Algorithm D, its Implementation in
Hacker’s Delight, and elsewhere
RunTime Library
Algorithm D (Division of nonnegative integers):
Note: I added an introductory step D0 providing definitions and descriptions.
Algorithm D (Division of nonnegative integers).
 D0. [Define]
 Let U be the dividend (also numerator) of m+n
digits, stored in an array of m+n+1 elements, onedigitper element, with the most significantdigitin element U[m+n−1] and the least significantdigitin element U[0]; Let V be the divisor (also denominator) of n
digits, stored in a second array of n elements, with n greater than 1, a nonzero most significantdigitin element V[n−1] and the least significantdigitin element V[0]; Let B be the base (also radix) of the
digits(alsolimbs,placesorwords), typically (a power of) a power of 2, with B greater than 1. (The algorithm computes the quotient Q of m+1
digitsas ⌊U ⁄ V⌋ (also U div V or U ÷ V), and the remainder R of ndigitsas U modulo V (also U mod V or U % V), using the followingprimitiveoperations:
¹ addition or subtraction of two singledigit integers, giving a singledigit sum and a carry, or a singledigit difference and a borrow;
²wideningmultiplication of two singledigit integers, giving a doubledigit product;
³narrowingdivision of a doubledigit integer by a singledigit integer, giving a singledigit quotient and a singledigit remainder.) D1. [Normalize]
 Set D to (B − 1) ÷ V[n−1];
 Multiply all
digitsof U and V by D. (On a binary computer, choose D to be a power of 2 instead of the value provided above; any value of D that results in V[n−1] not less than B ÷ 2 will suffice. If D is greater than 1, the eventual overflow
digitof the dividend U goes into element U[m+n].) D2. [Initialize j]
 Set the loop counter j to m.
 D3. [Calculate Q̂]
 Set Q̂ to (U[n+j] × B + U[n−1+j]) ÷ V[n−1];
 Set R̂ to (U[n+j] × B + U[n−1+j]) % V[n−1];
 Test if Q̂ equals B or Q̂ × V[n−2] is greater than R̂ × B + U[n−2+j];
 If yes, then decrease Q̂ by 1, increase R̂ by V[n−1], and repeat this test while R is less than B.
 D4. [Multiply and subtract]
 Replace (U[n+j]U[n−1+j]…U[j]) by (U[n+j]U[n−1+j]…U[j]) − Q̂ × (V[n−1]…V[1]V[0]).
 (The
digits(U[n+j]…U[j]) should be kept positive; if the result of this step is actually negative, (U[n+j]…U[j]) should be left as the true value plus B^{n+1}, namely as the B’s complement of the true value, and a borrow to the left should be remembered.) D5. [Test remainder]
 Set Q[j] to Q̂;
 If the result of step D4 was negative, i.e. the subtraction needed a borrow, then proceed with step D6; otherwise proceed with step D7.
 D6. [Add back]
 Decrease Q[j] by 1 and add (0V[n−1]…V[1]V[0]) to (U[n+j]U[n−1+j]…U[1+j]U[j]).
 (A carry will occur to the left of U[n+j], and it should be ignored since it cancels with the borrow that occurred in step D4.)
 D7. [Loop on j]
 Decrease j by 1;
 Test if j is not less than 0;
 If yes, go back to step D3.
 D8. [Unnormalize]
 Now (Q[m]…Q[1]Q[0]) is the desired quotient Q, and the desired remainder R may be obtained by dividing (U[n−1]…U[1]U[0]) by D.
Algorithm Din ANSI C.
Note: I removed some parts of the code which are not relevant here.
/* This divides an nword dividend by an mword divisor, giving an
nm+1word quotient and mword remainder. The bignums are in arrays of
words. Here a "word" is 32 bits. This routine is designed for a 64bit
machine which has a 64/64 division instruction. */ 1
…
/* q[0], r[0], u[0], and v[0] contain the LEAST significant words.
