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// Copyright (c) 2018 Google LLC.
// 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
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// See the License for the specific language governing permissions and
// limitations under the License.
#include "source/opt/scalar_analysis.h"
#include <functional>
#include <map>
#include <memory>
#include <set>
#include <unordered_set>
#include <utility>
#include <vector>
// Simplifies scalar analysis DAGs.
// 1. Given a node passed to SimplifyExpression we first simplify the graph by
// calling SimplifyPolynomial. This groups like nodes following basic arithmetic
// rules, so multiple adds of the same load instruction could be grouped into a
// single multiply of that instruction. SimplifyPolynomial will traverse the DAG
// and build up an accumulator buffer for each class of instruction it finds.
// For example take the loop:
// for (i=0, i<N; i++) { i+B+23+4+B+C; }
// In this example the expression "i+B+23+4+B+C" has four classes of
// instruction, induction variable i, the two value unknowns B and C, and the
// constants. The accumulator buffer is then used to rebuild the graph using
// the accumulation of each type. This example would then be folded into
// i+2*B+C+27.
// This new graph contains a single add node (or if only one type found then
// just that node) with each of the like terms (or multiplication node) as a
// child.
// 2. FoldRecurrentAddExpressions is then called on this new DAG. This will take
// RecurrentAddExpressions which are with respect to the same loop and fold them
// into a single new RecurrentAddExpression with respect to that same loop. An
// expression can have multiple RecurrentAddExpression's with respect to
// different loops in the case of nested loops. These expressions cannot be
// folded further. For example:
// for (i=0; i<N;i++) for(j=0,k=1; j<N;++j,++k)
// The 'j' and 'k' are RecurrentAddExpression with respect to the second loop
// and 'i' to the first. If 'j' and 'k' are used in an expression together then
// they will be folded into a new RecurrentAddExpression with respect to the
// second loop in that expression.
// 3. If the DAG now only contains a single RecurrentAddExpression we can now
// perform a final optimization SimplifyRecurrentAddExpression. This will
// transform the entire DAG into a RecurrentAddExpression. Additions to the
// RecurrentAddExpression are added to the offset field and multiplications to
// the coefficient.
namespace spvtools {
namespace opt {
// Implementation of the functions which are used to simplify the graph. Graphs
// of unknowns, multiplies, additions, and constants can be turned into a linear
// add node with each term as a child. For instance a large graph built from, X
// + X*2 + Y - Y*3 + 4 - 1, would become a single add expression with the
// children X*3, -Y*2, and the constant 3. Graphs containing a recurrent
// expression will be simplified to represent the entire graph around a single
// recurrent expression. So for an induction variable (i=0, i++) if you add 1 to
// i in an expression we can rewrite the graph of that expression to be a single
// recurrent expression of (i=1,i++).
class SENodeSimplifyImpl {
SENodeSimplifyImpl(ScalarEvolutionAnalysis* analysis,
SENode* node_to_simplify)
: analysis_(*analysis),
constant_accumulator_(0) {}
// Return the result of the simplification.
SENode* Simplify();
// Recursively descend through the graph to build up the accumulator objects
// which are used to flatten the graph. |child| is the node currently being
// traversed and the |negation| flag is used to signify that this operation
// was preceded by a unary negative operation and as such the result should be
// negated.
void GatherAccumulatorsFromChildNodes(SENode* new_node, SENode* child,
bool negation);
// Given a |multiply| node add to the accumulators for the term type within
// the |multiply| expression. Will return true if the accumulators could be
// calculated successfully. If the |multiply| is in any form other than
// unknown*constant then we return false. |negation| signifies that the
// operation was preceded by a unary negative.
bool AccumulatorsFromMultiply(SENode* multiply, bool negation);
SERecurrentNode* UpdateCoefficient(SERecurrentNode* recurrent,
int64_t coefficient_update) const;
// If the graph contains a recurrent expression, ie, an expression with the
// loop iterations as a term in the expression, then the whole expression
// can be rewritten to be a recurrent expression.
SENode* SimplifyRecurrentAddExpression(SERecurrentNode* node);
// Simplify the whole graph by linking like terms together in a single flat
// add node. So X*2 + Y -Y + 3 +6 would become X*2 + 9. Where X and Y are a
// ValueUnknown node (i.e, a load) or a recurrent expression.
