- input parser recognizes "sqrt()", which is also used in the output
[ginac.git] / ginac / normal.cpp
index 59323798b1cfa10fcf6faf223126deeff7ca7068..a161f82133a7a448ede06ff831868e9bb31e91d2 100644 (file)
@@ -6,7 +6,7 @@
  *  computation, square-free factorization and rational function normalization. */
 
 /*
- *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2001 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
@@ -23,7 +23,6 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <stdexcept>
 #include <algorithm>
 #include <map>
 
 #include "constant.h"
 #include "expairseq.h"
 #include "fail.h"
-#include "indexed.h"
 #include "inifcns.h"
 #include "lst.h"
 #include "mul.h"
-#include "ncmul.h"
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
+#include "matrix.h"
 #include "pseries.h"
 #include "symbol.h"
 #include "utils.h"
 
-#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_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
@@ -59,7 +55,8 @@ namespace GiNaC {
 #define USE_REMEMBER 0
 
 // Set this if you want divide_in_z() to use trial division followed by
-// polynomial interpolation (usually slower except for very large problems)
+// 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
@@ -77,10 +74,10 @@ static int heur_gcd_failed = 0;
 static struct _stat_print {
        _stat_print() {}
        ~_stat_print() {
-               cout << "gcd() called " << gcd_called << " times\n";
-               cout << "sr_gcd() called " << sr_gcd_called << " times\n";
-               cout << "heur_gcd() called " << heur_gcd_called << " times\n";
-               cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+               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
@@ -95,18 +92,18 @@ static struct _stat_print {
  *  @return "false" if no symbol was found, "true" otherwise */
 static bool get_first_symbol(const ex &e, const symbol *&x)
 {
-    if (is_ex_exactly_of_type(e, symbol)) {
-        x = static_cast<symbol *>(e.bp);
-        return true;
-    } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
-        for (unsigned i=0; i<e.nops(); i++)
-            if (get_first_symbol(e.op(i), x))
-                return true;
-    } else if (is_ex_exactly_of_type(e, power)) {
-        if (get_first_symbol(e.op(0), x))
-            return true;
-    }
-    return false;
+       if (is_ex_exactly_of_type(e, symbol)) {
+               x = &ex_to<symbol>(e);
+               return true;
+       } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
+               for (unsigned i=0; i<e.nops(); i++)
+                       if (get_first_symbol(e.op(i), x))
+                               return true;
+       } else if (is_ex_exactly_of_type(e, power)) {
+               if (get_first_symbol(e.op(0), x))
+                       return true;
+       }
+       return false;
 }
 
 
@@ -121,56 +118,65 @@ static bool get_first_symbol(const ex &e, const symbol *&x)
  *
  *  @see get_symbol_stats */
 struct sym_desc {
-    /** Pointer to symbol */
-    const symbol *sym;
+       /** Pointer to symbol */
+       const symbol *sym;
 
-    /** Highest degree of symbol in polynomial "a" */
-    int deg_a;
+       /** Highest degree of symbol in polynomial "a" */
+       int deg_a;
 
-    /** Highest degree of symbol in polynomial "b" */
-    int deg_b;
+       /** 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 "a" */
+       int ldeg_a;
 
-    /** Lowest degree of symbol in polynomial "b" */
-    int ldeg_b;
+       /** Lowest degree of symbol in polynomial "b" */
+       int ldeg_b;
 
-    /** Maximum of deg_a and deg_b (Used for sorting) */
-    int max_deg;
+       /** Maximum of deg_a and deg_b (Used for sorting) */
+       int max_deg;
 
-    /** Commparison operator for sorting */
-    bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
+       /** Maximum number of terms of leading coefficient of symbol in both polynomials */
+       int max_lcnops;
+
+       /** Commparison 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 vector<sym_desc> sym_desc_vec;
+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 symbol *s, sym_desc_vec &v)
 {
-    sym_desc_vec::iterator it = v.begin(), itend = v.end();
-    while (it != itend) {
-        if (it->sym->compare(*s) == 0)  // If it's already in there, don't add it a second time
-            return;
-        it++;
-    }
-    sym_desc d;
-    d.sym = s;
-    v.push_back(d);
+       sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
+       while (it != itend) {
+               if (it->sym->compare(*s) == 0)  // If it's already in there, don't add it a second time
+                       return;
+               ++it;
+       }
+       sym_desc d;
+       d.sym = s;
+       v.push_back(d);
 }
 
 // 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_ex_exactly_of_type(e, symbol)) {
-        add_symbol(static_cast<symbol *>(e.bp), v);
-    } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
-        for (unsigned i=0; i<e.nops(); i++)
-            collect_symbols(e.op(i), v);
-    } else if (is_ex_exactly_of_type(e, power)) {
-        collect_symbols(e.op(0), v);
-    }
+       if (is_ex_exactly_of_type(e, symbol)) {
+               add_symbol(&ex_to<symbol>(e), v);
+       } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
+               for (unsigned i=0; i<e.nops(); i++)
+                       collect_symbols(e.op(i), v);
+       } else if (is_ex_exactly_of_type(e, power)) {
+               collect_symbols(e.op(0), v);
+       }
 }
 
 /** Collect statistical information about symbols in polynomials.
@@ -187,27 +193,28 @@ static void collect_symbols(const ex &e, sym_desc_vec &v)
  *  @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.eval(), v);   // eval() to expand assigned symbols
-    collect_symbols(b.eval(), v);
-    sym_desc_vec::iterator it = v.begin(), itend = v.end();
-    while (it != itend) {
-        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 = max(deg_a, deg_b);
-        it->ldeg_a = a.ldegree(*(it->sym));
-        it->ldeg_b = b.ldegree(*(it->sym));
-        it++;
-    }
-    sort(v.begin(), v.end());
+       collect_symbols(a.eval(), v);   // eval() to expand assigned symbols
+       collect_symbols(b.eval(), v);
+       sym_desc_vec::iterator it = v.begin(), itend = v.end();
+       while (it != itend) {
+               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));
+               ++it;
+       }
+       std::sort(v.begin(), v.end());
 #if 0
-       clog << "Symbols:\n";
+       std::clog << "Symbols:\n";
        it = v.begin(); itend = v.end();
        while (it != itend) {
-               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 << endl;
-               clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
-               it++;
+               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
 }
@@ -221,21 +228,25 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
 // 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_ex_exactly_of_type(e, add)) {
-        numeric c = _num1();
-        for (unsigned i=0; i<e.nops(); i++)
-            c = lcmcoeff(e.op(i), c);
-        return lcm(c, l);
-    } else if (is_ex_exactly_of_type(e, mul)) {
-        numeric c = _num1();
-        for (unsigned i=0; i<e.nops(); i++)
-            c *= lcmcoeff(e.op(i), _num1());
-        return lcm(c, l);
-    } else if (is_ex_exactly_of_type(e, power))
-        return pow(lcmcoeff(e.op(0), l), ex_to_numeric(e.op(1)));
-    return l;
+       if (e.info(info_flags::rational))
+               return lcm(ex_to<numeric>(e).denom(), l);
+       else if (is_ex_exactly_of_type(e, add)) {
+               numeric c = _num1;
+               for (unsigned i=0; i<e.nops(); i++)
+                       c = lcmcoeff(e.op(i), c);
+               return lcm(c, l);
+       } else if (is_ex_exactly_of_type(e, mul)) {
+               numeric c = _num1;
+               for (unsigned i=0; i<e.nops(); i++)
+                       c *= lcmcoeff(e.op(i), _num1);
+               return lcm(c, l);
+       } else if (is_ex_exactly_of_type(e, power)) {
+               if (is_ex_exactly_of_type(e.op(0), symbol))
+                       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.
@@ -247,7 +258,7 @@ static numeric lcmcoeff(const ex &e, const numeric &l)
  *  @return LCM of denominators of coefficients */
 static numeric lcm_of_coefficients_denominators(const ex &e)
 {
-    return lcmcoeff(e, _num1());
+       return lcmcoeff(e, _num1);
 }
 
 /** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
@@ -258,22 +269,27 @@ static numeric lcm_of_coefficients_denominators(const ex &e)
 static ex multiply_lcm(const ex &e, const numeric &lcm)
 {
        if (is_ex_exactly_of_type(e, mul)) {
-               ex c = _ex1();
-               numeric lcm_accum = _num1();
+               unsigned num = e.nops();
+               exvector v; v.reserve(num + 1);
+               numeric lcm_accum = _num1;
                for (unsigned i=0; i<e.nops(); i++) {
-                       numeric op_lcm = lcmcoeff(e.op(i), _num1());
-                       c *= multiply_lcm(e.op(i), op_lcm);
+                       numeric op_lcm = lcmcoeff(e.op(i), _num1);
+                       v.push_back(multiply_lcm(e.op(i), op_lcm));
                        lcm_accum *= op_lcm;
                }
-               c *= lcm / lcm_accum;
-               return c;
+               v.push_back(lcm / lcm_accum);
+               return (new mul(v))->setflag(status_flags::dynallocated);
        } else if (is_ex_exactly_of_type(e, add)) {
-               ex c = _ex0();
-               for (unsigned i=0; i<e.nops(); i++)
-                       c += multiply_lcm(e.op(i), lcm);
-               return c;
+               unsigned num = e.nops();
+               exvector v; v.reserve(num);
+               for (unsigned i=0; i<num; i++)
+                       v.push_back(multiply_lcm(e.op(i), lcm));
+               return (new add(v))->setflag(status_flags::dynallocated);
        } else if (is_ex_exactly_of_type(e, power)) {
-               return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
+               if (is_ex_exactly_of_type(e.op(0), symbol))
+                       return e * lcm;
+               else
+                       return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
        } else
                return e * lcm;
 }
@@ -286,48 +302,48 @@ static ex multiply_lcm(const ex &e, const numeric &lcm)
  *  @return integer content */
 numeric ex::integer_content(void) const
 {
-    GINAC_ASSERT(bp!=0);
-    return bp->integer_content();
+       GINAC_ASSERT(bp!=0);
+       return bp->integer_content();
 }
 
