QuickPi v4.5 Documentation Copyright (c) 2000-2008 S. Pagliarulo HOW TO USE THIS DOCUMENT To properly view this document in Notepad, maximize the Notepad window and make sure a fixed-width font such as Courier is selected. To print this document, open it in Notepad or any other word processor and select Print from the File menu. INSTALLATION Installation of QuickPi is easy. Simply unzip the QPI45.ZIP file in the directory of your choosing. No changes are made to the Windows registry or any other part of the system. QuickPi is intended for use on Windows XP or Windows Vista with a Pentium 4 or newer processor. The 64-bit (x64) version of QuickPi requires Windows x64 edition and an AMD processor with AMD64 instruction set or an Intel processor with EM64T technology (Intel 64). DISTRIBUTION NOTES The files that comprise the QPI45.ZIP file including the QPI v4.5 executable are freely available for distribution. If you plan to distribute QuickPi online, you must distribute the entire file set as originally received and not the individual files. ----------------- QUICKPI OVERVIEW QuickPi is designed to quickly compute Pi to a very high precision. The program is one of the fastest available on the Windows platform. In addition to its speed, QuickPi offers 6 different methods for computing Pi and can also compute other constants such as e and the square root of 2 with equal precision and speed. USAGE INFORMATION QuickPi is a command line program and as such you will need to start a Command Prompt window to use it. Refer to your Windows help guide for information on how to start a Command Prompt window. Once a Command Prompt window has been started, be sure the QuickPi application QPI.EXE (or QPI64.EXE) is located in your system path. If it's not, then change the current working directory to that chosen when the program was first installed. Again, refer to your Windows help guide for information on PATHs and the CHDIR command. Entering QPI on the command line without any parameters will produce the following information: QPI-QuickPi v4.5, (c) 2000-2008 S. Pagliarulo Freely distributable, email: s_pagliarulo@hotmail.com usage: qpi digits[k|m|g] [destfile] [options|-h] Information in square brackets [] is optional. Items separated by a vertical bar | indicate an either-or choice. The k, m and g suffix following "digits" allows you to specify kilo (1,024), mega (1,048,576) or giga (1,073,741,824) digits respectively. Other suffixes include th, mi, and bi for a thousand (1,000), million (1,000,000) or billion (1,000,000,000) digits respectively. For example, to specify 1,000,000 digits of pi, enter: qpi 1mi The output will be sent to the console screen. To save the output to a text file, simply specify a destination file name following the digits specification: qpi 1mi pi.txt This will save the output to the file "pi.txt" as well as providing progress information to the console screen. Note, the output of QuickPi 4.5 can also be "redirected" with the > operator. In this case however, no progress information will be provided to the console screen. Entering QPI on the command line with the -h option will produce the following help information: -OPTIONS- -PI COMPUTATION- -OTHER CONSTANTS- -agm[1|2|3] AGM methods (1,2,3) -e, -exp natural log base, e -borwein Borweins' quartic -2, -sqrt2 Pythagoras' constant -chudnovsky Chudnovskys' method -phi golden mean -ramanujan Ramanujan's method -zeta3 Apery's constant -machin Machin's arctan formula -gamma Euler's constant -stormer Stormer arctan formula -catalan Catalan's constant -ln(n) natural log of (n) -sqrt(n) square root of (n) -cbrt(n) cube root of (n) -OUTPUT FORMAT- -fancy:dpc,dpr, dpc - digits per column [default=10] dpb,bs,dc, dpr - digits per row [default=50] fl dpb - digits per block [default=500] bs - block spacing [default=1] dc - digit counting (none,row,block,line) [default=row] fl - flags (noindent,indent) [default=noindent] -COMPUTATION- -formula use alternate formula (1-n) for computation -inverse compute multiplicative inverse -MEMORY MANAGEMENT- -memory: set upper memory limit (affects performance) -swap: specify the location of swap files -noswap disable swap file creation -MULTITHREADING- -threads collect and output thread usage information -threads: set maximum number of threads (affects performance) -nomt disable multithreading -MISCELLANEOUS- -train train program for optimal performance -binary compress