rewrite_simplify.cc 69.1 KB
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/*
 * Licensed to the Apache Software Foundation (ASF) under one
 * or more contributor license agreements.  See the NOTICE file
 * distributed with this work for additional information
 * regarding copyright ownership.  The ASF licenses this file
 * to you 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
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 *
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 *   http://www.apache.org/licenses/LICENSE-2.0
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 *
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 * 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.
 */

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/*!
 * \file rewrite_simplify.cc
 * \brief Rewrite-rule based simplification.
 */
// Acknowledgement: Most rewrite-rules are from Halide.
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#include <tvm/arith/analyzer.h>
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#include <tvm/tir/op.h>
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#include <algorithm>
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#include "const_fold.h"
#include "pattern_match.h"
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#include "rewrite_simplify.h"
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namespace tvm {
namespace arith {

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using namespace tir;
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// macro for doing simple rewrite
#define TVM_TRY_REWRITE(SrcExpr, ResExpr)       \
  if ((SrcExpr).Match(ret)) {                   \
    return (ResExpr).Eval();                    \
  }

// macro for rewrite + recursively rewrite ResExpr
#define TVM_TRY_RECURSIVE_REWRITE(SrcExpr, ResExpr) \
  if ((SrcExpr).Match(ret)) {                       \
    return RecursiveRewrite((ResExpr).Eval());      \
  }

// macro rewrite only if CondExor is true after match.
#define TVM_TRY_REWRITE_IF(SrcExpr, ResExpr, CondExpr)  \
  if ((SrcExpr).Match(ret) && (CondExpr)) {             \
    return (ResExpr).Eval();                            \
  }

// macro rewrite + recursive_rewrite only if CondExor is true after match.
#define TVM_TRY_RECURSIVE_REWRITE_IF(SrcExpr, ResExpr, CondExpr)  \
  if ((SrcExpr).Match(ret) && (CondExpr)) {                       \
    return RecursiveRewrite((ResExpr).Eval());                    \
  }

// NOTE for developers:
//
// We mainly focus on index expression simplification.
// Besides the RewriteSimplifier, some cases can be better
// handled by CanonicalSimplifier.
//
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// try to prove x equals val
RewriteSimplifier::Impl::CompareResult RewriteSimplifier::Impl::
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TryCompare(const PrimExpr& x, int64_t val) {
  PrimExpr diff = this->VisitExpr(x);
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  if (const auto* ptr = diff.as<IntImmNode>()) {
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    if (ptr->value == val) {
      return kEQ;
    } else if (ptr->value > val) {
      return kGT;
    } else if (ptr->value < val) {
      return kLT;
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    }
  }
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  ConstIntBound dbound = analyzer_->const_int_bound(diff);
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  if (dbound->min_value > val) {
    return kGT;
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  }
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  if (dbound->max_value < val) {
    return kLT;
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  }
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  if (dbound->min_value >= val) {
    return kGE;
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  }
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  if (dbound->max_value <= val) {
    return kLE;
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  }
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  if (val == 0) {
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    ModularSet dmod = analyzer_->modular_set(diff);
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    if (dmod->base != 0) {
      return kNE;
    }
  }
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  return kUnknown;
}
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void RewriteSimplifier::Impl::
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Update(const Var& var, const PrimExpr& info, bool override) {
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  if (!override) {
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    auto it = var_map_.find(var);
    if (it != var_map_.end()) {
      CHECK(Equal(it->second, info))
          << "Trying to update var \'" << var << "\'"
          << " with a different value: "
          << "original=" << it->second
          << ", new=" << info;
    }
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  }
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  var_map_[var] = info;
}
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PrimExpr RewriteSimplifier::Impl::
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VisitExpr_(const AddNode* op) {
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  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
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  op = ret.as<AddNode>();
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  PrimExpr const_res = TryConstFold<AddNode>(op->a, op->b);
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  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
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  PVar<PrimExpr> x, y, z, b1, b2, s1, s2;
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  // Pattern var match IntImm
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  PVar<IntImm> c1, c2, c3;
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  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;
  // Vector rules
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  if (op->dtype.lanes() != 1) {
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    TVM_TRY_REWRITE(ramp(b1, s1, lanes) + ramp(b2, s2, lanes),
                    ramp(b1 + b2, s1 + s2, lanes));
    TVM_TRY_REWRITE(ramp(b1, s1, lanes) + broadcast(x, lanes),
                    ramp(b1 + x, s1, lanes));
    TVM_TRY_REWRITE(broadcast(x, lanes) + ramp(b1, s1, lanes),
                    ramp(x + b1, s1, lanes));
    TVM_TRY_REWRITE(broadcast(x, lanes) + broadcast(y, lanes),
                    broadcast(x + y, lanes));
  }

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  if (IsIndexType(op->dtype)) {
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    // Index rules
    // cancelation rules
    TVM_TRY_REWRITE((x - y) + y, x);
    TVM_TRY_REWRITE(x + (y - x), y);

    TVM_TRY_REWRITE((x - y) + (y - z), x - z);
    TVM_TRY_REWRITE((x - y) + (z - x), z - y);

    TVM_TRY_REWRITE(min(x, y - z) + z, min(x + z, y));
    TVM_TRY_REWRITE(min(x - z, y) + z, min(x, y + z));
    TVM_TRY_REWRITE(max(x, y - z) + z, max(x + z, y));
    TVM_TRY_REWRITE(max(x - z, y) + z, max(x, y + z));
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    TVM_TRY_REWRITE_IF(min(x, y + z * c1) + z * c2, min(x + z * c2, y),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(max(x, y + z * c1) + z * c2, max(x + z * c2, y),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(min(y + z * c1, x) + z * c2, min(x + z * c2, y),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(max(y + z * c1, x) + z * c2, max(x + z * c2, y),
                       c1.Eval()->value == -c2.Eval()->value);

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    TVM_TRY_REWRITE(max(x, y) + min(x, y), x + y);
    TVM_TRY_REWRITE(min(x, y) + max(x, y), x + y);
    TVM_TRY_REWRITE(max(x, y) + min(y, x), x + y);
    TVM_TRY_REWRITE(min(x, y) + max(y, x), x + y);

    TVM_TRY_REWRITE_IF(min(x, y + c1) + c2, min(x + c2, y),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(min(x + c1, y) + c2, min(x, y + c2),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(max(x, y + c1) + c2, max(x + c2, y),
                       c1.Eval()->value == -c2.Eval()->value);
    TVM_TRY_REWRITE_IF(max(x + c1, y) + c2, max(x, y + c2),
                       c1.Eval()->value == -c2.Eval()->value);

    // constant folding
    // NOTE: canonicalization might better at this.
    TVM_TRY_REWRITE((x + c1) + c2, x + (c1 + c2));

    // mul co-efficient folding
    TVM_TRY_REWRITE(x + x, x * 2);
    TVM_TRY_REWRITE(x * y + x, x * (y + 1));
    TVM_TRY_REWRITE(y * x + x, x * (y + 1));
    TVM_TRY_REWRITE(x + y * x, x * (1 + y));
    TVM_TRY_REWRITE(x + x * y, x * (1 + y));
    TVM_TRY_REWRITE(x * y + x * z, x * (y + z));
    TVM_TRY_REWRITE(y * x + x * z, x * (y + z));
    TVM_TRY_REWRITE(x * y + z * x, x * (y + z));
    TVM_TRY_REWRITE(y * x + z * x, x * (y + z));

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    // DivMod rules
    // truc div
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    TVM_TRY_REWRITE(truncdiv(x, c1) * c1 + truncmod(x, c1), x);
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    // floor div
    TVM_TRY_REWRITE(floordiv(x, c1) * c1 + floormod(x, c1), x);
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    // canonicalization rule
    // will try rewrite again after canonicalization.
    TVM_TRY_RECURSIVE_REWRITE(x + (c1 - y), (x - y) + c1);
    TVM_TRY_RECURSIVE_REWRITE(x + c1 + y, (x + y) + c1);
    TVM_TRY_RECURSIVE_REWRITE(x + (c1 + y), (x + y) + c1);
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    TVM_TRY_RECURSIVE_REWRITE(x + max(y, z), max(y, z) + x);
    TVM_TRY_RECURSIVE_REWRITE(x + min(y, z), min(y, z) + x);
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    // DivMod rules
    // truc div
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    TVM_TRY_RECURSIVE_REWRITE(truncmod(y, c1) + x * c1, x * c1 + truncmod(y, c1));
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    // floor div
    TVM_TRY_RECURSIVE_REWRITE(floormod(y, c1) + x * c1, x * c1 + floormod(y, c1));
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  }

  // condition rules.
  TVM_TRY_REWRITE(select(x, b1, b2) + select(x, s1, s2),
                  select(x, b1 + s1, b2 + s2));
  // default value
  return ret;
}

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std::function<void()> RewriteSimplifier::Impl::EnterConstraint(const PrimExpr& constraint) {
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  size_t old_literal_size = literal_constraints_.size();
  literal_constraints_.push_back(constraint);
  size_t new_literal_size = literal_constraints_.size();
  auto frecover = [old_literal_size, new_literal_size, this]() {
    CHECK_EQ(literal_constraints_.size(), new_literal_size);
    literal_constraints_.resize(old_literal_size);
  };
  return frecover;
}

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PrimExpr RewriteSimplifier::Impl::
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VisitExpr_(const SubNode* op) {
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  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
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  op = ret.as<SubNode>();
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  PrimExpr const_res = TryConstFold<SubNode>(op->a, op->b);
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  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
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  PVar<PrimExpr> x, y, z, b1, b2, s1, s2;
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  // Pattern var match IntImm
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  PVar<IntImm> c1, c2, c3;
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  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;
  // Vector rules
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  if (op->dtype.lanes() != 1) {
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    TVM_TRY_REWRITE(ramp(b1, s1, lanes) - ramp(b2, s2, lanes),
                    ramp(b1 - b2, s1 - s2, lanes));
    TVM_TRY_REWRITE(ramp(b1, s1, lanes) - broadcast(x, lanes),
                    ramp(b1 - x, s1, lanes));
    TVM_TRY_REWRITE(broadcast(x, lanes) - ramp(b1, s1, lanes),
                    ramp(x - b1, 0 - s1, lanes));
    TVM_TRY_REWRITE(broadcast(x, lanes) - broadcast(y, lanes),
                    broadcast(x - y, lanes));
  }

