power.cpp

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/*
 *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */

#include "power.h"
#include "expairseq.h"
#include "add.h"
#include "mul.h"
#include "ncmul.h"
#include "numeric.h"
#include "constant.h"
#include "operators.h"
#include "inifcns.h" // for log() in power::derivative()
#include "matrix.h"
#include "indexed.h"
#include "symbol.h"
#include "lst.h"
#include "archive.h"
#include "utils.h"
#include "relational.h"
#include "compiler.h"

#include <iostream>
#include <limits>
#include <stdexcept>
#include <vector>

namespace GiNaC {

GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  print_func<print_dflt>(&power::do_print_dflt).
  print_func<print_latex>(&power::do_print_latex).
  print_func<print_csrc>(&power::do_print_csrc).
  print_func<print_python>(&power::do_print_python).
  print_func<print_python_repr>(&power::do_print_python_repr).
  print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))

typedef std::vector<int> intvector;

// default constructor

power::power() { }

// other constructors

// all inlined

// archiving

void power::read_archive(const archive_node &n, lst &sym_lst)
{
    inherited::read_archive(n, sym_lst);
    n.find_ex("basis", basis, sym_lst);
    n.find_ex("exponent", exponent, sym_lst);
}

void power::archive(archive_node &n) const
{
    inherited::archive(n);
    n.add_ex("basis", basis);
    n.add_ex("exponent", exponent);
}

// functions overriding virtual functions from base classes

// public

void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
{
    // Ordinary output of powers using '^' or '**'
    if (precedence() <= level)
        c.s << openbrace << '(';
    basis.print(c, precedence());
    c.s << powersymbol;
    c.s << openbrace;
    exponent.print(c, precedence());
    c.s << closebrace;
    if (precedence() <= level)
        c.s << ')' << closebrace;
}

void power::do_print_dflt(const print_dflt & c, unsigned level) const
{
    if (exponent.is_equal(_ex1_2)) {

        // Square roots are printed in a special way
        c.s << "sqrt(";
        basis.print(c);
        c.s << ')';

    } else
        print_power(c, "^", "", "", level);
}

void power::do_print_latex(const print_latex & c, unsigned level) const
{
    if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {

        // Powers with negative numeric exponents are printed as fractions
        c.s << "\\frac{1}{";
        power(basis, -exponent).eval().print(c);
        c.s << '}';

    } else if (exponent.is_equal(_ex1_2)) {

        // Square roots are printed in a special way
        c.s << "\\sqrt{";
        basis.print(c);
        c.s << '}';

    } else
        print_power(c, "^", "{", "}", level);
}

static void print_sym_pow(const print_context & c, const symbol &x, int exp)
{
    // Optimal output of integer powers of symbols to aid compiler CSE.
    // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
    // to learn why such a parenthesation is really necessary.
    if (exp == 1) {
        x.print(c);
    } else if (exp == 2) {
        x.print(c);
        c.s << "*";
        x.print(c);
    } else if (exp & 1) {
        x.print(c);
        c.s << "*";
        print_sym_pow(c, x, exp-1);
    } else {
        c.s << "(";
        print_sym_pow(c, x, exp >> 1);
        c.s << ")*(";
        print_sym_pow(c, x, exp >> 1);
        c.s << ")";
    }
}

void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
{
    if (exponent.is_equal(_ex_1)) {
        c.s << "recip(";
        basis.print(c);
        c.s << ')';
        return;
    }
    c.s << "expt(";
    basis.print(c);
    c.s << ", ";
    exponent.print(c);
    c.s << ')';
}

void power::do_print_csrc(const print_csrc & c, unsigned level) const
{
    // Integer powers of symbols are printed in a special, optimized way
    if (exponent.info(info_flags::integer)
     && (is_a<symbol>(basis) || is_a<constant>(basis))) {
        int exp = ex_to<numeric>(exponent).to_int();
        if (exp > 0)
            c.s << '(';
        else {
            exp = -exp;
            c.s << "1.0/(";
        }
        print_sym_pow(c, ex_to<symbol>(basis), exp);
        c.s << ')';

    // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
    } else if (exponent.is_equal(_ex_1)) {
        c.s << "1.0/(";
        basis.print(c);
        c.s << ')';

