src/lang/buffer.cc - tvm - Git at Google

 /*!
  *  Copyright (c) 2016 by Contributors
  * \file buffer.cc
  */
 #include <tvm/buffer.h>
 #include <tvm/runtime/device_api.h>
 #include <tvm/ir.h>
 #include <tvm/ir_pass.h>
 #include <iterator>
 #include "../arithmetic/compute_expr.h"

 namespace tvm {

 Array<Expr> SimplifyArray(Array<Expr> array) {
   for (size_t i = 0; i < array.size(); ++i) {
     array.Set(i, ir::Simplify(array[i]));
   }
   return array;
 }

 Buffer decl_buffer(Array<Expr> shape,
                    Type dtype,
                    std::string name) {
   return BufferNode::make(
       Var(name, Handle()),
       dtype,
       shape,
       Array<Expr>(),
       Expr(),
       name,
       "",
       0, 0);
 }

 // Split the given expression w.r.t the add operator
 inline std::vector<const Expr*> ExprSplitAddition(const Expr &expr) {
   using namespace ir;
   std::vector<const Expr*> ret;
   std::stack<const Expr*> split_buffer;
   split_buffer.push(&expr);
   while (!split_buffer.empty()) {
     const Expr* top_ele = split_buffer.top();
     split_buffer.pop();
     auto expr_add_match = top_ele->as<Add>();
     if (expr_add_match) {
       split_buffer.push(&expr_add_match->b);
       split_buffer.push(&expr_add_match->a);
     } else {
       ret.emplace_back(top_ele);
     }
   }
   return ret;
 }


 // Searches for the following types of expr:
 //   mult_expr = (a1 + a2 + ... + aj + c / (k1 * k2 * ... * ki) * k1 * ... * kt-1 ) * kt * ... * ki
 //   mod_l_expr = c
 //   mod_r_expr = k1 * k2 * ... * ki
 // If it can be optimized, returns (true, (a1 + a2 + ... + aj) * kt * ... * ki + c)
 // Currently the we will not search the add/mult combinations exhaustively
 //   as it will take too much computation.
 inline std::pair<bool, Expr> MergeMulModInner(const Expr &mult_expr,
                                               const Expr &mod_l_expr,
                                               const Expr &mod_r_expr) {
   using namespace ir;
   const Mul* mult_ptr = mult_expr.as<Mul>();
   if (!mult_ptr) return std::make_pair(false, Expr());
   Expr mult_outer = mult_ptr->b;
   const Expr* inner = &(mult_ptr->a);
   // 1. Calculate the outer multiplier
   while (true) {
     mult_ptr = inner->as<Mul>();
     if (mult_ptr) {
       inner = &(mult_ptr->a);
       mult_outer = mult_ptr->b * mult_outer;
     } else {
       break;
     }
   }
   // 2. Search for the pattern c / (...) * (...) + c % (...)
   // We match the search element with Add, Mul and Div.
   //   If Add is found, we need to continue our search for the rhs
   //   If Mult is found, we will expand the inner multiplication factor
   //   If Div is found, we will go on testing whether lhs matches the lhs of mod expr
   //      and returns the optimization result.
   const Expr* search_ptr = inner;
   Expr mult_inner;  // The inner multiplication factor
   Expr no_opt_sum;  // Sum of the exprs that cannot be optimized
   while (true) {
     auto inner_div_ptr = search_ptr->as<Div>();
     auto inner_mult_ptr = search_ptr->as<Mul>();
     auto inner_add_ptr = search_ptr->as<Add>();
     if (!inner_div_ptr && !inner_mult_ptr && !inner_add_ptr) {
       return std::make_pair(false, Expr());
     } else if (inner_div_ptr) {
       Expr overall_mult = mult_inner.get() ? mult_inner * mult_outer : mult_outer;
       if (Equal(overall_mult, inner_div_ptr->b)
           && Equal(overall_mult, mod_r_expr)
           && Equal(inner_div_ptr->a, mod_l_expr)) {
         // Found!
         Expr ret = no_opt_sum.get() ? no_opt_sum * mult_outer + mod_l_expr : mod_l_expr;
         return std::make_pair(true, ret);
       } else {
         return std::make_pair(false, Expr());
       }
     } else if (inner_mult_ptr) {
       mult_inner = mult_inner.get() ? inner_mult_ptr->b * mult_inner : inner_mult_ptr->b;
       search_ptr = &(inner_mult_ptr->a);
     } else if (inner_add_ptr) {
       if (mult_inner.get()) {
         return std::make_pair(false, Expr());
       }
       no_opt_sum = no_opt_sum.get() ? no_opt_sum + inner_add_ptr->a : inner_add_ptr->a;
       search_ptr = &(inner_add_ptr->b);
     } else {
       LOG(FATAL) << "Unexpected search result!";
       break;
     }
   }
   return std::make_pair(false, Expr());
 }