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  if (IsIndexType(op->dtype)) {
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    // Index rules
    // cancelation rules
    TVM_TRY_REWRITE((x + y) - y, x);
    TVM_TRY_REWRITE((x + y) - x, y);
    TVM_TRY_REWRITE(x - (y + x), 0 - y);
    TVM_TRY_REWRITE(x - (x + y), 0 - y);

    TVM_TRY_REWRITE(min(x, y) - x, min(0, y - x));
    TVM_TRY_REWRITE(min(x, y) - y, min(x - y, 0));
    TVM_TRY_REWRITE(max(x, y) - x, max(0, y - x));
    TVM_TRY_REWRITE(max(x, y) - y, max(x - y, 0));

    TVM_TRY_REWRITE(x - max(x, y), min(0, x - y));
    TVM_TRY_REWRITE(y - max(x, y), min(y - x, 0));
    TVM_TRY_REWRITE(x - min(x, y), max(0, x - y));
    TVM_TRY_REWRITE(y - min(x, y), max(y - x, 0));

    // mul co-efficient folding
    TVM_TRY_REWRITE(x - x, ZeroWithTypeLike(x));
    TVM_TRY_REWRITE(x * y - x, x * (y - 1));
    TVM_TRY_REWRITE(y * x - x, x * (y - 1));
    TVM_TRY_REWRITE(x - y * x, x * (1 - y));
    TVM_TRY_REWRITE(x - x * y, x * (1 - y));
    TVM_TRY_REWRITE(x * y - x * z, x * (y - z));
    TVM_TRY_REWRITE(y * x - x * z, x * (y - z));
    TVM_TRY_REWRITE(x * y - z * x, x * (y - z));
    TVM_TRY_REWRITE(y * x - z * x, x * (y - z));

    // constant cancelation
    TVM_TRY_REWRITE((x + c1) - c2, x + (c1 - c2));
    TVM_TRY_REWRITE((c1 - x) - (c2 - y), (y - x) + (c1 - c2));

    // cancelization rule involving 4 operands
    TVM_TRY_REWRITE((x + y) - (x + z), y - z);
    TVM_TRY_REWRITE((x + y) - (z + x), y - z);
    TVM_TRY_REWRITE((y + x) - (z + x), y - z);
    TVM_TRY_REWRITE((y + x) - (x + z), y - z);

    TVM_TRY_REWRITE(min(x + y, z) - x,  min(y, z - x));
    TVM_TRY_REWRITE(min(y + x, z) - x,  min(y, z - x));
    TVM_TRY_REWRITE(min(z, x + y) - x,  min(z - x, y));
    TVM_TRY_REWRITE(min(z, y + x) - x,  min(z - x, y));

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    TVM_TRY_REWRITE(max(x + y, z) - x,  max(y, z - x));
    TVM_TRY_REWRITE(max(y + x, z) - x,  max(y, z - x));
    TVM_TRY_REWRITE(max(z, x + y) - x,  max(z - x, y));
    TVM_TRY_REWRITE(max(z, y + x) - x,  max(z - x, y));

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    TVM_TRY_REWRITE(x - min(x + y, z),  max(0 - y, x - z));
    TVM_TRY_REWRITE(x - min(y + x, z),  max(0 - y, x - z));
    TVM_TRY_REWRITE(x - min(z, x + y),  max(x - z, 0 - y));
    TVM_TRY_REWRITE(x - min(z, y + x),  max(x - z, 0 - y));

    TVM_TRY_REWRITE(min(x, y) - min(y, x), ZeroWithTypeLike(x));
    TVM_TRY_REWRITE(max(x, y) - max(y, x), ZeroWithTypeLike(x));

    TVM_TRY_REWRITE_IF(min(b1, b2) - min(s1, s2), b1 - s1,
                       CanProveEqual(((b1 - s1) - (b2 - s2)).Eval(), 0));

    TVM_TRY_REWRITE_IF(min(b1, b2) - min(s1, s2), b1 - s2,
                       CanProveEqual(((b1 - s2) - (b2 - s1)).Eval(), 0));
    TVM_TRY_REWRITE_IF(max(b1, b2) - max(s1, s2), b1 - s1,
                       CanProveEqual(((b1 - s1) - (b2 - s2)).Eval(), 0));
    TVM_TRY_REWRITE_IF(max(b1, b2) - max(s1, s2), b1 - s2,
                       CanProveEqual(((b1 - s2) - (b2 - s1)).Eval(), 0));

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    // DivMod rules
    // trucdiv
    // NOTE: c*(x/c) + x % c == x is true all division mode.
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    TVM_TRY_REWRITE_IF(x - truncdiv(x, c1) * c1, truncmod(x, c1),
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                       c1.Eval()->value != 0);
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    TVM_TRY_REWRITE_IF(truncdiv(x, c1) * c1 - x, 0 - truncmod(x, c1),
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                       c1.Eval()->value != 0);
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    TVM_TRY_REWRITE_IF(x - (truncdiv(x + y, c1)) * c1, truncmod(x + y, c1) - y,
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                       c1.Eval()->value != 0);
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    TVM_TRY_REWRITE_IF((truncdiv(x + y, c1)) * c1 - x, y - truncmod(x + y, c1),
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                       c1.Eval()->value != 0);
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    TVM_TRY_REWRITE_IF(x - truncdiv(x - y, c1) * c1, truncmod(x - y, c1) + y,
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                       c1.Eval()->value != 0);
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    TVM_TRY_REWRITE_IF(truncdiv(x - y, c1) * c1 - x, 0 - truncmod(x - y, c1) - y,
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                       c1.Eval()->value != 0);

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    TVM_TRY_REWRITE_IF(x * c2 - truncdiv(x, c1) * c3, truncmod(x, c1) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
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    TVM_TRY_REWRITE_IF(truncdiv(x, c1) * c3 - x * c2, 0 - truncmod(x, c1) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
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    TVM_TRY_REWRITE_IF(x * c2 - truncdiv(x + y, c1) * c3, (truncmod(x + y, c1) - y) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
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    TVM_TRY_REWRITE_IF(truncdiv(x + y, c1) * c3 - x * c2, (y - truncmod(x + y, c1)) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
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    TVM_TRY_REWRITE_IF(x * c2 - truncdiv(x - y, c1) * c3, (truncmod(x - y, c1) + y) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
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    TVM_TRY_REWRITE_IF(truncdiv(x - y, c1) * c3 - x * c2, (0 - truncmod(x - y, c1) - y) * c2,
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                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);

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    // Proof in the case of floordiv, need positive condition.
    // let x = a * c3 + r
    // (x + c1) / c3 - x / c3 => (r + c1) / c3
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    // NOTE: the use of floormod(c2, c3) was intentional to simplify the const.
    TVM_TRY_REWRITE_IF(truncdiv(x + c1, c3)  - truncdiv(x + c2, c3),
                       truncdiv(truncmod(x + floormod(c2, c3), c3) + (c1 - c2), c3),
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                       CanProveGreaterEqual(x.Eval(), -c2.Eval()->value) &&
                       c1.Eval()->value >= c2.Eval()->value &&
                       c3.Eval()->value > 0);
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    TVM_TRY_REWRITE_IF(truncdiv(x + c1, c3)  - truncdiv(x, c3),
                       truncdiv(truncmod(x, c3) + c1, c3),
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                       CanProveGreaterEqual(x.Eval(), 0) &&
                       c1.Eval()->value >= 0 &&
                       c3.Eval()->value > 0);
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    // floordiv
    TVM_TRY_REWRITE_IF(x - floordiv(x, c1) * c1, floormod(x, c1),
                       c1.Eval()->value != 0);
    TVM_TRY_REWRITE_IF(floordiv(x, c1) * c1 - x, 0 - floormod(x, c1),
                       c1.Eval()->value != 0);
    TVM_TRY_REWRITE_IF(x - floordiv(x + y, c1) * c1, floormod(x + y, c1) - y,
                       c1.Eval()->value != 0);
    TVM_TRY_REWRITE_IF(floordiv(x + y, c1) * c1 - x, y - floormod(x + y, c1),
                       c1.Eval()->value != 0);
    TVM_TRY_REWRITE_IF(x - floordiv(x - y, c1) * c1, floormod(x - y, c1) + y,
                       c1.Eval()->value != 0);
    TVM_TRY_REWRITE_IF(floordiv(x - y, c1) * c1 - x, 0 - floormod(x - y, c1) - y,
                       c1.Eval()->value != 0);

    TVM_TRY_REWRITE_IF(x * c2 - floordiv(x, c1) * c3, floormod(x, c1) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
    TVM_TRY_REWRITE_IF(floordiv(x, c1) * c3 - x * c2, 0 - floormod(x, c1) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
    TVM_TRY_REWRITE_IF(x * c2 - floordiv(x + y, c1) * c3, (floormod(x + y, c1) - y) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
    TVM_TRY_REWRITE_IF(floordiv(x + y, c1) * c3 - x * c2, (y - floormod(x + y, c1)) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
    TVM_TRY_REWRITE_IF(x * c2 - floordiv(x - y, c1) * c3, (floormod(x - y, c1) + y) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);
    TVM_TRY_REWRITE_IF(floordiv(x - y, c1) * c3 - x * c2, (0 - floormod(x - y, c1) - y) * c2,
                       c1.Eval()->value != 0 &&
                       c3.Eval()->value == c1.Eval()->value * c2.Eval()->value);

    TVM_TRY_REWRITE_IF(floordiv(x + c1, c3) - floordiv(x + c2, c3),
                       floordiv(floormod(x + floormod(c2, c3), c3) + (c1 - c2), c3),
                       c3.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x + c1, c3)  - floordiv(x, c3),
                       floordiv(floormod(x, c3) + c1, c3),
                       c3.Eval()->value > 0);