    // Otherwise, use the pow() function
    } else {
        c.s << "pow(";
        basis.print(c);
        c.s << ',';
        exponent.print(c);
        c.s << ')';
    }
}

void power::do_print_python(const print_python & c, unsigned level) const
{
    print_power(c, "**", "", "", level);
}

void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
{
    c.s << class_name() << '(';
    basis.print(c);
    c.s << ',';
    exponent.print(c);
    c.s << ')';
}

bool power::info(unsigned inf) const
{
    switch (inf) {
        case info_flags::polynomial:
        case info_flags::integer_polynomial:
        case info_flags::cinteger_polynomial:
        case info_flags::rational_polynomial:
        case info_flags::crational_polynomial:
            return exponent.info(info_flags::nonnegint) &&
                   basis.info(inf);
        case info_flags::rational_function:
            return exponent.info(info_flags::integer) &&
                   basis.info(inf);
        case info_flags::algebraic:
            return !exponent.info(info_flags::integer) ||
                   basis.info(inf);
        case info_flags::expanded:
            return (flags & status_flags::expanded);
        case info_flags::positive:
            return basis.info(info_flags::positive) && exponent.info(info_flags::real);
        case info_flags::has_indices: {
            if (flags & status_flags::has_indices)
                return true;
            else if (flags & status_flags::has_no_indices)
                return false;
            else if (basis.info(info_flags::has_indices)) {
                setflag(status_flags::has_indices);
                clearflag(status_flags::has_no_indices);
                return true;
            } else {
                clearflag(status_flags::has_indices);
                setflag(status_flags::has_no_indices);
                return false;
            }
        }
    }
    return inherited::info(inf);
}

size_t power::nops() const
{
    return 2;
}

ex power::op(size_t i) const
{
    GINAC_ASSERT(i<2);

    return i==0 ? basis : exponent;
}

ex power::map(map_function & f) const
{
    const ex &mapped_basis = f(basis);
    const ex &mapped_exponent = f(exponent);

    if (!are_ex_trivially_equal(basis, mapped_basis)
     || !are_ex_trivially_equal(exponent, mapped_exponent))
        return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
    else
        return *this;
}

bool power::is_polynomial(const ex & var) const
{
    if (exponent.has(var))
        return false;
    if (!exponent.info(info_flags::nonnegint))
        return false;
    return basis.is_polynomial(var);
}

int power::degree(const ex & s) const
{
    if (is_equal(ex_to<basic>(s)))
        return 1;
    else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
        if (basis.is_equal(s))
            return ex_to<numeric>(exponent).to_int();
        else
            return basis.degree(s) * ex_to<numeric>(exponent).to_int();
    } else if (basis.has(s))
        throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
    else
        return 0;
}

int power::ldegree(const ex & s) const 
{
    if (is_equal(ex_to<basic>(s)))
        return 1;
    else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
        if (basis.is_equal(s))
            return ex_to<numeric>(exponent).to_int();
        else
            return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
    } else if (basis.has(s))
        throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
    else
        return 0;
}

ex power::coeff(const ex & s, int n) const
{
    if (is_equal(ex_to<basic>(s)))
        return n==1 ? _ex1 : _ex0;
    else if (!basis.is_equal(s)) {
        // basis not equal to s
        if (n == 0)
            return *this;
        else
            return _ex0;
    } else {
        // basis equal to s
        if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
            // integer exponent
            int int_exp = ex_to<numeric>(exponent).to_int();
            if (n == int_exp)
                return _ex1;
            else
                return _ex0;
        } else {
            // non-integer exponents are treated as zero
            if (n == 0)
                return *this;
            else
                return _ex0;
        }
    }
}

ex power::eval(int level) const
{
    if ((level==1) && (flags & status_flags::evaluated))
        return *this;
    else if (level == -max_recursion_level)
        throw(std::runtime_error("max recursion level reached"));
    
    const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
    const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
    
    const numeric *num_basis = NULL;
    const numeric *num_exponent = NULL;
    
    if (is_exactly_a<numeric>(ebasis)) {
        num_basis = &ex_to<numeric>(ebasis);
    }
    if (is_exactly_a<numeric>(eexponent)) {
        num_exponent = &ex_to<numeric>(eexponent);
    }
    
    // ^(x,0) -> 1  (0^0 also handled here)
    if (eexponent.is_zero()) {
        if (ebasis.is_zero())
            throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
        else
            return _ex1;
    }
    
    // ^(x,1) -> x
    if (eexponent.is_equal(_ex1))
        return ebasis;

    // ^(0,c1) -> 0 or exception  (depending on real value of c1)
    if ( ebasis.is_zero() && num_exponent ) {
        if ((num_exponent->real()).is_zero())
            throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
        else if ((num_exponent->real()).is_negative())
            throw (pole_error("power::eval(): division by zero",1));
        else
            return _ex0;
    }