 // Insert the elements into the corresponding mult_exprs and mod_exprs.
 // If the element is found to match Mul, it will be pushed to the mult_exprs.
 // If the element it found to match Mod, it will be pused to the mod_exprs.
 // Otherwise, the elements will be added to the no_opt_sum variable
 inline void MergeMulModInsertElements(const std::vector<const Expr*>& eles,
                                       std::list<Expr>* mult_exprs,
                                       std::list<std::pair<Expr, Expr> >* mod_exprs,
                                       Expr* no_opt_sum,
                                       bool* has_mult,
                                       bool* has_mod) {
   using namespace ir;
   *has_mult = false;
   *has_mod = false;
   for (const Expr* ele : eles) {
     auto mod_ptr = ele->as<Mod>();
     auto mult_ptr = ele->as<Mul>();
     if (mod_ptr) {
       *has_mod = true;
       mod_exprs->emplace_back(std::make_pair(std::move(mod_ptr->a), std::move(mod_ptr->b)));
     } else if (mult_ptr) {
       *has_mult = true;
       mult_exprs->emplace_back(*ele);
     } else {
       *no_opt_sum = no_opt_sum->get() ? *no_opt_sum + *ele : *ele;
     }
   }
 }

 // Searches for this types of expr:
 //   (a1 + a2 + ... + aj + c / (k1 * k2 * ... * ki) * k1 * ... * kt-1 ) * kt * ... * ki
 //   + c % (k1 * k2 * ... * ki)
 // and simplifies to (a1 + a2 + ... + aj) * kt * ... * ki + c
 // The search will be performed repeatively until no pattern is found.
 // Return: a pair with (false, Expr()) if cannot be optimized.
 //         a pair with (true, optimized_expr) if can be optimized
 inline Expr MergeMulMod(const Expr &base) {
   using namespace ir;
   // 1. Prepare the lists.
   // We store two lists, a list that contain all the elements that match Mul and
   //                     a list that contain all the elements that match Mod.
   // The elements in the Mod will be used to match against the elements in Mul.
   // The result will then be split and pushed back to these two lists.
   Expr simplified_base = Simplify(base);
   std::vector<const Expr*> eles = ExprSplitAddition(simplified_base);
   std::list<Expr> mult_exprs;
   std::list<std::pair<Expr, Expr> > mod_exprs;
   Expr no_opt_sum;
   bool has_mult;
   bool has_mod;
   MergeMulModInsertElements(eles, &mult_exprs, &mod_exprs,
                             &no_opt_sum, &has_mult, &has_mod);
   bool find_opt = false;
   std::list<std::pair<Expr, Expr> >::iterator search_mod_it = mod_exprs.begin();
   // 2. Exhaustive Search
   while (search_mod_it != mod_exprs.end()) {
     std::list<Expr>::iterator mult_it = mult_exprs.begin();
     bool inner_find_opt = false;
     while (mult_it != mult_exprs.end()) {
       std::pair<bool, Expr> ret = MergeMulModInner(*mult_it,
                                                    search_mod_it->first,
                                                    search_mod_it->second);
       if (ret.first) {
         inner_find_opt = true;
         auto temp_mod_it = search_mod_it;
         ++search_mod_it;
         mod_exprs.erase(temp_mod_it);
         mult_exprs.erase(mult_it);
         std::vector<const Expr*> ret_eles = ExprSplitAddition(ret.second);
         MergeMulModInsertElements(ret_eles, &mult_exprs, &mod_exprs,
                                   &no_opt_sum, &has_mult, &has_mod);
         if (has_mult) {
           search_mod_it = mod_exprs.begin();
         } else if (has_mod && search_mod_it == mod_exprs.end()) {
           search_mod_it--;
         }
         break;
       } else {
         ++mult_it;
       }
     }
     find_opt = find_opt || inner_find_opt;
     if (!inner_find_opt) {
       ++search_mod_it;
     }
   }
   if (!find_opt) {
     return simplified_base;
   }
   for (std::list<Expr>::iterator it = mult_exprs.begin(); it != mult_exprs.end(); ++it) {
     no_opt_sum = no_opt_sum.get() ? no_opt_sum + *it : *it;
   }
   for (std::list<std::pair<Expr, Expr> >::iterator it = mod_exprs.begin();
                                                    it != mod_exprs.end(); ++it) {
     no_opt_sum = no_opt_sum.get() ? no_opt_sum + it->first % it->second : it->first % it->second;
   }
   return no_opt_sum;
 }

