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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