Bits, Bytes, and Integers. Data Representa+on: Base- 2 number representa+on. Binary Representa+ons. Encoding Byte Values. Example Data Representa+ons

Transcription

1 Bits, Bytes, and Data Representa+on: Bits, Bytes, and CSci 1 - Machine Architectre and Organiza+on Professor Pen- Chng Yew Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary With sides from Randy Bryant and Dave O Hallaron 1 Binary Representa+ons Each bit is or 1 By encoding/interpre+ng sets of bits in varios ways Compters determine what to do (instrc9ons), and represent and maniplate nmbers, sets, strings, etc Why bits? Electronic Implementa+on Easy to store with bistable elements, e.g. switches, transistors, Reliably transmiked on noisy and inaccrate wires 1.1V.9V 1 Base- nmber representa+on Decimal to Binary Represent 15 1 as 111 Represent 1. 1 as 1.11[11] Represent 1.51 X 1 as 1.11 X 7 Binary to Decimal Represent 111 as 1 Represent as Represent X 3 as 1.35 X 1 1.V.V 3 Encoding Byte Vales Byte bits Binary to Decimal: 1 to 551 Hexadecimal: 1 to 1 Base 1 nmber representa9on Use characters to 9 and A to F Write FA1D37B1 in C as xfa1d37b xfa1d37b xfa1d37b Hex Decimal Binary A 1 11 B C 1 11 D E F Example Data Representa+ons C Data Type Typical 3- bit Typical - bit x- char short int long float doble long doble 1/1 pointer 5 1

2 Today: Bits, Bytes, and Boolean Algebra Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary Developed by George Boole in 19th Centry Algebraic representa9on of logic Encode Tre as 1 and False as And Or n A&B 1 when both A1 and B1 n A B 1 when either A1 or B1 Not Exclsive- Or (Xor) n ~A 1 when A n A^B 1 when either A1 or B1, bt not both 7 General Boolean Algebras Represen+ng & Manipla+ng Sets Operate on Bit Vectors Opera9ons applied bitwise 1111 & ^ ~ All of the Proper+es of Boolean Algebra Apply 9 Representa+on Width w bit vector represents sbsets of {,, w 1} a j 1 if j A 1111 A {, 3, 5, } A {,,, } 7531 Opera+ons & Intersec9on 11 {, } Union {,, 3,, 5, } ^ Symmetric difference 1111 {, 3,, 5 } ~ Complement 1111 { 1, 3, 5, 7 } 1 Bit- Level Opera+ons in C Opera+ons &,, ~, ^ Available in C Apply to any integral data type long, int, short, char, nsigned View argments as bit vectors Argments applied bit- wise Examples (char data type 1 byte) ~x1 xbe ~ ~x x ~ x9 & x55 x & x9 x55 x7d Contrast: Logic Opera+ons in C Contrast to Logical Operators &&,,! View as False Anything nonzero as Tre Always retrn or 1 Examples (char data type)!x1 x!x x1!!x1 x1 x9 && x55 x1 x9 x55 x1 11 1

