Subversion Repositories openrisc_2011-10-31

/* Gimple Represented as Polyhedra.

   Copyright (C) 2009, 2010 Free Software Foundation, Inc.

   Contributed by Sebastian Pop <sebastian.pop@amd.com>

   and Tobias Grosser <grosser@fim.uni-passau.de>

This file is part of GCC.

GCC is free software; you can redistribute it and/or modify

it under the terms of the GNU General Public License as published by

the Free Software Foundation; either version 3, or (at your option)

any later version.

GCC is distributed in the hope that it will be useful,

but WITHOUT ANY WARRANTY; without even the implied warranty of

MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the

GNU General Public License for more details.

You should have received a copy of the GNU General Public License

along with GCC; see the file COPYING3.  If not see

<http://www.gnu.org/licenses/>.  */

#include "config.h"

#include "system.h"

#include "coretypes.h"

#include "tm.h"

#include "ggc.h"

#ifdef HAVE_cloog

#include "ppl_c.h"

#include "cloog/cloog.h"

#include "graphite-ppl.h"

/* Translates row ROW of the CloogMatrix MATRIX to a PPL Constraint.  */

static ppl_Constraint_t

cloog_matrix_to_ppl_constraint (CloogMatrix *matrix, int row)

{

  int j;

  ppl_Constraint_t cstr;

  ppl_Coefficient_t coef;

  ppl_Linear_Expression_t expr;

  ppl_dimension_type dim = matrix->NbColumns - 2;

  ppl_new_Coefficient (&coef);

  ppl_new_Linear_Expression_with_dimension (&expr, dim);

  for (j = 1; j < matrix->NbColumns - 1; j++)

    {

      ppl_assign_Coefficient_from_mpz_t (coef, matrix->p[row][j]);

      ppl_Linear_Expression_add_to_coefficient (expr, j - 1, coef);

    }

  ppl_assign_Coefficient_from_mpz_t (coef,

                                     matrix->p[row][matrix->NbColumns - 1]);

  ppl_Linear_Expression_add_to_inhomogeneous (expr, coef);

  ppl_delete_Coefficient (coef);

  if (value_zero_p (matrix->p[row][0]))

    ppl_new_Constraint (&cstr, expr, PPL_CONSTRAINT_TYPE_EQUAL);

  else

    ppl_new_Constraint (&cstr, expr, PPL_CONSTRAINT_TYPE_GREATER_OR_EQUAL);

  ppl_delete_Linear_Expression (expr);

  return cstr;

}

/* Creates a PPL constraint system from MATRIX.  */

static void

new_Constraint_System_from_Cloog_Matrix (ppl_Constraint_System_t *pcs,

                                         CloogMatrix *matrix)

{

  int i;

  ppl_new_Constraint_System (pcs);

  for (i = 0; i < matrix->NbRows; i++)

    {

      ppl_Constraint_t c = cloog_matrix_to_ppl_constraint (matrix, i);

      ppl_Constraint_System_insert_Constraint (*pcs, c);

      ppl_delete_Constraint (c);

    }

}

/* Creates a PPL Polyhedron from MATRIX.  */

void

new_C_Polyhedron_from_Cloog_Matrix (ppl_Polyhedron_t *ph,

                                      CloogMatrix *matrix)

{

  ppl_Constraint_System_t cs;

  new_Constraint_System_from_Cloog_Matrix (&cs, matrix);

  ppl_new_C_Polyhedron_recycle_Constraint_System (ph, cs);

}

/* Counts the number of constraints in PCS.  */

static int

ppl_Constrain_System_number_of_constraints (ppl_const_Constraint_System_t pcs)

{

  ppl_Constraint_System_const_iterator_t cit, end;

  int num = 0;

  ppl_new_Constraint_System_const_iterator (&cit);

  ppl_new_Constraint_System_const_iterator (&end);

  for (ppl_Constraint_System_begin (pcs, cit),

        ppl_Constraint_System_end (pcs, end);

       !ppl_Constraint_System_const_iterator_equal_test (cit, end);

       ppl_Constraint_System_const_iterator_increment (cit))

    num++;

  ppl_delete_Constraint_System_const_iterator (cit);

  ppl_delete_Constraint_System_const_iterator (end);

  return num;

}

static void

oppose_constraint (CloogMatrix *m, int row)

