Bit OperationsBit Operations

Horton pp. 70 - 76

Why we need to work with bitsWhy we need to work with bitsSometimes one bit is enough to store your data: say the gender of the

student (e.g. 0 for men, 1 for women). We don’t have a 1-bit variable type in C++, so for gender, you will have to use a unsigned char or char type variable.

But if you need say 8 such 1-bit variables, say to record if the student were present in the times when attendance was taken, then you can actually combine all into one char variable.

class student { … private:

unsigned char attendance; //now I can fit 8 bits into this, we will see how}

In C++, there is special container class, called bitset, that is designed to store bits. But we will not see and use it. And you are not allowed to use it in this course.

Packing bitsPacking bits“Packing” 8 1-bit variables into 1 char variable is easy.

Say you know that the student were present in the first 3 class and not in the last 5. The variable attendance can be set as:

unsigned char attendance = 7;

0 0 0 0 0 1 1 1

where the lest significant 3 bits represent the first 3 attendances (just a choice, it could be the other way around as well).

But in addition to being able to set a variable’s value, we need to be able to handle each bit separately, for which we need bit operators.

Hexadecimal literals and outputHexadecimal literals and output

In C++ a hexadecimal literal (number) is represented by preceeding 0x before the literal.

E.g. 0xb4a2 // equivalent to 11x163 + 4x162 + 10x16 + 2 = 46242 (decimal)

char ch = 0x7c; // ch contains 124 (7x16+ 12)

When you want to display a numeric value in hexadecimal, use hex in cout before the value to be displayed.

–This affects all future integer outputs to be displayed in hexadecimal, so if you want to switch back to decimal, use dec later.

short nums = -200;

cout << "hexadecimal: " << hex << nums << " decimal: " << dec << nums << endl;

Output: hexadecimal: ff38 decimal: -200

unsigned int numi = 0xfd04e0e4;

cout << "hexadecimal: " << hex << numi << " decimal: " << dec << numi << endl;

Output: hexadecimal: fd04e0e4 decimal: 4244955364

Bit OperatorsBit Operators

& Bitwise and

| Bitwise or

^ Bitwise exclusive or

~ Complement (unary operator, takes only one operand)

<< shift left

>> shift right

Do not confuse && with &

|| with |

Bitwise OperationsBitwise Operations: : ANDAND

Take the AND of the two numbers, bit by bit.

1 1 0 1 0 1 0 1

0 1 0 0 0 1 0 0

x

y

z

char x,y,z;x = 0xd5;y = 0x6c;z = x&y;

0 1 1 0 1 1 0 0

a b a & b

0 0 0

0 1 0

1 0 0

1 1 1

Bitwise Bitwise ANDAND

unsigned char c1, c2, c3;

c1 = 0x45;c2 = 0x71;c3 = c1&c2;

c1 : 0100 0101c2 : 0111 0001c3 : 0100 0001 (=0x41 = 4x16+1 = 65 10)

Bitwise Bitwise OROR

Take the OR of the two numbers, bit by bit.

unsigned char c1, c2, c3;

c1 = 0x45;c2 = 0x71;c3 = c1 | c2;

c1 : 0100 0101c2 : 0111 0001c3 : 0111 0101 (=0x75 = 7x16+5 = 11710 )

a b a | b

0 0 0

0 1 1

1 0 1

1 1 1

Bitwise Bitwise XORXOR

Take the Exclusive OR of the two numbers, bit by bit.

unsigned char c1, c2, c3;

c1 = 0x45;c2 = 0x71;c3 = c1 ^ c2;

c1 : 0100 0101c2 : 0111 0001c3 : 0011 0100 (=0x34 = 3x16+4 = 5210 )

a b a ^ b

0 0 0

0 1 1

1 0 1

1 1 0

BitBitwise wise ComplementComplement

Complement operation (~) converts bits with value 0 to 1 and bits with value 1 to 0.

unsigned char b1 = 0xc1; // 1100 0001

unsigned char b4 = 0x08; // 0000 1000

b4 = ~b1; // 0011 1110

a ~a

0 1

1 0

WWhat do these hat do these two two statements do?statements do?

char x = 0xA5;

if ( x & 0x08 )//what does this mean?

x = x | 0x02;//what happened to x?

These are two of the most important bit operations! We will see more later, but basically you can access a particular bit and you can set a particular bit with these two operations.

Logic OperatorsLogic Operators versus Bitwise Logic Ops versus Bitwise Logic Ops..

The operators && , || and ! are not bitwise logic operators!

– The result of ! , && and || operations is an integral data type with the value 0 (every bit is a zero) or 1 (Least Significant bit is 1, all the others are 0).

