DOCSLIB.ORG

Unit 2 Skills & Outcomes Binary Arithmetic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Unit 2 Skills & Outcomes Binary Arithmetic 2.1 2.2 Skills & Outcomes • You should know and be able to apply the following skills with confidence Unit 2 – Perform addition & subtraction in unsigned & 2's complement system Integer Operations – Determine if overflow has occurred (Arithmetic, Overflow, Bitwise Logic, Shifting) – Perform bitwise operations on numbers – Perform logic and arithmetic shifts and understand how they can be used for multiplication/division – Understand arithmetic in binary and hex 2.3 2.4 Binary Arithmetic • Can perform all arithmetic operations (+,-,*, ÷) on binary numbers • Can use same methods as in decimal – Still use carries and borrows, etc. – Only now we carry when sum is __ or more rather than 10 or more (decimal) – We borrow ___’s not 10’s from other columns UNSIGNED BINARY ARITHMETIC • Easiest method is to add bits in your head in decimal (1+1 = 2) then convert the answer to binary (2 10 = 10 2) 2.5 2.6 Binary Addition Binary Addition & Subtraction • In decimal addition we carry when the sum is 10 or more • In binary addition we carry when the sum is 2 or more 0 1 1 1 (7) 1 0 1 0 (10) • Add bits in binary to produce a sum bit and a carry bit + 0 0 1 1 (3) - 0 1 0 1 (5) 1 0 0 1 1 + 0 + 1 + 0 + 1 00 01 01 10 no need sum bit no need sum bit no need sum bit carry 1 sum bit to carry to carry to carry into next column of bits 2.7 2.8 Binary Addition Binary Addition 1 2 0 10 0 0 011 0 (6) 01 10 (6) 1 + 1 110 + 011 1 (7) + 01 11 (7) + 1 01 0110 (6) 110 1 (13) 11 01 (13) 10 8 4 2 1 + 0111 (7) carry bit sum bit carry bit sum bit 1101 (13) 3 110 1 110 1 4 0110 (6) 0 0110 (6) 1 + 0111 (7) + 0 + 0111 (7) + 1 1101 (13) 01 1101 (13) 11 carry bit sum bit carry bit sum bit 2.9 2.10 Hexadecimal Arithmetic Binary Multiplication • Same style of operations • Like decimal multiplication, find each partial product – Carry when sum is 16 or more, etc. and _________ them, then sum them up • Multiplying two n-bit numbers yields at most a ______ -bit product 0 1 1 0 (6) 4 D 16 * 0 1 0 1 (5) 16 1 + B 5 16 16 1 Partial Products + Sum of the partial products 2.11 2.12 Binary Division • Use the same long division techniques as in decimal • Dividing two n-bit numbers may yield an n-bit quotient and n-bit remainder 0 1 0 1 r.1 (5 r.1) 10 "Taking the 2's complement" (2) 10 10 1 0 1 1 (11) 10 -1 0 SUBTRACTION THE EASY WAY 0 1 -0 0 1 1 -1 0 0 1 2.13 2.14 Modulo Arithmetic Taking the Negative CS:APP 2.3.3 • The primary difference between how humans and computers perform arithmetic is the finite • Question : Given a number in 2’s complement how _______________ of computers do we find its negative (i.e. -1 * X) – As humans we can use more digits (precision) as • Answer : By " __________________________ " needed – 0110 = +6 => -6 = 1010 – Computers can only used a _________ set of bits • Much like the odometer on your car once you go too – Operation defined as: many miles the values will wrap from 999999 to 000000 1. ______________________________ • Essentially all computer arithmetic is ___________ arithmetic 2. ______________________________ • If we have a width of w bits, then all operations are (i.e. finish with the same # of bits as we start with) module ______ – See next slides for example • This leads to alternate approaches to arithmetic – Example: Consider how you could change the clock time from 5 p.m. to 3 p.m. if you can't ________ hours 2.15 2.16 Taking the 2’s Complement Taking the 2’s Complement 1 -32 16 8 4 2 1 2 • Invert (flip) each bit Original # = 0 -32 16 8 4 2 1 101010 Original number = -22 0000 (take the 1’s 010011 Original number = +19 Take the Take the 2’s complement 2’s complement complement) yields the negative of a number Bit flip is called the 1’s – 1’s become 0’s 2’s comp. of 0 is __ complement of a number Resulting number = +22 – 0’s become 1’s 3 Resulting number = -19 Taking the 2’s complement 1000 Original # = -8 • Add 1 (drop final again yields the original number (the operation is carry-out, if any) symmetric) Take the 2’s complement Back to original = -22 Negative of -8 is ___ Important: Taking the 2’s complement is equivalent to (i.e. no positive taking the negative (negating) equivalent, but this is not a huge problem) 2.17 2.18 Radix Complement 12 . 