private void LongDivide3( Integer ToDivide,
Integer DivideBy,
Integer Quotient,
Integer Remainder )
{
int TestIndex = ToDivide.GetIndex() - DivideBy.GetIndex();
if( TestIndex < 0 )
throw( new Exception( "TestIndex < 0 in Divide3." ));
if( TestIndex != 0 )
{
// Is 1 too high?
TestForDivide1.SetDigitAndClear( TestIndex, 1 );
MultiplyTopOne( TestForDivide1, DivideBy );
if( ToDivide.ParamIsGreater( TestForDivide1 ))
TestIndex--;
}
// Keep a copy of the originals.
ToDivideKeep.Copy( ToDivide );
DivideByKeep.Copy( DivideBy );
ulong TestBits = DivideBy.GetD( DivideBy.GetIndex());
int ShiftBy = FindShiftBy( TestBits );
ToDivide.ShiftLeft( ShiftBy ); // Multiply the numerator and the denominator
DivideBy.ShiftLeft( ShiftBy ); // by the same amount.
ulong MaxValue;
if( (ToDivide.GetIndex() - 1) > (DivideBy.GetIndex() + TestIndex) )
{
MaxValue = ToDivide.GetD( ToDivide.GetIndex());
}
else
{
MaxValue = ToDivide.GetD( ToDivide.GetIndex()) << 32;
MaxValue |= ToDivide.GetD( ToDivide.GetIndex() - 1 );
}
ulong Denom = DivideBy.GetD( DivideBy.GetIndex());
if( Denom != 0 )
MaxValue = MaxValue / Denom;
else
MaxValue = 0xFFFFFFFF;
if( MaxValue > 0xFFFFFFFF )
MaxValue = 0xFFFFFFFF;
if( MaxValue == 0 )
throw( new Exception( "MaxValue is zero at the top in LongDivide3()." ));
Quotient.SetDigitAndClear( TestIndex, 1 );
Quotient.SetD( TestIndex, 0 );
TestForDivide1.Copy( Quotient );
TestForDivide1.SetD( TestIndex, MaxValue );
MultiplyTop( TestForDivide1, DivideBy );
/*
Test2.Copy( Quotient );
Test2.SetD( TestIndex, MaxValue );
Multiply( Test2, DivideBy );
if( !Test2.IsEqual( TestForDivide1 ))
throw( new Exception( "In Divide3() !IsEqual( Test2, TestForDivide1 )" ));
*/
if( TestForDivide1.ParamIsGreaterOrEq( ToDivide ))
{
// ToMatchExactCount++;
// Most of the time (roughly 5 out of every 6 times)
// this MaxValue estimate is exactly right:
Quotient.SetD( TestIndex, MaxValue );
}
else
{
// MaxValue can't be zero here. If it was it would
// already be low enough before it got here.
MaxValue--;
if( MaxValue == 0 )
throw( new Exception( "After decrement: MaxValue is zero in LongDivide3()." ));
TestForDivide1.Copy( Quotient );
TestForDivide1.SetD( TestIndex, MaxValue );
MultiplyTop( TestForDivide1, DivideBy );
/*
Test2.Copy( Quotient );
Test2.SetD( TestIndex, MaxValue );
Multiply( Test2, DivideBy );
if( !Test2.IsEqual( Test1 ))
throw( new Exception( "Top one. !Test2.IsEqual( Test1 ) in LongDivide3()" ));
*/
if( TestForDivide1.ParamIsGreaterOrEq( ToDivide ))
{
// ToMatchDecCount++;
Quotient.SetD( TestIndex, MaxValue );
}
else
{
// TestDivideBits is done as a last resort, but it's rare.
// But it does at least limit it to a worst case scenario
// of trying 32 bits, rather than 4 billion or so decrements.
TestDivideBits( MaxValue,
true,
TestIndex,
ToDivide,
DivideBy,
Quotient,
Remainder );
}
// TestGap = MaxValue - LgQuotient.D[TestIndex];
// if( TestGap > HighestToMatchGap )
// HighestToMatchGap = TestGap;
// HighestToMatchGap: 4,294,967,293
// uint size: 4,294,967,295 uint
}
// If it's done.
if( TestIndex == 0 )
{
TestForDivide1.Copy( Quotient );
Multiply( TestForDivide1, DivideByKeep );
Remainder.Copy( ToDivideKeep );
Subtract( Remainder, TestForDivide1 );
//if( DivideByKeep.ParamIsGreater( Remainder ))
// throw( new Exception( "Remainder > DivideBy in LongDivide3()." ));
return;
}
// Now do the rest of the digits.
TestIndex--;
while( true )
{
TestForDivide1.Copy( Quotient );
// First Multiply() for each digit.
