我竟在arm汇编除法算法里找到了leetcode某道题的解法

By | 2021年1月14日

今天讲讲arm汇编中除法的底层实现。汇编代码本身比较长了,如需参考请直接拉到文末。

下面我直接把arm的除法算法的汇编代码转译成C语言的代码贴出来,并进行解析。

因为篇幅有限,所以在此只解析无符号整型的除法运算,关于无符号除法和有符号除法的区别请参考上一篇推送

代码较长如下,电脑端看效果更佳,如无耐心请直接拉下去看讲解即可:

#include

unsigned int count_leading_zeros(unsigned int num)
{
    unsigned int cnt = 0;
    while(!(num & 0x80000000) && cnt > (31 - count_leading_zeros(divisor));
    }

    divisor_lz = count_leading_zeros(divisor);
    dividend_lz = count_leading_zeros(dividend);
    printf("dividend[0x%x], dividend_lz[%d], divisor[0x%x], divisor_lz[%d]n", dividend, dividend_lz, divisor, divisor_lz);
    cc = divisor_lz - dividend_lz;
    while(cc >= 0){
        answer = (divisor 

2次幂和移位运算

在以上代码中我们终于看到了移位运算对除法运算的优化:
当除数是2的N次幂时,可以直接对被除数做右移运算来代替除法, 比如除数是2即(2的1次幂),此时只需要对被除数做一次右移即可,同理如果除数是8则对被除数做三次右移。

而判断一个数字是不是2的N次幂只需要一行代码:

    if ((divisor & (divisor - 1)) == 0){

这一行代码也几乎就是leetcode的第231题2的幂的答案:

2^x n n – 1 n & (n – 1)
2^0 0001 0000 (0001) & (0000) == 0
2^1 0010 0001 (0010) & (0001) == 0
2^2 0100 0011 (0100) & (0011) == 0
2^3 1000 0111 (1000) & (0111) == 0

如有疑问请继续参考leetcode的题解:https://leetcode-cn.com/problems/power-of-two/solution/power-of-two-er-jin-zhi-ji-jian-by-jyd/

而计算2的N次幂中的N,也只需要这一句即可:

(31 - count_leading_zeros(divisor))

count_leading_zeros即为一个32bit的数字以二进制呈现的时候,从高位向低位数开始数有连续多少个0的数量。

比如数字2的二进制是: 0000 0000 0000 0000 0000 0000 0000 0010
在第一个bit1出现之前有30个0。

判断是否是2的N次幂,并且计算出N的大小并进行右移也只需要以下三行代码。

    if ((divisor & (divisor - 1)) == 0){
        return dividend >> (31 - count_leading_zeros(divisor));
    }

为什么要使用count_leading_zeros这种方法呢,虽然我在上面的代码中定义了函数count_leading_zeros,但是在arm汇编中只需要一条指令clz即可,计算2的N次幂的N加上右移也只需要三条指令即可,非常高效:

clz     r2, r1 //计算leading zeros的数量
rsb     r2, r2, #31    //31 - count_leading_zeros(divisor)
lsr.w   r0, r0, r2     // 进行右移

二进制的除法解析

那么更多情况下,除数也并不是2的N次幂。如果除数是3,那么还是要做一下正规的除法了。

我做了一张图来对比8/3的十进制和二进制的除法。

在二进制时,任何一个bit不可能大于1,所以当两个数字的leading zeros相同时,被除数不可能会整除除数超过或者等于两次。也就是说leading zeros相同时,被除数要么能整除除数一次,要么是0次。

