Threaded Mode | Linear Mode

Thomas Klemm · 11-02-2022, 10:22 PM

Recently I stumbled upon the Log-Arcsine Algorithm in Peter Henrici's book: Computational Analysis with the HP-25 Pocket Calculator. (pp. 207)

The following recurrence relation is used:

\(
\begin{align}
s_{n+1} = s_n \sqrt{\frac{2 s_n}{s_n + s_{n-1}}}
\end{align}
\)

This little program can be used with most HP calculators:

Code:

ENTER

ENTER

+

LASTX

R↑

+

÷

SQRT

×

Example

0 ENTER 1
R/S
R/S
R/S
…

1.41421356237
1.53073372946
1.56072257613
1.56827424527
1.57016557848
1.57063862547
1.57075690057
1.57078647018
1.57079386264
1.57079571076
1.57079617279
1.57079628829
1.57079631717
1.57079632439
1.57079632619
1.57079632664
1.57079632676
1.57079632679
1.57079632679
1.57079632679
…

Let's slap a loop around it and we have yet another program for the next \(\pi\)-day:

Code:

00 { 26-Byte Prgm }

01▸LBL "pi"

02 0

03 1

04▸LBL 00

05 ENTER

06 ENTER

07 +

08 LASTX

09 R↑

10 +

11 ÷

12 SQRT

13 ×

14 X≠Y?

15 GTO 00

16 2

17 ×

18 END

This allows us to calculate both \(\log\) and \(\arcsin\) on an HP-16C:

Code:

001 - 43,22, A  LBL A      

002 -       36  ENTER      

003 -       36  ENTER      

004 -    43 26  1/x        

005 -       40  +          

006 -        2  2          

007 -       10  /          

008 -       34  x<>y       

009 -       36  ENTER      

010 -    43 26  1/x        

011 -       30  -          

012 -        2  2          

013 -       10  /          

014 -       20  x          

015 -    43 36  LSTx       

016 -    22  0  GTO 0      

017 - 43,22, b  LBL B      

018 -       36  ENTER      

019 -       36  ENTER      

020 -       20  x          

021 -        1  1          

022 -       34  x<>y       

023 -       30  -          

024 -    43 25  SQRT       

025 -       34  x<>y       

026 -       20  x          

027 -    43 36  LSTx       

028 - 43,22, 0  LBL 0      

029 -       36  ENTER      

030 -       36  ENTER      

031 -       40  +          

032 -    43 36  LSTx       

033 -    43 33  R^         

034 -       40  +          

035 -       10  /          

036 -    43 25  SQRT       

037 -       20  x          

038 -    43 34  PSE        

039 -    43  0  x!=y       

040 -    22  0  GTO 0

Examples

10
GSB A

2.302585094

0.5
GSB B

0.523598775

Here is the same program for the HP-42S:

Code:

00 { 56-Byte Prgm }

01▸LBL "log"

02 ENTER

03 ENTER

04 1/X

05 +

06 2

07 ÷

08 X<>Y

09 ENTER

10 1/X

11 -

12 2

13 ÷

14 ×

15 LASTX

16 GTO 00

17▸LBL "asin"

18 ENTER

19 X↑2

20 1

21 X<>Y

22 -

23 SQRT

24 X<>Y

25 ×

26 LASTX

27▸LBL 00

28 ENTER

29 ENTER

30 +

31 LASTX

32 R↑

33 +

34 ÷

35 SQRT

36 ×

37 X≠Y?

38 GTO 00

39 END

References