Understanding Binary Numbers and Negative Number Representations

1. Chapter Three

2. Numbers • Bits are just bits (no inherent meaning) — conventions define relationship between bits and numbers • Binary numbers (base 2) 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001... decimal: 0...2n-1 • Of course it gets more complicated: numbers are finite (overflow) fractions and real numbers negative numbers e.g., no MIPS subi instruction; addi can add a negative number • How do we represent negative numbers? i.e., which bit patterns will represent which numbers?

3. Possible Representations • Sign Magnitude: One's Complement Two's Complement 000 = +0 000 = +0 000 = +0 001 = +1 001 = +1 001 = +1 010 = +2 010 = +2 010 = +2 011 = +3 011 = +3 011 = +3 100 = -0 100 = -3 100 = -4 101 = -1 101 = -2 101 = -3 110 = -2 110 = -1 110 = -2 111 = -3 111 = -0 111 = -1 • Issues: balance, number of zeros, ease of operations • Which one is best? Why?

4. maxint minint MIPS • 32 bit signed numbers:0000 0000 0000 0000 0000 0000 0000 0000two = 0ten0000 0000 0000 0000 0000 0000 0000 0001two = + 1ten0000 0000 0000 0000 0000 0000 0000 0010two = + 2ten...0111 1111 1111 1111 1111 1111 1111 1110two = + 2,147,483,646ten0111 1111 1111 1111 1111 1111 1111 1111two = + 2,147,483,647ten1000 0000 0000 0000 0000 0000 0000 0000two = – 2,147,483,648ten1000 0000 0000 0000 0000 0000 0000 0001two = – 2,147,483,647ten1000 0000 0000 0000 0000 0000 0000 0010two = – 2,147,483,646ten...1111 1111 1111 1111 1111 1111 1111 1101two = – 3ten1111 1111 1111 1111 1111 1111 1111 1110two = – 2ten1111 1111 1111 1111 1111 1111 1111 1111two = – 1ten

5. Two's Complement Operations • Negating a two's complement number: invert all bits and add 1 • remember: “negate” and “invert” are quite different! • Converting n bit numbers into numbers with more than n bits: • MIPS 16 bit immediate gets converted to 32 bits for arithmetic • copy the most significant bit (the sign bit) into the other bits 0010 -> 0000 0010 1010 -> 1111 1010 • "sign extension" (lbu vs. lb)

6. Addition & Subtraction • Just like in grade school (carry/borrow 1s) 0111 0111 0110+ 0110 - 0110 - 0101 • Two's complement operations easy • subtraction using addition of negative numbers 0111 + 1010 • Overflow (result too large for finite computer word): • e.g., adding two n-bit numbers does not yield an n-bit number 0111 + 0001 note that overflow term is somewhat misleading, 1000 it does not mean a carry “overflowed”

7. Detecting Overflow • No overflow when adding a positive and a negative number • No overflow when signs are the same for subtraction • Overflow occurs when the value affects the sign: • overflow when adding two positives yields a negative • or, adding two negatives gives a positive • or, subtract a negative from a positive and get a negative • or, subtract a positive from a negative and get a positive

8. Effects of Overflow • An exception (interrupt) occurs • Control jumps to predefined address for exception • Interrupted address is saved for possible resumption • Details based on software system / language • example: flight control vs. homework assignment • Don't always want to detect overflow — new MIPS instructions: addu, addiu, subu note: addiu still sign-extends! note: sltu, sltiu for unsigned comparisons

9. Multiplication • More complicated than addition • accomplished via shifting and addition • More time and more area • Let's look at 3 versions based on a gradeschool algorithm 0010 (multiplicand) __x_1011 (multiplier) • Negative numbers: convert and multiply • there are better techniques, we won’t look at them

10. Multiplication: Implementation Datapath Control

11. Final Version • Multiplier starts in right half of product What goes here?

12. Floating Point (a brief look) • We need a way to represent • numbers with fractions, e.g., 3.1416 • very small numbers, e.g., .000000001 • very large numbers, e.g., 3.15576 ´ 109 • Representation: • sign, exponent, significand: (–1)sign´ significand ´ 2exponent • more bits for significand gives more accuracy • more bits for exponent increases range • IEEE 754 floating point standard: • single precision: 8 bit exponent, 23 bit significand • double precision: 11 bit exponent, 52 bit significand

