optionally dump effectively used tile, grid and block sizes

[ppcg.git] / gpu.c

/*
 * Copyright 2010-2011 INRIA Saclay
 * Copyright 2012-2013 Ecole Normale Superieure
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
 * 91893 Orsay, France
 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 */

#include <assert.h>
#include <stdlib.h>
#include <string.h>

#include <isl/polynomial.h>
#include <isl/union_set.h>
#include <isl/aff.h>
#include <isl/ilp.h>
#include <isl/flow.h>
#include <isl/band.h>
#include <isl/schedule.h>
#include <isl/options.h>
#include <isl/ast_build.h>

#include "cpu.h"
#include "gpu.h"
#include "schedule.h"
#include "ppcg_options.h"
#include "print.h"

/* The fields stride, shift and shift_map only contain valid information
 * if shift != NULL.
 * If so, they express that current index is such that if you add shift,
 * then the result is always a multiple of stride.
 * shift_map contains the mapping
 *
 *      i -> (i + shift)/stride
 *
 * Let D represent the initial shared_len dimensions of the computed schedule.
 * The spaces of "lb" and "shift" are of the form
 *
 *      D -> [b]
 *
 * "shift_map" is of the form
 *
 *      [D -> i] -> [D -> (i + shift(D))/stride]
 */
struct gpu_array_bound {
        isl_val *size;
        isl_aff *lb;

        isl_val *stride;
        isl_aff *shift;
        isl_basic_map *shift_map;
};

/* A tile of an array.
 *
 * n is the dimension of the array.
 * bound is an array of size "n" representing the lower bound
 *      and size for each index.
 *
 * tiling maps a tile in the global array to the corresponding
 * shared/private memory tile and is of the form
 *
 *      { [D[i] -> A[a]] -> T[(a + shift(i))/stride - lb(i)] }
 *
 * where D represents the initial shared_len dimensions
 * of the computed schedule.
 */
struct gpu_array_tile {
        int n;
        struct gpu_array_bound *bound;
        isl_multi_aff *tiling;
};

struct gpu_array_info;

/* A group of array references in a kernel that should be handled together.
 * If private_tile is not NULL, then it is mapped to registers.
 * Otherwise, if shared_tile is not NULL, it is mapped to shared memory.
 * Otherwise, it is accessed from global memory.
 */
struct gpu_array_ref_group {
        /* The references in this group access this array. */
        struct gpu_array_info *array;
        /* Position of this group in the list of reference groups of array. */
        int nr;

        /* The following fields are use during the construction of the groups.
         * access is the combined access relation relative to the shared
         * memory tiling.  In particular, the domain of the map corresponds
         * to the first shared_len dimensions of the computed schedule.
         * write is set if any access in the group is a write.
         * exact_write is set if all writes are definite writes.
         */
        isl_map *access;
        int write;
        int exact_write;

        /* The shared memory tile, NULL if none. */
        struct gpu_array_tile *shared_tile;

        /* The private memory tile, NULL if none. */
        struct gpu_array_tile *private_tile;

        /* References in this group; point to elements of a linked list. */
        int n_ref;
        struct gpu_stmt_access **refs;

        /* Last shared memory tile dimension that affects tile of this group. */
        int last_shared;
};

struct gpu_gen {
        isl_ctx *ctx;
        struct ppcg_options *options;

        /* Callback for printing of AST in appropriate format. */
        __isl_give isl_printer *(*print)(__isl_take isl_printer *p,
                struct gpu_prog *prog, __isl_keep isl_ast_node *tree,
                struct gpu_types *types, void *user);
        void *print_user;

        struct gpu_prog *prog;
        /* The generated AST. */
        isl_ast_node *tree;

        /* The sequence of types for which a definition has been printed. */
        struct gpu_types types;

        /* User specified tile, grid and block sizes for each kernel */
        isl_union_map *sizes;

        /* Effectively used tile, grid and block sizes for each kernel */
        isl_union_map *used_sizes;

        /* Identifier of current kernel. */
        int kernel_id;
        /* Pointer to the current kernel. */
        struct ppcg_kernel *kernel;
        /* Does the computed schedule exhibit any parallelism? */
        int any_parallelism;

        /* First tile dimension. */
        int tile_first;
        /* Number of tile dimensions. */
        int tile_len;
        /* Number of initial parallel loops among tile dimensions. */
        int n_parallel;

        /* Number of dimensions determining shared memory. */
        int shared_len;

        /* Number of rows in the untiled schedule. */
        int untiled_len;
        /* Number of rows in the tiled schedule. */
        int tiled_len;
        /* Number of rows in schedule after tiling/wrapping over threads. */
        int thread_tiled_len;

        /* Global untiled schedule. */
        isl_union_map *sched;
        /* Local (per kernel launch) tiled schedule. */
        isl_union_map *tiled_sched;
        /* Local schedule per shared memory tile loop iteration. */
        isl_union_map *local_sched;

        /* Local tiled schedule projected onto the shared tile loops and
         * the loops that will be wrapped over the threads,
         * with all shared tile loops parametrized.
         */
        isl_union_map *shared_sched;
        /* Projects out the loops that will be wrapped over the threads
         * from shared_sched.
         */
        isl_union_map *shared_proj;

        /* A map that takes the range of shared_sched as input,
         * wraps the appropriate loops over the threads and then projects
         * out these loops.
         */
        isl_map *privatization;

        /* A map from the shared memory tile loops and the thread indices
         * (as parameters) to the set of accessed memory elements that
         * will be accessed through private copies.
         */
        isl_union_map *private_access;

        /* The schedule for the current private/shared access
         * (within print_private_access or print_shared_access).
         */
        isl_map *copy_sched;
        /* The array reference group corresponding to copy_sched. */
        struct gpu_array_ref_group *copy_group;

        /* Is any array in the current kernel marked force_private? */
        int any_force_private;

        /* First loop to unroll (or -1 if none) in the current part of the
         * schedule.
         */
        int first_unroll;

        int n_grid;
        int n_block;
        /* Note: in the input file, the sizes of the grid and the blocks
         * are specified in the order x, y, z, but internally, the sizes
         * are stored in reverse order, so that the last element always
         * refers to the x dimension.
         */
        int grid_dim[2];
        int block_dim[3];
        int *tile_size;
};

/* Print the name of the local copy of a given group of array references.
 */
static __isl_give isl_printer *print_array_name(__isl_take isl_printer *p,
        struct gpu_array_ref_group *group)
{
        int global = 0;

        if (group->private_tile)
                p = isl_printer_print_str(p, "private_");
        else if (group->shared_tile)
                p = isl_printer_print_str(p, "shared_");
        else
                global = 1;
        p = isl_printer_print_str(p, group->array->name);
        if (!global && group->array->n_group > 1) {
                p = isl_printer_print_str(p, "_");
                p = isl_printer_print_int(p, group->nr);
        }

        return p;
}

/* Collect all references to the given array and store pointers to them
 * in array->refs.
 *
 * If the array contains structures, then there is no need to collect
 * the references since we will not be computing any reference groups.
 */
static void collect_references(struct gpu_prog *prog,
        struct gpu_array_info *array)
{
        int i;
        int n;

        if (array->has_compound_element)
                return;

        n = 0;
        for (i = 0; i < prog->n_stmts; ++i) {
                struct gpu_stmt *stmt = &prog->stmts[i];
                struct gpu_stmt_access *access;

                for (access = stmt->accesses; access; access = access->next) {
                        const char *name;
                        name = isl_map_get_tuple_name(access->access,
                                                      isl_dim_out);
                        if (name && !strcmp(array->name, name))
                                n++;
                }
        }

        array->n_ref = n;
        array->refs = isl_alloc_array(prog->ctx, struct gpu_stmt_access *, n);
        assert(array->refs);

        n = 0;
        for (i = 0; i < prog->n_stmts; ++i) {
                struct gpu_stmt *stmt = &prog->stmts[i];
                struct gpu_stmt_access *access;

                for (access = stmt->accesses; access; access = access->next) {
                        const char *name;
                        name = isl_map_get_tuple_name(access->access,
                                                      isl_dim_out);
                        if (!name || strcmp(array->name, name))
                                continue;

                        array->refs[n++] = access;
                }
        }
}

/* Create a gpu_array_tile for an array of dimension "n_index".
 */
static struct gpu_array_tile *create_tile(isl_ctx *ctx, int n_index)
{
        int i;
        struct gpu_array_tile *tile;

        tile = isl_calloc_type(ctx, struct gpu_array_tile);
        assert(tile);

        tile->n = n_index;

        tile->bound = isl_alloc_array(ctx, struct gpu_array_bound, n_index);
        assert(tile->bound);

        for (i = 0; i < n_index; ++i) {
                tile->bound[i].size = NULL;
                tile->bound[i].lb = NULL;
                tile->bound[i].stride = NULL;
                tile->bound[i].shift = NULL;
                tile->bound[i].shift_map = NULL;
        }

        return tile;
}

static void *free_tile(struct gpu_array_tile *tile)
{
        int j;

        if (!tile)
                return NULL;

        for (j = 0; j < tile->n; ++j) {
                isl_val_free(tile->bound[j].size);
                isl_val_free(tile->bound[j].stride);
                isl_aff_free(tile->bound[j].lb);
                isl_aff_free(tile->bound[j].shift);
                isl_basic_map_free(tile->bound[j].shift_map);
        }
        free(tile->bound);
        isl_multi_aff_free(tile->tiling);
        free(tile);

        return NULL;
}

static struct pet_array *find_array(struct ppcg_scop *scop,
        __isl_keep isl_set *accessed)
{
        int i;
        isl_id *id;

        id = isl_set_get_tuple_id(accessed);

        for (i = 0; i < scop->n_array; ++i) {
                isl_id *id_i;

                id_i = isl_set_get_tuple_id(scop->arrays[i]->extent);
                isl_id_free(id_i);
                if (id == id_i)
                        break;
        }
        isl_id_free(id);

        return i < scop->n_array ? scop->arrays[i] : NULL;
}

/* Compute and return the extent of "array", taking into account the set of
 * accessed elements.
 *
 * In particular, the extent in the outer dimension is taken
 * from "accessed", while then extent in the remaing dimensions
 * are taken from array->extent.
 *
 * The extent in the outer dimension cannot be taken from array->extent
 * because that may be unbounded.  Furthermore, even if it is bounded,
 * it may be larger than the piece of the array that is being accessed.
 */
static __isl_give isl_set *compute_extent(struct pet_array *array,
        __isl_keep isl_set *accessed)
{
        int n_index;
        isl_id *id;
        isl_set *outer;
        isl_set *extent;

        extent = isl_set_copy(array->extent);

        n_index = isl_set_dim(accessed, isl_dim_set);
        if (n_index == 0)
                return extent;

        extent = isl_set_project_out(extent, isl_dim_set, 0, 1);
        outer = isl_set_copy(accessed);
        outer = isl_set_project_out(outer, isl_dim_set, 1, n_index - 1);
        extent = isl_set_flat_product(outer, extent);
        id = isl_set_get_tuple_id(accessed);
        extent = isl_set_set_tuple_id(extent, id);

        return extent;
}

/* Is the array "array" being extracted a read-only scalar?
 *
 * That is, is "array" a scalar that is never possibly written to.
 * An array containing structures is never considered to be a scalar.
 */
static int is_read_only_scalar(struct gpu_array_info *array,
        struct gpu_prog *prog)
{
        isl_set *space;
        isl_union_map *write;
        int empty;

        if (array->has_compound_element)
                return 0;
        if (array->n_index != 0)
                return 0;

        write = isl_union_map_copy(prog->may_write);
        space = isl_set_universe(isl_space_copy(array->space));
        write = isl_union_map_intersect_range(write,
                                                isl_union_set_from_set(space));
        empty = isl_union_map_is_empty(write);
        isl_union_map_free(write);

        return empty;
}

/* Compute bounds on the host arrays based on the accessed elements
 * and collect all references to the array.
 *
 * If the array is zero-dimensional and does not contain structures,
 * i.e., if the array is a scalar, we check whether it is read-only.
 */
static int extract_array_info(__isl_take isl_set *array, void *user)
{
        int i;
        struct gpu_prog *prog = (struct gpu_prog *)user;
        const char *name;
        int n_index;
        isl_pw_aff **bounds;
        struct pet_array *pa;
        struct gpu_array_info *info;
        isl_set *extent;

        info = &prog->array[prog->n_array];
        prog->n_array++;

        n_index = isl_set_dim(array, isl_dim_set);
        name = isl_set_get_tuple_name(array);
        bounds = isl_alloc_array(isl_set_get_ctx(array),
                                 isl_pw_aff *, n_index);
        if (!bounds)
                goto error;

        info->space = isl_set_get_space(array);
        info->name = strdup(name);
        info->n_index = n_index;
        info->bound = bounds;
        info->linearize = prog->scop->options->linearize_device_arrays;

        pa = find_array(prog->scop, array);
        if (!pa)
                isl_die(isl_set_get_ctx(array), isl_error_internal,
                        "unable to find array in scop", goto error);

        info->type = strdup(pa->element_type);
        info->size = pa->element_size;
        info->local = pa->declared && !pa->exposed;
        info->has_compound_element = pa->element_is_record;
        info->read_only_scalar = is_read_only_scalar(info, prog);

        extent = compute_extent(pa, array);
        info->extent = extent;
        for (i = 0; i < n_index; ++i) {
                isl_set *dom;
                isl_local_space *ls;
                isl_aff *one;
                isl_pw_aff *bound;

                dom = isl_set_copy(extent);
                dom = isl_set_project_out(dom, isl_dim_set, i + 1,
                                            n_index - (i + 1));
                dom = isl_set_project_out(dom, isl_dim_set, 0, i);
                if (!isl_set_dim_has_upper_bound(dom, isl_dim_set, 0)) {
                        fprintf(stderr, "unable to determine extent of '%s' "
                                "in dimension %d\n", info->name, i);
                        dom = isl_set_free(dom);
                }
                bound = isl_set_dim_max(dom, 0);
                dom = isl_pw_aff_domain(isl_pw_aff_copy(bound));
                ls = isl_local_space_from_space(isl_set_get_space(dom));
                one = isl_aff_zero_on_domain(ls);
                one = isl_aff_add_constant_si(one, 1);
                bound = isl_pw_aff_add(bound, isl_pw_aff_alloc(dom, one));
                bound = isl_pw_aff_gist(bound, isl_set_copy(prog->context));

                bounds[i] = bound;
                if (!isl_pw_aff_is_cst(bound))
                        info->linearize = 1;
        }

        collect_references(prog, info);

        isl_set_free(array);
        return 0;
error:
        isl_set_free(array);
        return -1;
}

/* Compute a mapping from all outer arrays (of structs) in scop
 * to their innermost arrays.
 *
 * In particular, for each array of a primitive type, the result
 * contains the identity mapping on that array.
 * For each array involving member accesses, the result
 * contains a mapping from the elements of the outer array of structs
 * to all corresponding elements of the innermost nested arrays.
 */
static __isl_give isl_union_map *compute_to_inner(struct ppcg_scop *scop)
{
        int i;
        isl_union_map *to_inner;

        to_inner = isl_union_map_empty(isl_set_get_space(scop->context));

        for (i = 0; i < scop->n_array; ++i) {
                struct pet_array *array = scop->arrays[i];
                isl_set *set;
                isl_map *map;

                if (array->element_is_record)
                        continue;

                set = isl_set_copy(array->extent);
                map = isl_set_identity(isl_set_copy(set));

                while (set && isl_set_is_wrapping(set)) {
                        isl_id *id;
                        isl_map *wrapped;

                        id = isl_set_get_tuple_id(set);
                        wrapped = isl_set_unwrap(set);
                        wrapped = isl_map_domain_map(wrapped);
                        wrapped = isl_map_set_tuple_id(wrapped, isl_dim_in, id);
                        map = isl_map_apply_domain(map, wrapped);
                        set = isl_map_domain(isl_map_copy(map));
                }

                map = isl_map_gist_domain(map, set);

                to_inner = isl_union_map_add_map(to_inner, map);
        }

        return to_inner;
}

/* Remove independence from the order constraints "order" on array "array".
 * Since the pairs of iterations in the filter relation of an independence
 * are guaranteed to be completely independent by the user, there is
 * no need to ensure that live ranges are ordered along thong pairs.
 * We make an exception for local variables, though, as the independence
 * guarantee does not apply to those.
 *
 * The order constraints are used in two places.
 * Those on scalars are used in check_scalar_live_ranges to check if
 * we need to force the scalar to be private.  Any non-local scalar
 * should not be forced scalar if it only appears in independent loops.
 * Those on non-scalars are added to the coincidence constraints
 * in compute_schedule because we do not support any array expansion.
 * Accesses to non-local arrays should not prevent a loop from being
 * considered coincident so we should indeed remove those constraints
 * from the order constraints.
 */
static __isl_give isl_union_map *remove_independences(struct gpu_prog *prog,
        struct gpu_array_info *array, __isl_take isl_union_map *order)
{
        int i;

        for (i = 0; i < prog->scop->n_independence; ++i) {
                struct pet_independence *pi = prog->scop->independences[i];
                if (isl_union_set_contains(pi->local, array->space))
                        continue;

                order = isl_union_map_subtract(order,
                                                isl_union_map_copy(pi->filter));
        }

        return order;
}

/* For each array in "prog", store the (untagged) order dependences
 * derived from the array in array->dep_order.
 * In particular, consider all references that access the given array
 * and take the order dependences that have one of these references
 * as source.  (Since an order dependence relates two references to
 * the same array, the target of these order dependences will also
 * be one of these references.)
 * Additionally, store the union of these array->dep_order relations
 * for all non-scalar arrays in prog->array_order.
 */
void collect_order_dependences(struct gpu_prog *prog)
{
        int i;
        isl_space *space;
        isl_union_map *accesses;

        space = isl_union_map_get_space(prog->read);
        prog->array_order = isl_union_map_empty(space);

        accesses = isl_union_map_copy(prog->scop->tagged_reads);
        accesses = isl_union_map_union(accesses,
                            isl_union_map_copy(prog->scop->tagged_may_writes));
        accesses = isl_union_map_universe(accesses);
        accesses = isl_union_map_apply_range(accesses,
                                            isl_union_map_copy(prog->to_outer));

        for (i = 0; i < prog->n_array; ++i) {
                struct gpu_array_info *array = &prog->array[i];
                isl_set *set;
                isl_union_set *uset;
                isl_union_map *order;

                set = isl_set_universe(isl_space_copy(array->space));
                uset = isl_union_set_from_set(set);
                uset = isl_union_map_domain(
                    isl_union_map_intersect_range(isl_union_map_copy(accesses),
                                                    uset));
                order = isl_union_map_copy(prog->scop->tagged_dep_order);
                order = isl_union_map_intersect_domain(order, uset);
                order = isl_union_map_zip(order);
                order = isl_union_set_unwrap(isl_union_map_domain(order));
                order = remove_independences(prog, array, order);
                array->dep_order = order;

                if (gpu_array_is_scalar(array))
                        continue;

                prog->array_order = isl_union_map_union(prog->array_order,
                                        isl_union_map_copy(array->dep_order));
        }

        isl_union_map_free(accesses);
}

/* Construct a gpu_array_info for each array possibly accessed by "prog" and
 * collect them in prog->array.
 *
 * If there are any member accesses involved, then they are first mapped
 * to the outer arrays of structs.
 *
 * If we are allowing live range reordering, then also set
 * the dep_order field.  Otherwise leave it NULL.
 */
static int collect_array_info(struct gpu_prog *prog)
{
        int r;
        isl_union_set *arrays;

        arrays = isl_union_map_range(isl_union_map_copy(prog->read));
        arrays = isl_union_set_union(arrays,
                    isl_union_map_range(isl_union_map_copy(prog->may_write)));

        arrays = isl_union_set_apply(arrays,
                                        isl_union_map_copy(prog->to_outer));

        arrays = isl_union_set_coalesce(arrays);

        prog->n_array = isl_union_set_n_set(arrays);
        prog->array = isl_calloc_array(prog->ctx,
                                     struct gpu_array_info, prog->n_array);
        assert(prog->array);
        prog->n_array = 0;
        r = isl_union_set_foreach_set(arrays, &extract_array_info, prog);
        isl_union_set_free(arrays);

        if (prog->scop->options->live_range_reordering)
                collect_order_dependences(prog);

        return r;
}

static void free_array_info(struct gpu_prog *prog)
{
        int i, j;

        for (i = 0; i < prog->n_array; ++i) {
                int n_index = prog->array[i].n_index;
                free(prog->array[i].type);
                free(prog->array[i].name);
                for (j = 0; j < n_index; ++j)
                        isl_pw_aff_free(prog->array[i].bound[j]);
                isl_space_free(prog->array[i].space);
                isl_set_free(prog->array[i].extent);
                free(prog->array[i].bound);
                free(prog->array[i].refs);
                isl_union_map_free(prog->array[i].dep_order);
        }
        free(prog->array);
}

/* Check if a gpu array is a scalar.  A scalar is a value that is not stored
 * as an array or through a pointer reference, but as a single data element.
 * At the moment, scalars are represented as zero-dimensional arrays.
 * A zero-dimensional array containing structures is not considered
 * to be a scalar.
 */
int gpu_array_is_scalar(struct gpu_array_info *array)
{
        return !array->has_compound_element && array->n_index == 0;
}

/* Is "array" a read-only scalar?
 */
int gpu_array_is_read_only_scalar(struct gpu_array_info *array)
{
        return array->read_only_scalar;
}

/* Return the set of parameter values for which the array has a positive
 * size in all dimensions.
 * If the sizes are only valid for some parameter values, then those
 * constraints are also taken into account.
 */
__isl_give isl_set *gpu_array_positive_size_guard(struct gpu_array_info *array)
{
        int i;
        isl_space *space;
        isl_set *guard;

        space = isl_space_params(isl_space_copy(array->space));
        guard = isl_set_universe(space);

        for (i = 0; i < array->n_index; ++i) {
                isl_pw_aff *bound;
                isl_set *guard_i, *zero;

                bound = isl_pw_aff_copy(array->bound[i]);
                guard_i = isl_pw_aff_nonneg_set(isl_pw_aff_copy(bound));
                zero = isl_pw_aff_zero_set(bound);
                guard_i = isl_set_subtract(guard_i, zero);
                guard = isl_set_intersect(guard, guard_i);
        }

        return guard;
}

/* Internal data structure for extract_size_of_type.
 * "type" specifies the name of the space that we want to extract.
 * "res" is used to store the subset of that space.
 */
struct ppcg_extract_size_data {
        const char *type;
        isl_set *res;
};

/* This function is called for each set in a union_set.
 * If the name of the set matches data->type, we store the
 * set in data->res.
 */
static int extract_size_of_type(__isl_take isl_set *size, void *user)
{
        struct ppcg_extract_size_data *data = user;
        const char *name;

        name = isl_set_get_tuple_name(size);
        if (name && !strcmp(name, data->type)) {
                data->res = size;
                return -1;
        }

        isl_set_free(size);
        return 0;
}

/* Given a union map { kernel[i] -> *[...] },
 * return the range in the space called "type" for the kernel with
 * sequence number "id".
 */
static __isl_give isl_set *extract_sizes(__isl_keep isl_union_map *sizes,
        const char *type, int id)
{
        isl_space *space;
        isl_set *dom;
        isl_union_set *local_sizes;
        struct ppcg_extract_size_data data = { type, NULL };

        if (!sizes)
                return NULL;

        space = isl_union_map_get_space(sizes);
        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, 1);
        space = isl_space_set_tuple_name(space, isl_dim_set, "kernel");
        dom = isl_set_universe(space);
        dom = isl_set_fix_si(dom, isl_dim_set, 0, id);

        local_sizes = isl_union_set_apply(isl_union_set_from_set(dom),
                                        isl_union_map_copy(sizes));
        isl_union_set_foreach_set(local_sizes, &extract_size_of_type, &data);
        isl_union_set_free(local_sizes);
        return data.res;
}

