Fix previous commit 'Speed up special cases of square-free...'.

[ginac.git] / ginac / normal.cpp

/** @file normal.cpp
 *
 *  This file implements several functions that work on univariate and
 *  multivariate polynomials and rational functions.
 *  These functions include polynomial quotient and remainder, GCD and LCM
 *  computation, square-free factorization and rational function normalization. */

/*
 *  GiNaC Copyright (C) 1999-2018 Johannes Gutenberg University Mainz, Germany
 *
 *  This program 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 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program 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 this program; if not, write to the Free Software
 *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */

#include "normal.h"
#include "basic.h"
#include "ex.h"
#include "add.h"
#include "constant.h"
#include "expairseq.h"
#include "fail.h"
#include "inifcns.h"
#include "lst.h"
#include "mul.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "operators.h"
#include "matrix.h"
#include "pseries.h"
#include "symbol.h"
#include "utils.h"
#include "polynomial/chinrem_gcd.h"

#include <algorithm>
#include <map>

namespace GiNaC {

// If comparing expressions (ex::compare()) is fast, you can set this to 1.
// Some routines like quo(), rem() and gcd() will then return a quick answer
// when they are called with two identical arguments.
#define FAST_COMPARE 1

// Set this if you want divide_in_z() to use remembering
#define USE_REMEMBER 0

// Set this if you want divide_in_z() to use trial division followed by
// polynomial interpolation (always slower except for completely dense
// polynomials)
#define USE_TRIAL_DIVISION 0

// Set this to enable some statistical output for the GCD routines
#define STATISTICS 0


#if STATISTICS
// Statistics variables
static int gcd_called = 0;
static int sr_gcd_called = 0;
static int heur_gcd_called = 0;
static int heur_gcd_failed = 0;

// Print statistics at end of program
static struct _stat_print {
        _stat_print() {}
        ~_stat_print() {
                std::cout << "gcd() called " << gcd_called << " times\n";
                std::cout << "sr_gcd() called " << sr_gcd_called << " times\n";
                std::cout << "heur_gcd() called " << heur_gcd_called << " times\n";
                std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
        }
} stat_print;
#endif


/** Return pointer to first symbol found in expression.  Due to GiNaC's
 *  internal ordering of terms, it may not be obvious which symbol this
 *  function returns for a given expression.
 *
 *  @param e  expression to search
 *  @param x  first symbol found (returned)
 *  @return "false" if no symbol was found, "true" otherwise */
static bool get_first_symbol(const ex &e, ex &x)
{
        if (is_a<symbol>(e)) {
                x = e;
                return true;
        } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
                for (size_t i=0; i<e.nops(); i++)
                        if (get_first_symbol(e.op(i), x))
                                return true;
        } else if (is_exactly_a<power>(e)) {
                if (get_first_symbol(e.op(0), x))
                        return true;
        }
        return false;
}


/*
 *  Statistical information about symbols in polynomials
 */

/** This structure holds information about the highest and lowest degrees
 *  in which a symbol appears in two multivariate polynomials "a" and "b".
 *  A vector of these structures with information about all symbols in
 *  two polynomials can be created with the function get_symbol_stats().
 *
 *  @see get_symbol_stats */
struct sym_desc {
        /** Initialize symbol, leave other variables uninitialized */
        sym_desc(const ex& s)
          : sym(s), deg_a(0), deg_b(0), ldeg_a(0), ldeg_b(0), max_deg(0), max_lcnops(0)
        { }

        /** Reference to symbol */
        ex sym;

        /** Highest degree of symbol in polynomial "a" */
        int deg_a;

        /** Highest degree of symbol in polynomial "b" */
        int deg_b;

        /** Lowest degree of symbol in polynomial "a" */
        int ldeg_a;

        /** Lowest degree of symbol in polynomial "b" */
        int ldeg_b;

        /** Maximum of deg_a and deg_b (Used for sorting) */
        int max_deg;

        /** Maximum number of terms of leading coefficient of symbol in both polynomials */
        size_t max_lcnops;

        /** Comparison operator for sorting */
        bool operator<(const sym_desc &x) const
        {
                if (max_deg == x.max_deg)
                        return max_lcnops < x.max_lcnops;
                else
                        return max_deg < x.max_deg;
        }
};

// Vector of sym_desc structures
typedef std::vector<sym_desc> sym_desc_vec;

// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
static void add_symbol(const ex &s, sym_desc_vec &v)
{
        for (auto & it : v)
                if (it.sym.is_equal(s))  // If it's already in there, don't add it a second time
                        return;

        v.push_back(sym_desc(s));
}

// Collect all symbols of an expression (used internally by get_symbol_stats())
static void collect_symbols(const ex &e, sym_desc_vec &v)
{
        if (is_a<symbol>(e)) {
                add_symbol(e, v);
        } else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
                for (size_t i=0; i<e.nops(); i++)
                        collect_symbols(e.op(i), v);
        } else if (is_exactly_a<power>(e)) {
                collect_symbols(e.op(0), v);
        }
}

/** Collect statistical information about symbols in polynomials.
 *  This function fills in a vector of "sym_desc" structs which contain
 *  information about the highest and lowest degrees of all symbols that
 *  appear in two polynomials. The vector is then sorted by minimum
 *  degree (lowest to highest). The information gathered by this
 *  function is used by the GCD routines to identify trivial factors
 *  and to determine which variable to choose as the main variable
 *  for GCD computation.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param v  vector of sym_desc structs (filled in) */
static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
{
        collect_symbols(a, v);
        collect_symbols(b, v);
        for (auto & it : v) {
                int deg_a = a.degree(it.sym);
                int deg_b = b.degree(it.sym);
                it.deg_a = deg_a;
                it.deg_b = deg_b;
                it.max_deg = std::max(deg_a, deg_b);
                it.max_lcnops = std::max(a.lcoeff(it.sym).nops(), b.lcoeff(it.sym).nops());
                it.ldeg_a = a.ldegree(it.sym);
                it.ldeg_b = b.ldegree(it.sym);
        }
        std::sort(v.begin(), v.end());

#if 0
        std::clog << "Symbols:\n";
        it = v.begin(); itend = v.end();
        while (it != itend) {
                std::clog << " " << it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl;
                std::clog << "  lcoeff_a=" << a.lcoeff(it->sym) << ", lcoeff_b=" << b.lcoeff(it->sym) << endl;
                ++it;
        }
#endif
}


/*
 *  Computation of LCM of denominators of coefficients of a polynomial
 */

// Compute LCM of denominators of coefficients by going through the
// expression recursively (used internally by lcm_of_coefficients_denominators())
static numeric lcmcoeff(const ex &e, const numeric &l)
{
        if (e.info(info_flags::rational))
                return lcm(ex_to<numeric>(e).denom(), l);
        else if (is_exactly_a<add>(e)) {
                numeric c = *_num1_p;
                for (size_t i=0; i<e.nops(); i++)
                        c = lcmcoeff(e.op(i), c);
                return lcm(c, l);
        } else if (is_exactly_a<mul>(e)) {
                numeric c = *_num1_p;
                for (size_t i=0; i<e.nops(); i++)
                        c *= lcmcoeff(e.op(i), *_num1_p);
                return lcm(c, l);
        } else if (is_exactly_a<power>(e)) {
                if (is_a<symbol>(e.op(0)))
                        return l;
                else
                        return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
        }
        return l;
}

/** Compute LCM of denominators of coefficients of a polynomial.
 *  Given a polynomial with rational coefficients, this function computes
 *  the LCM of the denominators of all coefficients. This can be used
 *  to bring a polynomial from Q[X] to Z[X].
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
        return lcmcoeff(e, *_num1_p);
}

/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
 *  determined LCM of the coefficient's denominators.
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @param lcm  LCM to multiply in */
static ex multiply_lcm(const ex &e, const numeric &lcm)
{
        if (lcm.is_equal(*_num1_p))
                // e * 1 -> e;
                return e;

        if (is_exactly_a<mul>(e)) {
                // (a*b*...)*lcm -> (a*lcma)*(b*lcmb)*...*(lcm/(lcma*lcmb*...))
                size_t num = e.nops();
                exvector v;
                v.reserve(num + 1);
                numeric lcm_accum = *_num1_p;
                for (size_t i=0; i<num; i++) {
                        numeric op_lcm = lcmcoeff(e.op(i), *_num1_p);
                        v.push_back(multiply_lcm(e.op(i), op_lcm));
                        lcm_accum *= op_lcm;
                }
                v.push_back(lcm / lcm_accum);
                return dynallocate<mul>(v);
        } else if (is_exactly_a<add>(e)) {
                // (a+b+...)*lcm -> a*lcm+b*lcm+...
                size_t num = e.nops();
                exvector v;
                v.reserve(num);
                for (size_t i=0; i<num; i++)
                        v.push_back(multiply_lcm(e.op(i), lcm));
                return dynallocate<add>(v);
        } else if (is_exactly_a<power>(e)) {
                if (!is_a<symbol>(e.op(0))) {
                        // (b^e)*lcm -> (b*lcm^(1/e))^e if lcm^(1/e) ∈ ℚ (i.e. not a float)
                        // but not for symbolic b, as evaluation would undo this again
                        numeric root_of_lcm = lcm.power(ex_to<numeric>(e.op(1)).inverse());
                        if (root_of_lcm.is_rational())
                                return pow(multiply_lcm(e.op(0), root_of_lcm), e.op(1));
                }
        }
        // can't recurse down into e
        return dynallocate<mul>(e, lcm);
}


