00 What are IEEE 754 floating point numbers?

IEEE 754 – Wikipedia

The IEEE 754 standard defines standard representations for binary and decimal floating-point numbers in computers and specifies precise procedures for performing mathematical operations, especially rounding.

History of the standard

Until the early 1970s, there were numerous different representation formats for floating point numbers, depending on the manufacturer or series of the processor. The display formats differed, for example, in the value range of the numbers that could be displayed, the precision, the support of tiny numbers close to 0, the rounding procedures or the precision of the internal intermediate results. These differences meant that computer programs delivered different results depending on the computer used.

IEEE 754

IEEE 754 defines 4 representations for binary floating point numbers, they are called binary16, binary32 (traditionally “single”), binary64 (traditionally “double”) and binary128. In addition, the standard also allows user-defined extended representations, which are constructed according to the same principles as the predefined representations.

In particular, the number of exponent bits determines the maximum and minimum numbers that can be represented. The number of mantissa bits determines the relative accuracy of these numbers (and only to a small extent the maximum and minimum).