simplify_algebra.cpp

/*
 * The MIT License (MIT)
 *
 * Copyright (c) 2015-2022 Advanced Micro Devices, Inc. All rights reserved.
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in
 * all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
 * THE SOFTWARE.
 */
#include <migraphx/simplify_algebra.hpp>
#include <migraphx/dead_code_elimination.hpp>
#include <migraphx/program.hpp>
#include <migraphx/op/concat.hpp>
#include <migraphx/op/slice.hpp>
#include <migraphx/op/convolution.hpp>
#include <migraphx/op/broadcast.hpp>
#include <migraphx/op/reshape.hpp>
#include <migraphx/op/transpose.hpp>
#include <migraphx/matcher.hpp>
#include <migraphx/literal.hpp>
#include <migraphx/make_op.hpp>
#include <migraphx/serialize.hpp>

#include <migraphx/algorithm.hpp>
#include <unordered_set>

namespace migraphx {
inline namespace MIGRAPHX_INLINE_NS {

auto lit_broadcast() { return match::any_of(match::is_constant(), match::name("broadcast")); }
auto not_lit_broadcast() { return match::none_of(match::is_constant(), match::name("broadcast")); }
auto op_lit_broadcast(std::string op, std::string x, std::string y)
{
    return match::name(std::move(op))(match::either_arg(0, 1)(
        lit_broadcast().bind(std::move(x)), not_lit_broadcast().bind(std::move(y))));
}

auto conv_const_weights()
{
    return match::name("convolution")(match::used_once(),
                                      match::args(match::any(), match::is_constant().bind("w")));
}

auto reduction() { return match::name_contains("reduce"); }

struct find_mul_conv
{
    auto matcher() const
    {
        return match::name("mul")(match::either_arg(0, 1)(conv_const_weights().bind("conv"),
                                                          match::name("broadcast").bind("a")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins      = r.result;
        auto conv_ins = r.instructions["conv"];
        auto a_ins    = r.instructions["a"];
        auto w_ins    = r.instructions["w"];

        auto broadcast_op = any_cast<op::broadcast>(a_ins->get_operator());
        if(broadcast_op.axis != 1)
            return;

        auto new_a = m.insert_instruction(
            ins,
            make_op("broadcast", {{"axis", 0}, {"out_lens", w_ins->get_shape().lens()}}),
            a_ins->inputs().front());
        auto new_mul  = m.insert_instruction(ins, make_op("mul"), new_a, w_ins);
        auto new_conv = m.insert_instruction(
            ins, conv_ins->get_operator(), conv_ins->inputs().front(), new_mul);
        m.replace_instruction(ins, new_conv);
    }
};

struct find_mul_slice_conv
{
    static auto conv()
    {
        return match::name("convolution")(
            match::all_of[match::outputs()](match::name("slice")),
            match::args(match::any(), match::is_constant().bind("w")));
    }
    auto matcher() const
    {
        return match::name("mul")(match::either_arg(0, 1)(
            match::name("slice")(match::used_once(), match::arg(0)(conv().bind("conv")))
                .bind("slice"),
            match::name("broadcast")(match::is_constant()).bind("a")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins       = r.result;
        auto slice_ins = r.instructions["slice"];
        auto conv_ins  = r.instructions["conv"];
        auto a_ins     = r.instructions["a"];
        auto w_ins     = r.instructions["w"];

        auto broadcast_op = any_cast<op::broadcast>(a_ins->get_operator());
        if(broadcast_op.axis != 1)
            return;

        auto slice_op = any_cast<op::slice>(slice_ins->get_operator());
        if(slice_op.axes.size() != 1)
            return;
        if(slice_op.axes.front() != 1)
            return;

        auto slice_idx = std::distance(conv_ins, slice_ins);
        if(std::any_of(conv_ins->outputs().begin(), conv_ins->outputs().end(), [&](auto i) {
               if(i == slice_ins)
                   return false;
               if(std::distance(conv_ins, i) < slice_idx)
                   return true;
               auto sop = any_cast<op::slice>(i->get_operator());
               if(sop.axes != slice_op.axes)
                   return true;
               if(std::max(sop.starts.front(), slice_op.starts.front()) <
                  std::min(sop.ends.front(), slice_op.ends.front()))
                   return true;
               return false;
           }))
            return;

        auto w_slice_op  = slice_op;
        w_slice_op.axes  = {0};
        auto slice_w_ins = m.insert_instruction(ins, w_slice_op, w_ins);

