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 flat assembler > Examples and Tutorials > Division by multiplication - revisited
Author
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
Division by multiplication - revisited
Back on the ASM Community message board, when there was a discussion about performing division by invariant divisor with a MUL instead of DIV, I looked to create an explanation of why this works through the theory of 2-adic numbers. But recently, when I played with these ideas again, I realized that it is possible to analyze it just on the basis of pure integer arithmetic. I also made a new (at least to me) variant of the algorithm, which works correctly for every possible input and is easily provable.

This is how the algorithm looks implemented in fasm/fasmg scripting language:
 Code: ; Preparation stage: BITNESS = 32    ; number of bits in word DIVISOR = 3     ; a number to divide by ; convert DIVISOR into MAGIC and SHIFT constants: SHIFT = bsr (DIVISOR-1) + 1 MAGIC = ((1 shl SHIFT - DIVISOR) shl BITNESS) / DIVISOR + 1 ; Division algorithm NUMBER = 7FFFFFFFh      ; an example number to divide PRODUCT = NUMBER * MAGIC + NUMBER shl BITNESS QUOTIENT = PRODUCT shr (SHIFT+BITNESS) REMAINDER = ( (PRODUCT and (1 shl (SHIFT+BITNESS) - 1)) * DIVISOR ) shr (SHIFT+BITNESS)
You need fasmg to get this running for larger input numbers, since fasm 1 overflows easily. It also shows how to use the low bits of the product to produce the remainder from the division, though this is not used the following example.

The implementation in assembly language is a bit more complex. Read further below to learn why it has additional quirks compared to the classic ones:
 Code: ; Preparation stage:         mov     rbx,[divisor]         mov     rcx,rbx         dec     rcx         bsr     rcx,rcx         inc     cl         mov     rdx,1         shl     rdx,cl         sub     rdx,rbx         xor     rax,rax         div     rbx         inc     rax         mov     [magic],rax         dec     cl         mov     [shift_m1],cl
 Code: ; Division:         mov     rbx,[number]         mov     rax,[magic]         mov     cl,[shift_m1]         mul     rbx         add     rdx,rbx         rcr     rdx,1         shr     rdx,cl         mov     [quotient],rdx
Note that it stores "shift minus one" value, because one of the needed bit shifts is made through RCR in order to not loose the carry bit when dividing large numbers. The 32-bit version would be exactly the same, just replace all the 64-bit register names with 32-bit ones.
_____________________________________

And now the explanation. To analyze the problem purely from the point of view of the integer arithmetic, we can formulate it as follows.

For a given invariant divisor D < 2^B (where B is the number of bits in word) we want to find a bit shift S and a "magic" number M such that:
M*D = 2^(B+S) + R
where 0 <= R <= 2^S

If we succeed, then to get the quotient Q = N / D, we can do the following calculation:
N*M = Q*2^(B+S) + L
with 0 <= L < 2^(B+S)
(this shows that we can drop the low B+S bits and take the high ones to get the Q)

Let's take a look at the N*M*D product and substitute M*D from the definition of M:
N*M*D = N*2^(B+S) + N*R
Because N < 2^B and R <= 2^S, we have N*R < 2^(B+S), so the high bits of N*M*D contain original number N.

If we look at the components of N*M, we can evaluate the same product in a different way:
N*M*D = (Q*2^(B+S) + L)*D = Q*D*2^(B+S) + L*D
now, as L < 2^(B+S), L*D < D*2^(B+S), therefore high bits of L*D contain a number smaller than D and we can represent it as:
L*D = X*2^(B+S) + Y
where X < D and Y < 2^(B+S)

Now:
N*M*D = Q*D*2^(B+S) + X*2^(B+S) + Y
therefore high bits of N*M*D contain just Q*D + X and we already know that they also equal the original number N, so:
N = Q*D + X
and because X is positive and smaller than D, this proves that the quotient obtained this way is correct.