output and create binary file -stats output computation statistics only (suppress digit data) -checksum include checksum of computation result -save save options as default values -license display software license information -pause pause when finished Most options can be abbreviated to their first letter. Where duplicates exist, using two letters will generally differentiate the options. OPTION DESCRIPTIONS -agm[1|2|3] This option causes QuickPi to compute Pi using one of the AGM (Arithmetic- Geometric-Mean) formulas first discovered by Gauss and Legendre and later independently by Brent and Salamin (1). QuickPi actually uses optimized algorithm variants of these formulas. Specifying -agm or -agm1 causes QuickPi to use a formula developed by Arnold Schonhage (2). This formula is currently QuickPi's fastest AGM method. Specifying -agm2 causes QuickPi to use a formula by Takuya Ooura (3). This is currently QuickPi's second-fastest AGM method. Specifying -agm3 causes QuickPi to use the Brent-Salamin formula. The AGM formulas are iterations that are quadratic in nature (the number of digits doubling with each iteration). To compute one million digits of Pi, the AGM typically uses 19 iterations. -borwein This option causes QuickPi to compute Pi using a quartic AGM iteration by Jonathan and Peter Borwein. A quartic iteration quadruples the number of digits at each iteration (at the cost of more operations per iteration). To compute two million digits of Pi, the Borwein quartic formula only requires 10 iterations. -chudnovsky This is the default option for QuickPi when no other options are specified. This option causes QuickPi to compute Pi using a Ramanujan-like series formula discovered by David and Gregory Chudnovsky. This formula provides approximately 14 correct digits per term. This is currently QuickPi's fastest method for computing Pi. -ramanujan This option causes QuickPi to compute Pi using one of the many series formulas discovered by Srinivasa Ramanujan. The particular formula used by QuickPi provides approximately 8 digits per term and runs predictably slower than the series by the Chudnovsky brothers. -machin This option causes QuickPi to compute Pi using the arctangent formula discovered by John Machin in 1706. Machin's refinement on the Gregory/Leibniz series yielded a formula with a significantly greater rate of convergence: pi/4 = 4 arctan(1/5) - arctan(1/239) -stormer This option causes QuickPi to compute Pi using an arctangent formula discovered by F.C.W. Stormer in 1896. This is one of the same formulas used by Yasumasa Kanada of the University of Tokyo (4) and his team to compute Pi to a record 1 trillion digits: pi/4 = 44 arctan(1/57) + 7 arctan(1/239) - 12 arctan(1/682) + 24 arctan(1/12943) -e, -exp This option causes QuickPi to compute the natural logarithmic base, e. The constant e = exp(1) is defined as: 1 + 1/1! + 1/2! + 1/3! + ... -2, -sqrt2 This option causes QuickPi to compute Pythagoras' constant, the square root of 2. QuickPi uses Newton's iteration for computing square roots. With an initial estimate (x), Newton's iteration for the square root of 2 converges quadratically using the sequence: x = (x/2 + 1/x). -phi This option causes QuickPi to compute the Golden Mean. Also known as the Golden Ratio or the Golden Section, it is usually represented by the Greek letter Phi. QuickPi uses Newton's iteration for computing Phi (see -sqrt2). The formula for Phi is: (1 + sqrt(5)) / 2. -zeta3 This option causes QuickPi to compute Apery's constant, Zeta(3). Zeta(3) is defined by the formula: 1 + 1/2^3 + 1/3^3 + 1/4^3 + ... QuickPi uses a faster series discovered by Amdeberhan and Zeilberger for this computation. -gamma This option causes QuickPi to compute Euler's constant (or Mascheroni's constant) and is defined as the limit of: 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n - ln(n). The constant is usually denoted by the lower-case Greek letter Gamma. QuickPi uses a faster method due to Brent and McMillan for this computation. -catalan This option causes QuickPi to compute Catalan's constant. Catalan is defined by the following series: 1/1^2 - 1/3^2 + 1/5^2 - 1/7^2 + 1/9^2 - ... and is usually denoted as K, G or sometimes C. QuickPi uses a faster series by Alexandru Lupas for this computation. -ln(n) This option causes QuickPi to compute the natural logarithm of an integer value (n). Natural logs are the solution to the equation: