IEEE binary floating-point arithmetic standard learning

See there is a project on the Internet is required to use the floating point number in the binary representation, need to use the IEEE754 standard, check the Chaviki and in-depth understanding of the computer system, re-learn the floating point in the machine's representation and memory storage,

Take a simple note first, and you need a deeper understanding behind it.

IEEE754 defines four ways to represent floating-point numbers: Single-precision (32bit), double-precision (64bit), extended single-precision (more than 43bit), extended double-precision (79bit above), the latter is rarely used, here is the first two.

Using binary to represent floating-point number three parts, the following are 32bit single-precision For example, double-precision similar can be inferred:

The three parts are: sign bit, exponent (exponent), Mantissa (significand, store binary fractional part), below is the graph on wiki:

In single-precision floating-point numbers, sign is 1bit,exponent to 8bit,significand 23bit and three parts are 32bit.

There is one more important concept of "exponential offset", in IEEE754, the exponential field in the floating-point representation (exponent) is the actual value of the exponent plus a fixed value, the fixed value of the algorithm is 2e-1-1

128-1 = 127 in a single-precision floating-point number.

The fractional part definition is f=0.fn-1fn-2...f0, the binary decimal point is to the left of the most significant bit, and the decimal is defined as M=1 + F, so M is represented as 1.fn-1...f0, by adjusting the exponent so that the effective number m is between two and one

Look at an example 8.25,

Convert to binary representation 1000.01 can be represented as 1.00001 *23

So index e = 3 + 127 = 130

So 8.25 in memory is represented as:

0 10000010 00001000000000000000000

Where the fractional part of how to convert to binary, you can click the way the fractional part, take the integer part value (0 or 1)

, and then continues to take the fractional part of the result, taking the integer portion, and looping until the desired number of digits is obtained,

such as 0.25 * = 0.5, the whole number is divided into 0, then 0.5*2=1.0, the whole number of parts is divided into 1, so the binary is represented as 0.0100000. 00

The index range is from -126~127, so when a floating-point number such as 0.25 can be expressed as 1.000*2-2

So the index offset value is-2 + 127=125

Represented in memory is

0 01111101 00000000000000000000000

These are the use of normalized values in floating-point representations, and other non-normalized and special values of two, later supplemented.

IEEE binary floating-point arithmetic standard learning