Ch 3-4
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Lectures for 3rd Edition Note: these lectures are often supplemented with other materials and also problems from the text worked out on the blackboard. You’ll want to customize these lectures for your class. The student audience for these lectures have had exposure to logic design and attend a hands-on assembly language programming lab that does not follow a typical lecture format. 2004 Morgan Kaufmann Publishers 1 Chapter Three 2004 Morgan Kaufmann Publishers 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? 2004 Morgan Kaufmann Publishers 3 Possible Representations • Sign Magnitude: 000 = +0 001 = +1 010 = +2 011 = +3 100 = -0 101 = -1 110 = -2 111 = -3 • • One's Complement Two's Complement 000 = +0 001 = +1 010 = +2 011 = +3 100 = -3 101 = -2 110 = -1 111 = -0 000 = +0 001 = +1 010 = +2 011 = +3 100 = -4 101 = -3 110 = -2 111 = -1 Issues: balance, number of zeros, ease of operations Which one is best? Why? 2004 Morgan Kaufmann Publishers 4 MIPS • 32 bit signed numbers: 0000 0000 0000 ... 0111 0111 1000 1000 1000 ... 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000two = 0ten 0000 0000 0000 0000 0000 0000 0001two = + 1ten 0000 0000 0000 0000 0000 0000 0010two = + 2ten 1111 1111 0000 0000 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 0000 1110two 1111two 0000two 0001two 0010two = = = = = + + – – – 2,147,483,646ten 2,147,483,647ten 2,147,483,648ten 2,147,483,647ten 2,147,483,646ten maxint minint 1111 1111 1111 1111 1111 1111 1101two = – 3ten 1111 1111 1111 1111 1111 1111 1110two = – 2ten 1111 1111 1111 1111 1111 1111 1111two = – 1ten 2004 Morgan Kaufmann Publishers 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) 2004 Morgan Kaufmann Publishers 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” 2004 Morgan Kaufmann Publishers 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 Consider the operations A + B, and A – B – Can overflow occur if B is 0 ? – Can overflow occur if A is 0 ? 2004 Morgan Kaufmann Publishers 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 2004 Morgan Kaufmann Publishers 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 __x_1011 • (multiplicand) (multiplier) Negative numbers: convert and multiply – there are better techniques, we won’t look at them 2004 Morgan Kaufmann Publishers 10 Multiplication: Implementation Start Multiplier0 = 1 1. Test Multiplier0 = 0 Multiplier0 1a. Add multiplicand to product and Multiplicand place the result in Product register Shift left 64 bits Multiplier Shift right 64-bit ALU 2. Shift the Multiplicand register left 1 bit 32 bits Product Write 3. Shift the Multiplier register right 1 bit Control test 64 bits No: < 32 repetitions 32nd repetition? Datapath Yes: 32 repetitions Control Done 2004 Morgan Kaufmann Publishers 11 Final Version Start •Multiplier starts in right half of product Product0 = 1 1. Test Product0 = 0 Product0 Multiplicand 32 bits 32-bit ALU Product Shift right Write Control test 3. Shift the Product register right 1 bit 64 bits No: < 32 repetitions 32nd repetition? What goes here? Yes: 32 repetitions Done 2004 Morgan Kaufmann Publishers 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 2004 Morgan Kaufmann Publishers 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 • Example: – decimal: -.75 = - ( ½ + ¼ ) – binary: -.11 = -1.1 x 2-1 – floating point: exponent = 126 = 01111110 – IEEE single precision: 10111111010000000000000000000000 2004 Morgan Kaufmann Publishers 14 Floating point addition • Sign Exponent Fraction Sign Exponent 1. Compare the exponents of the two numbers. Shift the smaller number to the right until its exponent would match the larger exponent Small ALU Exponent difference 0 Start Fraction 2. Add the significands 1 0 1 0 1 3. Normalize the sum, either shifting right and incrementing the exponent or shifting left and decrementing the exponent Shift right Control Overflow or underflow? Big ALU Yes No 0 0 1 Increment or decrement Exception 1 4. Round the significand to the appropriate number of bits Shift left or right No Rounding hardware Still normalized? Yes Sign Exponent Fraction Done 2004 Morgan Kaufmann Publishers 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! 