[ginac.git] / ginac / numeric.cpp

/** @file numeric.cpp
 *
 *  This file contains the interface to the underlying bignum package.
 *  Its most important design principle is to completely hide the inner
 *  working of that other package from the user of GiNaC.  It must either 
 *  provide implementation of arithmetic operators and numerical evaluation
 *  of special functions or implement the interface to the bignum package. */

/*
 *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 */

#include "config.h"

#include <vector>
#include <stdexcept>
#include <string>

#if defined(HAVE_SSTREAM)
#include <sstream>
#elif defined(HAVE_STRSTREAM)
#include <strstream>
#else
#error Need either sstream or strstream
#endif

#include "numeric.h"
#include "ex.h"
#include "archive.h"
#include "debugmsg.h"
#include "utils.h"

// CLN should pollute the global namespace as little as possible.  Hence, we
// include most of it here and include only the part needed for properly
// declaring cln::cl_number in numeric.h.  This can only be safely done in
// namespaced versions of CLN, i.e. version > 1.1.0.  Also, we only need a
// subset of CLN, so we don't include the complete <cln/cln.h> but only the
// essential stuff:
#include <cln/output.h>
#include <cln/integer_io.h>
#include <cln/integer_ring.h>
#include <cln/rational_io.h>
#include <cln/rational_ring.h>
#include <cln/lfloat_class.h>
#include <cln/lfloat_io.h>
#include <cln/real_io.h>
#include <cln/real_ring.h>
#include <cln/complex_io.h>
#include <cln/complex_ring.h>
#include <cln/numtheory.h>

#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
#endif  // ndef NO_NAMESPACE_GINAC

GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)

//////////
// default constructor, destructor, copy constructor assignment
// operator and helpers
//////////

// public

/** default ctor. Numerically it initializes to an integer zero. */
numeric::numeric() : basic(TINFO_numeric)
{
        debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
        value = cln::cl_I(0);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}

// protected

void numeric::copy(const numeric & other)
{
        basic::copy(other);
        value = other.value;
}

void numeric::destroy(bool call_parent)
{
        if (call_parent) basic::destroy(call_parent);
}

//////////
// other constructors
//////////

// public

numeric::numeric(int i) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
        // Not the whole int-range is available if we don't cast to long
        // first.  This is due to the behaviour of the cl_I-ctor, which
        // emphasizes efficiency.  However, if the integer is small enough, 
        // i.e. satisfies cl_immediate_p(), we save space and dereferences by
        // using an immediate type:
        if (cln::cl_immediate_p(i))
                value = cln::cl_I(i);
        else
                value = cln::cl_I((long) i);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}


numeric::numeric(unsigned int i) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
        // Not the whole uint-range is available if we don't cast to ulong
        // first.  This is due to the behaviour of the cl_I-ctor, which
        // emphasizes efficiency.  However, if the integer is small enough, 
        // i.e. satisfies cl_immediate_p(), we save space and dereferences by
        // using an immediate type:
        if (cln::cl_immediate_p(i))
                value = cln::cl_I(i);
        else
                value = cln::cl_I((unsigned long) i);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}


numeric::numeric(long i) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
        value = cln::cl_I(i);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}


numeric::numeric(unsigned long i) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
        value = cln::cl_I(i);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}

/** Ctor for rational numerics a/b.
 *
 *  @exception overflow_error (division by zero) */
numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
        if (!denom)
                throw std::overflow_error("division by zero");
        value = cln::cl_I(numer) / cln::cl_I(denom);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}


numeric::numeric(double d) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
        // We really want to explicitly use the type cl_LF instead of the
        // more general cl_F, since that would give us a cl_DF only which
        // will not be promoted to cl_LF if overflow occurs:
        value = cln::cl_float(d, cln::default_float_format);
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}

/** ctor from C-style string.  It also accepts complex numbers in GiNaC
 *  notation like "2+5*I". */
numeric::numeric(const char *s) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
        cln::cl_N ctorval = 0;
        // parse complex numbers (functional but not completely safe, unfortunately
        // std::string does not understand regexpese):
        // ss should represent a simple sum like 2+5*I
        std::string ss(s);
        // make it safe by adding explicit sign
        if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
                ss = '+' + ss;
        std::string::size_type delim;
        do {
                // chop ss into terms from left to right
                std::string term;
                bool imaginary = false;
                delim = ss.find_first_of(std::string("+-"),1);
                // Do we have an exponent marker like "31.415E-1"?  If so, hop on!
                if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
                        delim = ss.find_first_of(std::string("+-"),delim+1);
                term = ss.substr(0,delim);
                if (delim != std::string::npos)
                        ss = ss.substr(delim);
                // is the term imaginary?
                if (term.find("I") != std::string::npos) {
                        // erase 'I':
                        term = term.replace(term.find("I"),1,"");
                        // erase '*':
                        if (term.find("*") != std::string::npos)
                                term = term.replace(term.find("*"),1,"");
                        // correct for trivial +/-I without explicit factor on I:
                        if (term.size() == 1)
                                term += "1";
                        imaginary = true;
                }
                if (term.find(".") != std::string::npos) {
                        // CLN's short type cl_SF is not very useful within the GiNaC
                        // framework where we are mainly interested in the arbitrary
                        // precision type cl_LF.  Hence we go straight to the construction
                        // of generic floats.  In order to create them we have to convert
                        // our own floating point notation used for output and construction
                        // from char * to CLN's generic notation:
                        // 3.14      -->   3.14e0_<Digits>
                        // 31.4E-1   -->   31.4e-1_<Digits>
                        // and s on.
                        // No exponent marker?  Let's add a trivial one.
                        if (term.find("E") == std::string::npos)
                                term += "E0";
                        // E to lower case
                        term = term.replace(term.find("E"),1,"e");
                        // append _<Digits> to term
#if defined(HAVE_SSTREAM)
                        std::ostringstream buf;
                        buf << unsigned(Digits) << std::ends;
                        term += "_" + buf.str();
#else
                        char buf[14];
                        std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
                        term += "_" + string(buf);
#endif
                        // construct float using cln::cl_F(const char *) ctor.
                        if (imaginary)
                                ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
                        else
                                ctorval = ctorval + cln::cl_F(term.c_str());
                } else {
                        // not a floating point number...
                        if (imaginary)
                                ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
                        else
                                ctorval = ctorval + cln::cl_R(term.c_str());
                }
        } while(delim != std::string::npos);
        value = ctorval;
        calchash();
        setflag(status_flags::evaluated |
                        status_flags::expanded |
                        status_flags::hash_calculated);
}

