# include <cmath>
# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <ctime>

using namespace std;

# include "cordic.hpp"

//****************************************************************************80

double angle_shift ( double alpha, double beta )

//****************************************************************************80
//
//  Purpose:
//
//    ANGLE_SHIFT shifts angle ALPHA to lie between BETA and BETA+2PI.
//
//  Discussion:
//
//    The input angle ALPHA is shifted by multiples of 2 * PI to lie
//    between BETA and BETA+2*PI.
//
//    The resulting angle GAMMA has all the same trigonometric function
//    values as ALPHA.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    14 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double ALPHA, the angle to be shifted.
//
//    Input, double BETA, defines the lower endpoint of
//    the angle range.
//
//    Output, double ANGLE_SHIFT, the shifted angle.
//
{
  double gamma;
  double pi = 3.141592653589793;

  if ( alpha < beta )
  {
    gamma = beta - fmod ( beta - alpha, 2.0 * pi ) + 2.0 * pi;
  }
  else
  {
    gamma = beta + fmod ( alpha - beta, 2.0 * pi );
  }

  return gamma;
}
//****************************************************************************80

double arccos_cordic ( double t, int n )

//****************************************************************************80
//
//  Purpose:
//
//    ARCCOS_CORDIC returns the arccosine of an angle using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jean-Michel Muller,
//    Elementary Functions: Algorithms and Implementation,
//    Second Edition,
//    Birkhaeuser, 2006,
//    ISBN13: 978-0-8176-4372-0,
//    LC: QA331.M866.
//
//  Parameters:
//
//    Input, double T, the cosine of an angle.  -1 <= T <= 1.
//
//    Input, int N, the number of iterations to take.
//    A value of 10 is low.  Good accuracy is achieved with 20 or more
//    iterations.
//
//    Output, double ARCCOS_CORDIC, an angle whose cosine is T.
//
//  Local Parameters:
//
//    Local, double ANGLES(60) = arctan ( (1/2)^(0:59) );
//
{
# define ANGLES_LENGTH 60

  double angle;
  double angles[ANGLES_LENGTH] = {
    7.8539816339744830962E-01, 
    4.6364760900080611621E-01, 
    2.4497866312686415417E-01, 
    1.2435499454676143503E-01, 
    6.2418809995957348474E-02, 
    3.1239833430268276254E-02, 
    1.5623728620476830803E-02, 
    7.8123410601011112965E-03, 
    3.9062301319669718276E-03, 
    1.9531225164788186851E-03, 
    9.7656218955931943040E-04, 
    4.8828121119489827547E-04, 
    2.4414062014936176402E-04, 
    1.2207031189367020424E-04, 
    6.1035156174208775022E-05, 
    3.0517578115526096862E-05, 
    1.5258789061315762107E-05, 
    7.6293945311019702634E-06, 
    3.8146972656064962829E-06, 
    1.9073486328101870354E-06, 
    9.5367431640596087942E-07, 
    4.7683715820308885993E-07, 
    2.3841857910155798249E-07, 
    1.1920928955078068531E-07, 
    5.9604644775390554414E-08, 
    2.9802322387695303677E-08, 
    1.4901161193847655147E-08, 
    7.4505805969238279871E-09, 
    3.7252902984619140453E-09, 
    1.8626451492309570291E-09, 
    9.3132257461547851536E-10, 
    4.6566128730773925778E-10, 
    2.3283064365386962890E-10, 
    1.1641532182693481445E-10, 
    5.8207660913467407226E-11, 
    2.9103830456733703613E-11, 
    1.4551915228366851807E-11, 
    7.2759576141834259033E-12, 
    3.6379788070917129517E-12, 
    1.8189894035458564758E-12, 
    9.0949470177292823792E-13, 
    4.5474735088646411896E-13, 
    2.2737367544323205948E-13, 
    1.1368683772161602974E-13, 
    5.6843418860808014870E-14, 
    2.8421709430404007435E-14, 
    1.4210854715202003717E-14, 
    7.1054273576010018587E-15, 
    3.5527136788005009294E-15, 
    1.7763568394002504647E-15, 
    8.8817841970012523234E-16, 
    4.4408920985006261617E-16, 
    2.2204460492503130808E-16, 
    1.1102230246251565404E-16, 
    5.5511151231257827021E-17, 
    2.7755575615628913511E-17, 
    1.3877787807814456755E-17, 
    6.9388939039072283776E-18, 
    3.4694469519536141888E-18, 
    1.7347234759768070944E-18 };
  int i;
  int j;
  double poweroftwo;
  double sigma;
  double sign_z2;
  double theta;
  double x1;
  double x2;
  double y1;
  double y2;

  if ( 1.0 < fabs ( t ) )
  {
    cerr << "\n";
    cerr << "ARCCOS_CORDIC - Fatal error!\n";
    cerr << "  1.0 < |T|.\n";
    exit ( 1 );
  }

  theta = 0.0;
  x1 = 1.0;
  y1 = 0.0;
  poweroftwo = 1.0;

  for ( j = 1; j <= n; j++ )
  {
    if ( y1 < 0.0 )
    {
      sign_z2 = -1.0;
    }
    else
    {
      sign_z2 = +1.0;
    }

    if ( t <= x1 )
    {
      sigma = + sign_z2;
    }
    else
    {
      sigma = - sign_z2;
    }

    if ( j <= ANGLES_LENGTH )
    {
      angle = angles[j-1];
    }
    else
    {
      angle = angle / 2.0;
    }

    for ( i = 1; i <= 2; i++ )
    {
      x2 =                      x1 - sigma * poweroftwo * y1;
      y2 = sigma * poweroftwo * x1 +                      y1;

      x1 = x2;
      y1 = y2;
    }

    theta  = theta + 2.0 * sigma * angle;

    t = t + t * poweroftwo * poweroftwo;

    poweroftwo = poweroftwo / 2.0;
  }

  return theta;
# undef ANGLES_LENGTH
}
//****************************************************************************80

void arccos_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    ARCCOS_VALUES returns some values of the arc cosine function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      ArcCos[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 12

  static double fx_vec[N_MAX] = {
    1.6709637479564564156,
    1.5707963267948966192,
    1.4706289056333368229,
    1.3694384060045658278,
    1.2661036727794991113,
    1.1592794807274085998,
    1.0471975511965977462,
    0.92729521800161223243,
    0.79539883018414355549,
    0.64350110879328438680,
    0.45102681179626243254,
    0.00000000000000000000 };

  static double x_vec[N_MAX] = {
    -0.1,
     0.0,
     0.1,
     0.2,
     0.3,
     0.4,
     0.5,
     0.6,
     0.7,
     0.8,
     0.9,
     1.0 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double arcsin_cordic ( double t, int n )

