xfunction.hpp

/***************************************************************************
 * Copyright (c) Johan Mabille, Sylvain Corlay and Wolf Vollprecht          *
 * Copyright (c) QuantStack                                                 *
 *                                                                          *
 * Distributed under the terms of the BSD 3-Clause License.                 *
 *                                                                          *
 * The full license is in the file LICENSE, distributed with this software. *
 ****************************************************************************/

#ifndef XTENSOR_FUNCTION_HPP
#define XTENSOR_FUNCTION_HPP

#include <algorithm>
#include <cstddef>
#include <iterator>
#include <numeric>
#include <tuple>
#include <type_traits>
#include <utility>

#include <xtl/xsequence.hpp>
#include <xtl/xtype_traits.hpp>

#include "xaccessible.hpp"
#include "xexpression_traits.hpp"
#include "xiterable.hpp"
#include "xiterator.hpp"
#include "xlayout.hpp"
#include "xscalar.hpp"
#include "xshape.hpp"
#include "xstrides.hpp"
#include "xtensor_simd.hpp"
#include "xutils.hpp"

namespace xt
{
    namespace detail
    {

        template <bool... B>
        using conjunction_c = xtl::conjunction<std::integral_constant<bool, B>...>;

        /************************
         * xfunction_cache_impl *
         ************************/

        template <class S, class is_shape_trivial>
        struct xfunction_cache_impl
        {
            S shape;
            bool is_trivial;
            bool is_initialized;

            xfunction_cache_impl()
                : shape(xtl::make_sequence<S>(0, std::size_t(0)))
                , is_trivial(false)
                , is_initialized(false)
            {
            }
        };

        template <std::size_t... N, class is_shape_trivial>
        struct xfunction_cache_impl<fixed_shape<N...>, is_shape_trivial>
        {
            XTENSOR_CONSTEXPR_ENHANCED_STATIC fixed_shape<N...> shape = fixed_shape<N...>();
            XTENSOR_CONSTEXPR_ENHANCED_STATIC bool is_trivial = is_shape_trivial::value;
            XTENSOR_CONSTEXPR_ENHANCED_STATIC bool is_initialized = true;
        };

#ifdef XTENSOR_HAS_CONSTEXPR_ENHANCED
        // Out of line definitions to prevent linker errors prior to C++17
        template <std::size_t... N, class is_shape_trivial>
        constexpr fixed_shape<N...> xfunction_cache_impl<fixed_shape<N...>, is_shape_trivial>::shape;

        template <std::size_t... N, class is_shape_trivial>
        constexpr bool xfunction_cache_impl<fixed_shape<N...>, is_shape_trivial>::is_trivial;

        template <std::size_t... N, class is_shape_trivial>
        constexpr bool xfunction_cache_impl<fixed_shape<N...>, is_shape_trivial>::is_initialized;
#endif

        template <class... CT>
        struct xfunction_bool_load_type
        {
            using type = xtl::promote_type_t<typename std::decay_t<CT>::bool_load_type...>;
        };

        template <class CT>
        struct xfunction_bool_load_type<CT>
        {
            using type = typename std::decay_t<CT>::bool_load_type;
        };

        template <class... CT>
        using xfunction_bool_load_type_t = typename xfunction_bool_load_type<CT...>::type;
    }

    /************************
     * xfunction extensions *
     ************************/

    namespace extension
    {

        template <class Tag, class F, class... CT>
        struct xfunction_base_impl;

        template <class F, class... CT>
        struct xfunction_base_impl<xtensor_expression_tag, F, CT...>
        {
            using type = xtensor_empty_base;
        };

        template <class F, class... CT>
        struct xfunction_base : xfunction_base_impl<xexpression_tag_t<CT...>, F, CT...>
        {
        };

        template <class F, class... CT>
        using xfunction_base_t = typename xfunction_base<F, CT...>::type;
    }

    template <class promote>
    struct xfunction_cache : detail::xfunction_cache_impl<typename promote::type, promote>
    {
    };

    template <class F, class... CT>
    class xfunction_iterator;

    template <class F, class... CT>
    class xfunction_stepper;

    template <class F, class... CT>
    class xfunction;

    template <class F, class... CT>
    struct xiterable_inner_types<xfunction<F, CT...>>
    {
        using inner_shape_type = promote_shape_t<typename std::decay_t<CT>::shape_type...>;
        using const_stepper = xfunction_stepper<F, CT...>;
        using stepper = const_stepper;
    };

