HP42S (Free42, DM42) minichallenge – Wallis Product

07042020, 04:42 PM
(This post was last modified: 07042020 04:43 PM by Gerson W. Barbosa.)
Post: #1




HP42S (Free42, DM42) minichallenge – Wallis Product
\(\pi \approx 2\left ( \frac{4}{3} \times \frac{16}{15}\times \frac{36}{35}\times\frac{64}{63} \times \cdots \times \frac{ 4n ^{2}}{ 4n ^{2}1}\right )\)
Write a program to evaluate the Wallis product on your HP42S or DM42. Restrictions: 1. Stackonly (X, Y, Z, T & L); 2. Keep at least one copy of the previous content of the X register on the stack (no need to leave it on Y); 3. No limitation for the value of n. Least number of steps then least number of bytes, including a 2character label, wins. Examples: 1 XEQ “WP” > 2.66666666667 10 > 3.06770380664 123 > 3.13523960409 1E6 > 3.141591868192120714596476681464745 (Free42 only) Occasional diferences in the last two significant digits will be irrelevant. Have fun! 

07042020, 04:57 PM
Post: #2




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
I should participate but I have a really busy weekend!
I’ll give it a try tonight instead of sleeping Thibault  not collector but in love with the few HP models I own  Also musician : http://walruspark.co 

07042020, 05:07 PM
Post: #3




RE: HP42S (Free42, DM42) minichallenge – Wallis Product  
07042020, 09:25 PM
Post: #4




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Without the END but including LBL "WP": 11 steps, 23 bytes.
Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

07042020, 09:41 PM
Post: #5




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Hi everyone,
here is one solution: 01 LBL "WP" 02 2 03 LBL 01 04 RCL Y 05 ST+ X 06 X^2 07 ST* Y 08 DSE X 09 / 10 DSE Y 11 GTO 01 12 END 23 bytes on free42 25 bytes on HP41 Happy Independence Day. 

07052020, 12:56 AM
Post: #6




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Pretty offtopic, but I enjoy doing these minichallenges in other programming languages as well
Code:
Keep'em coming! Remember kids, "In a democracy, you get the government you deserve." 

07052020, 02:07 AM
Post: #7




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
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Hi, Gerson: (07042020 04:42 PM)Gerson W. Barbosa Wrote: Least number of steps then least number of bytes, including a 2character label, wins. A pity, should you have included a "fastest running time" win condition this "let'sthinkoutofthebox" concoction o'mine would've surely won, as its running time is about the same regardless of the input: [19step, 36byte program] 01 LBL "WP" 02 STO ST Y 03 N! 04 X^2 05 RCL ST Y 06 STO+ ST X 07 N! 08 / 09 X^2 10 2 11 2 12 RCLx ST T 13 STO+ ST X 14 Y^X 15 * 16 0.5 17 RCL+ ST Z 18 / 19 END Some Examples (run in Free42) 1 XEQ “WP” > 2.66666666667 10 > 3.06770380664 123 > 3.13523960403 which virtually match your results. As stated, the time is about the same whatever the argument, i.e., extremely fast, and it leaves the original contents of X when the program is run (i.e., the argument keyed in) in stack Y, Z and T. However, it's limited to those arguments that won't overflow the calculation, mostly the factorials, and a few steps can surely be shaved here and there, but I'm just too sleepy right now. :O Thanks for the nice minichallenge, I saw it at 4:00 a.m. and it was just what I needed before going to bed. Have a nice weekend. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

07052020, 03:43 AM
Post: #8




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Hello, Valentín!
(07052020 02:07 AM)Valentin Albillo Wrote: . Sorry, but the most important restriction here was evaluating the product for n as large as 1E6 or 1E7, which is impossible with that approach. Anyway, thanks for calling my attention to alternative methods as they may be useful for another application I have in mind for which a much more narrow range will suffice. More or less in the same vein: Code:
Examples: 414 > 3.13969841676 5102 > 3.141438733173932101937205797737162 These are the maximum values of n for the HP42S and Free42, respectively, that will not cause overflow. (07052020 02:07 AM)Valentin Albillo Wrote: Thanks for the nice minichallenge, I saw it at 4:00 a.m. and it was just what I needed before going to bed. You too! You are most welcome. Thanks again for you contribution. Gerson. 

07052020, 03:50 AM
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RE: HP42S (Free42, DM42) minichallenge – Wallis Product  
07052020, 03:52 AM
Post: #10




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
I came up with the same formula, Valentin.
It can be simplified a lot: Code: 00 { 27Byte Prgm } Blame it on the 4 am ;) Cheers, Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

07052020, 05:32 AM
Post: #11




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
(07052020 03:50 AM)Gerson W. Barbosa Wrote:APL indeed, and thank you for all your posts over the years!(07052020 12:56 AM)mfleming Wrote: Pretty offtopic, but I enjoy doing these minichallenges in other programming languages as well Remember kids, "In a democracy, you get the government you deserve." 

