ScalarEvolution.cpp

Go to the documentation of this file.

//===- ScalarEvolution.cpp - Scalar Evolution Analysis --------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// This 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
//
//===----------------------------------------------------------------------===//

#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/ADT/APInt.h"
#include "llvm/ADT/ArrayRef.h"
#include "llvm/ADT/DenseMap.h"
#include "llvm/ADT/DepthFirstIterator.h"
#include "llvm/ADT/EquivalenceClasses.h"
#include "llvm/ADT/FoldingSet.h"
#include "llvm/ADT/None.h"
#include "llvm/ADT/Optional.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/ADT/ScopeExit.h"
#include "llvm/ADT/Sequence.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/ADT/SmallSet.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/ADT/StringRef.h"
#include "llvm/Analysis/AssumptionCache.h"
#include "llvm/Analysis/ConstantFolding.h"
#include "llvm/Analysis/InstructionSimplify.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Analysis/TargetLibraryInfo.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/Config/llvm-config.h"
#include "llvm/IR/Argument.h"
#include "llvm/IR/BasicBlock.h"
#include "llvm/IR/CFG.h"
#include "llvm/IR/CallSite.h"
#include "llvm/IR/Constant.h"
#include "llvm/IR/ConstantRange.h"
#include "llvm/IR/Constants.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/DerivedTypes.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/Function.h"
#include "llvm/IR/GlobalAlias.h"
#include "llvm/IR/GlobalValue.h"
#include "llvm/IR/GlobalVariable.h"
#include "llvm/IR/InstIterator.h"
#include "llvm/IR/InstrTypes.h"
#include "llvm/IR/Instruction.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/IntrinsicInst.h"
#include "llvm/IR/Intrinsics.h"
#include "llvm/IR/LLVMContext.h"
#include "llvm/IR/Metadata.h"
#include "llvm/IR/Operator.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/IR/Type.h"
#include "llvm/IR/Use.h"
#include "llvm/IR/User.h"
#include "llvm/IR/Value.h"
#include "llvm/IR/Verifier.h"
#include "llvm/Pass.h"
#include "llvm/Support/Casting.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/Compiler.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/ErrorHandling.h"
#include "llvm/Support/KnownBits.h"
#include "llvm/Support/SaveAndRestore.h"
#include "llvm/Support/raw_ostream.h"
#include <algorithm>
#include <cassert>
#include <climits>
#include <cstddef>
#include <cstdint>
#include <cstdlib>
#include <map>
#include <memory>
#include <tuple>
#include <utility>
#include <vector>

using namespace llvm;

#define DEBUG_TYPE "scalar-evolution"

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 EXPENSIVE_CHECKS when the test suite is clean.
static cl::opt<bool> VerifySCEV(
    "verify-scev", cl::Hidden,
    cl::desc("Verify ScalarEvolution's backedge taken counts (slow)"));
static cl::opt<bool>
    VerifySCEVMap("verify-scev-maps", cl::Hidden,
                  cl::desc("Verify no dangling value in ScalarEvolution's "
                           "ExprValueMap (slow)"));

static cl::opt<bool> VerifyIR(
    "scev-verify-ir", cl::Hidden,
    cl::desc("Verify IR correctness when making sensitive SCEV queries (slow)"),
    cl::init(false));

static cl::opt<unsigned> MulOpsInlineThreshold(
    "scev-mulops-inline-threshold", cl::Hidden,
    cl::desc("Threshold for inlining multiplication operands into a SCEV"),
    cl::init(32));

static cl::opt<unsigned> AddOpsInlineThreshold(
    "scev-addops-inline-threshold", cl::Hidden,
    cl::desc("Threshold for inlining addition operands into a SCEV"),
    cl::init(500));

static cl::opt<unsigned> MaxSCEVCompareDepth(
    "scalar-evolution-max-scev-compare-depth", cl::Hidden,
    cl::desc("Maximum depth of recursive SCEV complexity comparisons"),
    cl::init(32));

static cl::opt<unsigned> MaxSCEVOperationsImplicationDepth(
    "scalar-evolution-max-scev-operations-implication-depth", cl::Hidden,
    cl::desc("Maximum depth of recursive SCEV operations implication analysis"),
    cl::init(2));

static cl::opt<unsigned> MaxValueCompareDepth(
    "scalar-evolution-max-value-compare-depth", cl::Hidden,
    cl::desc("Maximum depth of recursive value complexity comparisons"),
    cl::init(2));

static cl::opt<unsigned>
    MaxArithDepth("scalar-evolution-max-arith-depth", cl::Hidden,
                  cl::desc("Maximum depth of recursive arithmetics"),
                  cl::init(32));

static cl::opt<unsigned> MaxConstantEvolvingDepth(
    "scalar-evolution-max-constant-evolving-depth", cl::Hidden,
    cl::desc("Maximum depth of recursive constant evolving"), cl::init(32));

static cl::opt<unsigned>
    MaxCastDepth("scalar-evolution-max-cast-depth", cl::Hidden,
                 cl::desc("Maximum depth of recursive SExt/ZExt/Trunc"),
                 cl::init(8));

static cl::opt<unsigned>
    MaxAddRecSize("scalar-evolution-max-add-rec-size", cl::Hidden,
                  cl::desc("Max coefficients in AddRec during evolving"),
                  cl::init(8));

static cl::opt<unsigned>
    HugeExprThreshold("scalar-evolution-huge-expr-threshold", cl::Hidden,
                  cl::desc("Size of the expression which is considered huge"),
                  cl::init(4096));

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

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

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

void SCEV::print(raw_ostream &OS) const {
  switch (static_cast<SCEVTypes>(getSCEVType())) {
  case scConstant:
    cast<SCEVConstant>(this)->getValue()->printAsOperand(OS, 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->hasNoUnsignedWrap())
      OS << "nuw><";
    if (AR->hasNoSignedWrap())
      OS << "nsw><";
    if (AR->hasNoSelfWrap() &&
        !AR->getNoWrapFlags((NoWrapFlags)(FlagNUW | FlagNSW)))
      OS << "nw><";
    AR->getLoop()->getHeader()->printAsOperand(OS, /*PrintType=*/false);
    OS << ">";
    return;
  }
  case scAddExpr:
  case scMulExpr:
  case scUMaxExpr:
  case scSMaxExpr:
  case scUMinExpr:
  case scSMinExpr: {
    const SCEVNAryExpr *NAry = cast<SCEVNAryExpr>(this);
    const char *OpStr = nullptr;
    switch (NAry->getSCEVType()) {
    case scAddExpr: OpStr = " + "; break;
    case scMulExpr: OpStr = " * "; break;
    case scUMaxExpr: OpStr = " umax "; break;
    case scSMaxExpr: OpStr = " smax "; break;
    case scUMinExpr:
      OpStr = " umin ";
      break;
    case scSMinExpr:
      OpStr = " smin ";
      break;
    }
    OS << "(";
    for (SCEVNAryExpr::op_iterator I = NAry->op_begin(), E = NAry->op_end();
         I != E; ++I) {
      OS << **I;
      if (std::next(I) != E)
        OS << OpStr;
    }
    OS << ")";
    switch (NAry->getSCEVType()) {
    case scAddExpr:
    case scMulExpr:
      if (NAry->hasNoUnsignedWrap())
        OS << "<nuw>";
      if (NAry->hasNoSignedWrap())
        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 << ", ";
      FieldNo->printAsOperand(OS, false);
      OS << ")";
      return;
    }

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

Type *SCEV::getType() const {
  switch (static_cast<SCEVTypes>(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:
  case scUMinExpr:
  case scSMinExpr:
    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!");
  }
  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()->isMinusOne();
  return false;
}

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->getAPInt().isNegative();
}

SCEVCouldNotCompute::SCEVCouldNotCompute() :
  SCEV(FoldingSetNodeIDRef(), scCouldNotCompute, 0) {}

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 = nullptr;
  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, computeExpressionSize(op)), Op(op), Ty(ty) {}

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

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

SCEVSignExtendExpr::SCEVSignExtendExpr(const FoldingSetNodeIDRef ID,
                                       const SCEV *op, Type *ty)
  : SCEVCastExpr(ID, scSignExtend, op, ty) {
  assert(Op->getType()->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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(nullptr);
}

void SCEVUnknown::allUsesReplacedWith(Value *New) {
  // 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
//===----------------------------------------------------------------------===//

/// Compare the two values \p LV and \p RV in terms of their "complexity" where
/// "complexity" is a partial (and somewhat ad-hoc) relation used to order
/// operands in SCEV expressions.  \p EqCache is a set of pairs of values that
/// have been previously deemed to be "equally complex" by this routine.  It is
/// intended to avoid exponential time complexity in cases like:
///
///   %a = f(%x, %y)
///   %b = f(%a, %a)
///   %c = f(%b, %b)
///
///   %d = f(%x, %y)
///   %e = f(%d, %d)
///   %f = f(%e, %e)
///
///   CompareValueComplexity(%f, %c)
///
/// Since we do not continue running this routine on expression trees once we
/// have seen unequal values, there is no need to track them in the cache.
static int
CompareValueComplexity(EquivalenceClasses<const Value *> &EqCacheValue,
                       const LoopInfo *const LI, Value *LV, Value *RV,
                       unsigned Depth) {
  if (Depth > MaxValueCompareDepth || EqCacheValue.isEquivalent(LV, RV))
    return 0;

  // 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 auto *LA = dyn_cast<Argument>(LV)) {
    const auto *RA = cast<Argument>(RV);
    unsigned LArgNo = LA->getArgNo(), RArgNo = RA->getArgNo();
    return (int)LArgNo - (int)RArgNo;
  }

  if (const auto *LGV = dyn_cast<GlobalValue>(LV)) {
    const auto *RGV = cast<GlobalValue>(RV);

    const auto IsGVNameSemantic = [&](const GlobalValue *GV) {
      auto LT = GV->getLinkage();
      return !(GlobalValue::isPrivateLinkage(LT) ||
               GlobalValue::isInternalLinkage(LT));
    };

    // Use the names to distinguish the two values, but only if the
    // names are semantically important.
    if (IsGVNameSemantic(LGV) && IsGVNameSemantic(RGV))
      return LGV->getName().compare(RGV->getName());
  }

  // For instructions, compare their loop depth, and their operand count.  This
  // is pretty loose.
  if (const auto *LInst = dyn_cast<Instruction>(LV)) {
    const auto *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();
    if (LNumOps != RNumOps)
      return (int)LNumOps - (int)RNumOps;

    for (unsigned Idx : seq(0u, LNumOps)) {
      int Result =
          CompareValueComplexity(EqCacheValue, LI, LInst->getOperand(Idx),
                                 RInst->getOperand(Idx), Depth + 1);
      if (Result != 0)
        return Result;
    }
  }

  EqCacheValue.unionSets(LV, RV);
  return 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.
static int CompareSCEVComplexity(
    EquivalenceClasses<const SCEV *> &EqCacheSCEV,
    EquivalenceClasses<const Value *> &EqCacheValue,
    const LoopInfo *const LI, const SCEV *LHS, const SCEV *RHS,
    DominatorTree &DT, unsigned Depth = 0) {
  // 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;

  if (Depth > MaxSCEVCompareDepth || EqCacheSCEV.isEquivalent(LHS, RHS))
    return 0;
  // 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 (static_cast<SCEVTypes>(LType)) {
  case scUnknown: {
    const SCEVUnknown *LU = cast<SCEVUnknown>(LHS);
    const SCEVUnknown *RU = cast<SCEVUnknown>(RHS);

    int X = CompareValueComplexity(EqCacheValue, LI, LU->getValue(),
                                   RU->getValue(), Depth + 1);
    if (X == 0)
      EqCacheSCEV.unionSets(LHS, RHS);
    return X;
  }

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

    // Compare constant values.
    const APInt &LA = LC->getAPInt();
    const APInt &RA = RC->getAPInt();
    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);

    // There is always a dominance between two recs that are used by one SCEV,
    // so we can safely sort recs by loop header dominance. We require such
    // order in getAddExpr.
    const Loop *LLoop = LA->getLoop(), *RLoop = RA->getLoop();
    if (LLoop != RLoop) {
      const BasicBlock *LHead = LLoop->getHeader(), *RHead = RLoop->getHeader();
      assert(LHead != RHead && "Two loops share the same header?");
      if (DT.dominates(LHead, RHead))
        return 1;
      else
        assert(DT.dominates(RHead, LHead) &&
               "No dominance between recurrences used by one SCEV?");
      return -1;
    }

    // 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) {
      int X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI,
                                    LA->getOperand(i), RA->getOperand(i), DT,
                                    Depth + 1);
      if (X != 0)
        return X;
    }
    EqCacheSCEV.unionSets(LHS, RHS);
    return 0;
  }

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

    // Lexicographically compare n-ary expressions.
    unsigned LNumOps = LC->getNumOperands(), RNumOps = RC->getNumOperands();
    if (LNumOps != RNumOps)
      return (int)LNumOps - (int)RNumOps;

    for (unsigned i = 0; i != LNumOps; ++i) {
      int X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI,
                                    LC->getOperand(i), RC->getOperand(i), DT,
                                    Depth + 1);
      if (X != 0)
        return X;
    }
    EqCacheSCEV.unionSets(LHS, RHS);
    return 0;
  }

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

    // Lexicographically compare udiv expressions.
    int X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LC->getLHS(),
                                  RC->getLHS(), DT, Depth + 1);
    if (X != 0)
      return X;
    X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LC->getRHS(),
                              RC->getRHS(), DT, Depth + 1);
    if (X == 0)
      EqCacheSCEV.unionSets(LHS, RHS);
    return X;
  }

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

    // Compare cast expressions by operand.
    int X = CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI,
                                  LC->getOperand(), RC->getOperand(), DT,
                                  Depth + 1);
    if (X == 0)
      EqCacheSCEV.unionSets(LHS, RHS);
    return X;
  }

  case scCouldNotCompute:
    llvm_unreachable("Attempt to use a SCEVCouldNotCompute object!");
  }
  llvm_unreachable("Unknown SCEV kind!");
}

/// 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, DominatorTree &DT) {
  if (Ops.size() < 2) return;  // Noop

  EquivalenceClasses<const SCEV *> EqCacheSCEV;
  EquivalenceClasses<const Value *> EqCacheValue;
  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 (CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, RHS, LHS, DT) < 0)
      std::swap(LHS, RHS);
    return;
  }

  // Do the rough sort by complexity.
  llvm::stable_sort(Ops, [&](const SCEV *LHS, const SCEV *RHS) {
    return CompareSCEVComplexity(EqCacheSCEV, EqCacheValue, LI, LHS, RHS, DT) <
           0;
  });

  // 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!
      }
    }
  }
}

// Returns the size of the SCEV S.
static inline int sizeOfSCEV(const SCEV *S) {
  struct FindSCEVSize {
    int Size = 0;

    FindSCEVSize() = default;

    bool follow(const SCEV *S) {
      ++Size;
      // Keep looking at all operands of S.
      return true;
    }

    bool isDone() const {
      return false;
    }
  };

  FindSCEVSize F;
  SCEVTraversal<FindSCEVSize> ST(F);
  ST.visitAll(S);
  return F.Size;
}

/// Returns true if the subtree of \p S contains at least HugeExprThreshold
/// nodes.
static bool isHugeExpression(const SCEV *S) {
  return S->getExpressionSize() >= HugeExprThreshold;
}

/// Returns true of \p Ops contains a huge SCEV (see definition above).
static bool hasHugeExpression(ArrayRef<const SCEV *> Ops) {
  return any_of(Ops, isHugeExpression);
}

namespace {

struct SCEVDivision : public SCEVVisitor<SCEVDivision, void> {
public:
  // Computes the Quotient and Remainder of the division of Numerator by
  // Denominator.
  static void divide(ScalarEvolution &SE, const SCEV *Numerator,
                     const SCEV *Denominator, const SCEV **Quotient,
                     const SCEV **Remainder) {
    assert(Numerator && Denominator && "Uninitialized SCEV");

    SCEVDivision D(SE, Numerator, Denominator);

    // Check for the trivial case here to avoid having to check for it in the
    // rest of the code.
    if (Numerator == Denominator) {
      *Quotient = D.One;
      *Remainder = D.Zero;
      return;
    }

    if (Numerator->isZero()) {
      *Quotient = D.Zero;
      *Remainder = D.Zero;
      return;
    }

    // A simple case when N/1. The quotient is N.
    if (Denominator->isOne()) {
      *Quotient = Numerator;
      *Remainder = D.Zero;
      return;
    }

    // Split the Denominator when it is a product.
    if (const SCEVMulExpr *T = dyn_cast<SCEVMulExpr>(Denominator)) {
      const SCEV *Q, *R;
      *Quotient = Numerator;
      for (const SCEV *Op : T->operands()) {
        divide(SE, *Quotient, Op, &Q, &R);
        *Quotient = Q;

        // Bail out when the Numerator is not divisible by one of the terms of
        // the Denominator.
        if (!R->isZero()) {
          *Quotient = D.Zero;
          *Remainder = Numerator;
          return;
        }
      }
      *Remainder = D.Zero;
      return;
    }

    D.visit(Numerator);
    *Quotient = D.Quotient;
    *Remainder = D.Remainder;
  }

