compiler/optimizing/induction_var_range.cc

/*
 * Copyright (C) 2015 The Android Open Source Project
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

#include "induction_var_range.h"

#include <limits>
#include "optimizing/nodes.h"

namespace art HIDDEN {

/** Returns true if 64-bit constant fits in 32-bit constant. */
static bool CanLongValueFitIntoInt(int64_t c) {
  return std::numeric_limits<int32_t>::min() <= c && c <= std::numeric_limits<int32_t>::max();
}

/** Returns true if 32-bit addition can be done safely. */
static bool IsSafeAdd(int32_t c1, int32_t c2) {
  return CanLongValueFitIntoInt(static_cast<int64_t>(c1) + static_cast<int64_t>(c2));
}

/** Returns true if 32-bit subtraction can be done safely. */
static bool IsSafeSub(int32_t c1, int32_t c2) {
  return CanLongValueFitIntoInt(static_cast<int64_t>(c1) - static_cast<int64_t>(c2));
}

/** Returns true if 32-bit multiplication can be done safely. */
static bool IsSafeMul(int32_t c1, int32_t c2) {
  return CanLongValueFitIntoInt(static_cast<int64_t>(c1) * static_cast<int64_t>(c2));
}

/** Returns true if 32-bit division can be done safely. */
static bool IsSafeDiv(int32_t c1, int32_t c2) {
  return c2 != 0 && CanLongValueFitIntoInt(static_cast<int64_t>(c1) / static_cast<int64_t>(c2));
}

/** Computes a * b for a,b > 0 (at least until first overflow happens). */
static int64_t SafeMul(int64_t a, int64_t b, /*out*/ bool* overflow) {
  if (a > 0 && b > 0 && a > (std::numeric_limits<int64_t>::max() / b)) {
    *overflow = true;
  }
  return a * b;
}

/** Returns b^e for b,e > 0. Sets overflow if arithmetic wrap-around occurred. */
static int64_t IntPow(int64_t b, int64_t e, /*out*/ bool* overflow) {
  DCHECK_LT(0, b);
  DCHECK_LT(0, e);
  int64_t pow = 1;
  while (e) {
    if (e & 1) {
      pow = SafeMul(pow, b, overflow);
    }
    e >>= 1;
    if (e) {
      b = SafeMul(b, b, overflow);
    }
  }
  return pow;
}

/** Hunts "under the hood" for a suitable instruction at the hint. */
static bool IsMaxAtHint(
    HInstruction* instruction, HInstruction* hint, /*out*/HInstruction** suitable) {
  if (instruction->IsMin()) {
    // For MIN(x, y), return most suitable x or y as maximum.
    return IsMaxAtHint(instruction->InputAt(0), hint, suitable) ||
           IsMaxAtHint(instruction->InputAt(1), hint, suitable);
  } else {
    *suitable = instruction;
    return HuntForDeclaration(instruction) == hint;
  }
}

/** Post-analysis simplification of a minimum value that makes the bound more useful to clients. */
static InductionVarRange::Value SimplifyMin(InductionVarRange::Value v) {
  if (v.is_known && v.a_constant == 1 && v.b_constant <= 0) {
    // If a == 1,  instruction >= 0 and b <= 0, just return the constant b.
    // No arithmetic wrap-around can occur.
    if (IsGEZero(v.instruction)) {
      return InductionVarRange::Value(v.b_constant);
    }
  }
  return v;
}

/** Post-analysis simplification of a maximum value that makes the bound more useful to clients. */
static InductionVarRange::Value SimplifyMax(InductionVarRange::Value v, HInstruction* hint) {
  if (v.is_known && v.a_constant >= 1) {
    // An upper bound a * (length / a) + b, where a >= 1, can be conservatively rewritten as
    // length + b because length >= 0 is true.
    int64_t value;
    if (v.instruction->IsDiv() &&
        v.instruction->InputAt(0)->IsArrayLength() &&
        IsInt64AndGet(v.instruction->InputAt(1), &value) && v.a_constant == value) {
      return InductionVarRange::Value(v.instruction->InputAt(0), 1, v.b_constant);
    }
    // If a == 1, the most suitable one suffices as maximum value.
    HInstruction* suitable = nullptr;
    if (v.a_constant == 1 && IsMaxAtHint(v.instruction, hint, &suitable)) {
      return InductionVarRange::Value(suitable, 1, v.b_constant);
    }
  }
  return v;
}

/** Tests for a constant value. */
static bool IsConstantValue(InductionVarRange::Value v) {
  return v.is_known && v.a_constant == 0;
}

/** Corrects a value for type to account for arithmetic wrap-around in lower precision. */
static InductionVarRange::Value CorrectForType(InductionVarRange::Value v, DataType::Type type) {
  switch (type) {
    case DataType::Type::kUint8:
    case DataType::Type::kInt8:
    case DataType::Type::kUint16:
    case DataType::Type::kInt16: {
      // Constants within range only.
      // TODO: maybe some room for improvement, like allowing widening conversions
      int32_t min = DataType::MinValueOfIntegralType(type);
      int32_t max = DataType::MaxValueOfIntegralType(type);
      return (IsConstantValue(v) && min <= v.b_constant && v.b_constant <= max)
          ? v
          : InductionVarRange::Value();
    }
    default:
      return v;
  }
}

/** Inserts an instruction. */
static HInstruction* Insert(HBasicBlock* block, HInstruction* instruction) {
  DCHECK(block != nullptr);
  DCHECK(block->GetLastInstruction() != nullptr) << block->GetBlockId();
  DCHECK(instruction != nullptr);
  block->InsertInstructionBefore(instruction, block->GetLastInstruction());
  return instruction;
}

/** Obtains loop's control instruction. */
static HInstruction* GetLoopControl(const HLoopInformation* loop) {
  DCHECK(loop != nullptr);
  return loop->GetHeader()->GetLastInstruction();
}

/** Determines whether the `context` is in the body of the `loop`. */
static bool IsContextInBody(const HBasicBlock* context, const HLoopInformation* loop) {
  DCHECK(loop != nullptr);
  // We're currently classifying trip count only for the exit condition from loop header.
  // All other blocks in the loop are considered loop body.
  return context != loop->GetHeader() && loop->Contains(*context);
}

/** Determines whether to use the full trip count for given `context`, `loop` and `is_min`. */
bool UseFullTripCount(const HBasicBlock* context, const HLoopInformation* loop, bool is_min) {
  // We're currently classifying trip count only for the exit condition from loop header.
  // So, we should call this helper function only if the loop control is an `HIf` with
  // one edge leaving the loop. The loop header is the only block that's both inside
  // the loop and not in the loop body.
  DCHECK(GetLoopControl(loop)->IsIf());
  DCHECK_NE(loop->Contains(*GetLoopControl(loop)->AsIf()->IfTrueSuccessor()),
            loop->Contains(*GetLoopControl(loop)->AsIf()->IfFalseSuccessor()));
  if (loop->Contains(*context)) {
    // Use the full trip count if determining the maximum and context is not in the loop body.
    DCHECK_NE(context == loop->GetHeader(), IsContextInBody(context, loop));
    return !is_min && context == loop->GetHeader();
  } else {
    // Trip count after the loop is always the maximum (ignoring `is_min`),
    // as long as the `context` is dominated by the loop control exit block.
    // If there are additional exit edges, the value is unknown on those paths.
    HInstruction* loop_control = GetLoopControl(loop);
    HBasicBlock* then_block = loop_control->AsIf()->IfTrueSuccessor();
    HBasicBlock* else_block = loop_control->AsIf()->IfFalseSuccessor();
    HBasicBlock* loop_exit_block = loop->Contains(*then_block) ? else_block : then_block;
    return loop_exit_block->Dominates(context);
  }
}

//
// Public class methods.
//

InductionVarRange::InductionVarRange(HInductionVarAnalysis* induction_analysis)
    : induction_analysis_(induction_analysis),
      chase_hint_(nullptr) {
  DCHECK(induction_analysis != nullptr);
}

bool InductionVarRange::GetInductionRange(const HBasicBlock* context,
                                          HInstruction* instruction,
                                          HInstruction* chase_hint,
                                          /*out*/Value* min_val,
                                          /*out*/Value* max_val,
                                          /*out*/bool* needs_finite_test) {
  const HLoopInformation* loop = nullptr;
  HInductionVarAnalysis::InductionInfo* info = nullptr;
  HInductionVarAnalysis::InductionInfo* trip = nullptr;
  if (!HasInductionInfo(context, instruction, &loop, &info, &trip)) {
    return false;
  }
  // Type int or lower (this is not too restrictive since intended clients, like
  // bounds check elimination, will have truncated higher precision induction
  // at their use point already).
  switch (info->type) {
    case DataType::Type::kUint8:
    case DataType::Type::kInt8:
    case DataType::Type::kUint16:
    case DataType::Type::kInt16:
    case DataType::Type::kInt32:
      break;
    default:
      return false;
  }
  // Find range.
  chase_hint_ = chase_hint;
  int64_t stride_value = 0;
  *min_val = SimplifyMin(GetVal(context, loop, info, trip, /*is_min=*/ true));
  *max_val = SimplifyMax(GetVal(context, loop, info, trip, /*is_min=*/ false), chase_hint);
  *needs_finite_test =
      NeedsTripCount(context, loop, info, &stride_value) && IsUnsafeTripCount(trip);
  chase_hint_ = nullptr;
  // Retry chasing constants for wrap-around (merge sensitive).
  if (!min_val->is_known && info->induction_class == HInductionVarAnalysis::kWrapAround) {
    *min_val = SimplifyMin(GetVal(context, loop, info, trip, /*is_min=*/ true));
  }
  return true;
}

