[amavect]

  00 ->  0.0
  01 -> +1.0
  10 ->  NaN
  11 -> -1.0

Arithmetic:

  0.0 + x = x
  NaN + x = NaN
  +1.0 + -1.0 = 0.0
  +1.0 + +1.0 = NaN
  -1.0 + -1.0 = NaN
  
  -0.0 = 0.0
  -(+1.0) = -1.0
  -(-1.0) = +1.0
  -NaN = NaN
  
  x - y = x + (-y)
  
  NaN * x = NaN
  +1.0 * x = x
  -1.0 * x = -x
  0.0 * 0.0 = 0.0
  
  /0.0 = NaN
  /+1.0 = +1.0
  /-1.0 = -1.0
  /NaN = NaN
  
  x / y = x * (/y)

More interestingly, how to implement in logic gates. Addition with a 2's complement full adder and NaN detector. Negation with a 2's complement negation circuit. Reciprocal with a 0.0 detector.

Multiplication with a unique logic circuit (use a Karnaugh map):

  (ab * cd) = (a&~b | c&~d | ~a&b&c | a&~c&d)(b & d)

[nivertech]

What about comparison operators?

[amavect]

I'll use custom notation =? ≤≥? <? ≤? for comparison to distinguish from = < ≤.

  x =? x = True
  Otherwise, a =? b = False
  
  NaN ≤≥? NaN = False
  Otherwise, a ≤≥? b = a =? b
  
  -1.0 <? 0.0 = True
  -1.0 <? +1.0 = True
  0.0 <? +1.0 = True
  Otherwise, a <? b = False
  
  a >? b = b <? a
  a ≤? b = (a <? b | a ≤≥? b)
  a ≥? b = (a >? b | a ≤≥? b)

In logic gates: For =?, bitwise equality. For ≤≥?, bitwise equality and a NaN detector. For <?, use:

  ab <? cd = a&b&~c | ~a&~b&~c&d

I separate =? from ≤≥?. =? compares value, while ≤≥? compares order. NaN has no ordering, so it compares false. IEEE float only uses ≤≥? and names it ==.

[nivertech]

It's better to first show truth tables, then K-maps, and only then logical formulas.

But the main question is: does this FP2 have any real applications? Maybe it could be useful when only one operand is FP2? Especially for vectorized math.

[amavect]

I'm just having fun. I wrote out the full truth tables and Karnaugh maps on paper, but I trust that you get the idea and can recreate it yourself. (Or, I can write a more detailed blog post, if you'd find that interesting.)

If I had to guess, we could use this for a very compact output of the sign function. [-Inf,0) maps to -1.0, 0 maps to 0.0, (0,Inf] maps to +1.0, and NaN maps to NaN. I don't know what application would need the sign function, though. I haven't needed it yet in my programming experience.