  // Except in the trivial case described above, we do not know how to divide
  // Expr by Denominator for the following functions with empty implementation.
  void visitTruncateExpr(const SCEVTruncateExpr *Numerator) {}
  void visitZeroExtendExpr(const SCEVZeroExtendExpr *Numerator) {}
  void visitSignExtendExpr(const SCEVSignExtendExpr *Numerator) {}
  void visitUDivExpr(const SCEVUDivExpr *Numerator) {}
  void visitSMaxExpr(const SCEVSMaxExpr *Numerator) {}
  void visitUMaxExpr(const SCEVUMaxExpr *Numerator) {}
  void visitSMinExpr(const SCEVSMinExpr *Numerator) {}
  void visitUMinExpr(const SCEVUMinExpr *Numerator) {}
  void visitUnknown(const SCEVUnknown *Numerator) {}
  void visitCouldNotCompute(const SCEVCouldNotCompute *Numerator) {}

  void visitConstant(const SCEVConstant *Numerator) {
    if (const SCEVConstant *D = dyn_cast<SCEVConstant>(Denominator)) {
      APInt NumeratorVal = Numerator->getAPInt();
      APInt DenominatorVal = D->getAPInt();
      uint32_t NumeratorBW = NumeratorVal.getBitWidth();
      uint32_t DenominatorBW = DenominatorVal.getBitWidth();

      if (NumeratorBW > DenominatorBW)
        DenominatorVal = DenominatorVal.sext(NumeratorBW);
      else if (NumeratorBW < DenominatorBW)
        NumeratorVal = NumeratorVal.sext(DenominatorBW);

      APInt QuotientVal(NumeratorVal.getBitWidth(), 0);
      APInt RemainderVal(NumeratorVal.getBitWidth(), 0);
      APInt::sdivrem(NumeratorVal, DenominatorVal, QuotientVal, RemainderVal);
      Quotient = SE.getConstant(QuotientVal);
      Remainder = SE.getConstant(RemainderVal);
      return;
    }
  }

  void visitAddRecExpr(const SCEVAddRecExpr *Numerator) {
    const SCEV *StartQ, *StartR, *StepQ, *StepR;
    if (!Numerator->isAffine())
      return cannotDivide(Numerator);
    divide(SE, Numerator->getStart(), Denominator, &StartQ, &StartR);
    divide(SE, Numerator->getStepRecurrence(SE), Denominator, &StepQ, &StepR);
    // Bail out if the types do not match.
    Type *Ty = Denominator->getType();
    if (Ty != StartQ->getType() || Ty != StartR->getType() ||
        Ty != StepQ->getType() || Ty != StepR->getType())
      return cannotDivide(Numerator);
    Quotient = SE.getAddRecExpr(StartQ, StepQ, Numerator->getLoop(),
                                Numerator->getNoWrapFlags());
    Remainder = SE.getAddRecExpr(StartR, StepR, Numerator->getLoop(),
                                 Numerator->getNoWrapFlags());
  }

  void visitAddExpr(const SCEVAddExpr *Numerator) {
    SmallVector<const SCEV *, 2> Qs, Rs;
    Type *Ty = Denominator->getType();

    for (const SCEV *Op : Numerator->operands()) {
      const SCEV *Q, *R;
      divide(SE, Op, Denominator, &Q, &R);

      // Bail out if types do not match.
      if (Ty != Q->getType() || Ty != R->getType())
        return cannotDivide(Numerator);

      Qs.push_back(Q);
      Rs.push_back(R);
    }

    if (Qs.size() == 1) {
      Quotient = Qs[0];
      Remainder = Rs[0];
      return;
    }

    Quotient = SE.getAddExpr(Qs);
    Remainder = SE.getAddExpr(Rs);
  }

  void visitMulExpr(const SCEVMulExpr *Numerator) {
    SmallVector<const SCEV *, 2> Qs;
    Type *Ty = Denominator->getType();

    bool FoundDenominatorTerm = false;
    for (const SCEV *Op : Numerator->operands()) {
      // Bail out if types do not match.
      if (Ty != Op->getType())
        return cannotDivide(Numerator);

      if (FoundDenominatorTerm) {
        Qs.push_back(Op);
        continue;
      }

      // Check whether Denominator divides one of the product operands.
      const SCEV *Q, *R;
      divide(SE, Op, Denominator, &Q, &R);
      if (!R->isZero()) {
        Qs.push_back(Op);
        continue;
      }

      // Bail out if types do not match.
      if (Ty != Q->getType())
        return cannotDivide(Numerator);

      FoundDenominatorTerm = true;
      Qs.push_back(Q);
    }

    if (FoundDenominatorTerm) {
      Remainder = Zero;
      if (Qs.size() == 1)
        Quotient = Qs[0];
      else
        Quotient = SE.getMulExpr(Qs);
      return;
    }

    if (!isa<SCEVUnknown>(Denominator))
      return cannotDivide(Numerator);

    // The Remainder is obtained by replacing Denominator by 0 in Numerator.
    ValueToValueMap RewriteMap;
    RewriteMap[cast<SCEVUnknown>(Denominator)->getValue()] =
        cast<SCEVConstant>(Zero)->getValue();
    Remainder = SCEVParameterRewriter::rewrite(Numerator, SE, RewriteMap, true);

    if (Remainder->isZero()) {
      // The Quotient is obtained by replacing Denominator by 1 in Numerator.
      RewriteMap[cast<SCEVUnknown>(Denominator)->getValue()] =
          cast<SCEVConstant>(One)->getValue();
      Quotient =
          SCEVParameterRewriter::rewrite(Numerator, SE, RewriteMap, true);
      return;
    }

    // Quotient is (Numerator - Remainder) divided by Denominator.
    const SCEV *Q, *R;
    const SCEV *Diff = SE.getMinusSCEV(Numerator, Remainder);
    // This SCEV does not seem to simplify: fail the division here.
    if (sizeOfSCEV(Diff) > sizeOfSCEV(Numerator))
      return cannotDivide(Numerator);
    divide(SE, Diff, Denominator, &Q, &R);
    if (R != Zero)
      return cannotDivide(Numerator);
    Quotient = Q;
  }

private:
  SCEVDivision(ScalarEvolution &S, const SCEV *Numerator,
               const SCEV *Denominator)
      : SE(S), Denominator(Denominator) {
    Zero = SE.getZero(Denominator->getType());
    One = SE.getOne(Denominator->getType());

    // We generally do not know how to divide Expr by Denominator. We
    // initialize the division to a "cannot divide" state to simplify the rest
    // of the code.
    cannotDivide(Numerator);
  }

  // Convenience function for giving up on the division. We set the quotient to
  // be equal to zero and the remainder to be equal to the numerator.
  void cannotDivide(const SCEV *Numerator) {
    Quotient = Zero;
    Remainder = Numerator;
  }

  ScalarEvolution &SE;
  const SCEV *Denominator, *Quotient, *Remainder, *Zero, *One;
};

} // end anonymous namespace

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

/// 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.lshrInPlace(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::getOneBitSet(CalculationBits, 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));
}

/// 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,
                                             unsigned Depth) {
  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 = nullptr;
  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, Depth + 1);

  // 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, Depth + 1);

  // 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, Depth + 1);

  if (Depth > MaxCastDepth) {
    SCEV *S =
        new (SCEVAllocator) SCEVTruncateExpr(ID.Intern(SCEVAllocator), Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }

  // trunc(x1 + ... + xN) --> trunc(x1) + ... + trunc(xN) and
  // trunc(x1 * ... * xN) --> trunc(x1) * ... * trunc(xN),
  // if after transforming we have at most one truncate, not counting truncates
  // that replace other casts.
  if (isa<SCEVAddExpr>(Op) || isa<SCEVMulExpr>(Op)) {
    auto *CommOp = cast<SCEVCommutativeExpr>(Op);
    SmallVector<const SCEV *, 4> Operands;
    unsigned numTruncs = 0;
    for (unsigned i = 0, e = CommOp->getNumOperands(); i != e && numTruncs < 2;
         ++i) {
      const SCEV *S = getTruncateExpr(CommOp->getOperand(i), Ty, Depth + 1);
      if (!isa<SCEVCastExpr>(CommOp->getOperand(i)) && isa<SCEVTruncateExpr>(S))
        numTruncs++;
      Operands.push_back(S);
    }
    if (numTruncs < 2) {
      if (isa<SCEVAddExpr>(Op))
        return getAddExpr(Operands);
      else if (isa<SCEVMulExpr>(Op))
        return getMulExpr(Operands);
      else
        llvm_unreachable("Unexpected SCEV type for Op.");
    }
    // Although we checked in the beginning that ID is not in the cache, it is
    // possible that during recursion and different modification ID was inserted
    // into the cache. So if we find it, just return it.
    if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP))
      return S;
  }

  // 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 (const SCEV *Op : AddRec->operands())
      Operands.push_back(getTruncateExpr(Op, Ty, Depth + 1));
    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);
  addToLoopUseLists(S);
  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 *getSignedOverflowLimitForStep(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->getSignedRangeMax(Step));
  }
  if (SE->isKnownNegative(Step)) {
    *Pred = ICmpInst::ICMP_SGT;
    return SE->getConstant(APInt::getSignedMaxValue(BitWidth) -
                           SE->getSignedRangeMin(Step));
  }
  return nullptr;
}

// Get the limit of a recurrence such that incrementing by Step cannot cause
// unsigned overflow as long as the value of the recurrence within the loop does
// not exceed this limit before incrementing.
static const SCEV *getUnsignedOverflowLimitForStep(const SCEV *Step,
                                                   ICmpInst::Predicate *Pred,
                                                   ScalarEvolution *SE) {
  unsigned BitWidth = SE->getTypeSizeInBits(Step->getType());
  *Pred = ICmpInst::ICMP_ULT;

  return SE->getConstant(APInt::getMinValue(BitWidth) -
                         SE->getUnsignedRangeMax(Step));
}

namespace {

struct ExtendOpTraitsBase {
  typedef const SCEV *(ScalarEvolution::*GetExtendExprTy)(const SCEV *, Type *,
                                                          unsigned);
};

// Used to make code generic over signed and unsigned overflow.
template <typename ExtendOp> struct ExtendOpTraits {
  // Members present:
  //
  // static const SCEV::NoWrapFlags WrapType;
  //
  // static const ExtendOpTraitsBase::GetExtendExprTy GetExtendExpr;
  //
  // static const SCEV *getOverflowLimitForStep(const SCEV *Step,
  //                                           ICmpInst::Predicate *Pred,
  //                                           ScalarEvolution *SE);
};

template <>
struct ExtendOpTraits<SCEVSignExtendExpr> : public ExtendOpTraitsBase {
  static const SCEV::NoWrapFlags WrapType = SCEV::FlagNSW;

  static const GetExtendExprTy GetExtendExpr;

  static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                             ICmpInst::Predicate *Pred,
                                             ScalarEvolution *SE) {
    return getSignedOverflowLimitForStep(Step, Pred, SE);
  }
};

const ExtendOpTraitsBase::GetExtendExprTy ExtendOpTraits<
    SCEVSignExtendExpr>::GetExtendExpr = &ScalarEvolution::getSignExtendExpr;

template <>
struct ExtendOpTraits<SCEVZeroExtendExpr> : public ExtendOpTraitsBase {
  static const SCEV::NoWrapFlags WrapType = SCEV::FlagNUW;

  static const GetExtendExprTy GetExtendExpr;

  static const SCEV *getOverflowLimitForStep(const SCEV *Step,
                                             ICmpInst::Predicate *Pred,
                                             ScalarEvolution *SE) {
    return getUnsignedOverflowLimitForStep(Step, Pred, SE);
  }
};

const ExtendOpTraitsBase::GetExtendExprTy ExtendOpTraits<
    SCEVZeroExtendExpr>::GetExtendExpr = &ScalarEvolution::getZeroExtendExpr;

} // end anonymous namespace

// The recurrence AR has been shown to have no signed/unsigned wrap or something
// close to it. Typically, if we can prove NSW/NUW for AR, then we can just as
// easily prove NSW/NUW for its preincrement or postincrement sibling. This
// allows normalizing a sign/zero extended AddRec as such: {sext/zext(Step +
// Start),+,Step} => {(Step + sext/zext(Start),+,Step} As a result, the
// expression "Step + sext/zext(PreIncAR)" is congruent with
// "sext/zext(PostIncAR)"
template <typename ExtendOpTy>
static const SCEV *getPreStartForExtend(const SCEVAddRecExpr *AR, Type *Ty,
                                        ScalarEvolution *SE, unsigned Depth) {
  auto WrapType = ExtendOpTraits<ExtendOpTy>::WrapType;
  auto GetExtendExpr = ExtendOpTraits<ExtendOpTy>::GetExtendExpr;

  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 nullptr;

  // 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 (const SCEV *Op : SA->operands())
    if (Op != Step)
      DiffOps.push_back(Op);

  if (DiffOps.size() == SA->getNumOperands())
    return nullptr;

  // Try to prove `WrapType` (SCEV::FlagNSW or SCEV::FlagNUW) on `PreStart` +
  // `Step`:

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

  // "{S,+,X} is <nsw>/<nuw>" and "the backedge is taken at least once" implies
  // "S+X does not sign/unsign-overflow".
  //

  const SCEV *BECount = SE->getBackedgeTakenCount(L);
  if (PreAR && PreAR->getNoWrapFlags(WrapType) &&
      !isa<SCEVCouldNotCompute>(BECount) && SE->isKnownPositive(BECount))
    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->*GetExtendExpr)(PreStart, WideTy, Depth),
                     (SE->*GetExtendExpr)(Step, WideTy, Depth));
  if ((SE->*GetExtendExpr)(Start, WideTy, Depth) == OperandExtendedStart) {
    if (PreAR && AR->getNoWrapFlags(WrapType)) {
      // If we know `AR` == {`PreStart`+`Step`,+,`Step`} is `WrapType` (FlagNSW
      // or FlagNUW) and that `PreStart` + `Step` is `WrapType` too, then
      // `PreAR` == {`PreStart`,+,`Step`} is also `WrapType`.  Cache this fact.
      const_cast<SCEVAddRecExpr *>(PreAR)->setNoWrapFlags(WrapType);
    }
    return PreStart;
  }

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

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

  return nullptr;
}

// Get the normalized zero or sign extended expression for this AddRec's Start.
template <typename ExtendOpTy>
static const SCEV *getExtendAddRecStart(const SCEVAddRecExpr *AR, Type *Ty,
                                        ScalarEvolution *SE,
                                        unsigned Depth) {
  auto GetExtendExpr = ExtendOpTraits<ExtendOpTy>::GetExtendExpr;

  const SCEV *PreStart = getPreStartForExtend<ExtendOpTy>(AR, Ty, SE, Depth);
  if (!PreStart)
    return (SE->*GetExtendExpr)(AR->getStart(), Ty, Depth);

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

// Try to prove away overflow by looking at "nearby" add recurrences.  A
// motivating example for this rule: if we know `{0,+,4}` is `ult` `-1` and it
// does not itself wrap then we can conclude that `{1,+,4}` is `nuw`.
//
// Formally:
//
//     {S,+,X} == {S-T,+,X} + T
//  => Ext({S,+,X}) == Ext({S-T,+,X} + T)
//
// If ({S-T,+,X} + T) does not overflow  ... (1)
//
//  RHS == Ext({S-T,+,X} + T) == Ext({S-T,+,X}) + Ext(T)
//
// If {S-T,+,X} does not overflow  ... (2)
//
//  RHS == Ext({S-T,+,X}) + Ext(T) == {Ext(S-T),+,Ext(X)} + Ext(T)
//      == {Ext(S-T)+Ext(T),+,Ext(X)}
//
// If (S-T)+T does not overflow  ... (3)
//
//  RHS == {Ext(S-T)+Ext(T),+,Ext(X)} == {Ext(S-T+T),+,Ext(X)}
//      == {Ext(S),+,Ext(X)} == LHS
//
// Thus, if (1), (2) and (3) are true for some T, then
//   Ext({S,+,X}) == {Ext(S),+,Ext(X)}
//
// (3) is implied by (1) -- "(S-T)+T does not overflow" is simply "({S-T,+,X}+T)
// does not overflow" restricted to the 0th iteration.  Therefore we only need
// to check for (1) and (2).
//
// In the current context, S is `Start`, X is `Step`, Ext is `ExtendOpTy` and T
// is `Delta` (defined below).
template <typename ExtendOpTy>
bool ScalarEvolution::proveNoWrapByVaryingStart(const SCEV *Start,
                                                const SCEV *Step,
                                                const Loop *L) {
  auto WrapType = ExtendOpTraits<ExtendOpTy>::WrapType;

  // We restrict `Start` to a constant to prevent SCEV from spending too much
  // time here.  It is correct (but more expensive) to continue with a
  // non-constant `Start` and do a general SCEV subtraction to compute
  // `PreStart` below.
  const SCEVConstant *StartC = dyn_cast<SCEVConstant>(Start);
  if (!StartC)
    return false;

  APInt StartAI = StartC->getAPInt();

  for (unsigned Delta : {-2, -1, 1, 2}) {
    const SCEV *PreStart = getConstant(StartAI - Delta);

    FoldingSetNodeID ID;
    ID.AddInteger(scAddRecExpr);
    ID.AddPointer(PreStart);
    ID.AddPointer(Step);
    ID.AddPointer(L);
    void *IP = nullptr;
    const auto *PreAR =
      static_cast<SCEVAddRecExpr *>(UniqueSCEVs.FindNodeOrInsertPos(ID, IP));

