#include "library.h"

double power2(const double sofar, const double apower, const int p)
{ if (p == 0)
    return sofar;
  if (p % 2 == 0)
    return power2(sofar, apower * apower, p / 2);
  else
    return power2(sofar * apower, apower * apower, p / 2); }

double power2(const double x, const int p)
{ return power2(1.0, x, p); }


double factorial(const int n)
{ if (n == 0)
    return 1.0;
  return n * factorial(n - 1); }


int plus_or_minus(const int n)
{ return 2 - n % 4; }


double sine2(const double sofar, const double x, const int p, const int maxp)
{ if (p > maxp)
    return sofar;
  return sine2(sofar + plus_or_minus(p) * power2(x, p) / factorial(p), x, p + 2, maxp); }

double sine2(const double x, const double steps)
{ return sine2(0.0, x, 1, steps); }


const int winw = 800, winh = 600;
const double pi = acos(-1.0);
const double maxx = 4 * pi;

void plot(const int steps, const double x)
{ if (x > maxx)
    return;
  const double screenx = x / maxx * winw;
  const double screeny = (1 - sine2(x, steps) / 2) * winh / 2;
  if (x == 0)
    move_to(screenx, screeny);
  else
    draw_to(screenx, screeny);
  plot(steps, x + maxx / winw); }

void plot_real(const double x)
{ if (x > maxx)
    return;
  const double screenx = x / maxx * winw;
  const double screeny = (1 - sin(x) / 2) * winh / 2;
  if (x == 0)
    move_to(screenx, screeny);
  else
    draw_to(screenx, screeny);
  plot_real(x + maxx / winw); }

void show_error(const double x, const int steps)
{ cout << "error at " << x << " = " << fabs(sin(x) - sine2(x, steps)) << "\n"; }

void test(const int steps)
{ set_pen_color(color::red);
  plot(steps, 0);
  cout << "maximum term " << steps << ", " << (steps + 1) / 2 << " computational steps\n";
  show_error(pi / 4, steps);
  show_error(pi / 2, steps);
  show_error(pi, steps);
  show_error(2 * pi, steps);
  cout << "\n";
  wait_for_key_typed();
  set_pen_color(color::blue);
  plot(steps, 0);
  test(steps + 2); }

void main()
{ make_window(winw, winh);
  set_pen_width(2);
  set_pen_color(color::green);
  plot_real(0);
  test(1); }