[SCEV] Try to prove predicates by splitting them

[oota-llvm.git] / include / llvm / Analysis / ScalarEvolution.h

//===- llvm/Analysis/ScalarEvolution.h - Scalar Evolution -------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// The ScalarEvolution class is an LLVM pass which can be used to analyze and
// categorize scalar expressions in loops.  It specializes in recognizing
// general induction variables, representing them with the abstract and opaque
// SCEV class.  Given this analysis, trip counts of loops and other important
// properties can be obtained.
//
// This analysis is primarily useful for induction variable substitution and
// strength reduction.
//
//===----------------------------------------------------------------------===//

#ifndef LLVM_ANALYSIS_SCALAREVOLUTION_H
#define LLVM_ANALYSIS_SCALAREVOLUTION_H

#include "llvm/ADT/DenseSet.h"
#include "llvm/ADT/FoldingSet.h"
#include "llvm/IR/ConstantRange.h"
#include "llvm/IR/Function.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/Operator.h"
#include "llvm/IR/PassManager.h"
#include "llvm/IR/ValueHandle.h"
#include "llvm/Pass.h"
#include "llvm/Support/Allocator.h"
#include "llvm/Support/DataTypes.h"
#include <map>

namespace llvm {
  class APInt;
  class AssumptionCache;
  class Constant;
  class ConstantInt;
  class DominatorTree;
  class Type;
  class ScalarEvolution;
  class DataLayout;
  class TargetLibraryInfo;
  class LLVMContext;
  class Loop;
  class LoopInfo;
  class Operator;
  class SCEVUnknown;
  class SCEVAddRecExpr;
  class SCEV;
  template<> struct FoldingSetTrait<SCEV>;

  /// This class represents an analyzed expression in the program.  These are
  /// opaque objects that the client is not allowed to do much with directly.
  ///
  class SCEV : public FoldingSetNode {
    friend struct FoldingSetTrait<SCEV>;

    /// A reference to an Interned FoldingSetNodeID for this node.  The
    /// ScalarEvolution's BumpPtrAllocator holds the data.
    FoldingSetNodeIDRef FastID;

    // The SCEV baseclass this node corresponds to
    const unsigned short SCEVType;

  protected:
    /// This field is initialized to zero and may be used in subclasses to store
    /// miscellaneous information.
    unsigned short SubclassData;

  private:
    SCEV(const SCEV &) = delete;
    void operator=(const SCEV &) = delete;

  public:
    /// NoWrapFlags are bitfield indices into SubclassData.
    ///
    /// Add and Mul expressions may have no-unsigned-wrap <NUW> or
    /// no-signed-wrap <NSW> properties, which are derived from the IR
    /// operator. NSW is a misnomer that we use to mean no signed overflow or
    /// underflow.
    ///
    /// AddRec expressions may have a no-self-wraparound <NW> property if, in
    /// the integer domain, abs(step) * max-iteration(loop) <=
    /// unsigned-max(bitwidth).  This means that the recurrence will never reach
    /// its start value if the step is non-zero.  Computing the same value on
    /// each iteration is not considered wrapping, and recurrences with step = 0
    /// are trivially <NW>.  <NW> is independent of the sign of step and the
    /// value the add recurrence starts with.
    ///
    /// Note that NUW and NSW are also valid properties of a recurrence, and
    /// either implies NW. For convenience, NW will be set for a recurrence
    /// whenever either NUW or NSW are set.
    enum NoWrapFlags { FlagAnyWrap = 0,          // No guarantee.
                       FlagNW      = (1 << 0),   // No self-wrap.
                       FlagNUW     = (1 << 1),   // No unsigned wrap.
                       FlagNSW     = (1 << 2),   // No signed wrap.
                       NoWrapMask  = (1 << 3) -1 };

    explicit SCEV(const FoldingSetNodeIDRef ID, unsigned SCEVTy) :
      FastID(ID), SCEVType(SCEVTy), SubclassData(0) {}

    unsigned getSCEVType() const { return SCEVType; }

    /// Return the LLVM type of this SCEV expression.
    ///
    Type *getType() const;

    /// Return true if the expression is a constant zero.
    ///
    bool isZero() const;

    /// Return true if the expression is a constant one.
    ///
    bool isOne() const;

    /// Return true if the expression is a constant all-ones value.
    ///
    bool isAllOnesValue() const;

    /// Return true if the specified scev is negated, but not a constant.
    bool isNonConstantNegative() const;

    /// Print out the internal representation of this scalar to the specified
    /// stream.  This should really only be used for debugging purposes.
    void print(raw_ostream &OS) const;

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
    /// This method is used for debugging.
    ///
    void dump() const;
#endif
  };

  // Specialize FoldingSetTrait for SCEV to avoid needing to compute
  // temporary FoldingSetNodeID values.
  template<> struct FoldingSetTrait<SCEV> : DefaultFoldingSetTrait<SCEV> {
    static void Profile(const SCEV &X, FoldingSetNodeID& ID) {
      ID = X.FastID;
    }
    static bool Equals(const SCEV &X, const FoldingSetNodeID &ID,
                       unsigned IDHash, FoldingSetNodeID &TempID) {
      return ID == X.FastID;
    }
    static unsigned ComputeHash(const SCEV &X, FoldingSetNodeID &TempID) {
      return X.FastID.ComputeHash();
    }
  };

  inline raw_ostream &operator<<(raw_ostream &OS, const SCEV &S) {
    S.print(OS);
    return OS;
  }

  /// An object of this class is returned by queries that could not be answered.
  /// For example, if you ask for the number of iterations of a linked-list
  /// traversal loop, you will get one of these.  None of the standard SCEV
  /// operations are valid on this class, it is just a marker.
  struct SCEVCouldNotCompute : public SCEV {
    SCEVCouldNotCompute();

    /// Methods for support type inquiry through isa, cast, and dyn_cast:
    static bool classof(const SCEV *S);
  };