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    // canonicalization rule
    // will try rewrite again after canonicalization.
    TVM_TRY_REWRITE(x - c1, x + (0 - c1));
    TVM_TRY_RECURSIVE_REWRITE((x + c1) - y, (x - y) + c1);
    TVM_TRY_RECURSIVE_REWRITE(x - (y - z), (x + z) - y);
    TVM_TRY_RECURSIVE_REWRITE(x - y * c1, x + y * (0 - c1));
  }

  // condition rules.
  TVM_TRY_REWRITE(select(x, b1, b2) - select(x, s1, s2),
                  select(x, b1 - s1, b2 - s2));
  TVM_TRY_REWRITE(select(x, y, z) - z,
                  select(x, y - z, ZeroWithTypeLike(z)));
  TVM_TRY_REWRITE(select(x, y, z) - y,
                  select(x, ZeroWithTypeLike(y), z - y));
  return ret;
}

432
PrimExpr RewriteSimplifier::Impl::
433
VisitExpr_(const MulNode* op) {
434
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
435
  op = ret.as<MulNode>();
436
  PrimExpr const_res = TryConstFold<MulNode>(op->a, op->b);
437 438
  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
439
  PVar<PrimExpr> x, y, z, b1, b2, s1, s2;
440
  // Pattern var match IntImm
441
  PVar<IntImm> c1, c2;
442 443 444
  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;
  // Vector rules
445
  if (op->dtype.lanes() != 1) {
446 447 448 449 450 451 452 453
    TVM_TRY_REWRITE(broadcast(x, lanes) * broadcast(y, lanes),
                    broadcast(x * y, lanes));
    TVM_TRY_REWRITE(ramp(b1, s1, lanes) * broadcast(x, lanes),
                    ramp(b1 * x, s1 * x, lanes));
    TVM_TRY_REWRITE(broadcast(x, lanes) * ramp(b1, s1, lanes),
                    ramp(b1 * x, s1 * x, lanes));
  }

454
  if (IsIndexType(op->dtype)) {
455 456 457 458 459 460 461 462
    // constant simplification rule
    TVM_TRY_REWRITE((x + c1) * c2, x * c2 + c1 * c2);
    TVM_TRY_REWRITE((x * c1) * c2, x * (c1 * c2));
    TVM_TRY_REWRITE(min(x, y) * max(x, y), x * y);
    TVM_TRY_REWRITE(max(x, y) * min(x, y), x * y);

    // canonicalization
    TVM_TRY_RECURSIVE_REWRITE(x * (c1 * y), (x * y) * c1);
463
    TVM_TRY_RECURSIVE_REWRITE(c1 * x, x * c1);
464 465 466 467 468 469 470
    TVM_TRY_RECURSIVE_REWRITE_IF(
        (x - y) * c1, (y - x) * (0 - c1),
        c1.Eval()->value < 0);
  }
  return ret;
}

471
PrimExpr RewriteSimplifier::Impl::
472
VisitExpr_(const DivNode* op) {
473
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
474
  op = ret.as<DivNode>();
475
  PrimExpr const_res = TryConstFold<DivNode>(op->a, op->b);
476 477
  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
478
  PVar<PrimExpr> x, y, z, b1;
479
  // Pattern var match IntImm
480
  PVar<IntImm> c1, c2, c3;
481 482 483
  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;

484
  // x / 2.0 = x * 0.5
485
  if (const FloatImmNode* ptr = op->b.as<FloatImmNode>()) {
486 487
    CHECK(op->dtype.is_float());
    return op->a * make_const(op->b.dtype(), 1.0 / ptr->value);
488 489
  }

490
  // Vector rules
491
  if (op->dtype.lanes() != 1) {
492 493 494
    // NOTE: use div as the pattern also works for float.
    TVM_TRY_REWRITE(div(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(div(x, y), lanes));
495
    // ramp / bcast
496
    if ((div(ramp(b1, c1, lanes), broadcast(c2, lanes))).Match(ret)) {
497 498 499
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val % c2val == 0) {
500
        return ramp(div(b1, c2), div(c1, c2), lanes).Eval();
501 502 503
      }
      // If all possible indices in ramp are the same.
      if (CanProveGreaterEqual(b1.Eval(), 0)) {
504
        ModularSet bmod = analyzer_->modular_set(b1.Eval());
505 506 507
        int64_t ramp_min = bmod->base / c2val;
        int64_t ramp_max = (bmod->base + (lanes.Eval() - 1) * c1val) / c2val;
        if (bmod->coeff % c2val == 0 && ramp_min == ramp_max) {
508
          return broadcast(div(b1, c2), lanes).Eval();
509 510 511 512 513
        }
      }
    }
  }

514
  if (IsIndexType(op->dtype)) {
515 516 517 518
    // Be-aware of the division rules:
    // We adopt the default C division uses truncation instead of floordiv.
    // This means most rules need to check non-negativeness of the operands.

519 520 521 522
    // TryConstFold doesn't work for negative cases because it is also used by legacy
    // parts of tvm which still assume euclidean div. In this simplifier we assume that the division
    // is truncated, so perform const folding again.
    // NOTE: trunc div required
523
    if (truncdiv(c1, c2).Match(ret)) {
524 525
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
526
      return make_const(op->dtype, truncdiv(c1val, c2val));
527 528
    }

529 530
    // while it is always true for trunc div
    // restrict to common case(positive div)
531
    TVM_TRY_REWRITE_IF(truncdiv(truncdiv(x, c1), c2), truncdiv(x, c1 * c2),
532 533
                       c1.Eval()->value > 0 && c2.Eval()->value > 0);

534
    TVM_TRY_REWRITE_IF(truncdiv(truncdiv(x, c1) + c2, c3), truncdiv(x + c1 * c2, c1 * c3),
535 536 537 538 539
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value >= 0 &&
                       c3.Eval()->value > 0 &&
                       CanProveGreaterEqual(x.Eval(), 0));

540
    if (truncdiv(x * c1, c2).Match(ret)) {
541 542 543
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val > 0 && c2val > 0) {
544 545
        if (c1val % c2val == 0) return (x * truncdiv(c1, c2)).Eval();
        if (c2val % c1val == 0) return truncdiv(x, truncdiv(c2, c1)).Eval();
546 547 548
      }
    }

549 550 551
    TVM_TRY_REWRITE(truncdiv(x, x), OneWithTypeLike(x));
    TVM_TRY_REWRITE(truncdiv(x * c1, x), c1);
    TVM_TRY_REWRITE(truncdiv(c1 * x, x), c1);
552

553
    // Rules involving 2-operands.
554 555
    TVM_TRY_REWRITE_IF(truncdiv(x * c1 + y, c2),
                       x * truncdiv(c1, c2) + truncdiv(y, c2),
556 557 558 559 560 561
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

562 563
    TVM_TRY_REWRITE_IF(truncdiv(min(x * c1, y), c2),
                       min(x * truncdiv(c1, c2), truncdiv(y, c2)),
564 565 566 567 568 569
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

570 571
    TVM_TRY_REWRITE_IF(truncdiv(max(x * c1, y), c2),
                       max(x * truncdiv(c1, c2), truncdiv(y, c2)),
572 573 574 575 576 577
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

578 579
    TVM_TRY_REWRITE_IF(truncdiv(y + x * c1, c2),
                       truncdiv(y, c2) + x * truncdiv(c1, c2),
580 581 582 583 584 585
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

586 587
    TVM_TRY_REWRITE_IF(truncdiv(min(y, x * c1), c2),
                       min(truncdiv(y, c2), x * truncdiv(c1, c2)),
588 589 590 591 592 593
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

594 595
    TVM_TRY_REWRITE_IF(truncdiv(max(y, x * c1), c2),
                       max(truncdiv(y, c2), x * truncdiv(c1, c2)),
596 597 598 599 600 601 602
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

    // Rules involving 3-operands.
603 604
    TVM_TRY_REWRITE_IF(truncdiv(x * c1 + y + z, c2),
                       x * truncdiv(c1, c2) + truncdiv(y + z, c2),
605 606 607 608 609 610
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));

611 612
    TVM_TRY_REWRITE_IF(truncdiv(x * c1 - y + z, c2),
                       x * truncdiv(c1, c2) + truncdiv(z - y, c2),
613 614 615 616 617 618
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((z - y).Eval(), 0));

619 620
    TVM_TRY_REWRITE_IF(truncdiv(x * c1 + y - z, c2),
                       x * truncdiv(c1, c2) + truncdiv(y - z, c2),
621 622 623 624 625 626
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y - z).Eval(), 0));

627 628
    TVM_TRY_REWRITE_IF(truncdiv(y + x * c1 + z, c2),
                       x * truncdiv(c1, c2) + truncdiv(y + z, c2),
629 630 631 632 633 634
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));

635 636
    TVM_TRY_REWRITE_IF(truncdiv(x + c1, c2),
                       truncdiv(x, c2) + truncdiv(c1, c2),
637 638 639 640 641
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0));

642
    TVM_TRY_REWRITE_IF(truncdiv(x + y, x), truncdiv(y, x) + 1,
643 644
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));
645
    TVM_TRY_REWRITE_IF(truncdiv(y + x, x), truncdiv(y, x) + 1,
646 647 648
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

649 650
    TVM_TRY_REWRITE_IF(truncdiv((x + y) + z, x),
                       truncdiv(y + z, x) + 1,
651 652
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));
653 654
    TVM_TRY_REWRITE_IF(truncdiv((y + x) + z, x),
                       truncdiv(y + z, x) + 1,
655 656
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));
657 658
    TVM_TRY_REWRITE_IF(truncdiv(y + (z + x), x),
                       truncdiv(y + z, x) + 1,
659 660
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));
661 662
    TVM_TRY_REWRITE_IF(truncdiv(y + (x + z), x),
                       truncdiv(y + z, x) + 1,
663 664 665
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual((y + z).Eval(), 0));

666
    TVM_TRY_REWRITE_IF(truncdiv(x * y, y), x,
667 668
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));
669
    TVM_TRY_REWRITE_IF(truncdiv(y * x, y), x,
670 671 672
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0));

