Introduction to Computer Systems Recitation 2 May 29, Marjorie Carlson Aditya Gupta Shailin Desai

Size: px
Start display at page:

Download "Introduction to Computer Systems Recitation 2 May 29, Marjorie Carlson Aditya Gupta Shailin Desai"

Transcription

1 Introduction to Computer Systems Recitation 2 May 29, 2014 Marjorie Carlson Aditya Gupta Shailin Desai 1

2 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 2

3 Integer * and /! You can multiply and divide by powers of 2 with bitshifting alone.! Your computer does a lot of math this way!! Multiply:! To multiply by 2 k, simply left shift by k.! What s * 2?! What s * 4? 3

4 Integer * and /! You can multiply and divide by powers of 2 with bitshifting alone.! Your computer does a lot of math this way!! Divide:! To multiply by 2 k, right shift by k.! Let s try this with 01111/2?! How about (signed) 10001/2?! Uh- oh!! Shifting rounds down, but we want to round toward zero.! Solution: biasing when the number is negative 4

5 Biasing Division Only bias when the k bits dividend is negative! Dividend: x 1 Divisor: +2 k / 2 k Before we discard the last k bits, we add a biasing term consisting of k 1 s. 1 This bit will increment if appropriate. k bits of 1 s Then we >> k. x / 2 k And voila! The number now rounds toward zero instead of down. 5

6 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 6

7 FloaCng Point FracCons in Binary bi bi- 1 b2 b1 b0 b- 1 b- 2 b- 3 b- j 1/2 1/4 1/8 2 - j 2 i 2 i

8 FloaCng Point FracCons in Binary! Convert binary to decimal:! 1.1! ! ! Convert decimal to binary:! 3 3/4! 2 3/32!

9 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits? 9

10 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits?! Hint: - _. _ * 10

11 How to Represent #s Efficiently?! Forget about binary for a minute.! What do we do if we want to convey in only ten digits? * sign mancssa exponent 11

12 FloaCng Point ScienCfic NotaCon! How can we put binary numbers into scientific notation? * 2 2 sign (S) mancssa (M) exponent (E)! Numerical form: ( 1) S M 2 E 12

13 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers! Biasing division! Floats! Binary fractions! IEEE standard! Normalized numbers! Denormalized numbers! Special values 13

14 FloaCng Point IEEE Standard! Floating points encode binary scientific notation. s exp frac 1 8 bits 23 bits! exp encodes E (the exponent).! frac encodes M (the mantissa).! But, due to optimizations, exp E and frac M.! (Note: for the next five slides, forget denormalized floats exist. We ll get back to them, I promise.) 14

15 FloaCng Point First OpCmizaCon! In decimal scientific notation, the digit before the decimal place can be any number from 1 to 9: ! But in binary scientific notation, that digit will always be 1! ! So, encoding the 1 is unnecessary.! Instead of representing all of M in the frac, we discard the leading 1 and only encode the part after the decimal ! ! !.1111! frac = M 1 M = 1 + frac 15

16 FloaCng Point Second OpCmizaCon! exp is an unsigned 8- bit number, so it can represent the numbers 0 (0x ) to 255 (0x ).! Do we want floats be able to represent number from around 2 0 to around 2 255?! It would actually be much more useful to represent numbers from, say, to ! So, to get a more useful range of possible exponents, we subtract a bias of 127 from E to get the exp.! exp = E + bias bias = 2 k- 1-1 E = exp - bias 16

17 ConverCng a Number to a Float * 2 2 sign (S) mancssa (M) exponent (E)! S = + so s = 0! E = 2 so exp = 129! M = so frac = Remember! M = 1 + frac E = exp bias bias =

18 ConverCng a Number to a Float * 2 2 sign (S) mancssa (M) exponent (E) converted to a 32- bit float looks like: s exp frac