        auto new_a = m.insert_instruction(
            ins,
            make_op("broadcast", {{"axis", 0}, {"out_lens", slice_w_ins->get_shape().lens()}}),
            a_ins->inputs().front());
        auto new_mul = m.insert_instruction(ins, make_op("mul"), new_a, slice_w_ins);

        std::vector<instruction_ref> sliced_weights;
        if(slice_op.starts.front() != 0)
            sliced_weights.push_back(m.insert_instruction(
                ins,
                make_op("slice", {{"axes", {0}}, {"starts", {0}}, {"ends", slice_op.starts}}),
                w_ins));
        sliced_weights.push_back(new_mul);
        int64_t end_axis = w_ins->get_shape().lens().at(0);
        if(slice_op.ends.front() != end_axis)
            sliced_weights.push_back(m.insert_instruction(
                ins,
                make_op("slice", {{"axes", {0}}, {"starts", slice_op.ends}, {"ends", {end_axis}}}),
                w_ins));

        auto new_weights =
            m.insert_instruction(ins, make_op("concat", {{"axis", 0}}), sliced_weights);

        auto new_conv = m.insert_instruction(
            ins, conv_ins->get_operator(), conv_ins->inputs().front(), new_weights);
        assert(conv_ins->get_shape() == new_conv->get_shape());

        auto slice1 = m.insert_instruction(ins, slice_op, new_conv);
        assert(ins->get_shape().lens() == slice1->get_shape().lens());
        m.replace_instruction(ins, slice1);
        // TODO: Check each slice doesn't overlap and that it occurs after slice_ins
        auto outputs = conv_ins->outputs();
        for(auto output : outputs)
            if(output != slice_ins)
                instruction::replace_argument(output, conv_ins, new_conv);
    }
};

// a * (x + b) => a * x + a * b
struct find_mul_add
{
    auto matcher() const
    {
        return match::name("mul")(match::either_arg(0, 1)(
            match::name("add")(
                match::either_arg(0, 1)(
                    match::any().bind("x"),
                    match::any_of(conv_const_weights(), match::is_constant()).bind("b")),
                match::none_of(match::args(match::is_constant(), match::is_constant())),
                match::used_once()),
            match::is_constant().bind("a")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto a_ins = r.instructions["a"];
        auto b_ins = r.instructions["b"];
        auto x_ins = r.instructions["x"];
        assert(x_ins != b_ins);

        auto ax_ins = m.insert_instruction(ins, make_op("mul"), a_ins, x_ins);
        auto ab_ins = m.insert_instruction(ins, make_op("mul"), a_ins, b_ins);
        m.replace_instruction(ins, make_op("add"), ax_ins, ab_ins);
    }
};

struct find_dot_add
{
    auto matcher() const
    {
        return match::name("dot")(match::either_arg(0, 1)(
            match::name("add")(
                match::either_arg(0, 1)(match::any().bind("x"),
                                        match::any_of(match::is_constant()).bind("b")),
                match::none_of(match::args(match::is_constant(), match::is_constant())),
                match::used_once()),
            match::is_constant().bind("a")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto a_ins = r.instructions["a"];
        auto b_ins = r.instructions["b"];
        auto x_ins = r.instructions["x"];
        assert(x_ins != b_ins);

        const bool flipped = a_ins == ins->inputs().back();

        auto insert_dot = [&](auto x, auto y) {
            if(flipped)
                return m.insert_instruction(ins, make_op("dot"), y, x);
            else
                return m.insert_instruction(ins, make_op("dot"), x, y);
        };

        auto ax_ins = insert_dot(a_ins, x_ins);
        auto ab_ins = insert_dot(a_ins, b_ins);
        m.replace_instruction(ins, make_op("add"), ax_ins, ab_ins);
    }
};

struct find_add_lit_broadcast
{
    auto matcher() const
    {
        return match::name("add")(
            match::either_arg(0, 1)(op_lit_broadcast("add", "a", "x"), lit_broadcast().bind("b")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto x_ins = r.instructions["x"];
        auto a_ins = r.instructions["a"];
        auto b_ins = r.instructions["b"];

        auto sumab = m.insert_instruction(ins, make_op("add"), a_ins, b_ins);
        m.replace_instruction(ins, make_op("add"), x_ins, sumab);
    }
};

