Single-precision Floating-point Format - IEEE 754 Single-precision Binary Floating-point Format: Binary32

IEEE 754 Single-precision Binary Floating-point Format: Binary32

The IEEE 754 standard specifies a binary32 as having:

• Sign bit: 1 bit
• Exponent width: 8 bits
• Significand precision: 24 (23 explicitly stored)

This gives from 6 to 9 significant decimal digits precision (if a decimal string with at most 6 significant decimal is converted to IEEE 754 single precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 single precision is converted to a decimal string with at least 9 significant decimal and then converted back to single, then the final number must match the original ).

Sign bit determines the sign of the number, which is the sign of the significand as well. Exponent is either an 8 bit signed integer from −128 to 127 (2's Complement) or an 8 bit unsigned integer from 0 to 255 which is the accepted biased form in IEEE 754 binary32 definition. For this case an exponent value of 127 represents the actual zero.

The true significand includes 23 fraction bits to the right of the binary point and an implicit leading bit (to the left of the binary point) with value 1 unless the exponent is stored with all zeros. Thus only 23 fraction bits of the significand appear in the memory format but the total precision is 24 bits (equivalent to log₁₀(224) ≈ 7.225 decimal digits). The bits are laid out as follows:

The real value assumed by a given 32 bit binary32 data with a given biased exponent e and a 23 bit fraction is where more precisely we have :

In this example :

thus :