//===- 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/Target/TargetData.h"
#include "llvm/ADT/STLExtras.h"
using namespace llvm;

/// InsertCastOfTo - Insert a cast of V to the specified type, doing what
/// we can to share the casts.
Value *SCEVExpander::InsertCastOfTo(Instruction::CastOps opcode, Value *V, 
                                    const Type *Ty) {
  // Short-circuit unnecessary bitcasts.
  if (opcode == Instruction::BitCast && V->getType() == Ty)
    return V;

  // Short-circuit unnecessary inttoptr<->ptrtoint casts.
  if ((opcode == Instruction::PtrToInt || opcode == 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);
  }

  // FIXME: keep track of the cast instruction.
  if (Constant *C = dyn_cast<Constant>(V))
    return ConstantExpr::getCast(opcode, 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() == opcode) {
            // 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(opcode, V, Ty, "",
                                 A->getParent()->getEntryBlock().begin());
              NewCI->takeName(CI);
              CI->replaceAllUsesWith(NewCI);
              return NewCI;
            }
            return CI;
          }

    Instruction *I = CastInst::Create(opcode, V, Ty, V->getName(),
                                      A->getParent()->getEntryBlock().begin());
    InsertedValues.insert(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() == opcode) {
          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 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(opcode, V, Ty, "", It);
            NewCI->takeName(CI);
            CI->replaceAllUsesWith(NewCI);
            return NewCI;
          }
          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(opcode, V, Ty, V->getName(), IP);
  InsertedValues.insert(CI);
  return CI;
}

/// InsertNoopCastOfTo - Insert a cast of V to the specified type,
/// which must be possible with a noop cast.
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!");
  return InsertCastOfTo(Op, V, Ty);
}

/// 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, BasicBlock::iterator InsertPt) {
  // 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 = InsertPt->getParent()->begin();
  if (InsertPt != BlockBegin) {
    // Scanning starts from the last instruction before InsertPt.
    BasicBlock::iterator IP = InsertPt;
    --IP;
    for (; ScanLimit; --IP, --ScanLimit) {
      if (IP->getOpcode() == (unsigned)Opcode && IP->getOperand(0) == LHS &&
          IP->getOperand(1) == RHS)
        return IP;
      if (IP == BlockBegin) break;
    }
  }
  
  // If we haven't found this binop, insert it.
  Instruction *BO = BinaryOperator::Create(Opcode, LHS, RHS, "tmp", InsertPt);
  InsertedValues.insert(BO);
  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 divisble 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 APInt &Factor,
                              ScalarEvolution &SE) {
  // Everything is divisible by one.
  if (Factor == 1)
    return true;

  // For a Constant, check for a multiple of the given factor.
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S)) {
    ConstantInt *CI =
      ConstantInt::get(C->getValue()->getValue().sdiv(Factor));
    // 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 (C->isZero() || !CI->isZero()) {
      const SCEV* Div = SE.getConstant(CI);
      S = Div;
      Remainder =
        SE.getAddExpr(Remainder,
                      SE.getConstant(C->getValue()->getValue().srem(Factor)));
      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 (const SCEVConstant *C = dyn_cast<SCEVConstant>(M->getOperand(0)))
      if (!C->getValue()->getValue().srem(Factor)) {
        const SmallVectorImpl<const SCEV*> &MOperands = M->getOperands();
        SmallVector<const SCEV*, 4> NewMulOps(MOperands.begin(), MOperands.end());
        NewMulOps[0] =
          SE.getConstant(C->getValue()->getValue().sdiv(Factor));
        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.getIntegerSCEV(0, Step->getType());
    if (!FactorOutConstant(Step, StepRem, Factor, SE))
      return false;
    if (!StepRem->isZero())
      return false;
    const SCEV* Start = A->getStart();
    if (!FactorOutConstant(Start, Remainder, Factor, SE))
      return false;
    S = SE.getAddRecExpr(Start, Step, A->getLoop());
    return true;
  }

  return false;
}

/// expandAddToGEP - Expand a SCEVAddExpr with a pointer type into a GEP
/// instead of using ptrtoint+arithmetic+inttoptr. This helps
/// BasicAliasAnalysis analyze the result. However, it suffers from the
/// underlying bug described in PR2831. Addition in LLVM currently always
/// has two's complement wrapping guaranteed. However, the semantics for
/// getelementptr overflow are ambiguous. In the common case though, this
/// expansion gets used when a GEP in the original code has been converted
/// into integer arithmetic, in which case the resulting code will be no
/// more undefined than it was originally.
///
/// 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;