07052020, 11:20 PM
(This post was last modified: 07052020 11:22 PM by Valentin Albillo.)
Post: #12




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
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Hi, Werner: (07052020 03:52 AM)Werner Wrote: I came up with the same formula, Valentin. It can be simplified a lot: Hehe, I certainly could, I was more or less a zombie at those wee hours but I actually did notice the COMB possibility. However I preferred to use N! instead because I thought that it added some extra value to the computation, thinking that N! behaved just like X! does in many HP models, i.e., if the argument is integer it would provide the factorial as expected, but if x wasn't integer it would return the equivalent Gamma function. Alas, in the morning I checked it and no, in the HP42S N! doesn't work for noninteger arguments, so you have to code the equivalent Gamma evaluation in its place, which for my code above merely consist in putting the two steps "1 +" before each of the two factorials and changing both factorials to GAMMA, regardless of further optimizations (which surely are possible). After both insertions my code runs fine and now allows for these interesting possibilites: a) What's the value of the Wallis Product if adding up 10.5 terms ? 10.5 XEQ "WP" > 3.07102204549 b) What's the value of the Wallis Product if adding up Pi terms ? PI XEQ "WP" > 2.93377409474 c) How many terms do we need to add up to get 3 ? 4.9132932375 (terms) XEQ "WP" > 3 Best regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

07062020, 02:31 AM
(This post was last modified: 07062020 03:31 AM by Gerson W. Barbosa.)
Post: #13




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
The Wallis Product converges very slowly towards \(\frac{\pi}{2}\). JM Baillard’s program above, which is about 30% faster than mine, takes 173.2 seconds on my HP42S to run 1000 iterations, giving only two correct decimal digits of \(\pi\): 3.14080774637. Quite disappointing, isn’t it? How long would it take to compute 12 significant digits? As you have figured out by now, 10^d iterations are required for d correct significant digits. How long would it take to return 3.14159265359 on the HP42S, or to return \(\pi\) to 34 digits +/ a couple of ULPs on your computer or smartphone running Free42? Just do the math and be astounded by the results.
I have a 141byte, 81step program that gives 12 digits of \(\pi\) in less than seven seconds on the real HP42s and 34 digits instantly on my smartphone: 3.141592653589793238462643383279504 (the last digit should be a ‘3’), using the Wallis Product as a basis. These are achieved with only 8 and 24 iterations, respectively. I’ve managed to save a few bytes and steps by shamelessly borrowing part of JM Baillard’s code. I will publish it here later, if you are interested. Thank you all for your participation, contributions and great insights so far. Gerson. Edited do fix a few typos. 

07062020, 06:32 AM
Post: #14




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
After a first try of 19 steps, I finally found a 12 steps solutions, the same than JMBaillard :/
Thibault  not collector but in love with the few HP models I own  Also musician : http://walruspark.co 

07062020, 10:47 AM
Post: #15




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Here is mine, 30 bytes, 15 steps:
Code:
Thank you all for the great threestep saving! Gerson. 

07062020, 11:06 AM
Post: #16




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
(07062020 02:31 AM)Gerson W. Barbosa Wrote: I have a 141byte, 81step program that gives 12 digits of \(\pi\) in less than seven seconds on the real HP42s and 34 digits instantly on my smartphone: 3.141592653589793238462643383279504 (the last digit should be a ‘3’), using the Wallis Product as a basis. These are achieved with only 8 and 24 iterations, respectively. I’ve managed to save a few bytes and steps by shamelessly borrowing part of JM Baillard’s code. I will publish it here later, if you are interested.Interested? But of course! 

07062020, 12:40 PM
Post: #17




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
(07062020 11:06 AM)EdS2 Wrote:(07062020 02:31 AM)Gerson W. Barbosa Wrote: ...Interested? But of course! So, here it is, based on the amazing, on the astounding WallisWasicki formula :) Code:
01: 3.151515151515151515151515151515152 02: 3.140740740740740740740740740740741 03: 3.141577263316393751176359872012044 04: 3.141593864074484229522989212911695 05: 3.141592682610236909552589409774771 06: 3.141592651452292998395352098655921 07: 3.141592653532218907530988374852819 08: 3.141592653593856174058282094810482 09: 3.141592653589909875031699539089613 10: 3.141592653589785247473875899289923 11: 3.141592653589792999646253245914736 12: 3.141592653589793254475588473668716 13: 3.141592653589793238954647785071681 14: 3.141592653589793238430189382757839 15: 3.141592653589793238461625819757986 16: 3.141592653589793238462709648277746 17: 3.141592653589793238462645493302460 18: 3.141592653589793238462643247285411 19: 3.141592653589793238462643378896107 20: 3.141592653589793238462643383559629 21: 3.141592653589793238462643383288622 22: 3.141592653589793238462643383278927 23: 3.141592653589793238462643383279480 24: 3.141592653589793238462643383279504 The HP71B program might be more readable: Code:
RUN ? 8 3.14159265359 For more efficiency it is better to process two terms at a time, but then the results for odd n are not accessible: Code:
RUN ? 8 3.14159265361 

07062020, 06:04 PM
Post: #18




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
Amazing!
I know it is off topic, but here is my attempt in RPL: only stack, no local vars, preserve stack, only 4 levels (just for fun!): Code:
Thibault  not collector but in love with the few HP models I own  Also musician : http://walruspark.co 

07062020, 06:53 PM
Post: #19




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
(07062020 06:04 PM)pinkman Wrote: I know it is off topic, but here is my attempt in RPL: only stack, no local vars, preserve stack, only 4 levels (just for fun!): Not offtopic at all, since it’s still on Wallis. 67.5 bytes and (de)counting! Not that I intend to do an attempt, at least for now. On the 50g it should run faster if you use real constants: 119.17 seconds for ten thousand iterations. Should we decide to try 1E11 iterations on the real 50g to possibly get 10 correct decimal digits, we would have to wait until the Summer of 2058 before eventually getting an answer. 

07062020, 08:39 PM
Post: #20




RE: HP42S (Free42, DM42) minichallenge – Wallis Product
2058... but we will all be dead because of Y2038 bug!
Thibault  not collector but in love with the few HP models I own  Also musician : http://walruspark.co 

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