introduce evalue_shift_variables

[barvinok.git] / bernoulli.c

#include <assert.h>
#include <barvinok/options.h>
#include <barvinok/util.h>
#include "bernoulli.h"
#include "lattice_point.h"

#define ALLOC(type) (type*)malloc(sizeof(type))
#define ALLOCN(type,n) (type*)malloc((n) * sizeof(type))
#define REALLOCN(ptr,type,n) (type*)realloc(ptr, (n) * sizeof(type))

static struct bernoulli_coef bernoulli_coef;
static struct poly_list bernoulli;
static struct poly_list faulhaber;

struct bernoulli_coef *bernoulli_coef_compute(int n)
{
    int i, j;
    Value factor, tmp;

    if (n < bernoulli_coef.n)
        return &bernoulli_coef;

    if (n >= bernoulli_coef.size) {
        int size = 3*(n + 5)/2;
        Vector *b;

        b = Vector_Alloc(size);
        if (bernoulli_coef.n) {
            Vector_Copy(bernoulli_coef.num->p, b->p, bernoulli_coef.n);
            Vector_Free(bernoulli_coef.num);
        }
        bernoulli_coef.num = b;
        b = Vector_Alloc(size);
        if (bernoulli_coef.n) {
            Vector_Copy(bernoulli_coef.den->p, b->p, bernoulli_coef.n);
            Vector_Free(bernoulli_coef.den);
        }
        bernoulli_coef.den = b;
        b = Vector_Alloc(size);
        if (bernoulli_coef.n) {
            Vector_Copy(bernoulli_coef.lcm->p, b->p, bernoulli_coef.n);
            Vector_Free(bernoulli_coef.lcm);
        }
        bernoulli_coef.lcm = b;

        bernoulli_coef.size = size;
    }
    value_init(factor);
    value_init(tmp);
    for (i = bernoulli_coef.n; i <= n; ++i) {
        if (i == 0) {
            value_set_si(bernoulli_coef.num->p[0], 1);
            value_set_si(bernoulli_coef.den->p[0], 1);
            value_set_si(bernoulli_coef.lcm->p[0], 1);
            continue;
        }
        value_set_si(bernoulli_coef.num->p[i], 0);
        value_set_si(factor, -(i+1));
        for (j = i-1; j >= 0; --j) {
            mpz_mul_ui(factor, factor, j+1);
            mpz_divexact_ui(factor, factor, i+1-j);
            value_division(tmp, bernoulli_coef.lcm->p[i-1],
                           bernoulli_coef.den->p[j]);
            value_multiply(tmp, tmp, bernoulli_coef.num->p[j]);
            value_multiply(tmp, tmp, factor);
            value_addto(bernoulli_coef.num->p[i], bernoulli_coef.num->p[i], tmp);
        }
        mpz_mul_ui(bernoulli_coef.den->p[i], bernoulli_coef.lcm->p[i-1], i+1);
        value_gcd(tmp, bernoulli_coef.num->p[i], bernoulli_coef.den->p[i]);
        if (value_notone_p(tmp)) {
            value_division(bernoulli_coef.num->p[i],
                            bernoulli_coef.num->p[i], tmp);
            value_division(bernoulli_coef.den->p[i],
                            bernoulli_coef.den->p[i], tmp);
        }
        value_lcm(bernoulli_coef.lcm->p[i],
                  bernoulli_coef.lcm->p[i-1], bernoulli_coef.den->p[i]);
    }
    bernoulli_coef.n = n+1;
    value_clear(factor);
    value_clear(tmp);

    return &bernoulli_coef;
}

/*
 * Compute either Bernoulli B_n or Faulhaber F_n polynomials.
 *
 * B_n =         sum_{k=0}^n {  n  \choose k } b_k x^{n-k}
 * F_n = 1/(n+1) sum_{k=0}^n { n+1 \choose k } b_k x^{n+1-k}
 */
static struct poly_list *bernoulli_faulhaber_compute(int n, struct poly_list *pl,
                                                     int faulhaber)
{
    int i, j;
    Value factor;
    struct bernoulli_coef *bc;

    if (n < pl->n)
        return pl;

    if (n >= pl->size) {
        int size = 3*(n + 5)/2;
        Vector **poly;

        poly = ALLOCN(Vector *, size);
        for (i = 0; i < pl->n; ++i)
            poly[i] = pl->poly[i];
        free(pl->poly);
        pl->poly = poly;

        pl->size = size;
    }

    bc = bernoulli_coef_compute(n);