 numeric basic::integer_content(void) const
 {
-    return _num1();
+       return _num1;
 }
 
 numeric numeric::integer_content(void) const
 {
-    return abs(*this);
+       return abs(*this);
 }
 
 numeric add::integer_content(void) const
 {
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    numeric c = _num0();
-    while (it != itend) {
-        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-        GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
-        c = gcd(ex_to_numeric(it->coeff), c);
-        it++;
-    }
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    c = gcd(ex_to_numeric(overall_coeff),c);
-    return c;
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       numeric c = _num0;
+       while (it != itend) {
+               GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+               GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
+               c = gcd(ex_to<numeric>(it->coeff), c);
+               it++;
+       }
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       c = gcd(ex_to<numeric>(overall_coeff),c);
+       return c;
 }
 
 numeric mul::integer_content(void) const
 {
 #ifdef DO_GINAC_ASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        ++it;
-    }
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       while (it != itend) {
+               GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+               ++it;
+       }
 #endif // def DO_GINAC_ASSERT
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    return abs(ex_to_numeric(overall_coeff));
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       return abs(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -346,42 +362,42 @@ numeric mul::integer_content(void) const
  *  @return quotient of a and b in Q[x] */
 ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
 {
-    if (b.is_zero())
-        throw(std::overflow_error("quo: division by zero"));
-    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
-        return a / b;
+       if (b.is_zero())
+               throw(std::overflow_error("quo: division by zero"));
+       if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
+               return a / b;
 #if FAST_COMPARE
-    if (a.is_equal(b))
-        return _ex1();
+       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 q = _ex0();
-    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_ex_exactly_of_type(blcoeff, numeric);
-    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 *new ex(fail());
-        }
-        term *= power(x, rdeg - bdeg);
-        q += term;
-        r -= (term * b).expand();
-        if (r.is_zero())
-            break;
-        rdeg = r.degree(x);
-    }
-    return q;
+       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_ex_exactly_of_type(blcoeff, numeric);
+       exvector v; v.reserve(rdeg - bdeg + 1);
+       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 (new fail())->setflag(status_flags::dynallocated);
+               }
+               term *= power(x, rdeg - bdeg);
+               v.push_back(term);
+               r -= (term * b).expand();
+               if (r.is_zero())
+                       break;
+               rdeg = r.degree(x);
+       }
+       return (new add(v))->setflag(status_flags::dynallocated);
 }
 
 
@@ -396,44 +412,62 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @return remainder of a(x) and b(x) in Q[x] */
 ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
 {
-    if (b.is_zero())
-        throw(std::overflow_error("rem: division by zero"));
-    if (is_ex_exactly_of_type(a, numeric)) {
-        if  (is_ex_exactly_of_type(b, numeric))
-            return _ex0();
-        else
-            return b;
-    }
+       if (b.is_zero())
+               throw(std::overflow_error("rem: division by zero"));
+       if (is_ex_exactly_of_type(a, numeric)) {
+               if  (is_ex_exactly_of_type(b, numeric))
+                       return _ex0;
+               else
+                       return a;
+       }
 #if FAST_COMPARE
-    if (a.is_equal(b))
-        return _ex0();
+       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_ex_exactly_of_type(blcoeff, numeric);
-    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 *new ex(fail());
-        }
-        term *= power(x, rdeg - bdeg);
-        r -= (term * b).expand();
-        if (r.is_zero())
-            break;
-        rdeg = r.degree(x);
-    }
-    return r;
+       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_ex_exactly_of_type(blcoeff, numeric);
+       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 (new fail())->setflag(status_flags::dynallocated);
+               }
+               term *= power(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 symbol &x)
+{
+       ex nd = numer_denom(a);
+       ex numer = nd.op(0), denom = nd.op(1);
+       ex q = quo(numer, denom, x);
+       if (is_ex_exactly_of_type(q, fail))
+               return a;
+       else
+               return q + rem(numer, denom, x) / denom;
 }
 
 
@@ -447,45 +481,45 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @return pseudo-remainder of a(x) and b(x) in Z[x] */
 ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
 {
-    if (b.is_zero())
-        throw(std::overflow_error("prem: division by zero"));
-    if (is_ex_exactly_of_type(a, numeric)) {
-        if (is_ex_exactly_of_type(b, numeric))
-            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 * power(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 = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
-        if (rdeg == 0)
-            r = _ex0();
-        else
-            r -= rlcoeff * power(x, rdeg);
-        r = (blcoeff * r).expand() - term;
-        rdeg = r.degree(x);
-        i++;
-    }
-    return power(blcoeff, delta - i) * r;
+       if (b.is_zero())
+               throw(std::overflow_error("prem: division by zero"));
+       if (is_ex_exactly_of_type(a, numeric)) {
+               if (is_ex_exactly_of_type(b, numeric))
+                       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 * power(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 = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
+               if (rdeg == 0)
+                       r = _ex0;
+               else
+                       r -= rlcoeff * power(x, rdeg);
+               r = (blcoeff * r).expand() - term;
+               rdeg = r.degree(x);
+               i++;
+       }
+       return power(blcoeff, delta - i) * r;
 }
 
 
@@ -497,46 +531,45 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @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 Z[x] */
-
 ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
 {
-    if (b.is_zero())
-        throw(std::overflow_error("prem: division by zero"));
-    if (is_ex_exactly_of_type(a, numeric)) {
-        if (is_ex_exactly_of_type(b, numeric))
-            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 * power(x, bdeg);
-    } else
-        blcoeff = _ex1();
-
-    while (rdeg >= bdeg && !r.is_zero()) {
-        ex rlcoeff = r.coeff(x, rdeg);
-        ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
-        if (rdeg == 0)
-            r = _ex0();
-        else
-            r -= rlcoeff * power(x, rdeg);
-        r = (blcoeff * r).expand() - term;
-        rdeg = r.degree(x);
-    }
-    return r;
+       if (b.is_zero())
+               throw(std::overflow_error("prem: division by zero"));
+       if (is_ex_exactly_of_type(a, numeric)) {
+               if (is_ex_exactly_of_type(b, numeric))
+                       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 * power(x, bdeg);
+       } else
+               blcoeff = _ex1;
+
+       while (rdeg >= bdeg && !r.is_zero()) {
+               ex rlcoeff = r.coeff(x, rdeg);
+               ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
+               if (rdeg == 0)
+                       r = _ex0;
+               else
+                       r -= rlcoeff * power(x, rdeg);
+               r = (blcoeff * r).expand() - term;
+               rdeg = r.degree(x);
+       }
+       return r;
 }
 
 
@@ -548,57 +581,63 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @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 */
+ *          "false" otherwise (q left untouched) */
 bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 {
-    q = _ex0();
-    if (b.is_zero())
-        throw(std::overflow_error("divide: division by zero"));
-    if (a.is_zero())
-        return true;
-    if (is_ex_exactly_of_type(b, numeric)) {
-        q = a / b;
-        return true;
-    } else if (is_ex_exactly_of_type(a, numeric))
-        return false;
+       if (b.is_zero())
+               throw(std::overflow_error("divide: division by zero"));
+       if (a.is_zero()) {
+               q = _ex0;
+               return true;
+       }
+       if (is_ex_exactly_of_type(b, numeric)) {
+               q = a / b;
+               return true;
+       } else if (is_ex_exactly_of_type(a, numeric))
+               return false;
 #if FAST_COMPARE
-    if (a.is_equal(b)) {
-        q = _ex1();
-        return true;
-    }
+       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
-    const symbol *x;
-    if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
-        throw(std::invalid_argument("invalid expression in divide()"));
-
-    // Polynomial long division (recursive)
-    ex r = a.expand();
-    if (r.is_zero())
-        return true;
-    int bdeg = b.degree(*x);
-    int rdeg = r.degree(*x);
-    ex blcoeff = b.expand().coeff(*x, bdeg);
-    bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
-    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 *= power(*x, rdeg - bdeg);
-        q += term;
-        r -= (term * b).expand();
-        if (r.is_zero())
-            return true;
-        rdeg = r.degree(*x);
-    }
-    return false;
+       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
+       const symbol *x;
+       if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
+               throw(std::invalid_argument("invalid expression in divide()"));
+
+       // 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_ex_exactly_of_type(blcoeff, numeric);
+       exvector v; v.reserve(rdeg - bdeg + 1);
+       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 *= power(*x, rdeg - bdeg);
+               v.push_back(term);
+               r -= (term * b).expand();
+               if (r.is_zero()) {
+                       q = (new add(v))->setflag(status_flags::dynallocated);
+                       return true;
+               }
+               rdeg = r.degree(*x);
+       }
+       return false;
 }
 
 
@@ -607,17 +646,18 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
  *  Remembering
  */
 
-typedef pair<ex, ex> ex2;
-typedef pair<ex, bool> exbool;
+typedef std::pair<ex, ex> ex2;
+typedef std::pair<ex, bool> exbool;
 
 struct ex2_less {
-    bool operator() (const ex2 p, const ex2 q) const 
-    {
-        return p.first.compare(q.first) < 0 || (!(q.first.compare(p.first) < 0) && p.second.compare(q.second) < 0);        
-    }
+       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 map<ex2, exbool, ex2_less> ex2_exbool_remember;
+typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
 #endif
 
