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A Foundations of the Exact Fixed Point Numbers
Let
- n e N be the number of components of limited range of an exact fixed
point number, and let
- Mi e N \{0,1} be the corresponding
"moduli", 0 < i < n.
The set of exact fixed point
numbers
EFPN := ({1,-1} ×N ×PROD_i=0^n-1 Z_i)
\
{(-1,0,...,0)}
consists of the (n+2)-tuples (v,zn,zn-1, ..., z0), where
- v e {1,-1},
- zn e N and
- zi e { 0,1,..., (i-1) },
where we allow only v=1 if all zi=0, 0 < i < n.
On the one hand there is a function g: EFPN -> Z
that maps any element of EFPN to an integer:
g(v,z_n,z_n-1, ..., z_0):=
v ·SUM_i=0^n (z_i ·PROD_j=0^i-1 M_j)
On the other hand the function f:Z -> EFPN maps
an integer z e Z to an element of EFPN:
f(z):= (v,z_n,z_n-1, ..., z_0),
where
-
v :=
{
.
- zn:=|z| div PRODj=0n-1 Mj,
- zi:=(|z| div PRODj=0i-1 Mj) mod Mi, where 0
< i < n.
In the next two sections we prove g=f-1. Thus Z and EFPN are isomorphic sets.
CoFI
Note: L-12 -- Version: 0.3 -- November 1999.
Comments to cofi@informatik.uni-bremen.de