    value_init(factor);
    for (i = pl->n; i <= n; ++i) {
        pl->poly[i] = Vector_Alloc(i+faulhaber+2);
        value_assign(pl->poly[i]->p[i+faulhaber], bc->lcm->p[i]);
        if (faulhaber)
            mpz_mul_ui(pl->poly[i]->p[i+2], bc->lcm->p[i], i+1);
        else
            value_assign(pl->poly[i]->p[i+1], bc->lcm->p[i]);
        value_set_si(factor, 1);
        for (j = 1; j <= i; ++j) {
            mpz_mul_ui(factor, factor, i+faulhaber+1-j);
            mpz_divexact_ui(factor, factor, j);
            value_division(pl->poly[i]->p[i+faulhaber-j],
                           bc->lcm->p[i], bc->den->p[j]);
            value_multiply(pl->poly[i]->p[i+faulhaber-j],
                           pl->poly[i]->p[i+faulhaber-j], bc->num->p[j]);
            value_multiply(pl->poly[i]->p[i+faulhaber-j],
                           pl->poly[i]->p[i+faulhaber-j], factor);
        }
        Vector_Normalize(pl->poly[i]->p, i+faulhaber+2);
    }
    value_clear(factor);
    pl->n = n+1;

    return pl;
}

struct poly_list *bernoulli_compute(int n)
{
    return bernoulli_faulhaber_compute(n, &bernoulli, 0);
}

struct poly_list *faulhaber_compute(int n)
{
    return bernoulli_faulhaber_compute(n, &faulhaber, 1);
}

static evalue *shifted_copy(const evalue *src)
{
    evalue *e = ALLOC(evalue);
    value_init(e->d);
    evalue_copy(e, src);
    evalue_shift_variables(e, -1);
    return e;
}

/* Computes the argument for the Faulhaber polynomial.
 * If we are computing a polynomial approximation (exact == 0),
 * then this is simply arg/denom.
 * Otherwise, if neg == 0, then we are dealing with an upper bound
 * and we need to compute floor(arg/denom) = arg/denom - { arg/denom }
 * If neg == 1, then we are dealing with a lower bound
 * and we need to compute ceil(arg/denom) = arg/denom + { -arg/denom }
 *
 * Modifies arg (if exact == 1).
 */
static evalue *power_sums_base(Vector *arg, Value denom, int neg, int exact)
{
    evalue *fract;
    evalue *base = affine2evalue(arg->p, denom, arg->Size-1);

    if (!exact || value_one_p(denom))
        return base;

    if (neg)
        Vector_Oppose(arg->p, arg->p, arg->Size);

    fract = fractional_part(arg->p, denom, arg->Size-1, NULL);
    if (!neg) {
        value_set_si(arg->p[0], -1);
        evalue_mul(fract, arg->p[0]);
    }
    eadd(fract, base);
    evalue_free(fract);

    return base;
}

static evalue *power_sums(struct poly_list *faulhaber, const evalue *poly,
                          Vector *arg, Value denom, int neg, int alt_neg,
                          int exact)
{
    int i;
    evalue *base = power_sums_base(arg, denom, neg, exact);
    evalue *sum = evalue_zero();

    for (i = 1; i < poly->x.p->size; ++i) {
        evalue *term = evalue_polynomial(faulhaber->poly[i], base);
        evalue *factor = shifted_copy(&poly->x.p->arr[i]);
        emul(factor, term);
        if (alt_neg && (i % 2))
            evalue_negate(term);
        eadd(term, sum);
        evalue_free(factor);
        evalue_free(term);
    }
    if (neg)
        evalue_negate(sum);
    evalue_free(base);

    return sum;
}

/* Given a constraint (cst_affine) a x + b y + c >= 0, compate a constraint (c)
 * +- (b y + c) +- a >=,> 0
 * ^            ^      ^
 * |            |      strict
 * sign_affine  sign_cst
 */
static void bound_constraint(Value *c, unsigned dim,
                             Value *cst_affine,
                             int sign_affine, int sign_cst, int strict)
{
    if (sign_affine == -1)
        Vector_Oppose(cst_affine+1, c, dim+1);
    else
        Vector_Copy(cst_affine+1, c, dim+1);

    if (sign_cst == -1)
        value_subtract(c[dim], c[dim], cst_affine[0]);
    else if (sign_cst == 1)
        value_addto(c[dim], c[dim], cst_affine[0]);