//****************************************************************************80
//
//  Purpose:
//
//    ARCSIN_CORDIC returns the arcsine of an angle using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jean-Michel Muller,
//    Elementary Functions: Algorithms and Implementation,
//    Second Edition,
//    Birkhaeuser, 2006,
//    ISBN13: 978-0-8176-4372-0,
//    LC: QA331.M866.
//
//  Parameters:
//
//    Input, double T, the sine of an angle.  -1 <= T <= 1.
//
//    Input, int N, the number of iterations to take.
//    A value of 10 is low.  Good accuracy is achieved with 20 or more
//    iterations.
//
//    Output, double ARCSIN_CORDIC, an angle whose sine is T.
//
//  Local Parameters:
//
//    Local, double ANGLES(60) = arctan ( (1/2)^(0:59) );
//
{
# define ANGLES_LENGTH 60

  double angle;
  double angles[ANGLES_LENGTH] = {
    7.8539816339744830962E-01, 
    4.6364760900080611621E-01, 
    2.4497866312686415417E-01, 
    1.2435499454676143503E-01, 
    6.2418809995957348474E-02, 
    3.1239833430268276254E-02, 
    1.5623728620476830803E-02, 
    7.8123410601011112965E-03, 
    3.9062301319669718276E-03, 
    1.9531225164788186851E-03, 
    9.7656218955931943040E-04, 
    4.8828121119489827547E-04, 
    2.4414062014936176402E-04, 
    1.2207031189367020424E-04, 
    6.1035156174208775022E-05, 
    3.0517578115526096862E-05, 
    1.5258789061315762107E-05, 
    7.6293945311019702634E-06, 
    3.8146972656064962829E-06, 
    1.9073486328101870354E-06, 
    9.5367431640596087942E-07, 
    4.7683715820308885993E-07, 
    2.3841857910155798249E-07, 
    1.1920928955078068531E-07, 
    5.9604644775390554414E-08, 
    2.9802322387695303677E-08, 
    1.4901161193847655147E-08, 
    7.4505805969238279871E-09, 
    3.7252902984619140453E-09, 
    1.8626451492309570291E-09, 
    9.3132257461547851536E-10, 
    4.6566128730773925778E-10, 
    2.3283064365386962890E-10, 
    1.1641532182693481445E-10, 
    5.8207660913467407226E-11, 
    2.9103830456733703613E-11, 
    1.4551915228366851807E-11, 
    7.2759576141834259033E-12, 
    3.6379788070917129517E-12, 
    1.8189894035458564758E-12, 
    9.0949470177292823792E-13, 
    4.5474735088646411896E-13, 
    2.2737367544323205948E-13, 
    1.1368683772161602974E-13, 
    5.6843418860808014870E-14, 
    2.8421709430404007435E-14, 
    1.4210854715202003717E-14, 
    7.1054273576010018587E-15, 
    3.5527136788005009294E-15, 
    1.7763568394002504647E-15, 
    8.8817841970012523234E-16, 
    4.4408920985006261617E-16, 
    2.2204460492503130808E-16, 
    1.1102230246251565404E-16, 
    5.5511151231257827021E-17, 
    2.7755575615628913511E-17, 
    1.3877787807814456755E-17, 
    6.9388939039072283776E-18, 
    3.4694469519536141888E-18, 
    1.7347234759768070944E-18 };
  int i;
  int j;
  double poweroftwo;
  double sigma;
  double sign_z1;
  double theta;
  double x1;
  double x2;
  double y1;
  double y2;

  if ( 1.0 < fabs ( t ) )
  {
    cerr << "\n";
    cerr << "ARCSIN_CORDIC - Fatal error!\n";
    cerr << "  1.0 < |T|.\n";
    exit ( 1 );
  }

  theta = 0.0;
  x1 = 1.0;
  y1 = 0.0;
  poweroftwo = 1.0;

  for ( j = 1; j <= n; j++ )
  {
    if ( x1 < 0.0 )
    {
      sign_z1 = -1.0;
    }
    else
    {
      sign_z1 = +1.0;
    }

    if ( y1 <= t )
    {
      sigma = + sign_z1;
    }
    else
    {
      sigma = - sign_z1;
    }

    if ( j <= ANGLES_LENGTH )
    {
      angle = angles[j-1];
    }
    else
    {
      angle = angle / 2.0;
    }

    for ( i = 1; i <= 2; i++ )
    {
      x2 =                      x1 - sigma * poweroftwo * y1;
      y2 = sigma * poweroftwo * x1 +                      y1;

      x1 = x2;
      y1 = y2;
    }

    theta  = theta + 2.0 * sigma * angle;

    t = t + t * poweroftwo * poweroftwo;

    poweroftwo = poweroftwo / 2.0;
  }

  return theta;
# undef ANGLES_LENGTH
}
//****************************************************************************80

void arcsin_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    ARCSIN_VALUES returns some values of the arc sine function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      ArcSin[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 12

  static double fx_vec[N_MAX] = {
    -0.10016742116155979635,
     0.00000000000000000000,
     0.10016742116155979635,
     0.20135792079033079146,
     0.30469265401539750797,
     0.41151684606748801938,
     0.52359877559829887308,
     0.64350110879328438680,
     0.77539749661075306374,
     0.92729521800161223243,
     1.1197695149986341867,
     1.5707963267948966192 };

  static double x_vec[N_MAX] = {
    -0.1,
     0.0,
     0.1,
     0.2,
     0.3,
     0.4,
     0.5,
     0.6,
     0.7,
     0.8,
     0.9,
     1.0 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double arctan_cordic ( double x, double y, int n )