    // ^(1,x) -> 1
    if (ebasis.is_equal(_ex1))
        return _ex1;

    // power of a function calculated by separate rules defined for this function
    if (is_exactly_a<function>(ebasis))
        return ex_to<function>(ebasis).power(eexponent);

    // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
    if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
        return power(ebasis.op(0), ebasis.op(1) * eexponent);

    if ( num_exponent ) {

        // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
        // except if c1,c2 are rational, but c1^c2 is not)
        if ( num_basis ) {
            const bool basis_is_crational = num_basis->is_crational();
            const bool exponent_is_crational = num_exponent->is_crational();
            if (!basis_is_crational || !exponent_is_crational) {
                // return a plain float
                return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
                                                                               status_flags::evaluated |
                                                                               status_flags::expanded);
            }

            const numeric res = num_basis->power(*num_exponent);
            if (res.is_crational()) {
                return res;
            }
            GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now

            // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
            if (basis_is_crational && exponent_is_crational
                && num_exponent->is_real()
                && !num_exponent->is_integer()) {
                const numeric n = num_exponent->numer();
                const numeric m = num_exponent->denom();
                numeric r;
                numeric q = iquo(n, m, r);
                if (r.is_negative()) {
                    r += m;
                    --q;
                }
                if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
                    if (num_basis->is_rational() && !num_basis->is_integer()) {
                        // try it for numerator and denominator separately, in order to
                        // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
                        const numeric bnum = num_basis->numer();
                        const numeric bden = num_basis->denom();
                        const numeric res_bnum = bnum.power(*num_exponent);
                        const numeric res_bden = bden.power(*num_exponent);
                        if (res_bnum.is_integer())
                            return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
                        if (res_bden.is_integer())
                            return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
                    }
                    return this->hold();
                } else {
                    // assemble resulting product, but allowing for a re-evaluation,
                    // because otherwise we'll end up with something like
                    //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
                    // instead of 7/16*7^(1/3).
                    ex prod = power(*num_basis,r.div(m));
                    return prod*power(*num_basis,q);
                }
            }
        }
    
        // ^(^(x,c1),c2) -> ^(x,c1*c2)
        // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
        // case c1==1 should not happen, see below!)
        if (is_exactly_a<power>(ebasis)) {
            const power & sub_power = ex_to<power>(ebasis);
            const ex & sub_basis = sub_power.basis;
            const ex & sub_exponent = sub_power.exponent;
            if (is_exactly_a<numeric>(sub_exponent)) {
                const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
                GINAC_ASSERT(num_sub_exponent!=numeric(1));
                if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
                    return power(sub_basis,num_sub_exponent.mul(*num_exponent));
                }
            }
        }
    
        // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
        if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
            return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
        }

        // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
        if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
            numeric icont = ebasis.integer_content();
            const numeric lead_coeff = 
                ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);

            const bool canonicalizable = lead_coeff.is_integer();
            const bool unit_normal = lead_coeff.is_pos_integer();
            if (canonicalizable && (! unit_normal))
                icont = icont.mul(*_num_1_p);
            
            if (canonicalizable && (icont != *_num1_p)) {
                const add& addref = ex_to<add>(ebasis);
                add* addp = new add(addref);
                addp->setflag(status_flags::dynallocated);
                addp->clearflag(status_flags::hash_calculated);
                addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
                for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
                    i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);

                const numeric c = icont.power(*num_exponent);
                if (likely(c != *_num1_p))
                    return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
                else
                    return power(*addp, *num_exponent);
            }
        }

        // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
        // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
        if (is_exactly_a<mul>(ebasis)) {
            GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
            const mul & mulref = ex_to<mul>(ebasis);
            if (!mulref.overall_coeff.is_equal(_ex1)) {
                const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
                if (num_coeff.is_real()) {
                    if (num_coeff.is_positive()) {
                        mul *mulp = new mul(mulref);
                        mulp->overall_coeff = _ex1;
                        mulp->setflag(status_flags::dynallocated);
                        mulp->clearflag(status_flags::evaluated);
                        mulp->clearflag(status_flags::hash_calculated);
                        return (new mul(power(*mulp,exponent),
                                        power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
                    } else {
                        GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
                        if (!num_coeff.is_equal(*_num_1_p)) {
                            mul *mulp = new mul(mulref);
                            mulp->overall_coeff = _ex_1;
                            mulp->setflag(status_flags::dynallocated);
                            mulp->clearflag(status_flags::evaluated);
                            mulp->clearflag(status_flags::hash_calculated);
                            return (new mul(power(*mulp,exponent),
                                            power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
                        }
                    }
                }
            }
        }

        // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
        if (num_exponent->is_pos_integer() &&
            ebasis.return_type() != return_types::commutative &&
            !is_a<matrix>(ebasis)) {
            return ncmul(exvector(num_exponent->to_int(), ebasis), true);
        }
    }
    
    if (are_ex_trivially_equal(ebasis,basis) &&
        are_ex_trivially_equal(eexponent,exponent)) {
        return this->hold();
    }
    return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
                                                   status_flags::evaluated);
}

ex power::evalf(int level) const
{
    ex ebasis;
    ex eexponent;
    
    if (level==1) {
        ebasis = basis;
        eexponent = exponent;
    } else if (level == -max_recursion_level) {
        throw(std::runtime_error("max recursion level reached"));
    } else {
        ebasis = basis.evalf(level-1);
        if (!is_exactly_a<numeric>(exponent))
            eexponent = exponent.evalf(level-1);
        else
            eexponent = exponent;
    }

    return power(ebasis,eexponent);
}

ex power::evalm() const
{
    const ex ebasis = basis.evalm();
    const ex eexponent = exponent.evalm();
    if (is_a<matrix>(ebasis)) {
        if (is_exactly_a<numeric>(eexponent)) {
            return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
        }
    }
    return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
}

bool power::has(const ex & other, unsigned options) const
{
    if (!(options & has_options::algebraic))
        return basic::has(other, options);
    if (!is_a<power>(other))
        return basic::has(other, options);
    if (!exponent.info(info_flags::integer)
            || !other.op(1).info(info_flags::integer))
        return basic::has(other, options);
    if (exponent.info(info_flags::posint)
            && other.op(1).info(info_flags::posint)
            && ex_to<numeric>(exponent).to_int()
                    > ex_to<numeric>(other.op(1)).to_int()
            && basis.match(other.op(0)))
        return true;
    if (exponent.info(info_flags::negint)
            && other.op(1).info(info_flags::negint)
            && ex_to<numeric>(exponent).to_int()
                    < ex_to<numeric>(other.op(1)).to_int()
            && basis.match(other.op(0)))
        return true;
    return basic::has(other, options);
}

// from mul.cpp
extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);

ex power::subs(const exmap & m, unsigned options) const
{   
    const ex &subsed_basis = basis.subs(m, options);
    const ex &subsed_exponent = exponent.subs(m, options);

    if (!are_ex_trivially_equal(basis, subsed_basis)
     || !are_ex_trivially_equal(exponent, subsed_exponent)) 
        return power(subsed_basis, subsed_exponent).subs_one_level(m, options);

    if (!(options & subs_options::algebraic))
        return subs_one_level(m, options);

    for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
        int nummatches = std::numeric_limits<int>::max();
        exmap repls;
        if (tryfactsubs(*this, it->first, nummatches, repls)) {
            ex anum = it->second.subs(repls, subs_options::no_pattern);
            ex aden = it->first.subs(repls, subs_options::no_pattern);
            ex result = (*this)*power(anum/aden, nummatches);
            return (ex_to<basic>(result)).subs_one_level(m, options);
        }
    }

    return subs_one_level(m, options);
}

ex power::eval_ncmul(const exvector & v) const
{
    return inherited::eval_ncmul(v);
}

ex power::conjugate() const
{
    // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
    // branch cut which runs along the negative real axis.
    if (basis.info(info_flags::positive)) {
        ex newexponent = exponent.conjugate();
        if (are_ex_trivially_equal(exponent, newexponent)) {
            return *this;
        }
        return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
    }
    if (exponent.info(info_flags::integer)) {
        ex newbasis = basis.conjugate();
        if (are_ex_trivially_equal(basis, newbasis)) {
            return *this;
        }
        return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
    }
    return conjugate_function(*this).hold();
}

ex power::real_part() const
{
    if (exponent.info(info_flags::integer)) {
        ex basis_real = basis.real_part();
        if (basis_real == basis)
            return *this;
        realsymbol a("a"),b("b");
        ex result;
        if (exponent.info(info_flags::posint))
            result = power(a+I*b,exponent);
        else
            result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
        result = result.expand();
        result = result.real_part();
        result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
        return result;
    }
    
    ex a = basis.real_part();
    ex b = basis.imag_part();
    ex c = exponent.real_part();
    ex d = exponent.imag_part();
    return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
}

ex power::imag_part() const
{
    if (exponent.info(info_flags::integer)) {
        ex basis_real = basis.real_part();
        if (basis_real == basis)
            return 0;
        realsymbol a("a"),b("b");
        ex result;
        if (exponent.info(info_flags::posint))
            result = power(a+I*b,exponent);
        else
            result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
        result = result.expand();
        result = result.imag_part();
        result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
        return result;
    }
    
    ex a=basis.real_part();
    ex b=basis.imag_part();
    ex c=exponent.real_part();
    ex d=exponent.imag_part();
    return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
}