/** Ctor from CLN types.  This is for the initiated user or internal use
 *  only. */
numeric::numeric(const cln::cl_N & z) : basic(TINFO_numeric)
{
        debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
        value = z;
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}

//////////
// archiving
//////////

/** Construct object from archive_node. */
numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
{
        debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
        cln::cl_N ctorval = 0;

        // Read number as string
        std::string str;
        if (n.find_string("number", str)) {
#ifdef HAVE_SSTREAM
                std::istringstream s(str);
#else
                std::istrstream s(str.c_str(), str.size() + 1);
#endif
                cln::cl_idecoded_float re, im;
                char c;
                s.get(c);
                switch (c) {
                        case 'R':    // Integer-decoded real number
                                s >> re.sign >> re.mantissa >> re.exponent;
                                ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
                                break;
                        case 'C':    // Integer-decoded complex number
                                s >> re.sign >> re.mantissa >> re.exponent;
                                s >> im.sign >> im.mantissa >> im.exponent;
                                ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
                                                       im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
                                break;
                        default:    // Ordinary number
                                s.putback(c);
                                s >> ctorval;
                                break;
                }
        }
        value = ctorval;
        calchash();
        setflag(status_flags::evaluated |
                status_flags::expanded |
                status_flags::hash_calculated);
}

/** Unarchive the object. */
ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
{
        return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
}

/** Archive the object. */
void numeric::archive(archive_node &n) const
{
        inherited::archive(n);

        // Write number as string
#ifdef HAVE_SSTREAM
        std::ostringstream s;
#else
        char buf[1024];
        std::ostrstream s(buf, 1024);
#endif
        if (this->is_crational())
                s << cln::the<cln::cl_N>(value);
        else {
                // Non-rational numbers are written in an integer-decoded format
                // to preserve the precision
                if (this->is_real()) {
                        cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
                        s << "R";
                        s << re.sign << " " << re.mantissa << " " << re.exponent;
                } else {
                        cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
                        cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
                        s << "C";
                        s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
                        s << im.sign << " " << im.mantissa << " " << im.exponent;
                }
        }
#ifdef HAVE_SSTREAM
        n.add_string("number", s.str());
#else
        s << ends;
        std::string str(buf);
        n.add_string("number", str);
#endif
}

//////////
// functions overriding virtual functions from bases classes
//////////

/** Helper function to print a real number in a nicer way than is CLN's
 *  default.  Instead of printing 42.0L0 this just prints 42.0 to ostream os
 *  and instead of 3.99168L7 it prints 3.99168E7.  This is fine in GiNaC as
 *  long as it only uses cl_LF and no other floating point types that we might
 *  want to visibly distinguish from cl_LF.
 *
 *  @see numeric::print() */
static void print_real_number(std::ostream & os, const cln::cl_R & num)
{
        cln::cl_print_flags ourflags;
        if (cln::instanceof(num, cln::cl_RA_ring)) {
                // case 1: integer or rational, nothing special to do:
                cln::print_real(os, ourflags, num);
        } else {
                // case 2: float
                // make CLN believe this number has default_float_format, so it prints
                // 'E' as exponent marker instead of 'L':
                ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
                cln::print_real(os, ourflags, num);
        }
        return;
}

/** This method adds to the output so it blends more consistently together
 *  with the other routines and produces something compatible to ginsh input.
 *  
 *  @see print_real_number() */
void numeric::print(std::ostream & os, unsigned upper_precedence) const
{
        debugmsg("numeric print", LOGLEVEL_PRINT);
        cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
        cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
        if (cln::zerop(i)) {
                // case 1, real:  x  or  -x
                if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
                        os << "(";
                        print_real_number(os, r);
                        os << ")";
                } else {
                        print_real_number(os, r);
                }
        } else {
                if (cln::zerop(r)) {
                        // case 2, imaginary:  y*I  or  -y*I
                        if ((precedence<=upper_precedence) && (i < 0)) {
                                if (i == -1) {
                                        os << "(-I)";
                                } else {
                                        os << "(";
                                        print_real_number(os, i);
                                        os << "*I)";
                                }
                        } else {
                                if (i == 1) {
                                        os << "I";
                                } else {
                                        if (i == -1) {
                                                os << "-I";
                                        } else {
                                                print_real_number(os, i);
                                                os << "*I";
                                        }
                                }
                        }
                } else {
                        // case 3, complex:  x+y*I  or  x-y*I  or  -x+y*I  or  -x-y*I
                        if (precedence <= upper_precedence)
                                os << "(";
                        print_real_number(os, r);
                        if (i < 0) {
                                if (i == -1) {
                                        os << "-I";
                                } else {
                                        print_real_number(os, i);
                                        os << "*I";
                                }
                        } else {
                                if (i == 1) {
                                        os << "+I";
                                } else {
                                        os << "+";
                                        print_real_number(os, i);
                                        os << "*I";
                                }
                        }
                        if (precedence <= upper_precedence)
                                os << ")";
                }
        }
}


void numeric::printraw(std::ostream & os) const
{
        // The method printraw doesn't do much, it simply uses CLN's operator<<()
        // for output, which is ugly but reliable. e.g: 2+2i
        debugmsg("numeric printraw", LOGLEVEL_PRINT);
        os << "numeric(" << cln::the<cln::cl_N>(value) << ")";
}


void numeric::printtree(std::ostream & os, unsigned indent) const
{
        debugmsg("numeric printtree", LOGLEVEL_PRINT);
        os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
           << " (numeric): "
           << "hash=" << hashvalue
           << " (0x" << std::hex << hashvalue << std::dec << ")"
           << ", flags=" << flags << std::endl;
}