(The sequence is in littleendian order).
This is a fairly precise implementation of Knuth's Algorithm D, for a
binary computer with base b = 2**32. The caller supplies:
1. Space q for the quotient, m  n + 1 words (at least one).
2. Space r for the remainder (optional), n words.
3. The dividend u, m words, m >= 1.
4. The divisor v, n words, n >= 2.
The most significant digit of the divisor, v[n1], must be nonzero. The
dividend u may have leading zeros; this just makes the algorithm take
longer and makes the quotient contain more leading zeros. A value of
NULL may be given for the address of the remainder to signify that the
caller does not want the remainder.
The program does not alter the input parameters u and v.
The quotient and remainder returned may have leading zeros. The
function itself returns a value of 0 for success and 1 for invalid
parameters (e.g., division by 0).
For now, we must have m >= n. Knuth's Algorithm D also requires
that the dividend be at least as long as the divisor. (In his terms,
m >= 0 (unstated). Therefore m+n >= n.) */
int divmnu(unsigned q[], unsigned r[],
const unsigned u[], const unsigned v[],
int m, int n) {
const unsigned long long b = 4294967296LL; // Number base (2**32).
unsigned *un, *vn; // Normalized form of u, v.
unsigned long long qhat; // Estimated quotient digit.
unsigned long long rhat; // A remainder.
unsigned long long p; // Product of two digits.
long long t, k;
int s, i, j;
if (m < n  n <= 1  v[n1] == 0)
return 1; // Return if invalid param.
…
/* Normalize by shifting v left just enough so that its highorder
bit is on, and shift u left the same amount. We may have to append a
highorder digit on the dividend; we do that unconditionally. */
s = nlz(v[n1]); // 0 <= s <= 31.
vn = (unsigned *)alloca(4*n);
for (i = n  1; i > 0; i)
vn[i] = (v[i] << s)  ((unsigned long long)v[i1] >> (32s));
vn[0] = v[0] << s;
un = (unsigned *)alloca(4*(m + 1));
un[m] = (unsigned long long)u[m1] >> (32s);
for (i = m  1; i > 0; i)
un[i] = (u[i] << s)  ((unsigned long long)u[i1] >> (32s));
un[0] = u[0] << s;
for (j = m  n; j >= 0; j) { // Main loop.
// Compute estimate qhat of q[j].
qhat = (un[j+n]*b + un[j+n1])/vn[n1];
#ifdef OPTIMIZE
rhat = (un[j+n]*b + un[j+n1])%vn[n1];
#else // ORIGINAL
2 rhat = (un[j+n]*b + un[j+n1])  qhat*vn[n1];
#endif
again:
4 if (qhat >= b 
#ifdef OPTIMIZE
(unsigned)qhat*(unsigned long long)vn[n2] > b*rhat + un[j+n2]) {
#else // ORIGINAL
3 qhat*vn[n2] > b*rhat + un[j+n2]) {
#endif
qhat = qhat  1;
rhat = rhat + vn[n1];
if (rhat < b) goto again;
}
// Multiply and subtract.
k = 0;
for (i = 0; i < n; i++) {
#ifdef OPTIMIZE
p = (unsigned)qhat*(unsigned long long)vn[i];
#else // ORIGINAL
3 p = qhat*vn[i];
#endif
t = un[i+j]  k  (p & 0xFFFFFFFFLL);
un[i+j] = t;
k = (p >> 32)  (t >> 32);
}
t = un[j+n]  k;
un[j+n] = t;
q[j] = qhat; // Store quotient digit.
if (t < 0) { // If we subtracted too
q[j] = q[j]  1; // much, add back.
k = 0;
for (i = 0; i < n; i++) {
t = (unsigned long long)un[i+j] + vn[i] + k;
un[i+j] = t;
k = t >> 32;
}
un[j+n] = un[j+n] + k;
}
} // End j.