    // Give up if we don't already have the add recurrence we need because
    // actually constructing an add recurrence is relatively expensive.
    if (PreAR && PreAR->getNoWrapFlags(WrapType)) {  // proves (2)
      const SCEV *DeltaS = getConstant(StartC->getType(), Delta);
      ICmpInst::Predicate Pred = ICmpInst::BAD_ICMP_PREDICATE;
      const SCEV *Limit = ExtendOpTraits<ExtendOpTy>::getOverflowLimitForStep(
          DeltaS, &Pred, this);
      if (Limit && isKnownPredicate(Pred, PreAR, Limit))  // proves (1)
        return true;
    }
  }

  return false;
}

// Finds an integer D for an expression (C + x + y + ...) such that the top
// level addition in (D + (C - D + x + y + ...)) would not wrap (signed or
// unsigned) and the number of trailing zeros of (C - D + x + y + ...) is
// maximized, where C is the \p ConstantTerm, x, y, ... are arbitrary SCEVs, and
// the (C + x + y + ...) expression is \p WholeAddExpr.
static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
                                            const SCEVConstant *ConstantTerm,
                                            const SCEVAddExpr *WholeAddExpr) {
  const APInt C = ConstantTerm->getAPInt();
  const unsigned BitWidth = C.getBitWidth();
  // Find number of trailing zeros of (x + y + ...) w/o the C first:
  uint32_t TZ = BitWidth;
  for (unsigned I = 1, E = WholeAddExpr->getNumOperands(); I < E && TZ; ++I)
    TZ = std::min(TZ, SE.GetMinTrailingZeros(WholeAddExpr->getOperand(I)));
  if (TZ) {
    // Set D to be as many least significant bits of C as possible while still
    // guaranteeing that adding D to (C - D + x + y + ...) won't cause a wrap:
    return TZ < BitWidth ? C.trunc(TZ).zext(BitWidth) : C;
  }
  return APInt(BitWidth, 0);
}

// Finds an integer D for an affine AddRec expression {C,+,x} such that the top
// level addition in (D + {C-D,+,x}) would not wrap (signed or unsigned) and the
// number of trailing zeros of (C - D + x * n) is maximized, where C is the \p
// ConstantStart, x is an arbitrary \p Step, and n is the loop trip count.
static APInt extractConstantWithoutWrapping(ScalarEvolution &SE,
                                            const APInt &ConstantStart,
                                            const SCEV *Step) {
  const unsigned BitWidth = ConstantStart.getBitWidth();
  const uint32_t TZ = SE.GetMinTrailingZeros(Step);
  if (TZ)
    return TZ < BitWidth ? ConstantStart.trunc(TZ).zext(BitWidth)
                         : ConstantStart;
  return APInt(BitWidth, 0);
}

const SCEV *
ScalarEvolution::getZeroExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
  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, Depth + 1);

  // 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 = nullptr;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  if (Depth > MaxCastDepth) {
    SCEV *S = new (SCEVAllocator) SCEVZeroExtendExpr(ID.Intern(SCEVAllocator),
                                                     Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    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, Depth);
  }

  // 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 (!AR->hasNoUnsignedWrap()) {
        auto NewFlags = proveNoWrapViaConstantRanges(AR);
        const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(NewFlags);
      }

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->hasNoUnsignedWrap())
        return getAddRecExpr(
            getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this, Depth + 1),
            getZeroExtendExpr(Step, Ty, Depth + 1), 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(), Depth);
        const SCEV *RecastedMaxBECount = getTruncateOrZeroExtend(
            CastedMaxBECount, MaxBECount->getType(), Depth);
        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,
                                        SCEV::FlagAnyWrap, Depth + 1);
          const SCEV *ZAdd = getZeroExtendExpr(getAddExpr(Start, ZMul,
                                                          SCEV::FlagAnyWrap,
                                                          Depth + 1),
                                               WideTy, Depth + 1);
          const SCEV *WideStart = getZeroExtendExpr(Start, WideTy, Depth + 1);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy, Depth + 1);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getZeroExtendExpr(Step, WideTy, Depth + 1),
                                  SCEV::FlagAnyWrap, Depth + 1),
                       SCEV::FlagAnyWrap, Depth + 1);
          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(
                getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getZeroExtendExpr(Step, Ty, Depth + 1), 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, Depth + 1),
                                  SCEV::FlagAnyWrap, Depth + 1),
                       SCEV::FlagAnyWrap, Depth + 1);
          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(
                getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getSignExtendExpr(Step, Ty, Depth + 1), L,
                AR->getNoWrapFlags());
          }
        }
      }

      // Normally, in the cases we can prove no-overflow via a
      // backedge guarding condition, we can also compute a backedge
      // taken count for the loop.  The exceptions are assumptions and
      // guards present in the loop -- SCEV is not great at exploiting
      // these to compute max backedge taken counts, but can still use
      // these to prove lack of overflow.  Use this fact to avoid
      // doing extra work that may not pay off.
      if (!isa<SCEVCouldNotCompute>(MaxBECount) || HasGuards ||
          !AC.assumptions().empty()) {
        // 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) -
                                      getUnsignedRangeMax(Step));
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_ULT, AR, N) ||
              isKnownOnEveryIteration(ICmpInst::ICMP_ULT, AR, 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(
                getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getZeroExtendExpr(Step, Ty, Depth + 1), L,
                AR->getNoWrapFlags());
          }
        } else if (isKnownNegative(Step)) {
          const SCEV *N = getConstant(APInt::getMaxValue(BitWidth) -
                                      getSignedRangeMin(Step));
          if (isLoopBackedgeGuardedByCond(L, ICmpInst::ICMP_UGT, AR, N) ||
              isKnownOnEveryIteration(ICmpInst::ICMP_UGT, AR, 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(
                getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getSignExtendExpr(Step, Ty, Depth + 1), L,
                AR->getNoWrapFlags());
          }
        }
      }

      // zext({C,+,Step}) --> (zext(D) + zext({C-D,+,Step}))<nuw><nsw>
      // if D + (C - D + Step * n) could be proven to not unsigned wrap
      // where D maximizes the number of trailing zeros of (C - D + Step * n)
      if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
        const APInt &C = SC->getAPInt();
        const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
        if (D != 0) {
          const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
          const SCEV *SResidual =
              getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
          const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
          return getAddExpr(SZExtD, SZExtR,
                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                            Depth + 1);
        }
      }

      if (proveNoWrapByVaryingStart<SCEVZeroExtendExpr>(Start, Step, L)) {
        const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNUW);
        return getAddRecExpr(
            getExtendAddRecStart<SCEVZeroExtendExpr>(AR, Ty, this, Depth + 1),
            getZeroExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
      }
    }

  // zext(A % B) --> zext(A) % zext(B)
  {
    const SCEV *LHS;
    const SCEV *RHS;
    if (matchURem(Op, LHS, RHS))
      return getURemExpr(getZeroExtendExpr(LHS, Ty, Depth + 1),
                         getZeroExtendExpr(RHS, Ty, Depth + 1));
  }

  // zext(A / B) --> zext(A) / zext(B).
  if (auto *Div = dyn_cast<SCEVUDivExpr>(Op))
    return getUDivExpr(getZeroExtendExpr(Div->getLHS(), Ty, Depth + 1),
                       getZeroExtendExpr(Div->getRHS(), Ty, Depth + 1));

  if (auto *SA = dyn_cast<SCEVAddExpr>(Op)) {
    // zext((A + B + ...)<nuw>) --> (zext(A) + zext(B) + ...)<nuw>
    if (SA->hasNoUnsignedWrap()) {
      // If the addition does not unsign overflow then we can, by definition,
      // commute the zero extension with the addition operation.
      SmallVector<const SCEV *, 4> Ops;
      for (const auto *Op : SA->operands())
        Ops.push_back(getZeroExtendExpr(Op, Ty, Depth + 1));
      return getAddExpr(Ops, SCEV::FlagNUW, Depth + 1);
    }

    // zext(C + x + y + ...) --> (zext(D) + zext((C - D) + x + y + ...))
    // if D + (C - D + x + y + ...) could be proven to not unsigned wrap
    // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
    //
    // Often address arithmetics contain expressions like
    // (zext (add (shl X, C1), C2)), for instance, (zext (5 + (4 * X))).
    // This transformation is useful while proving that such expressions are
    // equal or differ by a small constant amount, see LoadStoreVectorizer pass.
    if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
      const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
      if (D != 0) {
        const SCEV *SZExtD = getZeroExtendExpr(getConstant(D), Ty, Depth);
        const SCEV *SResidual =
            getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
        const SCEV *SZExtR = getZeroExtendExpr(SResidual, Ty, Depth + 1);
        return getAddExpr(SZExtD, SZExtR,
                          (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                          Depth + 1);
      }
    }
  }

  if (auto *SM = dyn_cast<SCEVMulExpr>(Op)) {
    // zext((A * B * ...)<nuw>) --> (zext(A) * zext(B) * ...)<nuw>
    if (SM->hasNoUnsignedWrap()) {
      // If the multiply does not unsign overflow then we can, by definition,
      // commute the zero extension with the multiply operation.
      SmallVector<const SCEV *, 4> Ops;
      for (const auto *Op : SM->operands())
        Ops.push_back(getZeroExtendExpr(Op, Ty, Depth + 1));
      return getMulExpr(Ops, SCEV::FlagNUW, Depth + 1);
    }

    // zext(2^K * (trunc X to iN)) to iM ->
    // 2^K * (zext(trunc X to i{N-K}) to iM)<nuw>
    //
    // Proof:
    //
    //     zext(2^K * (trunc X to iN)) to iM
    //   = zext((trunc X to iN) << K) to iM
    //   = zext((trunc X to i{N-K}) << K)<nuw> to iM
    //     (because shl removes the top K bits)
    //   = zext((2^K * (trunc X to i{N-K}))<nuw>) to iM
    //   = (2^K * (zext(trunc X to i{N-K}) to iM))<nuw>.
    //
    if (SM->getNumOperands() == 2)
      if (auto *MulLHS = dyn_cast<SCEVConstant>(SM->getOperand(0)))
        if (MulLHS->getAPInt().isPowerOf2())
          if (auto *TruncRHS = dyn_cast<SCEVTruncateExpr>(SM->getOperand(1))) {
            int NewTruncBits = getTypeSizeInBits(TruncRHS->getType()) -
                               MulLHS->getAPInt().logBase2();
            Type *NewTruncTy = IntegerType::get(getContext(), NewTruncBits);
            return getMulExpr(
                getZeroExtendExpr(MulLHS, Ty),
                getZeroExtendExpr(
                    getTruncateExpr(TruncRHS->getOperand(), NewTruncTy), Ty),
                SCEV::FlagNUW, Depth + 1);
          }
  }

  // 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);
  addToLoopUseLists(S);
  return S;
}

const SCEV *
ScalarEvolution::getSignExtendExpr(const SCEV *Op, Type *Ty, unsigned Depth) {
  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, Depth + 1);

  // sext(zext(x)) --> zext(x)
  if (const SCEVZeroExtendExpr *SZ = dyn_cast<SCEVZeroExtendExpr>(Op))
    return getZeroExtendExpr(SZ->getOperand(), Ty, Depth + 1);

  // 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 = nullptr;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  // Limit recursion depth.
  if (Depth > MaxCastDepth) {
    SCEV *S = new (SCEVAllocator) SCEVSignExtendExpr(ID.Intern(SCEVAllocator),
                                                     Op, Ty);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
    return S;
  }

  // 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, Depth);
  }

  if (auto *SA = dyn_cast<SCEVAddExpr>(Op)) {
    // sext((A + B + ...)<nsw>) --> (sext(A) + sext(B) + ...)<nsw>
    if (SA->hasNoSignedWrap()) {
      // If the addition does not sign overflow then we can, by definition,
      // commute the sign extension with the addition operation.
      SmallVector<const SCEV *, 4> Ops;
      for (const auto *Op : SA->operands())
        Ops.push_back(getSignExtendExpr(Op, Ty, Depth + 1));
      return getAddExpr(Ops, SCEV::FlagNSW, Depth + 1);
    }

    // sext(C + x + y + ...) --> (sext(D) + sext((C - D) + x + y + ...))
    // if D + (C - D + x + y + ...) could be proven to not signed wrap
    // where D maximizes the number of trailing zeros of (C - D + x + y + ...)
    //
    // For instance, this will bring two seemingly different expressions:
    //     1 + sext(5 + 20 * %x + 24 * %y)  and
    //         sext(6 + 20 * %x + 24 * %y)
    // to the same form:
    //     2 + sext(4 + 20 * %x + 24 * %y)
    if (const auto *SC = dyn_cast<SCEVConstant>(SA->getOperand(0))) {
      const APInt &D = extractConstantWithoutWrapping(*this, SC, SA);
      if (D != 0) {
        const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
        const SCEV *SResidual =
            getAddExpr(getConstant(-D), SA, SCEV::FlagAnyWrap, Depth);
        const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
        return getAddExpr(SSExtD, SSExtR,
                          (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                          Depth + 1);
      }
    }
  }
  // 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 (!AR->hasNoSignedWrap()) {
        auto NewFlags = proveNoWrapViaConstantRanges(AR);
        const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(NewFlags);
      }

      // If we have special knowledge that this addrec won't overflow,
      // we don't need to do any further analysis.
      if (AR->hasNoSignedWrap())
        return getAddRecExpr(
            getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
            getSignExtendExpr(Step, Ty, Depth + 1), 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(), Depth);
        const SCEV *RecastedMaxBECount = getTruncateOrZeroExtend(
            CastedMaxBECount, MaxBECount->getType(), Depth);
        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,
                                        SCEV::FlagAnyWrap, Depth + 1);
          const SCEV *SAdd = getSignExtendExpr(getAddExpr(Start, SMul,
                                                          SCEV::FlagAnyWrap,
                                                          Depth + 1),
                                               WideTy, Depth + 1);
          const SCEV *WideStart = getSignExtendExpr(Start, WideTy, Depth + 1);
          const SCEV *WideMaxBECount =
            getZeroExtendExpr(CastedMaxBECount, WideTy, Depth + 1);
          const SCEV *OperandExtendedAdd =
            getAddExpr(WideStart,
                       getMulExpr(WideMaxBECount,
                                  getSignExtendExpr(Step, WideTy, Depth + 1),
                                  SCEV::FlagAnyWrap, Depth + 1),
                       SCEV::FlagAnyWrap, Depth + 1);
          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(
                getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getSignExtendExpr(Step, Ty, Depth + 1), 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, Depth + 1),
                                  SCEV::FlagAnyWrap, Depth + 1),
                       SCEV::FlagAnyWrap, Depth + 1);
          if (SAdd == OperandExtendedAdd) {
            // If AR wraps around then
            //
            //    abs(Step) * MaxBECount > unsigned-max(AR->getType())
            // => SAdd != OperandExtendedAdd
            //
            // Thus (AR is not NW => SAdd != OperandExtendedAdd) <=>
            // (SAdd == OperandExtendedAdd => AR is NW)

            const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNW);

            // Return the expression with the addrec on the outside.
            return getAddRecExpr(
                getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this,
                                                         Depth + 1),
                getZeroExtendExpr(Step, Ty, Depth + 1), L,
                AR->getNoWrapFlags());
          }
        }
      }

      // Normally, in the cases we can prove no-overflow via a
      // backedge guarding condition, we can also compute a backedge
      // taken count for the loop.  The exceptions are assumptions and
      // guards present in the loop -- SCEV is not great at exploiting
      // these to compute max backedge taken counts, but can still use
      // these to prove lack of overflow.  Use this fact to avoid
      // doing extra work that may not pay off.

      if (!isa<SCEVCouldNotCompute>(MaxBECount) || HasGuards ||
          !AC.assumptions().empty()) {
        // 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 =
            getSignedOverflowLimitForStep(Step, &Pred, this);
        if (OverflowLimit &&
            (isLoopBackedgeGuardedByCond(L, Pred, AR, OverflowLimit) ||
             isKnownOnEveryIteration(Pred, AR, OverflowLimit))) {
          // Cache knowledge of AR NSW, then propagate NSW to the wide AddRec.
          const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
          return getAddRecExpr(
              getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
              getSignExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
        }
      }

      // sext({C,+,Step}) --> (sext(D) + sext({C-D,+,Step}))<nuw><nsw>
      // if D + (C - D + Step * n) could be proven to not signed wrap
      // where D maximizes the number of trailing zeros of (C - D + Step * n)
      if (const auto *SC = dyn_cast<SCEVConstant>(Start)) {
        const APInt &C = SC->getAPInt();
        const APInt &D = extractConstantWithoutWrapping(*this, C, Step);
        if (D != 0) {
          const SCEV *SSExtD = getSignExtendExpr(getConstant(D), Ty, Depth);
          const SCEV *SResidual =
              getAddRecExpr(getConstant(C - D), Step, L, AR->getNoWrapFlags());
          const SCEV *SSExtR = getSignExtendExpr(SResidual, Ty, Depth + 1);
          return getAddExpr(SSExtD, SSExtR,
                            (SCEV::NoWrapFlags)(SCEV::FlagNSW | SCEV::FlagNUW),
                            Depth + 1);
        }
      }

      if (proveNoWrapByVaryingStart<SCEVSignExtendExpr>(Start, Step, L)) {
        const_cast<SCEVAddRecExpr *>(AR)->setNoWrapFlags(SCEV::FlagNSW);
        return getAddRecExpr(
            getExtendAddRecStart<SCEVSignExtendExpr>(AR, Ty, this, Depth + 1),
            getSignExtendExpr(Step, Ty, Depth + 1), L, AR->getNoWrapFlags());
      }
    }

  // If the input value is provably positive and we could not simplify
  // away the sext build a zext instead.
  if (isKnownNonNegative(Op))
    return getZeroExtendExpr(Op, Ty, Depth + 1);

  // 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);
  addToLoopUseLists(S);
  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->getAPInt().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 (const SCEV *Op : AR->operands())
      Ops.push_back(getAnyExtendExpr(Op, 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;
}

/// 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,
                             SmallVectorImpl<const SCEV *> &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->getAPInt();
  }

  // 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))->getAPInt();
      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);
        auto Pair = M.insert({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({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;
}

