inifcns_trans.cpp

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

#include "inifcns.h"
#include "ex.h"
#include "constant.h"
#include "numeric.h"
#include "power.h"
#include "operators.h"
#include "relational.h"
#include "symbol.h"
#include "pseries.h"
#include "utils.h"

#include <stdexcept>
#include <vector>

namespace GiNaC {

// exponential function

static ex exp_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return exp(ex_to<numeric>(x));
    
    return exp(x).hold();
}

static ex exp_eval(const ex & x)
{
    // exp(0) -> 1
    if (x.is_zero()) {
        return _ex1;
    }

    // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
    const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
    if (TwoExOverPiI.info(info_flags::integer)) {
        const numeric z = mod(ex_to<numeric>(TwoExOverPiI),*_num4_p);
        if (z.is_equal(*_num0_p))
            return _ex1;
        if (z.is_equal(*_num1_p))
            return ex(I);
        if (z.is_equal(*_num2_p))
            return _ex_1;
        if (z.is_equal(*_num3_p))
            return ex(-I);
    }

    // exp(log(x)) -> x
    if (is_ex_the_function(x, log))
        return x.op(0);
    
    // exp(float) -> float
    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
        return exp(ex_to<numeric>(x));
    
    return exp(x).hold();
}

static ex exp_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);

    // d/dx exp(x) -> exp(x)
    return exp(x);
}

static ex exp_real_part(const ex & x)
{
    return exp(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex exp_imag_part(const ex & x)
{
    return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex exp_conjugate(const ex & x)
{
    // conjugate(exp(x))==exp(conjugate(x))
    return exp(x.conjugate());
}

REGISTER_FUNCTION(exp, eval_func(exp_eval).
                       evalf_func(exp_evalf).
                       derivative_func(exp_deriv).
                       real_part_func(exp_real_part).
                       imag_part_func(exp_imag_part).
                       conjugate_func(exp_conjugate).
                       latex_name("\\exp"));

// natural logarithm

static ex log_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return log(ex_to<numeric>(x));
    
    return log(x).hold();
}

static ex log_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {
        if (x.is_zero())         // log(0) -> infinity
            throw(pole_error("log_eval(): log(0)",0));
        if (x.info(info_flags::rational) && x.info(info_flags::negative))
            return (log(-x)+I*Pi);
        if (x.is_equal(_ex1))  // log(1) -> 0
            return _ex0;
        if (x.is_equal(I))       // log(I) -> Pi*I/2
            return (Pi*I*_ex1_2);
        if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
            return (Pi*I*_ex_1_2);

        // log(float) -> float
        if (!x.info(info_flags::crational))
            return log(ex_to<numeric>(x));
    }

    // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
    if (is_ex_the_function(x, exp)) {
        const ex &t = x.op(0);
        if (t.info(info_flags::real))
            return t;
    }
    
    return log(x).hold();
}

static ex log_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx log(x) -> 1/x
    return power(x, _ex_1);
}

static ex log_series(const ex &arg,
                     const relational &rel,
                     int order,
                     unsigned options)
{
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    ex arg_pt;
    bool must_expand_arg = false;
    // maybe substitution of rel into arg fails because of a pole
    try {
        arg_pt = arg.subs(rel, subs_options::no_pattern);
    } catch (pole_error) {
        must_expand_arg = true;
    }
    // or we are at the branch point anyways
    if (arg_pt.is_zero())
        must_expand_arg = true;
    
    if (must_expand_arg) {
        // method:
        // This is the branch point: Series expand the argument first, then
        // trivially factorize it to isolate that part which has constant
        // leading coefficient in this fashion:
        //   x^n + x^(n+1) +...+ Order(x^(n+m))  ->  x^n * (1 + x +...+ Order(x^m)).
        // Return a plain n*log(x) for the x^n part and series expand the
        // other part.  Add them together and reexpand again in order to have
        // one unnested pseries object.  All this also works for negative n.
        pseries argser;          // series expansion of log's argument
        unsigned extra_ord = 0;  // extra expansion order
        do {
            // oops, the argument expanded to a pure Order(x^something)...
            argser = ex_to<pseries>(arg.series(rel, order+extra_ord, options));
            ++extra_ord;
        } while (!argser.is_terminating() && argser.nops()==1);

        const symbol &s = ex_to<symbol>(rel.lhs());
        const ex &point = rel.rhs();
        const int n = argser.ldegree(s);
        epvector seq;
        // construct what we carelessly called the n*log(x) term above
        const ex coeff = argser.coeff(s, n);
        // expand the log, but only if coeff is real and > 0, since otherwise
        // it would make the branch cut run into the wrong direction
        if (coeff.info(info_flags::positive))
            seq.push_back(expair(n*log(s-point)+log(coeff), _ex0));
        else
            seq.push_back(expair(log(coeff*pow(s-point, n)), _ex0));

