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

//===- ScalarEvolution.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 analysis
// engine, which is used primarily to analyze expressions involving induction
// variables in loops.
//
// There are several aspects to this library.  First is the representation of
// scalar expressions, which are represented as subclasses of the SCEV class.
// These classes are used to represent certain types of subexpressions that we
// can handle. We only create one SCEV of a particular shape, so
// pointer-comparisons for equality are legal.
//
// One important aspect of the SCEV objects is that they are never cyclic, even
// if there is a cycle in the dataflow for an expression (ie, a PHI node).  If
// the PHI node is one of the idioms that we can represent (e.g., a polynomial
// recurrence) then we represent it directly as a recurrence node, otherwise we
// represent it as a SCEVUnknown node.
//
// In addition to being able to represent expressions of various types, we also
// have folders that are used to build the *canonical* representation for a
// particular expression.  These folders are capable of using a variety of
// rewrite rules to simplify the expressions.
//
// Once the folders are defined, we can implement the more interesting
// higher-level code, such as the code that recognizes PHI nodes of various
// types, computes the execution count of a loop, etc.
//
// TODO: We should use these routines and value representations to implement
// dependence analysis!
//
//===----------------------------------------------------------------------===//
//
// There are several good references for the techniques used in this analysis.
//
//  Chains of recurrences -- a method to expedite the evaluation
//  of closed-form functions
//  Olaf Bachmann, Paul S. Wang, Eugene V. Zima
//
//  On computational properties of chains of recurrences
//  Eugene V. Zima
//
//  Symbolic Evaluation of Chains of Recurrences for Loop Optimization
//  Robert A. van Engelen
//
//  Efficient Symbolic Analysis for Optimizing Compilers
//  Robert A. van Engelen
//
//  Using the chains of recurrences algebra for data dependence testing and
//  induction variable substitution
//  MS Thesis, Johnie Birch
//
//===----------------------------------------------------------------------===//

#define DEBUG_TYPE "scalar-evolution"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/Analysis/ConstantFolding.h"
#include "llvm/Analysis/Dominators.h"
#include "llvm/Analysis/InstructionSimplify.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/Assembly/Writer.h"
#include "llvm/IR/Constants.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/DerivedTypes.h"
#include "llvm/IR/GlobalAlias.h"
#include "llvm/IR/GlobalVariable.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/LLVMContext.h"
#include "llvm/IR/Operator.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/ConstantRange.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/ErrorHandling.h"
#include "llvm/Support/GetElementPtrTypeIterator.h"
#include "llvm/Support/InstIterator.h"
#include "llvm/Support/MathExtras.h"
#include "llvm/Support/raw_ostream.h"
#include "llvm/Target/TargetLibraryInfo.h"
#include <algorithm>
using namespace llvm;

STATISTIC(NumArrayLenItCounts,
          "Number of trip counts computed with array length");
STATISTIC(NumTripCountsComputed,
          "Number of loops with predictable loop counts");
STATISTIC(NumTripCountsNotComputed,
          "Number of loops without predictable loop counts");
STATISTIC(NumBruteForceTripCountsComputed,
          "Number of loops with trip counts computed by force");

static cl::opt<unsigned>
MaxBruteForceIterations("scalar-evolution-max-iterations", cl::ReallyHidden,
                        cl::desc("Maximum number of iterations SCEV will "
                                 "symbolically execute a constant "
                                 "derived loop"),
                        cl::init(100));

// FIXME: Enable this with XDEBUG when the test suite is clean.
static cl::opt<bool>
VerifySCEV("verify-scev",
           cl::desc("Verify ScalarEvolution's backedge taken counts (slow)"));

INITIALIZE_PASS_BEGIN(ScalarEvolution, "scalar-evolution",
                "Scalar Evolution Analysis", false, true)
INITIALIZE_PASS_DEPENDENCY(LoopInfo)
INITIALIZE_PASS_DEPENDENCY(DominatorTree)
INITIALIZE_PASS_DEPENDENCY(TargetLibraryInfo)
INITIALIZE_PASS_END(ScalarEvolution, "scalar-evolution",
                "Scalar Evolution Analysis", false, true)
char ScalarEvolution::ID = 0;

//===----------------------------------------------------------------------===//
//                           SCEV class definitions
//===----------------------------------------------------------------------===//

//===----------------------------------------------------------------------===//
// Implementation of the SCEV class.
//

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
void SCEV::dump() const {
  print(dbgs());
  dbgs() << '\n';
}
#endif

void SCEV::print(raw_ostream &OS) const {
  switch (getSCEVType()) {
  case scConstant:
    WriteAsOperand(OS, cast<SCEVConstant>(this)->getValue(), false);
    return;
  case scTruncate: {
    const SCEVTruncateExpr *Trunc = cast<SCEVTruncateExpr>(this);
    const SCEV *Op = Trunc->getOperand();
    OS << "(trunc " << *Op->getType() << " " << *Op << " to "
       << *Trunc->getType() << ")";
    return;
  }
  case scZeroExtend: {
    const SCEVZeroExtendExpr *ZExt = cast<SCEVZeroExtendExpr>(this);
    const SCEV *Op = ZExt->getOperand();
    OS << "(zext " << *Op->getType() << " " << *Op << " to "
       << *ZExt->getType() << ")";
    return;
  }
  case scSignExtend: {
    const SCEVSignExtendExpr *SExt = cast<SCEVSignExtendExpr>(this);
    const SCEV *Op = SExt->getOperand();
    OS << "(sext " << *Op->getType() << " " << *Op << " to "
       << *SExt->getType() << ")";
    return;
  }
  case scAddRecExpr: {
    const SCEVAddRecExpr *AR = cast<SCEVAddRecExpr>(this);
    OS << "{" << *AR->getOperand(0);
    for (unsigned i = 1, e = AR->getNumOperands(); i != e; ++i)
      OS << ",+," << *AR->getOperand(i);
    OS << "}<";
    if (AR->getNoWrapFlags(FlagNUW))
      OS << "nuw><";
    if (AR->getNoWrapFlags(FlagNSW))
      OS << "nsw><";
    if (AR->getNoWrapFlags(FlagNW) &&
        !AR->getNoWrapFlags((NoWrapFlags)(FlagNUW | FlagNSW)))
      OS << "nw><";
    WriteAsOperand(OS, AR->getLoop()->getHeader(), /*PrintType=*/false);
    OS << ">";
    return;
  }
  case scAddExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr: {
    const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(this);
    const char *OpStr = 0;
    switch (NAry->getSCEVType()) {
    case scAddExpr: OpStr = " + "; break;
    case scMulExpr: OpStr = " * "; break;
    case scUMaxExpr: OpStr = " umax "; break;
    case scSMaxExpr: OpStr = " smax "; break;
    }
    OS << "(";
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      OS << **I;
      if (llvm::next(I) != E)
        OS << OpStr;
    }
    OS << ")";
    switch (NAry->getSCEVType()) {
    case scAddExpr:
    case scMulExpr:
      if (NAry->getNoWrapFlags(FlagNUW))
        OS << "<nuw>";
      if (NAry->getNoWrapFlags(FlagNSW))
        OS << "<nsw>";
    }
    return;
  }
  case scUDivExpr: {
    const SCEVUDivExpr *UDiv = cast<SCEVUDivExpr>(this);
    OS << "(" << *UDiv->getLHS() << " /u " << *UDiv->getRHS() << ")";
    return;
  }
  case scUnknown: {
    const SCEVUnknown *U = cast<SCEVUnknown>(this);
    Type *AllocTy;
    if (U->isSizeOf(AllocTy)) {
      OS << "sizeof(" << *AllocTy << ")";
      return;
    }
    if (U->isAlignOf(AllocTy)) {
      OS << "alignof(" << *AllocTy << ")";
      return;
    }

    Type *CTy;
    Constant *FieldNo;
    if (U->isOffsetOf(CTy, FieldNo)) {
      OS << "offsetof(" << *CTy << ", ";
      WriteAsOperand(OS, FieldNo, false);
      OS << ")";
      return;
    }

    // Otherwise just print it normally.
    WriteAsOperand(OS, U->getValue(), false);
    return;
  }
  case scCouldNotCompute:
    OS << "***COULDNOTCOMPUTE***";
    return;
  default: break;
  }
  llvm_unreachable("Unknown SCEV kind!");
}

Type *SCEV::getType() const {
  switch (getSCEVType()) {
  case scConstant:
    return cast<SCEVConstant>(this)->getType();
  case scTruncate:
  case scZeroExtend:
  case scSignExtend:
    return cast<SCEVCastExpr>(this)->getType();
  case scAddRecExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr:
    return cast<SCEVNAryExpr>(this)->getType();
  case scAddExpr:
    return cast<SCEVAddExpr>(this)->getType();
  case scUDivExpr:
    return cast<SCEVUDivExpr>(this)->getType();
  case scUnknown:
    return cast<SCEVUnknown>(this)->getType();
  case scCouldNotCompute:
    llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
  default:
    llvm_unreachable("Unknown SCEV kind!");
  }
}

bool SCEV::isZero() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isZero();
  return false;
}

bool SCEV::isOne() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isOne();
  return false;
}

bool SCEV::isAllOnesValue() const {
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(this))
    return SC->getValue()->isAllOnesValue();
  return false;
}

/// isNonConstantNegative - Return true if the specified scev is negated, but
/// not a constant.
bool SCEV::isNonConstantNegative() const {
  const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(this);
  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();
}

SCEVCouldNotCompute::SCEVCouldNotCompute() :
  SCEV(FoldingSetNodeIDRef(), scCouldNotCompute) {}

bool SCEVCouldNotCompute::classof(const SCEV *S) {
  return S->getSCEVType() == scCouldNotCompute;
}

const SCEV *ScalarEvolution::getConstant(ConstantInt *V) {
  FoldingSetNodeID ID;
  ID.AddInteger(scConstant);
  ID.AddPointer(V);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVConstant(ID.Intern(SCEVAllocator), V);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getConstant(const APInt& Val) {
  return getConstant(ConstantInt::get(getContext(), Val));
}

const SCEV *
ScalarEvolution::getConstant(Type *Ty, uint64_t V, bool isSigned) {
  IntegerType *ITy = cast<IntegerType>(getEffectiveSCEVType(Ty));
  return getConstant(ConstantInt::get(ITy, V, isSigned));
}

SCEVCastExpr::SCEVCastExpr(const FoldingSetNodeIDRef ID,
                           unsigned SCEVTy, const SCEV *op, Type *ty)
  : SCEV(ID, SCEVTy), Op(op), Ty(ty) {}

SCEVTruncateExpr::SCEVTruncateExpr(const FoldingSetNodeIDRef ID,
                                   const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scTruncate, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate non-integer value!");
}

SCEVZeroExtendExpr::SCEVZeroExtendExpr(const FoldingSetNodeIDRef ID,
                                       const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scZeroExtend, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot zero extend non-integer value!");
}

SCEVSignExtendExpr::SCEVSignExtendExpr(const FoldingSetNodeIDRef ID,
                                       const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scSignExtend, op, ty) {
  assert((Op->getType()->isIntegerTy() || Op->getType()->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot sign extend non-integer value!");
}

void SCEVUnknown::deleted() {
  // Clear this SCEVUnknown from various maps.
  SE->forgetMemoizedResults(this);

  // Remove this SCEVUnknown from the uniquing map.
  SE->UniqueSCEVs.RemoveNode(this);

  // Release the value.
  setValPtr(0);
}

void SCEVUnknown::allUsesReplacedWith(Value *New) {
  // Clear this SCEVUnknown from various maps.
  SE->forgetMemoizedResults(this);

  // Remove this SCEVUnknown from the uniquing map.
  SE->UniqueSCEVs.RemoveNode(this);

  // Update this SCEVUnknown to point to the new value. This is needed
  // because there may still be outstanding SCEVs which still point to
  // this SCEVUnknown.
  setValPtr(New);
}

bool SCEVUnknown::isSizeOf(Type *&AllocTy) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getOperand(0)->isNullValue() &&
            CE->getNumOperands() == 2)
          if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(1)))
            if (CI->isOne()) {
              AllocTy = cast<PointerType>(CE->getOperand(0)->getType())
                                 ->getElementType();
              return true;
            }

  return false;
}

bool SCEVUnknown::isAlignOf(Type *&AllocTy) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getOperand(0)->isNullValue()) {
          Type *Ty =
            cast<PointerType>(CE->getOperand(0)->getType())->getElementType();
          if (StructType *STy = dyn_cast<StructType>(Ty))
            if (!STy->isPacked() &&
                CE->getNumOperands() == 3 &&
                CE->getOperand(1)->isNullValue()) {
              if (ConstantInt *CI = dyn_cast<ConstantInt>(CE->getOperand(2)))
                if (CI->isOne() &&
                    STy->getNumElements() == 2 &&
                    STy->getElementType(0)->isIntegerTy(1)) {
                  AllocTy = STy->getElementType(1);
                  return true;
                }
            }
        }

  return false;
}

bool SCEVUnknown::isOffsetOf(Type *&CTy, Constant *&FieldNo) const {
  if (ConstantExpr *VCE = dyn_cast<ConstantExpr>(getValue()))
    if (VCE->getOpcode() == Instruction::PtrToInt)
      if (ConstantExpr *CE = dyn_cast<ConstantExpr>(VCE->getOperand(0)))
        if (CE->getOpcode() == Instruction::GetElementPtr &&
            CE->getNumOperands() == 3 &&
            CE->getOperand(0)->isNullValue() &&
            CE->getOperand(1)->isNullValue()) {
          Type *Ty =
            cast<PointerType>(CE->getOperand(0)->getType())->getElementType();
          // Ignore vector types here so that ScalarEvolutionExpander doesn't
          // emit getelementptrs that index into vectors.
          if (Ty->isStructTy() || Ty->isArrayTy()) {
            CTy = Ty;
            FieldNo = CE->getOperand(2);
            return true;
          }
        }

  return false;
}

//===----------------------------------------------------------------------===//
//                               SCEV Utilities
//===----------------------------------------------------------------------===//

namespace {
  /// SCEVComplexityCompare - Return true if the complexity of the LHS is less
  /// than the complexity of the RHS.  This comparator is used to canonicalize
  /// expressions.
  class SCEVComplexityCompare {
    const LoopInfo *const LI;
  public:
    explicit SCEVComplexityCompare(const LoopInfo *li) : LI(li) {}

    // Return true or false if LHS is less than, or at least RHS, respectively.
    bool operator()(const SCEV *LHS, const SCEV *RHS) const {
      return compare(LHS, RHS) < 0;
    }

    // Return negative, zero, or positive, if LHS is less than, equal to, or
    // greater than RHS, respectively. A three-way result allows recursive
    // comparisons to be more efficient.
    int compare(const SCEV *LHS, const SCEV *RHS) const {
      // Fast-path: SCEVs are uniqued so we can do a quick equality check.
      if (LHS == RHS)
        return 0;

      // Primarily, sort the SCEVs by their getSCEVType().
      unsigned LType = LHS->getSCEVType(), RType = RHS->getSCEVType();
      if (LType != RType)
        return (int)LType - (int)RType;

      // Aside from the getSCEVType() ordering, the particular ordering
      // isn't very important except that it's beneficial to be consistent,
      // so that (a + b) and (b + a) don't end up as different expressions.
      switch (LType) {
      case scUnknown: {
        const SCEVUnknown *LU = cast<SCEVUnknown>(LHS);
        const SCEVUnknown *RU = cast<SCEVUnknown>(RHS);

        // Sort SCEVUnknown values with some loose heuristics. TODO: This is
        // not as complete as it could be.
        const Value *LV = LU->getValue(), *RV = RU->getValue();

        // Order pointer values after integer values. This helps SCEVExpander
        // form GEPs.
        bool LIsPointer = LV->getType()->isPointerTy(),
             RIsPointer = RV->getType()->isPointerTy();
        if (LIsPointer != RIsPointer)
          return (int)LIsPointer - (int)RIsPointer;

        // Compare getValueID values.
        unsigned LID = LV->getValueID(),
                 RID = RV->getValueID();
        if (LID != RID)
          return (int)LID - (int)RID;

        // Sort arguments by their position.
        if (const Argument *LA = dyn_cast<Argument>(LV)) {
          const Argument *RA = cast<Argument>(RV);
          unsigned LArgNo = LA->getArgNo(), RArgNo = RA->getArgNo();
          return (int)LArgNo - (int)RArgNo;
        }

        // For instructions, compare their loop depth, and their operand
        // count.  This is pretty loose.
        if (const Instruction *LInst = dyn_cast<Instruction>(LV)) {
          const Instruction *RInst = cast<Instruction>(RV);

          // Compare loop depths.
          const BasicBlock *LParent = LInst->getParent(),
                           *RParent = RInst->getParent();
          if (LParent != RParent) {
            unsigned LDepth = LI->getLoopDepth(LParent),
                     RDepth = LI->getLoopDepth(RParent);
            if (LDepth != RDepth)
              return (int)LDepth - (int)RDepth;
          }

