# Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction

Save this PDF as:

Size: px
Start display at page:

Download "Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction"

## Transcription

1 1 Floating Point The World is Not Just Integers Programming languages support numbers with fraction Called floating-point numbers Examples: (π) (e) or (seconds in a nanosecond) 86,400,000,000,000 or (nanoseconds in a day) last number is a large integer that cannot fit in a 32-bit integer We use a scientific notation to represent Very small numbers (e.g ) Very large numbers (e.g ) Scientific notation: ± d. f 1 f 2 f 3 f 4 10 ± e 1 e 2 e 3

2 2 Floating-Point Numbers Examples of floating-point numbers in base , , , Examples of floating-point numbers in base , , , Exponents are kept in decimal for clarity binary point The binary number ( ) 2 = = Floating-point numbers should be normalized decimal point Exactly one non-zero digit should appear before the point In a decimal number, this digit can be from 1 to 9 In a binary number, this digit should be 1 Normalized FP Numbers: and NOT Normalized: and Floating-Point Representation A floating-point number is represented by the triple S is the Sign bit (0 is positive and 1 is negative) Representation is called sign and magnitude E is the Exponent field (signed) Very large numbers have large positive exponents Very small close-to-zero numbers have negative exponents More bits in exponent field increases range of values F is the Fraction field (fraction after binary point) More bits in fraction field improves the precision of FP numbers S Exponent Fraction Value of a floating-point number = (-1) S val(f) 2 val(e)

3 3 IEEE Floating-Point Standard Single Precision Floating Point Numbers (32 bits) 1-bit sign + 8-bit exponent + 23-bit fraction Double Precision Floating Point Numbers (64 bits) 1-bit sign + 11-bit exponent + 52-bit fraction S Exponent 8 Fraction 23 S Exponent 11 Fraction 52 (continued) Normalized Floating Point Numbers For a normalized floating point number (S, E, F) S E F = f 1 f 2 f 3 f 4 Significand is equal to (1.F) 2 = (1.f 1 f 2 f 3 f 4 ) 2 IEEE assumes hidden 1. (not stored) for normalized numbers Significand is 1 bit longer than fraction Value of a Normalized Floating Point Number is ( 1) S (1.F) 2 2 val(e) ( 1) S (1.f 1 f 2 f 3 f 4 ) 2 2 val(e) ( 1) S (1 + f f f f ) 2 2 val(e) ( 1) S is 1 when S is 0 (positive), and 1 when S is 1 (negative)

4 4 Biased Exponent Representation How to represent a signed exponent? Choices are Sign + magnitude representation for the exponent Two s complement representation Biased representation IEEE uses biased representation for the exponent Value of exponent = val(e) = E Bias (Bias is a constant) Recall that exponent field is 8 bits for single precision E can be in the range 0 to 255 E = 0 and E = 255 are reserved for special use E = 1 to 254 are used for normalized floating point numbers Bias = 127 (half of 254), val(e) = E 127 val(e=1) = 126, val(e=127) = 0, val(e=254) = 127 Biased Exponent Cont d For double precision, exponent field is 11 bits E can be in the range 0 to 2047 E = 0 and E = 2047 are reserved for special use E = 1 to 2046 are used for normalized floating point numbers Bias = 1023 (half of 2046), val(e) = E 1023 val(e=1) = 1022, val(e=1023) = 0, val(e=2046) = 1023 Value of a Normalized Floating Point Number is ( 1) S (1.F) 2 2 E Bias ( 1) S (1.f 1 f 2 f 3 f 4 ) 2 2 E Bias ( 1) S (1 + f f f f ) 2 2 E Bias