 
@@ -639,127 +679,129 @@ typedef map<ex2, exbool, ex2_less> ex2_exbool_remember;
  *  @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_ex_exactly_of_type(a, numeric)) {
-        if (is_ex_exactly_of_type(b, numeric)) {
-            q = a / b;
-            return q.info(info_flags::integer);
-        } else
-            return false;
-    }
+       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_ex_exactly_of_type(a, numeric)) {
+               if (is_ex_exactly_of_type(b, numeric)) {
+                       q = a / b;
+                       return q.info(info_flags::integer);
+               } else
+                       return false;
+       }
 #if FAST_COMPARE
-    if (a.is_equal(b)) {
-        q = _ex1();
-        return true;
-    }
+       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;
-    }
+       // 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
 
-    // Main symbol
-    const symbol *x = var->sym;
+       // Main symbol
+       const symbol *x = var->sym;
 
-    // Compare degrees
-    int adeg = a.degree(*x), bdeg = b.degree(*x);
-    if (bdeg > adeg)
-        return false;
+       // 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();
-    ex c;
-    for (i=0; i<=adeg; i++) {
-        ex bs = b.subs(*x == point);
-        while (bs.is_zero()) {
-            point += _num1();
-            bs = b.subs(*x == point);
-        }
-        if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
-            return false;
-        alpha.push_back(point);
-        u.push_back(c);
-        point += _num1();
-    }
-
-    // Compute inverses
-    vector<numeric> rcp; rcp.reserve(adeg + 1);
-    rcp.push_back(_num0());
-    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;
+       // 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;
+       ex c;
+       for (i=0; i<=adeg; i++) {
+               ex bs = b.subs(*x == point);
+               while (bs.is_zero()) {
+                       point += _num1;
+                       bs = b.subs(*x == point);
+               }
+               if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
+                       return false;
+               alpha.push_back(point);
+               u.push_back(c);
+               point += _num1;
+       }
+
+       // Compute inverses
+       vector<numeric> rcp; rcp.reserve(adeg + 1);
+       rcp.push_back(_num0);
+       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);
-    while (rdeg >= bdeg) {
-        ex term, rcoeff = r.coeff(*x, rdeg);
-        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
-            break;
-        term = (term * power(*x, rdeg - bdeg)).expand();
-        q += term;
-        r -= (term * eb).expand();
-        if (r.is_zero()) {
+       // 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(rdeg - bdeg + 1);
+       while (rdeg >= bdeg) {
+               ex term, rcoeff = r.coeff(*x, rdeg);
+               if (!divide_in_z(rcoeff, blcoeff, term, var+1))
+                       break;
+               term = (term * power(*x, rdeg - bdeg)).expand();
+               v.push_back(term);
+               r -= (term * eb).expand();
+               if (r.is_zero()) {
+                       q = (new add(v))->setflag(status_flags::dynallocated);
 #if USE_REMEMBER
-            dr_remember[ex2(a, b)] = exbool(q, true);
+                       dr_remember[ex2(a, b)] = exbool(q, true);
 #endif
-            return true;
-        }
-        rdeg = r.degree(*x);
-    }
+                       return true;
+               }
+               rdeg = r.degree(*x);
+       }
 #if USE_REMEMBER
-    dr_remember[ex2(a, b)] = exbool(q, false);
+       dr_remember[ex2(a, b)] = exbool(q, false);
 #endif
-    return false;
+       return false;
 
 #endif
 }
@@ -778,16 +820,16 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
  *  @see ex::content, ex::primpart */
 ex ex::unit(const symbol &x) const
 {
-    ex c = expand().lcoeff(x);
-    if (is_ex_exactly_of_type(c, numeric))
-        return c < _ex0() ? _ex_1() : _ex1();
-    else {
-        const symbol *y;
-        if (get_first_symbol(c, y))
-            return c.unit(*y);
-        else
-            throw(std::invalid_argument("invalid expression in unit()"));
-    }
+       ex c = expand().lcoeff(x);
+       if (is_ex_exactly_of_type(c, numeric))
+               return c < _ex0 ? _ex_1 : _ex1;
+       else {
+               const symbol *y;
+               if (get_first_symbol(c, y))
+                       return c.unit(*y);
+               else
+                       throw(std::invalid_argument("invalid expression in unit()"));
+       }
 }
 
 
@@ -800,30 +842,30 @@ ex ex::unit(const symbol &x) const
  *  @see ex::unit, ex::primpart */
 ex ex::content(const symbol &x) const
 {
-    if (is_zero())
-        return _ex0();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return info(info_flags::negative) ? -*this : *this;
-    ex e = expand();
-    if (e.is_zero())
-        return _ex0();
-
-    // First, try the integer content
-    ex c = e.integer_content();
-    ex r = e / c;
-    ex lcoeff = r.lcoeff(x);
-    if (lcoeff.info(info_flags::integer))
-        return c;
-
-    // GCD of all coefficients
-    int deg = e.degree(x);
-    int ldeg = e.ldegree(x);
-    if (deg == ldeg)
-        return e.lcoeff(x) / e.unit(x);
-    c = _ex0();
-    for (int i=ldeg; i<=deg; i++)
-        c = gcd(e.coeff(x, i), c, NULL, NULL, false);
-    return c;
+       if (is_zero())
+               return _ex0;
+       if (is_ex_exactly_of_type(*this, numeric))
+               return info(info_flags::negative) ? -*this : *this;
+       ex e = expand();
+       if (e.is_zero())
+               return _ex0;
+
+       // First, try the integer content
+       ex c = e.integer_content();
+       ex r = e / c;
+       ex lcoeff = r.lcoeff(x);
+       if (lcoeff.info(info_flags::integer))
+               return c;
+
+       // GCD of all coefficients
+       int deg = e.degree(x);
+       int ldeg = e.ldegree(x);
+       if (deg == ldeg)
+               return e.lcoeff(x) / e.unit(x);
+       c = _ex0;
+       for (int i=ldeg; i<=deg; i++)
+               c = gcd(e.coeff(x, i), c, NULL, NULL, false);
+       return c;
 }
 
 
@@ -836,19 +878,19 @@ ex ex::content(const symbol &x) const
  *  @see ex::unit, ex::content */
 ex ex::primpart(const symbol &x) const
 {
-    if (is_zero())
-        return _ex0();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return _ex1();
-
-    ex c = content(x);
-    if (c.is_zero())
-        return _ex0();
-    ex u = unit(x);
-    if (is_ex_exactly_of_type(c, numeric))
-        return *this / (c * u);
-    else
-        return quo(*this, c * u, x, false);
+       if (is_zero())
+               return _ex0;
+       if (is_ex_exactly_of_type(*this, numeric))
+               return _ex1;
+
+       ex c = content(x);
+       if (c.is_zero())
+               return _ex0;
+       ex u = unit(x);
+       if (is_ex_exactly_of_type(c, numeric))
+               return *this / (c * u);
+       else
+               return quo(*this, c * u, x, false);
 }
 
 
@@ -861,18 +903,18 @@ ex ex::primpart(const symbol &x) const
  *  @return primitive part */
 ex ex::primpart(const symbol &x, const ex &c) const
 {
-    if (is_zero())
-        return _ex0();
-    if (c.is_zero())
-        return _ex0();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return _ex1();
-
-    ex u = unit(x);
-    if (is_ex_exactly_of_type(c, numeric))
-        return *this / (c * u);
-    else
-        return quo(*this, c * u, x, false);
+       if (is_zero())
+               return _ex0;
+       if (c.is_zero())
+               return _ex0;
+       if (is_ex_exactly_of_type(*this, numeric))
+               return _ex1;
+
+       ex u = unit(x);
+       if (is_ex_exactly_of_type(c, numeric))
+               return *this / (c * u);
+       else
+               return quo(*this, c * u, x, false);
 }
 
 
@@ -892,33 +934,33 @@ ex ex::primpart(const symbol &x, const ex &c) const
 
 static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
 {
-//clog << "eu_gcd(" << a << "," << b << ")\n";
-
-    // Sort c and d so that c has higher degree
-    ex c, d;
-    int adeg = a.degree(*x), bdeg = b.degree(*x);
-    if (adeg >= bdeg) {
-        c = a;
-        d = b;
-    } else {
-        c = b;
-        d = a;
-    }
+//std::clog << "eu_gcd(" << a << "," << b << ")\n";
+
+       // Sort c and d so that c has higher degree
+       ex c, d;
+       int adeg = a.degree(*x), bdeg = b.degree(*x);
+       if (adeg >= bdeg) {
+               c = a;
+               d = b;
+       } else {
+               c = b;
+               d = a;
+       }
 
        // Normalize in Q[x]
        c = c / c.lcoeff(*x);
        d = d / d.lcoeff(*x);
 
        // Euclidean algorithm
-    ex r;
-    for (;;) {
-//clog << " d = " << d << endl;
-        r = rem(c, d, *x, false);
-        if (r.is_zero())
-            return d / d.lcoeff(*x);
-        c = d;
+       ex r;
+       for (;;) {
+//std::clog << " d = " << d << endl;
+               r = rem(c, d, *x, false);
+               if (r.is_zero())
+                       return d / d.lcoeff(*x);
+               c = d;
                d = r;
-    }
+       }
 }
 
