// $Id: atan_op.hpp 3757 2015-11-30 12:03:07Z bradbell $
# ifndef CPPAD_ATAN_OP_HPP
# define CPPAD_ATAN_OP_HPP

/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the
                    GNU General Public License Version 3.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */


namespace CppAD { // BEGIN_CPPAD_NAMESPACE
/*!
\file atan_op.hpp
Forward and reverse mode calculations for z = atan(x).
*/


/*!
Forward mode Taylor coefficient for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
	z = atan(x)
\endverbatim
The auxillary result is
\verbatim
	y = 1 + x * x
\endverbatim
The value of y, and its derivatives, are computed along with the value
and derivatives of z.

\copydetails forward_unary2_op
*/
template <class Base>
inline void forward_atan_op(
	size_t p           ,
	size_t q           ,
	size_t i_z         ,
	size_t i_x         ,
	size_t cap_order   ,
	Base*  taylor      )
{
	// check assumptions
	CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
	CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
	CPPAD_ASSERT_UNKNOWN( q < cap_order );
	CPPAD_ASSERT_UNKNOWN( p <= q );

	// Taylor coefficients corresponding to argument and result
	Base* x = taylor + i_x * cap_order;
	Base* z = taylor + i_z * cap_order;
	Base* b = z      -       cap_order;  // called y in documentation

	size_t k;
	if( p == 0 )
	{	z[0] = atan( x[0] );
		b[0] = Base(1) + x[0] * x[0];
		p++;
	}
	for(size_t j = p; j <= q; j++)
	{
		b[j] = Base(2) * x[0] * x[j];
		z[j] = Base(0);
		for(k = 1; k < j; k++)
		{	b[j] += x[k] * x[j-k];
			z[j] -= Base(k) * z[k] * b[j-k];
		}
		z[j] /= Base(j);
		z[j] += x[j];
		z[j] /= b[0];
	}
}

/*!
Multiple direction Taylor coefficient for op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
	z = atan(x)
\endverbatim
The auxillary result is
\verbatim
	y = 1 + x * x
\endverbatim
The value of y, and its derivatives, are computed along with the value
and derivatives of z.

\copydetails forward_unary2_op_dir
*/
template <class Base>
inline void forward_atan_op_dir(
	size_t q           ,
	size_t r           ,
	size_t i_z         ,
	size_t i_x         ,
	size_t cap_order   ,
	Base*  taylor      )
{
	// check assumptions
	CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
	CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
	CPPAD_ASSERT_UNKNOWN( 0 < q );
	CPPAD_ASSERT_UNKNOWN( q < cap_order );

	// Taylor coefficients corresponding to argument and result
	size_t num_taylor_per_var = (cap_order-1) * r + 1;
	Base* x = taylor + i_x * num_taylor_per_var;
	Base* z = taylor + i_z * num_taylor_per_var;
	Base* b = z      -       num_taylor_per_var; // called y in documentation

	size_t m = (q-1) * r + 1;
	for(size_t ell = 0; ell < r; ell++)
	{	b[m+ell] = Base(2) * x[m+ell] * x[0];
		z[m+ell] = Base(q) * x[m+ell];
		for(size_t k = 1; k < q; k++)
		{	b[m+ell] += x[(k-1)*r+1+ell] * x[(q-k-1)*r+1+ell];
			z[m+ell] -= Base(k) * z[(k-1)*r+1+ell] * b[(q-k-1)*r+1+ell];
		}
		z[m+ell] /= ( Base(q) * b[0] );
	}
}

/*!
Zero order forward mode Taylor coefficient for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
	z = atan(x)
\endverbatim
The auxillary result is
\verbatim
	y = 1 + x * x
\endverbatim
The value of y is computed along with the value of z.

\copydetails forward_unary2_op_0
*/
template <class Base>
inline void forward_atan_op_0(
	size_t i_z         ,
	size_t i_x         ,
	size_t cap_order   ,
	Base*  taylor      )
{
	// check assumptions
	CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
	CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
	CPPAD_ASSERT_UNKNOWN( 0 < cap_order );

	// Taylor coefficients corresponding to argument and result
	Base* x = taylor + i_x * cap_order;
	Base* z = taylor + i_z * cap_order;
	Base* b = z      -       cap_order; // called y in documentation

	z[0] = atan( x[0] );
	b[0] = Base(1) + x[0] * x[0];
}
/*!
Reverse mode partial derivatives for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
	z = atan(x)
\endverbatim
The auxillary result is
\verbatim
	y = 1 + x * x
\endverbatim
The value of y is computed along with the value of z.

\copydetails reverse_unary2_op
*/

template <class Base>
inline void reverse_atan_op(
	size_t      d            ,
	size_t      i_z          ,
	size_t      i_x          ,
	size_t      cap_order    ,
	const Base* taylor       ,
	size_t      nc_partial   ,
	Base*       partial      )
{
	// check assumptions
	CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
	CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
	CPPAD_ASSERT_UNKNOWN( d < cap_order );
	CPPAD_ASSERT_UNKNOWN( d < nc_partial );

	// Taylor coefficients and partials corresponding to argument
	const Base* x  = taylor  + i_x * cap_order;
	Base* px       = partial + i_x * nc_partial;

	// Taylor coefficients and partials corresponding to first result
	const Base* z  = taylor  + i_z * cap_order;
	Base* pz       = partial + i_z * nc_partial;

	// Taylor coefficients and partials corresponding to auxillary result
	const Base* b  = z  - cap_order; // called y in documentation
	Base* pb       = pz - nc_partial;

	Base inv_b0 = Base(1) / b[0];

	// number of indices to access
	size_t j = d;
	size_t k;
	while(j)
	{	// scale partials w.r.t z[j] and b[j]
		pz[j]  = azmul(pz[j], inv_b0);
		pb[j] *= Base(2);

		pb[0] -= azmul(pz[j], z[j]);
		px[j] += pz[j] + azmul(pb[j], x[0]);
		px[0] += azmul(pb[j], x[j]);

		// more scaling of partials w.r.t z[j]
		pz[j] /= Base(j);

		for(k = 1; k < j; k++)
		{	pb[j-k] -= Base(k) * azmul(pz[j], z[k]);
			pz[k]   -= Base(k) * azmul(pz[j], b[j-k]);
			px[k]   += azmul(pb[j], x[j-k]);
		}
		--j;
	}
	px[0] += azmul(pz[0], inv_b0) + Base(2) * azmul(pb[0], x[0]);
}

} // END_CPPAD_NAMESPACE
# endif
