private static void CalculateEOnly(int numBits)
{
PrecisionSpec extendedPrecision = new PrecisionSpec(numBits + 1, PrecisionSpec.BaseType.BIN);
PrecisionSpec normalPrecision = new PrecisionSpec(numBits, PrecisionSpec.BaseType.BIN);
int iExponent = (int)(Math.Sqrt(numBits));
BigFloat factorial = new BigFloat(1, extendedPrecision);
BigFloat constant = new BigFloat(1, extendedPrecision);
constant.exponent -= iExponent;
BigFloat numerator = new BigFloat(constant);
BigFloat reciprocal;
//Calculate the 2^iExponent th root of e
BigFloat e = new BigFloat(1, extendedPrecision);
int i;
for (i = 1; i < Int32.MaxValue; i++)
{
BigFloat number = new BigFloat(i, extendedPrecision);
factorial.Mul(number);
reciprocal = factorial.Reciprocal();
reciprocal.Mul(numerator);
if (-reciprocal.exponent > numBits) break;
e.Add(reciprocal);
numerator.Mul(constant);
System.GC.Collect();
}
for (i = 0; i < iExponent; i++)
{
numerator.Assign(e);
e.Mul(numerator);
}
//Set the cached static values.
eCache = new BigFloat(e, normalPrecision);
eRCPCache = new BigFloat(e.Reciprocal(), normalPrecision);
}
/// <summary>
/// Uses the Gauss-Legendre formula for pi
/// Taken from http://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm
/// </summary>
/// <param name="numBits"></param>
private static void CalculatePi(int numBits)
{
int bits = numBits + 32;
//Precision extend taken out.
PrecisionSpec normalPres = new PrecisionSpec(numBits, PrecisionSpec.BaseType.BIN);
PrecisionSpec extendedPres = new PrecisionSpec(bits, PrecisionSpec.BaseType.BIN);
if (scratch.Precision.NumBits != bits)
{
scratch = new BigInt(extendedPres);
}
//a0 = 1
BigFloat an = new BigFloat(1, extendedPres);
//b0 = 1/sqrt(2)
BigFloat bn = new BigFloat(2, extendedPres);
bn.Sqrt();
bn.exponent--;
//to = 1/4
BigFloat tn = new BigFloat(1, extendedPres);
tn.exponent -= 2;
int pn = 0;
BigFloat anTemp = new BigFloat(extendedPres);
int iteration = 0;
int cutoffBits = numBits >> 5;
for (iteration = 0; ; iteration++)
{
//Save a(n)
anTemp.Assign(an);
//Calculate new an
an.Add(bn);
an.exponent--;
//Calculate new bn
bn.Mul(anTemp);
bn.Sqrt();
//Calculate new tn
anTemp.Sub(an);
anTemp.mantissa.SquareHiFast(scratch);
anTemp.exponent += anTemp.exponent + pn + 1 - anTemp.mantissa.Normalise();
tn.Sub(anTemp);
anTemp.Assign(an);
anTemp.Sub(bn);
if (anTemp.exponent < -(bits - cutoffBits)) break;
//New pn
pn++;
}
an.Add(bn);
an.mantissa.SquareHiFast(scratch);
an.exponent += an.exponent + 1 - an.mantissa.Normalise();
tn.exponent += 2;
an.Div(tn);
pi = new BigFloat(an, normalPres);
piBy2 = new BigFloat(pi);
piBy2.exponent--;
twoPi = new BigFloat(pi, normalPres);
twoPi.exponent++;
piRecip = new BigFloat(an.Reciprocal(), normalPres);
twoPiRecip = new BigFloat(piRecip);
twoPiRecip.exponent--;
//1/3 is going to be useful for sin.
