private BigInteger EvaluateInternal(BigInteger n, BigInteger x0, BigInteger xmax)
{
this.n = n;
x0 = T1(x0 + 1);
xmax = T1(xmax);
if (x0 > xmax)
{
return(0);
}
var ymin = YFloor(xmax);
var xmin = IntegerMath.Max(x0, IntegerMath.Min(T1(C1 * IntegerMath.CeilingRoot(2 * n, 3)), xmax));
#if DIAG
Console.WriteLine("n = {0}, xmin = {1}, xmax = {2}", n, xmin, xmax);
#endif
var s = (BigInteger)0;
var a2 = (BigInteger)1;
var x2 = xmax;
var y2 = ymin;
var c2 = a2 * x2 + y2;
while (true)
{
var a1 = a2 + 1;
var x4 = YTan(a1);
var y4 = YFloor(x4);
var c4 = a1 * x4 + y4;
var x5 = x4 + 1;
var y5 = YFloor(x5);
var c5 = a1 * x5 + y5;
if (x4 <= xmin)
{
break;
}
s += Triangle(c4 - c2 - x0) - Triangle(c4 - c2 - x5) + Triangle(c5 - c2 - x5);
if (threads == 0)
{
s += ProcessRegion(0, a1 * x2 + y2 - c5, a2 * x5 + y5 - c2, a1, 1, c5, a2, 1, c2);
while (queue.Count > 0)
{
var r = queue.Take();
s += ProcessRegion(0, r.w, r.h, r.a1, r.b1, r.c1, r.a2, r.b2, r.c2);
}
}
else
{
Enqueue(new Region(a1 * x2 + y2 - c5, a2 * x5 + y5 - c2, a1, 1, c5, a2, 1, c2));
}
a2 = a1;
x2 = x4;
y2 = y4;
c2 = c4;
}
s += (xmax - x0 + 1) * ymin + Triangle(xmax - x0);
var rest = x2 - x0;
s -= y2 * rest + a2 * Triangle(rest);
xmanual = x2;
return(s);
}
public BigInteger Evaluate(BigInteger n)
{
t3Map.Clear();
var jmax = IntegerMath.FloorLog(n, 2);
var dmax = IntegerMath.FloorRoot(n, 3);
mobius = new MobiusCollection((int)(IntegerMath.Max(jmax, dmax) + 1), 0);
return(Pi3(n));
}
public Int128 Evaluate(UInt128 n)
{
if (n == 0)
{
return(0);
}
this.n = n;
u = (long)IntegerMath.Max(IntegerMath.FloorPower(n, 2, 3) * C1 / C2, IntegerMath.CeilingSquareRoot(n));
imax = (int)(n / (ulong)u);
mobius = new MobiusRangeAdditive(u + 1, threads);
var batchSize = Math.Min(u, maximumBatchSize);
mu = new sbyte[maximumSmallBatchSize];
m = new int[batchSize];
mx = new Int128[imax + 1];
r = new int[imax + 1];
var lmax = 0;
for (var i = 1; i <= imax; i += 2)
{
if (wheelInclude[(i % wheelSize) >> 1])
{
r[lmax++] = i;
}
}
Array.Resize(ref r, lmax);
niLarge = new UInt128[imax + 1];
niSmall = new long[imax + 1];
var buckets = Math.Max(1, threads);
var costs = new double[buckets];
var bucketListsLarge = Enumerable.Range(0, buckets).Select(i => new List <int>()).ToArray();
var bucketListsSmall = Enumerable.Range(0, buckets).Select(i => new List <int>()).ToArray();
for (var l = 0; l < lmax; l++)
{
var i = r[l];
var ni = n / (uint)i;
var large = ni > largeLimit;
var cost = Math.Sqrt((double)n / i) * (large ? C7 : 1);
var addto = 0;
var mincost = costs[0];
for (var bucket = 0; bucket < buckets; bucket++)
{
if (costs[bucket] < mincost)
{
mincost = costs[bucket];
addto = bucket;
}
}
niLarge[i] = ni;
if (large)
{
bucketListsLarge[addto].Add(i);
}
else
{
niSmall[i] = (long)ni;
bucketListsSmall[addto].Add(i);
}
costs[addto] += cost;
}
bucketsLarge = bucketListsLarge.Select(bucket => bucket.ToArray()).ToArray();
bucketsSmall = bucketListsSmall.Select(bucket => bucket.ToArray()).ToArray();
var m0 = 0;
var xmed = Math.Min((long)IntegerMath.FloorRoot(n, 2) * C5 / C6, u);
for (var x = (long)1; x <= xmed; x += maximumSmallBatchSize)
{
var xstart = x;
var xend = Math.Min(xstart + maximumSmallBatchSize - 1, xmed);
m0 = mobius.GetValuesAndSums(xstart, xend + 1, mu, m, m0);
ProcessBatch(xstart, xend);
}
for (var x = xmed + 1; x <= u; x += maximumBatchSize)
{
var xstart = x;
var xend = Math.Min(xstart + maximumBatchSize - 1, u);
m0 = mobius.GetSums(xstart, xend + 1, m, m0);
ProcessBatch(xstart, xend);
}
ComputeMx();
return(mx[1]);
}
public Integer Evaluate(Integer n, BigInteger xfirst, BigInteger xlast)
{
this.n = n;
// Count lattice points under the hyperbola x*y = n.
