lines 7-454 of file: example/atomic_three/tangent.cpp

{xrst_begin atomic_three_tangent.cpp}
{xrst_spell
   tanh
}

Tan and Tanh as User Atomic Operations: Example and Test
########################################################

Discussion
**********
The code below uses the :ref:`tan_forward-name` and :ref:`tan_reverse-name`
to implement the tangent and hyperbolic tangent
functions as atomic function operations.
It also uses ``AD<float>`` ,
while most atomic examples use ``AD<double>`` .

Start Class Definition
**********************
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{xrst_code cpp} */
# include <cppad/cppad.hpp>
namespace { // Begin empty namespace
using CppAD::vector;
//
class atomic_tangent : public CppAD::atomic_three<float> {
/* {xrst_code}
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Constructor
***********
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{xrst_code cpp} */
private:
   const bool hyperbolic_; // is this hyperbolic tangent
public:
   // constructor
   atomic_tangent(const char* name, bool hyperbolic)
   : CppAD::atomic_three<float>(name),
   hyperbolic_(hyperbolic)
   { }
private:
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for_type
********
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{xrst_code cpp} */
   // calculate type_y
   bool for_type(
      const vector<float>&                parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      vector<CppAD::ad_type_enum>&        type_y      ) override
   {  assert( parameter_x.size() == type_x.size() );
      bool ok = type_x.size() == 1; // n
      ok     &= type_y.size() == 2; // m
      if( ! ok )
         return false;
      type_y[0] = type_x[0];
      type_y[1] = type_x[0];
      return true;
   }
/* {xrst_code}
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forward
*******
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{xrst_code cpp} */
   // forward mode routine called by CppAD
   bool forward(
      const vector<float>&               parameter_x ,
      const vector<CppAD::ad_type_enum>& type_x      ,
      size_t                             need_y      ,
      size_t                             p           ,
      size_t                             q           ,
      const vector<float>&               tx          ,
      vector<float>&                     tzy         ) override
   {  size_t q1 = q + 1;
# ifndef NDEBUG
      size_t n  = tx.size()  / q1;
      size_t m  = tzy.size() / q1;
# endif
      assert( type_x.size() == n );
      assert( n == 1 );
      assert( m == 2 );
      assert( p <= q );
      size_t j, k;

      if( p == 0 )
      {  // z^{(0)} = tan( x^{(0)} ) or tanh( x^{(0)} )
         if( hyperbolic_ )
            tzy[0] = float( tanh( tx[0] ) );
         else
            tzy[0] = float( tan( tx[0] ) );

         // y^{(0)} = z^{(0)} * z^{(0)}
         tzy[q1 + 0] = tzy[0] * tzy[0];

         p++;
      }
      for(j = p; j <= q; j++)
      {  float j_inv = 1.f / float(j);
         if( hyperbolic_ )
            j_inv = - j_inv;

         // z^{(j)} = x^{(j)} +- sum_{k=1}^j k x^{(k)} y^{(j-k)} / j
         tzy[j] = tx[j];
         for(k = 1; k <= j; k++)
            tzy[j] += tx[k] * tzy[q1 + j-k] * float(k) * j_inv;

         // y^{(j)} = sum_{k=0}^j z^{(k)} z^{(j-k)}
         tzy[q1 + j] = 0.;
         for(k = 0; k <= j; k++)
            tzy[q1 + j] += tzy[k] * tzy[j-k];
      }

      // All orders are implemented and there are no possible errors
      return true;
   }
/* {xrst_code}
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reverse
*******
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{xrst_code cpp} */
   // reverse mode routine called by CppAD
   bool reverse(
      const vector<float>&               parameter_x ,
      const vector<CppAD::ad_type_enum>& type_x      ,
      size_t                             q           ,
      const vector<float>&               tx          ,
      const vector<float>&               tzy         ,
      vector<float>&                     px          ,
      const vector<float>&               pzy         ) override
   {  size_t q1 = q + 1;
# ifndef NDEBUG
      size_t n  = tx.size()  / q1;
      size_t m  = tzy.size() / q1;
# endif
      assert( px.size()  == n * q1 );
      assert( pzy.size() == m * q1 );
      assert( n == 1 );
      assert( m == 2 );

      size_t j, k;

      // copy because partials w.r.t. y and z need to change
      vector<float> qzy = pzy;

      // initialize accumultion of reverse mode partials
      for(k = 0; k < q1; k++)
         px[k] = 0.;

      // eliminate positive orders
      for(j = q; j > 0; j--)
      {  float j_inv = 1.f / float(j);
         if( hyperbolic_ )
            j_inv = - j_inv;

