normal.cpp

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/*
 *  GiNaC Copyright (C) 1999-2020 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
 
 
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
 */
 
struct sym_desc {
    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)
    { }
 
    ex sym;
 
    int deg_a;
 
    int deg_b;
 
    int ldeg_a;
 
    int ldeg_b;
 
    int max_deg;
 
    size_t max_lcnops;
 
    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);
    }
}
 
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";
    auto 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 << std::endl;
        std::clog << "  lcoeff_a=" << a.lcoeff(it->sym) << ", lcoeff_b=" << b.lcoeff(it->sym) << std::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;
}
 
static numeric lcm_of_coefficients_denominators(const ex &e)
{
    return lcmcoeff(e, *_num1_p);
}
 
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);
}
 
 
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
 */
 
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);
}
 
 
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;
}
 
 
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;
}
 
 
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;
}
 
 
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;
}
 
 
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
 
 
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
 */
 
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()"));
    }
}
 
 
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;
}
 
 
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;
}
 
 
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);
}
 
 
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
 */
 
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;
    }
}
 
 
numeric ex::max_coefficient() const
{
    return bp->max_coefficient();
}
 
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));
}
 
 
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;
}
 
 
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);
}
 
class 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;
}
 
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);
 
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();
    if (aex.is_zero()) {
        if (ca)
            *ca = _ex0;
        if (cb)
            *cb = _ex1;
        return b;
    }
    ex bex = b.expand();
    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 symbol 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;
    }
 
    // 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, exp_a - 1);
        if (cb)
            *cb = _ex1;
        return p;
    }
    if (is_a<symbol>(p)) {
        // Cancel trivial common factor
        int ldeg_a = ex_to<numeric>(exp_a).to_int();
        int ldeg_b = b.ldegree(p);
        int min_ldeg = std::min(ldeg_a, ldeg_b);
        if (min_ldeg > 0) {
            ex common = pow(p, min_ldeg);
            return gcd(pow(p, ldeg_a - min_ldeg), (b / common).expand(), ca, cb, false) * common;
        }
    }
 
    ex p_co, bpart_co;
    ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
 
    if (p_gcd.is_equal(_ex1)) {
        // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
        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);
}
 
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
 */
 
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()) {
        // manifest zero or hidden zero
        return {};
    }
    if (g.is_equal(_ex1)) {
        // w(x) and w'(x) share no factors: w(x) is square-free
        return {expair(a, _ex1)};
    }
 
    epvector factors;
    ex i = 0;  // exponent
    do {
        w = quo(w, g, x);
        if (w.is_zero()) {
            // hidden zero
            break;
        }
        z = quo(z, g, x) - w.diff(x);
        i += 1;
        if (w.is_equal(x)) {
            // shortcut for x^n with n ∈ ℕ
            i += quo(z, w.diff(x), x);
            factors.push_back(expair(w, i));
            break;
        }
        g = gcd(w, z);
        if (!g.is_equal(_ex1)) {
            factors.push_back(expair(g, i));
        }
    } while (!z.is_zero());
 
    // correct for lost factor
    // (being based on GCDs, Yun's algorithm only finds factors up to a unit)
    const ex lost_factor = quo(a, mul{factors}, x);
    if (lost_factor.is_equal(_ex1)) {
        // trivial lost factor
        return factors;
    }
    if (!factors.empty() && factors[0].coeff.is_equal(1)) {
        // multiply factor^1 with lost_factor
        factors[0].rest *= lost_factor;
        return factors;
    }
    // no factor^1: prepend lost_factor^1 to the results
    epvector results = {expair(lost_factor, 1)};
    std::move(factors.begin(), factors.end(), std::back_inserter(results));
    return results;
}
 
 
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);
    if (factors.empty()) {
        // the polynomial was a hidden zero
        return _ex0;
    }
 
    // 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 = mul(factors);
 
    // Put in the rational overall factor again and return
    return result * lcm.inverse();
}
 
 
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);
//std::clog << "numer = " << numer << ", denom = " << denom << std::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();
//std::clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << std::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();
//std::clog << "factors  : " << exprseq(factor) << std::endl;
//std::clog << "cofactors: " << exprseq(cofac) << std::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);
    }
//std::clog << "coeffs: " << sys << std::endl;
//std::clog << "rhs   : " << rhs << std::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.
 */
 
 
static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup, lst & modifier)
{
    // 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;
 