        if (!argser.is_terminating() || argser.nops()!=1) {
            // in this case n more (or less) terms are needed
            // (sadly, to generate them, we have to start from the beginning)
            if (n == 0 && coeff == 1) {
                epvector epv;
                ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
                epv.reserve(2);
                epv.push_back(expair(-1, _ex0));
                epv.push_back(expair(Order(_ex1), order));
                ex rest = pseries(rel, epv).add_series(argser);
                for (int i = order-1; i>0; --i) {
                    epvector cterm;
                    cterm.reserve(1);
                    cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
                    acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
                    acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
                }
                return acc;
            }
            const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
            return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
        } else  // it was a monomial
            return pseries(rel, seq);
    }
    if (!(options & series_options::suppress_branchcut) &&
         arg_pt.info(info_flags::negative)) {
        // method:
        // This is the branch cut: assemble the primitive series manually and
        // then add the corresponding complex step function.
        const symbol &s = ex_to<symbol>(rel.lhs());
        const ex &point = rel.rhs();
        const symbol foo;
        const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
        epvector seq;
        seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
        seq.push_back(expair(Order(_ex1), order));
        return series(replarg - I*Pi + pseries(rel, seq), rel, order);
    }
    throw do_taylor();  // caught by function::series()
}

static ex log_real_part(const ex & x)
{
    if (x.info(info_flags::nonnegative))
        return log(x).hold();
    return log(abs(x));
}

static ex log_imag_part(const ex & x)
{
    if (x.info(info_flags::nonnegative))
        return 0;
    return atan2(GiNaC::imag_part(x), GiNaC::real_part(x));
}

static ex log_conjugate(const ex & x)
{
    // conjugate(log(x))==log(conjugate(x)) unless on the branch cut which
    // runs along the negative real axis.
    if (x.info(info_flags::positive)) {
        return log(x);
    }
    if (is_exactly_a<numeric>(x) &&
        !x.imag_part().is_zero()) {
        return log(x.conjugate());
    }
    return conjugate_function(log(x)).hold();
}

REGISTER_FUNCTION(log, eval_func(log_eval).
                       evalf_func(log_evalf).
                       derivative_func(log_deriv).
                       series_func(log_series).
                       real_part_func(log_real_part).
                       imag_part_func(log_imag_part).
                       conjugate_func(log_conjugate).
                       latex_name("\\ln"));

// sine (trigonometric function)

static ex sin_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return sin(ex_to<numeric>(x));
    
    return sin(x).hold();
}

static ex sin_eval(const ex & x)
{
    // sin(n/d*Pi) -> { all known non-nested radicals }
    const ex SixtyExOverPi = _ex60*x/Pi;
    ex sign = _ex1;
    if (SixtyExOverPi.info(info_flags::integer)) {
        numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
        if (z>=*_num60_p) {
            // wrap to interval [0, Pi)
            z -= *_num60_p;
            sign = _ex_1;
        }
        if (z>*_num30_p) {
            // wrap to interval [0, Pi/2)
            z = *_num60_p-z;
        }
        if (z.is_equal(*_num0_p))  // sin(0)       -> 0
            return _ex0;
        if (z.is_equal(*_num5_p))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
            return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
        if (z.is_equal(*_num6_p))  // sin(Pi/10)   -> sqrt(5)/4-1/4
            return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
        if (z.is_equal(*_num10_p)) // sin(Pi/6)    -> 1/2
            return sign*_ex1_2;
        if (z.is_equal(*_num15_p)) // sin(Pi/4)    -> sqrt(2)/2
            return sign*_ex1_2*sqrt(_ex2);
        if (z.is_equal(*_num18_p)) // sin(3/10*Pi) -> sqrt(5)/4+1/4
            return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
        if (z.is_equal(*_num20_p)) // sin(Pi/3)    -> sqrt(3)/2
            return sign*_ex1_2*sqrt(_ex3);
        if (z.is_equal(*_num25_p)) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
            return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
        if (z.is_equal(*_num30_p)) // sin(Pi/2)    -> 1
            return sign;
    }

    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // sin(asin(x)) -> x
        if (is_ex_the_function(x, asin))
            return t;

        // sin(acos(x)) -> sqrt(1-x^2)
        if (is_ex_the_function(x, acos))
            return sqrt(_ex1-power(t,_ex2));

        // sin(atan(x)) -> x/sqrt(1+x^2)
        if (is_ex_the_function(x, atan))
            return t*power(_ex1+power(t,_ex2),_ex_1_2);
    }
    
    // sin(float) -> float
    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
        return sin(ex_to<numeric>(x));

    // sin() is odd
    if (x.info(info_flags::negative))
        return -sin(-x);
    
    return sin(x).hold();
}

static ex sin_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx sin(x) -> cos(x)
    return cos(x);
}

static ex sin_real_part(const ex & x)
{
    return cosh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
}

static ex sin_imag_part(const ex & x)
{
    return sinh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
}

static ex sin_conjugate(const ex & x)
{
    // conjugate(sin(x))==sin(conjugate(x))
    return sin(x.conjugate());
}

