subs() and normal() use maps instead of lists, resulting in a huge performance
[ginac.git] / ginac / mul.cpp
index a4bb802..2c84565 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's products of expressions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2003 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
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
+#include <iostream>
 #include <vector>
 #include <stdexcept>
+#include <limits>
 
 #include "mul.h"
 #include "add.h"
 #include "power.h"
+#include "operators.h"
 #include "matrix.h"
+#include "lst.h"
 #include "archive.h"
-#include "debugmsg.h"
 #include "utils.h"
 
 namespace GiNaC {
@@ -36,54 +39,46 @@ namespace GiNaC {
 GINAC_IMPLEMENT_REGISTERED_CLASS(mul, expairseq)
 
 //////////
-// default ctor, dctor, copy ctor assignment operator and helpers
+// default constructor
 //////////
 
 mul::mul()
 {
-       debugmsg("mul default ctor",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
 }
 
-DEFAULT_COPY(mul)
-DEFAULT_DESTROY(mul)
-
 //////////
-// other ctors
+// other constructors
 //////////
 
 // public
 
 mul::mul(const ex & lh, const ex & rh)
 {
-       debugmsg("mul ctor from ex,ex",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
-       overall_coeff = _ex1();
+       overall_coeff = _ex1;
        construct_from_2_ex(lh,rh);
        GINAC_ASSERT(is_canonical());
 }
 
 mul::mul(const exvector & v)
 {
-       debugmsg("mul ctor from exvector",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
-       overall_coeff = _ex1();
+       overall_coeff = _ex1;
        construct_from_exvector(v);
        GINAC_ASSERT(is_canonical());
 }
 
 mul::mul(const epvector & v)
 {
-       debugmsg("mul ctor from epvector",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
-       overall_coeff = _ex1();
+       overall_coeff = _ex1;
        construct_from_epvector(v);
        GINAC_ASSERT(is_canonical());
 }
 
 mul::mul(const epvector & v, const ex & oc)
 {
-       debugmsg("mul ctor from epvector,ex",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
        overall_coeff = oc;
        construct_from_epvector(v);
@@ -92,7 +87,6 @@ mul::mul(const epvector & v, const ex & oc)
 
 mul::mul(epvector * vp, const ex & oc)
 {
-       debugmsg("mul ctor from epvector *,ex",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
        GINAC_ASSERT(vp!=0);
        overall_coeff = oc;
@@ -103,14 +97,13 @@ mul::mul(epvector * vp, const ex & oc)
 
 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 {
-       debugmsg("mul ctor from ex,ex,ex",LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_mul;
        exvector factors;
        factors.reserve(3);
        factors.push_back(lh);
        factors.push_back(mh);
        factors.push_back(rh);
-       overall_coeff = _ex1();
+       overall_coeff = _ex1;
        construct_from_exvector(factors);
        GINAC_ASSERT(is_canonical());
 }
@@ -122,53 +115,57 @@ mul::mul(const ex & lh, const ex & mh, const ex & rh)
 DEFAULT_ARCHIVING(mul)
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
 // public
-
 void mul::print(const print_context & c, unsigned level) const
 {
-       debugmsg("mul print", LOGLEVEL_PRINT);
-
-       if (is_of_type(c, print_tree)) {
+       if (is_a<print_tree>(c)) {
 
                inherited::print(c, level);
 
-       } else if (is_of_type(c, print_csrc)) {
+       } else if (is_a<print_csrc>(c)) {
 
                if (precedence() <= level)
                        c.s << "(";
 
-               if (!overall_coeff.is_equal(_ex1())) {
-                       overall_coeff.bp->print(c, precedence());
+               if (!overall_coeff.is_equal(_ex1)) {
+                       overall_coeff.print(c, precedence());
                        c.s << "*";
                }
-       
+
                // Print arguments, separated by "*" or "/"
                epvector::const_iterator it = seq.begin(), itend = seq.end();
                while (it != itend) {
 
                        // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
-                       if (it == seq.begin() && ex_to_numeric(it->coeff).is_integer() && it->coeff.compare(_num0()) < 0) {
-                               if (is_of_type(c, print_csrc_cl_N))
+                       bool needclosingparenthesis = false;
+                       if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
+                               if (is_a<print_csrc_cl_N>(c)) {
                                        c.s << "recip(";
-                               else
+                                       needclosingparenthesis = true;
+                               } else
                                        c.s << "1.0/";
                        }
 
