How to get many places of pi from Machin's Equation?

Richard D. Moores rdmoores at gmail.com
Sat Jan 9 17:32:48 CET 2010


On Sat, Jan 9, 2010 at 07:57, Mark Dickinson <dickinsm at gmail.com> wrote:
> On Jan 9, 11:31 am, "Richard D. Moores" <rdmoo... at gmail.com> wrote:
>> Machin's Equation is
>>
>> 4 arctan (1/5) - arctan(1/239) = pi/4
>> [...]
>>
>> Is there a way in Python 3.1 to calculate pi to greater accuracy using
>> Machin's Equation? Even to an arbitrary number of places?
>
> Here's some crude code (no error bounds,  possibility of infinite
> loops, ...) that computes pi to 1000 places using Machin's formula and
> the decimal module.  The last few digits will be bogus, and should be
> ignored.
>
> from decimal import Decimal, getcontext
>
> def atan(x):
>    # reductions
>    reductions = 0
>    while 100*abs(x) > 1:
>        reductions += 1
>        x /= 1 + (1+x*x).sqrt()
>
>    # Taylor series
>    sum = 0
>    xpow = x
>    x2 = x*x
>    k = 1
>    while True:
>        term = xpow/k
>        oldsum = sum
>        sum += term
>        if sum == oldsum:
>            break
>        k += 2
>        xpow *= -x2
>
>    return sum * 2**reductions
>
> getcontext().prec = 1000
> one = Decimal(1)
> print(16*atan(one/5) - 4*atan(one/239))
> --
> http://mail.python.org/mailman/listinfo/python-list
>

Great! Done in Python 3 with arctan and the decimal module. And the
first 997 digits were accurate.

Just what I was after.

I don't believe the Chudnovsky algorithm has been mentioned. It isn't
what I wanted, but it is amazing. (<http://pastebin.com/f2a77629f>)

Thanks, everyone!

Dick Moores



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