bool InductionVarRange::CanGenerateRange(const HBasicBlock* context,
                                         HInstruction* instruction,
                                         /*out*/bool* needs_finite_test,
                                         /*out*/bool* needs_taken_test) {
  bool is_last_value = false;
  int64_t stride_value = 0;
  return GenerateRangeOrLastValue(context,
                                  instruction,
                                  is_last_value,
                                  nullptr,
                                  nullptr,
                                  nullptr,
                                  nullptr,
                                  nullptr,  // nothing generated yet
                                  &stride_value,
                                  needs_finite_test,
                                  needs_taken_test) &&
         (stride_value == -1 ||
          stride_value == 0 ||
          stride_value == 1);  // avoid arithmetic wrap-around anomalies.
}

void InductionVarRange::GenerateRange(const HBasicBlock* context,
                                      HInstruction* instruction,
                                      HGraph* graph,
                                      HBasicBlock* block,
                                      /*out*/HInstruction** lower,
                                      /*out*/HInstruction** upper) {
  bool is_last_value = false;
  int64_t stride_value = 0;
  bool b1, b2;  // unused
  if (!GenerateRangeOrLastValue(context,
                                instruction,
                                is_last_value,
                                graph,
                                block,
                                lower,
                                upper,
                                nullptr,
                                &stride_value,
                                &b1,
                                &b2) ||
      (stride_value != -1 &&
       stride_value != 0 &&
       stride_value != 1)) {
    LOG(FATAL) << "Failed precondition: CanGenerateRange()";
  }
}

HInstruction* InductionVarRange::GenerateTakenTest(HInstruction* loop_control,
                                                   HGraph* graph,
                                                   HBasicBlock* block) {
  const HBasicBlock* context = loop_control->GetBlock();
  HInstruction* taken_test = nullptr;
  bool is_last_value = false;
  int64_t stride_value = 0;
  bool b1, b2;  // unused
  if (!GenerateRangeOrLastValue(context,
                                loop_control,
                                is_last_value,
                                graph,
                                block,
                                nullptr,
                                nullptr,
                                &taken_test,
                                &stride_value,
                                &b1,
                                &b2) ||
      (stride_value != -1 &&
       stride_value != 0 &&
       stride_value != 1)) {
    LOG(FATAL) << "Failed precondition: CanGenerateRange()";
  }
  return taken_test;
}

bool InductionVarRange::CanGenerateLastValue(HInstruction* instruction) {
  const HBasicBlock* context = instruction->GetBlock();
  bool is_last_value = true;
  int64_t stride_value = 0;
  bool needs_finite_test = false;
  bool needs_taken_test = false;
  return GenerateRangeOrLastValue(context,
                                  instruction,
                                  is_last_value,
                                  nullptr,
                                  nullptr,
                                  nullptr,
                                  nullptr,
                                  nullptr,  // nothing generated yet
                                  &stride_value,
                                  &needs_finite_test,
                                  &needs_taken_test)
      && !needs_finite_test && !needs_taken_test;
}

HInstruction* InductionVarRange::GenerateLastValue(HInstruction* instruction,
                                                   HGraph* graph,
                                                   HBasicBlock* block) {
  const HBasicBlock* context = instruction->GetBlock();
  HInstruction* last_value = nullptr;
  bool is_last_value = true;
  int64_t stride_value = 0;
  bool needs_finite_test = false;
  bool needs_taken_test = false;
  if (!GenerateRangeOrLastValue(context,
                                instruction,
                                is_last_value,
                                graph,
                                block,
                                &last_value,
                                &last_value,
                                nullptr,
                                &stride_value,
                                &needs_finite_test,
                                &needs_taken_test) ||
      needs_finite_test ||
      needs_taken_test) {
    LOG(FATAL) << "Failed precondition: CanGenerateLastValue()";
  }
  return last_value;
}

void InductionVarRange::Replace(HInstruction* instruction,
                                HInstruction* fetch,
                                HInstruction* replacement) {
  for (HLoopInformation* lp = instruction->GetBlock()->GetLoopInformation();  // closest enveloping loop
       lp != nullptr;
       lp = lp->GetPreHeader()->GetLoopInformation()) {
    // Update loop's InductionInfo about fetch.
    ReplaceInduction(induction_analysis_->LookupInfo(lp, fetch), fetch, replacement);
    // Update loop's trip-count information.
    ReplaceInduction(induction_analysis_->LookupInfo(lp, GetLoopControl(lp)), fetch, replacement);
  }
}

bool InductionVarRange::IsFinite(const HLoopInformation* loop, /*out*/ int64_t* trip_count) const {
  bool is_constant_unused = false;
  return CheckForFiniteAndConstantProps(loop, &is_constant_unused, trip_count);
}

bool InductionVarRange::HasKnownTripCount(const HLoopInformation* loop,
                                          /*out*/ int64_t* trip_count) const {
  bool is_constant = false;
  CheckForFiniteAndConstantProps(loop, &is_constant, trip_count);
  // Set negative trip counts as 0, since it means that no trips would happen. Note that if the
  // `is_constant` value is false, `trip_count` would be disregareded.
  if (*trip_count < 0) {
    *trip_count = 0;
  }
  return is_constant;
}

bool InductionVarRange::IsUnitStride(const HBasicBlock* context,
                                     HInstruction* instruction,
                                     HGraph* graph,
                                     /*out*/ HInstruction** offset) const {
  const HLoopInformation* loop = nullptr;
  HInductionVarAnalysis::InductionInfo* info = nullptr;
  HInductionVarAnalysis::InductionInfo* trip = nullptr;
  if (HasInductionInfo(context, instruction, &loop, &info, &trip)) {
    if (info->induction_class == HInductionVarAnalysis::kLinear &&
        !HInductionVarAnalysis::IsNarrowingLinear(info)) {
      int64_t stride_value = 0;
      if (IsConstant(context, loop, info->op_a, kExact, &stride_value) && stride_value == 1) {
        int64_t off_value = 0;
        if (IsConstant(context, loop, info->op_b, kExact, &off_value)) {
          *offset = graph->GetConstant(info->op_b->type, off_value);
        } else if (info->op_b->operation == HInductionVarAnalysis::kFetch) {
          *offset = info->op_b->fetch;
        } else {
          return false;
        }
        return true;
      }
    }
  }
  return false;
}

HInstruction* InductionVarRange::GenerateTripCount(const HLoopInformation* loop,
                                                   HGraph* graph,
                                                   HBasicBlock* block) {
  HInstruction* loop_control = GetLoopControl(loop);
  HInductionVarAnalysis::InductionInfo *trip = induction_analysis_->LookupInfo(loop, loop_control);
  if (trip != nullptr && !IsUnsafeTripCount(trip)) {
    const HBasicBlock* context = loop_control->GetBlock();
    HInstruction* taken_test = nullptr;
    HInstruction* trip_expr = nullptr;
    if (IsBodyTripCount(trip)) {
      if (!GenerateCode(context,
                        loop,
                        trip->op_b,
                        /*trip=*/ nullptr,
                        graph,
                        block,
                        /*is_min=*/ false,
                        &taken_test)) {
        return nullptr;
      }
    }
    if (GenerateCode(context,
                     loop,
                     trip->op_a,
                     /*trip=*/ nullptr,
                     graph,
                     block,
                     /*is_min=*/ false,
                     &trip_expr)) {
      if (taken_test != nullptr) {
        HInstruction* zero = graph->GetConstant(trip->type, 0);
        ArenaAllocator* allocator = graph->GetAllocator();
        trip_expr = Insert(block, new (allocator) HSelect(taken_test, trip_expr, zero, kNoDexPc));
      }
      return trip_expr;
    }
  }
  return nullptr;
}

