# mith College Computer Science Fixed & Floating Point Formats CSC231 Fall 2017 Week #15 Dominique Thiébaut

Save this PDF as:

Size: px
Start display at page:

Download "mith College Computer Science Fixed & Floating Point Formats CSC231 Fall 2017 Week #15 Dominique Thiébaut"

## Transcription

1 mith College Computer Science Fixed & Floating Point CSC231 Fall 2017 Week #15 Formats Dominique Thiébaut

2 Trick Question for ( double d = 0; d!= 0.3; d += 0.1 ) System.out.println( d );

3 We stopped here last time

4 Normalization (in decimal) y = (normal = standard form) y = x 10 2 y = x 10 11

5 Normalization (in binary) y = ( d) y = x 2 3 y = x 10 11

6 Normalization (in binary) y = decimal y = x 2 3

7 Normalization (in binary) y = decimal y = x 2 3 y = x binary

8 x sign mantissa exponent

9 But, remember, all* numbers have a leading 1, so, we can pack the bits even more efficiently! *really?

10 implied bit! x sign mantissa exponent

11 IEEE Format S Exponent (8) Mantissa (23) 24 bits stored in 23 bits!

12 y = y = y = bbbbbbbb

13 y = x bbbbbbbb Why not ?

14 How is the exponent coded?

15 bbbbbbbb bias of 127

16 bbbbbbbb

17 y = x * 2^3 Ah! 3 represented by 130 = =

18 Verification in IEEE FP

19 Exercises How is 1.0 coded as a 32-bit floating point number? What about 0.5? 1.5? -1.5? what floating-point value is stored in the 32-bit number below?

20 1.0 = = = = = = - (stored exp = 131 > real exp = 4) = x 2^4 = x 16 = -31

22 0.1 decimal, in 32-bit precision, IEEE Format:

23 0.1 decimal, in 32-bit precision, IEEE Format: Value in double-precision:

24 NEVER NEVER COMPARE FLOATS OR DOUBLEs FOR EQUALITY! N-E-V-E-R!

25 for ( double d = 0; d!= 0.3; d += 0.1 ) System.out.println( d );

26 bbbbbbbb

27 Zero Why is it special? 0.0 =

28 bbbbbbbb {if mantissa is 0: number = 0.0

29 Very Small Numbers Smallest numbers have stored exponent of 0. In this case, the implied 1 is omitted, and the exponent is -126 (not -127!)

30 bbbbbbbb {if mantissa is 0: number = 0.0 if mantissa is!0: no hidden 1

31 Very Small Numbers Example: (2-126 ) x ( ) binary + (2-126 ) x ( ) =

32 bbbbbbbb?

33 Very large exponents stored exponent = if the mantissa is = 0 ==> +/-

34 Very large exponents stored exponent = if the mantissa is = 0 ==> +/-

35 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN

36 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN = Not-a-Number

37 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN

38 = = = NaN

39 Operations that create NaNs ( The divisions 0/0 and ± /± The multiplications 0 ± and ± 0 The additions + ( ), ( ) + and equivalent subtractions The square root of a negative number. The logarithm of a negative number The inverse sine or cosine of a number that is less than 1 or greater than +1

40 // import java.util.*; import static java.lang.double.nan; import static java.lang.double.positive_infinity; import static java.lang.double.negative_infinity; Generating NaNs public class GenerateNaN { public static void main(string args[]) { double[] allnans = { 0D / 0D, POSITIVE_INFINITY / POSITIVE_INFINITY, POSITIVE_INFINITY / NEGATIVE_INFINITY, NEGATIVE_INFINITY / POSITIVE_INFINITY, NEGATIVE_INFINITY / NEGATIVE_INFINITY, 0 * POSITIVE_INFINITY, 0 * NEGATIVE_INFINITY, Math.pow(1, POSITIVE_INFINITY), POSITIVE_INFINITY + NEGATIVE_INFINITY, NEGATIVE_INFINITY + POSITIVE_INFINITY, POSITIVE_INFINITY - POSITIVE_INFINITY, NEGATIVE_INFINITY - NEGATIVE_INFINITY, Math.sqrt(-1), Math.log(-1), Math.asin(-2), Math.acos(+2), }; System.out.println(Arrays.toString(allNaNs)); System.out.println(NaN == NaN); System.out.println(Double.isNaN(NaN)); } } [NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN] false true

41 NaN is sticky!

42 Range of Floating-Point Numbers

43 Range of Floating-Point Numbers Remember that!

44 Resolution of a Floating-Point Format Check out table here:

45

46 What does it have to do with Art?

47 Rosetta Landing on Comet 10-year trajectory

48 We stopped here last time

49 Why not using 2 s Complement for the Exponent? = = = =

50 Exercises Does this converter support NaN, and? Are there several different representations of +? What is the largest float representable with the 32-bit format? What is the smallest normalized float (i.e. a float which has an implied leading 1. bit)?