19 FloaCng Point Example! Consider the following 5- bit floating point representation based on the IEEE floating point format. This format does not have a sign bit; it can only represent nonnegative numbers.! There are k=3 exponent bits.! There are n=2 fraction bits ! What s the bias?! What does represent?! What does represent?! How would you represent 6?! How would you represent ¼? exp frac 19

20 FloaCng Point Denormalized Range! Given what we ve just discussed, the smallest representable number would be 1.0 * 2 - bias, which is not really that small.! It d be represented as all zeros. There d be no way to represent zero as a float!! IEEE uses a trick to give us numbers closer to 0, and 0 itself: for really small numbers (i.e., exp = 0), drop the implied leading 1.! This basically sacrifices a little precision for a wider range. 20

21 FloaCng Point Denormalized Range Normalized Denormalized exp 0 exp = 0 implied leading 1 no implied leading 1 E = exp - bias denser near origin represents most numbers E = 1 bias evenly spaced represents tiny numbers Why not 0 bias? Because we have to increment the exponent to counteract the missing leading 1. 21

22 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac! What does represent?! What s the smallest representable nonzero value?! What s the largest representable finite number?! What s the smallest normalized number?! What s the largest denormalized number? 22

23 FloaCng Point Special Cases! OK, denormalizing got us our zero. Now how about infinity? How about NaN (not a number)? 23

24 Last Two Tips to Convert Anything! 1. The tricky part about dec! float conversion is figuring out whether your number should be encoded as normalized or denormalized.! Strategy 1: compute the smallest possible noramlized number, then compare your number to it.! Strategy 2: try to encode it as normalized; if your exponent doesn t fit in exp, change exp to 0 and shift your decimal point accordingly. 2. You need to know how to round! 24

25 FloaCng Point Rounding Floats round to even if precisely between two options. (Avoids statistical bias from always rounding the same way.) truncate below half; round down interescng case; round to even above half; round up truncate below half; round down interescng case; round to even above half; round up truncate

26 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac Value Floating Point Bits Rounded Value 9/

27 FloaCng Point Examples! Back to our mini- floats:! There are k=3 exponent bits.! There are n=2 fraction bits.! Bias = exp frac Value Floating Point Bits Rounded Value 9/ /

28 CongratulaCons! You can now translate any real number (plus infinity and NaN) into and out of machine representation! 28

29 QuesCons? 29

Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Lecture 3

Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Lecture 3 Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Instructor: Nicole Hynes nicole.hynes@rutgers.edu 1 Fixed Point Numbers Fixed point number: integer part

More information

ECE232: Hardware Organization and Design

ECE232: Hardware Organization and Design ECE232: Hardware Organization and Design Lecture 11: Floating Point & Floating Point Addition Adapted from Computer Organization and Design, Patterson & Hennessy, UCB Last time: Single Precision Format

More information

Data Representation Floating Point

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:

More information

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. 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

More information

Data Representation Floating Point

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:

More information

Chapter 03: Computer Arithmetic. Lesson 09: Arithmetic using floating point numbers

Chapter 03: Computer Arithmetic. Lesson 09: Arithmetic using floating point numbers Chapter 03: Computer Arithmetic Lesson 09: Arithmetic using floating point numbers Objective To understand arithmetic operations in case of floating point numbers 2 Multiplication of Floating Point Numbers

More information

Foundations of Computer Systems

Foundations of Computer Systems 18-600 Foundations of Computer Systems Lecture 4: Floating Point Required Reading Assignment: Chapter 2 of CS:APP (3 rd edition) by Randy Bryant & Dave O Hallaron Assignments for This Week: Lab 1 18-600

More information

Floating Point. CSE 351 Autumn Instructor: Justin Hsia

Floating Point. CSE 351 Autumn Instructor: Justin Hsia Floating Point CSE 351 Autumn 2016 Instructor: Justin Hsia Teaching Assistants: Chris Ma Hunter Zahn John Kaltenbach Kevin Bi Sachin Mehta Suraj Bhat Thomas Neuman Waylon Huang Xi Liu Yufang Sun http://xkcd.com/899/