//
// Private class methods.
//

bool InductionVarRange::CheckForFiniteAndConstantProps(const HLoopInformation* loop,
                                                       /*out*/ bool* is_constant,
                                                       /*out*/ int64_t* trip_count) const {
  HInstruction* loop_control = GetLoopControl(loop);
  HInductionVarAnalysis::InductionInfo *trip = induction_analysis_->LookupInfo(loop, loop_control);
  if (trip != nullptr && !IsUnsafeTripCount(trip)) {
    const HBasicBlock* context = loop_control->GetBlock();
    *is_constant = IsConstant(context, loop, trip->op_a, kExact, trip_count);
    return true;
  }
  return false;
}

bool InductionVarRange::IsConstant(const HBasicBlock* context,
                                   const HLoopInformation* loop,
                                   HInductionVarAnalysis::InductionInfo* info,
                                   ConstantRequest request,
                                   /*out*/ int64_t* value) const {
  if (info != nullptr) {
    // A direct 32-bit or 64-bit constant fetch. This immediately satisfies
    // any of the three requests (kExact, kAtMost, and KAtLeast).
    if (info->induction_class == HInductionVarAnalysis::kInvariant &&
        info->operation == HInductionVarAnalysis::kFetch) {
      if (IsInt64AndGet(info->fetch, value)) {
        return true;
      }
    }
    // Try range analysis on the invariant, only accept a proper range
    // to avoid arithmetic wrap-around anomalies.
    Value min_val = GetVal(context, loop, info, /*trip=*/ nullptr, /*is_min=*/ true);
    Value max_val = GetVal(context, loop, info, /*trip=*/ nullptr, /*is_min=*/ false);
    if (IsConstantValue(min_val) &&
        IsConstantValue(max_val) && min_val.b_constant <= max_val.b_constant) {
      if ((request == kExact && min_val.b_constant == max_val.b_constant) || request == kAtMost) {
        *value = max_val.b_constant;
        return true;
      } else if (request == kAtLeast) {
        *value = min_val.b_constant;
        return true;
      }
    }
  }
  return false;
}

bool InductionVarRange::HasInductionInfo(
    const HBasicBlock* context,
    HInstruction* instruction,
    /*out*/ const HLoopInformation** loop,
    /*out*/ HInductionVarAnalysis::InductionInfo** info,
    /*out*/ HInductionVarAnalysis::InductionInfo** trip) const {
  DCHECK(context != nullptr);
  HLoopInformation* lp = context->GetLoopInformation();  // closest enveloping loop
  if (lp != nullptr) {
    HInductionVarAnalysis::InductionInfo* i = induction_analysis_->LookupInfo(lp, instruction);
    if (i != nullptr) {
      *loop = lp;
      *info = i;
      *trip = induction_analysis_->LookupInfo(lp, GetLoopControl(lp));
      return true;
    }
  }
  return false;
}

bool InductionVarRange::IsWellBehavedTripCount(const HBasicBlock* context,
                                               const HLoopInformation* loop,
                                               HInductionVarAnalysis::InductionInfo* trip) const {
  if (trip != nullptr) {
    // Both bounds that define a trip-count are well-behaved if they either are not defined
    // in any loop, or are contained in a proper interval. This allows finding the min/max
    // of an expression by chasing outward.
    InductionVarRange range(induction_analysis_);
    HInductionVarAnalysis::InductionInfo* lower = trip->op_b->op_a;
    HInductionVarAnalysis::InductionInfo* upper = trip->op_b->op_b;
    int64_t not_used = 0;
    return
        (!HasFetchInLoop(lower) || range.IsConstant(context, loop, lower, kAtLeast, &not_used)) &&
        (!HasFetchInLoop(upper) || range.IsConstant(context, loop, upper, kAtLeast, &not_used));
  }
  return true;
}

bool InductionVarRange::HasFetchInLoop(HInductionVarAnalysis::InductionInfo* info) const {
  if (info != nullptr) {
    if (info->induction_class == HInductionVarAnalysis::kInvariant &&
        info->operation == HInductionVarAnalysis::kFetch) {
      return info->fetch->GetBlock()->GetLoopInformation() != nullptr;
    }
    return HasFetchInLoop(info->op_a) || HasFetchInLoop(info->op_b);
  }
  return false;
}

bool InductionVarRange::NeedsTripCount(const HBasicBlock* context,
                                       const HLoopInformation* loop,
                                       HInductionVarAnalysis::InductionInfo* info,
                                       int64_t* stride_value) const {
  if (info != nullptr) {
    if (info->induction_class == HInductionVarAnalysis::kLinear) {
      return IsConstant(context, loop, info->op_a, kExact, stride_value);
    } else if (info->induction_class == HInductionVarAnalysis::kPolynomial) {
      return NeedsTripCount(context, loop, info->op_a, stride_value);
    } else if (info->induction_class == HInductionVarAnalysis::kWrapAround) {
      return NeedsTripCount(context, loop, info->op_b, stride_value);
    }
  }
  return false;
}

bool InductionVarRange::IsBodyTripCount(HInductionVarAnalysis::InductionInfo* trip) const {
  if (trip != nullptr) {
    if (trip->induction_class == HInductionVarAnalysis::kInvariant) {
      return trip->operation == HInductionVarAnalysis::kTripCountInBody ||
             trip->operation == HInductionVarAnalysis::kTripCountInBodyUnsafe;
    }
  }
  return false;
}

bool InductionVarRange::IsUnsafeTripCount(HInductionVarAnalysis::InductionInfo* trip) const {
  if (trip != nullptr) {
    if (trip->induction_class == HInductionVarAnalysis::kInvariant) {
      return trip->operation == HInductionVarAnalysis::kTripCountInBodyUnsafe ||
             trip->operation == HInductionVarAnalysis::kTripCountInLoopUnsafe;
    }
  }
  return false;
}

InductionVarRange::Value InductionVarRange::GetLinear(const HBasicBlock* context,
                                                      const HLoopInformation* loop,
                                                      HInductionVarAnalysis::InductionInfo* info,
                                                      HInductionVarAnalysis::InductionInfo* trip,
                                                      bool is_min) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kLinear);
  // Detect common situation where an offset inside the trip-count cancels out during range
  // analysis (finding max a * (TC - 1) + OFFSET for a == 1 and TC = UPPER - OFFSET or finding
  // min a * (TC - 1) + OFFSET for a == -1 and TC = OFFSET - UPPER) to avoid losing information
  // with intermediate results that only incorporate single instructions.
  if (trip != nullptr) {
    HInductionVarAnalysis::InductionInfo* trip_expr = trip->op_a;
    if (trip_expr->type == info->type && trip_expr->operation == HInductionVarAnalysis::kSub) {
      int64_t stride_value = 0;
      if (IsConstant(context, loop, info->op_a, kExact, &stride_value)) {
        if (!is_min && stride_value == 1) {
          // Test original trip's negative operand (trip_expr->op_b) against offset of induction.
          if (HInductionVarAnalysis::InductionEqual(trip_expr->op_b, info->op_b)) {
            // Analyze cancelled trip with just the positive operand (trip_expr->op_a).
            HInductionVarAnalysis::InductionInfo cancelled_trip(
                trip->induction_class,
                trip->operation,
                trip_expr->op_a,
                trip->op_b,
                nullptr,
                trip->type);
            return GetVal(context, loop, &cancelled_trip, trip, is_min);
          }
        } else if (is_min && stride_value == -1) {
          // Test original trip's positive operand (trip_expr->op_a) against offset of induction.
          if (HInductionVarAnalysis::InductionEqual(trip_expr->op_a, info->op_b)) {
            // Analyze cancelled trip with just the negative operand (trip_expr->op_b).
            HInductionVarAnalysis::InductionInfo neg(
                HInductionVarAnalysis::kInvariant,
                HInductionVarAnalysis::kNeg,
                nullptr,
                trip_expr->op_b,
                nullptr,
                trip->type);
            HInductionVarAnalysis::InductionInfo cancelled_trip(
                trip->induction_class, trip->operation, &neg, trip->op_b, nullptr, trip->type);
            return SubValue(Value(0), GetVal(context, loop, &cancelled_trip, trip, !is_min));
          }
        }
      }
    }
  }
  // General rule of linear induction a * i + b, for normalized 0 <= i < TC.
  return AddValue(GetMul(context, loop, info->op_a, trip, trip, is_min),
                  GetVal(context, loop, info->op_b, trip, is_min));
}

