Coping with Fixed Point
|
|
- Simon Greene
- 7 years ago
- Views:
Transcription
1 Coping with Fixed Point Mik BRY CEO Copyright Khronos Group, Page 1
2 Overview Fixed Point theory and history Floating Point to Fixed Point Maintain accuracy avoid overflows and optimization tricks Effective use of Fixed Point in OpenGL ES Copyright Khronos Group, Page 2
3 Fixed Point theory and history Widely used in software 3D Fixed Form Fast and simple to implement Copyright Khronos Group, Page 3
4 Widely used in software 3D Fixed Point number is represented by a real number in an integer format with an imaginary radix point separating the integer and fractional part. Prior to HW 3D, fixed point was widely used since the beginning of realtime 3D. But in small handheld devices, we need to go back to a constrained world where we have to take care of limited memory, tiny screen. And also on certain targets avoids floating computings. Copyright Khronos Group, Page 4
5 Fixed Form used in OpenGL ES Signed format : s15.16 used in OpenGL ES GLFixed = number*2^16 GLFixed = number<<16 #define FNUM int // Convert from int to fixed number #define INT2FNUM(x) x<<16 // Convert from fixed math to int #define FNUM2INT(x) x>>16 Copyright Khronos Group, Page 5
6 Fast and simple to implement Basic math operations Adding/Sub : same as int opts FNUM a = INT2FNUM(1); FNUM b = INT2FNUM(2); a += b; Multiply/div (avoid it see later) #define FMUL(x,y) (x*y) >>16 #define FDIV(x,y)(x/y)>>16 Comparisons same as int excepting zero compare Copyright Khronos Group, Page 6
7 Floating Point to Fixed Point Floats format : standardized Maths errors support Convert Fixed and Float numbers Copyright Khronos Group, Page 7
8 Floats format Floating format is now the commonly used number format for 3D operations. Instead of always multiply by a fixed exponent as in Fixed Point, It uses Exponent: Float a = mantissa*2^(exponent-127) Copyright Khronos Group, Page 8
9 So : Float = mantissa*2^(exponent-127) Fixed = number*2^16 Copyright Khronos Group, Page 9
10 Representation errors support Floats IEEE Standard has a full range of exception handling Floats support for NaN and infinite Overflow and Underflow are checked Copyright Khronos Group, Page 10
11 Convert Fixed and float numbers In preprocessing and constants vars it is transparent #define FNUM_PI (int)(3.14f*65536) // 2^16=65536 Simple implementation, not fast at all: Float to Fixed: fixednum = (int)(floatnum*65536 Fixed to Float: floatnum = ((float)fixednum)/65536 Without Floating support at all in C compiler it is a little bit more tricky Copyright Khronos Group, Page 11
12 Effective use of Fixed Point in OpenGL ES Why Fixed Points in OpenGL ES CL? ARM chipset : no floating point support Easy to implements in embedded systems But not as perfect as Floating math so 2 OpenGL ES profiles Copyright Khronos Group, Page 12
13 ARM architecture Designed for low power consumption so : No floating point support in mainstream ARM7 ARM9 cores No divide support in ARM7 Small cache memory L1 and no L2 Copyright Khronos Group, Page 13
14 Easy to implements So Fixed Points fit perfectly to ARM cores No needs for DSP or FPU Caution of Divide support And cache memory Copyright Khronos Group, Page 14
15 OpenGL ES support for Fixed Math Common Lite Profile is a strict fixed point implementation GLFixed : a 32 bits integer Clampx : GL Commands mapping using fixed math: x instead of f glclearcolorx(0, 0, 0, 0); OES_Fixed_Point extension Copyright Khronos Group, Page 15
16 Maintain accuracy and overflows and optimization tricks Accuracy and range Overflow underflow Avoiding pitfalls Optimization tricks Copyright Khronos Group, Page 16
17 Accuracy and Range A fixed point number has a limited integer range. It is not possible to represent very large and very small numbers. integer range in GLFixed : < integer part < (2^15) Fractionnal accuracy : 0, // 1/(2^16) A fixed point number has limited accuracy. You must choose small numbers and inputs a normalize data as possible. Copyright Khronos Group, Page 17
18 Overflow underflow Overflow - An "overflow" will occur when the result of a arithmetic operation is too large to fit into the fixed representation of a fixed point number a = 0x7FFF << 16 b = 0x20 << 16 a += b // An overflow Underflow When you used a not enough accurate value for fraction, like in trigo maths But in fixed point no exception handling Using 64 intermediate numbers but speed consuming Copyright Khronos Group, Page 18
