The GNU libc atanh is correctly rounded

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.

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.

> 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.

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.

Mixed precision computations need correctly rounded functions.

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.

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

> HAL is the default.

HAL is an institutional requirement, I believe.

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.)