// Complete Elliptic Integral of the First Kind
//
// ellint_1 - K(m)
//
// K(m) = ∫₀^(π/2) dθ / √(1 - m*sin²(θ)), computed via AGM
//
// Inputs:    1 (parameter m)
// Outputs:   1 (K(m))
// Constants: 0

#include <math.h>
#include "ts.h"


ts_parameter_t const ts_ellint_1_p = { .n_inputs = 1, .n_outputs = 1, .n_constants = 0, .params_size = 0 };

uint_t ts_ellint_1_offset( ts_constant_t const *c )
{
	( void )c;
	return ( 0 );
}

int ts_ellint_1(
	fp_t * const *y, uint_t y_n,
	fp_t const * const *x, uint_t x_n,
	uint_t n, ts_constant_t const *c
)
{
	TS_PROLOGUE( ts_ellint_1_p );
	(void)c;
    /* Complete elliptic integral of the first kind K(m) */
    for (uint_t i = 0; i < n; ++i)
    {
        fp_t m = x[0][i];
        if (m <= 0.0)
        {
            y[0][i] = PI / 2.0;
        }
        else if (m >= 1.0)
        {
            y[0][i] = INFINITY;
        }
        else
        {
            /* Arithmetic-geometric mean method */
            fp_t a = 1.0;
            fp_t b = sqrt(1.0 - m);
            fp_t cn = (a - b) / 2.0;
            for (uint_t j = 0; j < 30; ++j)
            {
                fp_t a_new = (a + b) / 2.0;
                fp_t b_new = sqrt(a * b);
                fp_t cn_new = (a_new - b_new) / 2.0;
                a = a_new;
                b = b_new;
                cn = cn_new;
                if (cn / a < 1e-15) break;
            }
            y[0][i] = PI / (2.0 * a);
        }
    }
	return 0;
}