CS 152 Lec 5.1 CS 152: Computer Architecture and Engineering Lecture 5 Hardware Description Languages, Multiple and Divide Randy H. Katz, Instructor Satrajit Chatterjee, Teaching Assistant George Porter, Teaching Assistant

CS 152 Lec 5.2 Hardware Representation Languages : Block Diagrams: FUs, Registers, & Dataflows Register Transfer Diagrams: Choice of busses to connect FUs, Regs Flowcharts State Diagrams Fifth Representation “Language”: Hardware Description Languages E.G., ISP' VHDL Verilog Descriptions in these languages can be used as input to simulation systems synthesis systems Representation Languages Two different ways to describe sequencing & microoperations hw modules described like programs with i/o ports, internal state, & parallel execution of assignment statements “software breadboard” generate hw from high level description "To Design is to Represent"

CS 152 Lec 5.3 Simulation Before Construction  "Physical Breadboarding” Discrete components/lower scale integration preceeds actual construction of prototype Verify initial design concept  Simulation Before Construction High level constructs implies faster to construct Play "what if" more easily Limited performance accuracy, however

CS 152 Lec 5.4 Levels of Description Architectural Simulation Functional/Behavioral Register Transfer Logic Circuit Models programmer's view at a high level; written in your favorite programming language More detailed model, like the block diagram view Commitment to datapath FUs, registers, busses; register xfer operations are clock phase accurate Model is in terms of logic gates; higher level MSI functions described in terms of these Electrical behavior; accurate waveforms Schematic capture + logic simulation package Special languages + simulation systems for describing the inherent parallel activity in hardware Less Abstract More Accurate Slower Simulation

CS 152 Lec 5.5 Verilog and VHDL (Hardware Description Languages)  Goals: Support design, documentation, and simulation of hardware -Verilog is C like, VHDL is ADA/Pascal like Digital system level to gate level “Technology Insertion”  Concepts: Verilog Module/VHDL Design Entity Time-based execution model. Module/Design Entity == Logical entity implemented by a piece of hardware e.g., logic gate, 32-bit adder, memory subsystem Interface == External Characteristics Architecture (Body ) == Internal Behavior or Structure

CS 152 Lec 5.6 Very Simple Verilog Model Example  Continuous assignment, combinational logic, behavioral description  Variables can be of type reg, wire, integer, real, event, time module add2bit (in1, in2, sum); input in1, in2; output[1:0] sum; wire in1, in2; reg[1:0] sum; or in2) begin sum = in1 + in2; $display (“The sum of %b and %b is %0d (time = %0d)”, in1, in2, sum, $time); end endmodule Interface: two inputs, one output Inputs are “wires”/interconnections, output is 2-bit “register”/mem elements Inputs are monitored, compute sum whenever they change print sum and simulation time

CS 152 Lec 5.7 Structural Descriptions  Instances of “wired” lower level module within the current module … fulladd f1 (cin0, a0, b0, sum0, cout0) fulladd f2 (cout0, a1, b1, sum1, cout2) … f1 a0 b0 cin0 sum0 cout0 f2 a1 b1 sum1 cout2

CS 152 Lec 5.8 Verilog Operators  Operators borrowed from C programming language + - * / >= < <= ! && || == != ?: {} % === !== ~ & | > Arithmetic Relational Logical Logical equality Conditional Concatenate Modulus Case equality Bit-wise Shift

CS 152 Lec 5.9 Control Constructs IF-THEN-ELSE if (condition) block; else block; FOR for (i = 0; i < 4; i = i + 1) block; WHILE begin i = 0; while (i < 4) begin statement; i = i + 1; end CASE case (i) 0: statement; 1: statement; 2: statement; default: $stop; endcase Also casex, casez, repeat, forever

CS 152 Lec 5.10 Clocking and Delays  Unlike conventional programs, not a serial model of execution  Global variable designates simulation time Multiple events scheduled for execution at the same time Execute all events for the current simulation time (in essentially a random order) before advancing to the next simulation time New events generated as a by-product of execution and are added to the simulation pending queue module repeat_loop (clock); input clock; initial begin repeat clock); $stop; end endmodule Initial: execute at start of sim Wait for 5 positive clock edges Stop simulation