    template <class F, class... CT>
    struct xcontainer_inner_types<xfunction<F, CT...>>
    {
        // Added indirection for MSVC 2017 bug with the operator value_type()
        using func_return_type = typename meta_identity<
            decltype(std::declval<F>()(std::declval<xvalue_type_t<std::decay_t<CT>>>()...))>::type;
        using value_type = std::decay_t<func_return_type>;
        using reference = func_return_type;
        using const_reference = reference;
        using size_type = common_size_type_t<std::decay_t<CT>...>;
    };

    template <class T, class F, class... CT>
    struct has_simd_interface<xfunction<F, CT...>, T> : xtl::conjunction<
                                                            has_simd_type<T>,
                                                            has_simd_apply<F, xt_simd::simd_type<T>>,
                                                            has_simd_interface<std::decay_t<CT>, T>...>
    {
    };

    /*************************************
     * overlapping_memory_checker_traits *
     *************************************/

    template <class E>
    struct overlapping_memory_checker_traits<
        E,
        std::enable_if_t<!has_memory_address<E>::value && is_specialization_of<xfunction, E>::value>>
    {
        template <std::size_t I = 0, class... T, std::enable_if_t<(I == sizeof...(T)), int> = 0>
        static bool check_tuple(const std::tuple<T...>&, const memory_range&)
        {
            return false;
        }

        template <std::size_t I = 0, class... T, std::enable_if_t<(I < sizeof...(T)), int> = 0>
        static bool check_tuple(const std::tuple<T...>& t, const memory_range& dst_range)
        {
            using ChildE = std::decay_t<decltype(std::get<I>(t))>;
            return overlapping_memory_checker_traits<ChildE>::check_overlap(std::get<I>(t), dst_range)
                   || check_tuple<I + 1>(t, dst_range);
        }

        static bool check_overlap(const E& expr, const memory_range& dst_range)
        {
            if (expr.size() == 0)
            {
                return false;
            }
            else
            {
                return check_tuple(expr.arguments(), dst_range);
            }
        }
    };

    /*************
     * xfunction *
     *************/

    /**
     * @class xfunction
     * @brief Multidimensional function operating on
     * xtensor expressions.
     *
     * The xfunction class implements a multidimensional function
     * operating on xtensor expressions.
     *
     * @tparam F the function type
     * @tparam CT the closure types for arguments of the function
     */
    template <class F, class... CT>
    class xfunction : private xconst_iterable<xfunction<F, CT...>>,
                      public xsharable_expression<xfunction<F, CT...>>,
                      private xconst_accessible<xfunction<F, CT...>>,
                      public extension::xfunction_base_t<F, CT...>
    {
    public:

        using self_type = xfunction<F, CT...>;
        using accessible_base = xconst_accessible<self_type>;
        using extension_base = extension::xfunction_base_t<F, CT...>;
        using expression_tag = typename extension_base::expression_tag;
        using only_scalar = all_xscalar<CT...>;
        using functor_type = typename std::remove_reference<F>::type;
        using tuple_type = std::tuple<CT...>;

        using inner_types = xcontainer_inner_types<self_type>;
        using value_type = typename inner_types::value_type;
        using reference = typename inner_types::reference;
        using const_reference = typename inner_types::const_reference;
        using pointer = value_type*;
        using const_pointer = const value_type*;
        using size_type = typename inner_types::size_type;
        using difference_type = common_difference_type_t<std::decay_t<CT>...>;

        using simd_value_type = xt_simd::simd_type<value_type>;

        // xtl::promote_type_t<typename std::decay_t<CT>::bool_load_type...>;
        using bool_load_type = detail::xfunction_bool_load_type_t<CT...>;

        template <class requested_type>
        using simd_return_type = xt_simd::simd_return_type<value_type, requested_type>;

        using iterable_base = xconst_iterable<xfunction<F, CT...>>;
        using inner_shape_type = typename iterable_base::inner_shape_type;
        using shape_type = inner_shape_type;

        using stepper = typename iterable_base::stepper;
        using const_stepper = typename iterable_base::const_stepper;

        static constexpr layout_type static_layout = compute_layout(std::decay_t<CT>::static_layout...);
        static constexpr bool contiguous_layout = static_layout != layout_type::dynamic;

        template <layout_type L>
        using layout_iterator = typename iterable_base::template layout_iterator<L>;
        template <layout_type L>
        using const_layout_iterator = typename iterable_base::template const_layout_iterator<L>;
        template <layout_type L>
        using reverse_layout_iterator = typename iterable_base::template reverse_layout_iterator<L>;
        template <layout_type L>
        using const_reverse_layout_iterator = typename iterable_base::template const_reverse_layout_iterator<L>;

