power.cpp

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/*
 *  GiNaC Copyright (C) 1999-2023 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>
#include <algorithm>
 
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))
 
 
// 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 14882:2011, 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 basis.info(inf) && exponent.info(info_flags::nonnegint);
        case info_flags::rational_function:
            return basis.info(inf) && exponent.info(info_flags::integer);
        case info_flags::real:
            return basis.info(inf) && exponent.info(info_flags::integer);
        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::nonnegative:
            return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
                   (basis.info(info_flags::real) && exponent.info(info_flags::even));
        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 dynallocate<power>(mapped_basis, mapped_exponent);
    else
        return *this;
}
 
bool power::is_polynomial(const ex & var) const
{
    if (basis.is_polynomial(var)) {
        if (basis.has(var))
            // basis is non-constant polynomial in var
            return exponent.info(info_flags::nonnegint);
        else
            // basis is constant in var
            return !exponent.has(var);
    }
    // basis is a non-polynomial function of var
    return false;
}
 
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() const
{
    if (flags & status_flags::evaluated)
        return *this;
 
    const numeric *num_basis = nullptr;
    const numeric *num_exponent = nullptr;
 
    if (is_exactly_a<numeric>(basis)) {
        num_basis = &ex_to<numeric>(basis);
    }
    if (is_exactly_a<numeric>(exponent)) {
        num_exponent = &ex_to<numeric>(exponent);
    }
    
    // ^(x,0) -> 1  (0^0 also handled here)
    if (exponent.is_zero()) {
        if (basis.is_zero())
            throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
        else
            return _ex1;
    }
    
    // ^(x,1) -> x
    if (exponent.is_equal(_ex1))
        return basis;
 
    // ^(0,c1) -> 0 or exception  (depending on real value of c1)
    if (basis.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 (basis.is_equal(_ex1))
        return _ex1;
 
    // power of a function calculated by separate rules defined for this function
    if (is_exactly_a<function>(basis))
        return ex_to<function>(basis).power(exponent);
 
    // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
    if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
        return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
 
    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 dynallocate<numeric>(num_basis->power(*num_exponent));
            }
 
            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 dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
                        if (res_bden.is_integer())
                            return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(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).
                    return pow(basis, r.div(m)) * pow(basis, q);
                }
            }
        }
    
        // ^(^(x,c1),c2) -> ^(x,c1*c2)
        // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
        // case c1==1 should not happen, see below!)
        if (is_exactly_a<power>(basis)) {
            const power & sub_power = ex_to<power>(basis);
            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() ||
                    (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
                    return dynallocate<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>(basis)) {
            return expand_mul(ex_to<mul>(basis), *num_exponent, false);
        }
 
        // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
        if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
            numeric icont = basis.integer_content();
            const numeric lead_coeff = 
                ex_to<numeric>(ex_to<add>(basis).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>(basis);
                add & addp = dynallocate<add>(addref);
                addp.clearflag(status_flags::hash_calculated);
                addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
                for (auto & i : addp.seq)
                    i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
 
                const numeric c = icont.power(*num_exponent);
                if (likely(c != *_num1_p))
                    return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
                else
                    return dynallocate<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>(basis)) {
            GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
            const mul & mulref = ex_to<mul>(basis);
            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 = dynallocate<mul>(mulref);
                        mulp.overall_coeff = _ex1;
                        mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
                        return dynallocate<mul>(dynallocate<power>(mulp, exponent),
                                                dynallocate<power>(num_coeff, *num_exponent));
                    } else {
                        GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
                        if (!num_coeff.is_equal(*_num_1_p)) {
                            mul & mulp = dynallocate<mul>(mulref);
                            mulp.overall_coeff = _ex_1;
                            mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
                            return dynallocate<mul>(dynallocate<power>(mulp, exponent),
                                                    dynallocate<power>(abs(num_coeff), *num_exponent));
                        }
                    }
                }
            }
        }
 
        // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
        if (num_exponent->is_pos_integer() &&
            basis.return_type() != return_types::commutative &&
            !is_a<matrix>(basis)) {
            return ncmul(exvector(num_exponent->to_int(), basis));
        }
    }
 
    return this->hold();
}
 
ex power::evalf() const
{
    ex ebasis = basis.evalf();
    ex eexponent;
    
    if (!is_exactly_a<numeric>(exponent))
        eexponent = exponent.evalf();
    else
        eexponent = exponent;
 
    return dynallocate<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 dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
        }
    }
    return dynallocate<power>(ebasis, eexponent);
}
 