//****************************************************************************80
//
//  Purpose:
//
//    ARCTAN_CORDIC returns the arctangent of an angle using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jean-Michel Muller,
//    Elementary Functions: Algorithms and Implementation,
//    Second Edition,
//    Birkhaeuser, 2006,
//    ISBN13: 978-0-8176-4372-0,
//    LC: QA331.M866.
//
//  Parameters:
//
//    Input, double X, Y, define the tangent of an angle as Y/X.
//
//    Input, int N, the number of iterations to take.
//    A value of 10 is low.  Good accuracy is achieved with 20 or more
//    iterations.
//
//    Output, double ARCTAN_CORDIC, the angle whose tangent is Y/X.
//
//  Local Parameters:
//
//    Local, double ANGLES(60) = arctan ( (1/2)^(0:59) );
//
{
# define ANGLES_LENGTH 60

  double angle;
  double angles[ANGLES_LENGTH] = {
    7.8539816339744830962E-01, 
    4.6364760900080611621E-01, 
    2.4497866312686415417E-01, 
    1.2435499454676143503E-01, 
    6.2418809995957348474E-02, 
    3.1239833430268276254E-02, 
    1.5623728620476830803E-02, 
    7.8123410601011112965E-03, 
    3.9062301319669718276E-03, 
    1.9531225164788186851E-03, 
    9.7656218955931943040E-04, 
    4.8828121119489827547E-04, 
    2.4414062014936176402E-04, 
    1.2207031189367020424E-04, 
    6.1035156174208775022E-05, 
    3.0517578115526096862E-05, 
    1.5258789061315762107E-05, 
    7.6293945311019702634E-06, 
    3.8146972656064962829E-06, 
    1.9073486328101870354E-06, 
    9.5367431640596087942E-07, 
    4.7683715820308885993E-07, 
    2.3841857910155798249E-07, 
    1.1920928955078068531E-07, 
    5.9604644775390554414E-08, 
    2.9802322387695303677E-08, 
    1.4901161193847655147E-08, 
    7.4505805969238279871E-09, 
    3.7252902984619140453E-09, 
    1.8626451492309570291E-09, 
    9.3132257461547851536E-10, 
    4.6566128730773925778E-10, 
    2.3283064365386962890E-10, 
    1.1641532182693481445E-10, 
    5.8207660913467407226E-11, 
    2.9103830456733703613E-11, 
    1.4551915228366851807E-11, 
    7.2759576141834259033E-12, 
    3.6379788070917129517E-12, 
    1.8189894035458564758E-12, 
    9.0949470177292823792E-13, 
    4.5474735088646411896E-13, 
    2.2737367544323205948E-13, 
    1.1368683772161602974E-13, 
    5.6843418860808014870E-14, 
    2.8421709430404007435E-14, 
    1.4210854715202003717E-14, 
    7.1054273576010018587E-15, 
    3.5527136788005009294E-15, 
    1.7763568394002504647E-15, 
    8.8817841970012523234E-16, 
    4.4408920985006261617E-16, 
    2.2204460492503130808E-16, 
    1.1102230246251565404E-16, 
    5.5511151231257827021E-17, 
    2.7755575615628913511E-17, 
    1.3877787807814456755E-17, 
    6.9388939039072283776E-18, 
    3.4694469519536141888E-18, 
    1.7347234759768070944E-18 };
  int j;
  double poweroftwo;
  double sigma;
  double sign_factor;
  double theta;
  double x1;
  double x2;
  double y1;
  double y2;

  x1 = x;
  y1 = y;
//
//  Account for signs.
//
  if ( x1 < 0.0 && y1 < 0.0 )
  {
    x1 = -x1;
    y1 = -y1;
  }

  if ( x1 < 0.0 )
  {
    x1 = -x1;
    sign_factor = -1.0;
  }
  else if ( y1 < 0.0 )
  {
    y1 = -y1;
    sign_factor = -1.0;
  }
  else
  {
    sign_factor = +1.0;
  }

  theta = 0.0;
  poweroftwo = 1.0;

  for ( j = 1; j <= n; j++ )
  {
    if ( y1 <= 0.0 )
    {
      sigma = +1.0;
    }
    else
    {
      sigma = -1.0;
    }

    if ( j <= ANGLES_LENGTH )
    {
      angle = angles[j-1];
    }
    else
    {
      angle = angle / 2.0;
    }

    x2 =                      x1 - sigma * poweroftwo * y1;
    y2 = sigma * poweroftwo * x1 +                      y1;
    theta  = theta - sigma * angle;

    x1 = x2;
    y1 = y2;

    poweroftwo = poweroftwo / 2.0;
  }

  theta = sign_factor * theta;

  return theta;
# undef ANGLES_LENGTH
}
//****************************************************************************80

void arctan_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    ARCTAN_VALUES returns some values of the arc tangent function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      ArcTan[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 11

  static double f_vec[N_MAX] = {
    0.00000000000000000000,
    0.24497866312686415417,
    0.32175055439664219340,
    0.46364760900080611621,
    0.78539816339744830962,
    1.1071487177940905030,
    1.2490457723982544258,
    1.3258176636680324651,
    1.3734007669450158609,
    1.4711276743037345919,
    1.5208379310729538578 };

  static double x_vec[N_MAX] = {
    0.00000000000000000000,
    0.25000000000000000000,
    0.33333333333333333333,
    0.50000000000000000000,
    1.0000000000000000000,
    2.0000000000000000000,
    3.0000000000000000000,
    4.0000000000000000000,
    5.0000000000000000000,
    10.000000000000000000,
    20.000000000000000000 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x = x_vec[n_data-1];
    fx = f_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double cbrt_cordic ( double x, int n )

//****************************************************************************80
//
//  Purpose:
//
//    CBRT_CORDIC returns the cube root of a value using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    23 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose cube root is desired.
//
//    Input, int N, the number of iterations to take.
//    This is essentially the number of binary digits of accuracy, and
//    might go as high as 53.
//
//    Output, double CBRT_CORDIC, the approximate cube root of X.
//
{
  int i;
  double poweroftwo;
  double x_mag;
  double y;

  x_mag = fabs ( x );

  if ( x_mag == 0.0 )
  {
    y = 0.0;
    return y;
  }

  if ( x_mag == 1.0 )
  {
    y = x;
    return y;
  }

  poweroftwo = 1.0;

  if ( x_mag < 1.0 )
  {
    while ( x_mag <= poweroftwo * poweroftwo * poweroftwo )
    {
      poweroftwo = poweroftwo / 2.0;
    }
    y = poweroftwo;
  }
  else if ( 1.0 < x_mag )
  {
    while ( poweroftwo * poweroftwo * poweroftwo <= x_mag )
    {
      poweroftwo = 2.0 * poweroftwo;
    }
    y = poweroftwo / 2.0;
  }

  for ( i = 1; i <= n; i++ )
  {
    poweroftwo = poweroftwo / 2.0;
    if ( ( y + poweroftwo ) * ( y + poweroftwo )* ( y + poweroftwo ) <= x_mag )
    {
      y = y + poweroftwo;
    }
  }

  if ( x < 0.0 )
  {
    y = - y;
  }

  return y;
}
//****************************************************************************80