/** Compute the integer content (= GCD of all numeric coefficients) of an
 *  expanded polynomial. For a polynomial with rational coefficients, this
 *  returns g/l where g is the GCD of the coefficients' numerators and l
 *  is the LCM of the coefficients' denominators.
 *
 *  @return integer content */
numeric ex::integer_content() const
{
        return bp->integer_content();
}

numeric basic::integer_content() const
{
        return *_num1_p;
}

numeric numeric::integer_content() const
{
        return abs(*this);
}

numeric add::integer_content() const
{
        numeric c = *_num0_p, l = *_num1_p;
        for (auto & it : seq) {
                GINAC_ASSERT(!is_exactly_a<numeric>(it.rest));
                GINAC_ASSERT(is_exactly_a<numeric>(it.coeff));
                c = gcd(ex_to<numeric>(it.coeff).numer(), c);
                l = lcm(ex_to<numeric>(it.coeff).denom(), l);
        }
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        c = gcd(ex_to<numeric>(overall_coeff).numer(), c);
        l = lcm(ex_to<numeric>(overall_coeff).denom(), l);
        return c/l;
}

numeric mul::integer_content() const
{
#ifdef DO_GINAC_ASSERT
        for (auto & it : seq) {
                GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(it)));
        }
#endif // def DO_GINAC_ASSERT
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        return abs(ex_to<numeric>(overall_coeff));
}


/*
 *  Polynomial quotients and remainders
 */

/** Quotient q(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return quotient of a and b in Q[x] */
ex quo(const ex &a, const ex &b, const ex &x, bool check_args)
{
        if (b.is_zero())
                throw(std::overflow_error("quo: division by zero"));
        if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
                return a / b;
#if FAST_COMPARE
        if (a.is_equal(b))
                return _ex1;
#endif
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));

        // Polynomial long division
        ex r = a.expand();
        if (r.is_zero())
                return r;
        int bdeg = b.degree(x);
        int rdeg = r.degree(x);
        ex blcoeff = b.expand().coeff(x, bdeg);
        bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
        exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(x, rdeg);
                if (blcoeff_is_numeric)
                        term = rcoeff / blcoeff;
                else {
                        if (!divide(rcoeff, blcoeff, term, false))
                                return dynallocate<fail>();
                }
                term *= pow(x, rdeg - bdeg);
                v.push_back(term);
                r -= (term * b).expand();
                if (r.is_zero())
                        break;
                rdeg = r.degree(x);
        }
        return dynallocate<add>(v);
}


/** Remainder r(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return remainder of a(x) and b(x) in Q[x] */
ex rem(const ex &a, const ex &b, const ex &x, bool check_args)
{
        if (b.is_zero())
                throw(std::overflow_error("rem: division by zero"));
        if (is_exactly_a<numeric>(a)) {
                if  (is_exactly_a<numeric>(b))
                        return _ex0;
                else
                        return a;
        }
#if FAST_COMPARE
        if (a.is_equal(b))
                return _ex0;
#endif
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));

        // Polynomial long division
        ex r = a.expand();
        if (r.is_zero())
                return r;
        int bdeg = b.degree(x);
        int rdeg = r.degree(x);
        ex blcoeff = b.expand().coeff(x, bdeg);
        bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(x, rdeg);
                if (blcoeff_is_numeric)
                        term = rcoeff / blcoeff;
                else {
                        if (!divide(rcoeff, blcoeff, term, false))
                                return dynallocate<fail>();
                }
                term *= pow(x, rdeg - bdeg);
                r -= (term * b).expand();
                if (r.is_zero())
                        break;
                rdeg = r.degree(x);
        }
        return r;
}


/** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x)
 *  with degree(n, x) < degree(D, x).
 *
 *  @param a rational function in x
 *  @param x a is a function of x
 *  @return decomposed function. */
ex decomp_rational(const ex &a, const ex &x)
{
        ex nd = numer_denom(a);
        ex numer = nd.op(0), denom = nd.op(1);
        ex q = quo(numer, denom, x);
        if (is_exactly_a<fail>(q))
                return a;
        else
                return q + rem(numer, denom, x) / denom;
}


/** Pseudo-remainder of polynomials a(x) and b(x) in Q[x].
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return pseudo-remainder of a(x) and b(x) in Q[x] */
ex prem(const ex &a, const ex &b, const ex &x, bool check_args)
{
        if (b.is_zero())
                throw(std::overflow_error("prem: division by zero"));
        if (is_exactly_a<numeric>(a)) {
                if (is_exactly_a<numeric>(b))
                        return _ex0;
                else
                        return b;
        }
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));

        // Polynomial long division
        ex r = a.expand();
        ex eb = b.expand();
        int rdeg = r.degree(x);
        int bdeg = eb.degree(x);
        ex blcoeff;
        if (bdeg <= rdeg) {
                blcoeff = eb.coeff(x, bdeg);
                if (bdeg == 0)
                        eb = _ex0;
                else
                        eb -= blcoeff * pow(x, bdeg);
        } else
                blcoeff = _ex1;

        int delta = rdeg - bdeg + 1, i = 0;
        while (rdeg >= bdeg && !r.is_zero()) {
                ex rlcoeff = r.coeff(x, rdeg);
                ex term = (pow(x, rdeg - bdeg) * eb * rlcoeff).expand();
                if (rdeg == 0)
                        r = _ex0;
                else
                        r -= rlcoeff * pow(x, rdeg);
                r = (blcoeff * r).expand() - term;
                rdeg = r.degree(x);
                i++;
        }
        return pow(blcoeff, delta - i) * r;
}


/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Q[x].
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return sparse pseudo-remainder of a(x) and b(x) in Q[x] */
ex sprem(const ex &a, const ex &b, const ex &x, bool check_args)
{
        if (b.is_zero())
                throw(std::overflow_error("prem: division by zero"));
        if (is_exactly_a<numeric>(a)) {
                if (is_exactly_a<numeric>(b))
                        return _ex0;
                else
                        return b;
        }
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));

        // Polynomial long division
        ex r = a.expand();
        ex eb = b.expand();
        int rdeg = r.degree(x);
        int bdeg = eb.degree(x);
        ex blcoeff;
        if (bdeg <= rdeg) {
                blcoeff = eb.coeff(x, bdeg);
                if (bdeg == 0)
                        eb = _ex0;
                else
                        eb -= blcoeff * pow(x, bdeg);
        } else
                blcoeff = _ex1;

        while (rdeg >= bdeg && !r.is_zero()) {
                ex rlcoeff = r.coeff(x, rdeg);
                ex term = (pow(x, rdeg - bdeg) * eb * rlcoeff).expand();
                if (rdeg == 0)
                        r = _ex0;
                else
                        r -= rlcoeff * pow(x, rdeg);
                r = (blcoeff * r).expand() - term;
                rdeg = r.degree(x);
        }
        return r;
}


/** Exact polynomial division of a(X) by b(X) in Q[X].
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return "true" when exact division succeeds (quotient returned in q),
 *          "false" otherwise (q left untouched) */
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
        if (b.is_zero())
                throw(std::overflow_error("divide: division by zero"));
        if (a.is_zero()) {
                q = _ex0;
                return true;
        }
        if (is_exactly_a<numeric>(b)) {
                q = a / b;
                return true;
        } else if (is_exactly_a<numeric>(a))
                return false;
#if FAST_COMPARE
        if (a.is_equal(b)) {
                q = _ex1;
                return true;
        }
#endif
        if (check_args && (!a.info(info_flags::rational_polynomial) ||
                           !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));

        // Find first symbol
        ex x;
        if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
                throw(std::invalid_argument("invalid expression in divide()"));

        // Try to avoid expanding partially factored expressions.
        if (is_exactly_a<mul>(b)) {
        // Divide sequentially by each term
                ex rem_new, rem_old = a;
                for (size_t i=0; i < b.nops(); i++) {
                        if (! divide(rem_old, b.op(i), rem_new, false))
                                return false;
                        rem_old = rem_new;
                }
                q = rem_new;
                return true;
        } else if (is_exactly_a<power>(b)) {
                const ex& bb(b.op(0));
                int exp_b = ex_to<numeric>(b.op(1)).to_int();
                ex rem_new, rem_old = a;
                for (int i=exp_b; i>0; i--) {
                        if (! divide(rem_old, bb, rem_new, false))
                                return false;
                        rem_old = rem_new;
                }
                q = rem_new;
                return true;
        } 
        
        if (is_exactly_a<mul>(a)) {
                // Divide sequentially each term. If some term in a is divisible 
                // by b we are done... and if not, we can't really say anything.
                size_t i;
                ex rem_i;
                bool divisible_p = false;
                for (i=0; i < a.nops(); ++i) {
                        if (divide(a.op(i), b, rem_i, false)) {
                                divisible_p = true;
                                break;
                        }
                }
                if (divisible_p) {
                        exvector resv;
                        resv.reserve(a.nops());
                        for (size_t j=0; j < a.nops(); j++) {
                                if (j==i)
                                        resv.push_back(rem_i);
                                else
                                        resv.push_back(a.op(j));
                        }
                        q = dynallocate<mul>(resv);
                        return true;
                }
        } else if (is_exactly_a<power>(a)) {
                // The base itself might be divisible by b, in that case we don't
                // need to expand a
                const ex& ab(a.op(0));
                int a_exp = ex_to<numeric>(a.op(1)).to_int();
                ex rem_i;
                if (divide(ab, b, rem_i, false)) {
                        q = rem_i * pow(ab, a_exp - 1);
                        return true;
                }
// code below is commented-out because it leads to a significant slowdown
//              for (int i=2; i < a_exp; i++) {
//                      if (divide(power(ab, i), b, rem_i, false)) {
//                              q = rem_i*power(ab, a_exp - i);
//                              return true;
//                      }
//              } // ... so we *really* need to expand expression.
        }
        