threeRecip = new BigFloat((new BigFloat(3, extendedPres)).Reciprocal(), normalPres);
}
private static BigFloat TenPow(int power, PrecisionSpec precision)
{
BigFloat acc = new BigFloat(1, precision);
BigFloat temp = new BigFloat(1, precision);
int powerTemp = power;
BigFloat multiplierToUse = new BigFloat(10, precision);
if (power < 0)
{
multiplierToUse = multiplierToUse.Reciprocal();
powerTemp = -power;
}
//Fast power function
while (powerTemp != 0)
{
temp.Mul(multiplierToUse);
multiplierToUse.Assign(temp);
if ((powerTemp & 1) != 0)
{
acc.Mul(temp);
}
powerTemp >>= 1;
}
return acc;
}
private static BigFloat R(BigFloat a0, BigFloat b0)
{
//Precision extend taken out.
int bits = a0.mantissa.Precision.NumBits;
PrecisionSpec extendedPres = new PrecisionSpec(bits, PrecisionSpec.BaseType.BIN);
BigFloat an = new BigFloat(a0, extendedPres);
BigFloat bn = new BigFloat(b0, extendedPres);
BigFloat sum = new BigFloat(extendedPres);
BigFloat term = new BigFloat(extendedPres);
BigFloat temp1 = new BigFloat(extendedPres);
BigFloat one = new BigFloat(1, extendedPres);
int iteration = 0;
for (iteration = 0; ; iteration++)
{
//Get the sum term for this iteration.
term.Assign(an);
term.Mul(an);
temp1.Assign(bn);
temp1.Mul(bn);
//term = an^2 - bn^2
term.Sub(temp1);
//term = 2^(n-1) * (an^2 - bn^2)
term.exponent += iteration - 1;
sum.Add(term);
if (term.exponent < -(bits - 8)) break;
//Calculate the new AGM estimates.
temp1.Assign(an);
an.Add(bn);
//a(n+1) = (an + bn) / 2
an.MulPow2(-1);
//b(n+1) = sqrt(an*bn)
bn.Mul(temp1);
bn.Sqrt();
}
one.Sub(sum);
one = one.Reciprocal();
return new BigFloat(one, a0.mantissa.Precision);
}
private void Exp(int numBits)
{
if (IsSpecialValue)
{
if (SpecialValue == SpecialValueType.ZERO)
{
//e^0 = 1
exponent = 0;
mantissa.SetHighDigit(0x80000000);
}
else if (SpecialValue == SpecialValueType.INF_MINUS)
{
//e^-inf = 0
SetZero();
}
return;
}
PrecisionSpec prec = new PrecisionSpec(numBits, PrecisionSpec.BaseType.BIN);
numBits = prec.NumBits;
if (scratch.Precision.NumBits != prec.NumBits)
{
scratch = new BigInt(prec);
}
if (inverseFactorialCache == null || invFactorialCutoff < numBits)
{
CalculateFactorials(numBits);
}
//let x = 1 * 'this'.mantissa (i.e. 1 <= x < 2)
//exp(2^n * x) = e^(2^n * x) = (e^x)^2n = exp(x)^2n
int oldExponent = 0;
if (exponent > -4)
{
oldExponent = exponent + 4;
exponent = -4;
}
BigFloat thisSave = new BigFloat(this, prec);
BigFloat temp = new BigFloat(1, prec);
BigFloat temp2 = new BigFloat(this, prec);
BigFloat res = new BigFloat(1, prec);
int length = inverseFactorialCache.Length;
int iterations;
for (int i = 1; i < length; i++)
{
//temp = x^i
temp.Mul(thisSave);
temp2.Assign(inverseFactorialCache[i]);
temp2.Mul(temp);
if (temp2.exponent < -(numBits + 4)) { iterations = i; break; }
res.Add(temp2);
}
//res = exp(x)
//Now... x^(2^n) = (x^2)^(2^(n - 1))
for (int i = 0; i < oldExponent; i++)
{
res.mantissa.SquareHiFast(scratch);
int shift = res.mantissa.Normalise();
res.exponent = res.exponent << 1;
res.exponent += 1 - shift;
}
//Deal with +/- inf
if (res.exponent == Int32.MaxValue)
{
res.mantissa.Zero();
}
Assign(res);
}
/// <summary>
/// Calculates ln(2) and returns -10^(n/2 + a bit) for reuse, using the AGM method as described in
/// http://lacim.uqam.ca/~plouffe/articles/log2.pdf
/// </summary>
/// <param name="numBits"></param>
/// <returns></returns>
private static void CalculateLog2(int numBits)
{
//Use the AGM method formula to get log2 to N digits.
