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Lab 7: Floating-Point Addition 0.0
Lab 7: Floating-Point Addition 0.0 Introduction In this lab, you will write a MIPS assembly language function that performs floating-point addition. You will then run your program using PCSpim (just as you did in Lab 6). For testing, you are provided a program that calls your function to compute the value of the mathematical constant e. For those with no assembly language experience, this will be a long lab, so plan your time accordingly. Background You should be familiar with the IEEE 754 Floating-Point Standard, which is described in Section 3.6 of your book. (Hopefully you have read that section carefully!) Here we will be dealing only with single precision floating- point values, which are formatted as follows (this is also described in the “Floating-Point Representation” subsec- tion of Section 3.6 in your book): Sign Exponent (8 bits) Significand (23 bits) 31 30 29 ... 24 23 22 21 ... 0 Remember that the exponent is biased by 127, which means that an exponent of zero is represented by 127 (01111111). The exponent is not encoded using 2s-complement. The significand is always positive, and the sign bit is kept separately. Note that the actual significand is 24 bits long; the first bit is always a 1 and thus does not need to be stored explicitly. This will be important to remember when you write your function! There are several details of IEEE 754 that you will not have to worry about in this lab. For example, the expo- nents 00000000 and 11111111 are reserved for special purposes that are described in your book (representing zero, denormalized numbers and NaNs). -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
Midterm-2020-Solution.Pdf
HONOR CODE Questions Sheet. A Lets C. [6 Points] 1. What type of address (heap,stack,static,code) does each value evaluate to Book1, Book1->name, Book1->author, &Book2? [4] 2. Will all of the print statements execute as expected? If NO, write print statement which will not execute as expected?[2] B. Mystery [8 Points] 3. When the above code executes, which line is modified? How many times? [2] 4. What is the value of register a6 at the end ? [2] 5. What is the value of register a4 at the end ? [2] 6. In one sentence what is this program calculating ? [2] C. C-to-RISC V Tree Search; Fill in the blanks below [12 points] D. RISCV - The MOD operation [8 points] 19. The data segment starts at address 0x10000000. What are the memory locations modified by this program and what are their values ? E Floating Point [8 points.] 20. What is the smallest nonzero positive value that can be represented? Write your answer as a numerical expression in the answer packet? [2] 21. Consider some positive normalized floating point number where p is represented as: What is the distance (i.e. the difference) between p and the next-largest number after p that can be represented? [2] 22. Now instead let p be a positive denormalized number described asp = 2y x 0.significand. What is the distance between p and the next largest number after p that can be represented? [2] 23. Sort the following minifloat numbers. [2] F. Numbers. [5] 24. What is the smallest number that this system can represent 6 digits (assume unsigned) ? [1] 25. -
2018-19 MAP 160 Byte File Layout Specifications
2018-19 MAP 160 Byte File Layout Specifications OVERVIEW: A) ISAC will provide an Eligibility Status File (ESF) record for each student to all schools listed as a college choice on the student’s Student Aid Report (SAR). The ESF records will be available daily as Record Type = 7. ESF records may be retrieved via the File Extraction option in MAP. B) Schools will transmit Payment Requests to ISAC via File Transfer Protocol (FTP) using the MAP 160byte layout and identify these with Record Type = 4. C) When payment requests are processed, ISAC will provide payment results to schools through the MAP system. The payment results records can be retrieved in the 160 byte format using the MAP Payment Results File Extraction Option. MAP results records have a Record Type = 5. The MAP Payment Results file contains some eligibility status data elements. Also, the same student record may appear on both the Payment Results and the Eligibility Status extract files. Schools may also use the Reports option in MAP to obtain payment results. D) To cancel Payment Requests, the school with the current Payment Results record on ISAC's Payment Database must transmit a matching record with MAP Payment Request Code = C, with the Requested Award Amount field equal to zero and the Enrollment Hours field equal to 0 along with other required data elements. These records must be transmitted to ISAC as Record Type = 4. E) Summary of Data Element Changes, revision (highlighted in grey) made to the 2018-19 layout. NONE 1 2018-19 MAP 160 Byte File Layout Specifications – 9/17 2018-19 MAP 160 Byte File Layout Specifications F) The following 160 byte record layout will be used for transmitting data between schools and ISAC. -
POINTER (IN C/C++) What Is a Pointer?