          // Compare the number of operands.
          unsigned LNumOps = LInst->getNumOperands(),
                   RNumOps = RInst->getNumOperands();
          return (int)LNumOps - (int)RNumOps;
        }

        return 0;
      }

      case scConstant: {
        const SCEVConstant *LC = cast<SCEVConstant>(LHS);
        const SCEVConstant *RC = cast<SCEVConstant>(RHS);

        // Compare constant values.
        const APInt &LA = LC->getValue()->getValue();
        const APInt &RA = RC->getValue()->getValue();
        unsigned LBitWidth = LA.getBitWidth(), RBitWidth = RA.getBitWidth();
        if (LBitWidth != RBitWidth)
          return (int)LBitWidth - (int)RBitWidth;
        return LA.ult(RA) ? -1 : 1;
      }

      case scAddRecExpr: {
        const SCEVAddRecExpr *LA = cast<SCEVAddRecExpr>(LHS);
        const SCEVAddRecExpr *RA = cast<SCEVAddRecExpr>(RHS);

        // Compare addrec loop depths.
        const Loop *LLoop = LA->getLoop(), *RLoop = RA->getLoop();
        if (LLoop != RLoop) {
          unsigned LDepth = LLoop->getLoopDepth(),
                   RDepth = RLoop->getLoopDepth();
          if (LDepth != RDepth)
            return (int)LDepth - (int)RDepth;
        }

        // Addrec complexity grows with operand count.
        unsigned LNumOps = LA->getNumOperands(), RNumOps = RA->getNumOperands();
        if (LNumOps != RNumOps)
          return (int)LNumOps - (int)RNumOps;

        // Lexicographically compare.
        for (unsigned i = 0; i != LNumOps; ++i) {
          long X = compare(LA->getOperand(i), RA->getOperand(i));
          if (X != 0)
            return X;
        }

        return 0;
      }

      case scAddExpr:
      case scMulExpr:
      case scSMaxExpr:
      case scUMaxExpr: {
        const SCEVNAryExpr *LC = cast<SCEVNAryExpr>(LHS);
        const SCEVNAryExpr *RC = cast<SCEVNAryExpr>(RHS);

        // Lexicographically compare n-ary expressions.
        unsigned LNumOps = LC->getNumOperands(), RNumOps = RC->getNumOperands();
        for (unsigned i = 0; i != LNumOps; ++i) {
          if (i >= RNumOps)
            return 1;
          long X = compare(LC->getOperand(i), RC->getOperand(i));
          if (X != 0)
            return X;
        }
        return (int)LNumOps - (int)RNumOps;
      }

      case scUDivExpr: {
        const SCEVUDivExpr *LC = cast<SCEVUDivExpr>(LHS);
        const SCEVUDivExpr *RC = cast<SCEVUDivExpr>(RHS);

        // Lexicographically compare udiv expressions.
        long X = compare(LC->getLHS(), RC->getLHS());
        if (X != 0)
          return X;
        return compare(LC->getRHS(), RC->getRHS());
      }

      case scTruncate:
      case scZeroExtend:
      case scSignExtend: {
        const SCEVCastExpr *LC = cast<SCEVCastExpr>(LHS);
        const SCEVCastExpr *RC = cast<SCEVCastExpr>(RHS);

        // Compare cast expressions by operand.
        return compare(LC->getOperand(), RC->getOperand());
      }

      default:
        llvm_unreachable("Unknown SCEV kind!");
      }
    }
  };
}

/// GroupByComplexity - Given a list of SCEV objects, order them by their
/// complexity, and group objects of the same complexity together by value.
/// When this routine is finished, we know that any duplicates in the vector are
/// consecutive and that complexity is monotonically increasing.
///
/// Note that we go take special precautions to ensure that we get deterministic
/// results from this routine.  In other words, we don't want the results of
/// this to depend on where the addresses of various SCEV objects happened to
/// land in memory.
///
static void GroupByComplexity(SmallVectorImpl<const SCEV *> &Ops,
                              LoopInfo *LI) {
  if (Ops.size() < 2) return;  // Noop
  if (Ops.size() == 2) {
    // This is the common case, which also happens to be trivially simple.
    // Special case it.
    const SCEV *&LHS = Ops[0], *&RHS = Ops[1];
    if (SCEVComplexityCompare(LI)(RHS, LHS))
      std::swap(LHS, RHS);
    return;
  }

  // Do the rough sort by complexity.
  std::stable_sort(Ops.begin(), Ops.end(), SCEVComplexityCompare(LI));

  // Now that we are sorted by complexity, group elements of the same
  // complexity.  Note that this is, at worst, N^2, but the vector is likely to
  // be extremely short in practice.  Note that we take this approach because we
  // do not want to depend on the addresses of the objects we are grouping.
  for (unsigned i = 0, e = Ops.size(); i != e-2; ++i) {
    const SCEV *S = Ops[i];
    unsigned Complexity = S->getSCEVType();

    // If there are any objects of the same complexity and same value as this
    // one, group them.
    for (unsigned j = i+1; j != e && Ops[j]->getSCEVType() == Complexity; ++j) {
      if (Ops[j] == S) { // Found a duplicate.
        // Move it to immediately after i'th element.
        std::swap(Ops[i+1], Ops[j]);
        ++i;   // no need to rescan it.
        if (i == e-2) return;  // Done!
      }
    }
  }
}



//===----------------------------------------------------------------------===//
//                      Simple SCEV method implementations
//===----------------------------------------------------------------------===//

/// BinomialCoefficient - Compute BC(It, K).  The result has width W.
/// Assume, K > 0.
static const SCEV *BinomialCoefficient(const SCEV *It, unsigned K,
                                       ScalarEvolution &SE,
                                       Type *ResultTy) {
  // Handle the simplest case efficiently.
  if (K == 1)
    return SE.getTruncateOrZeroExtend(It, ResultTy);

  // We are using the following formula for BC(It, K):
  //
  //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / K!
  //
  // Suppose, W is the bitwidth of the return value.  We must be prepared for
  // overflow.  Hence, we must assure that the result of our computation is
  // equal to the accurate one modulo 2^W.  Unfortunately, division isn't
  // safe in modular arithmetic.
  //
  // However, this code doesn't use exactly that formula; the formula it uses
  // is something like the following, where T is the number of factors of 2 in
  // K! (i.e. trailing zeros in the binary representation of K!), and ^ is
  // exponentiation:
  //
  //   BC(It, K) = (It * (It - 1) * ... * (It - K + 1)) / 2^T / (K! / 2^T)
  //
  // This formula is trivially equivalent to the previous formula.  However,
  // this formula can be implemented much more efficiently.  The trick is that
  // K! / 2^T is odd, and exact division by an odd number *is* safe in modular
  // arithmetic.  To do exact division in modular arithmetic, all we have
  // to do is multiply by the inverse.  Therefore, this step can be done at
  // width W.
  //
  // The next issue is how to safely do the division by 2^T.  The way this
  // is done is by doing the multiplication step at a width of at least W + T
  // bits.  This way, the bottom W+T bits of the product are accurate. Then,
  // when we perform the division by 2^T (which is equivalent to a right shift
  // by T), the bottom W bits are accurate.  Extra bits are okay; they'll get
  // truncated out after the division by 2^T.
  //
  // In comparison to just directly using the first formula, this technique
  // is much more efficient; using the first formula requires W * K bits,
  // but this formula less than W + K bits. Also, the first formula requires
  // a division step, whereas this formula only requires multiplies and shifts.
  //
  // It doesn't matter whether the subtraction step is done in the calculation
  // width or the input iteration count's width; if the subtraction overflows,
  // the result must be zero anyway.  We prefer here to do it in the width of
  // the induction variable because it helps a lot for certain cases; CodeGen
  // isn't smart enough to ignore the overflow, which leads to much less
  // efficient code if the width of the subtraction is wider than the native
  // register width.
  //
  // (It's possible to not widen at all by pulling out factors of 2 before
  // the multiplication; for example, K=2 can be calculated as
  // It/2*(It+(It*INT_MIN/INT_MIN)+-1). However, it requires
  // extra arithmetic, so it's not an obvious win, and it gets
  // much more complicated for K > 3.)

  // Protection from insane SCEVs; this bound is conservative,
  // but it probably doesn't matter.
  if (K > 1000)
    return SE.getCouldNotCompute();

  unsigned W = SE.getTypeSizeInBits(ResultTy);

  // Calculate K! / 2^T and T; we divide out the factors of two before
  // multiplying for calculating K! / 2^T to avoid overflow.
  // Other overflow doesn't matter because we only care about the bottom
  // W bits of the result.
  APInt OddFactorial(W, 1);
  unsigned T = 1;
  for (unsigned i = 3; i <= K; ++i) {
    APInt Mult(W, i);
    unsigned TwoFactors = Mult.countTrailingZeros();
    T += TwoFactors;
    Mult = Mult.lshr(TwoFactors);
    OddFactorial *= Mult;
  }

  // We need at least W + T bits for the multiplication step
  unsigned CalculationBits = W + T;

  // Calculate 2^T, at width T+W.
  APInt DivFactor = APInt(CalculationBits, 1).shl(T);

  // Calculate the multiplicative inverse of K! / 2^T;
  // this multiplication factor will perform the exact division by
  // K! / 2^T.
  APInt Mod = APInt::getSignedMinValue(W+1);
  APInt MultiplyFactor = OddFactorial.zext(W+1);
  MultiplyFactor = MultiplyFactor.multiplicativeInverse(Mod);
  MultiplyFactor = MultiplyFactor.trunc(W);

  // Calculate the product, at width T+W
  IntegerType *CalculationTy = IntegerType::get(SE.getContext(),
                                                      CalculationBits);
  const SCEV *Dividend = SE.getTruncateOrZeroExtend(It, CalculationTy);
  for (unsigned i = 1; i != K; ++i) {
    const SCEV *S = SE.getMinusSCEV(It, SE.getConstant(It->getType(), i));
    Dividend = SE.getMulExpr(Dividend,
                             SE.getTruncateOrZeroExtend(S, CalculationTy));
  }

  // Divide by 2^T
  const SCEV *DivResult = SE.getUDivExpr(Dividend, SE.getConstant(DivFactor));

  // Truncate the result, and divide by K! / 2^T.

  return SE.getMulExpr(SE.getConstant(MultiplyFactor),
                       SE.getTruncateOrZeroExtend(DivResult, ResultTy));
}

/// evaluateAtIteration - Return the value of this chain of recurrences at
/// the specified iteration number.  We can evaluate this recurrence by
/// multiplying each element in the chain by the binomial coefficient
/// corresponding to it.  In other words, we can evaluate {A,+,B,+,C,+,D} as:
///
///   A*BC(It, 0) + B*BC(It, 1) + C*BC(It, 2) + D*BC(It, 3)
///
/// where BC(It, k) stands for binomial coefficient.
///
const SCEV *SCEVAddRecExpr::evaluateAtIteration(const SCEV *It,
                                                ScalarEvolution &SE) const {
  const SCEV *Result = getStart();
  for (unsigned i = 1, e = getNumOperands(); i != e; ++i) {
    // The computation is correct in the face of overflow provided that the
    // multiplication is performed _after_ the evaluation of the binomial
    // coefficient.
    const SCEV *Coeff = BinomialCoefficient(It, i, SE, getType());
    if (isa<SCEVCouldNotCompute>(Coeff))
      return Coeff;

    Result = SE.getAddExpr(Result, SE.getMulExpr(getOperand(i), Coeff));
  }
  return Result;
}

//===----------------------------------------------------------------------===//
//                    SCEV Expression folder implementations
//===----------------------------------------------------------------------===//

const SCEV *ScalarEvolution::getTruncateExpr(const SCEV *Op,
                                             Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) > getTypeSizeInBits(Ty) &&
         "This is not a truncating conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  FoldingSetNodeID ID;
  ID.AddInteger(scTruncate);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getTrunc(SC->getValue(), Ty)));

  // trunc(trunc(x)) --> trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op))
    return getTruncateExpr(ST->getOperand(), Ty);

  // trunc(sext(x)) --> sext(x) if widening or trunc(x) if narrowing
  if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
    return getTruncateOrSignExtend(SS->getOperand(), Ty);

  // trunc(zext(x)) --> zext(x) if widening or trunc(x) if narrowing
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getTruncateOrZeroExtend(SZ->getOperand(), Ty);

  // trunc(x1+x2+...+xN) --> trunc(x1)+trunc(x2)+...+trunc(xN) if we can
  // eliminate all the truncates.
  if (const SCEVAddExpr *SA = dyn_cast<SCEVAddExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    bool hasTrunc = false;
    for (unsigned i = 0, e = SA->getNumOperands(); i != e && !hasTrunc; ++i) {
      const SCEV *S = getTruncateExpr(SA->getOperand(i), Ty);
      hasTrunc = isa<SCEVTruncateExpr>(S);
      Operands.push_back(S);
    }
    if (!hasTrunc)
      return getAddExpr(Operands);
    UniqueSCEVs.FindNodeOrInsertPos(ID, IP);  // Mutates IP, returns NULL.
  }

  // trunc(x1*x2*...*xN) --> trunc(x1)*trunc(x2)*...*trunc(xN) if we can
  // eliminate all the truncates.
  if (const SCEVMulExpr *SM = dyn_cast<SCEVMulExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    bool hasTrunc = false;
    for (unsigned i = 0, e = SM->getNumOperands(); i != e && !hasTrunc; ++i) {
      const SCEV *S = getTruncateExpr(SM->getOperand(i), Ty);
      hasTrunc = isa<SCEVTruncateExpr>(S);
      Operands.push_back(S);
    }
    if (!hasTrunc)
      return getMulExpr(Operands);
    UniqueSCEVs.FindNodeOrInsertPos(ID, IP);  // Mutates IP, returns NULL.
  }

  // If the input value is a chrec scev, truncate the chrec's operands.
  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(Op)) {
    SmallVector<const SCEV *, 4> Operands;
    for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
      Operands.push_back(getTruncateExpr(AddRec->getOperand(i), Ty));
    return getAddRecExpr(Operands, AddRec->getLoop(), SCEV::FlagAnyWrap);
  }

  // The cast wasn't folded; create an explicit cast node. We can reuse
  // the existing insert position since if we get here, we won't have
  // made any changes which would invalidate it.
  SCEV *S = new (SCEVAllocator) SCEVTruncateExpr(ID.Intern(SCEVAllocator),
                                                 Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getZeroExtendExpr(const SCEV *Op,
                                               Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getZExt(SC->getValue(), Ty)));

  // zext(zext(x)) --> zext(x)
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getZeroExtendExpr(SZ->getOperand(), Ty);

  // Before doing any expensive analysis, check to see if we've already
  // computed a SCEV for this Op and Ty.
  FoldingSetNodeID ID;
  ID.AddInteger(scZeroExtend);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // zext(trunc(x)) --> zext(x) or x or trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
    // It's possible the bits taken off by the truncate were all zero bits. If
    // so, we should be able to simplify this further.
    const SCEV *X = ST->getOperand();
    ConstantRange CR = getUnsignedRange(X);
    unsigned TruncBits = getTypeSizeInBits(ST->getType());
    unsigned NewBits = getTypeSizeInBits(Ty);
    if (CR.truncate(TruncBits).zeroExtend(NewBits).contains(
            CR.zextOrTrunc(NewBits)))
      return getTruncateOrZeroExtend(X, Ty);
  }

  // If the input value is a chrec scev, and we can prove that the value
  // did not overflow the old, smaller, value, we can zero extend all of the
  // operands (often constants).  This allows analysis of something like
  // this:  for (unsigned char X = 0; X < 100; ++X) { int Y = X; }
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
    if (AR->isAffine()) {
      const SCEV *Start = AR->getStart();
      const SCEV *Step = AR->getStepRecurrence(*this);
      unsigned BitWidth = getTypeSizeInBits(AR->getType());
      const Loop *L = AR->getLoop();

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->getNoWrapFlags(SCEV::FlagNUW))
        return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                             getZeroExtendExpr(Step, Ty),
                             L, AR->getNoWrapFlags());

      // Check whether the backedge-taken count is SCEVCouldNotCompute.
      // Note that this serves two purposes: It filters out loops that are
      // simply not analyzable, and it covers the case where this code is
      // being called from within backedge-taken count analysis, such that
      // attempting to ask for the backedge-taken count would likely result
      // in infinite recursion. In the later case, the analysis code will
      // cope with a conservative value, and it will take care to purge
      // that value once it has finished.
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(L);
      if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
        // Manually compute the final value for AR, checking for
        // overflow.