More information

Floating Point (with contributions from Dr. Bin Ren, William & Mary Computer Science)

Floating Point (with contributions from Dr. Bin Ren, William & Mary Computer Science) Floating Point (with contributions from Dr. Bin Ren, William & Mary Computer Science) Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition Example and properties

More information

Floating Point January 24, 2008

Floating Point January 24, 2008 15-213 The course that gives CMU its Zip! Floating Point January 24, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties class04.ppt 15-213, S 08 Floating

More information

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. ! 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

More information

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

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

More information

Floating Point Puzzles. Lecture 3B Floating Point. IEEE Floating Point. Fractional Binary Numbers. Topics. IEEE Standard 754

Floating Point Puzzles. Lecture 3B Floating Point. IEEE Floating Point. Fractional Binary Numbers. Topics. IEEE Standard 754 Floating Point Puzzles Topics Lecture 3B Floating Point IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties For each of the following C expressions, either: Argue that

More information

Floating Point Arithmetic

Floating Point Arithmetic Floating Point Arithmetic Computer Systems, Section 2.4 Abstraction Anything that is not an integer can be thought of as . e.g. 391.1356 Or can be thought of as + /

More information

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

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

More information

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

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

More information

Floating Point Numbers

Floating Point Numbers Floating Point Numbers Computer Systems Organization (Spring 2016) CSCI-UA 201, Section 2 Instructor: Joanna Klukowska Slides adapted from Randal E. Bryant and David R. O Hallaron (CMU) Mohamed Zahran

More information

Floating Point Numbers

Floating Point Numbers Floating Point Numbers Computer Systems Organization (Spring 2016) CSCI-UA 201, Section 2 Fractions in Binary Instructor: Joanna Klukowska Slides adapted from Randal E. Bryant and David R. O Hallaron (CMU)

More information

15213 Recitation 2: Floating Point

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

More information

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 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

More information

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. 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

More information

Module 2: Computer Arithmetic

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

More information

Floating Point Puzzles. Lecture 3B Floating Point. IEEE Floating Point. Fractional Binary Numbers. Topics. IEEE Standard 754

Floating Point Puzzles. Lecture 3B Floating Point. IEEE Floating Point. Fractional Binary Numbers. Topics. IEEE Standard 754 Floating Point Puzzles Topics Lecture 3B Floating Point IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties For each of the following C expressions, either: Argue that

More information

Floating Point. CSE 351 Autumn Instructor: Justin Hsia

Floating Point. CSE 351 Autumn Instructor: Justin Hsia Floating Point CSE 351 Autumn 2017 Instructor: Justin Hsia Teaching Assistants: Lucas Wotton Michael Zhang Parker DeWilde Ryan Wong Sam Gehman Sam Wolfson Savanna Yee Vinny Palaniappan Administrivia Lab

More information

Representing and Manipulating Floating Points

Representing and Manipulating Floating Points Representing and Manipulating Floating Points Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu The Problem How to represent fractional values with

More information

Representing and Manipulating Floating Points. Computer Systems Laboratory Sungkyunkwan University

Representing and Manipulating Floating Points. Computer Systems Laboratory Sungkyunkwan University Representing and Manipulating Floating Points Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu The Problem How to represent fractional values with

More information

Giving credit where credit is due

Giving credit where credit is due CSCE 230J Computer Organization Floating Point Dr. Steve Goddard goddard@cse.unl.edu http://cse.unl.edu/~goddard/courses/csce230j Giving credit where credit is due Most of slides for this lecture are based

More information

Bryant and O Hallaron, Computer Systems: A Programmer s Perspective, Third Edition. Carnegie Mellon