2004 Morgan Kaufmann Publishers 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!) 2004 Morgan Kaufmann Publishers 17 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) operation a 32 ALU result 32 b 32 2004 Morgan Kaufmann Publishers 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. 2004 Morgan Kaufmann Publishers 19 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 operation a op a b res result b • Possible Implementation (sum-of-products): 2004 Morgan Kaufmann Publishers 20 Review: The Multiplexor • Selects one of the inputs to be the output, based on a control input S • A 0 B 1 C note: we call this a 2-input mux even though it has 3 inputs! Lets build our ALU using a MUX: 2004 Morgan Kaufmann Publishers 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: CarryIn a Sum b cout = a b + a cin + b cin sum = a xor b xor cin CarryOut • How could we build a 1-bit ALU for add, and, and or? • How could we build a 32-bit ALU? 2004 Morgan Kaufmann Publishers 22 Building a 32 bit ALU CarryIn a0 b0 Operation CarryIn ALU0 Result0 CarryOut Operation CarryIn a1 a 0 b1 CarryIn ALU1 Result1 CarryOut 1 Result a2 2 b b2 CarryIn ALU2 Result2 CarryOut CarryOut a31 b31 CarryIn ALU31 Result31 2004 Morgan Kaufmann Publishers 23 What about subtraction (a – b) ? • • Two's complement approach: just negate b and add. How do we negate? • A very clever solution: Binvert Operation CarryIn a 0 1 b 0 Result 2 1 CarryOut 2004 Morgan Kaufmann Publishers 24 Adding a NOR function • Can also choose to invert a. How do we get “a NOR b” ? Ainvert Operation Binvert a CarryIn 0 0 1 1 b 0 + Result 2 1 CarryOut 2004 Morgan Kaufmann Publishers 25 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 2004 Morgan Kaufmann Publishers 26 Supporting slt • Can we figure out the idea? Binvert a Binvert CarryIn a 0 Operation Ainvert Operation Ainvert CarryIn 0 0 0 1 1 1 1 Result b 0 + Result b 0 2 + 2 1 1 Less Less 3 3 Set CarryOut Overflow detection Overflow Use this ALU for most significant bit all other bits 2004 Morgan Kaufmann Publishers 27 Supporting slt Operation Binvert Ainvert CarryIn a0 b0 CarryIn ALU0 Less CarryOut Result0 a1 b1 0 CarryIn ALU1 Less CarryOut Result1 a2 b2 0 CarryIn ALU2 Less CarryOut Result2 .. . a31 b31 0 .. . CarryIn CarryIn ALU31 Less .. . Result31 Set Overflow 2004 Morgan Kaufmann Publishers 28 Test for equality • Notice control lines: Operation Bnegate Ainvert 0000 0001 0010 0110 0111 1100 = = = = = = and or add subtract slt NOR •Note: zero is a 1 when the result is zero! a0 b0 CarryIn ALU0 Less CarryOut a1 b1 0 CarryIn ALU1 Less CarryOut a2 b2 0 CarryIn ALU2 Less CarryOut .. . a31 b31 0 Result0 Result1 .. . Result2 .. . CarryIn CarryIn ALU31 Less Zero .. . .. . Result31 Set Overflow 2004 Morgan Kaufmann Publishers 29 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 2004 Morgan Kaufmann Publishers 30 Problem: ripple carry adder is slow • • Is a 32-bit ALU as fast as a 1-bit ALU? Is there more than one way to do addition? – two extremes: ripple carry and sum-of-products Can you see the ripple? How could you get rid of it? c1 c2 c3 c4 = = = = b0c0 b1c1 b2c2 b3c3 + + + + a0c0 a1c1 a2c2 a3c3 + + + + a0b0 a1b1c2 = a2b2 a3b3 c3 = c4 = Not feasible! Why? 2004 Morgan Kaufmann Publishers 31 Carry-lookahead adder • • An approach in-between our two extremes Motivation: – If we didn't know the value of carry-in, what could we do? – When would we always generate a carry? gi = ai bi – When would we propagate the carry? pi = ai + bi • Did we get rid of the ripple? c1 c2 c3 c4 = = = = g0 g1 g2 g3 + + + + p0c0 p1c1 c2 = p2c2 c3 = p3c3 c4 = Feasible! Why? 2004 Morgan Kaufmann Publishers 32 Use principle to build bigger adders CarryIn a0 b0 a1 b1 a2 b2 a3 b3 a4 b4 a5 b5 a6 b6 a7 b7 a8 b8 a9 b9 a10 b10 a11 b11 a12 b12 a13 b13 a14 b14 a15 b15 CarryIn Result0–3 ALU0 P0 G0 pi gi C1 ci + 1 CarryIn Carry-lookahead unit Result4–7 ALU1 P1 G1 • • • pi + 1 gi + 1 C2 ci + 2 CarryIn Can’t build a 16 bit adder this way... (too big) Could use ripple carry of 4-bit CLA adders Better: use the CLA principle again! Result8–11 ALU2 P2 G2 pi + 2 gi + 2 C3 ci + 3 CarryIn Result12–15 ALU3 P3 G3 pi + 3 gi + 3 C4 ci + 4 CarryOut 2004 Morgan Kaufmann Publishers 33 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! 