var sum = (Integer)0;
// Compute the range of values over which we will apply the
// geometric algorithm.
xmax = (Integer)xlast;
xmin = IntegerMath.Max(xfirst, IntegerMath.Min(IntegerMath.FloorRoot(n, 3) * minimumMultiplier, xmax));
// Calculate the line tangent to the hyperbola at the x = sqrt(n).
var m0 = (Integer)1;
var x0 = xmax;
var y0 = n / x0;
var r0 = y0 + m0 * x0;
Debug.Assert(r0 - m0 * x0 == y0);
// Add the bottom rectangle.
var width = x0 - xfirst;
sum += (width + 1) * y0;
// Add the isosceles right triangle corresponding to the initial
// line L0 with -slope = 1.
sum += width * (width + 1) / 2;
// Process regions between tangent lines with integral slopes 1 & 2,
// 2 & 3, etc. until we reach xmin. This provides a first
// approximation to the hyperbola and accounts for the majority
// of the lattice points between xmin and max. The remainder of
// the points are computed by processing the regions bounded
// by the two tangent lines and the hyperbola itself.
while (true)
{
// Find the largest point (x1a, y1a) where -H'(X) >= the new slope.
var m1 = m0 + 1;
var x1a = IntegerMath.FloorSquareRoot(n / m1);
var y1a = n / x1a;
var r1a = y1a + m1 * x1a;
var x1b = x1a + 1;
var y1b = n / x1b;
var r1b = y1b + m1 * x1b;
Debug.Assert(r1a - m1 * x1a == y1a);
Debug.Assert(r1b - m1 * x1b == y1b);
// Handle left-overs.
if (x1a < xmin)
{
// Remove all the points we added between xfirst and x0.
var rest = x0 - xfirst;
sum -= (r0 - m0 * x0) * rest + m0 * rest * (rest + 1) / 2;
xmin = x0;
break;
}
// Invariants:
// The value before x1a along L1a is on or below the hyperbola.
// The value after x1b along L2b is on or below the hyperbola.
// The new slope is one greater than the old slope.
Debug.Assert((x1a - 1) * (r1a - m1 * (x1a - 1)) <= n);
Debug.Assert((x1b + 1) * (r1b - m1 * (x1b + 1)) <= n);
Debug.Assert(m1 - m0 == 1);
// Add the triangular wedge above the previous slope and below the new one
// and bounded on the left by xfirst.
var x0a = r1a - r0;
width = x0a - xfirst;
sum += width * (width + 1) / 2;
// Account for a drop or rise from L1a to L1b.
if (r1a != r1b && x1a < x0a)
{
// Remove the old triangle and add the new triangle.
// The formula is (ow+dr)*(ow+dr+1)/2 - ow*(ow+1)/2.
var ow = x1a - x0a;
var dr = r1a - r1b;
sum += dr * (2 * ow + dr + 1) / 2;
}
// Determine intersection of L0 and L1b.
var x0b = r1b - r0;
var y0b = r0 - m0 * x0b;
Debug.Assert(r0 - m0 * x0b == r1b - m1 * x0b);
// Calculate width and height of parallelogram counting only lattice points.
var w = (y0 - y0b) + m1 * (x0 - x0b);
var h = (y1b - y0b) + m0 * (x1b - x0b);
// Process the hyperbolic region bounded by L1b and L0.
sum += ProcessRegion(w, h, m1, 1, m0, 1, x0b, y0b);
// Advance to the next region.
m0 = m1;
x0 = x1a;
y0 = y1a;
r0 = r1a;
}
// Process values from xfirst up to xmin.
sum += manualAlgorithm.Evaluate(n, xfirst, xmin - 1);
return(sum);
}
