alt Hacker NewsA decades-old bug in Knuth's long division (TAOCP Vol II, Algorithm 4.3.1D)
Comments
This may very well be the most epic post in HN history.
EDIT : I recall fondly algorithm D... one of my first programming experiences in the 90s was trying to implement knuth's arithmetic algorithms for addition, substraction, etc. in 8086 asm. Got them right up until long multiplication (that one was a tremendous effort). Algorithm D was too formidable to even dare me attempt. Feel extremely happy to see people in 2026 looking at these algorithms closely.
Back in the 1980s, before there was a DIV instruction, I implemented integer division.
I used the same long division algorithm I was taught in 3rd grade, except in binary rather than base 10. Shift and subtract.
It was also the basis for implementing FDIV (floating point division) for those who did not have an x87 chip.
Nobody ever reported a bug in it.
Great find and write up. This year if I remember correctly 40+ people got the check, ~1000 have an account at the bank. I got mine this year, an exercise I revisit every few years since 2012 to learn a new programming lang or approach had a newer update that made it have 2 offbyone errors. I had extra time this year so went to the beginning of the chapter to attempt an open problem and in the preliminaries another off by two error. I was quite surprised.
Really made me appreciate how unlikely it is to find an error. It feels as if it was planned just for me to find it. Just like the author studied cryptography and then decided to do some exercises to hone his skills, an unlikely journey towards a check.
Congratulations! It's funny that the reward schedule is not based on importance. It's just 0x$1.00 for an error and 0x$0.20 for a suggestion, no matter what. Personally I have 0x$4.40 in the bank, more than the author's 0x$1.00, but none of my four errors and two suggestions were as important as this one. Getting your name in the book is pretty cool though!
This bug is only in the English description of the algorithm, right? No bug in either the MIX or MMIX implementations?
A story from my life about not judging a book by its cover...and Algorithm D:
Some years after the turn of the millennium I was a CS student at UC Santa Cruz. I was taking various classes for my major and I ended up in a Comparative Programming Languages class, which was a quarter-long survey of different modalities - I remember Haskell, OCaml, C++, and there were maybe two others.
Anyway I had started noticing a particular student showing up in some of my classes. He stood out. Firstly because he was always asking questions, sometimes to the point of annoying other students. And then because he was older than the rest of us - in hindsight he probably wasn't older than his early 40s - but I was ~20 and as I came to learn, he'd lived hard. He had a stout, platinum blonde beard that seemed yellowed from the hand-rolled cigarettes I always saw him smoking outside the computer lab.
After class one day I started chatting with him. I wasn't much of a question-asker, and I found his willingness to do so in the face of obvious annoyance to actually be kind of brave, so I think I probably opened by complimenting him and asking if the reactions from other students bothered him. His answer, gravely-voiced, was clear: he was paying for these classes same as anyone else, and he wanted to get his money's worth. I found it a refreshingly self-centered take. I decided I liked the guy.
Over time we became lab-mates, working on projects together. He always reeked of tobacco; his fingers too were yellowed from those rollies. I learned that he'd never finished college his first time around, instead getting hired into industry and riding the wave of the dot-com boom. When the crash eventually landed, he washed out and found himself living the surf bum life in Mexico, soaked in alcohol and seawater. When he eventually decided he had to get his life together, he sobered up and moved back to the States. But he was unemployed, homeless - living out of his VW van - and a 40-something college student. He was a misfit.
So, let's see..right, Algorithm D. So for our Comparative Languages class, the OCaml project was an arbitrary-precision calculator. We worked through addition, subtraction, and multiplication, and then as the project deadline approached we turned our sights towards division. Me, I took one look at Knuth and decided to start instead with a brute-force implementation. But once that worked, we began tackling Algorithm D. Around 2am, still not done, I threw up my hands and said I was going home - I'd take whatever grade was coming. My partner also went home - to his van parked in the Engineering lot. I knew he didn't own his own computer, so imagine my surprise when I saw him the next day and he told me he had finished the Algorithm D implementation overnight.
Turned out he had gone back to his van that night with a pen and a ream of paper and worked the code out by hand, only typing it up in the morning. We were lab partners but I wasn't going to copy something I'd had no hand in; I got whatever grade I deserved and he got the perfect score. I'm sure the older heads have plenty of stories of coding by hand, but even by that time, circa 2005, such a thing seemed arcane, almost unheard of. I was duly impressed.
I occasionally wonder what happened to him - he was a smart guy and a good engineer, and I learned some important lessons from him. I hope he found his footing. And for the sake of his cubicle mates, maybe also kicked the cigarette habit.
That is so cool. Exceptional work, friend.
That makes this thread a bit more interesting now.
https://stackoverflow.com/questions/60479571/is-there-a-bug-...
> "I'm especially glad to have this correction, because I think the readers of TAOCP Vol 2 look at Algorithm 4.3.1 D more than any other algorithm!"
If you look at the fore edge of my copy of vol 2 will see a noticeably grubby line. Open the book at that page and you do indeed arrive at Algorithm D!
I've implemented multiple-precision arithmetic at least a couple of times. I'm tempted to dig up an old project I haven't touched for over a decade and make the correction...
Nice work! I don't find all implementations using "while" or "goto loop" surprising though.
"Now test if q̂ ≥ b or q̂·vₙ₋₂ > b·r̂ + uₙ₋₂; if so, decrease q̂ by 1, increase r̂ by vₙ₋₁, and repeat this test if r̂ < b."
That clearly a while loop. Lather, rinse, repeat.
I got through only part of reducing long division to medium division before I couldn’t understand the nomenclature being used in the algorithms. If I picked up TAOCP from the begging does it help you get to the point of reading this or is there other prerequisites you need.
The typesetting of this looks very broken on firefox with extreme gaps between lines of text. Seems to render fine on chromium.
I found a bug in Algorithm D, the long division algorithm in Knuth's "The Art of Computer Programming". It was discussed on HN a couple of times https://news.ycombinator.com/item?id=26562819 as well as on other websites. I sent a letter to Knuth and received a check and an annotated reply. The updated Theorem B, which was unchanged since 1969 is now dated 2026.
While searching for vulnerable implementations I also found a "bug" in llvm, so I expanded a bit on that too.