A natural logarithm function that uses the arithmetic-geometric mean.

agm_log.c

double
agm_log(double x)
{
	double a, b, e, e0 = DBL_MAX;

	assert(x > 0.0);

	for (a = (x + 1.0) * 0.5, b = sqrt(x); (e = fabs(a - b)) < e0; e0 = e) {
		a = 0.5 * (a + b);
		b = sqrt(a * b);
	}
	return (x - 1.0) / a;
}