3 Contrast: Logic Opera+ons in C Contrast to Logical Operators &&,,! View as False Anything nonzero as Tre Always retrn or 1 Early termina9on Examples (char data type)!x1 x!x x1!!x1 x1 Watch ot for && vs. & (and vs. ) one of the more common oopsies in C programming x9 && x55 x1 x9 x55 x1 ShiZ Opera+ons LeZ ShiZ: x << y ShiZ bit- vector x lez y posi+ons Throw away extra bits on lez Fill with s on right Right ShiZ: x >> y ShiZ bit- vector x right y posi+ons Throw away extra bits on right Logical shiz Fill with s on lez Arithme+c shiz Replicate most significant bit on lez Undefined Behavior ShiZ amont < or word size Argment x 111 << 3 1 Log. >> 11 Arith. >> 11 Argment x 111 << 3 1 Log. >> 11 Arith. >> Today: Bits, Bytes, and Bit- level manipla+ons Representa+on: nsigned and signed Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary Smmary Encoding Unsigned and Signed Unsigned (BU) w 1 BU(X) x i i i short int x 1513; short int y -1513; Sign Bit For s complement, most significant bit indicates sign for nonnega+ve 1 for nega+ve In C, data type short is bytes long Two s Complement (BT) w BT(X) x w 1 w 1 + x i i i Decimal Hex Binary x B D y C Sign Bit 15 1 Encoding Example (Binary to Unsigned BU ) x 1513: y -1513: Weight Sm w 1 BU(X) x i i i w BT(X) x w 1 w 1 + x i i i 17 Binary Weight Table x 1 K 1 +1 x 1 K x 1 K 1 +1 x 1 1K x 1 3K 1 +1 x 1 K x 1 1K 1 +1 x 1 5K x 1 51K 1 3

4 Conver+ng Unsigned to Binary (UB) Dividing the nmber repeatedly by n+l the nmber becomes 9? w 1 BU(X) x i i i Same nota+on to ANY nmber system Divide by Nmber Remainder 9 1 Nmeric Ranges Unsigned Vales UMin UMax w Two s Complement Vales Tmin w 1 1 Tmax w Other Vales Mins 1 w Vales for W 1 Decimal Hex Binary UMax TMax 377 7F TMin x x + x 3 + x + x x 9 19 Vales for Different Word Sizes Observa+ons TMin TMax + 1 Asymmetric range UMax * TMax + 1 W 1 3 UMax 55 5,535,9,97,95 1,,7,73,79,551,15 TMax 17 3,77,17,3,7 9,3,37,3,5,775,7 TMin - 1-3,7 -,17,3, - 9,3,37,3,5,775, C Programming #inclde <limits.h> Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Vales plaiorm specific Need to know - > to avoid overflow 1 Unsigned & Signed Integer Vales X BU(X) BT(X) Eqivalence Same encodings for nonnega9ve integer vales Uniqeness Every bit pakern represents niqe integer vale Each representable integer has niqe bit encoding Can Invert Mappings UB(x) BU - 1 (x) Bit pakern for nsigned integer TB(x) BT - 1 (x) Bit pakern for two s comp integer Today: Bits, Bytes, and Bit- level manipla+ons Representa9on: nsigned and signed Conversion, cas+ng Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary Mapping Between Signed & Unsigned Two s Complement TU Unsigned x TB X BU x Maintain Same Bit Palern Unsigned UT Two s Complement x UB X BT x Maintain Same Bit Palern Mappings between nsigned and two s complement nmbers: keep bit representa+ons and reinterpret 3

5 Mapping Signed Unsigned Mapping Signed Unsigned Bits Signed Unsigned Bits Signed Unsigned TU UT / Rela+on between Signed & Unsigned Two s Complement w 1 x x Large nega+ve weight becomes Large posi+ve weight x TU TB X BU Maintain Same Bit Palern x Unsigned x x x x + w x < 7 Conversion Visalized s Comp. Unsigned Ordering Inversion Nega9ve Big Posi9ve Warning: Can case a lot of confsion and bgs in C!!! s Complement Range TMax 1 TMin UMax UMax 1 TMax + 1 TMax Unsigned Range Signed vs. Unsigned in C Constants By defalt are considered to be signed integers Unsigned if have U as sffix U, 99759U Cas+ng (i.e. conversion) Explicit cas+ng between signed & nsigned same as UT and TU int tx, ty; nsigned x, y; tx (int) x; y (nsigned) ty; Smmary Cas+ng Signed Unsigned: Basic Rles Bit palern is maintained Bt reinterpreted Can have nexpected effects: adding or sbtrac+ng w Expression containing signed and nsigned int int is cast to nsigned!! Implicit cas+ng also occrs via assignments and procedre calls tx x; y ty;