51 How do we add 2 FP numbers?

52 1.111 x x x x 2 8 after expansion x x x 2 8 locate largest number shift mantissa of smaller compute sum x 2 8 = x 2 8 round & truncate normalize = x 2 9

53 fp1 = s1 m1 e1 fp2 = s2 m2 e2 fp1 + fp2 =? denormalize both numbers (restore hidden 1) assume fp1 has largest exponent e1: make e2 equal to e1 and shift decimal point in m2 > m2 compute sum m1 + m2 truncate & round result renormalize result (after checking for special cases)

54 What are the "dangers" of adding 2 FP numbers?

55 Algorithm for adding a Series of FP numbers?

56 How do we Multiply 2 FP Numbers?

57 1.111 x 2 5 x x x 2 5 x x x = x 2 1 after expansion multiply mantissas normalize round & truncate = x 2 1 normalize = 14 add exponents =

58 fp1 = s1 m1 e1 fp2 = s2 m2 e2 fp1 x fp2 =? Test for multiplication by special numbers (0, NaN, ) denormalize both numbers (restore hidden 1) compute product of m1 x m2 compute sum e1 + e2 truncate & round m1 x m2 adjust e1+e2 and normalize.

59 How do we compare two FP numbers?

60 Check sign bits first, the compare as unsigned integers! No unpacking necessary!

61 Programming FP Operations in Assembly

62 Pentium EAX EBX ECX EDX ALU

63 EAX EBX ECX EDX ALU Cannot do FP computation

64

65

66 SP0 SP1 SP2 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

67 Operation: ( )/9.0 SP0 SP1 SP2 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

68 SP0 7 Operation: (7+10)/9 fpush 7 SP1 SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

69 SP0 SP Operation: (7+10)/9 fpush 7 fpush 10 SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

70 Operation: (7+10)/9 SP0 SP1 10 fpush 7 fpush 10 fadd SP2 7 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

71 SP0 SP1 17 Operation: (7+10)/9 fpush 7 fpush 10 fadd SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

72 SP0 SP1 SP Operation: (7+10)/9 fpush 7 fpush 10 fadd fpush 9 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

73 Operation: (7+10)/9 SP0 fpush 7 SP1 SP fpush 10 fadd fpush 9 fdiv SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

74 SP0 SP1 SP2 SP Operation: (7+10)/9 fpush 7 fpush 10 fadd fpush 9 fdiv SP4 SP5 SP6 SP7 FLOATING POINT UNIT

75 The Pentium computes FP expressions using RPN!

76 The Pentium computes FP expressions using RPN! Reverse Polish Notation

77 Nasm Example: z = x+y SECTION.data x dd 1.5 y dd 2.5 z dd 0 ; compute z = x + y SECTION.text fld fld fadd fstp dword [x] dword [y] dword [z]

78 Printing floats in C #include "stdio.h" int main() { float z = e10; printf( "z = %e\n\n", z ); return 0; }

79 Printing floats in C #include "stdio.h" int main() { float z = e10; printf( "z = %e\n\n", z ); return 0; } gcc -o printfloat printfloat.c./printfloat z = e+10

80 More code examples here: CSC231_An_Introduction_to_Fixed-_and_Floating- Point_Numbers#Assembly_Language_Programs

81 THE END!

### mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Dominique Thiébaut

mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Dominique Thiébaut dthiebaut@smith.edu Reference http://cs.smith.edu/dftwiki/index.php/ CSC231_An_Introduction_to_Fixed-_and_Floating-

### mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Assembly Language Week #14 Dominique Thiébaut

mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Assembly Language Week #14 Dominique Thiébaut dthiebaut@smith.edu Reference http://cs.smith.edu/dftwiki/index.php/ CSC231_An_Introduction_to_Fixed-_and_Floating-

### mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Assembly Language Week #13 Dominique Thiébaut

mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Assembly Language Week #13 Dominique Thiébaut dthiebaut@smith.edu Reference http://cs.smith.edu/dftwiki/index.php/ CSC231_An_Introduction_to_Fixed-_and_Floating-