CS 152 Lec 5.11 Clocking and Delays Synchronous: #expression – suspend execution for indicated time – suspend execution until event occurs Level: wait (expression) – suspend execution until expression true module time_control; reg[1:0] r; initial #70 $stop; initial begin : b1 //named block b1 #10 r = 1; //wait for 10 time units #20 r = 1; //wait for 20 time units #30 r = 1; //wait for 30 time units end initial begin : b2 //named block b2 #5 r = 2; //wait for 5 time units #20 r = 2; //wait for 20 time units #30 r = 2; //wait for 30 time units end begin $display (“r= %0d at time %0d”, r, $time); end endmodule

CS 152 Lec 5.12 Clocking and Delays Synchronous: #expression – suspend execution for indicated time – suspend execution until event occurs Level: wait (expression) – suspend execution until expression true module event_control; event e1, e2; begin $display (“I am in the middle.”); -> e2; end $display (“I am supposed to execute last.”); initial begin $display (“I am the first.”); -> e1; end endmodule

CS 152 Lec 5.13 MIPS Arithmetic Instructions InstructionExampleMeaningComments add add $1,$2,$3$1 = $2 + $33 operands; exception possible subtractsub $1,$2,$3$1 = $2 – $33 operands; exception possible add immediateaddi $1,$2,100$1 = $ constant; exception possible add unsignedaddu $1,$2,$3$1 = $2 + $33 operands; no exceptions subtract unsignedsubu $1,$2,$3$1 = $2 – $33 operands; no exceptions add imm. unsign.addiu $1,$2,100 $1 = $ constant; no exceptions multiply mult $2,$3Hi, Lo = $2 x $364-bit signed product multiply unsignedmultu$2,$3Hi, Lo = $2 x $3 64-bit unsigned product divide div $2,$3Lo = $2 ÷ $3,Lo = quotient, Hi = remainder Hi = $2 mod $3 divide unsigned divu $2,$3Lo = $2 ÷ $3,Unsigned quotient & remainder Hi = $2 mod $3 Move from Himfhi $1$1 = HiUsed to get copy of Hi Move from Lomflo $1$1 = LoUsed to get copy of Lo

CS 152 Lec 5.14 MULTIPLY (unsigned)  Paper and pencil example (unsigned): Multiplicand 1000 Multiplier Product  m bits x n bits = m+n bit product  Binary makes it easy: 0 => place 0 ( 0 x multiplicand) 1 => place a copy ( 1 x multiplicand)  Four versions of multiply hardware & algorithm: Successive refinement

CS 152 Lec 5.15 Unsigned Combinational Multiplier  Stage i accumulates A * 2 i if B i == 1  Q: How much hardware for 32 bit multiplier? Critical path? B0B0 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 0000

CS 152 Lec 5.16 Carry Save Addition of Four Integers  Add Columns first, then rows!  Can be used to reduce critical path of multiply  Example: 53 bit multiply (for floating point): At least 53 levels with naïve technique Only 9 with Carry save addition!

CS 152 Lec 5.17 How Does It Work?  At each stage shift A left ( x 2)  Use next bit of B to determine whether to add in shifted multiplicand  Accumulate 2n bit partial product at each stage B0B0 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P

CS 152 Lec 5.18 Unsigned Shift-Add Multiplier (Version 1)  64-bit Multiplicand reg, 64-bit ALU, 64-bit Product reg, 32-bit multiplier reg Product Multiplier Multiplicand 64-bit ALU Shift Left Shift Right Write Control 32 bits 64 bits Multiplier = datapath + control

CS 152 Lec 5.19 Multiply Algorithm Version 1  ProductMultiplierMultiplicand    Shift the Multiplier register right 1 bit. Done Yes: 32 repetitions 2. Shift the Multiplicand register left 1 bit. No: < 32 repetitions 1. Test Multiplier0 Multiplier0 = 0 Multiplier0 = 1 1a. Add multiplicand to product & place the result in Product register 32nd repetition? Start

CS 152 Lec 5.20 Observations on Multiply Version 1  1 clock per cycle =>  100 clocks per multiply Ratio of multiply to add 5:1 to 100:1  1/2 bits in multiplicand always 0 => 64-bit adder is wasted  0’s inserted in left of multiplicand as shifted => least significant bits of product never changed once formed  Instead of shifting multiplicand to left, shift product to right?