        // Polynomial long division (recursive)
        ex r = a.expand();
        if (r.is_zero()) {
                q = _ex0;
                return true;
        }
        int bdeg = b.degree(x);
        int rdeg = r.degree(x);
        ex blcoeff = b.expand().coeff(x, bdeg);
        bool blcoeff_is_numeric = is_exactly_a<numeric>(blcoeff);
        exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(x, rdeg);
                if (blcoeff_is_numeric)
                        term = rcoeff / blcoeff;
                else
                        if (!divide(rcoeff, blcoeff, term, false))
                                return false;
                term *= pow(x, rdeg - bdeg);
                v.push_back(term);
                r -= (term * b).expand();
                if (r.is_zero()) {
                        q = dynallocate<add>(v);
                        return true;
                }
                rdeg = r.degree(x);
        }
        return false;
}


#if USE_REMEMBER
/*
 *  Remembering
 */

typedef std::pair<ex, ex> ex2;
typedef std::pair<ex, bool> exbool;

struct ex2_less {
        bool operator() (const ex2 &p, const ex2 &q) const 
        {
                int cmp = p.first.compare(q.first);
                return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
        }
};

typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
#endif


/** Exact polynomial division of a(X) by b(X) in Z[X].
 *  This functions works like divide() but the input and output polynomials are
 *  in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
 *  divide(), it doesn't check whether the input polynomials really are integer
 *  polynomials, so be careful of what you pass in. Also, you have to run
 *  get_symbol_stats() over the input polynomials before calling this function
 *  and pass an iterator to the first element of the sym_desc vector. This
 *  function is used internally by the heur_gcd().
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param var  iterator to first element of vector of sym_desc structs
 *  @return "true" when exact division succeeds (the quotient is returned in
 *          q), "false" otherwise.
 *  @see get_symbol_stats, heur_gcd */
static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
{
        q = _ex0;
        if (b.is_zero())
                throw(std::overflow_error("divide_in_z: division by zero"));
        if (b.is_equal(_ex1)) {
                q = a;
                return true;
        }
        if (is_exactly_a<numeric>(a)) {
                if (is_exactly_a<numeric>(b)) {
                        q = a / b;
                        return q.info(info_flags::integer);
                } else
                        return false;
        }
#if FAST_COMPARE
        if (a.is_equal(b)) {
                q = _ex1;
                return true;
        }
#endif

#if USE_REMEMBER
        // Remembering
        static ex2_exbool_remember dr_remember;
        ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
        if (remembered != dr_remember.end()) {
                q = remembered->second.first;
                return remembered->second.second;
        }
#endif

        if (is_exactly_a<power>(b)) {
                const ex& bb(b.op(0));
                ex qbar = a;
                int exp_b = ex_to<numeric>(b.op(1)).to_int();
                for (int i=exp_b; i>0; i--) {
                        if (!divide_in_z(qbar, bb, q, var))
                                return false;
                        qbar = q;
                }
                return true;
        }

        if (is_exactly_a<mul>(b)) {
                ex qbar = a;
                for (const auto & it : b) {
                        sym_desc_vec sym_stats;
                        get_symbol_stats(a, it, sym_stats);
                        if (!divide_in_z(qbar, it, q, sym_stats.begin()))
                                return false;

                        qbar = q;
                }
                return true;
        }

        // Main symbol
        const ex &x = var->sym;

        // Compare degrees
        int adeg = a.degree(x), bdeg = b.degree(x);
        if (bdeg > adeg)
                return false;

#if USE_TRIAL_DIVISION

        // Trial division with polynomial interpolation
        int i, k;

        // Compute values at evaluation points 0..adeg
        vector<numeric> alpha; alpha.reserve(adeg + 1);
        exvector u; u.reserve(adeg + 1);
        numeric point = *_num0_p;
        ex c;
        for (i=0; i<=adeg; i++) {
                ex bs = b.subs(x == point, subs_options::no_pattern);
                while (bs.is_zero()) {
                        point += *_num1_p;
                        bs = b.subs(x == point, subs_options::no_pattern);
                }
                if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1))
                        return false;
                alpha.push_back(point);
                u.push_back(c);
                point += *_num1_p;
        }

        // Compute inverses
        vector<numeric> rcp; rcp.reserve(adeg + 1);
        rcp.push_back(*_num0_p);
        for (k=1; k<=adeg; k++) {
                numeric product = alpha[k] - alpha[0];
                for (i=1; i<k; i++)
                        product *= alpha[k] - alpha[i];
                rcp.push_back(product.inverse());
        }

        // Compute Newton coefficients
        exvector v; v.reserve(adeg + 1);
        v.push_back(u[0]);
        for (k=1; k<=adeg; k++) {
                ex temp = v[k - 1];
                for (i=k-2; i>=0; i--)
                        temp = temp * (alpha[k] - alpha[i]) + v[i];
                v.push_back((u[k] - temp) * rcp[k]);
        }

        // Convert from Newton form to standard form
        c = v[adeg];
        for (k=adeg-1; k>=0; k--)
                c = c * (x - alpha[k]) + v[k];

        if (c.degree(x) == (adeg - bdeg)) {
                q = c.expand();
                return true;
        } else
                return false;

#else

        // Polynomial long division (recursive)
        ex r = a.expand();
        if (r.is_zero())
                return true;
        int rdeg = adeg;
        ex eb = b.expand();
        ex blcoeff = eb.coeff(x, bdeg);
        exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(x, rdeg);
                if (!divide_in_z(rcoeff, blcoeff, term, var+1))
                        break;
                term = (term * pow(x, rdeg - bdeg)).expand();
                v.push_back(term);
                r -= (term * eb).expand();
                if (r.is_zero()) {
                        q = dynallocate<add>(v);
#if USE_REMEMBER
                        dr_remember[ex2(a, b)] = exbool(q, true);
#endif
                        return true;
                }
                rdeg = r.degree(x);
        }
#if USE_REMEMBER
        dr_remember[ex2(a, b)] = exbool(q, false);
#endif
        return false;

#endif
}


/*
 *  Separation of unit part, content part and primitive part of polynomials
 */

/** Compute unit part (= sign of leading coefficient) of a multivariate
 *  polynomial in Q[x]. The product of unit part, content part, and primitive
 *  part is the polynomial itself.
 *
 *  @param x  main variable
 *  @return unit part
 *  @see ex::content, ex::primpart, ex::unitcontprim */
ex ex::unit(const ex &x) const
{
        ex c = expand().lcoeff(x);
        if (is_exactly_a<numeric>(c))
                return c.info(info_flags::negative) ?_ex_1 : _ex1;
        else {
                ex y;
                if (get_first_symbol(c, y))
                        return c.unit(y);
                else
                        throw(std::invalid_argument("invalid expression in unit()"));
        }
}


/** Compute content part (= unit normal GCD of all coefficients) of a
 *  multivariate polynomial in Q[x]. The product of unit part, content part,
 *  and primitive part is the polynomial itself.
 *
 *  @param x  main variable
 *  @return content part
 *  @see ex::unit, ex::primpart, ex::unitcontprim */
ex ex::content(const ex &x) const
{
        if (is_exactly_a<numeric>(*this))
                return info(info_flags::negative) ? -*this : *this;

        ex e = expand();
        if (e.is_zero())
                return _ex0;

        // First, divide out the integer content (which we can calculate very efficiently).
        // If the leading coefficient of the quotient is an integer, we are done.
        ex c = e.integer_content();
        ex r = e / c;
        int deg = r.degree(x);
        ex lcoeff = r.coeff(x, deg);
        if (lcoeff.info(info_flags::integer))
                return c;

        // GCD of all coefficients
        int ldeg = r.ldegree(x);
        if (deg == ldeg)
                return lcoeff * c / lcoeff.unit(x);
        ex cont = _ex0;
        for (int i=ldeg; i<=deg; i++)
                cont = gcd(r.coeff(x, i), cont, nullptr, nullptr, false);
        return cont * c;
}


/** Compute primitive part of a multivariate polynomial in Q[x]. The result
 *  will be a unit-normal polynomial with a content part of 1. The product
 *  of unit part, content part, and primitive part is the polynomial itself.
 *
 *  @param x  main variable
 *  @return primitive part
 *  @see ex::unit, ex::content, ex::unitcontprim */
ex ex::primpart(const ex &x) const
{
        // We need to compute the unit and content anyway, so call unitcontprim()
        ex u, c, p;
        unitcontprim(x, u, c, p);
        return p;
}


/** Compute primitive part of a multivariate polynomial in Q[x] when the
 *  content part is already known. This function is faster in computing the
 *  primitive part than the previous function.
 *
 *  @param x  main variable
 *  @param c  previously computed content part
 *  @return primitive part */
ex ex::primpart(const ex &x, const ex &c) const
{
        if (is_zero() || c.is_zero())
                return _ex0;
        if (is_exactly_a<numeric>(*this))
                return _ex1;

        // Divide by unit and content to get primitive part
        ex u = unit(x);
        if (is_exactly_a<numeric>(c))
                return *this / (c * u);
        else
                return quo(*this, c * u, x, false);
}


/** Compute unit part, content part, and primitive part of a multivariate
 *  polynomial in Q[x]. The product of the three parts is the polynomial
 *  itself.
 *
 *  @param x  main variable
 *  @param u  unit part (returned)
 *  @param c  content part (returned)
 *  @param p  primitive part (returned)
 *  @see ex::unit, ex::content, ex::primpart */
void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const
{
        // Quick check for zero (avoid expanding)
        if (is_zero()) {
                u = _ex1;
                c = p = _ex0;
                return;
        }

        // Special case: input is a number
        if (is_exactly_a<numeric>(*this)) {
                if (info(info_flags::negative)) {
                        u = _ex_1;
                        c = abs(ex_to<numeric>(*this));
                } else {
                        u = _ex1;
                        c = *this;
                }
                p = _ex1;
                return;
        }

        // Expand input polynomial
        ex e = expand();
        if (e.is_zero()) {
                u = _ex1;
                c = p = _ex0;
                return;
        }

        // Compute unit and content
        u = unit(x);
        c = content(x);

