/// <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>
/// <remarks>This is quite a fast function, as elementary functions go. You can expect it to take
/// about twice as long as a floating-point division.
/// </remarks>
public static BigFloat Sqrt(BigFloat n1)
{
BigFloat res = new BigFloat(n1);
res.Sqrt();
return res;
}
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>
/// 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>
/// 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>
/// 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 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;
}