  /// The main scalar evolution driver. Because client code (intentionally)
  /// can't do much with the SCEV objects directly, they must ask this class
  /// for services.
  class ScalarEvolution {
  public:
    /// An enum describing the relationship between a SCEV and a loop.
    enum LoopDisposition {
      LoopVariant,    ///< The SCEV is loop-variant (unknown).
      LoopInvariant,  ///< The SCEV is loop-invariant.
      LoopComputable  ///< The SCEV varies predictably with the loop.
    };

    /// An enum describing the relationship between a SCEV and a basic block.
    enum BlockDisposition {
      DoesNotDominateBlock,  ///< The SCEV does not dominate the block.
      DominatesBlock,        ///< The SCEV dominates the block.
      ProperlyDominatesBlock ///< The SCEV properly dominates the block.
    };

    /// Convenient NoWrapFlags manipulation that hides enum casts and is
    /// visible in the ScalarEvolution name space.
    static SCEV::NoWrapFlags LLVM_ATTRIBUTE_UNUSED_RESULT
    maskFlags(SCEV::NoWrapFlags Flags, int Mask) {
      return (SCEV::NoWrapFlags)(Flags & Mask);
    }
    static SCEV::NoWrapFlags LLVM_ATTRIBUTE_UNUSED_RESULT
    setFlags(SCEV::NoWrapFlags Flags, SCEV::NoWrapFlags OnFlags) {
      return (SCEV::NoWrapFlags)(Flags | OnFlags);
    }
    static SCEV::NoWrapFlags LLVM_ATTRIBUTE_UNUSED_RESULT
    clearFlags(SCEV::NoWrapFlags Flags, SCEV::NoWrapFlags OffFlags) {
      return (SCEV::NoWrapFlags)(Flags & ~OffFlags);
    }

  private:
    /// A CallbackVH to arrange for ScalarEvolution to be notified whenever a
    /// Value is deleted.
    class SCEVCallbackVH final : public CallbackVH {
      ScalarEvolution *SE;
      void deleted() override;
      void allUsesReplacedWith(Value *New) override;
    public:
      SCEVCallbackVH(Value *V, ScalarEvolution *SE = nullptr);
    };

    friend class SCEVCallbackVH;
    friend class SCEVExpander;
    friend class SCEVUnknown;

    /// The function we are analyzing.
    ///
    Function &F;

    /// The target library information for the target we are targeting.
    ///
    TargetLibraryInfo &TLI;

    /// The tracker for @llvm.assume intrinsics in this function.
    AssumptionCache &AC;

    /// The dominator tree.
    ///
    DominatorTree &DT;

    /// The loop information for the function we are currently analyzing.
    ///
    LoopInfo &LI;

    /// This SCEV is used to represent unknown trip counts and things.
    std::unique_ptr<SCEVCouldNotCompute> CouldNotCompute;

    /// The typedef for ValueExprMap.
    ///
    typedef DenseMap<SCEVCallbackVH, const SCEV *, DenseMapInfo<Value *> >
      ValueExprMapType;

    /// This is a cache of the values we have analyzed so far.
    ///
    ValueExprMapType ValueExprMap;

    /// Mark predicate values currently being processed by isImpliedCond.
    DenseSet<Value*> PendingLoopPredicates;

    /// Set to true by isLoopBackedgeGuardedByCond when we're walking the set of
    /// conditions dominating the backedge of a loop.
    bool WalkingBEDominatingConds;

    /// Set to true by isKnownPredicateViaSplitting when we're trying to prove a
    /// predicate by splitting it into a set of independent predicates.
    bool ProvingSplitPredicate;

    /// Information about the number of loop iterations for which a loop exit's
    /// branch condition evaluates to the not-taken path.  This is a temporary
    /// pair of exact and max expressions that are eventually summarized in
    /// ExitNotTakenInfo and BackedgeTakenInfo.
    struct ExitLimit {
      const SCEV *Exact;
      const SCEV *Max;

      /*implicit*/ ExitLimit(const SCEV *E) : Exact(E), Max(E) {}

      ExitLimit(const SCEV *E, const SCEV *M) : Exact(E), Max(M) {}

      /// Test whether this ExitLimit contains any computed information, or
      /// whether it's all SCEVCouldNotCompute values.
      bool hasAnyInfo() const {
        return !isa<SCEVCouldNotCompute>(Exact) ||
          !isa<SCEVCouldNotCompute>(Max);
      }
    };

    /// Information about the number of times a particular loop exit may be
    /// reached before exiting the loop.
    struct ExitNotTakenInfo {
      AssertingVH<BasicBlock> ExitingBlock;
      const SCEV *ExactNotTaken;
      PointerIntPair<ExitNotTakenInfo*, 1> NextExit;

      ExitNotTakenInfo() : ExitingBlock(nullptr), ExactNotTaken(nullptr) {}

      /// Return true if all loop exits are computable.
      bool isCompleteList() const {
        return NextExit.getInt() == 0;
      }

      void setIncomplete() { NextExit.setInt(1); }

      /// Return a pointer to the next exit's not-taken info.
      ExitNotTakenInfo *getNextExit() const {
        return NextExit.getPointer();
      }

      void setNextExit(ExitNotTakenInfo *ENT) { NextExit.setPointer(ENT); }
    };

    /// Information about the backedge-taken count of a loop. This currently
    /// includes an exact count and a maximum count.
    ///
    class BackedgeTakenInfo {
      /// A list of computable exits and their not-taken counts.  Loops almost
      /// never have more than one computable exit.
      ExitNotTakenInfo ExitNotTaken;

      /// An expression indicating the least maximum backedge-taken count of the
      /// loop that is known, or a SCEVCouldNotCompute.
      const SCEV *Max;

    public:
      BackedgeTakenInfo() : Max(nullptr) {}

      /// Initialize BackedgeTakenInfo from a list of exact exit counts.
      BackedgeTakenInfo(
        SmallVectorImpl< std::pair<BasicBlock *, const SCEV *> > &ExitCounts,
        bool Complete, const SCEV *MaxCount);