5 5 Examples of Single Precision Float What is the decimal value of this Single Precision float? Solution: Sign = 1 is negative Exponent = ( ) 2 = 124, E bias = = 3 Significand = ( ) 2 = = 1.25 (1. is implicit) Value in decimal = = What is the decimal value of? Solution: implicit Value in decimal = +( ) = ( ) = ( ) 2 = Examples of Double Precision Float What is the decimal value of this Double Precision float? Solution: Value of exponent = ( ) 2 Bias = = 6 Value of double float = ( ) (1. is implicit) = ( ) 2 = 74.5 What is the decimal value of? Do it yourself! (answer should be = )

6 6 Converting FP Decimal to Binary Convert to binary in single and double precision Solution: Fraction bits can be obtained using multiplication by = = = (0.1101) = = ½ + ¼ + 1/16 = 13/ = 1.0 Stop when fractional part is 0 Fraction = (0.1101) 2 = (1.101) (Normalized) Exponent = 1 + Bias = 126 (single precision) and 1022 (double) Single Precision Double Precision Floating Point Addition Example Consider adding: (1.111) (1.011) For simplicity, we assume 4 bits of precision (or 3 bits of fraction) Cannot add significands Why? Because exponents are not equal How to make exponents equal? Shift the significand of the lesser exponent right until its exponent matches the larger number (1.011) = (0.1011) = ( ) Difference between the two exponents = 1 ( 3) = 2 So, shift right by 2 bits Now, add the significands: Carry

7 7 Addition Example cont d So, (1.111) (1.011) = ( ) However, result ( ) is NOT normalized Normalize result: ( ) = ( ) In this example, we have a carry So, shift right by 1 bit and increment the exponent Round the significand to fit in appropriate number of bits We assumed 4 bits of precision or 3 bits of fraction Round to nearest: ( ) 2 (1.001) 2 Renormalize if rounding generates a carry Detect overflow / underflow If exponent becomes too large (overflow) or too small (underflow) + Floating Point Subtraction Example Consider: (1.000) (1.000) We assume again: 4 bits of precision (or 3 bits of fraction) Shift significand of the lesser exponent right Difference between the two exponents = 2 ( 3) = 5 Shift right by 5 bits: (1.000) = ( ) Convert subtraction into addition to 2's complement Sign 2 s Complement Since result is negative, convert result from 2's complement to sign-magnitude 2 s Complement

8 8 Subtraction Example cont d So, (1.000) (1.000) = Normalize result: = For subtraction, we can have leading zeros Count number z of leading zeros (in this case z = 1) Shift left and decrement exponent by z Round the significand to fit in appropriate number of bits We assumed 4 bits of precision or 3 bits of fraction Round to nearest: (1.1111) 2 (10.000) 2 Renormalize: rounding generated a carry = Result would have been accurate if more fraction bits are used + Floating Point Multiplication Example Consider multiplying: by As before, we assume 4 bits of precision (or 3 bits of fraction) Unlike addition, we add the exponents of the operands Result exponent value = ( 1) + ( 2) = 3 Using the biased representation: E Z = E X + E Y Bias E X = ( 1) = 126 (Bias = 127 for SP) E Y = ( 2) = 125 E Z = = 124 (value = 3) Now, multiply the significands: (1.010) 2 (1.110) 2 = ( ) 2 3-bit fraction 3-bit fraction 6-bit fraction

9 9 Multiplication Example cont d Since sign S X S Y, sign of product S Z = 1 (negative) So, = However, result: is NOT normalized Normalize: = Shift right by 1 bit and increment the exponent At most 1 bit can be shifted right Why? Round the significand to nearest: (3-bit fraction) Result (normalized) Detect overflow / underflow No overflow / underflow because exponent is within range + Extra Bits to Maintain Precision Floating-point numbers are approximations for Real numbers that they cannot represent Infinite variety of real numbers exist between 1.0 and 2.0 However, exactly 2 23 fractions can be represented in SP, and Exactly 2 52 fractions can be represented in DP (double precision) Extra bits are generated in intermediate results when Shifting and adding/subtracting a p-bit significand Multiplying two p-bit significands (product can be 2p bits) But when packing result fraction, extra bits are discarded We only need few extra bits in an intermediate result Minimizing hardware but without compromising precision