        // Divide by unit and content to get primitive part
        if (c.is_zero()) {
                p = _ex0;
                return;
        }
        if (is_exactly_a<numeric>(c))
                p = *this / (c * u);
        else
                p = quo(e, c * u, x, false);
}


/*
 *  GCD of multivariate polynomials
 */

/** Compute GCD of multivariate polynomials using the subresultant PRS
 *  algorithm. This function is used internally by gcd().
 *
 *  @param a   first multivariate polynomial
 *  @param b   second multivariate polynomial
 *  @param var iterator to first element of vector of sym_desc structs
 *  @return the GCD as a new expression
 *  @see gcd */

static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
{
#if STATISTICS
        sr_gcd_called++;
#endif

        // The first symbol is our main variable
        const ex &x = var->sym;

        // Sort c and d so that c has higher degree
        ex c, d;
        int adeg = a.degree(x), bdeg = b.degree(x);
        int cdeg, ddeg;
        if (adeg >= bdeg) {
                c = a;
                d = b;
                cdeg = adeg;
                ddeg = bdeg;
        } else {
                c = b;
                d = a;
                cdeg = bdeg;
                ddeg = adeg;
        }

        // Remove content from c and d, to be attached to GCD later
        ex cont_c = c.content(x);
        ex cont_d = d.content(x);
        ex gamma = gcd(cont_c, cont_d, nullptr, nullptr, false);
        if (ddeg == 0)
                return gamma;
        c = c.primpart(x, cont_c);
        d = d.primpart(x, cont_d);

        // First element of subresultant sequence
        ex r = _ex0, ri = _ex1, psi = _ex1;
        int delta = cdeg - ddeg;

        for (;;) {

                // Calculate polynomial pseudo-remainder
                r = prem(c, d, x, false);
                if (r.is_zero())
                        return gamma * d.primpart(x);

                c = d;
                cdeg = ddeg;
                if (!divide_in_z(r, ri * pow(psi, delta), d, var))
                        throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
                ddeg = d.degree(x);
                if (ddeg == 0) {
                        if (is_exactly_a<numeric>(r))
                                return gamma;
                        else
                                return gamma * r.primpart(x);
                }

                // Next element of subresultant sequence
                ri = c.expand().lcoeff(x);
                if (delta == 1)
                        psi = ri;
                else if (delta)
                        divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
                delta = cdeg - ddeg;
        }
}


/** Return maximum (absolute value) coefficient of a polynomial.
 *  This function is used internally by heur_gcd().
 *
 *  @return maximum coefficient
 *  @see heur_gcd */
numeric ex::max_coefficient() const
{
        return bp->max_coefficient();
}

/** Implementation ex::max_coefficient().
 *  @see heur_gcd */
numeric basic::max_coefficient() const
{
        return *_num1_p;
}

numeric numeric::max_coefficient() const
{
        return abs(*this);
}

numeric add::max_coefficient() const
{
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        numeric cur_max = abs(ex_to<numeric>(overall_coeff));
        for (auto & it : seq) {
                numeric a;
                GINAC_ASSERT(!is_exactly_a<numeric>(it.rest));
                a = abs(ex_to<numeric>(it.coeff));
                if (a > cur_max)
                        cur_max = a;
        }
        return cur_max;
}

numeric mul::max_coefficient() const
{
#ifdef DO_GINAC_ASSERT
        for (auto & it : seq) {
                GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(it)));
        }
#endif // def DO_GINAC_ASSERT
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        return abs(ex_to<numeric>(overall_coeff));
}


/** Apply symmetric modular homomorphism to an expanded multivariate
 *  polynomial.  This function is usually used internally by heur_gcd().
 *
 *  @param xi  modulus
 *  @return mapped polynomial
 *  @see heur_gcd */
ex basic::smod(const numeric &xi) const
{
        return *this;
}

ex numeric::smod(const numeric &xi) const
{
        return GiNaC::smod(*this, xi);
}

ex add::smod(const numeric &xi) const
{
        epvector newseq;
        newseq.reserve(seq.size()+1);
        for (auto & it : seq) {
                GINAC_ASSERT(!is_exactly_a<numeric>(it.rest));
                numeric coeff = GiNaC::smod(ex_to<numeric>(it.coeff), xi);
                if (!coeff.is_zero())
                        newseq.push_back(expair(it.rest, coeff));
        }
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
        return dynallocate<add>(std::move(newseq), coeff);
}

ex mul::smod(const numeric &xi) const
{
#ifdef DO_GINAC_ASSERT
        for (auto & it : seq) {
                GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(it)));
        }
#endif // def DO_GINAC_ASSERT
        mul & mulcopy = dynallocate<mul>(*this);
        GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
        mulcopy.overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
        mulcopy.clearflag(status_flags::evaluated);
        mulcopy.clearflag(status_flags::hash_calculated);
        return mulcopy;
}


/** xi-adic polynomial interpolation */
static ex interpolate(const ex &gamma, const numeric &xi, const ex &x, int degree_hint = 1)
{
        exvector g; g.reserve(degree_hint);
        ex e = gamma;
        numeric rxi = xi.inverse();
        for (int i=0; !e.is_zero(); i++) {
                ex gi = e.smod(xi);
                g.push_back(gi * pow(x, i));
                e = (e - gi) * rxi;
        }
        return dynallocate<add>(g);
}

/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};

/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
 *  get_symbol_stats() must have been called previously with the input
 *  polynomials and an iterator to the first element of the sym_desc vector
 *  passed in. This function is used internally by gcd().
 *
 *  @param a  first integer multivariate polynomial (expanded)
 *  @param b  second integer multivariate polynomial (expanded)
 *  @param ca  cofactor of polynomial a (returned), nullptr to suppress
 *             calculation of cofactor
 *  @param cb  cofactor of polynomial b (returned), nullptr to suppress
 *             calculation of cofactor
 *  @param var iterator to first element of vector of sym_desc structs
 *  @param res the GCD (returned)
 *  @return true if GCD was computed, false otherwise.
 *  @see gcd
 *  @exception gcdheu_failed() */
static bool heur_gcd_z(ex& res, const ex &a, const ex &b, ex *ca, ex *cb,
                       sym_desc_vec::const_iterator var)
{
#if STATISTICS
        heur_gcd_called++;
#endif

        // Algorithm only works for non-vanishing input polynomials
        if (a.is_zero() || b.is_zero())
                return false;

        // GCD of two numeric values -> CLN
        if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
                numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
                if (ca)
                        *ca = ex_to<numeric>(a) / g;
                if (cb)
                        *cb = ex_to<numeric>(b) / g;
                res = g;
                return true;
        }

        // The first symbol is our main variable
        const ex &x = var->sym;

        // Remove integer content
        numeric gc = gcd(a.integer_content(), b.integer_content());
        numeric rgc = gc.inverse();
        ex p = a * rgc;
        ex q = b * rgc;
        int maxdeg =  std::max(p.degree(x), q.degree(x));
        
        // Find evaluation point
        numeric mp = p.max_coefficient();
        numeric mq = q.max_coefficient();
        numeric xi;
        if (mp > mq)
                xi = mq * (*_num2_p) + (*_num2_p);
        else
                xi = mp * (*_num2_p) + (*_num2_p);

        // 6 tries maximum
        for (int t=0; t<6; t++) {
                if (xi.int_length() * maxdeg > 100000) {
                        throw gcdheu_failed();
                }

                // Apply evaluation homomorphism and calculate GCD
                ex cp, cq;
                ex gamma;
                bool found = heur_gcd_z(gamma,
                                        p.subs(x == xi, subs_options::no_pattern),
                                        q.subs(x == xi, subs_options::no_pattern),
                                        &cp, &cq, var+1);
                if (found) {
                        gamma = gamma.expand();
                        // Reconstruct polynomial from GCD of mapped polynomials
                        ex g = interpolate(gamma, xi, x, maxdeg);

                        // Remove integer content
                        g /= g.integer_content();

                        // If the calculated polynomial divides both p and q, this is the GCD
                        ex dummy;
                        if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
                                g *= gc;
                                res = g;
                                return true;
                        }
                }

                // Next evaluation point
                xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
        }
        return false;
}

/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
 *  get_symbol_stats() must have been called previously with the input
 *  polynomials and an iterator to the first element of the sym_desc vector
 *  passed in. This function is used internally by gcd().
 *
 *  @param a  first rational multivariate polynomial (expanded)
 *  @param b  second rational multivariate polynomial (expanded)
 *  @param ca  cofactor of polynomial a (returned), nullptr to suppress
 *             calculation of cofactor
 *  @param cb  cofactor of polynomial b (returned), nullptr to suppress
 *             calculation of cofactor
 *  @param var iterator to first element of vector of sym_desc structs
 *  @param res the GCD (returned)
 *  @return true if GCD was computed, false otherwise.
 *  @see heur_gcd_z
 *  @see gcd
 */
static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb,
                     sym_desc_vec::const_iterator var)
{
        if (a.info(info_flags::integer_polynomial) && 
            b.info(info_flags::integer_polynomial)) {
                try {
                        return heur_gcd_z(res, a, b, ca, cb, var);
                } catch (gcdheu_failed) {
                        return false;
                }
        }

        // convert polynomials to Z[X]
        const numeric a_lcm = lcm_of_coefficients_denominators(a);
        const numeric ab_lcm = lcmcoeff(b, a_lcm);

        const ex ai = a*ab_lcm;
        const ex bi = b*ab_lcm;
        if (!ai.info(info_flags::integer_polynomial))
                throw std::logic_error("heur_gcd: not an integer polynomial [1]");

        if (!bi.info(info_flags::integer_polynomial))
                throw std::logic_error("heur_gcd: not an integer polynomial [2]");

        bool found = false;
        try {
                found = heur_gcd_z(res, ai, bi, ca, cb, var);
        } catch (gcdheu_failed) {
                return false;
        }
        