2004 Morgan Kaufmann Publishers 34 Chapter 4 2004 Morgan Kaufmann Publishers 35 Performance • • • • Measure, Report, and Summarize Make intelligent choices See through the marketing hype Key to understanding underlying organizational motivation Why is some hardware better than others for different programs? What factors of system performance are hardware related? (e.g., Do we need a new machine, or a new operating system?) How does the machine's instruction set affect performance? 2004 Morgan Kaufmann Publishers 36 Which of these airplanes has the best performance? Airplane Passengers Boeing 737-100 Boeing 747 BAC/Sud Concorde Douglas DC-8-50 101 470 132 146 Range (mi) Speed (mph) 630 4150 4000 8720 598 610 1350 544 •How much faster is the Concorde compared to the 747? •How much bigger is the 747 than the Douglas DC-8? 2004 Morgan Kaufmann Publishers 37 Computer Performance: TIME, TIME, TIME • Response Time (latency) — How long does it take for my job to run? — How long does it take to execute a job? — How long must I wait for the database query? • Throughput — How many jobs can the machine run at once? — What is the average execution rate? — How much work is getting done? • If we upgrade a machine with a new processor what do we increase? • If we add a new machine to the lab what do we increase? 2004 Morgan Kaufmann Publishers 38 Execution Time • • • Elapsed Time – counts everything (disk and memory accesses, I/O , etc.) – a useful number, but often not good for comparison purposes CPU time – doesn't count I/O or time spent running other programs – can be broken up into system time, and user time Our focus: user CPU time – time spent executing the lines of code that are "in" our program 2004 Morgan Kaufmann Publishers 39 Book's Definition of Performance • For some program running on machine X, PerformanceX = 1 / Execution timeX • "X is n times faster than Y" PerformanceX / PerformanceY = n • Problem: – machine A runs a program in 20 seconds – machine B runs the same program in 25 seconds 2004 Morgan Kaufmann Publishers 40 Clock Cycles • Instead of reporting execution time in seconds, we often use cycles seconds cycles seconds program program cycle • Clock “ticks” indicate when to start activities (one abstraction): time • • cycle time = time between ticks = seconds per cycle clock rate (frequency) = cycles per second (1 Hz. = 1 cycle/sec) A 4 Ghz. clock has a 1 4 109 1012 250 picosecond s (ps) cycle time 2004 Morgan Kaufmann Publishers 41 How to Improve Performance seconds cycles seconds program program cycle So, to improve performance (everything else being equal) you can either (increase or decrease?) ________ the # of required cycles for a program, or ________ the clock cycle time or, said another way, ________ the clock rate. 2004 Morgan Kaufmann Publishers 42 How many cycles are required for a program? ... 6th 5th 4th 3rd instruction 2nd instruction Could assume that number of cycles equals number of instructions 1st instruction • time This assumption is incorrect, different instructions take different amounts of time on different machines. Why? hint: remember that these are machine instructions, not lines of C code 2004 Morgan Kaufmann Publishers 43 Different numbers of cycles for different instructions time • Multiplication takes more time than addition • Floating point operations take longer than integer ones • Accessing memory takes more time than accessing registers • Important point: changing the cycle time often