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.
not completely coincidentally, March was also the first month of the year in many historical calendars. Afaik that also explains why the month names have offset to them (sept, oct, nov, dec)
edit: I just love that there are like 5 different comments pointing out this same thing
Also: March is named after the Roman god of war, Mars.
This is because March is when they would begin to mobilize armies for campaigns. The timing is chosen by when winter wheat will be ready for harvest, so soldiers will have nearby high-calorie foot to pilfer while on campaign.
One tricky part of pre-industrial armying is that you can mostly only bring what food you can carry. Things that carry food (e.g. donkeys) require food themselves. So then you have to bring feed as well, which requires more donkeys… etc.
Instead, they would “forage” local areas. If they got there too soon, there is nothing en masse to take!
I've read that not only March was the first month, but the number of months was only ten: winter months did not need to be counted because there was no agricultural work to be done (which was the primary purpose of the calendar). So after the tenth month there was a strange unmapped period.
The first article in this blog post series has a little section talking briefly about this history, and there's a representation of this that I think sheds a lot of light on the original design. See the heading "Side-Note on Month / Day Determination" in the below link [1].
Displaying the months like the following helps see the regularity at a glance. Columns 1, 3 and 5 are the long months, others being shorter:
+-----+-----+-----+-----+-----+
| 31 | 30 | 31 | 30 | 31 |
| I | II | III | IV | V |
| MAR | APR | MAY | JUN | JUL |
|-----+-----+-----+-----+-----+
| 31 | 30 | 31 | 30 | 31 |
| VI |VII |VIII | IX | X |
| AUG | SEP | OCT | NOV | DEC |
+-----+-----+-----+-----+-----+
| 31 |28/29|
| XI |XII |
| Jan | FEB |
+-----+-----+
> To a person who natively thinks in Roman numerals, remembering that the short months are: II, VII, XII, along with IV & IX would be much easier than the way us modern folks have to memorise it.
> not completely coincidentally, March was also the first month of the year in many historical calendars.
And often the last month too. The early modern English calendar began the year on March 25.
This is coincidental in relation to the offset in the names of the months. The Romans started their year in January just like we do today.
(Though in a very broad sense, it's common to begin the year with the new spring. That's the timing of Chinese new year and Persian new year. I believe I've read that the Roman shift two months backward was an administrative reform so that the consuls for the year would have time to prepare for the year's upcoming military campaigns before it was time to march off to war.)
I just added a link to the code with a brief comment. Basically, it simplifies the leap year date calculation. If February is the last month of the year, then the possibly-existing leap day is the last day of the year. If you do it the normal way your calculations for March through December need to know whether February is a leap year. Now none of that is needed. You don’t even need explicit code to calculate whether a given year is a leap year: it’s implicit in the constants 146097, 36524, and 1461.
Not so relevant, but some fun history, the Roman calendar did start in March, so tacking on the leap years was done at the finale. This also meant that the root of the words - the "oct" in october means 8 was also the eighth month of the year.
As well as the leap year stuff people have mentioned, there was something else that I've got a vague memory of (from an old SciAm article, IIRC, which was about using March as the first month for calculations) which pointed out that if you use March as 0, you can multiple the month number by (I forget exactly what but it was around 30.4ish?) and, if you round the fraction up, you get the day number of the start of that month and it all works out correctly for the right 31-30-31 etc sequence.
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 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.
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%.
Not really. It looks like that in the C code, but in the generated machine code it'll just be a single `MULH` instruction giving (only) the upper 64 bits of the result, no shift needed.
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.
std::floor was made constexpr in C++23, which is pretty recent as far as C++ standards go. It's possible the author didn't think using C++23 was worth the constraints it places on who could use the code.
That's a mathematical expression, not a C++ expression. And floor here isn't the C++ floor function, it's just describing the usual integer division semantics. The challenge here is that you need 128-bit integers to avoid overflowing.
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.
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.
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?
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.
Could you elaborate on what you mean? I think it's already impossible to accurately turn a future timestamp into a local time, leap seconds or not, because of timezone shenanigans. So I'm probably misunderstanding what you're talking about.
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.
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.
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.
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.
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.
[0]: https://github.com/kccqzy/smartcal/blob/9cfddf7e85c2c65aa6de...
not completely coincidentally, March was also the first month of the year in many historical calendars. Afaik that also explains why the month names have offset to them (sept, oct, nov, dec)
edit: I just love that there are like 5 different comments pointing out this same thing
Also: March is named after the Roman god of war, Mars.
This is because March is when they would begin to mobilize armies for campaigns. The timing is chosen by when winter wheat will be ready for harvest, so soldiers will have nearby high-calorie foot to pilfer while on campaign.
One tricky part of pre-industrial armying is that you can mostly only bring what food you can carry. Things that carry food (e.g. donkeys) require food themselves. So then you have to bring feed as well, which requires more donkeys… etc.
Instead, they would “forage” local areas. If they got there too soon, there is nothing en masse to take!
I've read that not only March was the first month, but the number of months was only ten: winter months did not need to be counted because there was no agricultural work to be done (which was the primary purpose of the calendar). So after the tenth month there was a strange unmapped period.
6 replies →
The first article in this blog post series has a little section talking briefly about this history, and there's a representation of this that I think sheds a lot of light on the original design. See the heading "Side-Note on Month / Day Determination" in the below link [1].