Moreover, this demonstrates that high bits of L*D contain the remainder.
_____________________________________

Now let's look at a classic example, this time for B=32:
For D = 3 we can take M = 0AAAAAAABh and S = 1, then
M*D = 200000001h = 2^(1+32) + 1
which fulfills the requirements, because 1 < 2 = 2^S. Therefore, to divide a number by 3, we can simply do this:
 Code: mov     eax,[number]         mov     edx,0AAAAAAABh         mul     edx         shr     edx,1         mov     [quotient],edx
And a couple more instructions could be added to recover the remainder from the low bits of the product, though this is more of a curiosity, since computation of remainder when the quotient is already present is a simple task anyway:
 Code: mov     ecx,3         mov     ebx,0         cmovc   ebx,ecx         mul     ecx         add     edx,ebx         shr     edx,1         mov     [remainder],edx
Note that the highest bit of the 33 low bits of the product landed in CF, and it is multiplied by 3 manually and then added the high bits of the second product.
_____________________________________

For some divisors finding M and S such that R < 2^S may be troublesome.
We can try the most straightforward route and just take smallest S such that D <= 2^S, then divide 2^(S+B) by D and increase it by one to cover the deficit caused by the truncation inherent in integer division:
M = 2^(S+B) / D + 1
then 2^(S+B) < M*D <= 2^(S+B) + D, so the requirements are fulfilled, the only problem is that M is going to be larger than 2^B, so it no longer fits in a machine word.

But, because we take S such that 2^S is the smallest power of two that is not smaller than D:
2^S / D = 1
and therefore M contains only a single bit set above the machine word. We can take M' = M - 2^B and get the "magic" number that fits in the machine word. But when multiplying by it, we have to remember that we need to actually multiply by M' + 2^B, which is the same as multiplying by M' and then adding N to the high bits.

This is how the algorithm I presented in the beginning works. The multiplier it computes is:
(2^(S+B) / D + 1) - 2^B = ((2^S - D) * 2^B) / D + 1
a then the product is adjusted by adding N*2^B before producing the quotient. This may produce carry, therefore the first of the right shifts that follows has to be RCL, to get the overflowing bit back into position.
_____________________________________

If we prefer to stick to the classic algorithm with M fitting in the machine word, it may not always be possible to find the right combination of M and S as defined above. But we can also search for a pair such that:
M*D = 2^(B+S) - R
where R <= 2^S, as previously.

What this modifies in the evaluations is that:
N*M*D = N*2^(B+S) - N*R
and since N*R again is smaller than 2^(B+S), the high bits of N*M*D this time are N-1.

So when we multiply by a number of this form, we actually compute (N-1) / D instead of N / D. The classic algorithm deals with this by simply increasing the number by one before multiplication.

For example, for D = 7 we can take M = 92492492h and S = 2. Then:
M*D = 3FFFFFFFEh = 2^(2+32) - 2
and 2 < 4 = 2^S. Therefore we can divide by 7 this way:
 Code: mov     eax,[number]         mov     edx,92492492h         inc     eax         mul     edx         shr     edx,2         mov     [quotient],edx
A tiny weak spot of this routine is that it fails for input 0FFFFFFFFh. This is the reason why I decided that the more complex algorithm I presented first is also worth sharing, as it should work correctly for any input and any divisor (if there is no mistake in my proof).
06 Sep 2017, 21:31
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
Now instead of taking the smallest power of two that is greater or equal to D, we can take the greatest power of two not greater than D and use it as S. If we now divide 2^(S+B) by D, we get:
2^(S+B) = M * D + P, P < D
Let's set R = D - P, then
M * D = 2^(S+B) - R
If R <= 2^S, we can use the algorithm (but we need INC instruction before MUL).
If R > 2^S, then P < 2^S, because we chose S such that D < 2^(S+1). Then we can take M' = M + 1 and use the algorithm without INC, because:
M' * D = 2^(S+B) - R + D = 2^(S+B) + P

This bring us to the following implementation of the classic algorithm:
 Code: ; Preparation stage:         mov     ebx,[divisor]         bsr     ecx,ebx         mov     [shift],cl         mov     edx,1         shl     edx,cl         xor     eax,eax         div     ebx         shr     edx,cl         setc    cl         mov     [increment],ecx         xor     cl,1         add     eax,ecx         mov     [magic],eax
 Code: ; Division:         mov     eax,[number]         mov     edx,[magic]         add     eax,[increment]         mul     edx         mov     cl,[shift]         shr     edx,cl         mov     [quotient],edx
The [integer] variable has value 0 or 1, it serves as a conditional INC instruction.
Again, no floating point calculations needed, the entire reasoning is solid just in terms of integer relations.
This algorithm does not work for divisors being powers of two, since M does not fit into machine word then. These divisors would need to be detected and division turned into a simple shift instead.
_____________________________________

