HashRecognize.cpp

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//===- HashRecognize.cpp ----------------------------------------*- C++ -*-===//
//
// 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
//
//===----------------------------------------------------------------------===//
//
// The HashRecognize analysis recognizes unoptimized polynomial hash functions
// with operations over a Galois field of characteristic 2, also called binary
// fields, or GF(2^n). 2^n is termed the order of the Galois field. This class
// of hash functions can be optimized using a lookup-table-driven
// implementation, or with target-specific instructions.
//
// Examples:
//
//  1. Cyclic redundancy check (CRC), which is a polynomial division in GF(2).
//  2. Rabin fingerprint, a component of the Rabin-Karp algorithm, which is a
//     rolling hash polynomial division in GF(2).
//  3. Rijndael MixColumns, a step in AES computation, which is a polynomial
//     multiplication in GF(2^3).
//  4. GHASH, the authentication mechanism in AES Galois/Counter Mode (GCM),
//     which is a polynomial evaluation in GF(2^128).
//
// All of them use an irreducible generating polynomial of degree m,
//
//    c_m * x^m + c_(m-1) * x^(m-1) + ... + c_0 * x^0
//
// where each coefficient c is can take values 0 or 1. The polynomial is simply
// represented by m+1 bits, corresponding to the coefficients. The different
// variants of CRC are named by degree of generating polynomial used: so CRC-32
// would use a polynomial of degree 32.
//
// The reason algorithms on GF(2^n) can be optimized with a lookup-table is the
// following: in such fields, polynomial addition and subtraction are identical
// and equivalent to XOR, polynomial multiplication is an AND, and polynomial
// division is identity: the XOR and AND operations in unoptimized
// implementations are performed bit-wise, and can be optimized to be performed
// chunk-wise, by interleaving copies of the generating polynomial, and storing
// the pre-computed values in a table.
//
// A generating polynomial of m bits always has the MSB set, so we usually
// omit it. An example of a 16-bit polynomial is the CRC-16-CCITT polynomial:
//
//   (x^16) + x^12 + x^5 + 1 = (1) 0001 0000 0010 0001 = 0x1021
//
// Transmissions are either in big-endian or little-endian form, and hash
// algorithms are written according to this. For example, IEEE 802 and RS-232
// specify little-endian transmission.
//
//===----------------------------------------------------------------------===//
//
// At the moment, we only recognize the CRC algorithm.
// Documentation on CRC32 from the kernel:
// https://www.kernel.org/doc/Documentation/crc32.txt
//
//
//===----------------------------------------------------------------------===//
 
#include "llvm/Analysis/HashRecognize.h"
#include "llvm/ADT/APInt.h"
#include "llvm/Analysis/LoopAnalysisManager.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/ScalarEvolution.h"
#include "llvm/Analysis/ScalarEvolutionPatternMatch.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/Support/KnownBits.h"
 
using namespace llvm;
using namespace PatternMatch;
using namespace SCEVPatternMatch;
 
#define DEBUG_TYPE "hash-recognize"
 
/// Checks if there's a stray instruction in the loop \p L outside of the
/// use-def chains from \p Roots, or if we escape the loop during the use-def
/// walk.
static bool containsUnreachable(const Loop &L,
                                ArrayRef<const Instruction *> Roots) {
  SmallPtrSet<const Instruction *, 16> Visited;
  BasicBlock *Latch = L.getLoopLatch();
 
  SmallVector<const Instruction *, 16> Worklist(Roots);
  while (!Worklist.empty()) {
    const Instruction *I = Worklist.pop_back_val();
    Visited.insert(I);
 
    if (isa<PHINode>(I))
      continue;
 
    for (const Use &U : I->operands()) {
      if (auto *UI = dyn_cast<Instruction>(U)) {
        if (!L.contains(UI))
          return true;
        Worklist.push_back(UI);
      }
    }
  }
  return std::distance(Latch->begin(), Latch->end()) != Visited.size();
}
 
/// A structure that can hold either a Simple Recurrence or a Conditional
/// Recurrence. Note that in the case of a Simple Recurrence, Step is an operand
/// of the BO, while in a Conditional Recurrence, it is a SelectInst.
struct RecurrenceInfo {
  const Loop &L;
  const PHINode *Phi = nullptr;
  BinaryOperator *BO = nullptr;
  Value *Start = nullptr;
  Value *Step = nullptr;
  std::optional<APInt> ExtraConst;
 
