# 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!

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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:

### 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

### 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.

### 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

### 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

### 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.

### 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

### 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 Fabián E. Bustamante, Spring 2010 IEEE Floating point Floating point

### 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 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

### 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) Does all work in CPU (aritmeettis-looginen

### 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

### 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

### 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

### 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

### 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,

### 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 >

### 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:

### 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

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

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

### 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

### 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

### 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.

### 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

### 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

### On a 64-bit CPU. Size/Range vary by CPU model and Word size.

On a 64-bit CPU. Size/Range vary by CPU model and Word size. unsigned short x; //range 0 to 65553 signed short x; //range ± 32767 short x; //assumed signed There are (usually) no unsigned floats or doubles.

### 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

### 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

### 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

### 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

### The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Comp 4 Computer Organization Fall 26 Solutions for Problem Set #7 Problem. Bits of Floating-Point Represent the following in single-precision IEEE floating

### 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

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

z+1 == z Computer arithmetics: integers, binary floating-point, and decimal floating-point Peter Sestoft 2013-02-18** x+1 < x p == n && 1/p!= 1/n 1 Computer arithmetics Computer numbers are cleverly designed,

### CO Computer Architecture and Programming Languages CAPL. Lecture 13 & 14

CO20-320241 Computer Architecture and Programming Languages CAPL Lecture 13 & 14 Dr. Kinga Lipskoch Fall 2017 Frame Pointer (1) The stack is also used to store variables that are local to function, but

### 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

### 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

### 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

### Assembly level Programming. 198:211 Computer Architecture. (recall) Von Neumann Architecture. Simplified hardware view. Lecture 10 Fall 2012

19:211 Computer Architecture Lecture 10 Fall 20 Topics:Chapter 3 Assembly Language 3.2 Register Transfer 3. ALU 3.5 Assembly level Programming We are now familiar with high level programming languages

### 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

### 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

### 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

### 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

### 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

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

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

### We can study computer architectures by starting with the basic building blocks. Adders, decoders, multiplexors, flip-flops, registers,...

COMPUTER ARCHITECTURE II: MICROPROCESSOR PROGRAMMING We can study computer architectures by starting with the basic building blocks Transistors and logic gates To build more complex circuits Adders, decoders,

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

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

### 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

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

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

### 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

### Section 1.4 Mathematics on the Computer: Floating Point Arithmetic

Section 1.4 Mathematics on the Computer: Floating Point Arithmetic Key terms Floating point arithmetic IEE Standard Mantissa Exponent Roundoff error Pitfalls of floating point arithmetic Structuring computations

### Solving Equations with Inverse Operations

Solving Equations with Inverse Operations Math 97 Supplement LEARNING OBJECTIVES 1. Solve equations by using inverse operations, including squares, square roots, cubes, and cube roots. The Definition of

### Signed umbers. Sign/Magnitude otation

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

### Basic Pentium Instructions. October 18

Basic Pentium Instructions October 18 CSC201 Section 002 Fall, 2000 The EFLAGS Register Bit 11 = Overflow Flag Bit 7 = Sign Flag Bit 6 = Zero Flag Bit 0 = Carry Flag "Sets the flags" means sets OF, ZF,

### CSC 8400: Computer Systems. Machine-Level Representation of Programs

CSC 8400: Computer Systems Machine-Level Representation of Programs Towards the Hardware High-level language (Java) High-level language (C) assembly language machine language (IA-32) 1 Compilation Stages

### Data Representation and Introduction to Visualization

Data Representation and Introduction to Visualization Massimo Ricotti ricotti@astro.umd.edu University of Maryland Data Representation and Introduction to Visualization p.1/18 VISUALIZATION Visualization

### More Programming Constructs -- Introduction

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

### 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

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

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

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

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

### Computer Architecture and Assembly Language. Practical Session 3

Computer Architecture and Assembly Language Practical Session 3 Advanced Instructions division DIV r/m - unsigned integer division IDIV r/m - signed integer division Dividend Divisor Quotient Remainder

### 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

### CS , Fall 2001 Exam 1

Andrew login ID: Full Name: CS 15-213, Fall 2001 Exam 1 October 9, 2001 Instructions: Make sure that your exam is not missing any sheets, then write your full name and Andrew login ID on the front. Write

