Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)

[Gerson W. Barbosa]

(01-14-2020 11:47 PM)Albert Chan Wrote:  BTW, if you change "⅗" to "217/360", above formula is always better than Ramanujan II

Except at the vicinity of h = 1, when Ramanujan II becomes better, albeit only slightly.

Here are the fixed expression for s and the new HP-42S/Free42 version:

s = 3/2 + h²/[32 - h²(217h²/360 + 4)]

00 { 66-Byte Prgm }
01▸LBL "MB"
02 COMPLEX
03 ENTER
04 COMPLEX
05 ENTER
06 X<> ST Z
07 STO- ST Z
08 +
09 ÷
10 X↑2
11 0.361
12 →HR
13 RCL× ST Y
14 4
15 +
16 RCL× ST Y
17 +/-
18 32
19 +
20 ÷
21 1.5
22 +
23 X<>Y
24 COMPLEX
25 RCL ST Z
26 Y↑X
27 X<>Y
28 LASTX
29 Y↑X
30 +
31 2
32 ÷
33 X<>Y
34 1/X
35 Y↑X
36 STO+ ST X
37 PI
38 ×
39 END


The error when computing the length of Halley’s comet is now 1146.924 km, about the flight distance from Milan to Warsaw („Z ziemi włoskiej do Polski”), but the results are overall better.