#include <cassert>
#include <cmath>
#include <iostream>
#include <numbers>

/***************************************************
 * Cubic Hermite approximation of sine
 *
 * We will approximate sin(x) on the interval
 *
 *     0 <= x <= pi/2
 *
 * with a cubic polynomial.
 *
 * Rather than simply requiring the polynomial to
 * pass through two points, we also require its
 * derivative to agree with the derivative of sine
 * at those points:
 *
 *     p(0)      = 0
 *     p'(0)     = 1
 *
 *     p(pi/2)   = 1
 *     p'(pi/2)  = 0.
 *
 * These are exactly the values and derivatives of
 * sin(x) at the two endpoints.
 *
 * A polynomial constructed by matching both values
 * and derivatives in this manner is called a
 * Hermite interpolating polynomial.
 *
 *
 * DEFAULT ARGUMENT
 *
 * By default, my_sin() assumes that its argument is
 * measured in radians:
 *
 *     my_sin( 0.5 )
 *
 * If the second argument is true, the argument is
 * instead interpreted as degrees:
 *
 *     my_sin( 30.0, true )
 *
 * The default argument must appear in the function
 * declaration so that the compiler knows what value
 * to supply when it encounters a function call with
 * only one argument.
 ***************************************************/

// Function declarations
int main();

double my_sin( double const x,
               bool const using_degrees = false );

// Function definitions
int main() {
    for ( int k{ 0 }; k <= 9; ++k ) {
        std::cout << "sin(" << 10*k << " deg) = "
                  << my_sin( 10.0*k, true )
                  << std::endl;
    }

    std::cout << std::endl;

    // The default value of using_degrees is false,
    // so these arguments are interpreted as radians.
    for ( int k{ 0 }; k <= 15; ++k ) {
        std::cout << "sin(" << 0.1*k << ") = "
                  << my_sin( 0.1*k )
                  << std::endl;
    }

    return 0;
}

/***************************************************
 * my_sin
 *
 * Return a cubic Hermite approximation of sin(x)
 * for
 *
 *     0 <= x <= pi/2.
 *
 * If using_degrees is true, x is interpreted as
 * degrees and converted to radians.
 *
 * Notice that the default argument is not repeated
 * in the function definition. It appeared only in
 * the function declaration above.
 ***************************************************/
double my_sin( double const x,
               bool const using_degrees ) {
    double const pi{ std::numbers::pi };
    double const radians_per_degree{ pi/180.0 };

    double const radians{
        using_degrees ? x*radians_per_degree : x
    };

    assert( (radians >= 0.0)
         && (radians <= pi/2.0) );

    return radians
         + 4.0*radians*radians/(pi*pi)
             *(radians - 4.0*radians/pi - pi + 3.0);
}