InductionVarRange::Value InductionVarRange::GetPolynomial(
    const HBasicBlock* context,
    const HLoopInformation* loop,
    HInductionVarAnalysis::InductionInfo* info,
    HInductionVarAnalysis::InductionInfo* trip,
    bool is_min) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kPolynomial);
  int64_t a = 0;
  int64_t b = 0;
  if (IsConstant(context, loop, info->op_a->op_a, kExact, &a) &&
      CanLongValueFitIntoInt(a) &&
      a >= 0 &&
      IsConstant(context, loop, info->op_a->op_b, kExact, &b) &&
      CanLongValueFitIntoInt(b) &&
      b >= 0) {
    // Evaluate bounds on sum_i=0^m-1(a * i + b) + c with a,b >= 0 for
    // maximum index value m as a * (m * (m-1)) / 2 + b * m + c.
    // Do not simply return `c` as minimum because the trip count may be non-zero
    // if the `context` is after the `loop` (and therefore ignoring `is_min`).
    Value c = GetVal(context, loop, info->op_b, trip, is_min);
    Value m = GetVal(context, loop, trip, trip, is_min);
    Value t = DivValue(MulValue(m, SubValue(m, Value(1))), Value(2));
    Value x = MulValue(Value(a), t);
    Value y = MulValue(Value(b), m);
    return AddValue(AddValue(x, y), c);
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetGeometric(const HBasicBlock* context,
                                                         const HLoopInformation* loop,
                                                         HInductionVarAnalysis::InductionInfo* info,
                                                         HInductionVarAnalysis::InductionInfo* trip,
                                                         bool is_min) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kGeometric);
  int64_t a = 0;
  int64_t f = 0;
  if (IsConstant(context, loop, info->op_a, kExact, &a) &&
      CanLongValueFitIntoInt(a) &&
      IsInt64AndGet(info->fetch, &f) && f >= 1) {
    // Conservative bounds on a * f^-i + b with f >= 1 can be computed without
    // trip count. Other forms would require a much more elaborate evaluation.
    const bool is_min_a = a >= 0 ? is_min : !is_min;
    if (info->operation == HInductionVarAnalysis::kDiv) {
      Value b = GetVal(context, loop, info->op_b, trip, is_min);
      return is_min_a ? b : AddValue(Value(a), b);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetFetch(const HBasicBlock* context,
                                                     const HLoopInformation* loop,
                                                     HInstruction* instruction,
                                                     HInductionVarAnalysis::InductionInfo* trip,
                                                     bool is_min) const {
  // Special case when chasing constants: single instruction that denotes trip count in the
  // loop-body is minimal 1 and maximal, with safe trip-count, max int,
  if (chase_hint_ == nullptr &&
      IsContextInBody(context, loop) &&
      trip != nullptr &&
      instruction == trip->op_a->fetch) {
    if (is_min) {
      return Value(1);
    } else if (!instruction->IsConstant() && !IsUnsafeTripCount(trip)) {
      return Value(std::numeric_limits<int32_t>::max());
    }
  }
  // Unless at a constant or hint, chase the instruction a bit deeper into the HIR tree, so that
  // it becomes more likely range analysis will compare the same instructions as terminal nodes.
  int64_t value;
  if (IsInt64AndGet(instruction, &value) && CanLongValueFitIntoInt(value)) {
    // Proper constant reveals best information.
    return Value(static_cast<int32_t>(value));
  } else if (instruction == chase_hint_) {
    // At hint, fetch is represented by itself.
    return Value(instruction, 1, 0);
  } else if (instruction->IsAdd()) {
    // Incorporate suitable constants in the chased value.
    if (IsInt64AndGet(instruction->InputAt(0), &value) && CanLongValueFitIntoInt(value)) {
      return AddValue(Value(static_cast<int32_t>(value)),
                      GetFetch(context, loop, instruction->InputAt(1), trip, is_min));
    } else if (IsInt64AndGet(instruction->InputAt(1), &value) && CanLongValueFitIntoInt(value)) {
      return AddValue(GetFetch(context, loop, instruction->InputAt(0), trip, is_min),
                      Value(static_cast<int32_t>(value)));
    }
  } else if (instruction->IsSub()) {
    // Incorporate suitable constants in the chased value.
    if (IsInt64AndGet(instruction->InputAt(0), &value) && CanLongValueFitIntoInt(value)) {
      return SubValue(Value(static_cast<int32_t>(value)),
                      GetFetch(context, loop, instruction->InputAt(1), trip, !is_min));
    } else if (IsInt64AndGet(instruction->InputAt(1), &value) && CanLongValueFitIntoInt(value)) {
      return SubValue(GetFetch(context, loop, instruction->InputAt(0), trip, is_min),
                      Value(static_cast<int32_t>(value)));
    }
  } else if (instruction->IsArrayLength()) {
    // Exploit length properties when chasing constants or chase into a new array declaration.
    if (chase_hint_ == nullptr) {
      return is_min ? Value(0) : Value(std::numeric_limits<int32_t>::max());
    } else if (instruction->InputAt(0)->IsNewArray()) {
      return GetFetch(
          context, loop, instruction->InputAt(0)->AsNewArray()->GetLength(), trip, is_min);
    }
  } else if (instruction->IsTypeConversion()) {
    // Since analysis is 32-bit (or narrower), chase beyond widening along the path.
    // For example, this discovers the length in: for (long i = 0; i < a.length; i++);
    if (instruction->AsTypeConversion()->GetInputType() == DataType::Type::kInt32 &&
        instruction->AsTypeConversion()->GetResultType() == DataType::Type::kInt64) {
      return GetFetch(context, loop, instruction->InputAt(0), trip, is_min);
    }
  }
  // Chase an invariant fetch that is defined by another loop if the trip-count used
  // so far is well-behaved in both bounds and the next trip-count is safe.
  // Example:
  //   for (int i = 0; i <= 100; i++)  // safe
  //     for (int j = 0; j <= i; j++)  // well-behaved
  //       j is in range [0, i  ] (if i is chase hint)
  //         or in range [0, 100] (otherwise)
  // Example:
  //   for (i = 0; i < 100; ++i)
  //     <some-code>
  //   for (j = 0; j < 10; ++j)
  //     sum += i;  // The `i` is a "fetch" of a loop Phi from the previous loop.
  const HLoopInformation* next_loop = nullptr;
  HInductionVarAnalysis::InductionInfo* next_info = nullptr;
  HInductionVarAnalysis::InductionInfo* next_trip = nullptr;
  if (HasInductionInfo(instruction->GetBlock(), instruction, &next_loop, &next_info, &next_trip) &&
      IsWellBehavedTripCount(context, next_loop, trip) &&
      !IsUnsafeTripCount(next_trip)) {
    return GetVal(context, next_loop, next_info, next_trip, is_min);
  }
  // Fetch is represented by itself.
  return Value(instruction, 1, 0);
}