Bryant and O Hallaron, Computer Systems: A Programmer s Perspective, Third Edition. Carnegie Mellon Carnegie Mellon Floating Point 15-213/18-213/14-513/15-513: Introduction to Computer Systems 4 th Lecture, Sept. 6, 2018 Today: Floating Point Background: Fractional binary numbers IEEE floating point

More information

System Programming CISC 360. Floating Point September 16, 2008

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:

More information

Giving credit where credit is due

Giving credit where credit is due JDEP 284H Foundations of Computer Systems Floating Point Dr. Steve Goddard goddard@cse.unl.edu Giving credit where credit is due Most of slides for this lecture are based on slides created by Drs. Bryant

More information

Data Representation Floating Point

Data Representation Floating Point Data Representation Floating Point CSCI 224 / ECE 317: Computer Architecture Instructor: Prof. Jason Fritts Slides adapted from Bryant & O Hallaron s slides Today: Floating Point Background: Fractional

More information

Floating Point. CSE 238/2038/2138: Systems Programming. Instructor: Fatma CORUT ERGİN. Slides adapted from Bryant & O Hallaron s slides

Floating Point. CSE 238/2038/2138: Systems Programming. Instructor: Fatma CORUT ERGİN. Slides adapted from Bryant & O Hallaron s slides Floating Point CSE 238/2038/2138: Systems Programming Instructor: Fatma CORUT ERGİN Slides adapted from Bryant & O Hallaron s slides Today: Floating Point Background: Fractional binary numbers IEEE floating

More information

Floating Point Numbers

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

More information

FLOATING POINT NUMBERS

FLOATING POINT NUMBERS Exponential Notation FLOATING POINT NUMBERS Englander Ch. 5 The following are equivalent representations of 1,234 123,400.0 x 10-2 12,340.0 x 10-1 1,234.0 x 10 0 123.4 x 10 1 12.34 x 10 2 1.234 x 10 3

More information

Floating Point : Introduction to Computer Systems 4 th Lecture, May 25, Instructor: Brian Railing. Carnegie Mellon

Floating Point : Introduction to Computer Systems 4 th Lecture, May 25, Instructor: Brian Railing. Carnegie Mellon Floating Point 15-213: Introduction to Computer Systems 4 th Lecture, May 25, 2018 Instructor: Brian Railing Today: Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition

More information

Representing and Manipulating Floating Points

Representing and Manipulating Floating Points Representing and Manipulating Floating Points Jinkyu Jeong (jinkyu@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu SSE23: Introduction to Computer Systems, Spring 218,

More information

Representing and Manipulating Floating Points

Representing and Manipulating Floating Points Representing and Manipulating Floating Points Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu The Problem How to represent fractional values with

More information

CS429: Computer Organization and Architecture

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

More information

COMP2611: Computer Organization. Data Representation

COMP2611: Computer Organization. Data Representation COMP2611: Computer Organization Comp2611 Fall 2015 2 1. Binary numbers and 2 s Complement Numbers 3 Bits: are the basis for binary number representation in digital computers What you will learn here: How

More information

Floating Point. CSE 351 Autumn Instructor: Justin Hsia

Floating Point. CSE 351 Autumn Instructor: Justin Hsia Floating Point CSE 351 Autumn 2017 Instructor: Justin Hsia Teaching Assistants: Lucas Wotton Michael Zhang Parker DeWilde Ryan Wong Sam Gehman Sam Wolfson Savanna Yee Vinny Palaniappan http://xkcd.com/571/

More information

Representing and Manipulating Floating Points. Jo, Heeseung

Representing and Manipulating Floating Points. Jo, Heeseung Representing and Manipulating Floating Points Jo, Heeseung The Problem How to represent fractional values with finite number of bits? 0.1 0.612 3.14159265358979323846264338327950288... 2 Fractional Binary

More information

Floating Point Numbers

Floating Point Numbers Floating Point Floating Point Numbers Mathematical background: tional binary numbers Representation on computers: IEEE floating point standard Rounding, addition, multiplication Kai Shen 1 2 Fractional