  // Decend 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 (;;) {
    APInt ElSize = APInt(SE.getTypeSizeInBits(Ty),
                         ElTy->isSized() ?  SE.TD->getTypeAllocSize(ElTy) : 0);
    SmallVector<const SCEV*, 8> NewOps;
    SmallVector<const SCEV*, 8> ScaledOps;
    for (unsigned i = 0, e = Ops.size(); i != e; ++i) {
      // Split AddRecs up into parts as either of the parts may be usable
      // without the other.
      if (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(Ops[i]))
        if (!A->getStart()->isZero()) {
          const SCEV* Start = A->getStart();
          Ops.push_back(SE.getAddRecExpr(SE.getIntegerSCEV(0, A->getType()),
                                         A->getStepRecurrence(SE),
                                         A->getLoop()));
          Ops[i] = Start;
          ++e;
        }
      // If the scale size is not 0, attempt to factor out a scale.
      if (ElSize != 0) {
        const SCEV* Op = Ops[i];
        const SCEV* Remainder = SE.getIntegerSCEV(0, Op->getType());
        if (FactorOutConstant(Op, Remainder, ElSize, SE)) {
          ScaledOps.push_back(Op); // Op now has ElSize factored out.
          NewOps.push_back(Remainder);
          continue;
        }
      }
      // If the operand was not divisible, add it to the list of operands
      // we'll scan next iteration.
      NewOps.push_back(Ops[i]);
    }
    Ops = NewOps;
    AnyNonZeroIndices |= !ScaledOps.empty();
    Value *Scaled = ScaledOps.empty() ?
                    Constant::getNullValue(Ty) :
                    expandCodeFor(SE.getAddExpr(ScaledOps), Ty);
    GepIndices.push_back(Scaled);

    // Collect struct field index operands.
    if (!Ops.empty())
      while (const StructType *STy = dyn_cast<StructType>(ElTy)) {
        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::Int32Ty, ElIdx));
              ElTy = STy->getTypeAtIndex(ElIdx);
              Ops[0] =
                SE.getConstant(Ty, FullOffset - SL.getElementOffset(ElIdx));
              AnyNonZeroIndices = true;
              continue;
            }
          }
        break;
      }

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

  // If none of the operands were convertable 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) {
    V = InsertNoopCastOfTo(V,
                           Type::Int8Ty->getPointerTo(PTy->getAddressSpace()));
    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 = InsertPt->getParent()->begin();
    if (InsertPt != BlockBegin) {
      // Scanning starts from the last instruction before InsertPt.
      BasicBlock::iterator IP = InsertPt;
      --IP;
      for (; ScanLimit; --IP, --ScanLimit) {
        if (IP->getOpcode() == Instruction::GetElementPtr &&
            IP->getOperand(0) == V && IP->getOperand(1) == Idx)
          return IP;
        if (IP == BlockBegin) break;
      }
    }

    Value *GEP = GetElementPtrInst::Create(V, Idx, "scevgep", InsertPt);
    InsertedValues.insert(GEP);
    return GEP;
  }

  // Insert a pretty getelementptr.
  Value *GEP = GetElementPtrInst::Create(V,
                                         GepIndices.begin(),
                                         GepIndices.end(),
                                         "scevgep", InsertPt);
  Ops.push_back(SE.getUnknown(GEP));
  InsertedValues.insert(GEP);
  return expand(SE.getAddExpr(Ops));
}

Value *SCEVExpander::visitAddExpr(const SCEVAddExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *V = expand(S->getOperand(S->getNumOperands()-1));

  // Turn things like ptrtoint+arithmetic+inttoptr into GEP. See the
  // comments on expandAddToGEP for details.
  if (SE.TD)
    if (const PointerType *PTy = dyn_cast<PointerType>(V->getType())) {
      const SmallVectorImpl<const SCEV*> &Ops = S->getOperands();
      return expandAddToGEP(&Ops[0], &Ops[Ops.size() - 1],
                            PTy, Ty, V);
    }

  V = InsertNoopCastOfTo(V, Ty);

  // Emit a bunch of add instructions
  for (int i = S->getNumOperands()-2; i >= 0; --i) {
    Value *W = expandCodeFor(S->getOperand(i), Ty);
    V = InsertBinop(Instruction::Add, V, W, InsertPt);
  }
  return V;
}

Value *SCEVExpander::visitMulExpr(const SCEVMulExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  int FirstOp = 0;  // Set if we should emit a subtract.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(S->getOperand(0)))
    if (SC->getValue()->isAllOnesValue())
      FirstOp = 1;

  int i = S->getNumOperands()-2;
  Value *V = expandCodeFor(S->getOperand(i+1), Ty);

  // Emit a bunch of multiply instructions
  for (; i >= FirstOp; --i) {
    Value *W = expandCodeFor(S->getOperand(i), Ty);
    V = InsertBinop(Instruction::Mul, V, W, InsertPt);
  }

  // -1 * ...  --->  0 - ...
  if (FirstOp == 1)
    V = InsertBinop(Instruction::Sub, Constant::getNullValue(Ty), V, InsertPt);
  return V;
}

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()),
                         InsertPt);
  }

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

/// 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.getIntegerSCEV(0, A->getType()),
                                          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);
  }
}