// We're trying to construct a SCEV of type `Type' with `Ops' as operands and
// `OldFlags' as can't-wrap behavior.  Infer a more aggressive set of
// can't-overflow flags for the operation if possible.
static SCEV::NoWrapFlags
StrengthenNoWrapFlags(ScalarEvolution *SE, SCEVTypes Type,
                      const ArrayRef<const SCEV *> Ops,
                      SCEV::NoWrapFlags Flags) {
  using namespace std::placeholders;

  using OBO = OverflowingBinaryOperator;

  bool CanAnalyze =
      Type == scAddExpr || Type == scAddRecExpr || Type == scMulExpr;
  (void)CanAnalyze;
  assert(CanAnalyze && "don't call from other places!");

  int SignOrUnsignMask = SCEV::FlagNUW | SCEV::FlagNSW;
  SCEV::NoWrapFlags SignOrUnsignWrap =
      ScalarEvolution::maskFlags(Flags, SignOrUnsignMask);

  // If FlagNSW is true and all the operands are non-negative, infer FlagNUW.
  auto IsKnownNonNegative = [&](const SCEV *S) {
    return SE->isKnownNonNegative(S);
  };

  if (SignOrUnsignWrap == SCEV::FlagNSW && all_of(Ops, IsKnownNonNegative))
    Flags =
        ScalarEvolution::setFlags(Flags, (SCEV::NoWrapFlags)SignOrUnsignMask);

  SignOrUnsignWrap = ScalarEvolution::maskFlags(Flags, SignOrUnsignMask);

  if (SignOrUnsignWrap != SignOrUnsignMask &&
      (Type == scAddExpr || Type == scMulExpr) && Ops.size() == 2 &&
      isa<SCEVConstant>(Ops[0])) {

    auto Opcode = [&] {
      switch (Type) {
      case scAddExpr:
        return Instruction::Add;
      case scMulExpr:
        return Instruction::Mul;
      default:
        llvm_unreachable("Unexpected SCEV op.");
      }
    }();

    const APInt &C = cast<SCEVConstant>(Ops[0])->getAPInt();

    // (A <opcode> C) --> (A <opcode> C)<nsw> if the op doesn't sign overflow.
    if (!(SignOrUnsignWrap & SCEV::FlagNSW)) {
      auto NSWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
          Opcode, C, OBO::NoSignedWrap);
      if (NSWRegion.contains(SE->getSignedRange(Ops[1])))
        Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNSW);
    }

    // (A <opcode> C) --> (A <opcode> C)<nuw> if the op doesn't unsign overflow.
    if (!(SignOrUnsignWrap & SCEV::FlagNUW)) {
      auto NUWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
          Opcode, C, OBO::NoUnsignedWrap);
      if (NUWRegion.contains(SE->getUnsignedRange(Ops[1])))
        Flags = ScalarEvolution::setFlags(Flags, SCEV::FlagNUW);
    }
  }

  return Flags;
}

bool ScalarEvolution::isAvailableAtLoopEntry(const SCEV *S, const Loop *L) {
  return isLoopInvariant(S, L) && properlyDominates(S, L->getHeader());
}

/// Get a canonical add expression, or something simpler if possible.
const SCEV *ScalarEvolution::getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags,
                                        unsigned Depth) {
  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

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

  Flags = StrengthenNoWrapFlags(this, scAddExpr, Ops, Flags);

  // 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->getAPInt() + RHSC->getAPInt());
      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];
  }

  // Limit recursion calls depth.
  if (Depth > MaxArithDepth || hasHugeExpression(Ops))
    return getOrCreateAddExpr(Ops, Flags);

  // 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], SCEV::FlagAnyWrap, Depth + 1);
      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, Depth + 1);

  // 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., n*trunc(x) + m*trunc(y) --> trunc(trunc(m)*x + trunc(n)*y)
  // if the contents of the resulting outer trunc fold to something simple.
  auto FindTruncSrcType = [&]() -> Type * {
    // We're ultimately looking to fold an addrec of truncs and muls of only
    // constants and truncs, so if we find any other types of SCEV
    // as operands of the addrec then we bail and return nullptr here.
    // Otherwise, we return the type of the operand of a trunc that we find.
    if (auto *T = dyn_cast<SCEVTruncateExpr>(Ops[Idx]))
      return T->getOperand()->getType();
    if (const auto *Mul = dyn_cast<SCEVMulExpr>(Ops[Idx])) {
      const auto *LastOp = Mul->getOperand(Mul->getNumOperands() - 1);
      if (const auto *T = dyn_cast<SCEVTruncateExpr>(LastOp))
        return T->getOperand()->getType();
    }
    return nullptr;
  };
  if (auto *SrcType = FindTruncSrcType()) {
    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 auto *C = dyn_cast<SCEVConstant>(M->getOperand(j))) {
            LargeMulOps.push_back(getAnyExtendExpr(C, SrcType));
          } else {
            Ok = false;
            break;
          }
        }
        if (Ok)
          LargeOps.push_back(getMulExpr(LargeMulOps, SCEV::FlagAnyWrap, Depth + 1));
      } else {
        Ok = false;
        break;
      }
    }
    if (Ok) {
      // Evaluate the expression in the larger type.
      const SCEV *Fold = getAddExpr(LargeOps, SCEV::FlagAnyWrap, Depth + 1);
      // If it folds to something simple, use it. Otherwise, don't.
      if (isa<SCEVConstant>(Fold) || isa<SCEVUnknown>(Fold))
        return getTruncateExpr(Fold, Ty);
    }
  }

  // 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 (Ops.size() > AddOpsInlineThreshold ||
          Add->getNumOperands() > AddOpsInlineThreshold)
        break;
      // 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, SCEV::FlagAnyWrap, Depth + 1);
  }

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

      // 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 (const SCEV *NewOp : NewOps)
        MulOpLists[M.find(NewOp)->second].push_back(NewOp);
      // Re-generate the operands list.
      Ops.clear();
      if (AccumulatedConstant != 0)
        Ops.push_back(getConstant(AccumulatedConstant));
      for (auto &MulOp : MulOpLists)
        if (MulOp.first != 0)
          Ops.push_back(getMulExpr(
              getConstant(MulOp.first),
              getAddExpr(MulOp.second, SCEV::FlagAnyWrap, Depth + 1),
              SCEV::FlagAnyWrap, Depth + 1));
      if (Ops.empty())
        return getZero(Ty);
      if (Ops.size() == 1)
        return Ops[0];
      return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
    }
  }

  // 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, SCEV::FlagAnyWrap, Depth + 1);
          }
          SmallVector<const SCEV *, 2> TwoOps = {getOne(Ty), InnerMul};
          const SCEV *AddOne = getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
          const SCEV *OuterMul = getMulExpr(AddOne, MulOpSCEV,
                                            SCEV::FlagAnyWrap, Depth + 1);
          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, SCEV::FlagAnyWrap, Depth + 1);
        }

      // 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, SCEV::FlagAnyWrap, Depth + 1);
            }
            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, SCEV::FlagAnyWrap, Depth + 1);
            }
            SmallVector<const SCEV *, 2> TwoOps = {InnerMul1, InnerMul2};
            const SCEV *InnerMulSum =
                getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
            const SCEV *OuterMul = getMulExpr(MulOpSCEV, InnerMulSum,
                                              SCEV::FlagAnyWrap, Depth + 1);
            if (Ops.size() == 2) return OuterMul;
            Ops.erase(Ops.begin()+Idx);
            Ops.erase(Ops.begin()+OtherMulIdx-1);
            Ops.push_back(OuterMul);
            return getAddExpr(Ops, SCEV::FlagAnyWrap, Depth + 1);
          }
      }
    }
  }

  // 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 (isAvailableAtLoopEntry(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());
      // This follows from the fact that the no-wrap flags on the outer add
      // expression are applicable on the 0th iteration, when the add recurrence
      // will be equal to its start value.
      AddRecOps[0] = getAddExpr(LIOps, Flags, Depth + 1);

      // 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, SCEV::FlagAnyWrap, Depth + 1);
    }

    // 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) {
      // We expect the AddRecExpr's to be sorted in reverse dominance order,
      // so that the 1st found AddRecExpr is dominated by all others.
      assert(DT.dominates(
           cast<SCEVAddRecExpr>(Ops[OtherIdx])->getLoop()->getHeader(),
           AddRec->getLoop()->getHeader()) &&
        "AddRecExprs are not sorted in reverse dominance order?");
      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) {
          const auto *OtherAddRec = 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;
              }
              SmallVector<const SCEV *, 2> TwoOps = {
                  AddRecOps[i], OtherAddRec->getOperand(i)};
              AddRecOps[i] = getAddExpr(TwoOps, SCEV::FlagAnyWrap, Depth + 1);
            }
            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, SCEV::FlagAnyWrap, Depth + 1);
      }
    }

    // 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.
  return getOrCreateAddExpr(Ops, Flags);
}

const SCEV *
ScalarEvolution::getOrCreateAddExpr(ArrayRef<const SCEV *> Ops,
                                    SCEV::NoWrapFlags Flags) {
  FoldingSetNodeID ID;
  ID.AddInteger(scAddExpr);
  for (const SCEV *Op : Ops)
    ID.AddPointer(Op);
  void *IP = nullptr;
  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);
    addToLoopUseLists(S);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

const SCEV *
ScalarEvolution::getOrCreateAddRecExpr(ArrayRef<const SCEV *> Ops,
                                       const Loop *L, SCEV::NoWrapFlags Flags) {
  FoldingSetNodeID ID;
  ID.AddInteger(scAddRecExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  ID.AddPointer(L);
  void *IP = nullptr;
  SCEVAddRecExpr *S =
      static_cast<SCEVAddRecExpr *>(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)
        SCEVAddRecExpr(ID.Intern(SCEVAllocator), O, Ops.size(), L);
    UniqueSCEVs.InsertNode(S, IP);
    addToLoopUseLists(S);
  }
  S->setNoWrapFlags(Flags);
  return S;
}

const SCEV *
ScalarEvolution::getOrCreateMulExpr(ArrayRef<const SCEV *> Ops,
                                    SCEV::NoWrapFlags Flags) {
  FoldingSetNodeID ID;
  ID.AddInteger(scMulExpr);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  void *IP = nullptr;
  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);
    addToLoopUseLists(S);
  }
  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;
}

/// Determine if any of the operands in this SCEV are a constant or if
/// any of the add or multiply expressions in this SCEV contain a constant.
static bool containsConstantInAddMulChain(const SCEV *StartExpr) {
  struct FindConstantInAddMulChain {
    bool FoundConstant = false;

    bool follow(const SCEV *S) {
      FoundConstant |= isa<SCEVConstant>(S);
      return isa<SCEVAddExpr>(S) || isa<SCEVMulExpr>(S);
    }

    bool isDone() const {
      return FoundConstant;
    }
  };

  FindConstantInAddMulChain F;
  SCEVTraversal<FindConstantInAddMulChain> ST(F);
  ST.visitAll(StartExpr);
  return F.FoundConstant;
}

/// Get a canonical multiply expression, or something simpler if possible.
const SCEV *ScalarEvolution::getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                                        SCEV::NoWrapFlags Flags,
                                        unsigned Depth) {
  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

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

  Flags = StrengthenNoWrapFlags(this, scMulExpr, Ops, Flags);

  // Limit recursion calls depth.
  if (Depth > MaxArithDepth || hasHugeExpression(Ops))
    return getOrCreateMulExpr(Ops, Flags);

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

    if (Ops.size() == 2)
      // C1*(C2+V) -> C1*C2 + C1*V
      if (const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Ops[1]))
        // If any of Add's ops are Adds or Muls with a constant, apply this
        // transformation as well.
        //
        // TODO: There are some cases where this transformation is not
        // profitable; for example, Add = (C0 + X) * Y + Z.  Maybe the scope of
        // this transformation should be narrowed down.
        if (Add->getNumOperands() == 2 && containsConstantInAddMulChain(Add))
          return getAddExpr(getMulExpr(LHSC, Add->getOperand(0),
                                       SCEV::FlagAnyWrap, Depth + 1),
                            getMulExpr(LHSC, Add->getOperand(1),
                                       SCEV::FlagAnyWrap, Depth + 1),
                            SCEV::FlagAnyWrap, Depth + 1);

    ++Idx;
    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold =
          ConstantInt::get(getContext(), LHSC->getAPInt() * RHSC->getAPInt());
      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()->isOne()) {
      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 (const SCEV *AddOp : Add->operands()) {
            const SCEV *Mul = getMulExpr(Ops[0], AddOp, SCEV::FlagAnyWrap,
                                         Depth + 1);
            if (!isa<SCEVMulExpr>(Mul)) AnyFolded = true;
            NewOps.push_back(Mul);
          }
          if (AnyFolded)
            return getAddExpr(NewOps, SCEV::FlagAnyWrap, Depth + 1);
        } else if (const auto *AddRec = dyn_cast<SCEVAddRecExpr>(Ops[1])) {
          // Negation preserves a recurrence's no self-wrap property.
          SmallVector<const SCEV *, 4> Operands;
          for (const SCEV *AddRecOp : AddRec->operands())
            Operands.push_back(getMulExpr(Ops[0], AddRecOp, SCEV::FlagAnyWrap,
                                          Depth + 1));

          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 (Ops.size() > MulOpsInlineThreshold)
        break;
      // 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, SCEV::FlagAnyWrap, Depth + 1);
  }

  // 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 (isAvailableAtLoopEntry(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, SCEV::FlagAnyWrap, Depth + 1);
      for (unsigned i = 0, e = AddRec->getNumOperands(); i != e; ++i)
        NewOps.push_back(getMulExpr(Scale, AddRec->getOperand(i),
                                    SCEV::FlagAnyWrap, Depth + 1));

      // 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, SCEV::FlagAnyWrap, Depth + 1);
    }

    // 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.

    // {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 (unsigned OtherIdx = Idx+1;
         OtherIdx != Ops.size() && isa<SCEVAddRecExpr>(Ops[OtherIdx]);
         ++OtherIdx) {
      const SCEVAddRecExpr *OtherAddRec =
        dyn_cast<SCEVAddRecExpr>(Ops[OtherIdx]);
      if (!OtherAddRec || OtherAddRec->getLoop() != AddRecLoop)
        continue;

      // Limit max number of arguments to avoid creation of unreasonably big
      // SCEVAddRecs with very complex operands.
      if (AddRec->getNumOperands() + OtherAddRec->getNumOperands() - 1 >
          MaxAddRecSize || isHugeExpression(AddRec) ||
          isHugeExpression(OtherAddRec))
        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) {
        SmallVector <const SCEV *, 7> SumOps;
        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);
            SumOps.push_back(getMulExpr(CoeffTerm, Term1, Term2,
                                        SCEV::FlagAnyWrap, Depth + 1));
          }
        }
        if (SumOps.empty())
          SumOps.push_back(getZero(Ty));
        AddRecOps.push_back(getAddExpr(SumOps, SCEV::FlagAnyWrap, Depth + 1));
      }
      if (!Overflow) {
        const SCEV *NewAddRec = getAddRecExpr(AddRecOps, AddRecLoop,
                                              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, SCEV::FlagAnyWrap, Depth + 1);

    // 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.
  return getOrCreateMulExpr(Ops, Flags);
}

/// Represents an unsigned remainder expression based on unsigned division.
const SCEV *ScalarEvolution::getURemExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  assert(getEffectiveSCEVType(LHS->getType()) ==
         getEffectiveSCEVType(RHS->getType()) &&
         "SCEVURemExpr operand types don't match!");

  // Short-circuit easy cases
  if (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(RHS)) {
    // If constant is one, the result is trivial
    if (RHSC->getValue()->isOne())
      return getZero(LHS->getType()); // X urem 1 --> 0

    // If constant is a power of two, fold into a zext(trunc(LHS)).
    if (RHSC->getAPInt().isPowerOf2()) {
      Type *FullTy = LHS->getType();
      Type *TruncTy =
          IntegerType::get(getContext(), RHSC->getAPInt().logBase2());
      return getZeroExtendExpr(getTruncateExpr(LHS, TruncTy), FullTy);
    }
  }

  // Fallback to %a == %x urem %y == %x -<nuw> ((%x udiv %y) *<nuw> %y)
  const SCEV *UDiv = getUDivExpr(LHS, RHS);
  const SCEV *Mult = getMulExpr(UDiv, RHS, SCEV::FlagNUW);
  return getMinusSCEV(LHS, Mult, SCEV::FlagNUW);
}

/// 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()->isOne())
      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->getAPInt().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->getAPInt().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->getAPInt();
          const APInt &DivInt = RHSC->getAPInt();
          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 (const SCEV *Op : AR->operands())
              Operands.push_back(getUDivExpr(Op, 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->getAPInt();
            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 (const SCEV *Op : M->operands())
          Operands.push_back(getZeroExtendExpr(Op, 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/(B*C) if safe and B*C can be folded.
      if (const SCEVUDivExpr *OtherDiv = dyn_cast<SCEVUDivExpr>(LHS)) {
        if (auto *DivisorConstant =
                dyn_cast<SCEVConstant>(OtherDiv->getRHS())) {
          bool Overflow = false;
          APInt NewRHS =
              DivisorConstant->getAPInt().umul_ov(RHSC->getAPInt(), Overflow);
          if (Overflow) {
            return getConstant(RHSC->getType(), 0, false);
          }
          return getUDivExpr(OtherDiv->getLHS(), getConstant(NewRHS));
        }
      }

