Core Library: Computing Pi

Computing Pi

We implement Machin's formula for computing

Pi = 3.14159265358979323846264338327950288419716939937510582097494459...
to any desired precision. It demonstrates the ease with which such high precision computations can be achieved using Core library's precision mechanism and its numerical output facility. Here is Machin's formula: 
    pi/4  = 4 arctan(1/5) - arctan(1/239)
          = \sum_{k=0} ^ \infty  (-1)k
                [   4 / ( (2k +1) 5^{2k+1} )
                  - 1 / ( (2k +1) 239^{2k+1} )  ]
Timing: Using this formula, a Sun Ultra 10 (440 MHz) can compute Pi to 2000 digits (=6644 bits) in about 1.51 seconds. (This is Core 1.3, June 2000) In some formulas (e.g., for erf(x)), there is also a need for high precision values of sqrt(Pi). The program also outputs sqrt(Pi). The program here also performs automatic self-verification of the computed values of Pi up to 250 digits, and sqrt(Pi) up to 100 digits.

Update for Core 1.7, Nov 2004

We originally implemented Brent's algorithm for Pi using Expr, and it turned out to be much slower than using Machin's formula. In "newPi.cpp", we re-implemented Brent's algorithm by replacing Expr by BigFloat. This became much faster than the old Machin's algorithm. We also reimplemented Machin's formula "newPi.cpp". In particular, we use binary splitting to compute arctan(1/x) where x is a bigInt. The BigFloat version of Brent algorithm is again slower than Machin's formula. Both are much faster than Maple. Using the same Solaris SunBlade as above to compute to 100,000 bits (30102 digits) (timing does not include printout of digits).
TEST TIME COMMENT
newPi 100000 0 3.35 sec Machin's formula using binary split
newPi 100000 1 33.3 sec Brent's algorithm using bigFloat

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