13. IEEE 754 floating-point standard • Leading “1” bit of significand is implicit • Exponent is “biased” to make sorting easier • all 0s is smallest exponent all 1s is largest • bias of 127 for single precision and 1023 for double precision • summary: (–1)sign´ (1+significand) ´ 2exponent – bias

14. Floating point addition

15. Floating Point Complexities • Operations are somewhat more complicated (see text) • In addition to overflow we can have “underflow” • Accuracy can be a big problem • IEEE 754 keeps two extra bits, guard and round • four rounding modes • positive divided by zero yields “infinity” • zero divide by zero yields “not a number” • other complexities • Implementing the standard can be tricky • Not using the standard can be even worse • see text for description of 80x86 and Pentium bug!

16. Chapter Three Summary • Computer arithmetic is constrained by limited precision • Bit patterns have no inherent meaning but standards do exist • two’s complement • IEEE 754 floating point • Computer instructions determine “meaning” of the bit patterns • Performance and accuracy are important so there are many complexities in real machines • Algorithm choice is important and may lead to hardware optimizations for both space and time (e.g., multiplication) • You may want to look back (Section 3.10 is great reading!)

17. operation a ALU 32 result 32 b 32 Lets Build a Processor • Almost ready to move into chapter 5 and start building a processor • First, let’s review Boolean Logic and build the ALU we’ll need (Material from Appendix B)

18. Review: Boolean Algebra & Gates • Problem: Consider a logic function with three inputs: A, B, and C. Output D is true if at least one input is true Output E is true if exactly two inputs are true Output F is true only if all three inputs are true • Show the truth table for these three functions. • Show the Boolean equations for these three functions. • Show an implementation consisting of inverters, AND, and OR gates.

19. operation op a b res result An ALU (arithmetic logic unit) • Let's build an ALU to support the andi and ori instructions • we'll just build a 1 bit ALU, and use 32 of them • Possible Implementation (sum-of-products): a b

20. S A C B Review: The Multiplexor • Selects one of the inputs to be the output, based on a control input • Lets build our ALU using a MUX: note: we call this a 2-input mux even though it has 3 inputs! 0 1

21. Different Implementations • Not easy to decide the “best” way to build something • Don't want too many inputs to a single gate • Don’t want to have to go through too many gates • for our purposes, ease of comprehension is important • Let's look at a 1-bit ALU for addition: • How could we build a 1-bit ALU for add, and, and or? • How could we build a 32-bit ALU? cout = a b + a cin + b cin sum = a xor b xor cin

22. Lecture 5

23. Building a 32 bit ALU

24. What about subtraction (a – b) ? • Two's complement approach: just negate b and add. • How do we negate? • A very clever solution:

25. Adding a NOR function • Can also choose to invert a. How do we get “a NOR b” ?

26. Tailoring the ALU to the MIPS • Need to support the set-on-less-than instruction (slt) • remember: slt is an arithmetic instruction • produces a 1 if rs < rt and 0 otherwise • use subtraction: (a-b) < 0 implies a < b • Need to support test for equality (beq $t5, $t6, $t7) • use subtraction: (a-b) = 0 implies a = b

27. Supporting slt • Can we figure out the idea? all other bits Use this ALU for most significant bit

28. Supporting slt

29. Test for equality • Notice control lines:0000 = and0001 = or0010 = add0110 = subtract0111 = slt1100 = NOR • Note: zero is a 1 when the result is zero!

