/// <summary>
/// Bitwise Xor
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
/// <returns></returns>
public static BigInt Xor(BigInt n1, BigInt n2)
{
if (n1.digitArray.Length != n2.digitArray.Length) MakeSafe(ref n1, ref n2);
BigInt res = new BigInt(n1);
res.And(n2);
return res;
}
/// <summary>
/// Floating-point helper function.
/// Squares the number and keeps the high bits of the calculation.
/// Takes a temporary BigInt as a working set.
/// </summary>
public void SquareHiFast(BigInt scratch)
{
int Length = digitArray.Length;
uint[] tempDigits = scratch.digitArray;
uint[] workingSet2 = scratch.workingSet;
//Temp storage for source (both working sets are used by the calculation)
for (int i = 0; i < Length; i++)
{
tempDigits[i] = digitArray[i];
digitArray[i] = 0;
}
for (int i = 0; i < Length; i++)
{
//Clear the working set
for (int j = i; j < Length; j++)
{
workingSet[j] = 0;
workingSet2[j] = 0;
}
if (i - 1 >= 0) workingSet[i - 1] = 0;
ulong digit = tempDigits[i];
if (digit == 0) continue;
for (int j = 0; j < Length; j++)
{
if (j + i + 1 < Length) continue;
//Multiply n1 by each of the integer digits of n2.
ulong temp = (ulong)tempDigits[j] * digit;
//n1.workingSet stores the low bits of each piecewise multiplication
if (j + i >= Length)
{
workingSet[j + i - Length] = (uint)(temp & 0xffffffff);
}
//n2.workingSet stores the high bits of each multiplication
workingSet2[j + i + 1 - Length] = (uint)(temp >> 32);
}
AddInternalBits(workingSet);
AddInternalBits(workingSet2);
}
sign = false;
}
/// <summary>
/// Calculates 'this' mod n2 (using the schoolbook division algorithm as above)
/// </summary>
/// <param name="n2"></param>
public void Mod(BigInt n2)
{
if (n2.digitArray.Length != digitArray.Length) MakeSafe(ref n2);
int OldLength = digitArray.Length;
//First, we need to prepare the operands for division using Div_32, which requires
//That the most significant digit of n2 be set. To do this, we need to shift n2 (and therefore n1) up.
//This operation can potentially increase the precision of the operands.
int shift = MakeSafeDiv(this, n2);
BigInt Q = new BigInt(this.pres);
BigInt R = new BigInt(this.pres);
Q.digitArray = new UInt32[this.digitArray.Length];
R.digitArray = new UInt32[this.digitArray.Length];
Div_32(this, n2, Q, R);
//Restore n2 to its pre-shift value
n2.RSH(shift);
R.RSH(shift);
R.sign = (sign != n2.sign);
AssignInt(R);
//Reset the lengths of the operands
SetNumDigits(OldLength);
n2.SetNumDigits(OldLength);
}
/// <summary>
/// Makes sure the numbers have matching precisions
/// </summary>
/// <param name="n2">the number to match to this</param>
private void MakeSafe(ref BigInt n2)
{
n2 = new BigInt(n2, pres);
n2.SetNumDigits(digitArray.Length);
}
/// <summary>
/// Subtraction and assignment - without intermediate memory allocation.
/// </summary>
/// <param name="n2"></param>
public uint Sub(BigInt n2)
{
if (n2.digitArray.Length != digitArray.Length) MakeSafe(ref n2);
if (sign != n2.sign)
{
return AddInternalBits(n2.digitArray);
}
else
{
bool lessThan = LtInt(this, n2);
if (lessThan)
{
int Length = digitArray.Length;
for (int i = 0; i < Length; i++)
{
workingSet[i] = digitArray[i];
digitArray[i] = n2.digitArray[i];
}
sign = !sign;
return SubInternalBits(workingSet);
}
else
{
return SubInternalBits(n2.digitArray);
}
}
}
private string BuildString(BigInt n, int m)
{
if (m == 0)
return n.ToString(10);
BigInt remainder;
BigInt quotient = BigInt.DivRem(n, powersOfTen[m], out remainder);
return BuildString(quotient, m - 1) + BuildString(remainder, m - 1);
}
/// <summary>
/// Sign-insensitive greater than comparison.