         // H_{x^{(k)}} += delta(j-k) +- H_{z^{(j)} y^{(j-k)} * k / j
         px[j] += qzy[j];
         for(k = 1; k <= j; k++)
            px[k] += qzy[j] * tzy[q1 + j-k] * float(k) * j_inv;

         // H_{y^{j-k)} += +- H_{z^{(j)} x^{(k)} * k / j
         for(k = 1; k <= j; k++)
            qzy[q1 + j-k] += qzy[j] * tx[k] * float(k) * j_inv;

         // H_{z^{(k)}} += H_{y^{(j-1)}} * z^{(j-k-1)} * 2.
         for(k = 0; k < j; k++)
            qzy[k] += qzy[q1 + j-1] * tzy[j-k-1] * 2.f;
      }

      // eliminate order zero
      if( hyperbolic_ )
         px[0] += qzy[0] * (1.f - tzy[q1 + 0]);
      else
         px[0] += qzy[0] * (1.f + tzy[q1 + 0]);

      return true;
   }
/* {xrst_code}
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jac_sparsity
************
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{xrst_code cpp} */
   // Jacobian sparsity routine called by CppAD
   bool jac_sparsity(
      const vector<float>&                parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      bool                                dependency  ,
      const vector<bool>&                 select_x    ,
      const vector<bool>&                 select_y    ,
      CppAD::sparse_rc< vector<size_t> >& pattern_out ) override
   {
      size_t n = select_x.size();
      size_t m = select_y.size();
      assert( parameter_x.size() == n );
      assert( n == 1 );
      assert( m == 2 );

      // number of non-zeros in sparsity pattern
      size_t nnz = 0;
      if( select_x[0] )
      {  if( select_y[0] )
            ++nnz;
         if( select_y[1] )
            ++nnz;
      }

      // sparsity pattern
      pattern_out.resize(m, n, nnz);
      size_t k = 0;
      if( select_x[0] )
      {  if( select_y[0] )
            pattern_out.set(k++, 0, 0);
         if( select_y[1] )
            pattern_out.set(k++, 1, 0);
      }
      assert( k == nnz );

      return true;
   }
/* {xrst_code}
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hes_sparsity
************
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{xrst_code cpp} */
   // Hessian sparsity routine called by CppAD
   bool hes_sparsity(
      const vector<float>&                parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      const vector<bool>&                 select_x    ,
      const vector<bool>&                 select_y    ,
      CppAD::sparse_rc< vector<size_t> >& pattern_out ) override
   {
      assert( parameter_x.size() == select_x.size() );
      assert( select_y.size() == 2 );
      size_t n = select_x.size();
      assert( n == 1 );

      // number of non-zeros in sparsity pattern
      size_t nnz = 0;
      if( select_x[0] & (select_y[0] | select_y[1]) )
         nnz = 1;
      // sparsity pattern
      pattern_out.resize(n, n, nnz);
      if( select_x[0] & (select_y[0] | select_y[1]) )
         pattern_out.set(0, 0, 0);

      return true;
   }
/* {xrst_code}
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End Class Definition
********************
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{xrst_code cpp} */
}; // End of atomic_tangent class
}  // End empty namespace

/* {xrst_code}
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Use Atomic Function
*******************
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{xrst_code cpp} */
bool tangent(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   float eps = 10.f * CppAD::numeric_limits<float>::epsilon();
/* {xrst_code}
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Constructor
===========
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{xrst_code cpp} */
   // --------------------------------------------------------------------
   // Creater a tan and tanh object
   atomic_tangent my_tan("my_tan", false), my_tanh("my_tanh", true);
/* {xrst_code}
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Recording
=========
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{xrst_code cpp} */
   // domain space vector
   size_t n  = 1;
   float  x0 = 0.5;
   CppAD::vector< AD<float> > ax(n);
   ax[0]     = x0;

   // declare independent variables and start tape recording
   CppAD::Independent(ax);

   // range space vector
   size_t m = 3;
   CppAD::vector< AD<float> > av(m);

   // temporary vector for computations
   // (my_tan and my_tanh computes tan or tanh and its square)
   CppAD::vector< AD<float> > au(2);

   // call atomic tan function and store tan(x) in f[0], ignore tan(x)^2
   my_tan(ax, au);
   av[0] = au[0];

   // call atomic tanh function and store tanh(x) in f[1], ignore tanh(x)^2
   my_tanh(ax, au);
   av[1] = au[0];

   // put a constant in f[2] = tanh(1.),  for sparsity pattern testing
   CppAD::vector< AD<float> > one(1);
   one[0] = 1.;
   my_tanh(one, au);
   av[2] = au[0];

   // create f: x -> v and stop tape recording
   CppAD::ADFun<float> f;
   f.Dependent(ax, av);
/* {xrst_code}
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forward
=======
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{xrst_code cpp} */
   // check function value
   float tan = std::tan(x0);
   ok &= NearEqual(av[0] , tan,  eps, eps);
   float tanh = std::tanh(x0);
   ok &= NearEqual(av[1] , tanh,  eps, eps);

