/// <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>
/// 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);
}
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 tan(x)
/// </summary>
public void Tan()
{
if (IsSpecialValue)
{
//Tan(x) has no limit as x->inf
if (SpecialValue == SpecialValueType.INF_MINUS || SpecialValue == SpecialValueType.INF_PLUS)
{
SetNaN();
}
else if (SpecialValue == SpecialValueType.ZERO)
{
SetZero();
}
return;
}
if (pi == null || pi.mantissa.Precision.NumBits != mantissa.Precision.NumBits)
{
CalculatePi(mantissa.Precision.NumBits);
}
//Work out the sign change (involves replicating some rescaling).
bool sign = mantissa.Sign;
mantissa.Sign = false;
if (mantissa.IsZero())
{
return;
}
//Rescale into 0 <= x < pi
if (GreaterThan(pi))
{
//There will be an inherent loss of precision doing this.
BigFloat newAngle = new BigFloat(this);
newAngle.Mul(piRecip);
newAngle.FPart();
newAngle.Mul(pi);
Assign(newAngle);
}
//Rescale to -pi/2 <= x < pi/2
if (!LessThan(piBy2))
{
Sub(pi);
}
//Now the sign of the sin determines the sign of the tan.
//tan(x) = sin(x) / sqrt(1 - sin^2(x))
Sin();
BigFloat denom = new BigFloat(this);
denom.Mul(this);
denom.Sub(new BigFloat(1, mantissa.Precision));
denom.mantissa.Sign = !denom.mantissa.Sign;
if (denom.mantissa.Sign)
{
denom.SetZero();
}
denom.Sqrt();
Div(denom);
if (sign) mantissa.Sign = !mantissa.Sign;
}
/// <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);
}
private static void CalculateFactorials(int numBits)
{
System.Collections.Generic.List<BigFloat> list = new System.Collections.Generic.List<BigFloat>(64);
System.Collections.Generic.List<BigFloat> list2 = new System.Collections.Generic.List<BigFloat>(64);
PrecisionSpec extendedPrecision = new PrecisionSpec(numBits + 1, PrecisionSpec.BaseType.BIN);
PrecisionSpec normalPrecision = new PrecisionSpec(numBits, PrecisionSpec.BaseType.BIN);
BigFloat factorial = new BigFloat(1, extendedPrecision);
BigFloat reciprocal;
//Calculate e while we're at it
BigFloat e = new BigFloat(1, extendedPrecision);
list.Add(new BigFloat(factorial, normalPrecision));
for (int i = 1; i < Int32.MaxValue; i++)
{
BigFloat number = new BigFloat(i, extendedPrecision);
factorial.Mul(number);
if (factorial.exponent > numBits) break;
list2.Add(new BigFloat(factorial, normalPrecision));
reciprocal = factorial.Reciprocal();
e.Add(reciprocal);
list.Add(new BigFloat(reciprocal, normalPrecision));
}
//Set the cached static values.
inverseFactorialCache = list.ToArray();
factorialCache = list2.ToArray();
invFactorialCutoff = numBits;
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);
}
/// <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);
}
/// <summary>
/// Raises a number to an integer power (positive or negative). This is a very accurate and fast function,
/// comparable to or faster than division (although it is slightly slower for
/// negative powers, obviously)
///
/// </summary>
/// <param name="power"></param>
public void Pow(int power)
{
BigFloat acc = new BigFloat(1, mantissa.Precision);
BigFloat temp = new BigFloat(1, mantissa.Precision);
int powerTemp = power;
if (power < 0)
{
Assign(Reciprocal());
powerTemp = -power;
}
//Fast power function
while (powerTemp != 0)
{
temp.Mul(this);
Assign(temp);
if ((powerTemp & 1) != 0)
{
acc.Mul(temp);
}
powerTemp >>= 1;
}
Assign(acc);
}
/// <summary>
/// Arcsinh(): the inverse sinh function
/// </summary>
public void Arcsinh()
{
//Just let all special values fall through
if (IsSpecialValue)
{
return;
}
BigFloat temp = new BigFloat(this);
temp.Mul(this);
temp.Add(new BigFloat(1, mantissa.Precision));
temp.Sqrt();
Add(temp);
Log();
}
/// <summary>
/// Arccosh(): the inverse cosh() function
/// </summary>
public void Arccosh()
{
//acosh isn't defined for x < 1
if (IsSpecialValue)
{
if (SpecialValue == SpecialValueType.INF_MINUS || SpecialValue == SpecialValueType.ZERO)
{
SetNaN();
return;
}
return;
}