More information

Today: Floating Point. Floating Point. Fractional Binary Numbers. Fractional binary numbers. bi bi 1 b2 b1 b0 b 1 b 2 b 3 b j

Today: Floating Point. Floating Point. Fractional Binary Numbers. Fractional binary numbers. bi bi 1 b2 b1 b0 b 1 b 2 b 3 b j Floating Point 15 213: Introduction to Computer Systems 4 th Lecture, Jan 24, 2013 Instructors: Seth Copen Goldstein, Anthony Rowe, Greg Kesden 2 Fractional binary numbers What is 1011.101 2? Fractional

More information

Floating Point Numbers. Lecture 9 CAP

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:

More information

Computer Organization: A Programmer's Perspective

Computer Organization: A Programmer's Perspective A Programmer's Perspective Representing Numbers Gal A. Kaminka galk@cs.biu.ac.il Fractional Binary Numbers 2 i 2 i 1 4 2 1 b i b i 1 b 2 b 1 b 0. b 1 b 2 b 3 b j 1/2 1/4 1/8 Representation Bits to right

More information

Numeric Encodings Prof. James L. Frankel Harvard University

Numeric Encodings Prof. James L. Frankel Harvard University Numeric Encodings Prof. James L. Frankel Harvard University Version of 10:19 PM 12-Sep-2017 Copyright 2017, 2016 James L. Frankel. All rights reserved. Representation of Positive & Negative Integral and

More information

Integers and Floating Point

Integers and Floating Point CMPE12 More about Numbers Integers and Floating Point (Rest of Textbook Chapter 2 plus more)" Review: Unsigned Integer A string of 0s and 1s that represent a positive integer." String is X n-1, X n-2,

More information

Computer Arithmetic Floating Point

Computer Arithmetic Floating Point Computer Arithmetic Floating Point Chapter 3.6 EEC7 FQ 25 About Floating Point Arithmetic Arithmetic basic operations on floating point numbers are: Add, Subtract, Multiply, Divide Transcendental operations

More information

Finite arithmetic and error analysis

Finite arithmetic and error analysis Finite arithmetic and error analysis Escuela de Ingeniería Informática de Oviedo (Dpto de Matemáticas-UniOvi) Numerical Computation Finite arithmetic and error analysis 1 / 45 Outline 1 Number representation:

More information

Computer Architecture and IC Design Lab. Chapter 3 Part 2 Arithmetic for Computers Floating Point

Computer Architecture and IC Design Lab. Chapter 3 Part 2 Arithmetic for Computers Floating Point Chapter 3 Part 2 Arithmetic for Computers Floating Point Floating Point Representation for non integral numbers Including very small and very large numbers 4,600,000,000 or 4.6 x 10 9 0.0000000000000000000000000166

More information

Chapter 3: Arithmetic for Computers

Chapter 3: Arithmetic for Computers Chapter 3: Arithmetic for Computers Objectives Signed and Unsigned Numbers Addition and Subtraction Multiplication and Division Floating Point Computer Architecture CS 35101-002 2 The Binary Numbering

More information

CS 261 Fall Floating-Point Numbers. Mike Lam, Professor. https://xkcd.com/217/

CS 261 Fall Floating-Point Numbers. Mike Lam, Professor. https://xkcd.com/217/ CS 261 Fall 2017 Mike Lam, Professor https://xkcd.com/217/ Floating-Point Numbers Floating-point Topics Binary fractions Floating-point representation Conversions and rounding error Binary fractions Now

More information

3.5 Floating Point: Overview

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

More information

Divide: Paper & Pencil

Divide: Paper & Pencil Divide: Paper & Pencil 1001 Quotient Divisor 1000 1001010 Dividend -1000 10 101 1010 1000 10 Remainder See how big a number can be subtracted, creating quotient bit on each step Binary => 1 * divisor or

More information

EE 109 Unit 19. IEEE 754 Floating Point Representation Floating Point Arithmetic

EE 109 Unit 19. IEEE 754 Floating Point Representation Floating Point Arithmetic 1 EE 109 Unit 19 IEEE 754 Floating Point Representation Floating Point Arithmetic 2 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadro s Number: +6.0247 * 10

More information

CS 261 Fall Floating-Point Numbers. Mike Lam, Professor.