6 Today: Bits, Bytes, and Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca+ng Addi9on, nega9on, ml9plica9on, shicing Smmary Sign Extension Task: Given w- bit signed integer x Convert it to w+k- bit integer with same vale Rle: Make k copies of sign bit: X ʹ x w 1,, x w 1, x w 1, x w,, x k copies of MSB X w 3 X ʹ k w 33 Sign Extension Example short int x 1513; int ix (int) x; short int y -1513; int iy (int) y; Conver+ng from smaller to larger integer data type C atoma+cally performs sign extension Decimal Hex Binary x B D ix B D y C iy C Smmary: Expanding, Trnca+ng: Basic Rles Expanding (e.g., short int to int) Unsigned: zeros added Signed: sign extension Both yield expected reslt Trnca+ng (e.g., nsigned to nsigned short) Unsigned/signed: bits are trncated Reslt reinterpreted Unsigned: mod opera9on Signed: similar to mod For small nmbers yields expected behavior 3 35 Today: Bits, Bytes, and Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca9ng Addi+on, nega+on, ml+plica+on, shizing Smmary Nega+on: Complement & Increment Claim: Following Holds for s Complement ~x + 1 -x Two s Complement Observa9on: ~x + x x ~x

7 Complement & Increment Examples x 1513 Decimal Hex Binary x B D ~x -151 C ~x C y C Unsigned Addi+on Operands: w bits + v Tre Sm: w+1 bits + v Discard Carry: w bits UAdd w (, v) x Decimal Hex Binary ~ ~+1 q What is - TMIN? (pp.37) Standard Addi+on Fnc+on Ignores carry otpt Implements Modlar Arithme+c s UAdd w (, v) ( + v) mod w + v + v < w UAdd w (,v) + v w + v w 3 39 Visalizing (Mathema+cal) Integer Addi+on Visalizing Unsigned Addi+on Integer Addi+on - bit integers, v Compte tre sm Add (, v) Vales increase linearly with and v Forms planar srface Add (, v) Integer Addition v Wraps Arond If tre sm w At most once Tre Sm w+1 w Overflow Modlar Sm Overflow UAdd (, v) v 1 Two s Complement Addi+on Operands: w bits Tre Sm: w+1 bits TAdd and UAdd have Iden+cal Bit- Level Behavior Signed vs. nsigned addi9on in C: int s, t,, v; s (int) ((nsigned) + (nsigned) v); t + v Will give s t + v + v Discard Carry: w bits TAdd w (, v) TAdd Overflow Fnc+onality Tre sm reqires w+1 bits Drop off Most Significant Bit (MSB) Treat remaining bits as s comp. integer (w) 1 (w 1) (w 1) 1 Tre Sm w PosOver NegOver TAdd Reslt

8 Visalizing s Complement Addi+on Characterizing TAdd Vales - bit two s comp. Range from - to +7 Wraps Arond If sm w 1 Becomes nega9ve At most once If sm < w 1 Becomes posi9ve At most once NegOver TAdd (, v) v PosOver Fnc+onality Tre sm reqires w+1 bits Drop off MSB Treat remaining bits as s comp. integer TAdd(, v) > v < Nega9ve Overflow < > Posi9ve Overflow + v + w 1 w + v < TMin w (NegOver) TAdd w (,v) + v TMin w + v TMax w + v w 1 TMax w < + v (PosOver) 5 Ml+plica+on Comp+ng Exact Prodct of w- bit nmbers x, y Either signed or nsigned Ranges Unsigned: Up to w bits Reslt range: x * y ( w 1) w w Two s complement min (nega9ve) : Up to w 1 bits Reslt range: x * y ( w 1 )*( w 1 1) w + w 1 Two s complement max (posi9ve): Up to w bits, bt only for (TMin w ) Reslt range: x * y ( w 1 ) w Maintaining Exact Reslts Wold need to keep expanding word size with each prodct compted Done in socware by arbitrary precision arithme9c packages Unsigned Ml+plica+on in C Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v Standard Ml+plica+on Fnc+on Ignores high order w bits Implements Modlar Arithme+c UMlt w (, v) v mod w * v UMlt w (, v) 7 Signed Ml+plica+on in C Power- of- Ml+ply with ShiZ Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v Standard Ml+plica+on Fnc+on Ignores high order w bits Some of which are different for signed vs. nsigned ml9plica9on Lower bits are the same * v TMlt w (, v) Opera+on << k gives * k Both signed and nsigned Operands: w bits Tre Prodct: w+k bits k k 1 Discard k bits: w bits UMlt w (, k ) TMlt w (, k ) Examples << 3 * << 5 - << 3 ( * 3) ( * ) * Most machines shic and add faster than ml9ply Compiler generates this code atoma9cally * k 9

