– 1 – CS 105 Computer Systems Introduction Topics: Class Intro Data Representation CS 105 “Tour of the Black Holes of Computing!” Geoff Kuenning Fall 2014

– 2 – CS 105 Course Theme Abstraction is good, but don’t forget reality! Many CS Courses emphasize abstraction Abstract data types Asymptotic analysis These abstractions have limits Especially in the presence of bugs Need to understand underlying implementations Useful outcomes Become more effective programmers Able to find and eliminate bugs efficiently Able to tune program performance Prepare for later “systems” classes in CS Compilers, Operating Systems, File Systems, Computer Architecture, Robotics, etc.

– 3 – CS 105 Textbooks Randal E. Bryant and David R. O’Hallaron, “Computer Systems: A Programmer’s Perspective”, 2 nd Edition, Prentice Hall, Brian Kernighan and Dennis Ritchie, “The C Programming Language, Second Edition”, Prentice Hall, 1988 Larry Miller and Alex Quilici The Joy of C, Wiley, 1997

– 4 – CS 105 Syllabus Syllabus on Web: Calendar defines due dates Labs: cs105submit for some, others have specific directions

– 5 – CS 105 Notes: Work groups You must work in pairs on all labs Honor-code violation to work without your partner! Corollary: showing up late doesn’t harm only youHandins Check calendar for due dates Electronic submissions only Grading Characteristics Lab scores tend to be high Serious handicap if you don’t hand a lab in Tests & quizzes typically have a wider range of scores I.e., they’re primary determinant of your grade »…but not the ONLY one Do your share of lab work and reading, or bomb tests Do practice problems in book

– 6 – CS 105 Facilities Assignments will use Intel computer systems Not all machines are created alike Even Intel Macs aren’t necessarily compatible Most modern machines (including Knuth) are 64-bit Wilkes: x86/Linux specifically set up for this class Log in on a Mac, then ssh to Wilkes If you want fancy programs, start X11 first Directories are cross-mounted, so you can edit on Knuth or your Mac, and Wilkes will see your files …or ssh into Wilkes from your dorm All programs must run on Wilkes: that’s where we grade

CS 105 “Tour of the Black Holes of Computing” Topics Numeric Encodings Unsigned & two’s complement Programming Implications C promotion rules Basic operations Addition, negation, multiplication Programming Implications Consequences of overflow Using shifts to perform power-of-2 multiply/divide CS 105 Integers

– 8 – CS 105 C Puzzles Taken from old exams Assume machine with 32-bit word size, two’s complement integers For each of the following C expressions, either: Argue that it is true for all argument values, or Give example where it is not true x < 0  ((x*2) < 0) ux >= 0 x & 7 == 7  (x<<30) < 0 ux > -1 x > y  -x < -y x * x >= 0 x > 0 && y > 0  x + y > 0 x >= 0  -x <= 0 x = 0 int x = foo(); int y = bar(); unsigned ux = x; unsigned uy = y; Initialization

– 9 – CS 105 Encoding Integers short int x = 15213; short int y = ; C short 2 bytes long Sign Bit For 2’s complement, most-significant bit indicates sign 0 for nonnegative 1 for negative Unsigned Two’s Complement Sign Bit

– 10 – CS 105 Encoding Integers (Cont.) x = 15213: y = :

– 11 – CS 105 Numeric Ranges Unsigned Values UMin=0 000…0 UMax = 2 w – 1 111…1 Two’s-Complement Values TMin= –2 w–1 100…0 TMax = 2 w–1 – 1 011…1 Other Values Minus 1 111…1 Values for W = 16

– 12 – CS 105 Values for Different Word Sizes Observations |TMin | = TMax + 1 Asymmetric range UMax=2 * TMax + 1 C Programming #include K&R Appendix B11 Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Values platform-specific

– 13 – CS 105 An Important Detail No Self-Identifying Data Looking at a bunch of bits doesn’t tell you what they mean Could be signed, unsigned integer Could be floating-point number Could be part of a string Only the Program (Instructions) Knows for Sure!

– 14 – CS 105 Unsigned & Signed Numeric Values XB2T(X)B2U(X) –88 –79 –610 –511 –412 –313 –214 – Equivalence Same encodings for nonnegative valuesUniqueness Every bit pattern represents unique integer value Each representable integer has unique bit encoding

– 15 – CS 105 short int x = 15213; unsigned short int ux = (unsigned short) x; short int y = ; unsigned short int uy = (unsigned short) y; Casting Signed to Unsigned C Allows Conversions from Signed to Unsigned Resulting Value No change in bit representation Nonnegative values unchanged ux = Negative values change into (large) positive values uy = 50323

– 16 – CS 105 Relation Between Signed & Unsigned uy = y + 2 * 32768=y

– 17 – CS 105 Signed vs. Unsigned in C Constants By default are considered to be signed integers Unsigned if have “U” as suffix 0U, uCasting Explicit casting between signed & unsigned same as U2T and T2U int tx, ty; unsigned ux, uy; tx = (int) ux; uy = (unsigned) ty; Implicit casting also occurs via assignments and procedure calls tx = ux; uy = ty;