InductionVarRange::Value InductionVarRange::GetVal(const HBasicBlock* context,
                                                   const HLoopInformation* loop,
                                                   HInductionVarAnalysis::InductionInfo* info,
                                                   HInductionVarAnalysis::InductionInfo* trip,
                                                   bool is_min) const {
  if (info != nullptr) {
    switch (info->induction_class) {
      case HInductionVarAnalysis::kInvariant:
        // Invariants.
        switch (info->operation) {
          case HInductionVarAnalysis::kAdd:
            return AddValue(GetVal(context, loop, info->op_a, trip, is_min),
                            GetVal(context, loop, info->op_b, trip, is_min));
          case HInductionVarAnalysis::kSub:  // second reversed!
            return SubValue(GetVal(context, loop, info->op_a, trip, is_min),
                            GetVal(context, loop, info->op_b, trip, !is_min));
          case HInductionVarAnalysis::kNeg:  // second reversed!
            return SubValue(Value(0),
                            GetVal(context, loop, info->op_b, trip, !is_min));
          case HInductionVarAnalysis::kMul:
            return GetMul(context, loop, info->op_a, info->op_b, trip, is_min);
          case HInductionVarAnalysis::kDiv:
            return GetDiv(context, loop, info->op_a, info->op_b, trip, is_min);
          case HInductionVarAnalysis::kRem:
            return GetRem(context, loop, info->op_a, info->op_b);
          case HInductionVarAnalysis::kXor:
            return GetXor(context, loop, info->op_a, info->op_b);
          case HInductionVarAnalysis::kFetch:
            return GetFetch(context, loop, info->fetch, trip, is_min);
          case HInductionVarAnalysis::kTripCountInLoop:
          case HInductionVarAnalysis::kTripCountInLoopUnsafe:
            if (UseFullTripCount(context, loop, is_min)) {
              // Return the full trip count (do not subtract 1 as we do in loop body).
              return GetVal(context, loop, info->op_a, trip, is_min);
            }
            FALLTHROUGH_INTENDED;
          case HInductionVarAnalysis::kTripCountInBody:
          case HInductionVarAnalysis::kTripCountInBodyUnsafe:
            if (is_min) {
              return Value(0);
            } else if (IsContextInBody(context, loop)) {
              return SubValue(GetVal(context, loop, info->op_a, trip, is_min), Value(1));
            }
            break;
          default:
            break;
        }
        break;
      case HInductionVarAnalysis::kLinear:
        return CorrectForType(GetLinear(context, loop, info, trip, is_min), info->type);
      case HInductionVarAnalysis::kPolynomial:
        return GetPolynomial(context, loop, info, trip, is_min);
      case HInductionVarAnalysis::kGeometric:
        return GetGeometric(context, loop, info, trip, is_min);
      case HInductionVarAnalysis::kWrapAround:
      case HInductionVarAnalysis::kPeriodic:
        return MergeVal(GetVal(context, loop, info->op_a, trip, is_min),
                        GetVal(context, loop, info->op_b, trip, is_min),
                        is_min);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetMul(const HBasicBlock* context,
                                                   const HLoopInformation* loop,
                                                   HInductionVarAnalysis::InductionInfo* info1,
                                                   HInductionVarAnalysis::InductionInfo* info2,
                                                   HInductionVarAnalysis::InductionInfo* trip,
                                                   bool is_min) const {
  // Constant times range.
  int64_t value = 0;
  if (IsConstant(context, loop, info1, kExact, &value)) {
    return MulRangeAndConstant(context, loop, value, info2, trip, is_min);
  } else if (IsConstant(context, loop, info2, kExact, &value)) {
    return MulRangeAndConstant(context, loop, value, info1, trip, is_min);
  }
  // Interval ranges.
  Value v1_min = GetVal(context, loop, info1, trip, /*is_min=*/ true);
  Value v1_max = GetVal(context, loop, info1, trip, /*is_min=*/ false);
  Value v2_min = GetVal(context, loop, info2, trip, /*is_min=*/ true);
  Value v2_max = GetVal(context, loop, info2, trip, /*is_min=*/ false);
  // Positive range vs. positive or negative range.
  if (IsConstantValue(v1_min) && v1_min.b_constant >= 0) {
    if (IsConstantValue(v2_min) && v2_min.b_constant >= 0) {
      return is_min ? MulValue(v1_min, v2_min) : MulValue(v1_max, v2_max);
    } else if (IsConstantValue(v2_max) && v2_max.b_constant <= 0) {
      return is_min ? MulValue(v1_max, v2_min) : MulValue(v1_min, v2_max);
    }
  }
  // Negative range vs. positive or negative range.
  if (IsConstantValue(v1_max) && v1_max.b_constant <= 0) {
    if (IsConstantValue(v2_min) && v2_min.b_constant >= 0) {
      return is_min ? MulValue(v1_min, v2_max) : MulValue(v1_max, v2_min);
    } else if (IsConstantValue(v2_max) && v2_max.b_constant <= 0) {
      return is_min ? MulValue(v1_max, v2_max) : MulValue(v1_min, v2_min);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetDiv(const HBasicBlock* context,
                                                   const HLoopInformation* loop,
                                                   HInductionVarAnalysis::InductionInfo* info1,
                                                   HInductionVarAnalysis::InductionInfo* info2,
                                                   HInductionVarAnalysis::InductionInfo* trip,
                                                   bool is_min) const {
  // Range divided by constant.
  int64_t value = 0;
  if (IsConstant(context, loop, info2, kExact, &value)) {
    return DivRangeAndConstant(context, loop, value, info1, trip, is_min);
  }
  // Interval ranges.
  Value v1_min = GetVal(context, loop, info1, trip, /*is_min=*/ true);
  Value v1_max = GetVal(context, loop, info1, trip, /*is_min=*/ false);
  Value v2_min = GetVal(context, loop, info2, trip, /*is_min=*/ true);
  Value v2_max = GetVal(context, loop, info2, trip, /*is_min=*/ false);
  // Positive range vs. positive or negative range.
  if (IsConstantValue(v1_min) && v1_min.b_constant >= 0) {
    if (IsConstantValue(v2_min) && v2_min.b_constant >= 0) {
      return is_min ? DivValue(v1_min, v2_max) : DivValue(v1_max, v2_min);
    } else if (IsConstantValue(v2_max) && v2_max.b_constant <= 0) {
      return is_min ? DivValue(v1_max, v2_max) : DivValue(v1_min, v2_min);
    }
  }
  // Negative range vs. positive or negative range.
  if (IsConstantValue(v1_max) && v1_max.b_constant <= 0) {
    if (IsConstantValue(v2_min) && v2_min.b_constant >= 0) {
      return is_min ? DivValue(v1_min, v2_min) : DivValue(v1_max, v2_max);
    } else if (IsConstantValue(v2_max) && v2_max.b_constant <= 0) {
      return is_min ? DivValue(v1_max, v2_min) : DivValue(v1_min, v2_max);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetRem(
    const HBasicBlock* context,
    const HLoopInformation* loop,
    HInductionVarAnalysis::InductionInfo* info1,
    HInductionVarAnalysis::InductionInfo* info2) const {
  int64_t v1 = 0;
  int64_t v2 = 0;
  // Only accept exact values.
  if (IsConstant(context, loop, info1, kExact, &v1) &&
      IsConstant(context, loop, info2, kExact, &v2) &&
      v2 != 0) {
    int64_t value = v1 % v2;
    if (CanLongValueFitIntoInt(value)) {
      return Value(static_cast<int32_t>(value));
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::GetXor(
    const HBasicBlock* context,
    const HLoopInformation* loop,
    HInductionVarAnalysis::InductionInfo* info1,
    HInductionVarAnalysis::InductionInfo* info2) const {
  int64_t v1 = 0;
  int64_t v2 = 0;
  // Only accept exact values.
  if (IsConstant(context, loop, info1, kExact, &v1) &&
      IsConstant(context, loop, info2, kExact, &v2)) {
    int64_t value = v1 ^ v2;
    if (CanLongValueFitIntoInt(value)) {
      return Value(static_cast<int32_t>(value));
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::MulRangeAndConstant(
    const HBasicBlock* context,
    const HLoopInformation* loop,
    int64_t value,
    HInductionVarAnalysis::InductionInfo* info,
    HInductionVarAnalysis::InductionInfo* trip,
    bool is_min) const {
  if (CanLongValueFitIntoInt(value)) {
    Value c(static_cast<int32_t>(value));
    return MulValue(GetVal(context, loop, info, trip, is_min == value >= 0), c);
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::DivRangeAndConstant(
    const HBasicBlock* context,
    const HLoopInformation* loop,
    int64_t value,
    HInductionVarAnalysis::InductionInfo* info,
    HInductionVarAnalysis::InductionInfo* trip,
    bool is_min) const {
  if (CanLongValueFitIntoInt(value)) {
    Value c(static_cast<int32_t>(value));
    return DivValue(GetVal(context, loop, info, trip, is_min == value >= 0), c);
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::AddValue(Value v1, Value v2) const {
  if (v1.is_known && v2.is_known && IsSafeAdd(v1.b_constant, v2.b_constant)) {
    int32_t b = v1.b_constant + v2.b_constant;
    if (v1.a_constant == 0) {
      return Value(v2.instruction, v2.a_constant, b);
    } else if (v2.a_constant == 0) {
      return Value(v1.instruction, v1.a_constant, b);
    } else if (v1.instruction == v2.instruction && IsSafeAdd(v1.a_constant, v2.a_constant)) {
      return Value(v1.instruction, v1.a_constant + v2.a_constant, b);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::SubValue(Value v1, Value v2) const {
  if (v1.is_known && v2.is_known && IsSafeSub(v1.b_constant, v2.b_constant)) {
    int32_t b = v1.b_constant - v2.b_constant;
    if (v1.a_constant == 0 && IsSafeSub(0, v2.a_constant)) {
      return Value(v2.instruction, -v2.a_constant, b);
    } else if (v2.a_constant == 0) {
      return Value(v1.instruction, v1.a_constant, b);
    } else if (v1.instruction == v2.instruction && IsSafeSub(v1.a_constant, v2.a_constant)) {
      return Value(v1.instruction, v1.a_constant - v2.a_constant, b);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::MulValue(Value v1, Value v2) const {
  if (v1.is_known && v2.is_known) {
    if (v1.a_constant == 0) {
      if (IsSafeMul(v1.b_constant, v2.a_constant) && IsSafeMul(v1.b_constant, v2.b_constant)) {
        return Value(v2.instruction, v1.b_constant * v2.a_constant, v1.b_constant * v2.b_constant);
      }
    } else if (v2.a_constant == 0) {
      if (IsSafeMul(v1.a_constant, v2.b_constant) && IsSafeMul(v1.b_constant, v2.b_constant)) {
        return Value(v1.instruction, v1.a_constant * v2.b_constant, v1.b_constant * v2.b_constant);
      }
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::DivValue(Value v1, Value v2) const {
  if (v1.is_known && v2.is_known && v1.a_constant == 0 && v2.a_constant == 0) {
    if (IsSafeDiv(v1.b_constant, v2.b_constant)) {
      return Value(v1.b_constant / v2.b_constant);
    }
  }
  return Value();
}

InductionVarRange::Value InductionVarRange::MergeVal(Value v1, Value v2, bool is_min) const {
  if (v1.is_known && v2.is_known) {
    if (v1.instruction == v2.instruction && v1.a_constant == v2.a_constant) {
      return Value(v1.instruction, v1.a_constant,
                   is_min ? std::min(v1.b_constant, v2.b_constant)
                          : std::max(v1.b_constant, v2.b_constant));
    }
  }
  return Value();
}