struct find_double_add_lit_broadcast
{
    auto matcher() const
    {
        return match::name("add")(
            match::args(op_lit_broadcast("add", "a", "x"), op_lit_broadcast("add", "b", "y")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto x_ins = r.instructions["x"];
        auto y_ins = r.instructions["y"];
        auto a_ins = r.instructions["a"];
        auto b_ins = r.instructions["b"];

        instruction_ref sumab;

        if(a_ins->name() == "broadcast" and b_ins->name() == "broadcast")
        {
            if(a_ins->inputs().at(0)->get_shape() != b_ins->inputs().at(0)->get_shape())
                return;
            auto op     = a_ins->get_operator();
            auto presum = m.insert_instruction(
                ins, make_op("add"), a_ins->inputs().at(0), b_ins->inputs().at(0));
            sumab = m.insert_instruction(ins, op, presum);
        }
        else
        {
            sumab = m.insert_instruction(ins, make_op("add"), a_ins, b_ins);
        }

        auto sumxy = m.insert_instruction(ins, make_op("add"), x_ins, y_ins);
        m.replace_instruction(ins, make_op("add"), sumxy, sumab);
    }
};

struct find_inner_broadcast
{
    auto matcher() const { return pointwise(match::all_of[match::inputs()](match::broadcast())); }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins        = r.result;
        auto broadcasts = ins->inputs();
        if(broadcasts.empty())
            return;
        std::vector<instruction_ref> inputs;
        std::transform(broadcasts.begin(),
                       broadcasts.end(),
                       std::back_inserter(inputs),
                       [](auto i) { return i->inputs().front(); });
        if(std::any_of(inputs.begin(), inputs.end(), [&](auto i) {
               return i->get_shape() != inputs.front()->get_shape();
           }))
            return;

        auto op = m.insert_instruction(ins, ins->get_operator(), inputs);
        m.replace_instruction(ins, broadcasts.front()->get_operator(), op);
    }
};

struct find_concat_op
{
    auto matcher() const
    {
        return match::name("concat")(match::any_of[match::inputs()](
            match::any_of(match::pointwise(), match::name("broadcast")), match::used_once()));
    }

    template <class Iterator>
    static std::vector<std::size_t> get_output_lens(Iterator start, Iterator last, std::size_t axis)
    {
        assert(start != last);
        std::size_t dim = 0;
        for(auto ins : range(start, last))
        {
            dim += ins->get_shape().lens().at(axis);
        }
        auto lens  = (*start)->get_shape().lens();
        lens[axis] = dim;
        return lens;
    }

    static bool is_valid_op(const operation& op)
    {
        return op.name() == "broadcast" or op.attributes().contains("pointwise");
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins  = r.result;
        auto axis = any_cast<op::concat>(ins->get_operator()).axis;

        auto each = [&](auto start, auto last) -> std::vector<instruction_ref> {
            if(std::distance(start, last) < 2)
                return {start, last};
            auto x = *start;
            if(x->inputs().size() > 2 or x->inputs().empty() or x->outputs().size() > 1)
                return {start, last};
            auto op = x->get_operator();
            if(not is_valid_op(op))
                return {start, last};
            auto iaxis = axis;
            // Adjust broadcast lens
            if(op.name() == "broadcast")
            {
                auto b = any_cast<op::broadcast>(op);
                if(b.axis != iaxis)
                    return {start, last};
                b.broadcast_lens = get_output_lens(start, last, iaxis);
                op               = b;
                iaxis            = 0;
            }

            std::vector<instruction_ref> concats;
            for(std::size_t i = 0; i < x->inputs().size(); i++)
            {
                std::vector<instruction_ref> inputs;
                std::transform(start, last, std::back_inserter(inputs), [&](auto j) {
                    return j->inputs().at(i);
                });
                auto concat =
                    m.insert_instruction(ins, make_op("concat", {{"axis", iaxis}}), inputs);
                concats.push_back(concat);
            }
            auto y = m.insert_instruction(ins, op, concats);
            return {y};
        };

        std::vector<instruction_ref> args;
        auto update_args = [&](auto start, auto last) {
            auto x = each(start, last);
            args.insert(args.end(), x.begin(), x.end());
        };
        auto pred = [](auto i, auto j) {
            return i->get_operator() == j->get_operator() and
                   i->inputs().size() == i->inputs().size() and
                   i->outputs().size() == i->outputs().size();
        };
        group_unique(ins->inputs().begin(), ins->inputs().end(), update_args, pred);
        if(args.size() == 1)
            m.replace_instruction(ins, args.front());
        else
            m.replace_instruction(ins, make_op("concat", {{"axis", axis}}), args);
    }
};