/// unsafe if n1 and n2 disagree in precision
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
/// <returns></returns>
private static bool GtInt(BigInt n1, BigInt n2)
{
//MakeSafe(ref n1, ref n2);
for (int i = n1.digitArray.Length - 1; i >= 0; i--)
{
if (n1.digitArray[i] > n2.digitArray[i]) return true;
if (n1.digitArray[i] < n2.digitArray[i]) return false;
}
//equal
return false;
}
/// <summary>
/// Gets the most significant bit
/// </summary>
/// <param name="value">the input to search for the MSB in</param>
/// <returns>-1 if the input was zero, the position of the MSB otherwise</returns>
public static int GetMSB(BigInt value)
{
int digit = value.GetMSD();
int bit = GetMSB(value.digitArray[digit]);
return (digit << 5) + bit;
}
/// <summary>
/// Shifts and optionally precision-extends the arguments to prepare for Div_32
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
private static int MakeSafeDiv(BigInt n1, BigInt n2)
{
int shift = n2.GetDivBitshift();
int n1MSD = n1.GetMSD();
uint temp = n1.digitArray[n1MSD];
if (n1MSD == n1.digitArray.Length - 1 && ((temp << shift) >> shift) != n1.digitArray[n1MSD])
{
//Precision-extend n1 and n2 if necessary
int digits = n1.digitArray.Length;
n1.SetNumDigits(digits + 1);
n2.SetNumDigits(digits + 1);
}
//Logical left-shift n1 and n2
n1.LSH(shift);
n2.LSH(shift);
return shift;
}
/// <summary>
/// Used for casting between BigFloats of different precisions - this assumes
/// that the number is a normalised mantissa.
/// </summary>
/// <param name="n2"></param>
/// <returns>true if a round-up caused the high bits to become zero</returns>
public bool AssignHigh(BigInt n2)
{
int length = digitArray.Length;
int length2 = n2.digitArray.Length;
int minWords = (length < length2) ? length : length2;
bool bRet = false;
for (int i = 1; i <= minWords; i++)
{
digitArray[length - i] = n2.digitArray[length2 - i];
}
if (length2 > length && n2.digitArray[length2 - (length + 1)] >= 0x80000000)
{
bRet = IncrementInt();
//Because we are assuming normalisation, we set the top bit (it will already be set if
//bRet is false.
digitArray[length - 1] = digitArray[length - 1] | 0x80000000;
}
sign = n2.sign;
return bRet;
}
/// <summary>
/// Constructor for copying length and precision
/// </summary>
/// <param name="inputToCopy">The BigInt to copy</param>
/// <param name="precision">The precision of the new BigInt</param>
/// <param name="bCopyLengthOnly">decides whether to copy the actual input, or just its digit length</param>
/// <example><code>//Create an integer
/// BigInt four = new BigInt(4, new PrecisionSpec(128, PrecisionSpec.BaseType.BIN));
///
/// //Pad four to double its usual number of digits (this does not affect the precision)
/// four.Pad();
///
/// //Create a new, empty integer with matching precision, also padded to twice the usual length
/// BigInt newCopy = new BigInt(four, four.Precision, true);</code></example>
public BigInt(BigInt inputToCopy, PrecisionSpec precision, bool bCopyLengthOnly)
{
digitArray = new uint[inputToCopy.digitArray.Length];
workingSet = new uint[inputToCopy.digitArray.Length];
if (!bCopyLengthOnly) Array.Copy(inputToCopy.digitArray, digitArray, digitArray.Length);
sign = inputToCopy.sign;
pres = inputToCopy.pres;
}
/// <summary>
/// Used to subtract n1 (true subtraction of digit arrays) - n1 must be less than or equal to this number
/// </summary>
/// <param name="n1"></param>
private uint SubInternal(BigInt n1)
{
return SubInternalBits(n1.digitArray);
}
/// <summary>
/// Used to add with matching signs (true addition of the digit arrays)
/// This is internal because it will fail spectacularly if n1 is negative.