        template <class S, layout_type L>
        using broadcast_iterator = typename iterable_base::template broadcast_iterator<S, L>;
        template <class S, layout_type L>
        using const_broadcast_iterator = typename iterable_base::template const_broadcast_iterator<S, L>;
        template <class S, layout_type L>
        using reverse_broadcast_iterator = typename iterable_base::template reverse_broadcast_iterator<S, L>;
        template <class S, layout_type L>
        using const_reverse_broadcast_iterator = typename iterable_base::template const_reverse_broadcast_iterator<S, L>;

        using const_linear_iterator = xfunction_iterator<F, CT...>;
        using linear_iterator = const_linear_iterator;
        using const_reverse_linear_iterator = std::reverse_iterator<const_linear_iterator>;
        using reverse_linear_iterator = std::reverse_iterator<linear_iterator>;

        using iterator = typename iterable_base::iterator;
        using const_iterator = typename iterable_base::const_iterator;
        using reverse_iterator = typename iterable_base::reverse_iterator;
        using const_reverse_iterator = typename iterable_base::const_reverse_iterator;

        template <class Func, class... CTA, class U = std::enable_if_t<!std::is_base_of<std::decay_t<Func>, self_type>::value>>
        xfunction(Func&& f, CTA&&... e) noexcept;

        template <class FA, class... CTA>
        xfunction(xfunction<FA, CTA...> xf) noexcept;

        ~xfunction() = default;

        xfunction(const xfunction&) = default;
        xfunction& operator=(const xfunction&) = default;

        xfunction(xfunction&&) = default;
        xfunction& operator=(xfunction&&) = default;

        using accessible_base::size;
        size_type dimension() const noexcept;
        const inner_shape_type& shape() const;
        layout_type layout() const noexcept;
        bool is_contiguous() const noexcept;
        using accessible_base::shape;

        template <class... Args>
        const_reference operator()(Args... args) const;

        template <class... Args>
        const_reference unchecked(Args... args) const;

        using accessible_base::at;
        using accessible_base::operator[];
        using accessible_base::back;
        using accessible_base::front;
        using accessible_base::in_bounds;
        using accessible_base::periodic;

        template <class It>
        const_reference element(It first, It last) const;

        template <class S>
        bool broadcast_shape(S& shape, bool reuse_cache = false) const;

        template <class S>
        bool has_linear_assign(const S& strides) const noexcept;

        using iterable_base::begin;
        using iterable_base::cbegin;
        using iterable_base::cend;
        using iterable_base::crbegin;
        using iterable_base::crend;
        using iterable_base::end;
        using iterable_base::rbegin;
        using iterable_base::rend;

        const_linear_iterator linear_begin() const noexcept;
        const_linear_iterator linear_end() const noexcept;
        const_linear_iterator linear_cbegin() const noexcept;
        const_linear_iterator linear_cend() const noexcept;

        const_reverse_linear_iterator linear_rbegin() const noexcept;
        const_reverse_linear_iterator linear_rend() const noexcept;
        const_reverse_linear_iterator linear_crbegin() const noexcept;
        const_reverse_linear_iterator linear_crend() const noexcept;

        template <class S>
        const_stepper stepper_begin(const S& shape) const noexcept;
        template <class S>
        const_stepper stepper_end(const S& shape, layout_type l) const noexcept;

        const_reference data_element(size_type i) const;

        const_reference flat(size_type i) const;

        template <class UT = self_type, class = typename std::enable_if<UT::only_scalar::value>::type>
        operator value_type() const;

        template <class align, class requested_type = value_type, std::size_t N = xt_simd::simd_traits<requested_type>::size>
        simd_return_type<requested_type> load_simd(size_type i) const;

        const tuple_type& arguments() const noexcept;

        const functor_type& functor() const noexcept;

    private:

        template <std::size_t... I>
        layout_type layout_impl(std::index_sequence<I...>) const noexcept;

        template <std::size_t... I, class... Args>
        const_reference access_impl(std::index_sequence<I...>, Args... args) const;

        template <std::size_t... I, class... Args>
        const_reference unchecked_impl(std::index_sequence<I...>, Args... args) const;

        template <std::size_t... I, class It>
        const_reference element_access_impl(std::index_sequence<I...>, It first, It last) const;

        template <std::size_t... I>
        const_reference data_element_impl(std::index_sequence<I...>, size_type i) const;

        template <class align, class requested_type, std::size_t N, std::size_t... I>
        auto load_simd_impl(std::index_sequence<I...>, size_type i) const;