void cbrt_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    CBRT_VALUES returns some values of the cube root function.
//
//  Discussion:
//
//    CBRT(X) = real number Y such that Y * Y * Y = X.
//
//    In Mathematica, the function can be evaluated by:
//
//      Sign[x] * ( Abs[x] )^(1/3)
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    22 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output double FX, the value of the function.
//
{
# define N_MAX 14

  static double fx_vec[N_MAX] = {
    0.0000000000000000E+00,
   -0.0020082988563383484484E+00,
    0.44814047465571647087E+00,
   -0.46415888336127788924E+00,
    0.73680629972807732116E+00,
   -1.0000000000000000000E+00,
    1.2599210498948731648E+00,
   -1.4422495703074083823E+00,
    1.4645918875615232630E+00,
   -2.6684016487219448673E+00,
    3.0723168256858472933E+00,
   -4.1408177494228532500E+00,
    4.5947008922070398061E+00,
   -497.93385921817447440E+00 };

  static double x_vec[N_MAX] = {
     0.0000000000000000E+00,
    -0.8100000073710001E-08,
     0.9000000000000000E-01,
    -0.1000000000000000E+00,
     0.4000000000000000E+00,
    -0.1000000000000000E+01,
     0.2000000000000000E+01,
    -0.3000000000000000E+01,
     0.3141592653589793E+01,
    -0.1900000000000000E+02,
     0.2900000000000000E+02,
    -0.7100000000000000E+02,
     0.9700000000000000E+02,
    -0.1234567890000000E+09 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

void cos_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    COS_VALUES returns some values of the cosine function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      Cos[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 13

  static double fx_vec[N_MAX] = {
     1.0000000000000000000,
     0.96592582628906828675,
     0.87758256189037271612,
     0.86602540378443864676,
     0.70710678118654752440,
     0.54030230586813971740,
     0.50000000000000000000,
     0.00000000000000000000,
    -0.41614683654714238700,
    -0.98999249660044545727,
    -1.0000000000000000000,
    -0.65364362086361191464,
     0.28366218546322626447 };

  static double x_vec[N_MAX] = {
    0.0000000000000000000,
    0.26179938779914943654,
    0.50000000000000000000,
    0.52359877559829887308,
    0.78539816339744830962,
    1.0000000000000000000,
    1.0471975511965977462,
    1.5707963267948966192,
    2.0000000000000000000,
    3.0000000000000000000,
    3.1415926535897932385,
    4.0000000000000000000,
    5.0000000000000000000 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

void cossin_cordic ( double beta, int n, double &c, double &s )

//****************************************************************************80
//
//  Purpose:
//
//    COSSIN_CORDIC returns the sine and cosine of an angle by the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    14 June 2007
//
//  Author:
//
//    Based on MATLAB code in a Wikipedia article.
//    C++ version by John Burkardt
//
//  Parameters:
//
//    Input, double BETA, the angle (in radians).
//
//    Input, int N, the number of iterations to take.
//    A value of 10 is low.  Good accuracy is achieved with 20 or more
//    iterations.
//
//    Output, double &C, &S, the cosine and sine of the angle.
//
//  Local Parameters:
//
//    Local, double ANGLES(60) = arctan ( (1/2)^(0:59) );
//
//    Local, double KPROD(33).  KPROD(j) = product ( 0 <= i <= j ) K(i)
//    where K(i) = 1 / sqrt ( 1 + (1/2)^(2i) ).
//
{
# define ANGLES_LENGTH 60
# define KPROD_LENGTH 33

  double angle;
  double angles[ANGLES_LENGTH] = {
    7.8539816339744830962E-01, 
    4.6364760900080611621E-01, 
    2.4497866312686415417E-01, 
    1.2435499454676143503E-01, 
    6.2418809995957348474E-02, 
    3.1239833430268276254E-02, 
    1.5623728620476830803E-02, 
    7.8123410601011112965E-03, 
    3.9062301319669718276E-03, 
    1.9531225164788186851E-03, 
    9.7656218955931943040E-04, 
    4.8828121119489827547E-04, 
    2.4414062014936176402E-04, 
    1.2207031189367020424E-04, 
    6.1035156174208775022E-05, 
    3.0517578115526096862E-05, 
    1.5258789061315762107E-05, 
    7.6293945311019702634E-06, 
    3.8146972656064962829E-06, 
    1.9073486328101870354E-06, 
    9.5367431640596087942E-07, 
    4.7683715820308885993E-07, 
    2.3841857910155798249E-07, 
    1.1920928955078068531E-07, 
    5.9604644775390554414E-08, 
    2.9802322387695303677E-08, 
    1.4901161193847655147E-08, 
    7.4505805969238279871E-09, 
    3.7252902984619140453E-09, 
    1.8626451492309570291E-09, 
    9.3132257461547851536E-10, 
    4.6566128730773925778E-10, 
    2.3283064365386962890E-10, 
    1.1641532182693481445E-10, 
    5.8207660913467407226E-11, 
    2.9103830456733703613E-11, 
    1.4551915228366851807E-11, 
    7.2759576141834259033E-12, 
    3.6379788070917129517E-12, 
    1.8189894035458564758E-12, 
    9.0949470177292823792E-13, 
    4.5474735088646411896E-13, 
    2.2737367544323205948E-13, 
    1.1368683772161602974E-13, 
    5.6843418860808014870E-14, 
    2.8421709430404007435E-14, 
    1.4210854715202003717E-14, 
    7.1054273576010018587E-15, 
    3.5527136788005009294E-15, 
    1.7763568394002504647E-15, 
    8.8817841970012523234E-16, 
    4.4408920985006261617E-16, 
    2.2204460492503130808E-16, 
    1.1102230246251565404E-16, 
    5.5511151231257827021E-17, 
    2.7755575615628913511E-17, 
    1.3877787807814456755E-17, 
    6.9388939039072283776E-18, 
    3.4694469519536141888E-18, 
    1.7347234759768070944E-18 };
  double c2;
  double factor;
  int j;
  double kprod[KPROD_LENGTH] = {
    0.70710678118654752440, 
    0.63245553203367586640, 
    0.61357199107789634961, 
    0.60883391251775242102, 
    0.60764825625616820093, 
    0.60735177014129595905, 
    0.60727764409352599905, 
    0.60725911229889273006, 
    0.60725447933256232972, 
    0.60725332108987516334, 
    0.60725303152913433540, 
    0.60725295913894481363, 
    0.60725294104139716351, 
    0.60725293651701023413, 
    0.60725293538591350073, 
    0.60725293510313931731, 
    0.60725293503244577146, 
    0.60725293501477238499, 
    0.60725293501035403837, 
    0.60725293500924945172, 
    0.60725293500897330506, 
    0.60725293500890426839, 
    0.60725293500888700922, 
    0.60725293500888269443, 
    0.60725293500888161574, 
    0.60725293500888134606, 
    0.60725293500888127864, 
    0.60725293500888126179, 
    0.60725293500888125757, 
    0.60725293500888125652, 
    0.60725293500888125626, 
    0.60725293500888125619, 
    0.60725293500888125617 };
  double pi = 3.141592653589793;
  double poweroftwo;
  double s2;
  double sigma;
  double sign_factor;
  double theta;
//
//  Shift angle to interval [-pi,pi].
//
  theta = angle_shift ( beta, -pi );
//
//  Shift angle to interval [-pi/2,pi/2] and account for signs.
//
  if ( theta < - 0.5 * pi )
  {
    theta = theta + pi;
    sign_factor = -1.0;
  }
  else if ( 0.5 * pi < theta )
  {
    theta = theta - pi;
    sign_factor = -1.0;
  }
  else
  {
    sign_factor = +1.0;
  }
//
//  Initialize loop variables:
//
  c = 1.0;
  s = 0.0;