/// </summary>
/// <param name="n1"></param>
private uint AddInternal(BigInt n1)
{
return AddInternalBits(n1.digitArray);
}
/// <summary>
/// The right-shift operator
/// </summary>
public static BigInt operator >>(BigInt n1, int n2)
{
BigInt res = new BigInt(n1);
res.RSH(n2);
return res;
}
/// <summary>
/// Constructs a bigint from the input, matching the new precision provided
/// </summary>
public BigInt(BigInt input, PrecisionSpec precision)
{
//Casts the input to the new precision.
Init(precision);
int Min = (input.digitArray.Length < digitArray.Length) ? input.digitArray.Length : digitArray.Length;
for (int i = 0; i < Min; i++)
{
digitArray[i] = input.digitArray[i];
}
sign = input.sign;
}
/// <summary>
/// Schoolbook division helper function.
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
/// <param name="Q">Quotient output value</param>
/// <param name="R">Remainder output value</param>
private static void Div_31(BigInt n1, BigInt n2, BigInt Q, BigInt R)
{
int digitsN1 = n1.GetMSD() + 1;
int digitsN2 = n2.GetMSD() + 1;
if ((digitsN1 > digitsN2))
{
BigInt n1New = new BigInt(n2);
n1New.DigitShiftSelfLeft(1);
//If n1 >= n2 * 2^32
if (!LtInt(n1, n1New))
{
n1New.sign = n1.sign;
SubFast(n1New, n1, n1New);
Div_32(n1New, n2, Q, R);
//Q = (A - B*2^32)/B + 2^32
Q.Add2Pow32Self();
return;
}
}
UInt32 q = 0;
if (digitsN1 >= 2)
{
UInt64 q64 = ((((UInt64)n1.digitArray[digitsN1 - 1]) << 32) + n1.digitArray[digitsN1 - 2]) / (UInt64)n2.digitArray[digitsN2 - 1];
if (q64 > 0xfffffffful)
{
q = 0xffffffff;
}
else
{
q = (UInt32)q64;
}
}
BigInt temp = Mul(n2, q);
if (GtInt(temp, n1))
{
temp.SubInternalBits(n2.digitArray);
q--;
if (GtInt(temp, n1))
{
temp.SubInternalBits(n2.digitArray);
q--;
}
}
Q.Zero();
Q.digitArray[0] = q;
R.Assign(n1);
R.SubInternalBits(temp.digitArray);
}
//********************* ToString members **********************
public String ToQuickString()
{
powersOfTen = new List<BigInt>();
powersOfTen.Add(new BigInt((int)1, new PrecisionSpec(1, PrecisionSpec.BaseType.DEC)));
for (BigInt i = new BigInt((int)10, new PrecisionSpec(1, PrecisionSpec.BaseType.DEC)); i.LessThan(this); i *= i)
{
powersOfTen.Add(i);
}
return BuildString(this, powersOfTen.Count - 1).TrimStart('0');
}
/// <summary>
/// Schoolbook division algorithm
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
/// <param name="Q"></param>
/// <param name="R"></param>
private static void Div_32(BigInt n1, BigInt n2, BigInt Q, BigInt R)
{
int digitsN1 = n1.GetMSD() + 1;
int digitsN2 = n2.GetMSD() + 1;
//n2 is bigger than n1
if (digitsN1 < digitsN2)
{
R.AssignInt(n1);
Q.Zero();
return;
}
if (digitsN1 == digitsN2)
{
//n2 is bigger than n1
if (LtInt(n1, n2))
{
R.AssignInt(n1);
Q.Zero();
return;
}
//n2 >= n1, but less the 2x n1 (initial conditions make this certain)
Q.Zero();
Q.digitArray[0] = 1;
R.Assign(n1);
R.SubInternalBits(n2.digitArray);
return;
}
int digits = digitsN1 - (digitsN2 + 1);
//Algorithm Div_31 can be used to get the answer in O(n) time.
