Here is a visualization I made recently on the density of float32. It seems that float32 is basically just PCM, which was a lossy audio compression exploiting the fact that human hearing has logarithmic sensitivity. I’m not sure why they needed the mantissa though. If you give all 31 bits to the exponent, then normalize it to +/-2^7, you get a continuous version of the same function.
float has higher accuracy around 1.0 than around 2*24. This makes it quite a bit different from PCM which is fully linear. Which is probably why floating point PCM keeps it's samples primarily between -1.0 and +1.0.
> which was a lossy audio compression
It's not lossy. Your bit depth simply defines the noise floor which is the smallest difference in volume you can represent. This may result in loss of information but at even 16 bits only the most sensitive of ears could even pretend to notice.
> If you give all 31 bits to the exponent, then normalize it to +/-2^7, you get a continuous version of the same function.
You'll extend the range but loose all the precision. This is probably the opposite of what any IEE754 user actually wants.
No, it's just it's more natural/intuitive to express algorithms in a normalized range if given the possibility.
Same with floating point RGBA (like in GPUs)
https://gist.github.com/deckar01/3f93802329debe116b0c3570bed...
exponent *= 2 ** (8 - E)
You also don't have subnormals, infinities or propagating NaNs. You manage to only retain the signed 0.
EDIT: And the midpoint of your system is 0.5. Which is a little uncomfortable.
Yes, exactly; the linear regions are needed to more evenly distribute precision, while the average precision remains the same. Alternatively, you can omit the mantissa, but use an exponent base much closer to 1 (perhaps 1 + 2⁻²³).