{

  int k;

  /* Do not oppose the first column: it is the eq/ineq one.  */

  for (k = 1; k < m->NbColumns; k++)

    value_oppose (m->p[row][k], m->p[row][k]);

}

/* Inserts constraint CSTR at row ROW of matrix M.  */

void

insert_constraint_into_matrix (CloogMatrix *m, int row,

                               ppl_const_Constraint_t cstr)

{

  ppl_Coefficient_t c;

  ppl_dimension_type i, dim, nb_cols = m->NbColumns;

  ppl_Constraint_space_dimension (cstr, &dim);

  ppl_new_Coefficient (&c);

  for (i = 0; i < dim; i++)

    {

      ppl_Constraint_coefficient (cstr, i, c);

      ppl_Coefficient_to_mpz_t (c, m->p[row][i + 1]);

    }

  for (i = dim; i < nb_cols - 1; i++)

    value_set_si (m->p[row][i + 1], 0);

  ppl_Constraint_inhomogeneous_term  (cstr, c);

  ppl_Coefficient_to_mpz_t (c, m->p[row][nb_cols - 1]);

  value_set_si (m->p[row][0], 1);

  switch (ppl_Constraint_type (cstr))

    {

    case PPL_CONSTRAINT_TYPE_LESS_THAN:

      oppose_constraint (m, row);

    case PPL_CONSTRAINT_TYPE_GREATER_THAN:

      value_sub_int (m->p[row][nb_cols - 1],

                     m->p[row][nb_cols - 1], 1);

      break;

    case PPL_CONSTRAINT_TYPE_LESS_OR_EQUAL:

      oppose_constraint (m, row);

    case PPL_CONSTRAINT_TYPE_GREATER_OR_EQUAL:

      break;

    case PPL_CONSTRAINT_TYPE_EQUAL:

      value_set_si (m->p[row][0], 0);

      break;

    default:

      /* Not yet implemented.  */

      gcc_unreachable();

    }

  ppl_delete_Coefficient (c);

}

/* Creates a CloogMatrix from constraint system PCS.  */

static CloogMatrix *

new_Cloog_Matrix_from_ppl_Constraint_System (ppl_const_Constraint_System_t pcs)

{

  CloogMatrix *matrix;

  ppl_Constraint_System_const_iterator_t cit, end;

  ppl_dimension_type dim;

  int rows;

  int row = 0;

  rows = ppl_Constrain_System_number_of_constraints (pcs);

  ppl_Constraint_System_space_dimension (pcs, &dim);

  matrix = cloog_matrix_alloc (rows, dim + 2);

  ppl_new_Constraint_System_const_iterator (&cit);

  ppl_new_Constraint_System_const_iterator (&end);

  for (ppl_Constraint_System_begin (pcs, cit),

        ppl_Constraint_System_end (pcs, end);

       !ppl_Constraint_System_const_iterator_equal_test (cit, end);

       ppl_Constraint_System_const_iterator_increment (cit))

    {

      ppl_const_Constraint_t c;

      ppl_Constraint_System_const_iterator_dereference (cit, &c);

      insert_constraint_into_matrix (matrix, row, c);

      row++;

    }

  ppl_delete_Constraint_System_const_iterator (cit);

  ppl_delete_Constraint_System_const_iterator (end);

  return matrix;

}

/* Creates a CloogMatrix from polyhedron PH.  */

CloogMatrix *

new_Cloog_Matrix_from_ppl_Polyhedron (ppl_const_Polyhedron_t ph)

{

  ppl_const_Constraint_System_t pcs;

  CloogMatrix *res;

  ppl_Polyhedron_get_constraints (ph, &pcs);

  res = new_Cloog_Matrix_from_ppl_Constraint_System (pcs);

  return res;

}

/* Creates a CloogDomain from polyhedron PH.  */

CloogDomain *

new_Cloog_Domain_from_ppl_Polyhedron (ppl_const_Polyhedron_t ph)

{

  CloogMatrix *mat = new_Cloog_Matrix_from_ppl_Polyhedron (ph);

  CloogDomain *res = cloog_domain_matrix2domain (mat);

  cloog_matrix_free (mat);

  return res;