        // GCD is not unique, it's defined up to a unit (i.e. invertible
        // element). If the coefficient ring is a field, every its element is
        // invertible, so one can multiply the polynomial GCD with any element
        // of the coefficient field. We use this ambiguity to make cofactors
        // integer polynomials.
        if (found)
                res /= ab_lcm;
        return found;
}


// gcd helper to handle partially factored polynomials (to avoid expanding
// large expressions). At least one of the arguments should be a power.
static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb);

// gcd helper to handle partially factored polynomials (to avoid expanding
// large expressions). At least one of the arguments should be a product.
static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb);

/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
 *  and b(X) in Z[X]. Optionally also compute the cofactors of a and b,
 *  defined by a = ca * gcd(a, b) and b = cb * gcd(a, b).
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param ca pointer to expression that will receive the cofactor of a, or nullptr
 *  @param cb pointer to expression that will receive the cofactor of b, or nullptr
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the GCD as a new expression */
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args, unsigned options)
{
#if STATISTICS
        gcd_called++;
#endif

        // GCD of numerics -> CLN
        if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
                numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
                if (ca || cb) {
                        if (g.is_zero()) {
                                if (ca)
                                        *ca = _ex0;
                                if (cb)
                                        *cb = _ex0;
                        } else {
                                if (ca)
                                        *ca = ex_to<numeric>(a) / g;
                                if (cb)
                                        *cb = ex_to<numeric>(b) / g;
                        }
                }
                return g;
        }

        // Check arguments
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) {
                throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
        }

        // Partially factored cases (to avoid expanding large expressions)
        if (!(options & gcd_options::no_part_factored)) {
                if (is_exactly_a<mul>(a) || is_exactly_a<mul>(b))
                        return gcd_pf_mul(a, b, ca, cb);
#if FAST_COMPARE
                if (is_exactly_a<power>(a) || is_exactly_a<power>(b))
                        return gcd_pf_pow(a, b, ca, cb);
#endif
        }

        // Some trivial cases
        ex aex = a.expand(), bex = b.expand();
        if (aex.is_zero()) {
                if (ca)
                        *ca = _ex0;
                if (cb)
                        *cb = _ex1;
                return b;
        }
        if (bex.is_zero()) {
                if (ca)
                        *ca = _ex1;
                if (cb)
                        *cb = _ex0;
                return a;
        }
        if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
                if (ca)
                        *ca = a;
                if (cb)
                        *cb = b;
                return _ex1;
        }
#if FAST_COMPARE
        if (a.is_equal(b)) {
                if (ca)
                        *ca = _ex1;
                if (cb)
                        *cb = _ex1;
                return a;
        }
#endif

        if (is_a<symbol>(aex)) {
                if (! bex.subs(aex==_ex0, subs_options::no_pattern).is_zero()) {
                        if (ca)
                                *ca = a;
                        if (cb)
                                *cb = b;
                        return _ex1;
                }
        }

        if (is_a<symbol>(bex)) {
                if (! aex.subs(bex==_ex0, subs_options::no_pattern).is_zero()) {
                        if (ca)
                                *ca = a;
                        if (cb)
                                *cb = b;
                        return _ex1;
                }
        }

        if (is_exactly_a<numeric>(aex)) {
                numeric bcont = bex.integer_content();
                numeric g = gcd(ex_to<numeric>(aex), bcont);
                if (ca)
                        *ca = ex_to<numeric>(aex)/g;
                if (cb)
                        *cb = bex/g;
                return g;
        }

        if (is_exactly_a<numeric>(bex)) {
                numeric acont = aex.integer_content();
                numeric g = gcd(ex_to<numeric>(bex), acont);
                if (ca)
                        *ca = aex/g;
                if (cb)
                        *cb = ex_to<numeric>(bex)/g;
                return g;
        }

        // Gather symbol statistics
        sym_desc_vec sym_stats;
        get_symbol_stats(a, b, sym_stats);

        // The symbol with least degree which is contained in both polynomials
        // is our main variable
        auto vari = sym_stats.begin();
        while ((vari != sym_stats.end()) && 
               (((vari->ldeg_b == 0) && (vari->deg_b == 0)) ||
                ((vari->ldeg_a == 0) && (vari->deg_a == 0))))
                vari++;

        // No common symbols at all, just return 1:
        if (vari == sym_stats.end()) {
                // N.B: keep cofactors factored
                if (ca)
                        *ca = a;
                if (cb)
                        *cb = b;
                return _ex1;
        }
        // move symbols which contained only in one of the polynomials
        // to the end:
        rotate(sym_stats.begin(), vari, sym_stats.end());

        sym_desc_vec::const_iterator var = sym_stats.begin();
        const ex &x = var->sym;

        // Cancel trivial common factor
        int ldeg_a = var->ldeg_a;
        int ldeg_b = var->ldeg_b;
        int min_ldeg = std::min(ldeg_a,ldeg_b);
        if (min_ldeg > 0) {
                ex common = pow(x, min_ldeg);
                return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
        }

        // Try to eliminate variables
        if (var->deg_a == 0 && var->deg_b != 0 ) {
                ex bex_u, bex_c, bex_p;
                bex.unitcontprim(x, bex_u, bex_c, bex_p);
                ex g = gcd(aex, bex_c, ca, cb, false);
                if (cb)
                        *cb *= bex_u * bex_p;
                return g;
        } else if (var->deg_b == 0 && var->deg_a != 0) {
                ex aex_u, aex_c, aex_p;
                aex.unitcontprim(x, aex_u, aex_c, aex_p);
                ex g = gcd(aex_c, bex, ca, cb, false);
                if (ca)
                        *ca *= aex_u * aex_p;
                return g;
        }

        // Try heuristic algorithm first, fall back to PRS if that failed
        ex g;
        if (!(options & gcd_options::no_heur_gcd)) {
                bool found = heur_gcd(g, aex, bex, ca, cb, var);
                if (found) {
                        // heur_gcd have already computed cofactors...
                        if (g.is_equal(_ex1)) {
                                // ... but we want to keep them factored if possible.
                                if (ca)
                                        *ca = a;
                                if (cb)
                                        *cb = b;
                        }
                        return g;
                }
#if STATISTICS
                else {
                        heur_gcd_failed++;
                }
#endif
        }
        if (options & gcd_options::use_sr_gcd) {
                g = sr_gcd(aex, bex, var);
        } else {
                exvector vars;
                for (std::size_t n = sym_stats.size(); n-- != 0; )
                        vars.push_back(sym_stats[n].sym);
                g = chinrem_gcd(aex, bex, vars);
        }

        if (g.is_equal(_ex1)) {
                // Keep cofactors factored if possible
                if (ca)
                        *ca = a;
                if (cb)
                        *cb = b;
        } else {
                if (ca)
                        divide(aex, g, *ca, false);
                if (cb)
                        divide(bex, g, *cb, false);
        }
        return g;
}

// gcd helper to handle partially factored polynomials (to avoid expanding
// large expressions). Both arguments should be powers.
static ex gcd_pf_pow_pow(const ex& a, const ex& b, ex* ca, ex* cb)
{
        ex p = a.op(0);
        const ex& exp_a = a.op(1);
        ex pb = b.op(0);
        const ex& exp_b = b.op(1);

        // a = p^n, b = p^m, gcd = p^min(n, m)
        if (p.is_equal(pb)) {
                if (exp_a < exp_b) {
                        if (ca)
                                *ca = _ex1;
                        if (cb)
                                *cb = pow(p, exp_b - exp_a);
                        return pow(p, exp_a);
                } else {
                        if (ca)
                                *ca = pow(p, exp_a - exp_b);
                        if (cb)
                                *cb = _ex1;
                        return pow(p, exp_b);
                }
        }

        ex p_co, pb_co;
        ex p_gcd = gcd(p, pb, &p_co, &pb_co, false);
        // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==> gcd(a,b) = 1
        if (p_gcd.is_equal(_ex1)) {
                        if (ca)
                                *ca = a;
                        if (cb)
                                *cb = b;
                        return _ex1;
                        // XXX: do I need to check for p_gcd = -1?
        }

        // there are common factors:
        // a(x) = g(x)^n A(x)^n, b(x) = g(x)^m B(x)^m ==>
        // gcd(a, b) = g(x)^n gcd(A(x)^n, g(x)^(n-m) B(x)^m
        if (exp_a < exp_b) {
                ex pg =  gcd(pow(p_co, exp_a), pow(p_gcd, exp_b-exp_a)*pow(pb_co, exp_b), ca, cb, false);
                return pow(p_gcd, exp_a)*pg;
        } else {
                ex pg = gcd(pow(p_gcd, exp_a - exp_b)*pow(p_co, exp_a), pow(pb_co, exp_b), ca, cb, false);
                return pow(p_gcd, exp_b)*pg;
        }
}

static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb)
{
        if (is_exactly_a<power>(a) && is_exactly_a<power>(b))
                return gcd_pf_pow_pow(a, b, ca, cb);

        if (is_exactly_a<power>(b) && (! is_exactly_a<power>(a)))
                return gcd_pf_pow(b, a, cb, ca);

        GINAC_ASSERT(is_exactly_a<power>(a));

        ex p = a.op(0);
        const ex& exp_a = a.op(1);
        if (p.is_equal(b)) {
                // a = p^n, b = p, gcd = p
                if (ca)
                        *ca = pow(p, a.op(1) - 1);
                if (cb)
                        *cb = _ex1;
                return p;
        } 

        ex p_co, bpart_co;
        ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);

        // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
        if (p_gcd.is_equal(_ex1)) {
                if (ca)
                        *ca = a;
                if (cb)
                        *cb = b;
                return _ex1;
        }
        // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
        ex rg = gcd(pow(p_gcd, exp_a-1)*pow(p_co, exp_a), bpart_co, ca, cb, false);
        return p_gcd*rg;
}

static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb)
{
        if (is_exactly_a<mul>(a) && is_exactly_a<mul>(b)
                                 && (b.nops() >  a.nops()))
                return gcd_pf_mul(b, a, cb, ca);

        if (is_exactly_a<mul>(b) && (!is_exactly_a<mul>(a)))
                return gcd_pf_mul(b, a, cb, ca);