二进制运算除法的时候,首先会对除数做左移操作,让除数和被除数进行“对齐”(即leading zeros数量相同),如果此时的被除数大于等于此时(左移后的)除数,那么在相应的答案位上置一,否则置0。然后对(左移后的)除数?做右移一位操作再继续和被除数做比较,直到除数恢复成原来的初始值(这时候会作最后一次运算)。如下代码所示:

    cc = divisor_lz - dividend_lz;
    while(cc >= 0){
        answer = (divisor 

所以在二进制整型数字的除法世界中,只需要减法和移位操作就能够满足除法运算的需求。最后我才发现,二进制的除法原本就是这么简单,比十进制的除法还要简单。

本文完,以下为参考资料。

arm的指令集查文档:
http://users.ece.utexas.edu/~valvano/Volume1/QuickReferenceCard.pdf
https://iitd-plos.github.io/col718/ref/arm-instructionset.pdf
div无符号整形的除法汇编如下:

00010490 :
   10490:       1e4a            subs    r2, r1, #1
   10492:       bf08            it      eq
   10494:       4770            bxeq    lr
   10496:       f0c0 8124       bcc.w   106e2 
   1049a:       4288            cmp     r0, r1
   1049c:       f240 8116       bls.w   106cc 
   104a0:       4211            tst     r1, r2
   104a2:       f000 8117       beq.w   106d4 
   104a6:       fab0 f380       clz     r3, r0
   104aa:       fab1 f281       clz     r2, r1
   104ae:       eba2 0303       sub.w   r3, r2, r3
   104b2:       f1c3 031f       rsb     r3, r3, #31
   104b6:       a204            add     r2, pc, #16     ; (adr r2, 104c8 )
   104b8:       eb02 1303       add.w   r3, r2, r3, lsl #4
   104bc:       f04f 0200       mov.w   r2, #0
   104c0:       469f            mov     pc, r3
   104c2:       bf00            nop
   104c4:       f3af 8000       nop.w
   104c8:       ebb0 7fc1       cmp.w   r0, r1, lsl #31
   104cc:       bf00            nop
   104ce:       eb42 0202       adc.w   r2, r2, r2
   104d2:       bf28            it      cs
   104d4:       eba0 70c1       subcs.w r0, r0, r1, lsl #31
   104d8:       ebb0 7f81       cmp.w   r0, r1, lsl #30
   104dc:       bf00            nop
   104de:       eb42 0202       adc.w   r2, r2, r2
   104e2:       bf28            it      cs
   104e4:       eba0 7081       subcs.w r0, r0, r1, lsl #30
   104e8:       ebb0 7f41       cmp.w   r0, r1, lsl #29
   104ec:       bf00            nop
   104ee:       eb42 0202       adc.w   r2, r2, r2
   104f2:       bf28            it      cs
   104f4:       eba0 7041       subcs.w r0, r0, r1, lsl #29
   104f8:       ebb0 7f01       cmp.w   r0, r1, lsl #28
   104fc:       bf00            nop
   104fe:       eb42 0202       adc.w   r2, r2, r2
   10502:       bf28            it      cs
   10504:       eba0 7001       subcs.w r0, r0, r1, lsl #28
   10508:       ebb0 6fc1       cmp.w   r0, r1, lsl #27
   1050c:       bf00            nop
   1050e:       eb42 0202       adc.w   r2, r2, r2
   10512:       bf28            it      cs
   10514:       eba0 60c1       subcs.w r0, r0, r1, lsl #27
   10518:       ebb0 6f81       cmp.w   r0, r1, lsl #26
   1051c:       bf00            nop
   1051e:       eb42 0202       adc.w   r2, r2, r2
   10522:       bf28            it      cs
   10524:       eba0 6081       subcs.w r0, r0, r1, lsl #26
   10528:       ebb0 6f41       cmp.w   r0, r1, lsl #25
   1052c:       bf00            nop
   1052e:       eb42 0202       adc.w   r2, r2, r2
   10532:       bf28            it      cs
   10534:       eba0 6041       subcs.w r0, r0, r1, lsl #25
   10538:       ebb0 6f01       cmp.w   r0, r1, lsl #24
   1053c:       bf00            nop
   1053e:       eb42 0202       adc.w   r2, r2, r2
   10542:       bf28            it      cs