### Summary: Direct Code Generation

Summary: Direct Code Generation 1 Direct Code Generation Code generation involves the generation of the target representation (object code) from the annotated parse tree (or Abstract Syntactic Tree, AST)

### Digital Arithmetic. Digital Arithmetic: Operations and Circuits Dr. Farahmand

Digital Arithmetic Digital Arithmetic: Operations and Circuits Dr. Farahmand Binary Arithmetic Digital circuits are frequently used for arithmetic operations Fundamental arithmetic operations on binary

### Fundamentals of Programming Session 2

Fundamentals of Programming Session 2 Instructor: Reza Entezari-Maleki Email: entezari@ce.sharif.edu 1 Fall 2013 Sharif University of Technology Outlines Programming Language Binary numbers Addition Subtraction

### FP_IEEE_DENORM_GET_ Procedure

FP_IEEE_DENORM_GET_ Procedure FP_IEEE_DENORM_GET_ Procedure The FP_IEEE_DENORM_GET_ procedure reads the IEEE floating-point denormalization mode. fp_ieee_denorm FP_IEEE_DENORM_GET_ (void); DeNorm The denormalization

### 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

### Assembly basics CS 2XA3. Term I, 2017/18

Assembly basics CS 2XA3 Term I, 2017/18 Outline What is Assembly Language? Assemblers NASM Program structure I/O First program Compiling Linking What is Assembly Language? In a high level language (HLL),

### Calculations with Sig Figs

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

### Floating Point. What can be represented in N bits? 0 to 2N-1. 9,349,398,989,787,762,244,859,087, x 1067

MIPS Floating Point Operations Cptr280 Dr Curtis Nelson Floating Point What can be represented in N bits? Unsigned 2 s Complement 0 to 2N-1-2N-1 to 2N-1-1 But, what about- Very large numbers? 9,349,398,989,787,762,244,859,087,678

### ANSI C Programming Simple Programs

ANSI C Programming Simple Programs /* This program computes the distance between two points */ #include #include #include main() { /* Declare and initialize variables */ double

### 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

### 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)

### C212 Early Evaluation Exam Mon Feb Name: Please provide brief (common sense) justifications with your answers below.

C212 Early Evaluation Exam Mon Feb 10 2014 Name: Please provide brief (common sense) justifications with your answers below. 1. What is the type (and value) of this expression: 5 * (7 + 4 / 2) 2. What

### Java provides a rich set of operators to manipulate variables. We can divide all the Java operators into the following groups:

Basic Operators Java provides a rich set of operators to manipulate variables. We can divide all the Java operators into the following groups: Arithmetic Operators Relational Operators Bitwise Operators

### 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

### LAB 1 Binary Numbers/ Introduction to C. Rajesh Rajamani. ME 4231 Department of Mechanical Engineering University Of Minnesota

LAB 1 Binary Numbers/ Introduction to C Rajesh Rajamani ME 431 Department of Mechanical Engineering University Of Minnesota OVERVIEW What is feedback control? Binary and Hexadecimal Numbers Working with

### CS , Spring 2004 Exam 1

Andrew login ID: Full Name: CS 15-213, Spring 2004 Exam 1 February 26, 2004 Instructions: Make sure that your exam is not missing any sheets (there should be 15), then write your full name and Andrew login

### 15-213/18-243, Spring 2011 Exam 1

Andrew login ID: Full Name: Section: 15-213/18-243, Spring 2011 Exam 1 Thursday, March 3, 2011 (v1) Instructions: Make sure that your exam is not missing any sheets, then write your Andrew login ID, full

### 1010 2?= ?= CS 64 Lecture 2 Data Representation. Decimal Numbers: Base 10. Reading: FLD Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

CS 64 Lecture 2 Data Representation Reading: FLD 1.2-1.4 Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) 1010 10?= 1010 2?= 1

### 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

### Arithmetic Operations

Arithmetic Operations Arithmetic Operations addition subtraction multiplication division Each of these operations on the integer representations: unsigned two's complement 1 Addition One bit of binary

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

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

### EECS 213 Fall 2007 Midterm Exam

Full Name: EECS 213 Fall 2007 Midterm Exam Instructions: Make sure that your exam is not missing any sheets, then write your full name on the front. Write your answers in the space provided below the problem.