11 1 11 12 1 10 2 10 2 Clock Analogy 9 3 9 3 4-2 = 4+10 8 4 8 4 5 7 6 7 6 5 99 00 01 99 00 01 98 02 98 02 03 10’s complement 03 04-02 = 04 + 98 The same algorithms regardless of unsigned or signed . 04 . 04 . ADDITION AND SUBTRACTION 0000 0000 1111 0001 1111 0001 1110 0010 1110 0010 0011 2’s complement 0011 . 0100 0100 - 0010 = 0100 + 1110 . 0100 . When using modulo arithmetic , subtraction can always be converted to addition. 2.19 2.20 2’s Complement Addition/Subtraction 2’s Complement Addition CS:APP 2.3.1 CS:APP 2.3.2 • Addition • No matter the sign of the operands just add as normal – Sign of the numbers _______________ • Drop any extra carry out – Add column by column – Drop any final ______________ 0011 (3) 1101 (-3) • The secret to modulo arithmetic + 0010 (2) + 0010 (2) • Subtraction – Any subtraction (A-B) can be converted to addition (_______) by taking the ______________of B – (A-B) becomes (A + _________ ) 0011 (3) 1101 (-3) – Drop any carry-out + 1110 (-2) + 1110 (-2) • The secret to modulo arithmetic 2.21 2.22 Unsigned and Signed Addition 2’s Complement Subtraction • Take the 2’s complement of the subtrahend (bottom #) • Addition process is the _________ for both and add to the original minuend (top #) unsigned and signed numbers • Drop any extra carry out – Add columns right to left 0011 (+3) 1101 (-3) • Examples: - 0010 (+2) - 1110 (-2) If unsigned If signed 1001 + 0011 2.23 2.24 Unsigned and Signed Subtraction Important Note • Subtraction process is the same for both • Almost all computers use 2's complement unsigned and signed numbers because… – Convert A – B to A + Comp. of B • The same addition and subtraction – Drop any final carry out ___________ can be used on unsigned and 2's • Examples: complement (signed) numbers If unsigned If signed • Thus we only need one set of 1100 (12) (-4) - 0010 (2) (2) _____________________________ to perform operations on both unsigned and signed numbers If unsigned If signed 2.25 2.26 CS:APP 2.3.1 Overflow CS:APP 2.3.2 • Overflow occurs when the result of an arithmetic operation is _____________ ________________________________ • Conditions and tests to determine overflow depend on the ________ being OVERFLOW used – Different algorithms for detecting overflow based on _____________________ 2.27 2.28 Unsigned Overflow 2’s Complement Overflow 0 Overflow occurs when you cross +1 -1 0000 this discontinuity 0 1111 0001 +15 -2 +2 0000 +1 1110 0010 1111 +14 0001 +2 -3 1110 0010 1101 0011 +3 +13 1101 +3 5 + 7 = +12 0011 -4 1100 0100 Plus 7 +4 10 + 7 = 17 +12 1100 0100 +4 -6 + -4 = -10 1011 0101 -5 +5 1010 1011 10 0101 With 4-bit 2’s complement 0110 With 4-bit unsigned numbers we +11 +5 numbers we can only represent -6 1001 0111 +6 can only represent 0 – 15. Thus, 1010 0110 1000 we say overflow has occurred. -8 to +7. Thus, we say overflow +7 +10 1001 0111 +6 -7 has occurred. -8 1000 +9 +7 +8 Overflow occurs when you cross this discontinuity 2.29 2.30 Overflow in Addition Overflow in Subtraction • Overflow occurs when the result of the addition • Overflow occurs when the result of the subtraction cannot be represented with the given number cannot be represented with the given number of of bits. bits. • Tests for overflow: • Tests for overflow: – Unsigned: if _______ [expect negative result] – Unsigned: if _________ [result __________ than inputs] – Signed: ________________ [result has inappropriate sign] – Signed: if _____________ [result has inappropriate sign] If unsigned If signed 0111_ 0111 (7) (7) 0111 A 1 1 If unsigned If signed 0 1 If unsigned If signed - 1000 (8) (-8) 0111 1’s comp. of B 1101 (13) (-3) 0110 (6) (6) (-1) (15) + 1 Add 1 + 0100 (4) (4) + 0101 (5) (5) 1111 (15) (-1) 0001 (17) (+1) 1011 (11) (-5) Desired Results If unsigned If signed Overflow No Overflow No Overflow Overflow Overflow Overflow Cout = 1 n + p Cout = 0 p + p = n Cout = 0 p + p = n 2.31 2.32 Binary Multiplication CS:APP 2.3.4 • Multiplying two n-bit numbers yields at most a 2*n -bit product • Multiplication operations on a modern processor can take ______ times longer than addition operations 0 1 1 0 (6) * 0 1 0 1 (5) 0 1 1 0 MULTIPLICATION AND DIVISION 0 0 0 0 Partial Products 0 1 1 0 + 0 0 0 0 0 0 1 1 1 1 0 Sum of the partial products 2.33 2.34 Binary Division Unsigned Multiplication Review • Dividing two n-bit numbers may yield an • Same rules as decimal multiplication n-bit quotient and n-bit remainder • Multiply each bit of Q by M shifting as you go • Division operations on a modern processor can take • An m-bit * n-bit mult.
