Division routines sometimes can be used for more than they were designed for! For example, it is possible to divide 16 or 32 bit dividend by 16 bit divisor to 16 bit quotient in just one routine.
The idea is simple: you do 16 bit division as usually, but when the dividend is 32 bit, the lower two bytes are used as dividend and the higher two bytes are used as initial value for remainder (which is zero when doing 16 bit division)
To understand how initializing remainder relates to dividend, let's examine the algorithms for two cases:
x q =  y
1) when x, y, and
q are 16 bit unsigned integers, and
2) when x and q
are 32 bit unsigned integers, but y is a 16 bit unsigned integer
Note that in some cases it is known that the result of 32 bit by 16 bit division is 16 bit maximum. The trick described on this page works only for 16 bit result.For example, when speed of a motor is calculated by a Hall sensor pulses (one pulse per revolution) the following formula is used:
60 x 1,000,000 (conversions for minutes and us, rpm =  timer is clocked by 1 MHz clock) 16 bit timer valueTimer is used to measure the Hall sensor pulse period. For this motor and timer the rpm range is within 91520,000 rpm, which fits perfectly into 16 bits.
x  16 bit dividend;
y  16 bit divisor;
q  16 bit quotient;
rem  16 bit remainder;
counter  loop counter.
1. Clear remainder, load counter=16
rem q   0000 ????   b1 b0 b1 b0
2. Shift left dividend x, and shift left remainder rem, by 1 bit, so that MSb of x was shifted to the remainder's LSb
rem x   r1r0 < x1x0 < shift left   b1 b0 b1 b0
3. Subtract divisor y from remainder. If subtraction successful (no borrow), next bit of q is 1. If borrow, next bit of q is 0, and remainder should be restored.
 r1r0 rem  b1 b0   y1y0 y  b1 b0
4. Shift left q and set its LSb according to subtraction result from previous step.
q carry(inverted borrow) flag   q1q0 < C shift in next result bit   b1 b0
5. Decrement counter and if it's not zero repeat from step 2 (16 iterations)
After that, q contains quotient, rem  remainder, x is garbage (it was shifted out completely), and y is untouched.
x  32 bit dividend;
y  16 bit divisor;
q  32 bit quotient;
rem  17 bit remainder (remainder may be bigger than 16 bit, for example,
when x=0x80000000 and y=0xFFFF);
counter  loop counter.
1. Clear remainder, load counter=32
rem q   000000 ????????   b2 b1 b0 b3 b2 b1 b0
2. Shift left dividend x, and shift left remainder rem, by 1 bit, so that MSb of x was shifted to the remainder's LSb
rem x   r2r1r0 < x3x2x1x0 < shift left   b2 b1 b0 b3 b2 b1 b0
3. Subtract divisor y from remainder. If subtraction successful (no borrow), next bit of q is 1. If borrow, next bit of q is 0, and remainder should be restored.
 r2r1r0 rem  b2 b1 b0   y1y0 y  b1 b0
4. Shift left q and set its LSb according to subtraction result from previous step.
q carry(inverted borrow) flag   q3q2q1q0 < C shift in next result bit   b3 b2 b1 b0
5. Decrement counter and if it's not zero repeat from step 2 (32 iterations)
After that, q contains quotient, rem  remainder (16 bit), x is garbage (it was shifted out completely), and y is untouched.
Now imagine that you know the result is always 16 bit (for example). That means that the first 16 iterations through steps 15 produce 16 zero bits in quotient (because we know that higher 2 bytes of quotient are zero in that case), x was shifted 16 times to remainder, remainder was never subtracted from. So we can say that with these 16 iterations we actually did the following:
From:
rem x    0 0 0 < x3x2x1x0 < shift left   b2 b1 b0 b3 b2 b1 b0  y q   y1y0 ????????   b1 b0 b3 b2 b1 b0
To:
rem x    0x3x2 < x1x0???? < shift left   b2 b1 b0 b3 b2 b1 b0  y q   y1y0 ???? 0 0   b1 b0 b3 b2 b1 b0
And after the next 16 iterations:
rem x    0r1r0 < ???????? < shift left   b2 b1 b0 b3 b2 b1 b0  y q   y1y0  0 0q1q0   b1 b0 b3 b2 b1 b0
This is very similar to 16 bit division:
From:
rem x    0 0 < x1x0 < shift left   b1 b0 b1 b0  y q   y1y0 ????   b1 b0 b1 b0
To:
rem x   r1r0 < ???? < shift left   b1 b0 b1 b0  y q   y1y0 q1q0   b1 b0 b1 b0
Except the only thing  we don't start with zero remainder, but instead from:
rem x   x3x2 < x1x0 < shift left   b1 b0 b1 b0  y q   y1y0  0 0   b1 b0 b1 b0
x3 should be less than 0x80 though, so that after first left shift the higher bit wouldn't disappear. So we actually can do 31by16to16 division with a 16by16to16 one!
This example uses slightly different algorithm from the one described above. It does not restore the remainder immidiately after a subtraction causes a borrow. However, the trick still aplies, and because this variant of division routine has an extended remainder (necessary for the nonrestoring method to hold the current remainder sign), it can do the full 32by16to16bit division.
; x = 60*1e6/y ; ; x, x+1  rpm ; y, y+1  pulse width in 1 us units ; FindRPM clr x mov W, #$87 mov x+1, W mov W, #$93 mov x+2, W mov W, #$03 mov x+3, W jmp div32by16to16 ; uint16 x = uint32 x / uint16 y ; ; Input: ; x, x+1, x+2, x+3  32 bit unsigned integer dividend (x  lsb, x+3  msb) ; y, y+1  16 bit unsigned integer divisor ; Output: ; x, x+1  16 bit unsigned integer quotient ; Temporary: ; counter ; temp  remainder extension ; ; Note: result must fit in 16 bits for routine to work ; correctly div32by16to16 jmp div16by16loopinit ;or just move the label ; uint16 x = uint16 x / uint16 y ; ; Input: ; x, x+1  16 bit unsigned integer dividend (x  lsb, x+1  msb) ; y, y+1  16 bit unsigned integer divisor ; Output: ; x, x+1  16 bit unsigned integer quotient ; Temporary: ; counter ; x+2, x+3  16 bit remainder ; temp  remainder extension ; Size: 36 instructions ; Max timing: 6+16*(5+14+4)2+2+3=377 cycles div16by16 clr x+2 ;clear clr x+3 ;remainder div16by16loopinit clr temp ;clear remainder extension mov W, #16 mov counter, W stc ;first iteration will be subtraction div16by16loop ;shift in next result bit and shift out next ;dividend bit to remainder rl x ;shift lsb rl x+1 ;shift msb rl x+2 rl x+3 rl temp mov W, y sb x.0 jmp div16by16add ;subtract divisor from remainder sub x+2, W mov W, y+1 sc movsz W, ++y+1 sub x+3, W mov W, #1 sc sub temp, W jmp div16by16next div16by16add ;add divisor to remainder add x+2, W mov W, y+1 snc movsz W, ++y+1 add x+3, W mov W, #1 snc add temp, W div16by16next ;carry is next result bit decsz counter jmp div16by16loop ;shift in last bit rl x rl x+1 ret
See also:
file: /Techref/scenix/lib/math/div/div16or32by16to16_sx.htm, 10KB, , updated: 2004/6/10 13:40, local time: 2022/1/23 17:45,
owner: NG944,

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