SENode* SimplifyPolynomial();
// Each recurrent expression is an expression with respect to a specific loop.
// If we have two different recurrent terms with respect to the same loop in a
// single expression then we can fold those terms into a single new term.
// For instance:
// induction i = 0, i++
// temp = i*10
// array[i+temp]
// We can fold the i + temp into a single expression. Rec(0,1) + Rec(0,10) can
// become Rec(0,11).
SENode* FoldRecurrentAddExpressions(SENode*);
// We can eliminate recurrent expressions which have a coefficient of zero by
// replacing them with their offset value. We are able to do this because a
// recurrent expression represents the equation coefficient*iterations +
// offset.
SENode* EliminateZeroCoefficientRecurrents(SENode* node);
// A reference the analysis which requested the simplification.
ScalarEvolutionAnalysis& analysis_;
// The node being simplified.
SENode* node_;
// An accumulator of the net result of all the constant operations performed
// in a graph.
int64_t constant_accumulator_;
// An accumulator for each of the non constant terms in the graph.
std::map<SENode*, int64_t> accumulators_;
// From a |multiply| build up the accumulator objects.
bool SENodeSimplifyImpl::AccumulatorsFromMultiply(SENode* multiply,
bool negation) {
if (multiply->GetChildren().size() != 2 ||
multiply->GetType() != SENode::Multiply)
return false;
SENode* operand_1 = multiply->GetChild(0);
SENode* operand_2 = multiply->GetChild(1);
SENode* value_unknown = nullptr;
SENode* constant = nullptr;
// Work out which operand is the unknown value.
if (operand_1->GetType() == SENode::ValueUnknown ||
operand_1->GetType() == SENode::RecurrentAddExpr)
value_unknown = operand_1;
else if (operand_2->GetType() == SENode::ValueUnknown ||
operand_2->GetType() == SENode::RecurrentAddExpr)
value_unknown = operand_2;
// Work out which operand is the constant coefficient.
if (operand_1->GetType() == SENode::Constant)
constant = operand_1;
else if (operand_2->GetType() == SENode::Constant)
constant = operand_2;
// If the expression is not a variable multiplied by a constant coefficient,
// exit out.
if (!(value_unknown && constant)) {
return false;
int64_t sign = negation ? -1 : 1;
auto iterator = accumulators_.find(value_unknown);
int64_t new_value = constant->AsSEConstantNode()->FoldToSingleValue() * sign;
// Add the result of the multiplication to the accumulators.
if (iterator != accumulators_.end()) {
(*iterator).second += new_value;
} else {
accumulators_.insert({value_unknown, new_value});
return true;
SENode* SENodeSimplifyImpl::Simplify() {
// We only handle graphs with an addition, multiplication, or negation, at the
// root.
if (node_->GetType() != SENode::Add && node_->GetType() != SENode::Multiply &&
node_->GetType() != SENode::Negative)
return node_;
SENode* simplified_polynomial = SimplifyPolynomial();
SERecurrentNode* recurrent_expr = nullptr;
node_ = simplified_polynomial;
// Fold recurrent expressions which are with respect to the same loop into a
// single recurrent expression.
simplified_polynomial = FoldRecurrentAddExpressions(simplified_polynomial);
simplified_polynomial =
// Traverse the immediate children of the new node to find the recurrent
// expression. If there is more than one there is nothing further we can do.
for (SENode* child : simplified_polynomial->GetChildren()) {
if (child->GetType() == SENode::RecurrentAddExpr) {
recurrent_expr = child->AsSERecurrentNode();
// We need to count the number of unique recurrent expressions in the DAG to
// ensure there is only one.
for (auto child_iterator = simplified_polynomial->graph_begin();
child_iterator != simplified_polynomial->graph_end(); ++child_iterator) {
if (child_iterator->GetType() == SENode::RecurrentAddExpr &&
recurrent_expr != child_iterator->AsSERecurrentNode()) {
return simplified_polynomial;
if (recurrent_expr) {
return SimplifyRecurrentAddExpression(recurrent_expr);
return simplified_polynomial;
// Traverse the graph to build up the accumulator objects.
