Question

In: Computer Science

Suppose we want to define an analogue of the IEEE 754 standard for 14 bits, with...

Suppose we want to define an analogue of the IEEE 754 standard for 14 bits, with 1 bit for sign, 6 bits for (biased) exponent, and 7 bits for the significand. Assume exponent 000000 and 111111 are reserved for 0, denormal, NaN and infinity, just like IEEE 754 standard.

a) What is the bias for the exponent? Express it in decimal.

b) What is the smallest positive denormalized number?

c) What is the smallest positive normalized number?

d) What is the largest positive normalized number (excluding infinity)?

Solutions

Expert Solution

Examples of IEEE 754 standards with normalized and De-normalized and other special numbers:

Required solution is highlighted in orange:

Type

Sign

Actual Exponent

Exp (biased)

Exponent field

Fraction field

Value

Zero

0

-126

0

0000 0000

000 0000 0000 0000 0000 0000

0.0

Negative zero

1

-126

0

0000 0000

000 0000 0000 0000 0000 0000

-0.0

One

0

0

127

0111 1111

000 0000 0000 0000 0000 0000

1.0

Minus One

1

0

127

0111 1111

000 0000 0000 0000 0000 0000

-1.0

Smallest denormalized number

*

-126

0

0000 0000

000 0000 0000 0000 0000 0001

±2−23 × 2−126 = ±2−149 ≈ ±1.4×10−45

"Middle" denormalized number

*

-126

0

0000 0000

100 0000 0000 0000 0000 0000

±2−1 × 2−126 = ±2−127 ≈ ±5.88×10−39

Largest denormalized number

*

-126

0

0000 0000

111 1111 1111 1111 1111 1111

±(1−2−23) × 2−126 ≈ ±1.18×10−38

Smallest normalized number

*

-126

1

0000 0001

000 0000 0000 0000 0000 0000

±2−126 ≈ ±1.18×10−38

Largest normalized number

*

127

254

1111 1110

111 1111 1111 1111 1111 1111

±(2−2−23) × 2127 ≈ ±3.4×1038

Positive infinity

0

128

255

1111 1111

000 0000 0000 0000 0000 0000

+∞

Negative infinity

1

128

255

1111 1111

000 0000 0000 0000 0000 0000

-∞

Not a number

*

128

255

1111 1111

non zero

NaN

* Sign bit can be either 0 or 1 .

Some examples of IEEE 754 standards for reference:

Sign

Exponent

Fraction

Value

0

00000000

00000000000000000000000

0

1

00000000

00000000000000000000000

-0

0

11111111

00000000000000000000000

Infinity

1

11111111

00000000000000000000000

-Infinity

0

11111111

00000100000000000000000

NaN (signaling)

1

11111111

00100010001001010101010

NaN (signaling)

0

11111111

10000000000000000000000

NaN (quiet)

1

11111111

10100011010101000001010

NaN (quiet)

0

10000000

00000000000000000000000

+1 * 2**(128-127) * 1.0 = 2

0

10000001

10100000000000000000000

+1 * 2**(129-127) * 1.101 = 6.5

1

10000001

10100000000000000000000

-1 * 2**(129-127) * 1.101 = -6.5

0

00000001

00000000000000000000000

+1 * 2**(1-127) * 1.0 = 2**(-126)

0

00000000

10000000000000000000000

+1 * 2**(-126) * 0.1 = 2**(-127)

0

00000000

00000000000000000000001

+1 * 2**(-126) * 0.00000000000000000000001

= 2**(-149) (Smallest positive value)


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