  poweroftwo = 1.0;
  angle = angles[0];
//
//  Iterations
//
  for ( j = 1; j <= n; j++ )
  {
    if ( theta < 0.0 )
    {
      sigma = -1.0;
    }
    else
    {
      sigma = 1.0;
    }

    factor = sigma * poweroftwo;

    c2 =          c - factor * s;
    s2 = factor * c +          s;

    c = c2;
    s = s2;
//
//  Update the remaining angle.
//
    theta = theta - sigma * angle;

    poweroftwo = poweroftwo / 2.0;
//
//  Update the angle from table, or eventually by just dividing by two.
//
    if ( ANGLES_LENGTH < j + 1 )
    {
      angle = angle / 2.0;
    }
    else
    {
      angle = angles[j];
    }
  }
//
//  Adjust length of output vector to be [cos(beta), sin(beta)]
//
//  KPROD is essentially constant after a certain point, so if N is
//  large, just take the last available value.
//
  if ( 0 < n )
  {
    c = c * kprod [ i4_min ( n, KPROD_LENGTH ) - 1 ];
    s = s * kprod [ i4_min ( n, KPROD_LENGTH ) - 1 ];
  }
//
//  Adjust for possible sign change because angle was originally
//  not in quadrant 1 or 4.
//
  c = sign_factor * c;
  s = sign_factor * s;

  return;
# undef ANGLES_LENGTH
# undef KPROD_LENGTH
}
//****************************************************************************80

double exp_cordic ( double x, int n )

//****************************************************************************80
//
//  Purpose:
//
//    EXP_CORDIC evaluates the exponential function using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    19 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Frederick Ruckdeschel,
//    BASIC Scientific Subroutines,
//    Volume II,
//    McGraw-Hill, 1980,
//    ISBN: 0-07-054202-3,
//    LC: QA76.95.R82.
//
//  Parameters:
//
//    Input, double X, the argument.
//
//    Input, int N, the number of steps to take.
//
//    Output, double FX, the exponential of X.
//
//  Local Parameters:
//
//    Local, double A(1:25) = exp ( (1/2)^(1:25) );
//
{
# define A_LENGTH 25

  double a[A_LENGTH] = {
    1.648721270700128, 
    1.284025416687742, 
    1.133148453066826, 
    1.064494458917859, 
    1.031743407499103, 
    1.015747708586686, 
    1.007843097206488, 
    1.003913889338348, 
    1.001955033591003, 
    1.000977039492417, 
    1.000488400478694, 
    1.000244170429748, 
    1.000122077763384, 
    1.000061037018933, 
    1.000030518043791, 
    1.0000152589054785, 
    1.0000076294236351, 
    1.0000038147045416, 
    1.0000019073504518, 
    1.0000009536747712, 
    1.0000004768372719, 
    1.0000002384186075, 
    1.0000001192092967, 
    1.0000000596046466, 
    1.0000000298023228 };
  double ai;
  double e = 2.718281828459045;
  double fx;
  int i;
  double poweroftwo;
  double *w;
  int x_int;
  double z;

  x_int = ( int ) ( floor ( x ) );
//
//  Determine the weights.
//
  poweroftwo = 0.5;
  z = x - ( double ) ( x_int );

  w = new double[n];

  for ( i = 0; i < n; i++ )
  {
    w[i] = 0.0;
    if ( poweroftwo < z )
    {
      w[i] = 1.0;;
      z = z - poweroftwo;
    }
    poweroftwo = poweroftwo / 2.0;
  }
//
//  Calculate products.
//
  fx = 1.0;

  for ( i = 0; i < n; i++ )
  {
    if ( i < A_LENGTH )
    {
      ai = a[i];
    }
    else
    {
      ai = 1.0 + ( ai - 1.0 ) / 2.0;
    }

    if ( 0.0 < w[i] )
    {
      fx = fx * ai;
    }
  }
//
//  Perform residual multiplication.
//
  fx = fx                     
    * ( 1.0 + z           
    * ( 1.0 + z / 2.0 
    * ( 1.0 + z / 3.0 
    * ( 1.0 + z / 4.0 ))));
//
//  Account for factor EXP(X_INT).
//
  if ( x_int < 0 )
  {
    for ( i = 1; i <= -x_int; i++ )
    {
      fx = fx / e;
    }
  }
  else
  {
    for ( i = 1; i <= x_int; i++ )
    {
      fx = fx * e;
    }
  }

  delete [] w;

  return fx;
# undef A_LENGTH
}
//****************************************************************************80

void exp_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    EXP_VALUES returns some values of the exponential function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      Exp[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    11 March 2008
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 24

  static double fx_vec[N_MAX] = {
    0.000045399929762484851536E+00,
    0.0067379469990854670966E+00,
    0.36787944117144232160E+00,
    1.0000000000000000000E+00,
    1.0000000100000000500E+00,
    1.0001000050001666708E+00,
    1.0010005001667083417E+00,
    1.0100501670841680575E+00,
    1.1051709180756476248E+00,
    1.2214027581601698339E+00,
    1.3498588075760031040E+00,
    1.4918246976412703178E+00,
    1.6487212707001281468E+00,
    1.8221188003905089749E+00,
    2.0137527074704765216E+00,
    2.2255409284924676046E+00,
    2.4596031111569496638E+00,
    2.7182818284590452354E+00,
    7.3890560989306502272E+00,
    23.140692632779269006E+00,
    148.41315910257660342E+00,
    22026.465794806716517E+00,
    4.8516519540979027797E+08,
    2.3538526683701998541E+17 };

  static double x_vec[N_MAX] = {
     -10.0E+00,
      -5.0E+00,
      -1.0E+00,
       0.0E+00,
       0.00000001E+00,
       0.0001E+00,
       0.001E+00,
       0.01E+00,
       0.1E+00,
       0.2E+00,
       0.3E+00,
       0.4E+00,
       0.5E+00,
       0.6E+00,
       0.7E+00,
       0.8E+00,
       0.9E+00,
       1.0E+00,
       2.0E+00,
       3.1415926535897932385E+00,
       5.0E+00,
      10.0E+00,
      20.0E+00,
      40.0E+00  };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

int i4_huge ( )