REGISTER_FUNCTION(sin, eval_func(sin_eval).
                       evalf_func(sin_evalf).
                       derivative_func(sin_deriv).
                       real_part_func(sin_real_part).
                       imag_part_func(sin_imag_part).
                       conjugate_func(sin_conjugate).
                       latex_name("\\sin"));

// cosine (trigonometric function)

static ex cos_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return cos(ex_to<numeric>(x));
    
    return cos(x).hold();
}

static ex cos_eval(const ex & x)
{
    // cos(n/d*Pi) -> { all known non-nested radicals }
    const ex SixtyExOverPi = _ex60*x/Pi;
    ex sign = _ex1;
    if (SixtyExOverPi.info(info_flags::integer)) {
        numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num120_p);
        if (z>=*_num60_p) {
            // wrap to interval [0, Pi)
            z = *_num120_p-z;
        }
        if (z>=*_num30_p) {
            // wrap to interval [0, Pi/2)
            z = *_num60_p-z;
            sign = _ex_1;
        }
        if (z.is_equal(*_num0_p))  // cos(0)       -> 1
            return sign;
        if (z.is_equal(*_num5_p))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
            return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex1_3*sqrt(_ex3));
        if (z.is_equal(*_num10_p)) // cos(Pi/6)    -> sqrt(3)/2
            return sign*_ex1_2*sqrt(_ex3);
        if (z.is_equal(*_num12_p)) // cos(Pi/5)    -> sqrt(5)/4+1/4
            return sign*(_ex1_4*sqrt(_ex5)+_ex1_4);
        if (z.is_equal(*_num15_p)) // cos(Pi/4)    -> sqrt(2)/2
            return sign*_ex1_2*sqrt(_ex2);
        if (z.is_equal(*_num20_p)) // cos(Pi/3)    -> 1/2
            return sign*_ex1_2;
        if (z.is_equal(*_num24_p)) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
            return sign*(_ex1_4*sqrt(_ex5)+_ex_1_4);
        if (z.is_equal(*_num25_p)) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
            return sign*_ex1_4*sqrt(_ex6)*(_ex1+_ex_1_3*sqrt(_ex3));
        if (z.is_equal(*_num30_p)) // cos(Pi/2)    -> 0
            return _ex0;
    }

    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // cos(acos(x)) -> x
        if (is_ex_the_function(x, acos))
            return t;

        // cos(asin(x)) -> sqrt(1-x^2)
        if (is_ex_the_function(x, asin))
            return sqrt(_ex1-power(t,_ex2));

        // cos(atan(x)) -> 1/sqrt(1+x^2)
        if (is_ex_the_function(x, atan))
            return power(_ex1+power(t,_ex2),_ex_1_2);
    }
    
    // cos(float) -> float
    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
        return cos(ex_to<numeric>(x));
    
    // cos() is even
    if (x.info(info_flags::negative))
        return cos(-x);
    
    return cos(x).hold();
}

static ex cos_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);

    // d/dx cos(x) -> -sin(x)
    return -sin(x);
}

static ex cos_real_part(const ex & x)
{
    return cosh(GiNaC::imag_part(x))*cos(GiNaC::real_part(x));
}

static ex cos_imag_part(const ex & x)
{
    return -sinh(GiNaC::imag_part(x))*sin(GiNaC::real_part(x));
}

static ex cos_conjugate(const ex & x)
{
    // conjugate(cos(x))==cos(conjugate(x))
    return cos(x.conjugate());
}

REGISTER_FUNCTION(cos, eval_func(cos_eval).
                       evalf_func(cos_evalf).
                       derivative_func(cos_deriv).
                       real_part_func(cos_real_part).
                       imag_part_func(cos_imag_part).
                       conjugate_func(cos_conjugate).
                       latex_name("\\cos"));

// tangent (trigonometric function)

static ex tan_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return tan(ex_to<numeric>(x));
    
    return tan(x).hold();
}

static ex tan_eval(const ex & x)
{
    // tan(n/d*Pi) -> { all known non-nested radicals }
    const ex SixtyExOverPi = _ex60*x/Pi;
    ex sign = _ex1;
    if (SixtyExOverPi.info(info_flags::integer)) {
        numeric z = mod(ex_to<numeric>(SixtyExOverPi),*_num60_p);
        if (z>=*_num60_p) {
            // wrap to interval [0, Pi)
            z -= *_num60_p;
        }
        if (z>=*_num30_p) {
            // wrap to interval [0, Pi/2)
            z = *_num60_p-z;
            sign = _ex_1;
        }
        if (z.is_equal(*_num0_p))  // tan(0)       -> 0
            return _ex0;
        if (z.is_equal(*_num5_p))  // tan(Pi/12)   -> 2-sqrt(3)
            return sign*(_ex2-sqrt(_ex3));
        if (z.is_equal(*_num10_p)) // tan(Pi/6)    -> sqrt(3)/3
            return sign*_ex1_3*sqrt(_ex3);
        if (z.is_equal(*_num15_p)) // tan(Pi/4)    -> 1
            return sign;
        if (z.is_equal(*_num20_p)) // tan(Pi/3)    -> sqrt(3)
            return sign*sqrt(_ex3);
        if (z.is_equal(*_num25_p)) // tan(5/12*Pi) -> 2+sqrt(3)
            return sign*(sqrt(_ex3)+_ex2);
        if (z.is_equal(*_num30_p)) // tan(Pi/2)    -> infinity
            throw (pole_error("tan_eval(): simple pole",1));
    }