                        // If the exponent is 1 or -1, it is left out
-                       if (it->coeff.compare(_ex1()) == 0 || it->coeff.compare(_num_1()) == 0)
+                       if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
                                it->rest.print(c, precedence());
-                       else {
-                               // Outer parens around ex needed for broken gcc-2.95 parser:
-                               (ex(power(it->rest, abs(ex_to_numeric(it->coeff))))).print(c, level);
-                       }
+                       else if (it->coeff.info(info_flags::negint))
+                               // Outer parens around ex needed for broken GCC parser:
+                               (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
+                       else
+                               // Outer parens around ex needed for broken GCC parser:
+                               (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
+
+                       if (needclosingparenthesis)
+                               c.s << ")";
 
                        // Separator is "/" for negative integer powers, "*" otherwise
                        ++it;
                        if (it != itend) {
-                               if (ex_to_numeric(it->coeff).is_integer() && it->coeff.compare(_num0()) < 0)
+                               if (it->coeff.info(info_flags::negint))
                                        c.s << "/";
                                else
                                        c.s << "*";
@@ -178,23 +175,29 @@ void mul::print(const print_context & c, unsigned level) const
                if (precedence() <= level)
                        c.s << ")";
 
+       } else if (is_a<print_python_repr>(c)) {
+               c.s << class_name() << '(';
+               op(0).print(c);
+               for (size_t i=1; i<nops(); ++i) {
+                       c.s << ',';
+                       op(i).print(c);
+               }
+               c.s << ')';
        } else {
 
                if (precedence() <= level) {
-                       if (is_of_type(c, print_latex))
+                       if (is_a<print_latex>(c))
                                c.s << "{(";
                        else
                                c.s << "(";
                }
 
-               bool first = true;
-
                // First print the overall numeric coefficient
-               numeric coeff = ex_to_numeric(overall_coeff);
+               const numeric &coeff = ex_to<numeric>(overall_coeff);
                if (coeff.csgn() == -1)
                        c.s << '-';
-               if (!coeff.is_equal(_num1()) &&
-                       !coeff.is_equal(_num_1())) {
+               if (!coeff.is_equal(_num1) &&
+                       !coeff.is_equal(_num_1)) {
                        if (coeff.is_rational()) {
                                if (coeff.is_negative())
                                        (-coeff).print(c);
@@ -206,7 +209,7 @@ void mul::print(const print_context & c, unsigned level) const
                                else
                                        coeff.print(c, precedence());
                        }
-                       if (is_of_type(c, print_latex))
+                       if (is_a<print_latex>(c))
                                c.s << ' ';
                        else
                                c.s << '*';
@@ -214,21 +217,55 @@ void mul::print(const print_context & c, unsigned level) const
 
                // Then proceed with the remaining factors
                epvector::const_iterator it = seq.begin(), itend = seq.end();
-               while (it != itend) {
-                       if (!first) {
-                               if (is_of_type(c, print_latex))
-                                       c.s << ' ';
+               if (is_a<print_latex>(c)) {
+
+                       // Separate factors into those with negative numeric exponent
+                       // and all others
+                       exvector neg_powers, others;
+                       while (it != itend) {
+                               GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
+                               if (ex_to<numeric>(it->coeff).is_negative())
+                                       neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
                                else
-                                       c.s << '*';
+                                       others.push_back(recombine_pair_to_ex(*it));
+                               ++it;
+                       }
+
+                       if (!neg_powers.empty()) {
+
+                               // Factors with negative exponent are printed as a fraction
+                               c.s << "\\frac{";
+                               mul(others).eval().print(c);
+                               c.s << "}{";
+                               mul(neg_powers).eval().print(c);
+                               c.s << "}";
+
                        } else {
-                               first = false;
+
+                               // All other factors are printed in the ordinary way
+                               exvector::const_iterator vit = others.begin(), vitend = others.end();
+                               while (vit != vitend) {
+                                       c.s << ' ';
+                                       vit->print(c, precedence());
+                                       ++vit;
+                               }
+                       }
+
+               } else {
+
+                       bool first = true;
+                       while (it != itend) {
+                               if (!first)
+                                       c.s << '*';
+                               else
+                                       first = false;
+                               recombine_pair_to_ex(*it).print(c, precedence());
+                               ++it;
                        }
-                       recombine_pair_to_ex(*it).print(c, precedence());
-                       it++;
                }
 
                if (precedence() <= level) {
-                       if (is_of_type(c, print_latex))
+                       if (is_a<print_latex>(c))
                                c.s << ")}";
                        else
                                c.s << ")";
@@ -245,16 +282,20 @@ bool mul::info(unsigned inf) const
                case info_flags::rational_polynomial:
                case info_flags::crational_polynomial:
                case info_flags::rational_function: {
-                       for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+                       epvector::const_iterator i = seq.begin(), end = seq.end();
+                       while (i != end) {
                                if (!(recombine_pair_to_ex(*i).info(inf)))
                                        return false;
+                               ++i;
                        }
                        return overall_coeff.info(inf);
                }
                case info_flags::algebraic: {
-                       for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+                       epvector::const_iterator i = seq.begin(), end = seq.end();
+                       while (i != end) {
                                if ((recombine_pair_to_ex(*i).info(inf)))
                                        return true;
+                               ++i;
                        }
                        return false;
                }
@@ -264,20 +305,26 @@ bool mul::info(unsigned inf) const
 
