Addition and Subtraction
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1 ddition and Subtraction Philipp Koehn 9 February 2018
2 1 addition
3 1-it dder 2 Let s start simple: dding two 1-it numbers Truth table
4 Really 2 Operations 3 Truth table for "position 0" bit xor Truth table for carry bit + carry and
5 Reminder: asic Gates 4 ND OR NOT NND NOR
6 Circuits 5 "Position 0" bit OUT xor OR NND ND OUT0 Carry bit OUTC and ND OUTC
7 Putting them Together 6 OR ND OUT0 NND ND CRRY
8 N-it ddition
9 N-it ddition = 0, carry the 1
10 N-it ddition = 1, carry the 1
11 N-it ddition copy carry bit
12 1-it dder 11 OR ND OUT0 NND ND CRRY Our adder cannot handle carry as input yet
13 Half 1-it dder 12 OR ND OUT0 NND ND CRRY 1-IT HLF DDER
14 uilding a 1-it Full dder 13 1-IT HLF DDER S C SUM CRRY
15 uilding a 1-it Full dder 14 CRRY 1-IT HLF DDER S C 1-IT HLF DDER CRRY S C SUM CRRY
16 uilding a 1-it Full dder 15 CRRY 1-IT HLF DDER S C 1-IT HLF DDER S C OR SUM CRRY
17 1-it Full dder 16 CRRY 1-IT HLF DDER S C 1-IT HLF DDER S C OR SUM CRRY 1-IT FULL DDER
18 N-it Full dder
19 N-it Full dder IT FULL DDER CO CI S SUM0
20 N-it Full dder IT FULL DDER CO CI S IT FULL DDER CO CI S SUM1 SUM0
21 N-it Full dder and so on... 1-IT FULL DDER CI 1-IT FULL DDER CI 1-IT FULL DDER CI CO S CO S CO S SUM2 SUM1 SUM0
22 21 subtraction
23 First, a Trick 22 Normally, we subtract like this:
24 Computing the Inverse 23 Now we use the inverse of the subtrahend
25 Subtraction by ddition 24 This allows us to carry our subtraction by addition Well, with minor corrections
26 lso Works in inary 25 Original problem Inverse of subtrahend ddition Corrections
27 Start with N-it dder 26 0 CO IT FULL DDER 3 CI S0 S1 S2 S3
28 Invert its of Subtrahend CO IT FULL DDER 3 CI S0 S1 S2 S3
29 dd One CO 4-IT FULL DDER CI V S0 S1 S2 S3 Trick: add one as carry in
30 Invert Overflow --- DONE CO 4-IT FULL DDER CI V S0 S1 S2 S3 OVERFLOW SUM mm...
31 30 unifying addition and subtraction machines
32 Goal 31 Not two machines for addition and subtraction Combined adder and subtractor Input:,, and subtraction flag SU Output if SU=0: + if SU=1: -
33 NOT only if SU CO 4-IT FULL DDER CI V S0 S1 S2 S3 OVERFLOW SUM mm...
34 NOT only if SU 33 Truth table SU X OUT Looks like XOR XOR
35 Combined Machine SU SU CO 4-IT FULL DDER S0 S1 S2 S3 CI SU OVERFLOW SUM
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