        template <class Func, std::size_t... I>
        const_stepper build_stepper(Func&& f, std::index_sequence<I...>) const noexcept;

        template <class Func, std::size_t... I>
        auto build_iterator(Func&& f, std::index_sequence<I...>) const noexcept;

        size_type compute_dimension() const noexcept;

        void compute_cached_shape() const;

        tuple_type m_e;
        functor_type m_f;
        mutable xfunction_cache<detail::promote_index<typename std::decay_t<CT>::shape_type...>> m_cache{};

        friend class xfunction_iterator<F, CT...>;
        friend class xfunction_stepper<F, CT...>;
        friend class xconst_iterable<self_type>;
        friend class xconst_accessible<self_type>;
    };

    /**********************
     * xfunction_iterator *
     **********************/

    template <class F, class... CT>
    class xfunction_iterator : public xtl::xrandom_access_iterator_base<
                                   xfunction_iterator<F, CT...>,
                                   typename xfunction<F, CT...>::value_type,
                                   typename xfunction<F, CT...>::difference_type,
                                   typename xfunction<F, CT...>::pointer,
                                   typename xfunction<F, CT...>::reference>
    {
    public:

        using self_type = xfunction_iterator<F, CT...>;
        using functor_type = typename std::remove_reference<F>::type;
        using xfunction_type = xfunction<F, CT...>;

        using value_type = typename xfunction_type::value_type;
        using reference = typename xfunction_type::value_type;
        using pointer = typename xfunction_type::const_pointer;
        using difference_type = typename xfunction_type::difference_type;
        using iterator_category = std::random_access_iterator_tag;

        template <class... It>
        xfunction_iterator(const xfunction_type* func, It&&... it) noexcept;

        self_type& operator++();
        self_type& operator--();

        self_type& operator+=(difference_type n);
        self_type& operator-=(difference_type n);

        difference_type operator-(const self_type& rhs) const;

        reference operator*() const;

        bool equal(const self_type& rhs) const;
        bool less_than(const self_type& rhs) const;

    private:

        using data_type = std::tuple<decltype(xt::linear_begin(std::declval<const std::decay_t<CT>>()))...>;

        template <std::size_t... I>
        reference deref_impl(std::index_sequence<I...>) const;

        template <std::size_t... I>
        difference_type
        tuple_max_diff(std::index_sequence<I...>, const data_type& lhs, const data_type& rhs) const;

        const xfunction_type* p_f;
        data_type m_it;
    };

    template <class F, class... CT>
    bool operator==(const xfunction_iterator<F, CT...>& it1, const xfunction_iterator<F, CT...>& it2);

    template <class F, class... CT>
    bool operator<(const xfunction_iterator<F, CT...>& it1, const xfunction_iterator<F, CT...>& it2);

    /*********************
     * xfunction_stepper *
     *********************/

    template <class F, class... CT>
    class xfunction_stepper
    {
    public:

        using self_type = xfunction_stepper<F, CT...>;
        using functor_type = typename std::remove_reference<F>::type;
        using xfunction_type = xfunction<F, CT...>;

        using value_type = typename xfunction_type::value_type;
        using reference = typename xfunction_type::reference;
        using pointer = typename xfunction_type::const_pointer;
        using size_type = typename xfunction_type::size_type;
        using difference_type = typename xfunction_type::difference_type;

        using shape_type = typename xfunction_type::shape_type;

        template <class requested_type>
        using simd_return_type = xt_simd::simd_return_type<value_type, requested_type>;

        template <class... St>
        xfunction_stepper(const xfunction_type* func, St&&... st) noexcept;

        void step(size_type dim);
        void step_back(size_type dim);
        void step(size_type dim, size_type n);
        void step_back(size_type dim, size_type n);
        void reset(size_type dim);
        void reset_back(size_type dim);

        void to_begin();
        void to_end(layout_type l);

        reference operator*() const;

        template <class T>
        simd_return_type<T> step_simd();

        void step_leading();

    private:

        template <std::size_t... I>
        reference deref_impl(std::index_sequence<I...>) const;

        template <class T, std::size_t... I>
        simd_return_type<T> step_simd_impl(std::index_sequence<I...>);

        const xfunction_type* p_f;
        std::tuple<typename std::decay_t<CT>::const_stepper...> m_st;
    };

    /*********************************
     * xfunction implementation *
     *********************************/