e^x = n. -sqrt(n) This option causes QuickPi to compute the square root of an integer value (n). -cbrt(n) This option causes QuickPi to compute the cube root of an integer value (n). -fancy: This option allows the formatted output of QuickPi to be customized. The default format is 10 digits per column, 50 digits per row and 500 digits per block. Blocks are separated by 1 blank line, digits are counted after each row and lines are not indented. Specifying each (in order) as follows can change these parameters: -fancy:dpc,dpr,dpb,bs,dc,fl For example, the following parameters cause QuickPi to output 5 digits per column, 50 digits per row and 1000 digits per block with lines indented. Note that any missing parameters remain unchanged. -fancy:5,50,1000,,,indent There are two other valid parameters for this option. They are -fancy:default and -fancy:raw. The first causes the QuickPi to revert to its original default parameters and the second causes QuickPi to produce unformatted output. Unformatted output is simply a stream of digits without spaces or line termination. -formula This option enables QuickPi to use an alternate formula for a computation. Specifying -formula1 causes QuickPi to use its primary (default) formula and specifying -formula2 thru -formula9 causes QuickPi to use an alternate (generally slower) formula. Alternate formulas are useful to verify the correctness of a computation. The following constants have alternate formulas: -exp -formula2 -phi -formula2 -catalan -formula2 -zeta3 -formula2 -sqrt(n) -formula2 -cbrt(n) -formula2 -ln(n) -formula2 -inverse This option causes QuickPi to compute the multiplicative inverse or reciprocal result. This option is not available for all constants. -memory: This option sets the upper limit of memory use for QuickPi. The size parameter may be followed by an m to indicate megabytes (1048576 bytes). The smallest value that can be specified is 32 megabytes, smaller values are ignored. When the upper limit of memory is reached, QuickPi will reduce its memory consumption and may swap memory to disk unless the creation of swap files is disabled (see -noswap option). Note, very large computations may still cause QuickPi to go over the specified memory limit. -swap: Normally, QuickPi creates swap files in the current directory. This option causes QuickPi to use the specified path as the location for creating swap files. A drive letter may be included as part of the path. The path must already exist. -noswap This option prevents QuickPi from creating swap files when running low on memory. -threads This option (without a value) causes QuickPi to collect and output information related to thread and processor usage. By default, QuickPi allocates one thread for each processor core or SMT execution context. -threads: This option (with a value) sets the maximum number of threads that QuickPi can use. Any value in the range from 1 to 32 is accepted. The actual number of threads used depends on the size of a computation, the amount of available memory and other internal factors. Increasing the number of threads does not always translate to improved performance. -nomt This option completely disables multithreading causing QuickPi to operate as a single-threaded application. -train This option causes QuickPi to train or configure itself for a specific computer environment. Once trained, QuickPi will usually run faster than its default configuration. Training will normally take several minutes to complete. Training information is saved to a QPI.INI file located in the same directory as QPI.EXE. QuickPi will then read the training information the next time it's run. It is recommended that all Windows applications be closed and any background tasks (including screensavers) be disabled prior to initiating training. -binary This option causes QuickPi to write its output as a compressed binary file. Compressed files are generally 1/2 to 1/3 the size of QuickPi's normal output. A secondary reader program QRD (included) is required to view compressed binary files. The QuickPi Reader program is described below. -stats This option causes QuickPi to output computation statistics only and suppress the output of digit data. This can be useful when only the statistics of a computation are desired for comparison purposes. -checksum This option causes QuickPi to calculate and display a checksum for a computation result. A checksum can