The floating-point representation of 12.25 is 01000001010001000000000000000000. Its sign is 0, its exponent is 100000102 = 130, and its fraction is 100010...0. Reconstructed to decimal, we get:

(−1)⁰ × 2^(130−127) × (1.10001)2
= 1 × 2³ × 1.53125
= 12.25

Any floating-point binary representation consists of three segments of bits: sign, exponent, and fraction.

• The sign (S) indicates positive or negative;
• The exponent (E) raises 2 to some power to scale the number; and
• The fraction (F) determines the number's exact digits.

The segments' lengths and exact formula applied to S, E, and F to recreate the number depend on the format's precision.

The IEEE754 floating-point format standard enables efficient storage and processing of numbers in computers.

• From a hardware perspective, many simplifications of floating-point operations can be made to significantly speed up arithmetic, thanks to the IEEE754 standard's specifications.

• For software, floats are very precise and typically lose a few millionths (if not less) per operation, which enables high-precision scientific and engineering applications.