        GINAC_ASSERT(is_exactly_a<mul>(a));
        size_t num = a.nops();
        exvector g; g.reserve(num);
        exvector acc_ca; acc_ca.reserve(num);
        ex part_b = b;
        for (size_t i=0; i<num; i++) {
                ex part_ca, part_cb;
                g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, false));
                acc_ca.push_back(part_ca);
                part_b = part_cb;
        }
        if (ca)
                *ca = dynallocate<mul>(acc_ca);
        if (cb)
                *cb = part_b;
        return dynallocate<mul>(g);
}

/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the LCM as a new expression */
ex lcm(const ex &a, const ex &b, bool check_args)
{
        if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
                return lcm(ex_to<numeric>(a), ex_to<numeric>(b));
        if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
                throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
        
        ex ca, cb;
        ex g = gcd(a, b, &ca, &cb, false);
        return ca * cb * g;
}


/*
 *  Square-free factorization
 */

/** Compute square-free factorization of multivariate polynomial a(x) using
 *  Yun's algorithm.  Used internally by sqrfree().
 *
 *  @param a  multivariate polynomial over Z[X], treated here as univariate
 *            polynomial in x (needs not be expanded).
 *  @param x  variable to factor in
 *  @return   vector of expairs (factor, exponent), sorted by exponents */
static epvector sqrfree_yun(const ex &a, const symbol &x)
{
        ex w = a;
        ex z = w.diff(x);
        ex g = gcd(w, z);
        if (g.is_zero()) {
                return epvector{};
        }
        if (g.is_equal(_ex1)) {
                return epvector{expair(a, _ex1)};
        }
        epvector results;
        ex exponent = _ex0;
        do {
                w = quo(w, g, x);
                if (w.is_zero()) {
                        return results;
                }
                z = quo(z, g, x) - w.diff(x);
                exponent = exponent + 1;
                if (w.is_equal(x)) {
                        // shortcut for x^n with n ∈ ℕ
                        exponent += quo(z, w.diff(x), x);
                        results.push_back(expair(w, exponent));
                        break;
                }
                g = gcd(w, z);
                if (!g.is_equal(_ex1)) {
                        results.push_back(expair(g, exponent));
                }
        } while (!z.is_zero());
        return results;
}


/** Compute a square-free factorization of a multivariate polynomial in Q[X].
 *
 *  @param a  multivariate polynomial over Q[X] (needs not be expanded)
 *  @param l  lst of variables to factor in, may be left empty for autodetection
 *  @return   a square-free factorization of \p a.
 *
 * \note
 * A polynomial \f$p(X) \in C[X]\f$ is said <EM>square-free</EM>
 * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$
 * are such that
 * \f[
 *     p(X) = q(X)^2 r(X),
 * \f]
 * we have \f$q(X) \in C\f$.
 * This means that \f$p(X)\f$ has no repeated factors, apart
 * eventually from constants.
 * Given a polynomial \f$p(X) \in C[X]\f$, we say that the
 * decomposition
 * \f[
 *   p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r}
 * \f]
 * is a <EM>square-free factorization</EM> of \f$p(X)\f$ if the
 * following conditions hold:
 * -#  \f$b \in C\f$ and \f$b \neq 0\f$;
 * -#  \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$;
 * -#  the degree of the polynomial \f$p_i\f$ is strictly positive
 *     for \f$i = 1, \ldots, r\f$;
 * -#  the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free.
 *
 * Square-free factorizations need not be unique.  For example, if
 * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$
 * into \f$-p_i(X)\f$.
 * Observe also that the factors \f$p_i(X)\f$ need not be irreducible
 * polynomials.
 */
ex sqrfree(const ex &a, const lst &l)
{
        if (is_exactly_a<numeric>(a) ||
            is_a<symbol>(a))        // shortcuts
                return a;

        // If no lst of variables to factorize in was specified we have to
        // invent one now.  Maybe one can optimize here by reversing the order
        // or so, I don't know.
        lst args;
        if (l.nops()==0) {
                sym_desc_vec sdv;
                get_symbol_stats(a, _ex0, sdv);
                for (auto & it : sdv)
                        args.append(it.sym);
        } else {
                args = l;
        }

        // Find the symbol to factor in at this stage
        if (!is_a<symbol>(args.op(0)))
                throw (std::runtime_error("sqrfree(): invalid factorization variable"));
        const symbol &x = ex_to<symbol>(args.op(0));

        // convert the argument from something in Q[X] to something in Z[X]
        const numeric lcm = lcm_of_coefficients_denominators(a);
        const ex tmp = multiply_lcm(a,lcm);

        // find the factors
        epvector factors = sqrfree_yun(tmp, x);

        // remove symbol x and proceed recursively with the remaining symbols
        args.remove_first();

        // recurse down the factors in remaining variables
        if (args.nops()>0) {
                for (auto & it : factors)
                        it.rest = sqrfree(it.rest, args);
        }

        // Done with recursion, now construct the final result
        ex result = _ex1;
        for (auto & it : factors)
                result *= pow(it.rest, it.coeff);

        // Yun's algorithm does not account for constant factors.  (For univariate
        // polynomials it works only in the monic case.)  We can correct this by
        // inserting what has been lost back into the result.  For completeness
        // we'll also have to recurse down that factor in the remaining variables.
        if (args.nops()>0)
                result *= sqrfree(quo(tmp, result, x), args);
        else
                result *= quo(tmp, result, x);

        // Put in the rational overall factor again and return
        return result * lcm.inverse();
}


/** Compute square-free partial fraction decomposition of rational function
 *  a(x).
 *
 *  @param a rational function over Z[x], treated as univariate polynomial
 *           in x
 *  @param x variable to factor in
 *  @return decomposed rational function */
ex sqrfree_parfrac(const ex & a, const symbol & x)
{
        // Find numerator and denominator
        ex nd = numer_denom(a);
        ex numer = nd.op(0), denom = nd.op(1);
//clog << "numer = " << numer << ", denom = " << denom << endl;

        // Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
        ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;

        // Factorize denominator and compute cofactors
        epvector yun = sqrfree_yun(denom, x);
        size_t yun_max_exponent = yun.empty() ? 0 : ex_to<numeric>(yun.back().coeff).to_int();
        exvector factor, cofac;
        for (size_t i=0; i<yun.size(); i++) {
                numeric i_exponent = ex_to<numeric>(yun[i].coeff);
                for (size_t j=0; j<i_exponent; j++) {
                        factor.push_back(pow(yun[i].rest, j+1));
                        ex prod = _ex1;
                        for (size_t k=0; k<yun.size(); k++) {
                                if (yun[k].coeff == i_exponent)
                                        prod *= pow(yun[k].rest, i_exponent-1-j);
                                else
                                        prod *= pow(yun[k].rest, yun[k].coeff);
                        }
                        cofac.push_back(prod.expand());
                }
        }
        size_t num_factors = factor.size();
//clog << "factors  : " << exprseq(factor) << endl;
//clog << "cofactors: " << exprseq(cofac) << endl;

        // Construct coefficient matrix for decomposition
        int max_denom_deg = denom.degree(x);
        matrix sys(max_denom_deg + 1, num_factors);
        matrix rhs(max_denom_deg + 1, 1);
        for (int i=0; i<=max_denom_deg; i++) {
                for (size_t j=0; j<num_factors; j++)
                        sys(i, j) = cofac[j].coeff(x, i);
                rhs(i, 0) = red_numer.coeff(x, i);
        }
//clog << "coeffs: " << sys << endl;
//clog << "rhs   : " << rhs << endl;

        // Solve resulting linear system
        matrix vars(num_factors, 1);
        for (size_t i=0; i<num_factors; i++)
                vars(i, 0) = symbol();
        matrix sol = sys.solve(vars, rhs);

        // Sum up decomposed fractions
        ex sum = 0;
        for (size_t i=0; i<num_factors; i++)
                sum += sol(i, 0) / factor[i];

        return red_poly + sum;
}


/*
 *  Normal form of rational functions
 */

/*
 *  Note: The internal normal() functions (= basic::normal() and overloaded
 *  functions) all return lists of the form {numerator, denominator}. This
 *  is to get around mul::eval()'s automatic expansion of numeric coefficients.
 *  E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
 *  the information that (a+b) is the numerator and 3 is the denominator.
 */


/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). The symbol and expression are appended to repl, for
 *  a later application of subs().
 *  @see ex::normal */
static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
{
        // Since the repl contains replaced expressions we should search for them
        ex e_replaced = e.subs(repl, subs_options::no_pattern);

        // Expression already replaced? Then return the assigned symbol
        auto it = rev_lookup.find(e_replaced);
        if (it != rev_lookup.end())
                return it->second;

        // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't itself contain symbols from repl,
        // because subs() is not recursive
        ex es = dynallocate<symbol>();
        repl.insert(std::make_pair(es, e_replaced));
        rev_lookup.insert(std::make_pair(e_replaced, es));
        return es;
}

/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). The symbol and expression are appended to repl, and the
 *  symbol is returned.
 *  @see basic::to_rational
 *  @see basic::to_polynomial */
static ex replace_with_symbol(const ex & e, exmap & repl)
{
        // Since the repl contains replaced expressions we should search for them
        ex e_replaced = e.subs(repl, subs_options::no_pattern);

        // Expression already replaced? Then return the assigned symbol
        for (auto & it : repl)
                if (it.second.is_equal(e_replaced))
                        return it.first;

        // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't itself contain symbols from repl,
        // because subs() is not recursive
        ex es = dynallocate<symbol>();
        repl.insert(std::make_pair(es, e_replaced));
        return es;
}