/* Given a singleton set, extract the first (at most *len) elements
 * of the single integer tuple into *sizes and update *len if needed.
 */
static void read_sizes_from_set(__isl_take isl_set *set, int *sizes, int *len)
{
        int i;
        int dim;

        if (!set)
                return;

        dim = isl_set_dim(set, isl_dim_set);
        if (dim < *len)
                *len = dim;

        for (i = 0; i < *len; ++i) {
                isl_val *v;

                v = isl_set_plain_get_val_if_fixed(set, isl_dim_set, i);
                assert(v);

                sizes[i] = isl_val_get_num_si(v);
                isl_val_free(v);
        }

        isl_set_free(set);
}

/* Add the map { kernel[id] -> type[sizes] } to gen->used_sizes,
 * if the option debug->dump_sizes is set.
 */
static void set_used_sizes(struct gpu_gen *gen, const char *type, int id,
        int *sizes, int len)
{
        int i;
        isl_space *space;
        isl_map *map;

        if (!gen->options->debug->dump_sizes)
                return;

        space = isl_union_map_get_space(gen->used_sizes);
        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, 1);
        space = isl_space_set_tuple_name(space, isl_dim_set, "kernel");
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, len);
        space = isl_space_set_tuple_name(space, isl_dim_out, type);

        map = isl_map_universe(space);
        map = isl_map_fix_si(map, isl_dim_in, 0, id);
        for (i = 0; i < len; ++i)
                map = isl_map_fix_si(map, isl_dim_out, i, sizes[i]);

        gen->used_sizes = isl_union_map_add_map(gen->used_sizes, map);
}

/* Extract user specified "tile" sizes from the "sizes" command line option,
 * defaulting to option->tile_size in each dimension.
 * Add the effectively used sizes to gen->used_sizes.
 */
static void read_tile_sizes(struct gpu_gen *gen)
{
        int n;
        isl_set *size;

        gen->tile_size = isl_alloc_array(gen->ctx, int, gen->tile_len);
        assert(gen->tile_size);
        for (n = 0; n < gen->tile_len; ++n)
                gen->tile_size[n] = gen->options->tile_size;

        size = extract_sizes(gen->sizes, "tile", gen->kernel_id);
        read_sizes_from_set(size, gen->tile_size, &gen->tile_len);
        set_used_sizes(gen, "tile", gen->kernel_id,
                        gen->tile_size, gen->tile_len);

        if (gen->n_parallel > gen->tile_len)
                gen->n_parallel = gen->tile_len;
}

/* Extract user specified "block" sizes from the "sizes" command line option,
 * after filling in some potentially useful defaults.
 * Add the effectively used sizes to gen->used_sizes.
 */
static void read_block_sizes(struct gpu_gen *gen)
{
        int n;
        isl_set *size;

        n = gen->n_parallel;
        gen->n_block = (n <= 3) ? n : 3;
        switch (gen->n_block) {
        case 1:
                gen->block_dim[0] = 512;
                break;
        case 2:
                gen->block_dim[0] = 32;
                gen->block_dim[1] = 16;
                break;
        default:
                gen->block_dim[0] = 32;
                gen->block_dim[1] = 4;
                gen->block_dim[2] = 4;
                break;
        }

        size = extract_sizes(gen->sizes, "block", gen->kernel_id);
        read_sizes_from_set(size, gen->block_dim, &gen->n_block);
        set_used_sizes(gen, "block", gen->kernel_id,
                        gen->block_dim, gen->n_block);
}

/* Extract user specified "grid" sizes from the "sizes" command line option,
 * after filling in some potentially useful defaults.
 * Add the effectively used sizes to gen->used_sizes.
 */
static void read_grid_sizes(struct gpu_gen *gen)
{
        int n = gen->n_parallel;
        isl_set *size;

        gen->n_grid = (n <= 2) ? n : 2;
        switch (gen->n_grid) {
        case 1:
                gen->grid_dim[0] = 32768;
                break;
        default:
                gen->grid_dim[0] = 256;
                gen->grid_dim[1] = 256;
                break;
        }

        size = extract_sizes(gen->sizes, "grid", gen->kernel_id);
        read_sizes_from_set(size, gen->grid_dim, &gen->n_grid);
        set_used_sizes(gen, "grid", gen->kernel_id, gen->grid_dim, gen->n_grid);
}

/* Extract user specified sizes from the "sizes" command line option
 * after filling in some potentially useful defaults.
 */
static void read_sizes(struct gpu_gen *gen)
{
        read_tile_sizes(gen);
        read_block_sizes(gen);
        read_grid_sizes(gen);
}

static void *free_stmts(struct gpu_stmt *stmts, int n)
{
        int i;

        if (!stmts)
                return NULL;

        for (i = 0; i < n; ++i) {
                struct gpu_stmt_access *access, *next;

                for (access = stmts[i].accesses; access; access = next) {
                        next = access->next;
                        isl_id_free(access->ref_id);
                        isl_map_free(access->access);
                        isl_map_free(access->tagged_access);
                        free(access);
                }

                isl_id_free(stmts[i].id);
        }
        free(stmts);

        return NULL;
}

/* Construct a map from a domain of dimensionality "len"
 * to a domain of dimensionality "len" + "tile_len" that tiles
 * the "tile_len" coordinates starting at "first".
 * In particular, [s_i] -> [s_i / tile_size[i], s_i % tile_size[i]].
 * "dim" prescribes the parameters.
 */
static __isl_give isl_map *tile(__isl_take isl_space *dim, int len,
        int first, int tile_len, int *tile_size)
{
        int i;
        isl_basic_map *bmap;
        isl_constraint *c;
        isl_local_space *ls;

        dim = isl_space_add_dims(dim, isl_dim_in, len);
        dim = isl_space_add_dims(dim, isl_dim_out, len + tile_len);
        bmap = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len - tile_len; ++i) {
                int j = i < first ? i : i + tile_len;
                int k = i < first ? i : i + 2 * tile_len;

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, j, -1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, k, 1);
                bmap = isl_basic_map_add_constraint(bmap, c);
        }

        for (i = 0; i < tile_len; ++i) {
                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in,
                                                first + i, -1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                first + i, tile_size[i]);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                first + i + tile_len, 1);
                bmap = isl_basic_map_add_constraint(bmap, c);

                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                   first + i + tile_len, 1);
                bmap = isl_basic_map_add_constraint(bmap, c);

                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                   first + i + tile_len, -1);
                c = isl_constraint_set_constant_si(c, tile_size[i] - 1);
                bmap = isl_basic_map_add_constraint(bmap, c);
        }

        isl_local_space_free(ls);

        return isl_map_from_basic_map(bmap);
}

/* Construct a map from a domain of dimensionality "len"
 * to a domain of dimensionality "len" + "wrap_len" that "wraps"
 * the "wrap_len" coordinates starting at "first" according to "wrap_size".
 * In particular, [s_i] -> [s_i, s_i % wrap_size[i]].
 * To do so, we need extra variables corresponding to [s_i / wrap_size[i]],
 * that are projected out at the end.
 * "dim" prescribes the parameters.
 */
static __isl_give isl_map *wrap(__isl_take isl_space *dim, int len,
        int first, int wrap_len, int *wrap_size)
{
        int i;
        isl_basic_map *bmap;
        isl_constraint *c;
        isl_local_space *ls;

        dim = isl_space_add_dims(dim, isl_dim_in, len);
        dim = isl_space_add_dims(dim, isl_dim_out, len + 2 * wrap_len);
        bmap = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len; ++i) {
                int k = i < first + wrap_len ? i : i + 2 * wrap_len;

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i, -1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, k, 1);
                bmap = isl_basic_map_add_constraint(bmap, c);
        }

        for (i = 0; i < wrap_len; ++i) {
                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                    first + i, -1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                    first + wrap_len + i, 1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                    first + 2 * wrap_len + i, wrap_size[i]);
                bmap = isl_basic_map_add_constraint(bmap, c);

                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                    first + wrap_len + i, 1);
                bmap = isl_basic_map_add_constraint(bmap, c);

                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_out,
                                                    first + wrap_len + i, -1);
                c = isl_constraint_set_constant_si(c, wrap_size[i] - 1);
                bmap = isl_basic_map_add_constraint(bmap, c);
        }

        isl_local_space_free(ls);

        bmap = isl_basic_map_project_out(bmap, isl_dim_out,
                                first + 2 * wrap_len, wrap_len);

        return isl_map_from_basic_map(bmap);
}

/* Add "n" parameters named prefix%d.
 */
static __isl_give isl_set *add_params( __isl_take isl_set *set,
        int n, const char *prefix)
{
        int i;
        unsigned nparam;
        char name[20];

        nparam = isl_set_dim(set, isl_dim_param);
        set = isl_set_add_dims(set, isl_dim_param, n);

        for (i = 0; i < n; ++i) {
                snprintf(name, sizeof(name), "%s%d", prefix, i);
                set = isl_set_set_dim_name(set, isl_dim_param,
                                            nparam + i, name);
        }

        return set;
}

/* Equate the "n" dimensions of "set" starting at "first" to
 * freshly created parameters named prefix%d.
 */
static __isl_give isl_set *parametrize(__isl_take isl_set *set,
        int first, int n, const char *prefix)
{
        int i;
        unsigned nparam;

        nparam = isl_set_dim(set, isl_dim_param);

        set = add_params(set, n, prefix);

        for (i = 0; i < n; ++i)
                set = isl_set_equate(set, isl_dim_param, nparam + i,
                                        isl_dim_set, first + i);

        return set;
}

/* Given a parameter space "space", create a set of dimension "len"
 * of which the "n" dimensions starting at "first" are equated to
 * freshly created parameters named prefix%d.
 */
static __isl_give isl_set *parametrization(__isl_take isl_space *space,
        int len, int first, int n, const char *prefix)
{
        isl_set *set;

        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, len);
        set = isl_set_universe(space);

        return parametrize(set, first, n, prefix);
}

/* Tile the B loops over the tile sizes and then tile/wrap
 * the T1 loops over the blocks.
 */
static __isl_give isl_union_map *tile_schedule(struct gpu_gen *gen,
        __isl_take isl_union_map *sched)
{
        isl_space *dim;
        isl_map *tiling, *block_tiling;

        dim = isl_union_map_get_space(sched);
        tiling = tile(isl_space_copy(dim), gen->untiled_len,
                      gen->tile_first, gen->tile_len, gen->tile_size);

        if (gen->options->wrap)
                block_tiling = wrap(dim, gen->untiled_len + gen->tile_len,
                                gen->tile_first, gen->n_grid, gen->grid_dim);
        else
                block_tiling = tile(dim, gen->untiled_len + gen->tile_len,
                                gen->tile_first, gen->n_grid, gen->grid_dim);

        gen->tiled_len = gen->untiled_len + gen->tile_len + gen->n_grid;

        tiling = isl_map_apply_range(tiling, block_tiling);

        sched = isl_union_map_apply_range(sched,
                                             isl_union_map_from_map(tiling));

        gen->shared_len = gen->tile_first + gen->tile_len + gen->n_grid;

        return sched;
}

/* Equate the "T1P" iterators in the tiled schedule "sched"
 * to the block dimensions.
 */
static __isl_give isl_union_map *parametrize_tiled_schedule(
        struct gpu_gen *gen, __isl_take isl_union_map *sched)
{
        isl_space *dim;
        isl_set *par;

        dim = isl_union_map_get_space(sched);
        par = parametrization(dim, gen->tiled_len,
                gen->tile_first + gen->n_grid, gen->n_grid, "b");
        sched = isl_union_map_intersect_range(sched,
                                                isl_union_set_from_set(par));

        return sched;
}

/* Tile/wrap the P1 loops over the threads.
 */
static __isl_give isl_union_map *thread_tile_schedule(struct gpu_gen *gen,
        __isl_take isl_union_map *sched)
{
        isl_space *dim;
        isl_map *tiling;
        isl_set *par;

        dim = isl_union_map_get_space(sched);

        if (gen->options->wrap)
                tiling = wrap(isl_space_copy(dim), gen->tiled_len,
                                gen->shared_len, gen->n_block, gen->block_dim);
        else
                tiling = tile(isl_space_copy(dim), gen->tiled_len,
                                gen->shared_len, gen->n_block, gen->block_dim);
        gen->thread_tiled_len = gen->tiled_len + gen->n_block;

        sched = isl_union_map_apply_range(sched,
                                             isl_union_map_from_map(tiling));

        par = parametrization(dim, gen->thread_tiled_len,
                gen->tile_first + gen->tile_len + gen->n_grid + gen->n_block,
                gen->n_block, "t");
        sched = isl_union_map_intersect_range(sched,
                                                isl_union_set_from_set(par));

        gen->shared_len = gen->tile_first + gen->tile_len + gen->n_grid;

        return sched;
}

/* If the user asked for it, scale the shared memory tile loops
 * (T1T and T2) of "sched" by gen->tile_size[i].
 * If we are not performing "wrapping", then additionally scale the T1P
 * loops by gen->grid_dim[i].
 */
static __isl_give isl_union_map *scale_tile_loops(struct gpu_gen *gen,
        __isl_take isl_union_map *sched)
{
        int i;
        isl_space *dim;
        isl_basic_map *scale;
        isl_constraint *c;
        isl_local_space *ls;

        if (!gen->options->scale_tile_loops)
                return sched;

        dim = isl_union_map_get_space(sched);
        dim = isl_space_add_dims(dim, isl_dim_in, gen->tiled_len);
        dim = isl_space_add_dims(dim, isl_dim_out, gen->tiled_len);
        scale = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < gen->tiled_len; ++i) {
                int f = 1;

                if (i >= gen->tile_first && i < gen->tile_first + gen->n_grid) {
                        f = gen->tile_size[i - gen->tile_first];
                        if (!gen->options->wrap)
                                f *= gen->grid_dim[i - gen->tile_first];
                } else if (i >= gen->tile_first + gen->n_grid &&
                           i < gen->tile_first + gen->n_grid + gen->tile_len) {
                        f = gen->tile_size[i - (gen->tile_first + gen->n_grid)];
                }

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i, f);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, i, -1);
                scale = isl_basic_map_add_constraint(scale, c);
        }

        isl_local_space_free(ls);

        sched = isl_union_map_apply_range(sched,
                isl_union_map_from_map(isl_map_from_basic_map(scale)));

        return sched;
}

/* If we are not performing "wrapping" and if the user asked for it,
 * scale the thread tile loops (P1T) of "sched" by gen->block_dim[i].
 */
static __isl_give isl_union_map *scale_thread_tile_loops(struct gpu_gen *gen,
        __isl_take isl_union_map *sched)
{
        int i;
        isl_space *dim;
        isl_basic_map *scale;
        isl_constraint *c;
        isl_local_space *ls;

        if (gen->options->wrap)
                return sched;
        if (!gen->options->scale_tile_loops)
                return sched;

        dim = isl_union_map_get_space(sched);
        dim = isl_space_add_dims(dim, isl_dim_in, gen->thread_tiled_len);
        dim = isl_space_add_dims(dim, isl_dim_out, gen->thread_tiled_len);
        scale = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < gen->thread_tiled_len; ++i) {
                int f = 1;

                if (i >= gen->shared_len &&
                    i < gen->shared_len + gen->n_block)
                        f = gen->block_dim[i - gen->shared_len];

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i, f);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, i, -1);
                scale = isl_basic_map_add_constraint(scale, c);
        }

        isl_local_space_free(ls);

        sched = isl_union_map_apply_range(sched,
                isl_union_map_from_map(isl_map_from_basic_map(scale)));

        return sched;
}

/* If we are not performing "wrapping" and if the user asked for it,
 * scale the "n_tile" loops starting at "first" of "sched" by gen->block_dim[i].
 */
static __isl_give isl_union_map *scale_access_tile_loops(struct gpu_gen *gen,
        __isl_take isl_union_map *sched, int len, int first, int n_tile)
{
        int i;
        isl_space *dim;
        isl_basic_map *scale;
        isl_constraint *c;
        isl_local_space *ls;

        if (gen->options->wrap)
                return sched;
        if (!gen->options->scale_tile_loops)
                return sched;

        dim = isl_union_map_get_space(sched);
        dim = isl_space_add_dims(dim, isl_dim_in, len);
        dim = isl_space_add_dims(dim, isl_dim_out, len);
        scale = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len; ++i) {
                int f = 1;

                if (i >= first && i < first + n_tile)
                        f = gen->kernel->block_dim[i - first];

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i, f);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, i, -1);
                scale = isl_basic_map_add_constraint(scale, c);
        }

        isl_local_space_free(ls);

        sched = isl_union_map_apply_range(sched,
                isl_union_map_from_map(isl_map_from_basic_map(scale)));

        return sched;
}

/* Add "len" parameters p[i] called prefix%d,
 * with bounds to 0 <= p[i] < size[i].
 */
__isl_give isl_set *add_bounded_parameters(__isl_take isl_set *set,
        int len, int *size, const char *prefix)
{
        int i;
        unsigned nparam;
        isl_space *dim;
        isl_basic_set *bset;
        isl_constraint *c;
        isl_local_space *ls;
        char name[20];

        nparam = isl_set_dim(set, isl_dim_param);
        set = isl_set_add_dims(set, isl_dim_param, len);

        for (i = 0; i < len; ++i) {
                snprintf(name, sizeof(name), "%s%d", prefix, i);
                set = isl_set_set_dim_name(set, isl_dim_param,
                                            nparam + i, name);
        }

        dim = isl_set_get_space(set);
        bset = isl_basic_set_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len; ++i) {
                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_param,
                                                        nparam + i, 1);
                bset = isl_basic_set_add_constraint(bset, c);
        
                c = isl_inequality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_param,
                                                        nparam + i, -1);
                c = isl_constraint_set_constant_si(c, size[i] - 1);
                bset = isl_basic_set_add_constraint(bset, c);
        }

        isl_local_space_free(ls);

        return isl_set_intersect(set, isl_set_from_basic_set(bset));
}

/* Add "len" parameters p[i] called prefix%d and intersect "set"
 * with
 *
 *      { : 0 <= p[i] < size[i] }
 *
 * or an overapproximation.
 */
static __isl_give isl_set *add_bounded_parameters_dynamic(
        __isl_take isl_set *set, __isl_keep isl_multi_pw_aff *size,
        const char *prefix)
{
        int i, len;
        unsigned nparam;
        isl_space *space;
        isl_local_space *ls;
        char name[20];

        len = isl_multi_pw_aff_dim(size, isl_dim_out);
        nparam = isl_set_dim(set, isl_dim_param);
        set = isl_set_add_dims(set, isl_dim_param, len);

        for (i = 0; i < len; ++i) {
                snprintf(name, sizeof(name), "%s%d", prefix, i);
                set = isl_set_set_dim_name(set, isl_dim_param,
                                            nparam + i, name);
        }

        space = isl_space_params(isl_set_get_space(set));
        ls = isl_local_space_from_space(space);
        for (i = 0; i < len; ++i) {
                isl_pw_aff *param, *size_i, *zero;
                isl_set *bound;

                param = isl_pw_aff_var_on_domain(isl_local_space_copy(ls),
                                                isl_dim_param, nparam + i);

                size_i = isl_multi_pw_aff_get_pw_aff(size, i);
                bound = isl_pw_aff_lt_set(isl_pw_aff_copy(param), size_i);
                bound = isl_set_from_basic_set(isl_set_simple_hull(bound));
                set = isl_set_intersect_params(set, bound);

                zero = isl_pw_aff_zero_on_domain(isl_local_space_copy(ls));
                bound = isl_pw_aff_ge_set(param, zero);
                set = isl_set_intersect_params(set, bound);
        }
        isl_local_space_free(ls);

        return set;
}

/* Construct a map from an access to group->array to the corresponding
 * shared/private memory tile.
 * The map is of the form
 *
 *      { [D[i] -> A[a]] -> T[t] }
 *
 * where D represents the initial shared_len dimensions
 * of the computed schedule.
 */
static __isl_give isl_map *shift_access(struct gpu_array_ref_group *group)
{
        struct gpu_array_tile *tile;
        isl_multi_aff *tiling;

        tile = group->private_tile;
        if (!tile)
                tile = group->shared_tile;

        tiling = isl_multi_aff_copy(tile->tiling);

        return isl_map_from_multi_aff(tiling);
}

/* Does "map" have an obviously fixed value at variable "pos" of "type"?
 */
static int map_plain_is_fixed(isl_map *map, enum isl_dim_type type,
        unsigned pos)
{
        isl_val *v;
        int fixed;

        v = isl_map_plain_get_val_if_fixed(map, type, pos);
        if (!v)
                return -1;
        fixed = isl_val_is_int(v);
        isl_val_free(v);

        return fixed;
}

/* Given a schedule that iterates over all elements in a piece of an array,
 * perform tiling/wrapping over the threads.
 *
 * In particular, we tile the final iterators so that the final thread
 * dimension runs over the final array dimension.
 * However, if those final iterators have only a single iteration,
 * we try to tile earlier iterators instead.
 */
static __isl_give isl_map *tile_access_schedule(struct gpu_gen *gen,
        __isl_take isl_map *sched)
{
        isl_space *dim;
        isl_union_map *usched;
        isl_map *tiling;
        isl_set *par;
        unsigned nvar = isl_map_dim(sched, isl_dim_out);
        int n_tile;
        int first;

        n_tile = gen->kernel->n_block;
        if (n_tile > nvar) {
                int i;
                sched = isl_map_insert_dims(sched,
                                                isl_dim_out, 0, n_tile - nvar);
                for (i = 0; i < n_tile - nvar; ++i)
                        sched = isl_map_fix_si(sched, isl_dim_out, i, 0);
                nvar = n_tile;
        }

        first = nvar - n_tile;

        for (; first > 0; first --)
                if (!map_plain_is_fixed(sched, isl_dim_out, first + n_tile - 1))
                        break;

        dim = isl_map_get_space(sched);
        dim = isl_space_params(dim);
        if (gen->options->wrap)
                tiling = wrap(isl_space_copy(dim), nvar, first,
                                n_tile, gen->kernel->block_dim);
        else
                tiling = tile(isl_space_copy(dim), nvar, first,
                                n_tile, gen->kernel->block_dim);
        sched = isl_map_apply_range(sched, tiling);

        par = parametrization(dim, nvar + n_tile, first + n_tile, n_tile, "t");
        sched = isl_map_intersect_range(sched, par);

        usched = isl_union_map_from_map(sched);
        usched = scale_access_tile_loops(gen, usched, nvar + n_tile,
                                         first, n_tile);
        sched = isl_map_from_union_map(usched);

        return sched;
}

/* Return the union of all read (read = 1) and/or write (write = 1)
 * access relations in the group.
 */
static __isl_give isl_union_map *group_access_relation(
        struct gpu_array_ref_group *group, int read, int write)
{
        int i;
        isl_union_map *access;

        access = isl_union_map_empty(isl_map_get_space(group->access));
        for (i = 0; i < group->n_ref; ++i) {
                isl_map *map_i;

                if (!((read && group->refs[i]->read) ||
                     (write && group->refs[i]->write)))
                        continue;
                map_i = isl_map_copy(group->refs[i]->access);
                access = isl_union_map_union(access,
                                            isl_union_map_from_map(map_i));
        }

        return access;
}

/* Return the union of all tagged access relations in the group.
 */
static __isl_give isl_union_map *group_tagged_access_relation(
        struct gpu_array_ref_group *group)
{
        int i;
        isl_union_map *access;

        access = isl_union_map_empty(isl_map_get_space(group->access));
        for (i = 0; i < group->n_ref; ++i) {
                isl_map *map_i;

                map_i = isl_map_copy(group->refs[i]->tagged_access);
                access = isl_union_map_union(access,
                                            isl_union_map_from_map(map_i));
        }

        return access;
}

/* Return the extent of "array", recomputed from the bounds.
 * The recomputed extent may be simpler than the original extent.
 */
static __isl_give isl_set *array_extent(struct gpu_array_info *array)
{
        int i;
        isl_id *id;
        isl_space *space;
        isl_local_space *ls;
        isl_set *extent;

        id = isl_set_get_tuple_id(array->extent);
        space = isl_set_get_space(array->extent);
        extent = isl_set_universe(isl_space_copy(space));
        ls = isl_local_space_from_space(space);
        for (i = 0; i < array->n_index; ++i) {
                isl_pw_aff *bound;
                isl_aff *aff;
                isl_pw_aff *index;
                isl_set *lt;