be useful to validate a result without having to perform a digit by digit comparison. -save This option causes QuickPi to save the settings of the above specified options to a QPI.INI file located in the same directory as QPI.EXE. The next time QuickPi is started, it will automatically read the QPI.INI file and default to the saved options. This is particularly useful for the -fancy option and its various parameters. -license This option displays the QuickPi license information. -pause This option causes QuickPi to pause and wait for a key to be pressed when its operation is complete. QuickPi will automatically pause if started from the Windows Desktop. ---------------- READER OVERVIEW The QuickPi Reader is designed to decompress a binary file written by QuickPi and output it in a variety of customizable formats. Using the Reader allows you to quickly output all or part of a QuickPi result without having to re-run a (perhaps lengthy) QuickPi computation. In addition, compressed binary files are up to 1/2 to 1/3 the size of QuickPi's normal output, saving a significant amount of disk space if result files are archived. USAGE INFORMATION The QuickPi Reader is a command line program and as such you will need to start a Command Prompt window to use it. Refer to your Windows help guide for information on how to start a Command Prompt window. Once a Command Prompt window has been started, be sure the Reader program QRD.EXE (or QRD64.EXE) is located in your system path. If it's not, then change the current working directory to that chosen when the program was first installed. Again, refer to your Windows help guide for information on PATHs and the CHDIR command. Entering QRD on the command line without any parameters will produce the following information: QRD-QuickPi binary reader v4.5, (c) 2000-2008 S. Pagliarulo Freely distributable, email: s_pagliarulo@hotmail.com usage: qrd sourcefile [destfile] [options|-h] Information in square brackets [] is optional. Items separated by a vertical bar | indicate an either-or choice. For example, to view the compressed binary file "pi.bin" created by QuickPi, enter: qrd pi.bin The output will be sent to the console screen. To save the output to a text file, simply specify a destination file name after the source file name: qrd pi.bin pi.txt This will save the output to the file "pi.txt". Entering QRD on the command line with the -h option will produce the following help information: options: -stats output computation statistics only -nostats output digit data only -range:start-end output a range of digits start - start of range (1-) end - end of range (1-) -skip:count,step skip over output count - output [count] digits step - every [step] digits -fancy:dpc,dpr, customize output format dpb,bs,dc, dpc - digits per column fl dpr - digits per row dpb - digits per block bs - block spacing dc - digit counting (none,row,block,line) fl - flags (noindent,indent) -inverse compute multiplicative inverse -square compute square -cube compute cube -ln compute natural log -binary create binary output file -license display software license information -pause pause when finished Most options can be abbreviated to their first letter. Where duplicates exist, using two letters will generally differentiate the options. OPTION DESCRIPTIONS -stats This option causes the QuickPi Reader to output computation statistics only and suppress the output of digit data. This option can be used to obtain a summary of a compressed binary file's contents. -nostats This option causes the QuickPi Reader to suppress the output of computation statistics that normally precede the digit data. This option should be used when only digit data is desired. -range: This option causes the QuickPi Reader to limit the range of digits output. A starting point, an ending point, or both starting and ending points can be entered, separated by a dash. Multiple ranges can be specified by separating them with a comma or by entering multiple -range options from the command line. For example, the following parameters will output the first and last 100 digits of 20k computed digits: -range:1-100,20381-20480 -skip: This option causes the QuickPi Reader to skip across the full range of computed digits and to only occasionally output a specified number of them. The first parameter specifies the number of digits to output. The second parameter specifies the skip amount. For example, the following parameters: -skip:100,10000 cause the QuickPi Reader to produce the following output with 20k computed digits of Pi: Listing Pi, 100 digits every 10,000 digits 3. 