Several formulae for calculating 100 million digits of Pi in less than 5 minutes and a Billion in an hour, using python and GMPY2
I wanted to see how long it would take to calulate pi to a million places to answer a kid's question.
Turns out a million is about 0.6 seconds.
You can use Chudnovsky (--alog 10) to calculate 1 billion digits in about 35 minutes. Others take around an hour or so. Manchin formulas take longer, I only had patience for 100 million at 35 minutes, Manchin would take about 9 hours for a billion I think. Memory gets to be a big deal for Manchin 30 or forty GIG. Don't even try a million with Bellard (--alog 8).
I found a page https://medium.com/@cosinekitty/how-to-calculate-a-million-digits-of-pi-d62ce3db8f58 that had a program using Machin's formula from 1706:
The article also included a link to several Machin-like formulae: https://en.wikipedia.org/wiki/Machin-like_formula I added several to the code. I also used GMPY2's mpfr() and mpz types to speed things along instead of just big integer support in python. The GMPY2 library has a wide range of arbitrary precision math functions. I also changed all of the Manchin type formulae use Multiproccessing, so the more CPUs the better.
I added 7 Machin formulae like:
Then I changed and added the Chudnovsky Bros. formula (The clear speed winner!) from Criag Wood's cool article https://www.craig-wood.com/nick/articles/pi-chudnovsky/:
Finaly I added a version of Arithmatic Geometric Mean by T.Ooura https://www.kurims.kyoto-u.ac.jp/~ooura/pi_fft.html:
The GMPY2.const_pi() function (--alog 11) seems to be about as fast as this AGM (--alog 9). Both are about half as fast as the Chudnovsky one for larger numbers of digits (> 100,000)
Here is a sample output of Binary Splitting Chudnovsky, about 3 minutes for 100 million digits on my old laptop:
python pi-pourri.py -d 100,000,000 -a 10 -v -f pi.txt
[INFO] 2022-09-25 15:59:14,361 <module>: MainProcess Computing π to 100,000,000 digits.
[DEBUG] 2022-09-25 15:59:24,276 compute: MainProcess Starting Chudnovsky brothers 1988
π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
formula to 100,000,000 decimal places
[DEBUG] 2022-09-25 15:59:29,742 __bs: MainProcess Chudnovsky ... 1,000,000 iterations and 5.47 seconds.
[DEBUG] 2022-09-25 15:59:36,543 __bs: MainProcess Chudnovsky ... 2,000,000 iterations and 12.27 seconds.
[DEBUG] 2022-09-25 15:59:42,042 __bs: MainProcess Chudnovsky ... 3,000,000 iterations and 17.77 seconds.
[DEBUG] 2022-09-25 15:59:51,825 __bs: MainProcess Chudnovsky ... 4,000,000 iterations and 27.55 seconds.
[DEBUG] 2022-09-25 15:59:57,436 __bs: MainProcess Chudnovsky ... 5,000,000 iterations and 33.16 seconds.
[DEBUG] 2022-09-25 16:00:04,441 __bs: MainProcess Chudnovsky ... 6,000,000 iterations and 40.17 seconds.
[DEBUG] 2022-09-25 16:00:10,074 __bs: MainProcess Chudnovsky ... 7,000,000 iterations and 45.80 seconds.
[DEBUG] 2022-09-25 16:00:26,668 __bs: MainProcess Chudnovsky ... 8,000,000 iterations and 62.39 seconds.
[DEBUG] 2022-09-25 16:00:33,767 __bs: MainProcess Chudnovsky ... 9,000,000 iterations and 69.49 seconds.
[DEBUG] 2022-09-25 16:00:39,648 __bs: MainProcess Chudnovsky ... 10,000,000 iterations and 75.37 seconds.
[DEBUG] 2022-09-25 16:00:49,520 __bs: MainProcess Chudnovsky ... 11,000,000 iterations and 85.24 seconds.
[DEBUG] 2022-09-25 16:00:55,810 __bs: MainProcess Chudnovsky ... 12,000,000 iterations and 91.53 seconds.
[DEBUG] 2022-09-25 16:01:02,876 __bs: MainProcess Chudnovsky ... 13,000,000 iterations and 98.60 seconds.
[DEBUG] 2022-09-25 16:01:08,859 __bs: MainProcess Chudnovsky ... 14,000,000 iterations and 104.58 seconds.
[DEBUG] 2022-09-25 16:01:51,402 compute: MainProcess Chudnovsky brothers 1988
π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
calulation Done! 14,102,733 iterations and 147.13 seconds.