    if (strict)
        value_decrement(c[dim], c[dim]);
}

struct Bernoulli_data {
    struct barvinok_options *options;
    struct evalue_section *s;
    int size;
    int ns;
    const evalue *e;
};

static evalue *compute_poly_u(evalue *poly_u, Value *upper, Vector *row,
                              unsigned dim, Value tmp,
                              struct poly_list *faulhaber,
                              struct Bernoulli_data *data)
{
    int exact = data->options->approximation_method == BV_APPROX_NONE;
    if (poly_u)
        return poly_u;
    Vector_Copy(upper+2, row->p, dim+1);
    value_oppose(tmp, upper[1]);
    value_addto(row->p[dim], row->p[dim], tmp);
    return power_sums(faulhaber, data->e, row, tmp, 0, 0, exact);
}

static evalue *compute_poly_l(evalue *poly_l, Value *lower, Vector *row,
                              unsigned dim,
                              struct poly_list *faulhaber,
                              struct Bernoulli_data *data)
{
    int exact = data->options->approximation_method == BV_APPROX_NONE;
    if (poly_l)
        return poly_l;
    Vector_Copy(lower+2, row->p, dim+1);
    value_addto(row->p[dim], row->p[dim], lower[1]);
    return power_sums(faulhaber, data->e, row, lower[1], 0, 1, exact);
}

/* Compute sum of constant term.
 */
static evalue *linear_term(const evalue *cst, Value *lower, Value *upper,
                           Vector *row, Value tmp, int exact)
{
    evalue *linear;
    unsigned dim = row->Size - 1;

    if (EVALUE_IS_ZERO(*cst))
        return evalue_zero();

    if (!exact) {
        value_absolute(tmp, upper[1]);
        /* upper - lower */
        Vector_Combine(lower+2, upper+2, row->p, tmp, lower[1], dim+1);
        value_multiply(tmp, tmp, lower[1]);
        /* upper - lower + 1 */
        value_addto(row->p[dim], row->p[dim], tmp);

        linear = affine2evalue(row->p, tmp, dim);
    } else {
        evalue *l;

        value_absolute(tmp, upper[1]);
        Vector_Copy(upper+2, row->p, dim+1);
        value_addto(row->p[dim], row->p[dim], tmp);
        /* floor(upper+1) */
        linear = power_sums_base(row, tmp, 0, 1);

        Vector_Copy(lower+2, row->p, dim+1);
        /* floor(-lower) */
        l = power_sums_base(row, lower[1], 0, 1);

        /* floor(upper+1) + floor(-lower) = floor(upper) - ceil(lower) + 1 */
        eadd(l, linear);
        evalue_free(l);
    }

    emul(cst, linear);
    return linear;
}

static void Bernoulli_init(unsigned n, void *cb_data)
{
    struct Bernoulli_data *data = (struct Bernoulli_data *)cb_data;
    int exact = data->options->approximation_method == BV_APPROX_NONE;
    int cases = exact ? 3 : 5;

    if (cases * n <= data->size)
        return;

    data->size = cases * (n + 16);
    data->s = REALLOCN(data->s, struct evalue_section, data->size);
}

static void Bernoulli_cb(Matrix *M, Value *lower, Value *upper, void *cb_data);

/*
 * This function requires that either the lower or the upper bound
 * represented by the constraints "lower" and "upper" is an integer
 * affine expression.
 * An affine substitution is performed to make this bound exactly
 * zero, ensuring that in the recursive call to Bernoulli_cb,
 * only one of the "cases" will apply.
 */
static void transform_to_single_case(Matrix *M, Value *lower, Value *upper,
                                     struct Bernoulli_data *data)
{
    unsigned dim = M->NbColumns-2;
    Vector *shadow;
    Value a, b;
    evalue **subs;
    const evalue *e = data->e;
    evalue *t;
    int i;

