1. Lecture 11
Advanced Dividers

2. Required Reading
Behrooz Parhami,
Computer Arithmetic: Algorithms and Hardware Design
Chapter 15 Variation in Dividers
15.3, Combinational and Array Dividers
Chapter 16, Division by Convergence

3. Division versus Multiplication
Division is more complex than multiplication:
Need for quotient digit selection or estimation
Overflow possibility: the high-order k bits of z
must be strictly less than d; this overflow check
also detects the divide-by-zero condition.
Pentium III latencies
Instruction Latency Cycles/Issue
Load / Store
Integer Multiply
Integer Divide
Double/Single FP Multiply
Double/Single FP Add
Double/Single FP Divide
The ratios haven’t changed much in
later Pentiums, Atom, or AMD products*
*Source: T. Granlund, “Instruction Latencies and Throughput for AMD and Intel x86 Processors,” Feb. 2012
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4. Classification of Dividers
Array
Dividers
Dividers
by Convergence
Sequential
Radix-2
High-radix
Restoring
Non-restoring
regular
SRT
using carry save adders
SRT using carry save adders

5. Fractional Division

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 for j=1..k
2k · sfrac = s(k)
sfrac = 2-k · s(k)

10. Fig. 13.2 Examples of sequential division with integer and fractional operands.

11. Array
Dividers

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

13. 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)

14. Restoring Array Divider
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15. 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

16. Nonrestoring Array Divider
Similarity to array multiplier is deceiving
Critical path
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17. Division by Convergence

