> How will your system represent ((lambda (x) (+ 4. (/ (abs x) x))) 0.)?

- It can't, it only attempts to deal with an ambiguous sign, not magnitude.

  (but have been convinced it's not worth the bother)

> Mathematically there is no reciprocal of zero.  It can be argued that
> (/ 1.0 0.0) should return #i0/0 or signal an error.  But (/ 1.0 +0.0)
> should return #i1/0 if +0.0 is greater than 0.

- if the convention that non-signed values are presumed positive, then
  by definition 0.0 is +0.0, and to eliminate any ordering ambiguity,
  there must be only 3 zeros: inexact -0.0, exact 0, inexact +0.0 (0.0)
  with the three corresponding reciprocals -1/0, 0/0, +1/0; where all exact
  zero's are equivalent: +0 => 0, -0 => 0, +0/0 => 0/0, -0/0 => 0/0, and
  only like signed inexact zero's are non-ambiguous: (+ +0.0 +0.0) => +0.0,
  (+ -0.0 -0.0) => -0.0, (+ -0.0 +0.0) => 0/0, (+ -1/0 +1/0) => 0/0

> The inexact real field must have an additive identity:
>
>   ForAll[x] x + 0.0 = x
>
> If -0.0 + 0.0 is not -0.0, then 0.0 is not the additive identity.
> What is it in your system?

All zero's are non-accumulating, therefore all additive identities for
values other than inexact zero's, however only exact 0 is an additive
identify for all values.