19 Optimization tricks Trigonometric operations Using LookupTable best with 1024 bytes size first quadrant and s8:24 for avoiding underflow. Square operations Using logarithm functions LUT are too big for small cache Use other fixed formats range s24:8 for larger numbers and convert to GLFixed: Fixed24_8Num = glfixednum>>8 // You lost some precision but higher integer range If (Fixed24_8Num&0x7F000000!= 0) error // to big number Else glfixednum = Fixed24_8Num<<8 // convert to GLFixed Copyright Khronos Group, Page 19
20 Avoiding pitfalls Overflow Integer Range Also divide errors (try to not use it) Copyright Khronos Group, Page 20
21 Fixed Point in OpenGL ES In all Profile Easy to implements in using same methods as floating ones and using GLFixed Take care of pitfalls and use small numbers Perfect for limited devices with small memory and screen and a simple ARM processor. Copyright Khronos Group, Page 21
22 Questions? Mik BRY Copyright Khronos Group, Page 22
This Unit: Floating Point Arithmetic. CIS 371 Computer Organization and Design. Readings. Floating Point (FP) Numbers
This Unit: Floating Point Arithmetic CIS 371 Computer Organization and Design Unit 7: Floating Point App App App System software Mem CPU I/O Formats Precision and range IEEE 754 standard Operations Addition
More informationDivide: Paper & Pencil. Computer Architecture ALU Design : Division and Floating Point. Divide algorithm. DIVIDE HARDWARE Version 1
Divide: Paper & Pencil Computer Architecture ALU Design : Division and Floating Point 1001 Quotient Divisor 1000 1001010 Dividend 1000 10 101 1010 1000 10 (or Modulo result) See how big a number can be
More informationBinary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1
Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1
More informationECE 0142 Computer Organization. Lecture 3 Floating Point Representations
ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur floating-point programming. Floating point greatly simplifies working with large (e.g.,
More informationBinary Division. Decimal Division. Hardware for Binary Division. Simple 16-bit Divider Circuit
Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend -480 41-36 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until
More informationNumerical Matrix Analysis
Numerical Matrix Analysis Lecture Notes #10 Conditioning and / Peter Blomgren, blomgren.peter@gmail.com Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x
More informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also
More informationNumbering Systems. InThisAppendix...
G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal
More informationMeasures of Error: for exact x and approximation x Absolute error e = x x. Relative error r = (x x )/x.
ERRORS and COMPUTER ARITHMETIC Types of Error in Numerical Calculations Initial Data Errors: from experiment, modeling, computer representation; problem dependent but need to know at beginning of calculation.
More informationFloating Point Fused Add-Subtract and Fused Dot-Product Units
Floating Point Fused Add-Subtract and Fused Dot-Product Units S. Kishor [1], S. P. Prakash [2] PG Scholar (VLSI DESIGN), Department of ECE Bannari Amman Institute of Technology, Sathyamangalam, Tamil Nadu,
More informationCHAPTER 5 Round-off errors
CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any
More informationThe string of digits 101101 in the binary number system represents the quantity
Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for
More informationCS321. Introduction to Numerical Methods
CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in
More informationFLOATING-POINT ARITHMETIC IN AMD PROCESSORS MICHAEL SCHULTE AMD RESEARCH JUNE 2015
FLOATING-POINT ARITHMETIC IN AMD PROCESSORS MICHAEL SCHULTE AMD RESEARCH JUNE 2015 AGENDA The Kaveri Accelerated Processing Unit (APU) The Graphics Core Next Architecture and its Floating-Point Arithmetic
More informationTo convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:
Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents
More informationDNA Data and Program Representation. Alexandre David 1.2.05 adavid@cs.aau.dk
DNA Data and Program Representation Alexandre David 1.2.05 adavid@cs.aau.dk Introduction Very important to understand how data is represented. operations limits precision Digital logic built on 2-valued
More informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationCSI 333 Lecture 1 Number Systems
CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...
More informationLSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology
LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (0-9) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 10-1 10-2 10-3
More informationLecture 2. Binary and Hexadecimal Numbers
Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations
More informationCOMPSCI 210. Binary Fractions. Agenda & Reading
COMPSCI 21 Binary Fractions Agenda & Reading Topics: Fractions Binary Octal Hexadecimal Binary -> Octal, Hex Octal -> Binary, Hex Decimal -> Octal, Hex Hex -> Binary, Octal Animation: BinFrac.htm Example
More informationBase Conversion written by Cathy Saxton
Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,
More informationwhat operations can it perform? how does it perform them? on what kind of data? where are instructions and data stored?
Inside the CPU how does the CPU work? what operations can it perform? how does it perform them? on what kind of data? where are instructions and data stored? some short, boring programs to illustrate the
More informationLecture 11: Number Systems
Lecture 11: Number Systems Numeric Data Fixed point Integers (12, 345, 20567 etc) Real fractions (23.45, 23., 0.145 etc.) Floating point such as 23. 45 e 12 Basically an exponent representation Any number
More informationNumber Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will
More informationAttention: This material is copyright 1995-1997 Chris Hecker. All rights reserved.
Attention: This material is copyright 1995-1997 Chris Hecker. All rights reserved. You have permission to read this article for your own education. You do not have permission to put it on your website
More information1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:
Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of
More informationNumber Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)
Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people
More information22S:295 Seminar in Applied Statistics High Performance Computing in Statistics
22S:295 Seminar in Applied Statistics High Performance Computing in Statistics Luke Tierney Department of Statistics & Actuarial Science University of Iowa August 30, 2007 Luke Tierney (U. of Iowa) HPC
More informationData Storage 3.1. Foundations of Computer Science Cengage Learning
3 Data Storage 3.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List five different data types used in a computer. Describe how
More informationQ-Format number representation. Lecture 5 Fixed Point vs Floating Point. How to store Q30 number to 16-bit memory? Q-format notation.
Lecture 5 Fixed Point vs Floating Point Objectives: Understand fixed point representations Understand scaling, overflow and rounding in fixed point Understand Q-format Understand TM32C67xx floating point
More information3D GRAPHICS OPTIMIZATIONS FOR ARM ARCHITECTURE
3D GRAPHICS OPTIMIZATIONS FOR ARM ARCHITECTURE Gopi K. Kolli gopi.k.kolli@intel.com Stephen Junkins mailto:stephen.junkins@intel.com Handheld Computing Division Emerging Platforms Lab Intel Corporation
More informationHOMEWORK # 2 SOLUTIO
HOMEWORK # 2 SOLUTIO Problem 1 (2 points) a. There are 313 characters in the Tamil language. If every character is to be encoded into a unique bit pattern, what is the minimum number of bits required to
More informationPrecision & Performance: Floating Point and IEEE 754 Compliance for NVIDIA GPUs
Precision & Performance: Floating Point and IEEE 754 Compliance for NVIDIA GPUs Nathan Whitehead Alex Fit-Florea ABSTRACT A number of issues related to floating point accuracy and compliance are a frequent
More informationFast Arithmetic Coding (FastAC) Implementations
Fast Arithmetic Coding (FastAC) Implementations Amir Said 1 Introduction This document describes our fast implementations of arithmetic coding, which achieve optimal compression and higher throughput by
More informationComputers. Hardware. The Central Processing Unit (CPU) CMPT 125: Lecture 1: Understanding the Computer
Computers CMPT 125: Lecture 1: Understanding the Computer Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 3, 2009 A computer performs 2 basic functions: 1.