Displaying the months like the following helps see the regularity at a glance. Columns 1, 3 and 5 are the long months, others being shorter:
> To a person who natively thinks in Roman numerals, remembering that the short months are: II, VII, XII, along with IV & IX would be much easier than the way us modern folks have to memorise it.
[1] https://www.benjoffe.com/fast-date
> explains why the month names have offset to them (sept, oct, nov, dec)
Everything now makes sense, I always wondered why September was the nine month with a 7 prefix.
I thought Sept, Oct, Nov, and Dec were shifted by the addition of July (Julius) and August (Augustus)?
4 replies →
> not completely coincidentally, March was also the first month of the year in many historical calendars.
And often the last month too. The early modern English calendar began the year on March 25.
This is coincidental in relation to the offset in the names of the months. The Romans started their year in January just like we do today.
(Though in a very broad sense, it's common to begin the year with the new spring. That's the timing of Chinese new year and Persian new year. I believe I've read that the Roman shift two months backward was an administrative reform so that the consuls for the year would have time to prepare for the year's upcoming military campaigns before it was time to march off to war.)
At this risk of me feeling stupid, could you briefly explain the benefit of this?
I just added a link to the code with a brief comment. Basically, it simplifies the leap year date calculation. If February is the last month of the year, then the possibly-existing leap day is the last day of the year. If you do it the normal way your calculations for March through December need to know whether February is a leap year. Now none of that is needed. You don’t even need explicit code to calculate whether a given year is a leap year: it’s implicit in the constants 146097, 36524, and 1461.
20 replies →
It's easy to know what day of the year it is because leap days are at the end.
Not so relevant, but some fun history, the Roman calendar did start in March, so tacking on the leap years was done at the finale. This also meant that the root of the words - the "oct" in october means 8 was also the eighth month of the year.
As well as the leap year stuff people have mentioned, there was something else that I've got a vague memory of (from an old SciAm article, IIRC, which was about using March as the first month for calculations) which pointed out that if you use March as 0, you can multiple the month number by (I forget exactly what but it was around 30.4ish?) and, if you round the fraction up, you get the day number of the start of that month and it all works out correctly for the right 31-30-31 etc sequence.
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.
Very cool algorithm and great write-up!
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.
That is an interesting question.
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.
Very nice writeup!
> Years are calculated backwards
How did that insight come about?
Thanks.
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...
Love the literate programming style explanation. Chapeau bas
Nicely done.
Interesting how it compares with the ClickHouse implementation, which uses a lookup table: https://github.com/ClickHouse/ClickHouse/blob/master/src/Com...
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.
Simply, ClickHouse only works on a 399 years span while OP's algorith parses any date, over 3 trillion years.
Nice to see the micro-optimising folks are still making progress on really foundational pieces of the programming stack
Yes, some sharing Vibe coded slop.
It took me a while to understand that internally it uses 128bit numbers, that `>> 64` in the pseudocode was super confusing until I saw the C++ code.
Neat code though!
Not really. It looks like that in the C code, but in the generated machine code it'll just be a single `MULH` instruction giving (only) the upper 64 bits of the result, no shift needed.
Good opportunity to plug this folklore legend: https://neosmart.net/forums/threads/an-extended-history-of-t...
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.
Nice to see that there are still some jewels left to be dug out from the algorithm land.
Well searching for strings, appending strings, comparing strings. All still unimplemented in standard libs. (Strings being unicode of course)
Perhaps nicer to avoid the comment and write:
as
std::floor was made constexpr in C++23, which is pretty recent as far as C++ standards go. It's possible the author didn't think using C++23 was worth the constraints it places on who could use the code.
That's a mathematical expression, not a C++ expression. And floor here isn't the C++ floor function, it's just describing the usual integer division semantics. The challenge here is that you need 128-bit integers to avoid overflowing.
1 reply →
Admittedly in a different league speed wise but also scope wise is my very fast timestamp library for Java https://github.com/williame/TimeMillis
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.
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.
1: https://naggum.no/lugm-time.html
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?
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.
Relevant Old New Thing: https://devblogs.microsoft.com/oldnewthing/20090306-00/?p=18...
https://stackoverflow.com/questions/10849717/what-is-the-sig...
TIL that Unix Time does not count leap seconds. If it did, it wouldn't have been possible to write routines that are this fast.
If Unix Time enumerated leap seconds, you couldn't convert future timestamps into localized times.
Could you elaborate on what you mean? I think it's already impossible to accurately turn a future timestamp into a local time, leap seconds or not, because of timezone shenanigans. So I'm probably misunderstanding what you're talking about.
4 replies →
> The algorithm provides accurate results over a period of ±1.89 Trillion years
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
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 question is if the calculation could be made even more efficient by reducing to 32 bits, or maybe even just 16 bits.
This is somewhat moot considering that 64-bits is the native width of most modern computers and Unix time will exceed 32-bits in just 12 years.
1 reply →
Shorter term the Gregorian calendar has the ratio for leap years just a tiny bit wrong which will be a day off by 3000 years or so.
> i'm placing my bets that in a few thousand years we'll have changed calendar system entirely haha
Given the chronostrife will occur in around 40_000 years (give or take 2_000) I somewhat doubt that </humor>
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.
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.
Or set 0 to be the Big Bang and make the type unsigned. Do it the same time we convert all temperature readings to Kelvin.
1 reply →
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.
This pretty much is assembly written as C++... there's not much the compiler can ruin.
Because that isn’t portable?
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.
[flagged]
Can this algorithm tell me how old I was last year?
Do these calculations take into account the lengthening of the days due to tidal friction?