    /**
     * @name Constructor
     */
    //@{
    /**
     * Constructs an xfunction applying the specified function to the given
     * arguments.
     * @param f the function to apply
     * @param e the \ref xexpression arguments
     */
    template <class F, class... CT>
    template <class Func, class... CTA, class U>
    inline xfunction<F, CT...>::xfunction(Func&& f, CTA&&... e) noexcept
        : m_e(std::forward<CTA>(e)...)
        , m_f(std::forward<Func>(f))
    {
    }

    /**
     * Constructs an xfunction applying the specified function given by another
     * xfunction with its arguments.
     * @param xf the xfunction to apply
     */
    template <class F, class... CT>
    template <class FA, class... CTA>
    inline xfunction<F, CT...>::xfunction(xfunction<FA, CTA...> xf) noexcept
        : m_e(xf.arguments())
        , m_f(xf.functor())
    {
    }

    //@}

    /**
     * @name Size and shape
     */
    //@{
    /**
     * Returns the number of dimensions of the function.
     */
    template <class F, class... CT>
    inline auto xfunction<F, CT...>::dimension() const noexcept -> size_type
    {
        size_type dimension = m_cache.is_initialized ? m_cache.shape.size() : compute_dimension();
        return dimension;
    }

    template <class F, class... CT>
    inline void xfunction<F, CT...>::compute_cached_shape() const
    {
        static_assert(!detail::is_fixed<shape_type>::value, "Calling compute_cached_shape on fixed!");

        m_cache.shape = uninitialized_shape<xindex_type_t<inner_shape_type>>(compute_dimension());
        m_cache.is_trivial = broadcast_shape(m_cache.shape, false);
        m_cache.is_initialized = true;
    }

    /**
     * Returns the shape of the xfunction.
     */
    template <class F, class... CT>
    inline auto xfunction<F, CT...>::shape() const -> const inner_shape_type&
    {
        xtl::mpl::static_if<!detail::is_fixed<inner_shape_type>::value>(
            [&](auto self)
            {
                if (!m_cache.is_initialized)
                {
                    self(this)->compute_cached_shape();
                }
            },
            [](auto /*self*/) {}
        );
        return m_cache.shape;
    }

    /**
     * Returns the layout_type of the xfunction.
     */
    template <class F, class... CT>
    inline layout_type xfunction<F, CT...>::layout() const noexcept
    {
        return layout_impl(std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline bool xfunction<F, CT...>::is_contiguous() const noexcept
    {
        return layout() != layout_type::dynamic
               && accumulate(
                   [](bool r, const auto& exp)
                   {
                       return r && exp.is_contiguous();
                   },
                   true,
                   m_e
               );
    }

    //@}

    /**
     * @name Data
     */

    /**
     * Returns a constant reference to the element at the specified position in the function.
     * @param args a list of indices specifying the position in the function. Indices
     * must be unsigned integers, the number of indices should be equal or greater than
     * the number of dimensions of the function.
     */
    template <class F, class... CT>
    template <class... Args>
    inline auto xfunction<F, CT...>::operator()(Args... args) const -> const_reference
    {
        // The static cast prevents the compiler from instantiating the template methods with signed integers,
        // leading to warning about signed/unsigned conversions in the deeper layers of the access methods
        return access_impl(std::make_index_sequence<sizeof...(CT)>(), static_cast<size_type>(args)...);
    }

    /**
     * @name Data
     */

    /**
     * Returns a constant reference to the element at the specified position of the underlying
     * contiguous storage of the function.
     * @param index index to underlying flat storage.
     */
    template <class F, class... CT>
    inline auto xfunction<F, CT...>::flat(size_type index) const -> const_reference
    {
        return data_element_impl(std::make_index_sequence<sizeof...(CT)>(), index);
    }

    /**
     * Returns a constant reference to the element at the specified position in the expression.
     * @param args a list of indices specifying the position in the expression. Indices
     * must be unsigned integers, the number of indices must be equal to the number of
     * dimensions of the expression, else the behavior is undefined.
     *
     * @warning This method is meant for performance, for expressions with a dynamic
     * number of dimensions (i.e. not known at compile time). Since it may have
     * undefined behavior (see parameters), operator() should be preferred whenever
     * it is possible.
     * @warning This method is NOT compatible with broadcasting, meaning the following
     * code has undefined behavior:
     * @code{.cpp}
     * xt::xarray<double> a = {{0, 1}, {2, 3}};
     * xt::xarray<double> b = {0, 1};
     * auto fd = a + b;
     * double res = fd.unchecked(0, 1);
     * @endcode
     */
    template <class F, class... CT>
    template <class... Args>
    inline auto xfunction<F, CT...>::unchecked(Args... args) const -> const_reference
    {
        // The static cast prevents the compiler from instantiating the template methods with signed integers,
        // leading to warning about signed/unsigned conversions in the deeper layers of the access methods
        return unchecked_impl(std::make_index_sequence<sizeof...(CT)>(), static_cast<size_type>(args)...);
    }