 
@@ -934,32 +976,32 @@ static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
 
 static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
 {
-//clog << "euprem_gcd(" << a << "," << b << ")\n";
-
-    // Sort c and d so that c has higher degree
-    ex c, d;
-    int adeg = a.degree(*x), bdeg = b.degree(*x);
-    if (adeg >= bdeg) {
-        c = a;
-        d = b;
-    } else {
-        c = b;
-        d = a;
-    }
+//std::clog << "euprem_gcd(" << a << "," << b << ")\n";
+
+       // Sort c and d so that c has higher degree
+       ex c, d;
+       int adeg = a.degree(*x), bdeg = b.degree(*x);
+       if (adeg >= bdeg) {
+               c = a;
+               d = b;
+       } else {
+               c = b;
+               d = a;
+       }
 
        // Calculate GCD of contents
        ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false);
 
        // Euclidean algorithm with pseudo-remainders
-    ex r;
-    for (;;) {
-//clog << " d = " << d << endl;
-        r = prem(c, d, *x, false);
-        if (r.is_zero())
-            return d.primpart(*x) * gamma;
-        c = d;
+       ex r;
+       for (;;) {
+//std::clog << " d = " << d << endl;
+               r = prem(c, d, *x, false);
+               if (r.is_zero())
+                       return d.primpart(*x) * gamma;
+               c = d;
                d = r;
-    }
+       }
 }
 
 
@@ -975,41 +1017,41 @@ static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
 
 static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
 {
-//clog << "peu_gcd(" << a << "," << b << ")\n";
-
-    // Sort c and d so that c has higher degree
-    ex c, d;
-    int adeg = a.degree(*x), bdeg = b.degree(*x);
-    int ddeg;
-    if (adeg >= bdeg) {
-        c = a;
-        d = b;
-        ddeg = bdeg;
-    } else {
-        c = b;
-        d = a;
-        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, NULL, NULL, false);
-    if (ddeg == 0)
-        return gamma;
-    c = c.primpart(*x, cont_c);
-    d = d.primpart(*x, cont_d);
-
-    // Euclidean algorithm with content removal
+//std::clog << "peu_gcd(" << a << "," << b << ")\n";
+
+       // Sort c and d so that c has higher degree
+       ex c, d;
+       int adeg = a.degree(*x), bdeg = b.degree(*x);
+       int ddeg;
+       if (adeg >= bdeg) {
+               c = a;
+               d = b;
+               ddeg = bdeg;
+       } else {
+               c = b;
+               d = a;
+               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, NULL, NULL, false);
+       if (ddeg == 0)
+               return gamma;
+       c = c.primpart(*x, cont_c);
+       d = d.primpart(*x, cont_d);
+
+       // Euclidean algorithm with content removal
        ex r;
-    for (;;) {
-//clog << " d = " << d << endl;
-        r = prem(c, d, *x, false);
-        if (r.is_zero())
-            return gamma * d;
-        c = d;
+       for (;;) {
+//std::clog << " d = " << d << endl;
+               r = prem(c, d, *x, false);
+               if (r.is_zero())
+                       return gamma * d;
+               c = d;
                d = r.primpart(*x);
-    }
+       }
 }
 
 
@@ -1024,59 +1066,59 @@ static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
 
 static ex red_gcd(const ex &a, const ex &b, const symbol *x)
 {
-//clog << "red_gcd(" << a << "," << b << ")\n";
-
-    // 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, NULL, NULL, false);
-    if (ddeg == 0)
-        return gamma;
-    c = c.primpart(*x, cont_c);
-    d = d.primpart(*x, cont_d);
-
-    // First element of divisor sequence
-    ex r, ri = _ex1();
-    int delta = cdeg - ddeg;
-
-    for (;;) {
-        // Calculate polynomial pseudo-remainder
-//clog << " d = " << d << endl;
-        r = prem(c, d, *x, false);
-        if (r.is_zero())
-            return gamma * d.primpart(*x);
-        c = d;
-        cdeg = ddeg;
-
-        if (!divide(r, pow(ri, delta), d, false))
-            throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
-        ddeg = d.degree(*x);
-        if (ddeg == 0) {
-            if (is_ex_exactly_of_type(r, numeric))
-                return gamma;
-            else
-                return gamma * r.primpart(*x);
-        }
-
-        ri = c.expand().lcoeff(*x);
-        delta = cdeg - ddeg;
-    }
+//std::clog << "red_gcd(" << a << "," << b << ")\n";
+
+       // 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, NULL, NULL, false);
+       if (ddeg == 0)
+               return gamma;
+       c = c.primpart(*x, cont_c);
+       d = d.primpart(*x, cont_d);
+
+       // First element of divisor sequence
+       ex r, ri = _ex1;
+       int delta = cdeg - ddeg;
+
+       for (;;) {
+               // Calculate polynomial pseudo-remainder
+//std::clog << " d = " << d << endl;
+               r = prem(c, d, *x, false);
+               if (r.is_zero())
+                       return gamma * d.primpart(*x);
+               c = d;
+               cdeg = ddeg;
+
+               if (!divide(r, pow(ri, delta), d, false))
+                       throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
+               ddeg = d.degree(*x);
+               if (ddeg == 0) {
+                       if (is_ex_exactly_of_type(r, numeric))
+                               return gamma;
+                       else
+                               return gamma * r.primpart(*x);
+               }
+
+               ri = c.expand().lcoeff(*x);
+               delta = cdeg - ddeg;
+       }
 }
 
 
@@ -1091,73 +1133,73 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x)
 
 static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
 {
-//clog << "sr_gcd(" << a << "," << b << ")\n";
+//std::clog << "sr_gcd(" << a << "," << b << ")\n";
 #if STATISTICS
        sr_gcd_called++;
 #endif
 
-    // The first symbol is our main variable
-    const symbol &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, NULL, NULL, false);
-    if (ddeg == 0)
-        return gamma;
-    c = c.primpart(x, cont_c);
-    d = d.primpart(x, cont_d);
-//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
-
-    // First element of subresultant sequence
-    ex r = _ex0(), ri = _ex1(), psi = _ex1();
-    int delta = cdeg - ddeg;
-
-    for (;;) {
-        // Calculate polynomial pseudo-remainder
-//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
-//clog << " d = " << d << endl;
-        r = prem(c, d, x, false);
-        if (r.is_zero())
-            return gamma * d.primpart(x);
-        c = d;
-        cdeg = ddeg;
-//clog << " dividing...\n";
-        if (!divide_in_z(r, ri * pow(psi, delta), d, var+1))
-            throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
-        ddeg = d.degree(x);
-        if (ddeg == 0) {
-            if (is_ex_exactly_of_type(r, numeric))
-                return gamma;
-            else
-                return gamma * r.primpart(x);
-        }
-
-        // Next element of subresultant sequence
-//clog << " calculating next subresultant...\n";
-        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;
-    }
+       // The first symbol is our main variable
+       const symbol &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, NULL, NULL, false);
+       if (ddeg == 0)
+               return gamma;
+       c = c.primpart(x, cont_c);
+       d = d.primpart(x, cont_d);
+//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
+
+       // First element of subresultant sequence
+       ex r = _ex0, ri = _ex1, psi = _ex1;
+       int delta = cdeg - ddeg;
+
+       for (;;) {
+               // Calculate polynomial pseudo-remainder
+//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//std::clog << " d = " << d << endl;
+               r = prem(c, d, x, false);
+               if (r.is_zero())
+                       return gamma * d.primpart(x);
+               c = d;
+               cdeg = ddeg;
+//std::clog << " dividing...\n";
+               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_ex_exactly_of_type(r, numeric))
+                               return gamma;
+                       else
+                               return gamma * r.primpart(x);
+               }
+
+               // Next element of subresultant sequence
+//std::clog << " calculating next subresultant...\n";
+               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;
+       }
 }
 
 
@@ -1169,128 +1211,121 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
  *  @see heur_gcd */
 numeric ex::max_coefficient(void) const
 {
-    GINAC_ASSERT(bp!=0);
-    return bp->max_coefficient();
+       GINAC_ASSERT(bp!=0);
+       return bp->max_coefficient();
 }
 
+/** Implementation ex::max_coefficient().
+ *  @see heur_gcd */
 numeric basic::max_coefficient(void) const
 {
-    return _num1();
+       return _num1;
 }
 
 numeric numeric::max_coefficient(void) const
 {
-    return abs(*this);
+       return abs(*this);
 }
 
 numeric add::max_coefficient(void) const
 {
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    numeric cur_max = abs(ex_to_numeric(overall_coeff));
-    while (it != itend) {
-        numeric a;
-        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-        a = abs(ex_to_numeric(it->coeff));
-        if (a > cur_max)
-            cur_max = a;
-        it++;
-    }
-    return cur_max;
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       numeric cur_max = abs(ex_to<numeric>(overall_coeff));
+       while (it != itend) {
+               numeric a;
+               GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+               a = abs(ex_to<numeric>(it->coeff));
+               if (a > cur_max)
+                       cur_max = a;
+               it++;
+       }
+       return cur_max;
 }
 
 numeric mul::max_coefficient(void) const
 {
 #ifdef DO_GINAC_ASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        it++;
-    }
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       while (it != itend) {
+               GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+               it++;
+       }
 #endif // def DO_GINAC_ASSERT
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    return abs(ex_to_numeric(overall_coeff));
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       return abs(ex_to<numeric>(overall_coeff));
 }
 
