Making all divisions slower to fix a bug in 0.000000000000000000542% of cases

# Making all divisions slower to fix a bug in 0.000000000000000000542% of cases Source: [https://futhark-lang.org/blog/2026-10-14-defining-division.html](https://futhark-lang.org/blog/2026-10-14-defining-division.html) Posted on October 14, 2026 Almost all modern machines use[two’s complement](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html)to represent integers\. This representation is asymmetric, in the sense that the most negative number has no corresponding positive number\. For example, with signed 8\-bit numbers, the most negative number is`\-128`, whilst the most positive number is`127`\. This quirk shows up in various cases, but for the purpose of this post, I am concerned with signed integer division, where`x/\-1`mathematically should produce`\-x`, but in the case where`x`is the most negative number, that is not possible\. For 32\-bit integers, which I arbitrarily decide are the most common, this means that one out of every 2⁶⁴ possible inputs is impossible to represent correctly \- leading to the somewhat clickbaity title of this post, and some recently fixed troubles in the Futhark compiler\. In this post I will explain what the actual problem is for a language implementation, how we fix it, and what the performance cost looks like\. The main audience of this post is other language implementers, but I think the concepts involved are simple enough that anyone curious about low level issues can follow along\. ### [The C issue](https://futhark-lang.org/blog/2026-10-14-defining-division.html#the-c-issue) The Futhark compiler has multiple backends, but the most important ones work by generating C code\. This has many practical advantages: 1. It is easy to debug the generated code because it is easy to read and modify by hand\. 2. Most systems have a C compiler \(or easy access to one\), meaning Futhark has fewer dependencies, is portable, and doesn’t need to bundle native code generators\. 3. Many languages can interoperate with C, so by exposing Futhark as a C API,[those languages can also call Futhark](https://futhark-lang.org/blog/2022-07-01-how-futhark-talks-to-its-friends.html)\. That said, there are also disadvantages: 1. Some code patterns are hard to express \(or express efficiently\) in C\. This is mainly the case for advanced language features such as coroutines, tail calls, or garbage collection\. Fortunately Futhark is[pretty boring from a semantic perspective](https://futhark-lang.org/blog/2025-09-26-the-biggest-semantic-mess.html), so this does not affect us too much\. 2. It is very easy to accidentally generate C code that is subtly wrong\. This does unfortunately keep pestering us\. C is of course renowned as a very safe and easy\-to\-use language, because C programmers never make coding mistakes\. The few times they do make mistakes, it is often due to mishandling of pointers or array sizes, leading to buffer overflows, heap smashing, etc\. These are actually not the main problem with generating C from Futhark, as Futhark itself is memory\-safe, and systematically inserting in bounds checks in generated code is easy \([although a little more interesting in GPU code](https://futhark-lang.org/blog/2020-07-13-bounds-checking.html)\)\. No, the main problem we’ve had is actually undefined behaviour\. As is by now well\-known, C specifies various events as explicitly undefined, meaning not only that a C compiler can assume it will never occur, but also that no particular result is guaranteed should a program perform one of the undefined operations\. Some causes of undefined behaviour are obvious and easy to avoid: reading arrays out\-of\-bounds, using freed memory, etc\. But C has a long list of more obscure cases\. My favourite example is that in C99, it is undefined behaviour if a source code file does not end with a final newline character\. Yet we have honestly not spent that much time dealing with that particular issue in Futhark\. No, the main problem, and now we are finally returning to the premise in the post, is that*signed*integer overflow is undefined behaviour in C\. This is particularly insidious since most programmers have a mental model of fixed\-size integer arithmetic where overflow is a natural event, and so does the actual machines we use, but a C compiler may assume it never happens\. The reason behind this specification is so the C compiler can optimise based on useful properties such as`x\+1\>x`, which are not true in the presence of overflow, but the motivation does not actually matter to us\. In a language like Futhark, where integer overflow*is*defined behaviour \(it wraps around\), this creates a very unfortunate mismatch between what we want to do, and what we can express in our target language\. Fortunately, C is merciful enough to specify that for*unsigned*numbers, overflow is well\-defined, and behaves just as you would expect \(on a two’s\-complement machine, anyway\)\. For this reason, Futhark’s C code generation actually represents all integer types using unsigned C types, \(so`i64`becomes`uint64\_t`\), and then performs the arithmetic on unsigned types\. This works due to the nice property of two’s complement representation that most arithmetic operations are done exactly the same, at the bit level, as for unsigned integers\. One exception is comparison operators, in which we cast to the signed type \(e\.g\.`int64\_t`\) before performing the comparison\. Such conversions are free, since they do not need to change the bit representation of the number, but only affect what instructions the C compiler emits for e\.g\. a`<=`operation\. Another operator where sign matters is division\. If we do`x/\-1`, then it is quite important whether we are dividing with a small negative number, or with the number that arises if we interpret the two’s complement bit pattern for`\-1`as an unsigned integer\. This means that for signed integer division, Futhark generates a signed C division\. This means that if we do`x/\-1`where`x`is the most negative number, we are stuck with the issue described in the post opener \- the correct result \(`\-x`\) cannot be represented as a signed integer, so this is a signed integer overflow, and that is undefined behaviour in C\. Futhark is a principled language, and therefore producing undefined behaviour is unacceptable\. \(I realise the hypocrisy of this puritan declaration whilst leaving in the bug for ten years\.