10 10 Guard Bit Guard bit: guards against loss of a significant bit Only one guard bit is needed to maintain accuracy of result Shifted left (if needed) during normalization as last fraction bit Example on the need of a guard bit: (subtraction) (shift right 7 bits) Guard bit do not discard (2's complement) (add significands) (normalized) Round and Sticky Bits Two extra bits are needed for rounding Just after normalizing a result significand Round bit: appears just after the normalized significand Sticky bit: appears after the round bit (OR of all additional bits) Reduce the hardware and still achieve accurate arithmetic As if result significand was computed exactly and rounded Consider the same example of previous slide: OR-reduce (2's complement) (sum) (normalized) Round bit Sticky bit

### Floating Point Arithmetic

Floating Point Arithmetic CS 365 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement -2 N-1 to 2 N-1-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678

### Floating Point Numbers

Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides

### Module 2: Computer Arithmetic

Module 2: Computer Arithmetic 1 B O O K : C O M P U T E R O R G A N I Z A T I O N A N D D E S I G N, 3 E D, D A V I D L. P A T T E R S O N A N D J O H N L. H A N N E S S Y, M O R G A N K A U F M A N N

### 3.5 Floating Point: Overview

3.5 Floating Point: Overview Floating point (FP) numbers Scientific notation Decimal scientific notation Binary scientific notation IEEE 754 FP Standard Floating point representation inside a computer

### Floating-point Arithmetic. where you sum up the integer to the left of the decimal point and the fraction to the right.

Floating-point Arithmetic Reading: pp. 312-328 Floating-Point Representation Non-scientific floating point numbers: A non-integer can be represented as: 2 4 2 3 2 2 2 1 2 0.2-1 2-2 2-3 2-4 where you sum

### Inf2C - Computer Systems Lecture 2 Data Representation

Inf2C - Computer Systems Lecture 2 Data Representation Boris Grot School of Informatics University of Edinburgh Last lecture Moore s law Types of computer systems Computer components Computer system stack

### Floating point. Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties. Next time. !

Floating point Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties Next time! The machine model Chris Riesbeck, Fall 2011 Checkpoint IEEE Floating point Floating

### Floating Point Numbers. Lecture 9 CAP

Floating Point Numbers Lecture 9 CAP 3103 06-16-2014 Review of Numbers Computers are made to deal with numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers:

### Computer Architecture and Organization

3-1 Chapter 3 - Arithmetic Computer Architecture and Organization Miles Murdocca and Vincent Heuring Chapter 3 Arithmetic 3-2 Chapter 3 - Arithmetic Chapter Contents 3.1 Fixed Point Addition and Subtraction

### Chapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation

Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation

### Number Systems and Computer Arithmetic

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time What do all those bits mean now? bits (011011011100010...01) instruction R-format I-format... integer data number text

### The Sign consists of a single bit. If this bit is '1', then the number is negative. If this bit is '0', then the number is positive.

IEEE 754 Standard - Overview Frozen Content Modified by on 13-Sep-2017 Before discussing the actual WB_FPU - Wishbone Floating Point Unit peripheral in detail, it is worth spending some time to look at

### Data Representation Floating Point

Data Representation Floating Point CSCI 2400 / ECE 3217: Computer Architecture Instructor: David Ferry Slides adapted from Bryant & O Hallaron s slides via Jason Fritts Today: Floating Point Background:

### CO Computer Architecture and Programming Languages CAPL. Lecture 15

CO20-320241 Computer Architecture and Programming Languages CAPL Lecture 15 Dr. Kinga Lipskoch Fall 2017 How to Compute a Binary Float Decimal fraction: 8.703125 Integral part: 8 1000 Fraction part: 0.703125

