3rd Lesson floating-point secret

1. Floating-point number in memory--storage mode: sign bit, exponent, mantissa symbol

Type	Sign bit	Index	Tail
Float	1-bit (31st bit)	8-bit (第23-30位)	23-bit (第0-22位)
Double	1-bit (63rd bit)	11-bit (第52-62位)	52-bit (第0-51位)

float The representation of a double type of data within a computer is the same, but because of the different storage space, the range and precision of the data values can be expressed separately.

2. example of storage of floating-point numbers

2.1 steps to convert floating-point numbers

(1) converting floating-point numbers to binary

(2) using scientific notation to represent binary floating-point numbers

(3) Calculate the value after the exponential offset: (with offset , float type: +127,double type: +1023)

Example: for exponential 6, the following values are offset:

float: 127 + 6→133

Double: 1023 +6 →1029

2.2 The conversion process is demonstrated using the real number 8.25 in the float type as an example in memory notation

(1) 8.25 binary notation: 1000.01, then into exponential form: 1.00001* (2^3)

① sign bit: 0

The ② index is 3: 127+3 =130→10000010

③ decimals: 00001. To turn to the mantissa, you need to fill 0

(2) 8.25 float in memory:0 0010 000000 0000 0000 0000 =0x41040000

(The Red part is the sign bit, the exponent is green , the mantissa is the Purple part plus 0 after the 23 bits)

Memory representation of the "Programming implementation 1" decimal floating-point number

#include <stdio.h>intMain () {floatf =8.25; //in order to display the binary value of the floating-point memory representation in 16 binary, the//segment memory displays that string of binary numbers as integers for human readingUnsignedint* p = (unsignedint*) &f;//integer pointer to f memoryprintf"0x%08x\n", *p);//output 0x41040000 with integral type    return 0;}

3. Secret of floating-point types

(1) Think: int and float both account for 4 bytes of memory, and the range of type int: [ -231,231-1],float type range: [ -3.4*1038,3.4*1038], float is much larger than Int's range?

(2) Secret

①Float can represent exactly the same number of numbers as int

②float can be expressed between the number is discontinuous , there is a jump

③float is just an approximate representation and cannot be used as an exact number.

④ because memory notation is relatively complex, float is much slower to operate than int

(3) Note: Double and float have the same memory notation, because double is not accurate. Because the double occupies more memory, it can represent a higher precision than float.

Imprecise example of "programmatic implementation" float type

#include <stdio.h>intMain () {floatF1 =3.1415f; floatF2 =123456789; //accurate to 10 digits after the decimal pointprintf"%0.10f\n", F1);//3.1414999962--not accurateprintf"%0.10f\n", F2);//123456792.0000000000 ---discontinuous//in the same vein, it is possible to experiment with a float type of 123456780-123456788 in memory//is represented by the number 123456784.0000000000.     return 0;}

4. Summary

(1) floating-point type differs from integer type in memory notation

(2) More complex memory representations of floating-point types

(3) A floating-point type can represent a larger range

(4) Floating-point type is an inexact type

(5) Slow operation of floating-point types

3rd Lesson floating-point secret