                extent = isl_set_lower_bound_si(extent, isl_dim_set, i, 0);

                aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
                                                isl_dim_set, i);
                index = isl_pw_aff_from_aff(aff);
                bound = isl_pw_aff_copy(array->bound[i]);
                bound = isl_pw_aff_from_range(bound);
                bound = isl_pw_aff_add_dims(bound, isl_dim_in, array->n_index);
                bound = isl_pw_aff_set_tuple_id(bound, isl_dim_in,
                                                isl_id_copy(id));
                lt = isl_pw_aff_lt_set(index, bound);
                extent = isl_set_intersect(extent, lt);
        }
        isl_local_space_free(ls);
        isl_id_free(id);

        return extent;
}

/* Return a map from the first shared_len dimensions of the computed
 * schedule to the array tile in
 * global memory that corresponds to the shared memory copy.
 *
 * In particular, return a map
 *
 *      { D[i] -> A[a] }
 *
 * with constraints
 *
 *      tile_offset(i) <= a <= tile_offset(i) + tile_size - 1           (1)
 *
 * and
 *
 *      0 <= a <= array_size - 1                                        (2)
 *
 * Note that if some stride has been detected (i.e., when
 * group->shared_tile->bound[i].shift is set), then a in (1) refers
 * to the shifted and scaled down version.
 *
 * Constraints (1) are obtained by mapping the size constraints on the
 * shared/private memory tile back to the access relation.
 * Constraints (2) are obtained from the (recomputed) extent.
 */
static __isl_give isl_map *group_tile(struct gpu_array_ref_group *group)
{
        int i;
        int n_index = group->array->n_index;
        isl_map *tile;
        isl_space *space;
        isl_set *local;
        isl_set *extent;

        space = isl_multi_aff_get_space(group->shared_tile->tiling);
        space = isl_space_range(space);
        local = isl_set_universe(space);
        for (i = 0; i < n_index; ++i) {
                isl_val *bound;

                local = isl_set_lower_bound_si(local, isl_dim_set, i, 0);
                bound = isl_val_copy(group->shared_tile->bound[i].size);
                bound = isl_val_sub_ui(bound, 1);
                local = isl_set_upper_bound_val(local, isl_dim_set, i, bound);
        }
        local = isl_set_preimage_multi_aff(local,
                                isl_multi_aff_copy(group->shared_tile->tiling));
        tile = isl_set_unwrap(local);
        extent = array_extent(group->array);
        tile = isl_map_intersect_range(tile, extent);

        return tile;
}

/* Given a mapping "iterator_map" from the AST schedule to a domain,
 * return the corresponding mapping from the AST schedule to
 * to the first shared_len dimensions of the schedule computed by PPCG.
 */
static __isl_give isl_pw_multi_aff *compute_sched_to_shared(struct gpu_gen *gen,
        __isl_take isl_pw_multi_aff *iterator_map)
{
        isl_union_map *umap;
        isl_space *space;
        isl_map *map, *sched;;

        space = isl_space_range(isl_pw_multi_aff_get_space(iterator_map));
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, gen->shared_len);

        umap = isl_union_map_copy(gen->shared_sched);
        umap = isl_union_map_apply_range(umap,
                        isl_union_map_copy(gen->shared_proj));
        map = isl_union_map_extract_map(umap, space);
        isl_union_map_free(umap);

        sched = isl_map_preimage_domain_pw_multi_aff(map, iterator_map);
        sched = isl_map_detect_equalities(sched);

        return isl_pw_multi_aff_from_map(sched);
}

/* Set unroll[j] if the input dimension j is involved in
 * the index expression represented by ma.
 */
static int check_unroll(__isl_take isl_set *set, __isl_take isl_multi_aff *ma,
        void *user)
{
        int i, j;
        int n_in = isl_multi_aff_dim(ma, isl_dim_in);
        int n_out = isl_multi_aff_dim(ma, isl_dim_out);
        int *unroll = user;

        for (i = 0; i < n_out; ++i) {
                isl_aff *aff;

                aff = isl_multi_aff_get_aff(ma, i);
                for (j = 0; j < n_in; ++j)
                        if (isl_aff_involves_dims(aff, isl_dim_in, j, 1))
                                unroll[j] = 1;
                isl_aff_free(aff);
        }

        isl_set_free(set);
        isl_multi_aff_free(ma);
        return 0;
}

/* Given an array pos mapping input dimensions to the corresponding
 * output dimension, construct the corresponding map.
 */
static __isl_give isl_map *permutation(__isl_take isl_space *dim,
        int *pos, int len)
{
        int i;
        isl_constraint *c;
        isl_basic_map *bmap;
        isl_local_space *ls;

        dim = isl_space_add_dims(dim, isl_dim_in, len);
        dim = isl_space_add_dims(dim, isl_dim_out, len);
        bmap = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len; ++i) {
                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i,
                                                      -1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, pos[i],
                                                      1);
                bmap = isl_basic_map_add_constraint(bmap, c);
        }
        isl_local_space_free(ls);

        return isl_map_from_basic_map(bmap);
}

/* Remove the private tiles from all array reference groups,
 * except for the groups of arrays that are marked force_private.
 */
static void remove_private_tiles(struct gpu_gen *gen)
{
        int i, j;

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                if (array->force_private)
                        continue;

                for (j = 0; j < array->n_group; ++j) {
                        struct gpu_array_ref_group *group = array->groups[j];

                        group->private_tile = free_tile(group->private_tile);
                }
        }
}

/* Find all loops involved in any of the index expressions for any of
 * the private accesses, move them innermost and then mark them as
 * requiring unrolling by setting gen->first_unroll.
 * The loops involved should all be parallel because of the checks
 * we performed in check_private_group_access.  Moving them innermost
 * is therefore a valid transformation.
 *
 * If any of the arrays are marked force_private, however, then
 * those loops may not be parallel with respect to the marked arrays.
 * If any of the loops would have to be moved innermost for the
 * (non forced) private accesses and if there are any force_private
 * arrays, then we revert the decision to map the selected arrays
 * to private memory.  An alternative solution would be to expand
 * the force_private arrays.
 *
 * Loops up to gen->shared_len are generated before the mapping to
 * threads is applied.  They should therefore be ignored.
 *
 * We compute the hidden equalities of the schedule first
 * since we will need them in our calls to isl_pw_multi_aff_from_map
 * and because we want to make sure that the same equalities
 * are also available to the code generator.
 */
static __isl_give isl_union_map *interchange_for_unroll(struct gpu_gen *gen,
        __isl_take isl_union_map *sched)
{
        int i, j;
        int unroll[gen->thread_tiled_len];
        int perm[gen->thread_tiled_len];
        isl_space *dim;
        isl_map *permute;
        int len = gen->shared_len + gen->n_parallel + gen->n_block;

        gen->first_unroll = -1;

        sched = isl_union_map_detect_equalities(sched);
        for (i = 0; i < gen->thread_tiled_len; ++i)
                unroll[i] = 0;
        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j) {
                        isl_union_map *access;
                        isl_map *acc;
                        isl_pw_multi_aff *pma;

                        if (!array->groups[j]->private_tile)
                                continue;

                        access = group_access_relation(array->groups[j], 1, 1);
                        access = isl_union_map_apply_domain(access,
                                                isl_union_map_copy(sched));

                        acc = isl_map_from_union_map(access);
                        pma = isl_pw_multi_aff_from_map(acc);
                        isl_pw_multi_aff_foreach_piece(pma,
                                                        &check_unroll, unroll);

                        isl_pw_multi_aff_free(pma);
                }
        }

        for (i = gen->shared_len; i < len; ++i)
                if (unroll[i])
                        break;

        if (i >= len)
                return sched;

        for (i = len; i < gen->thread_tiled_len; ++i)
                if (unroll[i])
                        return sched;

        if (gen->any_force_private) {
                remove_private_tiles(gen);
                return sched;
        }

        j = 0;
        for (i = 0; i < gen->shared_len; ++i)
                perm[i] = j++;
        for (i = gen->shared_len; i < gen->thread_tiled_len; ++i)
                if (!unroll[i])
                        perm[i] = j++;
        gen->first_unroll = j - gen->shared_len;
        for (i = gen->shared_len; i < len; ++i)
                if (unroll[i])
                        perm[i] = j++;

        dim = isl_union_map_get_space(sched);
        permute = permutation(dim, perm, gen->thread_tiled_len);
        sched = isl_union_map_apply_range(sched,
                                          isl_union_map_from_map(permute));

        return sched;
}

/* Given a constraint
 *
 *              a(p,i) + j = g f(e)
 *
 * or -a(p,i) - j = g f(e) if sign < 0,
 * store a(p,i) in bound->shift and g (stride) in bound->stride.
 * a(p,i) is assumed to be an expression in only the parameters
 * and the input dimensions.
 */
static void extract_stride(__isl_keep isl_constraint *c,
        struct gpu_array_bound *bound, __isl_keep isl_val *stride, int sign)
{
        int i;
        isl_val *v;
        isl_space *space;
        unsigned nparam;
        unsigned nvar;
        isl_aff *aff;

        isl_val_free(bound->stride);
        bound->stride = isl_val_copy(stride);

        space = isl_constraint_get_space(c);
        space = isl_space_domain(space);

        nparam = isl_space_dim(space, isl_dim_param);
        nvar = isl_space_dim(space, isl_dim_set);

        v = isl_constraint_get_constant_val(c);
        if (sign < 0)
                v = isl_val_neg(v);
        aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
        aff = isl_aff_set_constant_val(aff, v);

        for (i = 0; i < nparam; ++i) {
                if (!isl_constraint_involves_dims(c, isl_dim_param, i, 1))
                        continue;
                v = isl_constraint_get_coefficient_val(c, isl_dim_param, i);
                if (sign < 0)
                        v = isl_val_neg(v);
                aff = isl_aff_add_coefficient_val(aff, isl_dim_param, i, v);
        }

        for (i = 0; i < nvar; ++i) {
                if (!isl_constraint_involves_dims(c, isl_dim_in, i, 1))
                        continue;
                v = isl_constraint_get_coefficient_val(c, isl_dim_in, i);
                if (sign < 0)
                        v = isl_val_neg(v);
                aff = isl_aff_add_coefficient_val(aff, isl_dim_in, i, v);
        }

        bound->shift = aff;
}

/* Given an equality constraint of a map with a single output dimension j,
 * check if the constraint is of the form
 *
 *              a(p,i) + j = g f(e)
 *
 * with a(p,i) an expression in the parameters and input dimensions
 * and f(e) an expression in the existentially quantified variables.
 * If so, and if g is larger than any such g from a previously considered
 * constraint, then call extract_stride to record the stride information
 * in bound.
 */
static int check_stride_constraint(__isl_take isl_constraint *c, void *user)
{
        int i;
        isl_ctx *ctx;
        isl_val *v;
        unsigned n_div;
        struct gpu_array_bound *bound = user;

        ctx = isl_constraint_get_ctx(c);
        n_div = isl_constraint_dim(c, isl_dim_div);
        v = isl_constraint_get_coefficient_val(c, isl_dim_out, 0);

        if (n_div && (isl_val_is_one(v) || isl_val_is_negone(v))) {
                int s = isl_val_sgn(v);
                isl_val *stride = isl_val_zero(ctx);

                isl_val_free(v);
                for (i = 0; i < n_div; ++i) {
                        v = isl_constraint_get_coefficient_val(c,
                                                                isl_dim_div, i);
                        stride = isl_val_gcd(stride, v);
                }
                if (!isl_val_is_zero(stride) &&
                    isl_val_gt(stride, bound->stride))
                        extract_stride(c, bound, stride, s);

                isl_val_free(stride);
        } else
                isl_val_free(v);

        isl_constraint_free(c);
        return 0;
}

/* Given contraints on an array index i, check if we can find
 * a shift a(p) and a stride g such that
 *
 *      a(p) + i = 0 mod g
 *
 * If so, record the information in bound and apply the mapping
 * i -> (i + a(p))/g to the array index in bounds and return
 * the new constraints.
 * If not, simply return the original constraints.
 *
 * If bounds is a subset of the space
 *
 *      D -> i
 *
 * then the bound recorded in bound->shift is of the form
 *
 *      D -> s(D)
 *
 * with s(D) equal to a(p) above.
 * The mapping recorded in bound->shift_map is of the form
 *
 *      [D -> i] -> [D -> (i + S(D))/g]
 *
 * This mapping is computed as follows.
 * We first introduce "i" in the domain through precomposition
 * with [D -> i] -> D obtaining
 *
 *      [D -> i] -> s(D)
 *
 * Adding [D -> i] -> i produces
 *
 *      [D -> i] -> i + s(D)
 *
 * and the domain product with [D -> i] -> D yields
 *
 *      [D -> i] -> [D -> i + s(D)]
 *
 * Composition with [D -> i] -> [D -> i/g] gives the desired result.
 */
static __isl_give isl_basic_map *check_stride(struct gpu_array_bound *bound,
        __isl_take isl_basic_map *bounds)
{
        isl_space *space;
        isl_basic_map *hull;
        isl_basic_map *shift, *id, *bmap, *scale;
        isl_basic_set *bset;
        isl_aff *aff;

        bound->stride = NULL;

        hull = isl_basic_map_affine_hull(isl_basic_map_copy(bounds));

        isl_basic_map_foreach_constraint(hull, &check_stride_constraint, bound);

        isl_basic_map_free(hull);

        if (!bound->stride)
                return bounds;

        shift = isl_basic_map_from_aff(isl_aff_copy(bound->shift));
        space = isl_basic_map_get_space(bounds);
        bmap = isl_basic_map_domain_map(isl_basic_map_universe(space));
        shift = isl_basic_map_apply_range(bmap, shift);
        space = isl_basic_map_get_space(bounds);
        id = isl_basic_map_range_map(isl_basic_map_universe(space));
        shift = isl_basic_map_sum(id, shift);
        space = isl_basic_map_get_space(bounds);
        id = isl_basic_map_domain_map(isl_basic_map_universe(space));
        shift = isl_basic_map_range_product(id, shift);

        space = isl_space_domain(isl_basic_map_get_space(bounds));
        id = isl_basic_map_identity(isl_space_map_from_set(space));
        space = isl_space_range(isl_basic_map_get_space(bounds));
        aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
        aff = isl_aff_add_coefficient_si(aff, isl_dim_in, 0, 1);
        aff = isl_aff_scale_down_val(aff, isl_val_copy(bound->stride));
        scale = isl_basic_map_from_aff(aff);
        scale = isl_basic_map_product(id, scale);

        bound->shift_map = isl_basic_map_apply_range(shift, scale);
        bmap = isl_basic_map_copy(bound->shift_map);
        bset = isl_basic_set_apply(isl_basic_map_wrap(bounds), bmap);
        bounds = isl_basic_set_unwrap(bset);

        return bounds;
}

/* Data used in compute_array_dim_size and compute_size_in_direction.
 *
 * pos is the position of the variable representing the array index,
 * i.e., the variable for which want to compute the size.  This variable
 * is also the last variable in the set.
 */
struct gpu_size_info {
        isl_basic_set *bset;
        struct gpu_array_bound *bound;
        int pos;
};

/* Given a constraint from the basic set describing the bounds on
 * an array index, check if it is a lower bound, say m i >= b(x), and,
 * if so, check whether the expression "i - ceil(b(x)/m) + 1" has a constant
 * upper bound.  If so, and if this bound is smaller than any bound
 * derived from earlier constraints, set the size to this bound on
 * the expression and the lower bound to ceil(b(x)/m).
 */
static int compute_size_in_direction(__isl_take isl_constraint *c, void *user)
{
        struct gpu_size_info *size = user;
        unsigned nparam;
        unsigned n_div;
        isl_val *v;
        isl_aff *aff;
        isl_aff *lb;

        nparam = isl_basic_set_dim(size->bset, isl_dim_param);
        n_div = isl_constraint_dim(c, isl_dim_div);

        if (isl_constraint_involves_dims(c, isl_dim_div, 0, n_div) ||
            !isl_constraint_is_lower_bound(c, isl_dim_set, size->pos)) {
                isl_constraint_free(c);
                return 0;
        }

        aff = isl_constraint_get_bound(c, isl_dim_set, size->pos);
        aff = isl_aff_ceil(aff);

        lb = isl_aff_copy(aff);

        aff = isl_aff_neg(aff);
        aff = isl_aff_add_coefficient_si(aff, isl_dim_in, size->pos, 1);

        v = isl_basic_set_max_val(size->bset, aff);
        isl_aff_free(aff);

        if (isl_val_is_int(v)) {
                v = isl_val_add_ui(v, 1);
                if (!size->bound->size || isl_val_lt(v, size->bound->size)) {
                        isl_val_free(size->bound->size);
                        size->bound->size = isl_val_copy(v);
                        lb = isl_aff_drop_dims(lb, isl_dim_in, size->pos, 1);
                        isl_aff_free(size->bound->lb);
                        size->bound->lb = isl_aff_copy(lb);
                }
        }
        isl_val_free(v);
        isl_aff_free(lb);

        isl_constraint_free(c);

        return 0;
}

/* Given a basic map "bounds" that maps parameters and input dimensions
 * to a single output dimension, look for an expression in the parameters
 * and input dimensions such that the range of the output dimension shifted
 * by this expression is a constant.
 *
 * In particular, we currently only consider lower bounds on the output
 * dimension as candidate expressions.
 */
static int compute_array_dim_size(struct gpu_array_bound *bound,
        __isl_take isl_basic_map *bounds)
{
        struct gpu_size_info size;

        bounds = isl_basic_map_detect_equalities(bounds);
        bounds = check_stride(bound, bounds);

        bound->size = NULL;
        bound->lb = NULL;

        size.bound = bound;
        size.pos = isl_basic_map_dim(bounds, isl_dim_in);
        size.bset = isl_basic_map_wrap(bounds);
        size.bset = isl_basic_set_flatten(size.bset);
        size.bset = isl_set_simple_hull(isl_basic_set_compute_divs(size.bset));
        isl_basic_set_foreach_constraint(size.bset, &compute_size_in_direction,
                                        &size);
        isl_basic_set_free(size.bset);

        return bound->size ? 0 : -1;
}

/* Check if we can find a memory tile for the given array
 * based on the given accesses, and if so, put the results in "tile".
 *
 * We project the accesses on each index in turn and look for a parametric
 * offset such that the size is constant.
 */
static int can_tile(__isl_keep isl_map *access, struct gpu_array_tile *tile)
{
        int i;

        for (i = 0; i < tile->n; ++i) {
                isl_map *access_i;
                isl_basic_map *hull;

                access_i = isl_map_copy(access);
                access_i = isl_map_project_out(access_i, isl_dim_out, 0, i);
                access_i = isl_map_project_out(access_i, isl_dim_out,
                                            1, tile->n - (i + 1));
                access_i = isl_map_compute_divs(access_i);
                hull = isl_map_simple_hull(access_i);
                if (compute_array_dim_size(&tile->bound[i], hull) < 0)
                        return 0;
        }

        return 1;
}

/* Construct a map with input the shared tile loops and the loops that
 * will be wrapped around the threads that relates these later loops
 * to the thread indices and then projects them out.
 */
static __isl_give isl_map *compute_privatization(struct gpu_gen *gen)
{
        isl_map *priv;
        isl_map *tiling;
        isl_map *proj;
        isl_set *par;
        isl_space *dim;

        dim = isl_union_map_get_space(gen->shared_sched);

        if (gen->options->wrap)
                tiling = wrap(isl_space_copy(dim), gen->shared_len + gen->n_block,
                                gen->shared_len, gen->n_block, gen->block_dim);
        else
                tiling = tile(isl_space_copy(dim), gen->shared_len + gen->n_block,
                                gen->shared_len, gen->n_block, gen->block_dim);

        priv = tiling;

        par = parametrization(dim, gen->shared_len + 2 * gen->n_block,
                gen->tile_first + gen->tile_len + gen->n_grid + gen->n_block,
                gen->n_block, "t");

        priv = isl_map_align_params(priv, isl_set_get_space(par));
        priv = isl_map_intersect_range(priv, par);

        dim = isl_map_get_space(priv);
        dim = isl_space_drop_dims(dim, isl_dim_in, 0, isl_space_dim(dim, isl_dim_in));
        dim = isl_space_drop_dims(dim, isl_dim_out, 0, isl_space_dim(dim, isl_dim_out));
        proj = projection(dim, gen->shared_len + 2 * gen->n_block,
                          gen->shared_len);

        priv = isl_map_apply_range(priv, proj);

        return priv;
}

/* Construct a map from domain_dim to domain_dim that increments
 * the dimension at position "pos" and leaves all other dimensions
 * constant.
 */
static __isl_give isl_map *next(__isl_take isl_space *domain_dim, int pos)
{
        int i;
        int len = isl_space_dim(domain_dim, isl_dim_set);
        isl_space *dim;
        isl_basic_map *next;
        isl_local_space *ls;

        dim = isl_space_map_from_set(domain_dim);
        next = isl_basic_map_universe(isl_space_copy(dim));
        ls = isl_local_space_from_space(dim);

        for (i = 0; i < len; ++i) {
                isl_constraint *c;

                c = isl_equality_alloc(isl_local_space_copy(ls));
                c = isl_constraint_set_coefficient_si(c, isl_dim_in, i, 1);
                c = isl_constraint_set_coefficient_si(c, isl_dim_out, i, -1);
                if (i == pos)
                        c = isl_constraint_set_constant_si(c, 1);
                next = isl_basic_map_add_constraint(next, c);
        }

        isl_local_space_free(ls);

        return isl_map_from_basic_map(next);
}

/* Check if the given access is coalesced.
 * That is, check whether incrementing the dimension that will get
 * wrapped over the last thread index results in incrementing
 * the last array index.
 *
 * This function is only called for access relations without reuse.
 */
static int access_is_coalesced(struct gpu_gen *gen,
        __isl_keep isl_union_map *access)
{
        isl_space *dim;
        isl_map *access_map;
        isl_map *next_thread_x;
        isl_map *next_element;
        isl_map *map;
        int coalesced;

        access = isl_union_map_copy(access);
        access = isl_union_map_apply_domain(access,
                                isl_union_map_copy(gen->tiled_sched));
        access_map = isl_map_from_union_map(access);

        dim = isl_map_get_space(access_map);
        dim = isl_space_domain(dim);
        next_thread_x = next(dim, gen->shared_len + gen->n_block - 1);

        dim = isl_map_get_space(access_map);
        dim = isl_space_range(dim);
        next_element = next(dim, isl_space_dim(dim, isl_dim_set) - 1);

        map = isl_map_apply_domain(next_thread_x, isl_map_copy(access_map));
        map = isl_map_apply_range(map, access_map);

        coalesced = isl_map_is_subset(map, next_element);

        isl_map_free(next_element);
        isl_map_free(map);

        return coalesced;
}

/* Given an access relation in terms of the first gen->shared_len + gen->n_block
 * dimensions of the computed schedule, check if it is bijective for
 * fixed values of the first gen->shared_len dimensions.
 * We perform this check by equating these dimensions to parameters.
 */
static int access_is_bijective(struct gpu_gen *gen, __isl_keep isl_map *access)
{
        int res;
        isl_set *par;
        isl_space *space;

        access = isl_map_copy(access);
        space = isl_space_params(isl_map_get_space(access));
        par = parametrization(space, gen->shared_len + gen->n_block,
                                0, gen->shared_len, "s");
        access = isl_map_intersect_domain(access, par);
        res = isl_map_is_bijective(access);
        isl_map_free(access);

        return res;
}

/* Look for the last shared tile loop that affects the offset of "tile"
 * and return the result.
 * If there is no such loop, then return the index of the loop
 * before the first shared tile loop, in particular gen->tile_first - 1.
 */
static int compute_tile_last_shared(struct gpu_gen *gen,
        struct gpu_array_tile *tile)
{
        int i, j;

        for (j = gen->shared_len - 1; j >= gen->tile_first; --j) {
                for (i = 0; i < tile->n; ++i) {
                        isl_aff *lb;
                        isl_aff *shift;