More informationCS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011
CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS
More informationNumber Representation
Number Representation CS10001: Programming & Data Structures Pallab Dasgupta Professor, Dept. of Computer Sc. & Engg., Indian Institute of Technology Kharagpur Topics to be Discussed How are numeric data
More informationProgramming languages C
INTERNATIONAL STANDARD ISO/IEC 9899:1999 TECHNICAL CORRIGENDUM 2 Published 2004-11-15 INTERNATIONAL ORGANIZATION FOR STANDARDIZATION МЕЖДУНАРОДНАЯ ОРГАНИЗАЦИЯ ПО СТАНДАРТИЗАЦИИ ORGANISATION INTERNATIONALE
More informationThis 3-digit ASCII string could also be calculated as n = (Data[2]-0x30) +10*((Data[1]-0x30)+10*(Data[0]-0x30));
Introduction to Embedded Microcomputer Systems Lecture 5.1 2.9. Conversions ASCII to binary n = 100*(Data[0]-0x30) + 10*(Data[1]-0x30) + (Data[2]-0x30); This 3-digit ASCII string could also be calculated
More informationCorrectly Rounded Floating-point Binary-to-Decimal and Decimal-to-Binary Conversion Routines in Standard ML. By Prashanth Tilleti
Correctly Rounded Floating-point Binary-to-Decimal and Decimal-to-Binary Conversion Routines in Standard ML By Prashanth Tilleti Advisor Dr. Matthew Fluet Department of Computer Science B. Thomas Golisano
More informationSIMPLIFYING SQUARE ROOTS
40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationChapter 4 -- Decimals
Chapter 4 -- Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789
More information1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12
C5 Solutions 1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12 To Compute 0 0 = 00000000 To Compute 1 Step 1. Convert 1 to binary 00000001 Step 2. Flip the bits 11111110
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationArithmetic in MIPS. Objectives. Instruction. Integer arithmetic. After completing this lab you will:
6 Objectives After completing this lab you will: know how to do integer arithmetic in MIPS know how to do floating point arithmetic in MIPS know about conversion from integer to floating point and from
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationBinary Numbering Systems
Binary Numbering Systems April 1997, ver. 1 Application Note 83 Introduction Binary numbering systems are used in virtually all digital systems, including digital signal processing (DSP), networking, and
More informationInstruction Set Architecture (ISA)
Instruction Set Architecture (ISA) * Instruction set architecture of a machine fills the semantic gap between the user and the machine. * ISA serves as the starting point for the design of a new machine
More informationData Storage. Chapter 3. Objectives. 3-1 Data Types. Data Inside the Computer. After studying this chapter, students should be able to:
Chapter 3 Data Storage Objectives After studying this chapter, students should be able to: List five different data types used in a computer. Describe how integers are stored in a computer. Describe how
More informationFloating Point Arithmetic Chapter 14
Thi d t t d ith F M k 4 0 2 Floating Point Arithmetic Chapter 14 Although integers provide an exact representation for numeric values, they suffer from two major drawbacks: the inability to represent fractional
More informationTMS320C67x FastRTS Library Programmer s Reference
TMS320C67x FastRTS Library Programmer s Reference SPRU100A October 2002 Printed on Recycled Paper IMPORTANT NOTICE Texas Instruments Incorporated and its subsidiaries (TI) reserve the right to make corrections,
More informationA High-Performance 8-Tap FIR Filter Using Logarithmic Number System
A High-Performance 8-Tap FIR Filter Using Logarithmic Number System Yan Sun and Min Sik Kim School of Electrical Engineering and Computer Science Washington State University Pullman, Washington 99164-2752,
More informationThe programming language C. sws1 1
The programming language C sws1 1 The programming language C invented by Dennis Ritchie in early 1970s who used it to write the first Hello World program C was used to write UNIX Standardised as K&C (Kernighan
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationA Programming Language for Processor Based Embedded Systems
A Programming Language for Processor Based Embedded Systems Akihiko Inoue Hiroyuki Tomiyama Eko Fajar Nurprasetyo Hiroto Yasuura Department of Computer Science and Communication Engineering, Kyushu University
More informationChapter 2. Binary Values and Number Systems
Chapter 2 Binary Values and Number Systems Numbers Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative numbers A
More informationARM Microprocessor and ARM-Based Microcontrollers
ARM Microprocessor and ARM-Based Microcontrollers Nguatem William 24th May 2006 A Microcontroller-Based Embedded System Roadmap 1 Introduction ARM ARM Basics 2 ARM Extensions Thumb Jazelle NEON & DSP Enhancement
More informationSome Functions Computable with a Fused-mac
Some Functions Computable with a Fused-mac Sylvie Boldo and Jean-Michel Muller Laboratoire LIP (CNRS/ENS Lyon/Inria/Univ. lyon ), Projet Arénaire, 46 allée d Italie, 69364 Lyon Cedex 07, FRANCE Sylvie.Boldo@ens-lyon.fr,
More informationChapter 7D The Java Virtual Machine
This sub chapter discusses another architecture, that of the JVM (Java Virtual Machine). In general, a VM (Virtual Machine) is a hypothetical machine (implemented in either hardware or software) that directly
More informationFloating-point control in the Intel compiler and libraries or Why doesn t my application always give the expected answer?