changes the number of cycles required for various instructions (more later) 2004 Morgan Kaufmann Publishers 44 Example • Our favorite program runs in 10 seconds on computer A, which has a 4 GHz. clock. We are trying to help a computer designer build a new machine B, that will run this program in 6 seconds. The designer can use new (or perhaps more expensive) technology to substantially increase the clock rate, but has informed us that this increase will affect the rest of the CPU design, causing machine B to require 1.2 times as many clock cycles as machine A for the same program. What clock rate should we tell the designer to target?" • Don't Panic, can easily work this out from basic principles 2004 Morgan Kaufmann Publishers 45 Now that we understand cycles • A given program will require – some number of instructions (machine instructions) – some number of cycles – some number of seconds • We have a vocabulary that relates these quantities: – cycle time (seconds per cycle) – clock rate (cycles per second) – CPI (cycles per instruction) a floating point intensive application might have a higher CPI – MIPS (millions of instructions per second) this would be higher for a program using simple instructions 2004 Morgan Kaufmann Publishers 46 Performance • • Performance is determined by execution time Do any of the other variables equal performance? – # of cycles to execute program? – # of instructions in program? – # of cycles per second? – average # of cycles per instruction? – average # of instructions per second? • Common pitfall: thinking one of the variables is indicative of performance when it really isn’t. 2004 Morgan Kaufmann Publishers 47 CPI Example • Suppose we have two implementations of the same instruction set architecture (ISA). For some program, Machine A has a clock cycle time of 250 ps and a CPI of 2.0 Machine B has a clock cycle time of 500 ps and a CPI of 1.2 What machine is faster for this program, and by how much? • If two machines have the same ISA which of our quantities (e.g., clock rate, CPI, execution time, # of instructions, MIPS) will always be identical? 2004 Morgan Kaufmann Publishers 48 # of Instructions Example • A compiler designer is trying to decide between two code sequences for a particular machine. Based on the hardware implementation, there are three different classes of instructions: Class A, Class B, and Class C, and they require one, two, and three cycles (respectively). The first code sequence has 5 instructions: 2 of A, 1 of B, and 2 of C The second sequence has 6 instructions: 4 of A, 1 of B, and 1 of C. Which sequence will be faster? How much? What is the CPI for each sequence? 2004 Morgan Kaufmann Publishers 49 MIPS example • Two different compilers are being tested for a 4 GHz. machine with three different classes of instructions: Class A, Class B, and Class C, which require one, two, and three cycles (respectively). Both compilers are used to produce code for a large piece of software. The first compiler's code uses 5 million Class A instructions, 1 million Class B instructions, and 1 million Class C instructions. The second compiler's code uses 10 million Class A instructions, 1 million Class B instructions, and 1 million Class C instructions. • • Which sequence will be faster according to MIPS? Which sequence will be faster according to execution time? 2004 Morgan Kaufmann Publishers 50 Benchmarks • • • Performance best determined by running a real application – Use programs typical of expected workload – Or, typical of expected class of applications e.g., compilers/editors, scientific applications, graphics, etc. Small benchmarks – nice for architects and designers – easy to standardize – can be abused SPEC (System Performance Evaluation Cooperative) – companies have agreed on a set of real program and inputs – valuable indicator of performance (and compiler technology) – can still be abused 2004 Morgan Kaufmann Publishers 51 Benchmark Games • An embarrassed Intel Corp. acknowledged