Arithmetic - Signal Processing Systems - page 621

Fig. 36 Bipartite table
approximation structure

X
k bits k bits k bits

X 0 X 1 X 2

T 0 ( X 0 , X 1 ) T 1 ( X 0 , X 2 )

2 − W s X 1 we get

Now taking the first-order Taylor expansion of f at X 0 +

2 − W s X 1 )+ 2 − 2 W s X 2 f ( 2 − W s X 1 )

f ( X ) ≈ f ( X 0 + X 0 + (66)

Again, we take the Taylor expansion, this time a zeroth-order expansion of f ( X 0 +

2 − W s X 1 ) at X 0 as

f ( 2 − W s X 1 ) ≈ f (

X 0 + X 0 ) (67)

This gives the bipartite approximation as

f ( x ) ≈ T 1 ( X 0 , X 1 )+ T 2 ( X 0 , X 2 ) (68)

where

2 − W s X 1 ) T 1 ( X 0 , X 1 )= f ( X 0 +

2 − 2 W s X 2 f ( T 2 ( X 0 , X 2 )= X 0 ) (69)

The functions T 1 and T 2 are tabulated and the results are added. The resulting
structure is shown in Fig. 36 .
The bipartite approximation can be seen as a piecewise linear approximation
where the same slope tables are used in several intervals. Here, T 1 contains the
offset values, and T 2 contains tabulated lines with slope f ( .

The accuracy of the bipartite approximation can be improved by instead per-
forming the first Taylor expansion at X 0 +

X 0 )

2 − W s X 1 + 2 − 2 W s − 1

and the second at

2 − W s − 1 [ 47 ] . It is also possible to split the input word into more subwords
yielding a multipartite table approximation [ 8 ] .

X 0 +

7 Further Reading

Several topics have been published on related subjects. For general digital arithmetic
we refer to [ 11 , 24 , 26 , 41 ] . For the specific cases of approximation of elemen-
tary functions and floating-point arithmetic, [ 36 , 37 ] provide broad overviews,
respectively.