CS 261 Fall Floating-Point Numbers. Mike Lam, Professor. CS 261 Fall 2018 Mike Lam, Professor https://xkcd.com/217/ Floating-Point Numbers Floating-point Topics Binary fractions Floating-point representation Conversions and rounding error Binary fractions Now

More information

Machine Arithmetic 8/31/2007

Machine Arithmetic 8/31/2007 Machine Arithmetic 8/31/2007 1 Opening Discussion Let's look at some interclass problems. If you played with your program some you probably found that it behaves oddly in some regards. Why is this? What

More information

EE 109 Unit 20. IEEE 754 Floating Point Representation Floating Point Arithmetic

EE 109 Unit 20. IEEE 754 Floating Point Representation Floating Point Arithmetic 1 EE 109 Unit 20 IEEE 754 Floating Point Representation Floating Point Arithmetic 2 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadro s Number: +6.0247 * 10

More information

CS 33. Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

CS 33. Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved. CS 33 Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved. Byte-Oriented Memory Organization 00 0 FF F Programs refer to data by address

More information

Computer Architecture Chapter 3. Fall 2005 Department of Computer Science Kent State University

Computer Architecture Chapter 3. Fall 2005 Department of Computer Science Kent State University Computer Architecture Chapter 3 Fall 2005 Department of Computer Science Kent State University Objectives Signed and Unsigned Numbers Addition and Subtraction Multiplication and Division Floating Point

More information

CSCI 402: Computer Architectures. Arithmetic for Computers (3) Fengguang Song Department of Computer & Information Science IUPUI.

CSCI 402: Computer Architectures. Arithmetic for Computers (3) Fengguang Song Department of Computer & Information Science IUPUI. CSCI 402: Computer Architectures Arithmetic for Computers (3) Fengguang Song Department of Computer & Information Science IUPUI 3.5 Today s Contents Floating point numbers: 2.5, 10.1, 100.2, etc.. How

More information

Floating Point Arithmetic

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

More information

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

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,

More information

EE260: Logic Design, Spring n Integer multiplication. n Booth s algorithm. n Integer division. n Restoring, non-restoring

EE260: Logic Design, Spring n Integer multiplication. n Booth s algorithm. n Integer division. n Restoring, non-restoring EE 260: Introduction to Digital Design Arithmetic II Yao Zheng Department of Electrical Engineering University of Hawaiʻi at Mānoa Overview n Integer multiplication n Booth s algorithm n Integer division

More information

Organisasi Sistem Komputer

Organisasi Sistem Komputer LOGO Organisasi Sistem Komputer OSK 8 Aritmatika Komputer 1 1 PT. Elektronika FT UNY Does the calculations Arithmetic & Logic Unit Everything else in the computer is there to service this unit Handles

More information

IEEE Floating Point Numbers Overview

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

More information

Systems Programming and Computer Architecture ( )

Systems Programming and Computer Architecture ( ) (252-0061-00) Session 9 Floating Point Systems Group Department of Computer Science ETH Zürich 1 Floating Point Recap for the Assignment 2 Floating Point Representation Numerical Form Scientific Notation

More information

CS101 Lecture 04: Binary Arithmetic

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

More information

CO212 Lecture 10: Arithmetic & Logical Unit

CO212 Lecture 10: Arithmetic & Logical Unit CO212 Lecture 10: Arithmetic & Logical Unit Shobhanjana Kalita, Dept. of CSE, Tezpur University Slides courtesy: Computer Architecture and Organization, 9 th Ed, W. Stallings Integer Representation For