### C NUMERIC FORMATS. Overview. IEEE Single-Precision Floating-point Data Format. Figure C-0. Table C-0. Listing C-0.

C NUMERIC FORMATS Figure C-. Table C-. Listing C-. Overview The DSP supports the 32-bit single-precision floating-point data format defined in the IEEE Standard 754/854. In addition, the DSP supports an

### 15213 Recitation 2: Floating Point

15213 Recitation 2: Floating Point 1 Introduction This handout will introduce and test your knowledge of the floating point representation of real numbers, as defined by the IEEE standard. This information

### IT 1204 Section 2.0. Data Representation and Arithmetic. 2009, University of Colombo School of Computing 1

IT 1204 Section 2.0 Data Representation and Arithmetic 2009, University of Colombo School of Computing 1 What is Analog and Digital The interpretation of an analog signal would correspond to a signal whose

### Floating point. Today. IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time.

Floating point Today IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time The machine model Fabián E. Bustamante, Spring 2010 IEEE Floating point Floating point

### Systems I. Floating Point. Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties

Systems I Floating Point Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for

### CS429: Computer Organization and Architecture

CS429: Computer Organization and Architecture Dr. Bill Young Department of Computer Sciences University of Texas at Austin Last updated: September 18, 2017 at 12:48 CS429 Slideset 4: 1 Topics of this Slideset

### Classes of Real Numbers 1/2. The Real Line

Classes of Real Numbers All real numbers can be represented by a line: 1/2 π 1 0 1 2 3 4 real numbers The Real Line { integers rational numbers non-integral fractions irrational numbers Rational numbers

### System Programming CISC 360. Floating Point September 16, 2008

System Programming CISC 360 Floating Point September 16, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Powerpoint Lecture Notes for Computer Systems:

### Computer Arithmetic Ch 8

Computer Arithmetic Ch 8 ALU Integer Representation Integer Arithmetic Floating-Point Representation Floating-Point Arithmetic 1 Arithmetic Logical Unit (ALU) (2) Does all work in CPU (aritmeettis-looginen

### Floating Point Puzzles The course that gives CMU its Zip! Floating Point Jan 22, IEEE Floating Point. Fractional Binary Numbers.

class04.ppt 15-213 The course that gives CMU its Zip! Topics Floating Point Jan 22, 2004 IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

### A complement number system is used to represent positive and negative integers. A complement number system is based on a fixed length representation

Complement Number Systems A complement number system is used to represent positive and negative integers A complement number system is based on a fixed length representation of numbers Pretend that integers

### CS101 Lecture 04: Binary Arithmetic

CS101 Lecture 04: Binary Arithmetic Binary Number Addition Two s complement encoding Briefly: real number representation Aaron Stevens (azs@bu.edu) 25 January 2013 What You ll Learn Today Counting in binary

### 1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM

1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number

### CS 265. Computer Architecture. Wei Lu, Ph.D., P.Eng.

CS 265 Computer Architecture Wei Lu, Ph.D., P.Eng. 1 Part 1: Data Representation Our goal: revisit and re-establish fundamental of mathematics for the computer architecture course Overview: what are bits

### UNIT IV: DATA PATH DESIGN

UNIT IV: DATA PATH DESIGN Agenda Introduc/on Fixed Point Arithme/c Addi/on Subtrac/on Mul/plica/on & Serial mul/plier Division & Serial Divider Two s Complement (Addi/on, Subtrac/on) Booth s algorithm

### Scientific Computing. Error Analysis

ECE257 Numerical Methods and Scientific Computing Error Analysis Today s s class: Introduction to error analysis Approximations Round-Off Errors Introduction Error is the difference between the exact solution

### Chapter 3 Arithmetic for Computers (Part 2)

Department of Electr rical Eng ineering, Chapter 3 Arithmetic for Computers (Part 2) 王振傑 (Chen-Chieh Wang) ccwang@mail.ee.ncku.edu.tw ncku edu Depar rtment of Electr rical Eng ineering, Feng-Chia Unive