### 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

### 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

### mith College Computer Science Week 14 Fixed & Floating Point Formats CSC231 Fall 2017 Week #14 Dominique Thiébaut

mith College Computer Science Week 14 Fixed & Floating Point CSC231 Fall 2017 Week #14 Formats Dominique Thiébaut dthiebaut@smith.edu Reminder! In C, strings are terminated by a byte containing 0 decimal,

### 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

### Floating-point representations

Lecture 10 Floating-point representations Methods of representing real numbers (1) 1. Fixed-point number system limited range and/or limited precision results must be scaled 100101010 1111010 100101010.1111010

### mith College Computer Science CSC231 Assembly Week #5 Fall 2017 Dominique Thiébaut

mith College Computer Science CSC231 Assembly Week #5 Fall 2017 Dominique Thiébaut dthiebaut@smith.edu 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 A 10

### Floating-point representations

Lecture 10 Floating-point representations Methods of representing real numbers (1) 1. Fixed-point number system limited range and/or limited precision results must be scaled 100101010 1111010 100101010.1111010

### 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

### 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

### mith College Computer Science CSC231 - Assembly Week #3 Dominique Thiébaut

mith College Computer Science CSC231 - Assembly Week #3 Dominique Thiébaut dthiebaut@smith.edu memory mov add registers hexdump listings number systems Let's Review Last Week's Material section.data

### 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

### 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

### 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

### Operations On Data CHAPTER 4. (Solutions to Odd-Numbered Problems) Review Questions

CHAPTER 4 Operations On Data (Solutions to Odd-Numbered Problems) Review Questions 1. Arithmetic operations interpret bit patterns as numbers. Logical operations interpret each bit as a logical values

### mith College Computer Science CSC231 - Assembly Week #8 Dominique Thiébaut

mith College Computer Science CSC231 - Assembly Week #8 Dominique Thiébaut dthiebaut@smith.edu Can't Sleep https://xkcd.com/571/ Logic Design Logic Design Image from http://www.willegal.net/appleii/images/motherboard.jpg

### mith College Computer Science CSC231 - Assembly Week #4 Dominique Thiébaut

mith College Computer Science CSC231 - Assembly Week #4 Dominique Thiébaut dthiebaut@smith.edu Homework Solutions Outline Review Hexdump Pentium Data Registers 32-bit, 16-bit and 8-bit quantities (registers

### Instruction Set extensions to X86. Floating Point SIMD instructions

Instruction Set extensions to X86 Some extensions to x86 instruction set intended to accelerate 3D graphics AMD 3D-Now! Instructions simply accelerate floating point arithmetic. Accelerate object transformations

### 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

### 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

### 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

### 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

### 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

### CS 3843 Final Exam Fall 2012

CS 3843 Final Exam Fall 2012 Name (Last), (First) ID Please indicate your session: Morning Afternoon You may use a calculator and two sheets of notes on this exam, but no other materials and no computer.

### 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

### 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

### 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

### Floating Point Arithmetic. Jin-Soo Kim Computer Systems Laboratory Sungkyunkwan University

Floating Point Arithmetic Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu Floating Point (1) Representation for non-integral numbers Including very

### 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

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

Introduction to Computer Systems Recitation 2 May 29, 2014 Marjorie Carlson Aditya Gupta Shailin Desai 1 Agenda! Goal: translate any real number (plus some!) into and out of machine representation.! Integers!

### 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

### 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

### 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

### 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

### Chapter 4. Operations on Data

Chapter 4 Operations on Data 1 OBJECTIVES After reading this chapter, the reader should be able to: List the three categories of operations performed on data. Perform unary and binary logic operations

### 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

### mith College Computer Science CSC231 Assembly Week #11 Fall 2017 Dominique Thiébaut

mith College Computer Science CSC231 Assembly Week #11 Fall 2017 Dominique Thiébaut dthiebaut@smith.edu Back to Conditional Jumps Review sub eax, 10 jz there xxx xxx there:yyy yyy Review cmp eax, 10 jz

### 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,

### The ALU consists of combinational logic. Processes all data in the CPU. ALL von Neuman machines have an ALU loop.

CS 320 Ch 10 Computer Arithmetic The ALU consists of combinational logic. Processes all data in the CPU. ALL von Neuman machines have an ALU loop. Signed integers are typically represented in sign-magnitude

