What is a normalized floating point representation

Converting a floating point number into the floating point representation

The decimal number 18,4 should be converted into the binary floating point representation. The process usually consists of 6 steps. The order of the individual steps or separate sub-steps can also be carried out differently.

In principle, the process consists of converting the number, normalizing it, determining the new exponent, forming the sign and then combining the values ​​of sign, characteristic (exponent) and mantissa.

1. Convert integer number

In the first step, the binary number is determined for the number 18 in front of the decimal point. The division method is used here.

2. Convert decimal number

In the second step, the binary number is determined for the number 0.4 after the decimal point. The multiplication method is used.
The result is a dual fractional number.

Result from steps 1 and 2: 18,4 = 10010,011001100110011001101100100...

3. Normalize or normalize (determine mantissa)

With normalization you move the comma so that you get a normalized number. For example 1.0101 (2) or 0.123 (10). The exponential notation is used so that the number retains its value.

10010,01100110011... * 20
1,001001100110011... * 24 (Normalization)

Normalization means that you only want to save the decimal places. In the binary representation of numbers, there is always a one (1) at the front. This one in front of the decimal point can be omitted when saving or when displaying (hidden bit), because there is always a one here. But you have one more place at the back for accuracy.

4. Convert exponent (determine characteristic)

The exponent is also known as a characteristic because it has an offset that is known as bias or excess. What is meant is that the range of values ​​is shifted. You do this so that the exponent is always positive or at least zero. This saves a bit for representing the sign of the exponent. The bias display also makes it easier to compare sizes. Disadvantage, there is a positive zero (+0) and a negative zero (-0).
Two's complement and one's complement are also rare alternatives.

How much the value range is shifted depends on the selected precision (number of bits).

  • Single precision (32 bits) means a bias of 127.
  • Double precision (64 bits) means a bias of 1023.

Since we are working with a single precision (32 bit), this corresponds to an offset (bias) of 127.

In principle, the bias is subtracted from the exponent determined (through normalization) and the characteristic is obtained in this way.

Characteristic (new exponent) = exponent + bias = 4 + 127 = 131

The characteristic is then converted into a dual number.

5. Determine the sign

6. Form floating point number (with single precision)

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