673
    TVM_TRY_REWRITE_IF(truncdiv(x * z + y, z), x + truncdiv(y, z),
674 675 676
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0) &&
                       CanProveGreaterEqual(z.Eval(), 0));
677
    TVM_TRY_REWRITE_IF(truncdiv(z * x + y, z), x + truncdiv(y, z),
678 679 680
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0) &&
                       CanProveGreaterEqual(z.Eval(), 0));
681
    TVM_TRY_REWRITE_IF(truncdiv(y + x * z, z), truncdiv(y, z) + x,
682 683 684
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0) &&
                       CanProveGreaterEqual(z.Eval(), 0));
685
    TVM_TRY_REWRITE_IF(truncdiv(y + z * x, z), truncdiv(y, z) + x,
686 687 688 689 690 691 692
                       CanProveGreaterEqual(x.Eval(), 0) &&
                       CanProveGreaterEqual(y.Eval(), 0) &&
                       CanProveGreaterEqual(z.Eval(), 0));
  }
  return ret;
}

693
PrimExpr RewriteSimplifier::Impl::
694
VisitExpr_(const ModNode* op) {
695
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
696
  op = ret.as<ModNode>();
697
  PrimExpr const_res = TryConstFold<ModNode>(op->a, op->b);
698 699 700
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
701
  PVar<PrimExpr> x, y, z, b1;
702
  // Pattern var match IntImm
703
  PVar<IntImm> c1, c2;
704 705 706 707
  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;

  // Vector rules
708
  if (op->dtype.lanes() != 1) {
709 710
    TVM_TRY_REWRITE(truncmod(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(truncmod(x, y), lanes));
711 712

    // ramp % bcast
713
    if (truncmod(ramp(b1, c1, lanes), broadcast(c2, lanes)).Match(ret)) {
714 715 716
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val % c2val == 0) {
717
        return broadcast(truncmod(b1, c2), lanes).Eval();
718 719 720
      }
      // If all possible indices in ramp are the same.
      if (CanProveGreaterEqual(b1.Eval(), 0)) {
721
        ModularSet bmod = analyzer_->modular_set(b1.Eval());
722 723 724 725
        int64_t ramp_min = bmod->base / c2val;
        int64_t ramp_max = (bmod->base + (lanes.Eval() - 1) * c1val) / c2val;
        if (bmod->coeff % c2val == 0) {
          if (ramp_min == ramp_max) {
726
            return ramp(truncmod(bmod->base, c2), c1, lanes).Eval();
727
          } else {
728 729
            return truncmod(ramp(truncmod(bmod->base, c2), c1, lanes),
                            broadcast(c2, lanes)).Eval();
730 731 732 733 734 735
          }
        }
      }
    }
  }

736
  if (IsIndexType(op->dtype)) {
737 738 739
    // Be-aware of the division rules:
    // We adopt the default C division uses truncation instead of floordiv.
    // This means most rules need to check non-negativeness of the operands.
740
    TVM_TRY_REWRITE_IF(truncmod(x * c1, c2), ZeroWithTypeLike(x),
741 742 743
                       c2.Eval()->value != 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

744
    TVM_TRY_REWRITE_IF(truncmod(x * c1 + y, c2), truncmod(y, c2),
745 746
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
747
                       CanProveGreaterEqual((x * c1).Eval(), 0) &&
748 749
                       CanProveGreaterEqual(y.Eval(), 0));

750
    TVM_TRY_REWRITE_IF(truncmod(x + c1, c2), truncmod(x, c2),
751
                       c2.Eval()->value > 0 &&
752
                       c1.Eval()->value >= 0 &&
753 754 755
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0));

756
    TVM_TRY_REWRITE_IF(truncmod(x + y * c1, c2), truncmod(x, c2),
757 758 759
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0 &&
                       CanProveGreaterEqual(x.Eval(), 0) &&
760
                       CanProveGreaterEqual((y * c1).Eval(), 0));
761

762 763
    // canonicalization: x % c == x % (-c) for truncated division
    // NOTE: trunc div required
764 765
    TVM_TRY_RECURSIVE_REWRITE_IF(
        truncmod(x, c1),
766
        truncmod(x, PConst<PrimExpr>(make_const(op->dtype, -c1.Eval()->value))),
767
        c1.Eval()->value < 0);
768

769
    // try modular analysis
770
    if (truncmod(x, c1).Match(ret)) {
771
      ModularSet mod = analyzer_->modular_set(x.Eval());
772 773
      int64_t c1val = c1.Eval()->value;
      if (mod->coeff % c1val == 0 &&
774
          c1val > 0 &&
775
          CanProveGreaterEqual(x.Eval(), 0)) {
776
        return truncmod(mod->base, c1).Eval();
777 778 779 780 781 782
      }
    }
  }
  return ret;
}

783
PrimExpr RewriteSimplifier::Impl::
784
VisitExpr_(const FloorDivNode* op) {
785
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
786
  op = ret.as<FloorDivNode>();
787
  PrimExpr const_res = TryConstFold<FloorDivNode>(op->a, op->b);
788 789
  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
790
  PVar<PrimExpr> x, y, z, b1;
791
  // Pattern var match IntImm
792
  PVar<IntImm> c1, c2, c3;
793 794 795 796
  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;

  // Vector rules
797
  if (op->dtype.lanes() != 1) {
798 799 800 801 802 803 804 805 806 807
    TVM_TRY_REWRITE(floordiv(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(floordiv(x, y), lanes));
    // ramp // bcast
    if (floordiv(ramp(b1, c1, lanes), broadcast(c2, lanes)).Match(ret)) {
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val % c2val == 0) {
        return ramp(floordiv(b1, c2), floordiv(c1, c2), lanes).Eval();
      }
      // If all possible indices in ramp are the same.
808
      ModularSet bmod = analyzer_->modular_set(b1.Eval());
809 810 811 812 813 814 815 816
      int64_t ramp_min = floordiv(bmod->base, c2val);
      int64_t ramp_max = floordiv(bmod->base + (lanes.Eval() - 1) * c1val, c2val);
      if (bmod->coeff % c2val == 0 && ramp_min == ramp_max) {
        return broadcast(floordiv(b1, c2), lanes).Eval();
      }
    }
  }

817
  if (IsIndexType(op->dtype)) {
818 819 820 821 822 823 824 825 826 827 828 829
    // Be-aware of the division rules: this is floor division.
    TVM_TRY_REWRITE_IF(floordiv(floordiv(x, c1), c2), floordiv(x, c1 * c2),
                       c1.Eval()->value > 0 && c2.Eval()->value > 0);

    TVM_TRY_REWRITE_IF(floordiv(floordiv(x, c1) + c2, c3), floordiv(x + c1 * c2, c1 * c3),
                       c1.Eval()->value > 0 && c3.Eval()->value > 0);

    if (floordiv(x * c1, c2).Match(ret)) {
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val > 0 && c2val > 0) {
        if (c1val % c2val == 0) return (x * floordiv(c1, c2)).Eval();
830
        if (c2val % c1val == 0) return floordiv(x, floordiv(c2, c1)).Eval();
831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926
      }
    }

    TVM_TRY_REWRITE(floordiv(x, x), OneWithTypeLike(x));
    TVM_TRY_REWRITE(floordiv(x * c1, x), c1);
    TVM_TRY_REWRITE(floordiv(c1 * x, x), c1);

    // Rules involving 2-operands.
    TVM_TRY_REWRITE_IF(floordiv(x * c1 + y, c2),
                       x * floordiv(c1, c2) + floordiv(y, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(min(x * c1, y), c2),
                       min(x * floordiv(c1, c2), floordiv(y, c2)),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(max(x * c1, y), c2),
                       max(x * floordiv(c1, c2), floordiv(y, c2)),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(y + x * c1, c2),
                       floordiv(y, c2) + x * floordiv(c1, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(min(y, x * c1), c2),
                       min(floordiv(y, c2), x * floordiv(c1, c2)),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(max(y, x * c1), c2),
                       max(floordiv(y, c2), x * floordiv(c1, c2)),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    // Rules involving 3-operands.
    TVM_TRY_REWRITE_IF(floordiv(x * c1 + y + z, c2),
                       x * floordiv(c1, c2) + floordiv(y + z, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(x * c1 - y + z, c2),
                       x * floordiv(c1, c2) + floordiv(z - y, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(x * c1 + y - z, c2),
                       x * floordiv(c1, c2) + floordiv(y - z, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(y + x * c1 + z, c2),
                       x * floordiv(c1, c2) + floordiv(y + z, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(x + c1, c2),
                       floordiv(x, c2) + floordiv(c1, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floordiv(x + y, x), floordiv(y, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));

    TVM_TRY_REWRITE_IF(floordiv(y + x, x), floordiv(y, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));

    TVM_TRY_REWRITE_IF(floordiv((x + y) + z, x), floordiv(y + z, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv((y + x) + z, x), floordiv(y + z, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(y + (z + x), x), floordiv(y + z, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(y + (x + z), x), floordiv(y + z, x) + 1,
                       CanProveGreaterEqual(x.Eval(), 0));

    TVM_TRY_REWRITE_IF(floordiv(x * y, y), x,
                       CanProveGreaterEqual(y.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(y * x, y), x,
                       CanProveGreaterEqual(y.Eval(), 0));

    TVM_TRY_REWRITE_IF(floordiv(x * z + y, z), x + floordiv(y, z),
                       CanProveGreaterEqual(z.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(z * x + y, z), x + floordiv(y, z),
                       CanProveGreaterEqual(z.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(y + x * z, z), floordiv(y, z) + x,
                       CanProveGreaterEqual(z.Eval(), 0));
    TVM_TRY_REWRITE_IF(floordiv(y + z * x, z), floordiv(y, z) + x,
                       CanProveGreaterEqual(z.Eval(), 0));
  }
  return ret;
}