[INFO] 2022-09-25 16:02:16,558 <module>: MainProcess Last 5 digits of π were 51592 as expected at offset 99,999,995
[INFO] 2022-09-25 16:02:17,075 <module>: MainProcess Calculated π to 100,000,000 digits using a formula of:
10 Chudnovsky brothers 1988
π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
[DEBUG] 2022-09-25 16:02:17,075 <module>: MainProcess Wrote 100,000,000 digits of π to file pi.txt in 0:00:00.516872
[INFO] 2022-09-25 16:02:17,075 <module>: MainProcess Calculation took 14,102,733 iterations and 0:02:52.269288.
Here is a version of Craig Wood's chart:
Here is the help for the program type python3 pi-pourri.py -h to see it
usage: pi-pourri.py [-h] [-f [FILENAME]] [-d [1 to 1,000,000,000]] [-a [1 to 12]] [--verbose] [--quiet]
pi-pourri.py runs an algoritym from a list to calulate Pi to a number of decimal places
Default: pi-pourri.py --digits 100000 --file pi.txt --alog 4
So -d 100,000,000 will take a while to finish, -d 1,000,000 very quickly
A last 5 digit validity check is done on powers of ten (10,...100,000,000)
eg. pi-pourri.py --file elbow.txt -d 1000000 -a 10
pi-pourri.py -f test.txt -d 123,456
List of Formulae:
1 John Machin 1706
π/4 = 4*arctan(1/5)
- arctan(1/239)
2 F. C. M. Störmer 1896
π/4 = 44*arctan(1/57)
+ 7*arctan(1/239)
- 12*arctan(1/682)
+ 24*arctan(1/12943)
3 Kikuo Takano 1982
π/4 = 12*arctan(1/49)
+ 32*arctan(1/57)
- 5*arctan(1/239)
+ 12*arctan(1/110443)
4 Hwang Chien-Lih, 1997
π/4 = 183*arctan(1/239)
+ 32*arctan(1/1023)
- 68*arctan(1/5832)
+ 12*arctan(1/110443)
- 12*arctan(1/4841182)
- 100*arctan(1/6826318)
5 Hwang Chien-Lih, 2003
π/4 = 183*arctan(1/239)
+ 32*arctan(1/1023)
- 68*arctan(1/5832)
+ 12*arctan(1/113021)
- 100*arctan(1/6826318)
- 12*arctan(1/33366019650)
+ 12*arctan(1/43599522992503626068)
6 Jörg Uwe Arndt 1993
π/4 = 36462*arctan(1/390112)
+ 135908*arctan(1/485298)
+ 274509*arctan(1/683982)
- 39581*arctan(1/1984933)
+ 178477*arctan(1/2478328)
- 114569*arctan(1/3449051)
- 146571*arctan(1/18975991)
+ 61914*arctan(1/22709274)
- 69044*arctan(1/24208144)
- 89431*arctan(1/201229582)
- 43938*arctan(1/2189376182)
7 Hwang Chien-Lih, 2004
π/4 = 36462*arctan(1/51387)
+ 26522*arctan(1/485298)
+ 19275*arctan(1/683982)
- 3119*arctan(1/1984933)
- 3833*arctan(1/2478328)
- 5183*arctan(1/3449051)
- 37185*arctan(1/18975991)
- 11010*arctan(1/22709274)
+ 3880*arctan(1/24208144)
- 16507*arctan(1/201229582)
- 7476*arctan(1/2189376182)
8 Radius Generator- Fabrice Bellard?, 1997
π = 126N∑n=0(−1)n210n(−254n+1−14n+3+2810n+1−2610n+3−2210n+5−2210n+7+110n+9)
9 The Square AGM - Salamin & Brent, 1976
π = limit as n goes to infinity (an+bn)**2/(4tn)
10 Chudnovsky brothers 1988
π = (Q(0, N) / 12T(0, N) + 12AQ(0, N))**(C**(3/2))
11 const_pi() function from the gmpy2 library
12 mp.pi() function from the mpmath library
options:
-h, --help show this help message and exit
-f [FILENAME], --file [FILENAME]
File Name to write Pi to.. Default is [No File]
-d [1 to 1,000,000,000], --digits [1 to 1,000,000,000]
How many digits to calculate. Default is [100000]
-a [1 to 12], --algo [1 to 12]
Which Machin(like) formula. Default is [4]
--verbose, -v
--quiet, -q