  RecurrenceInfo(const Loop &L) : L(L) {}
  operator bool() const { return BO; }
 
  void print(raw_ostream &OS, unsigned Indent = 0) const {
    OS.indent(Indent) << "Phi: ";
    Phi->print(OS);
    OS << "\n";
    OS.indent(Indent) << "BinaryOperator: ";
    BO->print(OS);
    OS << "\n";
    OS.indent(Indent) << "Start: ";
    Start->print(OS);
    OS << "\n";
    OS.indent(Indent) << "Step: ";
    Step->print(OS);
    OS << "\n";
    if (ExtraConst) {
      OS.indent(Indent) << "ExtraConst: ";
      ExtraConst->print(OS, false);
      OS << "\n";
    }
  }
 
#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  LLVM_DUMP_METHOD void dump() const { print(dbgs()); }
#endif
 
  bool matchSimpleRecurrence(const PHINode *P);
  bool matchConditionalRecurrence(
      const PHINode *P,
      Instruction::BinaryOps BOWithConstOpToMatch = Instruction::BinaryOpsEnd);
 
private:
  BinaryOperator *digRecurrence(
      Instruction *V,
      Instruction::BinaryOps BOWithConstOpToMatch = Instruction::BinaryOpsEnd);
};
 
/// Check the well-formedness of the (most|least) significant bit check given \p
/// ConditionalRecurrence, \p SimpleRecurrence, depending on \p
/// ByteOrderSwapped. We check that ConditionalRecurrence.Step is a
/// Select(Cmp()) where the compare is `>= 0` in the big-endian case, and `== 0`
/// in the little-endian case (or the inverse, in which case the branches of the
/// compare are swapped). We check that the LHS is (ConditionalRecurrence.Phi
/// [xor SimpleRecurrence.Phi]) in the big-endian case, and additionally check
/// for an AND with one in the little-endian case. We then check AllowedByR
/// against CheckAllowedByR, which is [0, smin) in the big-endian case, and is
/// [0, 1) in the little-endian case. CheckAllowedByR checks for
/// significant-bit-clear, and we match the corresponding arms of the select
/// against bit-shift and bit-shift-and-xor-gen-poly.
static bool
isSignificantBitCheckWellFormed(const RecurrenceInfo &ConditionalRecurrence,
                                const RecurrenceInfo &SimpleRecurrence,
                                bool ByteOrderSwapped) {
  auto *SI = cast<SelectInst>(ConditionalRecurrence.Step);
  CmpPredicate Pred;
  const Value *L;
  const APInt *R;
  Instruction *TV, *FV;
  if (!match(SI, m_Select(m_ICmp(Pred, m_Value(L), m_APInt(R)),
                          m_Instruction(TV), m_Instruction(FV))))
    return false;
 
  // Match predicate with or without a SimpleRecurrence (the corresponding data
  // is LHSAux).
  auto MatchPred = m_CombineOr(
      m_Specific(ConditionalRecurrence.Phi),
      m_c_Xor(m_ZExtOrTruncOrSelf(m_Specific(ConditionalRecurrence.Phi)),
              m_ZExtOrTruncOrSelf(m_Specific(SimpleRecurrence.Phi))));
  bool LWellFormed = ByteOrderSwapped ? match(L, MatchPred)
                                      : match(L, m_c_And(MatchPred, m_One()));
  if (!LWellFormed)
    return false;
 
  KnownBits KnownR = KnownBits::makeConstant(*R);
  unsigned BW = KnownR.getBitWidth();
  auto RCR = ConstantRange::fromKnownBits(KnownR, false);
  auto AllowedByR = ConstantRange::makeAllowedICmpRegion(Pred, RCR);
  ConstantRange CheckAllowedByR(APInt::getZero(BW),
                                ByteOrderSwapped ? APInt::getSignedMinValue(BW)
                                                 : APInt(BW, 1));
 