if (digits == 0)
{
Div_31(n1, n2, Q, R);
return;
}
BigInt n1New = DigitShiftRight(n1, digits);
BigInt s = DigitTruncate(n1, digits);
BigInt Q2 = new BigInt(n1, n1.pres, true);
BigInt R2 = new BigInt(n1, n1.pres, true);
Div_31(n1New, n2, Q2, R2);
R2.DigitShiftSelfLeft(digits);
R2.Add(s);
Div_32(R2, n2, Q, R);
Q2.DigitShiftSelfLeft(digits);
Q.Add(Q2);
}
/// <summary>
/// Converts this to a string, in the specified base
/// </summary>
/// <param name="numberBase">the base to use (min 2, max 16)</param>
/// <returns>a string representation of the number</returns>
public string ToString(int numberBase)
{
char[] digitChars = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F' };
string output = "";
BigInt clone = new BigInt(this);
clone.sign = false;
int numDigits = 0;
while (!clone.IsZero())
{
BigInt div, mod;
DivMod(clone, (uint)numberBase, out div, out mod);
int iMod = (int)mod;
output = String.Concat(digitChars[(int)mod], output);
numDigits++;
clone = div;
}
if (output.Length == 0) output = "0";
if (sign) output = String.Concat("-", output);
return output;
}
/// <summary>
/// Assigns n2 to 'this'
/// </summary>
/// <param name="n2"></param>
public void Assign(BigInt n2)
{
if (digitArray.Length != n2.digitArray.Length) MakeSafe(ref n2);
sign = n2.sign;
AssignInt(n2);
}
/// <summary>
/// Makes sure the numbers have matching precisions
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
private static void MakeSafe(ref BigInt n1, ref BigInt n2)
{
if (n1.digitArray.Length == n2.digitArray.Length)
{
return;
}
else if (n1.digitArray.Length > n2.digitArray.Length)
{
n2 = new BigInt(n2, n1.pres);
}
else
{
n1 = new BigInt(n1, n2.pres);
}
}
/// <summary>
/// Assign n2 to 'this', safe only if precision-matched
/// </summary>
/// <param name="n2"></param>
/// <returns></returns>
private void AssignInt(BigInt n2)
{
int Length = digitArray.Length;
for (int i = 0; i < Length; i++)
{
digitArray[i] = n2.digitArray[i];
}
}
//*************** Utility Functions **************
/// <summary>
/// Casts a BigInt to the new precision provided.
/// Note: This will return the input if the precision already matches.
/// </summary>
/// <param name="input"></param>
/// <param name="precision"></param>
/// <returns></returns>
public static BigInt CastToPrecision(BigInt input, PrecisionSpec precision)
{
if (input.pres == precision) return input;
return new BigInt(input, precision);
}
private static BigInt DigitShiftRight(BigInt n1, int digits)
{
BigInt res = new BigInt(n1);
int Length = res.digitArray.Length;
for (int i = 0; i < Length - digits; i++)
{
res.digitArray[i] = res.digitArray[i + digits];
}
for (int i = Length - digits; i < Length; i++)
{
res.digitArray[i] = 0;
}
return res;
}
/// <summary>
/// Used for floating-point multiplication
/// Stores the high bits of the multiplication only (the carry bit from the
/// lower bits is missing, so the true answer might be 1 greater).