 int mul::degree(const ex & s) const
 {
+       // Sum up degrees of factors
        int deg_sum = 0;
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               if (ex_to_numeric(cit->coeff).is_integer())
-                       deg_sum+=cit->rest.degree(s) * ex_to_numeric(cit->coeff).to_int();
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               if (ex_to<numeric>(i->coeff).is_integer())
+                       deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
+               ++i;
        }
        return deg_sum;
 }
 
 int mul::ldegree(const ex & s) const
 {
+       // Sum up degrees of factors
        int deg_sum = 0;
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               if (ex_to_numeric(cit->coeff).is_integer())
-                       deg_sum+=cit->rest.ldegree(s) * ex_to_numeric(cit->coeff).to_int();
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               if (ex_to<numeric>(i->coeff).is_integer())
+                       deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
+               ++i;
        }
        return deg_sum;
 }
@@ -290,97 +337,103 @@ ex mul::coeff(const ex & s, int n) const
        if (n==0) {
                // product of individual coeffs
                // if a non-zero power of s is found, the resulting product will be 0
-               epvector::const_iterator it = seq.begin();
-               while (it!=seq.end()) {
-                       coeffseq.push_back(recombine_pair_to_ex(*it).coeff(s,n));
-                       ++it;
+               epvector::const_iterator i = seq.begin(), end = seq.end();
+               while (i != end) {
+                       coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
+                       ++i;
                }
                coeffseq.push_back(overall_coeff);
                return (new mul(coeffseq))->setflag(status_flags::dynallocated);
        }
        
-       epvector::const_iterator it=seq.begin();
-       bool coeff_found = 0;
-       while (it!=seq.end()) {
-               ex t = recombine_pair_to_ex(*it);
-               ex c = t.coeff(s,n);
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       bool coeff_found = false;
+       while (i != end) {
+               ex t = recombine_pair_to_ex(*i);
+               ex c = t.coeff(s, n);
                if (!c.is_zero()) {
                        coeffseq.push_back(c);
                        coeff_found = 1;
                } else {
                        coeffseq.push_back(t);
                }
-               ++it;
+               ++i;
        }
        if (coeff_found) {
                coeffseq.push_back(overall_coeff);
                return (new mul(coeffseq))->setflag(status_flags::dynallocated);
        }
        
-       return _ex0();
+       return _ex0;
 }
 
+/** Perform automatic term rewriting rules in this class.  In the following
+ *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
+ *  stand for such expressions that contain a plain number.
+ *  - *(...,x;0) -> 0
+ *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
+ *  - *(x;1) -> x
+ *  - *(;c) -> c
+ *
+ *  @param level cut-off in recursive evaluation */
 ex mul::eval(int level) const
 {
-       // simplifications  *(...,x;0) -> 0
-       //                  *(+(x,y,...);c) -> *(+(*(x,c),*(y,c),...)) (c numeric())
-       //                  *(x;1) -> x
-       //                  *(;c) -> c
-       
-       debugmsg("mul eval",LOGLEVEL_MEMBER_FUNCTION);
-       
-       epvector * evaled_seqp = evalchildren(level);
-       if (evaled_seqp!=0) {
+       epvector *evaled_seqp = evalchildren(level);
+       if (evaled_seqp) {
                // do more evaluation later
                return (new mul(evaled_seqp,overall_coeff))->
                           setflag(status_flags::dynallocated);
        }
        
 #ifdef DO_GINAC_ASSERT
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               GINAC_ASSERT((!is_ex_exactly_of_type((*cit).rest,mul)) ||
-                            (!(ex_to_numeric((*cit).coeff).is_integer())));
-               GINAC_ASSERT(!(cit->is_canonical_numeric()));
-               if (is_ex_exactly_of_type(recombine_pair_to_ex(*cit),numeric))
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
+                            (!(ex_to<numeric>(i->coeff).is_integer())));
+               GINAC_ASSERT(!(i->is_canonical_numeric()));
+               if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
                    print(print_tree(std::cerr));
-               GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*cit),numeric));
+               GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
                /* for paranoia */
-               expair p = split_ex_to_pair(recombine_pair_to_ex(*cit));
-               GINAC_ASSERT(p.rest.is_equal((*cit).rest));
-               GINAC_ASSERT(p.coeff.is_equal((*cit).coeff));
+               expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
+               GINAC_ASSERT(p.rest.is_equal(i->rest));
+               GINAC_ASSERT(p.coeff.is_equal(i->coeff));
                /* end paranoia */
+               ++i;
        }
 #endif // def DO_GINAC_ASSERT
        