/** Function object to be applied by basic::normal(). */
struct normal_map_function : public map_function {
        ex operator()(const ex & e) override { return normal(e); }
};

/** Default implementation of ex::normal(). It normalizes the children and
 *  replaces the object with a temporary symbol.
 *  @see ex::normal */
ex basic::normal(exmap & repl, exmap & rev_lookup) const
{
        if (nops() == 0)
                return dynallocate<lst>({replace_with_symbol(*this, repl, rev_lookup), _ex1});

        normal_map_function map_normal;
        return dynallocate<lst>({replace_with_symbol(map(map_normal), repl, rev_lookup), _ex1});
}


/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
 *  @see ex::normal */
ex symbol::normal(exmap & repl, exmap & rev_lookup) const
{
        return dynallocate<lst>({*this, _ex1});
}


/** Implementation of ex::normal() for a numeric. It splits complex numbers
 *  into re+I*im and replaces I and non-rational real numbers with a temporary
 *  symbol.
 *  @see ex::normal */
ex numeric::normal(exmap & repl, exmap & rev_lookup) const
{
        numeric num = numer();
        ex numex = num;

        if (num.is_real()) {
                if (!num.is_integer())
                        numex = replace_with_symbol(numex, repl, rev_lookup);
        } else { // complex
                numeric re = num.real(), im = num.imag();
                ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup);
                ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup);
                numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup);
        }

        // Denominator is always a real integer (see numeric::denom())
        return dynallocate<lst>({numex, denom()});
}


/** Fraction cancellation.
 *  @param n  numerator
 *  @param d  denominator
 *  @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
        ex num = n;
        ex den = d;
        numeric pre_factor = *_num1_p;

//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;

        // Handle trivial case where denominator is 1
        if (den.is_equal(_ex1))
                return dynallocate<lst>({num, den});

        // Handle special cases where numerator or denominator is 0
        if (num.is_zero())
                return dynallocate<lst>({num, _ex1});
        if (den.expand().is_zero())
                throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));

        // Bring numerator and denominator to Z[X] by multiplying with
        // LCM of all coefficients' denominators
        numeric num_lcm = lcm_of_coefficients_denominators(num);
        numeric den_lcm = lcm_of_coefficients_denominators(den);
        num = multiply_lcm(num, num_lcm);
        den = multiply_lcm(den, den_lcm);
        pre_factor = den_lcm / num_lcm;

        // Cancel GCD from numerator and denominator
        ex cnum, cden;
        if (gcd(num, den, &cnum, &cden, false) != _ex1) {
                num = cnum;
                den = cden;
        }

        // Make denominator unit normal (i.e. coefficient of first symbol
        // as defined by get_first_symbol() is made positive)
        if (is_exactly_a<numeric>(den)) {
                if (ex_to<numeric>(den).is_negative()) {
                        num *= _ex_1;
                        den *= _ex_1;
                }
        } else {
                ex x;
                if (get_first_symbol(den, x)) {
                        GINAC_ASSERT(is_exactly_a<numeric>(den.unit(x)));
                        if (ex_to<numeric>(den.unit(x)).is_negative()) {
                                num *= _ex_1;
                                den *= _ex_1;
                        }
                }
        }

        // Return result as list
//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
        return dynallocate<lst>({num * pre_factor.numer(), den * pre_factor.denom()});
}


/** Implementation of ex::normal() for a sum. It expands terms and performs
 *  fractional addition.
 *  @see ex::normal */
ex add::normal(exmap & repl, exmap & rev_lookup) const
{
        // Normalize children and split each one into numerator and denominator
        exvector nums, dens;
        nums.reserve(seq.size()+1);
        dens.reserve(seq.size()+1);
        for (auto & it : seq) {
                ex n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup);
                nums.push_back(n.op(0));
                dens.push_back(n.op(1));
        }
        ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup);
        nums.push_back(n.op(0));
        dens.push_back(n.op(1));
        GINAC_ASSERT(nums.size() == dens.size());

        // Now, nums is a vector of all numerators and dens is a vector of
        // all denominators
//std::clog << "add::normal uses " << nums.size() << " summands:\n";

        // Add fractions sequentially
        auto num_it = nums.begin(), num_itend = nums.end();
        auto den_it = dens.begin(), den_itend = dens.end();
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
        ex num = *num_it++, den = *den_it++;
        while (num_it != num_itend) {
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
                ex next_num = *num_it++, next_den = *den_it++;

                // Trivially add sequences of fractions with identical denominators
                while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
                        next_num += *num_it;
                        num_it++; den_it++;
                }

                // Addition of two fractions, taking advantage of the fact that
                // the heuristic GCD algorithm computes the cofactors at no extra cost
                ex co_den1, co_den2;
                ex g = gcd(den, next_den, &co_den1, &co_den2, false);
                num = ((num * co_den2) + (next_num * co_den1)).expand();
                den *= co_den2;         // this is the lcm(den, next_den)
        }
//std::clog << " common denominator = " << den << std::endl;

        // Cancel common factors from num/den
        return frac_cancel(num, den);
}


/** Implementation of ex::normal() for a product. It cancels common factors
 *  from fractions.
 *  @see ex::normal() */
ex mul::normal(exmap & repl, exmap & rev_lookup) const
{
        // Normalize children, separate into numerator and denominator
        exvector num; num.reserve(seq.size());
        exvector den; den.reserve(seq.size());
        ex n;
        for (auto & it : seq) {
                n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup);
                num.push_back(n.op(0));
                den.push_back(n.op(1));
        }
        n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup);
        num.push_back(n.op(0));
        den.push_back(n.op(1));

        // Perform fraction cancellation
        return frac_cancel(dynallocate<mul>(num), dynallocate<mul>(den));
}


/** Implementation of ex::normal([B) for powers. It normalizes the basis,
 *  distributes integer exponents to numerator and denominator, and replaces
 *  non-integer powers by temporary symbols.
 *  @see ex::normal */
ex power::normal(exmap & repl, exmap & rev_lookup) const
{
        // Normalize basis and exponent (exponent gets reassembled)
        ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup);
        ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup);
        n_exponent = n_exponent.op(0) / n_exponent.op(1);

        if (n_exponent.info(info_flags::integer)) {

                if (n_exponent.info(info_flags::positive)) {

                        // (a/b)^n -> {a^n, b^n}
                        return dynallocate<lst>({pow(n_basis.op(0), n_exponent), pow(n_basis.op(1), n_exponent)});

                } else if (n_exponent.info(info_flags::negative)) {

                        // (a/b)^-n -> {b^n, a^n}
                        return dynallocate<lst>({pow(n_basis.op(1), -n_exponent), pow(n_basis.op(0), -n_exponent)});
                }

        } else {

                if (n_exponent.info(info_flags::positive)) {

                        // (a/b)^x -> {sym((a/b)^x), 1}
                        return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1});

                } else if (n_exponent.info(info_flags::negative)) {

                        if (n_basis.op(1).is_equal(_ex1)) {

                                // a^-x -> {1, sym(a^x)}
                                return dynallocate<lst>({_ex1, replace_with_symbol(pow(n_basis.op(0), -n_exponent), repl, rev_lookup)});

                        } else {

                                // (a/b)^-x -> {sym((b/a)^x), 1}
                                return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(1) / n_basis.op(0), -n_exponent), repl, rev_lookup), _ex1});
                        }
                }
        }

        // (a/b)^x -> {sym((a/b)^x, 1}
        return dynallocate<lst>({replace_with_symbol(pow(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1});
}


/** Implementation of ex::normal() for pseries. It normalizes each coefficient
 *  and replaces the series by a temporary symbol.
 *  @see ex::normal */
ex pseries::normal(exmap & repl, exmap & rev_lookup) const
{
        epvector newseq;
        for (auto & it : seq) {
                ex restexp = it.rest.normal();
                if (!restexp.is_zero())
                        newseq.push_back(expair(restexp, it.coeff));
        }
        ex n = pseries(relational(var,point), std::move(newseq));
        return dynallocate<lst>({replace_with_symbol(n, repl, rev_lookup), _ex1});
}


/** Normalization of rational functions.
 *  This function converts an expression to its normal form
 *  "numerator/denominator", where numerator and denominator are (relatively
 *  prime) polynomials. Any subexpressions which are not rational functions
 *  (like non-rational numbers, non-integer powers or functions like sin(),
 *  cos() etc.) are replaced by temporary symbols which are re-substituted by
 *  the (normalized) subexpressions before normal() returns (this way, any
 *  expression can be treated as a rational function). normal() is applied
 *  recursively to arguments of functions etc.
 *
 *  @return normalized expression */
ex ex::normal() const
{
        exmap repl, rev_lookup;

        ex e = bp->normal(repl, rev_lookup);
        GINAC_ASSERT(is_a<lst>(e));

        // Re-insert replaced symbols
        if (!repl.empty())
                e = e.subs(repl, subs_options::no_pattern);

        // Convert {numerator, denominator} form back to fraction
        return e.op(0) / e.op(1);
}

/** Get numerator of an expression. If the expression is not of the normal
 *  form "numerator/denominator", it is first converted to this form and
 *  then the numerator is returned.
 *
 *  @see ex::normal
 *  @return numerator */
ex ex::numer() const
{
        exmap repl, rev_lookup;

        ex e = bp->normal(repl, rev_lookup);
        GINAC_ASSERT(is_a<lst>(e));

        // Re-insert replaced symbols
        if (repl.empty())
                return e.op(0);
        else
                return e.op(0).subs(repl, subs_options::no_pattern);
}

/** Get denominator of an expression. If the expression is not of the normal
 *  form "numerator/denominator", it is first converted to this form and
 *  then the denominator is returned.
 *
 *  @see ex::normal
 *  @return denominator */
ex ex::denom() const
{
        exmap repl, rev_lookup;

        ex e = bp->normal(repl, rev_lookup);
        GINAC_ASSERT(is_a<lst>(e));