/// </summary>
/// <param name="n2"></param>
public void MulHi(BigInt n2)
{
if (n2.digitArray.Length != digitArray.Length) MakeSafe(ref n2);
int Length = n2.digitArray.Length;
//Inner loop zero-mul avoidance
int maxDigit = 0;
for (int i = Length - 1; i >= 0; i--)
{
if (digitArray[i] != 0)
{
maxDigit = i + 1;
break;
}
}
//Result is zero, 'this' is unchanged
if (maxDigit == 0) return;
//Temp storage for source (both working sets are used by the calculation)
uint[] thisTemp = new uint[Length];
for (int i = 0; i < Length; i++)
{
thisTemp[i] = digitArray[i];
digitArray[i] = 0;
}
for (int i = 0; i < Length; i++)
{
//Clear the working set
for (int j = 0; j < Length; j++)
{
workingSet[j] = 0;
n2.workingSet[j] = 0;
}
n2.workingSet[i] = 0;
ulong digit = n2.digitArray[i];
if (digit == 0) continue;
//Only the high bits
if (maxDigit + i < Length - 1) continue;
for (int j = 0; j < maxDigit; j++)
{
if (j + i + 1 < Length) continue;
//Multiply n1 by each of the integer digits of n2.
ulong temp = (ulong)thisTemp[j] * digit;
//n1.workingSet stores the low bits of each piecewise multiplication
if (j + i >= Length)
{
workingSet[j + i - Length] = (uint)(temp & 0xffffffff);
}
//n2.workingSet stores the high bits of each multiplication
n2.workingSet[j + i + 1 - Length] = (uint)(temp >> 32);
}
AddInternalBits(workingSet);
AddInternalBits(n2.workingSet);
}
sign = (sign != n2.sign);
}
/// <summary>
/// Constructs a bigint from the input.
/// </summary>
/// <param name="input"></param>
public BigInt(BigInt input)
{
digitArray = new uint[input.digitArray.Length];
workingSet = new uint[input.digitArray.Length];
Array.Copy(input.digitArray, digitArray, digitArray.Length);
sign = input.sign;
pres = input.pres;
}
/// <summary>
/// This function is used for floating-point division.
/// </summary>
/// <param name="n2"></param>
//Given two numbers:
// In floating point 1 <= a, b < 2, meaning that both numbers have their top bits set.
// To calculate a / b, maintaining precision, we:
// 1. Double the number of digits available to both numbers.
// 2. set a = a * 2^d (where d is the number of digits)
// 3. calculate the quotient a <- q: 2^(d-1) <= q < 2^(d+1)
// 4. if a >= 2^d, s = 1, else s = 0
// 6. shift a down by s, and undo the precision extension
// 7. return 1 - shift (change necessary to exponent)
public int DivAndShift(BigInt n2)
{
if (n2.IsZero()) return -1;
if (digitArray.Length != n2.digitArray.Length) MakeSafe(ref n2);
int oldLength = digitArray.Length;
//Double the number of digits, and shift a into the higher digits.
Pad();
n2.Extend();
//Do the divide (at double precision, ouch!)
Div(n2);
//Shift down if 'this' >= 2^d
int ret = 1;
if (digitArray[oldLength] != 0)
{
RSH(1);
ret--;
}
SetNumDigits(oldLength);
n2.SetNumDigits(oldLength);
return ret;
}
private static BigInt DigitTruncate(BigInt n1, int digits)
{
BigInt res = new BigInt(n1);
for (int i = res.digitArray.Length - 1; i >= digits; i--)
{
res.digitArray[i] = 0;
}
return res;
}
/// <summary>
/// Calculates 'this'^power
/// </summary>
/// <param name="power"></param>
public void Power(BigInt power)
{
if (power.IsZero() || power.sign)
{
Zero();
digitArray[0] = 1;
return;
}
BigInt pow = new BigInt(power);
BigInt temp = new BigInt(this);
BigInt powTerm = new BigInt(this);
pow.Decrement();
for (; !pow.IsZero(); pow.RSH(1))
{
if ((pow.digitArray[0] & 1) == 1)
{
temp.Mul(powTerm);
}
powTerm.Square();
}
Assign(temp);
}
/// <summary>
/// true iff n1 == n2
/// </summary>
/// <param name="n1"></param>
/// <param name="n2"></param>
/// <returns></returns>
public static bool Equals(BigInt n1, BigInt n2)
{
return n1.Equals(n2);
}