void move_instructions_back(module& m, instruction_ref pos, std::vector<instruction_ref> inss)
{
    auto start = range(m.begin(), pos);
    for(auto ins : iterator_for(start))
    {
        auto it = std::find(inss.begin(), inss.end(), ins);
        if(it != inss.end())
            inss.erase(it);
    }
    for(auto ins : inss)
    {
        if(not m.has_instruction(ins))
            continue;
        move_instructions_back(m, pos, ins->inputs());
        m.move_instruction(ins, pos);
    }
}

std::vector<instruction_ref> get_splits(instruction_ref ins)
{
    std::vector<instruction_ref> result;
    std::copy_if(ins->outputs().begin(),
                 ins->outputs().end(),
                 std::back_inserter(result),
                 [&](auto i) { return i->name() == "slice"; });
    if(result.size() < 2)
        return {};
    auto get_slice = [](auto& i) -> auto& { return any_cast<op::slice>(i->get_operator()); };
    auto&& axes    = get_slice(result.front()).axes;
    if(std::any_of(result.begin(), result.end(), [&](auto i) { return get_slice(i).axes != axes; }))
        return {};
    auto get_start = [&](auto& i) -> auto& { return get_slice(i).starts; };
    auto get_end   = [&](auto& i) -> auto& { return get_slice(i).ends; };
    std::sort(
        result.begin(), result.end(), [&](auto x, auto y) { return get_start(x) < get_start(y); });
    if(std::any_of(get_start(result.front()).begin(), get_start(result.front()).end(), [&](auto i) {
           return i != 0;
       }))
        return {};
    auto it = std::adjacent_find(
        result.begin(), result.end(), [&](auto x, auto y) { return get_end(x) != get_start(y); });
    if(it != result.end())
        return {};
    for(std::size_t i = 0; i < axes.size(); i++)
    {
        auto axis = axes[i];
        if(ins->get_shape().lens()[axis] != get_slice(result.back()).ends[i])
            return {};
    }
    return result;
}

struct find_splits
{
    auto matcher() const
    {
        return match::any(
            match::any_of[match::outputs()](match::name("slice")(match::any_of[match::outputs()](
                match::pointwise(match::any_of(match::nargs(1), match::nargs(2))), reduction()))));
    }

    static bool is_dependent(const module& m, instruction_ref ins1, instruction_ref ins2)
    {

        std::unordered_set<instruction_ref> traversed;
        return fix<bool>([&](auto self, auto ins) -> bool {
            if(ins == ins2)
                return true;

            if(contains(traversed, ins))
                return false;

            traversed.insert(ins);
            const auto& inputs = ins->inputs();
            return std::any_of(inputs.begin(), inputs.end(), [&](auto in) {
                return m.has_instruction(in) and self(in);
            });
        })(ins1);
    }

    static std::vector<std::vector<instruction_ref>>
    get_split_groups(const module& m, const std::vector<instruction_ref>& splits)
    {
        std::vector<std::vector<instruction_ref>> groups;
        for(auto out : splits.front()->outputs())
        {
            if(out->name() == "slice")
                continue;
            std::vector<instruction_ref> group;
            for(auto split : splits)
            {
                auto it =
                    std::find_if(split->outputs().begin(), split->outputs().end(), [&](auto i) {
                        return i->get_operator() == out->get_operator();
                    });
                if(it == split->outputs().end())
                    break;
                assert((*it)->name() != "slice");

                // If there is a duplicate bail
                // there are should be no dependency between instructions in the group
                if(std::any_of(group.begin(), group.end(), [&](auto i) {
                       return is_dependent(m, *it, i) or is_dependent(m, i, *it);
                   }))
                {
                    return {};
                }

                group.push_back(*it);
            }
            if(group.size() != splits.size())
                continue;
            groups.push_back(group);
        }
        return groups;
    }

    bool is_fusable(instruction_ref start, instruction_ref split_front) const
    {
        auto op = start->get_operator();
        if(contains(op.name(), "reduce"))
        {
            auto slc         = any_cast<op::slice>(split_front->get_operator());
            auto slc_axes    = slc.axes;
            auto reduce_axes = start->get_operator().to_value()["axes"].to_vector<int64_t>();
            // axes of slice and reduce op cannot have overlap
            if(std::any_of(slc_axes.begin(), slc_axes.end(), [&](auto axis) {
                   return (std::find(reduce_axes.begin(), reduce_axes.end(), axis) !=
                           reduce_axes.end());
               }))
            {
                return false;
            }
        }
        else if(not op.attributes().contains("pointwise"))
        {
            return false;
        }

        return true;
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins    = r.result;
        auto splits = get_splits(ins);
        if(splits.empty())
            return;

        for(const auto& group : get_split_groups(m, splits))
        {
            auto start       = group.front();
            auto split_front = splits.front();
            auto op          = start->get_operator();
            if(not is_fusable(start, split_front))
            {
                continue;
            }