//****************************************************************************80
//
//  Purpose:
//
//    I4_HUGE returns a "huge" I4.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 May 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, int I4_HUGE, a "huge" I4.
//
{
  return 2147483647;
}
//****************************************************************************80

int i4_min ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
  int value;

  if ( i1 < i2 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

double ln_cordic ( double x, int n )

//****************************************************************************80
//
//  Purpose:
//
//    LN_CORDIC evaluates the natural logarithm using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    19 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Frederick Ruckdeschel,
//    BASIC Scientific Subroutines,
//    Volume II,
//    McGraw-Hill, 1980,
//    ISBN: 0-07-054202-3,
//    LC: QA76.95.R82.
//
//  Parameters:
//
//    Input, double X, the argument.
//
//    Input, int N, the number of steps to take.
//
//    Output, double FX, the natural logarithm of X.
//
//  Local Parameters:
//
//    Local, double A(1:25) = exp ( (1/2)^(1:25) );
//
{
# define A_LENGTH 25

  double a[A_LENGTH] = {
    1.648721270700128, 
    1.284025416687742, 
    1.133148453066826, 
    1.064494458917859, 
    1.031743407499103, 
    1.015747708586686, 
    1.007843097206488, 
    1.003913889338348, 
    1.001955033591003, 
    1.000977039492417, 
    1.000488400478694, 
    1.000244170429748, 
    1.000122077763384, 
    1.000061037018933, 
    1.000030518043791, 
    1.0000152589054785, 
    1.0000076294236351, 
    1.0000038147045416, 
    1.0000019073504518, 
    1.0000009536747712, 
    1.0000004768372719, 
    1.0000002384186075, 
    1.0000001192092967, 
    1.0000000596046466, 
    1.0000000298023228 };
  double ai;
  double e = 2.718281828459045;
  double fx;
  int i;
  int k;
  double poweroftwo;
  double *w;

  if ( x <= 0.0 )
  {
    cerr << "\n";
    cerr << "LN_CORDIC - Fatal error!\n";
    cerr << "  Input argument X <= 0.0\n";
    exit ( 1 );
  }

  k = 0;

  while ( e <= x )
  {
    k = k + 1;
    x = x / e;
  }

  while ( x < 1.0 )
  {
    k = k - 1;
    x = x * e;
  }
//
//  Determine the weights.
//
  w = new double[n];

  for ( i = 0; i < n; i++ )
  {
    w[i] = 0.0;

    if ( i < A_LENGTH )
    {
      ai = a[i];
    }
    else
    {
      ai = 1.0 + ( ai - 1.0 ) / 2.0;
    }

    if ( ai < x )
    {
      w[i] = 1.0;
      x = x / ai;
    }
  }

  x = x - 1.0;

  x = x
    * ( 1.0 - ( x / 2.0 ) 
    * ( 1.0 + ( x / 3.0 ) 
    * ( 1.0 -   x / 4.0 )));
//
//  Assemble
//
  poweroftwo = 0.5;

  for ( i = 0; i < n; i++ )
  {
    x = x + w[i] * poweroftwo;
    poweroftwo = poweroftwo / 2.0;
  }

  fx = x + ( double ) ( k );

  delete [] w;

  return fx;
# undef A_LENGTH
}
//****************************************************************************80

void ln_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    LN_VALUES returns some values of the natural logarithm function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      Log[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    19 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 20

  static double fx_vec[N_MAX] = {
    -11.512925464970228420E+00,
     -4.6051701859880913680E+00,
     -2.3025850929940456840E+00,
     -1.6094379124341003746E+00,
     -1.2039728043259359926E+00,
     -0.91629073187415506518E+00,
     -0.69314718055994530942E+00,
     -0.51082562376599068321E+00,
     -0.35667494393873237891E+00,
     -0.22314355131420975577E+00,
     -0.10536051565782630123E+00,
      0.00000000000000000000E+00,
      0.69314718055994530942E+00,
      1.0986122886681096914E+00,
      1.1447298858494001741E+00,
      1.6094379124341003746E+00,
      2.3025850929940456840E+00,
      2.9957322735539909934E+00,
      4.6051701859880913680E+00,
      18.631401766168018033E+00 };

  static double x_vec[N_MAX] = {
    1.0E-05,
    1.0E-02,
    0.1E+00,
    0.2E+00,
    0.3E+00,
    0.4E+00,
    0.5E+00,
    0.6E+00,
    0.7E+00,
    0.8E+00,
    0.9E+00,
    1.0E+00,
    2.0E+00,
    3.0E+00,
    3.1415926535897932385E+00,
    5.0E+00,
    10.0E+00,
    20.0E+00,
    100.0E+00,
    123456789.0E+00 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double multiply_cordic ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    MULTIPLY_CORDIC computes Z=X*Y the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    27 June 2018
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jean-Michel Muller,
//    Elementary Functions: Algorithms and Implementation,
//    Second Edition,
//    Birkhaeuser, 2006,
//    ISBN13: 978-0-8176-4372-0,
//    LC: QA331.M866.
//
//  Parameters:
//
//    Input, double X, Y, the factors.
//
//    Output, double Z, the product.
//
{
  double two_power;
  double x_abs;
  double x_log2;
  double x_sign;
  double z;

  z = 0.0;
//
//  Easy result if X or Y is zero.
//
  if ( x == 0.0 )
  {
    return z;
  }

  if ( y == 0.0 )
  {
    return z;
  }
//
//  X = X_SIGN * X_ABS.
//
  if ( x < 0.0 )
  {
    x_sign = -1.0;
    x_abs = - x;
  }
  else
  {
    x_sign = +1.0;
    x_abs = x;
  }
//
//  X = X_SIGN * X_ABS * 2^X_LOG2
//
  x_log2 = 0;
  while ( 2.0 <= x_abs )
  {
    x_abs = x_abs / 2.0;
    x_log2 = x_log2 + 1;
  }

