Storage of floating-point numbers in memory

1. Introduction to floating point numbers

1. Understanding of floating point numbers

• First of all, we need to understand that floating-point numbers are decimals in mathematics .
  ——————
  But why are they called floating-point numbers in computer languages?

1. First of all, know that [12.3] is a decimal
2. But it can also be written as [1.23*10^1]
3. After comparing these two numbers, it is not difficult to find that when the same number is expressed in different scientific notation,The decimal point is floating, so this is called a floating point number .

2. Introduction to floating point types

1. Floating point types include the following types:

float 、double 、long double

2. The range of floating-point numbers is defined inflaot.hMiddle:
see its definition:

How floating-point numbers are stored in memory

• Above we know how to express a floating point number, but how to store it in memory? Is it still the same as plastic surgery?
  Is == still the same as shaping? ==Let’s explore this question: If you still don’t know how plastic surgery is stored in memory, you can take a look at: My last blog post Plastic surgery is stored in memory .
  You can see the code to explore it:

#include<stdio.h>

int main()
{
    
    
	int i = 9;
	float* p =(float*) & i;

	printf("%d\n", i);
	printf("%f\n", i);

	return 0;
}

1. Of course, if it is different from the storage method of shaping, then is it stored in this way?
   Next is our main event：

Storage regulations:

According to the international standard IEEE 754, any base floating-point number V can be expressed in the following form:

V = (-1)^S * M * 2^E

1. Where (-1)^S represents a valid number, when S is 0, V is a positive number, and when S is 1, V is a negative number.
2. M represents a valid number, and the range is specified: greater than 1 and less than 2.
3. 2^E represents the exponent.

To write a floating-point number in base form: the values ​​​​of S, M, and E must be required。

————

So let's see how a floating point number is converted into base form:

It can be seen from this: V = (-1)^S * M * 2^E
5.5 --> 101.1 —> 1.011 * 2^2
so S = 0, M = 1.011, E = 2;

Note: It is also conceivable that as long as there are S, M, and E values, this floating-point number can be restored.

1. Storage of floating-point numbers

• The international standard IEEE 754 stipulates
  that when storing single-precision floating-point numbers in memory:

Of course, in addition to these, there are some special storage regulations for the significant number M and exponent E in the international standard IEEE 754

1. As we know above, M is a valid number, the range: 1<=M<2 , when saving M in the computer memory, the first digit of this number is always 1 by default, and can be discarded (when reading, put The first digit is added), and only the decimal point is reserved; for example: when 101.1 is reserved: only 011 is saved, for easier understanding, as shown in the figure:

2. The deposit of index E is a bit special:
First E is an unsigned number (unsigned int).
Because when it is 8 bits, it has a range of 0 ~ 255; and when it is 11 bits, its range is 0 ~ 2047, but it is known from scientific notation that there can be negative numbers at this time, (when storing 0.5, E = -1), so IEEE 754 stipulates that the real value of E needs to be added with an intermediate value when stored in memory. For 8 bits, the intermediate value is 127, and for 11 bits, the intermediate value is 1024.

And for the storage of double-precision floating-point type:

There are three types of exponent E taken from the memory:
1. When E is all 0: that is to say, the real value of the exponent E is equal to -127, and the effective number M is no longer 1, but a decimal such as 0.xxxxxx , the purpose of doing this is to represent such an almost very small number, close to 0.
——
2. When E is all 1: that is to say, such a number represents infinity (positive or negative depends on S).
——
3. When E has 1 and 0: subtract 127 (or 1024) from the calculated value of exponent E to get the real value, and then it is stipulated that the positive part must be 1 (that is, add 1 after M is taken out).

This is the case for the storage of floating-point types in memory. Below I will give an example to facilitate your understanding of the number:

int main()
{
    
    
	float n = 9.0;
	int* p = (int*)&n;

	printf("%f\n", n);
	printf("%d ", *p);

	return 0;
}

Finally, I hope that comrades can receive their own things after reading my explanation, and finally come on, mount! ! !