      // (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 (const SCEV *Op : A->operands())
          Operands.push_back(getZeroExtendExpr(Op, 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 = nullptr;
  if (const SCEV *S = UniqueSCEVs.FindNodeOrInsertPos(ID, IP)) return S;
  SCEV *S = new (SCEVAllocator) SCEVUDivExpr(ID.Intern(SCEVAllocator),
                                             LHS, RHS);
  UniqueSCEVs.InsertNode(S, IP);
  addToLoopUseLists(S);
  return S;
}

static const APInt gcd(const SCEVConstant *C1, const SCEVConstant *C2) {
  APInt A = C1->getAPInt().abs();
  APInt B = C2->getAPInt().abs();
  uint32_t ABW = A.getBitWidth();
  uint32_t BBW = B.getBitWidth();

  if (ABW > BBW)
    B = B.zext(ABW);
  else if (ABW < BBW)
    A = A.zext(BBW);

  return APIntOps::GreatestCommonDivisor(std::move(A), std::move(B));
}

/// Get a canonical unsigned division expression, or something simpler if
/// possible. There is no representation for an exact udiv in SCEV IR, but we
/// can attempt to remove factors from the LHS and RHS.  We can't do this when
/// it's not exact because the udiv may be clearing bits.
const SCEV *ScalarEvolution::getUDivExactExpr(const SCEV *LHS,
                                              const SCEV *RHS) {
  // TODO: we could try to find factors in all sorts of things, but for now we
  // just deal with u/exact (multiply, constant). See SCEVDivision towards the
  // end of this file for inspiration.

  const SCEVMulExpr *Mul = dyn_cast<SCEVMulExpr>(LHS);
  if (!Mul || !Mul->hasNoUnsignedWrap())
    return getUDivExpr(LHS, RHS);

  if (const SCEVConstant *RHSCst = dyn_cast<SCEVConstant>(RHS)) {
    // If the mulexpr multiplies by a constant, then that constant must be the
    // first element of the mulexpr.
    if (const auto *LHSCst = dyn_cast<SCEVConstant>(Mul->getOperand(0))) {
      if (LHSCst == RHSCst) {
        SmallVector<const SCEV *, 2> Operands;
        Operands.append(Mul->op_begin() + 1, Mul->op_end());
        return getMulExpr(Operands);
      }

      // We can't just assume that LHSCst divides RHSCst cleanly, it could be
      // that there's a factor provided by one of the other terms. We need to
      // check.
      APInt Factor = gcd(LHSCst, RHSCst);
      if (!Factor.isIntN(1)) {
        LHSCst =
            cast<SCEVConstant>(getConstant(LHSCst->getAPInt().udiv(Factor)));
        RHSCst =
            cast<SCEVConstant>(getConstant(RHSCst->getAPInt().udiv(Factor)));
        SmallVector<const SCEV *, 2> Operands;
        Operands.push_back(LHSCst);
        Operands.append(Mul->op_begin() + 1, Mul->op_end());
        LHS = getMulExpr(Operands);
        RHS = RHSCst;
        Mul = dyn_cast<SCEVMulExpr>(LHS);
        if (!Mul)
          return getUDivExactExpr(LHS, RHS);
      }
    }
  }

  for (int i = 0, e = Mul->getNumOperands(); i != e; ++i) {
    if (Mul->getOperand(i) == RHS) {
      SmallVector<const SCEV *, 2> Operands;
      Operands.append(Mul->op_begin(), Mul->op_begin() + i);
      Operands.append(Mul->op_begin() + i + 1, Mul->op_end());
      return getMulExpr(Operands);
    }
  }

  return getUDivExpr(LHS, RHS);
}

/// 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);
}

/// 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).

  Flags = StrengthenNoWrapFlags(this, scAddRecExpr, Operands, Flags);

  // 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 = all_of(
          Operands, [&](const SCEV *Op) { return isLoopInvariant(Op, L); });

      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 = all_of(NestedOperands, [&](const SCEV *Op) {
          return isLoopInvariant(Op, NestedLoop);
        });

        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.
  return getOrCreateAddRecExpr(Operands, L, Flags);
}

const SCEV *
ScalarEvolution::getGEPExpr(GEPOperator *GEP,
                            const SmallVectorImpl<const SCEV *> &IndexExprs) {
  const SCEV *BaseExpr = getSCEV(GEP->getPointerOperand());
  // getSCEV(Base)->getType() has the same address space as Base->getType()
  // because SCEV::getType() preserves the address space.
  Type *IntPtrTy = getEffectiveSCEVType(BaseExpr->getType());
  // FIXME(PR23527): Don't blindly transfer the inbounds flag from the GEP
  // instruction to its SCEV, because the Instruction may be guarded by control
  // flow and the no-overflow bits may not be valid for the expression in any
  // context. This can be fixed similarly to how these flags are handled for
  // adds.
  SCEV::NoWrapFlags Wrap = GEP->isInBounds() ? SCEV::FlagNSW
                                             : SCEV::FlagAnyWrap;

  const SCEV *TotalOffset = getZero(IntPtrTy);
  // The array size is unimportant. The first thing we do on CurTy is getting
  // its element type.
  Type *CurTy = ArrayType::get(GEP->getSourceElementType(), 0);
  for (const SCEV *IndexExpr : IndexExprs) {
    // Compute the (potentially symbolic) offset in bytes for this index.
    if (StructType *STy = dyn_cast<StructType>(CurTy)) {
      // For a struct, add the member offset.
      ConstantInt *Index = cast<SCEVConstant>(IndexExpr)->getValue();
      unsigned FieldNo = Index->getZExtValue();
      const SCEV *FieldOffset = getOffsetOfExpr(IntPtrTy, STy, FieldNo);

      // Add the field offset to the running total offset.
      TotalOffset = getAddExpr(TotalOffset, FieldOffset);

      // Update CurTy to the type of the field at Index.
      CurTy = STy->getTypeAtIndex(Index);
    } else {
      // Update CurTy to its element type.
      CurTy = cast<SequentialType>(CurTy)->getElementType();
      // For an array, add the element offset, explicitly scaled.
      const SCEV *ElementSize = getSizeOfExpr(IntPtrTy, CurTy);
      // Getelementptr indices are signed.
      IndexExpr = getTruncateOrSignExtend(IndexExpr, IntPtrTy);

      // Multiply the index by the element size to compute the element offset.
      const SCEV *LocalOffset = getMulExpr(IndexExpr, ElementSize, Wrap);

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

  // Add the total offset from all the GEP indices to the base.
  return getAddExpr(BaseExpr, TotalOffset, Wrap);
}

std::tuple<const SCEV *, FoldingSetNodeID, void *>
ScalarEvolution::findExistingSCEVInCache(int SCEVType,
                                         ArrayRef<const SCEV *> Ops) {
  FoldingSetNodeID ID;
  void *IP = nullptr;
  ID.AddInteger(SCEVType);
  for (unsigned i = 0, e = Ops.size(); i != e; ++i)
    ID.AddPointer(Ops[i]);
  return std::tuple<const SCEV *, FoldingSetNodeID, void *>(
      UniqueSCEVs.FindNodeOrInsertPos(ID, IP), std::move(ID), IP);
}

const SCEV *ScalarEvolution::getMinMaxExpr(unsigned Kind,
                                           SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "Cannot get empty (u|s)(min|max)!");
  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 &&
           "Operand types don't match!");
#endif

  bool IsSigned = Kind == scSMaxExpr || Kind == scSMinExpr;
  bool IsMax = Kind == scSMaxExpr || Kind == scUMaxExpr;

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

  // Check if we have created the same expression before.
  if (const SCEV *S = std::get<0>(findExistingSCEVInCache(Kind, Ops))) {
    return S;
  }

  // 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());
    auto FoldOp = [&](const APInt &LHS, const APInt &RHS) {
      if (Kind == scSMaxExpr)
        return APIntOps::smax(LHS, RHS);
      else if (Kind == scSMinExpr)
        return APIntOps::smin(LHS, RHS);
      else if (Kind == scUMaxExpr)
        return APIntOps::umax(LHS, RHS);
      else if (Kind == scUMinExpr)
        return APIntOps::umin(LHS, RHS);
      llvm_unreachable("Unknown SCEV min/max opcode");
    };

    while (const SCEVConstant *RHSC = dyn_cast<SCEVConstant>(Ops[Idx])) {
      // We found two constants, fold them together!
      ConstantInt *Fold = ConstantInt::get(
          getContext(), FoldOp(LHSC->getAPInt(), RHSC->getAPInt()));
      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]);
    }

    bool IsMinV = LHSC->getValue()->isMinValue(IsSigned);
    bool IsMaxV = LHSC->getValue()->isMaxValue(IsSigned);

    if (IsMax ? IsMinV : IsMaxV) {
      // If we are left with a constant minimum(/maximum)-int, strip it off.
      Ops.erase(Ops.begin());
      --Idx;
    } else if (IsMax ? IsMaxV : IsMinV) {
      // If we have a max(/min) with a constant maximum(/minimum)-int,
      // it will always be the extremum.
      return LHSC;
    }

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

  // Find the first operation of the same kind
  while (Idx < Ops.size() && Ops[Idx]->getSCEVType() < Kind)
    ++Idx;

  // Check to see if one of the operands is of the same kind. If so, expand its
  // operands onto our operand list, and recurse to simplify.
  if (Idx < Ops.size()) {
    bool DeletedAny = false;
    while (Ops[Idx]->getSCEVType() == Kind) {
      const SCEVMinMaxExpr *SMME = cast<SCEVMinMaxExpr>(Ops[Idx]);
      Ops.erase(Ops.begin()+Idx);
      Ops.append(SMME->op_begin(), SMME->op_end());
      DeletedAny = true;
    }

    if (DeletedAny)
      return getMinMaxExpr(Kind, 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.
  llvm::CmpInst::Predicate GEPred =
      IsSigned ? ICmpInst::ICMP_SGE : ICmpInst::ICMP_UGE;
  llvm::CmpInst::Predicate LEPred =
      IsSigned ? ICmpInst::ICMP_SLE : ICmpInst::ICMP_ULE;
  llvm::CmpInst::Predicate FirstPred = IsMax ? GEPred : LEPred;
  llvm::CmpInst::Predicate SecondPred = IsMax ? LEPred : GEPred;
  for (unsigned i = 0, e = Ops.size() - 1; i != e; ++i) {
    if (Ops[i] == Ops[i + 1] ||
        isKnownViaNonRecursiveReasoning(FirstPred, Ops[i], Ops[i + 1])) {
      //  X op Y op Y  -->  X op Y
      //  X op Y       -->  X, if we know X, Y are ordered appropriately
      Ops.erase(Ops.begin() + i + 1, Ops.begin() + i + 2);
      --i;
      --e;
    } else if (isKnownViaNonRecursiveReasoning(SecondPred, Ops[i],
                                               Ops[i + 1])) {
      //  X op Y       -->  Y, if we know X, Y are ordered appropriately
      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 expr.  Check to see if we
  // already have one, otherwise create a new one.
  const SCEV *ExistingSCEV;
  FoldingSetNodeID ID;
  void *IP;
  std::tie(ExistingSCEV, ID, IP) = findExistingSCEVInCache(Kind, Ops);
  if (ExistingSCEV)
    return ExistingSCEV;
  const SCEV **O = SCEVAllocator.Allocate<const SCEV *>(Ops.size());
  std::uninitialized_copy(Ops.begin(), Ops.end(), O);
  SCEV *S = new (SCEVAllocator) SCEVMinMaxExpr(
      ID.Intern(SCEVAllocator), static_cast<SCEVTypes>(Kind), O, Ops.size());

  UniqueSCEVs.InsertNode(S, IP);
  addToLoopUseLists(S);
  return S;
}

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

const SCEV *ScalarEvolution::getSMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  return getMinMaxExpr(scSMaxExpr, Ops);
}

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

const SCEV *ScalarEvolution::getUMaxExpr(SmallVectorImpl<const SCEV *> &Ops) {
  return getMinMaxExpr(scUMaxExpr, Ops);
}

const SCEV *ScalarEvolution::getSMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
  return getSMinExpr(Ops);
}

const SCEV *ScalarEvolution::getSMinExpr(SmallVectorImpl<const SCEV *> &Ops) {
  return getMinMaxExpr(scSMinExpr, Ops);
}

const SCEV *ScalarEvolution::getUMinExpr(const SCEV *LHS,
                                         const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
  return getUMinExpr(Ops);
}

const SCEV *ScalarEvolution::getUMinExpr(SmallVectorImpl<const SCEV *> &Ops) {
  return getMinMaxExpr(scUMinExpr, Ops);
}

const SCEV *ScalarEvolution::getSizeOfExpr(Type *IntTy, Type *AllocTy) {
  // We can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  return getConstant(IntTy, getDataLayout().getTypeAllocSize(AllocTy));
}

const SCEV *ScalarEvolution::getOffsetOfExpr(Type *IntTy,
                                             StructType *STy,
                                             unsigned FieldNo) {
  // We can bypass creating a target-independent
  // constant expression and then folding it back into a ConstantInt.
  // This is just a compile-time optimization.
  return getConstant(
      IntTy, getDataLayout().getStructLayout(STy)->getElementOffset(FieldNo));
}

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 = nullptr;
  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
//

/// 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->isIntOrPtrTy();
}

/// 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 (Ty->isPointerTy())
    return getDataLayout().getIndexTypeSizeInBits(Ty);
  return getDataLayout().getTypeSizeInBits(Ty);
}

/// 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!");
  return getDataLayout().getIntPtrType(Ty);
}

Type *ScalarEvolution::getWiderType(Type *T1, Type *T2) const {
  return  getTypeSizeInBits(T1) >= getTypeSizeInBits(T2) ? T1 : T2;
}

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

bool ScalarEvolution::checkValidity(const SCEV *S) const {
  bool ContainsNulls = SCEVExprContains(S, [](const SCEV *S) {
    auto *SU = dyn_cast<SCEVUnknown>(S);
    return SU && SU->getValue() == nullptr;
  });

  return !ContainsNulls;
}

bool ScalarEvolution::containsAddRecurrence(const SCEV *S) {
  HasRecMapType::iterator I = HasRecMap.find(S);
  if (I != HasRecMap.end())
    return I->second;

  bool FoundAddRec = SCEVExprContains(S, isa<SCEVAddRecExpr, const SCEV *>);
  HasRecMap.insert({S, FoundAddRec});
  return FoundAddRec;
}

/// Try to split a SCEVAddExpr into a pair of {SCEV, ConstantInt}.
/// If \p S is a SCEVAddExpr and is composed of a sub SCEV S' and an
/// offset I, then return {S', I}, else return {\p S, nullptr}.
static std::pair<const SCEV *, ConstantInt *> splitAddExpr(const SCEV *S) {
  const auto *Add = dyn_cast<SCEVAddExpr>(S);
  if (!Add)
    return {S, nullptr};

  if (Add->getNumOperands() != 2)
    return {S, nullptr};

  auto *ConstOp = dyn_cast<SCEVConstant>(Add->getOperand(0));
  if (!ConstOp)
    return {S, nullptr};

  return {Add->getOperand(1), ConstOp->getValue()};
}

/// Return the ValueOffsetPair set for \p S. \p S can be represented
/// by the value and offset from any ValueOffsetPair in the set.
SetVector<ScalarEvolution::ValueOffsetPair> *
ScalarEvolution::getSCEVValues(const SCEV *S) {
  ExprValueMapType::iterator SI = ExprValueMap.find_as(S);
  if (SI == ExprValueMap.end())
    return nullptr;
#ifndef NDEBUG
  if (VerifySCEVMap) {
    // Check there is no dangling Value in the set returned.
    for (const auto &VE : SI->second)
      assert(ValueExprMap.count(VE.first));
  }
#endif
  return &SI->second;
}

/// Erase Value from ValueExprMap and ExprValueMap. ValueExprMap.erase(V)
/// cannot be used separately. eraseValueFromMap should be used to remove
/// V from ValueExprMap and ExprValueMap at the same time.
void ScalarEvolution::eraseValueFromMap(Value *V) {
  ValueExprMapType::iterator I = ValueExprMap.find_as(V);
  if (I != ValueExprMap.end()) {
    const SCEV *S = I->second;
    // Remove {V, 0} from the set of ExprValueMap[S]
    if (SetVector<ValueOffsetPair> *SV = getSCEVValues(S))
      SV->remove({V, nullptr});

    // Remove {V, Offset} from the set of ExprValueMap[Stripped]
    const SCEV *Stripped;
    ConstantInt *Offset;
    std::tie(Stripped, Offset) = splitAddExpr(S);
    if (Offset != nullptr) {
      if (SetVector<ValueOffsetPair> *SV = getSCEVValues(Stripped))
        SV->remove({V, Offset});
    }
    ValueExprMap.erase(V);
  }
}

/// Check whether value has nuw/nsw/exact set but SCEV does not.
/// TODO: In reality it is better to check the poison recursively
/// but this is better than nothing.
static bool SCEVLostPoisonFlags(const SCEV *S, const Value *V) {
  if (auto *I = dyn_cast<Instruction>(V)) {
    if (isa<OverflowingBinaryOperator>(I)) {
      if (auto *NS = dyn_cast<SCEVNAryExpr>(S)) {
        if (I->hasNoSignedWrap() && !NS->hasNoSignedWrap())
          return true;
        if (I->hasNoUnsignedWrap() && !NS->hasNoUnsignedWrap())
          return true;
      }
    } else if (isa<PossiblyExactOperator>(I) && I->isExact())
      return true;
  }
  return false;
}

/// 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!");

  const SCEV *S = getExistingSCEV(V);
  if (S == nullptr) {
    S = createSCEV(V);
    // During PHI resolution, it is possible to create two SCEVs for the same
    // V, so it is needed to double check whether V->S is inserted into
    // ValueExprMap before insert S->{V, 0} into ExprValueMap.
    std::pair<ValueExprMapType::iterator, bool> Pair =
        ValueExprMap.insert({SCEVCallbackVH(V, this), S});
    if (Pair.second && !SCEVLostPoisonFlags(S, V)) {
      ExprValueMap[S].insert({V, nullptr});