– 18 – CS 105 Casting Surprises Expression Evaluation If you mix unsigned and signed in single expression, signed values are implicitly cast to unsigned Including comparison operations, ==, = Examples for W = 32 Constant 1 Constant 2 RelationEvaluation 00u u u (unsigned) u (int) u

– 19 – CS U== unsigned -10< signed -10U> unsigned > signed U < unsigned -1-2 > signed (unsigned) -1-2 > unsigned U < unsigned (int) U> signed Casting Surprises Expression Evaluation If you mix unsigned and signed in single expression, signed values are implicitly cast to unsigned Including comparison operations, ==, = Examples for W = 32 Constant 1 Constant 2 RelationEvaluation 00u u u (unsigned) u (int) u

– 20 – CS TMax TMin –1 –2 0 UMax UMax – 1 TMax TMax + 1 2’s Comp. Range Unsigned Range Explanation of Casting Surprises 2’s Comp.  Unsigned Ordering Inversion Negative  Big Positive

– 21 – CS 105 Sign Extension Task: Given w-bit signed integer x Convert it to w+k-bit integer with same valueRule: Make k copies of sign bit: X = x w–1,…, x w–1, x w–1, x w–2,…, x 0 k copies of MSB X X w w k

– 22 – CS 105 Sign Extension Example Converting from smaller to larger integer data type C automatically performs sign extension short int x = 15213; int ix = (int) x; short int y = ; int iy = (int) y;

– 23 – CS 105 Why Should I Use Unsigned? Be careful using C compilers on some machines generate less efficient code unsigned i; for (i = 1; i < cnt; i++) a[i] += a[i-1]; Easy to make mistakes for (i = cnt-2; i >= 0; i--) a[i] += a[i+1]; Do use when performing modular arithmetic Multiprecision arithmetic Other esoteric stuff Do use when need extra bit’s worth of range Working right up to limit of word size E.g., array sizes

– 24 – CS 105 Negating with Complement & Increment Claim: Following holds for 2’s complement ~x + 1 == -xComplement Observation: ~x + x == 1111…11 2 == -1Increment ~x + x + (-x + 1)==-1 + (-x + 1) ~x + 1==-x Warning: Be cautious treating int ’s as integers OK here (associativity holds) x ~x

– 25 – CS 105 Comp. & Incr. Examples x =

– 26 – CS 105 Unsigned Addition Standard Addition Function Ignores carry output Implements Modular Arithmetic s= UAdd w (u, v)=u + v mod 2 w u v + u + v True Sum: w+1 bits Operands: w bits Discard Carry: w bits UAdd w (u, v)

– 27 – CS 105 Two’s-Complement Addition TAdd and UAdd have identical bit-level behavior Signed vs. unsigned addition in C: int s, t, u, v; s = (int) ((unsigned) u + (unsigned) v); t = u + v Will give s == t u v + u + v True Sum: w+1 bits Operands: w bits Discard Carry: w bits TAdd w (u, v)

– 28 – CS 105 Detecting 2’s-Comp. Overflow Task Given s = TAdd w (u, v) Determine if s = Add w (u, v) Example int s, u, v; s = u + v;Claim Overflow iff either: u, v < 0, s  0(NegOver) u, v  0, s < 0(PosOver) 0 2 w –1 PosOver NegOver

– 29 – CS 105 A Fun Fact Official C standard says overflow is “undefined” Intention was to let machine define what happens Recently compiler writers have decided “undefined” means “we get to choose” We can generate 0, biggest integer, or anything else Or if we’re sure it’ll overflow, we can optimize out completely This can introduce some lovely bugs (e.g., you can’t check for overflow) Currently big fight between compiler community and security community over this issue

– 30 – CS 105 Multiplication Computing exact product of w-bit numbers x, y Either signed or unsignedRanges Unsigned: 0 ≤ x * y ≤ (2 w – 1) 2 = 2 2w – 2 w Up to 2w bits Two’s complement min: x * y ≥ (–2 w–1 )*(2 w–1 –1) = –2 2w–2 + 2 w–1 Up to 2w–1 bits (including 1 for sign) Two’s complement max: x * y ≤ (–2 w–1 ) 2 = 2 2w–2 Up to 2w bits, but only for (TMin w ) 2 Maintaining exact results Would need to keep expanding word size with each product computed Done in software by “arbitrary-precision” arithmetic packages

– 31 – CS 105 Power-of-2 Multiply by Shifting Operation u << k gives u * 2 k Both signed and unsignedExamples u << 3==u * 8 u << 5 - u << 3==u * 24 Most machines shift and add much faster than multiply Compiler generates this code automatically u 2k2k * u · 2 k True Product: w+k bits Operands: w bits Discard k bits: w bits UMult w (u, 2 k ) k 000 TMult w (u, 2 k ) 000

– 32 – CS 105 Unsigned Power-of-2 Divide by Shifting Quotient of unsigned by power of 2 u >> k gives  u / 2 k  Uses logical shift u 2k2k / u / 2 k Division: Operands: k 0  u / 2 k  Result:. Binary Point 0