                        lb = tile->bound[i].lb;
                        if (isl_aff_involves_dims(lb, isl_dim_in, j, 1))
                                break;

                        shift = tile->bound[i].shift;
                        if (!shift)
                                continue;
                        if (isl_aff_involves_dims(shift, isl_dim_in, j, 1))
                                break;
                }
                if (i < tile->n)
                        break;
        }

        return j;
}

/* Look for the last shared tile loop that affects the offset of the
 * shared or private tile and store the result in group->last_shared.
 * If there is no such loop, then group->last_shared is set to a value
 * before the first shared tile loop, in particular gen->tile_first - 1.
 * If there is no tile defined on the array reference group,
 * then set group->last_shared to gen->shared_len - 1.
 */
static void set_last_shared(struct gpu_gen *gen,
        struct gpu_array_ref_group *group)
{
        struct gpu_array_tile *tile;

        group->last_shared = gen->shared_len - 1;

        tile = group->private_tile;
        if (!tile)
                tile = group->shared_tile;
        if (!tile)
                return;

        group->last_shared = compute_tile_last_shared(gen, tile);
}

/* Compute a privatized copy of all access relations from reference groups that
 * are mapped to private memory and store the result in gen->privatization.
 *
 * Read-only scalars and arrays containing structures are not mapped
 * to private memory.
 */
static void compute_private_access(struct gpu_gen *gen)
{
        int i, j;
        isl_union_map *private;

        if (!gen->options->use_private_memory)
                return;

        private = isl_union_map_empty(isl_union_map_get_space(gen->shared_sched));

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                if (gpu_array_is_read_only_scalar(array))
                        continue;
                if (array->has_compound_element)
                        continue;

                for (j = 0; j < array->n_group; ++j) {
                        if (!array->groups[j]->private_tile)
                                continue;

                        private = isl_union_map_union(private,
                                group_access_relation(array->groups[j], 1, 1));
                }
        }

        if (isl_union_map_is_empty(private))
                isl_union_map_free(private);
        else {
                isl_union_map *priv;

                private = isl_union_map_apply_domain(private,
                                        isl_union_map_copy(gen->shared_sched));
                priv = isl_union_map_from_map(isl_map_copy(gen->privatization));
                private = isl_union_map_apply_domain(private, priv);
                gen->private_access = private;
        }
}

/* Compute the size of the tile specified by "tile"
 * in number of elements and return the result.
 */
static __isl_give isl_val *tile_size(isl_ctx *ctx, struct gpu_array_tile *tile)
{
        int i;
        isl_val *size;

        size = isl_val_one(ctx);

        for (i = 0; i < tile->n; ++i)
                size = isl_val_mul(size, isl_val_copy(tile->bound[i].size));

        return size;
}

/* If max_shared_memory is not set to infinity (-1), then make
 * sure that the total amount of shared memory required by the
 * array reference groups mapped to shared memory is no larger
 * than this maximum.
 *
 * We apply a greedy approach and discard (keep in global memory)
 * those groups that would result in a total memory size that
 * is larger than the maximum.
 *
 * This function should be called after any function that may
 * affect the decision on whether to place a reference group
 * in private, shared or global memory.
 */
static void check_shared_memory_bound(struct gpu_gen *gen)
{
        int i, j;
        isl_val *left, *size;

        if (gen->options->max_shared_memory < 0)
                return;

        left = isl_val_int_from_si(gen->ctx, gen->options->max_shared_memory);

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j) {
                        struct gpu_array_ref_group *group;

                        group = array->groups[j];
                        if (group->private_tile)
                                continue;
                        if (!group->shared_tile)
                                continue;

                        size = tile_size(gen->ctx, group->shared_tile);
                        size = isl_val_mul_ui(size, array->size);

                        if (isl_val_le(size, left)) {
                                left = isl_val_sub(left, size);
                                continue;
                        }
                        isl_val_free(size);

                        group->shared_tile = free_tile(group->shared_tile);
                }
        }

        isl_val_free(left);
}

/* Given a description of an array tile "tile" and the "space"
 *
 *      { D -> A }
 *
 * where D represents the first shared_len schedule dimensions
 * and A represents the array, construct an isl_multi_aff
 *
 *      { [D[i] -> A[a]] -> A'[a'] }
 *
 * with A' a scaled down copy of A according to the shifts and strides
 * in "tile".  In particular,
 *
 *      a' = (a + shift(i))/stride
 *
 * "insert_array" represents
 *
 *      { [D -> A] -> D }
 *
 * and is used to insert A into the domain of functions that only
 * reference D.
 */
static __isl_give isl_multi_aff *strided_tile(
        struct gpu_array_tile *tile, __isl_keep isl_space *space,
        __isl_keep isl_multi_aff *insert_array)
{
        int i;
        isl_ctx *ctx;
        isl_multi_aff *shift;
        isl_multi_val *stride;
        isl_space *space2;
        isl_local_space *ls;
        isl_multi_aff *tiling;

        ctx = isl_space_get_ctx(space);
        space2 = isl_space_domain(isl_space_copy(space));
        ls = isl_local_space_from_space(space2);
        space2 = isl_space_range(isl_space_copy(space));
        stride = isl_multi_val_zero(space2);
        shift = isl_multi_aff_zero(isl_space_copy(space));

        for (i = 0; i < tile->n; ++i) {
                struct gpu_array_bound *bound = &tile->bound[i];
                isl_val *stride_i;
                isl_aff *shift_i;

                if (tile->bound[i].shift) {
                        stride_i = isl_val_copy(bound->stride);
                        shift_i = isl_aff_copy(bound->shift);
                } else {
                        stride_i = isl_val_one(ctx);
                        shift_i = isl_aff_zero_on_domain(
                                        isl_local_space_copy(ls));
                }

                stride = isl_multi_val_set_val(stride, i, stride_i);
                shift = isl_multi_aff_set_aff(shift, i, shift_i);
        }
        isl_local_space_free(ls);

        shift = isl_multi_aff_pullback_multi_aff(shift,
                                    isl_multi_aff_copy(insert_array));

        tiling = isl_multi_aff_range_map(isl_space_copy(space));
        tiling = isl_multi_aff_add(tiling, shift);
        tiling = isl_multi_aff_scale_down_multi_val(tiling, stride);

        return tiling;
}

/* Compute a tiling for the array reference group "group".
 *
 * The tiling is of the form
 *
 *      { [D[i] -> A[a]] -> T[t] }
 *
 * where D represents the first shared_len schedule dimensions,
 * A represents the global array and T represents the shared or
 * private memory tile.  The name of T is the name of the local
 * array.
 *
 * If there is any stride in the accesses, then the mapping is
 *
 *      t = (a + shift(i))/stride - lb(i)
 *
 * otherwise, it is simply
 *
 *      t = a - lb(i)
 */
static void compute_group_tiling(struct gpu_array_ref_group *group)
{
        int i;
        struct gpu_array_tile *tile;
        struct gpu_array_info *array = group->array;
        isl_space *space;
        isl_multi_aff *tiling, *lb, *insert_array;
        isl_printer *p;
        char *local_name;

        tile = group->private_tile;
        if (!tile)
                tile = group->shared_tile;
        if (!tile)
                return;

        space = isl_map_get_space(group->access);
        insert_array = isl_multi_aff_domain_map(isl_space_copy(space));

        for (i = 0; i < tile->n; ++i)
                if (tile->bound[i].shift)
                        break;

        if (i < tile->n)
                tiling = strided_tile(tile, space, insert_array);
        else
                tiling = isl_multi_aff_range_map(isl_space_copy(space));

        lb = isl_multi_aff_zero(space);
        for (i = 0; i < tile->n; ++i) {
                isl_aff *lb_i = isl_aff_copy(tile->bound[i].lb);
                lb = isl_multi_aff_set_aff(lb, i, lb_i);
        }
        lb = isl_multi_aff_pullback_multi_aff(lb, insert_array);

        tiling = isl_multi_aff_sub(tiling, lb);

        p = isl_printer_to_str(isl_multi_aff_get_ctx(tiling));
        p = print_array_name(p, group);
        local_name = isl_printer_get_str(p);
        isl_printer_free(p);
        tiling = isl_multi_aff_set_tuple_name(tiling, isl_dim_out, local_name);
        free(local_name);

        tile->tiling = tiling;
}

/* Compute a tiling for all the array reference groups.
 */
static void compute_group_tilings(struct gpu_gen *gen)
{
        int i, j;

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j)
                        compute_group_tiling(array->groups[j]);
        }
}

/* Fill up the groups array with singleton groups, i.e., one group
 * per reference, initializing the array, access, write, n_ref and refs fields.
 * In particular the access field is initialized to the scheduled
 * access relation of the array reference.
 *
 * Return the number of elements initialized, i.e., the number of
 * active references in the current kernel.
 */
static int populate_array_references(struct gpu_array_info *array,
        __isl_keep isl_union_map *sched, struct gpu_array_ref_group **groups)
{
        int i;
        int n;
        isl_ctx *ctx = isl_union_map_get_ctx(sched);

        n = 0;
        for (i = 0; i < array->n_ref; ++i) {
                isl_union_map *umap;
                isl_map *map;
                struct gpu_array_ref_group *group;
                struct gpu_stmt_access *access = array->refs[i];

                map = isl_map_copy(access->access);
                umap = isl_union_map_from_map(map);
                umap = isl_union_map_apply_domain(umap,
                                isl_union_map_copy(sched));

                if (isl_union_map_is_empty(umap)) {
                        isl_union_map_free(umap);
                        continue;
                }

                map = isl_map_from_union_map(umap);
                map = isl_map_detect_equalities(map);

                group = isl_calloc_type(ctx, struct gpu_array_ref_group);
                assert(group);
                group->array = array;
                group->access = map;
                group->write = access->write;
                group->exact_write = access->exact_write;
                group->refs = &array->refs[i];
                group->n_ref = 1;

                groups[n++] = group;
        }

        return n;
}

/* If group->n_ref == 1, then group->refs was set by
 * populate_array_references to point directly into
 * group->array->refs and should not be freed.
 * If group->n_ref > 1, then group->refs was set by join_groups
 * to point to a newly allocated array.
 */
static void free_array_ref_group(struct gpu_array_ref_group *group)
{
        if (!group)
                return;
        free_tile(group->shared_tile);
        free_tile(group->private_tile);
        isl_map_free(group->access);
        if (group->n_ref > 1)
                free(group->refs);
        free(group);
}

/* Given a map where the input dimensions represent the tile loops,
 * eliminate the innermost of those that have a fixed value
 * until we reach one that does not (obviously) have a fixed value.
 */
static __isl_give isl_map *eliminate_fixed_inner_loops(
        __isl_take isl_map *access)
{
        int i, n;

        n = isl_map_dim(access, isl_dim_in);

        for (i = n - 1; i >= 0; --i) {
                if (!map_plain_is_fixed(access, isl_dim_in, i))
                        break;
                access = isl_map_eliminate(access, isl_dim_in, i, 1);
        }
        return access;
}

/* Check if the access relations of group1 and group2 overlap within
 * the innermost loop.  In particular, ignore any inner dimension
 * with a fixed value.
 * The copying to and from shared memory will be performed within
 * the innermost actual loop so we are only allowed to consider
 * the dimensions up to that innermost loop while checking whether
 * two access relations overlap.
 */
static int accesses_overlap(struct gpu_array_ref_group *group1,
        struct gpu_array_ref_group *group2)
{
        int empty;
        isl_map *access1, *access2;

        access1 = isl_map_copy(group1->access);
        access1 = eliminate_fixed_inner_loops(access1);
        access2 = isl_map_copy(group2->access);
        access2 = eliminate_fixed_inner_loops(access2);
        access1 = isl_map_intersect(access1, access2);
        empty = isl_map_is_empty(access1);
        isl_map_free(access1);

        return !empty;
}

/* Combine the given two groups into a single group, containing
 * the references of both groups.
 */
static struct gpu_array_ref_group *join_groups(
        struct gpu_array_ref_group *group1,
        struct gpu_array_ref_group *group2)
{
        int i;
        isl_ctx *ctx;
        struct gpu_array_ref_group *group;

        ctx = isl_map_get_ctx(group1->access);
        group = isl_calloc_type(ctx, struct gpu_array_ref_group);
        assert(group);
        group->array = group1->array;
        group->access = isl_map_union(isl_map_copy(group1->access),
                                        isl_map_copy(group2->access));
        group->write = group1->write || group2->write;
        group->exact_write = group1->exact_write && group2->exact_write;
        group->n_ref = group1->n_ref + group2->n_ref;
        group->refs = isl_alloc_array(ctx, struct gpu_stmt_access *,
                                        group->n_ref);
        assert(group->refs);
        for (i = 0; i < group1->n_ref; ++i)
                group->refs[i] = group1->refs[i];
        for (i = 0; i < group2->n_ref; ++i)
                group->refs[group1->n_ref + i] = group2->refs[i];

        return group;
}

/* Combine the given two groups into a single group and free
 * the original two groups.
 */
static struct gpu_array_ref_group *join_groups_and_free(
        struct gpu_array_ref_group *group1,
        struct gpu_array_ref_group *group2)
{
        struct gpu_array_ref_group *group;

        group = join_groups(group1, group2);
        free_array_ref_group(group1);
        free_array_ref_group(group2);
        return group;
}

/* Compute the private and/or shared memory tiles for the array
 * reference group "group" of array "array".
 * Return 0 on success and -1 on error.
 *
 * If the array is a read-only scalar or if the user requested
 * not to use shared or private memory, then we do not need to do anything.
 *
 * If the array group involves any may writes (that are not must writes),
 * then we would have to make sure that we load the data into shared/private
 * memory first in case the data is not written by the kernel
 * (but still written back out to global memory).
 * Since we don't have any such mechanism at the moment, we don't
 * compute shared/private tiles for groups involving may writes.
 *
 * We only try to compute a shared memory tile if there is any reuse
 * or if the access is not coalesced.
 *
 * For computing a private memory tile, we also require that there is
 * some reuse.  Moreover, we require that the access is private
 * to the thread.  That is, we check that any given array element
 * is only accessed by a single thread.
 * We compute an access relation that maps the shared tile loop iterators
 * and the shared point loop iterators that will be wrapped over the
 * threads to the array elements.
 * We actually check that those iterators that will be wrapped
 * partition the array space.  This check is stricter than necessary
 * since several iterations may be mapped onto the same thread
 * and then they could be allowed to access the same memory elements,
 * but our check does not allow this situation.
 *
 * We also check that the index expression only depends on parallel
 * loops.  That way, we can move those loops innermost and unroll them.
 * Again, we use a test that is stricter than necessary.
 * We actually check whether the index expression only depends
 * on the iterators that are wrapped over the threads.
 * These are necessarily parallel, but there may be more parallel loops.
 *
 * Combining the injectivity of the first test with the single-valuedness
 * of the second test, we simply test for bijectivity.
 *
 * If the array is marked force_private, then we bypass all checks
 * and assume we can (and should) use registers.
 *
 * If it turns out we can (or have to) use registers, we compute
 * the private memory tile size using can_tile, after introducing a dependence
 * on the thread indices.
 */
static int compute_group_bounds_core(struct gpu_gen *gen,
        struct gpu_array_ref_group *group)
{
        isl_ctx *ctx = isl_space_get_ctx(group->array->space);
        isl_union_map *access;
        int n_index = group->array->n_index;
        int no_reuse;
        isl_map *acc;
        int force_private = group->array->force_private;
        int use_shared = gen->options->use_shared_memory;
        int use_private = force_private || gen->options->use_private_memory;

        if (!use_shared && !use_private)
                return 0;
        if (gpu_array_is_read_only_scalar(group->array))
                return 0;
        if (!force_private && !group->exact_write)
                return 0;

        access = group_access_relation(group, 1, 1);
        no_reuse = isl_union_map_is_injective(access);

        if (use_shared && (!no_reuse || !access_is_coalesced(gen, access))) {
                group->shared_tile = create_tile(ctx, group->array->n_index);
                if (!can_tile(group->access, group->shared_tile))
                        group->shared_tile = free_tile(group->shared_tile);
        }

        if (!force_private && (!use_private || no_reuse)) {
                isl_union_map_free(access);
                return 0;
        }

        access = isl_union_map_apply_domain(access,
                                        isl_union_map_copy(gen->shared_sched));

        acc = isl_map_from_union_map(access);

        if (!force_private && !access_is_bijective(gen, acc)) {
                isl_map_free(acc);
                return 0;
        }

        group->private_tile = create_tile(gen->ctx, n_index);
        acc = isl_map_apply_domain(acc, isl_map_copy(gen->privatization));
        if (!can_tile(acc, group->private_tile))
                group->private_tile = free_tile(group->private_tile);

        isl_map_free(acc);

        if (force_private && !group->private_tile)
                isl_die(ctx, isl_error_internal,
                        "unable to map array reference group to registers",
                        return -1);

        return 0;
}

/* Compute the private and/or shared memory tiles for the array
 * reference group "group" of array "array" and set last_shared.
 * Return 0 on success and -1 on error.
 */
static int compute_group_bounds(struct gpu_gen *gen,
        struct gpu_array_ref_group *group)
{
        if (compute_group_bounds_core(gen, group) < 0)
                return -1;
        set_last_shared(gen, group);

        return 0;
}

/* If two groups have overlapping access relations (as determined by
 * the "overlap" function) and if one of them involves a write,
 * then merge the two groups into one.
 * If "compute_bounds" is set, then call compute_group_bounds
 * on the merged groups.
 *
 * Return the updated number of groups.
 * Return -1 on error.
 */
static int group_writes(struct gpu_gen *gen,
        int n, struct gpu_array_ref_group **groups,
        int (*overlap)(struct gpu_array_ref_group *group1,
                struct gpu_array_ref_group *group2), int compute_bounds)
{
        int i, j;

        for (i = 0; i < n; ++i) {
                for (j = n - 1; j > i; --j) {
                        if (!groups[i]->write && !groups[j]->write)
                                continue;

                        if (!overlap(groups[i], groups[j]))
                                continue;

                        groups[i] = join_groups_and_free(groups[i], groups[j]);
                        if (compute_bounds &&
                            compute_group_bounds(gen, groups[i]) < 0)
                                return -1;
                        if (j != n - 1)
                                groups[j] = groups[n - 1];
                        groups[n - 1] = NULL;
                        n--;
                }
        }

        return n;
}

/* If two groups have overlapping access relations (within the innermost
 * loop) and if one of them involves a write, then merge the two groups
 * into one.
 *
 * Return the updated number of groups.
 */
static int group_overlapping_writes(struct gpu_gen *gen,
        int n, struct gpu_array_ref_group **groups)
{
        return group_writes(gen, n, groups, &accesses_overlap, 0);
}

/* Check if the access relations of group1 and group2 overlap within
 * the outermost min(group1->last_shared, group2->last_shared) loops.
 */
static int last_shared_accesses_overlap(struct gpu_array_ref_group *group1,
        struct gpu_array_ref_group *group2)
{
        int last_shared;
        int dim;
        int empty;
        isl_map *map_i, *map_j, *map;

        last_shared = group1->last_shared;
        if (group2->last_shared < last_shared)
                last_shared = group2->last_shared;
        map_i = isl_map_copy(group1->access);
        dim = isl_map_dim(map_i, isl_dim_in);
        map_i = isl_map_eliminate(map_i, isl_dim_in,
                                last_shared + 1, dim - (last_shared + 1));
        map_j = isl_map_copy(group2->access);
        map_j = isl_map_eliminate(map_j, isl_dim_in,
                                last_shared + 1, dim - (last_shared + 1));
        map = isl_map_intersect(map_i, map_j);
        empty = isl_map_is_empty(map);
        isl_map_free(map);

        return !empty;
}

/* If two groups have overlapping access relations (within the outer
 * last_shared loops) and if one of them involves a write,
 * then merge the two groups into one.
 *
 * Return the updated number of groups.
 */
static int group_last_shared_overlapping_writes(struct gpu_gen *gen, int n,
        struct gpu_array_ref_group **groups)
{
        return group_writes(gen, n, groups, &last_shared_accesses_overlap, 1);
}

/* Is the size of the tile specified by "tile" smaller than the sum of
 * the sizes of the tiles specified by "tile1" and "tile2"?
 */
static int smaller_tile(isl_ctx *ctx, struct gpu_array_tile *tile,
        struct gpu_array_tile *tile1, struct gpu_array_tile *tile2)
{
        int smaller;
        isl_val *size, *size1, *size2;

        size = tile_size(ctx, tile);
        size1 = tile_size(ctx, tile1);
        size2 = tile_size(ctx, tile2);

        size = isl_val_sub(size, size1);
        size = isl_val_sub(size, size2);
        smaller = isl_val_is_neg(size);

        isl_val_free(size);

        return smaller;
}

/* Given an initial grouping of array references and shared memory tiles
 * for each group that allows for a shared memory tile, merge two groups
 * if both have a shared memory tile, the merged group also has
 * a shared memory tile and the size of the tile for the merge group
 * is smaller than the sum of the tile sizes of the individual groups.
 *
 * If merging two groups decreases the "last_shared" dimension of
 * one or both of the two groups, then we need to check for overlapping
 * writes again.
 *
 * Return the number of groups after merging.
 * Return -1 on error.
 */
static int group_common_shared_memory_tile(struct gpu_gen *gen,
        struct gpu_array_info *array, int n,
        struct gpu_array_ref_group **groups)
{
        int i, j;
        int recompute_overlap = 0;
        isl_ctx *ctx = isl_space_get_ctx(array->space);

        for (i = 0; i < n; ++i) {
                if (!groups[i]->shared_tile)
                        continue;
                for (j = n - 1; j > i; --j) {
                        isl_map *map;
                        int empty;
                        struct gpu_array_ref_group *group;

                        if (!groups[j]->shared_tile)
                                continue;

                        map = isl_map_intersect(isl_map_copy(groups[i]->access),
                                            isl_map_copy(groups[j]->access));
                        empty = isl_map_is_empty(map);
                        isl_map_free(map);

                        if (empty)
                                continue;

                        group = join_groups(groups[i], groups[j]);
                        if (compute_group_bounds(gen, group) < 0) {
                                free_array_ref_group(group);
                                return -1;
                        }
                        if (!group->shared_tile ||
                            !smaller_tile(ctx, group->shared_tile,
                                        groups[i]->shared_tile,
                                        groups[j]->shared_tile)) {
                                free_array_ref_group(group);
                                continue;
                        }

                        if (group->last_shared < groups[i]->last_shared ||
                            group->last_shared < groups[j]->last_shared)
                                recompute_overlap = 1;
                        free_array_ref_group(groups[i]);
                        free_array_ref_group(groups[j]);
                        groups[i] = group;
                        if (j != n - 1)
                                groups[j] = groups[n - 1];
                        n--;
                }
        }

        if (recompute_overlap)
                n = group_last_shared_overlapping_writes(gen, n, groups);
        return n;
}

/* Set array->n_group and array->groups to n and groups.
 *
 * Additionally, set the "nr" field of each group
 * and the "group" field of each reference in each group.
 */
static void set_array_groups(struct gpu_array_info *array,
        int n, struct gpu_array_ref_group **groups)
{
        int i, j;

        array->n_group = n;
        array->groups = groups;

        for (i = 0; i < n; ++i) {
                groups[i]->nr = i;

                for (j = 0; j < groups[i]->n_ref; ++j)
                        groups[i]->refs[j]->group = i;
        }
}