//R(a0, b0) = 1 / (1 - Sum(2^-n*(an^2 - bn^2)))
//log(1/2) = R(1, 10^-n) - R(1, 10^-n/2)
PrecisionSpec normalPres = new PrecisionSpec(numBits, PrecisionSpec.BaseType.BIN);
PrecisionSpec extendedPres = new PrecisionSpec(numBits + 1, PrecisionSpec.BaseType.BIN);
BigFloat a0 = new BigFloat(1, extendedPres);
BigFloat b0 = TenPow(-(int)((double)((numBits >> 1) + 2) * 0.302), extendedPres);
BigFloat pow10saved = new BigFloat(b0);
BigFloat firstAGMcacheSaved = new BigFloat(extendedPres);
//save power of 10 (in normal precision)
pow10cache = new BigFloat(b0, normalPres);
ln2cache = R(a0, b0);
//save the first half of the log calculation
firstAGMcache = new BigFloat(ln2cache, normalPres);
firstAGMcacheSaved.Assign(ln2cache);
b0.MulPow2(-1);
ln2cache.Sub(R(a0, b0));
//Convert to log(2)
ln2cache.mantissa.Sign = false;
//Save magic constant for newton log
//First guess in range 1 <= x < 2 is x0 = ln2 * (x - 1) + C
logNewtonConstant = new BigFloat(ln2cache);
logNewtonConstant.Mul(new BigFloat(3, extendedPres));
logNewtonConstant.exponent--;
logNewtonConstant.Sub(new BigFloat(1, extendedPres));
logNewtonConstant = new BigFloat(logNewtonConstant, normalPres);
//Save the inverse.
log2ecache = new BigFloat(ln2cache);
log2ecache = new BigFloat(log2ecache.Reciprocal(), normalPres);
//Now cache log10
//Because the log functions call this function to the precision to which they
//are called, we cannot call them without causing an infinite loop, so we need
//to inline the code.
log10recip = new BigFloat(10, extendedPres);
{
int power2 = log10recip.exponent + 1;
log10recip.exponent = -1;
//BigFloat res = new BigFloat(firstAGMcache);
BigFloat ax = new BigFloat(1, extendedPres);
BigFloat bx = new BigFloat(pow10saved);
bx.Mul(log10recip);
BigFloat r = R(ax, bx);
log10recip.Assign(firstAGMcacheSaved);
log10recip.Sub(r);
ax.Assign(ln2cache);
ax.Mul(new BigFloat(power2, log10recip.mantissa.Precision));
log10recip.Add(ax);
}
log10recip = log10recip.Reciprocal();
log10recip = new BigFloat(log10recip, normalPres);
//Trim to n bits
ln2cache = new BigFloat(ln2cache, normalPres);
}
/// <summary>
/// Tried the newton method for logs, but the exponential function is too slow to do it.
/// </summary>
private void LogNewton()
{
if (mantissa.IsZero() || mantissa.Sign)
{
return;
}
//Compute ln2.
if (ln2cache == null || mantissa.Precision.NumBits > ln2cache.mantissa.Precision.NumBits)
{
CalculateLog2(mantissa.Precision.NumBits);
}
int numBits = mantissa.Precision.NumBits;
//Use inverse exp function with Newton's method.
