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)?

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)

venereology answered 5 months ago

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