[notpushkin]

Some trivial cases produce... interesting results.

For x in [−1.79e308, 1.79e308]:

Initial Program: 100.0% accurate, 1.0× speedup

  def code(x):
      return math.sqrt((x + 1.0))

Alternative 1: 67.5% accurate, 5.6× speedup

  def code(x):
      return 1.0

[lifthrasiir]

That does make sense, because a half of all available fp numbers are less than 1 in their magnitude. In particular there should be a plenty of numbers x such that |x| << 1 so x + 1 ~= 1; in fact, the proportion should be just shy of 50%.

[andrepd]

That is indeed one of the problems with IEEE floats. There are only 10^80 atoms in the universe, and a Planck length is 1^-60th of the radius of the universe. But 64-bit floats have an absurd range of over 10^±300! Worse than that, notice that there are as many bit patterns in the never-used range between 10^300 and 10^301 as there are in the super-important range between 1 and 10! Super wasteful. Not to mention the quadrillions of values reserved to represent "NaN"...

This is one of the problems that alternative formats such as the Posit aim to solve. It's quite interesting: I've got an implementation in rust here if you want to play with it https://github.com/andrepd/posit-rust

[AlotOfReading]

Note that the logarithmic distribution of float density is also key to certain kinds of efficient hardware float implementations, because it means you can use the fixed mantissa bits alone as table indices. Unums have proven difficult to build efficient HW implementations for.

IEEE floats have a few warts like any other 1980s standard, but they're a fantastic design.

[andrepd]

> Unums have proven difficult to build efficient HW implementations for

Valid point, but not quite true anymore. It comes down basically to the latency of count_leading_ones/zeros for decoding the regime, on which everything else depends. But work has been done in the past ~2ish years and we can have posit units with lower latency than FP units of the same width! https://arxiv.org/abs/2603.01615

> IEEE floats have a few warts like any other 1980s standard, but they're a fantastic design.

Hmm I don't know if I would call it a fantastic design x) The "standard" is less a standard than a rough formalisation of a specific FPU design from back in the 1980s, and that design was in turn not really the product of a forward thinking visionary but something to fit the technical and business constraints of that specific piece of hardware.

It has more than a few warts and we can probably do much better nowadays. That's not really a diss on IEEE floats or their designers, it's just a matter of fact (which honestly applies to very many things which are 40 years old, let alone those designed under the constraints of IEEE754).

[AlotOfReading]

I'm sure you're much more knowledgeable about this than I am, but that's kind of my point. A month old preprint is the first thing to compare to implementations of a mildly evolved, warty old standard from 40 years ago. I consider that fantastic.

Thanks for the paper though. Looking forward to reading it more closely when I have time.

[andrepd]

Absolutely! IEEE floats are ubiquitous in software and hardware. I bet good money they will still be around 40 more years into the future :) and any alternative has to fight their ubiquity.

Better alternatives have been proposed for a long time though. Posits are very nice, and even they are almost 10 years old now :p

[cocoto]

But I guess using the density distribution of floating points is rarely useful in a problem. Your actual distribution will almost surely be way different. Imo, the tool presented here should provide a way to manually provide a custom density function (with some common presets like uniform and normal distributions).

[pavpanchekha]

Author here! Yes, the float distribution isn't what you want in practice, but distribution selector isn't really the right thing either, because a low probability bad result can still be pretty bad! Hence the range selector; the float distribution is good at picking extreme values that trigger FP error.

We usually recommend looking for 90%+ accuracy or carefully examining the accuracy plot

[FabHK]

Not really. 1+x/2, however, would be a good approximation to sqrt(1+x) for small (in absolute value) x.

[irchans]

How about:

(* Mathematica Notation, Assume x>0 ) If[ x < 10^(-10), 1+ x/2, ( order x^2 error ) If[ x> 10^10, Sqrt[x], ( order 1/Sqrt[x] error ) (else) Sqrt[x +1] ] ] ( I guess the If statements take too much time. *)

[notpushkin]

HN formatter ate up the asterisks in your code comments :) To paste code blocks here, prepend each line with 2 or more spaces:

  (* Mathematica Notation, Assume x>0 *)
  If[ x < 10^(-10),  1 + x/2,   (* order x^2 error *)
  If[ x > 10^10,     Sqrt[x],   (* order 1/Sqrt[x] error *)
  (* else *)         Sqrt[x+1] ] ]
  
  (* I guess the If statements take too much time. *)