Index

1.Outline of 4.2 billion caluculation of pi

2.History of pi calculation with computer

3.Outline of Super PI

4.Comparison of supercomputer and personal computer

5.Calculation time of Super PI

6.The form of output file "pi_data.txt"

7.Gauss-Legendre algorithm

8.Borwein's 4-th order convergent algorithm

9.How to obtain 4.2 billion digits of pi

10.Introduction of Kanada Lab.


Outline of 4.2 billion calculation of pi

Our latest record was established as the followings:

Declared record:

 4,294,460,000 decimal digits

 Two independent calculation based on two different algorithms generated 4,294,967,296 (=2^32) decimal digits of pi and comparison of two generated sequences matched 4,294,967,286 decimal digits, e.g., 10 decimal digits difference.  Then we are declaring 4,294,960,000 decimal digits as the new world record.

Main program run:

 Job start(1-st)          	: 28th July 1995 22:59

 Job end (1-st)           	: 31st July 1995 07:42

 Elapsed time (1-st)       	: 56:44:24

 Vector CPU (1-st)        	: 67:33:04

 Job start (2-ed)          	: 04th Aug. 1995 20:38

 Job end (2-ed)           	: 07st Aug. 1995 05:35

 Elapsed time (2-ed)      	: 56:57:11

 Vector CPU (2-nd)        	: 66:15:37

 Main memory              	: 1792.75 MB

 ES memory                	: 25120 MB

 Disk storage            	: 2 GB * 2 = 4 GB (as for the intermediate data storage)

 Algorithm                	: Borwein's 4-th order convergent algorithm

Verification program run:

 Job start (1-st)        	: 11th Aug. 1995 21:31

 Job end (1-st)          	: 14st Aug. 1995 15:54

 Elapsed time (1-st)       	: 66:23:29

 Vector CPU (1-st)        	: 77:14:59

 Job start (2-ed)          	: 25th Aug. 1995 23:12

 Job end (2-ed)           	: 28st Aug. 1995 15:09

 Elapsed time(2-ed)       	: 63:57:22

 Vector CPU               	: 75:10:29

 Main memory              	: 1792.75 MB

 ES memory                	: 26152 MB

 Disk storage            	: 2 GB * 3 = 6 GB (as for the intermediate data storage)

 Algorithm                 	: Gauss-Legendre algorithm

4,000,000,000-th digits of pi and 1/pi:

 pi  : 94375 34306 22684 47216

 1/pi: 71480 70425 69013 58924

                             ^

    Last number is 4,000,000,000-th.

(First digit '3' for pi or '0' for 1/pi is not included in the above count.)

Frequency distribution for pi-3 up to 4,000,000,000 decimal places:

 '0' : 400001233; '1' : 400002285; '2' : 399965405; '3' : 399984469

 '4' : 400006936; '5' : 399989052; '6' : 400033035; '7' : 399996122

 '8' : 400004741; '9' : 400016722; Chi square = 7.55

Frequency distribution for 1/pi up to 4,000,000,000 decimal places:

 '0' : 400003502; '1' : 400016211; '2' : 400002106; '3' : 399991609

 '4' : 399987516; '5' : 399972897; '6' : 400003565; '7' : 400018249

 '8' : 399999949; '9' : 400004396; Chi square = 4.01

4,294,960,000-th digits of pi and 1/pi:

 pi  : 55675 13149 35865 45528

 1/pi: 96350 29339 14953 51156

                             ^

    Last number is 4,294,960,000-th.

(First digit '3' for pi or '0' for 1/pi is not included in the above count.)

A developer and calculating machine:

 Programs were written by Mr. Daisuke TAKAHASHI, a member of Kanada Lab. CPU used was HITAC S-3800/480 at the Computer Centre, University of Tokyo. Two CPU were definitely used through single job parallel processing for total of four programs run.

Contact:

 Yasumasa KANADA

 Computer Centre. University of Tokyo

 Tokyo 113 Japan

 Fax	: +81-3-3814-7231 (office)

 E-mail	: kanada@pi.cc.u-tokyo.ac.jp

Outline of 3.2 billion calculation of pi


Outline of 3.2 billion calculation of pi

Declared record:

 3,221,220,000 decimal digits

 Two independent calculation based on two different algorithms generated 3,221,220,000 (=3*2^30) decimal digits of pi and comparison of two generated sequences matched 3,221,225,466 decimal digits, e.g., 6 decimal digits difference. Then we are declaring 3,221,220,000 decimal digits as the new world record.