POINTER (IN C/C++) What is a pointer? Variable in a program is something with a name, the value of which can vary. The way the compiler and linker handles this is that it assigns a specific block of memory within the computer to hold the value of that variable. • The left side is the value in memory. • The right side is the address of that memory Dereferencing: • int bar = *foo_ptr; • *foo_ptr = 42; // set foo to 42 which is also effect bar = 42 • To dereference ted, go to memory address of 1776, the value contain in that is 25 which is what we need. Differences between & and * & is the reference operator and can be read as "address of“ * is the dereference operator and can be read as "value pointed by" A variable referenced with & can be dereferenced with *. • Andy = 25; • Ted = &andy; All expressions below are true: • andy == 25 // true • &andy == 1776 // true • ted == 1776 // true • *ted == 25 // true How to declare pointer? • Type + “*” + name of variable. • Example: int * number; • char * c; • • number or c is a variable is called a pointer variable How to use pointer? • int foo; • int *foo_ptr = &foo; • foo_ptr is declared as a pointer to int. We have initialized it to point to foo. • foo occupies some memory. Its location in memory is called its address. &foo is the address of foo Assignment and pointer: • int *foo_pr = 5; // wrong • int foo = 5; • int *foo_pr = &foo; // correct way Change the pointer to the next memory block: • int foo = 5; • int *foo_pr = &foo; • foo_pr ++; Pointer arithmetics • char *mychar; // sizeof 1 byte • short *myshort; // sizeof 2 bytes • long *mylong; // sizeof 4 byts • mychar++; // increase by 1 byte • myshort++; // increase by 2 bytes • mylong++; // increase by 4 bytes Increase pointer is different from increase the dereference • *P++; // unary operation: go to the address of the pointer then increase its address and return a value • (*P)++; // get the value from the address of p then increase the value by 1 Arrays: • int array[] = {45,46,47}; • we can call the first element in the array by saying: *array or array[0]. -
Virtual Memory
Topic 18: Virtual Memory COS / ELE 375 Computer Architecture and Organization Princeton University Fall 2015 Prof. David August 1 Virtual Memory Any time you see virtual, think “using a level of indirection” Virtual memory: level of indirection to physical memory • Program uses virtual memory addresses • Virtual address is converted to a physical address • Physical address indicates physical location of data • Physical location can be memory or disk! 0x800 Disk:Mem: 0x803C4 0x3C00 Virtual address Physical address 2 Virtual Memory 3 Virtual Memory 1 4 Virtual Memory: Take 1 Main memory may not be large enough for a task • Programmers turn to overlays and disk • Many programs would have to do this • Programmers should have to worry about main memory size across machines Use virtual memory to make memory look bigger for all programs 5 A System with Only Physical Memory Examples: Most Cray machines, early PCs, nearly all current embedded systems, etc. Memory 0: 1: Store 0x10 CPU Load 0xf0 N-1: CPU’s load or store addresses used directly to access memory. 6 A System with Virtual Memory Examples: modern workstations, servers, PCs, etc. Memory 0: Page Table 1: Virtual Physical Addresses 0: Addresses 1: Store 0x10 CPU Load 0xf0 P-1: N-1: Disk Address Translation: the hardware converts virtual addresses into physical addresses via an OS-managed lookup table (page table) 7 Page Faults (Similar to “Cache Misses”) What if an object is on disk rather than in memory? 1. Page table indicates that the virtual address is not in memory 2. OS trap handler is invoked, moving data from disk into memory • Current process suspends, others can resume • OS has full control over placement, etc. -
Bits and Bytes
BITS AND BYTES To understand how a computer works, you need to understand the BINARY SYSTEM. The binary system is a numbering system that uses only two digits—0 and 1. Although this may seem strange to humans, it fits the computer perfectly! A computer chip is made up of circuits. For each circuit, there are two possibilities: An electric current flows through the circuit (ON), or An electric current does not flow through the circuit (OFF) The number 1 represents an “on” circuit. The number 0 represents an “off” circuit. The two digits, 0 and 1, are called bits. The word bit comes from binary digit: Binary digit = bit Every time the computer “reads” an instruction, it translates that instruction into a series of bits (0’s and 1’s). In most computers every letter, number, and symbol is translated