    // The expression can be the base of substituted power, which requires a more careful search
    if (! is_a<numeric>(e_replaced))
        for (auto & it : repl)
            if (is_a<power>(it.second) && e_replaced.is_equal(it.second.op(0))) {
                ex degree = pow(it.second.op(1), _ex_1);
                if (is_a<numeric>(degree) && ex_to<numeric>(degree).is_integer())
                    return pow(it.first, degree);
            }
 
    // We treat powers and the exponent functions differently because
    // they can be rationalised more efficiently
    if (is_a<function>(e_replaced) && is_ex_the_function(e_replaced, exp)) {
        for (auto & it : repl) {
            if (is_a<function>(it.second) && is_ex_the_function(e_replaced, exp)) {
                ex ratio = normal(e_replaced.op(0) / it.second.op(0));
                if (is_a<numeric>(ratio) && ex_to<numeric>(ratio).is_rational()) {
                    // Different exponents can be treated as powers of the same basic equation
                    if (ex_to<numeric>(ratio).is_integer()) {
                        // If ratio is an integer then this is simply the power of the existing symbol.
                        // std::clog << e_replaced << " is a " << ratio << " power of " << it.first << std::endl;
                        return dynallocate<power>(it.first, ratio);
                    } else {
                        // otherwise we need to give the replacement pattern to change
                        // the previous expression...
                        ex es = dynallocate<symbol>();
                        ex Num = numer(ratio);
                        modifier.append(it.first == power(es, denom(ratio)));
                        // std::clog << e_replaced << " is power " << Num << " and "
                        //        << it.first << " is power " << denom(ratio) << " of the common base "
                        //        << exp(e_replaced.op(0)/Num) << std::endl;
                        // ... and  modify the replacement tables
                        rev_lookup.erase(it.second);
                        rev_lookup.insert({exp(e_replaced.op(0)/Num), es});
                        repl.erase(it.first);
                        repl.insert({es, exp(e_replaced.op(0)/Num)});
                        return dynallocate<power>(es, Num);
                    }
                }
            }
        }
    } else if (is_a<power>(e_replaced) && !is_a<numeric>(e_replaced.op(0)) // We do not replace simple monomials like x^3 or sqrt(2)
               && ! (is_a<symbol>(e_replaced.op(0))
                   && is_a<numeric>(e_replaced.op(1)) && ex_to<numeric>(e_replaced.op(1)).is_integer())) {
        for (auto & it : repl) {
            if (e_replaced.op(0).is_equal(it.second) // The base is an allocated symbol or base of power
                || (is_a<power>(it.second) && e_replaced.op(0).is_equal(it.second.op(0)))) {
                ex ratio; // We bind together two above cases
                if (is_a<power>(it.second))
                    ratio = normal(e_replaced.op(1) / it.second.op(1));
                else
                    ratio = e_replaced.op(1);
                if (is_a<numeric>(ratio) && ex_to<numeric>(ratio).is_rational())  {
                    // Different powers can be treated as powers of the same basic equation
                    if (ex_to<numeric>(ratio).is_integer()) {
                        // If ratio is an integer then this is simply the power of the existing symbol.
                        //std::clog << e_replaced << " is a " << ratio << " power of " << it.first << std::endl;
                        return dynallocate<power>(it.first, ratio);
                    } else {
                        // otherwise we need to give the replacement pattern to change
                        // the previous expression...
                        ex es = dynallocate<symbol>();
                        ex Num = numer(ratio);
                        modifier.append(it.first == power(es, denom(ratio)));
                        //std::clog << e_replaced << " is power " << Num << " and "
                        //        << it.first << " is power " << denom(ratio) << " of the common base "
                        //        << pow(e_replaced.op(0), e_replaced.op(1)/Num) << std::endl;
                        // ... and  modify the replacement tables
                        rev_lookup.erase(it.second);
                        rev_lookup.insert({pow(e_replaced.op(0), e_replaced.op(1)/Num), es});
                        repl.erase(it.first);
                        repl.insert({es, pow(e_replaced.op(0), e_replaced.op(1)/Num)});
                        return dynallocate<power>(es, Num);
                    }
                }
            }
        }
        // There is no existing substitution, thus we are creating a new one.
        // This needs to be done separately to treat possible occurrences of
        // b = e_replaced.op(0) elsewhere in the expression as pow(b, 1).
        ex degree = pow(e_replaced.op(1), _ex_1);
        if (is_a<numeric>(degree) && ex_to<numeric>(degree).is_integer()) {
            ex es = dynallocate<symbol>();
            modifier.append(e_replaced.op(0) == power(es, degree));
            repl.insert({es, e_replaced});
            rev_lookup.insert({e_replaced, es});
            return es;
        }
    }
 
