The following is a scheme for floating point number representation using 16…
2003
The following is a scheme for floating point number representation using 16 bits.
Bit position 15 14 . . . 9 8 . . . . .0
s e m
Sign Exponent MantissaLet s,e, and m be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is:
What is the maximum difference between two successive real numbers representable in this system?
- A.
2-40
- B.
2-9
- C.
222
- D.
231
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Correct answer: C
Given: 1 sign bit, 6 exponent bits, 9 mantissa bits. For normalized numbers the value is
(-1)^s (1 + m·2^-9) · 2^{e-31},
where m changes in steps of 1. Therefore the difference between two successive representable numbers for a fixed exponent e is
Spacing Δ = 2^-9 · 2^{e-31} = 2^{e-40}.
The exponent field is 6 bits. The bit pattern 111111 (decimal 63) is reserved, so the largest finite exponent is e = 62.
Plugging e = 62 gives the maximum spacing: Δ_max = 2^{62-40} = 2^{22}.
Answer: 2^22