        if (flags & status_flags::evaluated) {
                GINAC_ASSERT(seq.size()>0);
-               GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1()));
+               GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
                return *this;
        }
        
        int seq_size = seq.size();
-       if (overall_coeff.is_equal(_ex0())) {
+       if (overall_coeff.is_zero()) {
                // *(...,x;0) -> 0
-               return _ex0();
+               return _ex0;
        } else if (seq_size==0) {
                // *(;c) -> c
                return overall_coeff;
-       } else if (seq_size==1 && overall_coeff.is_equal(_ex1())) {
+       } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
                // *(x;1) -> x
                return recombine_pair_to_ex(*(seq.begin()));
        } else if ((seq_size==1) &&
-                  is_ex_exactly_of_type((*seq.begin()).rest,add) &&
-                  ex_to_numeric((*seq.begin()).coeff).is_equal(_num1())) {
+                  is_exactly_a<add>((*seq.begin()).rest) &&
+                  ex_to<numeric>((*seq.begin()).coeff).is_equal(_num1)) {
                // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
-               const add & addref = ex_to_add((*seq.begin()).rest);
-               epvector distrseq;
-               distrseq.reserve(addref.seq.size());
-               for (epvector::const_iterator cit=addref.seq.begin(); cit!=addref.seq.end(); ++cit) {
-                       distrseq.push_back(addref.combine_pair_with_coeff_to_pair(*cit, overall_coeff));
+               const add & addref = ex_to<add>((*seq.begin()).rest);
+               epvector *distrseq = new epvector();
+               distrseq->reserve(addref.seq.size());
+               epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
+               while (i != end) {
+                       distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
+                       ++i;
                }
                return (new add(distrseq,
-                               ex_to_numeric(addref.overall_coeff).
-                               mul_dyn(ex_to_numeric(overall_coeff))))
+                               ex_to<numeric>(addref.overall_coeff).
+                               mul_dyn(ex_to<numeric>(overall_coeff))))
                      ->setflag(status_flags::dynallocated | status_flags::evaluated);
        }
        return this->hold();
@@ -394,22 +447,25 @@ ex mul::evalf(int level) const
        if (level==-max_recursion_level)
                throw(std::runtime_error("max recursion level reached"));
        
-       epvector s;
-       s.reserve(seq.size());
-       
+       epvector *s = new epvector();
+       s->reserve(seq.size());
+
        --level;
-       for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
-               s.push_back(combine_ex_with_coeff_to_pair((*it).rest.evalf(level),
-                                                         (*it).coeff));
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
+                                                          i->coeff));
+               ++i;
        }
-       return mul(s,overall_coeff.evalf(level));
+       return mul(s, overall_coeff.evalf(level));
 }
 
-ex mul::evalm(void) const
+ex mul::evalm() const
 {
        // numeric*matrix
-       if (seq.size() == 1 && is_ex_of_type(seq[0].rest, matrix))
-               return ex_to_matrix(seq[0].rest).mul(ex_to_numeric(overall_coeff));
+       if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
+        && is_a<matrix>(seq[0].rest))
+               return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 
        // Evaluate children first, look whether there are any matrices at all
        // (there can be either no matrices or one matrix; if there were more
@@ -420,22 +476,22 @@ ex mul::evalm(void) const
        bool have_matrix = false;
        epvector::iterator the_matrix;
 
-       epvector::const_iterator it = seq.begin(), itend = seq.end();
-       while (it != itend) {
-               const ex &m = recombine_pair_to_ex(*it).evalm();
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               const ex &m = recombine_pair_to_ex(*i).evalm();
                s->push_back(split_ex_to_pair(m));
-               if (is_ex_of_type(m, matrix)) {
+               if (is_a<matrix>(m)) {
                        have_matrix = true;
                        the_matrix = s->end() - 1;
                }
-               it++;
+               ++i;
        }
 
        if (have_matrix) {
 
                // The product contained a matrix. We will multiply all other factors
                // into that matrix.
-               matrix m = ex_to_matrix(the_matrix->rest);
+               matrix m = ex_to<matrix>(the_matrix->rest);
                s->erase(the_matrix);
                ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
                return m.mul_scalar(scalar);
@@ -444,18 +500,143 @@ ex mul::evalm(void) const
                return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 }
 