    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // tan(atan(x)) -> x
        if (is_ex_the_function(x, atan))
            return t;

        // tan(asin(x)) -> x/sqrt(1+x^2)
        if (is_ex_the_function(x, asin))
            return t*power(_ex1-power(t,_ex2),_ex_1_2);

        // tan(acos(x)) -> sqrt(1-x^2)/x
        if (is_ex_the_function(x, acos))
            return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
    }
    
    // tan(float) -> float
    if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
        return tan(ex_to<numeric>(x));
    }
    
    // tan() is odd
    if (x.info(info_flags::negative))
        return -tan(-x);
    
    return tan(x).hold();
}

static ex tan_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx tan(x) -> 1+tan(x)^2;
    return (_ex1+power(tan(x),_ex2));
}

static ex tan_real_part(const ex & x)
{
    ex a = GiNaC::real_part(x);
    ex b = GiNaC::imag_part(x);
    return tan(a)/(1+power(tan(a),2)*power(tan(b),2));
}

static ex tan_imag_part(const ex & x)
{
    ex a = GiNaC::real_part(x);
    ex b = GiNaC::imag_part(x);
    return tanh(b)/(1+power(tan(a),2)*power(tan(b),2));
}

static ex tan_series(const ex &x,
                     const relational &rel,
                     int order,
                     unsigned options)
{
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    // method:
    // Taylor series where there is no pole falls back to tan_deriv.
    // On a pole simply expand sin(x)/cos(x).
    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    if (!(2*x_pt/Pi).info(info_flags::odd))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a simple pole
    return (sin(x)/cos(x)).series(rel, order, options);
}

static ex tan_conjugate(const ex & x)
{
    // conjugate(tan(x))==tan(conjugate(x))
    return tan(x.conjugate());
}

REGISTER_FUNCTION(tan, eval_func(tan_eval).
                       evalf_func(tan_evalf).
                       derivative_func(tan_deriv).
                       series_func(tan_series).
                       real_part_func(tan_real_part).
                       imag_part_func(tan_imag_part).
                       conjugate_func(tan_conjugate).
                       latex_name("\\tan"));

// inverse sine (arc sine)

static ex asin_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return asin(ex_to<numeric>(x));
    
    return asin(x).hold();
}

static ex asin_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // asin(0) -> 0
        if (x.is_zero())
            return x;

        // asin(1/2) -> Pi/6
        if (x.is_equal(_ex1_2))
            return numeric(1,6)*Pi;

        // asin(1) -> Pi/2
        if (x.is_equal(_ex1))
            return _ex1_2*Pi;

        // asin(-1/2) -> -Pi/6
        if (x.is_equal(_ex_1_2))
            return numeric(-1,6)*Pi;

        // asin(-1) -> -Pi/2
        if (x.is_equal(_ex_1))
            return _ex_1_2*Pi;

        // asin(float) -> float
        if (!x.info(info_flags::crational))
            return asin(ex_to<numeric>(x));

        // asin() is odd
        if (x.info(info_flags::negative))
            return -asin(-x);
    }
    
    return asin(x).hold();
}

static ex asin_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx asin(x) -> 1/sqrt(1-x^2)
    return power(1-power(x,_ex2),_ex_1_2);
}

static ex asin_conjugate(const ex & x)
{
    // conjugate(asin(x))==asin(conjugate(x)) unless on the branch cuts which
    // run along the real axis outside the interval [-1, +1].
    if (is_exactly_a<numeric>(x) &&
        (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
        return asin(x.conjugate());
    }
    return conjugate_function(asin(x)).hold();
}

REGISTER_FUNCTION(asin, eval_func(asin_eval).
                        evalf_func(asin_evalf).
                        derivative_func(asin_deriv).
                        conjugate_func(asin_conjugate).
                        latex_name("\\arcsin"));

// inverse cosine (arc cosine)

static ex acos_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return acos(ex_to<numeric>(x));
    
    return acos(x).hold();
}

static ex acos_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // acos(1) -> 0
        if (x.is_equal(_ex1))
            return _ex0;

        // acos(1/2) -> Pi/3
        if (x.is_equal(_ex1_2))
            return _ex1_3*Pi;

        // acos(0) -> Pi/2
        if (x.is_zero())
            return _ex1_2*Pi;

        // acos(-1/2) -> 2/3*Pi
        if (x.is_equal(_ex_1_2))
            return numeric(2,3)*Pi;

