# Data Representation Floating Point

Save this PDF as:

Size: px
Start display at page:

## Transcription

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

2 Today: Floating Point Background: Fractional binary numbers Example and properties IEEE floating point standard: Definition Floating point in C Summary 2

3 Fractional binary numbers What is ? How can we express fractions like ¼ in binary? 3

4 Place-Value Fractional Binary Numbers 2 i 2 i = = = 1 bi bi-1 b2 b1 b0 b-1 b-2 b-3 b-j 2-1 = 1/2 2-2 = 1/4 2-3 = 1/8 2 -j Representation Bits to right of binary point represent fractional powers of 2 Represents rational number: 4

5 Fractional Binary Numbers: Examples Value Representation = = = = = = Observations Divide by 2 by shifting right Multiply by 2 by shifting left Limitations Can only exactly represent numbers of the form x/2 k Other rational numbers have repeating bit representations Value Representation 1/ [01] 2 1/ [0011] 2 Limited range when used with fixed point representations 5

6 Quick Check Convert: to binary to decimal 6

7 Quick Check Suppose an 8-bit fixed-point representation with: One sign bit Four integer bits Three fractional bits s integer fractional 1 4-bits 3-bits Convert: to binary to binary to decimal What bit pattern(s) have the largest positive value? What is it? What bit pattern(s) have the value closest to zero? What bit pattern(s) have the value of zero? 7

8 Today: Floating Point Background: Fractional binary numbers Example and properties IEEE floating point standard: Definition Floating point in C Summary 8

9 Floating Point Representation Similar to scientific notation E.g.: = 1,250 E.g.: = FP is this concept but with an efficient binary format! But Uses base 2 instead of base 10 Places restrictions on how certain values are represented Deals with finiteness of representation 9

10 Floating Point Representation Numerical Form: ( 1) s M 2 E Sign bit s determines whether number is negative or positive Significand (mantissa) M normally a fractional value in range [1.0, 2.0) Exponent E weights value by power of two Encoding s is sign bit s exp field encodes E (but is not equal to E) frac field encodes M (but is not equal to M) s exp frac 10

11 Tiny Floating Point Example s exp frac 1 4-bits 3-bits 8-bit Floating Point Representation the sign bit is in the most significant bit the next four bits are the exponent (exp), with a bias of = 7 the last three bits are the fraction (frac) Exponent bias enable exponent to represent both positive and negative powers of 2 use half of range for positive and half for negative power given k exponent bits, bias is then 2 k

12 Floating Point Encodings and Visualization Five encodings: Two general forms: Three special values: normalized, denormalized zero, infinity, NaN (not a number) Name Exponent(exp) Fraction(frac) zero exp == 0000 frac == 000 denormalized exp == 0000 frac!= 000 normalized 0000 < exp < 1111 frac!= 000 infinity exp == 1111 frac == 000 NaN exp == 1111 frac!= 000 Normalized Denorm +Denorm +Normalized + NaN 0 +0 NaN 12

13 Dynamic Range (Positives) Denormalized numbers Normalized numbers s exp frac E Value /8*1/64 = 1/ /8*1/64 = 2/ /8*1/64 = 6/ /8*1/64 = 7/ /8*1/64 = 8/ /8*1/64 = 9/ /8*1/2 = 14/ /8*1/2 = 15/ /8*1 = /8*1 = 9/ /8*1 = 10/ /8*128 = /8*128 = n/a inf xxx n/a NaN v = ( 1) s M 2 E norm: E = Exp Bias denorm: E = 1 Bias closest to zero largest denorm smallest norm closest to 1 below closest to 1 above largest norm infinity NaN (not a number) 13

14 Distribution of Values 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is = 3 s exp frac (reduced format from 8 bits to 6 bits for visualization) 1 3-bits 2-bits Notice how the distribution gets denser toward zero. 8 denormalized values (blowup of -1 1) Denormalized Normalized Infinity 14

15 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? 15

16 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? = 15 16

17 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? = 15 How many denormalized numbers are there? 17

18 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? = 15 How many denormalized numbers are there? Exponent = 00000, so 2 4 positive and 2 4 negative 18

19 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? = 15 How many denormalized numbers are there? Exponent = 00000, so 2 4 positive and 2 4 negative What is the bit pattern of the maximum value number? What is the bit pattern of the number closest to zero? 19

20 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the exponent bias? = 15 How many denormalized numbers are there? Exponent = 00000, so 2 4 positive and 2 4 negative What is the bit pattern of the maximum value number? Sign = 0, Exponent = 11110, frac=1111, so What is the bit pattern of the number closest to zero? Sign =?, Exponent = 00000, frac=0001, so?

