1. Lecture 10
Fast
Dividers

2. Computer Arithmetic: Algorithms and Hardware Design
Required Reading
Behrooz Parhami,
Computer Arithmetic: Algorithms and Hardware Design
Chapter 14, High-Radix Dividers
Note errata at:

3. Recommended Reading
J-P. Deschamps, G. Bioul, G. Sutter,
Synthesis of Arithmetic Circuits: FPGA, ASIC and
Embedded Systems
Chapter 6, Arithmetic Operations: Division
6.2.4, SRT Radix-2 Division
6.2.5, SRT Radix-2 Division with Stored Carry Encoding
6.2.6, P-D Diagram
6.2.7, SRT-4 Division
Chapter 13, Dividers
13.2.3, SRT Dividers
13.2.4, SRT-4 Divider

4. Classification of Dividers
Array
Dividers
Dividers
by Convergence
Sequential
Radix-2
High-radix
Restoring
Non-restoring
regular
SRT
regular using carry save adders
SRT using carry save adders

5. Array
Dividers

6. Unsigned Fractional Division
zfrac Dividend z-1z z-(2k-1)z-2k
dfrac Divisor d-1d d-(k-1) d-k
qfrac Quotient q-1q q-(k-1) q-k
sfrac Remainder …0s-(k+1) s-(2k-1) s-2k
k bits

7. Integer vs. Fractional Division
For Integers:
z = q d + s
 2-2k
z 2-2k = (q 2-k) (d 2-k) + s (2-2k)
For Fractions:
zfrac = qfrac dfrac + sfrac
where
zfrac = z 2-2k
dfrac = d 2-k
qfrac = q 2-k
sfrac = s 2-2k

8. Unsigned Fractional Division Overflow
Condition for no overflow:
zfrac < dfrac

9. Sequential Fractional Division
Basic Equations
s(0) = zfrac
s(j) = 2 s(j-1) - q-j dfrac
2k · sfrac = s(k)
sfrac = 2-k · s(k)

10. Restoring Unsigned Fractional Division
s(0) = z
for j = 1 to k
if 2 s(j-1) - d > 0
q-j = 1
s(j) = 2 s(j-1) - d
else
q-j = 0
s(j) = 2 s(j-1)

11. Fig. 15.7 Restoring array divider composed of controlled subtractor cells.

12. Non-Restoring Unsigned Fractional Division
s(-1) = z-d
for j = 0 to k-1
if s(j-1)  0
q-j = 1
s(j) = 2 s(j-1) - d
else
q-j = 0
s(j) = 2 s(j-1) + d
end for if s(k-1)  0
q-k = 1
q-k = 0
Correction step

13. Fig. 15.8 Nonrestoring array divider built of controlled add/subtract cells.

14. Sequential
Dividers

15. Sequential Fractional Division
Basic Equations
s(0) = zfrac
s(j) = 2 s(j-1) - q-j dfrac
2k · sfrac = s(k)
sfrac = 2-k · s(k)

16. Non-restoring Fractional Division
s(0) = z
for j = 1 to k
if 2s(j-1)  0
q-j = 1
s(j) = 2 s(j-1) - d
else
q-j = -1
s(j) = 2 s(j-1) + d
end for
q = BSD_2’s_comp_conversion(q)
Correction_step

17. Integer Division Correction step We have: z = q d + s
We need: sign(s) = sign (z)
z = (q-1) d + (s+d)
z = q’ d + s’
z = (q+1) d + (s-d)
z = q” d + s”

18. Fractional Division Correction step
We have: zfrac = qfrac dfrac + sfrac
We need: sign(sfrac) = sign(zfrac)
zfrac = (qfrac–2-k) dfrac + (sfrac+dfrac 2-k)
zfrac = q’frac dfrac + s’frac
zfrac = (qfrac+2-k) dfrac + (sfrac – dfrac 2-k)
zfrac = q”frac dfrac + s”frac

19. Non-restoring Fractional Division
s(0) = z
for j = 1 to k
if 2s(j-1)  0
q-j = 1
s(j) = 2 s(j-1) - d
else
q-j = -1
s(j) = 2 s(j-1) + d
end for
q = BSD_2’s_comp_conversion(q)
Correction_step

20. Fig The new partial remainder, s(j), as a function of the shifted old partial remainder, 2s(j–1), in radix-2 nonrestoring division.