 
-/** Apply symmetric modular homomorphism to a multivariate polynomial.
- *  This function is used internally by heur_gcd().
+/** Apply symmetric modular homomorphism to an expanded multivariate
+ *  polynomial.  This function is usually used internally by heur_gcd().
  *
- *  @param e  expanded multivariate polynomial
  *  @param xi  modulus
  *  @return mapped polynomial
  *  @see heur_gcd */
-ex ex::smod(const numeric &xi) const
-{
-    GINAC_ASSERT(bp!=0);
-    return bp->smod(xi);
-}
-
 ex basic::smod(const numeric &xi) const
 {
-    return *this;
+       return *this;
 }
 
 ex numeric::smod(const numeric &xi) const
 {
-#ifndef NO_NAMESPACE_GINAC
-    return GiNaC::smod(*this, xi);
-#else // ndef NO_NAMESPACE_GINAC
-    return ::smod(*this, xi);
-#endif // ndef NO_NAMESPACE_GINAC
+       return GiNaC::smod(*this, xi);
 }
 
 ex add::smod(const numeric &xi) const
 {
-    epvector newseq;
-    newseq.reserve(seq.size()+1);
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-#ifndef NO_NAMESPACE_GINAC
-        numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
-#else // ndef NO_NAMESPACE_GINAC
-        numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
-#endif // ndef NO_NAMESPACE_GINAC
-        if (!coeff.is_zero())
-            newseq.push_back(expair(it->rest, coeff));
-        it++;
-    }
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_NAMESPACE_GINAC
-    numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
-#else // ndef NO_NAMESPACE_GINAC
-    numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
-#endif // ndef NO_NAMESPACE_GINAC
-    return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
+       epvector newseq;
+       newseq.reserve(seq.size()+1);
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       while (it != itend) {
+               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));
+               it++;
+       }
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
+       return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
 ex mul::smod(const numeric &xi) const
 {
 #ifdef DO_GINAC_ASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        it++;
-    }
+       epvector::const_iterator it = seq.begin();
+       epvector::const_iterator itend = seq.end();
+       while (it != itend) {
+               GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+               it++;
+       }
 #endif // def DO_GINAC_ASSERT
-    mul * mulcopyp=new mul(*this);
-    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_NAMESPACE_GINAC
-    mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
-#else // ndef NO_NAMESPACE_GINAC
-    mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
-#endif // ndef NO_NAMESPACE_GINAC
-    mulcopyp->clearflag(status_flags::evaluated);
-    mulcopyp->clearflag(status_flags::hash_calculated);
-    return mulcopyp->setflag(status_flags::dynallocated);
+       mul * mulcopyp = new mul(*this);
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
+       mulcopyp->clearflag(status_flags::evaluated);
+       mulcopyp->clearflag(status_flags::hash_calculated);
+       return mulcopyp->setflag(status_flags::dynallocated);
 }
 
 
+/** xi-adic polynomial interpolation */
+static ex interpolate(const ex &gamma, const numeric &xi, const symbol &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 * power(x, i));
+               e = (e - gi) * rxi;
+       }
+       return (new add(g))->setflag(status_flags::dynallocated);
+}
+
 /** Exception thrown by heur_gcd() to signal failure. */
 class gcdheu_failed {};
 
@@ -1311,80 +1346,106 @@ class gcdheu_failed {};
  *  @exception gcdheu_failed() */
 static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
 {
-//clog << "heur_gcd(" << a << "," << b << ")\n";
+//std::clog << "heur_gcd(" << a << "," << b << ")\n";
 #if STATISTICS
        heur_gcd_called++;
 #endif
 
+       // Algorithm only works for non-vanishing input polynomials
+       if (a.is_zero() || b.is_zero())
+               return (new fail())->setflag(status_flags::dynallocated);
+
        // GCD of two numeric values -> CLN
-    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
-        numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
-        numeric rg;
-        if (ca || cb)
-            rg = g.inverse();
-        if (ca)
-            *ca = ex_to_numeric(a).mul(rg);
-        if (cb)
-            *cb = ex_to_numeric(b).mul(rg);
-        return g;
-    }
-
-    // The first symbol is our main variable
-    const symbol &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 = max(p.degree(x), q.degree(x));
-
-    // Find evaluation point
-    numeric mp = p.max_coefficient(), mq = q.max_coefficient();
-    numeric xi;
-    if (mp > mq)
-        xi = mq * _num2() + _num2();
-    else
-        xi = mp * _num2() + _num2();
-
-    // 6 tries maximum
-    for (int t=0; t<6; t++) {
-        if (xi.int_length() * maxdeg > 100000) {
-//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
-            throw gcdheu_failed();
+       if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+               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;
+               return g;
+       }
+
+       // The first symbol is our main variable
+       const symbol &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 + _num2;
+       else
+               xi = mp * _num2 + _num2;
+
+       // 6 tries maximum
+       for (int t=0; t<6; t++) {
+               if (xi.int_length() * maxdeg > 100000) {
+//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
+                       throw gcdheu_failed();
                }
 
-        // Apply evaluation homomorphism and calculate GCD
-        ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), NULL, NULL, var+1).expand();
-        if (!is_ex_exactly_of_type(gamma, fail)) {
-
-            // Reconstruct polynomial from GCD of mapped polynomials
-            ex g = _ex0();
-            numeric rxi = xi.inverse();
-            for (int i=0; !gamma.is_zero(); i++) {
-                ex gi = gamma.smod(xi);
-                g += gi * power(x, i);
-                gamma = (gamma - gi) * rxi;
-            }
-            // Remove integer content
-            g /= g.integer_content();
-
-            // If the calculated polynomial divides both a and b, 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;
-                ex lc = g.lcoeff(x);
-                if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
-                    return -g;
-                else
-                    return g;
-            }
-        }
-
-        // Next evaluation point
-        xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
-    }
-    return *new ex(fail());
+               // Apply evaluation homomorphism and calculate GCD
+               ex cp, cq;
+               ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand();
+               if (!is_ex_exactly_of_type(gamma, fail)) {
+
+                       // 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;
+                               ex lc = g.lcoeff(x);
+                               if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+                                       return -g;
+                               else
+                                       return g;
+                       }
+#if 0
+                       cp = interpolate(cp, xi, x);
+                       if (divide_in_z(cp, p, g, var)) {
+                               if (divide_in_z(g, q, cb ? *cb : dummy, var)) {
+                                       g *= gc;
+                                       if (ca)
+                                               *ca = cp;
+                                       ex lc = g.lcoeff(x);
+                                       if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+                                               return -g;
+                                       else
+                                               return g;
+                               }
+                       }
+                       cq = interpolate(cq, xi, x);
+                       if (divide_in_z(cq, q, g, var)) {
+                               if (divide_in_z(g, p, ca ? *ca : dummy, var)) {
+                                       g *= gc;
+                                       if (cb)
+                                               *cb = cq;
+                                       ex lc = g.lcoeff(x);
+                                       if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+                                               return -g;
+                                       else
+                                               return g;
+                               }
+                       }
+#endif
+               }
+
+               // Next evaluation point
+               xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
+       }
+       return (new fail())->setflag(status_flags::dynallocated);
 }
 
 
@@ -1398,63 +1459,74 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
  *  @return the GCD as a new expression */
 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 {
-//clog << "gcd(" << a << "," << b << ")\n";
+//std::clog << "gcd(" << a << "," << b << ")\n";
 #if STATISTICS
        gcd_called++;
 #endif
 
        // GCD of numerics -> CLN
-    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
-        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;
-        return g;
-    }
+       if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+               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"));
-    }
+       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 (is_ex_exactly_of_type(a, mul)) {
                if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
                        goto factored_b;
 factored_a:
-               ex g = _ex1();
-               ex acc_ca = _ex1();
+               unsigned num = a.nops();
+               exvector g; g.reserve(num);
+               exvector acc_ca; acc_ca.reserve(num);
                ex part_b = b;
-               for (unsigned i=0; i<a.nops(); i++) {
+               for (unsigned i=0; i<num; i++) {
                        ex part_ca, part_cb;
-                       g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
-                       acc_ca *= part_ca;
+                       g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
+                       acc_ca.push_back(part_ca);
                        part_b = part_cb;
                }
                if (ca)
-                       *ca = acc_ca;
+                       *ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
                if (cb)
                        *cb = part_b;
-               return g;
+               return (new mul(g))->setflag(status_flags::dynallocated);
        } else if (is_ex_exactly_of_type(b, mul)) {
                if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
                        goto factored_a;
 factored_b:
-               ex g = _ex1();
-               ex acc_cb = _ex1();
+               unsigned num = b.nops();
+               exvector g; g.reserve(num);
+               exvector acc_cb; acc_cb.reserve(num);
                ex part_a = a;
-               for (unsigned i=0; i<b.nops(); i++) {
+               for (unsigned i=0; i<num; i++) {
                        ex part_ca, part_cb;
-                       g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
-                       acc_cb *= part_cb;
+                       g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
+                       acc_cb.push_back(part_cb);
                        part_a = part_ca;
                }
                if (ca)
                        *ca = part_a;
                if (cb)
-                       *cb = acc_cb;
-               return g;
+                       *cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
+               return (new mul(g))->setflag(status_flags::dynallocated);
        }
 
