Negative accuracy numerics ( 0``-128 notation )

Question

What does it mean to have a negative accuracy number?

I understand 0``128 to mean "the number zero to 128 decimal points". Mathematica corroborates this:

In[1]:= 0``128 < 1
(* Out[1]= True *)
In[2]:= 0``128 < 0
(* Out[2]= False *)
In[3]:= Accuracy[0``128]
(* Out[3]= 128. *)

But then I have no idea what "negative accuracy" is:

In[1]:= 0``-128 < 1
(* Out[1]= False *)
In[2]:= 0``-128 > 1
(* Out[2]= False *)
In[3]:= 0``-128 == 1
(* Out[3]= True *)     (* Why????? *)
In[4]:= Accuracy[0``-128]
(* Out[4]= -128. *)

Context: these values are showing up in a system of ~10000 equations, as the result of computing 100s of expressions of the form

$$ \sum_i \frac{\textrm{numer}}{\textrm{denom}} \textrm{basis}_i, $$

where both numer and denom are randomly generated complex numbers, with 200 digit precision. I then collect coefficients of the (symbolic) $\textrm{basis}_i$'s to form algebraic equations.

The 0``128 objects appear in only a handful of the total number of equations, so I don't yet know what causes the 0``-128's to appear.

Answer 1

For the sake of an answer:

From the docs: "...The accuracy of a number can be negative, indicating that all the correctly known digits are to the left of the decimal point..." – ciao May 6 '15 at 0:29

In other words, the error in 0``a is at most 10^-a. So 0``-128 represents a numeric quantity at most 10^128 in magnitude.

Example:

N[10^148 (1 + 9*^-21), 20] - 10^148 // InputForm
(*  0``-128.  *)

And the error is less than 10^128:

10^148 * 9 * 10^-21 // N
(*  9.*10^127  *)