# CS101 Introduction to computing Floating Point Numbers

Save this PDF as:

Size: px
Start display at page:

Download "CS101 Introduction to computing Floating Point Numbers"

## Transcription

1 CS101 Introduction to computing Floating Point Numbers A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1

2 Outline Need to floating point number Number representation : IEEE 754 Floating point range Floating point density Accuracy Arithmetic and Logical Operation on FP Conversions and type casting in C 2

3 Need to go beyond integers integer 7 rational 5/8 real 3 complex 2 3 i complex real rational integer Extremely large and small values: distance pluto sun = m mass of electron = x gm

4 Representing fractions Integer pairs (for rational numbers) 5 8 = 5/8 Strings with explicit it decimal point Implicit point at a fixed position Floating point fraction x base power implicit point

5 Numbers with binary point = 1x x x x x2 2 = = = x 2 = x 2 = x 2 = x 2 = 1 +.6

6 Numeric Data Type char, short, int, long int char : 8 bit number (1 byte=1b) short: 16 bit number (2 byte) int : 32 bit number (4B) long int : 64 bit number (8B) float, double, long double float : 32 bit number (4B) double : 64 bit number (8B) long double : 128 bit number (16B) 6

7 Numeric Data Type unsigned char char unsigned short short Unsigned int int 7

8 Numeric Data Type char, short, int, long int We have : Signed and unsigned version char (8 bit) char : 128 to 127, we have +0 and 00 Fun unsigned char: 0 to 255 int : 2 31 to unsigned int : 0 to float, double, long double For fractional, real number data All these numberedare signed and get stored in different format 8

9 Sign bit Numeric Data Type Exponent Mantissa float Exponent Mantiss 1 Mantissa 2 double 9

10 FP numbers with base = 10 ( 1) S x F x 10 E S = Sign F = Fraction (fixed point number) usually called Mantissa or Significand E = Exponent (positive or negative integer) Example 5.9x10 12, 2.6x x Only one non zero digit left to the point

11 FP numbers with base = 2 ( 1) S x F x 2 E S = Sign F = Fraction (fixed point number) usually called Mantissa or Significand E = Exponent (positive or negative integer) How to divide a word into S, F and E? How to represent S, F and E? Example x2 12, x x 2 18 Only one non zero digit left to the point: default it will be 1 incase of binary So no need to store this

12 IEEE 754 standard Single precision numbers S E F Double precision numbers S E F

13 Representing F in IEEE 754 Single precision numbers F Double precision numbers F Only one non zero digit left to the point: default it will be 1 incase of binary. So no need to store this bit

14 Value Range for F Single precision numbers 1 F or 1 F <2 Double precision numbers 1 F or 1 F <2 These are normalized.

15 Representing E in IEEE 754 Single precision numbers E bias 127 Double precision numbers E bias 1023

16 Floating point values E=E 127, V =( 1) s x 1.M x 2 E 127 V= x 2 (40 127) = x 2 87 Single precision numbers S E F 16

17 Floating point values E=E 127, V =( 1) s x 1.M x 2 E 127 V= 1.1 x 2 ( ) = 1.1 x 2 1 = 0.11x2 0 = 0.11 = 11/2 2 10= 3/4 10 = Single precision numbers S E F 17

18 Value Range for E Single precision numbers 126 E 127 (all 0 s and all 1 s have special meanings) Double precision numbers 1022 E 1023 (ll0 (all 0 s and all 1 s have special meanings)

19 Floating point demo applet on the web schmidt.net/floatconverter/ieee754.html Google Float applet to get the above link 19

20 Overflow and underflow largest positive/negative number (SP) = ±( ) x ±2 x smallest positive/negative number (SP) = ± 1 x ±2 x Largest positive/negative number (DP) = ±(2 ( 2 52 ) x ± 2 x Smallest positive/negative number (DP) = ± 1 x ± 2 x

21 Density of int vs float Int : 32 bit Exponent Mantissa Float : 32 bit Number of number can be represented Both the cases (float, int) : 2 32 Range int ( 2 31 to ) float Large ±(2 ( 2 23 ) x Small± 1 x % of float numbers are Small (less then ±1 ) 21

22 Density of Floating Points 256 Persons in Room of Capacity 256 (Range) 8 bit integer : 256/256 = person in Room of Capacity (Range) 1 st Row should be filled with 128 person 50% number with negative power are 1 < N > +1 Density of Floating point number is Dense towards 0 Sparse towards

23 Expressible Numbers(int and float) Expressible integers overflow + overflow Expressible Float underflow + underflow overflow ( )x overflow 0 0.5x x2 127 ( )x2 128

