Summary of Floating Points

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Advantages of IEEE 754 Standard
IEEE 754 standard is used predominantly by the industry because of the following advantages, to name a few:

• Encoding of exponent and fraction simplifies comparison.


• Includes special exceptional values: NaN and ±∞ They are for special cases such as 0/0=NaN, sqrt(–1)=NaN, 1/0=∞, and 1/∞=0.


• Denormalized numbers fill the gap between smallest normalized number 1.0×2Emin and zero because denormalized numbers, values 0.F×2Emin, are closer to zero.

Floating Point Complexities
The floating points include the following features:

• Operations are somewhat more complicated.


• In addition to overflow, we can have underflow.


• Division by zero yields infinity and zero divide by zero yields Not-a-Number.

Implementing the standard can be tricky. See text for description of 80x86 and Pentium bug! However, not using the standard can be even worse.

Question: The 16-bit IEEE 754-2008 floating-point format is with five exponent bits. What would be the likely range of numbers it could represent?   1.0000 0000 00 × 20   to   1.1111 1111 11 × 231, 0 ±1.0000 0000 0   × 2-14 to ±1.1111 1111 1   × 215, ±0, ±∞, NaN ±1.0000 0000 00 × 2-14 to ±1.1111 1111 11 × 215, ±0, ±∞, NaN ±1.0000 0000 00 × 2-15 to ±1.1111 1111 11 × 214, ±0, ±∞, NaN