Note that both the algorithms given use the large values of S even if there may exist a smaller S for which the requirements would be fulfilled. An example for B=32 is number 641, where the algorithm works for S = 0 and M = 663D81h:
663D81h*641 = 2^32 + 1
R = 1 <= 2^S

So 32-bit divison by 641 is a rare case when we can do without the shift instruction:
 Code: mov     eax,[number]         mov     edx,663D81h         mul     edx         mov     [quotient],edx
07 Sep 2017, 11:12
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 15241
Location: 1I/ʻOumuamua
Re: Division by multiplication - revisited

Tomasz Grysztar wrote:
Therefore we can divide by 7 this way:
 Code: mov     eax,[number]         mov     edx,92492492h         inc     eax         mul     edx         shr     edx,2         mov     [quotient],edx
A tiny weak spot of this routine is that it fails for input 0FFFFFFFFh. This is the reason why I decided that the more complex algorithm I presented first is also worth sharing, as it should work correctly for any input and any divisor (if there is no mistake in my proof).

If branches are cheap then we can add a single jz:
 Code: mov     eax,[number]         mov     edx,92492492h         inc     eax         jz      @f         mul     edx     @@: shr     edx,2         mov     [quotient],edx
I use this method with ARM code with a conditional mul instruction. Although only some divisors need this, so it is somewhat wasteful to make it generic. Also if you know your numerator doesn't need it then leave out the jz.

The most common divisor 10 where M=CCC...CCD and S=3 can use the normal N*M >> S approach.
07 Sep 2017, 11:28
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
Re: Division by multiplication - revisited
 revolution wrote: If branches are cheap then we can add a single jz
You are right, this correctly handles the overflow (just like RCL in my first algorithm). It only gives the correct upper half of the product, though - so if you wanted to recover remainder, you'd need to do it some other way.

With this correction in mind, the only remaining advantage of the first algorithm is that it does not require any assumptions concerning divisors. But since the problematic divisors are the powers of two, we'd be better off handling them separately anyway.
07 Sep 2017, 12:01
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 15241
Location: 1I/ʻOumuamua
Re: Division by multiplication - revisited
 Tomasz Grysztar wrote: It only gives the correct upper half of the product, though ...
eax will be zero, so I thought it is the required full 64-bit result of (N+1)*M?
07 Sep 2017, 12:09
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
Re: Division by multiplication - revisited

revolution wrote:
 Tomasz Grysztar wrote: It only gives the correct upper half of the product, though ...
eax will be zero, so I thought it is the required full 64-bit result of (N+1)*M?

Ah, you got me there. It may not even be a coincidence that all the values end up being correct.
07 Sep 2017, 12:21
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
 revolution wrote: Also if you know your numerator doesn't need it then leave out the jz.
If we consider additional assumptions about the numerator, then there is one more interesting thing that can be deduced from my proof.

The requirement that R <= 2^S is only needed to show that N*R < 2^(S+B). So if we can get the same condition through the assumptions on N, it may enable new choices of S, even S = 0.

So if we know that N is always small enough that N*D <= 2^B, we can take S = 0, M = 2^B / D + 1 (except when D is a power of 2, when we should drop "+ 1"). Then multiplying by M gives correct N/D in the high bits for all N < M.

For example for D = 10, if we can safely assume that N is smaller than M = 2^32 / 10 + 1 = 1999999Ah, we can just multiply by it and take the high 32 bits as a result.
07 Sep 2017, 12:21
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
What about dividing numbers larger than the machine word? With DIV instruction it is possible to chain multiple divisions together, the remainder from the previous division is already in EDX/RDX ready to become the high word (smaller than the divisor, so overflow never happens) of the subsequent division in chain. With such chaining it is possible to divide arbitrarily large number by a small divisor.

The high word (EDX or RDX) must be smaller than the divisor, otherwise an overflow would occur (with DIV instruction it would be an exception 0). If we take V = bsr D, then:
N < 2^(B+V+1), because high word of N is smaller than the divisor
R < D < 2^(V+1)
N*R < 2^(B+2*V+2)
So we can take S = 2*V+2. Obviously M = 2^(B+S) / D + 1 is going to be larger than machine word, so the the multiplication is no longer going to be simple.