      // If S == Stripped + Offset, add Stripped -> {V, Offset} into
      // ExprValueMap.
      const SCEV *Stripped = S;
      ConstantInt *Offset = nullptr;
      std::tie(Stripped, Offset) = splitAddExpr(S);
      // If stripped is SCEVUnknown, don't bother to save
      // Stripped -> {V, offset}. It doesn't simplify and sometimes even
      // increase the complexity of the expansion code.
      // If V is GetElementPtrInst, don't save Stripped -> {V, offset}
      // because it may generate add/sub instead of GEP in SCEV expansion.
      if (Offset != nullptr && !isa<SCEVUnknown>(Stripped) &&
          !isa<GetElementPtrInst>(V))
        ExprValueMap[Stripped].insert({V, Offset});
    }
  }
  return S;
}

const SCEV *ScalarEvolution::getExistingSCEV(Value *V) {
  assert(isSCEVable(V->getType()) && "Value is not SCEVable!");

  ValueExprMapType::iterator I = ValueExprMap.find_as(V);
  if (I != ValueExprMap.end()) {
    const SCEV *S = I->second;
    if (checkValidity(S))
      return S;
    eraseValueFromMap(V);
    forgetMemoizedResults(S);
  }
  return nullptr;
}

/// Return a SCEV corresponding to -V = -1*V
const SCEV *ScalarEvolution::getNegativeSCEV(const SCEV *V,
                                             SCEV::NoWrapFlags Flags) {
  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))), Flags);
}

/// If Expr computes ~A, return A else return nullptr
static const SCEV *MatchNotExpr(const SCEV *Expr) {
  const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Expr);
  if (!Add || Add->getNumOperands() != 2 ||
      !Add->getOperand(0)->isAllOnesValue())
    return nullptr;

  const SCEVMulExpr *AddRHS = dyn_cast<SCEVMulExpr>(Add->getOperand(1));
  if (!AddRHS || AddRHS->getNumOperands() != 2 ||
      !AddRHS->getOperand(0)->isAllOnesValue())
    return nullptr;

  return AddRHS->getOperand(1);
}

/// 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())));

  // Fold ~(u|s)(min|max)(~x, ~y) to (u|s)(max|min)(x, y)
  if (const SCEVMinMaxExpr *MME = dyn_cast<SCEVMinMaxExpr>(V)) {
    auto MatchMinMaxNegation = [&](const SCEVMinMaxExpr *MME) {
      SmallVector<const SCEV *, 2> MatchedOperands;
      for (const SCEV *Operand : MME->operands()) {
        const SCEV *Matched = MatchNotExpr(Operand);
        if (!Matched)
          return (const SCEV *)nullptr;
        MatchedOperands.push_back(Matched);
      }
      return getMinMaxExpr(
          SCEVMinMaxExpr::negate(static_cast<SCEVTypes>(MME->getSCEVType())),
          MatchedOperands);
    };
    if (const SCEV *Replaced = MatchMinMaxNegation(MME))
      return Replaced;
  }

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

const SCEV *ScalarEvolution::getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                                          SCEV::NoWrapFlags Flags,
                                          unsigned Depth) {
  // Fast path: X - X --> 0.
  if (LHS == RHS)
    return getZero(LHS->getType());

  // We represent LHS - RHS as LHS + (-1)*RHS. This transformation
  // makes it so that we cannot make much use of NUW.
  auto AddFlags = SCEV::FlagAnyWrap;
  const bool RHSIsNotMinSigned =
      !getSignedRangeMin(RHS).isMinSignedValue();
  if (maskFlags(Flags, SCEV::FlagNSW) == SCEV::FlagNSW) {
    // Let M be the minimum representable signed value. Then (-1)*RHS
    // signed-wraps if and only if RHS is M. That can happen even for
    // a NSW subtraction because e.g. (-1)*M signed-wraps even though
    // -1 - M does not. So to transfer NSW from LHS - RHS to LHS +
    // (-1)*RHS, we need to prove that RHS != M.
    //
    // If LHS is non-negative and we know that LHS - RHS does not
    // signed-wrap, then RHS cannot be M. So we can rule out signed-wrap
    // either by proving that RHS > M or that LHS >= 0.
    if (RHSIsNotMinSigned || isKnownNonNegative(LHS)) {
      AddFlags = SCEV::FlagNSW;
    }
  }

  // FIXME: Find a correct way to transfer NSW to (-1)*M when LHS -
  // RHS is NSW and LHS >= 0.
  //
  // The difficulty here is that the NSW flag may have been proven
  // relative to a loop that is to be found in a recurrence in LHS and
  // not in RHS. Applying NSW to (-1)*M may then let the NSW have a
  // larger scope than intended.
  auto NegFlags = RHSIsNotMinSigned ? SCEV::FlagNSW : SCEV::FlagAnyWrap;

  return getAddExpr(LHS, getNegativeSCEV(RHS, NegFlags), AddFlags, Depth);
}

const SCEV *ScalarEvolution::getTruncateOrZeroExtend(const SCEV *V, Type *Ty,
                                                     unsigned Depth) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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, Depth);
  return getZeroExtendExpr(V, Ty, Depth);
}

const SCEV *ScalarEvolution::getTruncateOrSignExtend(const SCEV *V, Type *Ty,
                                                     unsigned Depth) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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, Depth);
  return getSignExtendExpr(V, Ty, Depth);
}

const SCEV *
ScalarEvolution::getNoopOrZeroExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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);
}

const SCEV *
ScalarEvolution::getNoopOrSignExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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);
}

const SCEV *
ScalarEvolution::getNoopOrAnyExtend(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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);
}

const SCEV *
ScalarEvolution::getTruncateOrNoop(const SCEV *V, Type *Ty) {
  Type *SrcTy = V->getType();
  assert(SrcTy->isIntOrPtrTy() && Ty->isIntOrPtrTy() &&
         "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);
}

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);
}

const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(const SCEV *LHS,
                                                        const SCEV *RHS) {
  SmallVector<const SCEV *, 2> Ops = { LHS, RHS };
  return getUMinFromMismatchedTypes(Ops);
}

const SCEV *ScalarEvolution::getUMinFromMismatchedTypes(
    SmallVectorImpl<const SCEV *> &Ops) {
  assert(!Ops.empty() && "At least one operand must be!");
  // Trivial case.
  if (Ops.size() == 1)
    return Ops[0];

  // Find the max type first.
  Type *MaxType = nullptr;
  for (auto *S : Ops)
    if (MaxType)
      MaxType = getWiderType(MaxType, S->getType());
    else
      MaxType = S->getType();

  // Extend all ops to max type.
  SmallVector<const SCEV *, 2> PromotedOps;
  for (auto *S : Ops)
    PromotedOps.push_back(getNoopOrZeroExtend(S, MaxType));

  // Generate umin.
  return getUMinExpr(PromotedOps);
}

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 = nullptr;
    for (const SCEV *NAryOp : NAry->operands()) {
      if (NAryOp->getType()->isPointerTy()) {
        // Cannot find the base of an expression with multiple pointer operands.
        if (PtrOp)
          return V;
        PtrOp = NAryOp;
      }
    }
    if (!PtrOp)
      return V;
    return getPointerBase(PtrOp);
  }
  return V;
}

/// 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 (User *U : I->users())
    Worklist.push_back(cast<Instruction>(U));
}

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).second)
      continue;

    auto 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)) {
        eraseValueFromMap(It->first);
        forgetMemoizedResults(Old);
      }
    }

    PushDefUseChildren(I, Worklist);
  }
}

namespace {

/// Takes SCEV S and Loop L. For each AddRec sub-expression, use its start
/// expression in case its Loop is L. If it is not L then
/// if IgnoreOtherLoops is true then use AddRec itself
/// otherwise rewrite cannot be done.
/// If SCEV contains non-invariant unknown SCEV rewrite cannot be done.
class SCEVInitRewriter : public SCEVRewriteVisitor<SCEVInitRewriter> {
public:
  static const SCEV *rewrite(const SCEV *S, const Loop *L, ScalarEvolution &SE,
                             bool IgnoreOtherLoops = true) {
    SCEVInitRewriter Rewriter(L, SE);
    const SCEV *Result = Rewriter.visit(S);
    if (Rewriter.hasSeenLoopVariantSCEVUnknown())
      return SE.getCouldNotCompute();
    return Rewriter.hasSeenOtherLoops() && !IgnoreOtherLoops
               ? SE.getCouldNotCompute()
               : Result;
  }

  const SCEV *visitUnknown(const SCEVUnknown *Expr) {
    if (!SE.isLoopInvariant(Expr, L))
      SeenLoopVariantSCEVUnknown = true;
    return Expr;
  }

  const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
    // Only re-write AddRecExprs for this loop.
    if (Expr->getLoop() == L)
      return Expr->getStart();
    SeenOtherLoops = true;
    return Expr;
  }

  bool hasSeenLoopVariantSCEVUnknown() { return SeenLoopVariantSCEVUnknown; }

  bool hasSeenOtherLoops() { return SeenOtherLoops; }

private:
  explicit SCEVInitRewriter(const Loop *L, ScalarEvolution &SE)
      : SCEVRewriteVisitor(SE), L(L) {}

  const Loop *L;
  bool SeenLoopVariantSCEVUnknown = false;
  bool SeenOtherLoops = false;
};

/// Takes SCEV S and Loop L. For each AddRec sub-expression, use its post
/// increment expression in case its Loop is L. If it is not L then
/// use AddRec itself.
/// If SCEV contains non-invariant unknown SCEV rewrite cannot be done.
class SCEVPostIncRewriter : public SCEVRewriteVisitor<SCEVPostIncRewriter> {
public:
  static const SCEV *rewrite(const SCEV *S, const Loop *L, ScalarEvolution &SE) {
    SCEVPostIncRewriter Rewriter(L, SE);
    const SCEV *Result = Rewriter.visit(S);
    return Rewriter.hasSeenLoopVariantSCEVUnknown()
        ? SE.getCouldNotCompute()
        : Result;
  }

  const SCEV *visitUnknown(const SCEVUnknown *Expr) {
    if (!SE.isLoopInvariant(Expr, L))
      SeenLoopVariantSCEVUnknown = true;
    return Expr;
  }

  const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
    // Only re-write AddRecExprs for this loop.
    if (Expr->getLoop() == L)
      return Expr->getPostIncExpr(SE);
    SeenOtherLoops = true;
    return Expr;
  }

  bool hasSeenLoopVariantSCEVUnknown() { return SeenLoopVariantSCEVUnknown; }

  bool hasSeenOtherLoops() { return SeenOtherLoops; }

private:
  explicit SCEVPostIncRewriter(const Loop *L, ScalarEvolution &SE)
      : SCEVRewriteVisitor(SE), L(L) {}

  const Loop *L;
  bool SeenLoopVariantSCEVUnknown = false;
  bool SeenOtherLoops = false;
};

/// This class evaluates the compare condition by matching it against the
/// condition of loop latch. If there is a match we assume a true value
/// for the condition while building SCEV nodes.
class SCEVBackedgeConditionFolder
    : public SCEVRewriteVisitor<SCEVBackedgeConditionFolder> {
public:
  static const SCEV *rewrite(const SCEV *S, const Loop *L,
                             ScalarEvolution &SE) {
    bool IsPosBECond = false;
    Value *BECond = nullptr;
    if (BasicBlock *Latch = L->getLoopLatch()) {
      BranchInst *BI = dyn_cast<BranchInst>(Latch->getTerminator());
      if (BI && BI->isConditional()) {
        assert(BI->getSuccessor(0) != BI->getSuccessor(1) &&
               "Both outgoing branches should not target same header!");
        BECond = BI->getCondition();
        IsPosBECond = BI->getSuccessor(0) == L->getHeader();
      } else {
        return S;
      }
    }
    SCEVBackedgeConditionFolder Rewriter(L, BECond, IsPosBECond, SE);
    return Rewriter.visit(S);
  }

  const SCEV *visitUnknown(const SCEVUnknown *Expr) {
    const SCEV *Result = Expr;
    bool InvariantF = SE.isLoopInvariant(Expr, L);

    if (!InvariantF) {
      Instruction *I = cast<Instruction>(Expr->getValue());
      switch (I->getOpcode()) {
      case Instruction::Select: {
        SelectInst *SI = cast<SelectInst>(I);
        Optional<const SCEV *> Res =
            compareWithBackedgeCondition(SI->getCondition());
        if (Res.hasValue()) {
          bool IsOne = cast<SCEVConstant>(Res.getValue())->getValue()->isOne();
          Result = SE.getSCEV(IsOne ? SI->getTrueValue() : SI->getFalseValue());
        }
        break;
      }
      default: {
        Optional<const SCEV *> Res = compareWithBackedgeCondition(I);
        if (Res.hasValue())
          Result = Res.getValue();
        break;
      }
      }
    }
    return Result;
  }

private:
  explicit SCEVBackedgeConditionFolder(const Loop *L, Value *BECond,
                                       bool IsPosBECond, ScalarEvolution &SE)
      : SCEVRewriteVisitor(SE), L(L), BackedgeCond(BECond),
        IsPositiveBECond(IsPosBECond) {}

  Optional<const SCEV *> compareWithBackedgeCondition(Value *IC);

  const Loop *L;
  /// Loop back condition.
  Value *BackedgeCond = nullptr;
  /// Set to true if loop back is on positive branch condition.
  bool IsPositiveBECond;
};

Optional<const SCEV *>
SCEVBackedgeConditionFolder::compareWithBackedgeCondition(Value *IC) {

  // If value matches the backedge condition for loop latch,
  // then return a constant evolution node based on loopback
  // branch taken.
  if (BackedgeCond == IC)
    return IsPositiveBECond ? SE.getOne(Type::getInt1Ty(SE.getContext()))
                            : SE.getZero(Type::getInt1Ty(SE.getContext()));
  return None;
}

class SCEVShiftRewriter : public SCEVRewriteVisitor<SCEVShiftRewriter> {
public:
  static const SCEV *rewrite(const SCEV *S, const Loop *L,
                             ScalarEvolution &SE) {
    SCEVShiftRewriter Rewriter(L, SE);
    const SCEV *Result = Rewriter.visit(S);
    return Rewriter.isValid() ? Result : SE.getCouldNotCompute();
  }

  const SCEV *visitUnknown(const SCEVUnknown *Expr) {
    // Only allow AddRecExprs for this loop.
    if (!SE.isLoopInvariant(Expr, L))
      Valid = false;
    return Expr;
  }

  const SCEV *visitAddRecExpr(const SCEVAddRecExpr *Expr) {
    if (Expr->getLoop() == L && Expr->isAffine())
      return SE.getMinusSCEV(Expr, Expr->getStepRecurrence(SE));
    Valid = false;
    return Expr;
  }

  bool isValid() { return Valid; }

private:
  explicit SCEVShiftRewriter(const Loop *L, ScalarEvolution &SE)
      : SCEVRewriteVisitor(SE), L(L) {}

  const Loop *L;
  bool Valid = true;
};

} // end anonymous namespace

SCEV::NoWrapFlags
ScalarEvolution::proveNoWrapViaConstantRanges(const SCEVAddRecExpr *AR) {
  if (!AR->isAffine())
    return SCEV::FlagAnyWrap;

  using OBO = OverflowingBinaryOperator;

  SCEV::NoWrapFlags Result = SCEV::FlagAnyWrap;

  if (!AR->hasNoSignedWrap()) {
    ConstantRange AddRecRange = getSignedRange(AR);
    ConstantRange IncRange = getSignedRange(AR->getStepRecurrence(*this));

    auto NSWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
        Instruction::Add, IncRange, OBO::NoSignedWrap);
    if (NSWRegion.contains(AddRecRange))
      Result = ScalarEvolution::setFlags(Result, SCEV::FlagNSW);
  }

  if (!AR->hasNoUnsignedWrap()) {
    ConstantRange AddRecRange = getUnsignedRange(AR);
    ConstantRange IncRange = getUnsignedRange(AR->getStepRecurrence(*this));

    auto NUWRegion = ConstantRange::makeGuaranteedNoWrapRegion(
        Instruction::Add, IncRange, OBO::NoUnsignedWrap);
    if (NUWRegion.contains(AddRecRange))
      Result = ScalarEvolution::setFlags(Result, SCEV::FlagNUW);
  }

  return Result;
}

namespace {

/// Represents an abstract binary operation.  This may exist as a
/// normal instruction or constant expression, or may have been
/// derived from an expression tree.
struct BinaryOp {
  unsigned Opcode;
  Value *LHS;
  Value *RHS;
  bool IsNSW = false;
  bool IsNUW = false;

  /// Op is set if this BinaryOp corresponds to a concrete LLVM instruction or
  /// constant expression.
  Operator *Op = nullptr;

  explicit BinaryOp(Operator *Op)
      : Opcode(Op->getOpcode()), LHS(Op->getOperand(0)), RHS(Op->getOperand(1)),
        Op(Op) {
    if (auto *OBO = dyn_cast<OverflowingBinaryOperator>(Op)) {
      IsNSW = OBO->hasNoSignedWrap();
      IsNUW = OBO->hasNoUnsignedWrap();
    }
  }

  explicit BinaryOp(unsigned Opcode, Value *LHS, Value *RHS, bool IsNSW = false,
                    bool IsNUW = false)
      : Opcode(Opcode), LHS(LHS), RHS(RHS), IsNSW(IsNSW), IsNUW(IsNUW) {}
};

} // end anonymous namespace

/// Try to map \p V into a BinaryOp, and return \c None on failure.
static Optional<BinaryOp> MatchBinaryOp(Value *V, DominatorTree &DT) {
  auto *Op = dyn_cast<Operator>(V);
  if (!Op)
    return None;