        // Check whether the backedge-taken count can be losslessly casted to
        // the addrec's type. The count is always unsigned.
        const SCEV *CastedMaxBECount =
          getTruncateOrZeroExtend(MaxBECount, Start->getType());
        const SCEV *RecastedMaxBECount =
          getTruncateOrZeroExtend(CastedMaxBECount, MaxBECount->getType());
        if (MaxBECount == RecastedMaxBECount) {
          Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
          // Check whether Start+Step*MaxBECount has no unsigned overflow.
          const SCEV *ZMul = getMulExpr(CastedMaxBECount, Step);
          const SCEV *ZAdd = getZeroExtendExpr(getAddExpr(Start, ZMul), WideTy);
          const SCEV *WideStart = getZeroExtendExpr(Start, WideTy);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getZeroExtendExpr(Step, WideTy)));
          if (ZAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NUW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
          // Similar to above, only this time treat the step value as signed.
          // This covers loops that count down.
          OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getSignExtendExpr(Step, WideTy)));
          if (ZAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NW, which is propagated to this AddRec.
            // Negative step causes unsigned wrap, but it still can't self-wrap.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }

        // If the backedge is guarded by a comparison with the pre-inc value
        // the addrec is safe. Also, if the entry is guarded by a comparison
        // with the start value and the backedge is guarded by a comparison
        // with the post-inc value, the addrec is safe.
        if (isKnownPositive(Step)) {
          const SCEV *N = getConstant(APInt::getMinValue(BitWidth) -
                                      getUnsignedRange(Step).getUnsignedMax());
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT, AR, N) ||
              (isLoopEntryGuardedByCond(L, ICmpInst::ICMP_ULT, Start, N) &&
               isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT,
                                           AR->getPostIncExpr(*this), N))) {
            // Cache knowledge of AR NUW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        } else if (isKnownNegative(Step)) {
          const SCEV *N = getConstant(APInt::getMaxValue(BitWidth) -
                                      getSignedRange(Step).getSignedMin());
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT, AR, N) ||
              (isLoopEntryGuardedByCond(L, ICmpInst::ICMP_UGT, Start, N) &&
               isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT,
                                           AR->getPostIncExpr(*this), N))) {
            // Cache knowledge of AR NW, which is propagated to this AddRec.
            // Negative step causes unsigned wrap, but it still can't self-wrap.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getZeroExtendExpr(Start, Ty),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }
      }
    }

  // The cast wasn't folded; create an explicit cast node.
  // Recompute the insert position, as it may have been invalidated.
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVZeroExtendExpr(ID.Intern(SCEVAllocator),
                                                   Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

// Get the limit of a recurrence such that incrementing by Step cannot cause
// signed overflow as long as the value of the recurrence within the loop does
// not exceed this limit before incrementing.
static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                           ICmpInst::Predicate *Pred,
                                           ScalarEvolution *SE) {
  unsigned BitWidth = SE->getTypeSizeInBits(Step->getType());
  if (SE->isKnownPositive(Step)) {
    *Pred = ICmpInst::ICMP_SLT;
    return SE->getConstant(APInt::getSignedMinValue(BitWidth) -
                           SE->getSignedRange(Step).getSignedMax());
  }
  if (SE->isKnownNegative(Step)) {
    *Pred = ICmpInst::ICMP_SGT;
    return SE->getConstant(APInt::getSignedMaxValue(BitWidth) -
                       SE->getSignedRange(Step).getSignedMin());
  }
  return 0;
}

// The recurrence AR has been shown to have no signed wrap. Typically, if we can
// prove NSW for AR, then we can just as easily prove NSW for its preincrement
// or postincrement sibling. This allows normalizing a sign extended AddRec as
// such: {sext(Step + Start),+,Step} => {(Step + sext(Start),+,Step} As a
// result, the expression "Step + sext(PreIncAR)" is congruent with
// "sext(PostIncAR)"
static const SCEV *getPreStartForSignExtend(const SCEVAddRecExpr *AR,
                                            Type *Ty,
                                            ScalarEvolution *SE) {
  const Loop *L = AR->getLoop();
  const SCEV *Start = AR->getStart();
  const SCEV *Step = AR->getStepRecurrence(*SE);

  // Check for a simple looking step prior to loop entry.
  const SCEVAddExpr *SA = dyn_cast<SCEVAddExpr>(Start);
  if (!SA)
    return 0;

  // Create an AddExpr for "PreStart" after subtracting Step. Full SCEV
  // subtraction is expensive. For this purpose, perform a quick and dirty
  // difference, by checking for Step in the operand list.
  SmallVector<const SCEV *, 4> DiffOps;
  for (SCEVAddExpr::op_iterator I = SA->op_begin(), E = SA->op_end();
       I != E; ++I) {
    if (*I != Step)
      DiffOps.push_back(*I);
  }
  if (DiffOps.size() == SA->getNumOperands())
    return 0;

  // This is a postinc AR. Check for overflow on the preinc recurrence using the
  // same three conditions that getSignExtendedExpr checks.

  // 1. NSW flags on the step increment.
  const SCEV *PreStart = SE->getAddExpr(DiffOps, SA->getNoWrapFlags());
  const SCEVAddRecExpr *PreAR = dyn_cast<SCEVAddRecExpr>(
    SE->getAddRecExpr(PreStart, Step, L, SCEV::FlagAnyWrap));

  if (PreAR && PreAR->getNoWrapFlags(SCEV::FlagNSW))
    return PreStart;

  // 2. Direct overflow check on the step operation's expression.
  unsigned BitWidth = SE->getTypeSizeInBits(AR->getType());
  Type *WideTy = IntegerType::get(SE->getContext(), BitWidth * 2);
  const SCEV *OperandExtendedStart =
    SE->getAddExpr(SE->getSignExtendExpr(PreStart, WideTy),
                   SE->getSignExtendExpr(Step, WideTy));
  if (SE->getSignExtendExpr(Start, WideTy) == OperandExtendedStart) {
    // Cache knowledge of PreAR NSW.
    if (PreAR)
      const_cast<SCEVAddRecExpr *>(PreAR)->setNoWrapFlags(SCEV::FlagNSW);
    // FIXME: this optimization needs a unit test
    DEBUG(dbgs() << "SCEV: untested prestart overflow check\n");
    return PreStart;
  }

  // 3. Loop precondition.
  ICmpInst::Predicate Pred;
  const SCEV *OverflowLimit = getOverflowLimitForStep(Step, &Pred, SE);

  if (OverflowLimit &&
      SE->isLoopEntryGuardedByCond(L, Pred, PreStart, OverflowLimit)) {
    return PreStart;
  }
  return 0;
}

// Get the normalized sign-extended expression for this AddRec's Start.
static const SCEV *getSignExtendAddRecStart(const SCEVAddRecExpr *AR,
                                            Type *Ty,
                                            ScalarEvolution *SE) {
  const SCEV *PreStart = getPreStartForSignExtend(AR, Ty, SE);
  if (!PreStart)
    return SE->getSignExtendExpr(AR->getStart(), Ty);

  return SE->getAddExpr(SE->getSignExtendExpr(AR->getStepRecurrence(*SE), Ty),
                        SE->getSignExtendExpr(PreStart, Ty));
}

const SCEV *ScalarEvolution::getSignExtendExpr(const SCEV *Op,
                                               Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Fold if the operand is constant.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    return getConstant(
      cast<ConstantInt>(ConstantExpr::getSExt(SC->getValue(), Ty)));

  // sext(sext(x)) --> sext(x)
  if (const SCEVSignExtendExpr *SS = dyn_cast<SCEVSignExtendExpr>(Op))
    return getSignExtendExpr(SS->getOperand(), Ty);

  // sext(zext(x)) --> zext(x)
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getZeroExtendExpr(SZ->getOperand(), Ty);

  // Before doing any expensive analysis, check to see if we've already
  // computed a SCEV for this Op and Ty.
  FoldingSetNodeID ID;
  ID.AddInteger(scSignExtend);
  ID.AddPointer(Op);
  ID.AddPointer(Ty);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;

  // If the input value is provably positive, build a zext instead.
  if (isKnownNonNegative(Op))
    return getZeroExtendExpr(Op, Ty);

  // sext(trunc(x)) --> sext(x) or x or trunc(x)
  if (const SCEVTruncateExpr *ST = dyn_cast<SCEVTruncateExpr>(Op)) {
    // It's possible the bits taken off by the truncate were all sign bits. If
    // so, we should be able to simplify this further.
    const SCEV *X = ST->getOperand();
    ConstantRange CR = getSignedRange(X);
    unsigned TruncBits = getTypeSizeInBits(ST->getType());
    unsigned NewBits = getTypeSizeInBits(Ty);
    if (CR.truncate(TruncBits).signExtend(NewBits).contains(
            CR.sextOrTrunc(NewBits)))
      return getTruncateOrSignExtend(X, Ty);
  }

  // If the input value is a chrec scev, and we can prove that the value
  // did not overflow the old, smaller, value, we can sign extend all of the
  // operands (often constants).  This allows analysis of something like
  // this:  for (signed char X = 0; X < 100; ++X) { int Y = X; }
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op))
    if (AR->isAffine()) {
      const SCEV *Start = AR->getStart();
      const SCEV *Step = AR->getStepRecurrence(*this);
      unsigned BitWidth = getTypeSizeInBits(AR->getType());
      const Loop *L = AR->getLoop();

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->getNoWrapFlags(SCEV::FlagNSW))
        return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                             getSignExtendExpr(Step, Ty),
                             L, SCEV::FlagNSW);

      // Check whether the backedge-taken count is SCEVCouldNotCompute.
      // Note that this serves two purposes: It filters out loops that are
      // simply not analyzable, and it covers the case where this code is
      // being called from within backedge-taken count analysis, such that
      // attempting to ask for the backedge-taken count would likely result
      // in infinite recursion. In the later case, the analysis code will
      // cope with a conservative value, and it will take care to purge
      // that value once it has finished.
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(L);
      if (!isa<SCEVCouldNotCompute>(MaxBECount)) {
        // Manually compute the final value for AR, checking for
        // overflow.

        // Check whether the backedge-taken count can be losslessly casted to
        // the addrec's type. The count is always unsigned.
        const SCEV *CastedMaxBECount =
          getTruncateOrZeroExtend(MaxBECount, Start->getType());
        const SCEV *RecastedMaxBECount =
          getTruncateOrZeroExtend(CastedMaxBECount, MaxBECount->getType());
        if (MaxBECount == RecastedMaxBECount) {
          Type *WideTy = IntegerType::get(getContext(), BitWidth * 2);
          // Check whether Start+Step*MaxBECount has no signed overflow.
          const SCEV *SMul = getMulExpr(CastedMaxBECount, Step);
          const SCEV *SAdd = getSignExtendExpr(getAddExpr(Start, SMul), WideTy);
          const SCEV *WideStart = getSignExtendExpr(Start, WideTy);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getSignExtendExpr(Step, WideTy)));
          if (SAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NSW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                                 getSignExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
          // Similar to above, only this time treat the step value as unsigned.
          // This covers loops that count up with an unsigned step.
          OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getZeroExtendExpr(Step, WideTy)));
          if (SAdd == OperandExtendedAdd) {
            // Cache knowledge of AR NSW, which is propagated to this AddRec.
            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
            // Return the expression with the addrec on the outside.
            return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                                 getZeroExtendExpr(Step, Ty),
                                 L, AR->getNoWrapFlags());
          }
        }

        // If the backedge is guarded by a comparison with the pre-inc value
        // the addrec is safe. Also, if the entry is guarded by a comparison
        // with the start value and the backedge is guarded by a comparison
        // with the post-inc value, the addrec is safe.
        ICmpInst::Predicate Pred;
        const SCEV *OverflowLimit = getOverflowLimitForStep(Step, &Pred, this);
        if (OverflowLimit &&
            (isLoopBackedgeGuardedByCond(L, Pred, AR, OverflowLimit) ||
             (isLoopEntryGuardedByCond(L, Pred, Start, OverflowLimit) &&
              isLoopBackedgeGuardedByCond(L, Pred, AR->getPostIncExpr(*this),
                                          OverflowLimit)))) {
          // Cache knowledge of AR NSW, then propagate NSW to the wide AddRec.
          const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
          return getAddRecExpr(getSignExtendAddRecStart(AR, Ty, this),
                               getSignExtendExpr(Step, Ty),
                               L, AR->getNoWrapFlags());
        }
      }
    }

  // The cast wasn't folded; create an explicit cast node.
  // Recompute the insert position, as it may have been invalidated.
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVSignExtendExpr(ID.Intern(SCEVAllocator),
                                                   Op, Ty);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

/// getAnyExtendExpr - Return a SCEV for the given operand extended with
/// unspecified bits out to the given type.
///
const SCEV *ScalarEvolution::getAnyExtendExpr(const SCEV *Op,
                                              Type *Ty) {
  assert(getTypeSizeInBits(Op->getType()) < getTypeSizeInBits(Ty) &&
         "This is not an extending conversion!");
  assert(isSCEVable(Ty) &&
         "This is not a conversion to a SCEVable type!");
  Ty = getEffectiveSCEVType(Ty);

  // Sign-extend negative constants.
  if (const SCEVConstant *SC = dyn_cast<SCEVConstant>(Op))
    if (SC->getValue()->getValue().isNegative())
      return getSignExtendExpr(Op, Ty);

  // Peel off a truncate cast.
  if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Op)) {
    const SCEV *NewOp = T->getOperand();
    if (getTypeSizeInBits(NewOp->getType()) < getTypeSizeInBits(Ty))
      return getAnyExtendExpr(NewOp, Ty);
    return getTruncateOrNoop(NewOp, Ty);
  }

  // Next try a zext cast. If the cast is folded, use it.
  const SCEV *ZExt = getZeroExtendExpr(Op, Ty);
  if (!isa<SCEVZeroExtendExpr>(ZExt))
    return ZExt;

  // Next try a sext cast. If the cast is folded, use it.
  const SCEV *SExt = getSignExtendExpr(Op, Ty);
  if (!isa<SCEVSignExtendExpr>(SExt))
    return SExt;

  // Force the cast to be folded into the operands of an addrec.
  if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Op)) {
    SmallVector<const SCEV *, 4> Ops;
    for (SCEVAddRecExpr::op_iterator I = AR->op_begin(), E = AR->op_end();
         I != E; ++I)
      Ops.push_back(getAnyExtendExpr(*I, Ty));
    return getAddRecExpr(Ops, AR->getLoop(), SCEV::FlagNW);
  }

  // If the expression is obviously signed, use the sext cast value.
  if (isa<SCEVSMaxExpr>(Op))
    return SExt;

  // Absent any other information, use the zext cast value.
  return ZExt;
}

/// CollectAddOperandsWithScales - Process the given Ops list, which is
/// a list of operands to be added under the given scale, update the given
/// map. This is a helper function for getAddRecExpr. As an example of
/// what it does, given a sequence of operands that would form an add
/// expression like this:
///
///    m + n + 13 + (A * (o + p + (B * q + m + 29))) + r + (-1 * r)
///
/// where A and B are constants, update the map with these values:
///
///    (m, 1+A*B), (n, 1), (o, A), (p, A), (q, A*B), (r, 0)
///
/// and add 13 + A*B*29 to AccumulatedConstant.
/// This will allow getAddRecExpr to produce this:
///
///    13+A*B*29 + n + (m * (1+A*B)) + ((o + p) * A) + (q * A*B)
///
/// This form often exposes folding opportunities that are hidden in
/// the original operand list.
///
/// Return true iff it appears that any interesting folding opportunities
/// may be exposed. This helps getAddRecExpr short-circuit extra work in
/// the common case where no interesting opportunities are present, and
/// is also used as a check to avoid infinite recursion.
///
static bool
CollectAddOperandsWithScales(DenseMap<const SCEV *, APInt> &M,
                             SmallVector<const SCEV *, 8> &NewOps,
                             APInt &AccumulatedConstant,
                             const SCEV *const *Ops, size_t NumOperands,
                             const APInt &Scale,
                             ScalarEvolution &SE) {
  bool Interesting = false;

  // Iterate over the add operands. They are sorted, with constants first.
  unsigned i = 0;
  while (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
    ++i;
    // Pull a buried constant out to the outside.
    if (Scale != 1 || AccumulatedConstant != 0 || C->getValue()->isZero())
      Interesting = true;
    AccumulatedConstant += Scale * C->getValue()->getValue();
  }

  // Next comes everything else. We're especially interested in multiplies
  // here, but they're in the middle, so just visit the rest with one loop.
  for (; i != NumOperands; ++i) {
    const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[i]);
    if (Mul && isa<SCEVConstant>(Mul->getOperand(0))) {
      APInt NewScale =
        Scale * cast<SCEVConstant>(Mul->getOperand(0))->getValue()->getValue();
      if (Mul->getNumOperands() == 2 && isa<SCEVAddExpr>(Mul->getOperand(1))) {
        // A multiplication of a constant with another add; recurse.
        const SCEVAddExpr *Add = cast<SCEVAddExpr>(Mul->getOperand(1));
        Interesting |=
          CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                       Add->op_begin(), Add->getNumOperands(),
                                       NewScale, SE);
      } else {
        // A multiplication of a constant with some other value. Update
        // the map.
        SmallVector<const SCEV *, 4> MulOps(Mul->op_begin()+1, Mul->op_end());
        const SCEV *Key = SE.getMulExpr(MulOps);
        std::pair<DenseMap<const SCEV *, APInt>::iterator, bool> Pair =
          M.insert(std::make_pair(Key, NewScale));
        if (Pair.second) {
          NewOps.push_back(Pair.first->first);
        } else {
          Pair.first->second += NewScale;
          // The map already had an entry for this value, which may indicate
          // a folding opportunity.
          Interesting = true;
        }
      }
    } else {
      // An ordinary operand. Update the map.
      std::pair<DenseMap<const SCEV *, APInt>::iterator, bool> Pair =
        M.insert(std::make_pair(Ops[i], Scale));
      if (Pair.second) {
        NewOps.push_back(Pair.first->first);
      } else {
        Pair.first->second += Scale;
        // The map already had an entry for this value, which may indicate
        // a folding opportunity.
        Interesting = true;
      }
    }
  }

  return Interesting;
}

namespace {
  struct APIntCompare {
    bool operator()(const APInt &LHS, const APInt &RHS) const {
      return LHS.ult(RHS);
    }
  };
}