\) Further, on x86 this kind of division will actually signal a CPU exception, crashing the program, so this is not merely a violation of the sanctity of the C specification\. Fixing it in principle is easy\. This is the definition of 32\-bit signed division in the Futhark runtime system: ``` int32_t squot32(int32_t x, int32_t y) { return x / y; } ``` And this is the version that avoids the 0\.000000000000000000542% undefined cases: ``` int32_t squot32(int32_t x, int32_t y) { return y == -1 ? (int32_t)(0 - (uint32_t)x) : x / y; } ``` This function simply checks whether the divisor is`\-1`, and if so does the negation with unsigned \(well\-defined\) arithmetic\. The cost is a branch on every single division\. The behaviour is still*implementation\-defined*, but produces the right results on the machines we care about\. \(Why`quot`instead of`div`? Futhark actually has*two*division operators \- one that rounds sensibly towards negative infinity, and one that rounds towards zero like C\. The function above is the one that follows C’s behaviour because it is less code, but the issue is the same either way\.\) ### [The cost](https://futhark-lang.org/blog/2026-10-14-defining-division.html#the-cost) Correctness is not really a thing we can compromise on, but in principle it is frustrating that*every*integer division needs an additional branch just to handle a*single*special case that may never actually happen, as it is highly unlikely that a program’s correctness*depends*on wraparound for signed integer division at the ends of the numeric scale\. But apart from the principal frustration, what is the actual overhead? Measured on the[Futhark benchmark suite](https://github.com/diku-dk/futhark-benchmarks)on GPU, I was unable to measure an impact\. This is likely because signed integer divisions are relatively rare, and even when they do occur, they likely occur due to index arithmetic, where the resulting memory loads have so much latency that they easily[hide](https://modal.com/gpu-glossary/perf/latency-hiding)the cost of doing the divisions\. But what if we try to construct a contrived program where the bottleneck is signed division throughput? That would demonstrate something close to worst\-case overhead of adding the branch\. Behold,`divisions\.fut`: ``` -- == -- random input { 1000000 [1000]i32 [1000]i32 } entry main [n] (k: i32) (xs: [n]i32) (ys: [n]i32) : [n]i32 = map2 (\x y -> loop z = x for _i < k do #[unsafe] z // y) xs ys ``` The program accepts a number`k`and two arrays of integers\. For each corresponding pair`x`and`y`, it`k`times divides`x`by`y`, using`//`to target the`squot32`function, and using`\#\[unsafe\]`to remove division\-by\-zero checks\. For most values of`k`, repeated division quickly produces zero, but the compiler cannot assume this, so it will have to actually perform the divisions\. You may reasonably argue that the program is not*purely*division bound, as the bookkeeping around the loop also involves costly things such as adding and comparing integers, but a program that contains a very long*unrolled*sequence of divisions stretches my credulity too far\. Using the handy[futhark bench](https://futhark-lang.org/examples/benchmarking.html)tool, we can then measure the impact using the compiler with a fixed implementation of`squot32`\([part of this pull request](https://github.com/diku-dk/futhark/pull/2579)\)\. First on sequential code, using the`c`backend: ``` $ futhark-old-and-busted bench divisions.fut ... divisions.fut (no tuning file): 100000i32 [10000]i32 [10000]i32: 3390148μs (95% CI: [ 3387613.7, 3396498.1]) $ futhark-new-hotness bench divisions.fut ... divisions.fut (no tuning file): 100000i32 [10000]i32 [10000]i32: 3389215μs (95% CI: [ 3387626.2, 3391036.4]) ``` There is no measurable difference, which is not a surprise on a modern speculatively executing CPU for a program with an easily predicted branch\. But what about GPUs? After all, “GPUs are bad at branching” is a common motto by those of my students who like to replace an`if`with ten lines of dense branchfree code\. Let’s try the experiment again with Futhark’s`cuda`backend: ``` $ futhark-old-and-busted bench --backend=cuda divisions.fut ... divisions.fut (no tuning file): 100000i32 [10000]i32 [10000]i32: 5548μs (95% CI: [ 5546.4, 5552.2]) $ futhark-new-hotness bench --backend=cuda divisions.fut ... divisions.fut (no tuning file): 100000i32 [10000]i32 [10000]i32: 12592μs (95% CI: [ 12563.5, 12622.1]) ``` Well then\! This really is a major difference\. Much more than I expected when I began this work, actually\. Reading the generated PTX \(CUDA machine code\), the old version is partially unrolled into four consecutive`div\.s32`instructions in a single basic block, whilst the fixed compiler still triggers unrolling, but with many branches between the divisions\. I observed a similar result on an AMD GPU with the`hip`backend, although the impact was slightly less there\. ### [Conclusions](https://futhark-lang.org/blog/2026-10-14-defining-division.html#conclusions) The cost of fixing those 0\.000000000000000000542% of divisions is real and unfortunate, and I would certainly like to have a cheaper fix\. Having undefined behaviour in the generated code is of course not acceptable, so we will have to live with this for now\. On non\-contrived benchmarks, it seems there is essentially no impact\. For programmers unwilling to pay this cost, you can always stick to*unsigned*divisions, if you know that you will never have negative numbers\. This is the last instance of undefined behaviour in Futhark’s generated C code that I remember\. There is however at least one instance of*implementation\-defined*behaviour, namely in the code generation for bit\-shifting integers by more bits than their size, but I have yet to gather the courage to fix it\.