            // Make sure there is no duplicates
            assert(std::none_of(
                std::next(group.begin()), group.end(), [&](auto i) { return i == start; }));

            auto split_idx    = 0;
            instruction_ref c = m.end();
            if(start->inputs().size() == 1)
            {
                c = m.insert_instruction(std::next(ins), op, ins);
            }
            else if(start->inputs().size() == 2)
            {
                assert(not std::none_of(start->inputs().begin(), start->inputs().end(), [](auto i) {
                    return i->name() == "slice";
                }) && "one argument must be a split");
                auto data_idx = 1;
                if(start->inputs().back()->name() == "slice")
                {
                    split_idx = 1;
                    data_idx  = 0;
                }

                std::vector<instruction_ref> data_args;
                std::transform(group.begin(),
                               group.end(),
                               std::back_inserter(data_args),
                               [&](auto i) { return i->inputs()[data_idx]; });

                // Data arguments must be a constant
                if(std::any_of(data_args.begin(), data_args.end(), [](auto i) {
                       return not i->can_eval();
                   }))
                    return;

                move_instructions_back(m, ins, data_args);

                auto slice_op = any_cast<op::slice>(splits.front()->get_operator());
                assert(not slice_op.axes.empty());
                if(slice_op.axes.size() > 1)
                    return;
                auto concat_axis = slice_op.axes.front();
                // TODO: Check if axises match
                auto concat = m.insert_instruction(
                    ins, make_op("concat", {{"axis", concat_axis}}), data_args);

                std::vector<instruction_ref> args;
                args.resize(2);
                args[split_idx] = ins;
                args[data_idx]  = concat;
                c               = m.insert_instruction(std::next(ins), op, args);
            }
            if(c != m.end())
            {
                for(auto i : group)
                {
                    auto split = i->inputs()[split_idx];
                    assert(split->name() == "slice");
                    // Insert contiguous for reshapes
                    auto outputs = i->outputs();
                    for(auto output : outputs)
                    {
                        if(output->name() != "reshape")
                            continue;
                        auto x = m.insert_instruction(output, make_op("contiguous"), i);
                        m.replace_instruction(output, output->get_operator(), x);
                    }

                    m.replace_instruction(i, split->get_operator(), c);
                }
            }
        }
    }
};

struct find_split_concat
{
    auto matcher() const
    {
        return match::any(match::any_of[match::outputs()](
            match::name("slice")(match::all_of[match::outputs()](match::name("concat")))));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins = r.result;

        auto splits = get_splits(ins);
        if(splits.empty())
            return;
        if(std::any_of(
               splits.begin(), splits.end(), [](auto i) { return i->outputs().size() != 1; }))
            return;
        // Check for concat operator
        auto concat = splits.front()->outputs().front();
        if(std::any_of(splits.begin(), splits.end(), [&](auto i) {
               return i->outputs().front() != concat;
           }))
            return;
        // Check axis match
        auto concat_op = any_cast<op::concat>(concat->get_operator());
        auto split_op  = any_cast<op::slice>(splits.front()->get_operator());
        if(split_op.axes.size() != 1)
            return;
        if(split_op.axes.front() != concat_op.axis)
            return;
        // Replace args
        auto args = concat->inputs();
        auto it =
            std::find_if(args.begin(), args.end(), [&](auto i) { return i == splits.front(); });
        if(std::distance(it, args.end()) < splits.size())
            return;
        // If the slices are not in order then stop
        if(not std::is_sorted(it, it + splits.size(), [](instruction_ref x, instruction_ref y) {
               auto xop = any_cast<op::slice>(x->get_operator());
               auto yop = any_cast<op::slice>(y->get_operator());
               return std::tie(xop.starts, xop.ends) < std::tie(yop.starts, yop.ends);
           }))
            return;
        *it = splits.front()->inputs().front();
        args.erase(std::next(it), it + splits.size());

