Timeline for How did the Apollo computers evaluate transcendental functions like sine, arctangent, log?
Current License: CC BY-SA 4.0
37 events
| when toggle format | what | by | license | comment | |
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| S Jun 27, 2022 at 21:06 | history | bounty ended | Starship | ||
| S Jun 27, 2022 at 21:06 | history | notice removed | Starship | ||
| S Jun 26, 2022 at 15:31 | history | bounty started | Starship | ||
| S Jun 26, 2022 at 15:31 | history | notice added | Starship | Reward existing answer | |
| Jun 11, 2020 at 12:40 | answer | added | sivizius | timeline score: 4 | |
| S Oct 4, 2018 at 20:05 | history | bounty ended | uhoh | ||
| S Oct 4, 2018 at 20:05 | history | notice removed | uhoh | ||
| S Oct 3, 2018 at 17:59 | history | bounty started | uhoh | ||
| S Oct 3, 2018 at 17:59 | history | notice added | uhoh | Reward existing answer | |
| Oct 1, 2018 at 12:46 | answer | added | dlatikay | timeline score: 42 | |
| Sep 30, 2018 at 19:47 | vote | accept | uhoh | ||
| S Sep 30, 2018 at 10:53 | history | suggested | Glorfindel | CC BY-SA 4.0 |
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| Sep 30, 2018 at 9:32 | review | Suggested edits | |||
| S Sep 30, 2018 at 10:53 | |||||
| Sep 29, 2018 at 13:57 | comment | added | Uwe | @phuclv: there were analog electronic circuits of that time for logarithm, exponentiation, adding, multiplication and square root but not for trigonometric functions like sin, cos and tan. | |
| Sep 29, 2018 at 12:04 | comment | added | Uwe | @Cris Straton: From wikipedia: "when a hardware multiplier is available (e.g., in a DSP microprocessor), table-lookup methods and power series are generally faster than CORDIC". The Apollo computer had a pretty fast multiplication, Memory Cycle time: 11.7 microseconds. Addition Time: 23.4 microseconds. Multiplication Time: 46.8 microseconds. | |
| Sep 29, 2018 at 7:50 | comment | added | phuclv | for log it's even possible to output the value to an analog computer, compute it and read the result back in an ADC. Those things can be done very quickly in an electric analog computer. I don't know if it works trigonometry but a mechanical analog computer can do that albeit slower | |
| Sep 28, 2018 at 17:45 | comment | added | Chris Stratton | @Uwe actually CORDIC was developed specifically for digital navigation computers in aircraft in 1956. | |
| Sep 28, 2018 at 6:54 | comment | added | uhoh | @JamieHanrahan there's SE site too! Coincidentally I've just asked What algorithm does (did) Excel use for Bessel functions that is discontinuous at x=8? | |
| Sep 28, 2018 at 6:51 | comment | added | Jamie Hanrahan | @Joshua exactly - every digital computer does it by this or similar methods. Another way to look at it is "where did the tables in the books we used to look this stuff up in come from?" In the old days, some poor schmucks had to grind through those series by hand, and later with the aid of mechanical calculators that could multiply and divide. The general field of study is called "Numerical methods" and you can easily find numerous books under that subject heading. | |
| Sep 28, 2018 at 5:19 | comment | added | uhoh | @Joshua all of these are unique to the specific set of constraints on this particular, one(few)-of-a-kind computer. To your reductionist use of "is the same as" the only thing I can say is "no it isn't", to which you could reply "yes it is" and we could continue ad infinitum | |
| Sep 28, 2018 at 5:13 | comment | added | uhoh | @Joshua while modern implementations may include one or more taylor expansions as a seed (depending on which function, sin is easy, found here), that's just the beginning of how modern computers do double precision transcendentals. What I learned from this answer is exactly how it was done in this case, the degree of resulting precision (~1E-04) they had to work with, the effort that went into pre- and post-scaling and why, and the spartan coding. | |
| Sep 28, 2018 at 3:12 | comment | added | NeutronStar | @uhoh, I agree the totality of the situation is compelling, but the meat of the answer (Taylor expansions) is the same as if I asked how my computer does it, at least how I understand these calculations happen today. | |
| Sep 27, 2018 at 21:03 | comment | added | Uwe | @MSalters: CORDIC needs tables of constants, about 50 values. Not very usefull for the Apollo computers with core rope memory for program and constants. | |
| Sep 27, 2018 at 20:19 | comment | added | uhoh | @Joshua consider that these were developed in the early 1960's. as well as considering the size and weight constraints for landing on the moon and returning to Earth, this isn't just "any computer". In fact, technologies developed for computers in the space program helped pave the way for "personal" scientific calculators a decade later. It's the totality of the situation that makes this particular question compelling. | |
| Sep 27, 2018 at 19:44 | comment | added | NeutronStar | One could ask this same question of e.g. calculators, or really any computer. | |
| Sep 27, 2018 at 17:13 | history | edited | uhoh | CC BY-SA 4.0 |
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| Sep 27, 2018 at 16:30 | comment | added | Russell Borogove | Storage (permanent and dynamic both) was at a huge premium so it's not surprising they didn't keep a table. | |
| Sep 27, 2018 at 15:01 | history | tweeted | twitter.com/StackSpaceExp/status/1045327490590429184 | ||
| Sep 27, 2018 at 11:43 | comment | added | MSalters | @Christoph: Already in 1956 we had a better algorithm than Taylor, namely CORDIC. I don't know if that was used in Apollo. | |
| Sep 27, 2018 at 11:30 | comment | added | uhoh | @Christoph have a look here: youtu.be/9YA7X5we8ng?t=1283 and also here youtu.be/YIBhPsyYCiM?t=273 | |
| Sep 27, 2018 at 11:27 | answer | added | supinf | timeline score: 304 | |
| Sep 27, 2018 at 11:17 | history | edited | uhoh |
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| Sep 27, 2018 at 11:16 | comment | added | uhoh |
@Christoph First sentence starts with two instruments that likely produce angular data. Also, what was done in the 1960's in computers in space, is not covered by how things are done now. I'll add the history tag to make that even clearer. You might also consider where tables would have to be stored, there was precious little memory in the Apollo computers.
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| Sep 27, 2018 at 11:16 | comment | added | SF. | ...or both, the taylor expansion preparing lookup tables on startup, in case permanent storage is less abundant than RAM. | |
| Sep 27, 2018 at 11:14 | comment | added | Christoph | Are you sure about the need to do trigonometry for gimbaling? I'm pretty sure it would be easier to stay within vector math if you use a linear actuator. Anyway trignonometry functions are usually implemented by table lookup or approximated as a taylor expansion. No source directly related to Apollo, sorry. | |
| Sep 27, 2018 at 10:59 | history | edited | uhoh | CC BY-SA 4.0 |
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| Sep 27, 2018 at 10:18 | history | asked | uhoh | CC BY-SA 4.0 |