Floating point number ieee 754

Convert decimal number to computer stored procedure

Take 9.625 single precision (32 bits) as an example

1. Convert decimal number to binary representation
   9.625 = 1001.101 = 1 × 2 3 + 0 × 22 + 0 × 21 + 1 × 20 + 1 × 2-1 + 0 × 2-2 + 1 × 2-3

2. Binary number normalization
   ieee 754 stipulates that binary representation must be in the format ±d.dd…d × βe , (0 ≤ d i < β)
   That is 1001.101 = 1.001 101 * 23

3. stored to computer
   Single precision total length 32 bits
   1-23 represents the mantissa: Because the binary number after normalization has only one 1 to the left of the decimal point, it can be omitted, so that 23 bits are used to represent 24 bits. That is, the mantissa is 001 101 in 1.001 101
   24-30 represents the exponent: 3 in 1.001 101 * 23
   31 represents the sign: 1 represents a negative number, 0 represents a positive number
   In addition, the exponent bit has 8 bits and can represent values ​​between 0-255. But the exponent may be negative (for example, 0.625 = 0.101 (binary) = 1.01 * 2-1). Therefore, IEEE 754 stipulates that the index is to increase the offset code: the normalized index value + 127. The range of the index becomes -127 - 128. Then the above index becomes 130

   According to the above rules, the binary decimal is stored in the computer as: [0] [1000 0010] [0011 0100 0000 0000 0000 000]

The precision is lost

1. It can be seen that after normalization The mantissa part of a binary number may be much longer than 23 bits, and the latter part will be lost when the computer stores it. Typical is 0.58.