// Calculating sines by approximating the sum of the infinite series,
// and comparing the quality of the different levels of approximation.

#include "library.h"
#include <iomanip>

     // <iomanip> is needed for access to setprecision()

double factorial(const int n)
{ double f = 1;
  int i = 1;
  while (i <= n)
  { f = f * i;
    i = i + 1; }
  return f; }

double power(const double x, const int n)
{ double p = 1;
  int i = 1;
  while (i <= n)
  { p = p * x;
    i = i + 1; }
  return p; }

double sineterm(const double x, const int n)
{ double r = power(x, n)/factorial(n);
  if (n % 4 == 1)
    return r;
  else
    return - r; }

double sine(const double x, const int last_term)
{ double sum = 0;
  int i = 1;
  while (i <= last_term)
  { sum = sum + sineterm(x, i);
    i = i + 2; }
  return sum; }

void exmain()
{ while (true)
  { const double x = read_double();
    const int n = read_int();
    cout << setprecision(17) << sine(x, n) << "    " 
         << setprecision(17) << sin(x) << "\n"; } }

void plot(const double start, const double end, const int which)
                          // which = 0 means use the standard sin function
                          // any other value says which level of approximation
{ if (end - start < 0.01)
  { const double x0 = start * 100,
                 x1 = end * 100;
    double y0, y1;
    if (which == 0)
    { y0 = sin(start);
      y1 = sin(end); }
    else
    { y0 = sine(start, which);
      y1 = sine(end, which); }
    y0 = y0 * 124 + 250,
    y1 = y1 * 124 + 250;
    move_to(x0, 500 - y0);
    draw_to(x1, 500 - y1); }
  else
  { const double middle = (start+end)/2;
    plot(start, middle, which);
    plot(middle, end, which); } }

void main()
{ make_window(629, 500);
  set_pen_width(3);
  set_pen_color(color::red);
  plot(0.0, 6.29, 0);        // This shows the official library sine function in red
  set_pen_width(2);
  set_pen_color(color::blue);
  int approx = 1;
  while (approx <= 15)       // now the first 8 approximations in blue
  { plot(0.0, 6.29, approx);
    approx = approx + 2; }
  set_pen_width(1);             // and the original in thin red again so we can see
  set_pen_color(color::red);    // where it was
  plot(0.0, 6.29, 0); }