# Arithmetic Operations

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1 Arithmetic Operations Arithmetic Operations addition subtraction multiplication division Each of these operations on the integer representations: unsigned two's complement 1

2 Addition One bit of binary addition carry out + a b carry in sum bit 2

3 Addition Truth Table 3 Carry In a b Carry Out Sum Bit

4 Addition Unsigned and 2's complement use the same addition algorithm Due to the fixed precision, throw away the carry out from the msb

5 Two s Complement Addition ( ) ( ) ( ) ( ) ( ) ( ) 5

6 Overflow The condition in which the result of an arithmetic operation cannot fit into the fixed number of bits available. For example: +8 cannot fit into a 3-bit, unsigned representation. It needs 4 bits:

7 Overflow Detection Most architectures have hardware that detects when overflow has occurred (for arithmetic operations). The detection algorithms are simple. 7

8 6-bit examples: Unsigned Overflow Detection Carry out from msbs is overflow in unsigned 8

9 Unsigned Overflow Detection 6-bit examples: No Overflow Overflow! Overflow! 9 Carry out from msbs is overflow in unsigned

10 Two s Complement Overflow Detection When adding 2 numbers of like sign + to + - to and the sign of the result is different! Overflow! Overflow! 10

11 Addition Overflow detection: 2 s complement 6-bit examples ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

12 Subtraction basic algorithm is like decimal 0 0 = = = =? BORROW!

13 Subtraction For two s complement representation The implementation redefines the operation: a b becomes a + (-b) This is a 2-step algorithm: 1. take the two s complement of b (common phrasing for: find the additive inverse of b) 2. do addition 13

14 Subtraction 6-bit, 2 s complement examples ( ) ( ) ( ) ( ) 14

15 Subtraction Overflow detection: 2 s complement If the addition causes overflow, so does the subtraction! ( ) ( ) 15

16 Multiplication 0 0 = = = = 1 Same algorithm as decimal There is a precision problem n bits * n bits n + n bits may be needed 16

17 In HW, space is always designated for a larger precision product. 32 bits * 32 bits 64 bits 17

18 Unsigned Multiplication *

19 Unsigned Multiplication *

20 Two s Complement Slightly trickier: must sign extend the partial products (sometimes!) 20

21 OR Sign extend multiplier and multiplicand to full width of product * product 21 And, use only exact number of lsbs of product

22 Multiplication OK sign ext. partial product reverse or additive inverses find additive inverses OK 22

23 Unsigned Division /3 23

24 Sign Extension The operation that allows the same 2's complement value to be represented, but using more bits (5 bits) _ (8 bits) (4 bits) (8 bits) 24

25 Zero Extension The same type of thing as sign extension, but used to represent the same unsigned value, but using more bits (5 bits) _ (8 bits) (4 bits) (8 bits) 25

26 Truth Table for a Few Logical Operations X Y X and Y X nand Y X or Y X xor Y

27 Logical Operations Logical operations are done bitwise on every computer Invented example: Assume that X,Y, and Z are 8-bit variables and Z, X, Y If X is Y is then Z is 27

28 To selectively clear bit(s) clear a bit means make it a 0 First, make a mask: (the generic description of a set of bits that do whatever you want them to) Within the mask, 1 s for unchanged bits 0 s for cleared bits To clear bits numbered 0,1, and 6 of variable X mask and use the instruction and result, X, mask 28

29 To selectively set bit(s) set a bit means make it a 1 First, make a mask: 0 s for unchanged bits 1 s for set bits To set bits numbered 2,3, and 4 of variable X mask and use the instruction or result, X, mask 29

30 Shift Moving bits around 1) arithmetic shift 2) logical shift 3) rotate Bits can move right or left 30

31 Arithmetic Shift Right sign extension! Left 31

32 Logical Shift Right Left Logical left is the same as arithmetic left. 32

33 Rotate Right Left 33 No bits lost, just moved

34 Assume a set of 4 chars. are in an integersized variable (X). Assume an instruction exists to print out the character all the way to the right putc X (prints D) Invent instructions, and write code to print ABCD, without changing X. 34

35 rotl X, 8 bits putc X # A rotl X, 8 bits putc X # B rotl X, 8 bits putc X # C rotl X, 8 bits putc X # D Karen s solution 35

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