CS140 Lecture 08: Data Representation: Bits and Ints John Magee - PowerPoint PPT Presentation

CS140 Lecture 08: Data Representation: Bits and Ints John Magee 13 February 2017 Material From Computer Systems: A Programmer's Perspective, 3/E (CS:APP3e) Randal E. Bryant and David R. O'Hallaron, Carnegie Mellon University 1

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 2

Binary Representations 0 1 0 3.3V 2.8V 0.5V 0.0V 3

Encoding Byte Values  Byte = 8 bits  Binary 00000000 2 to 11111111 2 0 0 0000  Decimal: 0 10 to 255 10 1 1 0001 2 2 0010  Hexadecimal 00 16 to FF 16 3 3 0011 4 4 0100  Base 16 number representation 5 5 0101 6 6 0110  Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ 7 7 0111  Write FA1D37B 16 in C as 8 8 1000 9 9 1001 – 0xFA1D37B A 10 1010 – 0xfa1d37b B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111 4

Byte-Oriented Memory Organization • • •  Programs Refer to Virtual Addresses  Conceptually very large array of bytes  Actually implemented with hierarchy of different memory types  System provides address space private to particular “process”  Program being executed  Program can clobber its own data, but not that of others  Compiler + Run-Time System Control Allocation  Where different program objects should be stored  All allocation within single virtual address space 5

Machine Words  Machine Has “Word Size”  Nominal size of integer-valued data  Including addresses  Recently most machines used 32 bits (4 bytes) words  Limits addresses to 4GB  Becoming too small for memory-intensive applications  High-end systems use 64 bits (8 bytes) words  Potential address space ≈ 1.8 X 10 19 bytes  x86-64 machines support 48-bit addresses: 256 Terabytes  Machines support multiple data formats  Fractions or multiples of word size  Always integral number of bytes 6

Word-Oriented Memory Organization 32-bit 64-bit Bytes Addr.  Addresses Specify Byte Words Words Locations 0000 Addr 0001  Address of first byte in word = 0002 0000 ??  Addresses of successive words differ Addr 0003 by 4 (32-bit) or 8 (64-bit) = 0004 0000 ?? Addr 0005 = 0006 0004 ?? 0007 0008 Addr 0009 = 0010 0008 ?? Addr 0011 = 0008 ?? 0012 Addr 0013 = 0014 0012 ?? 0015 7

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 8

Byte Ordering  How should bytes within a multi-byte word be ordered in memory?  Conventions  Big Endian: Sun, PPC Mac, Internet  Least significant byte has highest address  Little Endian: x86  Least significant byte has lowest address 9

Byte Ordering Example  Big Endian  Least significant byte has highest address  Little Endian  Least significant byte has lowest address  Example  Variable x has 4-byte representation 0x01234567  Address given by &x is 0x100 Big Endian 0x100 0x101 0x102 0x103 01 23 45 67 01 23 45 67 Little Endian 0x100 0x101 0x102 0x103 67 45 23 01 67 45 23 01 10

Reading Byte-Reversed Listings  Disassembly  Text representation of binary machine code  Generated by program that reads the machine code  Example Fragment Address Instruction Code Assembly Rendition 8048365: 5b pop %ebx 8048366: 81 c3 ab 12 00 00 add $0x12ab,%ebx 804836c: 83 bb 28 00 00 00 00 cmpl $0x0,0x28(%ebx)  Deciphering Numbers  Value: 0x12ab  Pad to 32 bits: 0x000012ab  Split into bytes: 00 00 12 ab  Reverse: ab 12 00 00 11

Decimal: 15213 Representing Integers 0011 1011 0110 1101 Binary: 3 B 6 D Hex: int A = 15213; long int C = 15213; IA32, x86-64 Sun IA32 x86-64 Sun 6D 00 6D 6D 00 3B 00 3B 3B 00 00 3B 00 00 3B 00 6D 00 00 6D 00 int B = -15213; 00 00 IA32, x86-64 Sun 00 93 FF C4 FF FF C4 Two’s complement representation FF 93 12

Representing Pointers int B = -15213; int *P = &B; Sun IA32 x86-64 EF D4 0C FF F8 89 FB FF EC 2C BF FF FF 7F 00 00 Different compilers & machines assign different locations to objects 13

Representing Strings char S[6] = "18243";  Strings in C  Represented by array of characters  Each character encoded in ASCII format Linux/Alpha Sun  Standard 7-bit encoding of character set 31 31  Character “0” has code 0x30 38 38 – Digit i has code 0x30+ i 32 32  String should be null-terminated 34 34  Final character = 0 33 33  Compatibility 00 00  Byte ordering not an issue 14