void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
{
        debugmsg("numeric print csrc", LOGLEVEL_PRINT);
        std::ios::fmtflags oldflags = os.flags();
        os.setf(std::ios::scientific);
        if (this->is_rational() && !this->is_integer()) {
                if (compare(_num0()) > 0) {
                        os << "(";
                        if (type == csrc_types::ctype_cl_N)
                                os << "cln::cl_F(\"" << numer().evalf() << "\")";
                        else
                                os << numer().to_double();
                } else {
                        os << "-(";
                        if (type == csrc_types::ctype_cl_N)
                                os << "cln::cl_F(\"" << -numer().evalf() << "\")";
                        else
                                os << -numer().to_double();
                }
                os << "/";
                if (type == csrc_types::ctype_cl_N)
                        os << "cln::cl_F(\"" << denom().evalf() << "\")";
                else
                        os << denom().to_double();
                os << ")";
        } else {
                if (type == csrc_types::ctype_cl_N)
                        os << "cln::cl_F(\"" << evalf() << "\")";
                else
                        os << to_double();
        }
        os.flags(oldflags);
}


bool numeric::info(unsigned inf) const
{
        switch (inf) {
                case info_flags::numeric:
                case info_flags::polynomial:
                case info_flags::rational_function:
                        return true;
                case info_flags::real:
                        return is_real();
                case info_flags::rational:
                case info_flags::rational_polynomial:
                        return is_rational();
                case info_flags::crational:
                case info_flags::crational_polynomial:
                        return is_crational();
                case info_flags::integer:
                case info_flags::integer_polynomial:
                        return is_integer();
                case info_flags::cinteger:
                case info_flags::cinteger_polynomial:
                        return is_cinteger();
                case info_flags::positive:
                        return is_positive();
                case info_flags::negative:
                        return is_negative();
                case info_flags::nonnegative:
                        return !is_negative();
                case info_flags::posint:
                        return is_pos_integer();
                case info_flags::negint:
                        return is_integer() && is_negative();
                case info_flags::nonnegint:
                        return is_nonneg_integer();
                case info_flags::even:
                        return is_even();
                case info_flags::odd:
                        return is_odd();
                case info_flags::prime:
                        return is_prime();
                case info_flags::algebraic:
                        return !is_real();
        }
        return false;
}

/** Disassemble real part and imaginary part to scan for the occurrence of a
 *  single number.  Also handles the imaginary unit.  It ignores the sign on
 *  both this and the argument, which may lead to what might appear as funny
 *  results:  (2+I).has(-2) -> true.  But this is consistent, since we also
 *  would like to have (-2+I).has(2) -> true and we want to think about the
 *  sign as a multiplicative factor. */
bool numeric::has(const ex & other) const
{
        if (!is_exactly_of_type(*other.bp, numeric))
                return false;
        const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
        if (this->is_equal(o) || this->is_equal(-o))
                return true;
        if (o.imag().is_zero())  // e.g. scan for 3 in -3*I
                return (this->real().is_equal(o) || this->imag().is_equal(o) ||
                        this->real().is_equal(-o) || this->imag().is_equal(-o));
        else {
                if (o.is_equal(I))  // e.g scan for I in 42*I
                        return !this->is_real();
                if (o.real().is_zero())  // e.g. scan for 2*I in 2*I+1
                        return (this->real().has(o*I) || this->imag().has(o*I) ||
                                this->real().has(-o*I) || this->imag().has(-o*I));
        }
        return false;
}


/** Evaluation of numbers doesn't do anything at all. */
ex numeric::eval(int level) const
{
        // Warning: if this is ever gonna do something, the ex ctors from all kinds
        // of numbers should be checking for status_flags::evaluated.
        return this->hold();
}


/** Cast numeric into a floating-point object.  For example exact numeric(1) is
 *  returned as a 1.0000000000000000000000 and so on according to how Digits is
 *  currently set.  In case the object already was a floating point number the
 *  precision is trimmed to match the currently set default.
 *
 *  @param level  ignored, only needed for overriding basic::evalf.
 *  @return  an ex-handle to a numeric. */
ex numeric::evalf(int level) const
{
        // level can safely be discarded for numeric objects.
        return numeric(cln::cl_float(1.0, cln::default_float_format) *
                                   (cln::the<cln::cl_N>(value)));
}

// protected

/** Implementation of ex::diff() for a numeric. It always returns 0.
 *
 *  @see ex::diff */
ex numeric::derivative(const symbol & s) const
{
        return _ex0();
}


int numeric::compare_same_type(const basic & other) const
{
        GINAC_ASSERT(is_exactly_of_type(other, numeric));
        const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
        
        return this->compare(o);
}


bool numeric::is_equal_same_type(const basic & other) const
{
        GINAC_ASSERT(is_exactly_of_type(other,numeric));
        const numeric *o = static_cast<const numeric *>(&other);
        
        return this->is_equal(*o);
}


unsigned numeric::calchash(void) const
{
        // Use CLN's hashcode.  Warning: It depends only on the number's value, not
        // its type or precision (i.e. a true equivalence relation on numbers).  As
        // a consequence, 3 and 3.0 share the same hashvalue.
        return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
}


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

// none

//////////
// non-virtual functions in this class
//////////

// public

/** Numerical addition method.  Adds argument to *this and returns result as
 *  a new numeric object. */
const numeric numeric::add(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral element by pointer.
        static const numeric * _num0p = &_num0();
        if (this==_num0p)
                return other;
        else if (&other==_num0p)
                return *this;
        
        return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
}


/** Numerical subtraction method.  Subtracts argument from *this and returns
 *  result as a new numeric object. */
const numeric numeric::sub(const numeric & other) const
{
        return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
}