18. Division by Convergence
Chapter Goals
Show how by using multiplication as the
basic operation in each division step,
the number of iterations can be reduced
Chapter Highlights
Digit-recurrence as convergence method
Convergence by Newton-Raphson iteration
Computing the reciprocal of a number
Hardware implementation and fine tuning
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19. 16.1 General Convergence Methods
Sequential digit-at-a-time (binary or high-radix) division
can be viewed as a convergence scheme
As each new digit of q = z / d is determined, the quotient value
is refined, until it reaches the final correct value
Convergence is from below in restoring division and oscillating
in nonrestoring division
Digit q 1
Meanwhile,
the remainder
s = z – q  d approaches 0;
the scaled remainder is kept in a certain range, such as [– d, d)
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20. Elaboration on Scaled Remainder in Division
The partial remainder s(j) in division recurrence isn’t the true remainder but a version scaled by 2j
Division with left shifts
s(j) = 2s(j–1) – qk–j (2k d) with s(0) = z and
|–shift–| s(k) = 2ks
|––– subtract –––|
Digit q 1
Quotient digit selection keeps the scaled remainder bounded (say, in the range
–d to d) to ensure the convergence of the true remainder to 0
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21. Recurrence Formulas for Convergence Methods
u (i+1) = f(u (i), v (i))
v (i+1) = g(u (i), v (i))
u (i+1) = f(u (i), v (i), w (i))
v (i+1) = g(u (i), v (i), w (i))
w (i+1) = h(u (i), v (i), w (i))
Constant
Desired
function
Guide the iteration such that one of the values converges to a constant (usually 0 or 1)
The other value then converges to the desired function
The complexity of this method depends on two factors:
a. Ease of evaluating f and g (and h)
b. Rate of convergence (number of iterations needed)
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22. 16.2 Division by Repeated Multiplications
Motivation: Suppose add takes 1 clock and multiply 3 clocks
64-bit divide takes 64 clocks in radix 2, 32 in radix 4
 Divide faster via multiplications faster if 10 or fewer needed
Idea:
Converges to q
Force to 1
Remainder often not needed, but can be obtained by another multiplication if desired: s = z – qd
To turn the identity into a division algorithm, we face three questions:
1. How to select the multipliers x(i) ?
2. How many iterations (pairs of multiplications)?
3. How to implement in hardware?
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23. Formulation as a Convergence Computation
Idea:
Force to 1
Converges to q
d (i+1) = d (i) x (i) Set d (0) = d; make d (m) converge to 1
z (i+1) = z (i) x (i) Set z (0) = z; obtain z/d = q  z (m)
Question 1: How to select the multipliers x (i) ? x (i) = 2 – d (i)
This choice transforms the recurrence equations into:
d (i+1) = d (i) (2 - d (i)) Set d (0) = d; iterate until d (m)  1
z (i+1) = z (i) (2 - d (i)) Set z (0) = z; obtain z/d = q  z (m)
u (i+1) = f(u (i), v (i))
v (i+1) = g(u (i), v (i))
Fits the general form
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24. Determining the Rate of Convergence
d (i+1) = d (i) x (i) Set d (0) = d; make d (m) converge to 1
z (i+1) = z (i) x (i) Set z (0) = z; obtain z/d = q  z (m)
Question 2: How quickly does d (i) converge to 1?
We can relate the error in step i + 1 to the error in step i:
d (i+1) = d (i) (2 - d (i)) = 1 – (1 – d (i))2
1 – d (i+1) = (1 – d (i))2
For 1 – d (i)  e, we get 1 – d (i+1)  e2: Quadratic convergence
In general, for k-bit operands, we need
2m – 1 multiplications and m 2’s complementations
where m = log2 k
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25. Quadratic Convergence
Table Quadratic convergence in computing z/d by repeated multiplications, where 1/2  d = 1 – y < 1
–––––––––––––––––––––––––––––––––––––––––––––––––––––––
i d (i) = d (i–1) x (i–1), with d (0) = d x (i) = 2 – d (i)
0 1 – y = (.1xxx xxxx xxxx xxxx)two  1/ y
1 1 – y 2 = (.11xx xxxx xxxx xxxx)two  3/ y 2
2 1 – y 4 = ( xxxx xxxx xxxx)two  15/ y 4
3 1 – y 8 = ( xxxx xxxx)two  255/ y 8
4 1 – y 16 = ( )two = 1 – ulp
Each iteration doubles the number of guaranteed leading 1s (convergence to 1 is from below)
Beginning with a single 1 (d  ½), after log2 k iterations we get as close to 1 as is possible in a fractional representation
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26. Graphical Depiction of Convergence to q
Fig Graphical representation of convergence in division by repeated multiplications.
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27. 16.5 Hardware Implementation
Repeated multiplications: Each pair of ops involves the same multiplier
d (i+1) = d (i) (2 - d (i)) Set d (0) = d; iterate until d (m)  1
z (i+1) = z (i) (2 - d (i)) Set z (0) = z; obtain z/d = q  z (m)
Fig Two multiplications fully overlapped in a 2-stage pipelined multiplier.
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28. 16.3 Division by Reciprocation
The Newton-Raphson method can be used for finding a root of f (x) = 0
Start with an initial estimate x(0) for the root
Iteratively refine the estimate via the recurrence
x(i+1) = x(i) – f (x(i)) / f (x(i))
Justification:
tan a(i) = f (x(i))
= f (x(i)) / (x(i) – x(i+1))
Fig Convergence to a root of
f(x) = 0 in the Newton-Raphson method.
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29. Computing 1/d by Convergence
1/d is the root of f (x) = 1/x – d
f (x) = –1/x2
Substitute in the Newton-Raphson
recurrence x(i+1) = x(i) – f (x(i)) / f (x(i)) to get:
x (i+1) = x (i) (2 - x (i)d)
One iteration = Two multiplications + One 2’s complementation
Error analysis: Let d (i) = 1/d – x(i) be the error at the ith iteration
d (i+1) = 1/d – x (i+1) = 1/d – x (i) (2 – x (i) d) = d (1/d – x (i))2 = d (d (i))2
Because d < 1, we have d (i+1) < (d (i))2 -d
1/d x
f(x)
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30. Choosing the Initial Approximation to 1/d
With x(0) in the range 0 < x(0) < 2/d, convergence is guaranteed
Justification: | d(0) | = | x(0) – 1/d | < 1/d
d(1) = | x(1) – 1/d | = d (d(0))2 = (d d(0)) d(0) < d(0)
For d in [1/2, 1):
Simple choice x(0) = 1.5
Max error = 0.5 < 1/d
Better approx x(0) = 4(3 – 1) – 2d
= – 2d
Max error  0.1 1 x
1/x 2
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31. 16.4 Speedup of Convergence Division
Compute y = 1/d
Do the multiplication yz
Division can be performed via 2 log2 k – 1 multiplications
This is not yet very impressive
64-bit numbers, 3-ns multiplier  33-ns division
Three types of speedup are possible:
Fewer multiplications (reduce m)
Narrower multiplications (reduce the width of some x(i)s)
Faster multiplications
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32. Initial Approximation via Table Lookup
Convergence is slow in the beginning: it takes 6 multiplications to get 8 bits of convergence and another 5 to go from 8 bits to 64 bits
d x(0) x(1) x(2) = ( )two
Better approx
Approx to 1/d
Read this value, x(0+), directly from a table, thereby reducing 6 multiplications to 2
A 2w  w lookup table is necessary and sufficient for w bits of convergence after 2 multiplications
Example with 4-bit lookup: d = xxxx (11/16  d < 12/16)
Inverses of the two extremes are 16/11  and 16/12 
So, is a good estimate for 1/d
 = (11/8)  (11/16) = 121/128 =
 = (11/8)  (3/4) = 33/32 =
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33. Visualizing the Convergence with Table Lookup
Fig Convergence in division by repeated multiplications with initial table lookup.
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34. Convergence Does Not Have to Be from Below
Fig Convergence in division by repeated multiplications with initial table lookup and the use of truncated multiplicative factors.
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35. Sequential
Dividers
with
Carry-Save Adders