Shift OperatorsShift Operators

Shift operators move bits left or right, discarding bits from one side and filling the other side with 0s.<< means shift left

>> means shift right

Do not confuse with stream operators how many times

it is shifted

Bit operationsBit operations: : left shiftleft shift

Suppose we want to shift the bits of a number N, k bits to the left– denoted N << k

• drop leftmost k bits

• append k zeros to the right

Ex:

y = x << 1;

Another example:unsigned char c2 = 0x9C; //10011100

c = c2 << 2 //01110000

1 1 0 1 0 1 0 1x

y 1 0 1 0 1 0 1 0

Bit Shifting as MultiplicationBit Shifting as Multiplication

Shift left by one position means multiplication by 2 (provided that the result is in the range)

- Works as multiplication for both unsigned & 2’s complement numbers

- May overflow.

– What is the effect of shifting a number left by 3?

0 0 1 1 = 19

0 1 1 0 = 38

1 1 1 1 = -3

1 1 1 1 = -6

0 0 0 1

0 0 1 0

1 1 0 1

1 0 1 0

Bit operationsBit operations: : right shiftright shift

As opposed to left shift, the right shift works differently for signed and unsigned numbers.

Suppose we want to shift N by k bits to the right (denoted N >> k):

– For unsigned numbers (i.e. unsigned char, unsigned int, etc.):• drop rightmost k bits

• append k zeros to the left

– For signed numbers (i.e. char, int, etc.):• drop the rightmost k bits

• append the sign bit k times to the left

right shift operatorright shift operator:examples:examples

Signed (all your variables are signed unless you specify “unsigned” specifically):– positives:

char c = 8; //0000 1000

c = c >> 2 //0000 0010 = 210

– negatives:char c = - 8; //1111 1000 in 2s comp.

c = c >> 2 //1111 1110 which is -210

– Called arithmetic shift

Unsigned:unsigned char d = 0xf8; //1111 1000 (24810)

d = d >> 2 //0011 1110 which is 6210

– Called logical shift

Bit Shifting as DivisionBit Shifting as Division: : summarysummary

0 1 1 1 = 7

0 0 1 1 = 3

1 0 0 1 = -7

1 1 0 0 = -4Always rounds down!

CAUTION: Regular division of negative integersrounds down magnitude-wise. E.g. -7/2 is -3

Arithmetic shift right for one bit divides by 2 for signed numbers

0 1 1 1 = 7

0 0 1 1 = 3

1 0 0 0 = 8

0 1 0 0 = 4Always rounds down!

Logical shift right for one bit divides by 2 for unsigned:

Bit Shifting as Multiplication & DivisionBit Shifting as Multiplication & Division

Why useful?– Simpler, thus faster, than general multiplication & division

Can shift multiple positions at once:– Multiplies or divides by corresponding power of 2.

– a << 5 (multiply by 25)

– a >> 5 (divide by 25)

Remember, when you need to use only positive numbers or you need to manipulate bits, use unsigned numbers.

Self-Self-QuizQuiz

What are the resulting values of x,y and z?

char x,y,z;

x = 0x33;

x = (x << 3) | 0x0F;

y = (x >> 1) & 0x0F;

z = x && y;

Example for Example for ssetting, testing, and clearing etting, testing, and clearing the bits of bytesthe bits of bytes

const char IO_ERROR = 0x01; //LSB (right-most bit) is 1, others are 0const char CHANNEL_DOWN = 0x10;char flags = 0;

//if ever CHANNEL_DOWN event happens, set its corresponding bit in the flags variable

flags = flags | CHANNEL_DOWN; // set the 5th bit

. . .

//to check what errors may have happened...if ((flags & IO_ERROR ) != 0) // check the 1st bit cout << "I/O error flag is set";else if ((flags & CHANNEL_DOWN) != 0) // check the 5th bit cout << "Channel down error flag is set";

. . .flags = flags & ~IO_ERROR; // clear the ERROR flag

This is also called masking

22

Example 2: Attendance Example 2: Attendance unsigned char attendance = 0x00;

//let the LSB be for quiz 1, the next one is for quiz 2, . . . //0 means unattended, 1 means attended//there are total of 8 quizzes; initally all unattended

unsigned char mask=0x01;

//Draft code showing how attendance array may be filled//if attended to quiz 1

attendance = attendance | mask; //if attended quiz 3 mask = mask << 2; // mask becomes 0000 0100

attendance = attendance | mask; . . .//to display attendance information mask=0x01;for (int i=1; i <= 8; i++){ if (attendance & mask)

cout << "Student attended quiz number " << i << endl; else

cout << "Student missed quiz number " << i << endl; mask = mask << 1;}