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) > ex_to<numeric>(other.op(1)) &&
        basis.match(other.op(0)))
        return true;
    if (exponent.info(info_flags::negint) &&
        other.op(1).info(info_flags::negint) &&
        ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
        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 dynallocate<power>(subsed_basis, subsed_exponent);
 
    if (!(options & subs_options::algebraic))
        return subs_one_level(m, options);
 
    for (auto & it : m) {
        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) * pow(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 dynallocate<power>(basis, newexponent);
    }
    if (exponent.info(info_flags::integer)) {
        ex newbasis = basis.conjugate();
        if (are_ex_trivially_equal(basis, newbasis)) {
            return *this;
        }
        return dynallocate<power>(newbasis, exponent);
    }
    return conjugate_function(*this).hold();
}
 
ex power::real_part() const
{
    // basis == a+I*b, exponent == c+I*d
    const ex a = basis.real_part();
    const ex c = exponent.real_part();
    if (basis.is_equal(a) && exponent.is_equal(c) &&
        (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) {
        // Re(a^c)
        return *this;
    }
 
    const ex b = basis.imag_part();
    if (exponent.info(info_flags::integer)) {
        // Re((a+I*b)^c)  w/  c ∈ ℤ
        long N = ex_to<numeric>(c).to_long();
        // Use real terms in Binomial expansion to construct
        // Re(expand(pow(a+I*b, N))).
        long NN = N > 0 ? N : -N;
        ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
        ex result = 0;
        for (long n = 0; n <= NN; n += 2) {
            ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
            if (n % 4 == 0) {
                result += term;  // sign: I^n w/ n == 4*m
            } else {
                result -= term;  // sign: I^n w/ n == 4*m+2
            }
        }
        return result;
    }
 
    // Re((a+I*b)^(c+I*d))
    const ex d = exponent.imag_part();
    return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
}
 
ex power::imag_part() const
{
    // basis == a+I*b, exponent == c+I*d
    const ex a = basis.real_part();
    const ex c = exponent.real_part();
    if (basis.is_equal(a) && exponent.is_equal(c) &&
        (a.info(info_flags::nonnegative) || c.info(info_flags::integer))) {
        // Im(a^c)
        return 0;
    }
 
    const ex b = basis.imag_part();
    if (exponent.info(info_flags::integer)) {
        // Im((a+I*b)^c)  w/  c ∈ ℤ
        long N = ex_to<numeric>(c).to_long();
        // Use imaginary terms in Binomial expansion to construct
        // Im(expand(pow(a+I*b, N))).
        long p = N > 0 ? 1 : 3;  // modulus for positive sign
        long NN = N > 0 ? N : -N;
        ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
        ex result = 0;
        for (long n = 1; n <= NN; n += 2) {
            ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
            if (n % 4 == p) {
                result += term;  // sign: I^n w/ n == 4*m+p
            } else {
                result -= term;  // sign: I^n w/ n == 4*m+2+p
            }
        }
        return result;
    }
 
    // Im((a+I*b)^(c+I*d))
    const ex d = exponent.imag_part();
    return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
}
 
// 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)
        const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
        return dynallocate<mul>(std::move(newseq), exponent);
    } else {
        // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
        return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(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;
    }
 
    // (x*p)^c -> x^c * p^c, if p>0
    // makes sense before expanding the basis
    if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
        const mul &m = ex_to<mul>(basis);
        exvector prodseq;
        epvector powseq;
        prodseq.reserve(m.seq.size() + 1);
        powseq.reserve(m.seq.size() + 1);
        bool possign = true;
 
        // search for positive/negative factors
        for (auto & cit : m.seq) {
            ex e=m.recombine_pair_to_ex(cit);
            if (e.info(info_flags::positive))
                prodseq.push_back(pow(e, exponent).expand(options));
            else if (e.info(info_flags::negative)) {
                prodseq.push_back(pow(-e, exponent).expand(options));
                possign = !possign;
            } else
                powseq.push_back(cit);
        }
 
        // take care on the numeric coefficient
        ex coeff=(possign? _ex1 : _ex_1);
        if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
            prodseq.push_back(pow(m.overall_coeff, exponent));
        else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1) {
            prodseq.push_back(pow(-m.overall_coeff, exponent));
            coeff = -coeff;
        } else
            coeff *= m.overall_coeff;
 
        // If positive/negative factors are found, then extract them.
        // In either case we set a flag to avoid the second run on a part
        // which does not have positive/negative terms.
        if (prodseq.size() > 0) {
            ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
            ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
            return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
        } else
            ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
    }
 
    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);
        for (auto & cit : a.seq) {
            distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(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);
            long 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(pow(expanded_basis, a.overall_coeff));
        } else
            distrseq.push_back(pow(expanded_basis, a.overall_coeff));
        
        // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
        ex r = dynallocate<mul>(distrseq);
        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 dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
        }
    }
    
    // integer numeric exponent
    const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
    long int_exponent = num_exponent.to_long();
    
    // (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 dynallocate<power>(expanded_basis, expanded_exponent).setflag(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, long n, unsigned options)
{
    // The special case power(+(x,...y;x),2) can be optimized better.
    if (n==2)
        return expand_add_2(a, options);
 
