Answered Essay: Consider the largest single precision floating point number,

Floating Point “Question”

1)Consider the largest single precision floating point number, 1.11…1 x 2127. What will happen the “gap” at that number if you change the type to double precision? DO NOT discuss the largest double precision number – just the largest single precision represented as a double type.

2)Now, consider the smallest single precision floating point number, 2-149. What will happen the “gap” at that number if you change the type to double precision? DO NOT discuss the smallest double precision number – just the largest single precision represented as a double type.

Expert Answer

 

Solution:

1)

Before proceeding let’s discuss double precision and how the sign, exponent, and Mantissa is categorized in it.

So double precision is 64-bit floating point and the details are given below:

So when largest single precision number will be represented as double precision it will look like this,

Sign Biased Exponent Mantissa
0 10001111110 1111111111111111111111100000000000000000000000000000

So the Bias for double precision number is +1023

and Biased Exponent= Actual exponent+Bias= 127+1023= 1150 (int binary: 10001111110)

and in the mantissa, after 23 1’s all the bits will be padded with 0’s.

2)

So the smallest single precision number 2^-149 will be represented as shown below:

Sign Biased Exponent Mantissa
0 01101101010 0000000000000000000000000000000000000000000000000000

So the Bias for double precision number is +1023

and Biased Exponent= Actual exponent+Bias= (-149)+1023= 874 (int binary 01101101010)

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