, IntTy));
    rememberInstruction(Result);
  }

  // Re-apply any non-loop-dominating offset.
  if (PostLoopOffset) {
    if (const PointerType *PTy = dyn_cast<PointerType>(ExpandTy)) {
      const SCEV *const OffsetArray[1] = { PostLoopOffset };
      Result = expandAddToGEP(OffsetArray, OffsetArray+1, PTy, IntTy, Result);
    } else {
      Result = InsertNoopCastOfTo(Result, IntTy);
      Result = Builder.CreateAdd(Result,
                                 expandCodeFor(PostLoopOffset, IntTy));
      rememberInstruction(Result);
    }
  }

  return Result;
}

Value *SCEVExpander::visitAddRecExpr(const SCEVAddRecExpr *S) {
  if (!CanonicalMode) return expandAddRecExprLiterally(S);

  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  const Loop *L = S->getLoop();

  // First check for an existing canonical IV in a suitable type.
  PHINode *CanonicalIV = 0;
  if (PHINode *PN = L->getCanonicalInductionVariable())
    if (SE.isSCEVable(PN->getType()) &&
        SE.getEffectiveSCEVType(PN->getType())->isIntegerTy() &&
        SE.getTypeSizeInBits(PN->getType()) >= SE.getTypeSizeInBits(Ty))
      CanonicalIV = PN;

  // Rewrite an AddRec in terms of the canonical induction variable, if
  // its type is more narrow.
  if (CanonicalIV &&
      SE.getTypeSizeInBits(CanonicalIV->getType()) >
      SE.getTypeSizeInBits(Ty)) {
    SmallVector<const SCEV *, 4> NewOps(S->getNumOperands());
    for (unsigned i = 0, e = S->getNumOperands(); i != e; ++i)
      NewOps[i] = SE.getAnyExtendExpr(S->op_begin()[i], CanonicalIV->getType());
    Value *V = expand(SE.getAddRecExpr(NewOps, S->getLoop()));
    BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
    BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();
    BasicBlock::iterator NewInsertPt =
      llvm::next(BasicBlock::iterator(cast<Instruction>(V)));
    while (isa<PHINode>(NewInsertPt)) ++NewInsertPt;
    V = expandCodeFor(SE.getTruncateExpr(SE.getUnknown(V), Ty), 0,
                      NewInsertPt);
    restoreInsertPoint(SaveInsertBB, SaveInsertPt);
    return V;
  }

  // {X,+,F} --> X + {0,+,F}
  if (!S->getStart()->isZero()) {
    SmallVector<const SCEV *, 4> NewOps(S->op_begin(), S->op_end());
    NewOps[0] = SE.getConstant(Ty, 0);
    const SCEV *Rest = SE.getAddRecExpr(NewOps, L);

    // Turn things like ptrtoint+arithmetic+inttoptr into GEP. See the
    // comments on expandAddToGEP for details.
    const SCEV *Base = S->getStart();
    const SCEV *RestArray[1] = { Rest };
    // Dig into the expression to find the pointer base for a GEP.
    ExposePointerBase(Base, RestArray[0], SE);
    // If we found a pointer, expand the AddRec with a GEP.
    if (const PointerType *PTy = dyn_cast<PointerType>(Base->getType())) {
      // Make sure the Base isn't something exotic, such as a multiplied
      // or divided pointer value. In those cases, the result type isn't
      // actually a pointer type.
      if (!isa<SCEVMulExpr>(Base) && !isa<SCEVUDivExpr>(Base)) {
        Value *StartV = expand(Base);
        assert(StartV->getType() == PTy && "Pointer type mismatch for GEP!");
        return expandAddToGEP(RestArray, RestArray+1, PTy, Ty, StartV);
      }
    }

    // Just do a normal add. Pre-expand the operands to suppress folding.
    return expand(SE.getAddExpr(SE.getUnknown(expand(S->getStart())),
                                SE.getUnknown(expand(Rest))));
  }

  // {0,+,1} --> Insert a canonical induction variable into the loop!
  if (S->isAffine() &&
      S->getOperand(1) == SE.getConstant(Ty, 1)) {
    // If there's a canonical IV, just use it.
    if (CanonicalIV) {
      assert(Ty == SE.getEffectiveSCEVType(CanonicalIV->getType()) &&
             "IVs with types different from the canonical IV should "
             "already have been handled!");
      return CanonicalIV;
    }

    // Create and insert the PHI node for the induction variable in the
    // specified loop.
    BasicBlock *Header = L->getHeader();
    PHINode *PN = PHINode::Create(Ty, "indvar", Header->begin());
    rememberInstruction(PN);