    /**
     * Returns a constant reference to the element at the specified position in the function.
     * @param first iterator starting the sequence of indices
     * @param last iterator ending the sequence of indices
     * The number of indices in the sequence should be equal to or greater
     * than the number of dimensions of the container.
     */
    template <class F, class... CT>
    template <class It>
    inline auto xfunction<F, CT...>::element(It first, It last) const -> const_reference
    {
        return element_access_impl(std::make_index_sequence<sizeof...(CT)>(), first, last);
    }

    //@}

    /**
     * @name Broadcasting
     */
    //@{
    /**
     * Broadcast the shape of the function to the specified parameter.
     * @param shape the result shape
     * @param reuse_cache boolean for reusing a previously computed shape
     * @return a boolean indicating whether the broadcasting is trivial
     */
    template <class F, class... CT>
    template <class S>
    inline bool xfunction<F, CT...>::broadcast_shape(S& shape, bool reuse_cache) const
    {
        if (m_cache.is_initialized && reuse_cache)
        {
            // Disable spurious warning when copying into a 1-sized `std::array`
            // in gcc >= 12.4.0; see https://gcc.gnu.org/bugzilla/show_bug.cgi?id=107852
            #pragma GCC diagnostic push
            #pragma GCC diagnostic ignored "-Warray-bounds"
            #pragma GCC diagnostic ignored "-Wstringop-overflow"
            std::copy(m_cache.shape.cbegin(), m_cache.shape.cend(), shape.begin());
            #pragma GCC diagnostic pop
            return m_cache.is_trivial;
        }
        else
        {
            // e.broadcast_shape must be evaluated even if b is false
            auto func = [&shape](bool b, auto&& e)
            {
                return e.broadcast_shape(shape) && b;
            };
            return accumulate(func, true, m_e);
        }
    }

    /**
     * Checks whether the xfunction can be linearly assigned to an expression
     * with the specified strides.
     * @return a boolean indicating whether a linear assign is possible
     */
    template <class F, class... CT>
    template <class S>
    inline bool xfunction<F, CT...>::has_linear_assign(const S& strides) const noexcept
    {
        auto func = [&strides](bool b, auto&& e)
        {
            return b && e.has_linear_assign(strides);
        };
        return accumulate(func, true, m_e);
    }

    //@}

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_begin() const noexcept -> const_linear_iterator
    {
        return linear_cbegin();
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_end() const noexcept -> const_linear_iterator
    {
        return linear_cend();
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_cbegin() const noexcept -> const_linear_iterator
    {
        auto f = [](const auto& e) noexcept
        {
            return xt::linear_begin(e);
        };
        return build_iterator(f, std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_cend() const noexcept -> const_linear_iterator
    {
        auto f = [](const auto& e) noexcept
        {
            return xt::linear_end(e);
        };
        return build_iterator(f, std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_rbegin() const noexcept -> const_reverse_linear_iterator
    {
        return linear_crbegin();
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_rend() const noexcept -> const_reverse_linear_iterator
    {
        return linear_crend();
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_crbegin() const noexcept -> const_reverse_linear_iterator
    {
        return const_reverse_linear_iterator(linear_cend());
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::linear_crend() const noexcept -> const_reverse_linear_iterator
    {
        return const_reverse_linear_iterator(linear_cbegin());
    }

    template <class F, class... CT>
    template <class S>
    inline auto xfunction<F, CT...>::stepper_begin(const S& shape) const noexcept -> const_stepper
    {
        auto f = [&shape](const auto& e) noexcept
        {
            return e.stepper_begin(shape);
        };
        return build_stepper(f, std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    template <class S>
    inline auto xfunction<F, CT...>::stepper_end(const S& shape, layout_type l) const noexcept -> const_stepper
    {
        auto f = [&shape, l](const auto& e) noexcept
        {
            return e.stepper_end(shape, l);
        };
        return build_stepper(f, std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::data_element(size_type i) const -> const_reference
    {
        return data_element_impl(std::make_index_sequence<sizeof...(CT)>(), i);
    }

    template <class F, class... CT>
    template <class UT, class>
    inline xfunction<F, CT...>::operator value_type() const
    {
        return operator()();
    }