  while ( x_abs < 1.0 )
  {
    x_abs = x_abs * 2.0;
    x_log2 = x_log2 - 1;
  }
//
//  X*Y = X_SIGN * sum ( 0 <= i) X_ABS(i) * 2^(-i) * Y ) * 2^X_LOG2
//  where X_ABS(I) is the I-th binary digit in expansion of X_ABS.
//
  two_power = 1.0;
  while ( 0.0 < x_abs )
  {
    if ( 1.0 <= x_abs )
    {
      x_abs = x_abs - 1.0;
      z = z + y * two_power;
    }
    x_abs = x_abs * 2.0;
    two_power = two_power / 2.0;
  }
//
//  Account for X_SIGN and X_LOG2.
//
  z = z * x_sign;

  while ( 0 < x_log2 )
  {
    z = z * 2.0;
    x_log2 = x_log2 - 1;
  }

  while ( x_log2 < 0 )
  {
    z = z / 2.0;
    x_log2 = x_log2 + 1;
  }
 
  return z;
}
//****************************************************************************80

double r8_uniform_01 ( int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_01 returns a unit pseudorandom R8.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//    If the initial seed is 12345, then the first three computations are
//
//      Input     Output      R8_UNIFORM_01
//      SEED      SEED
//
//         12345   207482415  0.096616
//     207482415  1790989824  0.833995
//    1790989824  2035175616  0.947702
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 April 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input/output, int &SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_01, a new pseudorandom variate, 
//    strictly between 0 and 1.
//
{
  const int i4_huge = 2147483647;
  int k;
  double r;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8_UNIFORM_01 - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  k = seed / 127773;

  seed = 16807 * ( seed - k * 127773 ) - k * 2836;

  if ( seed < 0 )
  {
    seed = seed + i4_huge;
  }
  r = ( double ) ( seed ) * 4.656612875E-10;

  return r;
}
//****************************************************************************80

void sin_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    SIN_VALUES returns some values of the sine function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      Sin[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 13

  static double fx_vec[N_MAX] = {
     0.00000000000000000000,
     0.25881904510252076235,
     0.47942553860420300027,
     0.50000000000000000000,
     0.70710678118654752440,
     0.84147098480789650665,
     0.86602540378443864676,
     1.00000000000000000000,
     0.90929742682568169540,
     0.14112000805986722210,
     0.00000000000000000000,
    -0.75680249530792825137,
    -0.95892427466313846889 };

  static double x_vec[N_MAX] = {
    0.0000000000000000000,
    0.26179938779914943654,
    0.50000000000000000000,
    0.52359877559829887308,
    0.78539816339744830962,
    1.0000000000000000000,
    1.0471975511965977462,
    1.5707963267948966192,
    2.0000000000000000000,
    3.0000000000000000000,
    3.1415926535897932385,
    4.0000000000000000000,
    5.0000000000000000000 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double sqrt_cordic ( double x, int n )

//****************************************************************************80
//
//  Purpose:
//
//    SQRT_CORDIC returns the square root of a value using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    21 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose square root is desired.
//
//    Input, int N, the number of iterations to take.
//    This is essentially the number of binary digits of accuracy, and
//    might go as high as 53.
//
//    Output, double SQRT_CORDIC, the approximate square root of X.
//
{
  int i;
  double poweroftwo;
  double y;

  if ( x < 0.0 )
  {
    cerr << "\n";
    cerr << "SQRT_CORDIC - Fatal error!\n";
    cerr << "  X < 0.\n";
    exit ( 1 );
  }

  if ( x == 0.0 )
  {
    y = 0.0;
    return y;
  }

  if ( x == 1.0 )
  {
    y = 1.0;
    return y;
  }

  poweroftwo = 1.0;

  if ( x < 1.0 )
  {
    while ( x <= poweroftwo * poweroftwo )
    {
      poweroftwo = poweroftwo / 2.0;
    }
    y = poweroftwo;
  }
  else if ( 1.0 < x )
  {
    while ( poweroftwo * poweroftwo <= x )
    {
      poweroftwo = 2.0 * poweroftwo;
    }
    y = poweroftwo / 2.0;
  }

  for ( i = 1; i <= n; i++ )
  {
    poweroftwo = poweroftwo / 2.0;
    if ( ( y + poweroftwo ) * ( y + poweroftwo ) <= x )
    {
      y = y + poweroftwo;
    }
  }

  return y;
}
//****************************************************************************80

void sqrt_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    SQRT_VALUES returns some values of the square root function.
//
//  Discussion:
//
//    SQRT(X) = positive real number Y such that Y * Y = X.
//
//    In Mathematica, the function can be evaluated by:
//
//      Sqrt[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    29 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output double FX, the value of the function.
//
{
# define N_MAX 14

  static double fx_vec[N_MAX] = {
     0.0000000000000000E+00,
     0.9000000040950000E-04,
     0.3000000000000000E+00,
     0.3162277660168379E+00,
     0.6324555320336759E+00,
     0.1000000000000000E+01,
     0.1414213562373095E+01,
     0.1732050807568877E+01,
     0.1772453850905516E+01,
     0.4358898943540674E+01,
     0.5385164807134504E+01,
     0.8426149773176359E+01,
     0.9848857801796105E+01,
     0.1111111106055556E+05 };

  static double x_vec[N_MAX] = {
     0.0000000000000000E+00,
     0.8100000073710001E-08,
     0.9000000000000000E-01,
     0.1000000000000000E+00,
     0.4000000000000000E+00,
     0.1000000000000000E+01,
     0.2000000000000000E+01,
     0.3000000000000000E+01,
     0.3141592653589793E+01,
     0.1900000000000000E+02,
     0.2900000000000000E+02,
     0.7100000000000000E+02,
     0.9700000000000000E+02,
     0.1234567890000000E+09 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

double tan_cordic ( double beta, int n )