24 Distribution of Values 6 bit IEEE like format e = 3 exponent bits f = 2 fraction bits Bias is 3 Notice how the distribution gets denser toward zero Denormalized Normalized Infinity

25 Distribution of Values 6 bit IEEE like format e = 3 exponent bits f = 2 fraction bits Bias is 3 (l (close up view) Denormalized Normalized Infinity

26 Density of 32 bit float SP Fraction/mantissa is 23 bit Number of different number can be stored for particular value of exponent Assume for exp=1, 2 23 =8x1024x1024 8x10 6 Between 1 2 we can store 8x10 6 numbers Similarly for exp=2, between 2 4, 8x10 6 number of number can be stored for exp=3, between 4 8, 8x10 6 number of number can be stored for exp=4, between 8 16, 8x10 6 number of number can be stored 26

27 Similarly Density of 32 bit float SP for exp=23, between , 810 8x10 6 number of number can be stored for exp=24, bt between , 810 8x10 6 number of number can be stored OK for exp=25, between , 8x10 6 number of number can be stored >8 x10 6 BAD for exp=127, between , 8x10 6 number of number can be stored WROST 27

28 Density of 32 bit float SP 2 23 =8x1024x1024 8x

29 Numbers in float format largest positive/negative number (SP) = ±( ) x ± 2 x Second largest number : ±( ) x Difference Largest FP 2 nd largest FP = ( )x2 127 =2x2 105 =2x10 32 Smallest positive/negative number (SP) = ± 1 x ±2 x

30 Addition/Sub of Floating Point Step I: 3.2 x 10 8 ± 2.8 x 10 6 Align Exponents 320 x 10 6 ± 2.8 x 10 6 Step 2: Add Mantissas x 10 6 Step 3: Normalize x

31 Floating point operations: ADD Add/subtract A = A1 ± A2 [( 1) S1 x F1 x 2 E1 ] ± [( 1) S2 x F2 x 2 E2 ] suppose E1 > E2, then we can write it as pp, [( 1) S1 x F1 x 2 E1 ] ± [( 1) S2 x F2 x 2 E1 ] where F2 = F2 / 2 E1 E2, 32x10 8 ± 28x x 10 8 ± 2.8 x x 10 6 ± 2.8 x 10 6 The result is x 10 6 ( 1) S1 x (F1 ± F2 ) x 2 E x 10 8 It may need to be normalized

32 Testing Associatively with FP X= 1.5x10 38, Y=1.5x10 38, z= X+(Y+Z) = 1.5x (1.5x ) = 1.5x x10 38 =0 (X+Y)+Z = ( 1.5x x10 38 ) = =

33 Multiply Floating Point Step I: x 10 8 X x 10 6 Multiply Mantissas 3.2 X 5.8 X 10 8 x 10 6 Step 2: Add Exponents x Step 3: Normalize x For 32 bit SP Float : one 23 bit multiplication and 8 bit addition For 32 bit int: one 32 bit multiplication Above example: 3.2x5.8 is simpler, also 6+8 is also simpler as compared to 32 bit multiplication 33

34 Floating point operations Multiply [( 1) S1 x F1 x 2 E1 ] x [( 1) S2 x F2 x 2 E2 ] = ( 1) S1 S2 x (F1xF2) x 2 E1+E2 Since 1 (F1xF2) < 4, the result may need to be normalized 32x X 58x X 5.8 X 10 8 x x x 10 15

35 Floating point operations Divide [( 1) S1 x F1 x 2 E1 ] [( 1) S2 x F2 x 2 E2 ] = ( 1) S1 S2 x (F1 F2) x 2 E1 E2 Since.5 < (F1 F2) < 2, the result may need to be normalized (assume F2 0)

36 Float and double Float : single precision floating point Double : Double precision floating point Floating points operation are slower But not in newer PC Double operation are even slower Precision/Accuracy in Calculation Integer Float Double Speed of Calculation 36

37 Floating point Comparison Three phases Phase I: Compare sign (give result) Phase II: If (sign of both numbers are same) Compare exponents and give result 90% of case it fall in this categories Faster as compare to integer comparison : Require only 8 bit comparison for float and 11 bit for double (Example : sorting of float numbers) ( p g ) Phase III: If (both sign and exponents are same) compare fraction/mantissa

38 Storing and Printing Floating Point float x=145.0,y; y=sqrt(sqrt((x))); x=(y*y)*(y*y); printf("\nx=%f",x); float x=1.0/3.0; if ( x==1.0/3.0) 0) printf( YES ); else printf( NO ); Many Round off cause loss of accuracy x= Value stored in x is not exactly same as 1.0/3.0 One is before round of and other (stored x) is after round of NO 38

39 Storing and Printing Floating Point float a= ; ; float b= ; float c= ; ; printf("\na=%8.6f, b=%8.6f c=%8.12f\n", a, b, c ); a= b= c= Big number with small fraction can not combined 39

