HP-15C (& LE!) 11-11-11 Mini-Challenge ORIGINAL SOLUTION & COMMENTS
Message #29 Posted by Valentin Albillo on 14 Nov 2011, 8:12 a.m.,
in response to message #1 by Valentin Albillo
Hi, all:
Thanks for your interest in my HP-15C (&LE!) Mini-challenge, I'm glad many of you enjoyed it and produced a number of correct solutions, some of which virtually duplicated my original one, as well as producing solutions for other models. That's the spirit of these challenges and mini-challenges and I thank you for joining in.
As for my original solution, this is it:
01 LBL A
02 EEX
03 6
04 CHS
05 STO I
06 I
07 GSB E
08 Re<>Im
09 CF 8
10 RCL/ I
11 RTN
which is 11 steps and meets all 5 requirements. I prefer to use h=1E-6 instead of smaller values and I think that CF 8 must be included lest you'll leave the machine in complex mode, which would be utterly unexpected to the user as both the input and the output are assumed and expected to be real.
Partially relaxing condition (1), i.e., supressing LBL and RTN, would save two steps, while relaxing condition (4), i.e., pre-storing 1E-6 in a register, would save three additional steps, leaving the grand total at a mere 6 steps or so. An additional step could be saved by supressing CF 8 but as stated above, I think this would seriously detract from the usability.
As for the register used to store the constant, I prefer using permanent register RI instead of R0 or R1 (also permanent) though, as the constant is less than one, the best solution is using register RAN#, which frees RI for indexing purposes and R0-R1 for general use or matrix operations. However, as there's no RCL\ RAN# instruction this means an additional step to perform the division.
Finally, I knew that for X arguments near 0 accuracy would get worse and so require extra steps to cater for that case, thus this is why I included the exception at condition (5) in order to keep it short and sweet. After all, this was intended as a Mini-challenge, not a full-fledged article on numerical derivation complete with industrial-strength code ! ... :D
When I was busy concocting this mini-challenge, I first implemented it for the HP-71B, where the code is as simple as
IMPT(FNF((X,H)))/H
where FNF is the user-defined function f(x) and H is a suitably small constat (1E-7, say). I conducted some experiments to see which H would be best, for instance:
10 DEF FNF(X)=5*COS(10*X)+((X-2)*X-6)*X+10
20 DEF FND(X)=-50*SIN(10*X)+3*X*X-4*X-6
30 FOR I=1 TO 9 @ H=10^(-I) @ DISP I;IMPT(FNF((1,H)))/H,FND(1) @ NEXT I
>RUN
1 24.9567129442 20.2010555444
2 20.24631331 20.2010555444
3 20.2015078976 20.2010555444
4 20.201060068 20.2010555444
5 20.2010555897 20.2010555444
6 20.2010555449 20.2010555444
7 20.2010555444 20.2010555444 << H=1E-7 first nails the exact derivative
8 20.2010555444 20.2010555444
9 20.2010555444 20.2010555444
and in virtually all cases H = 1E-7 was optimal (except for X near 0, of course, where a smaller H is needed). For the HP-15C, which only carries 10 (13) digits instead of 12 (15), H=1E-6 does the trick.
That's all. I've got (hopefully) interesting ideas for some dozen challenges and mini-challenges as well as for another dozen full-fledged articles but regrettably next to no or very little time to write them down & post them hera, and actually no suitable place to publish the long articles so it might be a while till I make another lengthy contribution ... :(
Thanks again for your interest and best regards from V.
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