BigFloat one = new BigFloat(1, mantissa.Precision);
if (LessThan(one))
{
SetNaN();
return;
}
BigFloat temp = new BigFloat(this);
temp.Mul(this);
temp.Sub(one);
temp.Sqrt();
Add(temp);
Log();
}
/// <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>
/// arcsin(): the inverse function of sin(), range of (-pi/2..pi/2)
/// </summary>
public void Arcsin()
{
if (IsSpecialValue)
{
if (SpecialValue == SpecialValueType.INF_MINUS || SpecialValue == SpecialValueType.INF_PLUS || SpecialValue == SpecialValueType.NAN)
{
SetNaN();
return;
}
return;
}
BigFloat one = new BigFloat(1, mantissa.Precision);
BigFloat plusABit = new BigFloat(1, mantissa.Precision);
plusABit.exponent -= (mantissa.Precision.NumBits - (mantissa.Precision.NumBits >> 6));
BigFloat onePlusABit = new BigFloat(1, mantissa.Precision);
onePlusABit.Add(plusABit);
bool sign = mantissa.Sign;
mantissa.Sign = false;
if (GreaterThan(onePlusABit))
{
SetNaN();
}
else if (LessThan(one))
{
BigFloat temp = new BigFloat(this);
temp.Mul(this);
temp.Sub(one);
temp.mantissa.Sign = !temp.mantissa.Sign;
temp.Sqrt();
temp.Add(one);
Div(temp);
Arctan();
exponent++;
mantissa.Sign = sign;
}
else
{
if (pi == null || pi.mantissa.Precision.NumBits != mantissa.Precision.NumBits)
{
CalculatePi(mantissa.Precision.NumBits);
}
Assign(piBy2);
if (sign) mantissa.Sign = true;
}
}
/// <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;
}
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;
}
/// <summary>
/// Multiplies two numbers and returns the result
/// </summary>
public static BigFloat Mul(BigFloat n1, BigFloat n2)
{
BigFloat ret = new BigFloat(n1);
ret.Mul(n2);
return ret;
}
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);
}
/// <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;
}
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>
/// Returns a base-10 string representing the number.
///
/// Note: This is inefficient and possibly inaccurate. Please use with enough
/// rounding digits (set using the RoundingDigits property) to ensure accuracy
/// </summary>
public override string ToString()
{
if (IsSpecialValue)
{
SpecialValueType s = SpecialValue;
if (s == SpecialValueType.ZERO)
{
return String.Format("0{0}0", System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NumberDecimalSeparator);
}
else if (s == SpecialValueType.INF_PLUS)
{
return System.Globalization.CultureInfo.CurrentCulture.NumberFormat.PositiveInfinitySymbol;
}
else if (s == SpecialValueType.INF_MINUS)
{
return System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NegativeInfinitySymbol;
}
else if (s == SpecialValueType.NAN)
{
return System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NaNSymbol;
}
else
{
return "Unrecognised special type";
}
}
if (scratch.Precision.NumBits != mantissa.Precision.NumBits)
{
scratch = new BigInt(mantissa.Precision);
}
//The mantissa expresses 1.xxxxxxxxxxx
//The highest possible value for the mantissa without the implicit 1. is 0.9999999...
scratch.Assign(mantissa);
//scratch.Round(3);
scratch.Sign = false;
BigInt denom = new BigInt("0", mantissa.Precision);
denom.SetBit(mantissa.Precision.NumBits - 1);
bool useExponentialNotation = false;
int halfBits = mantissa.Precision.NumBits / 2;
if (halfBits > 60) halfBits = 60;
int precDec = 10;
if (exponent > 0)
{
if (exponent < halfBits)
{
denom.RSH(exponent);
}
else
{
useExponentialNotation = true;
}
}
else if (exponent < 0)
{
int shift = -(exponent);
if (shift < precDec)
{
scratch.RSH(shift);
}
else
{
useExponentialNotation = true;
}
}
string output;
if (useExponentialNotation)
{
int absExponent = exponent;
if (absExponent < 0) absExponent = -absExponent;
int powerOf10 = (int)((double)absExponent * Math.Log10(2.0));
//Use 1 extra digit of precision (this is actually 32 bits more, nb)
BigFloat thisFloat = new BigFloat(this, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
thisFloat.mantissa.Sign = false;
//Multiplicative correction factor to bring number into range.