        // acos(-1) -> Pi
        if (x.is_equal(_ex_1))
            return Pi;

        // acos(float) -> float
        if (!x.info(info_flags::crational))
            return acos(ex_to<numeric>(x));

        // acos(-x) -> Pi-acos(x)
        if (x.info(info_flags::negative))
            return Pi-acos(-x);
    }
    
    return acos(x).hold();
}

static ex acos_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx acos(x) -> -1/sqrt(1-x^2)
    return -power(1-power(x,_ex2),_ex_1_2);
}

static ex acos_conjugate(const ex & x)
{
    // conjugate(acos(x))==acos(conjugate(x)) unless on the branch cuts which
    // run along the real axis outside the interval [-1, +1].
    if (is_exactly_a<numeric>(x) &&
        (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
        return acos(x.conjugate());
    }
    return conjugate_function(acos(x)).hold();
}

REGISTER_FUNCTION(acos, eval_func(acos_eval).
                        evalf_func(acos_evalf).
                        derivative_func(acos_deriv).
                        conjugate_func(acos_conjugate).
                        latex_name("\\arccos"));

// inverse tangent (arc tangent)

static ex atan_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return atan(ex_to<numeric>(x));
    
    return atan(x).hold();
}

static ex atan_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // atan(0) -> 0
        if (x.is_zero())
            return _ex0;

        // atan(1) -> Pi/4
        if (x.is_equal(_ex1))
            return _ex1_4*Pi;

        // atan(-1) -> -Pi/4
        if (x.is_equal(_ex_1))
            return _ex_1_4*Pi;

        if (x.is_equal(I) || x.is_equal(-I))
            throw (pole_error("atan_eval(): logarithmic pole",0));

        // atan(float) -> float
        if (!x.info(info_flags::crational))
            return atan(ex_to<numeric>(x));

        // atan() is odd
        if (x.info(info_flags::negative))
            return -atan(-x);
    }
    
    return atan(x).hold();
}

static ex atan_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);

    // d/dx atan(x) -> 1/(1+x^2)
    return power(_ex1+power(x,_ex2), _ex_1);
}

static ex atan_series(const ex &arg,
                      const relational &rel,
                      int order,
                      unsigned options)
{
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    // method:
    // Taylor series where there is no pole or cut falls back to atan_deriv.
    // There are two branch cuts, one runnig from I up the imaginary axis and
    // one running from -I down the imaginary axis.  The points I and -I are
    // poles.
    // On the branch cuts and the poles series expand
    //     (log(1+I*x)-log(1-I*x))/(2*I)
    // instead.
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!(I*arg_pt).info(info_flags::real))
        throw do_taylor();     // Re(x) != 0
    if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
        throw do_taylor();     // Re(x) == 0, but abs(x)<1
    // care for the poles, using the defining formula for atan()...
    if (arg_pt.is_equal(I) || arg_pt.is_equal(-I))
        return ((log(1+I*arg)-log(1-I*arg))/(2*I)).series(rel, order, options);
    if (!(options & series_options::suppress_branchcut)) {
        // method:
        // This is the branch cut: assemble the primitive series manually and
        // then add the corresponding complex step function.
        const symbol &s = ex_to<symbol>(rel.lhs());
        const ex &point = rel.rhs();
        const symbol foo;
        const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
        ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
        if ((I*arg_pt)<_ex0)
            Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
        else
            Order0correction += log((I*arg_pt+_ex1)/(I*arg_pt+_ex_1))*I*_ex1_2;
        epvector seq;
        seq.push_back(expair(Order0correction, _ex0));
        seq.push_back(expair(Order(_ex1), order));
        return series(replarg - pseries(rel, seq), rel, order);
    }
    throw do_taylor();
}

static ex atan_conjugate(const ex & x)
{
    // conjugate(atan(x))==atan(conjugate(x)) unless on the branch cuts which
    // run along the imaginary axis outside the interval [-I, +I].
    if (x.info(info_flags::real))
        return atan(x);
    if (is_exactly_a<numeric>(x)) {
        const numeric x_re = ex_to<numeric>(x.real_part());
        const numeric x_im = ex_to<numeric>(x.imag_part());
        if (!x_re.is_zero() ||
            (x_im > *_num_1_p && x_im < *_num1_p))
            return atan(x.conjugate());
    }
    return conjugate_function(atan(x)).hold();
}

REGISTER_FUNCTION(atan, eval_func(atan_eval).
                        evalf_func(atan_evalf).
                        derivative_func(atan_deriv).
                        series_func(atan_series).
                        conjugate_func(atan_conjugate).
                        latex_name("\\arctan"));

// inverse tangent (atan2(y,x))

static ex atan2_evalf(const ex &y, const ex &x)
{
    if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
        return atan(ex_to<numeric>(y), ex_to<numeric>(x));
    
    return atan2(y, x).hold();
}

static ex atan2_eval(const ex & y, const ex & x)
{
    if (y.is_zero()) {

        // atan2(0, 0) -> 0
        if (x.is_zero())
            return _ex0;