    value_init(a);
    value_init(b);
    subs = ALLOCN(evalue *, dim+1);
    for (i = 0; i < dim; ++i)
        subs[1+i] = evalue_var(1+i);
    shadow = Vector_Alloc(2 * (2+dim+1));
    if (value_one_p(lower[1])) {
        /* Replace i by i + l' when b = 1 */
        value_set_si(shadow->p[0], 1);
        Vector_Oppose(lower+2, shadow->p+1, dim+1);
        subs[0] = affine2evalue(shadow->p, shadow->p[0], dim+1);
        /* new lower
         *              i >= 0
         * new upper
         *              (-a i + u') + a (-l') >= 0
         */
        value_assign(shadow->p[2+dim+1+1], upper[1]);
        value_oppose(a, upper[1]);
        value_set_si(b, 1);
        Vector_Combine(upper+2, lower+2, shadow->p+2+dim+1+2,
                       b, a, dim+1);
        upper = shadow->p+2+dim+1;
        lower = shadow->p;
        value_set_si(lower[1], 1);
        Vector_Set(lower+2, 0, dim+1);
    } else {
        /* Replace i by i + u' when a = 1 */
        value_set_si(shadow->p[0], 1);
        Vector_Copy(upper+2, shadow->p+1, dim+1);
        subs[0] = affine2evalue(shadow->p, shadow->p[0], dim+1);
        /* new lower
         *              (b i - l') + b u' >= 0
         * new upper
         *              -i >= 0
         */
        value_assign(shadow->p[1], lower[1]);
        value_set_si(a, 1);
        value_assign(b, lower[1]);
        Vector_Combine(upper+2, lower+2, shadow->p+2,
                       b, a, dim+1);
        upper = shadow->p+2+dim+1;
        lower = shadow->p;
        value_set_si(upper[1], -1);
        Vector_Set(upper+2, 0, dim+1);
    }
    value_clear(a);
    value_clear(b);

    t = evalue_dup(data->e);
    evalue_substitute(t, subs);
    reduce_evalue(t);
    data->e = t;
    for (i = 0; i < dim+1; ++i)
        evalue_free(subs[i]);
    free(subs);

    Bernoulli_cb(M, lower, upper, data);

    evalue_free(t);
    data->e = e;
    Vector_Free(shadow);
}

static void Bernoulli_cb(Matrix *M, Value *lower, Value *upper, void *cb_data)
{
    struct Bernoulli_data *data = (struct Bernoulli_data *)cb_data;
    Matrix *M2;
    Polyhedron *T;
    const evalue *factor = NULL;
    evalue *linear = NULL;
    int constant = 0;
    Value tmp;
    unsigned dim = M->NbColumns-2;
    Vector *row;
    int exact = data->options->approximation_method == BV_APPROX_NONE;
    int cases = exact ? 3 : 5;

    assert(lower);
    assert(upper);
    assert(data->ns + cases <= data->size);

    M2 = Matrix_Copy(M);
    T = Constraints2Polyhedron(M2, data->options->MaxRays);
    Matrix_Free(M2);

    POL_ENSURE_VERTICES(T);
    if (emptyQ(T)) {
        Polyhedron_Free(T);
        return;
    }

    constant = value_notzero_p(data->e->d) ||
                data->e->x.p->type != polynomial ||
                data->e->x.p->pos != 1;
    if (!constant && (value_one_p(lower[1]) || value_mone_p(upper[1]))) {
        int single_case;
        int lower_cst, upper_cst;

        lower_cst = First_Non_Zero(lower+2, dim) == -1;
        upper_cst = First_Non_Zero(upper+2, dim) == -1;
        single_case =
            (lower_cst && value_negz_p(lower[2+dim])) ||
            (upper_cst && value_negz_p(upper[2+dim])) ||
            (lower_cst && upper_cst &&
             value_posz_p(lower[2+dim]) && value_posz_p(upper[2+dim]));
        if (!single_case) {
            transform_to_single_case(M, lower, upper, data);
            Polyhedron_Free(T);
            return;
        }
    }

    assert(lower != upper);

    row = Vector_Alloc(dim+1);
    value_init(tmp);
    if (value_notzero_p(data->e->d)) {
        factor = data->e;
        constant = 1;
    } else {
        if (data->e->x.p->type == polynomial && data->e->x.p->pos == 1)
            factor = shifted_copy(&data->e->x.p->arr[0]);
        else {
            factor = shifted_copy(data->e);
            constant = 1;
        }
    }
    linear = linear_term(factor, lower, upper, row, tmp, exact);

