![Page 1: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/1.jpg)
US008209370B2
(12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 Lablans (45) Date of Patent: *Jun. 26, 2012
(54) MULTI-VALUE DIGITAL CALCULATING 3,3 2 1i; glam?kawa 1 , , ontoye eta .
CIRCUITS, INCLUDING MULTIPLIERS 4,982,356 A M991 Ando
5,222,035 A 6/1993 Naka t l. (75) Inventor: Peter Lablans, Morris Township, N] 5227993 A 7/1993 yamaslfajvs
(US) 5,280,440 A 1/1994 Sugimura 5,289,399 A 2/1994 Yoshida
- , - - 5,379,245 A * 1/1995 Ueda ........................... .. 708/624 (73) Asslgnee. Ternaryloglc LLC, MornstoWn, NC 5,438,533 A 8/1995 Yoshida
(Us) 5,644,253 A 7/1997 Takatsu 6,535,902 B2 3/2003 Goto
( * ) Notice: Subject to any disclaimer, the term of this 6,567,835 B1 5/2003 Blomgren et a1. patent is extended or adjusted under 35 6,816,877 B2: 11/2004 Park et al' ~~~~~~~~~~~~~~~ " 708/628 U S C 154(1)) b 507 da S 7,562,106 B2 7/2009 Lablans ...................... .. 708/493
' ' ' y y ' 2001/0016864 A1 8/2001 Noeske
patent is subject IO a terminal (115 c a1mer.
Non-Final Of?ce Action in US. Appl. No. 12/267,900, mailed Sep.
(21) Appl. No.: 12/472,731 26, 2011, 6 PgS~ O. Sentieys et al. ‘Conception d’un processeur ternaire a faible
- . (e)nergie. (text in English)’, Colloque Faible Tension Faible (22) F?ed' May 27’ 2009 Consommation (FTFC ’03), 2003, 7 pages, Rennes, France.
_ _ _ C.S. Wallace; ‘A suggestion for a Fast Multiplier’ IEEE Trans. Elec
PI‘lOI‘ Publlcatlml Data tron‘ comput‘ Us 2009/0234900 A1 Sell 17 2009 R. Brent, H. Kung; ‘A regular lay-out for parallel adders.’ IEEE ’ Transactions on Computers C-31 (1982).
- - Gary W. Bewick; ‘Fast multiplication: algorithms and implementa Related U's' Apphcatlon Data tion.’ PhD. Thesis Stanford University. Feb. 1994. 170pg.
(63) Continuation of application No. 11/018,956, ?led on C t. d Dec. 20, 2004, noW Pat. No. 7,562,106. ( on “we )
(51) Int Cl Primary Examiner * Chuong D Ngo
Attorney, Agent, 0}’ i G06F 7/52 (2006.01)
(52) US. Cl. ...................................... .. 708/493; 708/620 (57) ABSTRACT
(58) Field of Classi?cation Search ................ .. 708/483, Apparatus and method for Performing multi-value arithmetic 708/620’ 623’ 624 operations are disclosed. Multi-Value signals can be added,
See application ?le for Complete Search history subtracted and multiplied using a ?rst truth table to generate a residue and a second truth table to generate a carry. Addi
(56) References Cited tionally, method and apparatus to ef?ciently perform the
US. PATENT DOCUMENTS
4,566,075 A 1/1986 Guttag 4,620,188 A 10/1986 Sengchanh 4,628,472 A 12/1986 Fensch
601 602
function a0b1+a1b0 on multi-Value signals are disclosed. Also an e?icient method of processing large binary signals is disclosed.
20 Claims, 34 Drawing Sheets
![Page 2: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/2.jpg)
US 8,209,370 B2 Page 2
OTHER PUBLICATIONS
Kai Hwang. ‘Computer Arithmetic. Principles, Architecture, and Design.’ John Wiley & Sons 1979. pp. 84-91. Abraham Peled, Bede Liu. ‘Digital Signal Processing. Theory, Design and Implementation.’ Abraham Wiley and Sons 1976, p. 18.
J. Sklansky. ‘Conditional-Sum Addition Logic.’ IRE Transactions on Electronic Computers (1960). 0.]. Bedrij. ‘Carry-Select Adder’, IRE Trans. Electr. Comp. Jun. 1962, 340-346.
