5 ms·

The article seems a bit backward to me. The reason they use 3.141..793 is because that's the value of pi in double precision floating point. And that was set by

by kens 6y ago

The article seems a bit backward to me. The reason they use 3.141..793 is because that's the value of pi in double precision floating point. And that was set by the Intel 8087 coprocessor. So to me, the real answer is that Intel (and Kahan) decided on the number of decimals in 1980, and NASA uses that because it's the standard. On the other hand, in the 1960s NASA figured out exactly how many bits of precision they needed to get to the Moon and came up with the 15-bit Apollo Guidance Computer; 14 bits wasn't enough and 16 bits was more than they needed. (The computer supports double precision and triple precision for tasks that needed more.) The point is that in the 1960s, aerospace systems would carefully consider exactly how many bits they needed and build systems with bizarre numbers of bits like 15 or 27. Now, they use standard systems, which generally have way more accuracy than needed.

ivanbakel 6y ago
The article doesn't try to justify the exact number of decimal places - and, by eye, the arguments it uses are likely to work to 14 decimal places as well, since the error would be similarly small. Instead, it tries to answer the thrust of the prompt question: given the massive numbers used in spaceflight, is pi calculated to the greatest possible practical accuracy? Going over the history of the 15-digit version would divert from the interesting part of the article (the effect of precision on calculation) and dilute a nice teachable moment. Though that fact about the Apollo Computer would make an interesting part of a follow-up.
- kens 6y ago
  The Apollo Guidance Computer wasn't 15 decimal places; it was 15 bits. The point is that they didn't use power-of-two word sizes back then; they used whatever word size fit the problem, even though that now seems bizarre.
  - deleted 6y ago
    [deleted]
  - ivanbakel 6y ago
    But the number in the article is to 15 decimal places. Pointing out that that precision comes from the size of the modern double-precision floating point representation doesn't really answer why that representation is enough.
    - CydeWeys 6y ago
      It's a fair point. They would use more decimal places than that if it were necessary, regardless of whatever a double-precision floating point does. Since it's not necessary, the double is adequate (and already widely available by default across programming languages and system architectures).
Const-me 6y ago
The coprocessor uses 80-bit floats, that’s more than double precision. It even has a special instruction, fldpi, to get π in the full precision: https://www.felixcloutier.com/x86/fld1:fldl2t:fldl2e:fldpi:fldlg2:fldln2:fldz https://www.felixcloutier.com/x86/fld1:fldl2t:fldl2e:fldpi:f...
- olliej 6y ago
  It does however completely bollocks up all the subsequent trigonometry functions. Even intel says not to use them. Also alas, 80bit is a bit of a misnomer - when comparing to the more regular ieee754 representations, it’s actually fp79 :D
  - jart 6y ago
    Intel's Manual says to use FLDPI. The x87 FPU represents pi internally at extended-extended-precision, so it's more like fp82 :D Were you thinking about the argument reduction gotcha? http://galaxy.agh.edu.pl/~amrozek/AK/x87.pdf http://galaxy.agh.edu.pl/~amrozek/AK/x87.pdf
    - olliej 6y ago
      Fldpi only loads the register, rounding according to the current rounding mode, so you still only have 63 bits of precision when you load the value. For intermediate calculations it incorporates the internal extended pi via the GRS bits. And yeah the train wreck of argument reduction was my reference
azhenley 6y ago