Main program run:

 Job start               	: 16th June 1995 22:41

 Job End                 	: 18th June 1995 11:33

 Elapsed time             	: 36:52:28

 Vector CPU              	: 44:30:50

 Main memory               	: 1888.5 MB

 ES memory              	: 26608 MB

 Algorithm                 	: Borwein's 4-th order convergent algorithm

Verification program run:

 Job start               	: 24th June 1995 1:13

 Job end                  	: 26th June 1995 6:56

 Elapsed time            	: 53:43:46

 Vector CPU              	: 48:28:43

 Main memory              	: 1891.25 MB

 ES memory                	: 26612 MB

 Disk storage             	: 164MB * 5 = 820 MB

 Algorithm                	: Gauss-Legendre algorithm

3,000,000,000-th digits of pi and 1/pi:

 pi  : 37608 19468 51598 04548

 1/pi: 43270 91443 13660 42701

                             ^

    Last number is 3.000,000,000-th.

(First digit '3' for pi or '0' for 1/pi is not included in the above count.)

Frequency distribution for pi-3 up to 3,000,000,000  decimal places:

 '0' : 299999143; '1' : 299995932; '2' : 299989126; '3' : 299992290

 '4' : 300002257; '5' : 299979016; '6' : 300025447; '7' : 299975510

 '8' : 300016550; '9' : 300024729; Chi square = 9.24

Frequency distribution for 1/pi up to 3,000,000,000 decimal places:

 '0' : 300009029; '1' : 300002431; '2' : 299992729; '3' : 299998519

 '4' : 299970883; '5' : 299980175; '6' : 300010123; '7' : 300025696

 '8' : 300008447; '9' : 300001968; Chi square = 7.40

3,221,220,000-th digits of pi and 1/pi:

 pi  : 34929 13958 77673 90319

 1/pi: 21355 75330 20722 29175

                             ^

    Last number is 3,221,220,000-th.

(First digit '3' for pi or '0' for 1/pi is not included in the above count.)

A developer and calculating machine:

 Programs were written by Mr. Daisuke TAKAHASHI, a member of Kanada Lab. CPU used was HITAC S-3800/480 at the Computer Centre, University of Tokyo. Two CPU were definitely used through single job parallel processing for total of four programs run.

Contact:

 Yasumasa KANADA

 Computer Centre. University of Tokyo

 Tokyo 113 Japan

 Fax	: +81-3-3814-7231 (office)

 E-mail	: kanada@pi.cc.u-tokyo.ac.jp


History of pi calculation with computer

 Brief history of pi calculation with computers.

Name	Year/Month	Calculated length	Machine used
G.W. Reitwiesner etc.	1949	2,037	ENIAC
S.C. Nicholson & J. Jeenel	1954	3,092	NORC
G.E. Felton	1957	7,480	Pegasus
F. Genuys	1958	10,000	IBM 704
G.E. Felton	1958	10,020	Pegasus
J. Guilloud	1959	16,167	IBM 704
W. Shanks & T.W. Wrench Jr.	1961	100,265	IBM 7090
J. Guilloud & J. Filliatre	1966	250,000	IBM 7030
J. Guilloud & M. Dichampt	1967	500,000	CDC 6600
J. Guilloud & M. Bouyer	1973	1,001,250	CDC 7600
K. Miyoshi & Y. Kanada	1981	2,000,036	FACOM M-200
J. Guilloud	1981-82	2,000,050	not known
Y. Tamura	1982	2,097,144	MELCOM 900II
Y. Tamura & Kanada	1982	4,194,288	HITAC M-280H
Y. Tamura & Kanada	1982	8,388,576	HITAC M-280H
Y. Kanada, S. Yoshino & Y. Tamura	1983	16,777,206	HITAC M-280H
Y. Ushiro & Y. Kanada	1983.10	(*)10,013,395	HITAC S-810/20
W. Gosper	1985.10	17,526,200	Symbolics 3670
D.H. Bailey	1986.1	29,360,111	CRAY-2
Y. Kanada & Y. Tamura	1986.9	33,554,414	HITAC S-810/20
Y. Kanada & Y. Tamura	1986.10	67,108,839	HITAC S-810/20
Y. Kanada, Y. Tamura, Y. Kubo, etc.	1987.1	134,214,700	NEC SX-2
Y. Kanada & Y. Tamura	1988.1	204,326,551	HITAC S-820/80
G.V. Chudnovsky & D.V. Chudnovsky	1989.5	480,000,000	CRAY-2 & IBM-3090/VF
G.V. Chudnovsky & D.V. Chudnovsky	1989.6	535,339,270	IBM 3090
Y. Kanada & Y. Tamura	1989.7	536,870,898	HITAC S-820/80
G.V. Chudnovsky & D.V. Chudnovsky	1989.8	1,011,196,691	IBM 3090
Y. Kanada & Y. Tamura	1989.11	1,073,740,799	HITAC S-820/80
G.V. Chudnovsky & D.V. Chudnovsky	1991,8	2,260,000,000	Home made parallel computer (detail unknown. not verified.)
D. Takahashi & Y. Kanada	1995.6	3,221,220,000	HITAC S-3800/480(uses 2 CPU)
G.V. Chudnovsky & D.V. Chudnovsky	1994,5	4,044,000,000	New home made parallel computer (detail unknown. not verified.)
D. Takahashi & Y. Kanada	1995.8	4,294,960,000	HITAC S-3800/480(uses 2 CPU)