/** Numerical multiplication method.  Multiplies *this and argument and returns
 *  result as a new numeric object. */
const numeric numeric::mul(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral element by pointer.
        static const numeric * _num1p = &_num1();
        if (this==_num1p)
                return other;
        else if (&other==_num1p)
                return *this;
        
        return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
}


/** Numerical division method.  Divides *this by argument and returns result as
 *  a new numeric object.
 *
 *  @exception overflow_error (division by zero) */
const numeric numeric::div(const numeric & other) const
{
        if (cln::zerop(cln::the<cln::cl_N>(other.value)))
                throw std::overflow_error("numeric::div(): division by zero");
        return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
}


const numeric numeric::power(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral exponent by pointer.
        static const numeric * _num1p = &_num1();
        if (&other==_num1p)
                return *this;
        
        if (cln::zerop(cln::the<cln::cl_N>(value))) {
                if (cln::zerop(cln::the<cln::cl_N>(other.value)))
                        throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
                else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
                        throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
                else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
                        throw std::overflow_error("numeric::eval(): division by zero");
                else
                        return _num0();
        }
        return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
}


const numeric & numeric::add_dyn(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral element by pointer.
        static const numeric * _num0p = &_num0();
        if (this==_num0p)
                return other;
        else if (&other==_num0p)
                return *this;
        
        return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
                                                                                setflag(status_flags::dynallocated));
}


const numeric & numeric::sub_dyn(const numeric & other) const
{
        return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
                                                                                setflag(status_flags::dynallocated));
}


const numeric & numeric::mul_dyn(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral element by pointer.
        static const numeric * _num1p = &_num1();
        if (this==_num1p)
                return other;
        else if (&other==_num1p)
                return *this;
        
        return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
                                                                                setflag(status_flags::dynallocated));
}


const numeric & numeric::div_dyn(const numeric & other) const
{
        if (cln::zerop(cln::the<cln::cl_N>(other.value)))
                throw std::overflow_error("division by zero");
        return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
                                                                                setflag(status_flags::dynallocated));
}


const numeric & numeric::power_dyn(const numeric & other) const
{
        // Efficiency shortcut: trap the neutral exponent by pointer.
        static const numeric * _num1p=&_num1();
        if (&other==_num1p)
                return *this;
        
        if (cln::zerop(cln::the<cln::cl_N>(value))) {
                if (cln::zerop(cln::the<cln::cl_N>(other.value)))
                        throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
                else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
                        throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
                else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
                        throw std::overflow_error("numeric::eval(): division by zero");
                else
                        return _num0();
        }
        return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
                                             setflag(status_flags::dynallocated));
}


const numeric & numeric::operator=(int i)
{
        return operator=(numeric(i));
}


const numeric & numeric::operator=(unsigned int i)
{
        return operator=(numeric(i));
}


const numeric & numeric::operator=(long i)
{
        return operator=(numeric(i));
}


const numeric & numeric::operator=(unsigned long i)
{
        return operator=(numeric(i));
}


const numeric & numeric::operator=(double d)
{
        return operator=(numeric(d));
}


const numeric & numeric::operator=(const char * s)
{
        return operator=(numeric(s));
}


/** Inverse of a number. */
const numeric numeric::inverse(void) const
{
        if (cln::zerop(cln::the<cln::cl_N>(value)))
                throw std::overflow_error("numeric::inverse(): division by zero");
        return numeric(cln::recip(cln::the<cln::cl_N>(value)));
}


/** Return the complex half-plane (left or right) in which the number lies.
 *  csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
 *  csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
 *
 *  @see numeric::compare(const numeric & other) */
int numeric::csgn(void) const
{
        if (cln::zerop(cln::the<cln::cl_N>(value)))
                return 0;
        cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
        if (!cln::zerop(r)) {
                if (cln::plusp(r))
                        return 1;
                else
                        return -1;
        } else {
                if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
                        return 1;
                else
                        return -1;
        }
}


/** This method establishes a canonical order on all numbers.  For complex
 *  numbers this is not possible in a mathematically consistent way but we need
 *  to establish some order and it ought to be fast.  So we simply define it
 *  to be compatible with our method csgn.
 *
 *  @return csgn(*this-other)
 *  @see numeric::csgn(void) */
int numeric::compare(const numeric & other) const
{
        // Comparing two real numbers?
        if (cln::instanceof(value, cln::cl_R_ring) &&
                cln::instanceof(other.value, cln::cl_R_ring))
                // Yes, so just cln::compare them
                return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
        else {
                // No, first cln::compare real parts...
                cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
                if (real_cmp)
                        return real_cmp;
                // ...and then the imaginary parts.
                return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
        }
}


bool numeric::is_equal(const numeric & other) const
{
        return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
}


/** True if object is zero. */
bool numeric::is_zero(void) const
{
        return cln::zerop(cln::the<cln::cl_N>(value));
}


/** True if object is not complex and greater than zero. */
bool numeric::is_positive(void) const
{
        if (this->is_real())
                return cln::plusp(cln::the<cln::cl_R>(value));
        return false;
}


/** True if object is not complex and less than zero. */
bool numeric::is_negative(void) const
{
        if (this->is_real())
                return cln::minusp(cln::the<cln::cl_R>(value));
        return false;
}


/** True if object is a non-complex integer. */
bool numeric::is_integer(void) const
{
        return cln::instanceof(value, cln::cl_I_ring);
}


/** True if object is an exact integer greater than zero. */
bool numeric::is_pos_integer(void) const
{
        return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
}


/** True if object is an exact integer greater or equal zero. */
bool numeric::is_nonneg_integer(void) const
{
        return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
}


/** True if object is an exact even integer. */
bool numeric::is_even(void) const
{
        return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
}


/** True if object is an exact odd integer. */
bool numeric::is_odd(void) const
{
        return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
}


/** Probabilistic primality test.
 *
 *  @return  true if object is exact integer and prime. */
bool numeric::is_prime(void) const
{
        return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
}


/** True if object is an exact rational number, may even be complex
 *  (denominator may be unity). */
bool numeric::is_rational(void) const
{
        return cln::instanceof(value, cln::cl_RA_ring);
}


/** True if object is a real integer, rational or float (but not complex). */
bool numeric::is_real(void) const
{
        return cln::instanceof(value, cln::cl_R_ring);
}


bool numeric::operator==(const numeric & other) const
{
        return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
}


bool numeric::operator!=(const numeric & other) const
{
        return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
}


/** True if object is element of the domain of integers extended by I, i.e. is
 *  of the form a+b*I, where a and b are integers. */
bool numeric::is_cinteger(void) const
{
        if (cln::instanceof(value, cln::cl_I_ring))
                return true;
        else if (!this->is_real()) {  // complex case, handle n+m*I
                if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
                    cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
                        return true;
        }
        return false;
}


/** True if object is an exact rational number, may even be complex
 *  (denominator may be unity). */
bool numeric::is_crational(void) const
{
        if (cln::instanceof(value, cln::cl_RA_ring))
                return true;
        else if (!this->is_real()) {  // complex case, handle Q(i):
                if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
                    cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
                        return true;
        }
        return false;
}


