A write-up of a new Gregorian date conversion algorithm.
It achieves a 30–40% speed improvement on x86-64 and ARM64 (Apple M4 Pro) by reversing the direction of the year count and reducing the operation count (4 multiplications instead of the usual 7+).
Paper-style explanation, benchmarks on multiple architectures, and full open-source C++ implementation.
I was fortunate enough to be programming on an ARM based device, which meant that the terms (x * 4 + 3) strongly stood out to me as highly inefficient, being 2 cycle prep for the more important division. On x64 computers, those two operations are calculated in only one operation by using the 'LEA' assembly instruction (which I wasn't aware of at the time), and so others using that type of computer might not have felt this step needed any simplification.
I tried everything under the sun to get rid of these steps. The technique noted in the article of using the year 101 BC was for a long time my strongest candidate, you can view the implementation of that attempt at the link below [1].
An epoch of 101 BC still meant that there was extra work required to re-normalise the timeline after the century calculation, but it was only a single addition of 365 in the calculation of `jul`. The explanation of how this works is probably a whole blog post in itself, but now that this algorithm has been discarded it's not worth the time to explain it fully.
I also had the year-modulus-bitshift technique developed at that time, but it couldn't be integrated cleanly with any of my algorithm attempts yet. My plan was to simply document it as an interesting but slower concept.
I don't know what sparked the idea of going backwards other than immersing myself deeply in the problem in my spare time for about a month. It finally came to me one evening, and I thought it was only going to save 1-cycle, but when it also meant the year-modulus-bitshift could be utilised, the entire thing fit together like a glove and the speed collapsed down from 20% time saving to 40%.
I was a bit confused initially about what your algorithm actually did, until I got to the pseudo-code. Ideally there would be a high level description of what the algorithm is supposed to do before that.
Something as simple as: “a date algorithm converts a number of days elapsed since the UNIX epoch (1970-01-01) to a Gregorian calendar date consisting of day, month, and year” would help readers understand what they're about to read.
Thanks, that is a good idea. This was originally a blog post series, and the first article gave a bit of an introduction.
When I started the blog series, I expected the first article to be the most noteworthy, with the 2nd and 3rd being lesser supplementary topics.
Now that the 3rd blog post ended up with a much larger result than I was expecting, it stands on its own and could do with some editing as you suggest.
How would this algorithm change on 16-bit or 8-bit devices? Or does some variety of the traditional naïve algorithm turn out to be optimal in that case? There's quite a bit of microcontroller software that might have to do date conversions, where performance might also matter. It's also worth exploring alternative epochs and how they would affect the calculation.
It might also come into play if developing SIMD alternatives for batch date processing, as one can have more lanes with 16-bit. I plan to make a blog post covering SIMD and if 16-bit algorithms have reasonable performance then that will be covered.
Wouldn’t it be accurate for that as well? Unless we change to base 10 time units or something. Then we all have a lot of work to do.
But if it’s just about starting over from 0 being the AI apocalypse or something, I’m sure it’ll be more manageable, and the fix could hopefully be done on a cave wall using a flint spear tip.
Maybe not in a few thousand years, but given the deceleration of the Earth’s rotation around its axis, mostly due to tidal friction with the moon, in a couple hundred thousand years our leap-day count will stop making sense. In roughly a million years, day length will have increased such that the year length will be close to 365.0 days.
I therefore agree that a trillion years of accuracy for broken-down date calculation has little practical relevance. The question is if the calculation could be made even more efficient by reducing to 32 bits, or maybe even just 16 bits.
The calendar system already changed. So this won't get correct dates, meaning the dates actually used, past that date. Well, those dates, as different countries changed at different times.
I wrote my own date calculation functions a while ago. And during that, I had an aha moment to treat March 1 as the beginning of the year during internal calculations[0]. I thought it was a stroke of genius. It turns out this article says that’s the traditional way.
Thank you for sharing. This is a great achievement not only in the ability to invent a novel algorithm with significant performance gains but also the presentation of the work. It's very thorough and detailed, and I appreciated reading it.