CS 152 Lec 5.21 MULTIPLY HARDWARE Version 2  32-bit Multiplicand reg, 32 -bit ALU, 64- bit Product reg, 32-bit Multiplier reg Product Multiplier Multiplicand 32-bit ALU Shift Right Write Control 32 bits 64 bits Shift Right

CS 152 Lec 5.22 How to think of this? B0B0 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P Remember original combinational multiplier:

CS 152 Lec 5.23 Simply Warp to Let Product Move Right...  Multiplicand stay’s still and product moves right B0B0 B1B1 B2B2 B3B3 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3 A0A0 A1A1 A2A2 A3A3

CS 152 Lec 5.24 Multiply Algorithm Version 2 3. Shift the Multiplier register right 1 bit. Done Yes: 32 repetitions 2. Shift the Product register right 1 bit. No: < 32 repetitions 1. Test Multiplier0 Multiplier0 = 0 Multiplier0 = 1 1a. Add multiplicand to the left half of product & place the result in the left half of Product register 32nd repetition? Start : : : : : : : : : : : : Product Multiplier Multiplicand

CS 152 Lec 5.25 Still More Wasted Space! 3. Shift the Multiplier register right 1 bit. Done Yes: 32 repetitions 2. Shift the Product register right 1 bit. No: < 32 repetitions 1. Test Multiplier0 Multiplier0 = 0 Multiplier0 = 1 1a. Add multiplicand to the left half of product & place the result in the left half of Product register 32nd repetition? Start : : : : : : : : : : : : Product Multiplier Multiplicand

CS 152 Lec 5.26 Observations on Multiply Version 2  Product register wastes space that exactly matches size of multiplier => combine Multiplier register and Product register

CS 152 Lec 5.27 MULTIPLY HARDWARE Version 3  32-bit Multiplicand reg, 32 -bit ALU, 64- bit Product reg, (0-bit Multiplier reg) Product (Multiplier) Multiplicand 32-bit ALU Write Control 32 bits 64 bits Shift Right

CS 152 Lec 5.28 Multiply Algorithm Version 3 MultiplicandProduct Done Yes: 32 repetitions 2. Shift the Product register right 1 bit. No: < 32 repetitions 1. Test Product0 Product0 = 0 Product0 = 1 1a. Add multiplicand to the left half of product & place the result in the left half of Product register 32nd repetition? Start

CS 152 Lec 5.29 Observations on Multiply Version 3  2 steps per bit because Multiplier & Product combined  MIPS registers Hi/Lo are left/right half of Product  Gives us MIPS instruction MultU  How can you make it faster?  What about signed multiplication? Easiest solution--make both positive & remember to comple- ment product if necessary when done (leave out the sign bit, run for 31 steps) Apply definition of 2’s complement -Need to sign-extend partial products and subtract at the end Booth’s Algorithm is elegant way to multiply signed numbers using same hardware as before and save cycles -Can handle multiple bits at a time

CS 152 Lec 5.30 Motivation for Booth’s Algorithm  Example 2 x 6 = 0010 x 0110: 0010 x shift (0 in multiplier) add (1 in multiplier) add (1 in multiplier) shift (0 in multiplier)  ALU with add or subtract gets same result in more than one way: 6= – = – =  For example  0010 x shift (0 in multiplier) – 0010 sub(first 1 in multpl.) shift (mid string of 1s) add (prior step had last 1)

CS 152 Lec 5.31 Booth’s Algorithm Current BitBit to the RightExplanationExampleOp 10Begins run of 1s sub 11Middle of run of 1s none 01End of run of 1s add 00Middle of run of 0s none Originally for Speed (when shift was faster than add)  Replace a string of 1s in multiplier with an initial subtract when we first see a one and then later add for the bit after the last one beginning of runend of run middle of run –