/** Numerical comparison: less.
 *
 *  @exception invalid_argument (complex inequality) */ 
bool numeric::operator<(const numeric & other) const
{
        if (this->is_real() && other.is_real())
                return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
        throw std::invalid_argument("numeric::operator<(): complex inequality");
}


/** Numerical comparison: less or equal.
 *
 *  @exception invalid_argument (complex inequality) */ 
bool numeric::operator<=(const numeric & other) const
{
        if (this->is_real() && other.is_real())
                return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
        throw std::invalid_argument("numeric::operator<=(): complex inequality");
}


/** Numerical comparison: greater.
 *
 *  @exception invalid_argument (complex inequality) */ 
bool numeric::operator>(const numeric & other) const
{
        if (this->is_real() && other.is_real())
                return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
        throw std::invalid_argument("numeric::operator>(): complex inequality");
}


/** Numerical comparison: greater or equal.
 *
 *  @exception invalid_argument (complex inequality) */  
bool numeric::operator>=(const numeric & other) const
{
        if (this->is_real() && other.is_real())
                return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
        throw std::invalid_argument("numeric::operator>=(): complex inequality");
}


/** Converts numeric types to machine's int.  You should check with
 *  is_integer() if the number is really an integer before calling this method.
 *  You may also consider checking the range first. */
int numeric::to_int(void) const
{
        GINAC_ASSERT(this->is_integer());
        return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
}


/** Converts numeric types to machine's long.  You should check with
 *  is_integer() if the number is really an integer before calling this method.
 *  You may also consider checking the range first. */
long numeric::to_long(void) const
{
        GINAC_ASSERT(this->is_integer());
        return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
}


/** Converts numeric types to machine's double. You should check with is_real()
 *  if the number is really not complex before calling this method. */
double numeric::to_double(void) const
{
        GINAC_ASSERT(this->is_real());
        return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
}


/** Returns a new CLN object of type cl_N, representing the value of *this.
 *  This method may be used when mixing GiNaC and CLN in one project.
 */
cln::cl_N numeric::to_cl_N(void) const
{
        return cln::cl_N(cln::the<cln::cl_N>(value));
}


/** Real part of a number. */
const numeric numeric::real(void) const
{
        return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
}


/** Imaginary part of a number. */
const numeric numeric::imag(void) const
{
        return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
}


/** Numerator.  Computes the numerator of rational numbers, rationalized
 *  numerator of complex if real and imaginary part are both rational numbers
 *  (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
 *  cases. */
const numeric numeric::numer(void) const
{
        if (this->is_integer())
                return numeric(*this);
        
        else if (cln::instanceof(value, cln::cl_RA_ring))
                return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
        
        else if (!this->is_real()) {  // complex case, handle Q(i):
                const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
                const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
                if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
                        return numeric(*this);
                if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
                        return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
                if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
                        return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
                if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
                        const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
                        return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
                                                            cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
                }
        }
        // at least one float encountered
        return numeric(*this);
}


/** Denominator.  Computes the denominator of rational numbers, common integer
 *  denominator of complex if real and imaginary part are both rational numbers
 *  (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
const numeric numeric::denom(void) const
{
        if (this->is_integer())
                return _num1();
        
        if (instanceof(value, cln::cl_RA_ring))
                return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
        
        if (!this->is_real()) {  // complex case, handle Q(i):
                const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
                const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
                if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
                        return _num1();
                if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
                        return numeric(cln::denominator(i));
                if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
                        return numeric(cln::denominator(r));
                if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
                        return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
        }
        // at least one float encountered
        return _num1();
}


/** Size in binary notation.  For integers, this is the smallest n >= 0 such
 *  that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
 *  2^(n-1) <= x < 2^n.
 *
 *  @return  number of bits (excluding sign) needed to represent that number
 *  in two's complement if it is an integer, 0 otherwise. */    
int numeric::int_length(void) const
{
        if (this->is_integer())
                return cln::integer_length(cln::the<cln::cl_I>(value));
        else
                return 0;
}


//////////
// static member variables
//////////

// protected

unsigned numeric::precedence = 30;

//////////
// global constants
//////////

/** Imaginary unit.  This is not a constant but a numeric since we are
 *  natively handing complex numbers anyways. */
const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));


/** Exponential function.
 *
 *  @return  arbitrary precision numerical exp(x). */
const numeric exp(const numeric & x)
{
        return cln::exp(x.to_cl_N());
}


/** Natural logarithm.
 *
 *  @param z complex number
 *  @return  arbitrary precision numerical log(x).
 *  @exception pole_error("log(): logarithmic pole",0) */
const numeric log(const numeric & z)
{
        if (z.is_zero())
                throw pole_error("log(): logarithmic pole",0);
        return cln::log(z.to_cl_N());
}


/** Numeric sine (trigonometric function).
 *
 *  @return  arbitrary precision numerical sin(x). */
const numeric sin(const numeric & x)
{
        return cln::sin(x.to_cl_N());
}


/** Numeric cosine (trigonometric function).
 *
 *  @return  arbitrary precision numerical cos(x). */
const numeric cos(const numeric & x)
{
        return cln::cos(x.to_cl_N());
}


/** Numeric tangent (trigonometric function).
 *
 *  @return  arbitrary precision numerical tan(x). */
const numeric tan(const numeric & x)
{
        return cln::tan(x.to_cl_N());
}
        

/** Numeric inverse sine (trigonometric function).
 *
 *  @return  arbitrary precision numerical asin(x). */
const numeric asin(const numeric & x)
{
        return cln::asin(x.to_cl_N());
}


/** Numeric inverse cosine (trigonometric function).
 *
 *  @return  arbitrary precision numerical acos(x). */
const numeric acos(const numeric & x)
{
        return cln::acos(x.to_cl_N());
}
        

/** Arcustangent.
 *
 *  @param z complex number
 *  @return atan(z)
 *  @exception pole_error("atan(): logarithmic pole",0) */
const numeric atan(const numeric & x)
{
        if (!x.is_real() &&
            x.real().is_zero() &&
            abs(x.imag()).is_equal(_num1()))
                throw pole_error("atan(): logarithmic pole",0);
        return cln::atan(x.to_cl_N());
}