    if (constant) {
        data->s[data->ns].E = linear;
        data->s[data->ns].D = T;
        ++data->ns;
        data->options->stats->bernoulli_sums++;
    } else {
        evalue *poly_u = NULL, *poly_l = NULL;
        Polyhedron *D;
        struct poly_list *faulhaber;
        assert(data->e->x.p->type == polynomial);
        assert(data->e->x.p->pos == 1);
        faulhaber = faulhaber_compute(data->e->x.p->size-1);
        /* lower is the constraint
         *                       b i - l' >= 0          i >= l'/b = l
         * upper is the constraint
         *                      -a i + u' >= 0          i <= u'/a = u
         */
        M2 = Matrix_Alloc(3, 2+T->Dimension);
        value_set_si(M2->p[0][0], 1);
        value_set_si(M2->p[1][0], 1);
        value_set_si(M2->p[2][0], 1);
        /* Case 1:
         *              1 <= l          l' - b >= 0
         *              0 < l           l' - 1 >= 0     if exact
         */
        if (exact)
            bound_constraint(M2->p[0]+1, T->Dimension, lower+1, -1, 0, 1);
        else
            bound_constraint(M2->p[0]+1, T->Dimension, lower+1, -1, -1, 0);
        D = AddConstraints(M2->p_Init, 1, T, data->options->MaxRays);
        POL_ENSURE_VERTICES(D);
        if (emptyQ2(D))
            Polyhedron_Free(D);
        else {
            evalue *extra;
            poly_u = compute_poly_u(poly_u, upper, row, dim, tmp,
                                        faulhaber, data);
            Vector_Oppose(lower+2, row->p, dim+1);
            extra = power_sums(faulhaber, data->e, row, lower[1], 1, 0, exact);
            eadd(poly_u, extra);
            eadd(linear, extra);

            data->s[data->ns].E = extra;
            data->s[data->ns].D = D;
            ++data->ns;
            data->options->stats->bernoulli_sums++;
        }

        /* Case 2:
         *              1 <= -u         -u' - a >= 0
         *              0 < -u          -u' - 1 >= 0    if exact
         */
        if (exact)
            bound_constraint(M2->p[0]+1, T->Dimension, upper+1, -1, 0, 1);
        else
            bound_constraint(M2->p[0]+1, T->Dimension, upper+1, -1, 1, 0);
        D = AddConstraints(M2->p_Init, 1, T, data->options->MaxRays);
        POL_ENSURE_VERTICES(D);
        if (emptyQ2(D))
            Polyhedron_Free(D);
        else {
            evalue *extra;
            poly_l = compute_poly_l(poly_l, lower, row, dim, faulhaber, data);
            Vector_Oppose(upper+2, row->p, dim+1);
            value_oppose(tmp, upper[1]);
            extra = power_sums(faulhaber, data->e, row, tmp, 1, 1, exact);
            eadd(poly_l, extra);
            eadd(linear, extra);

            data->s[data->ns].E = extra;
            data->s[data->ns].D = D;
            ++data->ns;
            data->options->stats->bernoulli_sums++;
        }

        /* Case 3:
         *              u >= 0          u' >= 0
         *              -l >= 0         -l' >= 0
         */
        bound_constraint(M2->p[0]+1, T->Dimension, upper+1, 1, 0, 0);
        bound_constraint(M2->p[1]+1, T->Dimension, lower+1, 1, 0, 0);
        D = AddConstraints(M2->p_Init, 2, T, data->options->MaxRays);
        POL_ENSURE_VERTICES(D);
        if (emptyQ2(D))
            Polyhedron_Free(D);
        else {
            poly_l = compute_poly_l(poly_l, lower, row, dim, faulhaber, data);
            poly_u = compute_poly_u(poly_u, upper, row, dim, tmp,
                                        faulhaber, data);
        
            data->s[data->ns].E = ALLOC(evalue);
            value_init(data->s[data->ns].E->d);
            evalue_copy(data->s[data->ns].E, poly_u);
            eadd(poly_l, data->s[data->ns].E);
            eadd(linear, data->s[data->ns].E);
            data->s[data->ns].D = D;
            ++data->ns;
            data->options->stats->bernoulli_sums++;
        }

        if (!exact) {
            /* Case 4:
             *          l < 1           -l' + b - 1 >= 0
             *          0 < l           l' - 1 >= 0
             *          u >= 1          u' - a >= 0
             */
            bound_constraint(M2->p[0]+1, T->Dimension, lower+1, 1, 1, 1);
            bound_constraint(M2->p[1]+1, T->Dimension, lower+1, -1, 0, 1);
            bound_constraint(M2->p[2]+1, T->Dimension, upper+1, 1, 1, 0);
            D = AddConstraints(M2->p_Init, 3, T, data->options->MaxRays);
            POL_ENSURE_VERTICES(D);
            if (emptyQ2(D))
                Polyhedron_Free(D);
            else {
                poly_u = compute_poly_u(poly_u, upper, row, dim, tmp,
                                            faulhaber, data);
                eadd(linear, poly_u);
                data->s[data->ns].E = poly_u;
                poly_u = NULL;
                data->s[data->ns].D = D;
                ++data->ns;
                data->options->stats->bernoulli_sums++;
            }

