The GNU libc atanh is correctly rounded | Hacker News Reader

The GNU libc atanh is correctly rounded

inria.hal.science

119 pointsmatt_d29 comments

One of the major projects that's ongoing in the current decade is moving the standard math library functions to fully correctly-rounded, as opposed to the traditional accuracy target of ~1 ULP (the last bit is off).

For single-precision unary functions, it's easy enough to just exhaustively test every single input (there's only 4 billion of them). But double precision has prohibitively many inputs to test, so you have to resort to actual proof techniques to prove correct rounding for double-precision functions.

> traditional accuracy target of ~1 ULP

I had to google this one…

ULP: “Unit in the Last Place” or “Unit of Least Precision: https://en.wikipedia.org/wiki/Unit_in_the_last_place

For what it’s worth, this is basically the first word you learn when discussing numerical precision; and I mean word—nobody thinks of it as an abbreviation, to the point that it’s very often written in lower case. So welcome to the club.

to me this feels like wasted effort due to solving the wrong problem. The extra half ulp error makes no difference to the accuracy of calculations. the problem is that languages traditionally rely on an OS provided libm leading to cross architecture differences. If instead, languages use a specific libm, all of these problems vanish.

Standardizing a particular libm essentially locks any further optimizations because that libm's implementation quirks have to be exactly followed. In comparison the "most correct" (0.5 ulp) answer is easy to standardize and agree upon.

Many of the conversions so far have been clearly faster. I don't think anything has been merged which shows a clear performance regression, at least not on CPUs with FMA support.

The bigger challenge is GPU/NPU. Branches for fast vs accurate path get costlier, among other things. On CPU this is less of a cost.

Most published libm on GPU/NPU side have a few ULP of error for the perf vs accuracy tradeoff. Eg, documented explicitly in the CUDA programming guide: https://docs.nvidia.com/cuda/cuda-programming-guide/05-appen... .

Prof. Zimmermann and collaborators have a great table at https://members.loria.fr/PZimmermann/papers/accuracy.pdf (Feb 2026) comparing various libm wrt accuracy.

using fma makes it possible to write faster libm functions, but going back to a 1 ulp world with the same fma optimizations would give you another 20% speedup at least. the other issue is that these functions tend to have much larger code size which tends not to be a significant problem in micro benchmarks, but means that in real applications you increase cache pressure allowing things down in aggregate

Mixed precision computations need correctly rounded functions.

no they don't... why would they?

> The extra half ulp error makes no difference to the accuracy of calculations

It absolutely does matter. The first, and most important reason, is one needs to know the guarantees of every operation in order to design numerical algorithms that meet some guarantee. Without knowing that the components provide, it's impossible to design algorithms on top with some guarantee. And this is needed in a massive amount of applications, from CAD, simulation, medical and financial items, control items, aerospace, and on and on.

And once one has a guarantee, making the lower components tighter allows higher components to do less work. This is a very low level component, so putting the guarantees there reduces work for tons of downstream work.

All this is precisely what drove IEEE 754 to become a thing and to become the standard in modern hardware.

> the problem is that languages traditionally rely on an OS provided libm leading to cross architecture differences

No, they don't not things like sqrt and atanh and related. They've relied on compiler provided libs since, well, as long as there have been languages. And the higher level libs, like BLAS, are built on specific compilers that provide guarantees by, again, libs the compiler used. I've not seen OS level calls describing the accuracy of the floating point items, but a lot of languages do, including C/C++ which underlies a lot of this code.

> The first, and most important reason, is one needs to know the guarantees of every operation in order to design numerical algorithms that meet some guarantee

sure, but a 1 ulp guarantee works just as well here while being substantially easier to provide.

> And the higher level libs, like BLAS, are built on specific compilers that provide guarantees

Sure, but Blas doesn't provide any accuracy guarantees so it being built on components that sort of do has pretty minimal value for it. For basically any real application, the error you experience is error from the composition of intrinsics, not the composed error of those intrinsic themselves, and that remains true even if those intrinsics have 10 ULP error or 0.5 ULP error

If only we switched to ternary, there rounding is simply truncating

Interesting: https://youtu.be/cb5r3r38O9c

Guy's world records get deleted due to changes in atanh over time

As that's a pretty long video would you mind giving a short summary of what happened? Was it a world record in a game?

yeah one of the trackmania games -- which feature a nominally deterministic physics engine, allowing for replays from a recorded sequence of inputs... except the physics engine relies on libc transcendental functions. players are generally on windows, but backend servers doing anti-cheat validations via replays are running linux. this resulted in false cheat positives when the linux server was running glibc prior to the glibc rounding fixes... and as a result the guy's world record kept being flagged as a cheat. it's a pretty good video with a lot of detail on how they narrowed it down to specific glibc versions/etc.

Pretty sure that's atan.

I don’t think I ever used atanh, but I always love some floating-point nerdery. These other documents by the same team are fantastic resources: https://inria.hal.science/hal-04714173v2/document for complex values and https://members.loria.fr/PZimmermann/papers/accuracy.pdf for real values.

Lots of good stuff here: https://members.loria.fr/PZimmermann/papers/ .

Tanh, and therefore atanh, are wonderful.

It's linear for small x, and exponential for large. Lots of applications of this:

Compressing data

Mapping (zoomed in near by, zoned out from afar)

There's a whole class of electronics amps for this.

tanh is a very pleasant sounding overdrive function for audio, for example.

Why not arxiv?

The author works at a French university. Some French researchers do choose to cross-post to arXiv (and Zimmermann may have too, I haven’t checked), but HAL is the default.

> HAL is the default.

HAL is an institutional requirement, I believe.

Good to know!

Who wrote it? Someone at Red Hat likely.

https://en.wikipedia.org/wiki/Paul_Zimmermann_(mathematician...

He doesn't work for Red Hat.

As the paper mentions, this particular routine was the work of Alexei Sibidanov, though Zimmermann seems to have been maintaining it since it was contributed. (Sibidanov doesn't work for Red Hat either, though.)

The CORE-MATH project authors, most of whom are French academics (including the author of the linked paper).

I don’t know of any interesting work in this space that came out of Red Hat, why do you suggest them?

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