 // The buffer offset in convention of number of elements of
 // original data ignoring number of lanes.
 // We also perform optimization to simplify the indexing expression.
 inline Expr ElemOffset(const BufferNode* n, Array<Expr> index) {
   Expr base = n->elem_offset;
   if (n->strides.size() == 0) {
     CHECK_EQ(n->shape.size(), index.size());
     if (index.size() > 0) {
       Expr offset = index[0];
       for (size_t i = 1; i < index.size(); ++i) {
         offset = MergeMulMod(offset * n->shape[i] + index[i]);
       }
       base = base + offset;
     }
   } else {
     CHECK_EQ(n->strides.size(), index.size());
     if (is_zero(base)) {
       base = MergeMulMod(index[0] * n->strides[0]);
     } else {
       base = MergeMulMod(base + index[0] * n->strides[0]);
     }
     for (size_t i = 1; i < index.size(); ++i) {
       base = MergeMulMod(base + index[i] * n->strides[i]);
     }
   }
   return base;
 }

 inline Expr BufferOffset(const BufferNode* n, Array<Expr> index, Type dtype) {
   Expr offset = ElemOffset(n, index);
   if (n->dtype.lanes() != 1) {
     offset = offset * make_const(offset.type(), dtype.lanes());
   }
   if (dtype.lanes() != 1) {
     return ir::Ramp::make(offset, make_const(offset.type(), 1), dtype.lanes());
   } else {
     return offset;
   }
 }

 Expr Buffer::vload(Array<Expr> begin, Type dtype) const {
   // specially handle bool, stored as Int(8)
   const BufferNode* n = operator->();
   CHECK(dtype.element_of() == n->dtype.element_of() &&
         dtype.lanes() % n->dtype.lanes() == 0)
       << "Cannot load " << dtype
       << " from buffer of " << n->dtype;
   if (dtype == Bool()) {
     return ir::Cast::make(
         Bool(),
         ir::Load::make(
             Int(8), n->data, BufferOffset(n, begin, Int(8)),
             const_true()));
   } else {
     return ir::Load::make(
         dtype, n->data, BufferOffset(n, begin, dtype),
         const_true(dtype.lanes()));
   }
 }

 Stmt Buffer::vstore(Array<Expr> begin, Expr value) const {
   // specially handle bool, stored as Int(8)
   const BufferNode* n = operator->();
   Type dtype = value.type();
   CHECK(dtype.element_of() == n->dtype.element_of() &&
         dtype.lanes() % n->dtype.lanes() == 0)
       << "Cannot load " << dtype
       << " from buffer of " << n->dtype;
   if (value.type() == Bool()) {
     return ir::Store::make(n->data,
                            ir::Cast::make(Int(8), value),
                            BufferOffset(n, begin, Int(8)),
                            const_true());
   } else {
     return ir::Store::make(n->data, value, BufferOffset(n, begin, dtype),
                            const_true(dtype.lanes()));
   }
 }