      /// Test whether this BackedgeTakenInfo contains any computed information,
      /// or whether it's all SCEVCouldNotCompute values.
      bool hasAnyInfo() const {
        return ExitNotTaken.ExitingBlock || !isa<SCEVCouldNotCompute>(Max);
      }

      /// Return an expression indicating the exact backedge-taken count of the
      /// loop if it is known, or SCEVCouldNotCompute otherwise. This is the
      /// number of times the loop header can be guaranteed to execute, minus
      /// one.
      const SCEV *getExact(ScalarEvolution *SE) const;

      /// Return the number of times this loop exit may fall through to the back
      /// edge, or SCEVCouldNotCompute. The loop is guaranteed not to exit via
      /// this block before this number of iterations, but may exit via another
      /// block.
      const SCEV *getExact(BasicBlock *ExitingBlock, ScalarEvolution *SE) const;

      /// Get the max backedge taken count for the loop.
      const SCEV *getMax(ScalarEvolution *SE) const;

      /// Return true if any backedge taken count expressions refer to the given
      /// subexpression.
      bool hasOperand(const SCEV *S, ScalarEvolution *SE) const;

      /// Invalidate this result and free associated memory.
      void clear();
    };

    /// Cache the backedge-taken count of the loops for this function as they
    /// are computed.
    DenseMap<const Loop*, BackedgeTakenInfo> BackedgeTakenCounts;

    /// This map contains entries for all of the PHI instructions that we
    /// attempt to compute constant evolutions for.  This allows us to avoid
    /// potentially expensive recomputation of these properties.  An instruction
    /// maps to null if we are unable to compute its exit value.
    DenseMap<PHINode*, Constant*> ConstantEvolutionLoopExitValue;

    /// This map contains entries for all the expressions that we attempt to
    /// compute getSCEVAtScope information for, which can be expensive in
    /// extreme cases.
    DenseMap<const SCEV *,
             SmallVector<std::pair<const Loop *, const SCEV *>, 2> > ValuesAtScopes;

    /// Memoized computeLoopDisposition results.
    DenseMap<const SCEV *,
             SmallVector<PointerIntPair<const Loop *, 2, LoopDisposition>, 2>>
        LoopDispositions;

    /// Compute a LoopDisposition value.
    LoopDisposition computeLoopDisposition(const SCEV *S, const Loop *L);

    /// Memoized computeBlockDisposition results.
    DenseMap<
        const SCEV *,
        SmallVector<PointerIntPair<const BasicBlock *, 2, BlockDisposition>, 2>>
        BlockDispositions;

    /// Compute a BlockDisposition value.
    BlockDisposition computeBlockDisposition(const SCEV *S, const BasicBlock *BB);

    /// Memoized results from getRange
    DenseMap<const SCEV *, ConstantRange> UnsignedRanges;

    /// Memoized results from getRange
    DenseMap<const SCEV *, ConstantRange> SignedRanges;

    /// Used to parameterize getRange
    enum RangeSignHint { HINT_RANGE_UNSIGNED, HINT_RANGE_SIGNED };

    /// Set the memoized range for the given SCEV.
    const ConstantRange &setRange(const SCEV *S, RangeSignHint Hint,
                                  const ConstantRange &CR) {
      DenseMap<const SCEV *, ConstantRange> &Cache =
          Hint == HINT_RANGE_UNSIGNED ? UnsignedRanges : SignedRanges;

      std::pair<DenseMap<const SCEV *, ConstantRange>::iterator, bool> Pair =
          Cache.insert(std::make_pair(S, CR));
      if (!Pair.second)
        Pair.first->second = CR;
      return Pair.first->second;
    }

    /// Determine the range for a particular SCEV.
    ConstantRange getRange(const SCEV *S, RangeSignHint Hint);

    /// We know that there is no SCEV for the specified value.  Analyze the
    /// expression.
    const SCEV *createSCEV(Value *V);

    /// Provide the special handling we need to analyze PHI SCEVs.
    const SCEV *createNodeForPHI(PHINode *PN);

    /// Provide the special handling we need to analyze GEP SCEVs.
    const SCEV *createNodeForGEP(GEPOperator *GEP);

    /// Implementation code for getSCEVAtScope; called at most once for each
    /// SCEV+Loop pair.
    ///
    const SCEV *computeSCEVAtScope(const SCEV *S, const Loop *L);

    /// This looks up computed SCEV values for all instructions that depend on
    /// the given instruction and removes them from the ValueExprMap map if they
    /// reference SymName. This is used during PHI resolution.
    void ForgetSymbolicName(Instruction *I, const SCEV *SymName);

    /// Return the BackedgeTakenInfo for the given loop, lazily computing new
    /// values if the loop hasn't been analyzed yet.
    const BackedgeTakenInfo &getBackedgeTakenInfo(const Loop *L);

    /// Compute the number of times the specified loop will iterate.
    BackedgeTakenInfo ComputeBackedgeTakenCount(const Loop *L);

    /// Compute the number of times the backedge of the specified loop will
    /// execute if it exits via the specified block.
    ExitLimit ComputeExitLimit(const Loop *L, BasicBlock *ExitingBlock);

    /// Compute the number of times the backedge of the specified loop will
    /// execute if its exit condition were a conditional branch of ExitCond,
    /// TBB, and FBB.
    ExitLimit ComputeExitLimitFromCond(const Loop *L,
                                       Value *ExitCond,
                                       BasicBlock *TBB,
                                       BasicBlock *FBB,
                                       bool IsSubExpr);

    /// Compute the number of times the backedge of the specified loop will
    /// execute if its exit condition were a conditional branch of the ICmpInst
    /// ExitCond, TBB, and FBB.
    ExitLimit ComputeExitLimitFromICmp(const Loop *L,
                                       ICmpInst *ExitCond,
                                       BasicBlock *TBB,
                                       BasicBlock *FBB,
                                       bool IsSubExpr);

