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Representing Numbers B261 Systems Architecture

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Previously Computer Architecture –Common misconceptions –Performance Instruction Set (MIPS) –How data is moved inside a computer –How instructions are executed

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Outline Numbers –Limitations of computer arithmetic How numbers are represented internally –Representing positive integers –Representing negative integers Addition Bit-wise operations

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Binary Representation Computers internally represent numbers in binary form –The sequence of binary digits »d n-1 d n-2... d 1 d 0 two –At its simplest, this represents the decimal number which is the sum of terms 2 i for each d i = 1. »Eg, two = 16 ten +4 ten +2 ten +1 ten = 23 ten »d 0 is called the least significant bit LSB »d n-1 is called the most significant bit MSB –But we will need to use different meanings for the bits in order to represent negative or floating-point numbers.

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Range of Positive Numbers and Word Lengths For an n-bit word length 2 n different numbers can be represented *including zero. 0,1,2,..., ((2 n ) -1) (if only want +ve numbers) A 3-bit word length would support 0,1,2,3,4,5,6,7 MIPS has a 32-bit word length 2 32 different numbers can be represented 0,1,2,..., ((2 32 )-1) (= 4,294,967,295).

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Negative Numbers The range afforded by the word length must simultaneously support both positive and negative numbers, equally. Two requirements: –There must be a way, within one word, to tell if it represents a positive or negative number, and its value. –A positive number x, and its complement (-x) must clearly add to zero »x + (-x) = 0.

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Twos Complement Example of n=3 bit word: 000 = = = = = = = = -1 For negative numbers the MSB (sign bit) is always 1 For positive numbers the MSB (sign bit) is always 0

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Twos complement In general for a n-bit word –There will be positive numbers (and zero) »0,1,2, …, ((2 n-1 ) - 1) –Negative numbers »-(2 n-1 ), …, -1 –The highest positive number is ((2 n-1 ) - 1) –The lowest negative numbers is -(2 n-1 ) –Note the slight imbalance of positive and negative.

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Conversion For an n-bit word, suppose that d n-1 is the sign bit. –Then the decimal value is: » (-d n-1 )* (2 n-1 ) + the usual conversion of the remainder of the word. –Eg, n=3, then »101 = (-1 * 4) = = -3 »011 = (0 * 4) = = 3

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Negating a Number There is a simple two-step process for negation (eg, turn 3 = 011 into -3 = 101): invert every bit (e.g. replace 011 by 100) add 1 (e.g = 101). Example, 4-bit word = = -3 »3 = 0011 »Invert bits: 1100 »add 1: 1101 = -3 as required.

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Sign Extension Turning an n-bit representation into one with more bits –Eg, in 3 bits the number -3 is 101 –In 4 bits the number -3 is 1101 Replicate the sign bit from the smaller word into all extra slots of larger word –For 5-bit word from 3 bits: = -3 –For 5-bit word from 4 bits: = -3

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Integer Types Some languages (e.g. C++) support two types of int: signed and unsigned. Signed integers have their MSB treated as a sign bit –so negative numbers can be represented Unsigned integers have their MSB treated without such an interpretation (no negative numbers).

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Signed/Unsigned Int Suppose we had a 4-bit word, then int x; »x could have range 0 to 7 positive »and -8 to -1 negative unsigned int x; »x has range 0 to 15 only.

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Addition Addition is carried out in the same way as decimal arithmetic: »0101 » »0110 Subtractions involve negating the relevant number: » = » = 0010

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Overflow Since there are only a fixed set of bits available for arithmetic things can go wrong: Consider n=4, and (6+5). »0110 » »1011 But 1011 = = -5 !

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Overflow Examples Consider (6 - (-5)) »= »= 1011 »= »= »= »= 0111 »= 7

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Overflow Situations A + B –where A>0, B>0, A+B too big, result negative A + B –where A<0, B<0, A+B too small, result positive A - B –where A>0, B<0, A-B too big, result negative A - B –where A 0, A-B too small, result positive.

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Bit Operations Shifts –shift a word x bits to the right or left: »0110 shift left by 1 is 1100 »0110 shift right by 1 is 0011 –In C++ shifts are expressed using > »unsigned int y = x >3; »means y is the value of x shifted 5 to left, and z is x shifted 3 to the right. –One application is multiplication/division by powers of 2.

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AND/OR AND is a bitwise operation on two words, where for each corresponding bit a and b: –a AND b = 1 only when a=1 and b=1, otherwise 0. OR is a bitwise operation, such that –a OR b = 0 only if both a=0 and b=0, otherwise 1. For example: –1011 AND 0010 = 0010 –1011 OR 0010 = 1011

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New MIPS Operations add unsigned –addu $1,$2,$3 »operands treated as unsigned ints. subtract unsigned –subu $1,$2,$3 »operands treated as unsigned ints add immediate unsigned –addiu $1,$2,10 »operands treated as unsigned ints

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Exceptions Where the signed versions (add, addi, etc) are used –if there is overflow »then an exception is raised by the hardware »control passes to a special procedure to handle the exception, and then returns to the next instruction after the one that raised the exception.

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More MIPS Load upper immediate –lui $1,100 »$1 = 2 16 * 100 »(100, shifted left by 16 = upper half of the word). and, or, and immediate (andi), or immediate (ori), shift left logical (sll), shift right logical (srl) –all of form $1,$2,$3 or $1,$2,10 (for immediate).

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Summary Basic number representation (integers) Representation of negatives Basic arithmetic (+ and -) Logical operations Next time - building an ALU.

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