Compute Pi to a ludicrously high precision
This sample demonstrates how to calculate Pi to a
specified number of digits and display the results in a
dialog box. It uses Fabrice Bellard's algorithm, and is based
on Chris Sells' C# version.
Run this sample with C# Programmable Calculator.
NOTE: To copy the following source code into C# Programmable Calculator,
right-click the code and select the "View Source" or "View Page Source"
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Calculator's code editor.
/*
* Computation of the n'th decimal digit of \pi with very little memory.
* Written by Fabrice Bellard on January 8, 1997.
*
* We use a slightly modified version of the method described by Simon
* Plouffe in "On the Computation of the n'th decimal digit of various
* transcendental numbers" (November 1996). We have modified the algorithm
* to get a running time of O(n^2) instead of O(n^3log(n)^3).
*
* This program uses mostly integer arithmetic. It may be slow on some
* hardwares where integer multiplications and divisons must be done
* by software. We have supposed that 'int' has a size of 32 bits. If
* your compiler supports 'long long' integers of 64 bits, you may use
* the integer version of 'mul_mod' (see HAS_LONG_LONG).
*/
private static int mul_mod(int a, int b, int m)
{
return (int) (((long) a * (long) b) % m);
}
/* return the inverse of x mod y */
private static int inv_mod(int x, int y)
{
int q,u,v,a,c,t;
u = x;
v = y;
c = 1;
a = 0;
do
{
q= v / u;
t = c;
c = a - q * c;
a = t;
t = u;
u = v - q * u;
v = t;
} while (u != 0);
a =a % y;
if (a < 0)
{
a = y + a;
}
return a;
}
/* return (a^b) mod m */
private static int pow_mod(int a, int b, int m)
{
int r, aa;
r = 1;
aa = a;
while (true)
{
if ((b & 1) != 0)
{
r = mul_mod(r, aa, m);
}
b = b >> 1;
if (b == 0)
{
break;
}
aa = mul_mod(aa, aa, m);
}
return r;
}
/* return true if n is prime */
private static bool is_prime(int n)
{
if ((n % 2) == 0)
{
return false;
}
int r = (int) Math.Sqrt(n);
for (int i = 3; i <= r; i += 2)
{
if ((n % i) == 0)
{
return false;
}
}
return true;
}
/* return the prime number immediatly after n */
private static int next_prime(int n)
{
do
{
n++;
} while (!is_prime(n));
return n;
}
private static String CalculatePiDigits(int n)
{
int av, vmax, num, den, s, t;
int N = (int) ((n + 20) * Math.Log(10) / Math.Log(2));
double sum = 0;
for (int a = 3; a <= (2 * N); a = next_prime(a))
{
vmax = (int) (Math.Log(2 * N) / Math.Log(a));
av = 1;
for (int i = 0; i < vmax; i++)
{
av = av * a;
}
s = 0;
num = 1;
den = 1;
int v = 0;
int kq = 1;
int kq2 = 1;
for (int k = 1; k <= N; k++)
{
t = k;
if (kq >= a)
{
do
{
t = t / a;
v--;
} while ((t % a) == 0);
kq = 0;
}
kq++;
num = mul_mod(num, t, av);
t = 2 * k - 1;
if (kq2 >= a)
{
if (kq2 == a)
{
do
{
t = t / a;
v++;
} while ((t % a) == 0);
}
kq2 -= a;
}
den = mul_mod(den, t, av);
kq2 += 2;
if (v > 0)
{
t = inv_mod(den, av);
t = mul_mod(t, num, av);
t = mul_mod(t, k, av);
for (int i = v; i < vmax; i++)
{
t = mul_mod(t, a, av);
}
s += t;
if (s >= av)
{
s -= av;
}
}
}
t = pow_mod(10, n - 1, av);
s = mul_mod(s, t, av);
sum = (sum + (double) s / (double) av) % 1.0;
}
int Result = (int) (sum * 1e9);
String StringResult = String.Format("{0:D9}", Result);
return StringResult;
}
// Put a space between every group of 10 digits.
private static String breakDigitsIntoGroupsOf10(String digits)
{
String result = "";
while (digits.Length > 10)
{
result += digits.Substring(0, 10) + " ";
digits = digits.Substring(10, digits.Length - 10);
}
result += digits;
return result;
}
[Button("Samples", "Calculate Pi")]
public static void CalculatePi(int digits)
{
ArrayList lines = new ArrayList();
lines.Add("3");
StringBuilder result = new StringBuilder();
DateTime StartTime = DateTime.Now;
if (digits > 0)
{
lines[0] += ".";
for (int i = 0; i < digits; i += 9)
{
String ds = CalculatePiDigits(i+1);
int digitCount = Math.Min(digits - i, 9);
while (ds.Length < 9)
{
ds = "0" + ds;
}
result.Append(ds.Substring(0, digitCount));
}
}
TimeSpan RunTime = DateTime.Now - StartTime;
const int charsPerLine = 100;
while (result.Length >= charsPerLine)
{
lines.Add(breakDigitsIntoGroupsOf10(result.ToString().Substring(0, charsPerLine)));
result = result.Remove(0, charsPerLine);
}
if (result.Length > 0)
{
lines.Add(breakDigitsIntoGroupsOf10(result.ToString()));
}
lines.Add("");
lines.Add("Calculated the first " + digits + " digits of pi in " + RunTime);
iGUIUtils.DisplayTextLines("The first " + digits + " digits of pi", (String[]) lines.ToArray(typeof(String)));
}