        // atan2(0, x), x real and positive -> 0
        if (x.info(info_flags::positive))
            return _ex0;

        // atan2(0, x), x real and negative -> Pi
        if (x.info(info_flags::negative))
            return Pi;
    }

    if (x.is_zero()) {

        // atan2(y, 0), y real and positive -> Pi/2
        if (y.info(info_flags::positive))
            return _ex1_2*Pi;

        // atan2(y, 0), y real and negative -> -Pi/2
        if (y.info(info_flags::negative))
            return _ex_1_2*Pi;
    }

    if (y.is_equal(x)) {

        // atan2(y, y), y real and positive -> Pi/4
        if (y.info(info_flags::positive))
            return _ex1_4*Pi;

        // atan2(y, y), y real and negative -> -3/4*Pi
        if (y.info(info_flags::negative))
            return numeric(-3, 4)*Pi;
    }

    if (y.is_equal(-x)) {

        // atan2(y, -y), y real and positive -> 3*Pi/4
        if (y.info(info_flags::positive))
            return numeric(3, 4)*Pi;

        // atan2(y, -y), y real and negative -> -Pi/4
        if (y.info(info_flags::negative))
            return _ex_1_4*Pi;
    }

    // atan2(float, float) -> float
    if (is_a<numeric>(y) && !y.info(info_flags::crational) &&
        is_a<numeric>(x) && !x.info(info_flags::crational))
        return atan(ex_to<numeric>(y), ex_to<numeric>(x));

    // atan2(real, real) -> atan(y/x) +/- Pi
    if (y.info(info_flags::real) && x.info(info_flags::real)) {
        if (x.info(info_flags::positive))
            return atan(y/x);

        if (x.info(info_flags::negative)) {
            if (y.info(info_flags::positive))
                return atan(y/x)+Pi;
            if (y.info(info_flags::negative))
                return atan(y/x)-Pi;
        }
    }

    return atan2(y, x).hold();
}    

static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param<2);
    
    if (deriv_param==0) {
        // d/dy atan2(y,x)
        return x*power(power(x,_ex2)+power(y,_ex2),_ex_1);
    }
    // d/dx atan2(y,x)
    return -y*power(power(x,_ex2)+power(y,_ex2),_ex_1);
}

REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
                         evalf_func(atan2_evalf).
                         derivative_func(atan2_deriv));

// hyperbolic sine (trigonometric function)

static ex sinh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return sinh(ex_to<numeric>(x));
    
    return sinh(x).hold();
}

static ex sinh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // sinh(0) -> 0
        if (x.is_zero())
            return _ex0;        

        // sinh(float) -> float
        if (!x.info(info_flags::crational))
            return sinh(ex_to<numeric>(x));

        // sinh() is odd
        if (x.info(info_flags::negative))
            return -sinh(-x);
    }
    
    if ((x/Pi).info(info_flags::numeric) &&
        ex_to<numeric>(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
        return I*sin(x/I);
    
    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // sinh(asinh(x)) -> x
        if (is_ex_the_function(x, asinh))
            return t;

        // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
        if (is_ex_the_function(x, acosh))
            return sqrt(t-_ex1)*sqrt(t+_ex1);

        // sinh(atanh(x)) -> x/sqrt(1-x^2)
        if (is_ex_the_function(x, atanh))
            return t*power(_ex1-power(t,_ex2),_ex_1_2);
    }
    
    return sinh(x).hold();
}

static ex sinh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx sinh(x) -> cosh(x)
    return cosh(x);
}

static ex sinh_real_part(const ex & x)
{
    return sinh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex sinh_imag_part(const ex & x)
{
    return cosh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex sinh_conjugate(const ex & x)
{
    // conjugate(sinh(x))==sinh(conjugate(x))
    return sinh(x.conjugate());
}

REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
                        evalf_func(sinh_evalf).
                        derivative_func(sinh_deriv).
                        real_part_func(sinh_real_part).
                        imag_part_func(sinh_imag_part).
                        conjugate_func(sinh_conjugate).
                        latex_name("\\sinh"));

// hyperbolic cosine (trigonometric function)

static ex cosh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return cosh(ex_to<numeric>(x));
    
    return cosh(x).hold();
}

static ex cosh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // cosh(0) -> 1
        if (x.is_zero())
            return _ex1;

        // cosh(float) -> float
        if (!x.info(info_flags::crational))
            return cosh(ex_to<numeric>(x));

        // cosh() is even
        if (x.info(info_flags::negative))
            return cosh(-x);
    }
    
    if ((x/Pi).info(info_flags::numeric) &&
        ex_to<numeric>(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
        return cos(x/I);
    
    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // cosh(acosh(x)) -> x
        if (is_ex_the_function(x, acosh))
            return t;

        // cosh(asinh(x)) -> sqrt(1+x^2)
        if (is_ex_the_function(x, asinh))
            return sqrt(_ex1+power(t,_ex2));