/// getAddExpr - Get a canonical add expression, or something simpler if
/// possible.
const SCEV *ScalarEvolution::getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags) {
  assert(!(Flags & ~(SCEV::FlagNUW | SCEV::FlagNSW)) &&
         "only nuw or nsw allowed");
  assert(!Ops.empty() && "Cannot get empty add!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVAddExpr operand types don't match!");
#endif

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Ops.begin(),
         E = Ops.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      Ops[0] = getConstant(LHSC->getValue()->getValue() +
                           RHSC->getValue()->getValue());
      if (Ops.size() == 2) return Ops[0];
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant zero being added, strip it off.
    if (LHSC->getValue()->isZero()) {
      Ops.erase(Ops.begin());
      --Idx;
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Okay, check to see if the same value occurs in the operand list more than
  // once.  If so, merge them together into an multiply expression.  Since we
  // sorted the list, these values are required to be adjacent.
  Type *Ty = Ops[0]->getType();
  bool FoundMatch = false;
  for (unsigned i = 0, e = Ops.size(); i != e-1; ++i)
    if (Ops[i] == Ops[i+1]) {      //  X + Y + Y  -->  X + Y*2
      // Scan ahead to count how many equal operands there are.
      unsigned Count = 2;
      while (i+Count != e && Ops[i+Count] == Ops[i])
        ++Count;
      // Merge the values into a multiply.
      const SCEV *Scale = getConstant(Ty, Count);
      const SCEV *Mul = getMulExpr(Scale, Ops[i]);
      if (Ops.size() == Count)
        return Mul;
      Ops[i] = Mul;
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+Count);
      --i; e -= Count - 1;
      FoundMatch = true;
    }
  if (FoundMatch)
    return getAddExpr(Ops, Flags);

  // Check for truncates. If all the operands are truncated from the same
  // type, see if factoring out the truncate would permit the result to be
  // folded. eg., trunc(x) + m*trunc(n) --> trunc(x + trunc(m)*n)
  // if the contents of the resulting outer trunc fold to something simple.
  for (; Idx < Ops.size() && isa<SCEVTruncateExpr>(Ops[Idx]); ++Idx) {
    const SCEVTruncateExpr *Trunc = cast<SCEVTruncateExpr>(Ops[Idx]);
    Type *DstType = Trunc->getType();
    Type *SrcType = Trunc->getOperand()->getType();
    SmallVector<const SCEV *, 8> LargeOps;
    bool Ok = true;
    // Check all the operands to see if they can be represented in the
    // source type of the truncate.
    for (unsigned i = 0, e = Ops.size(); i != e; ++i) {
      if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(Ops[i])) {
        if (T->getOperand()->getType() != SrcType) {
          Ok = false;
          break;
        }
        LargeOps.push_back(T->getOperand());
      } else if (const SCEVConstant *C = dyn_cast<SCEVConstant>(Ops[i])) {
        LargeOps.push_back(getAnyExtendExpr(C, SrcType));
      } else if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(Ops[i])) {
        SmallVector<const SCEV *, 8> LargeMulOps;
        for (unsigned j = 0, f = M->getNumOperands(); j != f && Ok; ++j) {
          if (const SCEVTruncateExpr *T =
                dyn_cast<SCEVTruncateExpr>(M->getOperand(j))) {
            if (T->getOperand()->getType() != SrcType) {
              Ok = false;
              break;
            }
            LargeMulOps.push_back(T->getOperand());
          } else if (const SCEVConstant *C =
                       dyn_cast<SCEVConstant>(M->getOperand(j))) {
            LargeMulOps.push_back(getAnyExtendExpr(C, SrcType));
          } else {
            Ok = false;
            break;
          }
        }
        if (Ok)
          LargeOps.push_back(getMulExpr(LargeMulOps));
      } else {
        Ok = false;
        break;
      }
    }
    if (Ok) {
      // Evaluate the expression in the larger type.
      const SCEV *Fold = getAddExpr(LargeOps, Flags);
      // If it folds to something simple, use it. Otherwise, don't.
      if (isa<SCEVConstant>(Fold) || isa<SCEVUnknown>(Fold))
        return getTruncateExpr(Fold, DstType);
    }
  }

  // Skip past any other cast SCEVs.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddExpr)
    ++Idx;

  // If there are add operands they would be next.
  if (Idx < Ops.size()) {
    bool DeletedAdd = false;
    while (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[Idx])) {
      // If we have an add, expand the add operands onto the end of the operands
      // list.
      Ops.erase(Ops.begin()+Idx);
      Ops.append(Add->op_begin(), Add->op_end());
      DeletedAdd = true;
    }

    // If we deleted at least one add, we added operands to the end of the list,
    // and they are not necessarily sorted.  Recurse to resort and resimplify
    // any operands we just acquired.
    if (DeletedAdd)
      return getAddExpr(Ops);
  }

  // Skip over the add expression until we get to a multiply.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
    ++Idx;

  // Check to see if there are any folding opportunities present with
  // operands multiplied by constant values.
  if (Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx])) {
    uint64_t BitWidth = getTypeSizeInBits(Ty);
    DenseMap<const SCEV *, APInt> M;
    SmallVector<const SCEV *, 8> NewOps;
    APInt AccumulatedConstant(BitWidth, 0);
    if (CollectAddOperandsWithScales(M, NewOps, AccumulatedConstant,
                                     Ops.data(), Ops.size(),
                                     APInt(BitWidth, 1), *this)) {
      // Some interesting folding opportunity is present, so its worthwhile to
      // re-generate the operands list. Group the operands by constant scale,
      // to avoid multiplying by the same constant scale multiple times.
      std::map<APInt, SmallVector<const SCEV *, 4>, APIntCompare> MulOpLists;
      for (SmallVector<const SCEV *, 8>::const_iterator I = NewOps.begin(),
           E = NewOps.end(); I != E; ++I)
        MulOpLists[M.find(*I)->second].push_back(*I);
      // Re-generate the operands list.
      Ops.clear();
      if (AccumulatedConstant != 0)
        Ops.push_back(getConstant(AccumulatedConstant));
      for (std::map<APInt, SmallVector<const SCEV *, 4>, APIntCompare>::iterator
           I = MulOpLists.begin(), E = MulOpLists.end(); I != E; ++I)
        if (I->first != 0)
          Ops.push_back(getMulExpr(getConstant(I->first),
                                   getAddExpr(I->second)));
      if (Ops.empty())
        return getConstant(Ty, 0);
      if (Ops.size() == 1)
        return Ops[0];
      return getAddExpr(Ops);
    }
  }

  // If we are adding something to a multiply expression, make sure the
  // something is not already an operand of the multiply.  If so, merge it into
  // the multiply.
  for (; Idx < Ops.size() && isa<SCEVMulExpr>(Ops[Idx]); ++Idx) {
    const SCEVMulExpr *Mul = cast<SCEVMulExpr>(Ops[Idx]);
    for (unsigned MulOp = 0, e = Mul->getNumOperands(); MulOp != e; ++MulOp) {
      const SCEV *MulOpSCEV = Mul->getOperand(MulOp);
      if (isa<SCEVConstant>(MulOpSCEV))
        continue;
      for (unsigned AddOp = 0, e = Ops.size(); AddOp != e; ++AddOp)
        if (MulOpSCEV == Ops[AddOp]) {
          // Fold W + X + (X * Y * Z)  -->  W + (X * ((Y*Z)+1))
          const SCEV *InnerMul = Mul->getOperand(MulOp == 0);
          if (Mul->getNumOperands() != 2) {
            // If the multiply has more than two operands, we must get the
            // Y*Z term.
            SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                Mul->op_begin()+MulOp);
            MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
            InnerMul = getMulExpr(MulOps);
          }
          const SCEV *One = getConstant(Ty, 1);
          const SCEV *AddOne = getAddExpr(One, InnerMul);
          const SCEV *OuterMul = getMulExpr(AddOne, MulOpSCEV);
          if (Ops.size() == 2) return OuterMul;
          if (AddOp < Idx) {
            Ops.erase(Ops.begin()+AddOp);
            Ops.erase(Ops.begin()+Idx-1);
          } else {
            Ops.erase(Ops.begin()+Idx);
            Ops.erase(Ops.begin()+AddOp-1);
          }
          Ops.push_back(OuterMul);
          return getAddExpr(Ops);
        }

      // Check this multiply against other multiplies being added together.
      for (unsigned OtherMulIdx = Idx+1;
           OtherMulIdx < Ops.size() && isa<SCEVMulExpr>(Ops[OtherMulIdx]);
           ++OtherMulIdx) {
        const SCEVMulExpr *OtherMul = cast<SCEVMulExpr>(Ops[OtherMulIdx]);
        // If MulOp occurs in OtherMul, we can fold the two multiplies
        // together.
        for (unsigned OMulOp = 0, e = OtherMul->getNumOperands();
             OMulOp != e; ++OMulOp)
          if (OtherMul->getOperand(OMulOp) == MulOpSCEV) {
            // Fold X + (A*B*C) + (A*D*E) --> X + (A*(B*C+D*E))
            const SCEV *InnerMul1 = Mul->getOperand(MulOp == 0);
            if (Mul->getNumOperands() != 2) {
              SmallVector<const SCEV *, 4> MulOps(Mul->op_begin(),
                                                  Mul->op_begin()+MulOp);
              MulOps.append(Mul->op_begin()+MulOp+1, Mul->op_end());
              InnerMul1 = getMulExpr(MulOps);
            }
            const SCEV *InnerMul2 = OtherMul->getOperand(OMulOp == 0);
            if (OtherMul->getNumOperands() != 2) {
              SmallVector<const SCEV *, 4> MulOps(OtherMul->op_begin(),
                                                  OtherMul->op_begin()+OMulOp);
              MulOps.append(OtherMul->op_begin()+OMulOp+1, OtherMul->op_end());
              InnerMul2 = getMulExpr(MulOps);
            }
            const SCEV *InnerMulSum = getAddExpr(InnerMul1,InnerMul2);
            const SCEV *OuterMul = getMulExpr(MulOpSCEV, InnerMulSum);
            if (Ops.size() == 2) return OuterMul;
            Ops.erase(Ops.begin()+Idx);
            Ops.erase(Ops.begin()+OtherMulIdx-1);
            Ops.push_back(OuterMul);
            return getAddExpr(Ops);
          }
      }
    }
  }

  // If there are any add recurrences in the operands list, see if any other
  // added values are loop invariant.  If so, we can fold them into the
  // recurrence.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
    ++Idx;

  // Scan over all recurrences, trying to fold loop invariants into them.
  for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
    // Scan all of the other operands to this add and add them to the vector if
    // they are loop invariant w.r.t. the recurrence.
    SmallVector<const SCEV *, 8> LIOps;
    const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
    const Loop *AddRecLoop = AddRec->getLoop();
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      if (isLoopInvariant(Ops[i], AddRecLoop)) {
        LIOps.push_back(Ops[i]);
        Ops.erase(Ops.begin()+i);
        --i; --e;
      }

    // If we found some loop invariants, fold them into the recurrence.
    if (!LIOps.empty()) {
      //  NLI + LI + {Start,+,Step}  -->  NLI + {LI+Start,+,Step}
      LIOps.push_back(AddRec->getStart());

      SmallVector<const SCEV *, 4> AddRecOps(AddRec->op_begin(),
                                             AddRec->op_end());
      AddRecOps[0] = getAddExpr(LIOps);

      // Build the new addrec. Propagate the NUW and NSW flags if both the
      // outer add and the inner addrec are guaranteed to have no overflow.
      // Always propagate NW.
      Flags = AddRec->getNoWrapFlags(setFlags(Flags, SCEV::FlagNW));
      const SCEV *NewRec = getAddRecExpr(AddRecOps, AddRecLoop, Flags);

      // If all of the other operands were loop invariant, we are done.
      if (Ops.size() == 1) return NewRec;

      // Otherwise, add the folded AddRec by the non-invariant parts.
      for (unsigned i = 0;; ++i)
        if (Ops[i] == AddRec) {
          Ops[i] = NewRec;
          break;
        }
      return getAddExpr(Ops);
    }

    // Okay, if there weren't any loop invariants to be folded, check to see if
    // there are multiple AddRec's with the same loop induction variable being
    // added together.  If so, we can fold them.
    for (unsigned OtherIdx = Idx+1;
         OtherIdx < Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
         ++OtherIdx)
      if (AddRecLoop == cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop()) {
        // Other + {A,+,B}<L> + {C,+,D}<L>  -->  Other + {A+C,+,B+D}<L>
        SmallVector<const SCEV *, 4> AddRecOps(AddRec->op_begin(),
                                               AddRec->op_end());
        for (; OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
             ++OtherIdx)
          if (const SCEVAddRecExpr *OtherAddRec =
                dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]))
            if (OtherAddRec->getLoop() == AddRecLoop) {
              for (unsigned i = 0, e = OtherAddRec->getNumOperands();
                   i != e; ++i) {
                if (i >= AddRecOps.size()) {
                  AddRecOps.append(OtherAddRec->op_begin()+i,
                                   OtherAddRec->op_end());
                  break;
                }
                AddRecOps[i] = getAddExpr(AddRecOps[i],
                                          OtherAddRec->getOperand(i));
              }
              Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
            }
        // Step size has changed, so we cannot guarantee no self-wraparound.
        Ops[Idx] = getAddRecExpr(AddRecOps, AddRecLoop, SCEV::FlagAnyWrap);
        return getAddExpr(Ops);
      }

    // Otherwise couldn't fold anything into this recurrence.  Move onto the
    // next one.
  }

  // Okay, it looks like we really DO need an add expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scAddExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  SCEVAddExpr *S =
    static_cast<SCEVAddExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
    std::uninitialized_copy(Ops.begin(), Ops.end(), O);
    S = new (SCEVAllocator) SCEVAddExpr(ID.Intern(SCEVAllocator),
                                        O, Ops.size());
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

static uint64_t umul_ov(uint64_t i, uint64_t j, bool &Overflow) {
  uint64_t k = i*j;
  if (j > 1 && k / j != i) Overflow = true;
  return k;
}

/// Compute the result of "n choose k", the binomial coefficient.  If an
/// intermediate computation overflows, Overflow will be set and the return will
/// be garbage. Overflow is not cleared on absence of overflow.
static uint64_t Choose(uint64_t n, uint64_t k, bool &Overflow) {
  // We use the multiplicative formula:
  //     n(n-1)(n-2)...(n-(k-1)) / k(k-1)(k-2)...1 .
  // At each iteration, we take the n-th term of the numeral and divide by the
  // (k-n)th term of the denominator.  This division will always produce an
  // integral result, and helps reduce the chance of overflow in the
  // intermediate computations. However, we can still overflow even when the
  // final result would fit.

  if (n == 0 || n == k) return 1;
  if (k > n) return 0;

  if (k > n/2)
    k = n-k;

  uint64_t r = 1;
  for (uint64_t i = 1; i <= k; ++i) {
    r = umul_ov(r, n-(i-1), Overflow);
    r /= i;
  }
  return r;
}

/// getMulExpr - Get a canonical multiply expression, or something simpler if
/// possible.
const SCEV *ScalarEvolution::getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags) {
  assert(Flags == maskFlags(Flags, SCEV::FlagNUW | SCEV::FlagNSW) &&
         "only nuw or nsw allowed");
  assert(!Ops.empty() && "Cannot get empty mul!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVMulExpr operand types don't match!");
#endif

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Ops.begin(),
         E = Ops.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {

    // C1*(C2+V) -> C1*C2 + C1*V
    if (Ops.size() == 2)
      if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1]))
        if (Add->getNumOperands() == 2 &&
            isa<SCEVConstant>(Add->getOperand(0)))
          return getAddExpr(getMulExpr(LHSC, Add->getOperand(0)),
                            getMulExpr(LHSC, Add->getOperand(1)));

    ++Idx;
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                                           LHSC->getValue()->getValue() *
                                           RHSC->getValue()->getValue());
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant one being multiplied, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->equalsInt(1)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isZero()) {
      // If we have a multiply of zero, it will always be zero.
      return Ops[0];
    } else if (Ops[0]->isAllOnesValue()) {
      // If we have a mul by -1 of an add, try distributing the -1 among the
      // add operands.
      if (Ops.size() == 2) {
        if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1])) {
          SmallVector<const SCEV *, 4> NewOps;
          bool AnyFolded = false;
          for (SCEVAddRecExpr::op_iterator I = Add->op_begin(),
                 E = Add->op_end(); I != E; ++I) {
            const SCEV *Mul = getMulExpr(Ops[0], *I);
            if (!isa<SCEVMulExpr>(Mul)) AnyFolded = true;
            NewOps.push_back(Mul);
          }
          if (AnyFolded)
            return getAddExpr(NewOps);
        }
        else if (const SCEVAddRecExpr *
                 AddRec = dyn_cast<SCEVAddRecExpr>(Ops[1])) {
          // Negation preserves a recurrence's no self-wrap property.
          SmallVector<const SCEV *, 4> Operands;
          for (SCEVAddRecExpr::op_iterator I = AddRec->op_begin(),
                 E = AddRec->op_end(); I != E; ++I) {
            Operands.push_back(getMulExpr(Ops[0], *I));
          }
          return getAddRecExpr(Operands, AddRec->getLoop(),
                               AddRec->getNoWrapFlags(SCEV::FlagNW));
        }
      }
    }

    if (Ops.size() == 1)
      return Ops[0];
  }

  // Skip over the add expression until we get to a multiply.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scMulExpr)
    ++Idx;