// protected

// protected

ex power::derivative(const symbol & s) const
{
    if (is_a<numeric>(exponent)) {
        // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
        epvector newseq;
        newseq.reserve(2);
        newseq.push_back(expair(basis, exponent - _ex1));
        newseq.push_back(expair(basis.diff(s), _ex1));
        return mul(newseq, exponent);
    } else {
        // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
        return mul(*this,
                   add(mul(exponent.diff(s), log(basis)),
                   mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
    }
}

int power::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_exactly_a<power>(other));
    const power &o = static_cast<const power &>(other);

    int cmpval = basis.compare(o.basis);
    if (cmpval)
        return cmpval;
    else
        return exponent.compare(o.exponent);
}

unsigned power::return_type() const
{
    return basis.return_type();
}

return_type_t power::return_type_tinfo() const
{
    return basis.return_type_tinfo();
}

ex power::expand(unsigned options) const
{
    if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
        // A special case worth optimizing.
        setflag(status_flags::expanded);
        return *this;
    }

    const ex expanded_basis = basis.expand(options);
    const ex expanded_exponent = exponent.expand(options);
    
    // x^(a+b) -> x^a * x^b
    if (is_exactly_a<add>(expanded_exponent)) {
        const add &a = ex_to<add>(expanded_exponent);
        exvector distrseq;
        distrseq.reserve(a.seq.size() + 1);
        epvector::const_iterator last = a.seq.end();
        epvector::const_iterator cit = a.seq.begin();
        while (cit!=last) {
            distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
            ++cit;
        }
        
        // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
        if (ex_to<numeric>(a.overall_coeff).is_integer()) {
            const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
            int int_exponent = num_exponent.to_int();
            if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
                distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
            else
                distrseq.push_back(power(expanded_basis, a.overall_coeff));
        } else
            distrseq.push_back(power(expanded_basis, a.overall_coeff));
        
        // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
        ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
        return r.expand(options);
    }
    
    if (!is_exactly_a<numeric>(expanded_exponent) ||
        !ex_to<numeric>(expanded_exponent).is_integer()) {
        if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
            return this->hold();
        } else {
            return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
        }
    }
    
    // integer numeric exponent
    const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
    int int_exponent = num_exponent.to_int();
    
    // (x+y)^n, n>0
    if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
        return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
    
    // (x*y)^n -> x^n * y^n
    if (is_exactly_a<mul>(expanded_basis))
        return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
    
    // cannot expand further
    if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
        return this->hold();
    else
        return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
}

// new virtual functions which can be overridden by derived classes

// none

// non-virtual functions in this class

ex power::expand_add(const add & a, int n, unsigned options) const
{
    if (n==2)
        return expand_add_2(a, options);

    const size_t m = a.nops();
    exvector result;
    // The number of terms will be the number of combinatorial compositions,
    // i.e. the number of unordered arrangements of m nonnegative integers
    // which sum up to n.  It is frequently written as C_n(m) and directly
    // related with binomial coefficients:
    result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
    intvector k(m-1);
    intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
    intvector upper_limit(m-1);

    for (size_t l=0; l<m-1; ++l) {
        k[l] = 0;
        k_cum[l] = 0;
        upper_limit[l] = n;
    }

    while (true) {
        exvector term;
        term.reserve(m+1);
        for (std::size_t l = 0; l < m - 1; ++l) {
            const ex & b = a.op(l);
            GINAC_ASSERT(!is_exactly_a<add>(b));
            GINAC_ASSERT(!is_exactly_a<power>(b) ||
                         !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
                         !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
                         !is_exactly_a<add>(ex_to<power>(b).basis) ||
                         !is_exactly_a<mul>(ex_to<power>(b).basis) ||
                         !is_exactly_a<power>(ex_to<power>(b).basis));
            if (is_exactly_a<mul>(b))
                term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
            else
                term.push_back(power(b,k[l]));
        }