21 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the bit pattern of the smallest positive normal number? 21

22 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the bit pattern of the smallest positive normal number? Sign = 0, exp = 00001, frac = 0000; so

23 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits What is the bit pattern of the smallest positive normal number? Sign = 0, exp = 00001, frac = 0000; so What is the value of the smallest positive normal number? 23

24 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits What is the bit pattern of the smallest positive normal number? Sign = 0, exp = 00001, frac = 0000; so What is the value of the smallest positive normal number? Value = ( 1) s M 2 E S = 0 Exponent bias = 15, so E = 1-15 = -14 M = = 1 s exp frac 1 5-bits 4-bits Value = ( 1) = =

25 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits Given a 32-bit floating point number, and a 32-bit integer, which can represent more discrete values? 25

26 Quick Check 10-bit IEEE-like format e = 5 exponent bits f = 4 fraction bits s exp frac 1 5-bits 4-bits Given a 32-bit floating point number, and a 32-bit integer, which can represent more discrete values? Both can represent 2 32 values, but some bit patterns duplicate values, e.g. +0/-0, + /-, and many NaNs (exponent = 11 1, frac!= 00 0) 26

27 Today: Floating Point Background: Fractional binary numbers Example and properties IEEE floating point standard: Definition Floating point in C Summary 27

28 IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for floating point arithmetic Before that, many idiosyncratic formats Supported by all major CPUs Driven by numerical concerns Nice standards for rounding, overflow, underflow Hard to make fast in hardware Numerical analysts predominated over hardware designers in defining standard 28

29 Floating Point Representation Numerical Form: ( 1) s M 2 E Sign bit s determines whether number is negative or positive Significand (mantissa) M normally a fractional value in range [1.0, 2.0) Exponent E weights value by power of two Encoding s is sign bit s exp field encodes E (but is not equal to E) frac field encodes M (but is not equal to M) s exp frac 29

30 Precisions Single precision: 32 bits (c type: float) s exp frac 1 8-bits 23-bits Double precision: 64 bits (c type: double) s exp frac 1 11-bits 52-bits Extended precision: 80 bits (Intel only) s exp frac 1 15-bits 63 or 64-bits 30

31 Normalized Values Condition: exp and exp Exponent coded as biased value: E = Exp Bias Exp: unsigned value of exp field Bias = 2 k-1-1, where k is number of exponent bits Single precision: 127 (exp: E: ) Double precision: 1023 (exp: E: ) Significand coded with implied leading 1: M = 1.xxx x2 xxx x: bits of frac Decimal value of normalized FP representations: Single-precision: Value 10 = 1 s 1. frac 2 exp 127 Double-precision: Value 10 = 1 s 1. frac 2 exp

32 Saint Louis Carnegie University Mellon Normalized Encoding Example Value: float F = ; = = x 2 13 Significand M = frac = shift binary point by K bits so that only one leading 1 bit remains on the left side of the binary point (here, shifted right by 13 bits, so K = 13), then multiply by 2 K (here, 2 13 ) Exponent (E = Exp Bias) E = 13 Bias = 127 Exp = E + Bias = 140 = s exp frac 32

33 Denormalized Values Condition: exp = Exponent value: E = Bias + 1 (instead of E = 0 Bias) Significand coded with implied leading 0: M = 0.xxx x2 xxx x: bits of frac Cases exp = 000 0, frac = Represents zero value Note distinct values: +0 and 0 (why?) exp = 000 0, frac Numbers very close to 0.0 Lose precision as get smaller Equispaced 33

34 Special Values Special condition: exp = Case: exp = 111 1, frac = Represents value (infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = 1.0/ 0.0 = +, 1.0/ 0.0 = Case: exp = 111 1, frac Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1),, 0 34

35 Interesting Numbers {single,double} Description exp frac Numeric Value Zero Smallest Pos. Denorm {23,52} x 2 {126,1022} Single 1.4 x Double 4.9 x Largest Denormalized (1.0 ε) x 2 {126,1022} Single 1.18 x Double 2.2 x Smallest Pos. Normalized x 2 {126,1022} Just larger than largest denormalized One Largest Normalized (2.0 ε) x 2 {127,1023} Single 3.4 x Double 1.8 x

36 Today: Floating Point Background: Fractional binary numbers Example and properties IEEE floating point standard: Definition Floating point in C Summary 36

37 Floating Point in C C Guarantees Two Levels float double single precision double precision Conversions/Casting Casting between int, float, and double changes bit representation double/float int Truncates fractional part Like rounding toward zero Not defined when out of range or NaN: Generally sets to TMin int double Exact conversion, as long as int has 53 bit word size int float Will round according to rounding mode 37