Friday that a bug in a software program known as a compiler had led the company to overstate the speed of its microprocessor chips on an industry benchmark by 10 percent. However, industry analysts said the coding error…was a sad commentary on a common industry practice of “cheating” on standardized performance tests…The error was pointed out to Intel two days ago by a competitor, Motorola …came in a test known as SPECint92…Intel acknowledged that it had “optimized” its compiler to improve its test scores. The company had also said that it did not like the practice but felt to compelled to make the optimizations because its competitors were doing the same thing…At the heart of Intel’s problem is the practice of “tuning” compiler programs to recognize certain computing problems in the test and then substituting special handwritten pieces of code… Saturday, January 6, 1996 New York Times 2004 Morgan Kaufmann Publishers 52 SPEC ‘89 Compiler “enhancements” and performance 800 700 600 SPEC performance ratio • 500 400 300 200 100 0 gcc espresso spice doduc nasa7 li eqntott matrix300 fpppp tomcatv Benchmark Compiler Enhanced compiler 2004 Morgan Kaufmann Publishers 53 SPEC CPU2000 2004 Morgan Kaufmann Publishers 54 SPEC 2000 Does doubling the clock rate double the performance? Can a machine with a slower clock rate have better performance? 1.6 Pentium M @ 1.6/0.6 GHz Pentium 4-M @ 2.4/1.2 GHz Pentium III-M @ 1.2/0.8 GHz 1400 1.4 1200 1.2 Pentium 4 CFP2000 1000 Pentium 4 CINT2000 1.0 800 0.8 600 0.6 Pentium III CINT2000 400 0.4 Pentium III CFP2000 200 0.2 0 0.0 500 1000 1500 2000 Clock rate in MHz 2500 3000 3500 SPECINT2000 SPECFP2000 SPECINT2000 SPECFP2000 SPECINT2000 SPECFP2000 Always on/maximum clock Laptop mode/adaptive clock Minimum power/minimum clock Benchmark and power mode 2004 Morgan Kaufmann Publishers 55 Experiment • Phone a major computer retailer and tell them you are having trouble deciding between two different computers, specifically you are confused about the processors strengths and weaknesses (e.g., Pentium 4 at 2Ghz vs. Celeron M at 1.4 Ghz ) • What kind of response are you likely to get? • What kind of response could you give a friend with the same question? 2004 Morgan Kaufmann Publishers 56 Amdahl's Law Execution Time After Improvement = Execution Time Unaffected +( Execution Time Affected / Amount of Improvement ) • Example: "Suppose a program runs in 100 seconds on a machine, with multiply responsible for 80 seconds of this time. How much do we have to improve the speed of multiplication if we want the program to run 4 times faster?" How about making it 5 times faster? • Principle: Make the common case fast 2004 Morgan Kaufmann Publishers 57 Example • Suppose we enhance a machine making all floating-point instructions run five times faster. If the execution time of some benchmark before the floating-point enhancement is 10 seconds, what will the speedup be if half of the 10 seconds is spent executing floating-point instructions? • We are looking for a benchmark to show off the new floating-point unit described above, and want the overall benchmark to show a speedup of 3. One benchmark we are considering runs for 100 seconds with the old floating-point hardware. How much of the execution time would floatingpoint instructions have to account for in this program in order to yield our desired speedup on this benchmark? 2004 Morgan Kaufmann Publishers 58 Remember • Performance is specific to a particular program/s – Total execution time is a consistent summary of performance • For a given architecture performance increases come from: – – – – • increases in clock rate (without adverse CPI affects) improvements in processor organization that lower CPI compiler enhancements that lower CPI and/or instruction count Algorithm/Language choices that affect instruction count Pitfall: expecting improvement in one aspect of a machine’s performance to affect the total performance 2004 Morgan Kaufmann Publishers 59
