assembly - Fast Division on GCC/ARM -

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as far know compilers fast division multiplying , bit shifting right. instance, if check this thread says when ask microsoft compiler division 10 multiply dividend 0x1999999a (which 2^32/10) , divide result 2^32 (using 32 shifts right).

so far good.

once tested same division 10 on arm using gcc, though, compiler did different. first multiplied dividend 0x66666667 (2^34/10), divided result 2^34. far it's same microsoft, except using higher multiplier. after that, however, subtracted (dividend/2^31) result.

my question: why on arm version there's subtraction? can give me numeric example without subtraction result wrong?

if want check generated code, it's below (with comments):

        ldr     r2, [r7, #4] @--this loads dividend memory r2         movw    r3, #:lower16:1717986919 @--moves lower 16 bits of constant          movt    r3, #:upper16:1717986919 @--moves upper 16 bits of constant         smull   r1, r3, r3, r2 @--multiply long, put lower 32 bits in r1, higher 32 in r3         asr     r1, r3, #2 @--r3>>2, store in r1 (effectively >>34, since r3 higher 32 bits of multiplication)         asr     r3, r2, #31 @--dividend>>31, store in r3         rsb     r3, r3, r1 @--r1 - r3, store in r3         str     r3, [r7, #0] @--this stores result in memory (from r3)

after that, however, subtracted (dividend/2^31) result.

actually, subtracts dividend >> 31, -1 negative dividend, , 0 non-negative dividend, when right-shifting negative integers arithmetic right-shift (and int 32 bits wide).

0x6666667 = (2^34 + 6)/10

so x < 0, have, writing x = 10*k + r -10 < r <= 0,

0x66666667 * (10*k+r) = (2^34+6)*k + (2^34 + 6)*r/10 = 2^34*k + 6*k + (2^34+6)*r/10

now, arithmetic right shift of negative integers yields floor of v / 2^n, so

(0x66666667 * x) >> 34

results in

k + floor((6*k + (2^34+6)*r/10) / 2^34)

so need see that

-2^34 < 6*k + (2^34+6)*r/10 < 0

the right inequality easy, both k , r non-positive, , not both 0.

for left inequality, bit more analysis needed.

r >= -9

so absolute value of (2^34+6)*r/10 @ 2^34+6 - (2^34+6)/10.

|k| <= 2^31/10,

so |6*k| <= 3*2^31/5.

and remains verify that

6 + 3*2^31/5 < (2^34+6)/10 1288490194   < 1717986919

yup, true.


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