* cited by examiner
![Page 3: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/3.jpg)
US. Patent Jun. 26, 2012 Sheet 1 0f 34 US 8,209,370 B2
‘I207
1204
\1211 (-1208 1202 1205
1215
‘I200
L
Figure 3
![Page 4: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/4.jpg)
US. Patent Jun. 26, 2012 Sheet 2 0f 34
3001 3016 3007
\ k": 3009 3003
3000
\O é) \ 3011
3002 3005 3008
\_ \: 51,3012
3006 \ 3013
Figure 4
\ ‘301 0 3004
US 8,209,370 B2
3014
3015
![Page 5: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/5.jpg)
US. Patent Jun. 26, 2012 Sheet 3 0f 34 US 8,209,370 B2
109 4f-102 l
/ / 104
106/
110 101 108
\107 103 —»C5
‘ \105 Figure 5
f 209
208 210
303
/ 304
\ 207
\ 205
Figure 7
n \J
/ 25 g) 203
F|gure6
201
![Page 6: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/6.jpg)
US. Patent Jun. 26, 2012 Sheet 4 0f 34 US 8,209,370 B2
ARRANGE TWO 4x4 DIGITS NUMBERS 310 _' a3a2a1a0
b3b2b1 b0
311 _ a3a2a1a0 312 — a3a2a1a0
b1b0 b3b200
313
\ V V / GENERATE 5 RESIDUES (R) GENERATE 5 RESIDUES (R)
\ V V / GENERATE 5 FIRST GENERATE 5 FIRST
CARRIES (C) CARRIES (C)
315 319 \ v /
GENERATE 3 SECOND GENERATE 3 SECOND CARRIES (K) CARRIES (K)
i l / 320 316 "’ CSA R+C+K CSA R+C+K
321 —— CSA
CLA — 322
Figure 7A
![Page 7: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/7.jpg)
US. Patent Jun. 26, 2012 Sheet 5 0f 34 US 8,209,370 B2
403
![Page 8: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/8.jpg)
US. Patent Jun. 26, 2012 Sheet 6 0f 34 US 8,209,370 B2
1001 \
2
/ 1 004 1003
Figure 9
![Page 9: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/9.jpg)
US. Patent Jun. 26, 2012 Sheet 7 0134 US 8,209,370 B2
501
\ ‘r502 504
Figure10
601 602
6030 6040 f f
607 6031 6041
Av’ r5’ 608 v \7
612 6032
M 446°” / 609 é, 1_ 603
6043 — M A’
610 2 i 6034
611 _d;/-— I605] A16044 606] 611 \Q/
Figure 11
![Page 10: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/10.jpg)
US. Patent Jun. 26, 2012 Sheet 8 0f 34 US 8,209,370 B2
905
901 \ \r 909
907 & fa \ 910
902 903 —‘é
\_ r‘ 909
902‘ H: 906
Figure 12
![Page 11: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/11.jpg)
US. Patent Jun. 26, 2012 Sheet 9 0f 34 US 8,209,370 B2
1401
1507
1400
\l
1513
1500
1604
1600
1610
1605
Figure 15
![Page 12: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/12.jpg)
US. Patent Jun. 26, 2012 Sheet 10 0f 34 US 8,209,370 B2
1700
1 —\ 1713
170917101711 1712 Figure16
1801 1802 1803 1806
1803 / 1807 \ 1810 1 f 1 f AI W — w
1805 1804 1808 1809 1811 1812
1800 \ \ \ \ \ 1813
Figure 17
![Page 13: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/13.jpg)
US. Patent Jun. 26, 2012 Sheet 11 0134 US 8,209,370 B2
701 702 703
\ \I 708 \ E 204
705 707 706 Figure ‘I8
820 808 X \ 802
O / k-/
803 1 80 Ah/
Figure 19
![Page 14: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/14.jpg)
US. Patent Jun. 26, 2012 Sheet 12 0134 US 8,209,370 B2
3106 3107 3108
3101 x x \ > T _ _ _ _ _
T T > @109
3102 3103 3104 3105 E
Figure 20
3210 3106 3107 3108
3200 \3101 \ x \ \ T T _ _ _ _ _ T 0 3109
\ 3212 1 162) J’ 3105/ 3110
3102 3103 3104 > [I
V
Figure 20A 3211
![Page 15: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/15.jpg)
US. Patent Jun. 26, 2012 Sheet 13 0f 34 US 8,209,370 B2
(1) a2
y(n)
Figure 208
![Page 16: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/16.jpg)
US. Patent Jun. 26, 2012 Sheet 14 0f 34 US 8,209,370 B2
N- switch a b
source drain T o— —o T
gate 8
Figure 21 Figure 22
a b
a a 2 b
S = 0
Figure 23 S = 1
P- switch source drain
0— —o a b
gate Figure 24 _ 8
Figure 25
a b a b
. A . .—/L. S = 0 S = 1
![Page 17: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/17.jpg)
US. Patent Jun. 26, 2012 Sheet 15 0f 34 US 8,209,370 B2
256
293
290 (J 298
291 \ / {J
299 (J 291 (J I {J
298 299
Figure 29
![Page 18: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/18.jpg)
US. Patent Jun. 26, 2012 Sheet 16 0f 34 US 8,209,370 B2
296
297
301 y/ 299 292
291
302 294 303 304
Flgure 30
301 291 290 305
/ \ {j (J 302 W = 0 0 /
x r’ V 295
![Page 19: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/19.jpg)
US. Patent Jun. 26, 2012 Sheet 17 0f 34 US 8,209,370 B2
Input Output_1
