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Comment by saulpw | Hacker News Reader

OTOH, it seems quite plausible that the most important numbers to represent are:

   +0
   -0
   +1
   -1
   +inf
   -inf

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In standard FP32, the infs are represented as a sign bit, all exponent bits=1, and all mantissa bits=0. The NaNs are represented as a sign bit, all exponent bits=1, and the mantissa is non-zero. If you used that interpretation with FP4, you'd get the table below, which restricts the representable range to +/- 3, and it feels less useful to me. If you're using FP4 you probably are space optimized and don't want to waste a quarter of your possible combinations on things that aren't actually numbers, and you'd likely focus your efforts on writing code that didn't need to represent inf and NaN.

  Bits s exp m  Value
  -------------------
  0000 0  00 0     +0
  0001 0  00 1   +0.5
  0010 0  01 0     +1
  0011 0  01 1   +1.5
  0100 0  10 0     +2
  0101 0  10 1     +3
  0110 0  11 0     +inf
  0111 0  11 1     NaN
  1000 1  00 0     -0
  1001 1  00 1   -0.5
  1010 1  01 0     -1
  1011 1  01 1   -1.5
  1100 1  10 0     -2
  1101 1  10 1     -3
  1110 1  11 0     -inf
  1111 1  11 1     NaN

I can see the most important values being:

   ± 0 (infinitesimal)
   ± 10^-2n
   ± 10^-n
   ± 1 (unity)
   ± 10^n
   ± 10^2n
   ± infinity

For fp4, this leaves 2 values. Maybe one of them should be NaN. What should the other one be?

Why waste a slot on -0?

Because it means "infinitesimal negative" which is distinct from "infinitesimal positive".

That sounds pretty niche. What's a use case where you have less than 8 bits and that distinction is more important than having an extra finite value? I don't think AI is one.

For neural net gradient descent, automatic differentiation etc, the widely used ReLU function has infornation carrying derivatives at +0 and –0 if those are infinitesimals.

Barely any information. After surviving RELU that signed zero is probably getting added to another value and then oops the information is gone. It sounds a lot worse than properly spaced values.

sign = most important bit of information

If you were looking at the entire number line, sign would roughly be the most important part.

But you still have all the other numbers carrying sign info. This is only the sign of denormals and that's way less valuable. Outside of particular equations it ends up added to something else and disappearing entirely. It would be way better to cut it and have either half the smallest existing positive value or double the largest existing value as a replacement. Or many other options.

You need it if you want the idea of total ordering over the extended Reals. There's +/- infinity--an affine closure, not projective (point at infinity)--so to make that math work you need to give 0 a sign.

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