BigFloat one = new BigFloat(1, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
BigFloat ten = new BigFloat(10, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
BigFloat tenRCP = ten.Reciprocal();
//Accumulator for the power of 10 calculation.
BigFloat acc = new BigFloat(1, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
BigFloat tenToUse;
if (exponent > 0)
{
tenToUse = new BigFloat(tenRCP, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
}
else
{
tenToUse = new BigFloat(ten, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
}
BigFloat tenToPower = new BigFloat(1, new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN));
int powerTemp = powerOf10;
//Fast power function
while (powerTemp != 0)
{
tenToPower.Mul(tenToUse);
tenToUse.Assign(tenToPower);
if ((powerTemp & 1) != 0)
{
acc.Mul(tenToPower);
}
powerTemp >>= 1;
}
thisFloat.Mul(acc);
//If we are out of range, correct.
if (thisFloat.GreaterThan(ten))
{
thisFloat.Mul(tenRCP);
if (exponent > 0)
{
powerOf10++;
}
else
{
powerOf10--;
}
}
else if (thisFloat.LessThan(one))
{
thisFloat.Mul(ten);
if (exponent > 0)
{
powerOf10--;
}
else
{
powerOf10++;
}
}
//Restore the precision and the sign.
BigFloat printable = new BigFloat(thisFloat, mantissa.Precision);
printable.mantissa.Sign = mantissa.Sign;
output = printable.ToString();
if (exponent < 0) powerOf10 = -powerOf10;
output = String.Format("{0}E{1}", output, powerOf10);
}
else
{
BigInt bigDigit = BigInt.Div(scratch, denom);
bigDigit.Sign = false;
scratch.Sub(BigInt.Mul(denom, bigDigit));
if (mantissa.Sign)
{
output = String.Format("-{0}.", bigDigit);
}
else
{
output = String.Format("{0}.", bigDigit);
}
denom = BigInt.Div(denom, 10u);
while (!denom.IsZero())
{
uint digit = (uint)BigInt.Div(scratch, denom);
if (digit == 10) digit--;
scratch.Sub(BigInt.Mul(denom, digit));
output = String.Format("{0}{1}", output, digit);
denom = BigInt.Div(denom, 10u);
}
output = RoundString(output, RoundingDigits);
}
return output;
}
/// <summary>
/// Constructs a BigFloat from a string
/// </summary>
/// <param name="value"></param>
/// <param name="mantissaPrec"></param>
public BigFloat(string value, PrecisionSpec mantissaPrec)
{
Init(mantissaPrec);
PrecisionSpec extendedPres = new PrecisionSpec(mantissa.Precision.NumBits + 1, PrecisionSpec.BaseType.BIN);
BigFloat ten = new BigFloat(10, extendedPres);
BigFloat iPart = new BigFloat(extendedPres);
BigFloat fPart = new BigFloat(extendedPres);
BigFloat tenRCP = ten.Reciprocal();
if (value.Contains(System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NaNSymbol))
{
SetNaN();
return;
}
else if (value.Contains(System.Globalization.CultureInfo.CurrentCulture.NumberFormat.PositiveInfinitySymbol))
{
SetInfPlus();
return;
}
else if (value.Contains(System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NegativeInfinitySymbol))
{
SetInfMinus();
return;
}
string decimalpoint = System.Globalization.CultureInfo.CurrentCulture.NumberFormat.NumberDecimalSeparator;
char[] digitChars = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', ',', '.' };
//Read in the integer part up the the decimal point.
bool sign = false;
value = value.Trim();
int i = 0;
if (value.Length > i && value[i] == '-')
{
sign = true;
i++;
}
if (value.Length > i && value[i] == '+')
{
i++;
}
for ( ; i < value.Length; i++)
{
//break on decimal point
if (value[i] == decimalpoint[0]) break;
int digit = Array.IndexOf(digitChars, value[i]);
if (digit < 0) break;
//Ignore place separators (assumed either , or .)