(*) First pi record calculated with supercomputer. (for the reference.)


Outline of Super PI

 Our record breaking program of 4.2 billion decimal digits was ported to WINDOWS OS.  Original program was written in FORTRAN, and we translated into C program with F2C program.

 Algorithm used is Gauss-Legendre.

Run environment:

 Windows NT, Windows 95.

 If you uses Windows 3.1, Win32s is needed for the program run.

Maximum length of calculation:

 33.55 million decimal digits.  340 MB of disk storage is needed for 33.55 million decimal calculation.

Calculation speed:

 35 minutes for 1 million decimal digits, 78 minutes for 2 million decimal digits, 183 minutes for 4 million decimal digits with Pentium 90MHz and 40MB or more main memory.  (Timing is elapsed time!)

Memory requirement:

 For the maximum calculation speed, 8 MB/1 million decimal digit is favorable, but 2 MB/1 million decimal digit is completely acceptable. Working memory is automatically adjusted by the software.   Anyway, available main memory size is crucial for the processing speed!

Disk storage:

 As for the working disk storage, 10.5 MB per 1 million decimal digit is needed.  Working disk storage is automaticall freed.  As for the permanent data storage, 1 MB per 1 million decimal digit is needed.  Elapsed time is very keen to the disk access time.  In order to short the elapsed time, you are better to equip high speed hard disk drive!

Source program:

 Source program is not opened to the public because we have to write papers for the new pi record of 4.2 billion decimal digits, first!

Others:

 We don't care the problems which may be caused by using or running Super PI program on your machine!   We don't recommend for running Super PI with your important tasks!


Comparison of supercomputer and personal computer

 With Super PI, you can calculate pi up to 33.55 million decimal digits which is comparable to the world record calculated by the Cray-2 supercomputer of 29 million decimal digits in September 1986.

 This does not mean that windows and higher clock Pentium based personal computer is comparable to the historical supercomputer.  More than 90% of calculation speed was atained with excellent algorithm for calculating pi up to 33.55 Million decimal digits.

 By using single CPU of HITAC S-3800/480 vector supercomputer at the

computer centre, University of Tokyo can generate 1 Million decimal digits of pi within 5 seconds and 33.55 million decimal digits of pi within 4 minutes.  How about your personal computer time?

 N.B. Current version of Super PI is limited and customized up to 33.55 Million decimal digits of pi calculation.


Gauss-Legendre algorithm

1. Initial value setting;

2. Repeat the following statements until the difference of a and b is within the desired accuracy;

3. Pi is approximated with a, b and t as;

 The algorithm has second order convergent nature.  Then if you want to calculate up to n digits, iteration count of the order log2 n is sufficient.  E.g. 19 times for 1 million decimal digits, 31 times for 3.2 billon decimal digits.


Borwein's 4-th order convergent algorithm

1. Initial value setting;

2. Repeat the followng statements until the difference of a[k] is within the desired accuracy;

3. Pi is approximated with a[n] as;


How to obtain 4.2 billion digits of pi

 Kanada Lab. operates FTP server with 4.2 billion digits of pi and 1/pi. 200 million digits of pi and 1/pi are free to all with anonymous-ftp and 4.2 billion digits of pi and 1/pi are free to all registered applicants.

200 million digits of pi and 1/pi:

  anonymous-ftp to the following address;

  ftp://www.cc.u-tokyo.ac.jp

  free to all.

4.2 billion digits of pi and 1/pi:

1) Please email to the following address with the subject of

 'PI registration'.  Since the emails are processed automatically,

 please do not inlcude your message in the mail body.

           ^^^

   Email: request@www.cc.u-tokyo.ac.jp

2) We will send you the registration form via email.  If you

 agree with the contents, write in the following data in the

 form, and send us the form via fax.