/* Group array references that should be considered together when
 * deciding whether to access them from private, shared or global memory.
 * Return -1 on error.
 *
 * In particular, if two array references overlap and if one of them
 * is a write, then the two references are grouped together.
 * We first perform an initial grouping based only on the access relation.
 * After computing shared and private memory tiles, we check for
 * overlapping writes again, but this time taking into account
 * the "last_shared" property.
 *
 * Furthermore, if two groups admit a shared memory tile and if the
 * combination of the two also admits a shared memory tile, we merge
 * the two groups.
 *
 * If the array contains structures, then there is no need to compute
 * reference groups since we do not map such arrays to private or shared
 * memory.
 */
static int group_array_references(struct gpu_gen *gen,
        struct gpu_array_info *array, __isl_keep isl_union_map *sched)
{
        int i;
        int n;
        isl_ctx *ctx = isl_union_map_get_ctx(sched);
        struct gpu_array_ref_group **groups;

        if (array->has_compound_element)
                return 0;

        groups = isl_calloc_array(ctx, struct gpu_array_ref_group *,
                                        array->n_ref);
        if (!groups)
                return -1;

        n = populate_array_references(array, sched, groups);

        n = group_overlapping_writes(gen, n, groups);

        for (i = 0; i < n; ++i)
                if (compute_group_bounds(gen, groups[i]) < 0)
                        n = -1;

        n = group_last_shared_overlapping_writes(gen, n, groups);

        n = group_common_shared_memory_tile(gen, array, n, groups);

        set_array_groups(array, n, groups);

        if (n >= 0)
                return 0;

        for (i = 0; i < array->n_ref; ++i)
                free_array_ref_group(groups[i]);
        return -1;
}

/* Take tiled_sched, project it onto the shared tile loops and
 * the loops that will be wrapped over the threads and
 * store the result in gen->shared_sched.
 * Also compute a projection that projects out the loops that will be
 * wrapped over the threads and store this projection in gen->shared_proj.
 */
static void compute_shared_sched(struct gpu_gen *gen)
{
        isl_space *dim;
        isl_map *proj;
        isl_set *par;
        isl_union_map *sched;

        sched = isl_union_map_copy(gen->tiled_sched);

        dim = isl_union_map_get_space(sched);
        proj = projection(dim, gen->tiled_len, gen->shared_len + gen->n_block);
        sched = isl_union_map_apply_range(sched, isl_union_map_from_map(proj));

        dim = isl_union_map_get_space(sched);
        proj = projection(dim, gen->shared_len + gen->n_block, gen->shared_len);

        gen->shared_sched = sched;
        gen->shared_proj = isl_union_map_from_map(proj);
}

/* For each scalar in the input program, check if there are any
 * order dependences active inside the current kernel, within
 * the same iteration of the host schedule.
 * If so, mark the scalar as force_private so that it will be
 * mapped to a register.
 */
static void check_scalar_live_ranges(struct gpu_gen *gen)
{
        int i;
        isl_map *proj;
        isl_union_map *sched;
        isl_union_set *domain;
        isl_union_map *same_host_iteration;

        gen->any_force_private = 0;

        if (!gen->options->live_range_reordering)
                return;

        sched = gen->shared_sched;
        sched = isl_union_map_universe(isl_union_map_copy(sched));
        domain = isl_union_map_domain(sched);

        sched = isl_union_map_copy(gen->sched);
        proj = projection(isl_union_map_get_space(sched),
                            gen->untiled_len, gen->tile_first);
        sched = isl_union_map_apply_range(sched, isl_union_map_from_map(proj));
        same_host_iteration = isl_union_map_apply_range(sched,
                            isl_union_map_reverse(isl_union_map_copy(sched)));

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];
                isl_union_map *order;

                array->force_private = 0;
                if (array->n_index != 0)
                        continue;
                order = isl_union_map_copy(array->dep_order);
                order = isl_union_map_intersect_domain(order,
                                                    isl_union_set_copy(domain));
                order = isl_union_map_intersect_range(order,
                                                    isl_union_set_copy(domain));
                order = isl_union_map_intersect(order,
                                    isl_union_map_copy(same_host_iteration));
                if (!isl_union_map_is_empty(order)) {
                        array->force_private = 1;
                        gen->any_force_private = 1;
                }
                isl_union_map_free(order);
        }

        isl_union_map_free(same_host_iteration);
        isl_union_set_free(domain);
}

/* Group references of all arrays in the program.
 */
static int group_references(struct gpu_gen *gen)
{
        int i;
        int r = 0;
        isl_union_map *sched;

        sched = isl_union_map_apply_range(isl_union_map_copy(gen->shared_sched),
                                          isl_union_map_copy(gen->shared_proj));

        for (i = 0; i < gen->prog->n_array; ++i) {
                r = group_array_references(gen, &gen->prog->array[i], sched);
                if (r < 0)
                        break;
        }

        isl_union_map_free(sched);

        return r;
}

/* Free all array information that is local to the current kernel.
 */
static void free_local_array_info(struct gpu_gen *gen)
{
        int i, j;

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j)
                        free_array_ref_group(array->groups[j]);
                free(array->groups);
        }
}

/* Compute the size of a bounding box around the origin and "set",
 * where "set" is assumed to contain only non-negative elements.
 * In particular, compute the maximal value of "set" in each direction
 * and add one.
 */
static __isl_give isl_multi_pw_aff *extract_size(__isl_take isl_set *set,
        __isl_keep isl_set *context)
{
        int i, n;
        isl_multi_pw_aff *mpa;

        n = isl_set_dim(set, isl_dim_set);
        mpa = isl_multi_pw_aff_zero(isl_set_get_space(set));
        for (i = 0; i < n; ++i) {
                isl_space *space;
                isl_aff *one;
                isl_pw_aff *bound;

                bound = isl_set_dim_max(isl_set_copy(set), i);
                bound = isl_pw_aff_coalesce(bound);
                bound = isl_pw_aff_gist(bound, isl_set_copy(context));

                space = isl_pw_aff_get_domain_space(bound);
                one = isl_aff_zero_on_domain(isl_local_space_from_space(space));
                one = isl_aff_add_constant_si(one, 1);
                bound = isl_pw_aff_add(bound, isl_pw_aff_from_aff(one));
                mpa = isl_multi_pw_aff_set_pw_aff(mpa, i, bound);
        }
        isl_set_free(set);

        return mpa;
}

/* Compute the effective grid size as a list of the sizes in each dimension.
 *
 * The grid size specified by the user or set by default
 * in read_grid_sizes() and applied in tile_schedule(),
 * may be too large for the given code in the sense that
 * it may contain blocks that don't need to execute anything.
 * We therefore don't return this grid size, but instead the
 * smallest grid size that ensures that all blocks that actually
 * execute code are included in the grid.
 *
 * We first extract a description of the grid, i.e., the possible values
 * of the block ids, from gen->tiled_sched.
 * The block ids are parameters in gen->tiled_sched.
 * We simply need to change them into set dimensions.
 *
 * Then, for each block dimension, we compute the maximal value of the block id
 * and add one.
 */
static __isl_give isl_multi_pw_aff *extract_grid_size(struct gpu_gen *gen,
        struct ppcg_kernel *kernel)
{
        int i;
        isl_set *grid;

        grid = isl_union_map_params(isl_union_map_copy(gen->tiled_sched));
        grid = isl_set_from_params(grid);
        grid = isl_set_add_dims(grid, isl_dim_set, gen->n_grid);
        for (i = 0; i < gen->n_grid; ++i) {
                int pos;
                char name[20];

                snprintf(name, sizeof(name), "b%d", i);
                pos = isl_set_find_dim_by_name(grid, isl_dim_param, name);
                assert(pos >= 0);
                grid = isl_set_equate(grid, isl_dim_param, pos, isl_dim_set, i);
                grid = isl_set_project_out(grid, isl_dim_param, pos, 1);
        }

        return extract_size(grid, kernel->context);
}

/* Compute the size of a fixed bounding box around the origin and "set",
 * where "set" is assumed to contain only non-negative elements,
 * and store the results in "size".
 * In particular, compute the maximal value of "set" in each direction
 * and add one.
 */
static void extract_fixed_size(__isl_take isl_set *set, int *size)
{
        int i, n;
        isl_local_space *ls;
        isl_aff *obj;

        n = isl_set_dim(set, isl_dim_set);
        ls = isl_local_space_from_space(isl_set_get_space(set));
        obj = isl_aff_zero_on_domain(ls);
        for (i = 0; i < n; ++i) {
                isl_val *max;

                obj = isl_aff_set_coefficient_si(obj, isl_dim_in, i, 1);
                max = isl_set_max_val(set, obj);
                size[i] = isl_val_get_num_si(max) + 1;
                isl_val_free(max);
                obj = isl_aff_set_coefficient_si(obj, isl_dim_in, i, 0);
        }
        isl_aff_free(obj);
        isl_set_free(set);
}

/* Compute the effective block size as a list of the sizes in each dimension
 * and store the sizes in kernel->block_dim.
 *
 * The block size specified by the user or set by default
 * in read_block_sizes() and applied in thread_tile_schedule(),
 * may be too large for the given code in the sense that
 * it may contain threads that don't need to execute anything.
 * We therefore don't store this block size in kernel->block_dim,
 * but instead the smallest block size that ensures that all threads
 * that actually execute code are included in the block.
 *
 * The current implementation eliminates all parameters, ensuring
 * that the size is a fixed constant in each dimension.
 * In principle we could also compute parametric sizes.
 * We would have to make sure to project out all b%d and t%d parameters,
 * however.
 */
static void extract_block_size(struct gpu_gen *gen, struct ppcg_kernel *kernel)
{
        int i;
        int nparam;
        isl_set *block;
        isl_multi_pw_aff *mpa;

        block = isl_union_map_params(isl_union_map_copy(gen->local_sched));
        block = isl_set_from_params(block);
        block = isl_set_add_dims(block, isl_dim_set, gen->n_block);
        kernel->n_block = gen->n_block;
        for (i = 0; i < gen->n_block; ++i) {
                int pos;
                char name[20];

                snprintf(name, sizeof(name), "t%d", i);
                pos = isl_set_find_dim_by_name(block, isl_dim_param, name);
                assert(pos >= 0);
                block = isl_set_equate(block, isl_dim_param, pos,
                                        isl_dim_set, i);
        }
        nparam = isl_set_dim(block, isl_dim_param);
        block = isl_set_project_out(block, isl_dim_param, 0, nparam);

        extract_fixed_size(block, kernel->block_dim);
}

void ppcg_kernel_free(void *user)
{
        struct ppcg_kernel *kernel = user;
        int i;

        if (!kernel)
                return;

        isl_multi_pw_aff_free(kernel->grid_size);
        isl_set_free(kernel->context);
        isl_union_set_free(kernel->arrays);
        isl_space_free(kernel->space);
        isl_ast_node_free(kernel->tree);

        for (i = 0; i < kernel->n_array; ++i)
                isl_pw_aff_list_free(kernel->array[i].bound);
        free(kernel->array);

        for (i = 0; i < kernel->n_var; ++i) {
                free(kernel->var[i].name);
                isl_vec_free(kernel->var[i].size);
        }
        free(kernel->var);

        free(kernel);
}

static void create_kernel_var(isl_ctx *ctx, struct gpu_array_ref_group *group,
        struct ppcg_kernel_var *var)
{
        int j;
        struct gpu_array_tile *tile;
        isl_printer *p;
        char *name;

        var->array = group->array;

        tile = group->private_tile;
        var->type = ppcg_access_private;
        if (!tile) {
                tile = group->shared_tile;
                var->type = ppcg_access_shared;
        }

        p = isl_printer_to_str(ctx);
        p = print_array_name(p, group);
        var->name = isl_printer_get_str(p);
        isl_printer_free(p);

        var->size = isl_vec_alloc(ctx, group->array->n_index);

        for (j = 0; j < group->array->n_index; ++j)
                var->size = isl_vec_set_element_val(var->size, j,
                                            isl_val_copy(tile->bound[j].size));
}

static void create_kernel_vars(struct gpu_gen *gen, struct ppcg_kernel *kernel)
{
        int i, j, n;

        n = 0;
        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j) {
                        struct gpu_array_ref_group *group = array->groups[j];
                        if (group->private_tile || group->shared_tile)
                                ++n;
                }
        }

        kernel->n_var = n;
        kernel->var = isl_calloc_array(gen->ctx, struct ppcg_kernel_var, n);
        assert(kernel->var);

        n = 0;
        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j) {
                        struct gpu_array_ref_group *group = array->groups[j];
                        if (!group->private_tile && !group->shared_tile)
                                continue;
                        create_kernel_var(gen->ctx, group, &kernel->var[n]);
                        ++n;
                }
        }
}

/* The sizes of the arrays on the host that have been computed by
 * extract_array_info may depend on the parameters.  Use the extra
 * constraints on the parameters that are valid at "host_domain"
 * to simplify these expressions and store the results in kernel->array.
 *
 * We only need these localized bounds for arrays that are accessed
 * by the current kernel.  If we have found at least one reference group
 * then the array is accessed by the kernel.  If the array has compound
 * elements then we skipped the construction of array reference groups.
 */
static void localize_bounds(struct gpu_gen *gen, struct ppcg_kernel *kernel,
        __isl_keep isl_set *host_domain)
{
        int i, j;
        isl_set *context;

        kernel->array = isl_calloc_array(gen->ctx,
                            struct gpu_local_array_info, gen->prog->n_array);
        assert(kernel->array);
        kernel->n_array = gen->prog->n_array;

        context = isl_set_copy(host_domain);
        context = isl_set_params(context);

        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];
                isl_pw_aff_list *local;

                if (array->n_group == 0 && !array->has_compound_element)
                        continue;

                local = isl_pw_aff_list_alloc(gen->ctx, array->n_index);

                for (j = 0; j < array->n_index; ++j) {
                        isl_pw_aff *pwaff;

                        pwaff = isl_pw_aff_copy(array->bound[j]);
                        pwaff = isl_pw_aff_gist(pwaff, isl_set_copy(context));
                        local = isl_pw_aff_list_add(local, pwaff);
                }

                kernel->array[i].bound = local;
        }
        isl_set_free(context);
}

/* Find the element in gen->stmt that has the given "id".
 * Return NULL if no such gpu_stmt can be found.
 */
static struct gpu_stmt *find_stmt(struct gpu_prog *prog, __isl_keep isl_id *id)
{
        int i;

        for (i = 0; i < prog->n_stmts; ++i) {
                if (id == prog->stmts[i].id)
                        break;
        }

        return i < prog->n_stmts ? &prog->stmts[i] : NULL;
}

/* Set gen->tile_len and gen->n_parallel to those of the statement
 * affected by the first map (part of the schedule)
 * on which this function is called.
 * Because of the way the schedule is constructed, the other statements
 * in the list, if any, should have the same values for these properties.
 */
static int extract_tile_len(__isl_take isl_map *map, void *user)
{
        struct gpu_gen *gen = (struct gpu_gen *) user;
        isl_id *id;
        struct gpu_stmt *stmt;

        id = isl_map_get_tuple_id(map, isl_dim_in);
        stmt = find_stmt(gen->prog, id);
        isl_id_free(id);

        isl_map_free(map);

        if (!stmt)
                isl_die(gen->ctx, isl_error_unknown,
                        "statement not found", return -1);

        gen->tile_len = stmt->tile_len;
        gen->n_parallel = stmt->n_parallel;

        return -1;
}

void ppcg_kernel_stmt_free(void *user)
{
        int i;
        struct ppcg_kernel_stmt *stmt = user;

        if (!stmt)
                return;

        switch (stmt->type) {
        case ppcg_kernel_copy:
                isl_ast_expr_free(stmt->u.c.index);
                isl_ast_expr_free(stmt->u.c.local_index);
                break;
        case ppcg_kernel_domain:
                isl_id_to_ast_expr_free(stmt->u.d.ref2expr);
                break;
        case ppcg_kernel_sync:
                break;
        }

        free(stmt);
}

/* Set the options of "context" to
 *
 *      { space -> [x] : x >= first }
 */
static __isl_give isl_ast_build *set_unroll(
        __isl_take isl_ast_build *build, __isl_take isl_space *space,
        int first)
{
        isl_ctx *ctx;
        isl_map *unroll;
        isl_union_map *opt;

        ctx = isl_ast_build_get_ctx(build);

        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, 1);
        space = isl_space_set_tuple_name(space, isl_dim_out, "unroll");
        unroll = isl_map_universe(space);
        unroll = isl_map_lower_bound_si(unroll, isl_dim_out, 0, first);
        opt = isl_union_map_from_map(unroll);

        build = isl_ast_build_set_options(build, opt);

        return build;
}

/* Return a list of isl_ids of the form "prefix%d".
 */
static __isl_give isl_id_list *generate_names(isl_ctx *ctx,
        int n, const char *prefix)
{
        int i;
        char name[10];
        isl_id_list *names;

        names = isl_id_list_alloc(ctx, n);
        for (i = 0; i < n; ++i) {
                isl_id *id;

                snprintf(name, sizeof(name), "%s%d", prefix, i);
                id = isl_id_alloc(ctx, name, NULL);
                names = isl_id_list_add(names, id);
        }

        return names;
}

/* Extend the schedule "schedule" with the part of "extension"
 * starting at "first" up to "len".
 */
static __isl_give isl_union_map *extend_schedule(
        __isl_take isl_union_map *schedule,
        __isl_take isl_union_map *extension, int first, int len)
{
        isl_space *space;
        isl_map *proj;
        isl_union_map *umap;
        isl_set *set;

        space = isl_union_map_get_space(schedule);
        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, len);
        proj = isl_set_identity(isl_set_universe(space));
        proj = isl_map_project_out(proj, isl_dim_out, 0, first);
        extension = isl_union_map_apply_range(extension,
                                                isl_union_map_from_map(proj));

        schedule = isl_union_map_range_product(schedule, extension);

        return schedule;
}

/* Return the gpu_stmt_access in the list "accesses" that corresponds
 * to "ref_id".
 */
static struct gpu_stmt_access *find_access(struct gpu_stmt_access *accesses,
        __isl_keep isl_id *ref_id)
{
        struct gpu_stmt_access *access;

        for (access = accesses; access; access = access->next)
                if (access->ref_id == ref_id)
                        return access;

        return NULL;
}

/* Return the index of the array called "name" in the list of arrays.
 */
static int find_array_index(struct gpu_gen *gen, const char *name)
{
        int i;

        for (i = 0; i < gen->prog->n_array; ++i)
                if (!strcmp(name, gen->prog->array[i].name))
                        return i;

        return -1;
}

/* Internal data structure for the index and AST expression transformation
 * callbacks for pet_stmt_build_ast_exprs.
 *
 * "accesses" is the list of gpu_stmt_access in the statement.
 * "iterator_map" expresses the statement iterators in terms of
 * the AST loop iterators.
 * "sched2shared" expresses the first shared_len dimensions of
 * the computed schedule in terms of the AST loop iterators.
 *
 * The following fields are set in transform_index and used in transform_expr.
 * "array" is the array that is being accessed.
 * "global" is set if the global array is accessed (rather than
 * shared/private memory).
 * "local_array" refers to information on the array specialized
 * to the current kernel.
 */
struct ppcg_transform_data {
        struct gpu_gen *gen;
        struct gpu_stmt_access *accesses;
        isl_pw_multi_aff *iterator_map;
        isl_pw_multi_aff *sched2shared;

        struct gpu_array_info *array;
        int global;
        struct gpu_local_array_info *local_array;
};

/* Return the name of the outer array (of structs) accessed by "access".
 */
static const char *get_outer_array_name(__isl_keep isl_map *access)
{
        isl_space *space;
        const char *name;

        space = isl_space_range(isl_map_get_space(access));
        while (space && isl_space_is_wrapping(space))
                space = isl_space_domain(isl_space_unwrap(space));
        name = isl_space_get_tuple_name(space, isl_dim_set);
        isl_space_free(space);

        return name;
}

/* Index transformation callback for pet_stmt_build_ast_exprs.
 *
 * "index" expresses the array indices in terms of statement iterators
 *
 * We first reformulate "index" in terms of the AST loop iterators.
 * Then we check if we are accessing the global array or
 * a shared/private copy.  In the former case, we simply return
 * the updated index.  If "index" is an affine expression rather
 * than an array access, then we also return the updated index here.
 *
 * If no reference groups have been computed for the array,
 * then we can only be accessing the global array.
 *
 * Otherwise, we apply the tiling to the index.
 * This tiling is of the form
 *
 *      [D -> A] -> T
 *
 * The index is of the form
 *
 *      L -> A
 *
 * We update the tiling to refer to the AST loop iteratos
 *
 *      [L -> A] -> T
 *
 * and modify index to keep track of those iterators
 *
 *      L -> [L -> A]
 *
 * Combining these two yields a tiled index expression in terms
 * of the AST loop iterators
 *
 *      L -> T
 */
static __isl_give isl_multi_pw_aff *transform_index(
        __isl_take isl_multi_pw_aff *index, __isl_keep isl_id *ref_id,
        void *user)
{
        struct ppcg_transform_data *data = user;
        struct gpu_stmt_access *access;
        struct gpu_array_ref_group *group;
        struct gpu_array_tile *tile;
        isl_pw_multi_aff *iterator_map;
        int i;
        const char *name;
        isl_space *space;
        isl_multi_pw_aff *tiling;
        isl_pw_multi_aff *pma;
        isl_multi_pw_aff *mpa;

        data->array = NULL;

        iterator_map = isl_pw_multi_aff_copy(data->iterator_map);
        index = isl_multi_pw_aff_pullback_pw_multi_aff(index, iterator_map);

        access = find_access(data->accesses, ref_id);
        if (!access)
                return index;
        if (!isl_map_has_tuple_name(access->access, isl_dim_out))
                return index;

        name = get_outer_array_name(access->access);
        i = find_array_index(data->gen, name);
        if (i < 0)
                isl_die(isl_multi_pw_aff_get_ctx(index), isl_error_internal,
                        "cannot find array",
                        return isl_multi_pw_aff_free(index));
        data->array = &data->gen->prog->array[i];
        data->local_array = &data->gen->kernel->array[i];

        if (access->group < 0) {
                data->global = 1;
                return index;
        }

        group = data->array->groups[access->group];
        tile = group->private_tile;
        if (!tile)
                tile = group->shared_tile;
        data->global = !tile;
        if (!tile)
                return index;

        space = isl_space_range(isl_multi_pw_aff_get_space(index));
        space = isl_space_map_from_set(space);
        pma = isl_pw_multi_aff_identity(space);
        pma = isl_pw_multi_aff_product(
                        isl_pw_multi_aff_copy(data->sched2shared), pma);
        tiling = isl_multi_pw_aff_from_multi_aff(
                                    isl_multi_aff_copy(tile->tiling));
        tiling = isl_multi_pw_aff_pullback_pw_multi_aff(tiling, pma);

        space = isl_space_domain(isl_multi_pw_aff_get_space(index));
        space = isl_space_map_from_set(space);
        mpa = isl_multi_pw_aff_identity(space);
        index = isl_multi_pw_aff_range_product(mpa, index);
        index = isl_multi_pw_aff_pullback_multi_pw_aff(tiling, index);

        return index;
}

/* Dereference "expr" by adding an index [0].
 * The original "expr" is assumed not to have any indices.
 *
 * If "expr" is a member access, then the dereferencing needs
 * to be applied to the structure argument of this member access.
 */
static __isl_give isl_ast_expr *dereference(__isl_take isl_ast_expr *expr)
{
        isl_ctx *ctx;
        isl_ast_expr *res;
        isl_ast_expr_list *list;

        if (isl_ast_expr_get_op_type(expr) == isl_ast_op_member) {
                isl_ast_expr *arg;

                arg = isl_ast_expr_get_op_arg(expr, 0);
                arg = dereference(arg);
                expr = isl_ast_expr_set_op_arg(expr, 0, arg);

                return expr;
        }

        ctx = isl_ast_expr_get_ctx(expr);
        res = isl_ast_expr_from_val(isl_val_zero(ctx));
        list = isl_ast_expr_list_from_ast_expr(res);
        res = isl_ast_expr_get_op_arg(expr, 0);
        res = isl_ast_expr_access(res, list);
        isl_ast_expr_free(expr);

        return res;
}

/* Linearize the index expression "expr" based on the array bounds
 * of "array".
 *
 * That is, transform expression
 *
 *      A[i_0][i_1]...[i_n]
 *
 * to
 *
 *      A[(..((i_0 * b_1 + i_1) ... ) * b_n + i_n]
 *
 * where b_0, b_1, ..., b_n are the bounds on the array.
 *
 * If the base of "expr" is a member access, then the linearization needs
 * to be applied to the structure argument of this member access.
 */
__isl_give isl_ast_expr *gpu_local_array_info_linearize_index(
        struct gpu_local_array_info *array, __isl_take isl_ast_expr *expr)
{
        int i, n;
        isl_ctx *ctx;
        isl_set *context;
        isl_ast_expr *arg0;
        isl_ast_expr *res;
        isl_ast_expr_list *list;
        isl_ast_build *build;