  // Implementation detail: all the cleverness here should happen without
  // creating new SCEV expressions -- our caller knowns tricks to avoid creating
  // SCEV expressions when possible, and we should not break that.

  switch (Op->getOpcode()) {
  case Instruction::Add:
  case Instruction::Sub:
  case Instruction::Mul:
  case Instruction::UDiv:
  case Instruction::URem:
  case Instruction::And:
  case Instruction::Or:
  case Instruction::AShr:
  case Instruction::Shl:
    return BinaryOp(Op);

  case Instruction::Xor:
    if (auto *RHSC = dyn_cast<ConstantInt>(Op->getOperand(1)))
      // If the RHS of the xor is a signmask, then this is just an add.
      // Instcombine turns add of signmask into xor as a strength reduction step.
      if (RHSC->getValue().isSignMask())
        return BinaryOp(Instruction::Add, Op->getOperand(0), Op->getOperand(1));
    return BinaryOp(Op);

  case Instruction::LShr:
    // Turn logical shift right of a constant into a unsigned divide.
    if (ConstantInt *SA = dyn_cast<ConstantInt>(Op->getOperand(1))) {
      uint32_t BitWidth = cast<IntegerType>(Op->getType())->getBitWidth();

      // If the shift count is not less than the bitwidth, the result of
      // the shift 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 (SA->getValue().ult(BitWidth)) {
        Constant *X =
            ConstantInt::get(SA->getContext(),
                             APInt::getOneBitSet(BitWidth, SA->getZExtValue()));
        return BinaryOp(Instruction::UDiv, Op->getOperand(0), X);
      }
    }
    return BinaryOp(Op);

  case Instruction::ExtractValue: {
    auto *EVI = cast<ExtractValueInst>(Op);
    if (EVI->getNumIndices() != 1 || EVI->getIndices()[0] != 0)
      break;

    auto *WO = dyn_cast<WithOverflowInst>(EVI->getAggregateOperand());
    if (!WO)
      break;

    Instruction::BinaryOps BinOp = WO->getBinaryOp();
    bool Signed = WO->isSigned();
    // TODO: Should add nuw/nsw flags for mul as well.
    if (BinOp == Instruction::Mul || !isOverflowIntrinsicNoWrap(WO, DT))
      return BinaryOp(BinOp, WO->getLHS(), WO->getRHS());

    // Now that we know that all uses of the arithmetic-result component of
    // CI are guarded by the overflow check, we can go ahead and pretend
    // that the arithmetic is non-overflowing.
    return BinaryOp(BinOp, WO->getLHS(), WO->getRHS(),
                    /* IsNSW = */ Signed, /* IsNUW = */ !Signed);
  }

  default:
    break;
  }

  return None;
}

/// Helper function to createAddRecFromPHIWithCasts. We have a phi
/// node whose symbolic (unknown) SCEV is \p SymbolicPHI, which is updated via
/// the loop backedge by a SCEVAddExpr, possibly also with a few casts on the
/// way. This function checks if \p Op, an operand of this SCEVAddExpr,
/// follows one of the following patterns:
/// Op == (SExt ix (Trunc iy (%SymbolicPHI) to ix) to iy)
/// Op == (ZExt ix (Trunc iy (%SymbolicPHI) to ix) to iy)
/// If the SCEV expression of \p Op conforms with one of the expected patterns
/// we return the type of the truncation operation, and indicate whether the
/// truncated type should be treated as signed/unsigned by setting
/// \p Signed to true/false, respectively.
static Type *isSimpleCastedPHI(const SCEV *Op, const SCEVUnknown *SymbolicPHI,
                               bool &Signed, ScalarEvolution &SE) {
  // The case where Op == SymbolicPHI (that is, with no type conversions on
  // the way) is handled by the regular add recurrence creating logic and
  // would have already been triggered in createAddRecForPHI. Reaching it here
  // means that createAddRecFromPHI had failed for this PHI before (e.g.,
  // because one of the other operands of the SCEVAddExpr updating this PHI is
  // not invariant).
  //
  // Here we look for the case where Op = (ext(trunc(SymbolicPHI))), and in
  // this case predicates that allow us to prove that Op == SymbolicPHI will
  // be added.
  if (Op == SymbolicPHI)
    return nullptr;

  unsigned SourceBits = SE.getTypeSizeInBits(SymbolicPHI->getType());
  unsigned NewBits = SE.getTypeSizeInBits(Op->getType());
  if (SourceBits != NewBits)
    return nullptr;

  const SCEVSignExtendExpr *SExt = dyn_cast<SCEVSignExtendExpr>(Op);
  const SCEVZeroExtendExpr *ZExt = dyn_cast<SCEVZeroExtendExpr>(Op);
  if (!SExt && !ZExt)
    return nullptr;
  const SCEVTruncateExpr *Trunc =
      SExt ? dyn_cast<SCEVTruncateExpr>(SExt->getOperand())
           : dyn_cast<SCEVTruncateExpr>(ZExt->getOperand());
  if (!Trunc)
    return nullptr;
  const SCEV *X = Trunc->getOperand();
  if (X != SymbolicPHI)
    return nullptr;
  Signed = SExt != nullptr;
  return Trunc->getType();
}

static const Loop *isIntegerLoopHeaderPHI(const PHINode *PN, LoopInfo &LI) {
  if (!PN->getType()->isIntegerTy())
    return nullptr;
  const Loop *L = LI.getLoopFor(PN->getParent());
  if (!L || L->getHeader() != PN->getParent())
    return nullptr;
  return L;
}

// Analyze \p SymbolicPHI, a SCEV expression of a phi node, and check if the
// computation that updates the phi follows the following pattern:
//   (SExt/ZExt ix (Trunc iy (%SymbolicPHI) to ix) to iy) + InvariantAccum
// which correspond to a phi->trunc->sext/zext->add->phi update chain.
// If so, try to see if it can be rewritten as an AddRecExpr under some
// Predicates. If successful, return them as a pair. Also cache the results
// of the analysis.
//
// Example usage scenario:
//    Say the Rewriter is called for the following SCEV:
//         8 * ((sext i32 (trunc i64 %X to i32) to i64) + %Step)
//    where:
//         %X = phi i64 (%Start, %BEValue)
//    It will visitMul->visitAdd->visitSExt->visitTrunc->visitUnknown(%X),
//    and call this function with %SymbolicPHI = %X.
//
//    The analysis will find that the value coming around the backedge has
//    the following SCEV:
//         BEValue = ((sext i32 (trunc i64 %X to i32) to i64) + %Step)
//    Upon concluding that this matches the desired pattern, the function
//    will return the pair {NewAddRec, SmallPredsVec} where:
//         NewAddRec = {%Start,+,%Step}
//         SmallPredsVec = {P1, P2, P3} as follows:
//           P1(WrapPred): AR: {trunc(%Start),+,(trunc %Step)}<nsw> Flags: <nssw>
//           P2(EqualPred): %Start == (sext i32 (trunc i64 %Start to i32) to i64)
//           P3(EqualPred): %Step == (sext i32 (trunc i64 %Step to i32) to i64)
//    The returned pair means that SymbolicPHI can be rewritten into NewAddRec
//    under the predicates {P1,P2,P3}.
//    This predicated rewrite will be cached in PredicatedSCEVRewrites:
//         PredicatedSCEVRewrites[{%X,L}] = {NewAddRec, {P1,P2,P3)}
//
// TODO's:
//
// 1) Extend the Induction descriptor to also support inductions that involve
//    casts: When needed (namely, when we are called in the context of the
//    vectorizer induction analysis), a Set of cast instructions will be
//    populated by this method, and provided back to isInductionPHI. This is
//    needed to allow the vectorizer to properly record them to be ignored by
//    the cost model and to avoid vectorizing them (otherwise these casts,
//    which are redundant under the runtime overflow checks, will be
//    vectorized, which can be costly).
//
// 2) Support additional induction/PHISCEV patterns: We also want to support
//    inductions where the sext-trunc / zext-trunc operations (partly) occur
//    after the induction update operation (the induction increment):
//
//      (Trunc iy (SExt/ZExt ix (%SymbolicPHI + InvariantAccum) to iy) to ix)
//    which correspond to a phi->add->trunc->sext/zext->phi update chain.
//
//      (Trunc iy ((SExt/ZExt ix (%SymbolicPhi) to iy) + InvariantAccum) to ix)
//    which correspond to a phi->trunc->add->sext/zext->phi update chain.
//
// 3) Outline common code with createAddRecFromPHI to avoid duplication.
Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
ScalarEvolution::createAddRecFromPHIWithCastsImpl(const SCEVUnknown *SymbolicPHI) {
  SmallVector<const SCEVPredicate *, 3> Predicates;

  // *** Part1: Analyze if we have a phi-with-cast pattern for which we can
  // return an AddRec expression under some predicate.

  auto *PN = cast<PHINode>(SymbolicPHI->getValue());
  const Loop *L = isIntegerLoopHeaderPHI(PN, LI);
  assert(L && "Expecting an integer loop header phi");

  // 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 = nullptr, *StartValueV = nullptr;
  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 = nullptr;
        break;
      }
    } else if (!StartValueV) {
      StartValueV = V;
    } else if (StartValueV != V) {
      StartValueV = nullptr;
      break;
    }
  }
  if (!BEValueV || !StartValueV)
    return None;

  const SCEV *BEValue = getSCEV(BEValueV);

  // If the value coming around the backedge is an add with the symbolic
  // value we just inserted, possibly with casts that we can ignore under
  // an appropriate runtime guard, then we found a simple induction variable!
  const auto *Add = dyn_cast<SCEVAddExpr>(BEValue);
  if (!Add)
    return None;

  // If there is a single occurrence of the symbolic value, possibly
  // casted, replace it with a recurrence.
  unsigned FoundIndex = Add->getNumOperands();
  Type *TruncTy = nullptr;
  bool Signed;
  for (unsigned i = 0, e = Add->getNumOperands(); i != e; ++i)
    if ((TruncTy =
             isSimpleCastedPHI(Add->getOperand(i), SymbolicPHI, Signed, *this)))
      if (FoundIndex == e) {
        FoundIndex = i;
        break;
      }

  if (FoundIndex == Add->getNumOperands())
    return None;

  // 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);

  // The runtime checks will not be valid if the step amount is
  // varying inside the loop.
  if (!isLoopInvariant(Accum, L))
    return None;

  // *** Part2: Create the predicates

  // Analysis was successful: we have a phi-with-cast pattern for which we
  // can return an AddRec expression under the following predicates:
  //
  // P1: A Wrap predicate that guarantees that Trunc(Start) + i*Trunc(Accum)
  //     fits within the truncated type (does not overflow) for i = 0 to n-1.
  // P2: An Equal predicate that guarantees that
  //     Start = (Ext ix (Trunc iy (Start) to ix) to iy)
  // P3: An Equal predicate that guarantees that
  //     Accum = (Ext ix (Trunc iy (Accum) to ix) to iy)
  //
  // As we next prove, the above predicates guarantee that:
  //     Start + i*Accum = (Ext ix (Trunc iy ( Start + i*Accum ) to ix) to iy)
  //
  //
  // More formally, we want to prove that:
  //     Expr(i+1) = Start + (i+1) * Accum
  //               = (Ext ix (Trunc iy (Expr(i)) to ix) to iy) + Accum
  //
  // Given that:
  // 1) Expr(0) = Start
  // 2) Expr(1) = Start + Accum
  //            = (Ext ix (Trunc iy (Start) to ix) to iy) + Accum :: from P2
  // 3) Induction hypothesis (step i):
  //    Expr(i) = (Ext ix (Trunc iy (Expr(i-1)) to ix) to iy) + Accum
  //
  // Proof:
  //  Expr(i+1) =
  //   = Start + (i+1)*Accum
  //   = (Start + i*Accum) + Accum
  //   = Expr(i) + Accum
  //   = (Ext ix (Trunc iy (Expr(i-1)) to ix) to iy) + Accum + Accum
  //                                                             :: from step i
  //
  //   = (Ext ix (Trunc iy (Start + (i-1)*Accum) to ix) to iy) + Accum + Accum
  //
  //   = (Ext ix (Trunc iy (Start + (i-1)*Accum) to ix) to iy)
  //     + (Ext ix (Trunc iy (Accum) to ix) to iy)
  //     + Accum                                                     :: from P3
  //
  //   = (Ext ix (Trunc iy ((Start + (i-1)*Accum) + Accum) to ix) to iy)
  //     + Accum                            :: from P1: Ext(x)+Ext(y)=>Ext(x+y)
  //
  //   = (Ext ix (Trunc iy (Start + i*Accum) to ix) to iy) + Accum
  //   = (Ext ix (Trunc iy (Expr(i)) to ix) to iy) + Accum
  //
  // By induction, the same applies to all iterations 1<=i<n:
  //

  // Create a truncated addrec for which we will add a no overflow check (P1).
  const SCEV *StartVal = getSCEV(StartValueV);
  const SCEV *PHISCEV =
      getAddRecExpr(getTruncateExpr(StartVal, TruncTy),
                    getTruncateExpr(Accum, TruncTy), L, SCEV::FlagAnyWrap);

  // PHISCEV can be either a SCEVConstant or a SCEVAddRecExpr.
  // ex: If truncated Accum is 0 and StartVal is a constant, then PHISCEV
  // will be constant.
  //
  //  If PHISCEV is a constant, then P1 degenerates into P2 or P3, so we don't
  // add P1.
  if (const auto *AR = dyn_cast<SCEVAddRecExpr>(PHISCEV)) {
    SCEVWrapPredicate::IncrementWrapFlags AddedFlags =
        Signed ? SCEVWrapPredicate::IncrementNSSW
               : SCEVWrapPredicate::IncrementNUSW;
    const SCEVPredicate *AddRecPred = getWrapPredicate(AR, AddedFlags);
    Predicates.push_back(AddRecPred);
  }

  // Create the Equal Predicates P2,P3:

  // It is possible that the predicates P2 and/or P3 are computable at
  // compile time due to StartVal and/or Accum being constants.
  // If either one is, then we can check that now and escape if either P2
  // or P3 is false.

  // Construct the extended SCEV: (Ext ix (Trunc iy (Expr) to ix) to iy)
  // for each of StartVal and Accum
  auto getExtendedExpr = [&](const SCEV *Expr,
                             bool CreateSignExtend) -> const SCEV * {
    assert(isLoopInvariant(Expr, L) && "Expr is expected to be invariant");
    const SCEV *TruncatedExpr = getTruncateExpr(Expr, TruncTy);
    const SCEV *ExtendedExpr =
        CreateSignExtend ? getSignExtendExpr(TruncatedExpr, Expr->getType())
                         : getZeroExtendExpr(TruncatedExpr, Expr->getType());
    return ExtendedExpr;
  };

  // Given:
  //  ExtendedExpr = (Ext ix (Trunc iy (Expr) to ix) to iy
  //               = getExtendedExpr(Expr)
  // Determine whether the predicate P: Expr == ExtendedExpr
  // is known to be false at compile time
  auto PredIsKnownFalse = [&](const SCEV *Expr,
                              const SCEV *ExtendedExpr) -> bool {
    return Expr != ExtendedExpr &&
           isKnownPredicate(ICmpInst::ICMP_NE, Expr, ExtendedExpr);
  };

  const SCEV *StartExtended = getExtendedExpr(StartVal, Signed);
  if (PredIsKnownFalse(StartVal, StartExtended)) {
    LLVM_DEBUG(dbgs() << "P2 is compile-time false\n";);
    return None;
  }

  // The Step is always Signed (because the overflow checks are either
  // NSSW or NUSW)
  const SCEV *AccumExtended = getExtendedExpr(Accum, /*CreateSignExtend=*/true);
  if (PredIsKnownFalse(Accum, AccumExtended)) {
    LLVM_DEBUG(dbgs() << "P3 is compile-time false\n";);
    return None;
  }

  auto AppendPredicate = [&](const SCEV *Expr,
                             const SCEV *ExtendedExpr) -> void {
    if (Expr != ExtendedExpr &&
        !isKnownPredicate(ICmpInst::ICMP_EQ, Expr, ExtendedExpr)) {
      const SCEVPredicate *Pred = getEqualPredicate(Expr, ExtendedExpr);
      LLVM_DEBUG(dbgs() << "Added Predicate: " << *Pred);
      Predicates.push_back(Pred);
    }
  };

  AppendPredicate(StartVal, StartExtended);
  AppendPredicate(Accum, AccumExtended);

  // *** Part3: Predicates are ready. Now go ahead and create the new addrec in
  // which the casts had been folded away. The caller can rewrite SymbolicPHI
  // into NewAR if it will also add the runtime overflow checks specified in
  // Predicates.
  auto *NewAR = getAddRecExpr(StartVal, Accum, L, SCEV::FlagAnyWrap);

  std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>> PredRewrite =
      std::make_pair(NewAR, Predicates);
  // Remember the result of the analysis for this SCEV at this locayyytion.
  PredicatedSCEVRewrites[{SymbolicPHI, L}] = PredRewrite;
  return PredRewrite;
}

Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
ScalarEvolution::createAddRecFromPHIWithCasts(const SCEVUnknown *SymbolicPHI) {
  auto *PN = cast<PHINode>(SymbolicPHI->getValue());
  const Loop *L = isIntegerLoopHeaderPHI(PN, LI);
  if (!L)
    return None;

  // Check to see if we already analyzed this PHI.
  auto I = PredicatedSCEVRewrites.find({SymbolicPHI, L});
  if (I != PredicatedSCEVRewrites.end()) {
    std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>> Rewrite =
        I->second;
    // Analysis was done before and failed to create an AddRec:
    if (Rewrite.first == SymbolicPHI)
      return None;
    // Analysis was done before and succeeded to create an AddRec under
    // a predicate:
    assert(isa<SCEVAddRecExpr>(Rewrite.first) && "Expected an AddRec");
    assert(!(Rewrite.second).empty() && "Expected to find Predicates");
    return Rewrite;
  }