1415926535 8979323846 2643383279 5028841971 6939937510 : 50 5820974944 5923078164 0628620899 8628034825 3421170679 : 100 5667227966 1988578279 4848855834 3975187445 4551296563 : 10050 4434803966 4205579829 3680435220 2770984294 2325330225 : 10100 2038565390 9910477594 1413215432 8440625030 1802757169 : 20050 6508209642 7348414695 7263978842 5600845312 1406593580 : 20100 -fancy: This option allows the formatted output of the QuickPi Reader to be customized. The default format is saved as part of the compressed binary file created by QuickPi. Unless changed when using QuickPi, the default format is 10 digits per column, 50 digits per row and 500 digits per block. Blocks are separated by 1 blank line, digits are counted after each row and lines are not indented. Specifying each (in order) as follows can change these parameters: -fancy:dpc,dpr,dpb,bs,dc,fl For example, the following parameters cause QuickPi Reader to output 5 digits per column, 50 digits per row and 1000 digits per block with lines indented. Note that any missing parameters remain unchanged. -fancy:5,50,1000,,,indent There are two other valid parameters for this option. They are -fancy:default and -fancy:raw. The first causes the QuickPi Reader to revert to its original default parameters and the second causes the QuickPi Reader to produce unformatted output. Unformatted output is simply a stream of digits without spaces or line termination. -inverse This option causes the QuickPi Reader to compute and output the multiplicative inverse or reciprocal of a result. -square This option causes the QuickPi Reader to compute and output the square of a result. -cube This option causes the QuickPi Reader to compute and output the cube of a result. -ln This option caused the QuickPi Reader to compute and output the natural logarithm of a result. -binary This option causes the QuickPi Reader to write its output as a compressed binary file. This option is useful when combined with the -inverse, -square, -cube or -ln options. -license This option displays the QuickPi Reader license information. -pause This option causes the QuickPi Reader to pause and wait for a key to be pressed when its operation is complete. The QuickPi Reader will automatically pause if started from the Windows Desktop. ------------------- PROGRAM LIMITATIONS The upper limit for the 32-bit version of QuickPi is 1 billion digits. The 64-bit version has been successfully tested with over 100 billion digits and has a theoretical upper limit of 500 billion digits. The exact upper limit depends on several factors including but not limited to the amount of memory and disk space available. Each new release of QuickPi should continue to perform faster and with higher precision. WARRANTIES AND DISCLAMERS THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, FUNCTIONALITY AND DATA INTEGRITY OR PROTECTION ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES WHICH MAY RESULT FROM THE USE OF THIS SOFTWARE, INCLUDING LOSS OR INTERRUPTION OF BUSINESS, ACCIDENTAL LOSS OF DATA OR DAMAGE TO COMPUTER EQUIPMENT. ACKNOWLEDGEMENTS My thanks and appreciation to all the Beta testers who have assisted with the testing of this version over the past many months. A special thanks to Shigeru Kondo for his extra dedication, enthusiasm, helpful feedback and suggestions. CONTACTING THE AUTHOR Questions and comments concerning QuickPi should be directed to Steve Pagliarulo. Steve can be contacted via email at: s_pagliarulo@hotmail.com WEBSITES DEDICATED TO PI Stu's pi page: http://home.istar.ca/~lyster/pi.html Xavier Gourdon: http://numbers.computation.free.fr/Constants/constants.html jasonp's home page: http://www.boo.net/~jasonp/ Shigeru's pi world: http://ja0hxv.calico.jp/pai/estart.html ------------------- (1) R.P. Brent, Fast multiple-precision evaluation of elementary functions, J. ACM 23 (1976), 242-251. (2) Arnold Schonhage, A.F.W. Grotefeld/E. Vetter, Fast Algorithms, A Multitape Turing Machine Implementation, BI Wissenschaftsverlag, Mannheim (1994), p.266. (3) Takuya Ooura maintains a website dedicated to Pi and the AGM at: http://www.kurims.kyoto-u.ac.jp/~ooura/pi_fft.html (4) The Kanada Laboratory maintains a website dedicated to computing Pi at: http://www.super-computing.org