Dave Angel wrote:
time echo "scale = 1010; 16 * a(1/5) - 4 * a(1/239)" |bc -lq
Wouldn't it be shorter to say:
time echo "scale = 1010; 4 * a(1)" |bc -lq
Well, you can check it out by doing the math... (its fun...)
... you will notice that 'time' is called first, which on *nix systems
clocks the processing, breaking out the system and user times... right?
... so try these 10,000 comparisons on your *nix system:
time echo "scale = 10010; 16 * a(1/5) - 4 * a(1/239)" |bc -lq
time echo "scale = 10010; 4 * a(1)" |bc -lq
(those will take a few minutes, if your processor is running 2-3Ghz...)
... then try these 100,000 runs:
time echo "scale = 100010; 16 * a(1/5) - 4 * a(1/239)" |bc -lq
time echo "scale = 100010; 4 * a(1)" |bc -lq
(Those will take some time, probably less than 20 - 30 minutes... )
After your time comparisons, tell me whether you want to use a(1)*4 ??
Leonard Euler came up with the formula I'm using here... and used it
himself for paper 'n pencil arithmetic... because the arctan(n) infinite
series converges much quicker (by orders of magnitude) for values of (n)
< 0. (the smaller the (n) the better)
We can make the entire function run even faster by using smp and
splitting the 'a(1/5)' and 'a(1/239)' across two cores, having the
arctan series' running in parallel. This is important if your 'big num'
is going to be hundreds of thousands or millions of places. On my baby
Beowulf cluster I have played a bit with running this on two separate
systems and then bringing the result together in the end... fun for
playtime... interesting to see how many digits (in how much time) can be
achieved *without* using a super-computer....
You can also try these formula for comparison sake:
PI = 20 * a(1/7) + 8 * a(3/79)
or
PI = 8 * a(1/3) + 4 * a(1/7)
or
PI = 24 * a(1/8) + 8 * a(1/57) + 4 * a(1/239)
Happy Easter, and have a slice of pie on me.... :)
kind regards,
m harris
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