I wish there were more details about the historical context in this. I recently went down a rabbit hole of trying to implement a cosine function from scratch and found that for most applications where I use cosine (low-resolution graphs or 2d games), I need a shockingly low level of precision. Even four decimals was overkill! For those that are interested, you can read about my adventures with precision and cosine: https://web.eecs.utk.edu/~azh/blog/cosine.html https://web.eecs.utk.edu/~azh/blog/cosine.html
- jandrese 6y ago
  In games it is very common to see lookup tables for trig functions. Not big ones either, maybe a few hundred entries.
  - richrichardsson 6y ago
    Audio plugin programming makes extensive use of lookup tables for expensive operations. Pre-calculating is a useful technique for such real-time sensitive work (like games that have to render a new frame every 16ms @ 60hz or audio plugins that need to return potentially a buffer every 0.72ms @ 32 samples/44.1kHz).
  - TwoBit 6y ago
    Calculators actually use lookup tables themselves. I don't know if FPUs do too, but I wouldn't be surprised, as it's faster than series expansion.
    - jacobolus 6y ago
      Handheld calculators generally use the CORDIC algorithm, https://en.wikipedia.org/wiki/CORDIC https://en.wikipedia.org/wiki/CORDIC
  - jacobolus 6y ago
    If you have access to modern processors (CPUs, GPUs) then a lookup table makes no sense. The polynomial is faster, more accurate, and needs less space to store precomputed values.
ojnabieoot 6y ago
I think you are misreading the article, or at least taking an oddly limited "developer-only" view instead of considering the audience. NASA is not trying to answer your question about how many digits of pi they use. They are trying to answer a non-technical person on Facebook (likely a kid) who is wondering, vaguely, if NASA needs to use a highly precise representation of pi to fly spaceships, or if something coarse will work. The answer the kid is looking for: NASA's most precise representation of pi is more precise than 3.14, but nowhere close to 500 digits like the question suggested. 15 is more than enough for most engineers at NASA, and anything that an astronomer might conceptually want to do would take at most 40 digits of pi to do with almost arbitrary precision. The fact that the current representation is architecturally convenient for modern FPUs is basically immaterial to the person's question, even if that's interesting for people with detailed knowledge about such things.
- Dylan16807 6y ago
  They're still only answering half the question. Okay, 15 digits is more than enough. But how much more? What would the minimum be, and why are we using more than that? Ideally the article would talk about what "more than enough" looks like (which it does), but also what minimums look like (which it doesn't), and then mention that they chose that specific size because most computers do two specific sizes really fast and that's the more accurate of the two.
olliej 6y ago
Double precision from ieee754 is 64bit, of which 53bits are the significand. X87 uses an 80bit format with functionally a 63 bit significand (they actually use a 64 mantissa, but for that actually doesn’t gain you anything, and adds many terrible edge cases). They use double precision values because that presumably provides more precision than they need, but 32 bit float would be woefully inadequate. They could use more (x87 or software float for instance) but there’s not really an any advantage to the increased precision you get, and there are many downsides.
kleiba 6y ago
This seems unlikely - if a specific number of bits in a standardized floating point representation would imply relevant calculation errors, Nasa would certainly not use it. Sure, if 24 bits were already sufficient, they would not go out of their way to avoid the extra 8 bits. So in that sense you're right of course. But it's not just "Hey, single precision floats are 32 bits, so why don't we just use that!"