BigFloat xn = new BigFloat(this);
BigFloat oldExponent = new BigFloat(xn.exponent, mantissa.Precision);
xn.exponent = 0;
this.exponent = 0;
//Hack to subtract 1
xn.mantissa.ClearBit(numBits - 1);
//x0 = (x - 1) * log2 - this is a straight line fit between log(1) = 0 and log(2) = ln2
xn.Mul(ln2cache);
//x0 = (x - 1) * log2 + C - this corrects for minimum error over the range.
xn.Add(logNewtonConstant);
BigFloat term = new BigFloat(mantissa.Precision);
BigFloat one = new BigFloat(1, mantissa.Precision);
int precision = 32;
int normalPrecision = mantissa.Precision.NumBits;
int iterations = 0;
while (true)
{
term.Assign(xn);
term.mantissa.Sign = true;
term.Exp(precision);
term.Mul(this);
term.Sub(one);
iterations++;
if (term.exponent < -((precision >> 1) - 4))
{
if (precision == normalPrecision)
{
if (term.exponent < -(precision - 4)) break;
}
else
{
precision = precision << 1;
if (precision > normalPrecision) precision = normalPrecision;
}
}
xn.Add(term);
}
//log(2^n*s) = log(2^n) + log(s) = nlog(2) + log(s)
term.Assign(ln2cache);
term.Mul(oldExponent);
this.Assign(xn);
this.Add(term);
}
/// <summary>
/// arctan(): the inverse function of sin(), range of (-pi/2..pi/2)
/// </summary>
public void Arctan()
{
//With 2 argument reductions, we increase precision by a minimum of 4 bits per term.
int numBits = mantissa.Precision.NumBits;
int maxTerms = numBits >> 2;
if (pi == null || pi.mantissa.Precision.NumBits != numBits)
{
CalculatePi(mantissa.Precision.NumBits);
}
//Make domain positive
bool sign = mantissa.Sign;
mantissa.Sign = false;
if (IsSpecialValue)
{
if (SpecialValue == SpecialValueType.INF_MINUS || SpecialValue == SpecialValueType.INF_PLUS)
{
Assign(piBy2);
mantissa.Sign = sign;
return;
}
return;
}
if (reciprocals == null || reciprocals[0].mantissa.Precision.NumBits != numBits || reciprocals.Length < maxTerms)
{
CalculateReciprocals(numBits, maxTerms);
}
bool invert = false;
BigFloat one = new BigFloat(1, mantissa.Precision);
//Invert if outside of convergence
if (GreaterThan(one))
{
invert = true;
Assign(Reciprocal());
}
//Reduce using half-angle formula:
//arctan(2x) = 2 arctan (x / (1 + sqrt(1 + x)))
//First reduction (guarantees 2 bits per iteration)
BigFloat temp = new BigFloat(this);
temp.Mul(this);
temp.Add(one);
temp.Sqrt();
temp.Add(one);
this.Div(temp);
//Second reduction (guarantees 4 bits per iteration)
temp.Assign(this);
temp.Mul(this);
temp.Add(one);
temp.Sqrt();
temp.Add(one);
this.Div(temp);
//Actual series calculation
int length = reciprocals.Length;
BigFloat term = new BigFloat(this);
//pow = x^2
BigFloat pow = new BigFloat(this);
pow.Mul(this);
BigFloat sum = new BigFloat(this);
for (int i = 1; i < length; i++)
{
//u(n) = u(n-1) * x^2
//t(n) = u(n) / (2n+1)
term.Mul(pow);
term.Sign = !term.Sign;
temp.Assign(term);
temp.Mul(reciprocals[i]);
if (temp.exponent < -numBits) break;
sum.Add(temp);
}
//Undo the reductions.
Assign(sum);
exponent += 2;
if (invert)
{
//Assign(Reciprocal());
mantissa.Sign = true;
Add(piBy2);
}
if (sign)
{
mantissa.Sign = sign;
}
}
/// <summary>
/// Multiplies two numbers and assigns the result to res.
/// </summary>
/// <param name="res">a pre-existing BigFloat to take the result</param>
/// <param name="n1">the first number</param>
/// <param name="n2">the second number</param>
/// <returns>a handle to res</returns>
public static BigFloat Mul(BigFloat res, BigFloat n1, BigFloat n2)
{
res.Assign(n1);
res.Mul(n2);
return res;
}
/// <summary>
/// Divides two numbers and assigns the result to res.