38 Today: Floating Point Background: Fractional binary numbers Example and properties IEEE floating point standard Rounding, addition, multiplication Floating point in C Summary 38

39 Floating Point Operations: Basic Idea x +f y = Round(x + y) x f y = Round(x y) Basic idea First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac 39

40 Rounding Rounding Modes (illustrate with \$ rounding) \$1.40 \$1.60 \$1.50 \$2.50 \$1.50 Towards zero \$1 \$1 \$1 \$2 \$1 Round down ( ) \$1 \$1 \$1 \$2 \$2 Round up (+ ) \$2 \$2 \$2 \$3 \$1 Nearest Even (default) \$1 \$2 \$2 \$2 \$2 40

41 Closer Look at Round-To-Even Default Rounding Mode Hard to get any other kind without dropping into assembly All others are statistically biased Sum of set of positive numbers will consistently be over- or underestimated Applying to Other Decimal Places / Bit Positions When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth (Less than half way) (Greater than half way) (Half way round up) (Half way round down) 41

42 Rounding Binary Numbers Binary Fractional Numbers Even when least significant bit is 0 Half way when bits to right of rounding position = Examples Round to nearest 1/4 (2 bits right of binary point) Value Binary Rounded Action Rounded Value 2 3/ (<1/2 down) 2 2 3/ (>1/2 up) 2 1/4 2 7/ ( 1/2 up) 3 2 5/ ( 1/2 down) 2 1/2 42

43 Scientific Notation Multiplication ( ) =? Compute result by pieces: Sign : sign left * sign right = 1*1 = 1 Significand : M left * M right = 2.5 * 3.0 = 7.5 Exponent : E left + E right = = 5 Result:

44 FP Multiplication ( 1) s1 M1 2 E1 x ( 1) s2 M2 2 E2 Exact Result: ( 1) s M 2 E Sign s: s1 ^ s2 Significand M: M1 x M2 Exponent E: E1 + E2 Fixing If M 2, shift M right, increment E If E out of range, overflow Round M to fit frac precision Implementation Biggest chore is multiplying significands 44

45 Scientific Notation Addition ( ) =? Assume E left is larger that E right Align by decimal point: Significand : Exponent : E = E left = 3 Result:

46 Floating Point Addition ( 1) s1 M1 2 E1 + (-1) s2 M2 2 E2 Assume E1 > E2 Get binary points lined up E1 E2 Exact Result: ( 1) s M 2 E Sign s, significand M: Result of signed align & add Exponent E: E1 + ( 1) s1 M1 ( 1) s M ( 1) s2 M2 Fixing If M 2, shift M right, increment E if M < 1, shift M left k positions, decrement E by k Overflow if E out of range Round M to fit frac precision 46

47 Mathematical Properties of FP Add Compare to those of Abelian Group Commutative? (a + b) = (b + a) Yes Associative? Overflow and inexactness of rounding No (3.14+1e10)-1e10 = 0, 3.14+(1e10-1e10) =

48 Mathematical Properties of FP Mult Compare to Commutative Ring Multiplication Commutative? Ex: (1e20*1e-20)=(1e-20*1e20) Yes Multiplication is Associative? Possibility of overflow, inexactness of rounding Ex: (1e20*1e20)*1e-20= inf, 1e20*(1e20*1e-20)= 1e20 Multiplication distributes over addition? Possibility of overflow, inexactness of rounding 1e20*(1e20-1e20)= 0.0, 1e20*1e20 1e20*1e20 = NaN No No 48

49 Summary Represents numbers of form M x 2 E One can reason about operations independent of implementation As if computed with perfect precision and then rounded Not the same as real arithmetic Violates associativity/distributivity Makes life difficult for compilers & serious numerical applications programmers 49

### 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:

### 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

### 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

### 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

### 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. 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 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

### 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

### 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 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

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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. 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. 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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:

### 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

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

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

### Floating Point Representation in Computers

Floating Point Representation in Computers Floating Point Numbers - What are they? Floating Point Representation Floating Point Operations Where Things can go wrong What are Floating Point Numbers? Any

### 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

### Bits and Bytes: Data Presentation Mohamed Zahran (aka Z)

CSCI-UA.0201 Computer Systems Organization Bits and Bytes: Data Presentation Mohamed Zahran (aka Z) mzahran@cs.nyu.edu http://www.mzahran.com Some slides adapted and modified from: Clark Barrett Jinyang

### 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

### 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

### 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

### 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,

### Floating-point representation

Lecture 3-4: Floating-point representation and arithmetic Floating-point representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However,

### 2 Computation with Floating-Point Numbers

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

### Arithmetic and Bitwise Operations on Binary Data

Arithmetic and Bitwise Operations on Binary Data CSCI 224 / ECE 317: Computer Architecture Instructor: Prof. Jason Fritts Slides adapted from Bryant & O Hallaron s slides 1 Boolean Algebra Developed by

