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Computer Organization COMP 210

Computer Organization COMP 210. Integers. Topics Numeric Encodings Unsigned & Two’s complement Programming Implications C promotion rules Basic operations Addition, negation, multiplication Programming Implications Consequences of overflow

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Computer Organization COMP 210

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  1. Computer OrganizationCOMP 210 Integers • 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

  2. C Puzzles • Answers on the last slide • Assume machine with 32 bit word size, two’s complement integers • For each of the following C expressions, either: • Argue that is true for all argument values • Give example where 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  -x >= 0 Initialization int x = foo(); int y = bar(); unsigned ux = x; unsigned uy = y;

  3. Real Life • Ariane 5 – European missile • On June 4, 1996 an unmanned Ariane 5 rocket launched by the European Space Agency exploded just forty seconds after its lift-off from Kourou, French Guiana. • first voyage, • a decade of development costing $7 billion. • rocket and its cargo were valued at $500 million • Cause: • a 64 bit floating point number relating to the horizontal velocity of the rocket with respect to the platform was converted to a 16 bit signed integer. • The number was larger than 32,767, the largest integer storeable in a 16 bit signed integer, • the conversion failed. • See http://www-users.math.umn.edu/~arnold/disasters/ariane.html

  4. How do we get to bits? We write programs in an abstract language like C How do we get to the binary representation? Compilers & Assemblers! We write x = 5. The compiler changes it into mov 5, 0x005F The assembler changes it into: 80483c7: 8b 45 5F Compiler figures out that this is an integer and changes it into 2’s Complement

  5. Negative numbers • Problem: how do we represent negative numbers? • Sign-magnitude. • Wastes space (range decreases) • Not used anymore (was used on a few IBM mainframes in the dark ages) • 2’s complement • Space efficient • Arithmetic works well

  6. The structure of a signed integer(if we’re using sign-magnitude) This is no longer used!

  7. Instead of using sign magnitude… We’ll break the number line in the middle

  8. two’s complement …and shift the right part of the number line to the left side to get… Note that negative numbers start with a 1, but starting with a 1 is not what makes the number negative. Result is called 2’s complement. There is an easy arithmetic way to convert numbers to their negative equivalent with 2’s complement.

  9. 2’s complement Rule: • To find the value represented by a binary number that starts with a 1 (i.e., a negative number) • Take the complement of the binary number. • Add 1 to the result. • determine the decimal value To find the value represented by 110 Take the Complement: 001 add 1 to the complement] + 1 find the decimal value 010 = +2 • To find the value represented by a binary number that starts with a 0 (i.e., a positive number) • Do nothing! This is the absolute value of 110

  10. 2’s complement range • There will be more negative numbers than positive numbers since 0 takes a positive space:

  11. The signed integers for a four-bit cell

  12. 2’s complement range • Range for any cell size: • Smallest number is 1000… 0 • Largest number is 0111…..1 • The magnitude of the smallest negative number is 1 greater than the magnitude of the largest positive number. • Example: 6-bit cell: -32 to 31

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

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

  15. Values for Different Word Sizes Observations • |TMin | = TMax + 1 • Asymmetric range • UMax = 2 * TMax + 1 C Programming • #include <limits.h> • K&R App. B11 • Declares constants, e.g., • ULONG_MAX • LONG_MAX • LONG_MIN • Values platform-specific

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