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CS253 Bit Manipulation

Kat Dennings as Max Black

Bases

• We’re used to base 10, because we have that many toes. Really.
• There’s nothing intrinsically good or bad about base 10.
• Vulcans use base 12. Well, in my head-canon they do.
• In base N, the digits are 0…N−1.
  • Base 8 has no digit 8, just as base 10 has no digit ⑩.
• Digits after 9 are usually letters: A…Z or a…z.

Writing numbers (mathematics)

Mathematics, not Computer Science

• When writing numbers, they are by default in base 10.
• A subscript indicates an explicit base: 27₈ + 43₆ = 32₁₆

Writing numbers (C++)

Computer Science, not Mathematics

• If a number starts with 0b or 0B, it’s binary (base 2).
• Otherwise, if a number starts with 0x or 0X, it’s hexadecimal (base 16).
  • Yes, “hexa” is Greek, whereas “decimal” is Latin. Oh, well.
• Otherwise, if a number starts with 0, it’s octal (base 8).
  • Hey, plain 0 is octal, not decimal!
• Otherwise, it’s decimal (base 10).

Place notation

Consider an ordinary, three-digit base 10 number. It has a hundreds place, a tens place, and a ones place. You know this so well that you’ve forgotten it.

So, 789₁₀ means:

• 7⋅10² + 8⋅10¹ + 9⋅10⁰
• 7⋅100 + 8⋅10 + 9⋅1
• 700 + 80 + 9
• 789

Place notation

Now, consider a base 8 number, such as 375₈. This means:

• 3⋅8² + 7⋅8¹ + 5⋅8⁰
• 3⋅64 + 7⋅8 + 5⋅1
• 192 + 56 + 5
• 253

Place notation

A binary number, such as 10110₂, means:

• 1⋅2⁴ + 0⋅2³ + 1⋅2² + 1⋅2¹ + 0⋅2⁰
• 1⋅16 + 0⋅8 + 1⋅4 + 1⋅2 + 0⋅1
• 16 + 0 + 4 + 2 + 0
• 22

Place notation

Of course, in practice, we wouldn’t bother writing down the terms with zero coefficients, since they contribute only zeroes to the sum.

A binary number, such as 10110₂, means:

• 1⋅2⁴ + 1⋅2² + 1⋅2¹
• 1⋅16 + 1⋅4 + 1⋅2
• 16 + 4 + 2
• 22

I’m thinking of a number…

• First we need a number for examples.
• How about 0644?
• The 0 in the beginning just means it is in octal.
• What is it though?
  • In C++, it’s an int, just as 42 is.

0644 in octal… okayyyyyy…

• What is it in different bases?
• So, first what does 644 mean in octal?
  • 6 in the 64’s place (64 = 8²)
  • 4 in the 8’s place (8 = 8¹)
  • 4 in the 1’s place (1 = 8⁰)

What is 0644 in binary?

Hardest way is convert to decimal, then convert to binary.

• Hard way:
  • 6⋅8² = 384
  • 4⋅8¹ = 32
  • 4⋅8⁰ = 4
  • Total = 420, or 256+128+32+4, or 110100100₂
• Easy Way: Break it into pieces of 3 each
  • Why 3? 8¹ = 2³, that’s why!
  • One digit in octal is three digits in binary!
  • 110 100 100 Look! 110₂ = 6 and 100₂ = 4

What is 0644 in octal?

• Duh
• 644
• ’Nuff said.

What is 0644 in decimal?

• Octal to decimal
• Not hard, we just need to do the math.
• It’s just arithmetic.
  • 6⋅8² + 4⋅8¹ + 4⋅8⁰
  • 6⋅64 + 4⋅8 + 4⋅1
  • 384 + 32 + 4
  • 420

What is 0644 in hexadecimal?

• Hard way:
  • Hexadecimal is base 16, so the places are 4096, 256, 16, 1
  • (16³, 16², 16¹, 16⁰)
• How many of each?
• 1⋅256 + 10⋅16 + 4⋅1
• So, it’s 0x1a4

What is 0644 in hexadecimal?

• Easy way:
  • We already know it is 110100100 in binary.
  • 16¹ = 2⁴
    • Each hex digit is 4 binary digits.
  • 1 1010 0100
  • 1 is easy: 1
  • 1010 is 8+2 = 10 = a
  • 0100 is 4 … so 4
  • 0x1a4

Why doesn’t the easy way always work?