  // If there are mul operands inline them all into this expression.
  if (Idx < Ops.size()) {
    bool DeletedMul = false;
    while (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(Ops[Idx])) {
      // If we have an mul, expand the mul operands onto the end of the operands
      // list.
      Ops.erase(Ops.begin()+Idx);
      Ops.append(Mul->op_begin(), Mul->op_end());
      DeletedMul = true;
    }

    // If we deleted at least one mul, we added operands to the end of the list,
    // and they are not necessarily sorted.  Recurse to resort and resimplify
    // any operands we just acquired.
    if (DeletedMul)
      return getMulExpr(Ops);
  }

  // If there are any add recurrences in the operands list, see if any other
  // added values are loop invariant.  If so, we can fold them into the
  // recurrence.
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scAddRecExpr)
    ++Idx;

  // Scan over all recurrences, trying to fold loop invariants into them.
  for (; Idx < Ops.size() && isa<SCEVAddRecExpr>(Ops[Idx]); ++Idx) {
    // Scan all of the other operands to this mul and add them to the vector if
    // they are loop invariant w.r.t. the recurrence.
    SmallVector<const SCEV *, 8> LIOps;
    const SCEVAddRecExpr *AddRec = cast<SCEVAddRecExpr>(Ops[Idx]);
    const Loop *AddRecLoop = AddRec->getLoop();
    for (unsigned i = 0, e = Ops.size(); i != e; ++i)
      if (isLoopInvariant(Ops[i], AddRecLoop)) {
        LIOps.push_back(Ops[i]);
        Ops.erase(Ops.begin()+i);
        --i; --e;
      }

    // If we found some loop invariants, fold them into the recurrence.
    if (!LIOps.empty()) {
      //  NLI * LI * {Start,+,Step}  -->  NLI * {LI*Start,+,LI*Step}
      SmallVector<const SCEV *, 4> NewOps;
      NewOps.reserve(AddRec->getNumOperands());
      const SCEV *Scale = getMulExpr(LIOps);
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
        NewOps.push_back(getMulExpr(Scale, AddRec->getOperand(i)));

      // Build the new addrec. Propagate the NUW and NSW flags if both the
      // outer mul and the inner addrec are guaranteed to have no overflow.
      //
      // No self-wrap cannot be guaranteed after changing the step size, but
      // will be inferred if either NUW or NSW is true.
      Flags = AddRec->getNoWrapFlags(clearFlags(Flags, SCEV::FlagNW));
      const SCEV *NewRec = getAddRecExpr(NewOps, AddRecLoop, Flags);

      // If all of the other operands were loop invariant, we are done.
      if (Ops.size() == 1) return NewRec;

      // Otherwise, multiply the folded AddRec by the non-invariant parts.
      for (unsigned i = 0;; ++i)
        if (Ops[i] == AddRec) {
          Ops[i] = NewRec;
          break;
        }
      return getMulExpr(Ops);
    }

    // Okay, if there weren't any loop invariants to be folded, check to see if
    // there are multiple AddRec's with the same loop induction variable being
    // multiplied together.  If so, we can fold them.
    for (unsigned OtherIdx = Idx+1;
         OtherIdx < Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
         ++OtherIdx) {
      if (AddRecLoop != cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop())
        continue;

      // {A1,+,A2,+,...,+,An}<L> * {B1,+,B2,+,...,+,Bn}<L>
      // = {x=1 in [ sum y=x..2x [ sum z=max(y-x, y-n)..min(x,n) [
      //       choose(x, 2x)*choose(2x-y, x-z)*A_{y-z}*B_z
      //   ]]],+,...up to x=2n}.
      // Note that the arguments to choose() are always integers with values
      // known at compile time, never SCEV objects.
      //
      // The implementation avoids pointless extra computations when the two
      // addrec's are of different length (mathematically, it's equivalent to
      // an infinite stream of zeros on the right).
      bool OpsModified = false;
      for (; OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
           ++OtherIdx) {
        const SCEVAddRecExpr *OtherAddRec =
          dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]);
        if (!OtherAddRec || OtherAddRec->getLoop() != AddRecLoop)
          continue;

        bool Overflow = false;
        Type *Ty = AddRec->getType();
        bool LargerThan64Bits = getTypeSizeInBits(Ty) > 64;
        SmallVector<const SCEV*, 7> AddRecOps;
        for (int x = 0, xe = AddRec->getNumOperands() +
               OtherAddRec->getNumOperands() - 1; x != xe && !Overflow; ++x) {
          const SCEV *Term = getConstant(Ty, 0);
          for (int y = x, ye = 2*x+1; y != ye && !Overflow; ++y) {
            uint64_t Coeff1 = Choose(x, 2*x - y, Overflow);
            for (int z = std::max(y-x, y-(int)AddRec->getNumOperands()+1),
                   ze = std::min(x+1, (int)OtherAddRec->getNumOperands());
                 z < ze && !Overflow; ++z) {
              uint64_t Coeff2 = Choose(2*x - y, x-z, Overflow);
              uint64_t Coeff;
              if (LargerThan64Bits)
                Coeff = umul_ov(Coeff1, Coeff2, Overflow);
              else
                Coeff = Coeff1*Coeff2;
              const SCEV *CoeffTerm = getConstant(Ty, Coeff);
              const SCEV *Term1 = AddRec->getOperand(y-z);
              const SCEV *Term2 = OtherAddRec->getOperand(z);
              Term = getAddExpr(Term, getMulExpr(CoeffTerm, Term1,Term2));
            }
          }
          AddRecOps.push_back(Term);
        }
        if (!Overflow) {
          const SCEV *NewAddRec = getAddRecExpr(AddRecOps, AddRec->getLoop(),
                                                SCEV::FlagAnyWrap);
          if (Ops.size() == 2) return NewAddRec;
          Ops[Idx] = NewAddRec;
          Ops.erase(Ops.begin() + OtherIdx); --OtherIdx;
          OpsModified = true;
          AddRec = dyn_cast<SCEVAddRecExpr>(NewAddRec);
          if (!AddRec)
            break;
        }
      }
      if (OpsModified)
        return getMulExpr(Ops);
    }

    // Otherwise couldn't fold anything into this recurrence.  Move onto the
    // next one.
  }

  // Okay, it looks like we really DO need an mul expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scMulExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  SCEVMulExpr *S =
    static_cast<SCEVMulExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
    std::uninitialized_copy(Ops.begin(), Ops.end(), O);
    S = new (SCEVAllocator) SCEVMulExpr(ID.Intern(SCEVAllocator),
                                        O, Ops.size());
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

/// getUDivExpr - Get a canonical unsigned division expression, or something
/// simpler if possible.
const SCEV *ScalarEvolution::getUDivExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  assert(getEffectiveSCEVType(LHS->getType()) ==
         getEffectiveSCEVType(RHS->getType()) &&
         "SCEVUDivExpr operand types don't match!");

  if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
    if (RHSC->getValue()->equalsInt(1))
      return LHS;                               // X udiv 1 --> x
    // If the denominator is zero, the result of the udiv is undefined. Don't
    // try to analyze it, because the resolution chosen here may differ from
    // the resolution chosen in other parts of the compiler.
    if (!RHSC->getValue()->isZero()) {
      // Determine if the division can be folded into the operands of
      // its operands.
      // TODO: Generalize this to non-constants by using known-bits information.
      Type *Ty = LHS->getType();
      unsigned LZ = RHSC->getValue()->getValue().countLeadingZeros();
      unsigned MaxShiftAmt = getTypeSizeInBits(Ty) - LZ - 1;
      // For non-power-of-two values, effectively round the value up to the
      // nearest power of two.
      if (!RHSC->getValue()->getValue().isPowerOf2())
        ++MaxShiftAmt;
      IntegerType *ExtTy =
        IntegerType::get(getContext(), getTypeSizeInBits(Ty) + MaxShiftAmt);
      if (const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(LHS))
        if (const SCEVConstant *Step =
            dyn_cast<SCEVConstant>(AR->getStepRecurrence(*this))) {
          // {X,+,N}/C --> {X/C,+,N/C} if safe and N/C can be folded.
          const APInt &StepInt = Step->getValue()->getValue();
          const APInt &DivInt = RHSC->getValue()->getValue();
          if (!StepInt.urem(DivInt) &&
              getZeroExtendExpr(AR, ExtTy) ==
              getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                            getZeroExtendExpr(Step, ExtTy),
                            AR->getLoop(), SCEV::FlagAnyWrap)) {
            SmallVector<const SCEV *, 4> Operands;
            for (unsigned i = 0, e = AR->getNumOperands(); i != e; ++i)
              Operands.push_back(getUDivExpr(AR->getOperand(i), RHS));
            return getAddRecExpr(Operands, AR->getLoop(),
                                 SCEV::FlagNW);
          }
          /// Get a canonical UDivExpr for a recurrence.
          /// {X,+,N}/C => {Y,+,N}/C where Y=X-(X%N). Safe when C%N=0.
          // We can currently only fold X%N if X is constant.
          const SCEVConstant *StartC = dyn_cast<SCEVConstant>(AR->getStart());
          if (StartC && !DivInt.urem(StepInt) &&
              getZeroExtendExpr(AR, ExtTy) ==
              getAddRecExpr(getZeroExtendExpr(AR->getStart(), ExtTy),
                            getZeroExtendExpr(Step, ExtTy),
                            AR->getLoop(), SCEV::FlagAnyWrap)) {
            const APInt &StartInt = StartC->getValue()->getValue();
            const APInt &StartRem = StartInt.urem(StepInt);
            if (StartRem != 0)
              LHS = getAddRecExpr(getConstant(StartInt - StartRem), Step,
                                  AR->getLoop(), SCEV::FlagNW);
          }
        }
      // (A*B)/C --> A*(B/C) if safe and B/C can be folded.
      if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(LHS)) {
        SmallVector<const SCEV *, 4> Operands;
        for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i)
          Operands.push_back(getZeroExtendExpr(M->getOperand(i), ExtTy));
        if (getZeroExtendExpr(M, ExtTy) == getMulExpr(Operands))
          // Find an operand that's safely divisible.
          for (unsigned i = 0, e = M->getNumOperands(); i != e; ++i) {
            const SCEV *Op = M->getOperand(i);
            const SCEV *Div = getUDivExpr(Op, RHSC);
            if (!isa<SCEVUDivExpr>(Div) && getMulExpr(Div, RHSC) == Op) {
              Operands = SmallVector<const SCEV *, 4>(M->op_begin(),
                                                      M->op_end());
              Operands[i] = Div;
              return getMulExpr(Operands);
            }
          }
      }
      // (A+B)/C --> (A/C + B/C) if safe and A/C and B/C can be folded.
      if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(LHS)) {
        SmallVector<const SCEV *, 4> Operands;
        for (unsigned i = 0, e = A->getNumOperands(); i != e; ++i)
          Operands.push_back(getZeroExtendExpr(A->getOperand(i), ExtTy));
        if (getZeroExtendExpr(A, ExtTy) == getAddExpr(Operands)) {
          Operands.clear();
          for (unsigned i = 0, e = A->getNumOperands(); i != e; ++i) {
            const SCEV *Op = getUDivExpr(A->getOperand(i), RHS);
            if (isa<SCEVUDivExpr>(Op) ||
                getMulExpr(Op, RHS) != A->getOperand(i))
              break;
            Operands.push_back(Op);
          }
          if (Operands.size() == A->getNumOperands())
            return getAddExpr(Operands);
        }
      }

      // Fold if both operands are constant.
      if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(LHS)) {
        Constant *LHSCV = LHSC->getValue();
        Constant *RHSCV = RHSC->getValue();
        return getConstant(cast<ConstantInt>(ConstantExpr::getUDiv(LHSCV,
                                                                   RHSCV)));
      }
    }
  }

  FoldingSetNodeID ID;
  ID.AddInteger(scUDivExpr);
  ID.AddPointer(LHS);
  ID.AddPointer(RHS);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVUDivExpr(ID.Intern(SCEVAllocator),
                                             LHS, RHS);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}


/// getAddRecExpr - Get an add recurrence expression for the specified loop.
/// Simplify the expression as much as possible.
const SCEV *ScalarEvolution::getAddRecExpr(const SCEV *Start, const SCEV *Step,
                                           const Loop *L,
                                           SCEV::NoWrapFlags Flags) {
  SmallVector<const SCEV *, 4> Operands;
  Operands.push_back(Start);
  if (const SCEVAddRecExpr *StepChrec = dyn_cast<SCEVAddRecExpr>(Step))
    if (StepChrec->getLoop() == L) {
      Operands.append(StepChrec->op_begin(), StepChrec->op_end());
      return getAddRecExpr(Operands, L, maskFlags(Flags, SCEV::FlagNW));
    }

  Operands.push_back(Step);
  return getAddRecExpr(Operands, L, Flags);
}

/// getAddRecExpr - Get an add recurrence expression for the specified loop.
/// Simplify the expression as much as possible.
const SCEV *
ScalarEvolution::getAddRecExpr(SmallVectorImpl<const SCEV *> &Operands,
                               const Loop *L, SCEV::NoWrapFlags Flags) {
  if (Operands.size() == 1) return Operands[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Operands[0]->getType());
  for (unsigned i = 1, e = Operands.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Operands[i]->getType()) == ETy &&
           "SCEVAddRecExpr operand types don't match!");
  for (unsigned i = 0, e = Operands.size(); i != e; ++i)
    assert(isLoopInvariant(Operands[i], L) &&
           "SCEVAddRecExpr operand is not loop-invariant!");
#endif

  if (Operands.back()->isZero()) {
    Operands.pop_back();
    return getAddRecExpr(Operands, L, SCEV::FlagAnyWrap); // {X,+,0}  -->  X
  }

  // It's tempting to want to call getMaxBackedgeTakenCount count here and
  // use that information to infer NUW and NSW flags. However, computing a
  // BE count requires calling getAddRecExpr, so we may not yet have a
  // meaningful BE count at this point (and if we don't, we'd be stuck
  // with a SCEVCouldNotCompute as the cached BE count).