Load more

Recommended publications
  • Bit Shifts Bit Operations, Logical Shifts, Arithmetic Shifts, Rotate Shifts
    Why bit operations Assembly languages all provide ways to manipulate individual bits in multi-byte values Some of the coolest “tricks” in assembly rely on Bit Shifts bit operations With only a few instructions one can do a lot very quickly using judicious bit operations And you can do them in almost all high-level ICS312 programming languages! Let’s look at some of the common operations, Machine-Level and starting with shifts Systems Programming logical shifts arithmetic shifts Henri Casanova ([email protected]) rotate shifts Shift Operations Logical Shifts The simplest shifts: bits disappear at one end A shift moves the bits around in some data and zeros appear at the other A shift can be toward the left (i.e., toward the most significant bits), or toward the right (i.e., original byte 1 0 1 1 0 1 0 1 toward the least significant bits) left log. shift 0 1 1 0 1 0 1 0 left log. shift 1 1 0 1 0 1 0 0 left log. shift 1 0 1 0 1 0 0 0 There are two kinds of shifts: right log. shift 0 1 0 1 0 1 0 0 Logical Shifts right log. shift 0 0 1 0 1 0 1 0 Arithmetic Shifts right log. shift 0 0 0 1 0 1 0 1 Logical Shift Instructions Shifts and Numbers Two instructions: shl and shr The common use for shifts: quickly multiply and divide by powers of 2 One specifies by how many bits the data is shifted In decimal, for instance: multiplying 0013 by 10 amounts to doing one left shift to obtain 0130 Either by just passing a constant to the instruction multiplying by 100=102 amounts to doing two left shifts to obtain 1300 Or by using whatever
    [Show full text]
  • Using LLVM for Optimized Lightweight Binary Re-Writing at Runtime
    Using LLVM for Optimized Lightweight Binary Re-Writing at Runtime Alexis Engelke, Josef Weidendorfer Department of Informatics Technical University of Munich Munich, Germany EMail: [email protected], [email protected] Abstract—Providing new parallel programming models/ab- helpful in reducing any overhead of provided abstractions. stractions as a set of library functions has the huge advantage But with new languages, porting of existing application that it allows for an relatively easy incremental porting path for code is required, being a high burden for adoption with legacy HPC applications, in contrast to the huge effort needed when novel concepts are only provided in new programming old code bases. Therefore, to provide abstractions such as languages or language extensions. However, performance issues PGAS for legacy codes, APIs implemented as libraries are are to be expected with fine granular usage of library functions. proposed [5], [6]. Libraries have the benefit that they easily In previous work, we argued that binary rewriting can bridge can be composed and can stay small and focused. However, the gap by tightly coupling application and library functions at libraries come with the caveat that fine granular usage of runtime. We showed that runtime specialization at the binary level, starting from a compiled, generic stencil code can help library functions in inner kernels will severely limit compiler in approaching performance of manually written, statically optimizations such as vectorization and thus, may heavily compiled version. reduce performance. In this paper, we analyze the benefits of post-processing the To this end, in previous work [7], we proposed a technique re-written binary code using standard compiler optimizations for lightweight code generation by re-combining existing as provided by LLVM.
