Viète à grande Vitesse or Accelerated Viète (kind of)

[Albert Chan]

(06-25-2024 12:41 AM)Albert Chan Wrote:  This version is going for simplicity, which turns out pretty good.

ε = asin(z)/z - 1 ≈ z^2/6 / (1 - 9/20*z^2) = 1 / (6/z^2 - 2.7)

Here is for next pade approximation

ε = asin(z)/z - 1 ≈ z^2/6 * (1 + 9/20*z^2 / (1 - 25/42*z^2))

ε ≈ (1 + 2.7/(x-25/7)) / x , where x = 6/z^2

Code:

00 { 50-Byte Prgm }
01▸LBL "Vh"
02 2
03 0
04 SIGN
05▸LBL 00
06 2
07 STO× ST Z
08 RCL+ ST L
09 SQRT
10 ×
11 DSE ST Z
12 GTO 00       @ 1/z   2^(n+1)
13 ÷
14 LASTX
15 X↑2
16 6           
17 ×            @ x = 6/z^2
18 DUP
19 .28
20 1/X          @ 25/7    x    x    pi'
21 -
22 2.7
23 X<>Y
24 ÷
25 LN1+X
26 E↑X
27 ÷            @ 1/eps   pi'  pi'  pi'
28 ÷
29 +
30 END

13 --> 3.141592653589793238462643383279481
14 --> 3.141592653589793238462643383279502