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 15

Boolean Algebra  Developed by George Boole in 19th Century  Algebraic representation of logic  Encode “True” as 1 and “False” as 0 And Or  A&B = 1 when both A=1 and B=1  A|B = 1 when either A=1 or B=1 Not Exclusive-Or (Xor)  ~A = 1 when A=0  A^B = 1 when either A=1 or B=1, but not both 16

General Boolean Algebras  Operate on Bit Vectors  Operations applied bitwise 01101001 01101001 01101001 & 01010101 | 01010101 ^ 01010101 ~ 01010101 01000001 01111101 00111100 10101010 01000001 01111101 00111100 10101010  All of the Properties of Boolean Algebra Apply 17

Bit-Level Operations in C  Operations & , | , ~ , ^ Available in C  Apply to any “integral” data type  long, int, short, char, unsigned  View arguments as bit vectors  Arguments applied bit-wise  Examples (Char data type)  ~0x41 & 0xBE  ~01000001 2 & 10111110 2  ~0x00 & 0xFF  ~00000000 2 & 11111111 2  0x69 & 0x55 & 0x41  01101001 2 & 01010101 2 & 01000001 2  0x69 | 0x55 | 0x7D  01101001 2 | 01010101 2 | 01111101 2 18

Contrast: Logic Operations in C  Contrast to Logical Operators  &&, ||, !  View 0 as “False”  Anything nonzero as “True”  Always return 0 or 1  Early termination  Examples (char data type)  !0x41 = 0x00  !0x00 = 0x01  !!0x41 = 0x01  0x69 && 0x55 && 0x01  0x69 || 0x55 || 0x01  p && *p (avoids null pointer access) 19

Contrast: Logic Operations in C  Contrast to Logical Operators  &&, ||, !  View 0 as “False”  Anything nonzero as “True”  Always return 0 or 1  Early termination Watch out for && vs. & (and || vs. |)…  Examples (char data type) one of the more common oopsies in  !0x41  0x00 C programming  !0x00  0x01  !!0x41  0x01  0x69 && 0x55  0x01  0x69 || 0x55  0x01  p && *p (avoids null pointer access) 20

Shift Operations  Left Shift: x << y Argument x 01100010  Shift bit-vector x left y positions << 3 00010 000 00010 000 00010 000 – Throw away extra bits on left Log. >> 2 00 011000 00 011000 00 011000  Fill with 0 ’s on right Arith. >> 2 00 011000 00 011000 00 011000  Right Shift: x >> y  Shift bit-vector x right y positions Argument x 10100010  Throw away extra bits on right  Logical shift << 3 00010 000 00010 000 00010 000  Fill with 0 ’s on left Log. >> 2 00 101000 00 101000 00 101000  Arithmetic shift Arith. >> 2 11 101000 11 101000 11 101000  Replicate most significant bit on left  Undefined Behavior  Shift amount < 0 or ≥ word size 21

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 22

Encoding Integers Unsigned Two’s Complement w − 1 w − 2 − x w − 1 ⋅ 2 w − 1 + ∑ ∑ = x i ⋅ 2 i = x i ⋅ 2 i B 2 U ( X ) B 2 T ( X ) i = 0 i = 0 short int x = 15213; Sign short int y = -15213; Bit  C short 2 bytes long Decimal Hex Binary x 3B 6D 00111011 01101101 15213 y C4 93 11000100 10010011 -15213  Sign Bit  For 2’s complement, most significant bit indicates sign  0 for nonnegative  1 for negative  B2U = Binary to Unsigned B2T = Binary to Two’s Complement 23

Conversion Visualized  2’s Comp. → Unsigned UMax  Ordering Inversion UMax – 1  Negative → Big Positive TMax + 1 Unsigned TMax TMax Range 2’s Complement 0 0 Range –1 –2 TMin 24

Numeric Ranges  Unsigned Values  Two’s Complement Values  UMin = 0  TMin –2 w –1 = 000…0 100…0 2 w – 1  UMax = 2 w –1 – 1  TMax = 111…1 011…1  Other Values  Minus 1 111…1 Values for W = 16 Decimal Hex Binary UMax FF FF 11111111 11111111 65535 TMax 7F FF 01111111 11111111 32767 TMin 80 00 10000000 00000000 -32768 -1 FF FF 11111111 11111111 -1 0 00 00 00000000 00000000 0 25