/// </summary>
/// <param name="res">a pre-existing BigFloat to take the result</param>
/// <param name="n1">the first number</param>
/// <param name="n2">the second number</param>
/// <returns>a handle to res</returns>
public static BigFloat Div(BigFloat res, BigFloat n1, BigFloat n2)
{
res.Assign(n1);
res.Div(n2);
return res;
}
/// <summary>
/// Subtracts two numbers and assigns the result to res.
/// </summary>
/// <param name="res">a pre-existing BigFloat to take the result</param>
/// <param name="n1">the first number</param>
/// <param name="n2">the second number</param>
/// <returns>a handle to res</returns>
public static BigFloat Sub(BigFloat res, BigFloat n1, BigFloat n2)
{
res.Assign(n1);
res.Sub(n2);
return res;
}
//******************* Fast static functions ********************
/// <summary>
/// Adds two numbers and assigns the result to res.
/// </summary>
/// <param name="res">a pre-existing BigFloat to take the result</param>
/// <param name="n1">the first number</param>
/// <param name="n2">the second number</param>
/// <returns>a handle to res</returns>
public static BigFloat Add(BigFloat res, BigFloat n1, BigFloat n2)
{
res.Assign(n1);
res.Add(n2);
return res;
}
/// <summary>
/// Two-variable iterative square root, taken from
/// http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#A_two-variable_iterative_method
/// </summary>
public void Sqrt()
{
if (mantissa.Sign || IsSpecialValue)
{
if (SpecialValue == SpecialValueType.ZERO)
{
return;
}
if (SpecialValue == SpecialValueType.INF_MINUS || mantissa.Sign)
{
SetNaN();
}
return;
}
BigFloat temp2;
BigFloat temp3 = new BigFloat(mantissa.Precision);
BigFloat three = new BigFloat(3, mantissa.Precision);
int exponentScale = 0;
//Rescale to 0.5 <= x < 2
if (exponent < -1)
{
int diff = -exponent;
if ((diff & 1) != 0)
{
diff--;
}
exponentScale = -diff;
exponent += diff;
}
else if (exponent > 0)
{
if ((exponent & 1) != 0)
{
exponentScale = exponent + 1;
exponent = -1;
}
else
{
exponentScale = exponent;
exponent = 0;
}
}
temp2 = new BigFloat(this);
temp2.Sub(new BigFloat(1, mantissa.Precision));
//if (temp2.mantissa.IsZero())
//{
// exponent += exponentScale;
// return;
//}
int numBits = mantissa.Precision.NumBits;
while ((exponent - temp2.exponent) < numBits && temp2.SpecialValue != SpecialValueType.ZERO)
{
//a(n+1) = an - an*cn / 2
temp3.Assign(this);
temp3.Mul(temp2);
temp3.MulPow2(-1);
this.Sub(temp3);
//c(n+1) = cn^2 * (cn - 3) / 4
temp3.Assign(temp2);
temp2.Sub(three);
temp2.Mul(temp3);
temp2.Mul(temp3);
temp2.MulPow2(-2);
}
exponent += (exponentScale >> 1);
}
//******************** Mathematical Constants *******************
/// <summary>
/// Gets pi to the indicated precision
/// </summary>
/// <param name="precision">The precision to perform the calculation to</param>
/// <returns>pi (the ratio of the area of a circle to its diameter)</returns>
public static BigFloat GetPi(PrecisionSpec precision)
{
if (pi == null || precision.NumBits <= pi.mantissa.Precision.NumBits)
{
CalculatePi(precision.NumBits);
}
BigFloat ret = new BigFloat (precision);
ret.Assign(pi);
return ret;
}
/// <summary>
/// Log(x) implemented as an Arithmetic-Geometric Mean. Fast for high precisions.