9 Unsigned Power- of- Divide with ShiZ Qo+ent of Unsigned by Power of >> k gives / k Uses logical shic k Operands: / k 1 Division: Reslt: / k / k Division Compted Hex Binary x B D x >> D B x >> B x >> B Binary Point Signed Power- of- Divide with ShiZ Qo+ent of Signed by Power of x >> k gives x / k Uses arithme9c shic Ok when >, bt ronds wrong direc9on when < k x Binary Point Operands: / k 1 Division: x / k. Reslt: RondDown(x / k ) Division Compted Hex Binary y C y >> E y >> FC y >> C Correct Power- of- Divide Qo+ent of Nega+ve Nmber by Power of Want x / k (Rond Toward ) Compte as (x+ k -1)/ k In C: (x + (1<<k)-1) >> k Biases dividend toward Case 1: No ronding k Dividend: 1 + k Divisor: / k / k Biasing has no effect Binary Point Correct Power- of- Divide (Cont.) Case : Ronding Dividend: Divisor: k x 1 + k / k x / k Biasing adds 1 to final reslt 1 Incremented by Binary Point Incremented by Today: Bits, Bytes, and Arithme+c: Basic Rles Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary Addi+on: Unsigned/signed: Normal addi9on followed by trncate, same opera9on on bit level Unsigned: addi9on mod w Mathema9cal addi9on + possible sbtrac9on of w Signed: modified addi9on mod w (reslt in proper range) Mathema9cal addi9on + possible addi9on or sbtrac9on of w Ml+plica+on: Unsigned/signed: Normal ml9plica9on followed by trncate, same opera9on on bit level Unsigned: ml9plica9on mod w Signed: modified ml9plica9on mod w (reslt in proper range)

10 Arithme+c: Basic Rles Why Shold I Use Unsigned? LeZ shiz (Ml+plica+on) Unsigned/signed: ml9plica9on by k Always logical lec shic Don t se withot nderstanding implica+ons Easy to make mistakes nsigned i; for (i cnt-; i > ; i--) a[i] + a[i+1]; Right shiz (Division) Unsigned: logical right shic, div (division + rond to zero) by k Signed: arithme9c shic Posi+ve nmbers: div (division + rond to zero) by k Nega+ve nmbers: div (division + rond away from zero) by k Use biasing to fix Can be very sbtle #define DELTA sizeof(int) int i; for (i CNT; i-delta > ; i- DELTA) Con+ng Down with Unsigned Why Shold I Use Unsigned? (cont.) Proper way to se nsigned as loop index nsigned i; for (i cnt-; i < cnt; i--) a[i] + a[i+1]; See Robert Seacord, Secre Coding in C and C++ C Standard garantees that nsigned addi9on will behave like modlar arithme9c 1 à UMax Even beler size_t i; for (i cnt-; i < cnt; i--) a[i] + a[i+1]; Data type size_t defined as nsigned vale with length word size Code will work even if cnt UMax What if cnt is signed and <? 5 Do Use When Performing Modlar Arithme+c Ml9precision arithme9c Do Use When Using Bits to Represent Sets Logical right shic, no sign extension 59 Today: Bits, Bytes, and Bit- level manipla+ons Representa9on: nsigned and signed Expanding, trnca9ng Addi9on, nega9on, ml9plica9on, shicing Smmary Byte- Oriented Memory Organiza+on Programs refer to data by address Conceptally, envision it as a very large array of bytes In reality, it s not, bt can think of it that way An address is like an index into that array and, a pointer variable stores an address F Note: system provides private address spaces to each process Think of a process as a program being exected So, a program can clobber its own data, bt not that of others 1 1