-ex mul::simplify_ncmul(const exvector & v) const
+ex mul::eval_ncmul(const exvector & v) const
 {
-       if (seq.size()==0) {
-               return inherited::simplify_ncmul(v);
+       if (seq.empty())
+               return inherited::eval_ncmul(v);
+
+       // Find first noncommutative element and call its eval_ncmul()
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               if (i->rest.return_type() == return_types::noncommutative)
+                       return i->rest.eval_ncmul(v);
+               ++i;
+       }
+       return inherited::eval_ncmul(v);
+}
+
+bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
+{      
+       ex origbase;
+       int origexponent;
+       int origexpsign;
+
+       if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
+               origbase = origfactor.op(0);
+               int expon = ex_to<numeric>(origfactor.op(1)).to_int();
+               origexponent = expon > 0 ? expon : -expon;
+               origexpsign = expon > 0 ? 1 : -1;
+       } else {
+               origbase = origfactor;
+               origexponent = 1;
+               origexpsign = 1;
+       }
+
+       ex patternbase;
+       int patternexponent;
+       int patternexpsign;
+
+       if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
+               patternbase = patternfactor.op(0);
+               int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
+               patternexponent = expon > 0 ? expon : -expon;
+               patternexpsign = expon > 0 ? 1 : -1;
+       } else {
+               patternbase = patternfactor;
+               patternexponent = 1;
+               patternexpsign = 1;
        }
 
-       // Find first noncommutative element and call its simplify_ncmul()
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               if (cit->rest.return_type() == return_types::noncommutative)
-                       return cit->rest.simplify_ncmul(v);
+       lst saverepls = repls;
+       if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
+               return false;
+       repls = saverepls;
+
+       int newnummatches = origexponent / patternexponent;
+       if (newnummatches < nummatches)
+               nummatches = newnummatches;
+       return true;
+}
+
+ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
+{      
+       std::vector<bool> subsed(seq.size(), false);
+       exvector subsresult(seq.size());
+
+       for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
+
+               if (is_exactly_a<mul>(it->first)) {
+
+                       int nummatches = std::numeric_limits<int>::max();
+                       std::vector<bool> currsubsed(seq.size(), false);
+                       bool succeed = true;
+                       lst repls;
+
+                       for (size_t j=0; j<it->first.nops(); j++) {
+                               bool found=false;
+                               for (size_t k=0; k<nops(); k++) {
+                                       if (currsubsed[k] || subsed[k])
+                                               continue;
+                                       if (tryfactsubs(op(k), it->first.op(j), nummatches, repls)) {
+                                               currsubsed[k] = true;
+                                               found = true;
+                                               break;
+                                       }
+                               }
+                               if (!found) {
+                                       succeed = false;
+                                       break;
+                               }
+                       }
+                       if (!succeed)
+                               continue;
+
+                       bool foundfirstsubsedfactor = false;
+                       for (size_t j=0; j<subsed.size(); j++) {
+                               if (currsubsed[j]) {
+                                       if (foundfirstsubsedfactor)
+                                               subsresult[j] = op(j);
+                                       else {
+                                               foundfirstsubsedfactor = true;
+                                               subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::subs_no_pattern) / it->first.subs(ex(repls), subs_options::subs_no_pattern), nummatches);
+                                       }
+                                       subsed[j] = true;
+                               }
+                       }
+
+               } else {
+
+                       int nummatches = std::numeric_limits<int>::max();
+                       lst repls;
+
+                       for (size_t j=0; j<this->nops(); j++) {
+                               if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
+                                       subsed[j] = true;
+                                       subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::subs_no_pattern) / it->first.subs(ex(repls), subs_options::subs_no_pattern), nummatches);
+                               }
+                       }
+               }
        }
-       return inherited::simplify_ncmul(v);
+
+       bool subsfound = false;
+       for (size_t i=0; i<subsed.size(); i++) {
+               if (subsed[i]) {
+                       subsfound = true;
+                       break;
+               }
+       }
+       if (!subsfound)
+               return subs_one_level(m, options | subs_options::subs_algebraic);
+
+       exvector ev; ev.reserve(nops());
+       for (size_t i=0; i<nops(); i++) {
+               if (subsed[i])
+                       ev.push_back(subsresult[i]);
+               else
+                       ev.push_back(op(i));
+       }
+
+       return (new mul(ev))->setflag(status_flags::dynallocated);
 }
 
 // protected
@@ -464,15 +645,21 @@ ex mul::simplify_ncmul(const exvector & v) const
  *  @see ex::diff */
 ex mul::derivative(const symbol & s) const
 {
+       size_t num = seq.size();
        exvector addseq;
-       addseq.reserve(seq.size());
+       addseq.reserve(num);
        
        // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
-       for (unsigned i=0; i!=seq.size(); ++i) {
-               epvector mulseq = seq;
-               mulseq[i] = split_ex_to_pair(power(seq[i].rest,seq[i].coeff - _ex1()) *
-                                            seq[i].rest.diff(s));
-               addseq.push_back((new mul(mulseq,overall_coeff*seq[i].coeff))->setflag(status_flags::dynallocated));
+       epvector mulseq = seq;
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       epvector::iterator i2 = mulseq.begin();
+       while (i != end) {
+               expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
+                                            i->rest.diff(s));
+               ep.swap(*i2);
+               addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
+               ep.swap(*i2);
+               ++i; ++i2;
        }
        return (new add(addseq))->setflag(status_flags::dynallocated);
 }
@@ -482,51 +669,50 @@ int mul::compare_same_type(const basic & other) const
        return inherited::compare_same_type(other);
 }
 