Value *SCEVExpander::visitAddRecExpr(const SCEVAddRecExpr *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()) &&
        isa<IntegerType>(SE.getEffectiveSCEVType(PN->getType())) &&
        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)) {
    const SCEV* Start = SE.getAnyExtendExpr(S->getStart(),
                                           CanonicalIV->getType());
    const SCEV* Step = SE.getAnyExtendExpr(S->getStepRecurrence(SE),
                                          CanonicalIV->getType());
    Value *V = expand(SE.getAddRecExpr(Start, Step, S->getLoop()));
    BasicBlock::iterator SaveInsertPt = getInsertionPoint();
    BasicBlock::iterator NewInsertPt =
      next(BasicBlock::iterator(cast<Instruction>(V)));
    while (isa<PHINode>(NewInsertPt)) ++NewInsertPt;
    V = expandCodeFor(SE.getTruncateExpr(SE.getUnknown(V), Ty), 0,
                      NewInsertPt);
    setInsertionPoint(SaveInsertPt);
    return V;
  }

  // {X,+,F} --> X + {0,+,F}
  if (!S->getStart()->isZero()) {
    const SmallVectorImpl<const SCEV*> &SOperands = S->getOperands();
    SmallVector<const SCEV*, 4> NewOps(SOperands.begin(), SOperands.end());
    NewOps[0] = SE.getIntegerSCEV(0, Ty);
    const SCEV* Rest = SE.getAddRecExpr(NewOps, L);

    // Turn things like ptrtoint+arithmetic+inttoptr into GEP. See the
    // comments on expandAddToGEP for details.
    if (SE.TD) {
      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.getIntegerSCEV(1, Ty)) {
    // 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());
    InsertedValues.insert(PN);
    PN->addIncoming(Constant::getNullValue(Ty), L->getLoopPreheader());

    pred_iterator HPI = pred_begin(Header);
    assert(HPI != pred_end(Header) && "Loop with zero preds???");
    if (!L->contains(*HPI)) ++HPI;
    assert(HPI != pred_end(Header) && L->contains(*HPI) &&
           "No backedge in loop?");

    // Insert a unit add instruction right before the terminator corresponding
    // to the back-edge.
    Constant *One = ConstantInt::get(Ty, 1);
    Instruction *Add = BinaryOperator::CreateAdd(PN, One, "indvar.next",
                                                 (*HPI)->getTerminator());
    InsertedValues.insert(Add);

    pred_iterator PI = pred_begin(Header);
    if (*PI == L->getLoopPreheader())
      ++PI;
    PN->addIncoming(Add, *PI);
    return PN;
  }

  // {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()));
  Instruction *I = new TruncInst(V, Ty, "tmp.", InsertPt);
  InsertedValues.insert(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()));
  Instruction *I = new ZExtInst(V, Ty, "tmp.", InsertPt);
  InsertedValues.insert(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()));
  Instruction *I = new SExtInst(V, Ty, "tmp.", InsertPt);
  InsertedValues.insert(I);
  return I;
}

Value *SCEVExpander::visitSMaxExpr(const SCEVSMaxExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *LHS = expandCodeFor(S->getOperand(0), Ty);
  for (unsigned i = 1; i < S->getNumOperands(); ++i) {
    Value *RHS = expandCodeFor(S->getOperand(i), Ty);
    Instruction *ICmp =
      new ICmpInst(ICmpInst::ICMP_SGT, LHS, RHS, "tmp", InsertPt);
    InsertedValues.insert(ICmp);
    Instruction *Sel = SelectInst::Create(ICmp, LHS, RHS, "smax", InsertPt);
    InsertedValues.insert(Sel);
    LHS = Sel;
  }
  return LHS;
}

Value *SCEVExpander::visitUMaxExpr(const SCEVUMaxExpr *S) {
  const Type *Ty = SE.getEffectiveSCEVType(S->getType());
  Value *LHS = expandCodeFor(S->getOperand(0), Ty);
  for (unsigned i = 1; i < S->getNumOperands(); ++i) {
    Value *RHS = expandCodeFor(S->getOperand(i), Ty);
    Instruction *ICmp =
      new ICmpInst(ICmpInst::ICMP_UGT, LHS, RHS, "tmp", InsertPt);
    InsertedValues.insert(ICmp);
    Instruction *Sel = SelectInst::Create(ICmp, LHS, RHS, "umax", InsertPt);
    InsertedValues.insert(Sel);
    LHS = Sel;
  }
  return LHS;
}

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

  // Compute an insertion point for this SCEV object. Hoist the instructions
  // as far out in the loop nest as possible.
  BasicBlock::iterator InsertPt = getInsertionPoint();
  BasicBlock::iterator SaveInsertPt = InsertPt;
  for (Loop *L = SE.LI->getLoopFor(InsertPt->getParent()); ;
       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))
        InsertPt = L->getHeader()->getFirstNonPHI();
      while (isInsertedInstruction(InsertPt)) ++InsertPt;
      break;
    }
  setInsertionPoint(InsertPt);

  Value *V = visit(S);

  setInsertionPoint(SaveInsertPt);
  InsertedExpressions[S] = V;
  return V;
}

/// 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->isInteger() && "Can only insert integer induction variables!");
  const SCEV* H = SE.getAddRecExpr(SE.getIntegerSCEV(0, Ty),
                                   SE.getIntegerSCEV(1, Ty), L);
  BasicBlock::iterator SaveInsertPt = getInsertionPoint();
  Value *V = expandCodeFor(H, 0, L->getHeader()->begin());
  setInsertionPoint(SaveInsertPt);
  return V;
}