 Buffer Buffer::MakeStrideView() const {
   if ((*this)->strides.size() != 0) return *this;
   if ((*this)->shape.size() == 0) return *this;
   std::vector<Expr> temp;
   auto n = make_node<BufferNode>(*operator->());
   Expr acc = make_const(n->DefaultIndexType(), 1);
   for (size_t i = n->shape.size(); i != 0 ; --i) {
     temp.push_back(acc);
     acc = acc * n->shape[i - 1];
   }
   for (size_t i = temp.size(); i != 0; --i) {
     n->strides.push_back(temp[i - 1]);
   }
   return Buffer(n);
 }

 Buffer Buffer::MakeSlice(Array<Expr> begins, Array<Expr> extents) const {
   const BufferNode* n = operator->();
   begins = SimplifyArray(begins);
   Expr elem_offset = ir::Simplify(ElemOffset(n, begins));
   Array<Expr> strides = n->strides;
   if (strides.size() == 0) {
     bool can_relax = true;
     bool need_stride = false;
     // check if stride is needed.
     for (size_t i = 0; i < extents.size(); ++i) {
       if (!can_relax) {
         if (!is_zero(begins[i]) ||
             !is_zero(ir::Simplify(extents[i] - n->shape[i]))) {
           need_stride = true;
         }
       }
       if (!is_one(extents[i])) can_relax = false;
     }
     // make stride.
     if (need_stride) {
       return MakeStrideView().MakeSlice(begins, extents);
     }
   }
   return BufferNode::make(n->data,
                           n->dtype,
                           extents,
                           strides,
                           elem_offset,
                           n->name + "_slice",
                           n->scope,
                           n->data_alignment,
                           0);
 }

 Expr Buffer::access_ptr(int access_mask, Type ptr_type, int content_lanes, Expr offset) const {
   const BufferNode* self = operator->();
   Expr e_dtype;
   Expr extent;
   if (self->shape.size() == 0) {
     extent = make_const(self->DefaultIndexType(), 1);
   } else if (self->strides.size() == self->shape.size()) {
     int highest_dim = 0;
     extent = arith::ComputeExpr<ir::Mul>(
         self->strides[highest_dim], self->shape[highest_dim]) - offset;
   } else {
     extent = arith::ComputeReduce<ir::Mul>(self->shape, Expr()) - offset;
   }
   Expr elem_offset = self->elem_offset + offset;
   if (content_lanes > 1) {
     e_dtype = ir::TypeAnnotation(self->dtype.with_lanes(content_lanes));
     extent = extent / make_const(self->elem_offset.type(), content_lanes);
     elem_offset = self->elem_offset / make_const(self->elem_offset.type(),
                                                  content_lanes);
   } else {
     e_dtype = ir::TypeAnnotation(self->dtype);
   }
   Array<Expr> acc_args{
     e_dtype, self->data, elem_offset,
         extent, make_const(Int(32), access_mask)};
   return ir::Call::make(
       ptr_type, ir::intrinsic::tvm_access_ptr, acc_args, ir::Call::Intrinsic);
 }

 Buffer BufferNode::make(Var data,
                         Type dtype,
                         Array<Expr> shape,
                         Array<Expr> strides,
                         Expr elem_offset,
                         std::string name,
                         std::string scope,
                         int data_alignment,
                         int offset_factor) {
   auto n = make_node<BufferNode>();
   n->data = std::move(data);
   n->dtype = dtype;
   n->shape = std::move(shape);
   n->strides = std::move(strides);
   n->name = std::move(name);
   if (scope.length() == 0) {
     scope = "global";
   }
   n->scope = std::move(scope);
   if (!elem_offset.defined()) {
     elem_offset = make_const(n->DefaultIndexType(), 0);
   }
   if (data_alignment <= 0) {
     data_alignment = runtime::kAllocAlignment;
   }
   if (offset_factor == 0) {
     offset_factor = 1;
   }
   n->elem_offset = std::move(elem_offset);
   n->data_alignment = data_alignment;
   n->offset_factor = offset_factor;
   return Buffer(n);
 }

 TVM_STATIC_IR_FUNCTOR(IRPrinter, vtable)
 .set_dispatch<BufferNode>([](const BufferNode *op, IRPrinter *p) {
     p->stream << "buffer(" << op->name << ", " << op << ")";
 });