    /// Compute the number of times the backedge of the specified loop will
    /// execute if its exit condition were a switch with a single exiting case
    /// to ExitingBB.
    ExitLimit
    ComputeExitLimitFromSingleExitSwitch(const Loop *L, SwitchInst *Switch,
                               BasicBlock *ExitingBB, bool IsSubExpr);

    /// Given an exit condition of 'icmp op load X, cst', try to see if we can
    /// compute the backedge-taken count.
    ExitLimit ComputeLoadConstantCompareExitLimit(LoadInst *LI,
                                                  Constant *RHS,
                                                  const Loop *L,
                                                  ICmpInst::Predicate p);

    /// If the loop is known to execute a constant number of times (the
    /// condition evolves only from constants), try to evaluate a few iterations
    /// of the loop until we get the exit condition gets a value of ExitWhen
    /// (true or false).  If we cannot evaluate the exit count of the loop,
    /// return CouldNotCompute.
    const SCEV *ComputeExitCountExhaustively(const Loop *L,
                                             Value *Cond,
                                             bool ExitWhen);

    /// Return the number of times an exit condition comparing the specified
    /// value to zero will execute.  If not computable, return CouldNotCompute.
    ExitLimit HowFarToZero(const SCEV *V, const Loop *L, bool IsSubExpr);

    /// Return the number of times an exit condition checking the specified
    /// value for nonzero will execute.  If not computable, return
    /// CouldNotCompute.
    ExitLimit HowFarToNonZero(const SCEV *V, const Loop *L);

    /// Return the number of times an exit condition containing the specified
    /// less-than comparison will execute.  If not computable, return
    /// CouldNotCompute. isSigned specifies whether the less-than is signed.
    ExitLimit HowManyLessThans(const SCEV *LHS, const SCEV *RHS,
                               const Loop *L, bool isSigned, bool IsSubExpr);
    ExitLimit HowManyGreaterThans(const SCEV *LHS, const SCEV *RHS,
                                  const Loop *L, bool isSigned, bool IsSubExpr);

    /// Return a predecessor of BB (which may not be an immediate predecessor)
    /// which has exactly one successor from which BB is reachable, or null if
    /// no such block is found.
    std::pair<BasicBlock *, BasicBlock *>
    getPredecessorWithUniqueSuccessorForBB(BasicBlock *BB);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the given FoundCondValue value evaluates to true.
    bool isImpliedCond(ICmpInst::Predicate Pred,
                       const SCEV *LHS, const SCEV *RHS,
                       Value *FoundCondValue,
                       bool Inverse);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the condition described by FoundPred, FoundLHS, FoundRHS is
    /// true.
    bool isImpliedCond(ICmpInst::Predicate Pred, const SCEV *LHS,
                       const SCEV *RHS, ICmpInst::Predicate FoundPred,
                       const SCEV *FoundLHS, const SCEV *FoundRHS);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
    /// true.
    bool isImpliedCondOperands(ICmpInst::Predicate Pred,
                               const SCEV *LHS, const SCEV *RHS,
                               const SCEV *FoundLHS, const SCEV *FoundRHS);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
    /// true.
    bool isImpliedCondOperandsHelper(ICmpInst::Predicate Pred,
                                     const SCEV *LHS, const SCEV *RHS,
                                     const SCEV *FoundLHS,
                                     const SCEV *FoundRHS);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
    /// true.  Utility function used by isImpliedCondOperands.
    bool isImpliedCondOperandsViaRanges(ICmpInst::Predicate Pred,
                                        const SCEV *LHS, const SCEV *RHS,
                                        const SCEV *FoundLHS,
                                        const SCEV *FoundRHS);

    /// Test whether the condition described by Pred, LHS, and RHS is true
    /// whenever the condition described by Pred, FoundLHS, and FoundRHS is
    /// true.
    ///
    /// This routine tries to rule out certain kinds of integer overflow, and
    /// then tries to reason about arithmetic properties of the predicates.
    bool isImpliedCondOperandsViaNoOverflow(ICmpInst::Predicate Pred,
                                            const SCEV *LHS, const SCEV *RHS,
                                            const SCEV *FoundLHS,
                                            const SCEV *FoundRHS);

    /// If we know that the specified Phi is in the header of its containing
    /// loop, we know the loop executes a constant number of times, and the PHI
    /// node is just a recurrence involving constants, fold it.
    Constant *getConstantEvolutionLoopExitValue(PHINode *PN, const APInt& BEs,
                                                const Loop *L);

    /// Test if the given expression is known to satisfy the condition described
    /// by Pred and the known constant ranges of LHS and RHS.
    ///
    bool isKnownPredicateWithRanges(ICmpInst::Predicate Pred,
                                    const SCEV *LHS, const SCEV *RHS);

    /// Try to split Pred LHS RHS into logical conjunctions (and's) and try to
    /// prove them individually.
    bool isKnownPredicateViaSplitting(ICmpInst::Predicate Pred, const SCEV *LHS,
                                      const SCEV *RHS);

    /// Drop memoized information computed for S.
    void forgetMemoizedResults(const SCEV *S);

    /// Return an existing SCEV for V if there is one, otherwise return nullptr.
    const SCEV *getExistingSCEV(Value *V);

    /// Return false iff given SCEV contains a SCEVUnknown with NULL value-
    /// pointer.
    bool checkValidity(const SCEV *S) const;

