private static void LehmerCore(ref BitsBuffer xBuffer,
ref BitsBuffer yBuffer,
long a, long b,
long c, long d)
{
Debug.Assert(xBuffer.GetLength() >= 1);
Debug.Assert(yBuffer.GetLength() >= 1);
Debug.Assert(xBuffer.GetLength() >= yBuffer.GetLength());
Debug.Assert(a <= 0x7FFFFFFF && b <= 0x7FFFFFFF);
Debug.Assert(c <= 0x7FFFFFFF && d <= 0x7FFFFFFF);
// Executes the combined calculation of Lehmer's step.
uint[] x = xBuffer.GetBits();
uint[] y = yBuffer.GetBits();
int length = yBuffer.GetLength();
long xCarry = 0L, yCarry = 0L;
for (int i = 0; i < length; i++)
{
long xDigit = a * x[i] - b * y[i] + xCarry;
long yDigit = d * y[i] - c * x[i] + yCarry;
xCarry = xDigit >> 32;
yCarry = yDigit >> 32;
x[i] = unchecked ((uint)xDigit);
y[i] = unchecked ((uint)yDigit);
}
xBuffer.Refresh(length);
yBuffer.Refresh(length);
}
private static void ExtractDigits(ref BitsBuffer xBuffer,
ref BitsBuffer yBuffer,
out ulong x, out ulong y)
{
Debug.Assert(xBuffer.GetLength() >= 3);
Debug.Assert(yBuffer.GetLength() >= 3);
Debug.Assert(xBuffer.GetLength() >= yBuffer.GetLength());
// Extracts the most significant bits of x and y,
// but ensures the quotient x / y does not change!
uint[] xBits = xBuffer.GetBits();
int xLength = xBuffer.GetLength();
uint[] yBits = yBuffer.GetBits();
int yLength = yBuffer.GetLength();
ulong xh = xBits[xLength - 1];
ulong xm = xBits[xLength - 2];
ulong xl = xBits[xLength - 3];
ulong yh, ym, yl;
// arrange the bits
switch (xLength - yLength)
{
case 0:
yh = yBits[yLength - 1];
ym = yBits[yLength - 2];
yl = yBits[yLength - 3];
break;
case 1:
yh = 0UL;
ym = yBits[yLength - 1];
yl = yBits[yLength - 2];
break;
case 2:
yh = 0UL;
ym = 0UL;
yl = yBits[yLength - 1];
break;
default:
yh = 0UL;
ym = 0UL;
yl = 0UL;
break;
}
// Use all the bits but one, see [hac] 14.58 (ii)
int z = LeadingZeros((uint)xh);
x = ((xh << 32 + z) | (xm << z) | (xl >> 32 - z)) >> 1;
y = ((yh << 32 + z) | (ym << z) | (yl >> 32 - z)) >> 1;
Debug.Assert(x >= y);
}
private static void Gcd(ref BitsBuffer left, ref BitsBuffer right)
{
Debug.Assert(left.GetLength() >= 2);
Debug.Assert(right.GetLength() >= 2);
Debug.Assert(left.GetLength() >= right.GetLength());
// Executes Lehmer's gcd algorithm, but uses the most
// significant bits to work with 64-bit (not 32-bit) values.
// Furthermore we're using an optimized version due to Jebelean.
// http://cacr.uwaterloo.ca/hac/about/chap14.pdf (see 14.4.2)
// ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz
while (right.GetLength() > 2)
{
ulong x, y;
ExtractDigits(ref left, ref right, out x, out y);
uint a = 1U, b = 0U;
uint c = 0U, d = 1U;
int iteration = 0;
// Lehmer's guessing
while (y != 0)
{
ulong q, r, s, t;
// odd iteration
q = x / y;
if (q > 0xFFFFFFFF)
break;
r = a + q * c;
s = b + q * d;
t = x - q * y;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
break;
if (t < s || t + r > y - c)
break;
a = (uint)r;
b = (uint)s;
x = t;
++iteration;
if (x == b)
break;
// even iteration
q = y / x;
if (q > 0xFFFFFFFF)
break;
r = d + q * b;
s = c + q * a;
t = y - q * x;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
break;
if (t < s || t + r > x - b)
break;
d = (uint)r;
c = (uint)s;
y = t;
++iteration;
if (y == c)
break;
}
if (b == 0)
{
// Euclid's step
left.Reduce(ref right);
BitsBuffer temp = left;
left = right;
right = temp;
}
else
{
// Lehmer's step
LehmerCore(ref left, ref right, a, b, c, d);
if (iteration % 2 == 1)
{
// ensure left is larger than right
BitsBuffer temp = left;
left = right;
right = temp;
}
}
}
if (right.GetLength() > 0)
{
// Euclid's step
left.Reduce(ref right);
uint[] xBits = right.GetBits();
uint[] yBits = left.GetBits();
ulong x = ((ulong)xBits[1] << 32) | xBits[0];
ulong y = ((ulong)yBits[1] << 32) | yBits[0];
left.Overwrite(Gcd(x, y));
right.Overwrite(0);
}
}
private static void LehmerCore(ref BitsBuffer xBuffer,
ref BitsBuffer yBuffer,
long a, long b,
long c, long d)
{
Debug.Assert(xBuffer.GetLength() >= 1);
Debug.Assert(yBuffer.GetLength() >= 1);
Debug.Assert(xBuffer.GetLength() >= yBuffer.GetLength());
Debug.Assert(a <= 0x7FFFFFFF && b <= 0x7FFFFFFF);
Debug.Assert(c <= 0x7FFFFFFF && d <= 0x7FFFFFFF);
// Executes the combined calculation of Lehmer's step.