### 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

### 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

### 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

### 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.,

### 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

### 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

### 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

### 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 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

### 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

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

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

### 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

### 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 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

### 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

### 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)

### 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

### Outline. What is Performance? Restating Performance Equation Time = Seconds. CPU Performance Factors

CS 61C: Great Ideas in Computer Architecture Performance and Floating-Point Arithmetic Instructors: Krste Asanović & Randy H. Katz http://inst.eecs.berkeley.edu/~cs61c/fa17 Outline Defining Performance

### 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

### 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,

### 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

### 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

### 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

### 95% of the folks out there are completely clueless about floating-point.

CS61C L11 Floating Point (1) Instructor Paul Pearce inst.eecs.berkeley.edu/~cs61c UCB CS61C : Machine Structures Well, not quite yet. A UC Davis student (shown left) is attempting to standardize hella

### 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.

### 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

### 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

### Chapter 2. Data Representation in Computer Systems

Chapter 2 Data Representation in Computer Systems Chapter 2 Objectives Understand the fundamentals of numerical data representation and manipulation in digital computers. Master the skill of converting

### MIPS Integer ALU Requirements

MIPS Integer ALU Requirements Add, AddU, Sub, SubU, AddI, AddIU: 2 s complement adder/sub with overflow detection. And, Or, Andi, Ori, Xor, Xori, Nor: Logical AND, logical OR, XOR, nor. SLTI, SLTIU (set

### 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

### 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

### Floating Point COE 308. Computer Architecture Prof. Muhamed Mudawar. Computer Engineering Department King Fahd University of Petroleum and Minerals

Floating Point COE 38 Computer Architecture Prof. Muhamed Mudawar Computer Engineering Department King Fahd University of Petroleum and Minerals Presentation Outline Floating-Point Numbers IEEE 754 Floating-Point

### Table : IEEE Single Format ± a a 2 a 3 :::a 8 b b 2 b 3 :::b 23 If exponent bitstring a :::a 8 is Then numerical value represented is ( ) 2 = (

Floating Point Numbers in Java by Michael L. Overton Virtually all modern computers follow the IEEE 2 floating point standard in their representation of floating point numbers. The Java programming language

### 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

### 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 =

### 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

### 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

### 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

### Number Systems and Computer Arithmetic

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

### The Perils of Floating Point

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

### Floating-Point Numbers in Digital Computers

POLYTECHNIC UNIVERSITY Department of Computer and Information Science Floating-Point Numbers in Digital Computers K. Ming Leung Abstract: We explain how floating-point numbers are represented and stored

### 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

### 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

### CS321. Introduction to Numerical Methods

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

### 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

### Midterm Exam Answers Instructor: Randy Shepherd CSCI-UA.0201 Spring 2017

Section 1: Multiple choice (select any that apply) - 20 points 01. Representing 10 using the 4 byte unsigned integer encoding and using 4 byte two s complements encoding yields the same bit pattern. (a)

### Floating Point Considerations

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

### Most nonzero floating-point numbers are normalized. This means they can be expressed as. x = ±(1 + f) 2 e. 0 f < 1

Floating-Point Arithmetic Numerical Analysis uses floating-point arithmetic, but it is just one tool in numerical computation. There is an impression that floating point arithmetic is unpredictable and

### 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

### Floating Point. CSC207 Fall 2017

Floating Point CSC207 Fall 2017 Ariane 5 Rocket Launch Ariane 5 rocket explosion In 1996, the European Space Agency s Ariane 5 rocket exploded 40 seconds after launch. During conversion of a 64-bit to

### Floating-Point Numbers in Digital Computers

POLYTECHNIC UNIVERSITY Department of Computer and Information Science Floating-Point Numbers in Digital Computers K. Ming Leung Abstract: We explain how floating-point numbers are represented and stored

### 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

### Physics 331 Introduction to Numerical Techniques in Physics

Physics 331 Introduction to Numerical Techniques in Physics Instructor: Joaquín Drut Lecture 2 Any logistics questions? Today: Number representation Sources of error Note: typo in HW! Two parts c. Call

### 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)

### 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

### Administrivia. CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries. Representing characters DATA REPRESENTATION

Administrivia CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries Jan Plane & Alan Sussman {jplane, als}@cs.umd.edu Project 6 due next Friday, 12/10 public tests posted

### fractional quantities are typically represented in computers using floating point format this approach is very much similar to scientific notation

Floating Point Arithmetic fractional quantities are typically represented in computers using floating point format this approach is very much similar to scientific notation for example, fixed point number