927
PrimExpr RewriteSimplifier::Impl::
928
VisitExpr_(const FloorModNode* op) {
929
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
930
  op = ret.as<FloorModNode>();
931
  PrimExpr const_res = TryConstFold<FloorModNode>(op->a, op->b);
932 933 934
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
935
  PVar<PrimExpr> x, y, z, b1;
936
  // Pattern var match IntImm
937
  PVar<IntImm> c1, c2;
938 939 940 941
  // Pattern var for lanes in broadcast and ramp
  PVar<int> lanes;

  // Vector rules
942
  if (op->dtype.lanes() != 1) {
943 944 945 946 947 948 949 950 951 952 953
    TVM_TRY_REWRITE(floormod(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(floormod(x, y), lanes));

    // floormod(ramp, bcast)
    if (floormod(ramp(b1, c1, lanes), broadcast(c2, lanes)).Match(ret)) {
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c1val % c2val == 0) {
        return broadcast(floormod(b1, c2), lanes).Eval();
      }
      // If all possible indices in ramp are the same.
954
      ModularSet bmod = analyzer_->modular_set(b1.Eval());
955 956 957 958 959 960 961 962 963 964 965 966
      int64_t ramp_min = floordiv(bmod->base, c2val);
      int64_t ramp_max = floordiv(bmod->base + (lanes.Eval() - 1) * c1val, c2val);
      if (bmod->coeff % c2val == 0) {
        if (ramp_min == ramp_max) {
          return ramp(floormod(bmod->base, c2), c1, lanes).Eval();
        } else {
          return floormod(ramp(floormod(bmod->base, c2), c1, lanes), broadcast(c2, lanes)).Eval();
        }
      }
    }
  }

967
  if (IsIndexType(op->dtype)) {
968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986
    // Be-aware of the division rules: we use floordiv/floormod here
    TVM_TRY_REWRITE_IF(floormod(x * c1, c2), ZeroWithTypeLike(x),
                       c2.Eval()->value != 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floormod(x * c1 + y, c2), floormod(y, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floormod(x + c1, c2), floormod(x, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    TVM_TRY_REWRITE_IF(floormod(x + y * c1, c2), floormod(x, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value % c2.Eval()->value == 0);

    // try modular analysis
    if (floormod(x, c1).Match(ret)) {
987
      ModularSet mod = analyzer_->modular_set(x.Eval());
988 989 990
      int64_t c1val = c1.Eval()->value;
      if (mod->coeff % c1val == 0 && c1val > 0) {
        return floormod(mod->base, c1).Eval();
991 992 993 994 995 996
      }
    }
  }
  return ret;
}

997
PrimExpr RewriteSimplifier::Impl::
998
VisitExpr_(const MinNode* op) {
999
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1000
  op = ret.as<MinNode>();
1001
  PrimExpr const_res = TryConstFold<MinNode>(op->a, op->b);
1002 1003 1004
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1005
  PVar<PrimExpr> x, y, z, s1, s2;
1006
  // Pattern var match IntImm
1007
  PVar<IntImm> c1, c2;
1008 1009 1010
  PVar<int> lanes;

  // vector rule
1011
  if (op->dtype.lanes() != 1) {
1012 1013 1014 1015 1016
    TVM_TRY_REWRITE(min(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(min(x, y), lanes));
    TVM_TRY_REWRITE(min(min(x, broadcast(y, lanes)), broadcast(z, lanes)),
                    min(x, broadcast(min(y, z), lanes)));
  }
1017
  if (IsIndexType(op->dtype)) {
1018 1019 1020
    TVM_TRY_REWRITE(min(x, x), x);

    // constant int bound
1021 1022
    ConstIntBound a_bound = analyzer_->const_int_bound(op->a);
    ConstIntBound b_bound = analyzer_->const_int_bound(op->b);
1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053
    if (a_bound->max_value <= b_bound->min_value) {
      return op->a;
    }
    if (b_bound->max_value <= a_bound->min_value) {
      return op->b;
    }

    // constant comparison
    if (min(x + c1, x + c2).Match(ret)) {
      if (c1.Eval()->value < c2.Eval()->value) {
        return (x + c1).Eval();
      } else {
        return (x + c2).Eval();
      }
    }
    if (min(x + c1, x).Match(ret) ||
        min(x, x + c1).Match(ret)) {
      if (c1.Eval()->value < 0) {
        return (x + c1).Eval();
      } else {
        return x.Eval();
      }
    }
    if (min(c1 - x, c2 - x).Match(ret)) {
      if (c1.Eval()->value < c2.Eval()->value) {
        return (c1 - x).Eval();
      } else {
        return (c2 - x).Eval();
      }
    }

1054 1055 1056
    // DivMod rules
    // Divide up rounding: truc div
    // NOTE: trucdiv(x, y) >= floordiv(x, y)
1057
    TVM_TRY_REWRITE_IF(min(truncdiv(x + c1, c2) * c2, x), x,
1058 1059
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
1060
    TVM_TRY_REWRITE_IF(min(truncdiv(x + c1, c2) * c2, max(x, c2)), max(x, c2),
1061 1062 1063 1064
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value &&
                       CanProveGreaterEqual(x.Eval(), 0));

1065
    TVM_TRY_REWRITE_IF(min(x, truncdiv(x + c1, c2) * c2), x,
1066 1067
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
1068
    TVM_TRY_REWRITE_IF(min(max(x, c2), truncdiv(x + c1, c2) * c2), max(x, c2),
1069 1070 1071 1072
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value &&
                       CanProveGreaterEqual(x.Eval(), 0));

1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092
    // Divide up rounding: floor div
    TVM_TRY_REWRITE_IF(min(floordiv(x + c1, c2) * c2, x), x,
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
    TVM_TRY_REWRITE_IF(min(floordiv(x + c1, c2) * c2, max(x, c2)), max(x, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);

    TVM_TRY_REWRITE_IF(min(x, floordiv(x + c1, c2) * c2), x,
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
    TVM_TRY_REWRITE_IF(min(max(x, c2), floordiv(x + c1, c2) * c2), max(x, c2),
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);

    TVM_TRY_REWRITE_IF(min(x, floordiv(x, c2) * c2), floordiv(x, c2) * c2,
                       c2.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(min(floordiv(x, c2) * c2, x), floordiv(x, c2) * c2,
                       c2.Eval()->value > 0);

1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135
    TVM_TRY_REWRITE(min(max(x, y), min(x, y)), min(x, y));
    TVM_TRY_REWRITE(min(max(x, y), min(y, x)), min(x, y));
    TVM_TRY_REWRITE(min(min(x, y), max(x, y)), min(x, y));
    TVM_TRY_REWRITE(min(min(x, y), max(y, x)), min(x, y));

    TVM_TRY_REWRITE(min(max(x, y), x), x);
    TVM_TRY_REWRITE(min(max(x, y), y), y);
    TVM_TRY_REWRITE(min(min(x, y), x), min(x, y));
    TVM_TRY_REWRITE(min(min(x, y), y), min(x, y));

    TVM_TRY_REWRITE(min(x, max(x, y)), x);
    TVM_TRY_REWRITE(min(y, max(x, y)), y);
    TVM_TRY_REWRITE(min(x, min(x, y)), min(x, y));
    TVM_TRY_REWRITE(min(y, min(x, y)), min(x, y));

    TVM_TRY_REWRITE(min(min(min(x, y), z), y), min(min(x, y), z));
    TVM_TRY_REWRITE(min(min(min(min(x, y), z), s1), y), min(min(min(x, y), z), s1));
    TVM_TRY_REWRITE(min(min(min(min(min(x, y), z), s1), s2), y),
                    min(min(min(min(x, y), z), s1), s2));

    TVM_TRY_REWRITE(min(max(x, y), max(x, z)), max(min(y, z), x));
    TVM_TRY_REWRITE(min(max(x, y), max(z, x)), max(min(y, z), x));
    TVM_TRY_REWRITE(min(max(y, x), max(x, z)), max(min(y, z), x));
    TVM_TRY_REWRITE(min(max(y, x), max(z, x)), max(min(y, z), x));

    TVM_TRY_REWRITE(min(min(x, y), min(x, z)), min(min(y, z), x));
    TVM_TRY_REWRITE(min(min(x, y), min(z, x)), min(min(y, z), x));
    TVM_TRY_REWRITE(min(min(y, x), min(x, z)), min(min(y, z), x));
    TVM_TRY_REWRITE(min(min(y, x), min(z, x)), min(min(y, z), x));

    TVM_TRY_REWRITE(min(y + x, z + x), min(y, z) + x);
    TVM_TRY_REWRITE(min(y + x, x + z), min(y, z) + x);
    TVM_TRY_REWRITE(min(x + y, x + z), min(y, z) + x);
    TVM_TRY_REWRITE(min(x + y, z + x), min(y, z) + x);

    // sub distribution
    TVM_TRY_REWRITE(min(y - x, z - x), min(y, z) - x);
    TVM_TRY_REWRITE(min(x - y, x - z), x - max(y, z));

    // constant folding rule.
    TVM_TRY_REWRITE(min(min(x, c1), c2), min(x, min(c1, c2)));

    // scaling rule
1136
    if (min(truncdiv(x, c1), truncdiv(y, c1)).Match(ret)) {
1137
      if (c1.Eval()->value > 0) {
1138
        return truncdiv(min(x, y), c1).Eval();
1139
      } else {
1140
        return truncdiv(max(x, y), c1).Eval();
1141 1142
      }
    }
1143 1144 1145 1146 1147 1148 1149
    if (min(floordiv(x, c1), floordiv(y, c1)).Match(ret)) {
      if (c1.Eval()->value > 0) {
        return floordiv(min(x, y), c1).Eval();
      } else {
        return floordiv(max(x, y), c1).Eval();
      }
    }
1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170
    if (min(x * c1, y * c1).Match(ret)) {
      if (c1.Eval()->value > 0) {
        return (min(x, y) * c1).Eval();
      } else {
        return (max(x, y) * c1).Eval();
      }
    }
    if (min(x * c1, c2).Match(ret)) {
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c2val % c1val == 0) {
        if (c2val / c1val >= 0) {
          return (min(x, c2val / c1val) * c1val).Eval();
        } else {
          return (max(x, c2val / c1val) * c1val).Eval();
        }
      }
    }