        // cosh(atanh(x)) -> 1/sqrt(1-x^2)
        if (is_ex_the_function(x, atanh))
            return power(_ex1-power(t,_ex2),_ex_1_2);
    }
    
    return cosh(x).hold();
}

static ex cosh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx cosh(x) -> sinh(x)
    return sinh(x);
}

static ex cosh_real_part(const ex & x)
{
    return cosh(GiNaC::real_part(x))*cos(GiNaC::imag_part(x));
}

static ex cosh_imag_part(const ex & x)
{
    return sinh(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
}

static ex cosh_conjugate(const ex & x)
{
    // conjugate(cosh(x))==cosh(conjugate(x))
    return cosh(x.conjugate());
}

REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
                        evalf_func(cosh_evalf).
                        derivative_func(cosh_deriv).
                        real_part_func(cosh_real_part).
                        imag_part_func(cosh_imag_part).
                        conjugate_func(cosh_conjugate).
                        latex_name("\\cosh"));

// hyperbolic tangent (trigonometric function)

static ex tanh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return tanh(ex_to<numeric>(x));
    
    return tanh(x).hold();
}

static ex tanh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // tanh(0) -> 0
        if (x.is_zero())
            return _ex0;

        // tanh(float) -> float
        if (!x.info(info_flags::crational))
            return tanh(ex_to<numeric>(x));

        // tanh() is odd
        if (x.info(info_flags::negative))
            return -tanh(-x);
    }
    
    if ((x/Pi).info(info_flags::numeric) &&
        ex_to<numeric>(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
        return I*tan(x/I);
    
    if (is_exactly_a<function>(x)) {
        const ex &t = x.op(0);

        // tanh(atanh(x)) -> x
        if (is_ex_the_function(x, atanh))
            return t;

        // tanh(asinh(x)) -> x/sqrt(1+x^2)
        if (is_ex_the_function(x, asinh))
            return t*power(_ex1+power(t,_ex2),_ex_1_2);

        // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
        if (is_ex_the_function(x, acosh))
            return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
    }
    
    return tanh(x).hold();
}

static ex tanh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx tanh(x) -> 1-tanh(x)^2
    return _ex1-power(tanh(x),_ex2);
}

static ex tanh_series(const ex &x,
                      const relational &rel,
                      int order,
                      unsigned options)
{
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    // method:
    // Taylor series where there is no pole falls back to tanh_deriv.
    // On a pole simply expand sinh(x)/cosh(x).
    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    if (!(2*I*x_pt/Pi).info(info_flags::odd))
        throw do_taylor();  // caught by function::series()
    // if we got here we have to care for a simple pole
    return (sinh(x)/cosh(x)).series(rel, order, options);
}

static ex tanh_real_part(const ex & x)
{
    ex a = GiNaC::real_part(x);
    ex b = GiNaC::imag_part(x);
    return tanh(a)/(1+power(tanh(a),2)*power(tan(b),2));
}

static ex tanh_imag_part(const ex & x)
{
    ex a = GiNaC::real_part(x);
    ex b = GiNaC::imag_part(x);
    return tan(b)/(1+power(tanh(a),2)*power(tan(b),2));
}

static ex tanh_conjugate(const ex & x)
{
    // conjugate(tanh(x))==tanh(conjugate(x))
    return tanh(x.conjugate());
}

REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
                        evalf_func(tanh_evalf).
                        derivative_func(tanh_deriv).
                        series_func(tanh_series).
                        real_part_func(tanh_real_part).
                        imag_part_func(tanh_imag_part).
                        conjugate_func(tanh_conjugate).
                        latex_name("\\tanh"));

// inverse hyperbolic sine (trigonometric function)

static ex asinh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return asinh(ex_to<numeric>(x));
    
    return asinh(x).hold();
}

static ex asinh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // asinh(0) -> 0
        if (x.is_zero())
            return _ex0;

        // asinh(float) -> float
        if (!x.info(info_flags::crational))
            return asinh(ex_to<numeric>(x));

        // asinh() is odd
        if (x.info(info_flags::negative))
            return -asinh(-x);
    }
    
    return asinh(x).hold();
}

static ex asinh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx asinh(x) -> 1/sqrt(1+x^2)
    return power(_ex1+power(x,_ex2),_ex_1_2);
}

static ex asinh_conjugate(const ex & x)
{
    // conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which
    // run along the imaginary axis outside the interval [-I, +I].
    if (x.info(info_flags::real))
        return asinh(x);
    if (is_exactly_a<numeric>(x)) {
        const numeric x_re = ex_to<numeric>(x.real_part());
        const numeric x_im = ex_to<numeric>(x.imag_part());
        if (!x_re.is_zero() ||
            (x_im > *_num_1_p && x_im < *_num1_p))
            return asinh(x.conjugate());
    }
    return conjugate_function(asinh(x)).hold();
}

REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
                         evalf_func(asinh_evalf).
                         derivative_func(asinh_deriv).
                         conjugate_func(asinh_conjugate));