### Signed umbers. Sign/Magnitude otation

Signed umbers So far we have discussed unsigned number representations. In particular, we have looked at the binary number system and shorthand methods in representing binary codes. With m binary digits,

### 10.1. Unit 10. Signed Representation Systems Binary Arithmetic

0. Unit 0 Signed Representation Systems Binary Arithmetic 0.2 BINARY REPRESENTATION SYSTEMS REVIEW 0.3 Interpreting Binary Strings Given a string of s and 0 s, you need to know the representation system

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

### Principles of Computer Architecture. Chapter 3: Arithmetic

3-1 Chapter 3 - Arithmetic Principles of Computer Architecture Miles Murdocca and Vincent Heuring Chapter 3: Arithmetic 3-2 Chapter 3 - Arithmetic 3.1 Overview Chapter Contents 3.2 Fixed Point Addition

### CS61C L10 MIPS Instruction Representation II, Floating Point I (6)

CS61C L1 MIPS Instruction Representation II, Floating Point I (1) inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #1 Instruction Representation II, Floating Point I 25-1-3 There is one

### CSCI 402: Computer Architectures. Arithmetic for Computers (4) Fengguang Song Department of Computer & Information Science IUPUI.

CSCI 402: Computer Architectures Arithmetic for Computers (4) Fengguang Song Department of Computer & Information Science IUPUI Homework 4 Assigned on Feb 22, Thursday Due Time: 11:59pm, March 5 on Monday

### Number Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example

Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example 1729 1 103 7 102 2 101 9 100 decimal = + + + There is no particular reason for the choice of 10,

### CS61C : Machine Structures

inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #10 Instruction Representation II, Floating Point I 2005-10-03 Lecturer PSOE, new dad Dan Garcia www.cs.berkeley.edu/~ddgarcia #9 bears

### Floating Point Arithmetic

Floating Point Arithmetic Floating point numbers are frequently used in many applications. Implementation of arithmetic units such as adder, multiplier, etc for Floating point numbers are more complex

### CSC201, SECTION 002, Fall 2000: Homework Assignment #2

1 of 7 11/8/2003 7:34 PM CSC201, SECTION 002, Fall 2000: Homework Assignment #2 DUE DATE Monday, October 2, at the start of class. INSTRUCTIONS FOR PREPARATION Neat, in order, answers easy to find. Staple

### Co-processor Math Processor. Richa Upadhyay Prabhu. NMIMS s MPSTME February 9, 2016

8087 Math Processor Richa Upadhyay Prabhu NMIMS s MPSTME richa.upadhyay@nmims.edu February 9, 2016 Introduction Need of Math Processor: In application where fast calculation is required Also where there

### FPSim: A Floating- Point Arithmetic Demonstration Applet

FPSim: A Floating- Point Arithmetic Demonstration Applet Jeffrey Ward Departmen t of Computer Science St. Cloud State University waje9901@stclouds ta te.edu Abstract An understan ding of IEEE 754 standar

### 1.2 Round-off Errors and Computer Arithmetic

1.2 Round-off Errors and Computer Arithmetic 1 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set

### Number Systems & Encoding

Number Systems & Encoding Lecturer: Sri Parameswaran Author: Hui Annie Guo Modified: Sri Parameswaran Week2 1 Lecture overview Basics of computing with digital systems Binary numbers Floating point numbers

### Floating-Point Arithmetic

Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } Reading: Study Chapter 3. L12 Multiplication 1 Approximating Real Numbers on Computers Thus far, we ve entirely ignored one of the most

### UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS

UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS (09 periods) Computer Arithmetic: Data Representation, Fixed Point Representation, Floating Point Representation, Addition and

### Chapter 2 Bits, Data Types, and Operations

Chapter 2 Bits, Data Types, and Operations How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of electrons Easy to recognize