/** Arcustangent.
 *
 *  @param x real number
 *  @param y real number
 *  @return atan(y/x) */
const numeric atan(const numeric & y, const numeric & x)
{
        if (x.is_real() && y.is_real())
                return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
                                 cln::the<cln::cl_R>(y.to_cl_N()));
        else
                throw std::invalid_argument("atan(): complex argument");        
}


/** Numeric hyperbolic sine (trigonometric function).
 *
 *  @return  arbitrary precision numerical sinh(x). */
const numeric sinh(const numeric & x)
{
        return cln::sinh(x.to_cl_N());
}


/** Numeric hyperbolic cosine (trigonometric function).
 *
 *  @return  arbitrary precision numerical cosh(x). */
const numeric cosh(const numeric & x)
{
        return cln::cosh(x.to_cl_N());
}


/** Numeric hyperbolic tangent (trigonometric function).
 *
 *  @return  arbitrary precision numerical tanh(x). */
const numeric tanh(const numeric & x)
{
        return cln::tanh(x.to_cl_N());
}
        

/** Numeric inverse hyperbolic sine (trigonometric function).
 *
 *  @return  arbitrary precision numerical asinh(x). */
const numeric asinh(const numeric & x)
{
        return cln::asinh(x.to_cl_N());
}


/** Numeric inverse hyperbolic cosine (trigonometric function).
 *
 *  @return  arbitrary precision numerical acosh(x). */
const numeric acosh(const numeric & x)
{
        return cln::acosh(x.to_cl_N());
}


/** Numeric inverse hyperbolic tangent (trigonometric function).
 *
 *  @return  arbitrary precision numerical atanh(x). */
const numeric atanh(const numeric & x)
{
        return cln::atanh(x.to_cl_N());
}


/*static cln::cl_N Li2_series(const ::cl_N & x,
                            const ::float_format_t & prec)
{
        // Note: argument must be in the unit circle
        // This is very inefficient unless we have fast floating point Bernoulli
        // numbers implemented!
        cln::cl_N c1 = -cln::log(1-x);
        cln::cl_N c2 = c1;
        // hard-wire the first two Bernoulli numbers
        cln::cl_N acc = c1 - cln::square(c1)/4;
        cln::cl_N aug;
        cln::cl_F pisq = cln::square(cln::cl_pi(prec));  // pi^2
        cln::cl_F piac = cln::cl_float(1, prec);  // accumulator: pi^(2*i)
        unsigned i = 1;
        c1 = cln::square(c1);
        do {
                c2 = c1 * c2;
                piac = piac * pisq;
                aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
                // aug = c2 * cln::cl_I(i%2 ? 1 : -1) / cln::cl_I(2*i+1) * cln::cl_zeta(2*i, prec) / piac / (cln::cl_I(1)<<(2*i-1));
                acc = acc + aug;
                ++i;
        } while (acc != acc+aug);
        return acc;
}*/

/** Numeric evaluation of Dilogarithm within circle of convergence (unit
 *  circle) using a power series. */
static cln::cl_N Li2_series(const cln::cl_N & x,
                            const cln::float_format_t & prec)
{
        // Note: argument must be in the unit circle
        cln::cl_N aug, acc;
        cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
        cln::cl_I den = 0;
        unsigned i = 1;
        do {
                num = num * x;
                den = den + i;  // 1, 4, 9, 16, ...
                i += 2;
                aug = num / den;
                acc = acc + aug;
        } while (acc != acc+aug);
        return acc;
}

/** Folds Li2's argument inside a small rectangle to enhance convergence. */
static cln::cl_N Li2_projection(const cln::cl_N & x,
                                const cln::float_format_t & prec)
{
        const cln::cl_R re = cln::realpart(x);
        const cln::cl_R im = cln::imagpart(x);
        if (re > cln::cl_F(".5"))
                // zeta(2) - Li2(1-x) - log(x)*log(1-x)
                return(cln::zeta(2)
                       - Li2_series(1-x, prec)
                       - cln::log(x)*cln::log(1-x));
        if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
                // -log(1-x)^2 / 2 - Li2(x/(x-1))
                return(- cln::square(cln::log(1-x))/2
                       - Li2_series(x/(x-1), prec));
        if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
                // Li2(x^2)/2 - Li2(-x)
                return(Li2_projection(cln::square(x), prec)/2
                       - Li2_projection(-x, prec));
        return Li2_series(x, prec);
}

/** Numeric evaluation of Dilogarithm.  The domain is the entire complex plane,
 *  the branch cut lies along the positive real axis, starting at 1 and
 *  continuous with quadrant IV.
 *
 *  @return  arbitrary precision numerical Li2(x). */
const numeric Li2(const numeric & x)
{
        if (x.is_zero())
                return _num0();
        
        // what is the desired float format?
        // first guess: default format
        cln::float_format_t prec = cln::default_float_format;
        const cln::cl_N value = x.to_cl_N();
        // second guess: the argument's format
        if (!x.real().is_rational())
                prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
        else if (!x.imag().is_rational())
                prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
        
        if (cln::the<cln::cl_N>(value)==1)  // may cause trouble with log(1-x)
                return cln::zeta(2, prec);
        
        if (cln::abs(value) > 1)
                // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
                return(- cln::square(cln::log(-value))/2
                       - cln::zeta(2, prec)
                       - Li2_projection(cln::recip(value), prec));
        else
                return Li2_projection(x.to_cl_N(), prec);
}


/** Numeric evaluation of Riemann's Zeta function.  Currently works only for
 *  integer arguments. */
const numeric zeta(const numeric & x)
{
        // A dirty hack to allow for things like zeta(3.0), since CLN currently
        // only knows about integer arguments and zeta(3).evalf() automatically
        // cascades down to zeta(3.0).evalf().  The trick is to rely on 3.0-3
        // being an exact zero for CLN, which can be tested and then we can just
        // pass the number casted to an int:
        if (x.is_real()) {
                const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
                if (cln::zerop(x.to_cl_N()-aux))
                        return cln::zeta(aux);
        }
        std::clog << "zeta(" << x
                          << "): Does anybody know good way to calculate this numerically?"
                          << std::endl;
        return numeric(0);
}