  BinaryOperator *BitShift = ConditionalRecurrence.BO;
  if (AllowedByR == CheckAllowedByR)
    return TV == BitShift &&
           match(FV, m_c_Xor(m_Specific(BitShift),
                             m_SpecificInt(*ConditionalRecurrence.ExtraConst)));
  if (AllowedByR.inverse() == CheckAllowedByR)
    return FV == BitShift &&
           match(TV, m_c_Xor(m_Specific(BitShift),
                             m_SpecificInt(*ConditionalRecurrence.ExtraConst)));
  return false;
}
 
/// Wraps llvm::matchSimpleRecurrence. Match a simple first order recurrence
/// cycle of the form:
///
/// loop:
///    %rec = phi [%start, %entry], [%BO, %loop]
///     ...
///     %BO = binop %rec, %step
///
/// or
///
/// loop:
///    %rec = phi [%start, %entry], [%BO, %loop]
///    ...
///    %BO = binop %step, %rec
///
bool RecurrenceInfo::matchSimpleRecurrence(const PHINode *P) {
  if (llvm::matchSimpleRecurrence(P, BO, Start, Step)) {
    Phi = P;
    return true;
  }
  return false;
}
 
/// Digs for a recurrence starting with \p V hitting the PHI node in a use-def
/// chain. Used by matchConditionalRecurrence.
BinaryOperator *
RecurrenceInfo::digRecurrence(Instruction *V,
                              Instruction::BinaryOps BOWithConstOpToMatch) {
  SmallVector<Instruction *> Worklist;
  Worklist.push_back(V);
  while (!Worklist.empty()) {
    Instruction *I = Worklist.pop_back_val();
 
    // Don't add a PHI's operands to the Worklist.
    if (isa<PHINode>(I))
      continue;
 
    // Find a recurrence over a BinOp, by matching either of its operands
    // with with the PHINode.
    if (match(I, m_c_BinOp(m_Value(), m_Specific(Phi))))
      return cast<BinaryOperator>(I);
 
    // Bind to ExtraConst, if we match exactly one.
    if (I->getOpcode() == BOWithConstOpToMatch) {
      if (ExtraConst)
        return nullptr;
      const APInt *C = nullptr;
      if (match(I, m_c_BinOp(m_APInt(C), m_Value())))
        ExtraConst = *C;
    }
 
    // Continue along the use-def chain.
    for (Use &U : I->operands())
      if (auto *UI = dyn_cast<Instruction>(U))
        if (L.contains(UI))
          Worklist.push_back(UI);
  }
  return nullptr;
}
 
/// A Conditional Recurrence is a recurrence of the form:
///
/// loop:
///    %rec = phi [%start, %entry], [%step, %loop]
///    ...
///    %step = select _, %tv, %fv
///
/// where %tv and %fv ultimately end up using %rec via the same %BO instruction,
/// after digging through the use-def chain.
///
/// ExtraConst is relevant if \p BOWithConstOpToMatch is supplied: when digging
/// the use-def chain, a BinOp with opcode \p BOWithConstOpToMatch is matched,
/// and ExtraConst is a constant operand of that BinOp. This peculiarity exists,
/// because in a CRC algorithm, the \p BOWithConstOpToMatch is an XOR, and the
/// ExtraConst ends up being the generating polynomial.
bool RecurrenceInfo::matchConditionalRecurrence(
    const PHINode *P, Instruction::BinaryOps BOWithConstOpToMatch) {
  Phi = P;
  if (Phi->getNumIncomingValues() != 2)
    return false;
 
  for (unsigned Idx = 0; Idx != 2; ++Idx) {
    Value *FoundStep = Phi->getIncomingValue(Idx);
    Value *FoundStart = Phi->getIncomingValue(!Idx);
 
    Instruction *TV, *FV;
    if (!match(FoundStep,
               m_Select(m_Cmp(), m_Instruction(TV), m_Instruction(FV))))
      continue;
 
    // For a conditional recurrence, both the true and false values of the
    // select must ultimately end up in the same recurrent BinOp.
    BinaryOperator *FoundBO = digRecurrence(TV, BOWithConstOpToMatch);
    BinaryOperator *AltBO = digRecurrence(FV, BOWithConstOpToMatch);
    if (!FoundBO || FoundBO != AltBO)
      return false;
 
    if (BOWithConstOpToMatch != Instruction::BinaryOpsEnd && !ExtraConst) {
      LLVM_DEBUG(dbgs() << "HashRecognize: Unable to match single BinaryOp "
                           "with constant in conditional recurrence\n");
      return false;
    }
 