### Lecture 5. Computer Arithmetic. Ch 9 [Stal10] Integer arithmetic Floating-point arithmetic

Lecture 5 Computer Arithmetic Ch 9 [Stal10] Integer arithmetic Floating-point arithmetic ALU ALU = Arithmetic Logic Unit (aritmeettis-looginen yksikkö) Actually performs operations on data Integer and

### International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering

An Efficient Implementation of Double Precision Floating Point Multiplier Using Booth Algorithm Pallavi Ramteke 1, Dr. N. N. Mhala 2, Prof. P. R. Lakhe M.Tech [IV Sem], Dept. of Comm. Engg., S.D.C.E, [Selukate],

### CS101 Introduction to computing Floating Point Numbers

CS101 Introduction to computing Floating Point Numbers A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1 Outline Need to floating point number Number representation

### CS321. Introduction to Numerical Methods

CS31 Introduction to Numerical Methods Lecture 1 Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506 0633 August 5, 017 Number

### FLOATING POINT NUMBERS

FLOATING POINT NUMBERS Robert P. Webber, Longwood University We have seen how decimal fractions can be converted to binary. For instance, we can write 6.25 10 as 4 + 2 + ¼ = 2 2 + 2 1 + 2-2 = 1*2 2 + 1*2

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### More Programming Constructs -- Introduction

More Programming Constructs -- Introduction We can now examine some additional programming concepts and constructs Chapter 5 focuses on: internal data representation conversions between one data type and

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### Computer Systems C S Cynthia Lee

Computer Systems C S 1 0 7 Cynthia Lee 2 Today s Topics LECTURE: Floating point! Real Numbers and Approximation MATH TIME! Some preliminary observations on approximation We know that some non-integer numbers

### EC2303-COMPUTER ARCHITECTURE AND ORGANIZATION

EC2303-COMPUTER ARCHITECTURE AND ORGANIZATION QUESTION BANK UNIT-II 1. What are the disadvantages in using a ripple carry adder? (NOV/DEC 2006) The main disadvantage using ripple carry adder is time delay.

### Arithmetic. Chapter 3 Computer Organization and Design

Arithmetic Chapter 3 Computer Organization and Design Addition Addition is similar to decimals 0000 0111 + 0000 0101 = 0000 1100 Subtraction (negate) 0000 0111 + 1111 1011 = 0000 0010 Over(under)flow For

### Computer Architecture. Chapter 3: Arithmetic for Computers

182.092 Computer Architecture Chapter 3: Arithmetic for Computers Adapted from Computer Organization and Design, 4 th Edition, Patterson & Hennessy, 2008, Morgan Kaufmann Publishers and Mary Jane Irwin

### IEEE-754 floating-point

IEEE-754 floating-point Real and floating-point numbers Real numbers R form a continuum - Rational numbers are a subset of the reals - Some numbers are irrational, e.g. π Floating-point numbers are an

### Hardware Modules for Safe Integer and Floating-Point Arithmetic

Hardware Modules for Safe Integer and Floating-Point Arithmetic A Thesis submitted to the Graduate School Of The University of Cincinnati In partial fulfillment of the requirements for the degree of Master

### Characters, Strings, and Floats

Characters, Strings, and Floats CS 350: Computer Organization & Assembler Language Programming 9/6: pp.8,9; 9/28: Activity Q.6 A. Why? We need to represent textual characters in addition to numbers. Floating-point

### 2 Computation with Floating-Point Numbers

2 Computation with Floating-Point Numbers 2.1 Floating-Point Representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However, real numbers

### The Perils of Floating Point

The Perils of Floating Point by Bruce M. Bush Copyright (c) 1996 Lahey Computer Systems, Inc. Permission to copy is granted with acknowledgement of the source. Many great engineering and scientific advances

### FLOATING POINT ADDERS AND MULTIPLIERS

Concordia University FLOATING POINT ADDERS AND MULTIPLIERS 1 Concordia University Lecture #4 In this lecture we will go over the following concepts: 1) Floating Point Number representation 2) Accuracy