More information

Number Systems. Both numbers are positive

Number Systems. Both numbers are positive Number Systems Range of Numbers and Overflow When arithmetic operation such as Addition, Subtraction, Multiplication and Division are performed on numbers the results generated may exceed the range of

More information

ANITA S SUPER AWESOME RECITATION SLIDES. 15/18-213: Introduction to Computer Systems Bit Logic and Floating Point, 27 January 2014 Anita Zhang

ANITA S SUPER AWESOME RECITATION SLIDES. 15/18-213: Introduction to Computer Systems Bit Logic and Floating Point, 27 January 2014 Anita Zhang ANITA S SUPER AWESOME RECITATION SLIDES 15/18-213: Introduction to Computer Systems Bit Logic and Floating Point, 27 January 2014 Anita Zhang WELCOME TO THE SPRING EDITION Data Lab due Thursday, 30 Jan

More information

Arithmetic for Computers. Hwansoo Han

Arithmetic for Computers. Hwansoo Han Arithmetic for Computers Hwansoo Han Arithmetic for Computers Operations on integers Addition and subtraction Multiplication and division Dealing with overflow Floating-point real numbers Representation

More information

COSC 243. Data Representation 3. Lecture 3 - Data Representation 3 1. COSC 243 (Computer Architecture)

COSC 243. Data Representation 3. Lecture 3 - Data Representation 3 1. COSC 243 (Computer Architecture) COSC 243 Data Representation 3 Lecture 3 - Data Representation 3 1 Data Representation Test Material Lectures 1, 2, and 3 Tutorials 1b, 2a, and 2b During Tutorial a Next Week 12 th and 13 th March If you

More information

In this lesson you will learn: how to add and multiply positive binary integers how to work with signed binary numbers using two s complement how fixed and floating point numbers are used to represent

More information

Floating-Point Arithmetic

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

More information

COMP Overview of Tutorial #2

COMP Overview of Tutorial #2 COMP 1402 Winter 2008 Tutorial #2 Overview of Tutorial #2 Number representation basics Binary conversions Octal conversions Hexadecimal conversions Signed numbers (signed magnitude, one s and two s complement,

More information

Number Systems and Binary Arithmetic. Quantitative Analysis II Professor Bob Orr

Number Systems and Binary Arithmetic. Quantitative Analysis II Professor Bob Orr Number Systems and Binary Arithmetic Quantitative Analysis II Professor Bob Orr Introduction to Numbering Systems We are all familiar with the decimal number system (Base 10). Some other number systems

More information

Chapter 2 Data Representations

Chapter 2 Data Representations Computer Engineering Chapter 2 Data Representations Hiroaki Kobayashi 4/21/2008 4/21/2008 1 Agenda in Chapter 2 Translation between binary numbers and decimal numbers Data Representations for Integers

More information

Number Systems. Decimal numbers. Binary numbers. Chapter 1 <1> 8's column. 1000's column. 2's column. 4's column

Number Systems. Decimal numbers. Binary numbers. Chapter 1 <1> 8's column. 1000's column. 2's column. 4's column 1's column 10's column 100's column 1000's column 1's column 2's column 4's column 8's column Number Systems Decimal numbers 5374 10 = Binary numbers 1101 2 = Chapter 1 1's column 10's column 100's

More information

Chapter 4 Section 2 Operations on Decimals

Chapter 4 Section 2 Operations on Decimals Chapter 4 Section 2 Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers.

More information

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

Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction 1 Floating Point The World is Not Just Integers Programming languages support numbers with fraction Called floating-point numbers Examples: 3.14159265 (π) 2.71828 (e) 0.000000001 or 1.0 10 9 (seconds in

More information

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

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

More information

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.