}

/* Creates a CloogDomain from a pointset powerset PS.  */

CloogDomain *

new_Cloog_Domain_from_ppl_Pointset_Powerset (

  ppl_Pointset_Powerset_C_Polyhedron_t ps)

{

  CloogDomain *res = NULL;

  ppl_Pointset_Powerset_C_Polyhedron_iterator_t it, end;

  ppl_new_Pointset_Powerset_C_Polyhedron_iterator (&it);

  ppl_new_Pointset_Powerset_C_Polyhedron_iterator (&end);

  for (ppl_Pointset_Powerset_C_Polyhedron_iterator_begin (ps, it),

       ppl_Pointset_Powerset_C_Polyhedron_iterator_end (ps, end);

       !ppl_Pointset_Powerset_C_Polyhedron_iterator_equal_test (it, end);

       ppl_Pointset_Powerset_C_Polyhedron_iterator_increment (it))

    {

      ppl_const_Polyhedron_t ph;

      CloogDomain *tmp;

      ppl_Pointset_Powerset_C_Polyhedron_iterator_dereference (it, &ph);

      tmp = new_Cloog_Domain_from_ppl_Polyhedron (ph);

      if (res == NULL)

        res = tmp;

      else

        res = cloog_domain_union (res, tmp);

    }

  ppl_delete_Pointset_Powerset_C_Polyhedron_iterator (it);

  ppl_delete_Pointset_Powerset_C_Polyhedron_iterator (end);

  gcc_assert (res != NULL);

  return res;

}

/* Set the inhomogeneous term of E to X.  */

void

ppl_set_inhomogeneous_gmp (ppl_Linear_Expression_t e, Value x)

{

  Value v0, v1;

  ppl_Coefficient_t c;

  value_init (v0);

  value_init (v1);

  ppl_new_Coefficient (&c);

  ppl_Linear_Expression_inhomogeneous_term (e, c);

  ppl_Coefficient_to_mpz_t (c, v1);

  value_oppose (v1, v1);

  value_assign (v0, x);

  value_addto (v0, v0, v1);

  ppl_assign_Coefficient_from_mpz_t (c, v0);

  ppl_Linear_Expression_add_to_inhomogeneous (e, c);

  value_clear (v0);

  value_clear (v1);

  ppl_delete_Coefficient (c);

}

/* Set E[I] to X.  */

void

ppl_set_coef_gmp (ppl_Linear_Expression_t e, ppl_dimension_type i, Value x)

{

  Value v0, v1;

  ppl_Coefficient_t c;

  value_init (v0);

  value_init (v1);

  ppl_new_Coefficient (&c);

  ppl_Linear_Expression_coefficient (e, i, c);

  ppl_Coefficient_to_mpz_t (c, v1);

  value_oppose (v1, v1);

  value_assign (v0, x);

  value_addto (v0, v0, v1);

  ppl_assign_Coefficient_from_mpz_t (c, v0);

  ppl_Linear_Expression_add_to_coefficient (e, i, c);

  value_clear (v0);

  value_clear (v1);

  ppl_delete_Coefficient (c);

}

/* Insert after X NB_NEW_DIMS empty dimensions into PH.

   With x = 3 and nb_new_dims = 4

   |  d0 d1 d2 d3 d4

   is transformed to

   |  d0 d1 d2 x0 x1 x2 x3 d3 d4

   | map = {0, 1, 2, 7, 8, 3, 4, 5, 6}

*/

void

ppl_insert_dimensions_pointset (ppl_Pointset_Powerset_C_Polyhedron_t ph, int x,

                                int nb_new_dims)

{

  ppl_dimension_type i, dim;

  ppl_dimension_type *map;

  ppl_dimension_type x_ppl, nb_new_dims_ppl;

  x_ppl = (ppl_dimension_type) x;

  nb_new_dims_ppl = (ppl_dimension_type) nb_new_dims;

  ppl_Pointset_Powerset_C_Polyhedron_space_dimension (ph, &dim);

  ppl_Pointset_Powerset_C_Polyhedron_add_space_dimensions_and_embed (ph, nb_new_dims);

  map = (ppl_dimension_type *) XNEWVEC (ppl_dimension_type, dim + nb_new_dims);

  for (i = 0; i < x_ppl; i++)

    map[i] = i;

  for (i = x_ppl; i < x_ppl + nb_new_dims_ppl; i++)

    map[dim + i - x_ppl] = i;

  for (i = x_ppl + nb_new_dims_ppl; i < dim + nb_new_dims_ppl; i++)

    map[i - nb_new_dims_ppl] = i;

  ppl_Pointset_Powerset_C_Polyhedron_map_space_dimensions (ph, map, dim + nb_new_dims);

  free (map);

}

/* Insert after X NB_NEW_DIMS empty dimensions into PH.