//****************************************************************************80
//
//  Purpose:
//
//    TAN_CORDIC returns the tangent of an angle using the CORDIC method.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 June 2007
//
//  Author:
//
//    Based on MATLAB code in a Wikipedia article.
//    C++ version by John Burkardt
//
//  Parameters:
//
//    Input, double BETA, the angle (in radians).
//
//    Input, int N, the number of iterations to take.
//    A value of 10 is low.  Good accuracy is achieved with 20 or more
//    iterations.
//
//    Output, double TAN_CORDIC, the tangent of the angle.
//
//  Local Parameters:
//
//    Local, double ANGLES(60) = arctan ( (1/2)^(0:59) );
//
{
# define ANGLES_LENGTH 60

  double angle;
  double angles[ANGLES_LENGTH] = {
    7.8539816339744830962E-01, 
    4.6364760900080611621E-01, 
    2.4497866312686415417E-01, 
    1.2435499454676143503E-01, 
    6.2418809995957348474E-02, 
    3.1239833430268276254E-02, 
    1.5623728620476830803E-02, 
    7.8123410601011112965E-03, 
    3.9062301319669718276E-03, 
    1.9531225164788186851E-03, 
    9.7656218955931943040E-04, 
    4.8828121119489827547E-04, 
    2.4414062014936176402E-04, 
    1.2207031189367020424E-04, 
    6.1035156174208775022E-05, 
    3.0517578115526096862E-05, 
    1.5258789061315762107E-05, 
    7.6293945311019702634E-06, 
    3.8146972656064962829E-06, 
    1.9073486328101870354E-06, 
    9.5367431640596087942E-07, 
    4.7683715820308885993E-07, 
    2.3841857910155798249E-07, 
    1.1920928955078068531E-07, 
    5.9604644775390554414E-08, 
    2.9802322387695303677E-08, 
    1.4901161193847655147E-08, 
    7.4505805969238279871E-09, 
    3.7252902984619140453E-09, 
    1.8626451492309570291E-09, 
    9.3132257461547851536E-10, 
    4.6566128730773925778E-10, 
    2.3283064365386962890E-10, 
    1.1641532182693481445E-10, 
    5.8207660913467407226E-11, 
    2.9103830456733703613E-11, 
    1.4551915228366851807E-11, 
    7.2759576141834259033E-12, 
    3.6379788070917129517E-12, 
    1.8189894035458564758E-12, 
    9.0949470177292823792E-13, 
    4.5474735088646411896E-13, 
    2.2737367544323205948E-13, 
    1.1368683772161602974E-13, 
    5.6843418860808014870E-14, 
    2.8421709430404007435E-14, 
    1.4210854715202003717E-14, 
    7.1054273576010018587E-15, 
    3.5527136788005009294E-15, 
    1.7763568394002504647E-15, 
    8.8817841970012523234E-16, 
    4.4408920985006261617E-16, 
    2.2204460492503130808E-16, 
    1.1102230246251565404E-16, 
    5.5511151231257827021E-17, 
    2.7755575615628913511E-17, 
    1.3877787807814456755E-17, 
    6.9388939039072283776E-18, 
    3.4694469519536141888E-18, 
    1.7347234759768070944E-18 };
  double c;
  double c2;
  double factor;
  int j;
  double pi = 3.141592653589793;
  double poweroftwo;
  double s;
  double s2;
  double sigma;
  double t;
  double theta;
//
//  Shift angle to interval [-pi,pi].
//
  theta = angle_shift ( beta, -pi );
//
//  Shift angle to interval [-pi/2,pi/2].
//
  if ( theta < - 0.5 * pi )
  {
    theta = theta + pi;
  }
  else if ( 0.5 * pi < theta )
  {
    theta = theta - pi;
  }

  c = 1.0;
  s = 0.0;

  poweroftwo = 1.0;
  angle = angles[0];

  for ( j = 1; j <= n; j++ )
  {
    if ( theta < 0.0 )
    {
      sigma = -1.0;
    }
    else
    {
      sigma = 1.0;
    }

    factor = sigma * poweroftwo;

    c2 =          c - factor * s;
    s2 = factor * c +          s;

    c = c2;
    s = s2;
//
//  Update the remaining angle.
//
    theta = theta - sigma * angle;

    poweroftwo = poweroftwo / 2.0;
//
//  Update the angle from table, or eventually by just dividing by two.
//
    if ( ANGLES_LENGTH < j + 1 )
    {
      angle = angle / 2.0;
    }
    else
    {
      angle = angles[j];
    }
  }

  t = s / c;

  return t;
# undef ANGLES_LENGTH
}
//****************************************************************************80

void tan_values ( int &n_data, double &x, double &fx )

//****************************************************************************80
//
//  Purpose:
//
//    TAN_VALUES returns some values of the tangent function.
//
//  Discussion:
//
//    In Mathematica, the function can be evaluated by:
//
//      Tan[x]
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 June 2007
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Milton Abramowitz, Irene Stegun,
//    Handbook of Mathematical Functions,
//    National Bureau of Standards, 1964,
//    ISBN: 0-486-61272-4,
//    LC: QA47.A34.
//
//    Stephen Wolfram,
//    The Mathematica Book,
//    Fourth Edition,
//    Cambridge University Press, 1999,
//    ISBN: 0-521-64314-7,
//    LC: QA76.95.W65.
//
//  Parameters:
//
//    Input/output, int &N_DATA.  The user sets N_DATA to 0 before the
//    first call.  On each call, the routine increments N_DATA by 1, and
//    returns the corresponding data; when there is no more data, the
//    output value of N_DATA will be 0 again.
//
//    Output, double &X, the argument of the function.
//
//    Output, double &FX, the value of the function.
//
{
# define N_MAX 15

  static double fx_vec[N_MAX] = {
     0.00000000000000000000,
     0.26794919243112270647,
     0.54630248984379051326,
     0.57735026918962576451,
     1.0000000000000000000,
     1.5574077246549022305,
     1.7320508075688772935,
     3.7320508075688772935,
     7.5957541127251504405,
    15.257051688265539110,
    -2.1850398632615189916,
    -0.14254654307427780530,
     0.0000000000000000000,
     1.1578212823495775831,
    -3.3805150062465856370 };

  static double x_vec[N_MAX] = {
    0.00000000000000000000,
    0.26179938779914943654,
    0.50000000000000000000,
    0.52359877559829887308,
    0.78539816339744830962,
    1.0000000000000000000,
    1.0471975511965977462,
    1.3089969389957471827,
    1.4398966328953219010,
    1.5053464798451092601,
    2.0000000000000000000,
    3.0000000000000000000,
    3.1415926535897932385,
    4.0000000000000000000,
    5.0000000000000000000 };

  if ( n_data < 0 )
  {
    n_data = 0;
  }

  n_data = n_data + 1;

  if ( N_MAX < n_data )
  {
    n_data = 0;
    x = 0.0;
    fx = 0.0;
  }
  else
  {
    x  = x_vec[n_data-1];
    fx = fx_vec[n_data-1];
  }

  return;
# undef N_MAX
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    24 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  time_t now;

  now = time ( NULL );
  tm = localtime ( &now );

  strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );

  cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}