For D = 10, we get S = 8 and M = 2^40 / 10 + 1 = 199999999Ah. This led me to the following 32-bit routine that can replace "DIV 10" (without checking for overflow, input EDX must be smaller than 10):
 Code: div10: ; this trashes EBX and ECX, you may need to preserve them         push    eax edx         imul    edx,19h         mov     ecx,edx         mov     edx,09999999Ah         mov     ebx,eax         mul     edx         mov     eax,ebx         mov     ebx,edx         mov     edx,19h         mul     edx         add     ebx,eax         adc     ecx,edx         pop     eax         mov     edx,09999999Ah         mul     edx         add     eax,ebx         adc     ecx,edx         shrd    eax,ecx,8         pop     edx         imul    ecx,eax,10         sub     edx,ecx         retn
This is a long function with five multiplication instructions, but when I took a routine for converting very large integers to decimal and replaced the DIV instruction with a call to "div10", it still gave a noticeable improvement on my machine. If anything, this demonstrates how expensive DIV can sometimes be.
08 Sep 2017, 15:05
tthsqe

Joined: 20 May 2009
Posts: 701
Since this is a thread on division, I though I would collect my thoughts on multiprecision division. Throughout this post, let be the word size of the machine - so usually , but any integer will do. Given word-sized positive integers , the x86 architecture has an instruction div for calculating

An exception is raised if this quotient is , i.e. not word sized.

The goal of division is to calculate for multi-word integers and . When is word-sized, this can be calculated by stitching together div's, so the problem is to solve the case when is bigger than a word. In the grade school algorithm, the words of the quotient are produced from most significant to least significant. This means that the grade school algorithm is only dependent upon a solution to the following problem:

Given big integers and with and , calculate the word-sized integer .

Now, by shifting and , this problem can also be represented as calculating

where and are integers with and , and and are real numbers with .
Fact: With the above assumptions, it holds that
.
This means that the difficult (n+1)-by-n word division can be reduced to the simpler 3-by-2 division . If the latter does not match the former, is must be over by only one. In the grade school algorithm, this corresponds to overestimating a quotient term, and thus having to 'fix it up'. The point is that if the quotient term is estimate incorrectly, it is overestimated and only overestimated by one.
Proof: Write where and . Then,

It is easy to see that

and some manipulation of the assumption shows that

This last quantity is greater than by the assumptions on .

So now the grade school division has been reduced to the division problem

with the assumptions that the and are word-sized and that .
This will be covered in a future post.
09 Sep 2017, 09:48
tthsqe

Joined: 20 May 2009
Posts: 701
In the previous post, large integer division was reduced to a sequence of 3-by-2 divisions followed by n-by-1 multiply-subtract operations, in true grade school fashion. The quotient of the 3-by-2 division was between and inclusive. Since the quotient terms for the grade school division are by definition word-sized, we need to calculate

where it is assumed that this floor is and of course all of the previous assumptions, including .

If you follow the proof given in the previous post for this problem you will see that

However, care is required as the division can overflow a word.
I propose that the following algorithm computes .
 Code: if B2 < A1     (q, r) = div(B2:B1, A1)     (lo, hi) = mul(q, A0)     T0 = sub(B0, lo)     T1 = sbb(r, hi)     T2 = sbb(0, 0)  ; copy carry flag across T2     if T2 <> 0         q = dec(q)         T0 = add(T0, A0)         T1 = adc(T1, A1)         T2 = adc(T2, 0)         if T2 <> 0             q = dec(q)             T0 = add(T0, A0)    ; only calculated             T1 = adc(T1, A1)    ; for the asserts             T2 = adc(T2, 0)     ; below         end if     end if     assert( T2 == 0)                    ; T2:T1:T0 is supposed     assert( T1 * X + T0 < A2 * X + A1)  ; to contain the remainder     return q else     q = X - 1     T0 = B0     T1 = sub(B1, A0)     T2 = sbb(B2, A1)     T3 = sbb(0, 0)     T0 = add(T0, A0)     T1 = adc(T1, A1)     T2 = adc(T2, 0)     T3 = adc(T3, 0)     if T3 <> 0         q = dec(q)     end if     return q end if

I do not know at this time why at most one fixup is required in the "else" block. Of course two fixups are required in general by the above inequality.