    BO = FoundBO;
    Start = FoundStart;
    Step = FoundStep;
    return true;
  }
  return false;
}
 
/// Iterates over all the phis in \p LoopLatch, and attempts to extract a
/// Conditional Recurrence and an optional Simple Recurrence.
static std::optional<std::pair<RecurrenceInfo, RecurrenceInfo>>
getRecurrences(BasicBlock *LoopLatch, const PHINode *IndVar, const Loop &L) {
  auto Phis = LoopLatch->phis();
  unsigned NumPhis = std::distance(Phis.begin(), Phis.end());
  if (NumPhis != 2 && NumPhis != 3)
    return {};
 
  RecurrenceInfo SimpleRecurrence(L);
  RecurrenceInfo ConditionalRecurrence(L);
  for (PHINode &P : Phis) {
    if (&P == IndVar)
      continue;
    if (!SimpleRecurrence)
      SimpleRecurrence.matchSimpleRecurrence(&P);
    if (!ConditionalRecurrence)
      ConditionalRecurrence.matchConditionalRecurrence(
          &P, Instruction::BinaryOps::Xor);
  }
  if (NumPhis == 3 && (!SimpleRecurrence || !ConditionalRecurrence))
    return {};
  return std::make_pair(SimpleRecurrence, ConditionalRecurrence);
}
 
PolynomialInfo::PolynomialInfo(unsigned TripCount, Value *LHS, const APInt &RHS,
                               Value *ComputedValue, bool ByteOrderSwapped,
                               Value *LHSAux)
    : TripCount(TripCount), LHS(LHS), RHS(RHS), ComputedValue(ComputedValue),
      ByteOrderSwapped(ByteOrderSwapped), LHSAux(LHSAux) {}
 
/// Generate a lookup table of 256 entries by interleaving the generating
/// polynomial. The optimization technique of table-lookup for CRC is also
/// called the Sarwate algorithm.
CRCTable HashRecognize::genSarwateTable(const APInt &GenPoly,
                                        bool ByteOrderSwapped) {
  unsigned BW = GenPoly.getBitWidth();
  CRCTable Table;
  Table[0] = APInt::getZero(BW);
 
  if (ByteOrderSwapped) {
    APInt CRCInit = APInt::getSignedMinValue(BW);
    for (unsigned I = 1; I < 256; I <<= 1) {
      CRCInit = CRCInit.shl(1) ^
                (CRCInit.isSignBitSet() ? GenPoly : APInt::getZero(BW));
      for (unsigned J = 0; J < I; ++J)
        Table[I + J] = CRCInit ^ Table[J];
    }
    return Table;
  }
 
  APInt CRCInit(BW, 1);
  for (unsigned I = 128; I; I >>= 1) {
    CRCInit = CRCInit.lshr(1) ^ (CRCInit[0] ? GenPoly : APInt::getZero(BW));
    for (unsigned J = 0; J < 256; J += (I << 1))
      Table[I + J] = CRCInit ^ Table[J];
  }
  return Table;
}
 
/// Checks that \p P1 and \p P2 are used together in an XOR in the use-def chain
/// of \p SI's condition, ignoring any casts. The purpose of this function is to
/// ensure that LHSAux from the SimpleRecurrence is used correctly in the CRC
/// computation.
///
/// In other words, it checks for the following pattern:
///
/// loop:
///   %P1 = phi [_, %entry], [%P1.next, %loop]
///   %P2 = phi [_, %entry], [%P2.next, %loop]
///   ...
///   %xor = xor (CastOrSelf %P1), (CastOrSelf %P2)
///
/// where %xor is in the use-def chain of \p SI's condition.
static bool isConditionalOnXorOfPHIs(const SelectInst *SI, const PHINode *P1,
                                     const PHINode *P2, const Loop &L) {
  SmallVector<const Instruction *> Worklist;
 