// inverse hyperbolic cosine (trigonometric function)

static ex acosh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return acosh(ex_to<numeric>(x));
    
    return acosh(x).hold();
}

static ex acosh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // acosh(0) -> Pi*I/2
        if (x.is_zero())
            return Pi*I*numeric(1,2);

        // acosh(1) -> 0
        if (x.is_equal(_ex1))
            return _ex0;

        // acosh(-1) -> Pi*I
        if (x.is_equal(_ex_1))
            return Pi*I;

        // acosh(float) -> float
        if (!x.info(info_flags::crational))
            return acosh(ex_to<numeric>(x));

        // acosh(-x) -> Pi*I-acosh(x)
        if (x.info(info_flags::negative))
            return Pi*I-acosh(-x);
    }
    
    return acosh(x).hold();
}

static ex acosh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
    return power(x+_ex_1,_ex_1_2)*power(x+_ex1,_ex_1_2);
}

static ex acosh_conjugate(const ex & x)
{
    // conjugate(acosh(x))==acosh(conjugate(x)) unless on the branch cut
    // which runs along the real axis from +1 to -inf.
    if (is_exactly_a<numeric>(x) &&
        (!x.imag_part().is_zero() || x > *_num1_p)) {
        return acosh(x.conjugate());
    }
    return conjugate_function(acosh(x)).hold();
}

REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
                         evalf_func(acosh_evalf).
                         derivative_func(acosh_deriv).
                         conjugate_func(acosh_conjugate));

// inverse hyperbolic tangent (trigonometric function)

static ex atanh_evalf(const ex & x)
{
    if (is_exactly_a<numeric>(x))
        return atanh(ex_to<numeric>(x));
    
    return atanh(x).hold();
}

static ex atanh_eval(const ex & x)
{
    if (x.info(info_flags::numeric)) {

        // atanh(0) -> 0
        if (x.is_zero())
            return _ex0;

        // atanh({+|-}1) -> throw
        if (x.is_equal(_ex1) || x.is_equal(_ex_1))
            throw (pole_error("atanh_eval(): logarithmic pole",0));

        // atanh(float) -> float
        if (!x.info(info_flags::crational))
            return atanh(ex_to<numeric>(x));

        // atanh() is odd
        if (x.info(info_flags::negative))
            return -atanh(-x);
    }
    
    return atanh(x).hold();
}

static ex atanh_deriv(const ex & x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);
    
    // d/dx atanh(x) -> 1/(1-x^2)
    return power(_ex1-power(x,_ex2),_ex_1);
}

static ex atanh_series(const ex &arg,
                       const relational &rel,
                       int order,
                       unsigned options)
{
    GINAC_ASSERT(is_a<symbol>(rel.lhs()));
    // method:
    // Taylor series where there is no pole or cut falls back to atanh_deriv.
    // There are two branch cuts, one runnig from 1 up the real axis and one
    // one running from -1 down the real axis.  The points 1 and -1 are poles
    // On the branch cuts and the poles series expand
    //     (log(1+x)-log(1-x))/2
    // instead.
    const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
    if (!(arg_pt).info(info_flags::real))
        throw do_taylor();     // Im(x) != 0
    if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
        throw do_taylor();     // Im(x) == 0, but abs(x)<1
    // care for the poles, using the defining formula for atanh()...
    if (arg_pt.is_equal(_ex1) || arg_pt.is_equal(_ex_1))
        return ((log(_ex1+arg)-log(_ex1-arg))*_ex1_2).series(rel, order, options);
    // ...and the branch cuts (the discontinuity at the cut being just I*Pi)
    if (!(options & series_options::suppress_branchcut)) {
        // method:
        // This is the branch cut: assemble the primitive series manually and
        // then add the corresponding complex step function.
        const symbol &s = ex_to<symbol>(rel.lhs());
        const ex &point = rel.rhs();
        const symbol foo;
        const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
        ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
        if (arg_pt<_ex0)
            Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
        else
            Order0correction += log((arg_pt+_ex1)/(arg_pt+_ex_1))*_ex_1_2;
        epvector seq;
        seq.push_back(expair(Order0correction, _ex0));
        seq.push_back(expair(Order(_ex1), order));
        return series(replarg - pseries(rel, seq), rel, order);
    }
    throw do_taylor();
}

static ex atanh_conjugate(const ex & x)
{
    // conjugate(atanh(x))==atanh(conjugate(x)) unless on the branch cuts which
    // run along the real axis outside the interval [-1, +1].
    if (is_exactly_a<numeric>(x) &&
        (!x.imag_part().is_zero() || (x > *_num_1_p && x < *_num1_p))) {
        return atanh(x.conjugate());
    }
    return conjugate_function(atanh(x)).hold();
}

REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
                         evalf_func(atanh_evalf).
                         derivative_func(atanh_deriv).
                         series_func(atanh_series).
                         conjugate_func(atanh_conjugate));


} // namespace GiNaC