2.85 Floating Point I

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Problem:

Given a floating-point format with a k-bit exponent and an n-bit fraction, write formulas for the exponenet E, significand M, the fraction f, and the value V for the quantities that follow. In addition, describe the bit representation.

Bias=2^(k-1)-1, V=2^E*M

A. The number 7.0

7.0=0B111.0, M=0B1.11, f=0B.11, E=2, e=2+Bias=2+2^(k-1)-1=2^(k-1)+1, V=7.0

bits:

0 10...01 110...

B. The largest odd integer that can be represented exactly

M=0B1.11111..., f=0B.11111...(n bits 1), E=e-bias=n, so e=n+bias=n+2^(k-1)-1, V=0B1111...(n+1 bits 1)

bits:

0 bias+n 1111...

C. The reciprocal of the smallest positive normalized value

Smallest positive normalized value:

0 00...01 000...

e=1, bias=2^(k-1)-1, M=1, V=2^(1-bias)

reciprocal is:

V=2^(bias-1)

E=bias-1, M=1.0, f=0.0, e=E+bias=2bias-1=2^k-3

bits:

0 111...101 000...