Lecture #5 – Floating Point Arithmetic

· Optimizing Integer Arithmetic

Ripple Adders use a series of 1-bit full adders to create a multi-bit adder. Ripple addition has a long propagation delay due to the carry being dependant on each previous stage.

· Floating Point Number

numbers with fractions, e.g., 3.1416

very small numbers, e.g., .000000001

very large numbers, e.g., 3.15576 x 10⁹

· Floating Point Representation:

sign, exponent, significand: (–1)^sign x significand x 2^exponent

more bits for significand gives more accuracy

more bits for exponent increases range

· Optimizing Integer Arithmetic

single precision: 8 bit exponent, 23 bit significand

double precision: 11 bit exponent, 52 bit significand

Leading “1” bit of significand is implicit

Exponent is “biased” to make sorting easier

– all 0s is smallest exponent all 1s is largest

– bias of 127 for single precision and 1023 for double precision

– summary: (–1)sign ´ (1+significand) ´ 2exponent – bias

· Example

decimal: -.75 = - ( ½ + ¼ )

binary: -.11 = -1.1 x 2^-1

floating point: exponent = 126 = 01111110

IEEE single precision: 1 01111110 10000000000000000000000

· Floating Point Addition