It’s all about the ratios:

From	To	Ratio	Difficulty
Hexadecimal	Binary	16 = 24	Easy
Hexadecimal	Octal	16 = 84/3	Not so bad
Hexadecimal	Decimal	16 = 101.20412	Hard
Decimal	Octal	10 = 81.1073	Hard
Decimal	Binary	10 = 23.3219	Hard
Octal	Binary	8 = 23	Easy

Everything’s simple, unless stupid decimal is involved.

So, our number is:

110100100₂ or 644₈ or 420₁₀ or 1a4₁₆

• What’s the difference?!
  • To a human? It’s the difference between a number and a headache.
  • To the computer? Nothing!

The computer doesn’t care!

cout << 0b10000 << ' ' << 020 << ' ' << 16 << ' ' << 0x10 << '\n';

16 16 16 16

• It stores it all in binary.
  • They all look the same in binary.
  • They’re all just ints.
• In fact, even if it DID care, it can’t tell the difference anyway!
  • If a memory cell contains 000110110000101100101, was it binary, octal, decimal, or hex to begin with?

How does it know?

It doesn’t know—you have to tell it!

int x = 060 - 0x00000000000000000000006;
cout << x << '\n'
     << oct << x << '\n'
     << hex << x << '\n'
     << dec << x << '\n';

42
52
2a
42

• Alas, there is no bin.
• I/O manipulators such as hex and oct are “sticky”; they persist until changed.

Refresher

C++ bit-manipulation tools:

• bitwise and (&), or (|), exclusive-or (^)
• shift left (<<), shift right (>>)
• assignment operator versions (&=, |=, ^=, <<=, >>=)
• octal constant (0377)
• hexadecimal constant (0x3FF)
• binary constant (0b1010110111)
• digit separator:
  • 0b1'010'110'111
  • 0x0123'4567'89ab'cdef
  • 123'456'789

Precedence matters

cout << (1|2) << '\n';

3

However, the binary bitwise operators are low precedence:

cout << 1|2 << '\n';

c.cc:1: error: no match for 'operator|' in '
   std::cout.std::basic_ostream<char>::operator<<(1) | (2 << 10)' (operand 
   types are 'std::basic_ostream<char>' and 'int')

Setting a bit

int n = 32;
cout << hex << n << '\n'
     << hex << (n|02) << '\n';

20
22

Or, using assignment and binary literals:

int n = 32;
cout << hex << n << '\n';
n |= 0b10;
cout << hex << n << '\n';

20
22

Clearing bits

cout << hex << 126 << '\n'
     << hex << (126 & ~0xF) << '\n';

7e
70

Testing bits

for (char c='A'; c<'N'; c++)
    cout << ((c & 0x01) ? "Odd:  " : "Even: ")
         << "'" << c << "' (" << int(c) << ")\n";

Odd:  'A' (65)
Even: 'B' (66)
Odd:  'C' (67)
Even: 'D' (68)
Odd:  'E' (69)
Even: 'F' (70)
Odd:  'G' (71)
Even: 'H' (72)
Odd:  'I' (73)
Even: 'J' (74)
Odd:  'K' (75)
Even: 'L' (76)
Odd:  'M' (77)

Shifting

int n = 42;
cout << oct << n << '\n';
n >>= 1;
cout << n << '\n';
n >>= 2;
cout << n << '\n';
n >>= 1;
cout << n << '\n';
n >>= 1;
cout << n << '\n';
n >>= 1;
cout << n << '\n';

52
25
5
2
1
0

Puzzle

int a = 1<<4;
cout << a << '\n';
cout << 1<<4 << '\n';

16
14

Why does using a variable give a different result?

Parsing! << is left-associative, so cout << 1<<4 parses as (cout << 1) << 4 and not as cout << (1 << 4).

Badness

The shift amount must be ≥ 0 and < word size in bits. That’s 0…31 for a 32-bit integer. Shifting a negative number of positions is not allowed.

static int con67 = 67, conm1 = -1;
cout << (1<<con67) << '\n';
cout << (0b1000 << conm1) << '\n';

8
0

Using static fooled the optimizer.

Badness

Java has two different right-shift operators:

• >> copies the sign bit
• >>> always shifts in a zero bit

System.out.println(42 >> 1);
System.out.println(-1 >> 1);
System.out.println(42 >>> 1);
System.out.println(-1 >>> 1);

21
-1
21
2147483647

In C++, when shifting negative values right, it is implementation-defined whether or not the sign bit is propagated:

cout << (-1 >> 1) << '\n';

-1

char vs. byte

• Most C++ programs use char variables to store raw bytes.
• The C++17 byte datatype is designed to replace that.
• A byte is a collection of bits, not a small integer.
  • Arithmetic operations (+ - * /) don’t work on a byte.
  • Bit operations (& | ^ << >>) do work.