bool InductionVarRange::GenerateRangeOrLastValue(const HBasicBlock* context,
                                                 HInstruction* instruction,
                                                 bool is_last_value,
                                                 HGraph* graph,
                                                 HBasicBlock* block,
                                                 /*out*/HInstruction** lower,
                                                 /*out*/HInstruction** upper,
                                                 /*out*/HInstruction** taken_test,
                                                 /*out*/int64_t* stride_value,
                                                 /*out*/bool* needs_finite_test,
                                                 /*out*/bool* needs_taken_test) const {
  const HLoopInformation* loop = nullptr;
  HInductionVarAnalysis::InductionInfo* info = nullptr;
  HInductionVarAnalysis::InductionInfo* trip = nullptr;
  if (!HasInductionInfo(context, instruction, &loop, &info, &trip) || trip == nullptr) {
    return false;  // codegen needs all information, including tripcount
  }
  // Determine what tests are needed. A finite test is needed if the evaluation code uses the
  // trip-count and the loop maybe unsafe (because in such cases, the index could "overshoot"
  // the computed range). A taken test is needed for any unknown trip-count, even if evaluation
  // code does not use the trip-count explicitly (since there could be an implicit relation
  // between e.g. an invariant subscript and a not-taken condition).
  *stride_value = 0;
  *needs_finite_test = NeedsTripCount(context, loop, info, stride_value) && IsUnsafeTripCount(trip);
  *needs_taken_test = IsBodyTripCount(trip);
  // Handle last value request.
  if (is_last_value) {
    DCHECK(!IsContextInBody(context, loop));
    switch (info->induction_class) {
      case HInductionVarAnalysis::kLinear:
        if (*stride_value > 0) {
          lower = nullptr;
          return GenerateLastValueLinear(
              context, loop, info, trip, graph, block, /*is_min=*/false, upper, needs_taken_test);
        } else {
          upper = nullptr;
          return GenerateLastValueLinear(
              context, loop, info, trip, graph, block, /*is_min=*/true, lower, needs_taken_test);
        }
      case HInductionVarAnalysis::kPolynomial:
        return GenerateLastValuePolynomial(context, loop, info, trip, graph, block, lower);
      case HInductionVarAnalysis::kGeometric:
        return GenerateLastValueGeometric(context, loop, info, trip, graph, block, lower);
      case HInductionVarAnalysis::kWrapAround:
        return GenerateLastValueWrapAround(context, loop, info, trip, graph, block, lower);
      case HInductionVarAnalysis::kPeriodic:
        return GenerateLastValuePeriodic(
            context, loop, info, trip, graph, block, lower, needs_taken_test);
      default:
        return false;
    }
  }
  // Code generation for taken test: generate the code when requested or otherwise analyze
  // if code generation is feasible when taken test is needed.
  if (taken_test != nullptr) {
    return GenerateCode(context,
                        loop,
                        trip->op_b,
                        /*trip=*/ nullptr,
                        graph,
                        block,
                        /*is_min=*/ false,
                        taken_test);
  } else if (*needs_taken_test) {
    if (!GenerateCode(context,
                      loop,
                      trip->op_b,
                      /*trip=*/ nullptr,
                      /*graph=*/ nullptr,
                      /*block=*/ nullptr,
                      /*is_min=*/ false,
                      /*result=*/ nullptr)) {
      return false;
    }
  }
  // Code generation for lower and upper.
  return
      // Success on lower if invariant (not set), or code can be generated.
      ((info->induction_class == HInductionVarAnalysis::kInvariant) ||
          GenerateCode(context, loop, info, trip, graph, block, /*is_min=*/ true, lower)) &&
      // And success on upper.
      GenerateCode(context, loop, info, trip, graph, block, /*is_min=*/ false, upper);
}

bool InductionVarRange::GenerateLastValueLinear(const HBasicBlock* context,
                                                const HLoopInformation* loop,
                                                HInductionVarAnalysis::InductionInfo* info,
                                                HInductionVarAnalysis::InductionInfo* trip,
                                                HGraph* graph,
                                                HBasicBlock* block,
                                                bool is_min,
                                                /*out*/ HInstruction** result,
                                                /*inout*/ bool* needs_taken_test) const {
  DataType::Type type = info->type;
  // Avoid any narrowing linear induction or any type mismatch between the linear induction and the
  // trip count expression.
  if (HInductionVarAnalysis::IsNarrowingLinear(info) || trip->type != type) {
    return false;
  }

  // Stride value must be a known constant that fits into int32. The stride will be the `i` in `a *
  // i + b`.
  int64_t stride_value = 0;
  if (!IsConstant(context, loop, info->op_a, kExact, &stride_value) ||
      !CanLongValueFitIntoInt(stride_value)) {
    return false;
  }

  // We require the calculation of `a` to not overflow.
  const bool is_min_a = stride_value >= 0 ? is_min : !is_min;
  HInstruction* opa;
  HInstruction* opb;
  if (!GenerateCode(context,
                    loop,
                    trip,
                    trip,
                    graph,
                    block,
                    is_min_a,
                    &opa,
                    /*allow_potential_overflow=*/false) ||
      !GenerateCode(context, loop, info->op_b, trip, graph, block, is_min, &opb)) {
    return false;
  }

  if (graph != nullptr) {
    ArenaAllocator* allocator = graph->GetAllocator();
    HInstruction* oper;
    // Emit instructions for `a * i + b`. These are fine to overflow as they would have overflown
    // also if we had kept the loop.
    if (stride_value == 1) {
      oper = new (allocator) HAdd(type, opa, opb);
    } else if (stride_value == -1) {
      oper = new (graph->GetAllocator()) HSub(type, opb, opa);
    } else {
      HInstruction* mul = new (allocator) HMul(type, graph->GetConstant(type, stride_value), opa);
      oper = new (allocator) HAdd(type, Insert(block, mul), opb);
    }
    *result = Insert(block, oper);
  }

  if (*needs_taken_test) {
    if (TryGenerateTakenTest(context, loop, trip->op_b, graph, block, result, opb)) {
      *needs_taken_test = false;  // taken care of
    } else {
      return false;
    }
  }

  return true;
}

bool InductionVarRange::GenerateLastValuePolynomial(const HBasicBlock* context,
                                                    const HLoopInformation* loop,
                                                    HInductionVarAnalysis::InductionInfo* info,
                                                    HInductionVarAnalysis::InductionInfo* trip,
                                                    HGraph* graph,
                                                    HBasicBlock* block,
                                                    /*out*/HInstruction** result) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kPolynomial);
  // Detect known coefficients and trip count (always taken).
  int64_t a = 0;
  int64_t b = 0;
  int64_t m = 0;
  if (IsConstant(context, loop, info->op_a->op_a, kExact, &a) &&
      IsConstant(context, loop, info->op_a->op_b, kExact, &b) &&
      IsConstant(context, loop, trip->op_a, kExact, &m) &&
      m >= 1) {
    // Evaluate bounds on sum_i=0^m-1(a * i + b) + c for known
    // maximum index value m as a * (m * (m-1)) / 2 + b * m + c.
    HInstruction* c = nullptr;
    if (GenerateCode(context,
                     loop,
                     info->op_b,
                     /*trip=*/ nullptr,
                     graph,
                     block,
                     /*is_min=*/ false,
                     graph ? &c : nullptr)) {
      if (graph != nullptr) {
        DataType::Type type = info->type;
        int64_t sum = a * ((m * (m - 1)) / 2) + b * m;
        if (type != DataType::Type::kInt64) {
          sum = static_cast<int32_t>(sum);  // okay to truncate
        }
        *result =
            Insert(block, new (graph->GetAllocator()) HAdd(type, graph->GetConstant(type, sum), c));
      }
      return true;
    }
  }
  return false;
}

bool InductionVarRange::GenerateLastValueGeometric(const HBasicBlock* context,
                                                   const HLoopInformation* loop,
                                                   HInductionVarAnalysis::InductionInfo* info,
                                                   HInductionVarAnalysis::InductionInfo* trip,
                                                   HGraph* graph,
                                                   HBasicBlock* block,
                                                   /*out*/HInstruction** result) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kGeometric);
  // Detect known base and trip count (always taken).
  int64_t f = 0;
  int64_t m = 0;
  if (IsInt64AndGet(info->fetch, &f) &&
      f >= 1 &&
      IsConstant(context, loop, trip->op_a, kExact, &m) &&
      m >= 1) {
    HInstruction* opa = nullptr;
    HInstruction* opb = nullptr;
    if (GenerateCode(
            context, loop, info->op_a, /*trip=*/ nullptr, graph, block, /*is_min=*/ false, &opa) &&
        GenerateCode(
            context, loop, info->op_b, /*trip=*/ nullptr, graph, block, /*is_min=*/ false, &opb)) {
      if (graph != nullptr) {
        DataType::Type type = info->type;
        // Compute f ^ m for known maximum index value m.
        bool overflow = false;
        int64_t fpow = IntPow(f, m, &overflow);
        if (info->operation == HInductionVarAnalysis::kDiv) {
          // For division, any overflow truncates to zero.
          if (overflow || (type != DataType::Type::kInt64 && !CanLongValueFitIntoInt(fpow))) {
            fpow = 0;
          }
        } else if (type != DataType::Type::kInt64) {
          // For multiplication, okay to truncate to required precision.
          DCHECK(info->operation == HInductionVarAnalysis::kMul);
          fpow = static_cast<int32_t>(fpow);
        }
        // Generate code.
        if (fpow == 0) {
          // Special case: repeated mul/div always yields zero.
          *result = graph->GetConstant(type, 0);
        } else {
          // Last value: a * f ^ m + b or a * f ^ -m + b.
          HInstruction* e = nullptr;
          ArenaAllocator* allocator = graph->GetAllocator();
          if (info->operation == HInductionVarAnalysis::kMul) {
            e = new (allocator) HMul(type, opa, graph->GetConstant(type, fpow));
          } else {
            e = new (allocator) HDiv(type, opa, graph->GetConstant(type, fpow), kNoDexPc);
          }
          *result = Insert(block, new (allocator) HAdd(type, Insert(block, e), opb));
        }
      }
      return true;
    }
  }
  return false;
}

