Arithmetic Operations for Computers: Addition, Subtraction, Multiplication, and Division

1. Lecture 9 Sept 28 Chapter 3 Arithmetic for Computers

2. Arithmetic for Computers §3.1 Introduction • Operations on integers • Addition and subtraction • Multiplication and division • Dealing with overflow • Floating-point real numbers • Representation and operations

3. Integer Addition Example: 7 + 6 §3.2 Addition and Subtraction • Overflow if result out of range • Adding +ve and –ve operands, no overflow • Adding two +ve operands • Overflow if result sign is 1 • Adding two –ve operands • Overflow if result sign is 0

4. Integer Subtraction Add negation of second operand Example: 7 – 6 = 7 + (–6) +7: 0000 0000 … 0000 0111–6: 1111 1111 … 1111 1010+1: 0000 0000 … 0000 0001 Overflow if result out of range • Subtracting two +ve or two –ve operands, no overflow • Subtracting +ve from –ve operand • Overflow if result sign is 0 • Subtracting –ve from +ve operand • Overflow if result sign is 1

5. Dealing with Overflow Some languages (e.g., C) ignore overflow • Use MIPS addu, addui, subu instructions Other languages (e.g., Ada, Fortran) require raising an exception • Use MIPS add, addi, sub instructions • On overflow, invoke exception handler • Save PC in exception program counter (EPC) register • Jump to predefined handler address • mfc0 (move from coprocessor reg) instruction can retrieve EPC value, to return after corrective action

6. Two’s-Complement Addition and Subtraction Binary adder used as 2’s-complement adder/subtractor.

7. Simple Adders Digit-set interpretation: {0, 1} + {0, 1} = {0, 2} + {0, 1} Digit-set interpretation: {0, 1} + {0, 1} + {0, 1} = {0, 2} + {0, 1} Binary half-adder (HA) and full-adder (FA).

8. Full-Adder Implementations Full adder implemented with two half-adders, by means of two 4-input multiplexers, and as two-level gate network.

9. Critical path Ripple-Carry Adder: Slow But Simple Ripple-carry binary adder with 32-bit inputs and output.

10. Carry Propagation Networks gi = xiyi pi = xiyi The main part of an adder is the carry network. The rest is just a set of gates to produce the g and p signals and the sum bits.

11. Carry-Lookahead Logic with 4-Bit Block Blocks needed in the design of carry-lookahead adders with four-way grouping of bits.

12. Another Speed-Up Method: Carry Select Allows doubling of adder width with a single-mux additional delay Carry-select addition circuit

13. Arithmetic for Multimedia • Graphics and media processing operates on vectors of 8-bit and 16-bit data • Use 64-bit adder, with partitioned carry chain • Operate on 8×8-bit, 4×16-bit, or 2×32-bit vectors • SIMD (single-instruction, multiple-data) • Saturating operations • On overflow, result is largest representable value • c.f. 2’s-complement modulo arithmetic • E.g., clipping in audio, saturation in video

14. Shift-Add Multiplication Multiplication of 4-bit numbers in dot notation z(j+1) = (z(j) + yjx 2k) 2–1 with z(0) = 0 and z(k) = z |––– add –––| |–– shift right ––|

15. 1000 × 1001 1000 0000 0000 1000 1001000 Multiplication §3.3 Multiplication • Start with long-multiplication approach multiplicand multiplier product Length of product is the sum of operand lengths

16. Multiplication Hardware Initially 0

17. Optimized Multiplier Perform steps in parallel: add/shift One cycle per partial-product addition That’s ok, if frequency of multiplications is low

18. Faster Multiplier Uses multiple adders • Cost/performance tradeoff Can be pipelined • Several multiplications performed in parallel

19. MIPS Multiplication Two 32-bit registers for product • HI: most-significant 32 bits • LO: least-significant 32-bits Instructions • mult rs, rt / multu rs, rt • 64-bit product in HI/LO • mfhi rd / mflo rd • Move from HI/LO to rd • Can test HI value to see if product overflows 32 bits • mul rd, rs, rt • Least-significant 32 bits of product –> rd

20. Division • Check for 0 divisor • Long division approach • If divisor ≤ dividend bits • 1 bit in quotient, subtract • Otherwise • 0 bit in quotient, bring down next dividend bit • Restoring division • Do the subtract, and if remainder goes < 0, add divisor back • Signed division • Divide using absolute values • Adjust sign of quotient and remainder as required quotient dividend 1001 1000 1001010 -1000 10 101 1010 -1000 10 divisor remainder n-bit operands yield n-bitquotient and remainder

21. Division Hardware Initially divisor in left half Initially dividend

22. Optimized Divider One cycle per partial-remainder subtraction • Looks a lot like a multiplier! • Same hardware can be used for both

23. MIPS Division • Use HI/LO registers for result • HI: 32-bit remainder • LO: 32-bit quotient • Instructions • div rs, rt / divu rs, rt • No overflow or divide-by-0 checking • Software must perform checks if required • Use mfhi, mflo to access result

24. Binary and Decimal Multiplication Position 7 6 5 4 3 2 1 0 Position 7 6 5 4 3 2 1 0 ========================= ========================= x24 1 0 1 0 x104 3 5 2 8 y 0 0 1 1 y 4 0 6 7 ========================= ========================= z(0) 0 0 0 0 z(0) 0 0 0 0 +y0x24 1 0 1 0 +y0x104 2 4 6 9 6 –––––––––––––––––––––––––– –––––––––––––––––––––––––– 2z(1) 0 1 0 1 0 10z(1) 2 4 6 9 6 z(1) 0 1 0 1 0 z(1) 0 2 4 6 9 6 +y1x24 1 0 1 0 +y1x104 2 1 1 6 8 –––––––––––––––––––––––––– –––––––––––––––––––––––––– 2z(2) 0 1 1 1 1 0 10z(2) 2 3 6 3 7 6 z(2) 0 1 1 1 1 0 z(2) 2 3 6 3 7 6 +y2x24 0 0 0 0 +y2x104 0 0 0 0 0 –––––––––––––––––––––––––– –––––––––––––––––––––––––– 2z(3) 0 0 1 1 1 1 0 10z(3) 0 2 3 6 3 7 6 z(3) 0 0 1 1 1 1 0 z(3) 0 2 3 6 3 7 6 +y3x24 0 0 0 0 +y3x104 1 4 1 1 2 –––––––––––––––––––––––––– –––––––––––––––––––––––––– 2z(4) 0 0 0 1 1 1 1 0 10z(4) 1 4 3 4 8 3 7 6 z(4) 0 0 0 1 1 1 1 0 z(4) 1 4 3 4 8 3 7 6 ========================= ========================= Step-by-step multiplication examples for 4-digit unsigned numbers.