text: # ifndef CPPAD_LOCAL_SQRT_OP_HPP
  # define CPPAD_LOCAL_SQRT_OP_HPP
  
  /* --------------------------------------------------------------------------
  CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-17 Bradley M. Bell
  
  CppAD is distributed under multiple licenses. This distribution is under
  the terms of the
                      Eclipse Public License Version 1.0.
  
  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 { namespace local { // BEGIN_CPPAD_LOCAL_NAMESPACE
  /*!
  \file sqrt_op.hpp
  Forward and reverse mode calculations for z = sqrt(x).
  */
  
  
  /*!
  Compute forward mode Taylor coefficient for result of op = SqrtOp.
  
  The C++ source code corresponding to this operation is
  \verbatim
  	z = sqrt(x)
  \endverbatim
  
  \copydetails CppAD::local::forward_unary1_op
  */
  template <class Base>
  #if defined(__GNUC__) && __GNUC__ * 100 + __GNUC_MINOR__ * 10 >= 490 && !defined(__INTEL_COMPILER)
  __attribute__((no_sanitize_undefined))
  #endif
  inline void forward_sqrt_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(SqrtOp) == 1 );
  	CPPAD_ASSERT_UNKNOWN( NumRes(SqrtOp) == 1 );
  	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;
  
  	size_t k;
  	if( p == 0 )
  	{	z[0] = sqrt( x[0] );
  		p++;
  	}
  	for(size_t j = p; j <= q; j++)
  	{
  		z[j] = Base(0.0);
  		for(k = 1; k < j; k++)
  			z[j] -= Base(double(k)) * z[k] * z[j-k];
  		z[j] /= Base(double(j));
  		z[j] += x[j] / Base(2.0);
  		z[j] /= z[0];
  	}
  }
  
  /*!
  Multiple direction forward mode Taylor coefficient for op = SqrtOp.
  
  The C++ source code corresponding to this operation is
  \verbatim
  	z = sqrt(x)
  \endverbatim
  
  \copydetails CppAD::local::forward_unary1_op_dir
  */
  template <class Base>
  inline void forward_sqrt_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(SqrtOp) == 1 );
  	CPPAD_ASSERT_UNKNOWN( NumRes(SqrtOp) == 1 );
  	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* z = taylor + i_z * num_taylor_per_var;
  	Base* x = taylor + i_x * num_taylor_per_var;
  
  	size_t m = (q-1) * r + 1;
  	for(size_t ell = 0; ell < r; ell++)
  	{	z[m+ell] = Base(0.0);
  		for(size_t k = 1; k < q; k++)
  			z[m+ell] -= Base(double(k)) * z[(k-1)*r+1+ell] * z[(q-k-1)*r+1+ell];
  		z[m+ell] /= Base(double(q));
  		z[m+ell] += x[m+ell] / Base(2.0);
  		z[m+ell] /= z[0];
  	}
  }
  
  /*!
  Compute zero order forward mode Taylor coefficient for result of op = SqrtOp.
  
  The C++ source code corresponding to this operation is
  \verbatim
  	z = sqrt(x)
  \endverbatim
  
  \copydetails CppAD::local::forward_unary1_op_0
  */
  template <class Base>
  inline void forward_sqrt_op_0(
  	size_t i_z         ,
  	size_t i_x         ,
  	size_t cap_order   ,
  	Base*  taylor      )
  {
  	// check assumptions
  	CPPAD_ASSERT_UNKNOWN( NumArg(SqrtOp) == 1 );
  	CPPAD_ASSERT_UNKNOWN( NumRes(SqrtOp) == 1 );
  	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;
  
  	z[0] = sqrt( x[0] );
  }
  /*!
  Compute reverse mode partial derivatives for result of op = SqrtOp.
  
  The C++ source code corresponding to this operation is
  \verbatim
  	z = sqrt(x)
  \endverbatim
  
  \copydetails CppAD::local::reverse_unary1_op
  */
  
  template <class Base>
  #if defined(__GNUC__) && __GNUC__ * 100 + __GNUC_MINOR__ * 10 >= 490 && !defined(__INTEL_COMPILER)
  __attribute__((no_sanitize_undefined))
  #endif
  inline void reverse_sqrt_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(SqrtOp) == 1 );
  	CPPAD_ASSERT_UNKNOWN( NumRes(SqrtOp) == 1 );
  	CPPAD_ASSERT_UNKNOWN( d < cap_order );
  	CPPAD_ASSERT_UNKNOWN( d < nc_partial );
  
  	// Taylor coefficients and partials corresponding to argument
  	Base* px       = partial + i_x * nc_partial;
  
  	// Taylor coefficients and partials corresponding to result
  	const Base* z  = taylor  + i_z * cap_order;
  	Base* pz       = partial + i_z * nc_partial;
  
  
  	Base inv_z0 = Base(1.0) / z[0];
  
  	// number of indices to access
  	size_t j = d;
  	size_t k;
  	while(j)
  	{
  
  		// scale partial w.r.t. z[j]
  		pz[j]    = azmul(pz[j], inv_z0);
  
  		pz[0]   -= azmul(pz[j], z[j]);
  		px[j]   += pz[j] / Base(2.0);
  		for(k = 1; k < j; k++)
  			pz[k]   -= azmul(pz[j], z[j-k]);
  		--j;
  	}
  	px[0] += azmul(pz[0], inv_z0) / Base(2.0);
  }
  
  } } // END_CPPAD_LOCAL_NAMESPACE
  # endif
  