Viète à grande Vitesse or Accelerated Viète (kind of)

[Albert Chan]

I just realized I had done this before, in Arc SOHCAHTOA method

(03-31-2022 11:07 PM)Albert Chan Wrote:  asinq(x) = 2 * asinq(x/2/(1+sqrt(1-x)))

lua> k, x = 4, .5 -- 4 * asinq(.5) = 4 * pi/4 = pi
lua> FMT = '%g * asinq(%.17g)\n'
lua> for i=1,5 do printf(FMT,k,x); k=k+k; x=0.5*x/(1+sqrt(1-x)) end
4 * asinq(0.5)
8 * asinq(0.14644660940672624)
16 * asinq(0.038060233744356624)
32 * asinq(0.009607359798384776)
64 * asinq(0.0024076366639015569)

Argument inside asinq, x = z^2 in Free42 code. Undo asinq, we have pi

lua> k*sqrt(x) -- = pi', with n=6
3.141277250932773
lua> _ * (1 + x/6*(1 + 9/20*x*(1 + 25/42*x*(1 + 49/72*x))))
3.141592653589793