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

More information

ecture 25 Floating Point Friedland and Weaver Computer Science 61C Spring 2017 March 17th, 2017

ecture 25 Floating Point Friedland and Weaver Computer Science 61C Spring 2017 March 17th, 2017 ecture 25 Computer Science 61C Spring 2017 March 17th, 2017 Floating Point 1 New-School Machine Structures (It s a bit more complicated!) Software Hardware Parallel Requests Assigned to computer e.g.,

More information

Floating Point. EE 109 Unit 20. Floating Point Representation. Fixed Point

Floating Point. EE 109 Unit 20. Floating Point Representation. Fixed Point 2.1 Floating Point 2.2 EE 19 Unit 2 IEEE 754 Floating Point Representation Floating Point Arithmetic Used to represent very numbers (fractions) and very numbers Avogadro s Number: +6.247 * 1 23 Planck

More information

Signed Multiplication Multiply the positives Negate result if signs of operand are different

Signed Multiplication Multiply the positives Negate result if signs of operand are different Another Improvement Save on space: Put multiplier in product saves on speed: only single shift needed Figure: Improved hardware for multiplication Signed Multiplication Multiply the positives Negate result

More information

CS 61C: Great Ideas in Computer Architecture Floating Point Arithmetic

CS 61C: Great Ideas in Computer Architecture Floating Point Arithmetic CS 61C: Great Ideas in Computer Architecture Floating Point Arithmetic Instructors: Vladimir Stojanovic & Nicholas Weaver http://inst.eecs.berkeley.edu/~cs61c/ New-School Machine Structures (It s a bit

More 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 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

More information

IEEE Standard for Floating-Point Arithmetic: 754

IEEE Standard for Floating-Point Arithmetic: 754 IEEE Standard for Floating-Point Arithmetic: 754 G.E. Antoniou G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: 754 1 / 34 Floating Point Standard: IEEE 754 1985/2008 Established in 1985 (2008)

More information

CHW 261: Logic Design

CHW 261: Logic Design CHW 261: Logic Design Instructors: Prof. Hala Zayed Dr. Ahmed Shalaby http://www.bu.edu.eg/staff/halazayed14 http://bu.edu.eg/staff/ahmedshalaby14# Slide 1 Slide 2 Slide 3 Digital Fundamentals CHAPTER

More information

Computer Systems C S Cynthia Lee

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

More information

The course that gives CMU its Zip! Floating Point Arithmetic Feb 17, 2000

The course that gives CMU its Zip! Floating Point Arithmetic Feb 17, 2000 15-213 The course that gives CMU its Zip! Floating Point Arithmetic Feb 17, 2000 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IA32 floating point Floating

More information

Data Representations & Arithmetic Operations

Data Representations & Arithmetic Operations Data Representations & Arithmetic Operations Hiroaki Kobayashi 7/13/2011 7/13/2011 Computer Science 1 Agenda Translation between binary numbers and decimal numbers Data Representations for Integers Negative

More information

Characters, Strings, and Floats

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

More information

Chapter 3 Arithmetic for Computers (Part 2)

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

More information

Lecture 13: (Integer Multiplication and Division) FLOATING POINT NUMBERS

Lecture 13: (Integer Multiplication and Division) FLOATING POINT NUMBERS Lecture 13: (Integer Multiplication and Division) FLOATING POINT NUMBERS Lecture 13 Floating Point I (1) Fall 2005 Integer Multiplication (1/3) Paper and pencil example (unsigned): Multiplicand 1000 8

More information

Floating-Point Arithmetic

Floating-Point Arithmetic Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } L11 Floating Point 1 What is the problem? Many numeric applications require numbers over a VERY large range. (e.g. nanoseconds to centuries)

More information

Inf2C - Computer Systems Lecture 2 Data Representation

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

More information

Introduction to Computers and Programming. Numeric Values

Introduction to Computers and Programming. Numeric Values Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture 5 Reading: B pp. 47-71 Sept 1 003 Numeric Values Storing the value of 5 10 using ASCII: 00110010 00110101 Binary notation: 00000000

More information

Number Systems and Computer Arithmetic

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

More information