            /* Case 5:
             *          -u < 1          u' + a - 1 >= 0
             *          0 < -u          -u' - 1 >= 0
             *          l <= -1         -l' - b >= 0
             */
            bound_constraint(M2->p[0]+1, T->Dimension, upper+1, 1, -1, 1);
            bound_constraint(M2->p[1]+1, T->Dimension, upper+1, -1, 0, 1);
            bound_constraint(M2->p[2]+1, T->Dimension, lower+1, 1, -1, 0);
            D = AddConstraints(M2->p_Init, 3, T, data->options->MaxRays);
            POL_ENSURE_VERTICES(D);
            if (emptyQ2(D))
                Polyhedron_Free(D);
            else {
                poly_l = compute_poly_l(poly_l, lower, row, dim,
                                        faulhaber, data);
                eadd(linear, poly_l);
                data->s[data->ns].E = poly_l;
                poly_l = NULL;
                data->s[data->ns].D = D;
                ++data->ns;
                data->options->stats->bernoulli_sums++;
            }
        }

        Matrix_Free(M2);
        Polyhedron_Free(T);
        if (poly_l)
            evalue_free(poly_l);
        if (poly_u)
            evalue_free(poly_u);
        evalue_free(linear);
    }
    if (factor != data->e)
        evalue_free((evalue *)factor);
    value_clear(tmp);
    Vector_Free(row);
}

/*
 * Move the variable at position pos to the front by exchanging
 * that variable with the first variable.
 */
static void more_var_to_front(Polyhedron **P_p , evalue **E_p, int pos)
{
    Polyhedron *P = *P_p;
    evalue *E = *E_p;
    unsigned dim = P->Dimension;

    assert(pos != 0);

    P = Polyhedron_Copy(P);
    Polyhedron_ExchangeColumns(P, 1, 1+pos);
    *P_p = P;

    if (value_zero_p(E->d)) {
        int j;
        evalue **subs;

        subs = ALLOCN(evalue *, dim);
        for (j = 0; j < dim; ++j)
            subs[j] = evalue_var(j);
        E = subs[0];
        subs[0] = subs[pos];
        subs[pos] = E;
        E = evalue_dup(*E_p);
        evalue_substitute(E, subs);
        for (j = 0; j < dim; ++j)
            evalue_free(subs[j]);
        free(subs);
        *E_p = E;
    }
}

static int type_offset(enode *p)
{
   return p->type == fractional ? 1 :
          p->type == flooring ? 1 : 0;
}

static void adjust_periods(evalue *e, unsigned nvar, Vector *periods)
{
    for (; value_zero_p(e->d); e = &e->x.p->arr[0]) {
        int pos;
        assert(e->x.p->type == polynomial);
        assert(e->x.p->size == 2);
        assert(value_notzero_p(e->x.p->arr[1].d));

        pos = e->x.p->pos - 1;
        if (pos >= nvar)
            break;

        value_lcm(periods->p[pos], periods->p[pos], e->x.p->arr[1].d);
    }
}

static void fractional_periods_r(evalue *e, unsigned nvar, Vector *periods)
{
    int i;

    if (value_notzero_p(e->d))
        return;

    assert(e->x.p->type == polynomial || e->x.p->type == fractional);

    if (e->x.p->type == fractional)
        adjust_periods(&e->x.p->arr[0], nvar, periods);

    for (i = type_offset(e->x.p); i < e->x.p->size; ++i)
        fractional_periods_r(&e->x.p->arr[i], nvar, periods);
}

/*
 * For each of the nvar variables, compute the lcm of the
 * denominators of the coefficients of that variable in
 * any of the fractional parts.
 */
static Vector *fractional_periods(evalue *e, unsigned nvar)
{
    int i;
    Vector *periods = Vector_Alloc(nvar);

    for (i = 0; i < nvar; ++i)
        value_set_si(periods->p[i], 1);

    fractional_periods_r(e, nvar, periods);

    return periods;
}

/* Move "best" variable to sum over into the first position,
 * possibly changing *P_p and *E_p.
 *
 * If there are any fractional parts (period != NULL), then we
 * first look for a variable with the smallest lcm of denominators
 * inside a fractional part.  This denominator is assigned to period
 * and corresponds to the number of "splinters" we need to construct
 * at this level.
 *
 * Of those with this denominator (all if period == NULL or if there
 * are no fractional parts), we select the variable with the smallest
 * maximal coefficient, as this coefficient will become a denominator
 * in new fractional parts (for an exact computation), which may
 * lead to splintering in the next step.
 */
static void move_best_to_front(Polyhedron **P_p, evalue **E_p, unsigned nvar,
                               Value *period)
{
    Polyhedron *P = *P_p;
    evalue *E = *E_p;
    int i, j, best_i = -1;
    Vector *periods;
    Value best, max;

