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In April 1983 TI PPC Notes (V8N2P4) reported that the French scientific journal Science et Vie had published a program by Renaud de La Taille which could deliver 1287 digits of pi on a TI-59. The solution occupies 13 digits per data register times 99 registers for the total of 1287 digits with only one data register (R00) reserved for dsz control. The program is painfully S - L - O - W ; it runs for 24.55 days! In June 1983 TI PPC Notes (V8N3P8) published the program and instructions for its use. In August 1983 TI PPC Notes (V8N4P21) published a Maurice Swinnen's translation of the Science et Vie article which explained the method of calculation as follows:
"The entire art consists of finding the right formula in order to design a short and simple program, and to use the largest possible number of memory registers. ... ... start with the series
pi/2 = 1 + 1/3 + (1*2)/(3*5) + (1*2*3)/(3*5*7) + (1*2*3*4)/(3*5*7*9) + ...
Thus, pi/2 = ((...((2n/(2n+1) + 2)(n-1)/(2n-1) + 2)(n-2)/(2n-3) + 2)(n-3)/...)1/3 +2
This method avoids the addition of terms in the series to the preceding sums. The resulting program contains only a multiplication and a division. ..."
The same issue of TI PPC Notes presented Palmer Hanson's modification of the Science et Vie program to run in fast mode and complete the calculations in 13.39 days. Palmer's modification also provided automatic calculation of the number of registers and iterations to be used when calculating fewer than 1287 digits. Correspondents of Science et Vie used similar techniques with other programmable calculators and reported the following results
4 ! Uses the formula: pi = 20*arctan(1/7) + 8*arctan(3/79)
5 !
6 ! and arctan(x) = (y/x) * (1 + 2/3*y + 2/3*4/5*y*y + 2/3*4/5*6/7*y*y*y + .....)
7 ! where y = x*x/(1+ x*x)
【 在 AGust2022 的大作中提到: 】
: 标 题: Re: 请教此递归为何等于π
: 发信站: 水木社区 (Sun Oct 29 17:22:19 2023), 站内
:
: 展开后:
: f(n)=1!/3+2!/(3*5)+3!/(3*5*7)+4!/(3*5*7*9)+...+n!/(奇序列乘积)
: =pi/2
:
: f(n)=sigma<n:1,N>{ n! / Π<k:1,n>(2k+1)} = pi/2
:
: f(n)=sigma<n:2,N>{ n! / Π<k:1,n>(2k-1)} = pi/2
: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~用此式
: :
: : π=2/3(2+3/5(2+4/7(2+5/9(2+...N/(2N-1)*2))))
: ※ 来源:·水木社区 mysmth.net·[FROM: 112.10.213.*]
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