   With x = 3 and nb_new_dims = 4

   |  d0 d1 d2 d3 d4

   is transformed to

   |  d0 d1 d2 x0 x1 x2 x3 d3 d4

   | map = {0, 1, 2, 7, 8, 3, 4, 5, 6}

*/

void

ppl_insert_dimensions (ppl_Polyhedron_t ph, int x,

                       int nb_new_dims)

{

  ppl_dimension_type i, dim;

  ppl_dimension_type *map;

  ppl_dimension_type x_ppl, nb_new_dims_ppl;

  x_ppl = (ppl_dimension_type) x;

  nb_new_dims_ppl = (ppl_dimension_type) nb_new_dims;

  ppl_Polyhedron_space_dimension (ph, &dim);

  ppl_Polyhedron_add_space_dimensions_and_embed (ph, nb_new_dims);

  map = (ppl_dimension_type *) XNEWVEC (ppl_dimension_type, dim + nb_new_dims);

  for (i = 0; i < x_ppl; i++)

    map[i] = i;

  for (i = x_ppl; i < x_ppl + nb_new_dims_ppl; i++)

    map[dim + i - x_ppl] = i;

  for (i = x_ppl + nb_new_dims_ppl; i < dim + nb_new_dims_ppl; i++)

    map[i - nb_new_dims_ppl] = i;

  ppl_Polyhedron_map_space_dimensions (ph, map, dim + nb_new_dims);

  free (map);

}

/* Based on the original polyhedron PH, returns a new polyhedron with

   an extra dimension placed at position LOOP + 1 that slices the

   dimension LOOP into strips of size STRIDE.  */

ppl_Polyhedron_t

ppl_strip_loop (ppl_Polyhedron_t ph, ppl_dimension_type loop, int stride)

{

  ppl_const_Constraint_System_t pcs;

  ppl_Constraint_System_const_iterator_t cit, end;

  ppl_const_Constraint_t cstr;

  ppl_Linear_Expression_t expr;

  int v;

  ppl_dimension_type dim;

  ppl_Polyhedron_t res;

  ppl_Coefficient_t c;

  Value val;

  value_init (val);

  ppl_new_Coefficient (&c);

  ppl_Polyhedron_space_dimension (ph, &dim);

  ppl_Polyhedron_get_constraints (ph, &pcs);

  /* Start from a copy of the constraints.  */

  ppl_new_C_Polyhedron_from_space_dimension (&res, dim + 1, 0);

  ppl_Polyhedron_add_constraints (res, pcs);

  /* Add an empty dimension for the strip loop.  */

  ppl_insert_dimensions (res, loop, 1);

  /* Identify the constraints that define the lower and upper bounds

     of the strip-mined loop, and add them to the strip loop.  */

  {

    ppl_Polyhedron_t tmp;

    ppl_new_C_Polyhedron_from_space_dimension (&tmp, dim + 1, 0);

    ppl_new_Constraint_System_const_iterator (&cit);

    ppl_new_Constraint_System_const_iterator (&end);

    for (ppl_Constraint_System_begin (pcs, cit),

           ppl_Constraint_System_end (pcs, end);

         !ppl_Constraint_System_const_iterator_equal_test (cit, end);

         ppl_Constraint_System_const_iterator_increment (cit))

      {

        ppl_Constraint_System_const_iterator_dereference (cit, &cstr);

        ppl_new_Linear_Expression_from_Constraint (&expr, cstr);

        ppl_Linear_Expression_coefficient (expr, loop, c);

        ppl_delete_Linear_Expression (expr);

        ppl_Coefficient_to_mpz_t (c, val);

        v = value_get_si (val);

        if (0 < v || v < 0)

          ppl_Polyhedron_add_constraint (tmp, cstr);