40 Storing and Printing Floating Point //15 S digits to store float a= ; //8 S digits to store float b= ; //6 S digits to store float c= ; Thumb rule: 8 to 9 significant digits of a number can be stored in a 32 bit number 40

41 Thanks 41

### & Control Flow of Program

CS101 Introduction to computing Recap of Floating Point & Control Flow of Program A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1 Outline Recap Operation on Floating

### 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 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

### 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:

### 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

### 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 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

### 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

### 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

### 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

### 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:

### 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

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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

### 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,

### 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

### 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 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

### 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

### 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

### 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

### 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

### 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

### CS61c Midterm Review (fa06) Number representation and Floating points From your friendly reader

CS61c Midterm Review (fa06) Number representation and Floating points From your friendly reader Number representation (See: Lecture 2, Lab 1, HW#1) KNOW: Kibi (2 10 ), Mebi(2 20 ), Gibi(2 30 ), Tebi(2

### 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

### 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

### 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

### 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)

### Up next. Midterm. Today s lecture. To follow

Up next Midterm Next Friday in class Exams page on web site has info + practice problems Excited for you to rock the exams like you have been the assignments! Today s lecture Back to numbers, bits, data

### 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

### 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

### Description Hex M E V smallest value > largest denormalized negative infinity number with hex representation 3BB0 ---

CSE2421 HOMEWORK #2 DUE DATE: MONDAY 11/5 11:59pm PROBLEM 2.84 Given a floating-point format with a k-bit exponent and an n-bit fraction, write formulas for the exponent E, significand M, the fraction

### 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

### REMEMBER TO REGISTER FOR THE EXAM.

REMEMBER TO REGISTER FOR THE EXAM http://tenta.angstrom.uu.se/tenta/ Floating point representation How are numbers actually stored? Some performance consequences and tricks Encoding Byte Values Byte =

### Declaration. Fundamental Data Types. Modifying the Basic Types. Basic Data Types. All variables must be declared before being used.

Declaration Fundamental Data Types All variables must be declared before being used. Tells compiler to set aside an appropriate amount of space in memory to hold a value. Enables the compiler to perform

### 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

### 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.,

### 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)

### 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

### 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 >

### Number Systems Standard positional representation of numbers: An unsigned number with whole and fraction portions is represented as:

N Number Systems Standard positional representation of numbers: An unsigned number with whole and fraction portions is represented as: a n a a a The value of this number is given by: = a n Ka a a a a a

### 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

### Simple Data Types in C. Alan L. Cox

Simple Data Types in C Alan L. Cox alc@rice.edu Objectives Be able to explain to others what a data type is Be able to use basic data types in C programs Be able to see the inaccuracies and limitations

### 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

### Chapter 4: Data Representations

Chapter 4: Data Representations Integer Representations o unsigned o sign-magnitude o one's complement o two's complement o bias o comparison o sign extension o overflow Character Representations Floating

### Floating Point Representation. CS Summer 2008 Jonathan Kaldor

Floating Point Representation CS3220 - Summer 2008 Jonathan Kaldor Floating Point Numbers Infinite supply of real numbers Requires infinite space to represent certain numbers We need to be able to represent

### ICS Instructor: Aleksandar Kuzmanovic TA: Ionut Trestian Recitation 2

ICS 2008 Instructor: Aleksandar Kuzmanovic TA: Ionut Trestian Recitation 2 Data Representations Sizes of C Objects (in Bytes) C Data Type Compaq Alpha Typical 32-bit Intel IA32 int 4 4 4 long int 8 4 4

### 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

### Data Types. Data Types. Integer Types. Signed Integers

Data Types Data Types Dr. TGI Fernando 1 2 The fundamental building blocks of any programming language. What is a data type? A data type is a set of values and a set of operations define on these values.

### 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

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

mith College Computer Science Fixed & Floating Point CSC231 Fall 2017 Week #15 Formats Dominique Thiébaut dthiebaut@smith.edu Trick Question for ( double d = 0; d!= 0.3; d += 0.1 ) System.out.println(

### 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

### l l l l l l l Base 2; each digit is 0 or 1 l Each bit in place i has value 2 i l Binary representation is used in computers

198:211 Computer Architecture Topics: Lecture 8 (W5) Fall 2012 Data representation 2.1 and 2.2 of the book Floating point 2.4 of the book Computer Architecture What do computers do? Manipulate stored information

### 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

### 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

### 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

### 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,

### false, import, new 1 class Lecture2 { 2 3 "Data types, Variables, and Operators" 4

1 class Lecture2 { 2 3 "Data types, Variables, and Operators" 4 5 } 6 7 // Keywords: 8 byte, short, int, long, char, float, double, boolean, true, false, import, new Zheng-Liang Lu Java Programming 44

### 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

### CS 101: Computer Programming and Utilization

CS 101: Computer Programming and Utilization Jul-Nov 2017 Umesh Bellur (cs101@cse.iitb.ac.in) Lecture 3: Number Representa.ons Representing Numbers Digital Circuits can store and manipulate 0 s and 1 s.