/// </summary>
private void LogAGM1()
{
if (mantissa.IsZero() || mantissa.Sign)
{
return;
}
//Compute ln2.
if (ln2cache == null || mantissa.Precision.NumBits > ln2cache.mantissa.Precision.NumBits)
{
CalculateLog2(mantissa.Precision.NumBits);
}
//Compute ln(x) using AGM formula
//1. Re-write the input as 2^n * (0.5 <= x < 1)
int power2 = exponent + 1;
exponent = -1;
//BigFloat res = new BigFloat(firstAGMcache);
BigFloat a0 = new BigFloat(1, mantissa.Precision);
BigFloat b0 = new BigFloat(pow10cache);
b0.Mul(this);
BigFloat r = R(a0, b0);
this.Assign(firstAGMcache);
this.Sub(r);
a0.Assign(ln2cache);
a0.Mul(new BigFloat(power2, mantissa.Precision));
this.Add(a0);
}
/// <summary>
/// Get e to the indicated precision
/// </summary>
/// <param name="precision">The preicision to perform the calculation to</param>
/// <returns>e (the number for which the d/dx(e^x) = e^x)</returns>
public static BigFloat GetE(PrecisionSpec precision)
{
if (eCache == null || eCache.mantissa.Precision.NumBits < precision.NumBits)
{
CalculateEOnly(precision.NumBits);
//CalculateFactorials(precision.NumBits);
}
BigFloat ret = new BigFloat(precision);
ret.Assign(eCache);
return ret;
}
/// <summary>
/// Calculates Sin(x):
/// This takes a little longer and is less accurate if the input is out of the range (-pi, pi].
/// </summary>
public void Sin()
{
if (IsSpecialValue)
{
//Sin(x) has no limit as x->inf
if (SpecialValue == SpecialValueType.INF_MINUS || SpecialValue == SpecialValueType.INF_PLUS)
{
SetNaN();
}
return;
}
//Convert to positive range (0 <= x < inf)
bool sign = mantissa.Sign;
mantissa.Sign = false;
if (pi == null || pi.mantissa.Precision.NumBits != mantissa.Precision.NumBits)
{
CalculatePi(mantissa.Precision.NumBits);
}
if (inverseFactorialCache == null || invFactorialCutoff != mantissa.Precision.NumBits)
{
CalculateFactorials(mantissa.Precision.NumBits);
}
//Rescale into 0 <= x < 2*pi
if (GreaterThan(twoPi))
{
//There will be an inherent loss of precision doing this.
BigFloat newAngle = new BigFloat(this);
newAngle.Mul(twoPiRecip);
newAngle.FPart();
newAngle.Mul(twoPi);
Assign(newAngle);
}
//Rescale into range 0 <= x < pi
if (GreaterThan(pi))
{
//sin(pi + a) = sin(pi)cos(a) + sin(a)cos(pi) = 0 - sin(a) = -sin(a)
Sub(pi);
sign = !sign;
}
BigFloat temp = new BigFloat(mantissa.Precision);
//Rescale into range 0 <= x < pi/2
if (GreaterThan(piBy2))
{
temp.Assign(this);
Assign(pi);
Sub(temp);
}
//Rescale into range 0 <= x < pi/6 to accelerate convergence.
//This is done using sin(3x) = 3sin(x) - 4sin^3(x)
Mul(threeRecip);
if (mantissa.IsZero())
{
exponent = 0;
return;
}
BigFloat term = new BigFloat(this);
BigFloat square = new BigFloat(this);
square.Mul(term);
BigFloat sum = new BigFloat(this);
bool termSign = true;
int length = inverseFactorialCache.Length;
int numBits = mantissa.Precision.NumBits;
for (int i = 3; i < length; i += 2)
{
term.Mul(square);
temp.Assign(inverseFactorialCache[i]);
temp.Mul(term);
temp.mantissa.Sign = termSign;
termSign = !termSign;
if (temp.exponent < -numBits) break;
sum.Add(temp);
}
//Restore the triple-angle: sin(3x) = 3sin(x) - 4sin^3(x)
Assign(sum);
sum.Mul(this);
sum.Mul(this);
Mul(new BigFloat(3, mantissa.Precision));
sum.exponent += 2;
Sub(sum);
//Restore the sign
mantissa.Sign = sign;
}