        if(args.size() == 1)
            m.replace_instruction(concat, args.front());
        else
            m.replace_instruction(concat, concat->get_operator(), args);
    }
};

bool axis_equal(const std::vector<std::size_t>& x,
                const std::vector<std::size_t>& y,
                std::size_t axis)
{
    return x.size() == y.size() and x.size() > axis and
           std::equal(x.begin(), x.begin() + axis, y.begin()) and
           std::equal(x.begin() + axis + 1, x.end(), y.begin() + axis + 1);
}

bool axis_shape_equal(const shape& x, const shape& y, std::size_t axis)
{
    // TODO: Check strides
    return axis_equal(x.lens(), y.lens(), axis);
}

struct find_add_convs
{
    auto matcher() const
    {
        return match::name("add")(
            match::args(conv_const_weights().bind("a"), conv_const_weights().bind("b")));
    }

    static bool symmetrical_strides(const op::convolution& op)
    {
        return op.stride[0] == op.stride[1];
    }

    static std::size_t compute_stride_factor(const op::convolution& x, const op::convolution& y)
    {
        if(not symmetrical_strides(x))
            return 0;
        if(not symmetrical_strides(y))
            return 0;
        if((x.stride[0] % y.stride[0]) != 0)
            return 0;
        return x.stride[0] / y.stride[0];
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins       = r.result;
        auto a_conv    = r.instructions["a"];
        auto a_input   = a_conv->inputs().at(0);
        auto a_weights = a_conv->inputs().at(1);
        auto b_conv    = r.instructions["b"];
        auto b_input   = b_conv->inputs().at(0);
        auto b_weights = b_conv->inputs().at(1);

        if(not axis_shape_equal(a_weights->get_shape(), b_weights->get_shape(), 1))
            return;

        auto a_op   = any_cast<op::convolution>(a_conv->get_operator());
        auto b_op   = any_cast<op::convolution>(b_conv->get_operator());
        auto new_op = a_op;

        if(a_op != b_op)
        {
            if(std::tie(a_op.padding, a_op.dilation, a_op.group) ==
                   std::tie(b_op.padding, b_op.dilation, b_op.group) and
               a_weights->get_shape().lens()[2] == 1 and a_weights->get_shape().lens()[3] == 1)
            {
                if(a_op.stride < b_op.stride)
                {
                    auto n = compute_stride_factor(b_op, a_op);
                    if(n == 0)
                        return;
                    new_op  = a_op;
                    b_input = m.insert_instruction(
                        ins, make_op("step", {{"axes", {2, 3}}, {"steps", {n, n}}}), b_input);
                }
                else if(b_op.stride < a_op.stride)
                {
                    auto n = compute_stride_factor(a_op, b_op);
                    if(n == 0)
                        return;
                    new_op  = b_op;
                    a_input = m.insert_instruction(
                        ins, make_op("step", {{"axes", {2, 3}}, {"steps", {n, n}}}), a_input);
                }
                else
                    return;
            }
            else
                return;
        }

        auto concat_input =
            m.insert_instruction(ins, make_op("concat", {{"axis", 1}}), a_input, b_input);
        auto concat_weights =
            m.insert_instruction(ins, make_op("concat", {{"axis", 1}}), a_weights, b_weights);
        m.replace_instruction(ins, new_op, concat_input, concat_weights);
    }
};

MIGRAPHX_PRED_MATCHER(horiz_conv_dot, instruction_ref ins)
{
    auto pred = [&](auto name) {
        return [=](auto i) {
            return i->name() == name and i->inputs().front() == ins and
                   i->inputs().at(1)->can_eval();
        };
    };
    auto dots  = std::count_if(ins->outputs().begin(), ins->outputs().end(), pred("dot"));
    auto convs = std::count_if(ins->outputs().begin(), ins->outputs().end(), pred("convolution"));
    return not(dots < 2 and convs < 2);
}

struct find_conv_dot_horiz_fusion
{
    auto matcher() const { return horiz_conv_dot(); }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins = r.result;

        auto pred = [](auto i, auto j) {
            if(i->get_operator() != j->get_operator())
                return false;
            if(not contains({"dot", "convolution"}, i->name()))
                return true;
            auto x = i->inputs()[1]->get_shape().lens();
            auto y = j->inputs()[1]->get_shape().lens();
            if(x.size() != y.size())
                return false;
            // Check that non-axes match
            int axis = 1;
            if(i->name() == "dot")
            {
                axis = x.size() - 1;
            }
            return axis_equal(x, y, axis);
        };

