Preface

This article is a supplement to the storage of data in memory , Explore floating point .

Floating point storage in memory

Common floating point numbers ：

3.14159
1E10
The family of floating-point numbers includes ：float、double、long double type .
The range represented by floating point numbers ：float.h In the definition of

The representation of floating-point numbers inside a computer ：

According to international standards , Any binary floating point number V It can be expressed in the following form ：
( -1 ) ^ S * M * 2 ^ E
( -1 ) ^ S The sign bit , When S=0 ,V Is a positive number ; When S=1, V It's a negative number .
M Represents a significant number , Greater than or equal to 1, Less than 2 .
2^E Indicates the index bit .
You can compare the scientific counting method of decimal system
-152.45 Namely - 1.5245 * 10 ^ 2

Example ：

#include<stdio.h>

int main()
{
    
	int n = 9;
	float* pFloat = (float*)&n;
	printf("n The value of is ：%d\n", n);
	printf("*pFloat The value of is ：%f\n", *pFloat);

	*pFloat = 9.0;
	printf("nume The value of is ：%d\n", n);
	printf("*pFloat The value of is ：%f\n", *pFloat);

	return 0;
}

num and *pFloat It's the same number in memory , Why are the interpretation results of floating-point numbers and integers so different ？
Another example ：
Decimal 5.0, Written as binary is 101.0, amount to 1.01x2^2.
that , According to the above V The format of , You can get S=0,M=1.01,E=2.
Decimal -5.0, Written as binary is -101.0, amount to -1.01x2^2.
that ,S=1.M=1.01,E=2.
IEEE 754 Regulations ：
about 32 Floating point number of bits , The highest 1 Bits are sign bits s, And then 8 Bits are exponents E, The rest 23 Bits are significant numbers M.

about 64 Floating point number of bits , The highest 1 Bits are sign bits S, And then 11 Bits are exponents E, The rest 52 Bits are significant numbers M.

IEEE 754 For significant figures M And the index E, There are some special rules .

As I said before ,1≤M<2, in other words ,M It can be written. 1.xxxxxx In the form of , among xxxxxx Represents the fractional part .
IEEE 754 Regulations , Keep it in the computer M when , By default, the first digit of this number is always 1, So it can be discarded , Save only the back xxxxxx part . For example preservation 1.01 When , Save only 01, Wait until you read , Put the first 1 Add . The purpose of this , It's saving 1 Significant digits . With 32 For example, a floating-point number , Leave to M Only 23 position , Will come first 1 After giving up , It's equivalent to being able to save 24 Significant digits .

As for the index E, The situation is more complicated .
First ,E For an unsigned integer （unsigned int）

It means , If E by 8 position , Its value range is 0_{255; If E by 11 position , Its value range is 0}2047. however , We know , In scientific counting E You can have negative numbers , therefore IEEE 754 Regulations , In memory E The true value of must be added with an intermediate number , about 8 Bit E, The middle number is 127; about 11 Bit E, The middle number is 1023. such as ,2^10 Of E yes 10, So save it as 32 When floating-point numbers are in place , Must be saved as 10+127=137, namely 10001001.

then , Index E Fetching from memory can be further divided into three cases ：
E Not all for 0 Or not all of them 1

At this time , Floating point numbers are represented by the following rules , The index E The calculated value of minus 127（ or 1023）, Get the real value , And then the significant number M Add the first 1.
for instance :
0.5 The binary form of is 0.1 , Since it is stipulated that the positive part must be 1 , That is to move the decimal point to the right 1 position , Then for
1.0 * 2 ^ ( - 1) ,S by 0 ; E by -1+127=126 , Expressed as 01111110 ; And the mantissa M = 1.0 Remove the integer part and make it 0 , A filling 0 To 23 position

E All for 0

At this time , The exponent of a floating point number E be equal to 1-127（ perhaps 1-1023） That's the true value , Significant figures M No more first 1, It's reduced to 0.xxxxxx Decimals of . This is to show that ±0, And close to 0 A very small number of .

E All for 1

At this time , If the significant number M All for 0, Express ± infinity （ It depends on the sign bit s）;