CSE 246: Computer Arithmetic Algorithms and Hardware Design Instructor: Prof. Chung-Kuan Cheng Lecture 5

CSE 2462 Topics: False Path: Cycles Multi-Operands Addition  Carry Save Addition Multiplication  2 ’ s Complement  Booth Recoding

CSE 2463 False Path: Cycles  Cycles of False Paths: Eg. 1 ’ s complement number addition Positive: x Negative: (2 n -1)-x  Addition (2 n -1)-x + (2 n -1)-y = 2 n +(2 n -1)-(x+y)-1 C out A 3,0 B 3,0 S 3,0 Adder C in

CSE 2464 Example  -3-5 =  0+0=

CSE 2465 Multi-Operand Addition  Carry save adder: a (3,2) counter

CSE 2466 Example  A (3,2) counter compresses X rows to 2/3X rows each time  Tree structure in implementation

CSE 2467 Other Counters  (5,3) counter CaCa CbCb S0S0  (7,3) counter S0S0 S2S2 S1S1  Design of (5,3) counter using full adders CaCa CbCb S0S0 CaCa CbCb S0S0

CSE 2468 Multiplication  Examples * * ’ s complement of -5

CSE 2469 Multiplication  Examples (Cont.) * * ’ s complement of 5 Sign bits

CSE Multiplication a 3 a 2 a 1 a 0 * b 3 b 2 b 1 b 0 -a 3 b 0 a 2 b 0 a 1 b 0 a 0 b 0 -a 3 b 1 a 2 b 1 a 1 b 1 a 0 b 1 -a 3 b 2 a 2 b 2 a 1 b 2 a 0 b 2 a 3 b 3 -a 2 b 3 -a 1 b 3 -a 0 b 3  Traditional multiplication of negative numbers

CSE Multiplication a 3 a 2 a 1 a 0 * b 3 b 2 b 1 b 0 a 3 b 0 a 2 b 0 a 1 b 0 a 0 b 0 a 3 b 1 a 2 b 1 a 1 b 1 a 0 b 1 a 3 b 2 a 2 b 2 a 1 b 2 a 0 b 2 a 3 b 3 a 2 b 3 a 1 b 3 a 0 b 3 0 -a 3 -a 3 -a 3 0 -a 2 -a 1 -a 0  -xy = x(1-y)-x = xy-x

CSE Multiplication a 3 a 2 a 1 a 0 * b 3 b 2 b 1 b 0 a 3 b 0 a 2 b 0 a 1 b 0 a 0 b 0 a 3 b 1 a 2 b 1 a 1 b 1 a 0 b 1 a 3 b 2 a 2 b 2 a 1 b 2 a 0 b 2 a 3 b 3 a 2 b 3 a 1 b 3 a 0 b 3 0 -a 3 -a 3 -a 3 0 -b 3 -b 3 -b 3  -xy = xy-x = xy-y

CSE Multiplication a 3 a 2 a 1 a 0 * b 3 b 2 b 1 b 0 a 3 b 0 a 2 b 0 a 1 b 0 a 0 b 0 a 3 b 1 a 2 b 1 a 1 b 1 a 0 b 1 a 3 b 2 a 2 b 2 a 1 b 2 a 0 b 2 a 3 b 3 a 2 b 3 a 1 b 3 a 0 b 3 a a 3 b b 3  Replay 0 -a -a -a by a 0 0 a

CSE Multiplication a 3 a 2 a 1 a 0 * b 3 b 2 b 1 b 0 a 3 b 0 a 2 b 0 a 1 b 0 a 0 b 0 a 3 b 1 a 2 b 1 a 1 b 1 a 0 b 1 a 3 b 2 a 2 b 2 a 1 b 2 a 0 b 2 a 3 b 3 a 2 b 3 a 1 b 3 a 0 b 3 a a 3 -1 b b 3  Fast multiplication: we avoid calculating the 2 ’ s complement of multiplicand, which means a potential long carry chain

CSE Booth Recoding  Multiplication: n rows by 2n-1 column matrix for addition  Objective: try to reduce n rows to log n rows  Radix 2 booth recoding  Radix 4 booth recoding

CSE Radix 2 Booth Recoding  Use 1,0,-1 to recode  Recoding table idxixi x i-1 y i =(x i-1 -x i ) 0000No change 1011End of string of 1 ’ s 210Begin of string of 1 ’ s 3110No change

CSE Example Assume here is 0 x 5 x 4 x 3 x 2 x 1 -x 5 x 4 -x 4 x 3 -x 3 x 2 -x 2 x 1 -x 1 x 4 -x 5 x 3 -x 4 x 2 -x 3 x 1 -x 2 0-x 1 y 5 y 4 y 3 y 2 y 1

CSE Radix 4 Booth Recoding  Use 2,1,0,-1,-2 to recode (z i/2 =2y i+1 +y i ) idx i+ 1 xixi x i-1 y i+1 yiyi z i/ No string of 1 ’ s End of string of 1 ’ s Isolated End of string of 1 ’ s Begin of string of 1 ’ s End of one string, begin new string Begin of string of 1 ’ s String of all 1 ’ s

CSE Examples Bin 0 1, 1 1, Radix 4: 2, 0, -1 2* = 31 2x 6 2x 4 2x 2 x 5 x 3 x 1 -2x 6 +x 5 +x 4 -2x 4 +x 3 +x 2 -2x 2 +x 1 +x 0 x 6 x 5 x 4 x 3 x 2 x 1 x 0