21. Fig. 14.4 The new partial remainder s(j) as a function of 2s(j–1), with q–j in {–1, 0, 1}.

22. Non-restoring Fractional Division with shifting over zeros
s(0) = z
for j = 1 to k
if 2s(j-1)  d
q-j = 1
s(j) = 2 s(j-1) - d
elseif 2s(j-1) < -d
q-j = -1
s(j) = 2 s(j-1) + d
else
q-j = 0
s(j) = 2 s(j-1)
end for
Conversion of q
Correction step

23. SRT Non-Restoring Fractional Division
Assumptions
d  1/2 (positive, bit-normalized divider)
-d ≤ -1/2 ≤ z, s(j) < 1/2 < d
If the latter condition not true:
z = z >> 1
perform k+1 instead of k steps of the algorithm
q = q << 1 and s = s << 1
z’=z/2 and z’=q’·d + s’
z = 2z’ = (2q’) ·d + 2s’=q ·d + s

24. Fig. 14.5 The relationship between new and old partial remainders in radix-2 SRT division.

25. SRT Non-Restoring Fractional Division
s(0) = z
for j = 1 to k
if 2s(j-1)  1/2
q-j = 1
s(j) = 2 s(j-1) - d
elseif 2s(j-1) < -1/2
q-j = -1
s(j) = 2 s(j-1) + d
else
q-j = 0
s(j) = 2 s(j-1)
end for
Conversion of q
Correction step

28. Fig. 14.6 Example of unsigned radix-2 SRT division.

29. Sequential
Dividers
with
Carry-Save Adders

30. Fig. 14.8 Block diagram of a radix-2 divider with partial remainder in stored-carry form.

31. Using Carry-Save Adders with the Dividers
sum = u = u1u0.u-1u-2u-3u-4….u-k
carry = v = v1v0.v-1v-2v-3v-4….v-k
t = u1u0.u-1u v1v0.v-1v-2
u + v - t = 00.00u-3u-4….u-k +
00.00v-3v-4….v-k 1 2
0 
<

32. Using Carry-Save Adders with the Dividers1 2 - t 1 2 1 2
t < - -
 t < 0
t  0 1 2 1 2 -
 u+v <
u+v  0
u+v < 0
q-j = -1
q-j = 0
q-j = 1

34. Fig Constant thresholds used for quotient digit selection in radix-2 division with qk–j in {–1, 0, 1}.

35. p-d Plot for Radix-2 Division
Fig A p-d plot for radix-2 division with d  [1/2,1), partial remainder in [–d, d), and quotient digits in [–1, 1].

36. High-Radix
Sequential
Dividers

37. Radix-4 division in dot notation

38. New Versus Shifted Old Partial Remainder in Radix-4 Division
Radix-4 fractional division with left shifts and q–j  [–3, 3]
s(j) = 4 s(j–1) – q–j d with s(0) = z and s(k) = 4k s
|–shift–|
|–– subtract ––|
Two difficulties:
How do you choose from among the 7 possible values for q-j?
If the choice is +3 or -3, how do you form 3d?

39. p-d Plot for Radix-4 SRT Division with Digit Set [-3,3]
Fig A p-d plot for radix-4 SRT division with quotient digit set [–3, 3].

40. Radix-4 SRT Divider with the Digit Set {-2, -1, 0, 1, 2}
Radix-4 fractional division with left shifts and q–j  [–2, 2]
s(j) = 4 s(j–1) – q–j d with s(0) = z and s(k) = 4k s
|–shift–|
|–– subtract ––|
Fig New versus shifted old partial remainder in radix-4 division with q–j in [–2, 2].
For this restriction to be feasible, we must have:
s  [-hd, hd) for some h < 1, and 4hd – 2d  hd
This yields h  2/3 (choose h = 2/3 to minimize the restriction)

41. p-d Plot for Radix-4 SRT Division with Digit Set [-2,2]
Fig A p-d plot for radix-4 SRT division with quotient digit set [–2, 2].

42. Radix r Divider
Process to derive the details:
Radix r
Digit set [–, ] for q–j
Number of bits of p (v and u) and d to be inspected
Quotient digit selection unit (table or logic)
Multiple generation/selection scheme
Conversion of redundant q to 2’s complement
Fig Block diagram of radix-r divider with partial remainder in stored-carry form.

43. Multiply/Divide
Unit

44. Multiply-Divide Unit
The control unit proceeds through necessary steps for multiplication or division
(including using the appropriate shift direction)
The slight speed penalty owing to a more complex control unit is insignificant
Fig Sequential radix-2 multiply/divide unit.