      }

    ppl_delete_Constraint_System_const_iterator (cit);

    ppl_delete_Constraint_System_const_iterator (end);

    ppl_insert_dimensions (tmp, loop + 1, 1);

    ppl_Polyhedron_get_constraints (tmp, &pcs);

    ppl_Polyhedron_add_constraints (res, pcs);

    ppl_delete_Polyhedron (tmp);

  }

  /* Lower bound of a tile starts at "stride * outer_iv".  */

  {

    ppl_Constraint_t new_cstr;

    ppl_new_Linear_Expression_with_dimension (&expr, dim + 1);

    ppl_set_coef (expr, loop + 1, 1);

    ppl_set_coef (expr, loop, -1 * stride);

    ppl_new_Constraint (&new_cstr, expr, PPL_CONSTRAINT_TYPE_GREATER_OR_EQUAL);

    ppl_delete_Linear_Expression (expr);

    ppl_Polyhedron_add_constraint (res, new_cstr);

    ppl_delete_Constraint (new_cstr);

  }

  /* Upper bound of a tile stops at "stride * outer_iv + stride - 1",

     or at the old upper bound that is not modified.  */

  {

    ppl_Constraint_t new_cstr;

    ppl_new_Linear_Expression_with_dimension (&expr, dim + 1);

    ppl_set_coef (expr, loop + 1, -1);

    ppl_set_coef (expr, loop, stride);

    ppl_set_inhomogeneous (expr, stride - 1);

    ppl_new_Constraint (&new_cstr, expr, PPL_CONSTRAINT_TYPE_GREATER_OR_EQUAL);

    ppl_delete_Linear_Expression (expr);

    ppl_Polyhedron_add_constraint (res, new_cstr);

    ppl_delete_Constraint (new_cstr);

  }

  value_clear (val);

  ppl_delete_Coefficient (c);

  return res;

}

/* Lexicographically compares two linear expressions A and B and

   returns negative when A < B, 0 when A == B and positive when A > B.  */

int

ppl_lexico_compare_linear_expressions (ppl_Linear_Expression_t a,

                                       ppl_Linear_Expression_t b)

{

  ppl_dimension_type min_length, length1, length2;

  ppl_dimension_type i;

  ppl_Coefficient_t c;

  int res;

  Value va, vb;

  ppl_Linear_Expression_space_dimension (a, &length1);

  ppl_Linear_Expression_space_dimension (b, &length2);

  ppl_new_Coefficient (&c);

  value_init (va);

  value_init (vb);

  if (length1 < length2)

    min_length = length1;

  else

    min_length = length2;

  for (i = 0; i < min_length; i++)

    {

      ppl_Linear_Expression_coefficient (a, i, c);

      ppl_Coefficient_to_mpz_t (c, va);

      ppl_Linear_Expression_coefficient (b, i, c);

      ppl_Coefficient_to_mpz_t (c, vb);

      res = value_compare (va, vb);

      if (res == 0)

        continue;

      value_clear (va);

      value_clear (vb);

      ppl_delete_Coefficient (c);

      return res;

    }

  value_clear (va);

  value_clear (vb);

  ppl_delete_Coefficient (c);

  return length1 - length2;

}

/* Print to FILE the polyhedron PH under its PolyLib matrix form.  */

void

ppl_print_polyhedron_matrix (FILE *file, ppl_const_Polyhedron_t ph)

{

  CloogMatrix *mat = new_Cloog_Matrix_from_ppl_Polyhedron (ph);

  cloog_matrix_print (file, mat);

  cloog_matrix_free (mat);

}

/* Print to FILE the linear expression LE.  */

void

ppl_print_linear_expr (FILE *file, ppl_Linear_Expression_t le)

{

  ppl_Constraint_t c;

  ppl_Polyhedron_t pol;

  ppl_dimension_type dim;

  ppl_Linear_Expression_space_dimension (le, &dim);

  ppl_new_C_Polyhedron_from_space_dimension (&pol, dim, 0);

  ppl_new_Constraint (&c, le, PPL_CONSTRAINT_TYPE_EQUAL);

  ppl_Polyhedron_add_constraint (pol, c);

  ppl_print_polyhedron_matrix (file, pol);

}

/* Print to STDERR the linear expression LE.  */

void

debug_ppl_linear_expr (ppl_Linear_Expression_t le)

{

  ppl_print_linear_expr (stderr, le);