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  // And vice-versa.
  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap = maskFlags(Flags, SignOrUnsignMask);
  if (SignOrUnsignWrap && (SignOrUnsignWrap != SignOrUnsignMask)) {
    bool All = true;
    for (SmallVectorImpl<const SCEV *>::const_iterator I = Operands.begin(),
         E = Operands.end(); I != E; ++I)
      if (!isKnownNonNegative(*I)) {
        All = false;
        break;
      }
    if (All) Flags = setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);
  }

  // Canonicalize nested AddRecs in by nesting them in order of loop depth.
  if (const SCEVAddRecExpr *NestedAR = dyn_cast<SCEVAddRecExpr>(Operands[0])) {
    const Loop *NestedLoop = NestedAR->getLoop();
    if (L->contains(NestedLoop) ?
        (L->getLoopDepth() < NestedLoop->getLoopDepth()) :
        (!NestedLoop->contains(L) &&
         DT->dominates(L->getHeader(), NestedLoop->getHeader()))) {
      SmallVector<const SCEV *, 4> NestedOperands(NestedAR->op_begin(),
                                                  NestedAR->op_end());
      Operands[0] = NestedAR->getStart();
      // AddRecs require their operands be loop-invariant with respect to their
      // loops. Don't perform this transformation if it would break this
      // requirement.
      bool AllInvariant = true;
      for (unsigned i = 0, e = Operands.size(); i != e; ++i)
        if (!isLoopInvariant(Operands[i], L)) {
          AllInvariant = false;
          break;
        }
      if (AllInvariant) {
        // Create a recurrence for the outer loop with the same step size.
        //
        // The outer recurrence keeps its NW flag but only keeps NUW/NSW if the
        // inner recurrence has the same property.
        SCEV::NoWrapFlags OuterFlags =
          maskFlags(Flags, SCEV::FlagNW | NestedAR->getNoWrapFlags());

        NestedOperands[0] = getAddRecExpr(Operands, L, OuterFlags);
        AllInvariant = true;
        for (unsigned i = 0, e = NestedOperands.size(); i != e; ++i)
          if (!isLoopInvariant(NestedOperands[i], NestedLoop)) {
            AllInvariant = false;
            break;
          }
        if (AllInvariant) {
          // Ok, both add recurrences are valid after the transformation.
          //
          // The inner recurrence keeps its NW flag but only keeps NUW/NSW if
          // the outer recurrence has the same property.
          SCEV::NoWrapFlags InnerFlags =
            maskFlags(NestedAR->getNoWrapFlags(), SCEV::FlagNW | Flags);
          return getAddRecExpr(NestedOperands, NestedLoop, InnerFlags);
        }
      }
      // Reset Operands to its original state.
      Operands[0] = NestedAR;
    }
  }

  // Okay, it looks like we really DO need an addrec expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scAddRecExpr);
  for (unsigned i = 0, e = Operands.size(); i != e; ++i)
    ID.AddPointer(Operands[i]);
  ID.AddPointer(L);
  void *IP = 0;
  SCEVAddRecExpr *S =
    static_cast<SCEVAddRecExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));
  if (!S) {
    const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Operands.size());
    std::uninitialized_copy(Operands.begin(), Operands.end(), O);
    S = new (SCEVAllocator) SCEVAddRecExpr(ID.Intern(SCEVAllocator),
                                           O, Operands.size(), L);
    UniqueSCEVs.InsertNode(S, IP);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

const SCEV *ScalarEvolution::getSMaxExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops;
  Ops.push_back(LHS);
  Ops.push_back(RHS);
  return getSMaxExpr(Ops);
}

const SCEV *
ScalarEvolution::getSMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "Cannot get empty smax!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVSMaxExpr operand types don't match!");
#endif

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                              APIntOps::smax(LHSC->getValue()->getValue(),
                                             RHSC->getValue()->getValue()));
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant minimum-int, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->isMinValue(true)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isMaxValue(true)) {
      // If we have an smax with a constant maximum-int, it will always be
      // maximum-int.
      return Ops[0];
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Find the first SMax
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scSMaxExpr)
    ++Idx;

  // Check to see if one of the operands is an SMax. If so, expand its operands
  // onto our operand list, and recurse to simplify.
  if (Idx < Ops.size()) {
    bool DeletedSMax = false;
    while (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(Ops[Idx])) {
      Ops.erase(Ops.begin()+Idx);
      Ops.append(SMax->op_begin(), SMax->op_end());
      DeletedSMax = true;
    }

    if (DeletedSMax)
      return getSMaxExpr(Ops);
  }

  // Okay, check to see if the same value occurs in the operand list twice.  If
  // so, delete one.  Since we sorted the list, these values are required to
  // be adjacent.
  for (unsigned i = 0, e = Ops.size()-1; i != e; ++i)
    //  X smax Y smax Y  -->  X smax Y
    //  X smax Y         -->  X, if X is always greater than Y
    if (Ops[i] == Ops[i+1] ||
        isKnownPredicate(ICmpInst::ICMP_SGE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+2);
      --i; --e;
    } else if (isKnownPredicate(ICmpInst::ICMP_SLE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i, Ops.begin()+i+1);
      --i; --e;
    }

  if (Ops.size() == 1) return Ops[0];

  assert(!Ops.empty() && "Reduced smax down to nothing!");

  // Okay, it looks like we really DO need an smax expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scSMaxExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
  std::uninitialized_copy(Ops.begin(), Ops.end(), O);
  SCEV *S = new (SCEVAllocator) SCEVSMaxExpr(ID.Intern(SCEVAllocator),
                                             O, Ops.size());
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getUMaxExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops;
  Ops.push_back(LHS);
  Ops.push_back(RHS);
  return getUMaxExpr(Ops);
}

const SCEV *
ScalarEvolution::getUMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "Cannot get empty umax!");
  if (Ops.size() == 1) return Ops[0];
#ifndef NDEBUG
  Type *ETy = getEffectiveSCEVType(Ops[0]->getType());
  for (unsigned i = 1, e = Ops.size(); i != e; ++i)
    assert(getEffectiveSCEVType(Ops[i]->getType()) == ETy &&
           "SCEVUMaxExpr operand types don't match!");
#endif

  // Sort by complexity, this groups all similar expression types together.
  GroupByComplexity(Ops, LI);

  // If there are any constants, fold them together.
  unsigned Idx = 0;
  if (const SCEVConstant *LHSC = dyn_cast<SCEVConstant>(Ops[0])) {
    ++Idx;
    assert(Idx < Ops.size());
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(getContext(),
                              APIntOps::umax(LHSC->getValue()->getValue(),
                                             RHSC->getValue()->getValue()));
      Ops[0] = getConstant(Fold);
      Ops.erase(Ops.begin()+1);  // Erase the folded element
      if (Ops.size() == 1) return Ops[0];
      LHSC = cast<SCEVConstant>(Ops[0]);
    }

    // If we are left with a constant minimum-int, strip it off.
    if (cast<SCEVConstant>(Ops[0])->getValue()->isMinValue(false)) {
      Ops.erase(Ops.begin());
      --Idx;
    } else if (cast<SCEVConstant>(Ops[0])->getValue()->isMaxValue(false)) {
      // If we have an umax with a constant maximum-int, it will always be
      // maximum-int.
      return Ops[0];
    }

    if (Ops.size() == 1) return Ops[0];
  }

  // Find the first UMax
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < scUMaxExpr)
    ++Idx;

  // Check to see if one of the operands is a UMax. If so, expand its operands
  // onto our operand list, and recurse to simplify.
  if (Idx < Ops.size()) {
    bool DeletedUMax = false;
    while (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(Ops[Idx])) {
      Ops.erase(Ops.begin()+Idx);
      Ops.append(UMax->op_begin(), UMax->op_end());
      DeletedUMax = true;
    }

    if (DeletedUMax)
      return getUMaxExpr(Ops);
  }

  // Okay, check to see if the same value occurs in the operand list twice.  If
  // so, delete one.  Since we sorted the list, these values are required to
  // be adjacent.
  for (unsigned i = 0, e = Ops.size()-1; i != e; ++i)
    //  X umax Y umax Y  -->  X umax Y
    //  X umax Y         -->  X, if X is always greater than Y
    if (Ops[i] == Ops[i+1] ||
        isKnownPredicate(ICmpInst::ICMP_UGE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i+1, Ops.begin()+i+2);
      --i; --e;
    } else if (isKnownPredicate(ICmpInst::ICMP_ULE, Ops[i], Ops[i+1])) {
      Ops.erase(Ops.begin()+i, Ops.begin()+i+1);
      --i; --e;
    }

  if (Ops.size() == 1) return Ops[0];

  assert(!Ops.empty() && "Reduced umax down to nothing!");

  // Okay, it looks like we really DO need a umax expr.  Check to see if we
  // already have one, otherwise create a new one.
  FoldingSetNodeID ID;
  ID.AddInteger(scUMaxExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = 0;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
  std::uninitialized_copy(Ops.begin(), Ops.end(), O);
  SCEV *S = new (SCEVAllocator) SCEVUMaxExpr(ID.Intern(SCEVAllocator),
                                             O, Ops.size());
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

const SCEV *ScalarEvolution::getSMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  // ~smax(~x, ~y) == smin(x, y).
  return getNotSCEV(getSMaxExpr(getNotSCEV(LHS), getNotSCEV(RHS)));
}

const SCEV *ScalarEvolution::getUMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  // ~umax(~x, ~y) == umin(x, y)
  return getNotSCEV(getUMaxExpr(getNotSCEV(LHS), getNotSCEV(RHS)));
}

const SCEV *ScalarEvolution::getSizeOfExpr(Type *AllocTy) {
  // If we have DataLayout, we can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  if (TD)
    return getConstant(TD->getIntPtrType(getContext()),
                       TD->getTypeAllocSize(AllocTy));

  Constant *C = ConstantExpr::getSizeOf(AllocTy);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;
  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(AllocTy));
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getAlignOfExpr(Type *AllocTy) {
  Constant *C = ConstantExpr::getAlignOf(AllocTy);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;
  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(AllocTy));
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getOffsetOfExpr(StructType *STy,
                                             unsigned FieldNo) {
  // If we have DataLayout, we can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  if (TD)
    return getConstant(TD->getIntPtrType(getContext()),
                       TD->getStructLayout(STy)->getElementOffset(FieldNo));

  Constant *C = ConstantExpr::getOffsetOf(STy, FieldNo);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;
  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(STy));
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getOffsetOfExpr(Type *CTy,
                                             Constant *FieldNo) {
  Constant *C = ConstantExpr::getOffsetOf(CTy, FieldNo);
  if (ConstantExpr *CE = dyn_cast<ConstantExpr>(C))
    if (Constant *Folded = ConstantFoldConstantExpression(CE, TD, TLI))
      C = Folded;
  Type *Ty = getEffectiveSCEVType(PointerType::getUnqual(CTy));
  return getTruncateOrZeroExtend(getSCEV(C), Ty);
}

const SCEV *ScalarEvolution::getUnknown(Value *V) {
  // Don't attempt to do anything other than create a SCEVUnknown object
  // here.  createSCEV only calls getUnknown after checking for all other
  // interesting possibilities, and any other code that calls getUnknown
  // is doing so in order to hide a value from SCEV canonicalization.

  FoldingSetNodeID ID;
  ID.AddInteger(scUnknown);
  ID.AddPointer(V);
  void *IP = 0;
  if (SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) {
    assert(cast<SCEVUnknown>(S)->getValue() == V &&
           "Stale SCEVUnknown in uniquing map!");
    return S;
  }
  SCEV *S = new (SCEVAllocator) SCEVUnknown(ID.Intern(SCEVAllocator), V, this,
                                            FirstUnknown);
  FirstUnknown = cast<SCEVUnknown>(S);
  UniqueSCEVs.InsertNode(S, IP);
  return S;
}

//===----------------------------------------------------------------------===//
//            Basic SCEV Analysis and PHI Idiom Recognition Code
//

/// isSCEVable - Test if values of the given type are analyzable within
/// the SCEV framework. This primarily includes integer types, and it
/// can optionally include pointer types if the ScalarEvolution class
/// has access to target-specific information.
bool ScalarEvolution::isSCEVable(Type *Ty) const {
  // Integers and pointers are always SCEVable.
  return Ty->isIntegerTy() || Ty->isPointerTy();
}

/// getTypeSizeInBits - Return the size in bits of the specified type,
/// for which isSCEVable must return true.
uint64_t ScalarEvolution::getTypeSizeInBits(Type *Ty) const {
  assert(isSCEVable(Ty) && "Type is not SCEVable!");

  // If we have a DataLayout, use it!
  if (TD)
    return TD->getTypeSizeInBits(Ty);

  // Integer types have fixed sizes.
  if (Ty->isIntegerTy())
    return Ty->getPrimitiveSizeInBits();

  // The only other support type is pointer. Without DataLayout, conservatively
  // assume pointers are 64-bit.
  assert(Ty->isPointerTy() && "isSCEVable permitted a non-SCEVable type!");
  return 64;
}

/// getEffectiveSCEVType - Return a type with the same bitwidth as
/// the given type and which represents how SCEV will treat the given
/// type, for which isSCEVable must return true. For pointer types,
/// this is the pointer-sized integer type.
Type *ScalarEvolution::getEffectiveSCEVType(Type *Ty) const {
  assert(isSCEVable(Ty) && "Type is not SCEVable!");

  if (Ty->isIntegerTy())
    return Ty;

  // The only other support type is pointer.
  assert(Ty->isPointerTy() && "Unexpected non-pointer non-integer type!");
  if (TD) return TD->getIntPtrType(getContext());

  // Without DataLayout, conservatively assume pointers are 64-bit.
  return Type::getInt64Ty(getContext());
}

const SCEV *ScalarEvolution::getCouldNotCompute() {
  return &CouldNotCompute;
}

/// getSCEV - Return an existing SCEV if it exists, otherwise analyze the
/// expression and create a new one.
const SCEV *ScalarEvolution::getSCEV(Value *V) {
  assert(isSCEVable(V->getType()) && "Value is not SCEVable!");

  ValueExprMapType::const_iterator I = ValueExprMap.find_as(V);
  if (I != ValueExprMap.end()) return I->second;
  const SCEV *S = createSCEV(V);

  // The process of creating a SCEV for V may have caused other SCEVs
  // to have been created, so it's necessary to insert the new entry
  // from scratch, rather than trying to remember the insert position
  // above.
  ValueExprMap.insert(std::make_pair(SCEVCallbackVH(V, this), S));
  return S;
}

/// getNegativeSCEV - Return a SCEV corresponding to -V = -1*V
///
const SCEV *ScalarEvolution::getNegativeSCEV(const SCEV *V) {
  if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
    return getConstant(
               cast<ConstantInt>(ConstantExpr::getNeg(VC->getValue())));

  Type *Ty = V->getType();
  Ty = getEffectiveSCEVType(Ty);
  return getMulExpr(V,
                  getConstant(cast<ConstantInt>(Constant::getAllOnesValue(Ty))));
}

/// getNotSCEV - Return a SCEV corresponding to ~V = -1-V
const SCEV *ScalarEvolution::getNotSCEV(const SCEV *V) {
  if (const SCEVConstant *VC = dyn_cast<SCEVConstant>(V))
    return getConstant(
                cast<ConstantInt>(ConstantExpr::getNot(VC->getValue())));

  Type *Ty = V->getType();
  Ty = getEffectiveSCEVType(Ty);
  const SCEV *AllOnes =
                   getConstant(cast<ConstantInt>(Constant::getAllOnesValue(Ty)));
  return getMinusSCEV(AllOnes, V);
}

/// getMinusSCEV - Return LHS-RHS.  Minus is represented in SCEV as A+B*-1.
const SCEV *ScalarEvolution::getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                                          SCEV::NoWrapFlags Flags) {
  assert(!maskFlags(Flags, SCEV::FlagNUW) && "subtraction does not have NUW");

  // Fast path: X - X --> 0.
  if (LHS == RHS)
    return getConstant(LHS->getType(), 0);

  // X - Y --> X + -Y
  return getAddExpr(LHS, getNegativeSCEV(RHS), Flags);
}

/// getTruncateOrZeroExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is zero
/// extended.
const SCEV *
ScalarEvolution::getTruncateOrZeroExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or zero extend with non-integer arguments!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
    return getTruncateExpr(V, Ty);
  return getZeroExtendExpr(V, Ty);
}

/// getTruncateOrSignExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is sign
/// extended.
const SCEV *
ScalarEvolution::getTruncateOrSignExtend(const SCEV *V,
                                         Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or zero extend with non-integer arguments!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  if (getTypeSizeInBits(SrcTy) > getTypeSizeInBits(Ty))
    return getTruncateExpr(V, Ty);
  return getSignExtendExpr(V, Ty);
}

/// getNoopOrZeroExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is zero
/// extended.  The conversion must not be narrowing.
const SCEV *
ScalarEvolution::getNoopOrZeroExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or zero extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrZeroExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getZeroExtendExpr(V, Ty);
}

/// getNoopOrSignExtend - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  If the type must be extended, it is sign
/// extended.  The conversion must not be narrowing.
const SCEV *
ScalarEvolution::getNoopOrSignExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or sign extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrSignExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getSignExtendExpr(V, Ty);
}

/// getNoopOrAnyExtend - Return a SCEV corresponding to a conversion of
/// the input value to the specified type. If the type must be extended,
/// it is extended with unspecified bits. The conversion must not be
/// narrowing.
const SCEV *
ScalarEvolution::getNoopOrAnyExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot noop or any extend with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) <= getTypeSizeInBits(Ty) &&
         "getNoopOrAnyExtend cannot truncate!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getAnyExtendExpr(V, Ty);
}

/// getTruncateOrNoop - Return a SCEV corresponding to a conversion of the
/// input value to the specified type.  The conversion must not be widening.
const SCEV *
ScalarEvolution::getTruncateOrNoop(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert((SrcTy->isIntegerTy() || SrcTy->isPointerTy()) &&
         (Ty->isIntegerTy() || Ty->isPointerTy()) &&
         "Cannot truncate or noop with non-integer arguments!");
  assert(getTypeSizeInBits(SrcTy) >= getTypeSizeInBits(Ty) &&
         "getTruncateOrNoop cannot extend!");
  if (getTypeSizeInBits(SrcTy) == getTypeSizeInBits(Ty))
    return V;  // No conversion
  return getTruncateExpr(V, Ty);
}

