# Floating Point/Floating Point Formats

## Floating-Point Formats

There are 4 different formats of floating point number representation in the IEEE 754 standard:

Single-Precision
Double-Precision
Single, Extended-Precision
Double, Extended-Precision

## Single-Precision

Single precision floating point numbers are 32 bits wide. The first bit (bit 31, the MSB) is a sign bit, the next 8 bits (bits 30-23) are the exponent, and the remaining 23 bits are for the significand. Note that even though 23 bits are stored for the significand, the precision(${\displaystyle p}$) is actually 24 bits. This is a trick made possible by a normalized floating point system with ${\displaystyle b=2}$. The exponent is biased by 127, so that negative exponents can be expressed.

## Double-Precision

Double-precision numbers are 64 bits wide. The MSB (bit 63) is the sign bit. The next 11 bits (bits 62-52) are the exponent, and the rest of the bits (bits 51-0) are for the significand. Again, the precision is actually 53 bits (not 52) because of the same normalization trick.

## Review

Format Width Precision Exponent Significand
Single 32 bits 23 bits bits 30-23 bits 22-0
Double 64 bits 52 bits bits 62-52 bits 51-0