  Optional<std::pair<const SCEV *, SmallVector<const SCEVPredicate *, 3>>>
    Rewrite = createAddRecFromPHIWithCastsImpl(SymbolicPHI);

  // Record in the cache that the analysis failed
  if (!Rewrite) {
    SmallVector<const SCEVPredicate *, 3> Predicates;
    PredicatedSCEVRewrites[{SymbolicPHI, L}] = {SymbolicPHI, Predicates};
    return None;
  }

  return Rewrite;
}

// FIXME: This utility is currently required because the Rewriter currently
// does not rewrite this expression:
// {0, +, (sext ix (trunc iy to ix) to iy)}
// into {0, +, %step},
// even when the following Equal predicate exists:
// "%step == (sext ix (trunc iy to ix) to iy)".
bool PredicatedScalarEvolution::areAddRecsEqualWithPreds(
    const SCEVAddRecExpr *AR1, const SCEVAddRecExpr *AR2) const {
  if (AR1 == AR2)
    return true;

  auto areExprsEqual = [&](const SCEV *Expr1, const SCEV *Expr2) -> bool {
    if (Expr1 != Expr2 && !Preds.implies(SE.getEqualPredicate(Expr1, Expr2)) &&
        !Preds.implies(SE.getEqualPredicate(Expr2, Expr1)))
      return false;
    return true;
  };

  if (!areExprsEqual(AR1->getStart(), AR2->getStart()) ||
      !areExprsEqual(AR1->getStepRecurrence(SE), AR2->getStepRecurrence(SE)))
    return false;
  return true;
}

/// A helper function for createAddRecFromPHI to handle simple cases.
///
/// This function tries to find an AddRec expression for the simplest (yet most
/// common) cases: PN = PHI(Start, OP(Self, LoopInvariant)).
/// If it fails, createAddRecFromPHI will use a more general, but slow,
/// technique for finding the AddRec expression.
const SCEV *ScalarEvolution::createSimpleAffineAddRec(PHINode *PN,
                                                      Value *BEValueV,
                                                      Value *StartValueV) {
  const Loop *L = LI.getLoopFor(PN->getParent());
  assert(L && L->getHeader() == PN->getParent());
  assert(BEValueV && StartValueV);

  auto BO = MatchBinaryOp(BEValueV, DT);
  if (!BO)
    return nullptr;

  if (BO->Opcode != Instruction::Add)
    return nullptr;

  const SCEV *Accum = nullptr;
  if (BO->LHS == PN && L->isLoopInvariant(BO->RHS))
    Accum = getSCEV(BO->RHS);
  else if (BO->RHS == PN && L->isLoopInvariant(BO->LHS))
    Accum = getSCEV(BO->LHS);

  if (!Accum)
    return nullptr;

  SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap;
  if (BO->IsNUW)
    Flags = setFlags(Flags, SCEV::FlagNUW);
  if (BO->IsNSW)
    Flags = setFlags(Flags, SCEV::FlagNSW);

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

  ValueExprMap[SCEVCallbackVH(PN, this)] = PHISCEV;

  // We can add Flags to the post-inc expression only if we
  // know that it is *undefined behavior* for BEValueV to
  // overflow.
  if (auto *BEInst = dyn_cast<Instruction>(BEValueV))
    if (isLoopInvariant(Accum, L) && isAddRecNeverPoison(BEInst, L))
      (void)getAddRecExpr(getAddExpr(StartVal, Accum, Flags), Accum, L, Flags);

  return PHISCEV;
}

const SCEV *ScalarEvolution::createAddRecFromPHI(PHINode *PN) {
  const Loop *L = LI.getLoopFor(PN->getParent());
  if (!L || L->getHeader() != PN->getParent())
    return nullptr;

  // 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 = nullptr, *StartValueV = nullptr;
  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 = nullptr;
        break;
      }
    } else if (!StartValueV) {
      StartValueV = V;
    } else if (StartValueV != V) {
      StartValueV = nullptr;
      break;
    }
  }
  if (!BEValueV || !StartValueV)
    return nullptr;

  assert(ValueExprMap.find_as(PN) == ValueExprMap.end() &&
         "PHI node already processed?");

  // First, try to find AddRec expression without creating a fictituos symbolic
  // value for PN.
  if (auto *S = createSimpleAffineAddRec(PN, BEValueV, StartValueV))
    return S;

  // Handle PHI node value symbolically.
  const SCEV *SymbolicName = getUnknown(PN);
  ValueExprMap.insert({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(SCEVBackedgeConditionFolder::rewrite(Add->getOperand(i),
                                                             L, *this));
      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 (auto BO = MatchBinaryOp(BEValueV, DT)) {
          if (BO->Opcode == Instruction::Add && BO->LHS == PN) {
            if (BO->IsNUW)
              Flags = setFlags(Flags, SCEV::FlagNUW);
            if (BO->IsNSW)
              Flags = setFlags(Flags, SCEV::FlagNSW);
          }
        } else if (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. We can guarantee that no unsigned wrap occurs if the
          // indices form a positive value.
          if (GEP->isInBounds() && GEP->getOperand(0) == PN) {
            Flags = setFlags(Flags, SCEV::FlagNW);

            const SCEV *Ptr = getSCEV(GEP->getPointerOperand());
            if (isKnownPositive(getMinusSCEV(getSCEV(GEP), Ptr)))
              Flags = setFlags(Flags, SCEV::FlagNUW);
          }

          // We cannot transfer nuw and nsw flags from subtraction
          // operations -- sub nuw X, Y is not the same as add nuw X, -Y
          // for instance.
        }

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

        // We can add Flags to the post-inc expression only if we
        // know that it is *undefined behavior* for BEValueV to
        // overflow.
        if (auto *BEInst = dyn_cast<Instruction>(BEValueV))
          if (isLoopInvariant(Accum, L) && isAddRecNeverPoison(BEInst, L))
            (void)getAddRecExpr(getAddExpr(StartVal, Accum), Accum, L, Flags);

        return PHISCEV;
      }
    }
  } else {
    // 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.
    //
    // We can generalize this saying that i is the shifted value of BEValue
    // by one iteration:
    //   PHI(f(0), f({1,+,1})) --> f({0,+,1})
    const SCEV *Shifted = SCEVShiftRewriter::rewrite(BEValue, L, *this);
    const SCEV *Start = SCEVInitRewriter::rewrite(Shifted, L, *this, false);
    if (Shifted != getCouldNotCompute() &&
        Start != getCouldNotCompute()) {
      const SCEV *StartVal = getSCEV(StartValueV);
      if (Start == StartVal) {
        // 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)] = Shifted;
        return Shifted;
      }
    }
  }

  // Remove the temporary PHI node SCEV that has been inserted while intending
  // to create an AddRecExpr for this PHI node. We can not keep this temporary
  // as it will prevent later (possibly simpler) SCEV expressions to be added
  // to the ValueExprMap.
  eraseValueFromMap(PN);

  return nullptr;
}

// Checks if the SCEV S is available at BB.  S is considered available at BB
// if S can be materialized at BB without introducing a fault.
static bool IsAvailableOnEntry(const Loop *L, DominatorTree &DT, const SCEV *S,
                               BasicBlock *BB) {
  struct CheckAvailable {
    bool TraversalDone = false;
    bool Available = true;

    const Loop *L = nullptr;  // The loop BB is in (can be nullptr)
    BasicBlock *BB = nullptr;
    DominatorTree &DT;

    CheckAvailable(const Loop *L, BasicBlock *BB, DominatorTree &DT)
      : L(L), BB(BB), DT(DT) {}

    bool setUnavailable() {
      TraversalDone = true;
      Available = false;
      return false;
    }

    bool follow(const SCEV *S) {
      switch (S->getSCEVType()) {
      case scConstant: case scTruncate: case scZeroExtend: case scSignExtend:
      case scAddExpr: case scMulExpr: case scUMaxExpr: case scSMaxExpr:
      case scUMinExpr:
      case scSMinExpr:
        // These expressions are available if their operand(s) is/are.
        return true;

      case scAddRecExpr: {
        // We allow add recurrences that are on the loop BB is in, or some
        // outer loop.  This guarantees availability because the value of the
        // add recurrence at BB is simply the "current" value of the induction
        // variable.  We can relax this in the future; for instance an add
        // recurrence on a sibling dominating loop is also available at BB.
        const auto *ARLoop = cast<SCEVAddRecExpr>(S)->getLoop();
        if (L && (ARLoop == L || ARLoop->contains(L)))
          return true;

        return setUnavailable();
      }

      case scUnknown: {
        // For SCEVUnknown, we check for simple dominance.
        const auto *SU = cast<SCEVUnknown>(S);
        Value *V = SU->getValue();

        if (isa<Argument>(V))
          return false;

        if (isa<Instruction>(V) && DT.dominates(cast<Instruction>(V), BB))
          return false;

        return setUnavailable();
      }

      case scUDivExpr:
      case scCouldNotCompute:
        // We do not try to smart about these at all.
        return setUnavailable();
      }
      llvm_unreachable("switch should be fully covered!");
    }

    bool isDone() { return TraversalDone; }
  };

  CheckAvailable CA(L, BB, DT);
  SCEVTraversal<CheckAvailable> ST(CA);

  ST.visitAll(S);
  return CA.Available;
}

// Try to match a control flow sequence that branches out at BI and merges back
// at Merge into a "C ? LHS : RHS" select pattern.  Return true on a successful
// match.
static bool BrPHIToSelect(DominatorTree &DT, BranchInst *BI, PHINode *Merge,
                          Value *&C, Value *&LHS, Value *&RHS) {
  C = BI->getCondition();

  BasicBlockEdge LeftEdge(BI->getParent(), BI->getSuccessor(0));
  BasicBlockEdge RightEdge(BI->getParent(), BI->getSuccessor(1));

  if (!LeftEdge.isSingleEdge())
    return false;

  assert(RightEdge.isSingleEdge() && "Follows from LeftEdge.isSingleEdge()");

  Use &LeftUse = Merge->getOperandUse(0);
  Use &RightUse = Merge->getOperandUse(1);

  if (DT.dominates(LeftEdge, LeftUse) && DT.dominates(RightEdge, RightUse)) {
    LHS = LeftUse;
    RHS = RightUse;
    return true;
  }

  if (DT.dominates(LeftEdge, RightUse) && DT.dominates(RightEdge, LeftUse)) {
    LHS = RightUse;
    RHS = LeftUse;
    return true;
  }

  return false;
}

const SCEV *ScalarEvolution::createNodeFromSelectLikePHI(PHINode *PN) {
  auto IsReachable =
      [&](BasicBlock *BB) { return DT.isReachableFromEntry(BB); };
  if (PN->getNumIncomingValues() == 2 && all_of(PN->blocks(), IsReachable)) {
    const Loop *L = LI.getLoopFor(PN->getParent());

    // We don't want to break LCSSA, even in a SCEV expression tree.
    for (unsigned i = 0, e = PN->getNumIncomingValues(); i != e; ++i)
      if (LI.getLoopFor(PN->getIncomingBlock(i)) != L)
        return nullptr;

    // Try to match
    //
    //  br %cond, label %left, label %right
    // left:
    //  br label %merge
    // right:
    //  br label %merge
    // merge:
    //  V = phi [ %x, %left ], [ %y, %right ]
    //
    // as "select %cond, %x, %y"

    BasicBlock *IDom = DT[PN->getParent()]->getIDom()->getBlock();
    assert(IDom && "At least the entry block should dominate PN");

    auto *BI = dyn_cast<BranchInst>(IDom->getTerminator());
    Value *Cond = nullptr, *LHS = nullptr, *RHS = nullptr;

    if (BI && BI->isConditional() &&
        BrPHIToSelect(DT, BI, PN, Cond, LHS, RHS) &&
        IsAvailableOnEntry(L, DT, getSCEV(LHS), PN->getParent()) &&
        IsAvailableOnEntry(L, DT, getSCEV(RHS), PN->getParent()))
      return createNodeForSelectOrPHI(PN, Cond, LHS, RHS);
  }

  return nullptr;
}

const SCEV *ScalarEvolution::createNodeForPHI(PHINode *PN) {
  if (const SCEV *S = createAddRecFromPHI(PN))
    return S;

  if (const SCEV *S = createNodeFromSelectLikePHI(PN))
    return S;

  // 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, {getDataLayout(), &TLI, &DT, &AC}))
    if (LI.replacementPreservesLCSSAForm(PN, V))
      return getSCEV(V);

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

const SCEV *ScalarEvolution::createNodeForSelectOrPHI(Instruction *I,
                                                      Value *Cond,
                                                      Value *TrueVal,
                                                      Value *FalseVal) {
  // Handle "constant" branch or select. This can occur for instance when a
  // loop pass transforms an inner loop and moves on to process the outer loop.
  if (auto *CI = dyn_cast<ConstantInt>(Cond))
    return getSCEV(CI->isOne() ? TrueVal : FalseVal);

  // Try to match some simple smax or umax patterns.
  auto *ICI = dyn_cast<ICmpInst>(Cond);
  if (!ICI)
    return getUnknown(I);

  Value *LHS = ICI->getOperand(0);
  Value *RHS = ICI->getOperand(1);

  switch (ICI->getPredicate()) {
  case ICmpInst::ICMP_SLT:
  case ICmpInst::ICMP_SLE:
    std::swap(LHS, RHS);
    LLVM_FALLTHROUGH;
  case ICmpInst::ICMP_SGT:
  case ICmpInst::ICMP_SGE:
    // a >s b ? a+x : b+x  ->  smax(a, b)+x
    // a >s b ? b+x : a+x  ->  smin(a, b)+x
    if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType())) {
      const SCEV *LS = getNoopOrSignExtend(getSCEV(LHS), I->getType());
      const SCEV *RS = getNoopOrSignExtend(getSCEV(RHS), I->getType());
      const SCEV *LA = getSCEV(TrueVal);
      const SCEV *RA = getSCEV(FalseVal);
      const SCEV *LDiff = getMinusSCEV(LA, LS);
      const SCEV *RDiff = getMinusSCEV(RA, RS);
      if (LDiff == RDiff)
        return getAddExpr(getSMaxExpr(LS, RS), LDiff);
      LDiff = getMinusSCEV(LA, RS);
      RDiff = getMinusSCEV(RA, LS);
      if (LDiff == RDiff)
        return getAddExpr(getSMinExpr(LS, RS), LDiff);
    }
    break;
  case ICmpInst::ICMP_ULT:
  case ICmpInst::ICMP_ULE:
    std::swap(LHS, RHS);
    LLVM_FALLTHROUGH;
  case ICmpInst::ICMP_UGT:
  case ICmpInst::ICMP_UGE:
    // a >u b ? a+x : b+x  ->  umax(a, b)+x
    // a >u b ? b+x : a+x  ->  umin(a, b)+x
    if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType())) {
      const SCEV *LS = getNoopOrZeroExtend(getSCEV(LHS), I->getType());
      const SCEV *RS = getNoopOrZeroExtend(getSCEV(RHS), I->getType());
      const SCEV *LA = getSCEV(TrueVal);
      const SCEV *RA = getSCEV(FalseVal);
      const SCEV *LDiff = getMinusSCEV(LA, LS);
      const SCEV *RDiff = getMinusSCEV(RA, RS);
      if (LDiff == RDiff)
        return getAddExpr(getUMaxExpr(LS, RS), LDiff);
      LDiff = getMinusSCEV(LA, RS);
      RDiff = getMinusSCEV(RA, LS);
      if (LDiff == RDiff)
        return getAddExpr(getUMinExpr(LS, RS), LDiff);
    }
    break;
  case ICmpInst::ICMP_NE:
    // n != 0 ? n+x : 1+x  ->  umax(n, 1)+x
    if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType()) &&
        isa<ConstantInt>(RHS) && cast<ConstantInt>(RHS)->isZero()) {
      const SCEV *One = getOne(I->getType());
      const SCEV *LS = getNoopOrZeroExtend(getSCEV(LHS), I->getType());
      const SCEV *LA = getSCEV(TrueVal);
      const SCEV *RA = getSCEV(FalseVal);
      const SCEV *LDiff = getMinusSCEV(LA, LS);
      const SCEV *RDiff = getMinusSCEV(RA, One);
      if (LDiff == RDiff)
        return getAddExpr(getUMaxExpr(One, LS), LDiff);
    }
    break;
  case ICmpInst::ICMP_EQ:
    // n == 0 ? 1+x : n+x  ->  umax(n, 1)+x
    if (getTypeSizeInBits(LHS->getType()) <= getTypeSizeInBits(I->getType()) &&
        isa<ConstantInt>(RHS) && cast<ConstantInt>(RHS)->isZero()) {
      const SCEV *One = getOne(I->getType());
      const SCEV *LS = getNoopOrZeroExtend(getSCEV(LHS), I->getType());
      const SCEV *LA = getSCEV(TrueVal);
      const SCEV *RA = getSCEV(FalseVal);
      const SCEV *LDiff = getMinusSCEV(LA, One);
      const SCEV *RDiff = getMinusSCEV(RA, LS);
      if (LDiff == RDiff)
        return getAddExpr(getUMaxExpr(One, LS), LDiff);
    }
    break;
  default:
    break;
  }

  return getUnknown(I);
}

/// 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 attempt to analyze GEPs over unsized objects.
  if (!GEP->getSourceElementType()->isSized())
    return getUnknown(GEP);

  SmallVector<const SCEV *, 4> IndexExprs;
  for (auto Index = GEP->