The IEEE standard double-precision floating-point operand format consists of 64 bits. The sign occupies 1 bit, the exponent has 11 bits, and the fraction occupies 52 bits. The exponent bias is 1023 and the base is 2. There is an implied bit to the left of the binary point in the fraction. Infinity is represented with a biased exponent equal to 2047 and a fraction of 0.

(a) Give the formula for finding the decimal value of a normalized number.

(b) List a few biased exponents in binary, as is done in Table 9-6.

(c) Calculate the largest and smallest positive normalized numbers that can be accommodated.

Table 9-6

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Evaluating Biased Exponents Exponent E in decimal -126 -001 000 +001 +126 +127 Biased exponent e = E + 127 Binary 00000001 01111110 01111111 10000000 11111101 11111110 Decimal -126 127 1 -001 + 127 126 000 + 127 = 127 001 + 127 128 126 + 127 = 253 127 + 127 = 254