    assert(nvar >= 1);

    if (period) {
        periods = fractional_periods(E, nvar);
        value_assign(*period, periods->p[0]);
        for (i = 1; i < nvar; ++i)
            if (value_lt(periods->p[i], *period))
                value_assign(*period, periods->p[i]);
    }

    value_init(best);
    value_init(max);

    for (i = 0; i < nvar; ++i) {
        if (period && value_ne(*period, periods->p[i]))
            continue;

        value_set_si(max, 0);

        for (j = 0; j < P->NbConstraints; ++j) {
            if (value_zero_p(P->Constraint[j][1+i]))
                continue;
            if (First_Non_Zero(P->Constraint[j]+1, i) == -1 &&
                First_Non_Zero(P->Constraint[j]+1+i+1, nvar-i-1) == -1)
                continue;
            if (value_abs_gt(P->Constraint[j][1+i], max))
                value_absolute(max, P->Constraint[j][1+i]);
        }

        if (best_i == -1 || value_lt(max, best)) {
            value_assign(best, max);
            best_i = i;
        }
    }

    value_clear(best);
    value_clear(max);

    if (period)
        Vector_Free(periods);

    if (best_i != 0)
        more_var_to_front(P_p, E_p, best_i);

    return;
}

static evalue *sum_over_polytope_base(Polyhedron *P, evalue *E, unsigned nvar,
                                      struct Bernoulli_data *data,
                                      struct barvinok_options *options)
{
    evalue *res;

    assert(P->NbEq == 0);

    data->ns = 0;
    data->e = E;

    for_each_lower_upper_bound(P, Bernoulli_init, Bernoulli_cb, data);

    res = evalue_from_section_array(data->s, data->ns);

    if (nvar > 1) {
        evalue *tmp = Bernoulli_sum_evalue(res, nvar-1, options);
        evalue_free(res);
        res = tmp;
    }

    return res;
}

static evalue *sum_over_polytope(Polyhedron *P, evalue *E, unsigned nvar,
                                 struct Bernoulli_data *data,
                                 struct barvinok_options *options);

static evalue *sum_over_polytope_with_equalities(Polyhedron *P, evalue *E,
                                 unsigned nvar,
                                 struct Bernoulli_data *data,
                                 struct barvinok_options *options)
{
    unsigned dim = P->Dimension;
    unsigned new_dim, new_nparam;
    Matrix *T = NULL, *CP = NULL;
    evalue **subs;
    evalue *sum;
    int j;

    if (emptyQ(P))
        return evalue_zero();

    assert(P->NbEq > 0);

    remove_all_equalities(&P, NULL, &CP, &T, dim-nvar, options->MaxRays);

    if (emptyQ(P)) {
        Polyhedron_Free(P);
        return evalue_zero();
    }

    new_nparam = CP ? CP->NbColumns-1 : dim - nvar;
    new_dim = T ? T->NbColumns-1 : nvar + new_nparam;

    /* We can avoid these substitutions if E is a constant */
    subs = ALLOCN(evalue *, dim);
    for (j = 0; j < nvar; ++j) {
        if (T)
            subs[j] = affine2evalue(T->p[j], T->p[nvar+new_nparam][new_dim],
                                    new_dim);
        else
            subs[j] = evalue_var(j);
    }
    for (j = 0; j < dim-nvar; ++j) {
        if (CP)
            subs[nvar+j] = affine2evalue(CP->p[j], CP->p[dim-nvar][new_nparam],
                                         new_nparam);
        else
            subs[nvar+j] = evalue_var(j);
        evalue_shift_variables(subs[nvar+j], new_dim-new_nparam);
    }