/** The Gamma function.
 *  This is only a stub! */
const numeric lgamma(const numeric & x)
{
        std::clog << "lgamma(" << x
                  << "): Does anybody know good way to calculate this numerically?"
                  << std::endl;
        return numeric(0);
}
const numeric tgamma(const numeric & x)
{
        std::clog << "tgamma(" << x
                  << "): Does anybody know good way to calculate this numerically?"
                  << std::endl;
        return numeric(0);
}


/** The psi function (aka polygamma function).
 *  This is only a stub! */
const numeric psi(const numeric & x)
{
        std::clog << "psi(" << x
                  << "): Does anybody know good way to calculate this numerically?"
                  << std::endl;
        return numeric(0);
}


/** The psi functions (aka polygamma functions).
 *  This is only a stub! */
const numeric psi(const numeric & n, const numeric & x)
{
        std::clog << "psi(" << n << "," << x
                  << "): Does anybody know good way to calculate this numerically?"
                  << std::endl;
        return numeric(0);
}


/** Factorial combinatorial function.
 *
 *  @param n  integer argument >= 0
 *  @exception range_error (argument must be integer >= 0) */
const numeric factorial(const numeric & n)
{
        if (!n.is_nonneg_integer())
                throw std::range_error("numeric::factorial(): argument must be integer >= 0");
        return numeric(cln::factorial(n.to_int()));
}


/** The double factorial combinatorial function.  (Scarcely used, but still
 *  useful in cases, like for exact results of tgamma(n+1/2) for instance.)
 *
 *  @param n  integer argument >= -1
 *  @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
 *  @exception range_error (argument must be integer >= -1) */
const numeric doublefactorial(const numeric & n)
{
        if (n == numeric(-1))
                return _num1();
        
        if (!n.is_nonneg_integer())
                throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
        
        return numeric(cln::doublefactorial(n.to_int()));
}


/** The Binomial coefficients.  It computes the binomial coefficients.  For
 *  integer n and k and positive n this is the number of ways of choosing k
 *  objects from n distinct objects.  If n is negative, the formula
 *  binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
const numeric binomial(const numeric & n, const numeric & k)
{
        if (n.is_integer() && k.is_integer()) {
                if (n.is_nonneg_integer()) {
                        if (k.compare(n)!=1 && k.compare(_num0())!=-1)
                                return numeric(cln::binomial(n.to_int(),k.to_int()));
                        else
                                return _num0();
                } else {
                        return _num_1().power(k)*binomial(k-n-_num1(),k);
                }
        }
        
        // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
        throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
}


/** Bernoulli number.  The nth Bernoulli number is the coefficient of x^n/n!
 *  in the expansion of the function x/(e^x-1).
 *
 *  @return the nth Bernoulli number (a rational number).
 *  @exception range_error (argument must be integer >= 0) */
const numeric bernoulli(const numeric & nn)
{
        if (!nn.is_integer() || nn.is_negative())
                throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
        
        // Method:
        //
        // The Bernoulli numbers are rational numbers that may be computed using
        // the relation
        //
        //     B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
        //
        // with B(0) = 1.  Since the n'th Bernoulli number depends on all the
        // previous ones, the computation is necessarily very expensive.  There are
        // several other ways of computing them, a particularly good one being
        // cl_I s = 1;
        // cl_I c = n+1;
        // cl_RA Bern = 0;
        // for (unsigned i=0; i<n; i++) {
        //     c = exquo(c*(i-n),(i+2));
        //     Bern = Bern + c*s/(i+2);
        //     s = s + expt_pos(cl_I(i+2),n);
        // }
        // return Bern;
        // 
        // But if somebody works with the n'th Bernoulli number she is likely to
        // also need all previous Bernoulli numbers. So we need a complete remember
        // table and above divide and conquer algorithm is not suited to build one
        // up.  The code below is adapted from Pari's function bernvec().
        // 
        // (There is an interesting relation with the tangent polynomials described
        // in `Concrete Mathematics', which leads to a program twice as fast as our
        // implementation below, but it requires storing one such polynomial in
        // addition to the remember table.  This doubles the memory footprint so
        // we don't use it.)
        
        // the special cases not covered by the algorithm below
        if (nn.is_equal(_num1()))
                return _num_1_2();
        if (nn.is_odd())
                return _num0();
        
        // store nonvanishing Bernoulli numbers here
        static std::vector< cln::cl_RA > results;
        static int highest_result = 0;
        // algorithm not applicable to B(0), so just store it
        if (results.size()==0)
                results.push_back(cln::cl_RA(1));
        
        int n = nn.to_long();
        for (int i=highest_result; i<n/2; ++i) {
                cln::cl_RA B = 0;
                long n = 8;
                long m = 5;
                long d1 = i;
                long d2 = 2*i-1;
                for (int j=i; j>0; --j) {
                        B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
                        n += 4;
                        m += 2;
                        d1 -= 1;
                        d2 -= 2;
                }
                B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
                results.push_back(B);
                ++highest_result;
        }
        return results[n/2];
}


/** Fibonacci number.  The nth Fibonacci number F(n) is defined by the
 *  recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
 *
 *  @param n an integer
 *  @return the nth Fibonacci number F(n) (an integer number)
 *  @exception range_error (argument must be an integer) */
const numeric fibonacci(const numeric & n)
{
        if (!n.is_integer())
                throw std::range_error("numeric::fibonacci(): argument must be integer");
        // Method:
        //
        // The following addition formula holds:
        //
        //      F(n+m)   = F(m-1)*F(n) + F(m)*F(n+1)  for m >= 1, n >= 0.
        //
        // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
        // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
        // agree.)
        // Replace m by m+1:
        //      F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1)      for m >= 0, n >= 0
        // Now put in m = n, to get
        //      F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
        //      F(2n+1) = F(n)^2 + F(n+1)^2
        // hence
        //      F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
        if (n.is_zero())
                return _num0();
        if (n.is_negative())
                if (n.is_even())
                        return -fibonacci(-n);
                else
                        return fibonacci(-n);
        
        cln::cl_I u(0);
        cln::cl_I v(1);
        cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L;  // floor(n/2);
        for (uintL bit=cln::integer_length(m); bit>0; --bit) {
                // Since a squaring is cheaper than a multiplication, better use
                // three squarings instead of one multiplication and two squarings.
                cln::cl_I u2 = cln::square(u);
                cln::cl_I v2 = cln::square(v);
                if (cln::logbitp(bit-1, m)) {
                        v = cln::square(u + v) - u2;
                        u = u2 + v2;
                } else {
                        u = v2 - cln::square(v - u);
                        v = u2 + v2;
                }
        }
        if (n.is_even())
                // Here we don't use the squaring formula because one multiplication
                // is cheaper than two squarings.
                return u * ((v << 1) - u);
        else
                return cln::square(u) + cln::square(v);    
}


