How does this assembly code divide by 1000 without calling idiv

I am trying to understand the trick assembly uses to divide numbers using multiplication and bit shifts. In this example I am working with a battery.

typedef struct{
  short volts;      
  char  percent;        

  char  mode;           
} batt_t;

Notice that this struct takes up 32 bits which means it can be stored compactly in %edi. I want to do volts/1000. The assembly code for this process is

    movswl  %di, %eax            #moves volts to %eax
    imull   $-31981, %eax, %edx  #multiplies volts by (2^25)/1000
    shrl    $16, %edx            #volts times (2^9)/1000
    addl    %edi, %edx           #Adding the original number???

    sarw    $9, %dx              #Dividing by 2^9
    movl    %edi, %r8d           #And from here it subtracts a 1
    sarw    $15, %r8w            #if the original short was negative???
    subl    %r8d, %edx
    movswl  %dx, %ecx            #stores result in %ecx

I don't understand the difference between shr and sar in this case. What I am especially confused about is why it decides to add the original number back on and why it subtracts a one at the end if it's dividing a negative number.

Note

The left 16 bits of %edi contains the two chars we don't need for this operation, but that's okay because the code is clever enough to prevent these bits from affecting the final answer.