/// getUMaxFromMismatchedTypes - Promote the operands to the wider of
/// the types using zero-extension, and then perform a umax operation
/// with them.
const SCEV *ScalarEvolution::getUMaxFromMismatchedTypes(const SCEV *LHS,
                                                        const SCEV *RHS) {
  const SCEV *PromotedLHS = LHS;
  const SCEV *PromotedRHS = RHS;

  if (getTypeSizeInBits(LHS->getType()) > getTypeSizeInBits(RHS->getType()))
    PromotedRHS = getZeroExtendExpr(RHS, LHS->getType());
  else
    PromotedLHS = getNoopOrZeroExtend(LHS, RHS->getType());

  return getUMaxExpr(PromotedLHS, PromotedRHS);
}

/// getUMinFromMismatchedTypes - Promote the operands to the wider of
/// the types using zero-extension, and then perform a umin operation
/// with them.
const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(const SCEV *LHS,
                                                        const SCEV *RHS) {
  const SCEV *PromotedLHS = LHS;
  const SCEV *PromotedRHS = RHS;

  if (getTypeSizeInBits(LHS->getType()) > getTypeSizeInBits(RHS->getType()))
    PromotedRHS = getZeroExtendExpr(RHS, LHS->getType());
  else
    PromotedLHS = getNoopOrZeroExtend(LHS, RHS->getType());

  return getUMinExpr(PromotedLHS, PromotedRHS);
}

/// getPointerBase - Transitively follow the chain of pointer-type operands
/// until reaching a SCEV that does not have a single pointer operand. This
/// returns a SCEVUnknown pointer for well-formed pointer-type expressions,
/// but corner cases do exist.
const SCEV *ScalarEvolution::getPointerBase(const SCEV *V) {
  // A pointer operand may evaluate to a nonpointer expression, such as null.
  if (!V->getType()->isPointerTy())
    return V;

  if (const SCEVCastExpr *Cast = dyn_cast<SCEVCastExpr>(V)) {
    return getPointerBase(Cast->getOperand());
  }
  else if (const SCEVNAryExpr *NAry = dyn_cast<SCEVNAryExpr>(V)) {
    const SCEV *PtrOp = 0;
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      if ((*I)->getType()->isPointerTy()) {
        // Cannot find the base of an expression with multiple pointer operands.
        if (PtrOp)
          return V;
        PtrOp = *I;
      }
    }
    if (!PtrOp)
      return V;
    return getPointerBase(PtrOp);
  }
  return V;
}

/// PushDefUseChildren - Push users of the given Instruction
/// onto the given Worklist.
static void
PushDefUseChildren(Instruction *I,
                   SmallVectorImpl<Instruction *> &Worklist) {
  // Push the def-use children onto the Worklist stack.
  for (Value::use_iterator UI = I->use_begin(), UE = I->use_end();
       UI != UE; ++UI)
    Worklist.push_back(cast<Instruction>(*UI));
}

/// ForgetSymbolicValue - This looks up computed SCEV values for all
/// instructions that depend on the given instruction and removes them from
/// the ValueExprMapType map if they reference SymName. This is used during PHI
/// resolution.
void
ScalarEvolution::ForgetSymbolicName(Instruction *PN, const SCEV *SymName) {
  SmallVector<Instruction *, 16> Worklist;
  PushDefUseChildren(PN, Worklist);

  SmallPtrSet<Instruction *, 8> Visited;
  Visited.insert(PN);
  while (!Worklist.empty()) {
    Instruction *I = Worklist.pop_back_val();
    if (!Visited.insert(I)) continue;

    ValueExprMapType::iterator It =
      ValueExprMap.find_as(static_cast<Value *>(I));
    if (It != ValueExprMap.end()) {
      const SCEV *Old = It->second;

      // Short-circuit the def-use traversal if the symbolic name
      // ceases to appear in expressions.
      if (Old != SymName && !hasOperand(Old, SymName))
        continue;

      // SCEVUnknown for a PHI either means that it has an unrecognized
      // structure, it's a PHI that's in the progress of being computed
      // by createNodeForPHI, or it's a single-value PHI. In the first case,
      // additional loop trip count information isn't going to change anything.
      // In the second case, createNodeForPHI will perform the necessary
      // updates on its own when it gets to that point. In the third, we do
      // want to forget the SCEVUnknown.
      if (!isa<PHINode>(I) ||
          !isa<SCEVUnknown>(Old) ||
          (I != PN && Old == SymName)) {
        forgetMemoizedResults(Old);
        ValueExprMap.erase(It);
      }
    }

    PushDefUseChildren(I, Worklist);
  }
}

/// createNodeForPHI - PHI nodes have two cases.  Either the PHI node exists in
/// a loop header, making it a potential recurrence, or it doesn't.
///
const SCEV *ScalarEvolution::createNodeForPHI(PHINode *PN) {
  if (const Loop *L = LI->getLoopFor(PN->getParent()))
    if (L->getHeader() == PN->getParent()) {
      // The loop may have multiple entrances or multiple exits; we can analyze
      // this phi as an addrec if it has a unique entry value and a unique
      // backedge value.
      Value *BEValueV = 0, *StartValueV = 0;
      for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i) {
        Value *V = PN->getIncomingValue(i);
        if (L->contains(PN->getIncomingBlock(i))) {
          if (!BEValueV) {
            BEValueV = V;
          } else if (BEValueV != V) {
            BEValueV = 0;
            break;
          }
        } else if (!StartValueV) {
          StartValueV = V;
        } else if (StartValueV != V) {
          StartValueV = 0;
          break;
        }
      }
      if (BEValueV && StartValueV) {
        // While we are analyzing this PHI node, handle its value symbolically.
        const SCEV *SymbolicName = getUnknown(PN);
        assert(ValueExprMap.find_as(PN) == ValueExprMap.end() &&
               "PHI node already processed?");
        ValueExprMap.insert(std::make_pair(SCEVCallbackVH(PN, this), SymbolicName));

        // Using this symbolic name for the PHI, analyze the value coming around
        // the back-edge.
        const SCEV *BEValue = getSCEV(BEValueV);

        // NOTE: If BEValue is loop invariant, we know that the PHI node just
        // has a special value for the first iteration of the loop.

        // If the value coming around the backedge is an add with the symbolic
        // value we just inserted, then we found a simple induction variable!
        if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(BEValue)) {
          // If there is a single occurrence of the symbolic value, replace it
          // with a recurrence.
          unsigned FoundIndex = Add->getNumOperands();
          for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
            if (Add->getOperand(i) == SymbolicName)
              if (FoundIndex == e) {
                FoundIndex = i;
                break;
              }

          if (FoundIndex != Add->getNumOperands()) {
            // Create an add with everything but the specified operand.
            SmallVector<const SCEV *, 8> Ops;
            for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
              if (i != FoundIndex)
                Ops.push_back(Add->getOperand(i));
            const SCEV *Accum = getAddExpr(Ops);

            // This is not a valid addrec if the step amount is varying each
            // loop iteration, but is not itself an addrec in this loop.
            if (isLoopInvariant(Accum, L) ||
                (isa<SCEVAddRecExpr>(Accum) &&
                 cast<SCEVAddRecExpr>(Accum)->getLoop() == L)) {
              SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;

              // If the increment doesn't overflow, then neither the addrec nor
              // the post-increment will overflow.
              if (const AddOperator *OBO = dyn_cast<AddOperator>(BEValueV)) {
                if (OBO->hasNoUnsignedWrap())
                  Flags = setFlags(Flags, SCEV::FlagNUW);
                if (OBO->hasNoSignedWrap())
                  Flags = setFlags(Flags, SCEV::FlagNSW);
              } else if (const GEPOperator *GEP =
                         dyn_cast<GEPOperator>(BEValueV)) {
                // If the increment is an inbounds GEP, then we know the address
                // space cannot be wrapped around. We cannot make any guarantee
                // about signed or unsigned overflow because pointers are
                // unsigned but we may have a negative index from the base
                // pointer.
                if (GEP->isInBounds())
                  Flags = setFlags(Flags, SCEV::FlagNW);
              }

              const SCEV *StartVal = getSCEV(StartValueV);
              const SCEV *PHISCEV = getAddRecExpr(StartVal, Accum, L, Flags);

              // Since the no-wrap flags are on the increment, they apply to the
              // post-incremented value as well.
              if (isLoopInvariant(Accum, L))
                (void)getAddRecExpr(getAddExpr(StartVal, Accum),
                                    Accum, L, Flags);

              // Okay, for the entire analysis of this edge we assumed the PHI
              // to be symbolic.  We now need to go back and purge all of the
              // entries for the scalars that use the symbolic expression.
              ForgetSymbolicName(PN, SymbolicName);
              ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
              return PHISCEV;
            }
          }
        } else if (const SCEVAddRecExpr *AddRec =
                     dyn_cast<SCEVAddRecExpr>(BEValue)) {
          // Otherwise, this could be a loop like this:
          //     i = 0;  for (j = 1; ..; ++j) { ....  i = j; }
          // In this case, j = {1,+,1}  and BEValue is j.
          // Because the other in-value of i (0) fits the evolution of BEValue
          // i really is an addrec evolution.
          if (AddRec->getLoop() == L && AddRec->isAffine()) {
            const SCEV *StartVal = getSCEV(StartValueV);

            // If StartVal = j.start - j.stride, we can use StartVal as the
            // initial step of the addrec evolution.
            if (StartVal == getMinusSCEV(AddRec->getOperand(0),
                                         AddRec->getOperand(1))) {
              // FIXME: For constant StartVal, we should be able to infer
              // no-wrap flags.
              const SCEV *PHISCEV =
                getAddRecExpr(StartVal, AddRec->getOperand(1), L,
                              SCEV::FlagAnyWrap);

              // Okay, for the entire analysis of this edge we assumed the PHI
              // to be symbolic.  We now need to go back and purge all of the
              // entries for the scalars that use the symbolic expression.
              ForgetSymbolicName(PN, SymbolicName);
              ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;
              return PHISCEV;
            }
          }
        }
      }
    }

  // If the PHI has a single incoming value, follow that value, unless the
  // PHI's incoming blocks are in a different loop, in which case doing so
  // risks breaking LCSSA form. Instcombine would normally zap these, but
  // it doesn't have DominatorTree information, so it may miss cases.
  if (Value *V = SimplifyInstruction(PN, TD, TLI, DT))
    if (LI->replacementPreservesLCSSAForm(PN, V))
      return getSCEV(V);

  // If it's not a loop phi, we can't handle it yet.
  return getUnknown(PN);
}

/// createNodeForGEP - Expand GEP instructions into add and multiply
/// operations. This allows them to be analyzed by regular SCEV code.
///
const SCEV *ScalarEvolution::createNodeForGEP(GEPOperator *GEP) {

  // Don't blindly transfer the inbounds flag from the GEP instruction to the
  // Add expression, because the Instruction may be guarded by control flow
  // and the no-overflow bits may not be valid for the expression in any
  // context.
  bool isInBounds = GEP->isInBounds();

  Type *IntPtrTy = getEffectiveSCEVType(GEP->getType());
  Value *Base = GEP->getOperand(0);
  // Don't attempt to analyze GEPs over unsized objects.
  if (!cast<PointerType>(Base->getType())->getElementType()->isSized())
    return getUnknown(GEP);
  const SCEV *TotalOffset = getConstant(IntPtrTy, 0);
  gep_type_iterator GTI = gep_type_begin(GEP);
  for (GetElementPtrInst::op_iterator I = llvm::next(GEP->op_begin()),
                                      E = GEP->op_end();
       I != E; ++I) {
    Value *Index = *I;
    // Compute the (potentially symbolic) offset in bytes for this index.
    if (StructType *STy = dyn_cast<StructType>(*GTI++)) {
      // For a struct, add the member offset.
      unsigned FieldNo = cast<ConstantInt>(Index)->getZExtValue();
      const SCEV *FieldOffset = getOffsetOfExpr(STy, FieldNo);

      // Add the field offset to the running total offset.
      TotalOffset = getAddExpr(TotalOffset, FieldOffset);
    } else {
      // For an array, add the element offset, explicitly scaled.
      const SCEV *ElementSize = getSizeOfExpr(*GTI);
      const SCEV *IndexS = getSCEV(Index);
      // Getelementptr indices are signed.
      IndexS = getTruncateOrSignExtend(IndexS, IntPtrTy);

      // Multiply the index by the element size to compute the element offset.
      const SCEV *LocalOffset = getMulExpr(IndexS, ElementSize,
                                           isInBounds ? SCEV::FlagNSW :
                                           SCEV::FlagAnyWrap);

      // Add the element offset to the running total offset.
      TotalOffset = getAddExpr(TotalOffset, LocalOffset);
    }
  }

  // Get the SCEV for the GEP base.
  const SCEV *BaseS = getSCEV(Base);

  // Add the total offset from all the GEP indices to the base.
  return getAddExpr(BaseS, TotalOffset,
                    isInBounds ? SCEV::FlagNSW : SCEV::FlagAnyWrap);
}

/// GetMinTrailingZeros - Determine the minimum number of zero bits that S is
/// guaranteed to end in (at every loop iteration).  It is, at the same time,
/// the minimum number of times S is divisible by 2.  For example, given {4,+,8}
/// it returns 2.  If S is guaranteed to be 0, it returns the bitwidth of S.
uint32_t
ScalarEvolution::GetMinTrailingZeros(const SCEV *S) {
  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return C->getValue()->getValue().countTrailingZeros();

  if (const SCEVTruncateExpr *T = dyn_cast<SCEVTruncateExpr>(S))
    return std::min(GetMinTrailingZeros(T->getOperand()),
                    (uint32_t)getTypeSizeInBits(T->getType()));

  if (const SCEVZeroExtendExpr *E = dyn_cast<SCEVZeroExtendExpr>(S)) {
    uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
    return OpRes == getTypeSizeInBits(E->getOperand()->getType()) ?
             getTypeSizeInBits(E->getType()) : OpRes;
  }

  if (const SCEVSignExtendExpr *E = dyn_cast<SCEVSignExtendExpr>(S)) {
    uint32_t OpRes = GetMinTrailingZeros(E->getOperand());
    return OpRes == getTypeSizeInBits(E->getOperand()->getType()) ?
             getTypeSizeInBits(E->getType()) : OpRes;
  }

  if (const SCEVAddExpr *A = dyn_cast<SCEVAddExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
    for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVMulExpr *M = dyn_cast<SCEVMulExpr>(S)) {
    // The result is the sum of all operands results.
    uint32_t SumOpRes = GetMinTrailingZeros(M->getOperand(0));
    uint32_t BitWidth = getTypeSizeInBits(M->getType());
    for (unsigned i = 1, e = M->getNumOperands();
         SumOpRes != BitWidth && i != e; ++i)
      SumOpRes = std::min(SumOpRes + GetMinTrailingZeros(M->getOperand(i)),
                          BitWidth);
    return SumOpRes;
  }

  if (const SCEVAddRecExpr *A = dyn_cast<SCEVAddRecExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(A->getOperand(0));
    for (unsigned i = 1, e = A->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(A->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVSMaxExpr *M = dyn_cast<SCEVSMaxExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
    for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVUMaxExpr *M = dyn_cast<SCEVUMaxExpr>(S)) {
    // The result is the min of all operands results.
    uint32_t MinOpRes = GetMinTrailingZeros(M->getOperand(0));
    for (unsigned i = 1, e = M->getNumOperands(); MinOpRes && i != e; ++i)
      MinOpRes = std::min(MinOpRes, GetMinTrailingZeros(M->getOperand(i)));
    return MinOpRes;
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    unsigned BitWidth = getTypeSizeInBits(U->getType());
    APInt Zeros(BitWidth, 0), Ones(BitWidth, 0);
    ComputeMaskedBits(U->getValue(), Zeros, Ones);
    return Zeros.countTrailingOnes();
  }

  // SCEVUDivExpr
  return 0;
}

/// getUnsignedRange - Determine the unsigned range for a particular SCEV.
///
ConstantRange
ScalarEvolution::getUnsignedRange(const SCEV *S) {
  // See if we've computed this range already.
  DenseMap<const SCEV *, ConstantRange>::iterator I = UnsignedRanges.find(S);
  if (I != UnsignedRanges.end())
    return I->second;

  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return setUnsignedRange(C, ConstantRange(C->getValue()->getValue()));

  unsigned BitWidth = getTypeSizeInBits(S->getType());
  ConstantRange ConservativeResult(BitWidth, /*isFullSet=*/true);

  // If the value has known zeros, the maximum unsigned value will have those
  // known zeros as well.
  uint32_t TZ = GetMinTrailingZeros(S);
  if (TZ != 0)
    ConservativeResult =
      ConstantRange(APInt::getMinValue(BitWidth),
                    APInt::getMaxValue(BitWidth).lshr(TZ).shl(TZ) + 1);

  if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(S)) {
    ConstantRange X = getUnsignedRange(Add->getOperand(0));
    for (unsigned i = 1, e = Add->getNumOperands(); i != e; ++i)
      X = X.add(getUnsignedRange(Add->getOperand(i)));
    return setUnsignedRange(Add, ConservativeResult.intersectWith(X));
  }

  if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(S)) {
    ConstantRange X = getUnsignedRange(Mul->getOperand(0));
    for (unsigned i = 1, e = Mul->getNumOperands(); i != e; ++i)
      X = X.multiply(getUnsignedRange(Mul->getOperand(i)));
    return setUnsignedRange(Mul, ConservativeResult.intersectWith(X));
  }

  if (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(S)) {
    ConstantRange X = getUnsignedRange(SMax->getOperand(0));
    for (unsigned i = 1, e = SMax->getNumOperands(); i != e; ++i)
      X = X.smax(getUnsignedRange(SMax->getOperand(i)));
    return setUnsignedRange(SMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(S)) {
    ConstantRange X = getUnsignedRange(UMax->getOperand(0));
    for (unsigned i = 1, e = UMax->getNumOperands(); i != e; ++i)
      X = X.umax(getUnsignedRange(UMax->getOperand(i)));
    return setUnsignedRange(UMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(S)) {
    ConstantRange X = getUnsignedRange(UDiv->getLHS());
    ConstantRange Y = getUnsignedRange(UDiv->getRHS());
    return setUnsignedRange(UDiv, ConservativeResult.intersectWith(X.udiv(Y)));
  }

  if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S)) {
    ConstantRange X = getUnsignedRange(ZExt->getOperand());
    return setUnsignedRange(ZExt,
      ConservativeResult.intersectWith(X.zeroExtend(BitWidth)));
  }

  if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S)) {
    ConstantRange X = getUnsignedRange(SExt->getOperand());
    return setUnsignedRange(SExt,
      ConservativeResult.intersectWith(X.signExtend(BitWidth)));
  }

  if (const SCEVTruncateExpr *Trunc = dyn_cast<SCEVTruncateExpr>(S)) {
    ConstantRange X = getUnsignedRange(Trunc->getOperand());
    return setUnsignedRange(Trunc,
      ConservativeResult.intersectWith(X.truncate(BitWidth)));
  }