        // Re-insert replaced symbols
        if (repl.empty())
                return e.op(1);
        else
                return e.op(1).subs(repl, subs_options::no_pattern);
}

/** Get numerator and denominator of an expression. If the expression is not
 *  of the normal form "numerator/denominator", it is first converted to this
 *  form and then a list [numerator, denominator] is returned.
 *
 *  @see ex::normal
 *  @return a list [numerator, denominator] */
ex ex::numer_denom() const
{
        exmap repl, rev_lookup;

        ex e = bp->normal(repl, rev_lookup);
        GINAC_ASSERT(is_a<lst>(e));

        // Re-insert replaced symbols
        if (repl.empty())
                return e;
        else
                return e.subs(repl, subs_options::no_pattern);
}


/** Rationalization of non-rational functions.
 *  This function converts a general expression to a rational function
 *  by replacing all non-rational subexpressions (like non-rational numbers,
 *  non-integer powers or functions like sin(), cos() etc.) to temporary
 *  symbols. This makes it possible to use functions like gcd() and divide()
 *  on non-rational functions by applying to_rational() on the arguments,
 *  calling the desired function and re-substituting the temporary symbols
 *  in the result. To make the last step possible, all temporary symbols and
 *  their associated expressions are collected in the map specified by the
 *  repl parameter, ready to be passed as an argument to ex::subs().
 *
 *  @param repl collects all temporary symbols and their replacements
 *  @return rationalized expression */
ex ex::to_rational(exmap & repl) const
{
        return bp->to_rational(repl);
}

ex ex::to_polynomial(exmap & repl) const
{
        return bp->to_polynomial(repl);
}

/** Default implementation of ex::to_rational(). This replaces the object with
 *  a temporary symbol. */
ex basic::to_rational(exmap & repl) const
{
        return replace_with_symbol(*this, repl);
}

ex basic::to_polynomial(exmap & repl) const
{
        return replace_with_symbol(*this, repl);
}


/** Implementation of ex::to_rational() for symbols. This returns the
 *  unmodified symbol. */
ex symbol::to_rational(exmap & repl) const
{
        return *this;
}

/** Implementation of ex::to_polynomial() for symbols. This returns the
 *  unmodified symbol. */
ex symbol::to_polynomial(exmap & repl) const
{
        return *this;
}


/** Implementation of ex::to_rational() for a numeric. It splits complex
 *  numbers into re+I*im and replaces I and non-rational real numbers with a
 *  temporary symbol. */
ex numeric::to_rational(exmap & repl) const
{
        if (is_real()) {
                if (!is_rational())
                        return replace_with_symbol(*this, repl);
        } else { // complex
                numeric re = real();
                numeric im = imag();
                ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl);
                ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl);
                return re_ex + im_ex * replace_with_symbol(I, repl);
        }
        return *this;
}

/** Implementation of ex::to_polynomial() for a numeric. It splits complex
 *  numbers into re+I*im and replaces I and non-integer real numbers with a
 *  temporary symbol. */
ex numeric::to_polynomial(exmap & repl) const
{
        if (is_real()) {
                if (!is_integer())
                        return replace_with_symbol(*this, repl);
        } else { // complex
                numeric re = real();
                numeric im = imag();
                ex re_ex = re.is_integer() ? re : replace_with_symbol(re, repl);
                ex im_ex = im.is_integer() ? im : replace_with_symbol(im, repl);
                return re_ex + im_ex * replace_with_symbol(I, repl);
        }
        return *this;
}


/** Implementation of ex::to_rational() for powers. It replaces non-integer
 *  powers by temporary symbols. */
ex power::to_rational(exmap & repl) const
{
        if (exponent.info(info_flags::integer))
                return pow(basis.to_rational(repl), exponent);
        else
                return replace_with_symbol(*this, repl);
}

/** Implementation of ex::to_polynomial() for powers. It replaces non-posint
 *  powers by temporary symbols. */
ex power::to_polynomial(exmap & repl) const
{
        if (exponent.info(info_flags::posint))
                return pow(basis.to_rational(repl), exponent);
        else if (exponent.info(info_flags::negint))
        {
                ex basis_pref = collect_common_factors(basis);
                if (is_exactly_a<mul>(basis_pref) || is_exactly_a<power>(basis_pref)) {
                        // (A*B)^n will be automagically transformed to A^n*B^n
                        ex t = pow(basis_pref, exponent);
                        return t.to_polynomial(repl);
                }
                else
                        return pow(replace_with_symbol(pow(basis, _ex_1), repl), -exponent);
        } 
        else
                return replace_with_symbol(*this, repl);
}


/** Implementation of ex::to_rational() for expairseqs. */
ex expairseq::to_rational(exmap & repl) const
{
        epvector s;
        s.reserve(seq.size());
        for (auto & it : seq)
                s.push_back(split_ex_to_pair(recombine_pair_to_ex(it).to_rational(repl)));

        ex oc = overall_coeff.to_rational(repl);
        if (oc.info(info_flags::numeric))
                return thisexpairseq(std::move(s), overall_coeff);
        else
                s.push_back(expair(oc, _ex1));
        return thisexpairseq(std::move(s), default_overall_coeff());
}

/** Implementation of ex::to_polynomial() for expairseqs. */
ex expairseq::to_polynomial(exmap & repl) const
{
        epvector s;
        s.reserve(seq.size());
        for (auto & it : seq)
                s.push_back(split_ex_to_pair(recombine_pair_to_ex(it).to_polynomial(repl)));

        ex oc = overall_coeff.to_polynomial(repl);
        if (oc.info(info_flags::numeric))
                return thisexpairseq(std::move(s), overall_coeff);
        else
                s.push_back(expair(oc, _ex1));
        return thisexpairseq(std::move(s), default_overall_coeff());
}


/** Remove the common factor in the terms of a sum 'e' by calculating the GCD,
 *  and multiply it into the expression 'factor' (which needs to be initialized
 *  to 1, unless you're accumulating factors). */
static ex find_common_factor(const ex & e, ex & factor, exmap & repl)
{
        if (is_exactly_a<add>(e)) {

                size_t num = e.nops();
                exvector terms; terms.reserve(num);
                ex gc;

                // Find the common GCD
                for (size_t i=0; i<num; i++) {
                        ex x = e.op(i).to_polynomial(repl);

                        if (is_exactly_a<add>(x) || is_exactly_a<mul>(x) || is_a<power>(x)) {
                                ex f = 1;
                                x = find_common_factor(x, f, repl);
                                x *= f;
                        }

                        if (i == 0)
                                gc = x;
                        else
                                gc = gcd(gc, x);

                        terms.push_back(x);
                }

                if (gc.is_equal(_ex1))
                        return e;

                // The GCD is the factor we pull out
                factor *= gc;

                // Now divide all terms by the GCD
                for (size_t i=0; i<num; i++) {
                        ex x;

                        // Try to avoid divide() because it expands the polynomial
                        ex &t = terms[i];
                        if (is_exactly_a<mul>(t)) {
                                for (size_t j=0; j<t.nops(); j++) {
                                        if (t.op(j).is_equal(gc)) {
                                                exvector v; v.reserve(t.nops());
                                                for (size_t k=0; k<t.nops(); k++) {
                                                        if (k == j)
                                                                v.push_back(_ex1);
                                                        else
                                                                v.push_back(t.op(k));
                                                }
                                                t = dynallocate<mul>(v);
                                                goto term_done;
                                        }
                                }
                        }

                        divide(t, gc, x);
                        t = x;
term_done:      ;
                }
                return dynallocate<add>(terms);

        } else if (is_exactly_a<mul>(e)) {

                size_t num = e.nops();
                exvector v; v.reserve(num);

                for (size_t i=0; i<num; i++)
                        v.push_back(find_common_factor(e.op(i), factor, repl));

                return dynallocate<mul>(v);

        } else if (is_exactly_a<power>(e)) {
                const ex e_exp(e.op(1));
                if (e_exp.info(info_flags::integer)) {
                        ex eb = e.op(0).to_polynomial(repl);
                        ex factor_local(_ex1);
                        ex pre_res = find_common_factor(eb, factor_local, repl);
                        factor *= pow(factor_local, e_exp);
                        return pow(pre_res, e_exp);
                        
                } else
                        return e.to_polynomial(repl);

        } else
                return e;
}


/** Collect common factors in sums. This converts expressions like
 *  'a*(b*x+b*y)' to 'a*b*(x+y)'. */
ex collect_common_factors(const ex & e)
{
        if (is_exactly_a<add>(e) || is_exactly_a<mul>(e) || is_exactly_a<power>(e)) {

                exmap repl;
                ex factor = 1;
                ex r = find_common_factor(e, factor, repl);
                return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern);

        } else
                return e;
}


/** Resultant of two expressions e1,e2 with respect to symbol s.
 *  Method: Compute determinant of Sylvester matrix of e1,e2,s.  */
ex resultant(const ex & e1, const ex & e2, const ex & s)
{
        const ex ee1 = e1.expand();
        const ex ee2 = e2.expand();
        if (!ee1.info(info_flags::polynomial) ||
            !ee2.info(info_flags::polynomial))
                throw(std::runtime_error("resultant(): arguments must be polynomials"));

        const int h1 = ee1.degree(s);
        const int l1 = ee1.ldegree(s);
        const int h2 = ee2.degree(s);
        const int l2 = ee2.ldegree(s);

        const int msize = h1 + h2;
        matrix m(msize, msize);

        for (int l = h1; l >= l1; --l) {
                const ex e = ee1.coeff(s, l);
                for (int k = 0; k < h2; ++k)
                        m(k, k+h1-l) = e;
        }
        for (int l = h2; l >= l2; --l) {
                const ex e = ee2.coeff(s, l);
                for (int k = 0; k < h1; ++k)
                        m(k+h2, k+h2-l) = e;
        }

        return m.determinant();
}


} // namespace GiNaC