    [Show full text]
  • Relational Representation of the LLVM Intermediate Language
    NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS SCHOOL OF SCIENCE DEPARTMENT OF INFORMATICS AND TELECOMMUNICATIONS UNDERGRADUATE STUDIES UNDERGRADUATE THESIS Relational Representation of the LLVM Intermediate Language Eirini I. Psallida Supervisors: Yannis Smaragdakis, Associate Professor NKUA Georgios Balatsouras, PhD Student NKUA ATHENS JANUARY 2014 ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ ΠΡΟΠΤΥΧΙΑΚΕΣ ΣΠΟΥΔΕΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Σχεσιακή Αναπαράσταση της Ενδιάμεσης Γλώσσας του LLVM Ειρήνη Ι. Ψαλλίδα Επιβλέποντες: Γιάννης Σμαραγδάκης, Αναπληρωτής Καθηγητής ΕΚΠΑ Γεώργιος Μπαλατσούρας, Διδακτορικός φοιτητής ΕΚΠΑ ΑΘΗΝΑ ΙΑΝΟΥΑΡΙΟΣ 2014 UNDERGRADUATE THESIS Relational Representation of the LLVM Intermediate Language Eirini I. Psallida R.N.: 1115200700272 Supervisor: Yannis Smaragdakis, Associate Professor NKUA Georgios Balatsouras, PhD Student NKUA ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Σχεσιακή Αναπαράσταση της ενδιάμεσης γλώσσας του LLVM Ειρήνη Ι. Ψαλλίδα Α.Μ.: 1115200700272 Επιβλέπων: Γιάννης Σμαραγδάκης, Αναπληρωτής Καθηγητής ΕΚΠΑ Γεώργιος Μπαλατσούρας, Διδακτορικός φοιτητής ΕΚΠΑ ΠΕΡΙΛΗΨΗ Περιγράφουμε τη σχεσιακή αναπαράσταση της ενδιάμεσης γλώσσας του LLVM, γνωστή ως LLVM IR. Η υλοποίηση μας παράγει σχέσεις από ένα πρόγραμμα εισόδου σε ενδιάμεση μορφή LLVM. Κάθε σχέση αποθηκεύεται σαν πίνακας βάσης δεδομένων σε ένα περιβάλλον εργασίας Datalog. Αναπαριστούμε το σύστημα τύπων καθώς και το σύνολο εντολών της γλώσσας του LLVM. Υποστηρίζουμε επίσης τους περιορισμούς της γλώσσας προσδιορίζοντάς τους με χρήση της προγραμματιστικής γλώσσας Datalog. ΘΕΜΑΤΙΚΗ ΠΕΡΙΟΧΗ: Μεταγλωτιστές, Γλώσσες Προγραμματισμού ΛΕΞΕΙΣ ΚΛΕΙΔΙΑ: σχεσιακή αναπαράσταση, ενδιάμεση αναπαράσταση, σύστημα τύπων, σύνολο εντολών, LLVM, Datalog ABSTRACT We describe the relational representation of the LLVM intermediate language, known as the LLVM IR. Our implementation produces the relation contents of an input program in the LLVM intermediate form. Each relation is stored as a database table into a Datalog workspace.
    [Show full text]
  • A Cross-Platform Programmer's Calculator
    – Independent Work Report Fall, 2015 – A Cross-Platform Programmer’s Calculator Brian Rosenfeld Advisor: Robert Dondero Abstract This paper details the development of the first cross-platform programmer’s calculator. As users of programmer’s calculators, we wanted to address limitations of existing, platform-specific options in order to make a new, better calculator for us and others. And rather than develop for only one- platform, we wanted to make an application that could run on multiple ones, maximizing its reach and utility. From the start, we emphasized software-engineering and human-computer-interaction best practices, prioritizing portability, robustness, and usability. In this paper, we explain the decision to build a Google Chrome Application and illustrate how using the developer-preview Chrome Apps for Mobile Toolchain enabled us to build an application that could also run as native iOS and Android applications [18]. We discuss how we achieved support of signed and unsigned 8, 16, 32, and 64-bit integral types in JavaScript, a language with only one numerical type [15], and we demonstrate how we adapted the user interface for different devices. Lastly, we describe our usability testing and explain how we addressed learnability concerns in a second version. The end result is a user-friendly and versatile calculator that offers value to programmers, students, and educators alike. 1. Introduction This project originated from a conversation with Dr. Dondero in which I expressed an interest in software engineering, and he mentioned a need for a good programmer’s calculator. Dr. Dondero uses a programmer’s calculator while teaching Introduction to Programming Systems at Princeton (COS 217), and he had found that the pre-installed Mac OS X calculator did not handle all of his use cases.
