Arithmetic for Computers
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1 MIPS Arithmetic Instructions Cptr280 Dr Curtis Nelson Arithmetic for Computers Operations on integers Addition and subtraction; Multiplication and division; Dealing with overflow; Signed vs. unsigned numbers. Floating-point numbers Representation and operations; Dealing with overflow and underflow. 1
2 MIPS Arithmetic Instructions MIPS arithmetic instructions Instruction Example Meaning Comments add add $1,$2,$3 $1 = $2 $3 3 operands; exception possible subtract sub $1,$2,$3 $1 = $2 $3 3 operands; exception possible add immediate addi $1,$2,100 $1 = $2 100 constant; exception possible add unsigned addu $1,$2,$3 $1 = $2 $3 3 operands; no exceptions subtract unsigned subu $1,$2,$3 $1 = $2 $3 3 operands; no exceptions add imm. unsign. addiu $1,$2,100 $1 = $2 100 constant; no exceptions multiply mult $2,$3 Hi, Lo = $2 x $3 64-bit signed product multiply unsigned multu $2,$3 Hi, Lo = $2 x $3 64-bit unsigned product divide div $2,$3 Lo = $2 $3, Lo = quotient, Hi = remainder Hi = $2 mod $3 divide unsigned divu $2,$3 Lo = $2 $3, Unsigned quotient & remainder Hi = $2 mod $3 Move from Hi mfhi $1 $1 = Hi Used to get copy of Hi Move from Lo mflo $1 $1 = Lo Used to get copy of Lo Note: Move from Hi and Move from Lo are really data transfers Details Usually math operands and the result have a fixed number of bits (8, 16, 32, or 64). These are the sizes that processors use to represent integers. To keep the result the same size as the operands, you may have to include zero bits in some of the leftmost columns (sign extension). Compute the carry-out of the leftmost column, but don't write it as part of the answer (because there is no room if you have a fixed number of bits.) When the operands are represented using the unsigned binary scheme, a carry-out of 1 from the leftmost column means the sum does not fit into the fixed number of bits. This is called Overflow. When the operands are represented using the two's complement scheme (which will be described later), then a carry-out of 1 from the leftmost column is not necessarily overflow. Integers may be represented using a scheme called unsigned binary or a scheme called two's complement binary. The same binary addition hardware is used, but to interpret the result you need to know what scheme is being used. 2
3 Unsigned Binary Addition Unsigned means no negative numbers are represented. Add and in 8-bit unsigned binary: Result is , carry = 0 Now add and in 8-bit unsigned binary: Result is , binary = ? Result is 11 10, carry = 1, indicating an overflow occurred Overflow in unsigned binary addition (assembly instructions addu, addiu) means the result is incorrect an error condition that is not flagged by the processor. Unsigned Binary Subtraction Subtract from in 8-bit unsigned binary. Result is 3 10, carry = 0 Now subtract from Result is 3 10, binary = (253 10?) Result is , carry = 1, i.e. underflow Underflow in unsigned binary subtraction (subu) means the result is incorrect an error condition that will not be flagged by the processor (or QTSpim). 3
4 Unsigned - Detecting Overflow For unsigned numbers, overflow occurs if there is a carry out of the most significant bit. For example, 1001 = = = 1 With the MIPS architecture Overflow exceptions occur for two s complement arithmetic add, sub, addi Overflow exceptions do not occur for unsigned arithmetic addu, subu, addiu Unsigned Vs. Signed Numbers b n 1 b 1 b 0 MSB Magnitude (a) Unsigned number bn 1bn 2 b 1 b 0 Sign 0 denotes 1 denotes MSB Magnitude (b) Signed number 4
5 Interpretation of Four-Bit Signed Integers abcd Sign and magnitude 1 s complement 2 s complement s Complement Representation To negate a two's complement integer, invert all the bits and add a one to the least significant bit. What are the two s complements of: 6 = =
6 Examples What is the value of the two's complement integer 1101 in decimal? What is the value of the unsigned integer 1101 in decimal? What is the negation of the two's complement integer 1010 in binary? Graphical Representation of Four-bit 2 s Complement Numbers
7 Two s Complement Addition To add two's complement numbers, add the corresponding bits of both numbers with carry between bits. For example, 3 = = = = = = = = Unsigned and signed two s complement addition are performed exactly the same way, but how they detect overflow differs. More 2 s Complement Addition Examples ( 5) ( 2) (5) ( 2) ( 7) (3) ( 5) ( 2) (5) ( 2) ( 3) (7) ignore ignore 7
8 Two s Complement Subtraction To subtract two's complement numbers, first negate the second number and then add the corresponding bits of both numbers. For example: 3 = = = = = = = = More 2 s Complement Subtraction Examples ( 5) ( 2) ( 3) (5) ( 2) (7) ignore ignore ( 5) (2) ( 7) (5) (2) (3)
9 Integer Addition Example: 7 6 Overflow if result out of range Adding a and - operand, no overflow possible. Adding two operands Overflow if sign of result is 1. Adding two operands Overflow if sign of result is 0. Integer Subtraction Add negation of second operand. Example: 7 6 = 7 (6) 7: : : Overflow if result out of range Subtracting two or two operands, no overflow. Subtracting from operand Overflow if result sign is 0. Subtracting from operand Overflow if result sign is 1. 9
10 Detecting Overflow Logically When adding two's complement numbers, overflow will only occur if: The numbers being added have the same sign; The sign of the result is different then the sign of the two operands. If we perform the addition a n-1 a n-2... a 1 a 0 b n-1 b n-2 b 1 b = s n-1 s n-2 s 1 s 0 Overflow can be detected as V = an - 1 bn - 1 sn-1 an-1 bn-1 sn - 1 Overflow can also be detected as V = c n Ä cn - 1 where c n-1 and c n are the carry in and carry out of the most significant bit. MIPS Arithmetic Logic Unit (ALU) zero ovf Must support the Arithmetic/Logic operations of the ISA A add, addi, addiu, addu sub, subu mult, multu, div, divu B sqrt and, andi, nor, or, ori, xor, xori beq, bne, slt, slti, sltiu, sltu ALU result 32 4 m (operation) With special handling for - Sign extend addi, addiu, slti, sltiu - Zero extend andi, ori, xori - Overflow detection add, addi, sub 10
11 Conclusions In this presentation you learned about: Representation of numbers in computers; Signed vs. unsigned numbers; Conditions which cause overflow; Some MIPS instructions. 11
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