private static double JacobiSn_ViaRangeReduction(double u, double k, out long u0, out double u1)
{
Debug.Assert(u >= 0.0);
double m = k * k;
// First we compare u to a quick-to-compute lower bound for K / 2.
// If it's below the bound, we can move directly to series for sn without having
// to compute K or perform range reduction.
double K = LowerBoundK(m);
if (u <= K / 2.0)
{
u0 = 0;
u1 = u;
}
else
{
// Too bad, we need to actually compute K and range reduce.
K = AdvancedMath.EllipticK(k);
double v = u / K;
double v0 = Math.Round(v);
if (v < Int64.MaxValue)
{
u0 = (long)v0;
u1 = (v - v0) * K;
}
else
{
u0 = Int64.MaxValue;
u1 = 0.0;
}
}
Debug.Assert(Math.Abs(u1) <= K / 2.0);
// Note that for u >> K, this has the same problem as naïve trig function calculations
// for x >> 2 \pi: we loose a lot of accuracy for u1 because it is computed via subtraction.
// There is not much we can do about this, though, short of moving to arbitrary-precision
// arithmetic, because K is different for each value of m.
// Compute sn of the reduced -K/2 < u1 < K/2
return(JacobiSn_ReduceToSeries(u1, m));
// We should be able to do even better: |u1| <= K / 4. Our first attempt had some problems,
// and the series still work for the larger range, so leave it as is for now.
}
