We can characterize a floating point number system (FPNS) based on the base, b,

Question

We can characterize a floating point number system (FPNS) based on the base, b, the number of bits in the significand ,s, the minimum exponent m, and the maximum exponent M. For the floating point system (b,s,m,M) =(2, 4, -3,4) , assuming that the system is normalized as in IEE standard( with an extra implied 1 in the significand when the exponent >3), and that infinity is represented as in IEEE system (largest exponent value reserved for infinity), what are the values of

a) the smallest normalized positve number

b) the smallest denormalized positive number

c) the floating representation of 1.0 in this system

d) the largest number representable in the system

Explanation / Answer

a) nsingle = realmin('single')

b) What's the smallest positive denormalizd number?

This looks like:

This bitstring pattern maps to the number 0.0221 x 2-126, which is 1.0 x 2-149. This number has 1 bit of precision. The 22 zeroes are merely place holders and do not affect the number of bits of precision.

You may not believe that this number has only 1 bit of precision, but it does. Consider the decimal number 123. This number has 3 digits of precision. Consider 00123. This also has 3 digits of precision. The leading 0's do not affect the number of digits of precision. Similary, if you have 0.000123, the zeroes are merely to place the 123 correctly, but are not significant digits. However, 0.01230 has 4 significant digits, because the rightmost 0 actually adds to the precision.

Thus, for our example, we have 22 zeroes followed by a 1 after the radix point, and the 22 zeroes have nothing to do with the number of significant bits.

By using denormalized numbers, we were able to make the smallest positive float to be 1.0 X 2-149, instead of 1.0 X 2-127, which we would have had if the number had been normalized.

c)

d) On a 32-bit computer with 7 bits for the exponent and 24 bits for the mantissa with base 2

Biggest number - +1

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

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