    template <class F, class... CT>
    template <class align, class requested_type, std::size_t N>
    inline auto xfunction<F, CT...>::load_simd(size_type i) const -> simd_return_type<requested_type>
    {
        return load_simd_impl<align, requested_type, N>(std::make_index_sequence<sizeof...(CT)>(), i);
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::arguments() const noexcept -> const tuple_type&
    {
        return m_e;
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::functor() const noexcept -> const functor_type&
    {
        return m_f;
    }

    template <class F, class... CT>
    template <std::size_t... I>
    inline layout_type xfunction<F, CT...>::layout_impl(std::index_sequence<I...>) const noexcept
    {
        return compute_layout(std::get<I>(m_e).layout()...);
    }

    template <class F, class... CT>
    template <std::size_t... I, class... Args>
    inline auto xfunction<F, CT...>::access_impl(std::index_sequence<I...>, Args... args) const
        -> const_reference
    {
        XTENSOR_TRY(check_index(shape(), args...));
        XTENSOR_CHECK_DIMENSION(shape(), args...);
        return m_f(std::get<I>(m_e)(args...)...);
    }

    template <class F, class... CT>
    template <std::size_t... I, class... Args>
    inline auto xfunction<F, CT...>::unchecked_impl(std::index_sequence<I...>, Args... args) const
        -> const_reference
    {
        return m_f(std::get<I>(m_e).unchecked(args...)...);
    }

    template <class F, class... CT>
    template <std::size_t... I, class It>
    inline auto xfunction<F, CT...>::element_access_impl(std::index_sequence<I...>, It first, It last) const
        -> const_reference
    {
        XTENSOR_TRY(check_element_index(shape(), first, last));
        return m_f((std::get<I>(m_e).element(first, last))...);
    }

    template <class F, class... CT>
    template <std::size_t... I>
    inline auto xfunction<F, CT...>::data_element_impl(std::index_sequence<I...>, size_type i) const
        -> const_reference
    {
        return m_f((std::get<I>(m_e).data_element(i))...);
    }

    template <class F, class... CT>
    template <class align, class requested_type, std::size_t N, std::size_t... I>
    inline auto xfunction<F, CT...>::load_simd_impl(std::index_sequence<I...>, size_type i) const
    {
        return m_f.simd_apply((std::get<I>(m_e).template load_simd<align, requested_type>(i))...);
    }

    template <class F, class... CT>
    template <class Func, std::size_t... I>
    inline auto xfunction<F, CT...>::build_stepper(Func&& f, std::index_sequence<I...>) const noexcept
        -> const_stepper
    {
        return const_stepper(this, f(std::get<I>(m_e))...);
    }

    template <class F, class... CT>
    template <class Func, std::size_t... I>
    inline auto xfunction<F, CT...>::build_iterator(Func&& f, std::index_sequence<I...>) const noexcept
    {
        return const_linear_iterator(this, f(std::get<I>(m_e))...);
    }

    template <class F, class... CT>
    inline auto xfunction<F, CT...>::compute_dimension() const noexcept -> size_type
    {
        auto func = [](size_type d, auto&& e) noexcept
        {
            return (std::max)(d, e.dimension());
        };
        return accumulate(func, size_type(0), m_e);
    }

    /*************************************
     * xfunction_iterator implementation *
     *************************************/

    template <class F, class... CT>
    template <class... It>
    inline xfunction_iterator<F, CT...>::xfunction_iterator(const xfunction_type* func, It&&... it) noexcept
        : p_f(func)
        , m_it(std::forward<It>(it)...)
    {
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator++() -> self_type&
    {
        auto f = [](auto& it)
        {
            ++it;
        };
        for_each(f, m_it);
        return *this;
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator--() -> self_type&
    {
        auto f = [](auto& it)
        {
            return --it;
        };
        for_each(f, m_it);
        return *this;
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator+=(difference_type n) -> self_type&
    {
        auto f = [n](auto& it)
        {
            it += n;
        };
        for_each(f, m_it);
        return *this;
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator-=(difference_type n) -> self_type&
    {
        auto f = [n](auto& it)
        {
            it -= n;
        };
        for_each(f, m_it);
        return *this;
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator-(const self_type& rhs) const -> difference_type
    {
        return tuple_max_diff(std::make_index_sequence<sizeof...(CT)>(), m_it, rhs.m_it);
    }