    /// Return true if `ExtendOpTy`({`Start`,+,`Step`}) can be proved to be
    /// equal to {`ExtendOpTy`(`Start`),+,`ExtendOpTy`(`Step`)}.  This is
    /// equivalent to proving no signed (resp. unsigned) wrap in
    /// {`Start`,+,`Step`} if `ExtendOpTy` is `SCEVSignExtendExpr`
    /// (resp. `SCEVZeroExtendExpr`).
    ///
    template<typename ExtendOpTy>
    bool proveNoWrapByVaryingStart(const SCEV *Start, const SCEV *Step,
                                   const Loop *L);

    bool isMonotonicPredicateImpl(const SCEVAddRecExpr *LHS,
                                  ICmpInst::Predicate Pred, bool &Increasing);

    /// Return true if, for all loop invariant X, the predicate "LHS `Pred` X"
    /// is monotonically increasing or decreasing.  In the former case set
    /// `Increasing` to true and in the latter case set `Increasing` to false.
    ///
    /// A predicate is said to be monotonically increasing if may go from being
    /// false to being true as the loop iterates, but never the other way
    /// around.  A predicate is said to be monotonically decreasing if may go
    /// from being true to being false as the loop iterates, but never the other
    /// way around.
    bool isMonotonicPredicate(const SCEVAddRecExpr *LHS,
                              ICmpInst::Predicate Pred, bool &Increasing);

    // Return SCEV no-wrap flags that can be proven based on reasoning
    // about how poison produced from no-wrap flags on this value
    // (e.g. a nuw add) would trigger undefined behavior on overflow.
    SCEV::NoWrapFlags getNoWrapFlagsFromUB(const Value *V);

  public:
    ScalarEvolution(Function &F, TargetLibraryInfo &TLI, AssumptionCache &AC,
                    DominatorTree &DT, LoopInfo &LI);
    ~ScalarEvolution();
    ScalarEvolution(ScalarEvolution &&Arg);

    LLVMContext &getContext() const { return F.getContext(); }

    /// 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 isSCEVable(Type *Ty) const;

    /// Return the size in bits of the specified type, for which isSCEVable must
    /// return true.
    uint64_t getTypeSizeInBits(Type *Ty) const;

    /// 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 *getEffectiveSCEVType(Type *Ty) const;

    /// Return a SCEV expression for the full generality of the specified
    /// expression.
    const SCEV *getSCEV(Value *V);

    const SCEV *getConstant(ConstantInt *V);
    const SCEV *getConstant(const APInt& Val);
    const SCEV *getConstant(Type *Ty, uint64_t V, bool isSigned = false);
    const SCEV *getTruncateExpr(const SCEV *Op, Type *Ty);
    const SCEV *getZeroExtendExpr(const SCEV *Op, Type *Ty);
    const SCEV *getSignExtendExpr(const SCEV *Op, Type *Ty);
    const SCEV *getAnyExtendExpr(const SCEV *Op, Type *Ty);
    const SCEV *getAddExpr(SmallVectorImpl<const SCEV *> &Ops,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap);
    const SCEV *getAddExpr(const SCEV *LHS, const SCEV *RHS,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap) {
      SmallVector<const SCEV *, 2> Ops;
      Ops.push_back(LHS);
      Ops.push_back(RHS);
      return getAddExpr(Ops, Flags);
    }
    const SCEV *getAddExpr(const SCEV *Op0, const SCEV *Op1, const SCEV *Op2,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap) {
      SmallVector<const SCEV *, 3> Ops;
      Ops.push_back(Op0);
      Ops.push_back(Op1);
      Ops.push_back(Op2);
      return getAddExpr(Ops, Flags);
    }
    const SCEV *getMulExpr(SmallVectorImpl<const SCEV *> &Ops,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap);
    const SCEV *getMulExpr(const SCEV *LHS, const SCEV *RHS,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap)
    {
      SmallVector<const SCEV *, 2> Ops;
      Ops.push_back(LHS);
      Ops.push_back(RHS);
      return getMulExpr(Ops, Flags);
    }
    const SCEV *getMulExpr(const SCEV *Op0, const SCEV *Op1, const SCEV *Op2,
                           SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap) {
      SmallVector<const SCEV *, 3> Ops;
      Ops.push_back(Op0);
      Ops.push_back(Op1);
      Ops.push_back(Op2);
      return getMulExpr(Ops, Flags);
    }
    const SCEV *getUDivExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getUDivExactExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getAddRecExpr(const SCEV *Start, const SCEV *Step,
                              const Loop *L, SCEV::NoWrapFlags Flags);
    const SCEV *getAddRecExpr(SmallVectorImpl<const SCEV *> &Operands,
                              const Loop *L, SCEV::NoWrapFlags Flags);
    const SCEV *getAddRecExpr(const SmallVectorImpl<const SCEV *> &Operands,
                              const Loop *L, SCEV::NoWrapFlags Flags) {
      SmallVector<const SCEV *, 4> NewOp(Operands.begin(), Operands.end());
      return getAddRecExpr(NewOp, L, Flags);
    }
    /// \brief Returns an expression for a GEP
    ///
    /// \p PointeeType The type used as the basis for the pointer arithmetics
    /// \p BaseExpr The expression for the pointer operand.
    /// \p IndexExprs The expressions for the indices.
    /// \p InBounds Whether the GEP is in bounds.
    const SCEV *getGEPExpr(Type *PointeeType, const SCEV *BaseExpr,
                           const SmallVectorImpl<const SCEV *> &IndexExprs,
                           bool InBounds = false);
    const SCEV *getSMaxExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getSMaxExpr(SmallVectorImpl<const SCEV *> &Operands);
    const SCEV *getUMaxExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getUMaxExpr(SmallVectorImpl<const SCEV *> &Operands);
    const SCEV *getSMinExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getUMinExpr(const SCEV *LHS, const SCEV *RHS);
    const SCEV *getUnknown(Value *V);
    const SCEV *getCouldNotCompute();

    /// \brief Return a SCEV for the constant 0 of a specific type.
    const SCEV *getZero(Type *Ty) { return getConstant(Ty, 0); }

    /// \brief Return a SCEV for the constant 1 of a specific type.
    const SCEV *getOne(Type *Ty) { return getConstant(Ty, 1); }