/** Absolute value. */
const numeric abs(const numeric& x)
{
        return cln::abs(x.to_cl_N());
}


/** Modulus (in positive representation).
 *  In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
 *  sign of a or is zero. This is different from Maple's modp, where the sign
 *  of b is ignored. It is in agreement with Mathematica's Mod.
 *
 *  @return a mod b in the range [0,abs(b)-1] with sign of b if both are
 *  integer, 0 otherwise. */
const numeric mod(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer())
                return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
                                cln::the<cln::cl_I>(b.to_cl_N()));
        else
                return _num0();
}


/** Modulus (in symmetric representation).
 *  Equivalent to Maple's mods.
 *
 *  @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
const numeric smod(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer()) {
                const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
                return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
                                cln::the<cln::cl_I>(b.to_cl_N())) - b2;
        } else
                return _num0();
}


/** Numeric integer remainder.
 *  Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
 *  In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
 *  sign of a or is zero.
 *
 *  @return remainder of a/b if both are integer, 0 otherwise. */
const numeric irem(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer())
                return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
                                cln::the<cln::cl_I>(b.to_cl_N()));
        else
                return _num0();
}


/** Numeric integer remainder.
 *  Equivalent to Maple's irem(a,b,'q') it obeyes the relation
 *  irem(a,b,q) == a - q*b.  In general, mod(a,b) has the sign of b or is zero,
 *  and irem(a,b) has the sign of a or is zero.  
 *
 *  @return remainder of a/b and quotient stored in q if both are integer,
 *  0 otherwise. */
const numeric irem(const numeric & a, const numeric & b, numeric & q)
{
        if (a.is_integer() && b.is_integer()) {
                const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
                                                               cln::the<cln::cl_I>(b.to_cl_N()));
                q = rem_quo.quotient;
                return rem_quo.remainder;
        } else {
                q = _num0();
                return _num0();
        }
}


/** Numeric integer quotient.
 *  Equivalent to Maple's iquo as far as sign conventions are concerned.
 *  
 *  @return truncated quotient of a/b if both are integer, 0 otherwise. */
const numeric iquo(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer())
                return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
                             cln::the<cln::cl_I>(b.to_cl_N()));
        else
                return _num0();
}


/** Numeric integer quotient.
 *  Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
 *  r == a - iquo(a,b,r)*b.
 *
 *  @return truncated quotient of a/b and remainder stored in r if both are
 *  integer, 0 otherwise. */
const numeric iquo(const numeric & a, const numeric & b, numeric & r)
{
        if (a.is_integer() && b.is_integer()) {
                const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
                                                               cln::the<cln::cl_I>(b.to_cl_N()));
                r = rem_quo.remainder;
                return rem_quo.quotient;
        } else {
                r = _num0();
                return _num0();
        }
}


/** Greatest Common Divisor.
 *   
 *  @return  The GCD of two numbers if both are integer, a numerical 1
 *  if they are not. */
const numeric gcd(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer())
                return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
                                cln::the<cln::cl_I>(b.to_cl_N()));
        else
                return _num1();
}


/** Least Common Multiple.
 *   
 *  @return  The LCM of two numbers if both are integer, the product of those
 *  two numbers if they are not. */
const numeric lcm(const numeric & a, const numeric & b)
{
        if (a.is_integer() && b.is_integer())
                return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
                                cln::the<cln::cl_I>(b.to_cl_N()));
        else
                return a.mul(b);
}


/** Numeric square root.
 *  If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
 *  should return integer 2.
 *
 *  @param z numeric argument
 *  @return square root of z. Branch cut along negative real axis, the negative
 *  real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
 *  where imag(z)>0. */
const numeric sqrt(const numeric & z)
{
        return cln::sqrt(z.to_cl_N());
}


/** Integer numeric square root. */
const numeric isqrt(const numeric & x)
{
        if (x.is_integer()) {
                cln::cl_I root;
                cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
                return root;
        } else
                return _num0();
}


/** Floating point evaluation of Archimedes' constant Pi. */
ex PiEvalf(void)
{ 
        return numeric(cln::pi(cln::default_float_format));
}


/** Floating point evaluation of Euler's constant gamma. */
ex EulerEvalf(void)
{ 
        return numeric(cln::eulerconst(cln::default_float_format));
}


/** Floating point evaluation of Catalan's constant. */
ex CatalanEvalf(void)
{
        return numeric(cln::catalanconst(cln::default_float_format));
}


_numeric_digits::_numeric_digits()
  : digits(17)
{
        // It initializes to 17 digits, because in CLN float_format(17) turns out
        // to be 61 (<64) while float_format(18)=65.  The reason is we want to
        // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
        assert(!too_late);
        too_late = true;
        cln::default_float_format = cln::float_format(17);
}


/** Assign a native long to global Digits object. */
_numeric_digits& _numeric_digits::operator=(long prec)
{
        digits = prec;
        cln::default_float_format = cln::float_format(prec); 
        return *this;
}


/** Convert global Digits object to native type long. */
_numeric_digits::operator long()
{
        // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
        return (long)digits;
}


/** Append global Digits object to ostream. */
void _numeric_digits::print(std::ostream & os) const
{
        debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
        os << digits;
}


std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
{
        e.print(os);
        return os;
}

//////////
// static member variables
//////////

// private

bool _numeric_digits::too_late = false;


/** Accuracy in decimal digits.  Only object of this type!  Can be set using
 *  assignment from C++ unsigned ints and evaluated like any built-in type. */
_numeric_digits Digits;

#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
#endif // ndef NO_NAMESPACE_GINAC