-bool mul::is_equal_same_type(const basic & other) const
-{
-       return inherited::is_equal_same_type(other);
-}
-
-unsigned mul::return_type(void) const
+unsigned mul::return_type() const
 {
-       if (seq.size()==0) {
+       if (seq.empty()) {
                // mul without factors: should not happen, but commutes
                return return_types::commutative;
        }
        
-       bool all_commutative = 1;
-       unsigned rt;
-       epvector::const_iterator cit_noncommutative_element; // point to first found nc element
+       bool all_commutative = true;
+       epvector::const_iterator noncommutative_element; // point to first found nc element
        
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               rt=(*cit).rest.return_type();
-               if (rt==return_types::noncommutative_composite) return rt; // one ncc -> mul also ncc
-               if ((rt==return_types::noncommutative)&&(all_commutative)) {
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               unsigned rt = i->rest.return_type();
+               if (rt == return_types::noncommutative_composite)
+                       return rt; // one ncc -> mul also ncc
+               if ((rt == return_types::noncommutative) && (all_commutative)) {
                        // first nc element found, remember position
-                       cit_noncommutative_element = cit;
-                       all_commutative = 0;
+                       noncommutative_element = i;
+                       all_commutative = false;
                }
-               if ((rt==return_types::noncommutative)&&(!all_commutative)) {
+               if ((rt == return_types::noncommutative) && (!all_commutative)) {
                        // another nc element found, compare type_infos
-                       if ((*cit_noncommutative_element).rest.return_type_tinfo()!=(*cit).rest.return_type_tinfo()) {
+                       if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
                                // diffent types -> mul is ncc
                                return return_types::noncommutative_composite;
                        }
                }
+               ++i;
        }
        // all factors checked
        return all_commutative ? return_types::commutative : return_types::noncommutative;
 }
    
-unsigned mul::return_type_tinfo(void) const
+unsigned mul::return_type_tinfo() const
 {
-       if (seq.size()==0)
+       if (seq.empty())
                return tinfo_key;  // mul without factors: should not happen
        
        // return type_info of first noncommutative element
-       for (epvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-               if ((*cit).rest.return_type()==return_types::noncommutative)
-                       return (*cit).rest.return_type_tinfo();
+       epvector::const_iterator i = seq.begin(), end = seq.end();
+       while (i != end) {
+               if (i->rest.return_type() == return_types::noncommutative)
+                       return i->rest.return_type_tinfo();
+               ++i;
        }
        // no noncommutative element found, should not happen
        return tinfo_key;
@@ -534,22 +720,22 @@ unsigned mul::return_type_tinfo(void) const
 
 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
 {
-       return (new mul(v,oc))->setflag(status_flags::dynallocated);
+       return (new mul(v, oc))->setflag(status_flags::dynallocated);
 }
 
 ex mul::thisexpairseq(epvector * vp, const ex & oc) const
 {
-       return (new mul(vp,oc))->setflag(status_flags::dynallocated);
+       return (new mul(vp, oc))->setflag(status_flags::dynallocated);
 }
 
 expair mul::split_ex_to_pair(const ex & e) const
 {
-       if (is_ex_exactly_of_type(e,power)) {
-               const power & powerref = ex_to_power(e);
-               if (is_ex_exactly_of_type(powerref.exponent,numeric))
+       if (is_exactly_a<power>(e)) {
+               const power & powerref = ex_to<power>(e);
+               if (is_exactly_a<numeric>(powerref.exponent))
                        return expair(powerref.basis,powerref.exponent);
        }
-       return expair(e,_ex1());
+       return expair(e,_ex1);
 }
        
 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
@@ -557,11 +743,11 @@ expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 {
        // to avoid duplication of power simplification rules,
        // we create a temporary power object
-       // otherwise it would be hard to correctly simplify
+       // otherwise it would be hard to correctly evaluate
        // expression like (4^(1/3))^(3/2)
-       if (are_ex_trivially_equal(c,_ex1()))
+       if (c.is_equal(_ex1))
                return split_ex_to_pair(e);
-       
+
        return split_ex_to_pair(power(e,c));
 }
        