        const ex & b = a.op(m - 1);
        GINAC_ASSERT(!is_exactly_a<add>(b));
        GINAC_ASSERT(!is_exactly_a<power>(b) ||
                     !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
                     !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
                     !is_exactly_a<add>(ex_to<power>(b).basis) ||
                     !is_exactly_a<mul>(ex_to<power>(b).basis) ||
                     !is_exactly_a<power>(ex_to<power>(b).basis));
        if (is_exactly_a<mul>(b))
            term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
        else
            term.push_back(power(b,n-k_cum[m-2]));

        numeric f = binomial(numeric(n),numeric(k[0]));
        for (std::size_t l = 1; l < m - 1; ++l)
            f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));

        term.push_back(f);

        result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));

        // increment k[]
        bool done = false;
        std::size_t l = m - 2;
        while ((++k[l]) > upper_limit[l]) {
            k[l] = 0;
            if (l != 0)
                --l;
            else {
                done = true;
                break;
            }
        }
        if (done)
            break;

        // recalc k_cum[] and upper_limit[]
        k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);

        for (size_t i=l+1; i<m-1; ++i)
            k_cum[i] = k_cum[i-1]+k[i];

        for (size_t i=l+1; i<m-1; ++i)
            upper_limit[i] = n-k_cum[i-1];
    }

    return (new add(result))->setflag(status_flags::dynallocated |
                                      status_flags::expanded);
}


ex power::expand_add_2(const add & a, unsigned options) const
{
    epvector sum;
    size_t a_nops = a.nops();
    sum.reserve((a_nops*(a_nops+1))/2);
    epvector::const_iterator last = a.seq.end();

    // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
    // first part: ignore overall_coeff and expand other terms
    for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
        const ex & r = cit0->rest;
        const ex & c = cit0->coeff;
        
        GINAC_ASSERT(!is_exactly_a<add>(r));
        GINAC_ASSERT(!is_exactly_a<power>(r) ||
                     !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
                     !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
                     !is_exactly_a<add>(ex_to<power>(r).basis) ||
                     !is_exactly_a<mul>(ex_to<power>(r).basis) ||
                     !is_exactly_a<power>(ex_to<power>(r).basis));
        
        if (c.is_equal(_ex1)) {
            if (is_exactly_a<mul>(r)) {
                sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
                                     _ex1));
            } else {
                sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
                                     _ex1));
            }
        } else {
            if (is_exactly_a<mul>(r)) {
                sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
                                     ex_to<numeric>(c).power_dyn(*_num2_p)));
            } else {
                sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
                                     ex_to<numeric>(c).power_dyn(*_num2_p)));
            }
        }

        for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
            const ex & r1 = cit1->rest;
            const ex & c1 = cit1->coeff;
            sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
                                                          _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
        }
    }
    
    GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
    
    // second part: add terms coming from overall_factor (if != 0)
    if (!a.overall_coeff.is_zero()) {
        epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
        while (i != end) {
            sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
            ++i;
        }
        sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
    }
    
    GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
    
    return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
}

ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
{
    GINAC_ASSERT(n.is_integer());

    if (n.is_zero()) {
        return _ex1;
    }

    // do not bother to rename indices if there are no any.
    if ((!(options & expand_options::expand_rename_idx)) 
            && m.info(info_flags::has_indices))
        options |= expand_options::expand_rename_idx;
    // Leave it to multiplication since dummy indices have to be renamed
    if ((options & expand_options::expand_rename_idx) &&
        (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
        ex result = m;
        exvector va = get_all_dummy_indices(m);
        sort(va.begin(), va.end(), ex_is_less());

        for (int i=1; i < n.to_int(); i++)
            result *= rename_dummy_indices_uniquely(va, m);
        return result;
    }

    epvector distrseq;
    distrseq.reserve(m.seq.size());
    bool need_reexpand = false;

    epvector::const_iterator last = m.seq.end();
    epvector::const_iterator cit = m.seq.begin();
    while (cit!=last) {
        expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
        if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
            // this happens when e.g. (a+b)^(1/2) gets squared and
            // the resulting product needs to be reexpanded
            need_reexpand = true;
        }
        distrseq.push_back(p);
        ++cit;
    }

    const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
    if (need_reexpand)
        return ex(result).expand(options);
    if (from_expand)
        return result.setflag(status_flags::expanded);
    return result;
}

GINAC_BIND_UNARCHIVER(power);

} // namespace GiNaC