36. Block diagram of a radix-2 SRT divider with partial remainder in stored-carry form

37. Pentium bug (1)
October 1994
Thomas Nicely, Lynchburg Collage, Virginia
finds an error in his computer calculations, and traces
it back to the Pentium processor
November 7, 1994
First press announcement, Electronic Engineering Times
Late 1994
Tim Coe, Vitesse Semiconductor
presents an example with the worst-case error
c = /
Pentium =
Correct result =

38. Pentium bug (2) Intel admits “subtle flaw” November 30, 1994
Intel’s white paper about the bug and its possible consequences
Intel - average spreadsheet user affected once in 27,000 years
IBM - average spreadsheet user affected once every 24 days
Replacements based on customer needs
December 20, 1994
Announcement of no-question-asked replacements

39. Pentium bug (3) Error traced back to the look-up table used by
the radix-4 SRT division algorithm
2048 cells, 1066 non-zero values {-2, -1, 1, 2}
5 non-zero values not downloaded correctly
to the lookup table due to an error in the C script

41. Follow-up
Courses

42. DIGITAL SYSTEMS DESIGN
ECE 681 VLSI Design for ASICs
(Fall semesters)
H. Homayoun, project/lab, front-end and back-end
ASIC design with Synopsys tools
2. ECE 699 Digital Signal Processing Hardware Architectures
(Spring semesters)
A. Cohen, project, FPGA design for DSP
ECE 682 VLSI Test Concepts
(Spring semesters) T. Storey, homework

43. NETWORK AND SYSTEM SECURITY
ECE 646 Cryptography and Computer Network Security
(Fall semesters)
K.Gaj, hardware, software, or analytical project
2. ECE 746 Advanced Applied Cryptography
(Spring semesters) J.-P. Kaps, hardware, software, or analytical project
ECE 899 Cryptographic Engineering
(Spring semesters) J.-P. Kaps, research-oriented project