into eight bits, a combination of eight 0’s and 1’s. For example the letter A is translated into 01000001. The letter B is 01000010. Every single keystroke on the keyboard translates into a different combination of eight bits. A group of eight bits is called a byte. Therefore, a byte is a combination of eight 0’s and 1’s. Eight bits = 1 byte Capacity of computer memory, storage such as USB devices, DVD’s are measured in bytes. For example a Word file might be 35 KB while a picture taken by a digital camera might be 4.5 MG. Hard drives normally are measured in GB or TB: Kilobyte (KB) approximately 1,000 bytes MegaByte (MB) approximately 1,000,000 (million) bytes Gigabtye (GB) approximately 1,000,000,000 (billion) bytes Terabyte (TB) approximately 1,000,000,000,000 (trillion) bytes The binary code that computers use is called the ASCII (American Standard Code for Information Interchange) code. -
Subtyping Recursive Types
ACM Transactions on Programming Languages and Systems, 15(4), pp. 575-631, 1993. Subtyping Recursive Types Roberto M. Amadio1 Luca Cardelli CNRS-CRIN, Nancy DEC, Systems Research Center Abstract We investigate the interactions of subtyping and recursive types, in a simply typed λ-calculus. The two fundamental questions here are whether two (recursive) types are in the subtype relation, and whether a term has a type. To address the first question, we relate various definitions of type equivalence and subtyping that are induced by a model, an ordering on infinite trees, an algorithm, and a set of type rules. We show soundness and completeness between the rules, the algorithm, and the tree semantics. We also prove soundness and a restricted form of completeness for the model. To address the second question, we show that to every pair of types in the subtype relation we can associate a term whose denotation is the uniquely determined coercion map between the two types. Moreover, we derive an algorithm that, when given a term with implicit coercions, can infer its least type whenever possible. 1This author's work has been supported in part by Digital Equipment Corporation and in part by the Stanford-CNR Collaboration Project. Page 1 Contents 1. Introduction 1.1 Types 1.2 Subtypes 1.3 Equality of Recursive Types 1.4 Subtyping of Recursive Types 1.5 Algorithm outline 1.6 Formal development 2. A Simply Typed λ-calculus with Recursive Types 2.1 Types 2.2 Terms 2.3 Equations 3. Tree Ordering 3.1 Subtyping Non-recursive Types 3.2 Folding and Unfolding 3.3 Tree Expansion 3.4 Finite Approximations 4. -
Concept-Oriented Programming: References, Classes and Inheritance Revisited
Concept-Oriented Programming: References, Classes and Inheritance Revisited Alexandr Savinov Database Technology Group, Technische Universität Dresden, Germany http://conceptoriented.org Abstract representation and access is supposed to be provided by the translator. Any object is guaranteed to get some kind The main goal of concept-oriented programming (COP) is of primitive reference and a built-in access procedure describing how objects are represented and accessed. It without a possibility to change them. Thus there is a makes references (object locations) first-class elements of strong asymmetry between the role of objects and refer- the program responsible for many important functions ences in OOP: objects are intended to implement domain- which are difficult to model via objects. COP rethinks and specific structure and behavior while references have a generalizes such primary notions of object-orientation as primitive form and are not modeled by the programmer. class and inheritance by introducing a novel construct, Programming means describing objects but not refer- concept, and a new relation, inclusion. An advantage is ences. that using only a few basic notions we are able to describe One reason for this asymmetry is that there is a very many general patterns of thoughts currently belonging to old and very strong belief that it is entity that should be in different programming paradigms: modeling object hier- the focus of modeling (including programming) while archies (prototype-based programming), precedence of identities simply serve entities. And even though object parent methods over child methods (inner methods in identity has been considered “a pillar of object orienta- Beta), modularizing cross-cutting concerns (aspect- tion” [Ken91] their support has always been much weaker oriented programming), value-orientation (functional pro- in comparison to that of entities. -
IEEE Standard 754 for Binary Floating-Point Arithmetic