Also, notice that this algorithm contains a call to div with a fixed divisor for each quotient word produced in the whole grade school algorithm. Thomasz would probably want to change this invariant division into some multiplications.
09 Sep 2017, 11:07
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 15241
Location: 1I/ʻOumuamua
When I was doing this stuff a few years ago I found that Barrett and Montgomery were the most used two methods for division and modulo for multi-precision operations. There are also whole books dedicated to this subject, and many PhD theses that attempt to improve the efficiency and performance in specific CPUs.
09 Sep 2017, 11:59
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
 tthsqe wrote: Since this is a thread on division, I though I would collect my thoughts on multiprecision division.
I wanted to continue my thoughts on "magic multipliers" but I see that you have moved the topic in a slightly different direction already. Perhaps we should split the thread?

Also, what I wanted to focus on here is the analysis of these tricks purely on the grounds of integer arithmetic, with no event slightest mention of real (or even p-adic) numbers. This is my personal hobby.
09 Sep 2017, 13:28
tthsqe

Joined: 20 May 2009
Posts: 701
I don't think it needs to be split. I am just giving you more to think about.
What you have been doing is the division of singe words (like the c "/" operator). You can interpret what I am saying as request for a division of two words by one word (like the x86 "div" instruction) by using multiplications with precomputed numbers.
09 Sep 2017, 14:50
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
Back to the beginning: the idea behind the division by multiplication can be presented as follows.

If we want to divide integer N by divisor D, what we can try is to find number such that and:
,
.
The same product can also be written as:

and because :
,
so:
,
,
.
Therefore and thus which gives us correct division with remainder, since . We get the result Q simply by taking the high bits of X. We can also get the remainder R by taking high bits of LD.

Now, to find the number X we can multiply N by so-called magic multiplier M, which is a number such that:
,

and then:

which gives the correct result as long as .
All the algorithms presented above made this work by finding n large enough that this inequality would be fulfilled for all N in the required range.

But there is still another approach we could try. When we compute M, we also know P, and we can use this knowledge to try to adjust the result even if NP ends up being large enough to meddle with the high bits of NMD.

We can compute NP and split into high and low part:

and then try to adjust for the error by taking:
.

Now, keeping in mind that:

we also need to ensure that:
.
Then the worst that may happen is that the high bits of N'MD might contain N - 1 instead of N. This can be easily fixed after computing the remainder - if remainder ends up being larger than D we can subtract D from it and increase the quotient by one.

Let's try it on an example, with D = 10, n = 32. Then M=1999999Ah and P = 4:

To simulate a DIV instruction we need to operate on range:

With this in mind we have:

so

and finally:

This is far from being dangerous for the high bits of N'MD.

It is time for an implementation:
 Code: div10: ;        cmp     edx,10 ;        jae     int0         push    ebx ecx         push    eax         mov     ebx,eax         mov     ecx,edx         shld    edx,eax,2         sub     ebx,edx         sbb     ecx,0         mov     eax,ebx         mov     ebx,1999999Ah         mul     ebx         mov     eax,ecx         imul    eax,ebx         add     eax,edx         pop     edx         imul    ecx,eax,10         sub     edx,ecx         cmp     edx,10         jb      .done         sub     edx,10         inc     eax     .done:         pop     ecx ebx         retn
This ends up being faster than the previous one, it only has three multiplications (and just one of them is the full-sized MUL) thanks to the fact that P=4 and multiplication by P can be done by shifting.
For 64-bit version just replace all 32-bit registers with their 64-bit counterparts and the constant with 199999999999999Ah (P is the same).
09 Sep 2017, 14:53
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6633
Location: Kraków, Poland
 tthsqe wrote: You can interpret what I am saying as request for a division of two words by one word (like the x86 "div" instruction) by using multiplications with precomputed numbers.
What I wrote above is about such division, though perhaps not general enough for your needs. It only works for divisors D such that:
.
But when dividing large integer by another large integer, you usually need to divide by a normalized divisor, such that:
.
I have not yet tried to solve this variant with my integer-based approach.

When a divisor is normalized, the algorithm I presented first has the shift equal to the size of machine word (the assembly implementation I provided does not work then, because SHRD sees such shift as zero), so the result can be taken straight from the third word after multiplication. Also, multiplying the lowest word seems pointless, because it can affect the result by no more than one, and this can be corrected when computing remainder, as in my previous post. However to extend the range of algorithm, some kind of compensation for the NP term would be needed. I don't know if I can solve this variant with integer arithmetic as nicely as the previous ones.
09 Sep 2017, 14:58
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