        auto each = [&](auto start, auto last) {
            if(std::distance(start, last) < 2)
                return;
            auto&& name = (*start)->name();
            if(not contains({"dot", "convolution"}, name))
                return;
            auto op   = (*start)->get_operator();
            int group = 1;
            if(name == "convolution")
                group = any_cast<op::convolution>(op).group;
            // Skip group convolution
            if(group != 1)
                return;
            auto input = (*start)->inputs().front();
            std::vector<instruction_ref> args;
            std::transform(
                start, last, std::back_inserter(args), [&](auto x) { return x->inputs().at(1); });
            int axis        = 1;
            int concat_axis = 0;
            if(name == "dot")
            {
                axis        = int(args.front()->get_shape().lens().size() - 1);
                concat_axis = axis;
            }

            move_instructions_back(m, input, args);
            // TODO: Check if axes match
            auto concat =
                m.insert_instruction(input, make_op("concat", {{"axis", concat_axis}}), args);
            auto fused     = m.insert_instruction(std::next(input), op, input, concat);
            int64_t offset = 0;
            for(auto arg : range(start, last))
            {
                auto outputs = arg->outputs();
                for(auto output : outputs)
                {
                    if(output->name() != "reshape")
                        continue;
                    auto x = m.insert_instruction(output, make_op("contiguous"), arg);
                    m.replace_instruction(output, output->get_operator(), x);
                }

                int64_t len = arg->get_shape().lens()[axis];
                m.replace_instruction(
                    arg,
                    make_op("slice",
                            {{"axes", {axis}}, {"starts", {offset}}, {"ends", {offset + len}}}),
                    fused);
                offset += len;
            }
        };

        auto outputs = ins->outputs();
        group_by(outputs.begin(), outputs.end(), each, pred);
    }
};

struct find_div_const
{
    auto matcher() const
    {
        return match::name("div")(match::arg(1)(match::is_constant().bind("c")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto c_ins = r.instructions["c"];

        auto recip = m.insert_instruction(std::next(c_ins), make_op("recip"), c_ins);

        auto args = ins->inputs();

        m.replace_instruction(ins, make_op("mul"), args.front(), recip);
    }
};

struct find_sub_const
{
    auto matcher() const
    {
        return match::name("sub")(match::arg(1)(match::is_constant().bind("c")));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto c_ins = r.instructions["c"];

        auto neg = m.insert_instruction(std::next(c_ins), make_op("neg"), c_ins);

        auto args = ins->inputs();

        m.replace_instruction(ins, make_op("add"), args.front(), neg);
    }
};

struct find_rsqrt
{
    auto matcher() const
    {
        return match::name("recip")(match::args(
            match::name("sqrt")(match::used_once(), match::args(match::any().bind("x")))));
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto ins   = r.result;
        auto x_ins = r.instructions["x"];

        m.replace_instruction(ins, make_op("rsqrt"), x_ins);
    }
};

static bool same_ops(const std::vector<instruction_ref>& vec_ins)
{
    return std::all_of(vec_ins.begin(), vec_ins.end(), [&](auto i) {
        return i->get_operator() == vec_ins.front()->get_operator();
    });
}

struct find_split_reshape
{
    auto matcher() const
    {
        return match::name("reshape")(match::arg(0)(match::name("contiguous")(
                                          match::arg(0)(match::name("slice").bind("slice")))))
            .bind("reshape");
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto slc = r.instructions["slice"];
        auto rsp = r.instructions["reshape"];

        auto input         = slc->inputs().front();
        auto split_outputs = get_splits(input);
        if(split_outputs.empty())
        {
            return;
        }

        std::vector<instruction_ref> vec_rsp(split_outputs.size());
        std::transform(split_outputs.begin(), split_outputs.end(), vec_rsp.begin(), [](auto i) {
            assert(i->outputs().size() == 1);
            auto cont = i->outputs().front();
            assert(cont->outputs().size() == 1);
            return cont->outputs().front();
        });

        // all outputs are reshape and of the same shape
        auto dims = any_cast<op::reshape>(rsp->get_operator()).dims;
        if(not same_ops(vec_rsp))
        {
            return;
        }

        // ensure reshape happens after the axis dimension
        auto axis         = any_cast<op::slice>(slc->get_operator()).axes[0];
        auto slc_lens     = slc->get_shape().lens();
        auto slc_dim_size = std::accumulate(
            slc_lens.begin() + axis, slc_lens.end(), 1, std::multiplies<std::size_t>());