    // canonicalization
    TVM_TRY_RECURSIVE_REWRITE(min(min(x, c1), y), min(min(x, y), c1));
1171 1172 1173
    TVM_TRY_RECURSIVE_REWRITE_IF(
        min(c1 - x, c2), c1 - max(x, c1 - c2),
        c2.Eval()->value != 0);
1174 1175 1176 1177 1178 1179 1180 1181
  }

  // condition rules.
  TVM_TRY_REWRITE(min(select(x, y, z), select(x, s1, s2)),
                  select(x, min(y, s1), min(z, s2)));
  return ret;
}

1182
PrimExpr RewriteSimplifier::Impl::
1183
VisitExpr_(const MaxNode* op) {
1184
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1185
  op = ret.as<MaxNode>();
1186
  PrimExpr const_res = TryConstFold<MaxNode>(op->a, op->b);
1187 1188 1189
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1190
  PVar<PrimExpr> x, y, z, s1, s2;
1191
  // Pattern var match IntImm
1192
  PVar<IntImm> c1, c2;
1193 1194 1195
  PVar<int> lanes;

  // vector rule
1196
  if (op->dtype.lanes() != 1) {
1197 1198 1199 1200 1201
    TVM_TRY_REWRITE(max(broadcast(x, lanes), broadcast(y, lanes)),
                    broadcast(max(x, y), lanes));
    TVM_TRY_REWRITE(max(max(x, broadcast(y, lanes)), broadcast(z, lanes)),
                    max(x, broadcast(max(y, z), lanes)));
  }
1202
  if (IsIndexType(op->dtype)) {
1203 1204 1205
    TVM_TRY_REWRITE(max(x, x), x);

    // constant int bound
1206 1207
    ConstIntBound a_bound = analyzer_->const_int_bound(op->a);
    ConstIntBound b_bound = analyzer_->const_int_bound(op->b);
1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238
    if (a_bound->min_value >= b_bound->max_value) {
      return op->a;
    }
    if (b_bound->min_value >= a_bound->max_value) {
      return op->b;
    }

    // constant comparison
    if (max(x + c1, x + c2).Match(ret)) {
      if (c1.Eval()->value > c2.Eval()->value) {
        return (x + c1).Eval();
      } else {
        return (x + c2).Eval();
      }
    }
    if (max(x + c1, x).Match(ret) ||
        max(x, x + c1).Match(ret)) {
      if (c1.Eval()->value > 0) {
        return (x + c1).Eval();
      } else {
        return x.Eval();
      }
    }
    if (max(c1 - x, c2 - x).Match(ret)) {
      if (c1.Eval()->value > c2.Eval()->value) {
        return (c1 - x).Eval();
      } else {
        return (c2 - x).Eval();
      }
    }

1239 1240 1241
    // DivMod rules
    // Divide up rounding: truc div
    // NOTE: trucdiv(x, y) >= floordiv(x, y)
1242 1243
    TVM_TRY_REWRITE_IF(max(truncdiv(x + c1, c2) * c2, x),
                       truncdiv(x + c1, c2) * c2,
1244 1245
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
1246 1247
    TVM_TRY_REWRITE_IF(max(x, truncdiv(x + c1, c2) * c2),
                       truncdiv(x + c1, c2) * c2,
1248 1249 1250
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);

1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263
    // Divide up rounding: floor div
    TVM_TRY_REWRITE_IF(max(floordiv(x + c1, c2) * c2, x), floordiv(x + c1, c2) * c2,
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);
    TVM_TRY_REWRITE_IF(max(x, floordiv(x + c1, c2) * c2), floordiv(x + c1, c2) * c2,
                       c2.Eval()->value > 0 &&
                       c1.Eval()->value + 1 == c2.Eval()->value);

    TVM_TRY_REWRITE_IF(max(floordiv(x, c2) * c2, x), x,
                       c2.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(max(x, floordiv(x, c2) * c2), x,
                       c2.Eval()->value > 0);

1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309
    TVM_TRY_REWRITE(max(min(x, y), max(x, y)), max(x, y));
    TVM_TRY_REWRITE(max(min(x, y), max(y, x)), max(x, y));
    TVM_TRY_REWRITE(max(max(x, y), min(x, y)), max(x, y));
    TVM_TRY_REWRITE(max(max(x, y), min(y, x)), max(x, y));

    TVM_TRY_REWRITE(max(min(x, y), x), x);
    TVM_TRY_REWRITE(max(min(x, y), y), y);
    TVM_TRY_REWRITE(max(max(x, y), x), max(x, y));
    TVM_TRY_REWRITE(max(max(x, y), y), max(x, y));

    TVM_TRY_REWRITE(max(x, min(x, y)), x);
    TVM_TRY_REWRITE(max(y, min(x, y)), y);
    TVM_TRY_REWRITE(max(x, max(x, y)), max(x, y));
    TVM_TRY_REWRITE(max(y, max(x, y)), max(x, y));

    TVM_TRY_REWRITE(max(max(max(x, y), z), y), max(max(x, y), z));
    TVM_TRY_REWRITE(max(max(max(max(x, y), z), s1), y), max(max(max(x, y), z), s1));
    TVM_TRY_REWRITE(max(max(max(max(max(x, y), z), s1), s2), y),
                    max(max(max(max(x, y), z), s1), s2));

    // max/max cancelation
    TVM_TRY_REWRITE(max(max(x, y), max(x, z)), max(max(y, z), x));
    TVM_TRY_REWRITE(max(max(x, y), max(z, x)), max(max(y, z), x));
    TVM_TRY_REWRITE(max(max(y, x), max(x, z)), max(max(y, z), x));
    TVM_TRY_REWRITE(max(max(y, x), max(z, x)), max(max(y, z), x));

    // max/min distribution
    TVM_TRY_REWRITE(max(min(x, y), min(x, z)), min(max(y, z), x));
    TVM_TRY_REWRITE(max(min(x, y), min(z, x)), min(max(y, z), x));
    TVM_TRY_REWRITE(max(min(y, x), min(x, z)), min(max(y, z), x));
    TVM_TRY_REWRITE(max(min(y, x), min(z, x)), min(max(y, z), x));

    // add distribution
    TVM_TRY_REWRITE(max(y + x, z + x), max(y, z) + x);
    TVM_TRY_REWRITE(max(y + x, x + z), max(y, z) + x);
    TVM_TRY_REWRITE(max(x + y, x + z), max(y, z) + x);
    TVM_TRY_REWRITE(max(x + y, z + x), max(y, z) + x);

    // sub distribution
    TVM_TRY_REWRITE(max(y - x, z - x), max(y, z) - x);
    TVM_TRY_REWRITE(max(x - y, x - z), x - min(y, z));

    // constant folding rule.
    TVM_TRY_REWRITE(max(max(x, c1), c2), max(x, max(c1, c2)));

    // scaling rule
1310
    if (max(truncdiv(x, c1), truncdiv(y, c1)).Match(ret)) {
1311
      if (c1.Eval()->value > 0) {
1312
        return truncdiv(max(x, y), c1).Eval();
1313
      } else {
1314
        return truncdiv(min(x, y), c1).Eval();
1315 1316
      }
    }
1317 1318 1319 1320 1321 1322 1323
    if (max(floordiv(x, c1), floordiv(y, c1)).Match(ret)) {
      if (c1.Eval()->value > 0) {
        return floordiv(max(x, y), c1).Eval();
      } else {
        return floordiv(min(x, y), c1).Eval();
      }
    }
1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344
    if (max(x * c1, y * c1).Match(ret)) {
      if (c1.Eval()->value > 0) {
        return (max(x, y) * c1).Eval();
      } else {
        return (min(x, y) * c1).Eval();
      }
    }
    if (max(x * c1, c2).Match(ret)) {
      int64_t c1val = c1.Eval()->value;
      int64_t c2val = c2.Eval()->value;
      if (c2val % c1val == 0) {
        if (c2val / c1val >= 0) {
          return (max(x, c2val / c1val) * c1val).Eval();
        } else {
          return (min(x, c2val / c1val) * c1val).Eval();
        }
      }
    }

    // canonicalization
    TVM_TRY_RECURSIVE_REWRITE(max(max(x, c1), y), max(max(x, y), c1));
1345 1346
    TVM_TRY_RECURSIVE_REWRITE_IF(
        max(c1 - x, c2), c1 - min(x, c1 - c2), c2.Eval()->value != 0);
1347 1348 1349 1350 1351 1352 1353 1354
  }

  // condition rules.
  TVM_TRY_REWRITE(max(select(x, y, z), select(x, s1, s2)),
                  select(x, max(y, s1), max(z, s2)));
  return ret;
}

1355
PrimExpr RewriteSimplifier::Impl::
1356
VisitExpr_(const EQNode* op) {
1357
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1358
  op = ret.as<EQNode>();
1359
  PrimExpr const_res = TryConstFold<EQNode>(op->a, op->b);
1360 1361 1362
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1363
  PVar<PrimExpr> x, y;
1364
  // Pattern var match IntImm
1365
  PVar<IntImm> c1;
1366 1367 1368
  PVar<int> lanes;

  // vector rule
1369
  if (op->dtype.lanes() != 1) {
1370 1371 1372 1373
    TVM_TRY_REWRITE(broadcast(x, lanes) == broadcast(y, lanes),
                    broadcast(x == y, lanes));
  }

1374
  if (IsIndexType(op->a.dtype())) {
1375
    CompareResult result = TryCompare(op->a - op->b, 0);
1376
    if (result == kEQ) {
1377
      return make_const(op->dtype, true);
1378
    } else if (result == kNE || result == kGT || result == kLT) {
1379
      return make_const(op->dtype, false);
1380 1381 1382 1383 1384 1385 1386 1387 1388
    }
    TVM_TRY_REWRITE(x - c1 == 0, x == c1);
    TVM_TRY_REWRITE(c1 - x == 0, x == c1);
    TVM_TRY_REWRITE(x + c1 == 0, x == 0 - c1);
    TVM_TRY_REWRITE(x * y == 0, x == 0 || y == 0);
  }
  return ret;
}