The Windows epoch starts on 1601-01-01. I always assumed that was because it slightly simplifies the calculation, as described in the article. But it's not as good as the article's method of counting backwards.
This focuses on string <-> timestamp and a few other utilities that are super common in data processing and where the native Java date functions are infamously slow.
I wrote it for some hot paths in some pipelines but was super pleased my employer let me share it. Hope it helps others.
For something this short that is pure math, why not just hand write asm for the most popular platforms? Prevents compiler from deoptimizing in the future.
Have a fallback with this algorithm for all other platforms.
An interesting writeup on using a different representation for time is here[1]. It can represent any specific second from March 1, 2000 +/-2.9Myears with 62 bits and can efficiently calculate Gregorian dates using only 32-bit arithmetic. An optimization involving a 156K lookup table is also discussed.
A few notes for those not familiar with Lisp:
1. Common Lisp defines a time called "universal time" that is similar to unix time, just with a different epoch
2. A "fixnum" is a signed-integer that is slightly (1-3 bits) smaller than the machine word size (32-bits at the time the article was written). The missing bits are used for run-time type tagging. Erik's math assumes 31-bits for a fixnum (2.9M years is approximately 2^30 days and fixnums are signed).
3. Anywhere he talks about "vectors of type (UNSIGNED-BYTE X)" this means a vector of x-bit unsigned values. Most lisp implementations will allow vectors of unboxed integers for reasonable values of X (e.g. 1, 8, 16, 32, 64), and some will pack bits for arbitrary values of X, doing the shift/masking for you.
That pseudo-code isn't very imprecise because there's no type information (64-bit or 128-bit integers? signed or unsigned?) and it doesn't account for results of overflow or underflow in the realm of UB. It's also inconsistent to introduce bit shifts instead of division and then use modulus instead of "and" masking; typically, pick one style or the other.
caldat is the third algorithm in the Numerical Recipes in Pascal (1986,89,90,92) book[0] (p. 13), where Julian days are easy to turn into days since the UNIX epoch. It uses 3 single-precision floating point divisions and 3 multiplications with pre-Gregorian support or 2 each respectively without, but is convertible to an algorithm using a mix of 8-bit and 16-bit signed fixed point integer math for microcontroller usage. 64-bit (or higher) integer math is not strictly required, but whatever's faster and correct for a given target is fine.
0: The last time I dug up the book was some time last year because I was hunting for an algorithm for the precise position of the Sun in the sky given a lat lon (WGS 84) and date time for a solar tracker that didn't need light sensors, only time and location that was already available for free.
For 64 bit timekeeping arguably for lots of uses counting nanoseconds makes a more sense than seconds. You can still cover decent usable range (2^64 ns > 584 years) and save the need for separate subsecond counter.
What would be the most efficient algorithm to handle such ns scale? I guess one option would be just to divide by 10^9 and run the code from the article, but can we do better?
35 comments
[ 2.4 ms ] story [ 67.0 ms ] threadIt achieves a 30–40% speed improvement on x86-64 and ARM64 (Apple M4 Pro) by reversing the direction of the year count and reducing the operation count (4 multiplications instead of the usual 7+).
Paper-style explanation, benchmarks on multiple architectures, and full open-source C++ implementation.
> Years are calculated backwards
How did that insight come about?
I was fortunate enough to be programming on an ARM based device, which meant that the terms (x * 4 + 3) strongly stood out to me as highly inefficient, being 2 cycle prep for the more important division. On x64 computers, those two operations are calculated in only one operation by using the 'LEA' assembly instruction (which I wasn't aware of at the time), and so others using that type of computer might not have felt this step needed any simplification.
I tried everything under the sun to get rid of these steps. The technique noted in the article of using the year 101 BC was for a long time my strongest candidate, you can view the implementation of that attempt at the link below [1].
An epoch of 101 BC still meant that there was extra work required to re-normalise the timeline after the century calculation, but it was only a single addition of 365 in the calculation of `jul`. The explanation of how this works is probably a whole blog post in itself, but now that this algorithm has been discarded it's not worth the time to explain it fully.