   10544:       eba0 6001       subcs.w r0, r0, r1, lsl #24
   10548:       ebb0 5fc1       cmp.w   r0, r1, lsl #23
   1054c:       bf00            nop
   1054e:       eb42 0202       adc.w   r2, r2, r2
   10552:       bf28            it      cs
   10554:       eba0 50c1       subcs.w r0, r0, r1, lsl #23
   10558:       ebb0 5f81       cmp.w   r0, r1, lsl #22
   1055c:       bf00            nop
   1055e:       eb42 0202       adc.w   r2, r2, r2
   10562:       bf28            it      cs
   10564:       eba0 5081       subcs.w r0, r0, r1, lsl #22
   10568:       ebb0 5f41       cmp.w   r0, r1, lsl #21
   1056c:       bf00            nop
   1056e:       eb42 0202       adc.w   r2, r2, r2
   10572:       bf28            it      cs
   10574:       eba0 5041       subcs.w r0, r0, r1, lsl #21
   10578:       ebb0 5f01       cmp.w   r0, r1, lsl #20
   1057c:       bf00            nop
   1057e:       eb42 0202       adc.w   r2, r2, r2
   10582:       bf28            it      cs
   10584:       eba0 5001       subcs.w r0, r0, r1, lsl #20
   10588:       ebb0 4fc1       cmp.w   r0, r1, lsl #19
   1058c:       bf00            nop
   1058e:       eb42 0202       adc.w   r2, r2, r2
   10592:       bf28            it      cs
   10594:       eba0 40c1       subcs.w r0, r0, r1, lsl #19
   10598:       ebb0 4f81       cmp.w   r0, r1, lsl #18
   1059c:       bf00            nop
   1059e:       eb42 0202       adc.w   r2, r2, r2
   105a2:       bf28            it      cs
   105a4:       eba0 4081       subcs.w r0, r0, r1, lsl #18
   105a8:       ebb0 4f41       cmp.w   r0, r1, lsl #17
   105ac:       bf00            nop
   105ae:       eb42 0202       adc.w   r2, r2, r2
   105b2:       bf28            it      cs
   105b4:       eba0 4041       subcs.w r0, r0, r1, lsl #17
   105b8:       ebb0 4f01       cmp.w   r0, r1, lsl #16
   105bc:       bf00            nop
   105be:       eb42 0202       adc.w   r2, r2, r2
   105c2:       bf28            it      cs
   105c4:       eba0 4001       subcs.w r0, r0, r1, lsl #16
   105c8:       ebb0 3fc1       cmp.w   r0, r1, lsl #15
   105cc:       bf00            nop
   105ce:       eb42 0202       adc.w   r2, r2, r2
   105d2:       bf28            it      cs
   105d4:       eba0 30c1       subcs.w r0, r0, r1, lsl #15
   105d8:       ebb0 3f81       cmp.w   r0, r1, lsl #14
   105dc:       bf00            nop
   105de:       eb42 0202       adc.w   r2, r2, r2
   105e2:       bf28            it      cs
   105e4:       eba0 3081       subcs.w r0, r0, r1, lsl #14
   105e8:       ebb0 3f41       cmp.w   r0, r1, lsl #13
   105ec:       bf00            nop
   105ee:       eb42 0202       adc.w   r2, r2, r2
   105f2:       bf28            it      cs
   105f4:       eba0 3041       subcs.w r0, r0, r1, lsl #13
   105f8:       ebb0 3f01       cmp.w   r0, r1, lsl #12
   105fc:       bf00            nop
   105fe:       eb42 0202       adc.w   r2, r2, r2
   10602:       bf28            it      cs
   10604:       eba0 3001       subcs.w r0, r0, r1, lsl #12
   10608:       ebb0 2fc1       cmp.w   r0, r1, lsl #11
   1060c:       bf00            nop
   1060e:       eb42 0202       adc.w   r2, r2, r2
   10612:       bf28            it      cs
   10614:       eba0 20c1       subcs.w r0, r0, r1, lsl #11
   10618:       ebb0 2f81       cmp.w   r0, r1, lsl #10