    [Show full text]
  • Moscow ML Library Documentation
    Moscow ML Library Documentation Version 2.00 of June 2000 Sergei Romanenko, Russian Academy of Sciences, Moscow, Russia Claudio Russo, Cambridge University, Cambridge, United Kingdom Peter Sestoft, Royal Veterinary and Agricultural University, Copenhagen, Denmark This document This manual describes the Moscow ML library, which includes parts of the SML Basis Library and several extensions. The manual has been generated automatically from the commented signature files. Alternative formats of this document Hypertext on the World-Wide Web The manual is available at http://www.dina.kvl.dk/~sestoft/mosmllib/ for online browsing. Hypertext in the Moscow ML distribution The manual is available for offline browsing at mosml/doc/mosmllib/index.html in the distribution. On-line help in the Moscow ML interactive system The manual is available also in interactive mosml sessions. Type help "lib"; for an overview of built-in function libraries. Type help "fromstring"; for help on a particular identifier, such as fromString. This will produce a menu of all library structures which contain the identifier fromstring (disregarding the lowercase/uppercase distinction): -------------------------------- | 1 | val Bool.fromString | | 2 | val Char.fromString | | 3 | val Date.fromString | | 4 | val Int.fromString | | 5 | val Path.fromString | | 6 | val Real.fromString | | 7 | val String.fromString | | 8 | val Time.fromString | | 9 | val Word.fromString | | 10 | val Word8.fromString | -------------------------------- Choosing a number from this menu will invoke the
    [Show full text]
  • Bitwise Instructions
    Bitwise Instructions CSE 30: Computer Organization and Systems Programming Dept. of Computer Science and Engineering University of California, San Diego Overview vBitwise Instructions vShifts and Rotates vARM Arithmetic Datapath Logical Operators vBasic logical operators: vAND: outputs 1 only if both inputs are 1 vOR: outputs 1 if at least one input is 1 vXOR: outputs 1 if exactly one input is 1 vIn general, can define them to accept >2 inputs, but in the case of ARM assembly, both of these accept exactly 2 inputs and produce 1 output vAgain, rigid syntax, simpler hardware Logical Operators vTruth Table: standard table listing all possible combinations of inputs and resultant output for each vTruth Table for AND, OR and XOR A B A AND B A OR B A XOR B A BIC B 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 Uses for Logical Operators vNote that ANDing a bit with 0 produces a 0 at the output while ANDing a bit with 1 produces the original bit. vThis can be used to create a mask. vExample: 1011 0110 1010 0100 0011 1101 1001 1010 mask: 0000 0000 0000 0000 0000 1111 1111 1111 vThe result of ANDing these: 0000 0000 0000 0000 0000 1101 1001 1010 mask last 12 bits Uses for Logical Operators vSimilarly, note that ORing a bit with 1 produces a 1 at the output while ORing a bit with 0 produces the original bit. vThis can be used to force certain bits of a string to 1s.
