kremen@aero.ARPA (Gary Kremen) (12/30/85)
Some comments on computation of PI 1) There was a recent posting on the exactness of pi/4 = 4*atan(1/5) - atan(1/239). This is a true relationship. You can easily prove this by taking the tangents of both sides and using the tan(a+b) relationship. Quick proof: tan x = 1 / 5 tan 2x = 2 tan x / (1 - tan x * tan x) = 5 / 12 tan 4x = 2 tan 2x / (1- tan 2x * tan 2x) = 120 / 119 > tan (4x - PI/4) = (tan 4x - 1) / (1 + tan 4x) = 1 / 239 >>> atan(1 / 239) = 4x - PI/4 = 4 atan(1/5) - PI/4 2) Sheldon Meth talked about the simplicity of using the expansion of atan(1) which is equal to pi/4. Yes, it is simple but it converges extremely slow. Machin's 1706 series is much better. 3) There is a new method of calculating PI discovered in 1976. It is asymetrically faster than using arctan methods. It is pretty complex. If there is any interest I will post info to the net. 4) In 1983 I computed PI to 225,000 digits using IBM 4341. No problem, but it took 1.1 days of machine time. It was lucky I was the machine's only operator. I wrote the program in very optimized FORTRAN/assembly code. Recently I rewrote it in C, for use on microcomputers. 5) There is a book "A History of Pi" by Peter Beckman. It is well worth reading even if you don't have a mathematics background. He explains how history of progress in Pi and mathematics mirrors man's history. For example - during the dark ages the most enlighted countries have the most mathematics progress. 6) Current Pi record is around 16 million places. -- Name: Gary Kremen Address 1: kremen@aerospace.ARPA Address 2: {sdcrdcf,trwrb,randvax}!aero!kremen.UUCP Address 3: BITNET, CSNET, MAILNET, others - through correct gateways Quote:"Everybody loves to see justice done...on someone else" - Bruce Cockburn Contrapositive: "To Live and Die to live and drive in LA" Disclaimer 1: "The company does not know what I am doing" Disclaimer 2: "Both the company and I have great lawyers"