1389
PrimExpr RewriteSimplifier::Impl::
1390 1391
VisitExpr_(const NENode* op) {
  return this->VisitExpr(NotNode::make(op->a == op->b));
1392 1393
}

1394
PrimExpr RewriteSimplifier::Impl::
1395 1396
VisitExpr_(const LENode* op) {
  return this->VisitExpr(NotNode::make(op->b < op->a));
1397 1398
}

1399
PrimExpr RewriteSimplifier::Impl::
1400
VisitExpr_(const GTNode* op) {
1401
  return this->VisitExpr(op->b < op->a);
1402 1403
}

1404
PrimExpr RewriteSimplifier::Impl::
1405 1406
VisitExpr_(const GENode* op) {
  return this->VisitExpr(NotNode::make(op->a < op->b));
1407 1408
}

1409
PrimExpr RewriteSimplifier::Impl::
1410
VisitExpr_(const LTNode* op) {
1411
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1412
  op = ret.as<LTNode>();
1413
  PrimExpr const_res = TryConstFold<LTNode>(op->a, op->b);
1414 1415 1416
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1417
  PVar<PrimExpr> x, y, z, s1, s2;
1418
  // Pattern var match IntImm
1419
  PVar<IntImm> c1, c2;
1420 1421 1422
  PVar<int> lanes;

  // vector rule
1423
  if (op->dtype.lanes() != 1) {
1424 1425 1426 1427 1428 1429
    TVM_TRY_REWRITE(broadcast(x, lanes) < broadcast(y, lanes),
                    broadcast(x < y, lanes));
    TVM_TRY_REWRITE(ramp(x, s1, lanes) < ramp(y, s1, lanes),
                    broadcast(x < y, lanes));
  }

1430
  if (IsIndexType(op->a.dtype())) {
1431 1432
    CompareResult result = TryCompare(op->a - op->b, 0);
    if (result == kLT) {
1433
      return make_const(op->dtype, true);
1434
    }
1435
    if (result == kEQ || result == kGT || result == kGE) {
1436
      return make_const(op->dtype, false);
1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456
    }

    TVM_TRY_REWRITE(x + y < x + z, y < z);
    TVM_TRY_REWRITE(x + y < z + x, y < z);
    TVM_TRY_REWRITE(y + x < x + z, y < z);
    TVM_TRY_REWRITE(y + x < z + x, y < z);
    TVM_TRY_REWRITE(y - x < z - x, y < z);
    TVM_TRY_REWRITE(x - y < x - z, z < y);

    TVM_TRY_REWRITE(x < x + z, 0 < z);
    TVM_TRY_REWRITE(x < z + x, 0 < z);
    TVM_TRY_REWRITE(x < x - z, z < 0);
    TVM_TRY_REWRITE(c1 < x + c2, c1 - c2 < x);
    TVM_TRY_REWRITE(c1 < c2 - x, x < c2 - c1);

    TVM_TRY_REWRITE_IF(x * c1 < y * c1, x < y,
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(x * c1 < y * c1, y < x,
                       c1.Eval()->value < 0);

1457 1458
    // constant cancelation: only need to make use of one mod
    // truc div
1459
    TVM_TRY_REWRITE_IF(x * c2 < c1, x < truncdiv(c1 - 1, c2) + 1,
1460 1461
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value > 0);
1462
    // NOTE: trunc div required
1463
    TVM_TRY_REWRITE_IF(x * c2 < c1, x < truncdiv(c1, c2),
1464 1465 1466
                       c1.Eval()->value <= 0 &&
                       c2.Eval()->value > 0);
    // NOTE: trunc div required (euclidean is ok too, floored is not)
1467
    TVM_TRY_REWRITE_IF(x * c2 < c1, truncdiv(c1 - 1, c2) - 1 < x,
1468
                       c1.Eval()->value > 0 &&
1469 1470
                       c2.Eval()->value < 0);
    // NOTE: trunc div required (floored is ok too, euclidean is not)
1471
    TVM_TRY_REWRITE_IF(x * c2 < c1, truncdiv(c1, c2) < x,
1472 1473 1474
                       c1.Eval()->value <= 0 &&
                       c2.Eval()->value < 0);
    // NOTE: trunc div required
1475
    TVM_TRY_REWRITE_IF(c1 < x * c2, truncdiv(c1 + 1, c2) - 1 < x,
1476
                       c1.Eval()->value < 0 &&
1477
                       c2.Eval()->value > 0);
1478
    TVM_TRY_REWRITE_IF(c1 < x * c2, truncdiv(c1, c2) < x,
1479 1480
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0);
1481
    // NOTE: trunc div required (floored is ok too, euclidean is not)
1482
    TVM_TRY_REWRITE_IF(c1 < x * c2, x < truncdiv(c1 + 1, c2) + 1,
1483 1484 1485
                       c1.Eval()->value < 0 &&
                       c2.Eval()->value < 0);
    // NOTE: trunc div required (euclidean is ok too, floored is not)
1486
    TVM_TRY_REWRITE_IF(c1 < x * c2, x < truncdiv(c1, c2),
1487 1488
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value < 0);
1489 1490
    // DivMod rules
    // trucdiv
1491
    TVM_TRY_REWRITE_IF(truncdiv(x, c1) < c2, x < c1 * c2,
1492 1493 1494
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value > 0);
    // NOTE: trunc div required
1495
    TVM_TRY_REWRITE_IF(truncdiv(x, c1) < c2, x < c1 * (c2 - 1) + 1,
1496 1497 1498
                       c1.Eval()->value > 0 &&
                       c2.Eval()->value <= 0);

1499
    TVM_TRY_REWRITE_IF(c1 < truncdiv(x, c2), (c1 + 1) * c2 - 1 < x,
1500 1501
                       c1.Eval()->value >= 0 &&
                       c2.Eval()->value > 0);
1502
    // NOTE: trunc div required
1503
    TVM_TRY_REWRITE_IF(c1 < truncdiv(x, c2), c1 * c2 < x,
1504 1505
                       c1.Eval()->value < 0 &&
                       c2.Eval()->value > 0);
1506 1507

    // invariance for any div mod: x - (x / c1) * c1 == x % c1
1508
    TVM_TRY_REWRITE_IF(truncdiv(x, c1) * c1 < x, 0 < truncmod(x, c1),
1509
                       c1.Eval()->value > 0);
1510
    TVM_TRY_REWRITE_IF(truncdiv(x, c1) * c1 < x + y, 0 < truncmod(x, c1) + y,
1511
                       c1.Eval()->value > 0);
1512
    TVM_TRY_REWRITE_IF(truncdiv(x, c1) * c1 < x - y, y < truncmod(x, c1),
1513 1514
                       c1.Eval()->value > 0);

1515 1516
    TVM_TRY_REWRITE_IF(truncdiv(x + c2, c1) * c1 < x,
                       c2 < truncmod(x + c2, c1),
1517
                       c1.Eval()->value > 0);
1518 1519
    TVM_TRY_REWRITE_IF(truncdiv(x + c2, c1) * c1 < x + y,
                       c2 < truncmod(x + c2, c1) + y,
1520
                       c1.Eval()->value > 0);
1521 1522
    TVM_TRY_REWRITE_IF(truncdiv(x + c2, c1) * c1 < x - y,
                       y < truncmod(x + c2, c1) + (0 - c2),
1523 1524
                       c1.Eval()->value > 0);

1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546
    // floordiv
    TVM_TRY_REWRITE_IF(floordiv(x, c1) < c2, x < c1 * c2,
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(c1 < floordiv(x, c2), (c1 + 1) * c2 - 1 < x,
                       c2.Eval()->value > 0);

    TVM_TRY_REWRITE_IF(floordiv(x, c1) * c1 < x, 0 < floormod(x, c1),
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x, c1) * c1 < x + y, 0 < floormod(x, c1) + y,
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x, c1) * c1 < x - y, y < floormod(x, c1),
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x + c2, c1) * c1 < x,
                       c2 < floormod(x + c2, c1),
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x + c2, c1) * c1 < x + y,
                       c2 < floormod(x + c2, c1) + y,
                       c1.Eval()->value > 0);
    TVM_TRY_REWRITE_IF(floordiv(x + c2, c1) * c1 < x - y,
                       y < floormod(x + c2, c1) + (0 - c2),
                       c1.Eval()->value > 0);

1547 1548 1549 1550 1551 1552
    // canonicalization rule
    TVM_TRY_RECURSIVE_REWRITE(min(x, y) < z, x < z || y < z);
    TVM_TRY_RECURSIVE_REWRITE(max(x, y) < z, x < z && y < z);
    TVM_TRY_RECURSIVE_REWRITE(z < min(x, y), z < x && z < y);
    TVM_TRY_RECURSIVE_REWRITE(z < max(x, y), z < x || z < y);

1553 1554 1555 1556 1557
    TVM_TRY_RECURSIVE_REWRITE(x < c1 - y, x + y < c1);
    TVM_TRY_RECURSIVE_REWRITE(x < c1 + y, x - y < c1);
    TVM_TRY_RECURSIVE_REWRITE(c1 - y < x, c1 < x + y);
    TVM_TRY_RECURSIVE_REWRITE(c1 + y < x, c1 < x - y);

1558 1559
    TVM_TRY_RECURSIVE_REWRITE(x + c1 < c2, x < c2 - c1);
    TVM_TRY_RECURSIVE_REWRITE(x - c1 < c2, x < c2 + c1);
1560 1561 1562 1563 1564
    TVM_TRY_REWRITE(x - c1 < 0, x < c1);
  }
  return ret;
}

1565
PrimExpr RewriteSimplifier::Impl::
1566
VisitExpr_(const NotNode* op) {
1567
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1568
  op = ret.as<NotNode>();
1569
  PrimExpr const_res = TryConstFold<NotNode>(op->a);
1570 1571
  if (const_res.defined()) return const_res;
  // Pattern var to match any expression
1572
  PVar<PrimExpr> x, y;
1573
  PVar<int> lanes;
1574
  if (op->dtype.lanes() != 1) {
1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589
    TVM_TRY_REWRITE(!broadcast(x, lanes), broadcast(!x, lanes));
  }