Work in Progress: Lecture Notes on the Status of IEEE 754 October 1, 1997 3:36 am Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic Prof. W. Kahan Elect. Eng. & Computer Science University of California Berkeley CA 94720-1776 Introduction: Twenty years ago anarchy threatened floating-point arithmetic. Over a dozen commercially significant arithmetics boasted diverse wordsizes, precisions, rounding procedures and over/underflow behaviors, and more were in the works. “Portable” software intended to reconcile that numerical diversity had become unbearably costly to develop. Thirteen years ago, when IEEE 754 became official, major microprocessor manufacturers had already adopted it despite the challenge it posed to implementors. With unprecedented altruism, hardware designers had risen to its challenge in the belief that they would ease and encourage a vast burgeoning of numerical software. They did succeed to a considerable extent. Anyway, rounding anomalies that preoccupied all of us in the 1970s afflict only CRAY X-MPs — J90s now. Now atrophy threatens features of IEEE 754 caught in a vicious circle: Those features lack support in programming languages and compilers, so those features are mishandled and/or practically unusable, so those features are little known and less in demand, and so those features lack support in programming languages and compilers. To help break that circle, those features are discussed in these notes under the following headings: Representable Numbers, Normal and Subnormal, Infinite -
A Variable Precision Hardware Acceleration for Scientific Computing Andrea Bocco
A variable precision hardware acceleration for scientific computing Andrea Bocco To cite this version: Andrea Bocco. A variable precision hardware acceleration for scientific computing. Discrete Mathe- matics [cs.DM]. Université de Lyon, 2020. English. NNT : 2020LYSEI065. tel-03102749 HAL Id: tel-03102749 https://tel.archives-ouvertes.fr/tel-03102749 Submitted on 7 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. N°d’ordre NNT : 2020LYSEI065 THÈSE de DOCTORAT DE L’UNIVERSITÉ DE LYON Opérée au sein de : CEA Grenoble Ecole Doctorale InfoMaths EDA N17° 512 (Informatique Mathématique) Spécialité de doctorat :Informatique Soutenue publiquement le 29/07/2020, par : Andrea Bocco A variable precision hardware acceleration for scientific computing Devant le jury composé de : Frédéric Pétrot Président et Rapporteur Professeur des Universités, TIMA, Grenoble, France Marc Dumas Rapporteur Professeur des Universités, École Normale Supérieure de Lyon, France Nathalie Revol Examinatrice Docteure, École Normale Supérieure de Lyon, France Fabrizio Ferrandi Examinateur Professeur associé, Politecnico di Milano, Italie Florent de Dinechin Directeur de thèse Professeur des Universités, INSA Lyon, France Yves Durand Co-directeur de thèse Docteur, CEA Grenoble, France Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2020LYSEI065/these.pdf © [A. -
Numerical Computing with IEEE Floating Point Arithmetic This Page Intentionally Left Blank Numerical Computing with IEEE Floating Point Arithmetic
Numerical Computing with IEEE Floating Point Arithmetic This page intentionally left blank Numerical Computing with IEEE Floating Point Arithmetic Including One Theorem, One Rule of Thumb, and One Hundred and One Exercises Michael L. Overton Courant Institute of Mathematical Sciences New York University New York, New York siam. Society for Industrial and Applied Mathematics Philadelphia Copyright © 2001 by the Society for Industrial and Applied Mathematics. 1098765432 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Overton, Michael L Numerical computing with IEEE floating point arithmetic / Michael L Overton. p. cm. Includes bibliographical references and index. ISBN 0-89871-571-7 I. Computer arithmetic. 2. Floating-point arithmetic. 3. Numerical calculations. I. Title. QA76.9.M35O94200I O04'.0l'5l--dc2l 00-067941 SlcLJTL is a registered trademark. Dedicated to girls who like math especially my daughter Eleuthera Overton Sa This page intentionally left blank Contents Preface ix Acknowledgments xi 1 Introduction 1 2 The Real Numbers 5 3 Computer Representation of Numbers 9 4 IEEE Floating Point Representation 17 5 Rounding 25 6 Correctly Rounded Floating Point Operations 31 7 Exceptions 41 8 The Intel Microprocessors 49 9 Programming Languages 55 10 Floating Point in C 59 11 Cancellation 71 12 Conditioning of Problems 77 13 Stability of Algorithms 83 14 Conclusion 97 Bibliography 101 vii This page intentionally left blank Preface Numerical computing is a vital part of the modern scientific infrastructure.