### 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:

### 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

### 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:

### Floating Point Arithmetic

Floating Point Arithmetic Clark N. Taylor Department of Electrical and Computer Engineering Brigham Young University clark.taylor@byu.edu 1 Introduction Numerical operations are something at which digital

### 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. Borrowed from the Carnegie Mellon (Thanks, guys!) Carnegie Mellon

Floating Point Borrowed from the 15-213 Team @ Carnegie Mellon (Thanks, guys!) Today: Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition Example and properties

### Floating-Point Arithmetic

ENEE446---Lectures-4/10-15/08 A. Yavuz Oruç Professor, UMD, College Park Copyright 2007 A. Yavuz Oruç. All rights reserved. Floating-Point Arithmetic Integer or fixed-point arithmetic provides a complete

### Computer Architecture and Assembly Language. Practical Session 5

Computer Architecture and Assembly Language Practical Session 5 Addressing Mode - "memory address calculation mode" An addressing mode specifies how to calculate the effective memory address of an operand.

### 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

### Floating Point Arithmetic

Floating Point Arithmetic Jinkyu Jeong (jinkyu@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu EEE3050: Theory on Computer Architectures, Spring 2017, Jinkyu Jeong (jinkyu@skku.edu)

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

### mith College Computer Science CSC231 Assembly Week #9 Fall 2017 Dominique Thiébaut

mith College Computer Science CSC231 Assembly Week #9 Fall 2017 Dominique Thiébaut dthiebaut@smith.edu Looping Through Arrays LOOP INSTRUCTION Looping Through Arrays INDIRECT ADDRESSING MODE Indirect Addressing

### Fixed Point. Basic idea: Choose a xed place in the binary number where the radix point is located. For the example above, the number is

Real Numbers How do we represent real numbers? Several issues: How many digits can we represent? What is the range? How accurate are mathematical operations? Consistency... Is a + b = b + a? Is (a + b)

### 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,

### 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

### 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:

### 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

### 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

### 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

### 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

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

### 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)

### 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

### 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

### 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

### Chapter Three. Arithmetic

Chapter Three 1 Arithmetic Where we've been: Performance (seconds, cycles, instructions) Abstractions: Instruction Set Architecture Assembly Language and Machine Language What's up ahead: Implementing

### 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.

### 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 + /

### 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

### 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

### 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

### Numerical computing. How computers store real numbers and the problems that result

Numerical computing How computers store real numbers and the problems that result The scientific method Theory: Mathematical equations provide a description or model Experiment Inference from data Test

### 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

### unused unused unused unused unused unused

BCD numbers. In some applications, such as in the financial industry, the errors that can creep in due to converting numbers back and forth between decimal and binary is unacceptable. For these applications

### 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

### 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

### 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

### Real Numbers. range -- need to deal with larger range, not just [not just -2^(n-1) to +2^(n-1)-1] - large numbers - small numbers (fractions)

Real Numbers range -- need to deal with larger range, not just [not just -2^(n-1) to +2^(n-1)-1] - large numbers - small numbers (fractions) Real Numbers von Neumann recommended using fixed point with

### 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

### Chapter 3. Arithmetic Text: P&H rev

Chapter 3 Arithmetic Text: P&H rev3.29.16 Arithmetic for Computers Operations on integers Addition and subtraction Multiplication and division Dealing with overflow Floating-point real numbers Representation

### 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

### 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

### 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)

### 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:

### 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

### 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

### 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

### Figure 8-1. x87 FPU Execution Environment

Sign 79 78 64 63 R7 Exponent R6 R5 R4 R3 R2 R1 R0 Data Registers Significand 0 15 Control Register 0 47 Last Instruction Pointer 0 Status Register Last Data (Operand) Pointer Tag Register 10 Opcode 0 Figure

### 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) (aritmeettis-looginen yksikkö) Does all

### 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

### 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

### Data Representation Type of Data Representation Integers Bits Unsigned 2 s Comp Excess 7 Excess 8

Data Representation At its most basic level, all digital information must reduce to 0s and 1s, which can be discussed as binary, octal, or hex data. There s no practical limit on how it can be interpreted

### 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

### 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

### 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

### Computer Systems Programming. Practice Midterm. Name:

Computer Systems Programming Practice Midterm Name: 1. (4 pts) (K&R Ch 1-4) What is the output of the following C code? main() { int i = 6; int j = -35; printf( %d %d\n,i++, ++j); i = i >

### 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