361 div.1. Computer Architecture EECS 361 Lecture 7: ALU Design : Division

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1 361 div.1 Computer Architecture EECS 361 Lecture 7: ALU Design : Division

2 Outline of Today s Lecture Introduction to Today s Lecture Divide Questions and Administrative Matters Introduction to Single cycle processor design 361 div.2

3 Divide: Paper & Pencil 1001 Quotient Divisor Dividend Remainder (or Modulo result) See how big a number can be subtracted, creating quotient bit on each step Binary => 1 * divisor or 0 * divisor Dividend = Quotient x Divisor + Remainder => Dividend = Quotient + Divisor 3 versions of divide, successive refinement 361 div.3

4 DIVIDE HARDWARE Version 1 64-bit Divisor reg, 64-bit ALU, 64-bit Remainder reg, 32-bit Quotient reg Divisor 64 bits Shift Right 64-bit ALU Quotient 32 bits Shift Left Remainder 64 bits Write Control 361 div.4

5 Divide Algorithm Version 1 Start: Place Dividend in Remainder Takes n+1 steps for n-bit Quotient & Rem. Remainder Quotient Divisor Remainder >= 0 1. Subtract the Divisor register from the Remainder register, and place the result in the Remainder register. Test Remainder Remainder < 0 2a. Shift the Quotient register to the left setting the new rightmost bit to 1. 2b. Restore the original value by adding the Divisor register to the Remainder register, & place the sum in the Remainder register. Also shift the Quotient register to the left, setting the new least significant bit to Shift the Divisor register right1 bit. 361 div.5 n+1 repetition? Done No: < n+1 repetitions Yes: n+1 repetitions (n = 4 here)

6 Observations on Divide Version 1 1/2 bits in divisor always 0 => 1/2 of 64-bit adder is wasted => 1/2 of divisor is wasted Instead of shifting divisor to right, shift remainder to left? 1st step cannot produce a 1 in quotient bit (otherwise too big) => switch order to shift first and then subtract, can save 1 iteration 361 div.6

7 DIVIDE HARDWARE Version 2 32-bit Divisor reg, 32-bit ALU, 64-bit Remainder reg, 32-bit Quotient reg Divisor 32-bit ALU 32 bits Quotient 32 bits Shift Left Remainder 64 bits Shift Left Write Control 361 div.7

8 Divide Algorithm Version 2 Remainder Quotient Divisor Start: Place Dividend in Remainder 1. Shift the Remainder register left 1 bit. 2. Subtract the Divisor register from the left half of the Remainder register, & place the result in the left half of the Remainder register. Remainder >= 0 Test Remainder Remainder < 0 3a. Shift the Quotient register to the left setting the new rightmost bit to 1. 3b. Restore the original value by adding the Divisor register to the left half of the Remainderregister, &place the sum in the left half of the Remainder register. Also shift the Quotient register to the left, setting the new least significant bit to div.8 nth repetition? Done No: < n repetitions Yes: n repetitions (n = 4 here)

9 Observations on Divide Version 2 Eliminate Quotient register by combining with Remainder as shifted left Start by shifting the Remainder left as before. Thereafter loop contains only two steps because the shifting of the Remainder register shifts both the remainder in the left half and the quotient in the right half The consequence of combining the two registers together and the new order of the operations in the loop is that the remainder will shifted left one time too many. Thus the final correction step must shift back only the remainder in the left half of the register 361 div.9

10 DIVIDE HARDWARE Version 3 32-bit Divisor reg, 32 -bit ALU, 64-bit Remainder reg, (0-bit Quotient reg) Divisor 32 bits 32-bit ALU HI LO Remainder (Quotient) 64 bits Shift Left Write Control 361 div.10

11 Divide Algorithm Version 3 Start: Place Dividend in Remainder Remainder Divisor Shift the Remainder register left 1 bit. 2. Subtract the Divisor register from the left half of the Remainder register, & place the result in the left half of the Remainder register. Remainder 0 Test Remainder Remainder < 0 3a. Shift the Remainder register to the left setting the new rightmost bit to 1. 3b. Restore the original value by adding the Divisor register to the left half of the Remainderregister, &place the sum in the left half of the Remainder register. Also shift the Remainder register to the left, setting the new least significant bit to div.11 nth repetition? No: < n repetitions Yes: n repetitions (n = 4 here) Done. Shift left half of Remainder right 1 bit.

12 Observations on Divide Version 3 Same Hardware as Multiply: just need ALU to add or subtract, and 63-bit register to shift left or shift right Hi and Lo registers in MIPS combine to act as 64-bit register for multiply and divide Signed Divides: Simplest is to remember signs, make positive, and complement quotient and remainder if necessary Note: Dividend and Remainder must have same sign Note: Quotient negated if Divisor sign & Dividend sign disagree e.g., 7 2 = 3, remainder = 1 Possible for quotient to be too large: if divide 64-bit interger by 1, quotient is 64 bits ( called saturation ) 361 div.12

13 Summary Bits have no inherent meaning: operations determine whether they are really ASCII characters, integers, floating point numbers Divide can use same hardware as multiply: Hi & Lo registers in MIPS 361 div.13

14 The Big Picture: Where are We Now? The Five Classic Components of a Computer Processor Control Memory Datapath Input Output Next Topic: Design a Single Cycle Processor machine design Arithmetic 361 div.14 inst. set design technology

15 The Big Picture: The Performance Perspective Performance of a machine is determined by: Instruction count Clock cycle time Clock cycles per instruction Processor design (datapath and control) will determine: Clock cycle time Clock cycles per instruction Next Class: 361 div.15 Single cycle processor: - Advantage: One clock cycle per instruction - Disadvantage: long cycle time Inst. Count CPI Cycle Time

16 How to Design a Processor: step-by-step 1. Analyze instruction set => datapath requirements the meaning of each instruction is given by the register transfers datapath must include storage element for ISA registers - possibly more datapath must support each register transfer 2. Select set of datapath components and establish clocking methodology 3. Assemble datapath meeting the requirements 4. Analyze implementation of each instruction to determine setting of control points that effects the register transfer. 5. Assemble the control logic 361 div.16

17 The MIPS Instruction Formats All MIPS instructions are 32 bits long. The three instruction formats: R-type op rs rt rd shamt funct 6 bits 5 bits 5 bits 5 bits 5 bits 6 bits I-type op rs rt immediate 6 bits 5 bits 5 bits 16 bits J-type op target address 6 bits 26 bits The different fields are: op: operation of the instruction rs, rt, rd: the source and destination register specifiers shamt: shift amount funct: selects the variant of the operation in the op field address / immediate: address offset or immediate value target address: target address of the jump instruction 361 div.17

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