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Comment by hyperman1 | Hacker News Reader

You know, even if I know plenty of digits, I can't think of any real world reason in my personal life for needing more decimals than 3.14 . If I add one more, that's a correction of less than 1/1000 and no measurement I do is that precise. NASA's rocket trajectories are less precise than that. Of course, extremely precise devices need better, but they are removed far from my everyday life.

Has anybody needed more than 3.14?

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This is an incredibly interesting question, and the answer is an emphatic YES! Many systems involve iterative schemes, where the output of one step is used as the input to the next step. Here, these precision errors can accumulate, and if there's a multiplicative term in your equations, they can explode!

These sorts of problems are actually very common in a lot of scientific computing and simulation contexts, which is why many in the scientific computing community look aghast at the rise of FP16 (and even fp32) in machine learning applications. Of course, those algorithms are often of a _very_ different nature from (say) the large-scale linear algebra or PDE solvers we're using, but still it's pretty shocking if you're used to worrying about machine precision!

Machine learning might also prefer focusing on the magnitude rather than an exact value. (With the lower precision number part more about being nudged between magnitude bins.) E.G. bfloat16

https://en.wikipedia.org/wiki/Bfloat16_floating-point_format

That's a thousandth of an inch. Lots of things are machined to a thousandth of an inch tolerance, and some things are machined to a significantly finer tolerance. And if you're drilling bolt holes on a diameter that's greater than an inch, you'll need to use a lot more than 3.14.

For work I needed the accuracy out to 15 decimal places. It was for calculating random jitter based on integrating and averaging a large number of phase noise measurements.

I can't think of any real world reason in my personal life for needing more decimals than 3.14

If you keep your feet on the ground, that's probably enough for most things. But your grandchildren will probably need more, as spaceflight becomes commoditized. A couple of decimal places might mean the difference between landing on Mars, and icy death in the vacuum of space.

> correction of less than 1/1000 and no measurement I do is that precise. NASA's rocket trajectories are less precise than that.

Source? If you make that big of an error you're gonna fail your orbital insertion at the other end entirely when going to places like Mars, let alone any farther.

I don't have a source to support the OP, but I would think there is a significant allowance for course correction that reduces the required precision. It's not a purely ballistic trajectory.

The mid-course correction is not because you used an imprecise value of pi. If that's all it were about, you'd just use a precise pi from the beginning and then not even need the mid-course correction. Missions have failed because the rocket engines failed to start up for the mid-course correction. Unnecessarily risking that because for some reason you insist on using far fewer digits of pi than your computer is capable of would be insanity.

The context of my statement/question was definitely "given the precision that can be achieved with available precision" and not with an arbitrary imprecise constant in the calculations. As you say, the mechanical aspect of the launch and transfer orbit insertion is going to be several orders of magnitude more imprecise than the mathematical goal - that is what the mid-course corrections are for (aka the difference between theory and reality)

The comment I was originally responding to said "Has anybody needed more [digits in pi] than 3.14?" and "NASA's rocket trajectories are less precise than [one part in a thousand]", both of which I take issue with. You then stepped in to defend those statements against my objections. I'm glad it turns out you don't actually agree with those statements, but hopefully you can see my confusion.

How precisely do you have to get it right up front? Maybe it's OK to get your initial trajectory off by a 1/1000 part, and make an adjustment when you're 3/4 of the way to Mars and your relative error is now more like 1/250.

There's no world in which it makes sense to save a few digits on pi and then risk running out of fuel while making your mid-course correction because your initial burn was too far off. The mid-course correction is because you can only be so precise when you're firing rocket engines, not because you're foolishly using imprecise constants for no good reason at all.

So I'll reiterate, I want to see a source that NASA uses only 3.14 as the value for pi in making their trajectory calculations. And I want to point out the absurdity of even so much as having this debate in the very comment thread for an article on nasa.gov where NASA itself is saying that they use 3.141592653589793 for pi. We already have as good as a source as we're going to find, and there's a lot more sig figs in it than 3!

Sure. But the question asked was not about what value NASA use, but what precision you need for space travel. The rule of thumb I use when looking at noisy estimates is that you should shoot for twice the precision (one extra bit of information) that you have in your noisiest other value (or of your total noise - they're usually about the same thing). Any less precision than that and your estimate gets worse, but more precision doesn't materially improve your estimate. I don't know a rigorous proof for this but you can hand wave it from the Nyquist-Shannon sampling theorem [0].

So how much uncertainty comes from other sources when deciding on an initial burn for your trip to Mars? Contributing factors could be your fine control over the thrust from the rocket engine, but also solar radiation, gravity or - I suspect this is biggest - attitude control.

I'd guess NASA can achieve a precision of better than 1/500 but probably not 1/50,000, so they'd need about 5 or 6 digits of pi. But I'm interested in hearing a more educated guess!

[0] https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_samp...

And so we're back to still needing a source to do better than guessing :/

Some factors are way more important than others for rocket navigation. For instance, the azimuth, the angle you're headed is very important, since a small error in angle gets you way off target. The Saturn V theodolite, used to calibrate this angle before launch had an accuracy of better than +/- 2 seconds of arc, i.e. 1 part in 648000. That suggests you'd need to use pi to at least 6 decimal places (probably more) for surveying and azimuth calibration.

The book "Inventing Accuracy" is about missile guidance (which has mostly the same issues) and discusses the various sources of error in detail and the various contributions to the "error budget".

Wow, calm down, CydeWeys, you're scaring me. To be clear: No source, imprecise sentence, should have added a ton of buts and ifs, but that would make the question twice as long and thrice as boring. I meant it more in a philosophical way.

For example: I don't think rockets start on course with a precision of 1e-3, and even if they do, small atmospheric changes will probably cause errors greater than that. I assume the feedback loop of the guidance systems easily smooth over errors this small. So while the theoretical trajectory calculations need more precision, the real world guidance might do not.

Another consideration: Is e.g. the length of the rocket know this precise? If temperature on a day changes over a range of e.g. 10 degrees, the materials might expand and contract more than 1e-3.

Now that's NASA. There are clearly things requiring much smaller errors. My CPU litography being off a factor 1e-3 will not end well.

I meant the quest in my everyday life. If I build a round table for my home, 3.14 will probably serve me well enough on the drawing table. At what point will random Joe Shmoe notice when pi is not exactly 3.14

Do you have any sources you can point to from research you've done on this subject? This is all coming off as uninformed speculation.

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