Computer Arithmetic Essentials: Integer Operations Overview

1. CprE 381 Computer Organization and Assembly Level Programming, Fall 2013 Chapter 3 Arithmetic for Computers Zhao Zhang Iowa State University Revised from original slides provided by MKP

2. Announcements • Midterm Exam 1 on Friday Oct. 3, 9:00-9:50am • 12.5% weight of the overall grade • No quizzes homework this week • Midterm review 1 on Wednesday §2.8 Supporting Procedures in Computer Hardware Chapter 2 — Instructions: Language of the Computer — 2

3. Review of Week 5 • Bubble sort code • FP instructions • FP pat of the call convention • ARM and x86 ISAs Chapter 1 — Computer Abstractions and Technology — 3

4. Exam 1 • Open book, open notes, calculator are allowed • E-book reader is allowed • Must be put in airplane mode • Coverage • Chapter 1, Computer Abstraction and Technology • Chapter 2, Instructions: Language of the Computer • Some contents from Appendix B • MIPS floating-point instructions Chapter 1 — Computer Abstractions and Technology — 4

5. Exam Question Types • Short conceptual questions • Calculation: speedup, power saving, CPI, etc. • MIPS assembly programming • Translate C statements to MIPS (arithmetic, load/store, branch and jump, others) • Translate C functions to MIPS (call convention) • Among others Suggestions: • Review slidesand textbook • Review homework and quizzes • Redo homework questions Chapter 1 — Computer Abstractions and Technology — 5

6. 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 Chapter 3 — Arithmetic for Computers — 6

7. Integer Addition • 1-bit adder • Inputs: A, B, C (carry input) • Output Sum, Cout (carry output) • Try drawing the truth table • Performance: 3 gate delays • 3 gate delays on Sum (three layers of gates) • 2 gate delays on Cout Chapter 1 — Computer Abstractions and Technology — 7

8. Integer Addition • N-bit Adder example: 7 + 6 • This is called ripple-carry adder • 32-bit adder has 31×2+3 = 65 gate delays • 31×2 for carry ripping, plus 3 for the last Sum • Faster design: Carry look-ahead adder (CLA) §3.2 Addition and Subtraction Chapter 3 — Arithmetic for Computers — 8

9. Addition Overflow • 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 Chapter 1 — Computer Abstractions and Technology — 9

10. Integer Subtraction • Add negation of second operand • Example: 7 – 6 = 7 + (–6) +7: 0000 0000 … 0000 0111 (+6: 0000 0000 … 0000 0110)–6: 1111 1111 … 1111 1010+1: 0000 0000 … 0000 0001 • An adder can do subtraction, with simple circuit extension • Invert every bit of the 2nd operand • Set the carry input bit to be 1 Chapter 3 — Arithmetic for Computers — 10

11. Overflow in Subtraction • 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 Chapter 1 — Computer Abstractions and Technology — 11

12. 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 Chapter 3 — Arithmetic for Computers — 12

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 • Example: 200 + 300 = 255 for 8-bit addition • c.f. 2s-complement modulo arithmetic • E.g., clipping in audio, saturation in video Chapter 3 — Arithmetic for Computers — 13

14. 1000 × 1001 00001000 00000000 00000000 01000000 01001000 Multiplication §3.3 Multiplication • Start with long-multiplication approach multiplicand multiplier product Length of product is the sum of operand lengths Chapter 3 — Arithmetic for Computers — 14

15. Control-Test Unit Design • A state machine of N states • Each state represents a step • State 0 => 1 => 2 => … => N-1 • Input • m(0): Bit 0 of multiplier • Outputs • Shift-right control to the multiplier • Shift-left control to the multiplicand • ALU operation type (select addition) • Product register write-enable: Enable if m(0) = ‘1’, disable if m(0) = ‘0” Chapter 3 — Arithmetic for Computers — 15

16. Optimized Multiplier • Perform steps in parallel: add/shift • Utilized the unused bits in product to save multiplier • Same performance: N additions per multiplication Chapter 3 — Arithmetic for Computers — 16

17. Faster Multiplier • Uses multiple adders • Cost/performance tradeoff • Use 32-bit adders instead of 64-bit ones • Can be pipelined: Several multiplication performed in parallel Chapter 3 — Arithmetic for Computers — 17

18. 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 Chapter 3 — Arithmetic for Computers — 18