public void Add(double y)
{
// Here's Shewchuk's solution...
double u; // hold exact sum as [s, t, u]
// Accumulate starting at least significant end
y = GeoMath.Sum(y, _t, out u);
_s = GeoMath.Sum(y, _s, out _t);
// Start is _s, _t decreasing and non-adjacent. Sum is now (s + t + u)
// exactly with s, t, u non-adjacent and in decreasing order (except for
// possible zeros). The following code tries to normalize the result.
// Ideally, we want _s = round(s+t+u) and _u = round(s+t+u - _s). The
// following does an approximate job (and maintains the decreasing
// non-adjacent property). Here are two "failures" using 3-bit floats:
//
// Case 1: _s is not equal to round(s+t+u) -- off by 1 ulp
// [12, -1] - 8 -> [4, 0, -1] -> [4, -1] = 3 should be [3, 0] = 3
//
// Case 2: _s+_t is not as close to s+t+u as it shold be
// [64, 5] + 4 -> [64, 8, 1] -> [64, 8] = 72 (off by 1)
// should be [80, -7] = 73 (exact)
//
// "Fixing" these problems is probably not worth the expense. The
// representation inevitably leads to small errors in the accumulated
// values. The additional errors illustrated here amount to 1 ulp of the
// less significant word during each addition to the Accumulator and an
// additional possible error of 1 ulp in the reported sum.
//
// Incidentally, the "ideal" representation described above is not
// canonical, because _s = round(_s + _t) may not be true. For example,
// with 3-bit floats:
//
// [128, 16] + 1 -> [160, -16] -- 160 = round(145).
// But [160, 0] - 16 -> [128, 16] -- 128 = round(144).
//
if (_s == 0) // This implies t == 0,
{
_s = u; // so result is u
}
else
{
_t += u; // otherwise just accumulate u to t.
}
}
