2's Complement and Floating-Point, Slides of Number Theory

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2’s Complement and

Floating-Point

CSE 351 Section 3

Two’s Complement

• An n-bit, two’s complement number can represent the range [−

, 2

− 1].

• Note the asymmetry of this

range about 0 – there’s one more

negative number than positive

• Note what happens when you overflow 4 - bit two’s complement range

Two’s Complement: Operations

1. Convert numbers to binary
   • 0xAB = 0b
   • 1710 = 0b
2. Compute x | y 0000 ... 1010 1011 | 0000 ... 0001 0001

---

0000 ... 1011 1011

3. Compute ~(x | y) (flip the bits) ~ 0000 0000 0000 0000 0000 0000 1011 1011 ----------------------------------------- 1111 1111 1111 1111 1111 1111 0100 0100 int x = 0xAB; int y = 17; int z = 5; int result = ~(x | y) + z; What is the value of result in decimal?

Two’s Complement: Operations

4. Compute ~(x | y) + z

• You can either convert to decimal before or after adding z
• 1710 = 0b
• Convert 111111111111111111111111010001002 to decimal (hint: - x = ~x + 1)
• ~11111111111111111111111101000100 2 = (^0000000000000000000000000101110112)
• ~0000……10111011 2 + 1 = 0000…… (^101111002)
• • x = 0000……10111100 2
• x = 1000……10111100 2 int x = 0xAB; int y = 17; int z = 5; int result = ~(x | y) + z; What is the value of result in decimal?

Floating Point

• value = (-1) S * M * 2 E
• Value: ± 1 × Mantissa × 2 Exponent
• Bit Fields: (-1) S × 1. M × 2 ( E +bias)
• Bias
  • Read exponent as unsigned, but with bias of – ( w-

• Representable exponents roughly ½ positive and ½ negative
• Exponent 0 (Exp = 0) is represented as E = 0b 0111 1111
• Why?
• Floating point arithmetic = easier
• Somewhat compatible with 2’s complement s exp mant

Floating Point

• Exponent overflow yields +∞ or - ∞
• Floats with value +∞, - ∞, and NaN can be used in operations
• Result usually still +∞, - ∞, or NaN; sometimes intuitive, sometimes not
• Floating point ops do not work like real math, due to rounding!
• Not associative:
  • (3.14 + 1e100) – 1e100 != 3.14 + (1e100 – 1e100)
• Not distributive:
  • 100 * (0.1 + 0.2) != 100 * 0.1 + 100 * 0.
• Not cumulative
  • Repeatedly adding a very small number to a large one may do nothing s exp mant

Floating Point Examples

• How do you represent 1.0 in floating point?
  • Sign bit: 0
  • First the integral part of the value: 1 = 0b
  • 1.0 10 => 1.0b
  • Don’t need to normalize because it’s already in scientific notation: 1.0 x 2 0
  • Exponent: 0 + 127 = 127 10 = 01111111 2
  • Mantissa: 00000000000000000000000 2
  • 1.010 = 0 01111111 00000000000000000000000 2

= 0x3F

s exp mant 1 8 23

Floating Point Addition (–1) s M1 E

• (-1) s M2 E (Assume E1 > E2)

• Exact Result: (–1) s M E
• Sign s, mantissa M:
• M = M1 + M2, result of signed align & add
• Exponent E: E
• Fixing
  • If M ≥ 2, shift M right, increment E
  • if M < 1, shift M left k positions, decrement E by k
  • Overflow if E out of range
  • Round M to fit frac precision s exp mant 1 8 23

What is floating point result of 12.3125 + 1.5?

In scientific notation: (1.1000101 x 2

3

) + (1.1 x 2

0

1100.0101 x 2 0

• 1.1 x 20

---

1101.1101 x 20

Adjusted mantissa: 1.1011101 => M = 1011101

Ans: 0 10000010 10111010000000000000000

Floating Point Worksheet