### Binary Addition & Subtraction. Unsigned and Sign & Magnitude numbers

Binary Addition & Subtraction Unsigned and Sign & Magnitude numbers Addition and subtraction of unsigned or sign & magnitude binary numbers by hand proceeds exactly as with decimal numbers. (In fact this

### Computer Arithmetic. L. Liu Department of Computer Science, ETH Zürich Fall semester, Reconfigurable Computing Systems ( L) Fall 2012

Reconfigurable Computing Systems (252-2210-00L) all 2012 Computer Arithmetic L. Liu Department of Computer Science, ETH Zürich all semester, 2012 Source: ixed-point arithmetic slides come from Prof. Jarmo

### Double Precision Floating-Point Arithmetic on FPGAs

MITSUBISHI ELECTRIC ITE VI-Lab Title: Double Precision Floating-Point Arithmetic on FPGAs Internal Reference: Publication Date: VIL04-D098 Author: S. Paschalakis, P. Lee Rev. A Dec. 2003 Reference: Paschalakis,

### Arithmetic Logic Unit

Arithmetic Logic Unit A.R. Hurson Department of Computer Science Missouri University of Science & Technology A.R. Hurson 1 Arithmetic Logic Unit It is a functional bo designed to perform "basic" arithmetic,

### COMPUTER HARDWARE. Instruction Set Architecture

COMPUTER HARDWARE Instruction Set Architecture Overview Computer architecture Operand addressing Addressing architecture Addressing modes Elementary instructions Data transfer instructions Data manipulation

### Significant Figures & Scientific Notation

Significant Figures & Scientific Notation Measurements are important in science (particularly chemistry!) Quantity that contains both a number and a unit Must be able to say how correct a measurement is

### H.1 Introduction H-2 H.2 Basic Techniques of Integer Arithmetic H-2 H.3 Floating Point H-13 H.4 Floating-Point Multiplication H-17 H.

H.1 Introduction H-2 H.2 Basic Techniques of Integer rithmetic H-2 H.3 Floating Point H-13 H.4 Floating-Point Multiplication H-17 H.5 Floating-Point ddition H-21 H.6 Division and Remainder H-27 H.7 More

### Course Schedule. CS 221 Computer Architecture. Week 3: Plan. I. Hexadecimals and Character Representations. Hexadecimal Representation

Course Schedule CS 221 Computer Architecture Week 3: Information Representation (2) Fall 2001 W1 Sep 11- Sep 14 Introduction W2 Sep 18- Sep 21 Information Representation (1) (Chapter 3) W3 Sep 25- Sep

### An Efficient Implementation of Floating Point Multiplier

An Efficient Implementation of Floating Point Multiplier Mohamed Al-Ashrafy Mentor Graphics Mohamed_Samy@Mentor.com Ashraf Salem Mentor Graphics Ashraf_Salem@Mentor.com Wagdy Anis Communications and Electronics

### Computer Representation of Numbers and Computer Arithmetic

Computer Representation of Numbers and Computer Arithmetic January 17, 2017 Contents 1 Binary numbers 6 2 Memory 9 3 Characters 12 1 c A. Sandu, 1998-2015. Distribution outside classroom not permitted.

### 4/8/17. Admin. Assignment 5 BINARY. David Kauchak CS 52 Spring 2017

4/8/17 Admin! Assignment 5 BINARY David Kauchak CS 52 Spring 2017 Diving into your computer Normal computer user 1 After intro CS After 5 weeks of cs52 What now One last note on CS52 memory address binary

### IEEE Standard 754 Floating Point Numbers

IEEE Standard 754 Floating Point Numbers Steve Hollasch / Last update 2005-Feb-24 IEEE Standard 754 floating point is the most common representation today for real numbers on computers, including Intel-based