  // matchConditionalRecurrence has already ensured that the SelectInst's
  // condition is an Instruction.
  Worklist.push_back(cast<Instruction>(SI->getCondition()));
 
  while (!Worklist.empty()) {
    const Instruction *I = Worklist.pop_back_val();
 
    // Don't add a PHI's operands to the Worklist.
    if (isa<PHINode>(I))
      continue;
 
    // If we match an XOR of the two PHIs ignoring casts, we're done.
    if (match(I, m_c_Xor(m_ZExtOrTruncOrSelf(m_Specific(P1)),
                         m_ZExtOrTruncOrSelf(m_Specific(P2)))))
      return true;
 
    // Continue along the use-def chain.
    for (const Use &U : I->operands())
      if (auto *UI = dyn_cast<Instruction>(U))
        if (L.contains(UI))
          Worklist.push_back(UI);
  }
  return false;
}
 
// Recognizes a multiplication or division by the constant two, using SCEV. By
// doing this, we're immune to whether the IR expression is mul/udiv or
// equivalently shl/lshr. Return false when it is a UDiv, true when it is a Mul,
// and std::nullopt otherwise.
static std::optional<bool> isBigEndianBitShift(Value *V, ScalarEvolution &SE) {
  if (!V->getType()->isIntegerTy())
    return {};
 
  const SCEV *E = SE.getSCEV(V);
  if (match(E, m_scev_UDiv(m_SCEV(), m_scev_SpecificInt(2))))
    return false;
  if (match(E, m_scev_Mul(m_scev_SpecificInt(2), m_SCEV())))
    return true;
  return {};
}
 
/// The main entry point for analyzing a loop and recognizing the CRC algorithm.
/// Returns a PolynomialInfo on success, and a StringRef on failure.
std::variant<PolynomialInfo, StringRef> HashRecognize::recognizeCRC() const {
  if (!L.isInnermost())
    return "Loop is not innermost";
  BasicBlock *Latch = L.getLoopLatch();
  BasicBlock *Exit = L.getExitBlock();
  const PHINode *IndVar = L.getCanonicalInductionVariable();
  if (!Latch || !Exit || !IndVar || L.getNumBlocks() != 1)
    return "Loop not in canonical form";
  unsigned TC = SE.getSmallConstantTripCount(&L);
  if (!TC || TC % 8)
    return "Unable to find a small constant byte-multiple trip count";
 
  auto R = getRecurrences(Latch, IndVar, L);
  if (!R)
    return "Found stray PHI";
  auto [SimpleRecurrence, ConditionalRecurrence] = *R;
  if (!ConditionalRecurrence)
    return "Unable to find conditional recurrence";
 
  // Make sure that all recurrences are either all SCEVMul with two or SCEVDiv
  // with two, or in other words, that they're single bit-shifts.
  std::optional<bool> ByteOrderSwapped =
      isBigEndianBitShift(ConditionalRecurrence.BO, SE);
  if (!ByteOrderSwapped)
    return "Loop with non-unit bitshifts";
  if (SimpleRecurrence) {
    if (isBigEndianBitShift(SimpleRecurrence.BO, SE) != ByteOrderSwapped)
      return "Loop with non-unit bitshifts";
 
    // Ensure that the PHIs have exactly two uses:
    // the bit-shift, and the XOR (or a cast feeding into the XOR).
    if (!ConditionalRecurrence.Phi->hasNUses(2) ||
        !SimpleRecurrence.Phi->hasNUses(2))
      return "Recurrences have stray uses";
 
    // Check that the SelectInst ConditionalRecurrence.Step is conditional on
    // the XOR of SimpleRecurrence.Phi and ConditionalRecurrence.Phi.
    if (!isConditionalOnXorOfPHIs(cast<SelectInst>(ConditionalRecurrence.Step),
                                  SimpleRecurrence.Phi,
                                  ConditionalRecurrence.Phi, L))
      return "Recurrences not intertwined with XOR";
  }
 
  // Make sure that the TC doesn't exceed the bitwidth of LHSAux, or LHS.
  Value *LHS = ConditionalRecurrence.Start;
  Value *LHSAux = SimpleRecurrence ? SimpleRecurrence.Start : nullptr;
  if (TC > (LHSAux ? LHSAux->getType()->getIntegerBitWidth()
                   : LHS->getType()->getIntegerBitWidth()))
    return "Loop iterations exceed bitwidth of data";
 