    template <class F, class... CT>
    inline auto xfunction_iterator<F, CT...>::operator*() const -> reference
    {
        return deref_impl(std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline bool xfunction_iterator<F, CT...>::equal(const self_type& rhs) const
    {
        // Optimization: no need to compare each subiterator since they all
        // are incremented decremented together.
        constexpr std::size_t temp = xtl::mpl::find_if<is_not_xdummy_iterator, data_type>::value;
        constexpr std::size_t index = (temp == std::tuple_size<data_type>::value) ? 0 : temp;
        return std::get<index>(m_it) == std::get<index>(rhs.m_it);
    }

    template <class F, class... CT>
    inline bool xfunction_iterator<F, CT...>::less_than(const self_type& rhs) const
    {
        // Optimization: no need to compare each subiterator since they all
        // are incremented decremented together.
        constexpr std::size_t temp = xtl::mpl::find_if<is_not_xdummy_iterator, data_type>::value;
        constexpr std::size_t index = (temp == std::tuple_size<data_type>::value) ? 0 : temp;
        return std::get<index>(m_it) < std::get<index>(rhs.m_it);
    }

    template <class F, class... CT>
    template <std::size_t... I>
    inline auto xfunction_iterator<F, CT...>::deref_impl(std::index_sequence<I...>) const -> reference
    {
        return (p_f->m_f)(*std::get<I>(m_it)...);
    }

    template <class F, class... CT>
    template <std::size_t... I>
    inline auto xfunction_iterator<F, CT...>::tuple_max_diff(
        std::index_sequence<I...>,
        const data_type& lhs,
        const data_type& rhs
    ) const -> difference_type
    {
        auto diff = std::make_tuple((std::get<I>(lhs) - std::get<I>(rhs))...);
        auto func = [](difference_type n, auto&& v)
        {
            return (std::max)(n, v);
        };
        return accumulate(func, difference_type(0), diff);
    }

    template <class F, class... CT>
    inline bool operator==(const xfunction_iterator<F, CT...>& it1, const xfunction_iterator<F, CT...>& it2)
    {
        return it1.equal(it2);
    }

    template <class F, class... CT>
    inline bool operator<(const xfunction_iterator<F, CT...>& it1, const xfunction_iterator<F, CT...>& it2)
    {
        return it1.less_than(it2);
    }

    /************************************
     * xfunction_stepper implementation *
     ************************************/

    template <class F, class... CT>
    template <class... St>
    inline xfunction_stepper<F, CT...>::xfunction_stepper(const xfunction_type* func, St&&... st) noexcept
        : p_f(func)
        , m_st(std::forward<St>(st)...)
    {
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::step(size_type dim)
    {
        auto f = [dim](auto& st)
        {
            st.step(dim);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::step_back(size_type dim)
    {
        auto f = [dim](auto& st)
        {
            st.step_back(dim);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::step(size_type dim, size_type n)
    {
        auto f = [dim, n](auto& st)
        {
            st.step(dim, n);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::step_back(size_type dim, size_type n)
    {
        auto f = [dim, n](auto& st)
        {
            st.step_back(dim, n);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::reset(size_type dim)
    {
        auto f = [dim](auto& st)
        {
            st.reset(dim);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::reset_back(size_type dim)
    {
        auto f = [dim](auto& st)
        {
            st.reset_back(dim);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::to_begin()
    {
        auto f = [](auto& st)
        {
            st.to_begin();
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::to_end(layout_type l)
    {
        auto f = [l](auto& st)
        {
            st.to_end(l);
        };
        for_each(f, m_st);
    }

    template <class F, class... CT>
    inline auto xfunction_stepper<F, CT...>::operator*() const -> reference
    {
        return deref_impl(std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    template <std::size_t... I>
    inline auto xfunction_stepper<F, CT...>::deref_impl(std::index_sequence<I...>) const -> reference
    {
        return (p_f->m_f)(*std::get<I>(m_st)...);
    }

    template <class F, class... CT>
    template <class T, std::size_t... I>
    inline auto xfunction_stepper<F, CT...>::step_simd_impl(std::index_sequence<I...>) -> simd_return_type<T>
    {
        return (p_f->m_f.simd_apply)(std::get<I>(m_st).template step_simd<T>()...);
    }

    template <class F, class... CT>
    template <class T>
    inline auto xfunction_stepper<F, CT...>::step_simd() -> simd_return_type<T>
    {
        return step_simd_impl<T>(std::make_index_sequence<sizeof...(CT)>());
    }

    template <class F, class... CT>
    inline void xfunction_stepper<F, CT...>::step_leading()
    {
        auto step_leading_lambda = [](auto&& st)
        {
            st.step_leading();
        };
        for_each(step_leading_lambda, m_st);
    }
}

#endif