   1061c:       bf00            nop
   1061e:       eb42 0202       adc.w   r2, r2, r2
   10622:       bf28            it      cs
   10624:       eba0 2081       subcs.w r0, r0, r1, lsl #10
   10628:       ebb0 2f41       cmp.w   r0, r1, lsl #9
   1062c:       bf00            nop
   1062e:       eb42 0202       adc.w   r2, r2, r2
   10632:       bf28            it      cs
   10634:       eba0 2041       subcs.w r0, r0, r1, lsl #9
   10638:       ebb0 2f01       cmp.w   r0, r1, lsl #8
   1063c:       bf00            nop
   1063e:       eb42 0202       adc.w   r2, r2, r2
   10642:       bf28            it      cs
   10644:       eba0 2001       subcs.w r0, r0, r1, lsl #8
   10648:       ebb0 1fc1       cmp.w   r0, r1, lsl #7
   1064c:       bf00            nop
   1064e:       eb42 0202       adc.w   r2, r2, r2
   10652:       bf28            it      cs
   10654:       eba0 10c1       subcs.w r0, r0, r1, lsl #7
   10658:       ebb0 1f81       cmp.w   r0, r1, lsl #6
   1065c:       bf00            nop
   1065e:       eb42 0202       adc.w   r2, r2, r2
   10662:       bf28            it      cs
   10664:       eba0 1081       subcs.w r0, r0, r1, lsl #6
   10668:       ebb0 1f41       cmp.w   r0, r1, lsl #5
   1066c:       bf00            nop
   1066e:       eb42 0202       adc.w   r2, r2, r2
   10672:       bf28            it      cs
   10674:       eba0 1041       subcs.w r0, r0, r1, lsl #5
   10678:       ebb0 1f01       cmp.w   r0, r1, lsl #4
   1067c:       bf00            nop
   1067e:       eb42 0202       adc.w   r2, r2, r2
   10682:       bf28            it      cs
   10684:       eba0 1001       subcs.w r0, r0, r1, lsl #4
   10688:       ebb0 0fc1       cmp.w   r0, r1, lsl #3
   1068c:       bf00            nop
   1068e:       eb42 0202       adc.w   r2, r2, r2
   10692:       bf28            it      cs
   10694:       eba0 00c1       subcs.w r0, r0, r1, lsl #3
   10698:       ebb0 0f81       cmp.w   r0, r1, lsl #2
   1069c:       bf00            nop
   1069e:       eb42 0202       adc.w   r2, r2, r2
   106a2:       bf28            it      cs
   106a4:       eba0 0081       subcs.w r0, r0, r1, lsl #2
   106a8:       ebb0 0f41       cmp.w   r0, r1, lsl #1
   106ac:       bf00            nop
   106ae:       eb42 0202       adc.w   r2, r2, r2
   106b2:       bf28            it      cs
   106b4:       eba0 0041       subcs.w r0, r0, r1, lsl #1
   106b8:       ebb0 0f01       cmp.w   r0, r1
   106bc:       bf00            nop
   106be:       eb42 0202       adc.w   r2, r2, r2
   106c2:       bf28            it      cs
   106c4:       eba0 0001       subcs.w r0, r0, r1
   106c8:       4610            mov     r0, r2
   106ca:       4770            bx      lr
   106cc:       bf0c            ite     eq
   106ce:       2001            moveq   r0, #1
   106d0:       2000            movne   r0, #0
   106d2:       4770            bx      lr
   106d4:       fab1 f281       clz     r2, r1
   106d8:       f1c2 021f       rsb     r2, r2, #31
   106dc:       fa20 f002       lsr.w   r0, r0, r2
   106e0:       4770            bx      lr
   106e2:       b108            cbz     r0, 106e8 
   106e4:       f04f 30ff       mov.w   r0, #4294967295 ; 0xffffffff
   106e8:       f000 b966       b.w     109b8 

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