bool InductionVarRange::GenerateLastValueWrapAround(const HBasicBlock* context,
                                                    const HLoopInformation* loop,
                                                    HInductionVarAnalysis::InductionInfo* info,
                                                    HInductionVarAnalysis::InductionInfo* trip,
                                                    HGraph* graph,
                                                    HBasicBlock* block,
                                                    /*out*/HInstruction** result) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kWrapAround);
  // Count depth.
  int32_t depth = 0;
  for (; info->induction_class == HInductionVarAnalysis::kWrapAround;
       info = info->op_b, ++depth) {}
  // Handle wrap(x, wrap(.., y)) if trip count reaches an invariant at end.
  // TODO: generalize, but be careful to adjust the terminal.
  int64_t m = 0;
  if (info->induction_class == HInductionVarAnalysis::kInvariant &&
      IsConstant(context, loop, trip->op_a, kExact, &m) &&
      m >= depth) {
    return GenerateCode(
        context, loop, info, /*trip=*/ nullptr, graph, block, /*is_min=*/ false, result);
  }
  return false;
}

bool InductionVarRange::GenerateLastValuePeriodic(const HBasicBlock* context,
                                                  const HLoopInformation* loop,
                                                  HInductionVarAnalysis::InductionInfo* info,
                                                  HInductionVarAnalysis::InductionInfo* trip,
                                                  HGraph* graph,
                                                  HBasicBlock* block,
                                                  /*out*/ HInstruction** result,
                                                  /*inout*/ bool* needs_taken_test) const {
  DCHECK(info != nullptr);
  DCHECK_EQ(info->induction_class, HInductionVarAnalysis::kPeriodic);
  // Count period and detect all-invariants.
  int64_t period = 1;
  bool all_invariants = true;
  HInductionVarAnalysis::InductionInfo* p = info;
  for (; p->induction_class == HInductionVarAnalysis::kPeriodic; p = p->op_b, ++period) {
    DCHECK_EQ(p->op_a->induction_class, HInductionVarAnalysis::kInvariant);
    if (p->op_a->operation != HInductionVarAnalysis::kFetch) {
      all_invariants = false;
    }
  }
  DCHECK_EQ(p->induction_class, HInductionVarAnalysis::kInvariant);
  if (p->operation != HInductionVarAnalysis::kFetch) {
    all_invariants = false;
  }
  // Don't rely on FP arithmetic to be precise, unless the full period
  // consist of pre-computed expressions only.
  if (info->type == DataType::Type::kFloat32 || info->type == DataType::Type::kFloat64) {
    if (!all_invariants) {
      return false;
    }
  }
  // Handle any periodic(x, periodic(.., y)) for known maximum index value m.
  int64_t m = 0;
  if (IsConstant(context, loop, trip->op_a, kExact, &m) && m >= 1) {
    int64_t li = m % period;
    for (int64_t i = 0; i < li; info = info->op_b, i++) {}
    if (info->induction_class == HInductionVarAnalysis::kPeriodic) {
      info = info->op_a;
    }
    return GenerateCode(
        context, loop, info, /*trip=*/ nullptr, graph, block, /*is_min=*/ false, result);
  }
  // Handle periodic(x, y) using even/odd-select on trip count. Enter trip count expression
  // directly to obtain the maximum index value t even if taken test is needed.
  HInstruction* x = nullptr;
  HInstruction* y = nullptr;
  HInstruction* t = nullptr;

  // Overflows when the stride is equal to `1` are fine since the periodicity is
  // `2` and the lowest bit is the same. Similar with `-1`.
  auto allow_potential_overflow = [&]() {
    int64_t stride_value = 0;
    return IsConstant(context, loop, trip->op_a->op_b, kExact, &stride_value) &&
           (stride_value == 1 || stride_value == -1);
  };

  if (period == 2 &&
      GenerateCode(context,
                   loop,
                   info->op_a,
                   /*trip=*/ nullptr,
                   graph,
                   block,
                   /*is_min=*/ false,
                   graph ? &x : nullptr) &&
      GenerateCode(context,
                   loop,
                   info->op_b,
                   /*trip=*/ nullptr,
                   graph,
                   block,
                   /*is_min=*/ false,
                   graph ? &y : nullptr) &&
      GenerateCode(context,
                   loop,
                   trip->op_a,
                   /*trip=*/ nullptr,
                   graph,
                   block,
                   /*is_min=*/ false,
                   graph ? &t : nullptr,
                   allow_potential_overflow())) {
    // During actual code generation (graph != nullptr), generate is_even ? x : y.
    if (graph != nullptr) {
      DataType::Type type = trip->type;
      ArenaAllocator* allocator = graph->GetAllocator();
      HInstruction* msk =
          Insert(block, new (allocator) HAnd(type, t, graph->GetConstant(type, 1)));
      HInstruction* is_even =
          Insert(block, new (allocator) HEqual(msk, graph->GetConstant(type, 0), kNoDexPc));
      *result = Insert(block, new (graph->GetAllocator()) HSelect(is_even, x, y, kNoDexPc));
    }

    if (*needs_taken_test) {
      if (TryGenerateTakenTest(context, loop, trip->op_b, graph, block, result, x)) {
        *needs_taken_test = false;  // taken care of
      } else {
        return false;
      }
    }
    return true;
  }
  return false;
}

bool InductionVarRange::GenerateCode(const HBasicBlock* context,
                                     const HLoopInformation* loop,
                                     HInductionVarAnalysis::InductionInfo* info,
                                     HInductionVarAnalysis::InductionInfo* trip,
                                     HGraph* graph,  // when set, code is generated
                                     HBasicBlock* block,
                                     bool is_min,
                                     /*out*/ HInstruction** result,
                                     bool allow_potential_overflow) const {
  if (info != nullptr) {
    // If during codegen, the result is not needed (nullptr), simply return success.
    if (graph != nullptr && result == nullptr) {
      return true;
    }
    // Handle current operation.
    DataType::Type type = info->type;
    HInstruction* opa = nullptr;
    HInstruction* opb = nullptr;
    switch (info->induction_class) {
      case HInductionVarAnalysis::kInvariant:
        // Invariants (note that since invariants only have other invariants as
        // sub expressions, viz. no induction, there is no need to adjust is_min).
        switch (info->operation) {
          case HInductionVarAnalysis::kAdd:
          case HInductionVarAnalysis::kSub:
          case HInductionVarAnalysis::kMul:
          case HInductionVarAnalysis::kDiv:
          case HInductionVarAnalysis::kRem:
          case HInductionVarAnalysis::kXor:
          case HInductionVarAnalysis::kLT:
          case HInductionVarAnalysis::kLE:
          case HInductionVarAnalysis::kGT:
          case HInductionVarAnalysis::kGE:
            if (GenerateCode(context,
                             loop,
                             info->op_a,
                             trip,
                             graph,
                             block,
                             is_min,
                             &opa,
                             allow_potential_overflow) &&
                GenerateCode(context,
                             loop,
                             info->op_b,
                             trip,
                             graph,
                             block,
                             is_min,
                             &opb,
                             allow_potential_overflow)) {
              // Check for potentially invalid operations.
              if (!allow_potential_overflow) {
                switch (info->operation) {
                  case HInductionVarAnalysis::kAdd:
                    return TryGenerateAddWithoutOverflow(
                        context, loop, info, graph, opa, opb, result);
                  case HInductionVarAnalysis::kSub:
                    return TryGenerateSubWithoutOverflow(context, loop, info, graph, opa, result);
                  default:
                    // The rest of the operations are not relevant in the cases where
                    // `allow_potential_overflow` is false. Fall through to the allowed overflow
                    // case.
                    break;
                }
              }