  TVM_TRY_REWRITE(!(!x), x);
  TVM_TRY_REWRITE(!(x <= y), y < x);
  TVM_TRY_REWRITE(!(x >= y), x < y);
  TVM_TRY_REWRITE(!(x < y), y <= x);
  TVM_TRY_REWRITE(!(x > y), x <= y);
  TVM_TRY_REWRITE(!(x == y), x != y);
  TVM_TRY_REWRITE(!(x != y), x == y);
  TVM_TRY_RECURSIVE_REWRITE(!(x || y), (!x) && (!y));
  TVM_TRY_RECURSIVE_REWRITE(!(x && y), (!x) || (!y));
  return ret;
}

1590
PrimExpr RewriteSimplifier::Impl::
1591
VisitExpr_(const AndNode* op) {
1592
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1593
  op = ret.as<AndNode>();
1594
  PrimExpr const_res = TryConstFold<AndNode>(op->a, op->b);
1595 1596 1597
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1598
  PVar<PrimExpr> x, y;
1599
  // Pattern var match IntImm
1600
  PVar<IntImm> c1, c2;
1601 1602
  PVar<int> lanes;

1603
  if (op->dtype.lanes() != 1) {
1604 1605 1606 1607
    TVM_TRY_REWRITE(broadcast(x, lanes) && broadcast(y, lanes),
                    broadcast(x && y, lanes));
  }

1608
  auto cfalse = PConst<PrimExpr>(make_const(op->dtype, false));
1609 1610 1611 1612
  TVM_TRY_REWRITE(x == y && x != y, cfalse);
  TVM_TRY_REWRITE(x != y && x == y, cfalse);
  TVM_TRY_REWRITE(x && !x, cfalse);
  TVM_TRY_REWRITE(x <= y && y < x, cfalse);
1613
  TVM_TRY_REWRITE(y < x && x <= y, cfalse);
1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638

  TVM_TRY_REWRITE_IF(x < c1 && c2 < x, cfalse,
                     c2.Eval()->value + 1 >= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 < x && x < c1, cfalse,
                     c2.Eval()->value + 1 >= c1.Eval()->value);

  TVM_TRY_REWRITE_IF(x < c1 && c2 <= x, cfalse,
                     c2.Eval()->value >= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 <= x && x < c1, cfalse,
                     c2.Eval()->value >= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(x <= c1 && c2 < x, cfalse,
                     c2.Eval()->value >= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 < x && x <= c1, cfalse,
                     c2.Eval()->value >= c1.Eval()->value);

  TVM_TRY_REWRITE_IF(x <= c1 && c2 <= x, cfalse,
                     c2.Eval()->value > c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 <= x && x <= c1, cfalse,
                     c2.Eval()->value > c1.Eval()->value);

  TVM_TRY_REWRITE(x == c1 && x != c2, x == c1 && c1 != c2);
  TVM_TRY_REWRITE(x != c2 && x == c1, x == c1 && c1 != c2);
  return ret;
}

1639
PrimExpr RewriteSimplifier::Impl::
1640
VisitExpr_(const OrNode* op) {
1641
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1642
  op = ret.as<OrNode>();
1643
  PrimExpr const_res = TryConstFold<OrNode>(op->a, op->b);
1644 1645 1646
  if (const_res.defined()) return const_res;

  // Pattern var to match any expression
1647
  PVar<PrimExpr> x, y;
1648
  // Pattern var match IntImm
1649
  PVar<IntImm> c1, c2;
1650 1651
  PVar<int> lanes;

1652
  if (op->dtype.lanes() != 1) {
1653 1654 1655 1656
    TVM_TRY_REWRITE(broadcast(x, lanes) || broadcast(y, lanes),
                    broadcast(x || y, lanes));
  }

1657
  auto ctrue = PConst<PrimExpr>(make_const(op->dtype, true));
1658 1659 1660 1661 1662

  TVM_TRY_REWRITE(x == y || x != y, ctrue);
  TVM_TRY_REWRITE(x != y || x == y, ctrue);
  TVM_TRY_REWRITE(x || !x, ctrue);
  TVM_TRY_REWRITE(x <= y || y < x, ctrue);
1663
  TVM_TRY_REWRITE(y < x || x <= y, ctrue);
1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688

  TVM_TRY_REWRITE_IF(x < c1 || c2 < x, ctrue,
                     c2.Eval()->value < c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 < x || x < c1, ctrue,
                     c2.Eval()->value < c1.Eval()->value);

  TVM_TRY_REWRITE_IF(x <= c1 || c2 < x, ctrue,
                     c2.Eval()->value <= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 < x || x <= c1, ctrue,
                     c2.Eval()->value <= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(x < c1 || c2 <= x, ctrue,
                     c2.Eval()->value <= c1.Eval()->value);
  TVM_TRY_REWRITE_IF(c2 <= x || x < c1, ctrue,
                     c2.Eval()->value <= c1.Eval()->value);

  TVM_TRY_REWRITE_IF(x <= c1 || c2 <= x, ctrue,
                     c2.Eval()->value <= c1.Eval()->value + 1);
  TVM_TRY_REWRITE_IF(c2 <= x || x <= c1, ctrue,
                     c2.Eval()->value <= c1.Eval()->value + 1);

  TVM_TRY_REWRITE(x != c1 || x == c2, x != c1 || c1 == c2);
  TVM_TRY_REWRITE(x == c2 || x != c1, x != c1 || c1 == c2);
  return ret;
}

1689
PrimExpr RewriteSimplifier::Impl::
1690
VisitExpr_(const SelectNode* op) {
1691
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1692
  op = ret.as<SelectNode>();
1693
  if (op == nullptr) return ret;
1694
  // Pattern var to match any expression
1695
  PVar<PrimExpr> x, y;
1696 1697 1698
  TVM_TRY_REWRITE(select(x, y, y), y);
  return ret;
}
1699

1700
PrimExpr RewriteSimplifier::Impl::
1701
VisitExpr_(const CallNode* op) {
1702
  // add condition context to if_then_else
1703
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1704
  op = ret.as<CallNode>();
1705
  if (op == nullptr) return ret;
1706
  if (op->is_intrinsic(CallNode::likely) && is_const(op->args[0])) {
1707
    return op->args[0];
1708 1709
  } else if (op->is_intrinsic(CallNode::shift_right)) {
    if (op->args[0].as<IntImmNode>() && op->args[1].as<IntImmNode>()) {
1710 1711 1712
      // the operator overload will eagerly constant fold.
      return op->args[0] >> op->args[1];
    }
1713 1714
  } else if (op->is_intrinsic(CallNode::bitwise_and)) {
    if (op->args[0].as<IntImmNode>() && op->args[1].as<IntImmNode>()) {
1715 1716 1717
      // the operator overload will eagerly constant fold.
      return op->args[0] & op->args[1];
    }
1718
  }
1719
  if (op->is_intrinsic(CallNode::likely)) {
1720 1721 1722
    for (const auto& constraint : literal_constraints_) {
      // Cases such as for (i, 0, bound) {if (likely(iter_var < bound)) { .. } }
      if (Equal(constraint, op->args[0])) {
1723
        return make_const(op->dtype, true);
1724 1725 1726
      }
    }
  }
1727 1728 1729
  return ret;
}

1730
PrimExpr RewriteSimplifier::Impl::
1731
VisitExpr_(const VarNode* op) {
1732 1733 1734 1735 1736
  Var var = GetRef<Var>(op);
  auto it = var_map_.find(var);
  if (it != var_map_.end()) {
    return it->second;
  }
1737
  return GetRef<PrimExpr>(op);
1738 1739
}

1740
PrimExpr RewriteSimplifier::Impl::
1741
VisitExpr_(const CastNode* op) {
1742
  PrimExpr ret = IRMutatorWithAnalyzer::VisitExpr_(op);
1743
  op = ret.as<CastNode>();
1744
  return cast(op->dtype, op->value);
1745 1746
}

1747
PrimExpr RewriteSimplifier::Impl::
1748
VisitExpr_(const LetNode* op) {
1749
  PrimExpr value = this->VisitExpr(op->value);
1750
  if (!tir::HasSideEffect(value)) {
1751 1752 1753
    // it is fine to discard the let binding
    // because the value will always be inlined in the simplifier.
    analyzer_->Bind(op->var, value);
1754
    return this->VisitExpr(op->body);
1755
  }
1756
  PrimExpr body = this->VisitExpr(op->body);
1757 1758
  if (value.same_as(op->value) &&
      body.same_as(op->body)) {
1759
    return GetRef<PrimExpr>(op);
1760
  } else {
1761
    return LetNode::make(op->var, value, body);
1762 1763 1764
  }
}

1765
PrimExpr RewriteSimplifier::operator()(const PrimExpr& expr) {
1766
  // Run simplification in post order
1767
  PrimExpr res = expr;
1768 1769
  int max_iter = 2;
  for (int i = 0; i < max_iter; ++i) {
1770
    PrimExpr new_expr = impl_->operator()(res);
1771 1772 1773 1774
    if (new_expr.same_as(res)) return res;
    res = new_expr;
  }
  return res;
1775 1776 1777
}

void RewriteSimplifier::Update(const Var& var,
1778
                               const PrimExpr& info,
1779 1780 1781 1782
                               bool override) {
  impl_->Update(var, info, override);
}

1783
std::function<void()> RewriteSimplifier::EnterConstraint(const PrimExpr& constraint) {
1784 1785 1786
  return impl_->EnterConstraint(constraint);
}

1787 1788 1789 1790 1791 1792 1793 1794 1795 1796
RewriteSimplifier::RewriteSimplifier(Analyzer* parent)
    : impl_(new Impl(parent)) {
}

RewriteSimplifier::~RewriteSimplifier() {
  delete impl_;
}

}  // namespace arith
}  // namespace tvm