    /// Return an expression for sizeof AllocTy that is type IntTy
    ///
    const SCEV *getSizeOfExpr(Type *IntTy, Type *AllocTy);

    /// Return an expression for offsetof on the given field with type IntTy
    ///
    const SCEV *getOffsetOfExpr(Type *IntTy, StructType *STy, unsigned FieldNo);

    /// Return the SCEV object corresponding to -V.
    ///
    const SCEV *getNegativeSCEV(const SCEV *V,
                                SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap);

    /// Return the SCEV object corresponding to ~V.
    ///
    const SCEV *getNotSCEV(const SCEV *V);

    /// Return LHS-RHS.  Minus is represented in SCEV as A+B*-1.
    const SCEV *getMinusSCEV(const SCEV *LHS, const SCEV *RHS,
                             SCEV::NoWrapFlags Flags = SCEV::FlagAnyWrap);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type.  If the type must be extended, it is zero extended.
    const SCEV *getTruncateOrZeroExtend(const SCEV *V, Type *Ty);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type.  If the type must be extended, it is sign extended.
    const SCEV *getTruncateOrSignExtend(const SCEV *V, Type *Ty);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type.  If the type must be extended, it is zero extended.  The
    /// conversion must not be narrowing.
    const SCEV *getNoopOrZeroExtend(const SCEV *V, Type *Ty);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type.  If the type must be extended, it is sign extended.  The
    /// conversion must not be narrowing.
    const SCEV *getNoopOrSignExtend(const SCEV *V, Type *Ty);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type. If the type must be extended, it is extended with
    /// unspecified bits. The conversion must not be narrowing.
    const SCEV *getNoopOrAnyExtend(const SCEV *V, Type *Ty);

    /// Return a SCEV corresponding to a conversion of the input value to the
    /// specified type.  The conversion must not be widening.
    const SCEV *getTruncateOrNoop(const SCEV *V, Type *Ty);

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

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

    /// Transitively follow the chain of pointer-type operands until reaching a
    /// SCEV that does not have a single pointer operand. This returns a
    /// SCEVUnknown pointer for well-formed pointer-type expressions, but corner
    /// cases do exist.
    const SCEV *getPointerBase(const SCEV *V);

    /// Return a SCEV expression for the specified value at the specified scope
    /// in the program.  The L value specifies a loop nest to evaluate the
    /// expression at, where null is the top-level or a specified loop is
    /// immediately inside of the loop.
    ///
    /// This method can be used to compute the exit value for a variable defined
    /// in a loop by querying what the value will hold in the parent loop.
    ///
    /// In the case that a relevant loop exit value cannot be computed, the
    /// original value V is returned.
    const SCEV *getSCEVAtScope(const SCEV *S, const Loop *L);

    /// This is a convenience function which does getSCEVAtScope(getSCEV(V), L).
    const SCEV *getSCEVAtScope(Value *V, const Loop *L);

    /// Test whether entry to the loop is protected by a conditional between LHS
    /// and RHS.  This is used to help avoid max expressions in loop trip
    /// counts, and to eliminate casts.
    bool isLoopEntryGuardedByCond(const Loop *L, ICmpInst::Predicate Pred,
                                  const SCEV *LHS, const SCEV *RHS);

    /// Test whether the backedge of the loop is protected by a conditional
    /// between LHS and RHS.  This is used to to eliminate casts.
    bool isLoopBackedgeGuardedByCond(const Loop *L, ICmpInst::Predicate Pred,
                                     const SCEV *LHS, const SCEV *RHS);

    /// \brief Returns the maximum trip count of the loop if it is a single-exit
    /// loop and we can compute a small maximum for that loop.
    ///
    /// Implemented in terms of the \c getSmallConstantTripCount overload with
    /// the single exiting block passed to it. See that routine for details.
    unsigned getSmallConstantTripCount(Loop *L);

    /// Returns the maximum trip count of this loop as a normal unsigned
    /// value. Returns 0 if the trip count is unknown or not constant. This
    /// "trip count" assumes that control exits via ExitingBlock. More
    /// precisely, it is the number of times that control may reach ExitingBlock
    /// before taking the branch. For loops with multiple exits, it may not be
    /// the number times that the loop header executes if the loop exits
    /// prematurely via another branch.
    unsigned getSmallConstantTripCount(Loop *L, BasicBlock *ExitingBlock);

    /// \brief Returns the largest constant divisor of the trip count of the
    /// loop if it is a single-exit loop and we can compute a small maximum for
    /// that loop.
    ///
    /// Implemented in terms of the \c getSmallConstantTripMultiple overload with
    /// the single exiting block passed to it. See that routine for details.
    unsigned getSmallConstantTripMultiple(Loop *L);

    /// Returns the largest constant divisor of the trip count of this loop as a
    /// normal unsigned value, if possible. This means that the actual trip
    /// count is always a multiple of the returned value (don't forget the trip
    /// count could very well be zero as well!). As explained in the comments
    /// for getSmallConstantTripCount, this assumes that control exits the loop
    /// via ExitingBlock.
    unsigned getSmallConstantTripMultiple(Loop *L, BasicBlock *ExitingBlock);

    /// Get the expression for the number of loop iterations for which this loop
    /// is guaranteed not to exit via ExitingBlock. Otherwise return
    /// SCEVCouldNotCompute.
    const SCEV *getExitCount(Loop *L, BasicBlock *ExitingBlock);

    /// If the specified loop has a predictable backedge-taken count, return it,
    /// otherwise return a SCEVCouldNotCompute object. The backedge-taken count
    /// is the number of times the loop header will be branched to from within
    /// the loop. This is one less than the trip count of the loop, since it
    /// doesn't count the first iteration, when the header is branched to from
    /// outside the loop.
    ///
    /// Note that it is not valid to call this method on a loop without a
    /// loop-invariant backedge-taken count (see
    /// hasLoopInvariantBackedgeTakenCount).
    ///
    const SCEV *getBackedgeTakenCount(const Loop *L);