CS 152 Lec 5.32 Booth’s Example (2 x 7) 1a. P = P - m shift P (sign ext) 1b > nop, shift > nop, shift > add 4a shift 4b done OperationMultiplicandProductnext? 0. initial value > sub

CS 152 Lec 5.33 Booth’s Example (2 x -3) 1a. P = P - m shift P (sign ext) 1b > add a shift P 2b > sub a shift 3b > nop 4a shift 4b done OperationMultiplicandProductnext? 0. initial value > sub

CS 152 Lec 5.34 Current Bit to theExplanationExampleRecode BitsRight 0 00Middle of zeros (0) 0 1 0Single one (1) 1 00Begins run of 1s (-2) 1 10Begins run of 1s (-1) 0 01Ends run of 1s (1) 0 11Ends run of 1s (2) 1 01Isolated (-1) 1 11Middle of run (0) Radix-4 Modified Booth’s Multiple Representations Once admit new symbols (i.e. 1), can have multiple representations of a number:

CS 152 Lec 5.35 MIPS Logical Instructions InstructionExampleMeaningComment and and $1,$2,$3$1 = $2 & $33 reg. operands; Logical AND or or $1,$2,$3$1 = $2 | $33 reg. operands; Logical OR xor xor $1,$2,$3$1 = $2  $33 reg. operands; Logical XOR nor nor $1,$2,$3$1 = ~($2 | $3)3 reg. operands; Logical NOR and immediate andi $1,$2,10$1 = $2 & 10Logical AND reg, constant or immediate ori $1,$2,10$1 = $2 | 10Logical OR reg, constant xor immediate xori $1, $2,10 $1 = ~$2 &~10Logical XOR reg, constant shift left logical sll $1,$2,10$1 = $2 << 10Shift left by constant shift right logical srl $1,$2,10$1 = $2 >> 10Shift right by constant shift right arithm. sra $1,$2,10$1 = $2 >> 10Shift right (sign extend) shift left logical sllv $1,$2,$3$1 = $2 << $3 Shift left by variable shift right logical srlv $1,$2, $3 $1 = $2 >> $3 Shift right by variable shift right arithm. srav $1,$2, $3 $1 = $2 >> $3 Shift right arith. by variable

CS 152 Lec 5.36 Shifters Two kinds: logical-- value shifted in is always "0" arithmetic-- on right shifts, sign extend msblsb"0" msblsb"0" Note: these are single bit shifts. A given instruction might request 0 to 32 bits to be shifted!

CS 152 Lec 5.37 Combinational Shifter from MUXes  What comes in the MSBs?  How many levels for 32-bit shifter?  What if we use 4-1 Muxes ? 1 0 sel A B D Basic Building Block 8-bit right shifter S 2 S 1 S 0 A0A0 A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 A7A7 R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7

CS 152 Lec 5.38 General Shift Right Scheme Using 16 bit Example If added Right-to-left connections could support Rotate (not in MIPS but found in ISAs)

CS 152 Lec 5.39 Funnel Shifter XY R  Shift A by i bits (sa= shift right amount)  Logical: Y = 0, X=A, sa=i  Arithmetic? Y = _, X=_, sa=_  Rotate? Y = _, X=_, sa=_  Left shifts? Y = _, X=_, sa=_ Instead Extract 32 bits of 64 Shift Right 32 Y X R

CS 152 Lec 5.40 Barrel Shifter Technology-dependent solutions: transistor per switch

CS 152 Lec 5.41 Summary  Introduction to Verilog (more in discussion sections!) High level language to describe hardware -Module describes behavior or structure -Behavior can be higher level, e.g., x = boolean_expression(A,B,C,D); Time units as an explicitly managed concept within description Can activate modules when inputs change, not specifically invoked Inherently parallel, need to understand implications of event scheduling  Multiply: successive refinement to see final design 32-bit Adder, 64-bit shift register, 32-bit Multiplicand Register Booth’s algorithm to handle signed multiplies There are algorithms that calculate many bits of multiply per cycle (see exercises 4.36 to 4.39 in COD)  Shifter: successive refinement of 1/bit at a time shift register to barrel shifter