 TVM_REGISTER_NODE_TYPE(BufferNode);

 }  // namespace tvm
	/*!
	* Copyright (c) 2016 by Contributors
	* \file buffer.cc
	*/
	#include <tvm/buffer.h>
	#include <tvm/runtime/device_api.h>
	#include <tvm/ir.h>
	#include <tvm/ir_pass.h>
	#include <iterator>
	#include "../arithmetic/compute_expr.h"

	namespace tvm {

	Array<Expr> SimplifyArray(Array<Expr> array) {
	for (size_t i = 0; i < array.size(); ++i) {
	array.Set(i, ir::Simplify(array[i]));
	}
	return array;
	}

	Buffer decl_buffer(Array<Expr> shape,
	Type dtype,
	std::string name) {
	return BufferNode::make(
	Var(name, Handle()),
	dtype,
	shape,
	Array<Expr>(),
	Expr(),
	name,
	"",
	0, 0);
	}

	// Split the given expression w.r.t the add operator
	inline std::vector<const Expr*> ExprSplitAddition(const Expr &expr) {
	using namespace ir;
	std::vector<const Expr*> ret;
	std::stack<const Expr*> split_buffer;
	split_buffer.push(&expr);
	while (!split_buffer.empty()) {
	const Expr* top_ele = split_buffer.top();
	split_buffer.pop();
	auto expr_add_match = top_ele->as<Add>();
	if (expr_add_match) {
	split_buffer.push(&expr_add_match->b);
	split_buffer.push(&expr_add_match->a);
	} else {
	ret.emplace_back(top_ele);
	}
	}
	return ret;
	}


	// Searches for the following types of expr:
	// mult_expr = (a1 + a2 + ... + aj + c / (k1 * k2 * ... * ki) * k1 * ... * kt-1 ) * kt * ... * ki
	// mod_l_expr = c
	// mod_r_expr = k1 * k2 * ... * ki
	// If it can be optimized, returns (true, (a1 + a2 + ... + aj) * kt * ... * ki + c)
	// Currently the we will not search the add/mult combinations exhaustively
	// as it will take too much computation.
	inline std::pair<bool, Expr> MergeMulModInner(const Expr &mult_expr,
	const Expr &mod_l_expr,
	const Expr &mod_r_expr) {
	using namespace ir;
	const Mul* mult_ptr = mult_expr.as<Mul>();
	if (!mult_ptr) return std::make_pair(false, Expr());
	Expr mult_outer = mult_ptr->b;
	const Expr* inner = &(mult_ptr->a);
	// 1. Calculate the outer multiplier
	while (true) {
	mult_ptr = inner->as<Mul>();
	if (mult_ptr) {
	inner = &(mult_ptr->a);
	mult_outer = mult_ptr->b * mult_outer;
	} else {
	break;
	}
	}
	// 2. Search for the pattern c / (...) * (...) + c % (...)
	// We match the search element with Add, Mul and Div.
	// If Add is found, we need to continue our search for the rhs
	// If Mult is found, we will expand the inner multiplication factor
	// If Div is found, we will go on testing whether lhs matches the lhs of mod expr
	// and returns the optimization result.
	const Expr* search_ptr = inner;
	Expr mult_inner; // The inner multiplication factor
	Expr no_opt_sum; // Sum of the exprs that cannot be optimized
	while (true) {
	auto inner_div_ptr = search_ptr->as<Div>();
	auto inner_mult_ptr = search_ptr->as<Mul>();
	auto inner_add_ptr = search_ptr->as<Add>();
	if (!inner_div_ptr && !inner_mult_ptr && !inner_add_ptr) {
	return std::make_pair(false, Expr());
	} else if (inner_div_ptr) {
	Expr overall_mult = mult_inner.get() ? mult_inner * mult_outer : mult_outer;
	if (Equal(overall_mult, inner_div_ptr->b)
	&& Equal(overall_mult, mod_r_expr)
	&& Equal(inner_div_ptr->a, mod_l_expr)) {
	// Found!
	Expr ret = no_opt_sum.get() ? no_opt_sum * mult_outer + mod_l_expr : mod_l_expr;
	return std::make_pair(true, ret);
	} else {
	return std::make_pair(false, Expr());
	}
	} else if (inner_mult_ptr) {
	mult_inner = mult_inner.get() ? inner_mult_ptr->b * mult_inner : inner_mult_ptr->b;
	search_ptr = &(inner_mult_ptr->a);
	} else if (inner_add_ptr) {
	if (mult_inner.get()) {
	return std::make_pair(false, Expr());
	}
	no_opt_sum = no_opt_sum.get() ? no_opt_sum + inner_add_ptr->a : inner_add_ptr->a;
	search_ptr = &(inner_add_ptr->b);
	} else {
	LOG(FATAL) << "Unexpected search result!";
	break;
	}
	}
	return std::make_pair(false, Expr());
	}