    Constant *One = ConstantInt::get(Ty, 1);
    for (pred_iterator HPI = pred_begin(Header), HPE = pred_end(Header);
         HPI != HPE; ++HPI)
      if (L->contains(*HPI)) {
        // Insert a unit add instruction right before the terminator
        // corresponding to the back-edge.
        Instruction *Add = BinaryOperator::CreateAdd(PN, One, "indvar.next",
                                                     (*HPI)->getTerminator());
        rememberInstruction(Add);
        PN->addIncoming(Add, *HPI);
      } else {
        PN->addIncoming(Constant::getNullValue(Ty), *HPI);
      }
  }

  // {0,+,F} --> {0,+,1} * F
  // Get the canonical induction variable I for this loop.
  Value *I = CanonicalIV ?
             CanonicalIV :
             getOrInsertCanonicalInductionVariable(L, Ty);

  // If this is a simple linear addrec, emit it now as a special case.
  if (S->isAffine())    // {0,+,F} --> i*F
    return
      expand(SE.getTruncateOrNoop(
        SE.getMulExpr(SE.getUnknown(I),
                      SE.getNoopOrAnyExtend(S->getOperand(1),
                                            I->getType())),
        Ty));

  // If this is a chain of recurrences, turn it into a closed form, using the
  // folders, then expandCodeFor the closed form.  This allows the folders to
  // simplify the expression without having to build a bunch of special code
  // into this folder.
  const SCEV *IH = SE.getUnknown(I);   // Get I as a "symbolic" SCEV.

  // Promote S up to the canonical IV type, if the cast is foldable.
  const SCEV *NewS = S;
  const SCEV *Ext = SE.getNoopOrAnyExtend(S, I->getType());
  if (isa<SCEVAddRecExpr>(Ext))
    NewS = Ext;

  const SCEV *V = cast<SCEVAddRecExpr>(NewS)->evaluateAtIteration(IH, SE);
  //cerr << "Evaluated: " << *this << "\n     to: " << *V << "\n";

  // Truncate the result down to the original type, if needed.
  const SCEV *T = SE.getTruncateOrNoop(V, Ty);
  return expand(T);
}

Value *SCEVExpander::visitTruncateExpr(const SCEVTruncateExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *V = expandCodeFor(S->getOperand(),
                           SE.getEffectiveSCEVType(S->getOperand()->getType()));
  Value *I = Builder.CreateTrunc(V, Ty, "tmp");
  rememberInstruction(I);
  return I;
}

Value *SCEVExpander::visitZeroExtendExpr(const SCEVZeroExtendExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *V = expandCodeFor(S->getOperand(),
                           SE.getEffectiveSCEVType(S->getOperand()->getType()));
  Value *I = Builder.CreateZExt(V, Ty, "tmp");
  rememberInstruction(I);
  return I;
}

Value *SCEVExpander::visitSignExtendExpr(const SCEVSignExtendExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *V = expandCodeFor(S->getOperand(),
                           SE.getEffectiveSCEVType(S->getOperand()->getType()));
  Value *I = Builder.CreateSExt(V, Ty, "tmp");
  rememberInstruction(I);
  return I;
}

Value *SCEVExpander::visitSMaxExpr(const SCEVSMaxExpr *S) {
  Value *LHS = expand(S->getOperand(S->getNumOperands()-1));
  const Type *Ty = LHS->getType();
  for (int i = S->getNumOperands()-2; i >= 0; --i) {
    // In the case of mixed integer and pointer types, do the
    // rest of the comparisons as integer.
    if (S->getOperand(i)->getType() != Ty) {
      Ty = SE.getEffectiveSCEVType(Ty);
      LHS = InsertNoopCastOfTo(LHS, Ty);
    }
    Value *RHS = expandCodeFor(S->getOperand(i), Ty);
    Value *ICmp = Builder.CreateICmpSGT(LHS, RHS, "tmp");
    rememberInstruction(ICmp);
    Value *Sel = Builder.CreateSelect(ICmp, LHS, RHS, "smax");
    rememberInstruction(Sel);
    LHS = Sel;
  }
  // In the case of mixed integer and pointer types, cast the
  // final result back to the pointer type.
  if (LHS->getType() != S->getType())
    LHS = InsertNoopCastOfTo(LHS, S->getType());
  return LHS;
}

Value *SCEVExpander::visitUMaxExpr(const SCEVUMaxExpr *S) {
  Value *LHS = expand(S->getOperand(S->getNumOperands()-1));
  const Type *Ty = LHS->getType();
  for (int i = S->getNumOperands()-2; i >= 0; --i) {
    // In the case of mixed integer and pointer types, do the
    // rest of the comparisons as integer.
    if (S->getOperand(i)->getType() != Ty) {
      Ty = SE.getEffectiveSCEVType(Ty);
      LHS = InsertNoopCastOfTo(LHS, Ty);
    }
    Value *RHS = expandCodeFor(S->getOperand(i), Ty);
    Value *ICmp = Builder.CreateICmpUGT(LHS, RHS, "tmp");
    rememberInstruction(ICmp);
    Value *Sel = Builder.CreateSelect(ICmp, LHS, RHS, "umax");
    rememberInstruction(Sel);
    LHS = Sel;
  }
  // In the case of mixed integer and pointer types, cast the
  // final result back to the pointer type.
  if (LHS->getType() != S->getType())
    LHS = InsertNoopCastOfTo(LHS, S->getType());
  return LHS;
}