Latch Inverter Output 2 A _
V
Figure 32.
660
input_data / |0g_1
dock _ output_s _ |og_3
Latch
Q |og_2 output_r
Figure 33
![Page 20: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/20.jpg)
US. Patent Jun. 26, 2012 Sheet 18 0f 34 US 8,209,370 B2
Input a b C d k I m n
I abcd efg h
k l m n
Figure 34
900 Input 0 1 2 j 0 1 2
I: m30 1 2 0 1 0
/ j 1 1 2 908 \ 901 2 2 2
904 f 905
Figure 35
Input 0 1 2 1 2 0 m31 2
|—— 0 1 0 1 2
2 0 1 1 2 0
2 0
Figure 36
![Page 21: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/21.jpg)
![Page 22: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/22.jpg)
![Page 23: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/23.jpg)
![Page 24: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/24.jpg)
![Page 25: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/25.jpg)
![Page 26: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/26.jpg)
![Page 27: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/27.jpg)
![Page 28: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/28.jpg)
![Page 29: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/29.jpg)
![Page 30: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/30.jpg)
![Page 31: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/31.jpg)
![Page 32: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/32.jpg)
![Page 33: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/33.jpg)
![Page 34: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/34.jpg)
![Page 35: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/35.jpg)
![Page 36: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/36.jpg)
![Page 37: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/37.jpg)
![Page 38: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/38.jpg)
![Page 39: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/39.jpg)
![Page 40: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/40.jpg)
![Page 41: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/41.jpg)
![Page 42: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/42.jpg)
![Page 43: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/43.jpg)
![Page 44: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/44.jpg)
![Page 45: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/45.jpg)
![Page 46: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/46.jpg)
![Page 47: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/47.jpg)
![Page 48: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/48.jpg)
![Page 49: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/49.jpg)
![Page 50: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/50.jpg)
![Page 51: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/51.jpg)
![Page 52: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/52.jpg)
![Page 53: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/53.jpg)
![Page 54: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/54.jpg)
![Page 55: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/55.jpg)
![Page 56: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/56.jpg)
![Page 57: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/57.jpg)
![Page 58: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/58.jpg)
![Page 59: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/59.jpg)
![Page 60: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/60.jpg)
![Page 61: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/61.jpg)
![Page 62: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/62.jpg)
![Page 63: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/63.jpg)
![Page 64: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/64.jpg)
![Page 65: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/65.jpg)
![Page 66: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/66.jpg)
![Page 67: (12) Ulllted States Patent (10) Patent N0.: US 8,209,370 B2 … · 2015. 2. 15. · US 8,209,370 B2 Page 2 OTHER PUBLICATIONS Kai Hwang. ‘Computer Arithmetic. Principles, Architecture,](https://reader036.vdocument.in/reader036/viewer/2022071500/611ed1fc7cb3317bce32e165/html5/thumbnails/67.jpg)