        arg0 = isl_ast_expr_get_op_arg(expr, 0);
        if (isl_ast_expr_get_type(arg0) == isl_ast_expr_op &&
            isl_ast_expr_get_op_type(arg0) == isl_ast_op_member) {
                isl_ast_expr *arg;

                arg = isl_ast_expr_get_op_arg(arg0, 0);
                arg = gpu_local_array_info_linearize_index(array, arg);
                arg0 = isl_ast_expr_set_op_arg(arg0, 0, arg);
                expr = isl_ast_expr_set_op_arg(expr, 0, arg0);

                return expr;
        }
        isl_ast_expr_free(arg0);

        ctx = isl_ast_expr_get_ctx(expr);
        context = isl_set_universe(isl_space_params_alloc(ctx, 0));
        build = isl_ast_build_from_context(context);

        n = isl_ast_expr_get_op_n_arg(expr);
        res = isl_ast_expr_get_op_arg(expr, 1);
        for (i = 2; i < n; ++i) {
                isl_pw_aff *bound_i;
                isl_ast_expr *expr_i;

                bound_i = isl_pw_aff_list_get_pw_aff(array->bound, i - 1);
                expr_i = isl_ast_build_expr_from_pw_aff(build, bound_i);
                res = isl_ast_expr_mul(res, expr_i);
                expr_i = isl_ast_expr_get_op_arg(expr, i);
                res = isl_ast_expr_add(res, expr_i);
        }

        isl_ast_build_free(build);

        list = isl_ast_expr_list_from_ast_expr(res);
        res = isl_ast_expr_get_op_arg(expr, 0);
        res = isl_ast_expr_access(res, list);

        isl_ast_expr_free(expr);

        return res;
}

/* AST expression transformation callback for pet_stmt_build_ast_exprs.
 *
 * If the AST expression refers to a global scalar that is not
 * a read-only scalar, then its address was passed to the kernel and
 * we need to dereference it.
 *
 * If the AST expression refers to an access to a global array,
 * then we linearize the access exploiting the bounds in data->local_array.
 */
static __isl_give isl_ast_expr *transform_expr(__isl_take isl_ast_expr *expr,
        __isl_keep isl_id *id, void *user)
{
        struct ppcg_transform_data *data = user;

        if (!data->array)
                return expr;
        if (gpu_array_is_read_only_scalar(data->array))
                return expr;
        if (!data->global)
                return expr;
        if (data->array->n_index == 0)
                return dereference(expr);
        if (!data->array->linearize)
                return expr;

        return gpu_local_array_info_linearize_index(data->local_array, expr);
}

/* This function is called for each instance of a user statement
 * in the kernel.
 *
 * We attach a struct ppcg_kernel_stmt to the "node", containing
 * a computed AST expression for each access.
 * These AST expressions are computed from iterator_map,
 * which expresses the domain
 * elements in terms of the generated loops, and sched2shared,
 * which expresses the first shared_len dimensions of the schedule
 * computed by PPCG in terms of the generated loops.
 */
static __isl_give isl_ast_node *at_each_domain(__isl_take isl_ast_node *node,
        __isl_keep isl_ast_build *build, void *user)
{
        struct ppcg_transform_data data;
        struct gpu_gen *gen = (struct gpu_gen *) user;
        struct ppcg_kernel_stmt *stmt;
        isl_id *id;
        isl_pw_multi_aff *sched2shared;
        isl_map *map;
        isl_pw_multi_aff *iterator_map;
        isl_ast_expr *expr, *arg;
        isl_union_map *schedule;
        int i, n;

        stmt = isl_calloc_type(gen->ctx, struct ppcg_kernel_stmt);
        if (!stmt)
                return isl_ast_node_free(node);

        expr = isl_ast_node_user_get_expr(node);
        arg = isl_ast_expr_get_op_arg(expr, 0);
        id = isl_ast_expr_get_id(arg);

        schedule = isl_ast_build_get_schedule(build);
        map = isl_map_reverse(isl_map_from_union_map(schedule));
        iterator_map = isl_pw_multi_aff_from_map(map);
        sched2shared = compute_sched_to_shared(gen,
                                        isl_pw_multi_aff_copy(iterator_map));

        stmt->type = ppcg_kernel_domain;
        stmt->u.d.stmt = find_stmt(gen->prog, id);
        if (!stmt->u.d.stmt)
                goto error;

        data.gen = gen;
        data.accesses = stmt->u.d.stmt->accesses;
        data.iterator_map = iterator_map;
        data.sched2shared = sched2shared;
        stmt->u.d.ref2expr = pet_stmt_build_ast_exprs(stmt->u.d.stmt->stmt,
                                            build, &transform_index, &data,
                                            &transform_expr, &data);

        isl_id_free(id);
        isl_pw_multi_aff_free(iterator_map);
        isl_pw_multi_aff_free(sched2shared);
        isl_ast_expr_free(arg);
        isl_ast_expr_free(expr);

        id = isl_id_alloc(gen->ctx, NULL, stmt);
        id = isl_id_set_free_user(id, &ppcg_kernel_stmt_free);
        return isl_ast_node_set_annotation(node, id);
error:
        isl_id_free(id);
        isl_pw_multi_aff_free(iterator_map);
        ppcg_kernel_stmt_free(stmt);
        isl_pw_multi_aff_free(sched2shared);
        return isl_ast_node_free(node);
}

/* This function is called when code has been generated for the shared
 * tile loops.  The "schedule" refers only to the original statements.
 *
 * We extend the schedule with that part of gen->local_sched that hasn't
 * been taken into account yet.  This introduces parameters referring
 * to thread ids in the schedule, so we add them (with the appropriate
 * bounds to the context as well).
 * Finally, we set the appropriate unrolling options
 * if gen->first_unroll is set.
 */
static __isl_give isl_ast_node *create_domain_leaf(
        __isl_take isl_union_map *schedule, __isl_take isl_ast_build *build,
        void *user)
{
        struct gpu_gen *gen = (struct gpu_gen *) user;
        isl_space *space;
        isl_union_map *sched;
        isl_ast_node *tree;
        isl_set *set;
        isl_id_list *iterators;
        int n;

        schedule = extend_schedule(schedule,
                        isl_union_map_copy(gen->local_sched),
                        gen->shared_len, gen->thread_tiled_len);

        space = isl_ast_build_get_schedule_space(build);
        set = isl_set_universe(space);
        set = add_bounded_parameters(set, gen->kernel->n_block,
                                        gen->kernel->block_dim, "t");
        build = isl_ast_build_restrict(build, set);

        n = gen->thread_tiled_len - gen->shared_len;

        if (gen->first_unroll >= 0) {
                space = isl_space_set_alloc(gen->ctx, 0, n);
                build = set_unroll(build, space, gen->first_unroll);
        }
        iterators = generate_names(gen->ctx, n, "c");
        build = isl_ast_build_set_iterators(build, iterators);
        build = isl_ast_build_set_at_each_domain(build, &at_each_domain, gen);
        tree = isl_ast_build_ast_from_schedule(build, schedule);
        isl_ast_build_free(build);

        return tree;
}

/* This function is called for each statement node in the AST of the code
 * for copying to or from shared/private memory.
 * Attach a pointer to a ppcg_kernel_stmt representing the copy
 * statement to the node.
 * The statement name is "read" or "write", depending on whether we are
 * reading from global memory or writing to global memory.
 * The name of the T space is {shared,private}_<array>.
 *
 * The schedule is of the form
 *
 *      type[A -> T] -> L
 *
 * where A refers to a piece of an array and T to the corresponding
 * shifted tile.  We split this schedule into mappings L -> A and L -> T
 * and store the corresponding expressions in stmt->index and stmt->local_index,
 * where stmt points to the ppcg_kernel_stmt that is attached to the node.
 */
static __isl_give isl_ast_node *attach_copy_stmt(__isl_take isl_ast_node *node,
        __isl_keep isl_ast_build *build, void *user)
{
        struct gpu_gen *gen = (struct gpu_gen *) user;
        struct ppcg_kernel_stmt *stmt;
        isl_id *id;
        isl_ast_expr *expr;
        isl_space *space;
        isl_map *access, *local_access, *map;
        isl_pw_multi_aff *pma;
        const char *type;
        int array_index;

        stmt = isl_calloc_type(gen->ctx, struct ppcg_kernel_stmt);
        if (!stmt)
                return isl_ast_node_free(node);

        access = isl_map_from_union_map(isl_ast_build_get_schedule(build));
        type = isl_map_get_tuple_name(access, isl_dim_in);
        stmt->u.c.read = !strcmp(type, "read");
        access = isl_map_reverse(access);
        space = isl_space_unwrap(isl_space_range(isl_map_get_space(access)));
        local_access = isl_map_copy(access);

        map = isl_map_domain_map(isl_map_universe(isl_space_copy(space)));
        id = isl_map_get_tuple_id(access, isl_dim_out);
        map = isl_map_set_tuple_id(map, isl_dim_in, id);
        access = isl_map_apply_range(access, map);
        pma = isl_pw_multi_aff_from_map(access);
        expr = isl_ast_build_access_from_pw_multi_aff(build, pma);
        stmt->u.c.index = expr;

        map = isl_map_range_map(isl_map_universe(space));
        id = isl_map_get_tuple_id(local_access, isl_dim_out);
        map = isl_map_set_tuple_id(map, isl_dim_in, id);
        local_access = isl_map_apply_range(local_access, map);
        pma = isl_pw_multi_aff_from_map(local_access);
        expr = isl_ast_build_access_from_pw_multi_aff(build, pma);
        stmt->u.c.local_index = expr;

        stmt->u.c.array = gen->copy_group->array;
        array_index = stmt->u.c.array - gen->prog->array;
        stmt->u.c.local_array = &gen->kernel->array[array_index];
        stmt->type = ppcg_kernel_copy;

        id = isl_id_alloc(gen->ctx, NULL, stmt);
        id = isl_id_set_free_user(id, &ppcg_kernel_stmt_free);
        return isl_ast_node_set_annotation(node, id);
}

/* Given a schedule of the form
 *
 *      [S -> A] -> L
 *
 * (with S the first shared_len dimensions of the computed schedule,
 * A the array and L the schedule correponding to the generated loops),
 * indicating where to copy the array elements that need to be copied,
 * construct code for performing the copying.
 *
 * "group" is the array reference group that is being copied
 * "type" is either "read" or "write"
 * private is set if copying needs to be performed to/from registers
 *
 * We first construct a mapping to a shifted tile of the array,
 *
 *      [S -> A] -> T(S,A)                                      (1)
 *
 * If private is set, then we also use this mapping as a schedule
 * (which is already thread-specific and will be completely unrolled).
 * Otherwise, we wrap/tile the range over the threads.
 * The result is
 *
 *      [S -> A] -> T'(S,A)
 *
 * Combined with the given schedule, we have
 *
 *      [S -> A] -> [L -> T'(S,A)]                              (2)
 *
 * From the shifted tile mapping, we construct a mapping
 *
 *      [S -> A] -> [A -> T(S,A)]
 *
 * and apply it to the schedule (2), obtaining
 *
 *      [A -> T(S(L),A)] -> [L -> T'(S(L),A)]
 *
 * Note that we can project out S because it is uniquely defined by L.
 */
static __isl_give isl_ast_node *copy_access(struct gpu_gen *gen,
        __isl_take isl_map *sched,
        const char *type, struct gpu_array_ref_group *group,
        __isl_take isl_ast_build *build, int private)
{
        isl_space *space;
        isl_ast_node *tree;
        isl_map *schedule, *shift, *map;
        isl_set *set;
        isl_id_list *iterators;
        int n;

        shift = shift_access(group);

        schedule = isl_map_copy(shift);
        schedule = isl_map_reset_tuple_id(schedule, isl_dim_out);
        if (!private)
                schedule = tile_access_schedule(gen, schedule);

        n = isl_map_dim(schedule, isl_dim_out);
        set = isl_set_universe(isl_ast_build_get_schedule_space(build));
        set = add_bounded_parameters(set, gen->kernel->n_block,
                                        gen->kernel->block_dim, "t");

        schedule = isl_map_range_product(sched, schedule);

        space = isl_space_domain(isl_map_get_space(shift));
        map = isl_map_range_map(isl_map_universe(isl_space_unwrap(space)));
        map = isl_map_range_product(map, shift);

        schedule = isl_map_apply_domain(schedule, map);

        schedule = isl_map_set_tuple_name(schedule, isl_dim_in, type);

        build = isl_ast_build_restrict(build, set);

        gen->copy_group = group;

        if (private) {
                space = isl_space_range(isl_map_get_space(schedule));
                space = isl_space_range(isl_space_unwrap(space));
                build = set_unroll(build, space, 0);
        }
        iterators = generate_names(gen->ctx, n, "c");
        build = isl_ast_build_set_iterators(build, iterators);
        build = isl_ast_build_set_at_each_domain(build, &attach_copy_stmt, gen);
        tree = isl_ast_build_ast_from_schedule(build,
                                            isl_union_map_from_map(schedule));
        isl_ast_build_free(build);

        return tree;
}

/* Return code for reading into or writing from shared memory
 * the given array reference group.
 *
 * If we are performing a read from global memory to shared memory and
 * if the array involved is not a scalar, then we copy
 * the entire tile to shared memory.  This may result in some extra
 * elements getting copied, but it should lead to simpler code
 * (which means that fewer registers may be needed) and less divergence.
 *
 * Otherwise, we only copy the elements that will be read or have been written
 * in the kernel.
 *
 *
 * The input "sched" is of the form.
 *
 *      type[S -> A] -> L
 *
 * with S the first shared_len dimensions of the computed schedule,
 * A the array and L the schedule correponding to the generated loops.
 *
 * We first drop "type",
 *
 *      [S -> A] -> L
 *
 * If the above conditions are satisfied, we project out A,
 * resulting in
 *
 *      S -> L
 *
 * and then introduce the group tile [S -> T], resulting in
 *
 *      [S -> T] -> L
 */
static __isl_give isl_ast_node *copy_group_shared_accesses(
        struct gpu_gen *gen, struct gpu_array_ref_group *group,
        __isl_take isl_map *sched, __isl_take isl_ast_build *build)
{
        const char *type;
        int read;
        isl_union_map *access;

        type = isl_map_get_tuple_name(sched, isl_dim_in);
        read = !strcmp(type, "read");

        sched = isl_map_reset_tuple_id(sched, isl_dim_in);

        if (read && !gpu_array_is_scalar(group->array)) {
                isl_space *space;
                isl_map *map;

                space = isl_space_domain(isl_map_get_space(sched));
                space = isl_space_unwrap(space);
                map = isl_map_domain_map(isl_map_universe(space));
                sched = isl_map_apply_domain(sched, map);

                map = group_tile(group);
                map = isl_map_reverse(isl_map_domain_map(map));
                sched = isl_map_apply_domain(sched, map);
        }

        return copy_access(gen, sched, type, group, build, 0);
}

/* Return code for reading into or writing from private memory
 * the given array reference group.
 *
 * Let S be the first shared_len dimensions of the computed schedule,
 * D the iteration domains, A the array and L the schedule correponding
 * to the generated loops.
 * "sched" is of the form
 *
 *      type[S -> A] -> L
 *
 * where type is either "read" or "write".
 * We apply the privatization D -> S(t), with t the thread ids,
 * to the access relation D -> A to obtain the privatized access relation
 *
 *      S(t) -> A
 *
 * We drop the type from "sched" and intersect with the privatized access
 * relation to obtain
 *
 *      [S(t) -> A] -> L
 */
static __isl_give isl_ast_node *copy_group_private_accesses(
        struct gpu_gen *gen, struct gpu_array_ref_group *group,
        __isl_take isl_map *sched, __isl_take isl_ast_build *build)
{
        const char *type;
        int read;
        isl_union_map *priv;
        isl_union_map *access;
        isl_map *access_map;

        type = isl_map_get_tuple_name(sched, isl_dim_in);
        read = !strcmp(type, "read");

        priv = isl_union_map_from_map(isl_map_copy(gen->privatization));
        priv = isl_union_map_apply_range(isl_union_map_copy(gen->shared_sched),
                                        priv);

        access = group_access_relation(group, read, !read);
        access = isl_union_map_apply_domain(access, priv);
        access_map = isl_map_from_union_map(access);

        sched = isl_map_reset_tuple_id(sched, isl_dim_in);
        sched = isl_map_intersect_domain(sched, isl_map_wrap(access_map));

        return copy_access(gen, sched, type, group, build, 1);
}

/* Return code for reading into or writing from shared or private memory.
 *
 * "schedule" is of the form
 *
 *      type[S -> A] -> L
 *
 * with S be the first shared_len dimensions of the computed schedule,
 * A the array and L the schedule correponding to the generated loops.
 * The array reference group is attached to "type".
 */
static __isl_give isl_ast_node *create_access_leaf(
        struct gpu_gen *gen, __isl_take isl_map *schedule,
        __isl_take isl_ast_build *build)
{
        struct gpu_array_ref_group *group;
        isl_id *id;

        id = isl_map_get_tuple_id(schedule, isl_dim_in);
        group = isl_id_get_user(id);
        isl_id_free(id);

        if (group->private_tile)
                return copy_group_private_accesses(gen, group, schedule,
                                                        build);
        else
                return copy_group_shared_accesses(gen, group, schedule,
                                                        build);
}

/* Create a domain node representing a synchronization.
 */
static __isl_give isl_ast_node *create_sync_leaf(
        struct gpu_gen *gen, __isl_take isl_map *schedule,
        __isl_take isl_ast_build *build)
{
        struct ppcg_kernel_stmt *stmt;
        isl_id *id;
        isl_space *space;
        isl_ast_node *node;
        isl_ast_expr *expr;

        isl_map_free(schedule);

        stmt = isl_calloc_type(gen->ctx, struct ppcg_kernel_stmt);
        if (!stmt)
                return NULL;

        stmt->type = ppcg_kernel_sync;

        space = isl_ast_build_get_schedule_space(build);
        space = isl_space_from_domain(space);
        space = isl_space_set_tuple_name(space, isl_dim_out, "sync");
        expr = isl_ast_build_call_from_pw_multi_aff(build,
                    isl_pw_multi_aff_from_multi_aff(isl_multi_aff_zero(space)));
        node = isl_ast_node_alloc_user(expr);
        isl_ast_build_free(build);

        id = isl_id_alloc(gen->ctx, NULL, stmt);
        id = isl_id_set_free_user(id, &ppcg_kernel_stmt_free);
        return isl_ast_node_set_annotation(node, id);
}

/* This function is called during the code generation at the point
 * where the schedule domain element is completely determined by
 * the generated code.  The input schedule contains the original
 * statements as well as synchronization and copy "statements".
 * The latter are scheduled at different points than any of the original
 * statements, so they will only arrive here in isolation.
 *
 * If the current schedule only refers to a single statement,
 * we check if it is a copy or synchronization statement and
 * call the appropriate functions.
 * Otherwise, we assume we are dealing with the original statements
 * and we call create_domain_leaf.
 */
static __isl_give isl_ast_node *create_kernel_leaf(
        __isl_take isl_ast_build *build, void *user)
{
        struct gpu_gen *gen = (struct gpu_gen *) user;
        isl_map *map;
        isl_union_map *schedule;
        const char *name;

        schedule = isl_ast_build_get_schedule(build);

        if (isl_union_map_n_map(schedule) != 1)
                return create_domain_leaf(schedule, build, user);

        map = isl_map_from_union_map(schedule);
        name = isl_map_get_tuple_name(map, isl_dim_in);
        if (!strcmp(name, "read") || !strcmp(name, "write"))
                return create_access_leaf(gen, map, build);
        if (!strcmp(name, "sync"))
                return create_sync_leaf(gen, map, build);

        return create_domain_leaf(isl_union_map_from_map(map), build, user);
}

/* Mark all odd schedule dimensions as "atomic" (when the even dimensions
 * have value 0) and all even schedule dimensions as "unroll".
 *
 * That is, the options look as follows
 *
 *      { [0, b, 0, d, ..., 0] -> atomic[i] : exists a : i = 2 a + 1;
 *        [a, b, c, d, ..., z] -> unroll[i] : exists a : i = 2 a }
 *
 * The even positions are used to be able to schedule copying blocks
 * and synchronization before or after each level of the shared memory
 * tile loops and we want to make sure that code for these is generated
 * separately (within each level).
 */
static __isl_give isl_ast_build *set_atomic_and_unroll(
        __isl_take isl_ast_build *build,
        __isl_take isl_space *space, int sched_len)
{
        isl_ctx *ctx;
        isl_map *map;
        isl_constraint *c;
        isl_union_map *opt;
        isl_local_space *ls;
        int i, n;

        ctx = isl_ast_build_get_ctx(build);

        space = isl_space_params(space);
        space = isl_space_add_dims(space, isl_dim_set, sched_len);
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, 2);
        map = isl_map_universe(isl_space_copy(space));
        for (i = 0; i < sched_len; i += 2)
                map = isl_map_fix_si(map, isl_dim_in, i, 0);
        ls = isl_local_space_from_space(isl_map_get_space(map));
        c = isl_equality_alloc(ls);
        c = isl_constraint_set_coefficient_si(c, isl_dim_out, 0, 1);
        c = isl_constraint_set_coefficient_si(c, isl_dim_out, 1, 2);
        c = isl_constraint_set_constant_si(c, 1);
        map = isl_map_add_constraint(map, c);
        map = isl_map_project_out(map, isl_dim_out, 1, 1);
        map = isl_map_set_tuple_name(map, isl_dim_out, "atomic");
        opt = isl_union_map_from_map(map);

        map = isl_map_universe(space);
        ls = isl_local_space_from_space(isl_map_get_space(map));
        c = isl_equality_alloc(ls);
        c = isl_constraint_set_coefficient_si(c, isl_dim_out, 0, 1);
        c = isl_constraint_set_coefficient_si(c, isl_dim_out, 1, 2);
        map = isl_map_add_constraint(map, c);
        map = isl_map_project_out(map, isl_dim_out, 1, 1);
        map = isl_map_set_tuple_name(map, isl_dim_out, "unroll");
        opt = isl_union_map_add_map(opt, map);

        build = isl_ast_build_set_options(build, opt);

        return build;
}

/* Return a map that maps a space of dimension gen->shared_len
 * to its last dimensions starting at gen->tile_first.
 * The range is of dimension
 *
 *      2 * (gen->shared_len - gen->tile_first) + 1
 *
 * The input dimensions are mapped to the odd dimensions in the output,
 * while the even dimensions (except 2*pos) are fixed to 0.
 * Output dimension 2*pos (if pos >= 0) is fixed to "val".
 * If pos >= 0, then only the pos first dimensions starting at gen->tile_first
 * are mapped to the output.  The remaining input dimensions are projected
 * out and the corresponding output dimensions are fixed to 0.
 */
static __isl_give isl_map *insert_even(struct gpu_gen *gen,
        __isl_take isl_space *space, int pos, int val)
{
        int i, n;
        isl_map *proj;

        space = isl_space_set_from_params(space);
        space = isl_space_add_dims(space, isl_dim_set, gen->shared_len);
        space = isl_space_map_from_set(space);
        proj = isl_map_identity(space);
        proj = isl_map_project_out(proj, isl_dim_out, 0, gen->tile_first);
        n = gen->shared_len - gen->tile_first;
        for (i = 0; i <= n; ++i) {
                proj = isl_map_insert_dims(proj, isl_dim_out, 2 * i, 1);
                if (i == pos)
                        proj = isl_map_fix_si(proj, isl_dim_out, 2 * i, val);
                else
                        proj = isl_map_fix_si(proj, isl_dim_out, 2 * i, 0);
        }

        if (pos < 0)
                return proj;