### Comparison of Adders for optimized Exponent Addition circuit in IEEE754 Floating point multiplier using VHDL

International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 11, Issue 07 (July 2015), PP.60-65 Comparison of Adders for optimized Exponent Addition

### 8/30/2016. In Binary, We Have A Binary Point. ECE 120: Introduction to Computing. Fixed-Point Representations Support Fractions

University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Fixed- and Floating-Point Representations In Binary, We Have A Binary Point Let

### IEEE Floating Point Numbers Overview

COMP 40: Machine Structure and Assembly Language Programming (Fall 2015) IEEE Floating Point Numbers Overview Noah Mendelsohn Tufts University Email: noah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah

### Floating Point Considerations

Chapter 6 Floating Point Considerations In the early days of computing, floating point arithmetic capability was found only in mainframes and supercomputers. Although many microprocessors designed in the

### Independent Representation

Independent Representation 1 Integer magnitude + 1 2 3 4 5 6 7 8 9 0 Float sign sign of exponent mantissa On standard 32 bit machines: INT_MAX = 2147483647 which gives 10 digits of precision, (i.e. the

Concordia University FLOATING POINT ADDERS AND MULTIPLIERS 1 Concordia University Lecture #4 In this lecture we will go over the following concepts: 1) Floating Point Number representation 2) Accuracy

### Calculations with Sig Figs

Calculations with Sig Figs When you make calculations using data with a specific level of uncertainty, it is important that you also report your answer with the appropriate level of uncertainty (i.e.,

### CS367 Test 1 Review Guide

CS367 Test 1 Review Guide This guide tries to revisit what topics we've covered, and also to briefly suggest/hint at types of questions that might show up on the test. Anything on slides, assigned reading,

### CMSC 313 Lecture 03 Multiple-byte data big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes

Multiple-byte data CMSC 313 Lecture 03 big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes UMBC, CMSC313, Richard Chang 4-5 Chapter

### LogiCORE IP Floating-Point Operator v6.2

LogiCORE IP Floating-Point Operator v6.2 Product Guide Table of Contents SECTION I: SUMMARY IP Facts Chapter 1: Overview Unsupported Features..............................................................

### Name: CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1. Question Points I. /34 II. /30 III.

CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1 Name: Question Points I. /34 II. /30 III. /36 TOTAL: /100 Instructions: 1. This is a closed-book, closed-notes exam. 2. You

### Declaring Floating Point Data

Declaring Floating Point Data There are three ways to declare floating point storage. These are E D L Single precision floating point, Double precision floating point, and Extended precision floating point.

### Numeric Variable Storage Pattern

Numeric Variable Storage Pattern Sreekanth Middela Srinivas Vanam Rahul Baddula Percept Pharma Services, Bridgewater, NJ ABSTRACT This paper presents the Storage pattern of Numeric Variables within the

### Computer Organization CS 206 T Lec# 2: Instruction Sets

Computer Organization CS 206 T Lec# 2: Instruction Sets Topics What is an instruction set Elements of instruction Instruction Format Instruction types Types of operations Types of operand Addressing mode

### Computer arithmetics: integers, binary floating-point, and decimal floating-point

n!= 0 && -n == n z+1 == z Computer arithmetics: integers, binary floating-point, and decimal floating-point v+w-w!= v x+1 < x Peter Sestoft 2010-02-16 y!= y p == n && 1/p!= 1/n 1 Computer arithmetics Computer

### Algebra 1 Review. Properties of Real Numbers. Algebraic Expressions

Algebra 1 Review Properties of Real Numbers Algebraic Expressions Real Numbers Natural Numbers: 1, 2, 3, 4,.. Numbers used for counting Whole Numbers: 0, 1, 2, 3, 4,.. Natural Numbers and 0 Integers:,

### Logic, Words, and Integers

Computer Science 52 Logic, Words, and Integers 1 Words and Data The basic unit of information in a computer is the bit; it is simply a quantity that takes one of two values, 0 or 1. A sequence of k bits