  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(S)) {
    // If there's no unsigned wrap, the value will never be less than its
    // initial value.
    if (AddRec->getNoWrapFlags(SCEV::FlagNUW))
      if (const SCEVConstant *C = dyn_cast<SCEVConstant>(AddRec->getStart()))
        if (!C->getValue()->isZero())
          ConservativeResult =
            ConservativeResult.intersectWith(
              ConstantRange(C->getValue()->getValue(), APInt(BitWidth, 0)));

    // TODO: non-affine addrec
    if (AddRec->isAffine()) {
      Type *Ty = AddRec->getType();
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(AddRec->getLoop());
      if (!isa<SCEVCouldNotCompute>(MaxBECount) &&
          getTypeSizeInBits(MaxBECount->getType()) <= BitWidth) {
        MaxBECount = getNoopOrZeroExtend(MaxBECount, Ty);

        const SCEV *Start = AddRec->getStart();
        const SCEV *Step = AddRec->getStepRecurrence(*this);

        ConstantRange StartRange = getUnsignedRange(Start);
        ConstantRange StepRange = getSignedRange(Step);
        ConstantRange MaxBECountRange = getUnsignedRange(MaxBECount);
        ConstantRange EndRange =
          StartRange.add(MaxBECountRange.multiply(StepRange));

        // Check for overflow. This must be done with ConstantRange arithmetic
        // because we could be called from within the ScalarEvolution overflow
        // checking code.
        ConstantRange ExtStartRange = StartRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtStepRange = StepRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtMaxBECountRange =
          MaxBECountRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtEndRange = EndRange.zextOrTrunc(BitWidth*2+1);
        if (ExtStartRange.add(ExtMaxBECountRange.multiply(ExtStepRange)) !=
            ExtEndRange)
          return setUnsignedRange(AddRec, ConservativeResult);

        APInt Min = APIntOps::umin(StartRange.getUnsignedMin(),
                                   EndRange.getUnsignedMin());
        APInt Max = APIntOps::umax(StartRange.getUnsignedMax(),
                                   EndRange.getUnsignedMax());
        if (Min.isMinValue() && Max.isMaxValue())
          return setUnsignedRange(AddRec, ConservativeResult);
        return setUnsignedRange(AddRec,
          ConservativeResult.intersectWith(ConstantRange(Min, Max+1)));
      }
    }

    return setUnsignedRange(AddRec, ConservativeResult);
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    APInt Zeros(BitWidth, 0), Ones(BitWidth, 0);
    ComputeMaskedBits(U->getValue(), Zeros, Ones, TD);
    if (Ones == ~Zeros + 1)
      return setUnsignedRange(U, ConservativeResult);
    return setUnsignedRange(U,
      ConservativeResult.intersectWith(ConstantRange(Ones, ~Zeros + 1)));
  }

  return setUnsignedRange(S, ConservativeResult);
}

/// getSignedRange - Determine the signed range for a particular SCEV.
///
ConstantRange
ScalarEvolution::getSignedRange(const SCEV *S) {
  // See if we've computed this range already.
  DenseMap<const SCEV *, ConstantRange>::iterator I = SignedRanges.find(S);
  if (I != SignedRanges.end())
    return I->second;

  if (const SCEVConstant *C = dyn_cast<SCEVConstant>(S))
    return setSignedRange(C, ConstantRange(C->getValue()->getValue()));

  unsigned BitWidth = getTypeSizeInBits(S->getType());
  ConstantRange ConservativeResult(BitWidth, /*isFullSet=*/true);

  // If the value has known zeros, the maximum signed value will have those
  // known zeros as well.
  uint32_t TZ = GetMinTrailingZeros(S);
  if (TZ != 0)
    ConservativeResult =
      ConstantRange(APInt::getSignedMinValue(BitWidth),
                    APInt::getSignedMaxValue(BitWidth).ashr(TZ).shl(TZ) + 1);

  if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(S)) {
    ConstantRange X = getSignedRange(Add->getOperand(0));
    for (unsigned i = 1, e = Add->getNumOperands(); i != e; ++i)
      X = X.add(getSignedRange(Add->getOperand(i)));
    return setSignedRange(Add, ConservativeResult.intersectWith(X));
  }

  if (const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(S)) {
    ConstantRange X = getSignedRange(Mul->getOperand(0));
    for (unsigned i = 1, e = Mul->getNumOperands(); i != e; ++i)
      X = X.multiply(getSignedRange(Mul->getOperand(i)));
    return setSignedRange(Mul, ConservativeResult.intersectWith(X));
  }

  if (const SCEVSMaxExpr *SMax = dyn_cast<SCEVSMaxExpr>(S)) {
    ConstantRange X = getSignedRange(SMax->getOperand(0));
    for (unsigned i = 1, e = SMax->getNumOperands(); i != e; ++i)
      X = X.smax(getSignedRange(SMax->getOperand(i)));
    return setSignedRange(SMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUMaxExpr *UMax = dyn_cast<SCEVUMaxExpr>(S)) {
    ConstantRange X = getSignedRange(UMax->getOperand(0));
    for (unsigned i = 1, e = UMax->getNumOperands(); i != e; ++i)
      X = X.umax(getSignedRange(UMax->getOperand(i)));
    return setSignedRange(UMax, ConservativeResult.intersectWith(X));
  }

  if (const SCEVUDivExpr *UDiv = dyn_cast<SCEVUDivExpr>(S)) {
    ConstantRange X = getSignedRange(UDiv->getLHS());
    ConstantRange Y = getSignedRange(UDiv->getRHS());
    return setSignedRange(UDiv, ConservativeResult.intersectWith(X.udiv(Y)));
  }

  if (const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(S)) {
    ConstantRange X = getSignedRange(ZExt->getOperand());
    return setSignedRange(ZExt,
      ConservativeResult.intersectWith(X.zeroExtend(BitWidth)));
  }

  if (const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(S)) {
    ConstantRange X = getSignedRange(SExt->getOperand());
    return setSignedRange(SExt,
      ConservativeResult.intersectWith(X.signExtend(BitWidth)));
  }

  if (const SCEVTruncateExpr *Trunc = dyn_cast<SCEVTruncateExpr>(S)) {
    ConstantRange X = getSignedRange(Trunc->getOperand());
    return setSignedRange(Trunc,
      ConservativeResult.intersectWith(X.truncate(BitWidth)));
  }

  if (const SCEVAddRecExpr *AddRec = dyn_cast<SCEVAddRecExpr>(S)) {
    // If there's no signed wrap, and all the operands have the same sign or
    // zero, the value won't ever change sign.
    if (AddRec->getNoWrapFlags(SCEV::FlagNSW)) {
      bool AllNonNeg = true;
      bool AllNonPos = true;
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i) {
        if (!isKnownNonNegative(AddRec->getOperand(i))) AllNonNeg = false;
        if (!isKnownNonPositive(AddRec->getOperand(i))) AllNonPos = false;
      }
      if (AllNonNeg)
        ConservativeResult = ConservativeResult.intersectWith(
          ConstantRange(APInt(BitWidth, 0),
                        APInt::getSignedMinValue(BitWidth)));
      else if (AllNonPos)
        ConservativeResult = ConservativeResult.intersectWith(
          ConstantRange(APInt::getSignedMinValue(BitWidth),
                        APInt(BitWidth, 1)));
    }

    // TODO: non-affine addrec
    if (AddRec->isAffine()) {
      Type *Ty = AddRec->getType();
      const SCEV *MaxBECount = getMaxBackedgeTakenCount(AddRec->getLoop());
      if (!isa<SCEVCouldNotCompute>(MaxBECount) &&
          getTypeSizeInBits(MaxBECount->getType()) <= BitWidth) {
        MaxBECount = getNoopOrZeroExtend(MaxBECount, Ty);

        const SCEV *Start = AddRec->getStart();
        const SCEV *Step = AddRec->getStepRecurrence(*this);

        ConstantRange StartRange = getSignedRange(Start);
        ConstantRange StepRange = getSignedRange(Step);
        ConstantRange MaxBECountRange = getUnsignedRange(MaxBECount);
        ConstantRange EndRange =
          StartRange.add(MaxBECountRange.multiply(StepRange));

        // Check for overflow. This must be done with ConstantRange arithmetic
        // because we could be called from within the ScalarEvolution overflow
        // checking code.
        ConstantRange ExtStartRange = StartRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtStepRange = StepRange.sextOrTrunc(BitWidth*2+1);
        ConstantRange ExtMaxBECountRange =
          MaxBECountRange.zextOrTrunc(BitWidth*2+1);
        ConstantRange ExtEndRange = EndRange.sextOrTrunc(BitWidth*2+1);
        if (ExtStartRange.add(ExtMaxBECountRange.multiply(ExtStepRange)) !=
            ExtEndRange)
          return setSignedRange(AddRec, ConservativeResult);

        APInt Min = APIntOps::smin(StartRange.getSignedMin(),
                                   EndRange.getSignedMin());
        APInt Max = APIntOps::smax(StartRange.getSignedMax(),
                                   EndRange.getSignedMax());
        if (Min.isMinSignedValue() && Max.isMaxSignedValue())
          return setSignedRange(AddRec, ConservativeResult);
        return setSignedRange(AddRec,
          ConservativeResult.intersectWith(ConstantRange(Min, Max+1)));
      }
    }

    return setSignedRange(AddRec, ConservativeResult);
  }

  if (const SCEVUnknown *U = dyn_cast<SCEVUnknown>(S)) {
    // For a SCEVUnknown, ask ValueTracking.
    if (!U->getValue()->getType()->isIntegerTy() && !TD)
      return setSignedRange(U, ConservativeResult);
    unsigned NS = ComputeNumSignBits(U->getValue(), TD);
    if (NS == 1)
      return setSignedRange(U, ConservativeResult);
    return setSignedRange(U, ConservativeResult.intersectWith(
      ConstantRange(APInt::getSignedMinValue(BitWidth).ashr(NS - 1),
                    APInt::getSignedMaxValue(BitWidth).ashr(NS - 1)+1)));
  }

  return setSignedRange(S, ConservativeResult);
}

/// createSCEV - We know that there is no SCEV for the specified value.
/// Analyze the expression.
///
const SCEV *ScalarEvolution::createSCEV(Value *V) {
  if (!isSCEVable(V->getType()))
    return getUnknown(V);

  unsigned Opcode = Instruction::UserOp1;
  if (Instruction *I = dyn_cast<Instruction>(V)) {
    Opcode = I->getOpcode();

    // Don't attempt to analyze instructions in blocks that aren't
    // reachable. Such instructions don't matter, and they aren't required
    // to obey basic rules for definitions dominating uses which this
    // analysis depends on.
    if (!DT->isReachableFromEntry(I->getParent()))
      return getUnknown(V);
  } else if (ConstantExpr *CE = dyn_cast<ConstantExpr>(V))
    Opcode = CE->getOpcode();
  else if (ConstantInt *CI = dyn_cast<ConstantInt>(V))
    return getConstant(CI);
  else if (isa<ConstantPointerNull>(V))
    return getConstant(V->getType(), 0);
  else if (GlobalAlias *GA = dyn_cast<GlobalAlias>(V))
    return GA->mayBeOverridden() ? getUnknown(V) : getSCEV(GA->getAliasee());
  else
    return getUnknown(V);

  Operator *U = cast<Operator>(V);
  switch (Opcode) {
  case Instruction::Add: {
    // The simple thing to do would be to just call getSCEV on both operands
    // and call getAddExpr with the result. However if we're looking at a
    // bunch of things all added together, this can be quite inefficient,
    // because it leads to N-1 getAddExpr calls for N ultimate operands.
    // Instead, gather up all the operands and make a single getAddExpr call.
    // LLVM IR canonical form means we need only traverse the left operands.
    //
    // Don't apply this instruction's NSW or NUW flags to the new
    // expression. The instruction may be guarded by control flow that the
    // no-wrap behavior depends on. Non-control-equivalent instructions can be
    // mapped to the same SCEV expression, and it would be incorrect to transfer
    // NSW/NUW semantics to those operations.
    SmallVector<const SCEV *, 4> AddOps;
    AddOps.push_back(getSCEV(U->getOperand(1)));
    for (Value *Op = U->getOperand(0); ; Op = U->getOperand(0)) {
      unsigned Opcode = Op->getValueID() - Value::InstructionVal;
      if (Opcode != Instruction::Add && Opcode != Instruction::Sub)
        break;
      U = cast<Operator>(Op);
      const SCEV *Op1 = getSCEV(U->getOperand(1));
      if (Opcode == Instruction::Sub)
        AddOps.push_back(getNegativeSCEV(Op1));
      else
        AddOps.push_back(Op1);
    }
    AddOps.push_back(getSCEV(U->getOperand(0)));
    return getAddExpr(AddOps);
  }
  case Instruction::Mul: {
    // Don't transfer NSW/NUW for the same reason as AddExpr.
    SmallVector<const SCEV *, 4> MulOps;
    MulOps.push_back(getSCEV(U->getOperand(1)));
    for (Value *Op = U->getOperand(0);
         Op->getValueID() == Instruction::Mul + Value::InstructionVal;
         Op = U->getOperand(0)) {
      U = cast<Operator>(Op);
      MulOps.push_back(getSCEV(U->getOperand(1)));
    }
    MulOps.push_back(getSCEV(U->getOperand(0)));
    return getMulExpr(MulOps);
  }
  case Instruction::UDiv:
    return getUDivExpr(getSCEV(U->getOperand(0)),
                       getSCEV(U->getOperand(1)));
  case Instruction::Sub:
    return getMinusSCEV(getSCEV(U->getOperand(0)),
                        getSCEV(U->getOperand(1)));
  case Instruction::And:
    // For an expression like x&255 that merely masks off the high bits,
    // use zext(trunc(x)) as the SCEV expression.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      if (CI->isNullValue())
        return getSCEV(U->getOperand(1));
      if (CI->isAllOnesValue())
        return getSCEV(U->getOperand(0));
      const APInt &A = CI->getValue();

      // Instcombine's ShrinkDemandedConstant may strip bits out of
      // constants, obscuring what would otherwise be a low-bits mask.
      // Use ComputeMaskedBits to compute what ShrinkDemandedConstant
      // knew about to reconstruct a low-bits mask value.
      unsigned LZ = A.countLeadingZeros();
      unsigned BitWidth = A.getBitWidth();
      APInt KnownZero(BitWidth, 0), KnownOne(BitWidth, 0);
      ComputeMaskedBits(U->getOperand(0), KnownZero, KnownOne, TD);

      APInt EffectiveMask = APInt::getLowBitsSet(BitWidth, BitWidth - LZ);

      if (LZ != 0 && !((~A & ~KnownZero) & EffectiveMask))
        return
          getZeroExtendExpr(getTruncateExpr(getSCEV(U->getOperand(0)),
                                IntegerType::get(getContext(), BitWidth - LZ)),
                            U->getType());
    }
    break;

  case Instruction::Or:
    // If the RHS of the Or is a constant, we may have something like:
    // X*4+1 which got turned into X*4|1.  Handle this as an Add so loop
    // optimizations will transparently handle this case.
    //
    // In order for this transformation to be safe, the LHS must be of the
    // form X*(2^n) and the Or constant must be less than 2^n.
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      const SCEV *LHS = getSCEV(U->getOperand(0));
      const APInt &CIVal = CI->getValue();
      if (GetMinTrailingZeros(LHS) >=
          (CIVal.getBitWidth() - CIVal.countLeadingZeros())) {
        // Build a plain add SCEV.
        const SCEV *S = getAddExpr(LHS, getSCEV(CI));
        // If the LHS of the add was an addrec and it has no-wrap flags,
        // transfer the no-wrap flags, since an or won't introduce a wrap.
        if (const SCEVAddRecExpr *NewAR = dyn_cast<SCEVAddRecExpr>(S)) {
          const SCEVAddRecExpr *OldAR = cast<SCEVAddRecExpr>(LHS);
          const_cast<SCEVAddRecExpr *>(NewAR)->setNoWrapFlags(
            OldAR->getNoWrapFlags());
        }
        return S;
      }
    }
    break;
  case Instruction::Xor:
    if (ConstantInt *CI = dyn_cast<ConstantInt>(U->getOperand(1))) {
      // If the RHS of the xor is a signbit, then this is just an add.
      //