                                ^^^

  - your name and pronounciation

  - affiliation

  - address

  - telphone number

  - fax number

  - email address

  - comment

  - signature

3) We will send you the temporary guest id and the password for it,

 via email.  The password for the guest ID will be changed in 1 month.


Introduction of Kanada Lab.

Field of Research:

 High Performance Computing, Virtual Workstation System, Research on Research, Computer Algebra

Name of Laboratory:

 Laboratory for Computational tools

Location:

 University of Tokyo. Information science.

Name of staff:

 Yasumasa KANADA (Associate Professor)

 Akira YOSHIOKA  (Research Associate)

 Graduate course students

The Details of Research:

 Our Laboratory studies the computer through computer application point of view. We particularly emphasize the studies on the computational power to the scientific research field.

The followings are some our our main research topics.

1. Research on the supercomputer. One of the most powerful research tools currently available is so called vector and parallel supercomputer. However, supercomputer is not a omnipotent machine but rather peculiar machine. Research on the efficient algorithms good for currently available vector and parallel supercomputer is one of the research topics to the supercomputer.

2. Research on the computer algebra system. REDUCE is one of the most succeeded artificial systems. They are heavily used at the scientific research fields in physics, engineering, mathematics, etc. Developments of efficient algorithms for the computer algebra system and powerful computer algebra system are two of research topics at our Lab.

3. Research on scientmetric. It is possible to analyse the scientific society through quantitative and qualitative figures through widely available database data. E.g. when is the peaked age in the average to the scientific research activity? are there life cycles to the research topics?

etc. Methodology to the research on scientmetric, research, is one of the most interesting research topics in this research field.


Graduate course students

Daisuke TAKAHASHI (D.1) <daisuke@pi.cc.u-tokyo.ac.jp>

 He wrote Super PI and world record established pi programs.

 Pi calculation : 13 years of experience.

 All machine is for the pi calculation!

 "Why do you calculate pi?"

 "Because I like pi most of all."

URL:   http://www.ecc.u-tokyo.ac.jp/~ss57011

Hisayasu KURODA (D.1) <kuroda@pi.cc.u-tokyo.ac.jp>

 Super PI was ported to Windows by him.

 Pi calculation : 11 years of experience.

 He is good Windows programmer. Please remember FPD95.

 "Why dou you calculate pi?"

 "I have nothing else." (not true)

URL:   http://www.ecc.u-tokyo.ac.jp/~ss57007

Tsuyoshi TOMIMATSU (M.2) <tomi@pi.cc.u-tokyo.ac.jp>

 He helped for porting Super PI to Windows system.

 Pi calculation : 10 years of experience.

 He is so eager for consuming all of personal computer power

 for pi calculations.

 He will go on to the next stage (= doctor course).

URL:   http://www.ecc.u-tokyo.ac.jp/~ss46017

Hiroaki WATANABE (M.1) <nabe@pi.cc.u-tokyo.ac.jp>

 Recentry, he learned how Kanada Lab. behaves.

 Pi calculation : none

 "Uum, in this laboratory those who calculate pi gathered.

  In that case, I have to do quite different thing...."

 He is about to be a novice LISPer.

URL:   http://www.ecc.u-tokyo.ac.jp/~ss56033


FPD95

Music Software which sufficiently emulates MIDI with PCM.

If you have MIDI data, you are better to get it.

Note: not related to pi calculation.


A form of output file "pi_data.txt"
The form of output file "pi_data.txt"

The form of output file "pi_data.txt" is as follows:

PI=3.

1415926535 8979323846 2643383279 5028841971 6939937510

5820974944 5923078164 0628620899 8628034825 3421170679

8214 ...

Every 10 digits, there is a space.

Every 50 digits, there is a carriage return.

Every 1000 digits, there is a line feed.


Calculation time of Super PI

 Uses HITACHI FLORA3010(Pentium 90MHz, Memory 40MB, Available disk storage 340MB).

 The digits	        Calculation time

        	 (Windows 95)	  (Windows 3.1+Win32s)

     16K	       0:12  	          0:12

     32K	       0:27  	          0:26

     64K	       1:02  	          0:56

    128K	       2:42  	          2:33

    256K	       6:30  	          5:53

    512K	      15:18  	         14:11

      1M	      35:32  	         34:02

      2M	    1:17:51  	       1:32:45

      4M	    3:02:35  	       3:26:50

      8M	    7:23:49  	       9:23:08

     16M	   22:27:47  	      30:11:22

     32M	    not yet  	      80:37:47

If you can make the faster program than our Super PI, please inform us!