 #if FAST_COMPARE
@@ -1467,7 +1539,7 @@ factored_b:
                                ex exp_a = a.op(1), exp_b = b.op(1);
                                if (exp_a < exp_b) {
                                        if (ca)
-                                               *ca = _ex1();
+                                               *ca = _ex1;
                                        if (cb)
                                                *cb = power(p, exp_b - exp_a);
                                        return power(p, exp_a);
@@ -1475,7 +1547,7 @@ factored_b:
                                        if (ca)
                                                *ca = power(p, exp_a - exp_b);
                                        if (cb)
-                                               *cb = _ex1();
+                                               *cb = _ex1;
                                        return power(p, exp_b);
                                }
                        }
@@ -1485,7 +1557,7 @@ factored_b:
                                if (ca)
                                        *ca = power(p, a.op(1) - 1);
                                if (cb)
-                                       *cb = _ex1();
+                                       *cb = _ex1;
                                return p;
                        }
                }
@@ -1494,7 +1566,7 @@ factored_b:
                if (p.is_equal(a)) {
                        // a = p, b = p^n, gcd = p
                        if (ca)
-                               *ca = _ex1();
+                               *ca = _ex1;
                        if (cb)
                                *cb = power(p, b.op(1) - 1);
                        return p;
@@ -1502,84 +1574,84 @@ factored_b:
        }
 #endif
 
-    // Some trivial cases
+       // 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 (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;
-    }
+       if (a.is_equal(b)) {
+               if (ca)
+                       *ca = _ex1;
+               if (cb)
+                       *cb = _ex1;
+               return a;
+       }
 #endif
 
-    // Gather symbol statistics
-    sym_desc_vec sym_stats;
-    get_symbol_stats(a, b, sym_stats);
-
-    // The symbol with least degree is our main variable
-    sym_desc_vec::const_iterator var = sym_stats.begin();
-    const symbol &x = *(var->sym);
-
-    // Cancel trivial common factor
-    int ldeg_a = var->ldeg_a;
-    int ldeg_b = var->ldeg_b;
-    int min_ldeg = min(ldeg_a, ldeg_b);
-    if (min_ldeg > 0) {
-        ex common = power(x, min_ldeg);
-//clog << "trivial common factor " << common << endl;
-        return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
-    }
-
-    // Try to eliminate variables
-    if (var->deg_a == 0) {
-//clog << "eliminating variable " << x << " from b" << endl;
-        ex c = bex.content(x);
-        ex g = gcd(aex, c, ca, cb, false);
-        if (cb)
-            *cb *= bex.unit(x) * bex.primpart(x, c);
-        return g;
-    } else if (var->deg_b == 0) {
-//clog << "eliminating variable " << x << " from a" << endl;
-        ex c = aex.content(x);
-        ex g = gcd(c, bex, ca, cb, false);
-        if (ca)
-            *ca *= aex.unit(x) * aex.primpart(x, c);
-        return g;
-    }
-
-    ex g;
+       // Gather symbol statistics
+       sym_desc_vec sym_stats;
+       get_symbol_stats(a, b, sym_stats);
+
+       // The symbol with least degree is our main variable
+       sym_desc_vec::const_iterator var = sym_stats.begin();
+       const symbol &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 = power(x, min_ldeg);
+//std::clog << "trivial common factor " << common << std::endl;
+               return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
+       }
+
+       // Try to eliminate variables
+       if (var->deg_a == 0) {
+//std::clog << "eliminating variable " << x << " from b" << std::endl;
+               ex c = bex.content(x);
+               ex g = gcd(aex, c, ca, cb, false);
+               if (cb)
+                       *cb *= bex.unit(x) * bex.primpart(x, c);
+               return g;
+       } else if (var->deg_b == 0) {
+//std::clog << "eliminating variable " << x << " from a" << std::endl;
+               ex c = aex.content(x);
+               ex g = gcd(c, bex, ca, cb, false);
+               if (ca)
+                       *ca *= aex.unit(x) * aex.primpart(x, c);
+               return g;
+       }
+
+       ex g;
 #if 1
-    // Try heuristic algorithm first, fall back to PRS if that failed
-    try {
-        g = heur_gcd(aex, bex, ca, cb, var);
-    } catch (gcdheu_failed) {
-        g = *new ex(fail());
-    }
-    if (is_ex_exactly_of_type(g, fail)) {
-//clog << "heuristics failed" << endl;
+       // Try heuristic algorithm first, fall back to PRS if that failed
+       try {
+               g = heur_gcd(aex, bex, ca, cb, var);
+       } catch (gcdheu_failed) {
+               g = fail();
+       }
+       if (is_ex_exactly_of_type(g, fail)) {
+//std::clog << "heuristics failed" << std::endl;
 #if STATISTICS
                heur_gcd_failed++;
 #endif
@@ -1590,21 +1662,21 @@ factored_b:
 //             g = peu_gcd(aex, bex, &x);
 //             g = red_gcd(aex, bex, &x);
                g = sr_gcd(aex, bex, var);
-               if (g.is_equal(_ex1())) {
+               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);
+                       if (ca)
+                               divide(aex, g, *ca, false);
+                       if (cb)
+                               divide(bex, g, *cb, false);
                }
 #if 1
-    } else {
-               if (g.is_equal(_ex1())) {
+       } else {
+               if (g.is_equal(_ex1)) {
                        // Keep cofactors factored if possible
                        if (ca)
                                *ca = a;
@@ -1613,7 +1685,7 @@ factored_b:
                }
        }
 #endif
-    return g;
+       return g;
 }
 
 
@@ -1626,14 +1698,14 @@ factored_b:
  *  @return the LCM as a new expression */
 ex lcm(const ex &a, const ex &b, bool check_args)
 {
-    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
-        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;
+       if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
+               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;
 }
 
 
@@ -1641,70 +1713,172 @@ ex lcm(const ex &a, const ex &b, bool check_args)
  *  Square-free factorization
  */
 
-// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()).
-// a and b can be multivariate polynomials but they are treated as univariate polynomials in x.
-static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
+/** 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.
+ *  @param x  variable to factor in
+ *  @return   vector of factors sorted in ascending degree */
+static exvector sqrfree_yun(const ex &a, const symbol &x)
 {
-    if (a.is_zero())
-        return b;
-    if (b.is_zero())
-        return a;
-    if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
-        return _ex1();
-    if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
-        return gcd(ex_to_numeric(a), ex_to_numeric(b));
-    if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
-        throw(std::invalid_argument("univariate_gcd: arguments must be polynomials over the rationals"));
-
-    // Euclidean algorithm
-    ex c, d, r;
-    if (a.degree(x) >= b.degree(x)) {
-        c = a;
-        d = b;
-    } else {
-        c = b;
-        d = a;
-    }
-    for (;;) {
-        r = rem(c, d, x, false);
-        if (r.is_zero())
-            break;
-        c = d;
-        d = r;
-    }
-    return d / d.lcoeff(x);
+       exvector res;
+       ex w = a;
+       ex z = w.diff(x);
+       ex g = gcd(w, z);
+       if (g.is_equal(_ex1)) {
+               res.push_back(a);
+               return res;
+       }
+       ex y;
+       do {
+               w = quo(w, g, x);
+               y = quo(z, g, x);
+               z = y - w.diff(x);
+               g = gcd(w, z);
+               res.push_back(g);
+       } while (!z.is_zero());
+       return res;
 }
 
+/** Compute square-free factorization of multivariate polynomial in Q[X].
+ *
+ *  @param a  multivariate polynomial over Q[X]
+ *  @param x  lst of variables to factor in, may be left empty for autodetection
+ *  @return   polynomial a in square-free factored form. */
+ex sqrfree(const ex &a, const lst &l)
+{
+       if (is_a<numeric>(a) ||     // algorithm does not trap a==0
+           is_a<symbol>(a))        // shortcut
+               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);
+               sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
+               while (it != itend) {
+                       args.append(*it->sym);
+                       ++it;
+               }
+       } else {
+               args = l;
+       }
 
-/** Compute square-free factorization of multivariate polynomial a(x) using
- *  Yun´s algorithm.
+       // Find the symbol to factor in at this stage
+       if (!is_ex_of_type(args.op(0), symbol))
+               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
+       exvector factors = sqrfree_yun(tmp,x);
+
+       // construct the next list of symbols with the first element popped
+       lst newargs = args;
+       newargs.remove_first();
+
+       // recurse down the factors in remaining variables
+       if (newargs.nops()>0) {
+               exvector::iterator i = factors.begin();
+               while (i != factors.end()) {
+                       *i = sqrfree(*i, newargs);
+                       ++i;
+               }
+       }
+
+       // Done with recursion, now construct the final result
+       ex result = _ex1;
+       exvector::const_iterator it = factors.begin(), itend = factors.end();
+       for (int p = 1; it!=itend; ++it, ++p)
+               result *= power(*it, p);
+
+       // 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 (newargs.nops()>0)
+               result *= sqrfree(quo(tmp, result, x), newargs);
+       else
+               result *= quo(tmp, result, x);
+
+       // Put in the reational overall factor again and return
+       return result * lcm.inverse();
+}
+
+/** Compute square-free partial fraction decomposition of rational function
+ *  a(x).
  *
- * @param a  multivariate polynomial
- * @param x  variable to factor in
- * @return factored polynomial */
-ex sqrfree(const ex &a, const symbol &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)
 {
-    int i = 1;
-    ex res = _ex1();
-    ex b = a.diff(x);
-    ex c = univariate_gcd(a, b, x);
-    ex w;
-    if (c.is_equal(_ex1())) {
-        w = a;
-    } else {
-        w = quo(a, c, x);
-        ex y = quo(b, c, x);
-        ex z = y - w.diff(x);
-        while (!z.is_zero()) {
-            ex g = univariate_gcd(w, z, x);
-            res *= power(g, i);
-            w = quo(w, g, x);
-            y = quo(z, g, x);
-            z = y - w.diff(x);
-            i++;
-        }
-    }
-    return res * power(w, i);
+       // 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
+       exvector yun = sqrfree_yun(denom, x);
+//clog << "yun factors: " << exprseq(yun) << endl;
+       unsigned num_yun = yun.size();
+       exvector factor; factor.reserve(num_yun);
+       exvector cofac; cofac.reserve(num_yun);
+       for (unsigned i=0; i<num_yun; i++) {
+               if (!yun[i].is_equal(_ex1)) {
+                       for (unsigned j=0; j<=i; j++) {
+                               factor.push_back(pow(yun[i], j+1));
+                               ex prod = _ex1;
+                               for (unsigned k=0; k<num_yun; k++) {
+                                       if (k == i)
+                                               prod *= pow(yun[k], i-j);
+                                       else
+                                               prod *= pow(yun[k], k+1);
+                               }
+                               cofac.push_back(prod.expand());
+                       }
+               }
+       }
+       unsigned 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 (unsigned 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 (unsigned i=0; i<num_factors; i++)
+               vars(i, 0) = symbol();
+       matrix sol = sys.solve(vars, rhs);
+
+       // Sum up decomposed fractions
+       ex sum = 0;
+       for (unsigned i=0; i<num_factors; i++)
+               sum += sol(i, 0) / factor[i];
+
+       return red_poly + sum;
 }
 