    /// Similar to getBackedgeTakenCount, except return the least SCEV value
    /// that is known never to be less than the actual backedge taken count.
    const SCEV *getMaxBackedgeTakenCount(const Loop *L);

    /// Return true if the specified loop has an analyzable loop-invariant
    /// backedge-taken count.
    bool hasLoopInvariantBackedgeTakenCount(const Loop *L);

    /// This method should be called by the client when it has changed a loop in
    /// a way that may effect ScalarEvolution's ability to compute a trip count,
    /// or if the loop is deleted.  This call is potentially expensive for large
    /// loop bodies.
    void forgetLoop(const Loop *L);

    /// This method should be called by the client when it has changed a value
    /// in a way that may effect its value, or which may disconnect it from a
    /// def-use chain linking it to a loop.
    void forgetValue(Value *V);

    /// \brief Called when the client has changed the disposition of values in
    /// this loop.
    ///
    /// We don't have a way to invalidate per-loop dispositions. Clear and
    /// recompute is simpler.
    void forgetLoopDispositions(const Loop *L) { LoopDispositions.clear(); }

    /// Determine the minimum number of zero bits that S is guaranteed to end in
    /// (at every loop iteration).  It is, at the same time, the minimum number
    /// of times S is divisible by 2.  For example, given {4,+,8} it returns 2.
    /// If S is guaranteed to be 0, it returns the bitwidth of S.
    uint32_t GetMinTrailingZeros(const SCEV *S);

    /// Determine the unsigned range for a particular SCEV.
    ///
    ConstantRange getUnsignedRange(const SCEV *S) {
      return getRange(S, HINT_RANGE_UNSIGNED);
    }

    /// Determine the signed range for a particular SCEV.
    ///
    ConstantRange getSignedRange(const SCEV *S) {
      return getRange(S, HINT_RANGE_SIGNED);
    }

    /// Test if the given expression is known to be negative.
    ///
    bool isKnownNegative(const SCEV *S);

    /// Test if the given expression is known to be positive.
    ///
    bool isKnownPositive(const SCEV *S);

    /// Test if the given expression is known to be non-negative.
    ///
    bool isKnownNonNegative(const SCEV *S);

    /// Test if the given expression is known to be non-positive.
    ///
    bool isKnownNonPositive(const SCEV *S);

    /// Test if the given expression is known to be non-zero.
    ///
    bool isKnownNonZero(const SCEV *S);

    /// Test if the given expression is known to satisfy the condition described
    /// by Pred, LHS, and RHS.
    ///
    bool isKnownPredicate(ICmpInst::Predicate Pred,
                          const SCEV *LHS, const SCEV *RHS);

    /// Return true if the result of the predicate LHS `Pred` RHS is loop
    /// invariant with respect to L.  Set InvariantPred, InvariantLHS and
    /// InvariantLHS so that InvariantLHS `InvariantPred` InvariantRHS is the
    /// loop invariant form of LHS `Pred` RHS.
    bool isLoopInvariantPredicate(ICmpInst::Predicate Pred, const SCEV *LHS,
                                  const SCEV *RHS, const Loop *L,
                                  ICmpInst::Predicate &InvariantPred,
                                  const SCEV *&InvariantLHS,
                                  const SCEV *&InvariantRHS);

    /// Simplify LHS and RHS in a comparison with predicate Pred. Return true
    /// iff any changes were made. If the operands are provably equal or
    /// unequal, LHS and RHS are set to the same value and Pred is set to either
    /// ICMP_EQ or ICMP_NE.
    ///
    bool SimplifyICmpOperands(ICmpInst::Predicate &Pred,
                              const SCEV *&LHS,
                              const SCEV *&RHS,
                              unsigned Depth = 0);

    /// Return the "disposition" of the given SCEV with respect to the given
    /// loop.
    LoopDisposition getLoopDisposition(const SCEV *S, const Loop *L);

    /// Return true if the value of the given SCEV is unchanging in the
    /// specified loop.
    bool isLoopInvariant(const SCEV *S, const Loop *L);

    /// Return true if the given SCEV changes value in a known way in the
    /// specified loop.  This property being true implies that the value is
    /// variant in the loop AND that we can emit an expression to compute the
    /// value of the expression at any particular loop iteration.
    bool hasComputableLoopEvolution(const SCEV *S, const Loop *L);

    /// Return the "disposition" of the given SCEV with respect to the given
    /// block.
    BlockDisposition getBlockDisposition(const SCEV *S, const BasicBlock *BB);

    /// Return true if elements that makes up the given SCEV dominate the
    /// specified basic block.
    bool dominates(const SCEV *S, const BasicBlock *BB);

    /// Return true if elements that makes up the given SCEV properly dominate
    /// the specified basic block.
    bool properlyDominates(const SCEV *S, const BasicBlock *BB);

    /// Test whether the given SCEV has Op as a direct or indirect operand.
    bool hasOperand(const SCEV *S, const SCEV *Op) const;

    /// Return the size of an element read or written by Inst.
    const SCEV *getElementSize(Instruction *Inst);

    /// Compute the array dimensions Sizes from the set of Terms extracted from
    /// the memory access function of this SCEVAddRecExpr.
    void findArrayDimensions(SmallVectorImpl<const SCEV *> &Terms,
                             SmallVectorImpl<const SCEV *> &Sizes,
                             const SCEV *ElementSize) const;

    void print(raw_ostream &OS) const;
    void verify() const;

    /// Collect parametric terms occurring in step expressions.
    void collectParametricTerms(const SCEV *Expr,
                                SmallVectorImpl<const SCEV *> &Terms);



    /// Return in Subscripts the access functions for each dimension in Sizes.
    void computeAccessFunctions(const SCEV *Expr,
                                SmallVectorImpl<const SCEV *> &Subscripts,
                                SmallVectorImpl<const SCEV *> &Sizes);