11 Machine Words Any Given Machine Has a Word Size Nominal size of integer- valed data Inclding addresses Many crrent machines se 3 bits ( bytes) words Limits addresses to 3 GB Becoming too small for memory- intensive applica9ons Most high- end systems se bits ( bytes) words Poten9al address space 1. X 1 19 bytes (exabytes) Machines spport ml9ple data formats Frac9ons or ml9ples of word size Always integral nmber of bytes 3 Word- Oriented Memory Organiza+on Addresses Specify Byte Loca+ons Address of first byte in a word Addresses of sccessive words differ by (3- bit) or (- bit) Need to dis+ngish the no+on of address vs. data stored in that address 3-bit Words Addr?? Addr?? Addr?? Addr 1?? -bit Words Addr?? Addr?? Bytes Addr Byte Ordering How shold bytes within a ml+- byte word be ordered in memory? Conven+ons Big Endian: Sn, PPC Mac, Internet Least significant byte has highest address, i.e Most significant byte first Lille Endian: x, ARM processors rnning Android, ios and Windows Least significant byte has lowest address, i.e. Least significant byte first Byte Ordering Example Big Endian Least significant byte has highest address Lille Endian Least significant byte has lowest address Example Variable x has - byte representa9on x1357 Address given by &x is x1 Big Endian Little Endian x1 x11 x1 x x1 x11 x1 x Reading Byte- Reversed Lis+ngs Disassembly Text representa9on of binary machine code Generated by program that reads the machine code Example Fragment Address Instrction Code Assembly Rendition 35: 5b pop %ebx 3: 1 c3 ab 1 add $x1ab,%ebx 3c: 3 bb cmpl $x,x(%ebx) Deciphering Nmbers Vale: x1ab Pad to 3 bits: x1ab Split into bytes: 1 ab Reverse: ab 1 7 Represen+ng int A 1513; IA3, x- D 3B IA3, x- 93 C Sn 3B D int B -1513; Sn C 93 Decimal: 1513 Binary: Hex: 3 B D long int C 1513; IA3 D 3B x- D 3B Two s complement representation (Covered later) Sn 3B D 11

12 Examining Data Representa+ons Code to Print Byte Representa+on of Data Cas9ng pointer to nsigned char * allows treatment as a byte array typedef nsigned char *pointer; void show_bytes(pointer start, size_t len){ size_t i; for (i ; i < len; i++) printf( %p\tx%.x\n",start+i, start[i]); printf("\n"); } Printf directives: %p: Print pointer %x: Print Hexadecimal show_bytes Exec+on Example int a 1513; printf("int a 1513;\n"); show_bytes((pointer) &a, sizeof(int)); Reslt (Linx x-): int a 1513; x7fffb7f71dbc d x7fffb7f71dbd 3b x7fffb7f71dbe x7fffb7f71dbf 9 7 Represen+ng Pointers int B -1513; int *P &B; Sn EF FB C IA3 D F BF x- C 9 EC 7F Represen+ng Strings char S[] "13"; Strings in C Represented by array of characters Each character encoded in ASCII format Linx/Alpha Sn Standard 7- bit encoding of character set Character has code x3 3 3 Digit i has code x3+i 3 3 String shold be nll- terminated 3 3 Final character Compa+bility Byte ordering not an isse Different compilers & machines assign different loca9ons to objects Even get different reslts each 9me rn program