}

/* Print to FILE the powerset PS in its PolyLib matrix form.  */

void

ppl_print_powerset_matrix (FILE *file,

                           ppl_Pointset_Powerset_C_Polyhedron_t ps)

{

  size_t nb_disjuncts;

  ppl_Pointset_Powerset_C_Polyhedron_iterator_t it, end;

  ppl_new_Pointset_Powerset_C_Polyhedron_iterator (&it);

  ppl_new_Pointset_Powerset_C_Polyhedron_iterator (&end);

  ppl_Pointset_Powerset_C_Polyhedron_size (ps, &nb_disjuncts);

  fprintf (file, "%d\n", (int) nb_disjuncts);

  for (ppl_Pointset_Powerset_C_Polyhedron_iterator_begin (ps, it),

       ppl_Pointset_Powerset_C_Polyhedron_iterator_end (ps, end);

       !ppl_Pointset_Powerset_C_Polyhedron_iterator_equal_test (it, end);

       ppl_Pointset_Powerset_C_Polyhedron_iterator_increment (it))

    {

      ppl_const_Polyhedron_t ph;

      ppl_Pointset_Powerset_C_Polyhedron_iterator_dereference (it, &ph);

      ppl_print_polyhedron_matrix (file, ph);

    }

  ppl_delete_Pointset_Powerset_C_Polyhedron_iterator (it);

  ppl_delete_Pointset_Powerset_C_Polyhedron_iterator (end);

}

/* Print to STDERR the polyhedron PH under its PolyLib matrix form.  */

void

debug_ppl_polyhedron_matrix (ppl_Polyhedron_t ph)

{

  ppl_print_polyhedron_matrix (stderr, ph);

}

/* Print to STDERR the powerset PS in its PolyLib matrix form.  */

void

debug_ppl_powerset_matrix (ppl_Pointset_Powerset_C_Polyhedron_t ps)

{

  ppl_print_powerset_matrix (stderr, ps);

}

/* Read from FILE a polyhedron under PolyLib matrix form and return a

   PPL polyhedron object.  */

void

ppl_read_polyhedron_matrix (ppl_Polyhedron_t *ph, FILE *file)

{

  CloogMatrix *mat = cloog_matrix_read (file);

  new_C_Polyhedron_from_Cloog_Matrix (ph, mat);

  cloog_matrix_free (mat);

}

/* Return in RES the maximum of the linear expression LE on the

   pointset powerset of polyhedra PS.  */

void

ppl_max_for_le_pointset (ppl_Pointset_Powerset_C_Polyhedron_t ps,

                         ppl_Linear_Expression_t le, Value res)

{

  ppl_Coefficient_t num, denom;

  Value dv, nv;

  int maximum, err;

  value_init (nv);

  value_init (dv);

  ppl_new_Coefficient (&num);

  ppl_new_Coefficient (&denom);

  err = ppl_Pointset_Powerset_C_Polyhedron_maximize (ps, le, num, denom, &maximum);

  if (err > 0)

    {

      ppl_Coefficient_to_mpz_t (num, nv);

      ppl_Coefficient_to_mpz_t (denom, dv);

      gcc_assert (value_notzero_p (dv));

      value_division (res, nv, dv);

    }

  value_clear (nv);

  value_clear (dv);

  ppl_delete_Coefficient (num);

  ppl_delete_Coefficient (denom);

}

/* Return in RES the maximum of the linear expression LE on the

   polyhedron POL.  */

void

ppl_min_for_le_pointset (ppl_Pointset_Powerset_C_Polyhedron_t ps,

                         ppl_Linear_Expression_t le, Value res)

{

  ppl_Coefficient_t num, denom;

  Value dv, nv;

  int minimum, err;

  value_init (nv);

  value_init (dv);

  ppl_new_Coefficient (&num);

  ppl_new_Coefficient (&denom);

  err = ppl_Pointset_Powerset_C_Polyhedron_minimize (ps, le, num, denom, &minimum);

  if (err > 0)

    {

      ppl_Coefficient_to_mpz_t (num, nv);

      ppl_Coefficient_to_mpz_t (denom, dv);

      gcc_assert (value_notzero_p (dv));

      value_division (res, nv, dv);

    }

  value_clear (nv);

  value_clear (dv);

  ppl_delete_Coefficient (num);

  ppl_delete_Coefficient (denom);

}