   // check zero order forward
   CppAD::vector<float> x(n), v(m);
   x[0] = x0;
   v    = f.Forward(0, x);
   ok &= NearEqual(v[0] , tan,  eps, eps);
   ok &= NearEqual(v[1] , tanh,  eps, eps);

   // tan'(x)   = 1 + tan(x)  * tan(x)
   // tanh'(x)  = 1 - tanh(x) * tanh(x)
   float tanp  = 1.f + tan * tan;
   float tanhp = 1.f - tanh * tanh;

   // compute first partial of f w.r.t. x[0] using forward mode
   CppAD::vector<float> dx(n), dv(m);
   dx[0] = 1.;
   dv    = f.Forward(1, dx);
   ok   &= NearEqual(dv[0] , tanp,   eps, eps);
   ok   &= NearEqual(dv[1] , tanhp,  eps, eps);
   ok   &= NearEqual(dv[2] , 0.f,    eps, eps);

   // tan''(x)   = 2 *  tan(x) * tan'(x)
   // tanh''(x)  = - 2 * tanh(x) * tanh'(x)
   // Note that second order Taylor coefficient for u half the
   // corresponding second derivative.
   float two    = 2;
   float tanpp  =   two * tan * tanp;
   float tanhpp = - two * tanh * tanhp;

   // compute second partial of f w.r.t. x[0] using forward mode
   CppAD::vector<float> ddx(n), ddv(m);
   ddx[0] = 0.;
   ddv    = f.Forward(2, ddx);
   ok   &= NearEqual(two * ddv[0], tanpp, eps, eps);
   ok   &= NearEqual(two * ddv[1], tanhpp, eps, eps);
   ok   &= NearEqual(two * ddv[2], 0.f, eps, eps);

/* {xrst_code}
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reverse
=======
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{xrst_code cpp} */
   // compute derivative of tan - tanh using reverse mode
   CppAD::vector<float> w(m), dw(n);
   w[0]  = 1.;
   w[1]  = 1.;
   w[2]  = 0.;
   dw    = f.Reverse(1, w);
   ok   &= NearEqual(dw[0], w[0]*tanp + w[1]*tanhp, eps, eps);

   // compute second derivative of tan - tanh using reverse mode
   CppAD::vector<float> ddw(2);
   ddw   = f.Reverse(2, w);
   ok   &= NearEqual(ddw[0], w[0]*tanp  + w[1]*tanhp , eps, eps);
   ok   &= NearEqual(ddw[1], w[0]*tanpp + w[1]*tanhpp, eps, eps);
/* {xrst_code}
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for_jac_sparsity
================
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{xrst_code cpp} */
   // forward mode Jacobian sparstiy pattern
   CppAD::sparse_rc< CPPAD_TESTVECTOR(size_t) > pattern_in, pattern_out;
   pattern_in.resize(1, 1, 1);
   pattern_in.set(0, 0, 0);
   bool transpose     = false;
   bool dependency    = false;
   bool internal_bool = false;
   f.for_jac_sparsity(
      pattern_in, transpose, dependency, internal_bool, pattern_out
   );
   // (0, 0) and (1, 0) are in sparsity pattern
   ok &= pattern_out.nnz() == 2;
   ok &= pattern_out.row()[0] == 0;
   ok &= pattern_out.col()[0] == 0;
   ok &= pattern_out.row()[1] == 1;
   ok &= pattern_out.col()[1] == 0;
/* {xrst_code}
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rev_sparse_hes
==============
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{xrst_code cpp} */
   // Hesian sparsity (using previous for_jac_sparsity call)
   CPPAD_TESTVECTOR(bool) select_y(m);
   select_y[0] = true;
   select_y[1] = false;
   select_y[2] = false;
   f.rev_hes_sparsity(
      select_y, transpose, internal_bool, pattern_out
   );
   ok &= pattern_out.nnz() == 1;
   ok &= pattern_out.row()[0] == 0;
   ok &= pattern_out.col()[0] == 0;
/* {xrst_code}
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Large x Values
==============
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{xrst_code cpp} */
   // check tanh results for a large value of x
   x[0]  = std::numeric_limits<float>::max() / two;
   v     = f.Forward(0, x);
   tanh  = 1.;
   ok   &= NearEqual(v[1], tanh, eps, eps);
   dv    = f.Forward(1, dx);
   tanhp = 0.;
   ok   &= NearEqual(dv[1], tanhp, eps, eps);

   return ok;
}
/* {xrst_code}
{xrst_spell_on}

{xrst_end atomic_three_tangent.cpp}