	// Insert the elements into the corresponding mult_exprs and mod_exprs.
	// If the element is found to match Mul, it will be pushed to the mult_exprs.
	// If the element it found to match Mod, it will be pused to the mod_exprs.
	// Otherwise, the elements will be added to the no_opt_sum variable
	inline void MergeMulModInsertElements(const std::vector<const Expr*>& eles,
	std::list<Expr>* mult_exprs,
	std::list<std::pair<Expr, Expr> >* mod_exprs,
	Expr* no_opt_sum,
	bool* has_mult,
	bool* has_mod) {
	using namespace ir;
	*has_mult = false;
	*has_mod = false;
	for (const Expr* ele : eles) {
	auto mod_ptr = ele->as<Mod>();
	auto mult_ptr = ele->as<Mul>();
	if (mod_ptr) {
	*has_mod = true;
	mod_exprs->emplace_back(std::make_pair(std::move(mod_ptr->a), std::move(mod_ptr->b)));
	} else if (mult_ptr) {
	*has_mult = true;
	mult_exprs->emplace_back(*ele);
	} else {
	no_opt_sum = no_opt_sum->get() ? no_opt_sum + ele : ele;
	}
	}
	}

	// Searches for this types of expr:
	// (a1 + a2 + ... + aj + c / (k1 * k2 * ... * ki) * k1 * ... * kt-1 ) * kt * ... * ki
	// + c % (k1 * k2 * ... * ki)
	// and simplifies to (a1 + a2 + ... + aj) * kt * ... * ki + c
	// The search will be performed repeatively until no pattern is found.
	// Return: a pair with (false, Expr()) if cannot be optimized.
	// a pair with (true, optimized_expr) if can be optimized
	inline Expr MergeMulMod(const Expr &base) {
	using namespace ir;
	// 1. Prepare the lists.
	// We store two lists, a list that contain all the elements that match Mul and
	// a list that contain all the elements that match Mod.
	// The elements in the Mod will be used to match against the elements in Mul.
	// The result will then be split and pushed back to these two lists.
	Expr simplified_base = Simplify(base);
	std::vector<const Expr*> eles = ExprSplitAddition(simplified_base);
	std::list<Expr> mult_exprs;
	std::list<std::pair<Expr, Expr> > mod_exprs;
	Expr no_opt_sum;
	bool has_mult;
	bool has_mod;
	MergeMulModInsertElements(eles, &mult_exprs, &mod_exprs,
	&no_opt_sum, &has_mult, &has_mod);
	bool find_opt = false;
	std::list<std::pair<Expr, Expr> >::iterator search_mod_it = mod_exprs.begin();
	// 2. Exhaustive Search
	while (search_mod_it != mod_exprs.end()) {
	std::list<Expr>::iterator mult_it = mult_exprs.begin();
	bool inner_find_opt = false;
	while (mult_it != mult_exprs.end()) {
	std::pair<bool, Expr> ret = MergeMulModInner(*mult_it,
	search_mod_it->first,
	search_mod_it->second);
	if (ret.first) {
	inner_find_opt = true;
	auto temp_mod_it = search_mod_it;
	++search_mod_it;
	mod_exprs.erase(temp_mod_it);
	mult_exprs.erase(mult_it);
	std::vector<const Expr*> ret_eles = ExprSplitAddition(ret.second);
	MergeMulModInsertElements(ret_eles, &mult_exprs, &mod_exprs,
	&no_opt_sum, &has_mult, &has_mod);
	if (has_mult) {
	search_mod_it = mod_exprs.begin();
	} else if (has_mod && search_mod_it == mod_exprs.end()) {
	search_mod_it--;
	}
	break;
	} else {
	++mult_it;
	}
	}
	find_opt = find_opt \|\| inner_find_opt;
	if (!inner_find_opt) {
	++search_mod_it;
	}
	}
	if (!find_opt) {
	return simplified_base;
	}
	for (std::list<Expr>::iterator it = mult_exprs.begin(); it != mult_exprs.end(); ++it) {
	no_opt_sum = no_opt_sum.get() ? no_opt_sum + it : it;
	}
	for (std::list<std::pair<Expr, Expr> >::iterator it = mod_exprs.begin();
	it != mod_exprs.end(); ++it) {
	no_opt_sum = no_opt_sum.get() ? no_opt_sum + it->first % it->second : it->first % it->second;
	}
	return no_opt_sum;
	}

