While searching for vulnerable implementations I also found a "bug" in llvm, so I expanded a bit on that too.
https://www-cs-faculty.stanford.edu/~knuth/boss.html
(unfortunately, I have yet to find another typo since getting my $2.88 for _Digital Typography_)
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.
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.
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...
That makes this thread a bit more interesting now.
https://stackoverflow.com/questions/60479571/is-there-a-bug-...
So at the time my conclusion had been that there was no mistake (interpreting the "repeat" as a loop), but it's arguable… maybe if the person who posted the question had asked Knuth instead, he'd have had a reward check?
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.
That is a rather nice ritual, I think I will use it now to motivate myself to learn Scala.
This bug is only in the English description of the algorithm, right? No bug in either the MIX or MMIX implementations?
Honestly, while waiting for the check I wondered what it would be, and 0x$1.00 feels just right. The name in the book came unexpectedly, it's really a reward on its own.
No bug in MIX, and there is no MMIX implementation yet. The transition of the first three volumes from MIX to MMIX is still far ahead. The MIX program actually implements Step D3 differently from what's written in Algorithm D, and this implementation, more aligned with 1st and 2nd editions of the book, avoids the error. The bug is due to a 1995 change in the trial quotient computation (that from my perspective came as part of a transition to MMIX). This broke correctness of Theorem B, which then led to overflows post Step D3 (in only one peculiar, rare, odd case).
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.
"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.
A loop would also call for additional run-time analysis. And Knuth changed Step D3, if it were a loop he wouldn't have had to. Additionally, there is no loop in Program D, his implementation of Algorithm D in MIX.
If you read the whole chapter and not just the statement itself (though I'd argue the statement is enough), it's very clearly two if's.
https://www.scribd.com/document/956350280/The-Art-of-Compute...
... on page 274 the MIX jumps in lines 058 and 060 to label 3H if the tests fail, where qhat is decremented again. I'm not at all a Mix expert, but where is the counter that the loop is only executed twice?
I introduce my own notation from medium->small division onwards. If it's any help u'', v'' are the limbs that the division instruction sees (the ones pertinent to qhat) and u',v' are all the lower limbs.
I have a passing interest in mathematics and especially comp sci but I routinely find myself stymied just by trying to understand the notation.
EDIT: I have added a different fallback font so it should work on your Firefox now.
Assuming OP didn't patch something, you may have a misbehaving extension.
If you have vol 2 you also might as well read at least the intro to chapter 3 which is beautiful.
I think Knuth originally wanted a loop but obviously needed a proof that it terminates and does not waste too many iterations. That is why he mentions <= 2 in the text..
I mean, as you say, every implementation apart from LLVM understood the text as a loop.
Anyway, extremely nice work to correct Theorem B to <= 3!
I don't think I interpreted it as strictly a loop, though, because normally that's a bigger deal and not casually hidden in one step of an algorithm. Algorithm M, for example has the loop parts annotated as such and always "go back to step M3" etc. My copy also has a comment in step D3 saying the test eliminates all cases where the guess is two too large, to it's completely understandable to not interpret it as a loop IMO.
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1. I know at least one of these came from porting lesser-known utilities to VM/CMS. By that time, DEK already had adopted a practice of having his secretary print his emails and he’d hand-write a reply and you’d get it in the mail.
> It turns out that only 9 of the first 275 checks that I've sent out since the beginning of 2006 have actually been cashed. The others have apparently been cached. So this change in policy will probably not affect too many people. On the other hand, I don't like to renege on promises, so I shall do my best to find a suitable way to send money to anyone who really prefers legal tender.
@bluGill, you owe me some money or this framing won't be professional. ;)