Consider a computer that uses the IEEE 754 standard to represent numbers in term

Question

Consider a computer that uses the IEEE 754 standard to represent numbers in terms of a 20-bit word: it stores numbers using the convention x = (-1)^s (1.f)(2^c - b) where s is the sign bit, f is the fraction, c is the characteristic and b is the bias, equal to 31. The 20 bits are allocated as follows: the first bit is the sign bit, the next 6 bits relate to the characteristic, and the last 13 give the fraction. In accordance with the IEEE standard, infinity corresponds to all the characteristic bits equal to 1 (c = 111111), and zero corresponds to all the characteristic bits equal to 0 (c = 000000). For this computer, determine: a) the largest positive number b) the smallest positive number c) the machine epsilon For a) and b), show the computer word as well as the equivalent decimal value, and show all your work to relate the two. Express the decimal values with 5 significant figures and use chopping. To simplify your calculations, you may find the following relationships useful: sigma^n_i = 0 2^i = 2^n + 1 - 1 and sigma^n_i = 0 2^-i = 2 - 2^-n

Explanation / Answer

Solution:

So we have

1 bit to represent the sign and 6 bits for exponent and 13 bis for the mantissa.

a)

The largest positive number

The number will be largest when it's exponent is highest and all mantissa bits are set to 1.

so,

Sign

Biased Exponent

Mantissa

0

111111

1111111111111

the bias= 31 (11111), and biased exponent= 63(1111111), Actual exponent= Biased exponent- Bias= 32(100000)

1.1111111111111*2^100000

in decimal 8589410304

2)

A smallest positive number,

Sign

Biased Exponent

Mantissa

0

000000

0000000000000

1.0000000000000*2^-31 will be the smallest positive number

and in decimal 0.00000000046566128731

3)

Machine epsilon( )= 2^-13= 0.00006103515625

I hope this helps if you find any problem. Please comment below. Don't forget to give a thumbs up if you liked it. :)

Sign

Biased Exponent

Mantissa

0

111111

1111111111111

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.