Question

What would be the IEEE 754 double precision floating point representation of 1.32487359893280124981233898124124 times 10^-17. For explanation, I want you to document the steps you perform, in this order: (1) What is n in decimal fixed point form (ddd.ddd,dd); (2) What is n in binary fixed point form (bbb.bbbb), storing the first 110 bits following the binary point); (3) What is the normalized binary number, written in the form 1.bbbbb...bbb times 2^e, storing 54 bits following the binary point) (4) What are the 52 mantissa bits, after the bits in bit positions -53, -54, ... are eliminated using the round to nearest, ties to even mode; exclude the 1. part; (5) What is the biased exponent in decimal and in binary? (6) Write the 64-bits of the number in the order: s e m, and (7) Write the final answer as a 16-hexdigit number.

Answer 1

First of all it will be converted to binary number and that is

0.0000000000000000000000000000000000000000000000000000000011110100011001010110001100000011110001110001

i.e, 1.1110100011001010110001100000011110001110001 * 2^-57, now exponent's value is -57.

sign is positive and mantissa is 1110100011001010110001100000011110001110001.

now we can write

Sign	Exponent	Mantissa
0	01111000110	1110100011001010110001100000011110001110001000000000

and in hexa decimal the number is

0x3C6E8CAC6078E200

​

Now, in order to write it in double precision floating point format we need to write this number in normalized form

So the answer will be

1) 132.487359893280127146769272484e-19

2) 111.1010001100101011000110000001111000111000100000000000000000000000000000000000000000000000000000000000000000​

3) 1.1110100011001010110001100000011110001110001 *2^-57

4) 1110100011001010110001100000011110001110001000000000

5) Biased exponent (in binary): 01111000110

   Biased exponent (in decimal): 966

6) 0011110001101110100011001010110001100000011110001110001000000000

7) 0x3C6E8CAC6078E200

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