### 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

### 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

### 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

### 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

### ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014

Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate

### CprE 281: Digital Logic

CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Floating Point Numbers CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev

### Floating-point numbers. Phys 420/580 Lecture 6

Floating-point numbers Phys 420/580 Lecture 6 Random walk CA Activate a single cell at site i = 0 For all subsequent times steps, let the active site wander to i := i ± 1 with equal probability Random

### What Every Programmer Should Know About Floating-Point Arithmetic

What Every Programmer Should Know About Floating-Point Arithmetic Last updated: October 15, 2015 Contents 1 Why don t my numbers add up? 3 2 Basic Answers 3 2.1 Why don t my numbers, like 0.1 + 0.2 add

### 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

### LESSON 5 FUNDAMENTAL DATA TYPES. char short int long unsigned char unsigned short unsigned unsigned long

LESSON 5 ARITHMETIC DATA PROCESSING The arithmetic data types are the fundamental data types of the C language. They are called "arithmetic" because operations such as addition and multiplication can be

### Computational Methods CMSC/AMSC/MAPL 460. Representing numbers in floating point Error Analysis. Ramani Duraiswami, Dept. of Computer Science

Computational Methods CMSC/AMSC/MAPL 460 Representing numbers in floating point Error Analysis Ramani Duraiswami, Dept. of Computer Science Class Outline Recap of floating point representation Matlab demos

### 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

### 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,

### 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

### 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

### Natural Numbers and Integers. Big Ideas in Numerical Methods. Overflow. Real Numbers 29/07/2011. Taking some ideas from NM course a little further

Natural Numbers and Integers Big Ideas in Numerical Methods MEI Conference 2011 Natural numbers can be in the range [0, 2 32 1]. These are known in computing as unsigned int. Numbers in the range [ (2

### Number Representation 3/2/01 Lecture #

Number Representation 3/2/01 Lecture #11 16.070 How are numbers represented in a computer? Data come in two basic types! Numbers Whole/Integer (1, -3, 0) Natural, Positive (1, 2, 3) Real/Floating-point

### Programming Using C Homework 4

Programming Using C Homewk 4 1. In Homewk 3 you computed the histogram of an array, in which each entry represents one value of the array. We can generalize the histogram so that each entry, called a bin,

### Computational Methods CMSC/AMSC/MAPL 460. Representing numbers in floating point and associated issues. Ramani Duraiswami, Dept. of Computer Science

Computational Methods CMSC/AMSC/MAPL 460 Representing numbers in floating point and associated issues Ramani Duraiswami, Dept. of Computer Science Class Outline Computations should be as accurate and as

### Binary representation of integer numbers Operations on bits

Outline Binary representation of integer numbers Operations on bits The Bitwise AND Operator The Bitwise Inclusive-OR Operator The Bitwise Exclusive-OR Operator The Ones Complement Operator The Left Shift

### 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)

### Floating-Point Arithmetic

Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } Reading: Study Chapter 4. L12 Multiplication 1 Why Floating Point? Aren t Integers enough? Many applications require numbers with a VERY

### 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

### GPU Floating Point Features

CSE 591: GPU Programming Floating Point Considerations Klaus Mueller Computer Science Department Stony Brook University Objective To understand the fundamentals of floating-point representation To know

### 3.1 DATA REPRESENTATION (PART C)

3.1 DATA REPRESENTATION (PART C) 3.1.3 REAL NUMBERS AND NORMALISED FLOATING-POINT REPRESENTATION In decimal notation, the number 23.456 can be written as 0.23456 x 10 2. This means that in decimal notation,

### Precision and Accuracy

inst.eecs.berkeley.edu/~cs61c UC Berkeley CS61C : Machine Structures Lecture 16 Floating Point II 2010-02-26 TA Michael Greenbaum www.cs.berkeley.edu/~cs61c-tf Research without Google would be like life

### 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

### Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit

Announcements Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit Quiz 2 Monday on Number System Conversions

### Architecture and Design of Generic IEEE-754 Based Floating Point Adder, Subtractor and Multiplier

Architecture and Design of Generic IEEE-754 Based Floating Point Adder, Subtractor and Multiplier Sahdev D. Kanjariya VLSI & Embedded Systems Design Gujarat Technological University PG School Ahmedabad,

### 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