@@ -570,38 +756,38 @@ expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 {
        // to avoid duplication of power simplification rules,
        // we create a temporary power object
-       // otherwise it would be hard to correctly simplify
+       // otherwise it would be hard to correctly evaluate
        // expression like (4^(1/3))^(3/2)
-       if (are_ex_trivially_equal(c,_ex1()))
+       if (c.is_equal(_ex1))
                return p;
-       
+
        return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 }
        
 ex mul::recombine_pair_to_ex(const expair & p) const
 {
-       if (ex_to_numeric(p.coeff).is_equal(_num1())) 
+       if (ex_to<numeric>(p.coeff).is_equal(_num1)) 
                return p.rest;
        else
-               return power(p.rest,p.coeff);
+               return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 }
 
 bool mul::expair_needs_further_processing(epp it)
 {
-       if (is_ex_exactly_of_type((*it).rest,mul) &&
-               ex_to_numeric((*it).coeff).is_integer()) {
+       if (is_exactly_a<mul>(it->rest) &&
+               ex_to<numeric>(it->coeff).is_integer()) {
                // combined pair is product with integer power -> expand it
                *it = split_ex_to_pair(recombine_pair_to_ex(*it));
                return true;
        }
-       if (is_ex_exactly_of_type((*it).rest,numeric)) {
-               expair ep=split_ex_to_pair(recombine_pair_to_ex(*it));
+       if (is_exactly_a<numeric>(it->rest)) {
+               expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
                if (!ep.is_equal(*it)) {
                        // combined pair is a numeric power which can be simplified
                        *it = ep;
                        return true;
                }
-               if (ex_to_numeric((*it).coeff).is_equal(_num1())) {
+               if (it->coeff.is_equal(_ex1)) {
                        // combined pair has coeff 1 and must be moved to the end
                        return true;
                }
@@ -609,73 +795,110 @@ bool mul::expair_needs_further_processing(epp it)
        return false;
 }       
 
-ex mul::default_overall_coeff(void) const
+ex mul::default_overall_coeff() const
 {
-       return _ex1();
+       return _ex1;
 }
 
 void mul::combine_overall_coeff(const ex & c)
 {
-       GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-       GINAC_ASSERT(is_ex_exactly_of_type(c,numeric));
-       overall_coeff = ex_to_numeric(overall_coeff).mul_dyn(ex_to_numeric(c));
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       GINAC_ASSERT(is_exactly_a<numeric>(c));
+       overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 }
 
 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 {
-       GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-       GINAC_ASSERT(is_ex_exactly_of_type(c1,numeric));
-       GINAC_ASSERT(is_ex_exactly_of_type(c2,numeric));
-       overall_coeff = ex_to_numeric(overall_coeff).mul_dyn(ex_to_numeric(c1).power(ex_to_numeric(c2)));
+       GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+       GINAC_ASSERT(is_exactly_a<numeric>(c1));
+       GINAC_ASSERT(is_exactly_a<numeric>(c2));
+       overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 }
 
 bool mul::can_make_flat(const expair & p) const
 {
-       GINAC_ASSERT(is_ex_exactly_of_type(p.coeff,numeric));
+       GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
        // this assertion will probably fail somewhere
        // it would require a more careful make_flat, obeying the power laws
        // probably should return true only if p.coeff is integer
-       return ex_to_numeric(p.coeff).is_equal(_num1());
+       return ex_to<numeric>(p.coeff).is_equal(_num1);
 }
 