    [Show full text]
  • Bit-Level Transformation and Optimization for Hardware
    Bit-Level Transformation and Optimization for Hardware Synthesis of Algorithmic Descriptions Jiyu Zhang*† , Zhiru Zhang+, Sheng Zhou+, Mingxing Tan*, Xianhua Liu*, Xu Cheng*, Jason Cong† *MicroProcessor Research and Development Center, Peking University, Beijing, PRC † Computer Science Department, University Of California, Los Angeles, CA 90095, USA +AutoESL Design Technologies, Los Angeles, CA 90064, USA {zhangjiyu, tanmingxing, liuxianhua, chengxu}@mprc.pku.edu.cn {zhiruz, zhousheng}@autoesl.com, [email protected] ABSTRACT and more popularity [3-6]. However, high-quality As the complexity of integrated circuit systems implementations are difficult to achieve automatically, increases, automated hardware design from higher- especially when the description of the functionality is level abstraction is becoming more and more important. written in a high-level software programming language. However, for many high-level programming languages, For bitwise computation-intensive applications, one of such as C/C++, the description of bitwise access and the main difficulties is the lack of bit-accurate computation is not as direct as hardware description descriptions in high-level software programming languages, and hardware synthesis of algorithmic languages. The wide use of bitwise operations in descriptions may generate sub-optimal implement- certain application domains calls for specific bit-level tations for bitwise computation-intensive applications. transformation and optimization to assist hardware In this paper we introduce a bit-level
    [Show full text]
  • Bitwise Operators
    Logical operations ANDORNOTXORAND,OR,NOT,XOR •Loggpical operations are the o perations that have its result as a true or false. • The logical operations can be: • Unary operations that has only one operand (NOT) • ex. NOT operand • Binary operations that has two operands (AND,OR,XOR) • ex. operand 1 AND operand 2 operand 1 OR operand 2 operand 1 XOR operand 2 1 Dr.AbuArqoub Logical operations ANDORNOTXORAND,OR,NOT,XOR • Operands of logical operations can be: - operands that have values true or false - operands that have binary digits 0 or 1. (in this case the operations called bitwise operations). • In computer programming ,a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. 2 Dr.AbuArqoub Truth tables • The following tables (truth tables ) that shows the result of logical operations that operates on values true, false. x y Z=x AND y x y Z=x OR y F F F F F F F T F F T T T F F T F T T T T T T T x y Z=x XOR y x NOT X F F F F T F T T T F T F T T T F 3 Dr.AbuArqoub Bitwise Operations • In computer programming ,a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. • Bitwise operators • NOT • The bitwise NOT, or complement, is an unary operation that performs logical negation on each bit, forming the ones' complement of the given binary value.
    [Show full text]
  • Bits, Bytes, and Integers Today: Bits, Bytes, and Integers
    Bits, Bytes, and Integers Today: Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers . Representation: unsigned and signed . Conversion, casting . Expanding, truncating . Addition, negation, multiplication, shifting . Summary Representations in memory, pointers, strings Everything is bits Each bit is 0 or 1 By encoding/interpreting sets of bits in various ways . Computers determine what to do (instructions) . … and represent and manipulate numbers, sets, strings, etc… Why bits? Electronic Implementation . Easy to store with bistable elements . Reliably transmitted on noisy and inaccurate wires 0 1 0 1.1V 0.9V 0.2V 0.0V For example, can count in binary Base 2 Number Representation . Represent 1521310 as 111011011011012 . Represent 1.2010 as 1.0011001100110011[0011]…2 4 13 . Represent 1.5213 X 10 as 1.11011011011012 X 2 Encoding Byte Values Byte = 8 bits . Binary 000000002 to 111111112 0 0 0000 . Decimal: 010 to 25510 1 1 0001 2 2 0010 . Hexadecimal 0016 to FF16 3 3 0011 . 4 4 0100 Base 16 number representation 5 5 0101 . Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ 6 6 0110 7 7 0111 . Write FA1D37B16 in C as 8 8 1000 – 0xFA1D37B 9 9 1001 A 10 1010 – 0xfa1d37b B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111 Example Data Representations C Data Type Typical 32-bit Typical 64-bit x86-64 char 1 1 1 short 2 2 2 int 4 4 4 long 4 8 8 float 4 4 4 double 8 8 8 long double − − 10/16 pointer 4 8 8 Today: Bits, Bytes, and Integers Representing information as bits Bit-level manipulations Integers .