  // Make sure that the computed value is used in the exit block: this should be
  // true even if it is only really used in an outer loop's exit block, since
  // the loop is in LCSSA form.
  auto *ComputedValue = cast<SelectInst>(ConditionalRecurrence.Step);
  if (none_of(ComputedValue->users(), [Exit](User *U) {
        auto *UI = dyn_cast<Instruction>(U);
        return UI && UI->getParent() == Exit;
      }))
    return "Unable to find use of computed value in loop exit block";
 
  assert(ConditionalRecurrence.ExtraConst &&
         "Expected ExtraConst in conditional recurrence");
  const APInt &GenPoly = *ConditionalRecurrence.ExtraConst;
 
  if (!isSignificantBitCheckWellFormed(ConditionalRecurrence, SimpleRecurrence,
                                       *ByteOrderSwapped))
    return "Malformed significant-bit check";
 
  SmallVector<const Instruction *> Roots(
      {ComputedValue,
       cast<Instruction>(IndVar->getIncomingValueForBlock(Latch)),
       L.getLatchCmpInst(), Latch->getTerminator()});
  if (SimpleRecurrence)
    Roots.push_back(SimpleRecurrence.BO);
  if (containsUnreachable(L, Roots))
    return "Found stray unvisited instructions";
 
  return PolynomialInfo(TC, LHS, GenPoly, ComputedValue, *ByteOrderSwapped,
                        LHSAux);
}
 
void CRCTable::print(raw_ostream &OS) const {
  for (unsigned I = 0; I < 256; I++) {
    (*this)[I].print(OS, false);
    OS << (I % 16 == 15 ? '\n' : ' ');
  }
}
 
#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
void CRCTable::dump() const { print(dbgs()); }
#endif
 
void HashRecognize::print(raw_ostream &OS) const {
  if (!L.isInnermost())
    return;
  OS << "HashRecognize: Checking a loop in '"
     << L.getHeader()->getParent()->getName() << "' from " << L.getLocStr()
     << "\n";
  auto Ret = recognizeCRC();
  if (!std::holds_alternative<PolynomialInfo>(Ret)) {
    OS << "Did not find a hash algorithm\n";
    if (std::holds_alternative<StringRef>(Ret))
      OS << "Reason: " << std::get<StringRef>(Ret) << "\n";
    return;
  }
 
  auto Info = std::get<PolynomialInfo>(Ret);
  OS << "Found" << (Info.ByteOrderSwapped ? " big-endian " : " little-endian ")
     << "CRC-" << Info.RHS.getBitWidth() << " loop with trip count "
     << Info.TripCount << "\n";
  OS.indent(2) << "Initial CRC: ";
  Info.LHS->print(OS);
  OS << "\n";
  OS.indent(2) << "Generating polynomial: ";
  Info.RHS.print(OS, false);
  OS << "\n";
  OS.indent(2) << "Computed CRC: ";
  Info.ComputedValue->print(OS);
  OS << "\n";
  if (Info.LHSAux) {
    OS.indent(2) << "Auxiliary data: ";
    Info.LHSAux->print(OS);
    OS << "\n";
  }
  OS.indent(2) << "Computed CRC lookup table:\n";
  genSarwateTable(Info.RHS, Info.ByteOrderSwapped).print(OS);
}
 
#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
void HashRecognize::dump() const { print(dbgs()); }
#endif
 
std::optional<PolynomialInfo> HashRecognize::getResult() const {
  auto Res = HashRecognize(L, SE).recognizeCRC();
  if (std::holds_alternative<PolynomialInfo>(Res))
    return std::get<PolynomialInfo>(Res);
  return std::nullopt;
}
 
HashRecognize::HashRecognize(const Loop &L, ScalarEvolution &SE)
    : L(L), SE(SE) {}
 
PreservedAnalyses HashRecognizePrinterPass::run(Loop &L,
                                                LoopAnalysisManager &AM,
                                                LoopStandardAnalysisResults &AR,
                                                LPMUpdater &) {
  HashRecognize(L, AR.SE).print(OS);
  return PreservedAnalyses::all();
}