    // 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;
}
 
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;
}
 
 
struct normal_map_function : public map_function {
    ex operator()(const ex & e) override { return normal(e); }
};
 
ex basic::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    if (nops() == 0)
        return dynallocate<lst>({replace_with_symbol(*this, repl, rev_lookup, modifier), _ex1});
 
    normal_map_function map_normal;
    int nmod = modifier.nops(); // To watch new modifiers to the replacement list
    lst result = dynallocate<lst>({replace_with_symbol(map(map_normal), repl, rev_lookup, modifier), _ex1});
    for (int imod = nmod; imod < modifier.nops(); ++imod) {
        exmap this_repl;
        this_repl.insert(std::make_pair(modifier.op(imod).op(0), modifier.op(imod).op(1)));
        result = ex_to<lst>(result.subs(this_repl, subs_options::no_pattern));
    }
 
    return result;
}
 
 
ex symbol::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    return dynallocate<lst>({*this, _ex1});
}
 
 
ex numeric::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    numeric num = numer();
    ex numex = num;
 
    if (num.is_real()) {
        if (!num.is_integer())
            numex = replace_with_symbol(numex, repl, rev_lookup, modifier);
    } else { // complex
        numeric re = num.real(), im = num.imag();
        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup, modifier);
        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup, modifier);
        numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup, modifier);
    }
 
    // Denominator is always a real integer (see numeric::denom())
    return dynallocate<lst>({numex, denom()});
}
 
 
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()});
}
 
 
ex add::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    // Normalize children and split each one into numerator and denominator
    exvector nums, dens;
    nums.reserve(seq.size()+1);
    dens.reserve(seq.size()+1);
    int nmod = modifier.nops(); // To watch new modifiers to the replacement list
    for (auto & it : seq) {
        ex n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup, modifier);
        nums.push_back(n.op(0));
        dens.push_back(n.op(1));
    }
    ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, modifier);
    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;
    for (int imod = nmod; imod < modifier.nops(); ++imod) {
        while (num_it != num_itend) {
            *num_it = num_it->subs(modifier.op(imod), subs_options::no_pattern);
            ++num_it;
            *den_it = den_it->subs(modifier.op(imod), subs_options::no_pattern);
            ++den_it;
        }
        // Reset iterators for the next round
        num_it = nums.begin();
        den_it = dens.begin();
    }
 
    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);
}
 
 
ex mul::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    // Normalize children, separate into numerator and denominator
    exvector num; num.reserve(seq.size());
    exvector den; den.reserve(seq.size());
    ex n;
    int nmod = modifier.nops(); // To watch new modifiers to the replacement list
    for (auto & it : seq) {
        n = ex_to<basic>(recombine_pair_to_ex(it)).normal(repl, rev_lookup, modifier);
        num.push_back(n.op(0));
        den.push_back(n.op(1));
    }
    n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, modifier);
    num.push_back(n.op(0));
    den.push_back(n.op(1));
    auto num_it = num.begin(), num_itend = num.end();
    auto den_it = den.begin(), den_itend = den.end();
    for (int imod = nmod; imod < modifier.nops(); ++imod) {
        while (num_it != num_itend) {
            *num_it = num_it->subs(modifier.op(imod), subs_options::no_pattern);
            ++num_it;
            *den_it = den_it->subs(modifier.op(imod), subs_options::no_pattern);
            ++den_it;
        }
        num_it = num.begin();
        den_it = den.begin();
    }
 