	// The buffer offset in convention of number of elements of
	// original data ignoring number of lanes.
	// We also perform optimization to simplify the indexing expression.
	inline Expr ElemOffset(const BufferNode* n, Array<Expr> index) {
	Expr base = n->elem_offset;
	if (n->strides.size() == 0) {
	CHECK_EQ(n->shape.size(), index.size());
	if (index.size() > 0) {
	Expr offset = index[0];
	for (size_t i = 1; i < index.size(); ++i) {
	offset = MergeMulMod(offset * n->shape[i] + index[i]);
	}
	base = base + offset;
	}
	} else {
	CHECK_EQ(n->strides.size(), index.size());
	if (is_zero(base)) {
	base = MergeMulMod(index[0] * n->strides[0]);
	} else {
	base = MergeMulMod(base + index[0] * n->strides[0]);
	}
	for (size_t i = 1; i < index.size(); ++i) {
	base = MergeMulMod(base + index[i] * n->strides[i]);
	}
	}
	return base;
	}

	inline Expr BufferOffset(const BufferNode* n, Array<Expr> index, Type dtype) {
	Expr offset = ElemOffset(n, index);
	if (n->dtype.lanes() != 1) {
	offset = offset * make_const(offset.type(), dtype.lanes());
	}
	if (dtype.lanes() != 1) {
	return ir::Ramp::make(offset, make_const(offset.type(), 1), dtype.lanes());
	} else {
	return offset;
	}
	}

	Expr Buffer::vload(Array<Expr> begin, Type dtype) const {
	// specially handle bool, stored as Int(8)
	const BufferNode* n = operator->();
	CHECK(dtype.element_of() == n->dtype.element_of() &&
	dtype.lanes() % n->dtype.lanes() == 0)
	<< "Cannot load " << dtype
	<< " from buffer of " << n->dtype;
	if (dtype == Bool()) {
	return ir::Cast::make(
	Bool(),
	ir::Load::make(
	Int(8), n->data, BufferOffset(n, begin, Int(8)),
	const_true()));
	} else {
	return ir::Load::make(
	dtype, n->data, BufferOffset(n, begin, dtype),
	const_true(dtype.lanes()));
	}
	}

	Stmt Buffer::vstore(Array<Expr> begin, Expr value) const {
	// specially handle bool, stored as Int(8)
	const BufferNode* n = operator->();
	Type dtype = value.type();
	CHECK(dtype.element_of() == n->dtype.element_of() &&
	dtype.lanes() % n->dtype.lanes() == 0)
	<< "Cannot load " << dtype
	<< " from buffer of " << n->dtype;
	if (value.type() == Bool()) {
	return ir::Store::make(n->data,
	ir::Cast::make(Int(8), value),
	BufferOffset(n, begin, Int(8)),
	const_true());
	} else {
	return ir::Store::make(n->data, value, BufferOffset(n, begin, dtype),
	const_true(dtype.lanes()));
	}
	}

