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Chapter 4計算機算數

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Outline

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Problem: Designing MIPS ALU

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Functional Specification

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A Bit-slice ALU

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A 1-bit ALU

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A 4-bit ALU

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How about Subtraction?

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Revised Diagram

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Overflow

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Overflow Detection

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Overflow Detection Logic

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Zero Detection Logic

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Putting It Altogether (I)

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Putting It Altogether (II)

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Problems with Ripple Carry Adder

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Carry Lookahead: Theory (I)

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Carry Lookahead: Theory (II)

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Cascaded Carry Lookahead

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Carry-select Adder

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Add XOR to ALU

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Shifters

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Combinational Shifter

What comes in the MSBs? how many levels for 32-bitshifter? what if we use 4-1 Muxes ?

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Outline

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Multiplication in MIPS

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Division in MIPS

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MIPS Multiply/Divide Summary

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Unsigned Multiply

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Unisigned Multiplier (Ver. 1)

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Observations: Multiply Ver. 1

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Unisigned Multiplier (Ver. 2)

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Unisigned Multiplier (Ver. 3)

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Observations: Multiply Ver. 3

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Booth’s Algorithm: Motivation

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Booth’s Algorithm: Rationale

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Booth’s Algorithm

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Booths Example (2 x 7)

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Booths Example (2 x -3)

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Combinational Multiplier

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How Does It Work?

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Outline

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Divide: Paper & Pencil

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Divide Hardware (Version 1)

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Observations: Divide Version 1

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Divide Hardware (Version 2)

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Observations: Divide Version 2

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Divide Hardware (Version 3)

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Observations: Divide Version 3

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Outline

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Floating-Point: Motivation

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Scientific Notation: Binary

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FP Representation

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Double Precision Representation

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IEEE 754 Standard (1/4)

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IEEE 754 Standard (2/4)

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IEEE 754 Standard (3/4)

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IEEE 754 Standard (4/4)

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Example: FP to Decimal

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Continuing Example: Binary to ???

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Big Idea: Type Not Associated withData

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Example: Decimal to FP

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Representation for 0

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Special Numbers

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Representation for +/- Infinity

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Representation for Not a Number

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Special Numbers (cont’d)

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Floating-Point Addition

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Floating-Point Multiplication

(4) set the sign of product

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MIPS Floating Point

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MIPS Floating Point Instructions

• Organized as a coprocessor

– Separate registers $f0-$f31– Separate operations

– Separate data transfer (to same memory)

• Basic operations– add.s - single add.d - double– sub.s - single sub.d - double– mul.s - single mul.d - double– div.s - single div.d - double

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MIPS Floating Point Instructions (cont’d)

• Data transfer– lwc1, swcl (l.s, s.s) - load/store float

to fp reg– l.d, s.d - load/store double to fp reg pair

• Testing / branching– c.lt.s, c.lt.d, c.eq.s, c.eq.d, …

compare and set condition bit if true– bclt - branch if condition true– bclf - branch if condition false

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Rounding

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Round to Even

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Floating Point Fallacy

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Summary