              // Overflows here are accepted.
              if (graph != nullptr) {
                HInstruction* operation = nullptr;
                switch (info->operation) {
                  case HInductionVarAnalysis::kAdd:
                    operation = new (graph->GetAllocator()) HAdd(type, opa, opb); break;
                  case HInductionVarAnalysis::kSub:
                    operation = new (graph->GetAllocator()) HSub(type, opa, opb); break;
                  case HInductionVarAnalysis::kMul:
                    operation = new (graph->GetAllocator()) HMul(type, opa, opb, kNoDexPc); break;
                  case HInductionVarAnalysis::kDiv:
                    operation = new (graph->GetAllocator()) HDiv(type, opa, opb, kNoDexPc); break;
                  case HInductionVarAnalysis::kRem:
                    operation = new (graph->GetAllocator()) HRem(type, opa, opb, kNoDexPc); break;
                  case HInductionVarAnalysis::kXor:
                    operation = new (graph->GetAllocator()) HXor(type, opa, opb); break;
                  case HInductionVarAnalysis::kLT:
                    operation = new (graph->GetAllocator()) HLessThan(opa, opb); break;
                  case HInductionVarAnalysis::kLE:
                    operation = new (graph->GetAllocator()) HLessThanOrEqual(opa, opb); break;
                  case HInductionVarAnalysis::kGT:
                    operation = new (graph->GetAllocator()) HGreaterThan(opa, opb); break;
                  case HInductionVarAnalysis::kGE:
                    operation = new (graph->GetAllocator()) HGreaterThanOrEqual(opa, opb); break;
                  default:
                    LOG(FATAL) << "unknown operation";
                }
                *result = Insert(block, operation);
              }
              return true;
            }
            break;
          case HInductionVarAnalysis::kNeg:
            if (GenerateCode(context,
                             loop,
                             info->op_b,
                             trip,
                             graph,
                             block,
                             !is_min,
                             &opb,
                             allow_potential_overflow)) {
              if (graph != nullptr) {
                *result = Insert(block, new (graph->GetAllocator()) HNeg(type, opb));
              }
              return true;
            }
            break;
          case HInductionVarAnalysis::kFetch:
            if (graph != nullptr) {
              *result = info->fetch;  // already in HIR
            }
            return true;
          case HInductionVarAnalysis::kTripCountInLoop:
          case HInductionVarAnalysis::kTripCountInLoopUnsafe:
            if (UseFullTripCount(context, loop, is_min)) {
              // Generate the full trip count (do not subtract 1 as we do in loop body).
              return GenerateCode(context,
                                  loop,
                                  info->op_a,
                                  trip,
                                  graph,
                                  block,
                                  is_min,
                                  result,
                                  allow_potential_overflow);
            }
            FALLTHROUGH_INTENDED;
          case HInductionVarAnalysis::kTripCountInBody:
          case HInductionVarAnalysis::kTripCountInBodyUnsafe:
            if (is_min) {
              if (graph != nullptr) {
                *result = graph->GetConstant(type, 0);
              }
              return true;
            } else if (IsContextInBody(context, loop) ||
                       (context == loop->GetHeader() && !allow_potential_overflow)) {
              if (GenerateCode(context,
                               loop,
                               info->op_a,
                               trip,
                               graph,
                               block,
                               is_min,
                               &opb,
                               allow_potential_overflow)) {
                if (graph != nullptr) {
                  if (IsContextInBody(context, loop)) {
                    ArenaAllocator* allocator = graph->GetAllocator();
                    *result =
                        Insert(block, new (allocator) HSub(type, opb, graph->GetConstant(type, 1)));
                  } else {
                    // We want to generate the full trip count since we want the last value. This
                    // will be combined with an `is_taken` test so we don't want to subtract one.
                    DCHECK(context == loop->GetHeader());
                    // TODO(solanes): Remove the !allow_potential_overflow restriction and allow
                    // other parts e.g. BCE to take advantage of this.
                    DCHECK(!allow_potential_overflow);
                    *result = opb;
                  }
                }
                return true;
              }
            }
            break;
          case HInductionVarAnalysis::kNop:
            LOG(FATAL) << "unexpected invariant nop";
        }  // switch invariant operation
        break;
      case HInductionVarAnalysis::kLinear: {
        // Linear induction a * i + b, for normalized 0 <= i < TC. For ranges, this should
        // be restricted to a unit stride to avoid arithmetic wrap-around situations that
        // are harder to guard against. For a last value, requesting min/max based on any
        // known stride yields right value. Always avoid any narrowing linear induction or
        // any type mismatch between the linear induction and the trip count expression.
        // TODO: careful runtime type conversions could generalize this latter restriction.
        if (!HInductionVarAnalysis::IsNarrowingLinear(info) && trip->type == type) {
          int64_t stride_value = 0;
          if (IsConstant(context, loop, info->op_a, kExact, &stride_value) &&
              CanLongValueFitIntoInt(stride_value)) {
            const bool is_min_a = stride_value >= 0 ? is_min : !is_min;
            if (GenerateCode(context,
                             loop,
                             trip,
                             trip,
                             graph,
                             block,
                             is_min_a,
                             &opa,
                             allow_potential_overflow) &&
                GenerateCode(context,
                             loop,
                             info->op_b,
                             trip,
                             graph,
                             block,
                             is_min,
                             &opb,
                             allow_potential_overflow)) {
              if (graph != nullptr) {
                ArenaAllocator* allocator = graph->GetAllocator();
                HInstruction* oper;
                if (stride_value == 1) {
                  oper = new (allocator) HAdd(type, opa, opb);
                } else if (stride_value == -1) {
                  oper = new (graph->GetAllocator()) HSub(type, opb, opa);
                } else {
                  HInstruction* mul =
                      new (allocator) HMul(type, graph->GetConstant(type, stride_value), opa);
                  oper = new (allocator) HAdd(type, Insert(block, mul), opb);
                }
                *result = Insert(block, oper);
              }
              return true;
            }
          }
        }
        break;
      }
      case HInductionVarAnalysis::kPolynomial:
      case HInductionVarAnalysis::kGeometric:
        break;
      case HInductionVarAnalysis::kWrapAround:
      case HInductionVarAnalysis::kPeriodic: {
        // Wrap-around and periodic inductions are restricted to constants only, so that extreme
        // values are easy to test at runtime without complications of arithmetic wrap-around.
        Value extreme = GetVal(context, loop, info, trip, is_min);
        if (IsConstantValue(extreme)) {
          if (graph != nullptr) {
            *result = graph->GetConstant(type, extreme.b_constant);
          }
          return true;
        }
        break;
      }
    }  // switch induction class
  }
  return false;
}

bool InductionVarRange::TryGenerateAddWithoutOverflow(const HBasicBlock* context,
                                                      const HLoopInformation* loop,
                                                      HInductionVarAnalysis::InductionInfo* info,
                                                      HGraph* graph,
                                                      /*in*/ HInstruction* opa,
                                                      /*in*/ HInstruction* opb,
                                                      /*out*/ HInstruction** result) const {
  // Calculate `a + b` making sure we can't overflow.
  int64_t val_a;
  const bool a_is_const = IsConstant(context, loop, info->op_a, kExact, &val_a);
  int64_t val_b;
  const bool b_is_const = IsConstant(context, loop, info->op_b, kExact, &val_b);
  if (a_is_const && b_is_const) {
    // Calculate `a + b` and use that. Note that even when the values are known,
    // their addition can still overflow.
    Value add_val = AddValue(Value(val_a), Value(val_b));
    if (add_val.is_known) {
      DCHECK(IsConstantValue(add_val));
      // Known value not overflowing.
      if (graph != nullptr) {
        *result = graph->GetConstant(info->type, add_val.b_constant);
      }
      return true;
    }
  }

  // When `a` is `0`, we can just use `b`.
  if (a_is_const && val_a == 0) {
    if (graph != nullptr) {
      *result = opb;
    }
    return true;
  }

  if (b_is_const && val_b == 0) {
    if (graph != nullptr) {
      *result = opa;
    }
    return true;
  }

  // Couldn't safely calculate the addition.
  return false;
}

bool InductionVarRange::TryGenerateSubWithoutOverflow(const HBasicBlock* context,
                                                      const HLoopInformation* loop,
                                                      HInductionVarAnalysis::InductionInfo* info,
                                                      HGraph* graph,
                                                      /*in*/ HInstruction* opa,
                                                      /*out*/ HInstruction** result) const {
  // Calculate `a - b` making sure we can't overflow.
  int64_t val_b;
  if (!IsConstant(context, loop, info->op_b, kExact, &val_b)) {
    // If b is unknown, a - b can potentially overflow for any value of a since b
    // can be Integer.MIN_VALUE.
    return false;
  }

  int64_t val_a;
  if (IsConstant(context, loop, info->op_a, kExact, &val_a)) {
    // Calculate `a - b` and use that. Note that even when the values are known,
    // their subtraction can still overflow.
    Value sub_val = SubValue(Value(val_a), Value(val_b));
    if (sub_val.is_known) {
      DCHECK(IsConstantValue(sub_val));
      // Known value not overflowing.
      if (graph != nullptr) {
        *result = graph->GetConstant(info->type, sub_val.b_constant);
      }
      return true;
    }
  }

  // When `b` is `0`, we can just use `a`.
  if (val_b == 0) {
    if (graph != nullptr) {
      *result = opa;
    }
    return true;
  }

  // Couldn't safely calculate the subtraction.
  return false;
}

bool InductionVarRange::TryGenerateTakenTest(const HBasicBlock* context,
                                             const HLoopInformation* loop,
                                             HInductionVarAnalysis::InductionInfo* info,
                                             HGraph* graph,
                                             HBasicBlock* block,
                                             /*inout*/ HInstruction** result,
                                             /*inout*/ HInstruction* not_taken_result) const {
  HInstruction* is_taken = nullptr;
  if (GenerateCode(context,
                   loop,
                   info,
                   /*trip=*/nullptr,
                   graph,
                   block,
                   /*is_min=*/false,
                   graph != nullptr ? &is_taken : nullptr)) {
    if (graph != nullptr) {
      ArenaAllocator* allocator = graph->GetAllocator();
      *result =
          Insert(block, new (allocator) HSelect(is_taken, *result, not_taken_result, kNoDexPc));
    }
    return true;
  } else {
    return false;
  }
}

void InductionVarRange::ReplaceInduction(HInductionVarAnalysis::InductionInfo* info,
                                         HInstruction* fetch,
                                         HInstruction* replacement) {
  if (info != nullptr) {
    if (info->induction_class == HInductionVarAnalysis::kInvariant &&
        info->operation == HInductionVarAnalysis::kFetch &&
        info->fetch == fetch) {
      info->fetch = replacement;
    }
    ReplaceInduction(info->op_a, fetch, replacement);
    ReplaceInduction(info->op_b, fetch, replacement);
  }
}

}  // namespace art