Micro-challenge: Special Event

[Albert Chan]

(01-04-2023 01:56 AM)Thomas Klemm Wrote:  Here’s a program for the HP-42S to calculate \(\pi\) ...
Example

40 R/S

3.14159265359

FYI, 40 meant 40 digits after radix point (total 41 digits)

lua> require'fun'()
lua> horner = fn's,a,b: s*b+a'
lua> terms = fn'n: range(n,1,-1)'
lua> pi_atan_euler = fn'x: 2, x/(2*x+1)'

x → ∞ produced (2, 1/2): 2*(1/2) + 2 = 3

(2.2222 ...)_b last digit = 3 converge to pi faster.

lua> terms(40) :map(pi_atan_euler) :reduce(horner, 2)
3.141592653589546
lua> terms(40) :map(pi_atan_euler) :reduce(horner, 3)
3.1415926535896728

---

Another example, pi = 6*asin(1/2) taylor series.

\(\displaystyle \pi = 3 +
\left(\frac{1^2}{4×2×3}\right) \left(3 +\;
\left(\frac{3^2}{4×4×5}\right) \left(3 +\;
\left(\frac{5^2}{4×6×7}\right) \left(3 +\;
\cdots \right) \right) \right) \)

lua> pi_asin_half = fn'x: x=2*x-1; 3, (x*x)/(4*(x+1)*(x+2))'

x → ∞ produced (3, 1/4): 3*(1/4) + 3 = 3.75

(3.3333 ...)_b last digit = 3.75 converge to pi faster.

(1/4) = (1/2)^2 --> about half terms needed for equivalent accuracy.

lua> terms(20) :map(pi_asin_half) :reduce(horner, 3)
3.141592653589791
lua> terms(20) :map(pi_asin_half) :reduce(horner, 3+.75)
3.1415926535897927

Or, we turn this into ratio of "integers", with top digit base = 4*2*3/1^2 = 24.
Note: range(20) mean 1 .. 20

lua> base = range(20): map(fn'x: x=2*x-1; (4*(x+1)*(x+2))/(x*x)')
lua> n = zip(xrepeat(3), base) :reduce(horner, 3)
lua> d = base :product()
lua> n/d, (n+.75)/d
3.141592653589791      3.141592653589793