        proj = isl_map_eliminate(proj, isl_dim_in, gen->tile_first + pos,
                                gen->shared_len - (gen->tile_first + pos));
        for (i = pos; i < n; ++i)
                proj = isl_map_fix_si(proj, isl_dim_out, 2 * i + 1, 0);

        return proj;
}

/* Given the AST context schedule "schedule" and the mapping from
 * domains to the shared tile loops "shared_sched", add a schedule
 * for a synchronization operation at position "val" of loop level "pos".
 *
 * schedule is of the form
 *
 *      D -> L
 *
 * (with D the iteration domains and L the already generated loops),
 * while shared_sched is of the form
 *
 *      D -> S
 *
 * We combine them into
 *
 *      L -> S
 *
 * apply a mapping
 *
 *      [s_0,...] -> [0,s_{tile_first},0,..., val, 0, 0, ... 0]
 *
 * and use the result as a schedule for "sync".
 */
static __isl_give isl_union_map *add_sync_schedule(struct gpu_gen *gen,
        __isl_take isl_union_map *res, __isl_keep isl_union_map *schedule,
        __isl_keep isl_union_map *shared_sched, int pos, int val)
{
        isl_space *space;
        isl_map *proj, *map;

        shared_sched = isl_union_map_copy(shared_sched);
        schedule = isl_union_map_copy(schedule);

        space = isl_union_map_get_space(shared_sched);
        schedule = isl_union_map_apply_domain(shared_sched, schedule);
        map = isl_map_from_union_map(schedule);

        proj = insert_even(gen, space, pos, val);
        map = isl_map_apply_range(map, proj);
        map = isl_map_from_range(isl_map_wrap(map));
        map = isl_map_set_tuple_name(map, isl_dim_in, "sync");

        res = isl_union_map_add_map(res, map);

        return res;
}

/* Given a set of wrapped references "ref", return the corresponding
 * access relations based on the tagged access relations "tagged".
 *
 * The elements of "ref" are of the form
 *
 *      [D -> R]
 *
 * with D an iteration domains and R a reference.
 * The elements of "tagged" are of the form
 *
 *      [D -> R] -> A
 *
 * with A an array.
 *
 * Extend "tagged" to include the iteration domain in the range, i.e.,
 *
 *      [D -> R] -> [D -> A]
 *
 * apply the result to "ref" and then unwrap the resulting set
 * to obtain relations of the form
 *
 *      D -> A
 */
static __isl_give isl_union_map *wrapped_reference_to_access(
        __isl_take isl_union_set *ref, __isl_take isl_union_map *tagged)
{
        isl_union_map *tag2access;

        tag2access = isl_union_map_copy(tagged);
        tag2access = isl_union_map_universe(tag2access);
        tag2access = isl_union_set_unwrap(isl_union_map_domain(tag2access));
        tag2access = isl_union_map_domain_map(tag2access);
        tag2access = isl_union_map_range_product(tag2access, tagged);

        ref = isl_union_set_coalesce(ref);
        ref = isl_union_set_apply(ref, tag2access);

        return isl_union_set_unwrap(ref);
}

/* Given an access relation "access" from "group", remove those reads
 * if ("read" is 1) or writes (if "read" is 0) that are only needed to
 * communicate data within the same iteration of the last_shared dimension
 * of the group.
 *
 * If the access is a read then it is necessarily an element of
 *
 *      live_in union (range flow)
 *
 * where live_in and flow may be overapproximations.
 * If the access is a write then it is necessarily an element of
 *
 *      live_out union (domain flow)
 *
 * In both cases, the access relation is also a subset of
 * the group access relation.
 *
 * Essentially, we compute the intersection of "access" with either
 *
 *      live_in union (range non-local-flow)
 *
 * or
 *
 *      live_out union (domain non-local-flow)
 *
 * We first construct a relation "local"
 *
 *      [[D -> R] -> [D' -> R']]
 *
 * of pairs of domain iterations accessing the reference group
 * and references in the group that are scheduled to the same iteration
 * of the last_shared dimension.
 *
 * If this relation does not intersect the dataflow dependences,
 * then there is nothing we can possibly remove and we simply
 * return the input.
 *
 * Otherwise, we remove the "local" dataflow dependences from
 * the set of all dataflow dependences.
 * Note that if the potential dataflow dependences are an overapproximation
 * of the actual dataflow dependences, then the result remains an
 * overapproximation of the non-local dataflow dependences.
 * Copying to/from global memory is only needed for the references
 * in the domain/range of the result or for accesses that are live out/in
 * for the entire scop.
 *
 * We therefore map the domain/range of the "external" relation
 * to the corresponding access relation and take the union with
 * the live out/in relation.
 */
static __isl_give isl_union_map *remove_local_accesses(struct gpu_gen *gen,
        struct gpu_array_ref_group *group, __isl_take isl_union_map *access,
        int read)
{
        int empty;
        isl_union_map *tagger;
        isl_union_set *domain;
        isl_space *space;
        isl_union_map *sched, *local, *tagged, *external;
        isl_union_set *tag_set;
        isl_map *proj;

        if (isl_union_map_is_empty(access))
                return access;

        tagged = group_tagged_access_relation(group);

        sched = isl_union_map_copy(gen->sched);

        space = isl_union_map_get_space(sched);
        proj = projection(space, gen->untiled_len, group->last_shared + 1);
        sched = isl_union_map_apply_range(sched, isl_union_map_from_map(proj));

        tagger = isl_union_map_copy(gen->prog->scop->tagger);
        domain = isl_union_map_domain(isl_union_map_copy(tagged));
        tagger = isl_union_map_intersect_range(tagger, domain);
        sched = isl_union_map_apply_domain(sched, tagger);

        local = isl_union_map_apply_range(sched,
                            isl_union_map_reverse(isl_union_map_copy(sched)));
        local = isl_union_map_intersect(local,
                        isl_union_map_copy(gen->prog->scop->tagged_dep_flow));

        empty = isl_union_map_is_empty(local);
        if (empty < 0 || empty) {
                isl_union_map_free(tagged);
                isl_union_map_free(local);
                if (empty < 0)
                        return isl_union_map_free(access);
                return access;
        }

        external = isl_union_map_copy(gen->prog->scop->tagged_dep_flow);
        external = isl_union_map_intersect_params(external,
                                isl_set_copy(gen->prog->scop->context));
        external = isl_union_map_subtract(external, local);

        if (read) {
                tag_set = isl_union_map_range(external);
                external = wrapped_reference_to_access(tag_set, tagged);
                external = isl_union_map_union(external,
                                isl_union_map_copy(gen->prog->scop->live_in));
        } else {
                tag_set = isl_union_map_domain(external);
                external = wrapped_reference_to_access(tag_set, tagged);
                external = isl_union_map_union(external,
                                isl_union_map_copy(gen->prog->scop->live_out));
        }

        access = isl_union_map_intersect(access, external);

        return access;
}

/* Given the AST context schedule "schedule" and the mapping from
 * domains to the shared tile loops "shared_sched", add a schedule
 * for copying an array reference group to/from shared/private memory.
 * "read" is set if data should be copied from global memory
 * to shared/private memory.
 * "k" represents the current group
 * "s" is the total number of groups
 *
 * We schedule an operation before or after the innermost loop
 * of "shared_sched" that affects the tile of the array reference group.
 *
 * schedule is of the form
 *
 *      D -> L
 *
 * (with D the iteration domains and L the already generated loops),
 * while shared_sched is of the form
 *
 *      D -> S
 *
 * We first compute the access relation for the reference group
 *
 *      D -> A
 *
 * and remove from this access relation those reads or writes
 * that only needed to communicate data within the same iteration
 * of the last_shared dimension of the group.
 * We then combine what is left with shared_sched into
 *
 *      D -> [S -> A]
 *
 * If this results in an empty relation, no copying needs to be performed
 * at this point.
 * Otherwise, we invert the relation and combine it with "schedule" into
 *
 *      [S -> A] -> L
 *
 * The actual additional piece of the schedule is obtained from combining
 *
 *      [S -> A] -> S
 *
 * with a mapping
 *
 *      [s_0,...] -> [0,s_{tile_first},0,..., val, 0, 0, ... 0]
 *
 * The position of "val" corresponds to the innermost loop that affects
 * the tile and the value indicates where the copying is scheduled
 * with respect to the actual kernel code (at value 0).
 * Reads are schedule before the code, writes to global memory from
 * private memory are scheduled at values 1 to s, writes to global
 * memory from shared memory are scheduled at values s + 2 to 2 * s + 1.
 *
 * If we are scheduling a read from global memory to shared memory,
 * we insert a synchronization before the kernel code (at the innermost
 * level).
 * If we are scheduling a write to global memory, then we add
 * a synchronization after all writes (at value 2 *s + 2).
 * However, there is no need for a synchronization after the outermost loop.
 * A write to global memory from private memory at the innermost level
 * does not require a synchronization, because it is covered by
 * the synchronization after the kernel inserted by body_schedule.
 */
static __isl_give isl_union_map *add_group_schedule(struct gpu_gen *gen,
        __isl_take isl_union_map *res, __isl_keep isl_union_map *schedule,
        __isl_keep isl_union_map *shared_sched,
        struct gpu_array_ref_group *group, int read, int k, int s)
{
        int n;
        int pos, val;
        isl_space *space;
        isl_union_map *access;
        isl_map *map, *proj, *access_map;
        isl_id *id;

        access = group_access_relation(group, read, !read);
        access = remove_local_accesses(gen, group, access, read);
        access = isl_union_map_range_product(isl_union_map_copy(shared_sched),
                                                access);

        if (isl_union_map_is_empty(access)) {
                isl_union_map_free(access);
                return res;
        }

        access = isl_union_map_reverse(access);
        access = isl_union_map_apply_range(access,
                                            isl_union_map_copy(schedule));
        access_map = isl_map_from_union_map(access);

        space = isl_space_copy(group->array->space);
        space = isl_space_from_range(space);
        space = isl_space_add_dims(space, isl_dim_in, gen->shared_len);
        map = isl_map_domain_map(isl_map_universe(space));

        space = isl_union_map_get_space(schedule);
        pos = group->last_shared + 1 - gen->tile_first;
        assert(pos >= 0);
        if (read)
                val = -2 - k;
        else if (group->private_tile)
                val = 1 + k;
        else
                val = 1 + s + 1 + k;
        proj = insert_even(gen, space, pos, val);
        map = isl_map_apply_range(map, proj);

        access_map = isl_map_range_product(access_map, map);

        id = isl_id_alloc(gen->ctx, read ? "read" : "write", group);
        access_map = isl_map_set_tuple_id(access_map, isl_dim_in, id);

        res = isl_union_map_add_map(res, access_map);

        n = gen->shared_len - gen->tile_first;
        if (read) {
                if (!group->private_tile)
                        res = add_sync_schedule(gen, res, schedule,
                                                shared_sched, n, -1);
        } else {
                if (pos == 0)
                        return res;
                if (pos == n && group->private_tile)
                        return res;
                res = add_sync_schedule(gen, res, schedule, shared_sched,
                                        pos, 2 * s + 2);
        }

        return res;
}

/* Return a schedule for the shared tile loops based on the current
 * AST context schedule.
 *
 * We create a "shared_sched" that maps the domains to the first
 * shared_len dimensions of the computed schedule, project out the
 * first tile_first dimensions (as these are already covered by
 * the host code) and insert "statement-level" dimensions at even
 * positions so that we can schedule copy blocks and synchronization
 * before/after each level.
 *
 * In particular, copy blocks are inserted inside the innermost
 * level that affect the tile.  For the copying to global memory,
 * those from private memory are scheduled before those from shared
 * memory such that synchronization can be inserted between the two
 * at the innermost level.
 * Synchronization is inserted at the innermost level before the
 * actual kernel code if there is any copying from global memory
 * to shared memory.  It is inserted unconditionally at the innermost
 * level after the actual kernel code and the copying to global memory
 * from private memory (if any).  Finally, it is inserted after
 * any copying to global memory, except at the outermost level
 * and at the innermost level if there is no copying from shared
 * memory.  The copying from private memory is covered by the unconditional
 * synchronization at the innermost level.
 */
static __isl_give isl_union_map *body_schedule(struct gpu_gen *gen,
        __isl_take isl_union_map *schedule)
{
        isl_space *space;
        isl_union_map *res;
        isl_union_map *shared_sched;
        isl_union_map *sched;
        isl_map *proj, *map;
        int i, j, k, s;

        shared_sched = isl_union_map_copy(gen->tiled_sched);
        proj = projection(isl_union_map_get_space(shared_sched),
                                gen->tiled_len, gen->shared_len);
        shared_sched = isl_union_map_apply_range(shared_sched,
                                isl_union_map_from_map(proj));
        space = isl_union_map_get_space(shared_sched);
        proj = insert_even(gen, space, -1, 0);
        sched = isl_union_map_apply_range(isl_union_map_copy(shared_sched),
                                isl_union_map_from_map(proj));

        res = isl_union_map_range_product(isl_union_map_copy(schedule), sched);

        s = 0;
        for (i = 0; i < gen->prog->n_array; ++i)
                s += gen->prog->array[i].n_group;

        k = 0;
        for (i = 0; i < gen->prog->n_array; ++i) {
                struct gpu_array_info *array = &gen->prog->array[i];

                for (j = 0; j < array->n_group; ++j) {
                        struct gpu_array_ref_group *group;

                        group = array->groups[j];
                        if (!group->private_tile && !group->shared_tile)
                                continue;
                        res = add_group_schedule(gen, res, schedule,
                                                shared_sched, group, 0, k, s);
                        res = add_group_schedule(gen, res, schedule,
                                                shared_sched, group, 1, k, s);
                        ++k;
                }
        }

        res = add_sync_schedule(gen, res, schedule, shared_sched,
                            gen->shared_len - gen->tile_first, 1 + s);

        isl_union_map_free(shared_sched);
        isl_union_map_free(schedule);

        return res;
}

/* Generate code for "kernel" in the given "context".
 *
 * We first generate code for the shared tile loops (T1T, T1P and T2)
 * in a context that includes the block ids.
 * Within each iteration of these loops an additional code generation
 * is performed (within create_kernel_leaf) for the rest of the schedule
 * in a context that includes the thread ids.
 */
static __isl_give isl_ast_node *generate_kernel(struct gpu_gen *gen,
        __isl_keep isl_ast_build *build, __isl_keep isl_set *host_domain,
        __isl_keep isl_multi_pw_aff *grid_size)
{
        isl_space *space;
        isl_set *set;
        isl_id_list *iterators;
        isl_union_map *schedule;
        isl_ast_node *tree;
        int sched_len;

        schedule = isl_ast_build_get_schedule(build);

        build = isl_ast_build_copy(build);
        build = isl_ast_build_restrict(build, isl_set_copy(host_domain));
        space = isl_ast_build_get_schedule_space(build);
        set = isl_set_universe(isl_space_copy(space));
        set = add_bounded_parameters_dynamic(set, grid_size, "b");
        build = isl_ast_build_restrict(build, set);

        schedule = body_schedule(gen, schedule);

        sched_len = 2 * (gen->shared_len - gen->tile_first) + 1;

        build = set_atomic_and_unroll(build, space, sched_len);
        iterators = generate_names(gen->ctx, sched_len, "g");
        build = isl_ast_build_set_iterators(build, iterators);
        build = isl_ast_build_set_create_leaf(build, &create_kernel_leaf, gen);
        tree = isl_ast_build_ast_from_schedule(build, schedule);
        isl_ast_build_free(build);

        return tree;
}

/* Attach "id" to the given node.
 */
static __isl_give isl_ast_node *attach_id(__isl_take isl_ast_node *node,
        __isl_keep isl_ast_build *build, void *user)
{
        isl_id *id = user;

        node = isl_ast_node_set_annotation(node, id);

        return node;
}

/* Construct an AST node for performing a kernel launch and attach
 * the information about the kernel to that node.
 *
 * The kernel AST has been constructed in the context of the range
 * of "schedule".  In particular, the grid size has been computed
 * in the context.  We therefore still need to make sure that these
 * constraints are expressed in the code.  We do this by creating a schedule
 *
 *      kernel[] -> [S -> []]
 *
 * where S is the schedule domain, i.e., the range of "schedule".
 * The AST generation will then create a single call surrounded by
 * all the condition in "S" that have not been expressed yet.
 *
 * The kernel information is attached to this node in attach_id.
 */
static __isl_give isl_ast_node *construct_launch(
        __isl_take isl_ast_build *build, __isl_take isl_union_map *schedule,
        __isl_take struct ppcg_kernel *kernel)
{
        isl_id *id;
        isl_ctx *ctx;
        isl_union_set *domain;
        isl_set *set;
        isl_map *map;
        isl_ast_node *node;

        ctx = isl_ast_build_get_ctx(build);

        id = isl_id_alloc(ctx, NULL, kernel);
        id = isl_id_set_free_user(id, &ppcg_kernel_free);

        domain = isl_union_map_range(schedule);
        set = isl_set_from_union_set(domain);
        map = isl_map_from_domain(set);
        map = isl_map_from_range(isl_map_wrap(map));
        map = isl_map_set_tuple_name(map, isl_dim_in, "kernel");
        schedule = isl_union_map_from_map(map);

        build = isl_ast_build_set_at_each_domain(build, &attach_id, id);
        node = isl_ast_build_ast_from_schedule(build, schedule);
        isl_ast_build_free(build);

        return node;
}

/* This function is called for each leaf in the AST of the host code.
 * We first specialize the schedule to the site of the leaf, compute
 * the size of shared memory and then construct the body of the host code
 * and the associated kernel.
 *
 * The necessary information for printing the kernel launch is
 * stored in a struct ppcg_kernel and attached to the leaf node
 * created to represent the launch.
 */
static __isl_give isl_ast_node *create_host_leaf(
        __isl_take isl_ast_build *build, void *user)
{
        struct gpu_gen *gen = (struct gpu_gen *) user;
        isl_id *id;
        isl_ast_node *node;
        struct ppcg_kernel *kernel;
        isl_set *host_domain;
        isl_union_map *schedule;
        isl_union_map *local_sched;
        isl_union_map *access;
        isl_union_set *domain;
        int i;

        schedule = isl_ast_build_get_schedule(build);

        isl_union_map_foreach_map(schedule, &extract_tile_len, gen);
        read_sizes(gen);

        domain = isl_union_map_domain(isl_union_map_copy(schedule));

        local_sched = isl_union_map_copy(gen->sched);
        local_sched = isl_union_map_intersect_domain(local_sched, domain);
        access = isl_union_map_union(isl_union_map_copy(gen->prog->read),
                                     isl_union_map_copy(gen->prog->may_write));
        access = isl_union_map_apply_domain(access,
                                            isl_union_map_copy(local_sched));

        gen->tiled_sched = tile_schedule(gen, local_sched);
        gen->tiled_sched = parametrize_tiled_schedule(gen, gen->tiled_sched);
        gen->tiled_sched = scale_tile_loops(gen, gen->tiled_sched);

        gen->local_sched = isl_union_map_copy(gen->tiled_sched);
        gen->local_sched = thread_tile_schedule(gen, gen->local_sched);
        gen->local_sched = scale_thread_tile_loops(gen, gen->local_sched);

        kernel = gen->kernel = isl_calloc_type(gen->ctx, struct ppcg_kernel);
        if (!kernel)
                goto error;

        kernel->id = gen->kernel_id++;
        kernel->context = isl_union_map_params(isl_union_map_copy(schedule));
        kernel->grid_size = extract_grid_size(gen, kernel);
        extract_block_size(gen, kernel);
        kernel->arrays = isl_union_map_range(access);
        kernel->arrays = isl_union_set_apply(kernel->arrays,
                                isl_union_map_copy(gen->prog->to_outer));
        kernel->space = isl_ast_build_get_schedule_space(build);

        gen->private_access = NULL;
        compute_shared_sched(gen);
        gen->privatization = compute_privatization(gen);
        check_scalar_live_ranges(gen);
        if (group_references(gen) < 0)
                schedule = isl_union_map_free(schedule);
        compute_private_access(gen);
        host_domain = isl_set_from_union_set(isl_union_map_range(
                                                isl_union_map_copy(schedule)));
        localize_bounds(gen, kernel, host_domain);

        gen->local_sched = interchange_for_unroll(gen, gen->local_sched);
        check_shared_memory_bound(gen);
        compute_group_tilings(gen);

        kernel->tree = generate_kernel(gen, build, host_domain,
                                        kernel->grid_size);
        create_kernel_vars(gen, kernel);

        free_local_array_info(gen);
        isl_map_free(gen->privatization);
        isl_union_map_free(gen->private_access);
        isl_union_map_free(gen->local_sched);
        isl_union_map_free(gen->tiled_sched);
        isl_union_map_free(gen->shared_sched);
        isl_union_map_free(gen->shared_proj);
        isl_set_free(host_domain);
        free(gen->tile_size);

        node = construct_launch(build, schedule, kernel);

        return node;
error:
        isl_union_map_free(schedule);
        return NULL;
}

/* Use isl to generate code for the outer gen->tile_first loops
 * of the global schedule in gen->sched, resulting in the host code.
 * Within each iteration of this partial schedule, i.e., for each kernel
 * launch, create_host_leaf takes care of generating the kernel code.
 */
static __isl_give isl_ast_node *generate_host_code(struct gpu_gen *gen)
{
        isl_ast_build *build;
        isl_ast_node *tree;
        isl_union_map *sched;
        isl_map *proj;
        isl_id_list *iterators;

        sched = isl_union_map_copy(gen->sched);
        proj = projection(isl_union_map_get_space(sched),
                            gen->untiled_len, gen->tile_first);
        sched = isl_union_map_apply_range(sched, isl_union_map_from_map(proj));

        isl_options_set_ast_build_group_coscheduled(gen->ctx, 1);
        build = isl_ast_build_from_context(isl_set_copy(gen->prog->context));
        iterators = generate_names(gen->ctx, gen->tile_first, "h");
        build = isl_ast_build_set_iterators(build, iterators);
        build = isl_ast_build_set_create_leaf(build, &create_host_leaf, gen);
        tree = isl_ast_build_ast_from_schedule(build, sched);
        isl_ast_build_free(build);

        return tree;
}

__isl_give isl_union_map *extract_sizes_from_str(isl_ctx *ctx, const char *str)
{
        if (!str)
                return NULL;
        return isl_union_map_read_from_str(ctx, str);
}

/* Information about the outermost tilable bands in the forest of bands.
 *
 * tile_len and n_parallel are only sets on band_info structures
 * that correspond to outermost bands.  For other bands (in particular,
 * ancestors of the outermost bands), n_parallal is set to 0.
 *
 * prefix is the (padded) schedule leading up to the outermost tilable bands.
 *
 * tile_first is the number of schedule dimensions in prefix.
 *
 * suffix is the schedule of the outermost tilable bands and their descendants.
 */
struct band_info {
        struct gpu_gen *gen;
        int tile_first;
        int tile_len;
        int n_parallel;
        isl_union_map *prefix;
        isl_union_map *suffix;
};

/* Set tile_len and n_parallel of the statement to that of
 * their outermost band, recorded in the band_info.
 */
static int set_stmt_tile_len(__isl_take isl_map *map, void *user)
{
        struct band_info *info = user;
        struct gpu_stmt *stmt;
        isl_id *id;

        id = isl_map_get_tuple_id(map, isl_dim_in);
        stmt = find_stmt(info->gen->prog, id);
        isl_id_free(id);

        stmt->tile_len = info->tile_len;
        stmt->n_parallel = info->n_parallel;

        isl_map_free(map);

        return 0;
}

static void list_select_outer_band(struct gpu_gen *gen,
        __isl_take isl_band_list *list, int pos, struct band_info *list_info);