uint[] x = xBuffer.GetBits();
uint[] y = yBuffer.GetBits();
int length = yBuffer.GetLength();
long xCarry = 0L, yCarry = 0L;
for (int i = 0; i < length; i++)
{
long xDigit = a * x[i] - b * y[i] + xCarry;
long yDigit = d * y[i] - c * x[i] + yCarry;
xCarry = xDigit >> 32;
yCarry = yDigit >> 32;
x[i] = (uint)xDigit;
y[i] = (uint)yDigit;
}
xBuffer.Refresh(length);
yBuffer.Refresh(length);
}
private static void ExtractDigits(ref BitsBuffer xBuffer,
ref BitsBuffer yBuffer,
out ulong x, out ulong y)
{
Debug.Assert(xBuffer.GetLength() >= 3);
Debug.Assert(yBuffer.GetLength() >= 3);
Debug.Assert(xBuffer.GetLength() >= yBuffer.GetLength());
// Extracts the most significant bits of x and y,
// but ensures the quotient x / y does not change!
uint[] xBits = xBuffer.GetBits();
int xLength = xBuffer.GetLength();
uint[] yBits = yBuffer.GetBits();
int yLength = yBuffer.GetLength();
ulong xh = xBits[xLength - 1];
ulong xm = xBits[xLength - 2];
ulong xl = xBits[xLength - 3];
ulong yh, ym, yl;
// arrange the bits
switch (xLength - yLength)
{
case 0:
yh = yBits[yLength - 1];
ym = yBits[yLength - 2];
yl = yBits[yLength - 3];
break;
case 1:
yh = 0UL;
ym = yBits[yLength - 1];
yl = yBits[yLength - 2];
break;
case 2:
yh = 0UL;
ym = 0UL;
yl = yBits[yLength - 1];
break;
default:
yh = 0UL;
ym = 0UL;
yl = 0UL;
break;
}
// use all the bits but one, see [hac] 14.58 (ii)
int z = LeadingZeros((uint)xh);
x = ((xh << 32 + z) | (xm << z) | (xl >> 32 - z)) >> 1;
y = ((yh << 32 + z) | (ym << z) | (yl >> 32 - z)) >> 1;
Debug.Assert(x >= y);
}
private static void Gcd(ref BitsBuffer left, ref BitsBuffer right)
{
Debug.Assert(left.GetLength() >= 2);
Debug.Assert(right.GetLength() >= 2);
Debug.Assert(left.GetLength() >= right.GetLength());
// Executes Lehmer's gcd algorithm, but uses the most
// significant bits to work with 64-bit (not 32-bit) values.
// Furthermore we're using an optimized version due to Jebelean.
// http://cacr.uwaterloo.ca/hac/about/chap14.pdf (see 14.4.2)
// ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz
while (right.GetLength() > 2)
{
ulong x, y;
ExtractDigits(ref left, ref right, out x, out y);
uint a = 1U, b = 0U;
uint c = 0U, d = 1U;
int iteration = 0;
// Lehmer's guessing
while (y != 0)
{
ulong q, r, s, t;
// Odd iteration
q = x / y;
if (q > 0xFFFFFFFF)
{
break;
}
r = a + q * c;
s = b + q * d;
t = x - q * y;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
{
break;
}
if (t < s || t + r > y - c)
{
break;
}
a = (uint)r;
b = (uint)s;
x = t;
++iteration;
if (x == b)
{
break;
}
// Even iteration
q = y / x;
if (q > 0xFFFFFFFF)
{
break;
}
r = d + q * b;
s = c + q * a;
t = y - q * x;
if (r > 0x7FFFFFFF || s > 0x7FFFFFFF)
{
break;
}
if (t < s || t + r > x - b)
{
break;
}
d = (uint)r;
c = (uint)s;
y = t;
++iteration;
if (y == c)
{
break;
}
}
if (b == 0)
{
// Euclid's step
left.Reduce(ref right);
BitsBuffer temp = left;
left = right;
right = temp;
}
else
{
// Lehmer's step
LehmerCore(ref left, ref right, a, b, c, d);
if (iteration % 2 == 1)
{
// Ensure left is larger than right
BitsBuffer temp = left;
left = right;
right = temp;
}
}
}
if (right.GetLength() > 0)
{
// Euclid's step
left.Reduce(ref right);
uint[] xBits = right.GetBits();
uint[] yBits = left.GetBits();
ulong x = ((ulong)xBits[1] << 32) | xBits[0];
ulong y = ((ulong)yBits[1] << 32) | yBits[0];
left.Overwrite(Gcd(x, y));
right.Overwrite(0);
}
}