	Buffer Buffer::MakeStrideView() const {
	if ((this)->strides.size() != 0) return this;
	if ((this)->shape.size() == 0) return this;
	std::vector<Expr> temp;
	auto n = make_node<BufferNode>(*operator->());
	Expr acc = make_const(n->DefaultIndexType(), 1);
	for (size_t i = n->shape.size(); i != 0 ; --i) {
	temp.push_back(acc);
	acc = acc * n->shape[i - 1];
	}
	for (size_t i = temp.size(); i != 0; --i) {
	n->strides.push_back(temp[i - 1]);
	}
	return Buffer(n);
	}

	Buffer Buffer::MakeSlice(Array<Expr> begins, Array<Expr> extents) const {
	const BufferNode* n = operator->();
	begins = SimplifyArray(begins);
	Expr elem_offset = ir::Simplify(ElemOffset(n, begins));
	Array<Expr> strides = n->strides;
	if (strides.size() == 0) {
	bool can_relax = true;
	bool need_stride = false;
	// check if stride is needed.
	for (size_t i = 0; i < extents.size(); ++i) {
	if (!can_relax) {
	if (!is_zero(begins[i]) \|\|
	!is_zero(ir::Simplify(extents[i] - n->shape[i]))) {
	need_stride = true;
	}
	}
	if (!is_one(extents[i])) can_relax = false;
	}
	// make stride.
	if (need_stride) {
	return MakeStrideView().MakeSlice(begins, extents);
	}
	}
	return BufferNode::make(n->data,
	n->dtype,
	extents,
	strides,
	elem_offset,
	n->name + "_slice",
	n->scope,
	n->data_alignment,
	0);
	}

	Expr Buffer::access_ptr(int access_mask, Type ptr_type, int content_lanes, Expr offset) const {
	const BufferNode* self = operator->();
	Expr e_dtype;
	Expr extent;
	if (self->shape.size() == 0) {
	extent = make_const(self->DefaultIndexType(), 1);
	} else if (self->strides.size() == self->shape.size()) {
	int highest_dim = 0;
	extent = arith::ComputeExpr<ir::Mul>(
	self->strides[highest_dim], self->shape[highest_dim]) - offset;
	} else {
	extent = arith::ComputeReduce<ir::Mul>(self->shape, Expr()) - offset;
	}
	Expr elem_offset = self->elem_offset + offset;
	if (content_lanes > 1) {
	e_dtype = ir::TypeAnnotation(self->dtype.with_lanes(content_lanes));
	extent = extent / make_const(self->elem_offset.type(), content_lanes);
	elem_offset = self->elem_offset / make_const(self->elem_offset.type(),
	content_lanes);
	} else {
	e_dtype = ir::TypeAnnotation(self->dtype);
	}
	Array<Expr> acc_args{
	e_dtype, self->data, elem_offset,
	extent, make_const(Int(32), access_mask)};
	return ir::Call::make(
	ptr_type, ir::intrinsic::tvm_access_ptr, acc_args, ir::Call::Intrinsic);
	}

	Buffer BufferNode::make(Var data,
	Type dtype,
	Array<Expr> shape,
	Array<Expr> strides,
	Expr elem_offset,
	std::string name,
	std::string scope,
	int data_alignment,
	int offset_factor) {
	auto n = make_node<BufferNode>();
	n->data = std::move(data);
	n->dtype = dtype;
	n->shape = std::move(shape);
	n->strides = std::move(strides);
	n->name = std::move(name);
	if (scope.length() == 0) {
	scope = "global";
	}
	n->scope = std::move(scope);
	if (!elem_offset.defined()) {
	elem_offset = make_const(n->DefaultIndexType(), 0);
	}
	if (data_alignment <= 0) {
	data_alignment = runtime::kAllocAlignment;
	}
	if (offset_factor == 0) {
	offset_factor = 1;
	}
	n->elem_offset = std::move(elem_offset);
	n->data_alignment = data_alignment;
	n->offset_factor = offset_factor;
	return Buffer(n);
	}

	TVM_STATIC_IR_FUNCTOR(IRPrinter, vtable)
	.set_dispatch<BufferNode>([](const BufferNode op, IRPrinter p) {
	p->stream << "buffer(" << op->name << ", " << op << ")";
	});

	TVM_REGISTER_NODE_TYPE(BufferNode);

	} // namespace tvm