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FastDividers

Lecture 10

Required Reading

Chapter 14, High-Radix Dividers

Note errata at:http://www.ece.ucsb.edu/~parhami/text_comp_arit_1ed.htm#errors

Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design

Recommended Reading

J-P. Deschamps, G. Bioul, G. Sutter, Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems

Chapter 6, Arithmetic Operations: Division6.2.4, SRT Radix-2 Division6.2.5, SRT Radix-2 Division with Stored Carry Encoding6.2.6, P-D Diagram6.2.7, SRT-4 Division

Chapter 13, Dividers13.2.3, SRT Dividers13.2.4, SRT-4 Divider

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Classification of Dividers

Sequential

Radix-2 High-radix

RestoringNon-restoring

• regular• SRT• regular using carry save adders• SRT using carry save adders

ArrayDividers

Dividersby Convergence

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ArrayDividers

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Unsigned Fractional Division

zfrac Dividend .z-1z-2 . . . z-(2k-1)z-2k

dfrac Divisor .d-1d-2 . . . d-(k-1) d-k

qfrac Quotient .q-1q-2 . . . q-(k-1) q-k

sfrac Remainder .000…0s-(k+1) . . . s-(2k-1) s-2kk bits

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Integer vs. Fractional Division

For Integers:

z = q d + s 2-2k

z 2-2k = (q 2-k) (d 2-k) + s (2-2k)

zfrac = qfrac dfrac + sfrac

For Fractions:

wherezfrac = z 2-2k

dfrac = d 2-k

qfrac = q 2-k

sfrac = s 2-2k

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Unsigned Fractional Division Overflow

Condition for no overflow:

zfrac < dfrac

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Sequential Fractional DivisionBasic Equations

s(0) = zfrac

s(j) = 2 s(j-1) - q-j dfrac

2k · sfrac = s(k)

sfrac = 2-k · s(k)

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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)

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Fig. 15.7 Restoring array divider composed of controlledsubtractor cells.

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Non-Restoring Unsigned Fractional Division

s(-1) = z-dfor 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) + dend forif s(k-1) 0 q-k = 1else

q-k = 0 Correction step

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Fig. 15.8 Nonrestoring array divider built of controlledadd/subtract cells.

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SequentialDividers

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Sequential Fractional DivisionBasic Equations

s(0) = zfrac

s(j) = 2 s(j-1) - q-j dfrac

2k · sfrac = s(k)

sfrac = 2-k · s(k)

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Non-restoring Fractional Division

s(0) = zfor 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) + dend for

q = BSD_2’s_comp_conversion(q)Correction_step

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Integer Division

Correction step

We have: z = q d + s

z = (q-1) d + (s+d)z = q’ d + s’

z = (q+1) d + (s-d)z = q” d + s”

We need: sign(s) = sign (z)

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Fractional Division

Correction step

We have: zfrac = qfrac dfrac + sfrac

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

We need: sign(sfrac) = sign(zfrac)

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Non-restoring Fractional Division

s(0) = zfor 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) + dend for

q = BSD_2’s_comp_conversion(q)Correction_step

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Fig. 14.3 The new partial remainder, s(j), as a function of the shifted old partial remainder, 2s(j–1), in radix-2 nonrestoring division.

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

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Non-restoring Fractional Division with shifting over zeros

s(0) = zfor 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 forConversion of qCorrection step

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SRT Non-Restoring Fractional DivisionAssumptions

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

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

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SRT Non-Restoring Fractional Division

s(0) = zfor 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 forConversion of qCorrection step

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27

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

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SequentialDividers

with Carry-Save Adders

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

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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-2 + v1v0.v-1v-2

u + v - t = 00.00u-3u-4….u-k + 00.00v-3v-4….v-k

< 0 1

2

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Using Carry-Save Adders with the Dividers

0 t1

2-

t 0

q-j = 1

t <

q-j = -1

1

2-

u+v <1

2-1

2

t < 01

2-

q-j = 0

u+v < 0 u+v 0

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Fig. 14.7 Constant thresholds used for quotient digitselection in radix-2 division with qk–j in {–1, 0, 1}.

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Fig. 14.10 A p-d plot for radix-2 division with d[1/2,1), partial remainder in[–d, d), and quotient digits in [–1, 1].

d

p

Infeasible region (p cannot be 2d)

Infeasible region (p cannot be < 2d)

.100 .101 .110 .111 1.

00.1

00.0

11.1

10.0

10.1

11.0

01.1

01.0

00.1

01.0

01.1

10.0

d

2d

2d

d

Worst-case error margin in comparison

Choose 1

Choose 1

Choose 0

1

1

1 max

1 min

1 min

1 max

0 max

0 min

Ove

rla

p

Ove

rla

p

0

p-d Plot for Radix-2 Division

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High-RadixSequentialDividers

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Radix-4 division in dot notation

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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 qj? If the choice is +3 or 3, how do you form 3d?

New Versus Shifted Old Partial Remainder in Radix-4 Division

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d

p

Infeasible region (p cannot be 4d)

.100 .101 .110 .111

10.1

10.0

01.1

00.0

00.1

01.0

11.1

11.0

d

2d

Choose 2

Choose 0

Choose 1

3

1

2 max

2 min

1 min

1 max

0 max

Ove

rlap

0

3d

4d

Choose 3

3 min

2

Ove

rlap

O

verla

p

Fig. 14.12 A p-d plot for radix-4 SRT division with quotient digit set [–3, 3].

p-d Plot for Radix-4 SRT Division with Digit Set [-3,3]

–4d 4d

d

–d

4s (j–1)

–3 –2 –1 0 +1 +2 +3

s (j)

2d/3

8d/3 –2d/3

–8d/3

Fig. 14.13 New versus shifted old partial remainder in radix-4 division with q–j in [–2, 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 ––|

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)

Radix-4 SRT Divider with the Digit Set {-2, -1, 0, 1, 2}

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d

p

.100 .101 .110 .111

10.1

10.0

01.1

00.0

00.1

01.0

11.1

11.0

Choose 2

Choose 0

Choose 1 1

2 min

1 min

2 max

1 max

0 max

0

2

Ove

rlap

O

verla

p

Infeasible region (p cannot be 8d/3)

8d/3

5d/3

4d/3

2d/3

d/3

Fig. 14.14 A p-d plot for radix-4 SRT division with quotient digit set [–2, 2].

p-d Plot for Radix-4 SRT Division with Digit Set [-2,2]

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Carry v

CSA tree

Adder

Divisor d

k k

Select q –j

Shift left

2s Sum u

Multiple generation /

selection

Carry Sum

q –j

. . . q –j | | d or its complement

Fig. 14.15 Block diagram of radix-r divider with partial remainder in stored-carry form.

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

Radix r Divider

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Multiply/DivideUnit

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The control unit proceeds through necessary steps for multiplication or division (including using the appropriate shift direction)

Fig. 15.9 Sequential radix-2 multiply/divide unit.

Multiplier x or quotient q

Mux

Adder out c

0 1

Partial product p or partial remainder s

Multiplicand a or divisor d

Shift control

Shift

Enable

in c

q k–j

MSB of 2s (j–1)

k

k

k

j x

MSB of p (j+1)

Divisor sign

Multiply/ divide control

Select

Mul Div

The slight speed penalty owing to a more complex control unit is insignificant

Multiply-Divide Unit