Value *SCEVExpander::expandCodeFor(const SCEV *SH, const Type *Ty,
                                   Instruction *I) {
  BasicBlock::iterator IP = I;
  while (isInsertedInstruction(IP) || isa<DbgInfoIntrinsic>(IP))
    ++IP;
  Builder.SetInsertPoint(IP->getParent(), IP);
  return expandCodeFor(SH, Ty);
}

Value *SCEVExpander::expandCodeFor(const SCEV *SH, const Type *Ty) {
  // Expand the code for this SCEV.
  Value *V = expand(SH);
  if (Ty) {
    assert(SE.getTypeSizeInBits(Ty) == SE.getTypeSizeInBits(SH->getType()) &&
           "non-trivial casts should be done with the SCEVs directly!");
    V = InsertNoopCastOfTo(V, Ty);
  }
  return V;
}

Value *SCEVExpander::expand(const SCEV *S) {
  // Compute an insertion point for this SCEV object. Hoist the instructions
  // as far out in the loop nest as possible.
  Instruction *InsertPt = Builder.GetInsertPoint();
  for (Loop *L = SE.LI->getLoopFor(Builder.GetInsertBlock()); ;
       L = L->getParentLoop())
    if (S->isLoopInvariant(L)) {
      if (!L) break;
      if (BasicBlock *Preheader = L->getLoopPreheader())
        InsertPt = Preheader->getTerminator();
    } else {
      // If the SCEV is computable at this level, insert it into the header
      // after the PHIs (and after any other instructions that we've inserted
      // there) so that it is guaranteed to dominate any user inside the loop.
      if (L && S->hasComputableLoopEvolution(L) && !PostIncLoops.count(L))
        InsertPt = L->getHeader()->getFirstNonPHI();
      while (isInsertedInstruction(InsertPt) || isa<DbgInfoIntrinsic>(InsertPt))
        InsertPt = llvm::next(BasicBlock::iterator(InsertPt));
      break;
    }

  // Check to see if we already expanded this here.
  std::map<std::pair<const SCEV *, Instruction *>,
           AssertingVH<Value> >::iterator I =
    InsertedExpressions.find(std::make_pair(S, InsertPt));
  if (I != InsertedExpressions.end())
    return I->second;

  BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
  BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();
  Builder.SetInsertPoint(InsertPt->getParent(), InsertPt);

  // Expand the expression into instructions.
  Value *V = visit(S);

  // Remember the expanded value for this SCEV at this location.
  if (PostIncLoops.empty())
    InsertedExpressions[std::make_pair(S, InsertPt)] = V;

  restoreInsertPoint(SaveInsertBB, SaveInsertPt);
  return V;
}

void SCEVExpander::rememberInstruction(Value *I) {
  if (PostIncLoops.empty())
    InsertedValues.insert(I);

  // If we just claimed an existing instruction and that instruction had
  // been the insert point, adjust the insert point forward so that
  // subsequently inserted code will be dominated.
  if (Builder.GetInsertPoint() == I) {
    BasicBlock::iterator It = cast<Instruction>(I);
    do { ++It; } while (isInsertedInstruction(It) ||
                        isa<DbgInfoIntrinsic>(It));
    Builder.SetInsertPoint(Builder.GetInsertBlock(), It);
  }
}

void SCEVExpander::restoreInsertPoint(BasicBlock *BB, BasicBlock::iterator I) {
  // If we acquired more instructions since the old insert point was saved,
  // advance past them.
  while (isInsertedInstruction(I) || isa<DbgInfoIntrinsic>(I)) ++I;

  Builder.SetInsertPoint(BB, I);
}

/// getOrInsertCanonicalInductionVariable - This method returns the
/// canonical induction variable of the specified type for the specified
/// loop (inserting one if there is none).  A canonical induction variable
/// starts at zero and steps by one on each iteration.
Value *
SCEVExpander::getOrInsertCanonicalInductionVariable(const Loop *L,
                                                    const Type *Ty) {
  assert(Ty->isIntegerTy() && "Can only insert integer induction variables!");
  const SCEV *H = SE.getAddRecExpr(SE.getConstant(Ty, 0),
                                   SE.getConstant(Ty, 1), L);
  BasicBlock *SaveInsertBB = Builder.GetInsertBlock();
  BasicBlock::iterator SaveInsertPt = Builder.GetInsertPoint();
  Value *V = expandCodeFor(H, 0, L->getHeader()->begin());
  if (SaveInsertBB)
    restoreInsertPoint(SaveInsertBB, SaveInsertPt);
  return V;
}