30. Conclusion • We can build an ALU to support the MIPS instruction set • key idea: use multiplexor to select the output we want • we can efficiently perform subtraction using two’s complement • we can replicate a 1-bit ALU to produce a 32-bit ALU • Important points about hardware • all of the gates are always working • the speed of a gate is affected by the number of inputs to the gate • the speed of a circuit is affected by the number of gates in series (on the “critical path” or the “deepest level of logic”) • Our primary focus: comprehension, however, • Clever changes to organization can improve performance (similar to using better algorithms in software) • We saw this in multiplication, let’s look at addition now

31. ALU Summary • We can build an ALU to support MIPS addition • Our focus is on comprehension, not performance • Real processors use more sophisticated techniques for arithmetic • Where performance is not critical, hardware description languages allow designers to completely automate the creation of hardware!

32. DONE

33. Chapter Five

34. The Processor: Datapath & Control • We're ready to look at an implementation of the MIPS • Simplified to contain only: • memory-reference instructions: lw, sw • arithmetic-logical instructions: add, sub, and, or, slt • control flow instructions: beq, j • Generic Implementation: • use the program counter (PC) to supply instruction address • get the instruction from memory • read registers • use the instruction to decide exactly what to do • All instructions use the ALU after reading the registers Why? memory-reference? arithmetic? control flow?

35. More Implementation Details • Abstract / Simplified View:Two types of functional units: • elements that operate on data values (combinational) • elements that contain state (sequential)

36. Five Execution Steps • Instruction Fetch • Instruction Decode and Register Fetch • Execution, Memory Address Computation, or Branch Completion • Memory Access or R-type instruction completion • Write-back step INSTRUCTIONS TAKE FROM 3 - 5 CYCLES!

37. State Elements • Unclocked vs. Clocked • Clocks used in synchronous logic • when should an element that contains state be updated? cycle time

38. An unclocked state element • The set-reset latch • output depends on present inputs and also on past inputs

39. Latches and Flip-flops • Output is equal to the stored value inside the element (don't need to ask for permission to look at the value) • Change of state (value) is based on the clock • Latches: whenever the inputs change, and the clock is asserted • Flip-flop: state changes only on a clock edge (edge-triggered methodology) "logically true", — could mean electrically low A clocking methodology defines when signals can be read and written — wouldn't want to read a signal at the same time it was being written

40. D-latch • Two inputs: • the data value to be stored (D) • the clock signal (C) indicating when to read & store D • Two outputs: • the value of the internal state (Q) and it's complement

41. D flip-flop • Output changes only on the clock edge

42. Our Implementation • An edge triggered methodology • Typical execution: • read contents of some state elements, • send values through some combinational logic • write results to one or more state elements

43. Register File • Built using D flip-flops Do you understand? What is the “Mux” above?

44. Abstraction • Make sure you understand the abstractions! • Sometimes it is easy to think you do, when you don’t

45. Register File • Note: we still use the real clock to determine when to write

46. Simple Implementation • Include the functional units we need for each instruction Why do we need this stuff?

47. P C S r c M A d d u x A L U A d d 4 r e s u l t S h i f t l e f t 2 R e a d A L U S r c A L U o p e r a t i o n R e a d 4 r e g i s t e r 1 P C R e a d a d d r e s s M e m W r i t e d a t a 1 R e a d M e m t o R e g Z e r o r e g i s t e r 2 I n s t r u c t i o n A L U R e g i s t e r s R e a d A L U R e a d A d d r e s s W r i t e d a t a r e s u l t d a t a 2 M I n s t r u c t i o n M r e g i s t e r u u m e m o r y x x W r i t e d a t a D a t a W r i t e m e m o r y R e g W r i t e d a t a M e m R e a d 1 6 3 2 S i g n e x t e n d Building the Datapath • Use multiplexors to stitch them together

48. DONE

49. Control • Selecting the operations to perform (ALU, read/write, etc.) • Controlling the flow of data (multiplexor inputs) • Information comes from the 32 bits of the instruction • Example: add $8, $17, $18 Instruction Format: 000000 10001 10010 01000 00000 100000 op rs rt rd shamt funct • ALU's operation based on instruction type and function code

50. Control • e.g., what should the ALU do with this instruction • Example: lw $1, 100($2) 35 2 1 100 op rs rt 16 bit offset • ALU control input0000 AND 0001 OR 0010 add 0110 subtract 0111 set-on-less-than 1100 NOR • Why is the code for subtract 0110 and not 0011?