void SENodeSimplifyImpl::GatherAccumulatorsFromChildNodes(SENode* new_node,
SENode* child,
bool negation) {
int32_t sign = negation ? -1 : 1;
if (child->GetType() == SENode::Constant) {
// Collect all the constants and add them together.
constant_accumulator_ +=
child->AsSEConstantNode()->FoldToSingleValue() * sign;
} else if (child->GetType() == SENode::ValueUnknown ||
child->GetType() == SENode::RecurrentAddExpr) {
// To rebuild the graph of X+X+X*2 into 4*X we count the occurrences of X
// and create a new node of count*X after. X can either be a ValueUnknown or
// a RecurrentAddExpr. The count for each X is stored in the accumulators_
// map.
auto iterator = accumulators_.find(child);
// If we've encountered this term before add to the accumulator for it.
if (iterator == accumulators_.end())
accumulators_.insert({child, sign});
iterator->second += sign;
} else if (child->GetType() == SENode::Multiply) {
if (!AccumulatorsFromMultiply(child, negation)) {
} else if (child->GetType() == SENode::Add) {
for (SENode* next_child : *child) {
GatherAccumulatorsFromChildNodes(new_node, next_child, negation);
} else if (child->GetType() == SENode::Negative) {
SENode* negated_node = child->GetChild(0);
GatherAccumulatorsFromChildNodes(new_node, negated_node, !negation);
} else {
// If we can't work out how to fold the expression just add it back into
// the graph.
SERecurrentNode* SENodeSimplifyImpl::UpdateCoefficient(
SERecurrentNode* recurrent, int64_t coefficient_update) const {
std::unique_ptr<SERecurrentNode> new_recurrent_node{new SERecurrentNode(
recurrent->GetParentAnalysis(), recurrent->GetLoop())};
SENode* new_coefficient = analysis_.CreateMultiplyNode(
// See if the node can be simplified.
SENode* simplified = analysis_.SimplifyExpression(new_coefficient);
if (simplified->GetType() != SENode::CanNotCompute)
new_coefficient = simplified;
if (coefficient_update < 0) {
} else {
return analysis_.GetCachedOrAdd(std::move(new_recurrent_node))
// Simplify all the terms in the polynomial function.
SENode* SENodeSimplifyImpl::SimplifyPolynomial() {
std::unique_ptr<SENode> new_add{new SEAddNode(node_->GetParentAnalysis())};
// Traverse the graph and gather the accumulators from it.
GatherAccumulatorsFromChildNodes(new_add.get(), node_, false);
// Fold all the constants into a single constant node.
if (constant_accumulator_ != 0) {
for (auto& pair : accumulators_) {
SENode* term = pair.first;
int64_t count = pair.second;
// We can eliminate the term completely.
if (count == 0) continue;
if (count == 1) {
} else if (count == -1 && term->GetType() != SENode::RecurrentAddExpr) {
// If the count is -1 we can just add a negative version of that node,
// unless it is a recurrent expression as we would rather the negative
// goes on the recurrent expressions children. This makes it easier to
// work with in other places.
} else {
// Output value unknown terms as count*term and output recurrent
// expression terms as rec(offset, coefficient + count) offset and
// coefficient are the same as in the original expression.
if (term->GetType() == SENode::ValueUnknown) {
SENode* count_as_constant = analysis_.CreateConstant(count);
analysis_.CreateMultiplyNode(count_as_constant, term));
} else {
assert(term->GetType() == SENode::RecurrentAddExpr &&
"We only handle value unknowns or recurrent expressions");
// Create a new recurrent expression by adding the count to the
// coefficient of the old one.
new_add->AddChild(UpdateCoefficient(term->AsSERecurrentNode(), count));
// If there is only one term in the addition left just return that term.
if (new_add->GetChildren().size() == 1) {
return new_add->GetChild(0);
// If there are no terms left in the addition just return 0.
if (new_add->GetChildren().size() == 0) {
return analysis_.CreateConstant(0);
return analysis_.GetCachedOrAdd(std::move(new_add));
SENode* SENodeSimplifyImpl::FoldRecurrentAddExpressions(SENode* root) {
std::unique_ptr<SEAddNode> new_node{new SEAddNode(&analysis_)};
// A mapping of loops to the list of recurrent expressions which are with
// respect to those loops.