Floating-point control in the Intel compiler and libraries or Why doesn t my application always give the expected answer? Software Solutions Group Intel Corporation 2012 *Other brands and names are the
More informationEmbedded Systems. Review of ANSI C Topics. A Review of ANSI C and Considerations for Embedded C Programming. Basic features of C
Embedded Systems A Review of ANSI C and Considerations for Embedded C Programming Dr. Jeff Jackson Lecture 2-1 Review of ANSI C Topics Basic features of C C fundamentals Basic data types Expressions Selection
More informationOverview. CISC Developments. RISC Designs. CISC Designs. VAX: Addressing Modes. Digital VAX
Overview CISC Developments Over Twenty Years Classic CISC design: Digital VAX VAXÕs RISC successor: PRISM/Alpha IntelÕs ubiquitous 80x86 architecture Ð 8086 through the Pentium Pro (P6) RJS 2/3/97 Philosophy
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationDigital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 04 Digital Logic II May, I before starting the today s lecture
More informationADVANCED PROCESSOR ARCHITECTURES AND MEMORY ORGANISATION Lesson-12: ARM
ADVANCED PROCESSOR ARCHITECTURES AND MEMORY ORGANISATION Lesson-12: ARM 1 The ARM architecture processors popular in Mobile phone systems 2 ARM Features ARM has 32-bit architecture but supports 16 bit
More informationToday. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic
Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1
More informationGoals. Unary Numbers. Decimal Numbers. 3,148 is. 1000 s 100 s 10 s 1 s. Number Bases 1/12/2009. COMP370 Intro to Computer Architecture 1
Number Bases //9 Goals Numbers Understand binary and hexadecimal numbers Be able to convert between number bases Understand binary fractions COMP37 Introduction to Computer Architecture Unary Numbers Decimal
More informationBinary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.
Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems
More informationLevent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC
Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n
More informationBachelors of Computer Application Programming Principle & Algorithm (BCA-S102T)
Unit- I Introduction to c Language: C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating
More information16. Recursion. COMP 110 Prasun Dewan 1. Developing a Recursive Solution
16. Recursion COMP 110 Prasun Dewan 1 Loops are one mechanism for making a program execute a statement a variable number of times. Recursion offers an alternative mechanism, considered by many to be more
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST
More informationA Static Analyzer for Large Safety-Critical Software. Considered Programs and Semantics. Automatic Program Verification by Abstract Interpretation
PLDI 03 A Static Analyzer for Large Safety-Critical Software B. Blanchet, P. Cousot, R. Cousot, J. Feret L. Mauborgne, A. Miné, D. Monniaux,. Rival CNRS École normale supérieure École polytechnique Paris
More informationZuse's Z3 Square Root Algorithm Talk given at Fall meeting of the Ohio Section of the MAA October 1999 - College of Wooster
Zuse's Z3 Square Root Algorithm Talk given at Fall meeting of the Ohio Section of the MAA October 1999 - College of Wooster Abstract Brian J. Shelburne Dept of Math and Comp Sci Wittenberg University In
More informationRadicals - Square Roots
8.1 Radicals - Square Roots Objective: Simplify expressions with square roots. Square roots are the most common type of radical used. A square root unsquares a number. For example, because 5 2 = 25 we
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationIntel 64 and IA-32 Architectures Software Developer s Manual
Intel 64 and IA-32 Architectures Software Developer s Manual Volume 1: Basic Architecture NOTE: The Intel 64 and IA-32 Architectures Software Developer's Manual consists of seven volumes: Basic Architecture,
More informationNext Generation GPU Architecture Code-named Fermi
Next Generation GPU Architecture Code-named Fermi The Soul of a Supercomputer in the Body of a GPU Why is NVIDIA at Super Computing? Graphics is a throughput problem paint every pixel within frame time
More informationNotes on Assembly Language
Notes on Assembly Language Brief introduction to assembly programming The main components of a computer that take part in the execution of a program written in assembly code are the following: A set of
More informationNEON. Support in Compilation Tools. Development Article. Copyright 2009 ARM Limited. All rights reserved. DHT 0004A (ID081609)
NEON Support in Compilation Tools Development Article Copyright 2009 ARM Limited. All rights reserved. DHT 0004A () NEON Support in Compilation Tools Development Article Copyright 2009 ARM Limited. All
More informationNUMBER SYSTEMS. William Stallings
NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationA single register, called the accumulator, stores the. operand before the operation, and stores the result. Add y # add y from memory to the acc
Other architectures Example. Accumulator-based machines A single register, called the accumulator, stores the operand before the operation, and stores the result after the operation. Load x # into acc
More informationSupplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm
More informationPerformance Optimization and Debug Tools for mobile games with PlayCanvas
Performance Optimization and Debug Tools for mobile games with PlayCanvas Jonathan Kirkham, Senior Software Engineer, ARM Will Eastcott, CEO, PlayCanvas 1 Introduction Jonathan Kirkham, ARM Worked with
More informationHigh-Performance Modular Multiplication on the Cell Processor
High-Performance Modular Multiplication on the Cell Processor Joppe W. Bos Laboratory for Cryptologic Algorithms EPFL, Lausanne, Switzerland joppe.bos@epfl.ch 1 / 19 Outline Motivation and previous work
More informationStack machines The MIPS assembly language A simple source language Stack-machine implementation of the simple language Readings: 9.1-9.
Code Generation I Stack machines The MIPS assembly language A simple source language Stack-machine implementation of the simple language Readings: 9.1-9.7 Stack Machines A simple evaluation model No variables
More informationCaml Virtual Machine File & data formats Document version: 1.4 http://cadmium.x9c.fr
Caml Virtual Machine File & data formats Document version: 1.4 http://cadmium.x9c.fr Copyright c 2007-2010 Xavier Clerc cadmium@x9c.fr Released under the LGPL version 3 February 6, 2010 Abstract: This
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationEE 261 Introduction to Logic Circuits. Module #2 Number Systems
EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook
More informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
More informationCSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential
More information6.087 Lecture 2 January 12, 2010
6.087 Lecture 2 January 12, 2010 Review Variables and data types Operators Epilogue 1 Review: C Programming language C is a fast, small,general-purpose,platform independent programming language. C is used
More informationARM Architecture. ARM history. Why ARM? ARM Ltd. 1983 developed by Acorn computers. Computer Organization and Assembly Languages Yung-Yu Chuang
ARM history ARM Architecture Computer Organization and Assembly Languages g Yung-Yu Chuang 1983 developed by Acorn computers To replace 6502 in BBC computers 4-man VLSI design team Its simplicity it comes
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationFast Logarithms on a Floating-Point Device
TMS320 DSP DESIGNER S NOTEBOOK Fast Logarithms on a Floating-Point Device APPLICATION BRIEF: SPRA218 Keith Larson Digital Signal Processing Products Semiconductor Group Texas Instruments March 1993 IMPORTANT
More information