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15-213 Recitation 1 – 2/19/01

This recitation covers lab 1 examples including endianness, integer representation, floating point numbers, and type casting in C. It also discusses the concepts of signed and unsigned representation, addition and overflow, and the basics of IEEE floating point format.

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15-213 Recitation 1 – 2/19/01

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  1. 15-213 Recitation 1 – 2/19/01 Outline • Lab 1 Examples • Endianness • Integer Representation • Floating Point • Type Casting Anusha e-mail: anusha@andrew.cmu.edu Office Hours: Tuesday 11:30 – 1:00 Wean Clusters • Reminders • Lab 1: Tuesday, 11:59

  2. Lab 1 Examples /* * not(x) returns ~x without * explicit use of ~ */ int not(int x) { } return x ^ -1;

  3. Lab 1 Examples /* * or_n(x,y,n) returns a result where * the least significant n bits * are x | y, and the remaining high * bits are x. */ int or_n(int x, int y, int n) { } int m = -1 << n; return (x & m) | ((x | y) & ~m);

  4. Lab 1 Examples Some bit-twiddling hacks: http://graphics.stanford.edu/~seander/bithacks.html

  5. Endianess • Little Endian • Least sig. byte has highest address • Compaq, Intel, PC’s • Big Endian • Most sig. byte has highest address • IBM, Sun’s, Motorola

  6. Endianess, con’t. Notice ordered by byte not bit. Ex. Assume variable x has a value of 0x7A3469F2. x starts at address 0x200. Big 7A 34 69 F2 Little F2 69 34 7A

  7. Integer Representation Signed vs. Unsigned Representation UnsignedTwo’s Complement B2U(X) = B2T(X) = • MSB acts as the sign bit • -x = ~x + 1

  8. Signed vs. Unsigned, con’t. Assume here w = 8. 11 11 139 -117 255 -1

  9. Range of Integers • Unsigned • [0 … 2w - 1] • Signed • [-2w-1 … 2w-1 - 1] • |Tmin| = Tmax + 1 For w = 8, 255 1111 1111 127 0111 1111 -128 1000 0000 -1 1111 1111

  10. Addition and Overflow • Assume x,y unsigned. • 0 <= x, y <= 2w - 1 • 0 <= x + y <= 2w+1 - 2 /* could require w+1 bits to represent */ • use addition modulo 2w Consider w = 8, x = 15010 = 1001 01102 y = 15010 = 1001 01102 x + y = 30010 = 1 0010 11002 => 0010 11002 = 4410

  11. Addition and Overflow, con’t. • Assume x,y signed. • -2w-1 <= x, y <= 2w-1 - 1 • -2w <= x + y <= 2w - 2 /* could require w+1 bits to represent */ • again just compute sum using addition modulo 2w

  12. Addition and Overflow, con’t. Consider w = 8, Underflow 127 Within range 4 Overflow -127

  13. More examples Use w = 8, 0000 0111 1111 0011 -88 254 1111 1110 -1 1111 1111 -1 1111 1111 0 0000 0000

  14. s exp frac Floating Point(Better known as “I’m going to kill the person that thought this up”) • IEEE Floating Point • Standard notation • Tons of features we won’t look at • Floating Point at a bit level: • s – sign bit (S) • exp – exponent (maps to E, has e bits) • frac – significand (maps to M, has f bits) • Numerical Equivalent: –1s M 2E • “Normalized” and “Denormalized” encoding

  15. “Normalized” Encoding • exp  0 andexp  111…1 • If exp = 111…1, it’sor NAN • E = exp – B • B is the “Bias” • Usually 2e-1 – 1, but can be different • exp: Unsigned integer value [1, 2e – 1] • M = 1.{frac} • {frac} are the bits of frac • frac is a fractional binary number • Normalized Numbers have range [21-B, 2B+1) • And their negatives

  16.  + -Normalized +Denorm +Normalized -Denorm NaN NaN 0 +0 “Denormalized” Encoding • exp = 0 • E = -B+1 • M = 0.{frac} • Denormalized Numbers have Range [0, 21-B) • And their negatives

  17. Examples • 8 bit FP, 1 bit sign, 4 bit exponent, 3 bit significand, Bias of 7 Representation -> Number 0 0101 011 0 0000 101 1 1011 110 0.34375 0.009765625 -28.0

  18. Examples • 8 bit FP, 1 bit sign, 4 bit exponent, 3 bit significand, Bias of 7 Number -> Representation 9 .6 -15 0 1010 001 0 0110 001 1 1010 111

  19. Typecasting • C has the property that some types can be converted (or typecast) to others. • When programming in C, it is often unavoidable and essential that data is cast from one type to another.

  20. Typecasting (cont.) • Syntax • Assume x has type type_a • Also assume y has some type that can be cast to type_a • x = (type_a)y; • /* cast y to type_a and assign x */

  21. Typecasting (cont.) • What types can be cast? • All primitive types in C can be cast to another, and pointers can be also be cast. • For now, concentrate on integer/floating point casting.

  22. Typecasting Examples int a = 1024; short s = (short)a; -or- short s = a; -> S = 1024;

  23. Typecasting Examples float a = -3.5; int i = (int)a; -> i = -3 (note round to zero behavior)

  24. Typecasting Examples int i = 4; double x = (double)i; int root = (int)sqrt(x); -> x = 4.0, root = 2

  25. Typecasting Cautions • Overflow • Casting from one integer type to a smaller integer type int x = 0x0034a140; short s = x; -> s = 0xa140 = -24256

  26. Typecasting Cautions • Sign Extension • Casting from a negative integer to a larger integer type • To preserve sign (and value), the sign bit is copied to every higher bit • Example: char c = 0xf0; int i = c; -> i = -16, NOT 240

  27. Typecasting Cautions • Floating Point • Some large integers can not be represented exactly in floating point. • Example: unsigned int i = 0xfffffff0; float f = (float)i; double d = (double)i; -> i = 4294967280, f = 4294967296.0, d = 4294967280.0

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