FWIW, both of those can be expressed exactly by floating-point numbers ;)

Whew. I almost put some non-zeros in there.

Right all integers up to 2^53 (or something like that) can be (in double precision).

I assume that's the reason they made the mantissa linear, even though having the whole thing logarithmic makes more sense.

Addition/subtraction are also much simpler/cheaper than they would be in an entirely logarithmic model. If floats were just 2^x with some 64 bit fixed point x, it's not clear to me how to do addition efficiently.

What. Mantissa is already logarithmic, bit number n has value 2^(n - N-1) for an N-bit mantissa.

This is how positional number systems work at all.

The mantissa is linear. It's unrelated to how positional number systems work. A floating point value is split into two numbers - the exponent and the mantissa. Normally they are used to represent a final number like:

    x = 2^e * (1 + m)

Where e is the exponent and m is the mantissa (varying linearly from 0 to 1).

But you could have a fully exponential number format:

    x = 2^(m + o)

As pointed out though, it makes addition much more complicated, you can't exactly represent integers, and someone told me it makes quantisation noise worse too. Bad idea.

I'd always wondered how automated these comments were