    /// Split this SCEVAddRecExpr into two vectors of SCEVs representing the
    /// subscripts and sizes of an array access.
    ///
    /// The delinearization is a 3 step process: the first two steps compute the
    /// sizes of each subscript and the third step computes the access functions
    /// for the delinearized array:
    ///
    /// 1. Find the terms in the step functions
    /// 2. Compute the array size
    /// 3. Compute the access function: divide the SCEV by the array size
    ///    starting with the innermost dimensions found in step 2. The Quotient
    ///    is the SCEV to be divided in the next step of the recursion. The
    ///    Remainder is the subscript of the innermost dimension. Loop over all
    ///    array dimensions computed in step 2.
    ///
    /// To compute a uniform array size for several memory accesses to the same
    /// object, one can collect in step 1 all the step terms for all the memory
    /// accesses, and compute in step 2 a unique array shape. This guarantees
    /// that the array shape will be the same across all memory accesses.
    ///
    /// FIXME: We could derive the result of steps 1 and 2 from a description of
    /// the array shape given in metadata.
    ///
    /// Example:
    ///
    /// A[][n][m]
    ///
    /// for i
    ///   for j
    ///     for k
    ///       A[j+k][2i][5i] =
    ///
    /// The initial SCEV:
    ///
    /// A[{{{0,+,2*m+5}_i, +, n*m}_j, +, n*m}_k]
    ///
    /// 1. Find the different terms in the step functions:
    /// -> [2*m, 5, n*m, n*m]
    ///
    /// 2. Compute the array size: sort and unique them
    /// -> [n*m, 2*m, 5]
    /// find the GCD of all the terms = 1
    /// divide by the GCD and erase constant terms
    /// -> [n*m, 2*m]
    /// GCD = m
    /// divide by GCD -> [n, 2]
    /// remove constant terms
    /// -> [n]
    /// size of the array is A[unknown][n][m]
    ///
    /// 3. Compute the access function
    /// a. Divide {{{0,+,2*m+5}_i, +, n*m}_j, +, n*m}_k by the innermost size m
    /// Quotient: {{{0,+,2}_i, +, n}_j, +, n}_k
    /// Remainder: {{{0,+,5}_i, +, 0}_j, +, 0}_k
    /// The remainder is the subscript of the innermost array dimension: [5i].
    ///
    /// b. Divide Quotient: {{{0,+,2}_i, +, n}_j, +, n}_k by next outer size n
    /// Quotient: {{{0,+,0}_i, +, 1}_j, +, 1}_k
    /// Remainder: {{{0,+,2}_i, +, 0}_j, +, 0}_k
    /// The Remainder is the subscript of the next array dimension: [2i].
    ///
    /// The subscript of the outermost dimension is the Quotient: [j+k].
    ///
    /// Overall, we have: A[][n][m], and the access function: A[j+k][2i][5i].
    void delinearize(const SCEV *Expr,
                     SmallVectorImpl<const SCEV *> &Subscripts,
                     SmallVectorImpl<const SCEV *> &Sizes,
                     const SCEV *ElementSize);

  private:
    /// Compute the backedge taken count knowing the interval difference, the
    /// stride and presence of the equality in the comparison.
    const SCEV *computeBECount(const SCEV *Delta, const SCEV *Stride,
                               bool Equality);

    /// Verify if an linear IV with positive stride can overflow when in a
    /// less-than comparison, knowing the invariant term of the comparison,
    /// the stride and the knowledge of NSW/NUW flags on the recurrence.
    bool doesIVOverflowOnLT(const SCEV *RHS, const SCEV *Stride,
                            bool IsSigned, bool NoWrap);

    /// Verify if an linear IV with negative stride can overflow when in a
    /// greater-than comparison, knowing the invariant term of the comparison,
    /// the stride and the knowledge of NSW/NUW flags on the recurrence.
    bool doesIVOverflowOnGT(const SCEV *RHS, const SCEV *Stride,
                            bool IsSigned, bool NoWrap);

  private:
    FoldingSet<SCEV> UniqueSCEVs;
    BumpPtrAllocator SCEVAllocator;

    /// The head of a linked list of all SCEVUnknown values that have been
    /// allocated. This is used by releaseMemory to locate them all and call
    /// their destructors.
    SCEVUnknown *FirstUnknown;
  };

  /// \brief Analysis pass that exposes the \c ScalarEvolution for a function.
  class ScalarEvolutionAnalysis {
    static char PassID;

  public:
    typedef ScalarEvolution Result;

    /// \brief Opaque, unique identifier for this analysis pass.
    static void *ID() { return (void *)&PassID; }

    /// \brief Provide a name for the analysis for debugging and logging.
    static StringRef name() { return "ScalarEvolutionAnalysis"; }

    ScalarEvolution run(Function &F, AnalysisManager<Function> *AM);
  };

  /// \brief Printer pass for the \c ScalarEvolutionAnalysis results.
  class ScalarEvolutionPrinterPass {
    raw_ostream &OS;

  public:
    explicit ScalarEvolutionPrinterPass(raw_ostream &OS) : OS(OS) {}
    PreservedAnalyses run(Function &F, AnalysisManager<Function> *AM);

    static StringRef name() { return "ScalarEvolutionPrinterPass"; }
  };

  class ScalarEvolutionWrapperPass : public FunctionPass {
    std::unique_ptr<ScalarEvolution> SE;

  public:
    static char ID;

    ScalarEvolutionWrapperPass();

    ScalarEvolution &getSE() { return *SE; }
    const ScalarEvolution &getSE() const { return *SE; }

    bool runOnFunction(Function &F) override;
    void releaseMemory() override;
    void getAnalysisUsage(AnalysisUsage &AU) const override;
    void print(raw_ostream &OS, const Module * = nullptr) const override;
    void verifyAnalysis() const override;
  };
}

#endif