if (digit > 9) continue;
if (i > 0) iPart.Mul(ten);
iPart.Add(new BigFloat(digit, extendedPres));
}
//If we've run out of characters, assign everything and return
if (i == value.Length)
{
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
return;
}
//Assign the characters after the decimal point to fPart
if (value[i] == '.' && i < value.Length - 1)
{
BigFloat RecipToUse = new BigFloat(tenRCP);
for (i++; i < value.Length; i++)
{
int digit = Array.IndexOf(digitChars, value[i]);
if (digit < 0) break;
BigFloat temp = new BigFloat(digit, extendedPres);
temp.Mul(RecipToUse);
RecipToUse.Mul(tenRCP);
fPart.Add(temp);
}
}
//If we're run out of characters, add fPart and iPart and return
if (i == value.Length)
{
iPart.Add(fPart);
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
return;
}
if (value[i] == '+' || value[i] == '-') i++;
if (i == value.Length)
{
iPart.Add(fPart);
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
return;
}
//Look for exponential notation.
if ((value[i] == 'e' || value[i] == 'E') && i < value.Length - 1)
{
//Convert the exponent to an int.
int exp;
try
{
exp = System.Convert.ToInt32(new string(value.ToCharArray(i + 1, value.Length - (i + 1))));
}
catch (Exception)
{
iPart.Add(fPart);
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
return;
}
//Raise or lower 10 to the power of the exponent
BigFloat acc = new BigFloat(1, extendedPres);
BigFloat temp = new BigFloat(1, extendedPres);
int powerTemp = exp;
BigFloat multiplierToUse;
if (exp < 0)
{
multiplierToUse = new BigFloat(tenRCP);
powerTemp = -exp;
}
else
{
multiplierToUse = new BigFloat(ten);
}
//Fast power function
while (powerTemp != 0)
{
temp.Mul(multiplierToUse);
multiplierToUse.Assign(temp);
if ((powerTemp & 1) != 0)
{
acc.Mul(temp);
}
powerTemp >>= 1;
}
iPart.Add(fPart);
iPart.Mul(acc);
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
return;
}
iPart.Add(fPart);
iPart.mantissa.Sign = sign;
exponent = iPart.exponent;
if (mantissa.AssignHigh(iPart.mantissa)) exponent++;
}
/// <summary>
/// Newton's method reciprocal, fastest for larger precisions over 15,000 bits.
/// </summary>
/// <returns>The reciprocal 1/this</returns>
private BigFloat ReciprocalNewton2()
{
if (mantissa.IsZero())
{
exponent = Int32.MaxValue;
return null;
}
bool oldSign = mantissa.Sign;
int oldExponent = exponent;
//Kill exponent for now (will re-institute later)
exponent = 0;
BigFloat reciprocal = new BigFloat(mantissa.Precision);
BigFloat constant2 = new BigFloat(mantissa.Precision);
BigFloat temp = new BigFloat(mantissa.Precision);
reciprocal.exponent = 1;
reciprocal.mantissa.SetHighDigit(3129112985u);
constant2.exponent = 1;
constant2.mantissa.SetHighDigit(0x80000000u);
//D is deliberately left negative for all the following operations.
mantissa.Sign = true;
//Initial estimate.
reciprocal.Add(this);
//mantissa.Sign = false;
//Shift down into 0.5 < this < 1 range
mantissa.RSH(1);
//Iteration.
int accuracyBits = 2;
int mantissaBits = mantissa.Precision.NumBits;
//Each iteration is a pass of newton's method for RCP.
//The is a substantial optimisation to be done here...
//You can double the number of bits for the calculations
//at each iteration, meaning that the whole process only
//takes some constant multiplier of the time for the
//full-scale multiplication.
while (accuracyBits < mantissaBits)
{
//temp = Xn
temp.exponent = reciprocal.exponent;
temp.mantissa.Assign(reciprocal.mantissa);
//temp = -Xn * D
temp.Mul(this);
//temp = -Xn * D + 2 (= 2 - Xn * D)
temp.Add(constant2);
//reciprocal = X(n+1) = Xn * (2 - Xn * D)
reciprocal.Mul(temp);
accuracyBits *= 2;
}
//'reciprocal' is now the reciprocal of the shifted down, zero-exponent mantissa of 'this'
//Restore the mantissa.
mantissa.LSH(1);
exponent = oldExponent;
mantissa.Sign = oldSign;
reciprocal.exponent = -(oldExponent + 1);
reciprocal.mantissa.Sign = oldSign;
return reciprocal;
}