I also had the year-modulus-bitshift technique developed at that time, but it couldn't be integrated cleanly with any of my algorithm attempts yet. My plan was to simply document it as an interesting but slower concept.
I don't know what sparked the idea of going backwards other than immersing myself deeply in the problem in my spare time for about a month. It finally came to me one evening, and I thought it was only going to save 1-cycle, but when it also meant the year-modulus-bitshift could be utilised, the entire thing fit together like a glove and the speed collapsed down from 20% time saving to 40%.
[1] https://github.com/benjoffe/fast-date-benchmarks/blob/218356...
I was a bit confused initially about what your algorithm actually did, until I got to the pseudo-code. Ideally there would be a high level description of what the algorithm is supposed to do before that.
Something as simple as: “a date algorithm converts a number of days elapsed since the UNIX epoch (1970-01-01) to a Gregorian calendar date consisting of day, month, and year” would help readers understand what they're about to read.
When I started the blog series, I expected the first article to be the most noteworthy, with the 2nd and 3rd being lesser supplementary topics.
Now that the 3rd blog post ended up with a much larger result than I was expecting, it stands on its own and could do with some editing as you suggest.
It might also come into play if developing SIMD alternatives for batch date processing, as one can have more lanes with 16-bit. I plan to make a blog post covering SIMD and if 16-bit algorithms have reasonable performance then that will be covered.
i'm placing my bets that in a few thousand years we'll have changed calendar system entirely haha
but, really interesting to see the insane methods used to achieve this
But if it’s just about starting over from 0 being the AI apocalypse or something, I’m sure it’ll be more manageable, and the fix could hopefully be done on a cave wall using a flint spear tip.
I therefore agree that a trillion years of accuracy for broken-down date calculation has little practical relevance. The question is if the calculation could be made even more efficient by reducing to 32 bits, or maybe even just 16 bits.
Given the chronostrife will occur in around 40_000 years (give or take 2_000) I somewhat doubt that </humor>
[0]: https://github.com/kccqzy/smartcal/blob/9cfddf7e85c2c65aa6de...
Neat code though!
This focuses on string <-> timestamp and a few other utilities that are super common in data processing and where the native Java date functions are infamously slow.
I wrote it for some hot paths in some pipelines but was super pleased my employer let me share it. Hope it helps others.
So that a day number can be directly mapped to year, month, and day, and the calendar date can be mapped back with a year-month LUT.
Have a fallback with this algorithm for all other platforms.
A few notes for those not familiar with Lisp:
1. Common Lisp defines a time called "universal time" that is similar to unix time, just with a different epoch
2. A "fixnum" is a signed-integer that is slightly (1-3 bits) smaller than the machine word size (32-bits at the time the article was written). The missing bits are used for run-time type tagging. Erik's math assumes 31-bits for a fixnum (2.9M years is approximately 2^30 days and fixnums are signed).
3. Anywhere he talks about "vectors of type (UNSIGNED-BYTE X)" this means a vector of x-bit unsigned values. Most lisp implementations will allow vectors of unboxed integers for reasonable values of X (e.g. 1, 8, 16, 32, 64), and some will pack bits for arbitrary values of X, doing the shift/masking for you.
1: https://naggum.no/lugm-time.html
caldat is the third algorithm in the Numerical Recipes in Pascal (1986,89,90,92) book[0] (p. 13), where Julian days are easy to turn into days since the UNIX epoch. It uses 3 single-precision floating point divisions and 3 multiplications with pre-Gregorian support or 2 each respectively without, but is convertible to an algorithm using a mix of 8-bit and 16-bit signed fixed point integer math for microcontroller usage. 64-bit (or higher) integer math is not strictly required, but whatever's faster and correct for a given target is fine.
0: The last time I dug up the book was some time last year because I was hunting for an algorithm for the precise position of the Sun in the sky given a lat lon (WGS 84) and date time for a solar tracker that didn't need light sensors, only time and location that was already available for free.
What would be the most efficient algorithm to handle such ns scale? I guess one option would be just to divide by 10^9 and run the code from the article, but can we do better?