    [Show full text]
  • Integer Arithmetic and Undefined Behavior in C Brad Karp UCL Computer Science
    Carnegie Mellon Integer Arithmetic and Undefined Behavior in C Brad Karp UCL Computer Science CS 3007 23rd January 2018 (lecture notes derived from material from Eddie Kohler, John Regehr, Phil Gibbons, Randy Bryant, and Dave O’Hallaron) 1 Carnegie Mellon Outline: Integer Arithmetic and Undefined Behavior in C ¢ C primitive data types ¢ C integer arithmetic § Unsigned and signed (two’s complement) representations § Maximum and minimum values; conversions and casts § Perils of C integer arithmetic, unsigned and especially signed ¢ Undefined behavior (UB) in C § As defined in the C99 language standard § Consequences for program behavior § Consequences for compiler and optimizer behavior § Examples § Prevalence of UB in real-world Linux application-level code ¢ Recommendations for defensive programming in C 2 Carnegie Mellon Example Data Representations C Data Type Typical 32-bit Typical 64-bit x86-64 char 1 1 1 signed short (default) 2 2 2 and int 4 4 4 unsigned variants long 4 8 8 float 4 4 4 double 8 8 8 pointer 4 8 8 3 Carnegie Mellon Portable C types with Fixed Sizes C Data Type all archs {u}int8_t 1 {u}int16_t 2 {u}int32_t 4 {u}int64_t 8 uintptr_t 4 or 8 Type definitions available in #include <stdint.h> 4 Carnegie Mellon Shift Operations ¢ Left Shift: x << y Argument x 01100010 § Shift bit-vector x left y positions << 3 00010000 – Throw away extra bits on left Log. >> 2 00011000 § Fill with 0’s on right Arith. >> 2 00011000 ¢ Right Shift: x >> y § Shift bit-vector x right y positions Argument 10100010 § Throw away extra bits on right x § Logical shift << 3 00010000 § Fill with 0’s on left Log.
    [Show full text]
  • Episode 7.03 – Coding Bitwise Operations
    Episode 7.03 – Coding Bitwise Operations Welcome to the Geek Author series on Computer Organization and Design Fundamentals. I’m David Tarnoff, and in this series, we are working our way through the topics of Computer Organization, Computer Architecture, Digital Design, and Embedded System Design. If you’re interested in the inner workings of a computer, then you’re in the right place. The only background you’ll need for this series is an understanding of integer math, and if possible, a little experience with a programming language such as Java. And one more thing. This episode has direct consequences for our code. You can find coding examples on the episode worksheet, a link to which can be found on the transcript page at intermation.com. Way back in Episode 2.2 – Unsigned Binary Conversion, we introduced three operators that allow us to manipulate integers at the bit level: the logical shift left (represented with two adjacent less-than operators, <<), the arithmetic shift right (represented with two adjacent greater-than operators, >>), and the logical shift right (represented with three adjacent greater-than operators, >>>). These special operators allow us to take all of the bits in a binary integer and move them left or right by a specified number of bit positions. The syntax of all three of these operators is to place the integer we wish to shift on the left side of the operator and the number of bits we wish to shift it by on the right side of the operator. In that episode, we introduced these operators to show how bit shifts could take the place of multiplication or division by powers of two.
    [Show full text]
  • Openvms Linker Utility Manual
    OpenVMS Linker Utility Manual Order Number: AA–PV6CD–TK April 2001 This manual describes the OpenVMS Linker utility. Revision/Update Information: This manual supersedes the OpenVMS Linker Utility Manual, Version 7.1 Software Version: OpenVMS Alpha Version 7.3 OpenVMS VAX Version 7.3 Compaq Computer Corporation Houston, Texas © 2001 Compaq Computer Corporation Compaq, VAX, VMS, and the Compaq logo Registered in U.S. Patent and Trademark Office. OpenVMS is a trademark of Compaq Information Technologies Group, L.P. in the United States and other countries. UNIX is a trademark of The Open Group. All other product names mentioned herein may be the trademarks or registered trademarks of their respective companies. Confidential computer software. Valid license from Compaq required for possession, use, or copying. Consistent with FAR 12.211 and 12.212, Commercial Computer Software, Computer Software Documentation, and Technical Data for Commercial Items are licensed to the U.S. Government under vendor’s standard commercial license. Compaq shall not be liable for technical or editorial errors or omissions contained herein. The information in this document is provided "as is" without warranty of any kind and is subject to change without notice. The warranties for Compaq products are set forth in the express limited warranty statements accompanying such products. Nothing herein should be construed as constituting an additional warranty. ZK4548 The Compaq OpenVMS documentation set is available on CD–ROM. This document was prepared using VAX DOCUMENT Version 2.1. Contents Preface ............................................................ xi Part I Linker Utility Description 1 Introduction 1.1 Overview . ................................................ 1–1 1.1.1 Linker Functions .
    [Show full text]