by kens 6y ago

ivanbakel 6y ago

The article doesn't try to justify the exact number of decimal places - and, by eye, the arguments it uses are likely to work to 14 decimal places as well, since the error would be similarly small. Instead, it tries to answer the thrust of the prompt question: given the massive numbers used in spaceflight, is pi calculated to the greatest possible practical accuracy? Going over the history of the 15-digit version would divert from the interesting part of the article (the effect of precision on calculation) and dilute a nice teachable moment. Though that fact about the Apollo Computer would make an interesting part of a follow-up.

kens 6y ago
The Apollo Guidance Computer wasn't 15 decimal places; it was 15 bits. The point is that they didn't use power-of-two word sizes back then; they used whatever word size fit the problem, even though that now seems bizarre.
- deleted 6y ago
  [deleted]
- ivanbakel 6y ago
  But the number in the article is to 15 decimal places. Pointing out that that precision comes from the size of the modern double-precision floating point representation doesn't really answer why that representation is enough.
  - CydeWeys 6y ago
    It's a fair point. They would use more decimal places than that if it were necessary, regardless of whatever a double-precision floating point does. Since it's not necessary, the double is adequate (and already widely available by default across programming languages and system architectures).

Const-me 6y ago

The coprocessor uses 80-bit floats, that’s more than double precision. It even has a special instruction, fldpi, to get π in the full precision: https://www.felixcloutier.com/x86/fld1:fldl2t:fldl2e:fldpi:fldlg2:fldln2:fldz https://www.felixcloutier.com/x86/fld1:fldl2t:fldl2e:fldpi:f...

olliej 6y ago
It does however completely bollocks up all the subsequent trigonometry functions. Even intel says not to use them. Also alas, 80bit is a bit of a misnomer - when comparing to the more regular ieee754 representations, it’s actually fp79 :D
- jart 6y ago
  Intel's Manual says to use FLDPI. The x87 FPU represents pi internally at extended-extended-precision, so it's more like fp82 :D Were you thinking about the argument reduction gotcha? http://galaxy.agh.edu.pl/~amrozek/AK/x87.pdf http://galaxy.agh.edu.pl/~amrozek/AK/x87.pdf
  - olliej 6y ago
    Fldpi only loads the register, rounding according to the current rounding mode, so you still only have 63 bits of precision when you load the value. For intermediate calculations it incorporates the internal extended pi via the GRS bits. And yeah the train wreck of argument reduction was my reference

azhenley 6y ago

I wish there were more details about the historical context in this. I recently went down a rabbit hole of trying to implement a cosine function from scratch and found that for most applications where I use cosine (low-resolution graphs or 2d games), I need a shockingly low level of precision. Even four decimals was overkill! For those that are interested, you can read about my adventures with precision and cosine: https://web.eecs.utk.edu/~azh/blog/cosine.html https://web.eecs.utk.edu/~azh/blog/cosine.html

jandrese 6y ago
In games it is very common to see lookup tables for trig functions. Not big ones either, maybe a few hundred entries.
- richrichardsson 6y ago
  Audio plugin programming makes extensive use of lookup tables for expensive operations. Pre-calculating is a useful technique for such real-time sensitive work (like games that have to render a new frame every 16ms @ 60hz or audio plugins that need to return potentially a buffer every 0.72ms @ 32 samples/44.1kHz).
- TwoBit 6y ago
  Calculators actually use lookup tables themselves. I don't know if FPUs do too, but I wouldn't be surprised, as it's faster than series expansion.
  - jacobolus 6y ago
    Handheld calculators generally use the CORDIC algorithm, https://en.wikipedia.org/wiki/CORDIC https://en.wikipedia.org/wiki/CORDIC
- jacobolus 6y ago
  If you have access to modern processors (CPUs, GPUs) then a lookup table makes no sense. The polynomial is faster, more accurate, and needs less space to store precomputed values.

ojnabieoot 6y ago

I think you are misreading the article, or at least taking an oddly limited "developer-only" view instead of considering the audience. NASA is not trying to answer your question about how many digits of pi they use. They are trying to answer a non-technical person on Facebook (likely a kid) who is wondering, vaguely, if NASA needs to use a highly precise representation of pi to fly spaceships, or if something coarse will work. The answer the kid is looking for: NASA's most precise representation of pi is more precise than 3.14, but nowhere close to 500 digits like the question suggested. 15 is more than enough for most engineers at NASA, and anything that an astronomer might conceptually want to do would take at most 40 digits of pi to do with almost arbitrary precision. The fact that the current representation is architecturally convenient for modern FPUs is basically immaterial to the person's question, even if that's interesting for people with detailed knowledge about such things.

Dylan16807 6y ago
They're still only answering half the question. Okay, 15 digits is more than enough. But how much more? What would the minimum be, and why are we using more than that? Ideally the article would talk about what "more than enough" looks like (which it does), but also what minimums look like (which it doesn't), and then mention that they chose that specific size because most computers do two specific sizes really fast and that's the more accurate of the two.

olliej 6y ago

Double precision from ieee754 is 64bit, of which 53bits are the significand. X87 uses an 80bit format with functionally a 63 bit significand (they actually use a 64 mantissa, but for that actually doesn’t gain you anything, and adds many terrible edge cases). They use double precision values because that presumably provides more precision than they need, but 32 bit float would be woefully inadequate. They could use more (x87 or software float for instance) but there’s not really an any advantage to the increased precision you get, and there are many downsides.

kleiba 6y ago

This seems unlikely - if a specific number of bits in a standardized floating point representation would imply relevant calculation errors, Nasa would certainly not use it. Sure, if 24 bits were already sufficient, they would not go out of their way to avoid the extra 8 bits. So in that sense you're right of course. But it's not just "Hey, single precision floats are 32 bits, so why don't we just use that!"