    // method:
    //
    // Consider base as the sum of all symbolic terms and the overall numeric
    // coefficient and apply the binomial theorem:
    // S = power(+(x,...,z;c),n)
    //   = power(+(+(x,...,z;0);c),n)
    //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
    // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
    // The multinomial theorem is computed by an outer loop over all
    // partitions of the exponent and an inner loop over all compositions of
    // that partition. This method makes the expansion a combinatorial
    // problem and allows us to directly construct the expanded sum and also
    // to re-use the multinomial coefficients (since they depend only on the
    // partition, not on the composition).
    // 
    // multinomial power(+(x,y,z;0),3) example:
    // partition : compositions                : multinomial coefficient
    // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
    // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
    // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
    //  =>  (x + y + z)^3 =
    //        x^3 + y^3 + z^3
    //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
    //      + 6*x*y*z
    //
    // multinomial power(+(x,y,z;0),4) example:
    // partition : compositions                : multinomial coefficient
    // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
    // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
    // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
    // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
    // (no [1,1,1,1] partition since it has too many parts)
    //  =>  (x + y + z)^4 =
    //        x^4 + y^4 + z^4
    //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
    //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
    //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
    //
    // Summary:
    // r = 0
    // for k from 0 to n:
    //     f = c^(n-k)*binomial(n,k)
    //     for p in all partitions of n with m parts (including zero parts):
    //         h = f * multinomial coefficient of p
    //         for c in all compositions of p:
    //             t = 1
    //             for e in all elements of c:
    //                 t = t * a[e]^e
    //             r = r + h*t
    // return r
 
    epvector 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: binomial(n+m-1,m-1).
    size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
    if (!a.overall_coeff.is_zero()) {
        // the result's overall_coeff is one of the terms
        --result_size;
    }
    result.reserve(result_size);
 
    // Iterate over all terms in binomial expansion of
    // S = power(+(x,...,z;c),n)
    //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
    for (int k = 1; k <= n; ++k) {
        numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
        if (a.overall_coeff.is_zero()) {
            // degenerate case with zero overall_coeff:
            // apply multinomial theorem directly to power(+(x,...z;0),n)
            binomial_coefficient = 1;
            if (k < n) {
                continue;
            }
        } else {
            binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
        }
 
        // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
        // Iterate over all partitions of k with exactly as many parts as
        // there are symbolic terms in the basis (including zero parts).
        partition_with_zero_parts_generator partitions(k, a.seq.size());
        do {
            const std::vector<unsigned>& partition = partitions.get();
            // All monomials of this partition have the same number of terms and the same coefficient.
            const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
            const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
 
            // Iterate over all compositions of the current partition.
            composition_generator compositions(partition);
            do {
                const std::vector<unsigned>& exponent = compositions.get();
                epvector monomial;
                monomial.reserve(msize);
                numeric factor = coeff;
                for (unsigned i = 0; i < exponent.size(); ++i) {
                    const ex & r = a.seq[i].rest;
                    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));
                    GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
                    const numeric & c = ex_to<numeric>(a.seq[i].coeff);
                    if (exponent[i] == 0) {
                        // optimize away
                    } else if (exponent[i] == 1) {
                        // optimized
                        monomial.emplace_back(expair(r, _ex1));
                        if (c != *_num1_p)
                            factor = factor.mul(c);
                    } else { // general case exponent[i] > 1
                        monomial.emplace_back(expair(r, exponent[i]));
                        if (c != *_num1_p)
                            factor = factor.mul(c.power(exponent[i]));
                    }
                }
                result.emplace_back(expair(mul(std::move(monomial)).expand(options), factor));
            } while (compositions.next());
        } while (partitions.next());
    }
 
    GINAC_ASSERT(result.size() == result_size);
    if (a.overall_coeff.is_zero()) {
        return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
    } else {
        return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
    }
}
 
 
ex power::expand_add_2(const add & a, unsigned options)
{
    epvector result;
    size_t result_size = (a.nops() * (a.nops()+1)) / 2;
    if (!a.overall_coeff.is_zero()) {
        // the result's overall_coeff is one of the terms
        --result_size;
    }
    result.reserve(result_size);
 
    auto 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 (auto 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)) {
                result.emplace_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
                                           _ex1));
            } else {
                result.emplace_back(expair(dynallocate<power>(r, _ex2),
                                           _ex1));
            }
        } else {
            if (is_exactly_a<mul>(r)) {
                result.emplace_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
                                           ex_to<numeric>(c).power_dyn(*_num2_p)));
            } else {
                result.emplace_back(expair(dynallocate<power>(r, _ex2),
                                           ex_to<numeric>(c).power_dyn(*_num2_p)));
            }
        }
 
        for (auto cit1=cit0+1; cit1!=last; ++cit1) {
            const ex & r1 = cit1->rest;
            const ex & c1 = cit1->coeff;
            result.emplace_back(expair(mul(r,r1).expand(options),
                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
        }
    }
    
    // second part: add terms coming from overall_coeff (if != 0)
    if (!a.overall_coeff.is_zero()) {
        for (auto & i : a.seq)
            result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
    }
 
    GINAC_ASSERT(result.size() == result_size);
 
    if (a.overall_coeff.is_zero()) {
        return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
    } else {
        return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
    }
}
 
ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
{
    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;
 
    for (auto & cit : m.seq) {
        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);
    }
 
    const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
    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