 
@@ -1720,55 +1894,75 @@ ex sqrfree(const ex &a, const symbol &x)
  *  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 is appended to sym_lst and returned, the
  *  expression is appended to repl_lst.
  *  @see ex::normal */
 static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
 {
-    // Expression already in repl_lst? Then return the assigned symbol
-    for (unsigned i=0; i<repl_lst.nops(); i++)
-        if (repl_lst.op(i).is_equal(e))
-            return sym_lst.op(i);
-    
-    // Otherwise create new symbol and add to list, taking care that the
+       // Expression already in repl_lst? Then return the assigned symbol
+       for (unsigned i=0; i<repl_lst.nops(); i++)
+               if (repl_lst.op(i).is_equal(e))
+                       return sym_lst.op(i);
+       
+       // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't contain symbols from the sym_lst
        // because subs() is not recursive
        symbol s;
        ex es(s);
        ex e_replaced = e.subs(sym_lst, repl_lst);
-    sym_lst.append(es);
-    repl_lst.append(e_replaced);
-    return es;
+       sym_lst.append(es);
+       repl_lst.append(e_replaced);
+       return es;
 }
 
 /** Create a symbol for replacing the expression "e" (or return a previously
  *  assigned symbol). An expression of the form "symbol == expression" is added
  *  to repl_lst and the symbol is returned.
- *  @see ex::to_rational */
+ *  @see basic::to_rational */
 static ex replace_with_symbol(const ex &e, lst &repl_lst)
 {
-    // Expression already in repl_lst? Then return the assigned symbol
-    for (unsigned i=0; i<repl_lst.nops(); i++)
-        if (repl_lst.op(i).op(1).is_equal(e))
-            return repl_lst.op(i).op(0);
-    
-    // Otherwise create new symbol and add to list, taking care that the
+       // Expression already in repl_lst? Then return the assigned symbol
+       for (unsigned i=0; i<repl_lst.nops(); i++)
+               if (repl_lst.op(i).op(1).is_equal(e))
+                       return repl_lst.op(i).op(0);
+       
+       // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't contain symbols from the sym_lst
        // because subs() is not recursive
        symbol s;
        ex es(s);
        ex e_replaced = e.subs(repl_lst);
-    repl_lst.append(es == e_replaced);
-    return es;
+       repl_lst.append(es == e_replaced);
+       return es;
 }
 
-/** Default implementation of ex::normal(). It replaces the object with a
- *  temporary symbol.
+
+/** Function object to be applied by basic::normal(). */
+struct normal_map_function : public map_function {
+       int level;
+       normal_map_function(int l) : level(l) {}
+       ex operator()(const ex & e) { return normal(e, level); }
+};
+
+/** Default implementation of ex::normal(). It normalizes the children and
+ *  replaces the object with a temporary symbol.
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+       if (nops() == 0)
+               return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+       else {
+               if (level == 1)
+                       return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+               else if (level == -max_recursion_level)
+                       throw(std::runtime_error("max recursion level reached"));
+               else {
+                       normal_map_function map_normal(level - 1);
+                       return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+               }
+       }
 }
 
 
@@ -1776,7 +1970,7 @@ ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+       return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1789,15 +1983,15 @@ ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
        numeric num = numer();
        ex numex = num;
 
-    if (num.is_real()) {
-        if (!num.is_integer())
-            numex = replace_with_symbol(numex, sym_lst, repl_lst);
-    } else { // complex
-        numeric re = num.real(), im = num.imag();
-        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
-        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
-        numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
-    }
+       if (num.is_real()) {
+               if (!num.is_integer())
+                       numex = replace_with_symbol(numex, sym_lst, repl_lst);
+       } else { // complex
+               numeric re = num.real(), im = num.imag();
+               ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
+               ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
+               numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+       }
 
        // Denominator is always a real integer (see numeric::denom())
        return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
@@ -1810,29 +2004,33 @@ ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @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();
-
-//clog << "frac_cancel num = " << num << ", den = " << den << endl;
-
-    // Handle special cases where numerator or denominator is 0
-    if (num.is_zero())
-               return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
-    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);
+       ex num = n;
+       ex den = d;
+       numeric pre_factor = _num1;
+
+//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
+
+       // Handle trivial case where denominator is 1
+       if (den.is_equal(_ex1))
+               return (new lst(num, den))->setflag(status_flags::dynallocated);
+
+       // Handle special cases where numerator or denominator is 0
+       if (num.is_zero())
+               return (new lst(num, _ex1))->setflag(status_flags::dynallocated);
+       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;
+       pre_factor = den_lcm / num_lcm;
 
-    // Cancel GCD from numerator and denominator
-    ex cnum, cden;
-    if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
+       // Cancel GCD from numerator and denominator
+       ex cnum, cden;
+       if (gcd(num, den, &cnum, &cden, false) != _ex1) {
                num = cnum;
                den = cden;
        }
@@ -1841,16 +2039,16 @@ static ex frac_cancel(const ex &n, const ex &d)
        // as defined by get_first_symbol() is made positive)
        const symbol *x;
        if (get_first_symbol(den, x)) {
-                GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
-               if (ex_to_numeric(den.unit(*x)).is_negative()) {
-                       num *= _ex_1();
-                       den *= _ex_1();
+               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
-//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
-    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
+//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
+       return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1860,78 +2058,56 @@ static ex frac_cancel(const ex &n, const ex &d)
 ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
        if (level == 1)
-               return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+               return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
        else if (level == -max_recursion_level)
-        throw(std::runtime_error("max recursion level reached"));
-
-    // Normalize and expand children, chop into summands
-    exvector o;
-    o.reserve(seq.size()+1);
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-
-               // Normalize and expand child
-        ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
-
-               // If numerator is a sum, chop into summands
-        if (is_ex_exactly_of_type(n.op(0), add)) {
-            epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
-            while (bit != bitend) {
-                o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
-                bit++;
-            }
-
-                       // The overall_coeff is already normalized (== rational), we just
-                       // split it into numerator and denominator
-                       GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
-                       numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
-            o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
-        } else
-            o.push_back(n);
-        it++;
-    }
-    o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
-
-       // o is now a vector of {numerator, denominator} lists
-
-    // Determine common denominator
-    ex den = _ex1();
-    exvector::const_iterator ait = o.begin(), aitend = o.end();
-//clog << "add::normal uses the following summands:\n";
-    while (ait != aitend) {
-//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
-        den = lcm(ait->op(1), den, false);
-        ait++;
-    }
-//clog << " common denominator = " << den << endl;
-
-    // Add fractions
-    if (den.is_equal(_ex1())) {
-
-               // Common denominator is 1, simply add all numerators
-        exvector num_seq;
-               for (ait=o.begin(); ait!=aitend; ait++) {
-                       num_seq.push_back(ait->op(0));
+               throw(std::runtime_error("max recursion level reached"));
+
+       // Normalize children and split each one into numerator and denominator
+       exvector nums, dens;
+       nums.reserve(seq.size()+1);
+       dens.reserve(seq.size()+1);
+       epvector::const_iterator it = seq.begin(), itend = seq.end();
+       while (it != itend) {
+               ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+               nums.push_back(n.op(0));
+               dens.push_back(n.op(1));
+               it++;
+       }
+       ex n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+       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
+       exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
+       exvector::const_iterator 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++;
                }
-               return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
 
-       } else {
+               // Additiion 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;
 
-               // Perform fractional addition
-        exvector num_seq;
-        for (ait=o.begin(); ait!=aitend; ait++) {
-            ex q;
-            if (!divide(den, ait->op(1), q, false)) {
-                // should not happen
-                throw(std::runtime_error("invalid expression in add::normal, division failed"));
-            }
-            num_seq.push_back((ait->op(0) * q).expand());
-        }
-        ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
-
-        // Cancel common factors from num/den
-        return frac_cancel(num, den);
-    }
+       // Cancel common factors from num/den
+       return frac_cancel(num, den);
 }
 
 
@@ -1941,27 +2117,28 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
        if (level == 1)
-               return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+               return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
        else if (level == -max_recursion_level)
-        throw(std::runtime_error("max recursion level reached"));
+               throw(std::runtime_error("max recursion level reached"));
 