std::map<const Loop*, std::vector<std::pair<SERecurrentNode*, bool>>>
bool has_multiple_same_loop_recurrent_terms = false;
for (SENode* child : *root) {
bool negation = false;
if (child->GetType() == SENode::Negative) {
child = child->GetChild(0);
negation = true;
if (child->GetType() == SENode::RecurrentAddExpr) {
const Loop* loop = child->AsSERecurrentNode()->GetLoop();
SERecurrentNode* rec = child->AsSERecurrentNode();
if (loops_to_recurrent.find(loop) == loops_to_recurrent.end()) {
loops_to_recurrent[loop] = {std::make_pair(rec, negation)};
} else {
loops_to_recurrent[loop].push_back(std::make_pair(rec, negation));
has_multiple_same_loop_recurrent_terms = true;
} else {
if (!has_multiple_same_loop_recurrent_terms) return root;
for (auto pair : loops_to_recurrent) {
std::vector<std::pair<SERecurrentNode*, bool>>& recurrent_expressions =
const Loop* loop = pair.first;
std::unique_ptr<SENode> new_coefficient{new SEAddNode(&analysis_)};
std::unique_ptr<SENode> new_offset{new SEAddNode(&analysis_)};
for (auto node_pair : recurrent_expressions) {
SERecurrentNode* node = node_pair.first;
bool negative = node_pair.second;
if (!negative) {
} else {
std::unique_ptr<SERecurrentNode> new_recurrent{
new SERecurrentNode(&analysis_, loop)};
SENode* new_coefficient_simplified =
SENode* new_offset_simplified =
if (new_coefficient_simplified->GetType() == SENode::Constant &&
new_coefficient_simplified->AsSEConstantNode()->FoldToSingleValue() ==
0) {
return new_offset_simplified;
// If we only have one child in the add just return that.
if (new_node->GetChildren().size() == 1) {
return new_node->GetChild(0);
return analysis_.GetCachedOrAdd(std::move(new_node));
SENode* SENodeSimplifyImpl::EliminateZeroCoefficientRecurrents(SENode* node) {
if (node->GetType() != SENode::Add) return node;
bool has_change = false;
std::vector<SENode*> new_children{};
for (SENode* child : *node) {
if (child->GetType() == SENode::RecurrentAddExpr) {
SENode* coefficient = child->AsSERecurrentNode()->GetCoefficient();
// If coefficient is zero then we can eliminate the recurrent expression
// entirely and just return the offset as the recurrent expression is
// representing the equation coefficient*iterations + offset.
if (coefficient->GetType() == SENode::Constant &&
coefficient->AsSEConstantNode()->FoldToSingleValue() == 0) {
has_change = true;
} else {
} else {
if (!has_change) return node;
std::unique_ptr<SENode> new_add{new SEAddNode(node_->GetParentAnalysis())};
for (SENode* child : new_children) {
return analysis_.GetCachedOrAdd(std::move(new_add));
SENode* SENodeSimplifyImpl::SimplifyRecurrentAddExpression(
SERecurrentNode* recurrent_expr) {
const std::vector<SENode*>& children = node_->GetChildren();
std::unique_ptr<SERecurrentNode> recurrent_node{new SERecurrentNode(
recurrent_expr->GetParentAnalysis(), recurrent_expr->GetLoop())};
// Create and simplify the new offset node.
std::unique_ptr<SENode> new_offset{
new SEAddNode(recurrent_expr->GetParentAnalysis())};
for (SENode* child : children) {
if (child->GetType() != SENode::RecurrentAddExpr) {
// Simplify the new offset.
SENode* simplified_child = analysis_.SimplifyExpression(new_offset.get());
// If the child can be simplified, add the simplified form otherwise, add it
// via the usual caching mechanism.
if (simplified_child->GetType() != SENode::CanNotCompute) {
} else {
return analysis_.GetCachedOrAdd(std::move(recurrent_node));
* Scalar Analysis simplification public methods.
SENode* ScalarEvolutionAnalysis::SimplifyExpression(SENode* node) {
SENodeSimplifyImpl impl{this, node};
return impl.Simplify();
} // namespace opt
} // namespace spvtools