Download presentation

1
**B261 Systems Architecture**

Representing Numbers B261 Systems Architecture

2
**Previously Computer Architecture Instruction Set (MIPS)**

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

3
**Outline Numbers How numbers are represented internally Addition**

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

4
**Binary Representation**

Computers internally represent numbers in binary form The sequence of binary digits dn-1 dn d1 d0 two At its simplest, this represents the decimal number which is the sum of terms 2i for each di = 1. Eg, 10111two = 16ten+4ten+2ten+1ten = 23ten d0 is called the ‘least significant bit’ LSB dn-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.

5
**Range of Positive Numbers and Word Lengths**

For an n-bit word length 2n different numbers can be represented including zero. 0,1,2,..., ((2n) -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 232 different numbers can be represented 0,1,2,..., ((232)-1) (= 4,294,967,295).

6
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.

7
**Two’s Complement Example of n=3 bit word: 000 = 0 001 = 1 010 = 2**

011 = 3 100 = -4 101 = -3 110 = -2 111 = -1 For positive numbers the MSB (‘sign bit’) is always 0 For negative numbers the MSB (‘sign bit’) is always 1

8
**Two’s complement In general for a n-bit word**

There will be positive numbers (and zero) 0,1,2, … , ((2n-1) - 1) Negative numbers -(2n-1), … , -1 The highest positive number is ((2n-1) - 1) The lowest negative numbers is -(2n-1 ) Note the slight imbalance of positive and negative.

9
**Conversion For an n-bit word, suppose that dn-1 is the sign bit.**

Then the decimal value is: (-dn-1)* (2n-1) + the usual conversion of the remainder of the word. Eg, n=3, then 101 = (-1 * 4) = = -3 011 = (0 * 4) = = 3

10
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 1101 1101 = = -3 3 = 0011 Invert bits: 1100 add 1: 1101 = -3 as required.

11
**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

12
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).

13
**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.

14
Addition Addition is carried out in the same way as decimal arithmetic: 0101 0001 + 0110 Subtractions involve negating the relevant number: = = 0010

15
Overflow Since there are only a fixed set of bits available for arithmetic things can go wrong: Consider n=4, and (6+5). 0110 0101 + 1011 But 1011 = = -5 !

16
**Overflow Examples Consider (6 - (-5)) -6 - 3 0110 - 1011 = 0110 + 0101**

= 1011 = = -5 -6 - 3 = = 0111 = 7

17
**Overflow Situations A + B A - B**

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

18
**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<<5, z = 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.

19
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

20
**New MIPS Operations add unsigned subtract unsigned**

operands treated as unsigned ints. subtract unsigned subu $1,$2,$3 operands treated as unsigned ints add immediate unsigned addiu $1,$2,10

21
**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.

22
**More MIPS Load upper immediate**

lui $1,100 $1 = 216 * 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).

23
**Summary Basic number representation (integers)**

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

Similar presentations

OK

CSCI-365 Computer Organization Lecture Note: Some slides and/or pictures in the following are adapted from: Computer Organization and Design, Patterson.

CSCI-365 Computer Organization Lecture Note: Some slides and/or pictures in the following are adapted from: Computer Organization and Design, Patterson.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google