-    // Normalize children, separate into numerator and denominator
-       ex num = _ex1();
-       ex den = _ex1(); 
+       // Normalize children, separate into numerator and denominator
+       exvector num; num.reserve(seq.size());
+       exvector den; den.reserve(seq.size());
        ex n;
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-               n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
-               num *= n.op(0);
-               den *= n.op(1);
-        it++;
-    }
-       n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
-       num *= n.op(0);
-       den *= n.op(1);
+       epvector::const_iterator it = seq.begin(), itend = seq.end();
+       while (it != itend) {
+               n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+               num.push_back(n.op(0));
+               den.push_back(n.op(1));
+               it++;
+       }
+       n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+       num.push_back(n.op(0));
+       den.push_back(n.op(1));
 
        // Perform fraction cancellation
-    return frac_cancel(num, den);
+       return frac_cancel((new mul(num))->setflag(status_flags::dynallocated),
+                          (new mul(den))->setflag(status_flags::dynallocated));
 }
 
 
@@ -1972,78 +2149,72 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
        if (level == 1)
-               return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+               return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
        else if (level == -max_recursion_level)
-        throw(std::runtime_error("max recursion level reached"));
+               throw(std::runtime_error("max recursion level reached"));
 
-       // Normalize basis
-    ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+       // Normalize basis and exponent (exponent gets reassembled)
+       ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
+       ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
+       n_exponent = n_exponent.op(0) / n_exponent.op(1);
 
-       if (exponent.info(info_flags::integer)) {
+       if (n_exponent.info(info_flags::integer)) {
 
-           if (exponent.info(info_flags::positive)) {
+               if (n_exponent.info(info_flags::positive)) {
 
                        // (a/b)^n -> {a^n, b^n}
-                       return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+                       return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);
 
-               } else if (exponent.info(info_flags::negative)) {
+               } else if (n_exponent.info(info_flags::negative)) {
 
                        // (a/b)^-n -> {b^n, a^n}
-                       return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
+                       return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
                }
 
        } else {
 
-               if (exponent.info(info_flags::positive)) {
+               if (n_exponent.info(info_flags::positive)) {
 
                        // (a/b)^x -> {sym((a/b)^x), 1}
-                       return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                       return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 
-               } else if (exponent.info(info_flags::negative)) {
+               } else if (n_exponent.info(info_flags::negative)) {
 
-                       if (n.op(1).is_equal(_ex1())) {
+                       if (n_basis.op(1).is_equal(_ex1)) {
 
                                // a^-x -> {1, sym(a^x)}
-                               return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+                               return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
 
                        } else {
 
                                // (a/b)^-x -> {sym((b/a)^x), 1}
-                               return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                               return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
                        }
 
-               } else {        // exponent not numeric
+               } else {        // n_exponent not numeric
 
                        // (a/b)^x -> {sym((a/b)^x, 1}
-                       return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                       return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
                }
-    }
+       }
 }
 
 
-/** Implementation of ex::normal() for pseries. It normalizes each coefficient and
- *  replaces the series by a temporary symbol.
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient
+ *  and replaces the series by a temporary symbol.
  *  @see ex::normal */
 ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    epvector new_seq;
-    new_seq.reserve(seq.size());
-
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        new_seq.push_back(expair(it->rest.normal(), it->coeff));
-        it++;
-    }
-    ex n = pseries(relational(var,point), new_seq);
-       return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
-}
-
-
-/** Implementation of ex::normal() for relationals. It normalizes both sides.
- *  @see ex::normal */
-ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
-{
-       return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated);
+       epvector newseq;
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               ex restexp = i->rest.normal();
+               if (!restexp.is_zero())
+                       newseq.push_back(expair(restexp, i->coeff));
+               ++i;
+       }
+       ex n = pseries(relational(var,point), newseq);
+       return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -2061,63 +2232,94 @@ ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @return normalized expression */
 ex ex::normal(int level) const
 {
-    lst sym_lst, repl_lst;
+       lst sym_lst, repl_lst;
 
-    ex e = bp->normal(sym_lst, repl_lst, level);
-       GINAC_ASSERT(is_ex_of_type(e, lst));
+       ex e = bp->normal(sym_lst, repl_lst, level);
+       GINAC_ASSERT(is_a<lst>(e));
 
        // Re-insert replaced symbols
-    if (sym_lst.nops() > 0)
-        e = e.subs(sym_lst, repl_lst);
+       if (sym_lst.nops() > 0)
+               e = e.subs(sym_lst, repl_lst);
 
        // Convert {numerator, denominator} form back to fraction
-    return e.op(0) / e.op(1);
+       return e.op(0) / e.op(1);
 }
 
-/** 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.
+/** 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(void) const
 {
-    lst sym_lst, repl_lst;
+       lst sym_lst, repl_lst;
 
-    ex e = bp->normal(sym_lst, repl_lst, 0);
-       GINAC_ASSERT(is_ex_of_type(e, lst));
+       ex e = bp->normal(sym_lst, repl_lst, 0);
+       GINAC_ASSERT(is_a<lst>(e));
 
        // Re-insert replaced symbols
-    if (sym_lst.nops() > 0)
-        return e.op(0).subs(sym_lst, repl_lst);
+       if (sym_lst.nops() > 0)
+               return e.op(0).subs(sym_lst, repl_lst);
        else
                return e.op(0);
 }
 
-/** 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.
+/** 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(void) const
 {
-    lst sym_lst, repl_lst;
+       lst sym_lst, repl_lst;
 
-    ex e = bp->normal(sym_lst, repl_lst, 0);
-       GINAC_ASSERT(is_ex_of_type(e, lst));
+       ex e = bp->normal(sym_lst, repl_lst, 0);
+       GINAC_ASSERT(is_a<lst>(e));
 
        // Re-insert replaced symbols
-    if (sym_lst.nops() > 0)
-        return e.op(1).subs(sym_lst, repl_lst);
+       if (sym_lst.nops() > 0)
+               return e.op(1).subs(sym_lst, repl_lst);
        else
                return e.op(1);
 }
 
+/** Get numerator and denominator of an expression. If the expresison 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(void) const
+{
+       lst sym_lst, repl_lst;
+
+       ex e = bp->normal(sym_lst, repl_lst, 0);
+       GINAC_ASSERT(is_a<lst>(e));
 
-/** Default implementation of ex::to_rational(). It replaces the object with a
- *  temporary symbol.
- *  @see ex::to_rational */
+       // Re-insert replaced symbols
+       if (sym_lst.nops() > 0)
+               return e.subs(sym_lst, repl_lst);
+       else
+               return e;
+}
+
+
+/** Rationalization of non-rational functions.
+ *  This function converts a general expression to a rational polynomial
+ *  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 list specified by the
+ *  repl_lst parameter in the form {symbol == expression}, ready to be passed
+ *  as an argument to ex::subs().
+ *
+ *  @param repl_lst collects a list of all temporary symbols and their replacements
+ *  @return rationalized expression */
 ex basic::to_rational(lst &repl_lst) const
 {
        return replace_with_symbol(*this, repl_lst);
@@ -2125,37 +2327,34 @@ ex basic::to_rational(lst &repl_lst) const
 
 
 /** Implementation of ex::to_rational() for symbols. This returns the
- *  unmodified symbol.
- *  @see ex::to_rational */
+ *  unmodified symbol. */
 ex symbol::to_rational(lst &repl_lst) const
 {
-    return *this;
+       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.
- *  @see ex::to_rational */
+ *  temporary symbol. */
 ex numeric::to_rational(lst &repl_lst) const
 {
-    if (is_real()) {
-        if (!is_rational())
-            return replace_with_symbol(*this, repl_lst);
-    } else { // complex
-        numeric re = real();
-        numeric im = imag();
-        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
-        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
-        return re_ex + im_ex * replace_with_symbol(I, repl_lst);
-    }
+       if (is_real()) {
+               if (!is_rational())
+                       return replace_with_symbol(*this, repl_lst);
+       } else { // complex
+               numeric re = real();
+               numeric im = imag();
+               ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
+               ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
+               return re_ex + im_ex * replace_with_symbol(I, repl_lst);
+       }
        return *this;
 }
 
 
 /** Implementation of ex::to_rational() for powers. It replaces non-integer
- *  powers by temporary symbols.
- *  @see ex::to_rational */
+ *  powers by temporary symbols. */
 ex power::to_rational(lst &repl_lst) const
 {
        if (exponent.info(info_flags::integer))
@@ -2165,44 +2364,23 @@ ex power::to_rational(lst &repl_lst) const
 }
 
 
-/** Implementation of ex::to_rational() for expairseqs.
- *  @see ex::to_rational */
+/** Implementation of ex::to_rational() for expairseqs. */
 ex expairseq::to_rational(lst &repl_lst) const
 {
-    epvector s;
-    s.reserve(seq.size());
-    for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
-        s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
-        // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
-    }
-    ex oc = overall_coeff.to_rational(repl_lst);
-    if (oc.info(info_flags::numeric))
-        return thisexpairseq(s, overall_coeff);
-    else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1()));
-    return thisexpairseq(s, default_overall_coeff());
-}
-
-
-/** Rationalization of non-rational functions.
- *  This function converts a general expression to a rational polynomial
- *  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 list specified by the
- *  repl_lst parameter in the form {symbol == expression}, ready to be passed
- *  as an argument to ex::subs().
- *
- *  @param repl_lst collects a list of all temporary symbols and their replacements
- *  @return rationalized expression */
-ex ex::to_rational(lst &repl_lst) const
-{
-       return bp->to_rational(repl_lst);
+       epvector s;
+       s.reserve(seq.size());
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst)));
+               ++i;
+       }
+       ex oc = overall_coeff.to_rational(repl_lst);
+       if (oc.info(info_flags::numeric))
+               return thisexpairseq(s, overall_coeff);
+       else
+               s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1));
+       return thisexpairseq(s, default_overall_coeff());
 }
 
 
-#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC