Question

4. Explain how the following computations are performed using both IEEE single precision and double precision floating point representation. a. 11716 +2A916 b. 1011.112-11.1510 c. 1.0010102 x 0.011012 d. Comment on any differences you see in accuracy and precision.

Answer 1

d.  

Precision: The smallest change that can be represented in floating point representation is called as precision. The fractional part of a single precision normalized number has exactly 23 bits of resolution, (24 bits with the implied bit). This corresponds to log₍₁₀₎ (2²³) = 6.924 = 7 (the characteristic of logarithm) decimal digits of accuracy. Similarly, in case of double precision numbers the precision is log₍₁₀₎ (2⁵²) = 15.654 = 16 decimal digits.

Accuracy: Accuracy in floating point representation is governed by number of significand bits, whereas range is limited by exponent. Not all real numbers can exactly be represented in floating point format. For any numberwhich is not floating point number, there are two options for floating point approximation, say, the closest floating point number less than x as x_ and the closest floating point number greater than x as x+. A rounding operation is performed on number of significant bits in the mantissa field based on the selected mode. The round down mode causes x set to x_, the round up mode causes x set to x+, the round towards zero mode causes x is either x_ or x+ whichever is between zero and. The round to nearest mode sets x to x_ or x+ whichever is nearest to x. Usually round to nearest is most used mode. The closeness of floating point representation to the actual value is called as accuracy.