    // Perform fraction cancellation
    return frac_cancel(dynallocate<mul>(num), dynallocate<mul>(den));
}
 
 
ex power::normal(exmap & repl, exmap & rev_lookup, lst & modifier) const
{
    // Normalize basis and exponent (exponent gets reassembled)
    int nmod = modifier.nops(); // To watch new modifiers to the replacement list
    ex n_basis = ex_to<basic>(basis).normal(repl, rev_lookup, modifier);
    for (int imod = nmod; imod < modifier.nops(); ++imod)
        n_basis = n_basis.subs(modifier.op(imod), subs_options::no_pattern);
 
    nmod = modifier.nops();
    ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, modifier);
    for (int imod = nmod; imod < modifier.nops(); ++imod)
        n_exponent = n_exponent.subs(modifier.op(imod), subs_options::no_pattern);
    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, modifier), _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, modifier)});
 
            } 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, modifier), _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, modifier), _ex1});
}
 
 
ex pseries::normal(exmap & repl, exmap & rev_lookup, lst & modifier) 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, modifier), _ex1});
}
 
 
ex ex::normal() const
{
    exmap repl, rev_lookup;
    lst modifier;
 
    ex e = bp->normal(repl, rev_lookup, modifier);
    GINAC_ASSERT(is_a<lst>(e));
 
    // Re-insert replaced symbols
    if (!repl.empty()) {
        for(int i=0; i < modifier.nops(); ++i)
            e = e.subs(modifier.op(i), subs_options::no_pattern);
        e = e.subs(repl, subs_options::no_pattern);
    }
 
    // Convert {numerator, denominator} form back to fraction
    return e.op(0) / e.op(1);
}
 
ex ex::numer() const
{
    exmap repl, rev_lookup;
    lst modifier;
 
    ex e = bp->normal(repl, rev_lookup, modifier);
    GINAC_ASSERT(is_a<lst>(e));
 
    // Re-insert replaced symbols
    if (repl.empty())
        return e.op(0);
    else {
        for(int i=0; i < modifier.nops(); ++i)
            e = e.subs(modifier.op(i), subs_options::no_pattern);
 
        return e.op(0).subs(repl, subs_options::no_pattern);
    }
}
 
ex ex::denom() const
{
    exmap repl, rev_lookup;
    lst modifier;
 
    ex e = bp->normal(repl, rev_lookup, modifier);
    GINAC_ASSERT(is_a<lst>(e));
 
    // Re-insert replaced symbols
    if (repl.empty())
        return e.op(1);
    else {
        for(int i=0; i < modifier.nops(); ++i)
            e = e.subs(modifier.op(i), subs_options::no_pattern);
 
        return e.op(1).subs(repl, subs_options::no_pattern);
    }
}
 
ex ex::numer_denom() const
{
    exmap repl, rev_lookup;
    lst modifier;
 
    ex e = bp->normal(repl, rev_lookup, modifier);
    GINAC_ASSERT(is_a<lst>(e));
 
    // Re-insert replaced symbols
    if (repl.empty())
        return e;
    else {
        for(int i=0; i < modifier.nops(); ++i)
            e = e.subs(modifier.op(i), subs_options::no_pattern);
 
        return e.subs(repl, subs_options::no_pattern);
    }
}
 
 
ex ex::to_rational(exmap & repl) const
{
    return bp->to_rational(repl);
}
 
ex ex::to_polynomial(exmap & repl) const
{
    return bp->to_polynomial(repl);
}
 
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);
}
 
 
ex symbol::to_rational(exmap & repl) const
{
    return *this;
}
 
ex symbol::to_polynomial(exmap & repl) const
{
    return *this;
}
 
 
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;
}
 
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;
}
 
 
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);
}
 
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);
}
 
 
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());
}
 
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());
}
 
 
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 (gc.is_zero())
                gc = x;
            else
                gc = gcd(gc, x);
 
            terms.push_back(x);
        }
 
        if (gc.is_equal(_ex1))
            return e;
 
        if (gc.is_zero())
            return _ex0;
 
        // 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;
}
 
 
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;
}
 
 
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