    E = evalue_dup(E);
    evalue_substitute(E, subs);
    reduce_evalue(E);

    for (j = 0; j < dim; ++j)
        evalue_free(subs[j]);
    free(subs);

    if (new_dim-new_nparam > 0) {
        sum = sum_over_polytope(P, E, new_dim-new_nparam, data, options);
        evalue_free(E);
        Polyhedron_Free(P);
    } else {
        sum = ALLOC(evalue);
        value_init(sum->d);
        sum->x.p = new_enode(partition, 2, new_dim);
        EVALUE_SET_DOMAIN(sum->x.p->arr[0], P);
        value_clear(sum->x.p->arr[1].d);
        sum->x.p->arr[1] = *E;
        free(E);
    }

    if (CP) {
        evalue_backsubstitute(sum, CP, options->MaxRays);
        Matrix_Free(CP);
    }

    if (T)
        Matrix_Free(T);

    return sum;
}

/* Splinter P into period slices along the first variable x = period y + c,
 * 0 <= c < perdiod, * ensuring no fractional part contains the first variable,
 * and sum over all slices.
 */
static evalue *sum_over_polytope_slices(Polyhedron *P, evalue *E,
                                        unsigned nvar,
                                        Value period,
                                        struct Bernoulli_data *data,
                                        struct barvinok_options *options)
{
    evalue *sum = evalue_zero();
    evalue **subs;
    unsigned dim = P->Dimension;
    Matrix *T;
    Value i;
    Value one;
    int j;

    value_init(i);
    value_init(one);
    value_set_si(one, 1);

    subs = ALLOCN(evalue *, dim);

    T = Matrix_Alloc(dim+1, dim+1);
    value_assign(T->p[0][0], period);
    for (j = 1; j < dim+1; ++j)
        value_set_si(T->p[j][j], 1);

    for (j = 0; j < dim; ++j)
        subs[j] = evalue_var(j);
    evalue_mul(subs[0], period);

    for (value_set_si(i, 0); value_lt(i, period); value_increment(i, i)) {
        evalue *tmp;
        Polyhedron *S = DomainPreimage(P, T, options->MaxRays);
        evalue *e = evalue_dup(E);
        evalue_substitute(e, subs);
        reduce_evalue(e);

        if (S->NbEq)
            tmp = sum_over_polytope_with_equalities(S, e, nvar, data, options);
        else
            tmp = sum_over_polytope_base(S, e, nvar, data, options);

        assert(tmp);
        eadd(tmp, sum);
        evalue_free(tmp);

        Domain_Free(S);
        evalue_free(e);

        value_increment(T->p[0][dim], T->p[0][dim]);
        evalue_add_constant(subs[0], one);
    }

    value_clear(i);
    value_clear(one);
    Matrix_Free(T);
    for (j = 0; j < dim; ++j)
        evalue_free(subs[j]);
    free(subs);

    reduce_evalue(sum);
    return sum;
}

static evalue *sum_over_polytope(Polyhedron *P, evalue *E, unsigned nvar,
                                 struct Bernoulli_data *data,
                                 struct barvinok_options *options)
{
    Polyhedron *P_orig = P;
    evalue *E_orig = E;
    Value period;
    evalue *sum;
    int exact = options->approximation_method == BV_APPROX_NONE;

    if (P->NbEq)
        return sum_over_polytope_with_equalities(P, E, nvar, data, options);

    value_init(period);

    move_best_to_front(&P, &E, nvar, exact ? &period : NULL);
    if (exact && value_notone_p(period))
        sum = sum_over_polytope_slices(P, E, nvar, period, data, options);
    else
        sum = sum_over_polytope_base(P, E, nvar, data, options);

    if (P != P_orig)
        Polyhedron_Free(P);
    if (E != E_orig)
        evalue_free(E);

    value_clear(period);

    return sum;
}

evalue *Bernoulli_sum_evalue(evalue *e, unsigned nvar,
                         struct barvinok_options *options)
{
    struct Bernoulli_data data;
    int i, j;
    evalue *sum = evalue_zero();

    if (EVALUE_IS_ZERO(*e))
        return sum;

    if (nvar == 0) {
        eadd(e, sum);
        return sum;
    }

    assert(value_zero_p(e->d));
    assert(e->x.p->type == partition);

    data.size = 16;
    data.s = ALLOCN(struct evalue_section, data.size);
    data.options = options;

    for (i = 0; i < e->x.p->size/2; ++i) {
        Polyhedron *D;
        for (D = EVALUE_DOMAIN(e->x.p->arr[2*i]); D; D = D->next) {
            Polyhedron *next = D->next;
            evalue *tmp;
            D->next = NULL;

            tmp = sum_over_polytope(D, &e->x.p->arr[2*i+1], nvar, &data, options);
            assert(tmp);
            eadd(tmp, sum);
            evalue_free(tmp);

            D->next = next;
        }
    }

    free(data.s);

    reduce_evalue(sum);
    return sum;
}