        // search the reshape output (standard shape) to decide which axis are
        // in its output corresponding to the slc_dim_size
        auto rsp_lens    = rsp->get_shape().lens();
        auto rsp_strides = rsp->get_shape().strides();
        rsp_strides.insert(rsp_strides.begin(), rsp_strides[0] * rsp_lens[0]);
        auto ait = std::find(rsp_strides.begin(), rsp_strides.end(), slc_dim_size);
        if(ait == rsp_strides.end())
        {
            return;
        }
        int rsp_axis = std::distance(rsp_strides.begin(), ait);

        // calculate reshape output shape
        std::vector<int64_t> vec_dims(vec_rsp.size());
        std::transform(vec_rsp.begin(), vec_rsp.end(), vec_dims.begin(), [&](auto is) {
            return is->get_shape().lens()[rsp_axis];
        });

        std::vector<int64_t> rsp_out_lens(rsp_lens.begin(), rsp_lens.end());
        rsp_out_lens[rsp_axis] = std::accumulate(vec_dims.begin(), vec_dims.end(), std::int64_t{0});

        // insert the reshape instruction and add contiguous if needed
        if(not input->get_shape().standard())
        {
            input = m.insert_instruction(std::next(input), make_op("contiguous"), input);
        }
        auto rsp_ins = m.insert_instruction(
            std::next(input), make_op("reshape", {{"dims", rsp_out_lens}}), input);

        // replace the original reshape with slice
        int64_t start = 0;
        for(std::size_t i = 0; i < vec_rsp.size(); ++i)
        {
            m.replace_instruction(
                vec_rsp[i],
                make_op(
                    "slice",
                    {{"axes", {rsp_axis}}, {"starts", {start}}, {"ends", {start + vec_dims[i]}}}),
                rsp_ins);
            start += vec_dims[i];
        }
    }
};

struct find_split_transpose
{
    auto matcher() const
    {
        return match::name("transpose")(match::arg(0)(match::name("slice").bind("slice")))
            .bind("trans");
    }

    void apply(module& m, const match::matcher_result& r) const
    {
        auto slc   = r.instructions["slice"];
        auto trans = r.instructions["trans"];

        auto input         = slc->inputs().front();
        auto split_outputs = get_splits(input);
        if(split_outputs.empty())
        {
            return;
        }

        std::vector<instruction_ref> vec_trans(split_outputs.size());
        std::transform(split_outputs.begin(), split_outputs.end(), vec_trans.begin(), [](auto i) {
            assert(i->outputs().size() == 1);
            return i->outputs().front();
        });

        // all transpose are the same
        auto perm = any_cast<op::transpose>(trans->get_operator()).dims;
        if(not same_ops(vec_trans))
        {
            return;
        }

        // insert an transpose instruction
        auto tr = m.insert_instruction(
            std::next(input), make_op("transpose", {{"permutation", perm}}), input);

        // compute the axis in the slice
        auto axis = any_cast<op::slice>(slc->get_operator()).axes.front();
        auto it   = std::find(perm.begin(), perm.end(), axis);
        assert(it != perm.end());
        int64_t axis_new = std::distance(perm.begin(), it);

        for(auto in : split_outputs)
        {
            auto oper    = any_cast<op::slice>(in->get_operator());
            auto starts  = oper.starts;
            auto ends    = oper.ends;
            auto tr_orig = in->outputs().front();
            m.replace_instruction(
                tr_orig,
                make_op("slice", {{"axes", {axis_new}}, {"starts", starts}, {"ends", ends}}),
                tr);
        }
    }
};

void simplify_algebra::apply(module& m) const
{
    // Run simplifications multiple times
    for(int i = 0; i < 8; i++)
    {
        match::find_matches(m,
                            find_inner_broadcast{},
                            find_double_add_lit_broadcast{},
                            find_add_lit_broadcast{},
                            find_add_convs{},
                            find_conv_dot_horiz_fusion{},
                            find_mul_conv{},
                            find_mul_slice_conv{},
                            find_mul_add{},
                            find_dot_add{},
                            find_div_const{},
                            find_sub_const{},
                            find_rsqrt{},
                            find_concat_op{},
                            find_split_concat{},
                            find_splits{},
                            find_split_reshape{},
                            find_split_transpose{});
        dead_code_elimination{}.apply(m);
    }
}

} // namespace MIGRAPHX_INLINE_NS
} // namespace migraphx