Multiply( TestForDivide1, DivideBy );
// if( ToDivide.ParamIsGreater( TestForDivide1 ))
// throw( new Exception( "Bug here in LongDivide3()." ));
Remainder.Copy( ToDivide );
Subtract( Remainder, TestForDivide1 );
MaxValue = Remainder.GetD( Remainder.GetIndex()) << 32;
int CheckIndex = Remainder.GetIndex() - 1;
if( CheckIndex > 0 )
MaxValue |= Remainder.GetD( CheckIndex );
Denom = DivideBy.GetD( DivideBy.GetIndex());
if( Denom != 0 )
MaxValue = MaxValue / Denom;
else
MaxValue = 0xFFFFFFFF;
if( MaxValue > 0xFFFFFFFF )
MaxValue = 0xFFFFFFFF;
TestForDivide1.Copy( Quotient );
TestForDivide1.SetD( TestIndex, MaxValue );
// There's a minimum of two full Multiply() operations per digit.
Multiply( TestForDivide1, DivideBy );
if( TestForDivide1.ParamIsGreaterOrEq( ToDivide ))
{
// Most of the time this MaxValue estimate is exactly right:
// ToMatchExactCount++;
Quotient.SetD( TestIndex, MaxValue );
}
else
{
MaxValue--;
TestForDivide1.Copy( Quotient );
TestForDivide1.SetD( TestIndex, MaxValue );
Multiply( TestForDivide1, DivideBy );
if( TestForDivide1.ParamIsGreaterOrEq( ToDivide ))
{
// ToMatchDecCount++;
Quotient.SetD( TestIndex, MaxValue );
}
else
{
TestDivideBits( MaxValue,
false,
TestIndex,
ToDivide,
DivideBy,
Quotient,
Remainder );
// TestGap = MaxValue - LgQuotient.D[TestIndex];
// if( TestGap > HighestToMatchGap )
// HighestToMatchGap = TestGap;
}
}
if( TestIndex == 0 )
break;
TestIndex--;
}
TestForDivide1.Copy( Quotient );
Multiply( TestForDivide1, DivideByKeep );
Remainder.Copy( ToDivideKeep );
Subtract( Remainder, TestForDivide1 );
// if( DivideByKeep.ParamIsGreater( Remainder ))
// throw( new Exception( "Remainder > DivideBy in LongDivide3()." ));
}
// Finding the square root of a number is similar to division since
// it is a search algorithm. The TestSqrtBits method shown next is
// very much like TestDivideBits(). It works the same as
// FindULSqrRoot(), but on a bigger scale.
/*
private void TestSqrtBits( int TestIndex, Integer Square, Integer SqrRoot )
{
Integer Test1 = new Integer();
uint BitTest = 0x80000000;
for( int BitCount = 31; BitCount >= 0; BitCount-- )
{
Test1.Copy( SqrRoot );
Test1.D[TestIndex] |= BitTest;
Test1.Square();
if( !Square.ParamIsGreater( Test1 ) )
SqrRoot.D[TestIndex] |= BitTest; // Use the bit.
BitTest >>= 1;
}
}
*/
// In the SquareRoot() method SqrRoot.Index is half of Square.Index.
// Compare this to the Square() method where the Carry might or
// might not increment the index to an odd number. (So if the Index
// was 5 its square root would have an Index of 5 / 2 = 2.)
// The SquareRoot1() method uses FindULSqrRoot() either to find the
// whole answer, if it's a small number, or it uses it to find the
// top part. Then from there it goes on to a bit by bit search
// with TestSqrtBits().
public bool SquareRoot( Integer Square, Integer SqrRoot )
{
ulong ToMatch;
if( Square.IsULong() )
{
ToMatch = Square.GetAsULong();
SqrRoot.SetD( 0, FindULSqrRoot( ToMatch ));
SqrRoot.SetIndex( 0 );
if( (SqrRoot.GetD(0 ) * SqrRoot.GetD( 0 )) == ToMatch )
return true;
else
return false;
}
int TestIndex = Square.GetIndex() >> 1; // LgSquare.Index / 2;
SqrRoot.SetDigitAndClear( TestIndex, 1 );
// if( (TestIndex * 2) > (LgSquare.Index - 1) )
if( (TestIndex << 1) > (Square.GetIndex() - 1) )
{
ToMatch = Square.GetD( Square.GetIndex());
}
else
{
// LgSquare.Index is at least 2 here.
ToMatch = Square.GetD( Square.GetIndex()) << 32;
ToMatch |= Square.GetD( Square.GetIndex() - 1 );
}
SqrRoot.SetD( TestIndex, FindULSqrRoot( ToMatch ));
TestIndex--;
while( true )
{
// TestSqrtBits( TestIndex, LgSquare, LgSqrRoot );
SearchSqrtXPart( TestIndex, Square, SqrRoot );
if( TestIndex == 0 )
break;
TestIndex--;
}
// Avoid squaring the whole thing to see if it's an exact square root:
if( ((SqrRoot.GetD( 0 ) * SqrRoot.GetD( 0 )) & 0xFFFFFFFF) != Square.GetD( 0 ))
return false;
TestForSquareRoot.Copy( SqrRoot );
DoSquare( TestForSquareRoot );
if( Square.IsEqual( TestForSquareRoot ))
return true;
else
return false;
}