The GNU libc atanh is correctly rounded

https://inria.hal.science/hal-05591661

by matt_d • 3 months ago

119 29 3 months ago

inria.hal.science

jonathrg• 3 months ago

Good to know!

kergonath• 3 months ago

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/ .

nmbrskeptix• 3 months ago

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.

cyco130• 3 months ago

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

jcranmer• 3 months ago

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.

WalterGR• 3 months ago

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

mananaysiempre2• 3 months ago

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.

adgjlsfhk14• 3 months ago

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.

lifthrasiir• 3 months ago

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.

fweimer• 3 months ago

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.

ghighi7878• 3 months ago

Mixed precision computations need correctly rounded functions.

SideQuark• 3 months ago

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.

incognito124• 3 months ago

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

brcmthrowaway2• 3 months ago

Who wrote it? Someone at Red Hat likely.

DiabloD3• 3 months ago

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

He doesn't work for Red Hat.

ameliaquining• 3 months ago

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.)

stephencanon• 3 months ago

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?

brcmthrowaway2• 3 months ago

GNU libc

RyJones• 3 months ago

Interesting: https://youtu.be/cb5r3r38O9c

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

im3w1l• 3 months ago

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?

dgaudet• 3 months ago

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.

ncruces• 3 months ago

Pretty sure that's atan.

RandomTeaParty• 3 months ago

Why not arxiv?

TimorousBestie• 3 months ago

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.

mananaysiempre2• 3 months ago

HAL is the default.

HAL is an institutional requirement, I believe.

The GNU libc atanh is correctly rounded

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