Four rounding modes in IEEE754 standard

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One 、 Preface

Recently, I was writing a project based on IEEE754 standard Of floating-point adder , One of the requirements is to meet IEEE754 standard Four rounding modes of .

When we normalize order or right , The operands with smaller order are shifted to the right , It will cause the lower bits of the mantissa to be lost , This will cause errors . That's why we need to meet our needs , One of the four rounding modes is used to round the mantissa to reduce the error .

Two 、IEEE754 Standard 4 Rounding mode

1、 Nearest rounding ：

Namely ten Base number Under the rounding . But there are also the following situations ：

• The extra number is 1001, It is greater than 0.5, Therefore, the lowest position goes into 1.
• The extra number is 0111, It's smaller than 0.5, Then discard the extra numbers directly .
• The extra number is 1000, Is exactly equal to 0.5 In special circumstances ; Then the lowest order is 0 Then we give up the extra places , The lowest is 1 Then carry 1.

Note that the digits here are binary numbers . Because this is the mantissa , So we are calculating these binary sums 0.5 In a relationship , That is to say 10 When in base , We multiply the weight of each bit by 2^(-i) Then sum up .

2、 the 0 Round off ： That is, round to the zero point of the number axis , So let's just cut the tail .

3、 Round to infinity ： For positive numbers , All the extra bits are 0 Then cut the tail directly , Not all for 0 Then carry... To the least significant bit 1; Negative numbers, no matter how many extra bits, can be truncated directly .

4、 Round to negative infinity ： For negative numbers , All the extra bits are 0 Then cut the tail directly , Not all for 0 Then carry... To the least significant bit 1; If it is a positive number, you can truncate it directly regardless of the number of extra bits .

3、 ... and 、 give an example

After the decimal point is required 3 position

1、 Nearest rounding ：

 about 1.001_1001, After rounding, it is 1.010（ Get rid of the excess 4 position , Add 0.001）
 about 1.001_0111, After rounding, it is 1.001（ Get rid of the excess 4 position ）
 about -1.001_1000, After rounding, it is -1.010（ Get rid of the excess 4 position , Add 0.001, Because the lowest order is 1）
 about -1.010_1000, After rounding, it is -1.010（ Just get rid of the excess 4 position , Because the lowest order is 0）

  
   

  
   

    

   

2、 the 0 Round off ：

// Positive numbers are truncated directly 
 about 1.001_1001, After rounding, it is 1.001（ Just get rid of the excess 4 position ）
 about 1.001_0111, After rounding, it is 1.001（ Just get rid of the excess 4 position ）

// Negative numbers are truncated directly
about -1.001_1000, After rounding, it is -1.001（ Just get rid of the excess 4 position ）
about -1.010_1000, After rounding, it is -1.010（ Just get rid of the excess 4 position ）

3、 Round to infinity ：

// The extra bits of a positive number are not all 0 carry 1
 about 1.001_1001, After rounding, it is 1.010（ Get rid of the excess 4 position , Add 0.001）
 about 1.001_0111, After rounding, it is 1.010（ Get rid of the excess 4 position , Add 0.001）

// All the extra bits of a positive number are 0 Straight tailing
about 1.001_0000, After rounding, it is 1.001（ Just get rid of the excess 4 position ）

// Negative numbers are truncated directly
about -1.001_1010, After rounding, it is -1.001（ Just get rid of the excess 4 position ）

4、 Round to negative infinity ：

// Positive numbers are truncated directly 
 about 1.001_1001, After rounding, it is 1.001（ Just get rid of the excess 4 position ）
 about 1.001_0111, After rounding, it is 1.001（ Just get rid of the excess 4 position ）

// All the extra bits of negative numbers are 0 Straight tailing
about -1.001_0000, After rounding, it is -1.001（ Just get rid of the excess 4 position ）

// The extra bits of negative numbers are not all 0 carry 1
about -1.001_1010, After rounding, it is -1.010（ Get rid of the excess 4 position , Add 0.001）