 ex mul::expand(unsigned options) const
 {
-       if (flags & status_flags::expanded)
-               return *this;
-       
-       exvector sub_expanded_seq;
-       
+       // First, expand the children
        epvector * expanded_seqp = expandchildren(options);
-       
-       const epvector & expanded_seq = expanded_seqp==0 ? seq : *expanded_seqp;
-       
+       const epvector & expanded_seq = (expanded_seqp == NULL) ? seq : *expanded_seqp;
+
+       // Now, look for all the factors that are sums and multiply each one out
+       // with the next one that is found while collecting the factors which are
+       // not sums
        int number_of_adds = 0;
+       ex last_expanded = _ex1;
        epvector non_adds;
        non_adds.reserve(expanded_seq.size());
-       epvector::const_iterator cit = expanded_seq.begin();
-       epvector::const_iterator last = expanded_seq.end();
-       ex last_expanded = _ex1();
-       while (cit!=last) {
-               if (is_ex_exactly_of_type((*cit).rest,add) &&
-                       ((*cit).coeff.is_equal(_ex1()))) {
+       epvector::const_iterator cit = expanded_seq.begin(), last = expanded_seq.end();
+       while (cit != last) {
+               if (is_exactly_a<add>(cit->rest) &&
+                       (cit->coeff.is_equal(_ex1))) {
                        ++number_of_adds;
-                       if (is_ex_exactly_of_type(last_expanded,add)) {
-                               // expand adds
-                               const add & add1 = ex_to_add(last_expanded);
-                               const add & add2 = ex_to_add((*cit).rest);
-                               int n1 = add1.nops();
-                               int n2 = add2.nops();
-                               exvector distrseq;
-                               distrseq.reserve(n1*n2);
-                               for (int i1=0; i1<n1; ++i1) {
-                                       for (int i2=0; i2<n2; ++i2) {
-                                               distrseq.push_back(add1.op(i1)*add2.op(i2));
+                       if (is_exactly_a<add>(last_expanded)) {
+
+                               // Expand a product of two sums, aggressive version.
+                               // Caring for the overall coefficients in separate loops can
+                               // sometimes give a performance gain of up to 15%!
+
+                               const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
+                               // add2 is for the inner loop and should be the bigger of the two sums
+                               // in the presence of asymptotically good sorting:
+                               const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
+                               const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
+                               const epvector::const_iterator add1begin = add1.seq.begin();
+                               const epvector::const_iterator add1end   = add1.seq.end();
+                               const epvector::const_iterator add2begin = add2.seq.begin();
+                               const epvector::const_iterator add2end   = add2.seq.end();
+                               epvector distrseq;
+                               distrseq.reserve(add1.seq.size()+add2.seq.size());
+                               // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
+                               if (!add1.overall_coeff.is_zero()) {
+                                       if (add1.overall_coeff.is_equal(_ex1))
+                                               distrseq.insert(distrseq.end(),add2begin,add2end);
+                                       else
+                                               for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
+                                                       distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
+                               }
+                               // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
+                               if (!add2.overall_coeff.is_zero()) {
+                                       if (add2.overall_coeff.is_equal(_ex1))
+                                               distrseq.insert(distrseq.end(),add1begin,add1end);
+                                       else
+                                               for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
+                                                       distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
+                               }
+                               // Compute the new overall coefficient and put it together:
+                               ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
+                               // Multiply explicitly all non-numeric terms of add1 and add2:
+                               for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
+                                       // We really have to combine terms here in order to compactify
+                                       // the result.  Otherwise it would become waayy tooo bigg.
+                                       numeric oc;
+                                       distrseq.clear();
+                                       for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
+                                               // Don't push_back expairs which might have a rest that evaluates to a numeric,
+                                               // since that would violate an invariant of expairseq:
+                                               const ex rest = (new mul(i1->rest, i2->rest))->setflag(status_flags::dynallocated);
+                                               if (is_exactly_a<numeric>(rest))
+                                                       oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
+                                               else
+                                                       distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
                                        }
+                                       tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
                                }
-                               last_expanded = (new add(distrseq))->setflag(status_flags::dynallocated | status_flags::expanded);
+                               last_expanded = tmp_accu;
+
                        } else {
                                non_adds.push_back(split_ex_to_pair(last_expanded));
-                               last_expanded = (*cit).rest;
+                               last_expanded = cit->rest;
                        }
                } else {
                        non_adds.push_back(*cit);
@@ -684,23 +907,26 @@ ex mul::expand(unsigned options) const
        }
        if (expanded_seqp)
                delete expanded_seqp;
-
-       if (is_ex_exactly_of_type(last_expanded,add)) {
-               add const & finaladd = ex_to_add(last_expanded);
+       
+       // Now the only remaining thing to do is to multiply the factors which
+       // were not sums into the "last_expanded" sum
+       if (is_exactly_a<add>(last_expanded)) {
+               const add & finaladd = ex_to<add>(last_expanded);
                exvector distrseq;
-               int n = finaladd.nops();
+               size_t n = finaladd.nops();
                distrseq.reserve(n);
-               for (int i=0; i<n; ++i) {
+               for (size_t i=0; i<n; ++i) {
                        epvector factors = non_adds;
                        factors.push_back(split_ex_to_pair(finaladd.op(i)));
-                       distrseq.push_back((new mul(factors,overall_coeff))->setflag(status_flags::dynallocated | status_flags::expanded));
+                       distrseq.push_back((new mul(factors, overall_coeff))->
+                                           setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
                }
                return ((new add(distrseq))->
-                       setflag(status_flags::dynallocated | status_flags::expanded));
+                       setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
        }
        non_adds.push_back(split_ex_to_pair(last_expanded));
-       return (new mul(non_adds,overall_coeff))->
-               setflag(status_flags::dynallocated | status_flags::expanded);
+       return (new mul(non_adds, overall_coeff))->
+               setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 }
 
   
@@ -724,7 +950,7 @@ ex mul::expand(unsigned options) const
  *  pointer, if sequence is unchanged. */
 epvector * mul::expandchildren(unsigned options) const
 {
-       epvector::const_iterator last = seq.end();
+       const epvector::const_iterator last = seq.end();
        epvector::const_iterator cit = seq.begin();
        while (cit!=last) {
                const ex & factor = recombine_pair_to_ex(*cit);