/* Check if this band has any parallel loops.  If so, take it as
 * the outermost tilable band.  If not, continue looking for the
 * outermost tilable band in the children of the current band.
 */
static void band_select_outer_band(struct gpu_gen *gen,
        __isl_take isl_band *band, int pos, struct band_info *info)
{
        int n = isl_band_n_member(band);
        int n_parallel;

        for (n_parallel = 0; n_parallel < n; ++n_parallel)
                if (!isl_band_member_is_coincident(band, n_parallel))
                        break;

        info->n_parallel = n_parallel;
        if (n_parallel) {
                gen->any_parallelism = 1;
                info->gen = gen;
                info->tile_first = pos;
                info->tile_len = n;
                info->prefix = isl_band_get_prefix_schedule(band);
                info->suffix = isl_union_map_flat_range_product(
                                isl_band_get_partial_schedule(band),
                                isl_band_get_suffix_schedule(band));
                isl_union_map_foreach_map(info->prefix,
                                            &set_stmt_tile_len, info);
        } else if (isl_band_has_children(band)) {
                isl_band_list *children;
                children = isl_band_get_children(band);
                list_select_outer_band(gen, children, pos + n, info);
        } else {
                info->gen = gen;
                info->tile_first = pos + n;
                info->tile_len = 0;
                info->prefix = isl_union_map_flat_range_product(
                                isl_band_get_prefix_schedule(band),
                                isl_band_get_partial_schedule(band));
                info->suffix = isl_band_get_suffix_schedule(band);
                isl_union_map_foreach_map(info->prefix,
                                            &set_stmt_tile_len, info);
        }

        isl_band_free(band);
}

/* Comparison function that returns a non-zero value for band_infos
 * with different tile_len fields or different n_parallel fields.
 */
static int cmp_band(const void *p1, const void *p2)
{
        const struct band_info *info1 = p1;
        const struct band_info *info2 = p2;

        if (info1->tile_len != info2->tile_len)
                return info1->tile_len - info2->tile_len;

        return info1->n_parallel - info2->n_parallel;
}

/* Extend "umap" with coordinates with fixed value "val"
 * to a total length of "dst_len", assuming the original dimension is "src_len".
 */
static __isl_give isl_union_map *extend_range(
        __isl_take isl_union_map *umap, int src_len, int dst_len, int val)
{
        isl_space *dim;
        isl_map *map;
        int i;

        dim = isl_union_map_get_space(umap);
        map = isl_map_reverse(projection(dim, dst_len, src_len));
        for (i = src_len; i < dst_len; ++i)
                map = isl_map_fix_si(map, isl_dim_out, i, val);

        umap = isl_union_map_apply_range(umap, isl_union_map_from_map(map));

        return umap;
}

/* Group bands with the same values for tile_len and n_parallel.
 * The prefix schedule is then extended with a fixed coordinate that
 * is different for each such group.
 * Note that the actual values for this coordinate are not important.
 * The bands have already been effectively separated at a higher level
 * or they are independent and may be executed in parallel.
 * The list of band_info has been sorted before this functions is called.
 */
static void separate_bands(struct band_info *info, int n)
{
        int i;
        int j = 0;

        for (i = 0; i < n; ++i) {
                int l = info[i].tile_first;

                if (i &&
                    (info[i].tile_len != info[i - 1].tile_len ||
                     info[i].n_parallel != info[i - 1].n_parallel))
                        j++;

                info[i].prefix = extend_range(info[i].prefix,
                                                l, l + 1, j);
                info[i].tile_first = l + 1;
        }
}

/* Select the outermost bands in the elements of the list, align
 * their prefix schedules, separate bands with different values
 * for tile_len and/or n_parallel and then combine the resulting
 * prefix and suffix schedules into a single pair of prefix and
 * suffix schedules for the entire list.
 */
static void list_select_outer_band(struct gpu_gen *gen,
        __isl_take isl_band_list *list, int pos, struct band_info *list_info)
{
        isl_band *band;
        int i;
        int n = isl_band_list_n_band(list);
        isl_ctx *ctx = isl_band_list_get_ctx(list);
        struct band_info *info;
        int max_tile_first;
        isl_union_map *prefix;
        isl_union_map *suffix;

        assert(n >= 1);
        info = isl_calloc_array(ctx, struct band_info, n);
        assert(info);

        max_tile_first = 0;
        for (i = 0; i < n; ++i) {
                band = isl_band_list_get_band(list, i);
                band_select_outer_band(gen, band, pos, &info[i]);
                if (info[i].tile_first > max_tile_first)
                        max_tile_first = info[i].tile_first;
        }

        for (i = 0; i < n; ++i) {
                if (info[i].tile_first == max_tile_first)
                        continue;
                info[i].prefix = extend_range(info[i].prefix,
                                        info[i].tile_first, max_tile_first, 0);
                info[i].tile_first = max_tile_first;
        }

        qsort(info, n, sizeof(struct band_info), &cmp_band);

        for (i = 0; i < n - 1; ++i)
                if (info[i].tile_len != info[i + 1].tile_len ||
                    info[i].n_parallel != info[i + 1].n_parallel)
                        break;

        if (i < n -1)
                separate_bands(info, n);

        prefix = info[0].prefix;
        suffix = info[0].suffix;

        for (i = 1; i < n; ++i) {
                prefix = isl_union_map_union(prefix, info[i].prefix);
                suffix = isl_union_map_union(suffix, info[i].suffix);
        }

        list_info->tile_first = info[0].tile_first;
        list_info->tile_len = -1;
        list_info->prefix = prefix;
        list_info->suffix = suffix;

        isl_band_list_free(list);
        free(info);
}

/* Select the outermost tilable band that (by construction)
 * has at least one parallel loop.
 * The starting position of the aligned band is stored in the pair
 * gen->tile_first.
 * The sizes and number of parallel loops may be different in different
 * parts of the band forest and are therefore stored in the gpu_stmts.
 *
 * Return the complete schedule, with the tilable bands aligned
 * at gen->tile_first and padded with zero, if needed.
 */
static __isl_give isl_union_map *select_outer_tilable_band(struct gpu_gen *gen,
        __isl_keep isl_schedule *schedule)
{
        isl_band_list *list;
        struct band_info info;

        gen->n_parallel = 0;
        gen->tile_len = -1;

        list = isl_schedule_get_band_forest(schedule);

        if (isl_band_list_n_band(list) == 0) {
                isl_band_list_free(list);
                return isl_schedule_get_map(schedule);
        }

        list_select_outer_band(gen, list, 0, &info);

        gen->tile_first = info.tile_first;
        info.suffix = align_range(info.suffix);

        return isl_union_map_flat_range_product(info.prefix, info.suffix);
}

/* Set gen->untiled_len to the number of scheduling dimensions
 * for the schedule of the first domain.
 * We assume here that this number is the same for all domains.
 */
static int set_untiled_len(__isl_take isl_map *map, void *user)
{
        unsigned *untiled_len = user;

        *untiled_len = isl_map_dim(map, isl_dim_out);

        isl_map_free(map);
        return -1;
}

/* Compute an appropriate schedule based on the accesses in
 * gen->read and gen->write.
 *
 * We use the dependences in gen->prog->scop to compute
 * a schedule that has a parallel loop in each tilable band.
 * Finally, we select the outermost tilable band.
 *
 * If live range reordering is allowed, then we need to make sure
 * that live ranges on arrays are not run in parallel since doing
 * so would require array expansion.  We therefore add the array
 * order dependences to the coincidence dependences.  Non-zero array
 * order dependences will then prevent a schedule dimension from being
 * considered parallel.
 * Live ranges derived from scalars are allowed to be run in parallel
 * since we force the scalars to be mapped to private memory in
 * check_scalar_live_ranges.
 * If live range reordering is allowed, then the false dependences
 * are not added to the validity constraints as that would prevent
 * reordering.  Instead, the external false dependences that enforce that reads
 * from potentially live-in data precede any later write and
 * that writes of potentially live-out data follow any other earlier write
 * are added to the validity and the coincidence constraints.
 * The false dependences are still added to the proximity constraints
 * for consistency with the case where live range reordering is not allowed.
 * The coincidence constraints then consist of flow dependences,
 * exernal false dependences and array order dependences.
 * The independences can be filtered out from the first two sets.
 * They have already been filtered out from the array order dependences
 * on a per array basis in collect_order_dependences.
 * There is no need for a per array handling of the other two sets
 * as there should be no flow or external false dependence on local
 * variables that can be filtered out.
 */
static void compute_schedule(struct gpu_gen *gen)
{
        isl_union_set *domain;
        isl_union_map *dep_raw, *dep;
        isl_union_map *validity, *proximity, *coincidence;
        isl_union_map *sched;
        isl_schedule_constraints *sc;
        isl_schedule *schedule;

        domain = isl_union_set_copy(gen->prog->scop->domain);
        domain = isl_union_set_intersect_params(domain,
                                isl_set_copy(gen->prog->scop->context));
        sc = isl_schedule_constraints_on_domain(isl_union_set_copy(domain));
        if (gen->options->live_range_reordering) {
                sc = isl_schedule_constraints_set_conditional_validity(sc,
                        isl_union_map_copy(gen->prog->scop->tagged_dep_flow),
                        isl_union_map_copy(gen->prog->scop->tagged_dep_order));
                proximity = isl_union_map_copy(gen->prog->scop->dep_flow);
                validity = isl_union_map_copy(proximity);
                validity = isl_union_map_union(validity,
                            isl_union_map_copy(gen->prog->scop->dep_external));
                proximity = isl_union_map_union(proximity,
                            isl_union_map_copy(gen->prog->scop->dep_false));
                coincidence = isl_union_map_copy(validity);
                coincidence = isl_union_map_subtract(coincidence,
                        isl_union_map_copy(gen->prog->scop->independence));
                coincidence = isl_union_map_union(coincidence,
                                isl_union_map_copy(gen->prog->array_order));
        } else {
                dep_raw = isl_union_map_copy(gen->prog->scop->dep_flow);
                dep = isl_union_map_copy(gen->prog->scop->dep_false);
                dep = isl_union_map_union(dep, dep_raw);
                dep = isl_union_map_coalesce(dep);
                proximity = isl_union_map_copy(dep);
                coincidence = isl_union_map_copy(dep);
                validity = dep;
        }
        sc = isl_schedule_constraints_set_validity(sc, validity);
        sc = isl_schedule_constraints_set_coincidence(sc, coincidence);
        sc = isl_schedule_constraints_set_proximity(sc, proximity);

        if (gen->options->debug->dump_schedule_constraints)
                isl_schedule_constraints_dump(sc);
        schedule = isl_schedule_constraints_compute_schedule(sc);
        if (gen->options->debug->dump_schedule)
                isl_schedule_dump(schedule);

        sched = select_outer_tilable_band(gen, schedule);

        isl_union_map_foreach_map(sched, &set_untiled_len, &gen->untiled_len);
        sched = isl_union_map_intersect_domain(sched, domain);
        gen->sched = sched;

        isl_schedule_free(schedule);
}

/* Compute the sets of outer array elements that need to be copied in and out.
 *
 * In particular, for each array that is possibly written anywhere in
 * gen->prog and that is visible outside the corresponding scop,
 * we copy out its entire extent.
 *
 * Any array elements that is read without first being written needs
 * to be copied in. Furthermore, if there are any array elements that
 * are copied out, but that may not be written inside gen->prog, then
 * they also need to be copied in to ensure that the value after execution
 * is the same as the value before execution.
 * In case the array elements are structures, we need to take into
 * account that all members of the structures need to be written
 * by gen->prog before we can avoid copying the data structure in.
 *
 * While computing the set of array elements that are copied out but
 * not necessarily written, we intersect both sets with the context.
 * This helps in those cases where the arrays are declared with a fixed size,
 * while the accesses are parametric and the context assigns a fixed value
 * to the parameters.
 *
 * If an element from a local array is read without first being written,
 * then there is no point in copying it in since it cannot have been
 * written prior to the scop.  Warn about the uninitialized read instead.
 */
static void compute_copy_in_and_out(struct gpu_gen *gen)
{
        int i;
        isl_union_set *local;
        isl_union_set *may_write, *must_write;
        isl_union_set *copy_in, *copy_out;
        isl_union_set *not_written;
        isl_union_map *uninitialized;
        isl_union_map *local_uninitialized;

        must_write = isl_union_map_range(
                                isl_union_map_copy(gen->prog->must_write));
        must_write = isl_union_set_intersect_params(must_write,
                                            isl_set_copy(gen->prog->context));
        may_write = isl_union_map_range(
                                isl_union_map_copy(gen->prog->may_write));
        may_write = isl_union_set_intersect_params(may_write,
                                            isl_set_copy(gen->prog->context));
        may_write = isl_union_set_universe(may_write);
        may_write = isl_union_set_apply(may_write,
                                    isl_union_map_copy(gen->prog->to_outer));
        copy_out = isl_union_set_empty(isl_union_set_get_space(may_write));
        local = isl_union_set_copy(copy_out);

        for (i = 0; i < gen->prog->n_array; ++i) {
                isl_space *space;
                isl_set *write_i;
                int empty;

                space = isl_space_copy(gen->prog->array[i].space);

                if (gen->prog->array[i].local) {
                        isl_set *set;

                        set = isl_set_universe(space);
                        local = isl_union_set_add_set(local, set);
                        continue;
                }

                write_i = isl_union_set_extract_set(may_write, space);
                empty = isl_set_fast_is_empty(write_i);
                isl_set_free(write_i);
                if (empty)
                        continue;

                write_i = isl_set_copy(gen->prog->array[i].extent);
                copy_out = isl_union_set_add_set(copy_out, write_i);
        }
        isl_union_set_free(may_write);

        copy_out = isl_union_set_intersect_params(copy_out,
                                            isl_set_copy(gen->prog->context));

        gen->prog->copy_out = isl_union_set_copy(copy_out);

        copy_out = isl_union_set_apply(copy_out,
                                    isl_union_map_copy(gen->prog->to_inner));
        not_written = isl_union_set_subtract(copy_out, must_write);

        uninitialized = isl_union_map_copy(gen->prog->scop->live_in);
        local_uninitialized = isl_union_map_copy(uninitialized);

        local = isl_union_set_apply(local,
                                    isl_union_map_copy(gen->prog->to_inner));
        local_uninitialized = isl_union_map_intersect_range(local_uninitialized,
                                                            local);
        if (!isl_union_map_is_empty(local_uninitialized)) {
                fprintf(stderr,
                        "possibly uninitialized reads (not copied in):\n");
                isl_union_map_dump(local_uninitialized);
        }
        uninitialized = isl_union_map_subtract(uninitialized,
                                                local_uninitialized);
        copy_in = isl_union_map_range(uninitialized);
        copy_in = isl_union_set_union(copy_in, not_written);
        copy_in = isl_union_set_apply(copy_in,
                                    isl_union_map_copy(gen->prog->to_outer));

        gen->prog->copy_in = copy_in;
}

/* Internal data structure for extract_access.
 * "next_access" points to the end of a linked list that is extended
 * by extract_access.
 * "single_expression" is set if the access expressions belong to
 * an expression statement (i.e., a statement without internal control).
 */
struct ppcg_extract_access_data {
        struct gpu_stmt_access **next_access;
        int single_expression;
};

/* Extract a gpu_stmt_access from "expr", append it to the list
 * that ends in *data->next_access and update the end of the list.
 * If the access expression performs a write, then it is considered
 * exact only if it appears in a single expression statement and
 * if its may access relation is equal to its must access relation.
 */
static int extract_access(__isl_keep pet_expr *expr, void *user)
{
        struct ppcg_extract_access_data *data = user;
        isl_map *may;
        struct gpu_stmt_access *access;
        isl_ctx *ctx;

        may = pet_expr_access_get_may_access(expr);
        ctx = isl_map_get_ctx(may);
        access = isl_alloc_type(ctx, struct gpu_stmt_access);
        assert(access);
        access->next = NULL;
        access->read = pet_expr_access_is_read(expr);
        access->write = pet_expr_access_is_write(expr);
        access->access = may;
        access->tagged_access = pet_expr_access_get_tagged_may_access(expr);
        if (!access->write) {
                access->exact_write = 1;
        } else if (!data->single_expression) {
                access->exact_write = 0;
        } else {
                isl_map *must;
                must = pet_expr_access_get_must_access(expr);
                access->exact_write = isl_map_is_equal(must, access->access);
                isl_map_free(must);
        }
        access->ref_id = pet_expr_access_get_ref_id(expr);
        access->group = -1;

        *data->next_access = access;
        data->next_access = &(*data->next_access)->next;

        return 0;
}

/* Construct a linked list of gpu_stmt_access objects,
 * one for each access expression in the statement body.
 */
static void pet_stmt_extract_accesses(struct gpu_stmt *stmt)
{
        struct ppcg_extract_access_data data;

        stmt->accesses = NULL;
        data.next_access = &stmt->accesses;
        data.single_expression =
                pet_tree_get_type(stmt->stmt->body) == pet_tree_expr;
        pet_tree_foreach_access_expr(stmt->stmt->body, &extract_access, &data);
}

/* Return an array of gpu_stmt representing the statements in "scop".
 */
static struct gpu_stmt *extract_stmts(isl_ctx *ctx, struct ppcg_scop *scop,
        __isl_keep isl_set *context)
{
        int i;
        struct gpu_stmt *stmts;

        stmts = isl_calloc_array(ctx, struct gpu_stmt, scop->n_stmt);
        if (!stmts)
                return NULL;

        for (i = 0; i < scop->n_stmt; ++i) {
                struct gpu_stmt *s = &stmts[i];

                s->id = isl_set_get_tuple_id(scop->stmts[i]->domain);
                s->stmt = scop->stmts[i];
                pet_stmt_extract_accesses(s);
        }

        return stmts;
}

/* Callback for ppcg_print_guarded that calls the callback for generate_gpu.
 */
static __isl_give isl_printer *print_gpu(__isl_take isl_printer *p, void *user)
{
        struct gpu_gen *gen = user;

        return gen->print(p, gen->prog, gen->tree, &gen->types,
                            gen->print_user);
}

/* Generate CUDA code for "scop" and print it to "p".
 * After generating an AST for the transformed scop as explained below,
 * we call "gen->print" to print the AST in the desired output format
 * to "p".
 *
 * If it turns out that it does not make sense to generate GPU code,
 * then we generate CPU code instead.
 *
 * The GPU code is generated in a context where at least one
 * statement instance is executed.  The corresponding guard (if any) is printed
 * around the entire generated GPU code, except for the declaration
 * of the arrays that are visible outside of the scop and that therefore
 * cannot be declared inside the body of any possible guard.
 *
 * We first compute a schedule that respects the dependences
 * of the original program and select the outermost band
 * of tilable dimensions that has at least one parallel loop.
 * We then have three blocks of dimensions
 *
 *      H               B                       G
 *
 * The tilable band "B" is first tiled according to "tile" sizes, resulting
 * in
 *
 *      H       T               P               G
 *
 * For each iteration of the T loop and for each array, we compute
 * the array elements accessed by that iteration, construct a rectangular
 * box around it and shift it to the origin.  The result is used
 * as shared memory for the array.
 *
 * We then split off at most 2 parallel loops from the T loops and
 * at most 3 parallel loops from the P loops
 *
 *      H       T1      T2      P1      P2      G
 *
 * The T1/P1 loops are then tiled or "wrapped" over the blocks/threads,
 * according to "grid"/"block" sizes.
 *
 *      H       T1T T1P T2      P1T P1P P2      G
 *
 * Finally, the T1P and P1P iterators are equated to the block and
 * thread dimensions respectively and so are effectively removed.
 * The H loops are run on the host.  The T1T, T2, P1T, P2 and G loops
 * are run on the GPU.
 *
 * Code is generated in three stages.  We first generate code for the
 * host (the H loops), with iterators h%d.  Then, for each leaf node
 * of the resulting AST, we generate code for the shared loops (up to
 * and including T2), with iterators g%d and after equating the H loops
 * to h%d parameters and the T1P loops to the block dimensions.
 * Finally, we generate code for the remaining loops in a similar fashion.
 */
static __isl_give isl_printer *generate(__isl_take isl_printer *p,
        struct gpu_gen *gen, struct ppcg_scop *scop,
        struct ppcg_options *options)
{
        struct gpu_prog *prog;
        isl_ctx *ctx;
        isl_set *context, *guard;

        if (!scop)
                return isl_printer_free(p);

        ctx = isl_printer_get_ctx(p);
        prog = gpu_prog_alloc(ctx, scop);
        if (!prog)
                return isl_printer_free(p);

        context = isl_set_copy(prog->context);
        guard = isl_union_set_params(isl_union_set_copy(prog->scop->domain));
        prog->context = isl_set_intersect(prog->context, isl_set_copy(guard));

        gen->prog = prog;
        gen->any_parallelism = 0;
        compute_schedule(gen);

        if (!gen->any_parallelism) {
                isl_set_free(context);
                isl_set_free(guard);
                p = print_cpu(p, scop, options);
        } else {
                compute_copy_in_and_out(gen);
                gen->tree = generate_host_code(gen);
                p = ppcg_print_exposed_declarations(p, prog->scop);
                p = ppcg_print_guarded(p, guard, context, &print_gpu, gen);
                isl_ast_node_free(gen->tree);
        }

        isl_union_map_free(gen->sched);

        gpu_prog_free(prog);

        return p;
}

/* Wrapper around generate for use as a ppcg_transform callback.
 */
static __isl_give isl_printer *generate_wrap(__isl_take isl_printer *p,
        struct ppcg_scop *scop, void *user)
{
        struct gpu_gen *gen = user;

        return generate(p, gen, scop, gen->options);
}

/* Transform the code in the file called "input" by replacing
 * all scops by corresponding GPU code and write the results to "out".
 */
int generate_gpu(isl_ctx *ctx, const char *input, FILE *out,
        struct ppcg_options *options,
        __isl_give isl_printer *(*print)(__isl_take isl_printer *p,
                struct gpu_prog *prog, __isl_keep isl_ast_node *tree,
                struct gpu_types *types, void *user), void *user)
{
        struct gpu_gen gen;
        int r;
        int i;

        gen.ctx = ctx;
        gen.sizes = extract_sizes_from_str(ctx, options->sizes);
        gen.options = options;
        gen.kernel_id = 0;
        gen.print = print;
        gen.print_user = user;
        gen.types.n = 0;
        gen.types.name = NULL;

        if (options->debug->dump_sizes) {
                isl_space *space = isl_space_params_alloc(ctx, 0);
                gen.used_sizes = isl_union_map_empty(space);
        }

        r = ppcg_transform(ctx, input, out, options, &generate_wrap, &gen);

        if (options->debug->dump_sizes) {
                isl_union_map_dump(gen.used_sizes);
                isl_union_map_free(gen.used_sizes);
        }

        isl_union_map_free(gen.sizes);
        for (i = 0; i < gen.types.n; ++i)
                free(gen.types.name[i]);
        free(gen.types.name);

        return r;
}

struct gpu_prog *gpu_prog_alloc(isl_ctx *ctx, struct ppcg_scop *scop)
{
        struct gpu_prog *prog;

        if (!scop)
                return NULL;

        prog = isl_calloc_type(ctx, struct gpu_prog);
        assert(prog);

        prog->ctx = ctx;
        prog->scop = scop;
        prog->context = isl_set_copy(scop->context);
        prog->n_stmts = scop->n_stmt;
        prog->stmts = extract_stmts(ctx, scop, prog->context);
        prog->read = isl_union_map_copy(scop->reads);
        prog->may_write = isl_union_map_copy(scop->may_writes);
        prog->must_write = isl_union_map_copy(scop->must_writes);
        prog->to_inner = compute_to_inner(scop);
        prog->to_outer = isl_union_map_copy(prog->to_inner);
        prog->to_outer = isl_union_map_reverse(prog->to_outer);

        if (!prog->stmts)
                return gpu_prog_free(prog);

        if (collect_array_info(prog) < 0)
                return gpu_prog_free(prog);

        return prog;
}

void *gpu_prog_free(struct gpu_prog *prog)
{
        if (!prog)
                return NULL;
        free_array_info(prog);
        free_stmts(prog->stmts, prog->n_stmts);
        isl_union_map_free(prog->to_outer);
        isl_union_map_free(prog->to_inner);
        isl_union_set_free(prog->copy_in);
        isl_union_set_free(prog->copy_out);
        isl_union_map_free(prog->read);
        isl_union_map_free(prog->may_write);
        isl_union_map_free(prog->must_write);
        isl_union_map_free(prog->array_order);
        isl_set_free(prog->context);
        free(prog);
        return NULL;
}