// If the caller wants the remainder, unnormalize
// it and pass it back.
if (r != NULL) {
for (i = 0; i < n1; i++)
r[i] = (un[i] >> s)  ((unsigned long long)un[i+1] << (32s));
r[n1] = un[n1] >> s;
}
return 0;
}
…
This implementation exhibits the following misrepresentation and
shortcomings:
64/64 division instructionis explicitly stated, the code doesn’t take full advantage of it: instead to use the
%
operator to for freeon 64bit machines, and almost for free with software implementations on 32bit machines), it but performs an extraneous 64×64bit multiplication — which is not free, especially on 32bit machines!
qhat*vn[n2]
and qhat*vn[i]
: while an
optimising compiler should detect that these are
32×32bit multiplications returning a 64bit product, I would
not count on it — better give the compiler the proper hints!
narrowingdivision overflows!
Algorithm Dbuilds upon a 64÷32bit division returning a 32bit quotient and a 32bit remainder, also known as
narrowingdivision, and a 32×32bit multiplication returning a 64bit product, also known as
wideningmultiplication, which both are but not supported in ANSI C!
Note: I removed some parts of the code which are not relevant here.
/* Long division, unsigned (64/32 ==> 32).
This procedure performs unsigned "long division" i.e., division of a
64bit unsigned dividend by a 32bit unsigned divisor, producing a
32bit quotient. In the overflow cases (divide by 0, or quotient
exceeds 32 bits), it returns a remainder of 0xFFFFFFFF (an impossible
value).
The dividend is u1 and u0, with u1 being the most significant word.
The divisor is parameter v. The value returned is the quotient.
[…] Several of the variables below could be
"short," but having them fullwords gives better code on gcc/Intel.
[…]
This is the version that's in the hacker book. */
unsigned divlu2(unsigned u1, unsigned u0, unsigned v,
unsigned *r) {
const unsigned b = 65536; // Number base (16 bits).
unsigned un1, un0, // Norm. dividend LSD's.
vn1, vn0, // Norm. divisor digits.
q1, q0, // Quotient digits.
un32, un21, un10,// Dividend digit pairs.
rhat; // A remainder.
int s; // Shift amount for norm.
if (u1 >= v) { // If overflow, set rem.
if (r != NULL) // to an impossible value,
*r = 0xFFFFFFFF; // and return the largest
return 0xFFFFFFFF;} // possible quotient.
s = nlz(v); // 0 <= s <= 31.
v = v << s; // Normalize divisor.
vn1 = v >> 16; // Break divisor up into
vn0 = v & 0xFFFF; // two 16bit digits.
un32 = (u1 << s)  (u0 >> 32  s) & (s >> 31);
un10 = u0 << s; // Shift dividend left.
un1 = un10 >> 16; // Break right half of
un0 = un10 & 0xFFFF; // dividend into two digits.
q1 = un32/vn1; // Compute the first
#ifdef OPTIMIZE
rhat = un32%vn1; // quotient digit, q1.
#else // ORIGINAL
rhat = un32  q1*vn1; // quotient digit, q1.
#endif
again1:
if (q1 >= b  q1*vn0 > b*rhat + un1) {
q1 = q1  1;
rhat = rhat + vn1;
if (rhat < b) goto again1;}
un21 = un32*b + un1  q1*v; // Multiply and subtract.
q0 = un21/vn1; // Compute the second
#ifdef OPTIMIZE
rhat = un21%vn1; // quotient digit, q0.
#else // ORIGINAL
rhat = un21  q0*vn1; // quotient digit, q0.
#endif
again2:
if (q0 >= b  q0*vn0 > b*rhat + un0) {
q0 = q0  1;
rhat = rhat + vn1;
if (rhat < b) goto again2;}
if (r != NULL) // If remainder is wanted,
*r = (un21*b + un0  q0*v) >> s; // return it.
return q1*b + q0;
}
…
Especially notice the highlighted part of the initial comment!
Warning: don’t use this function; it
demonstrates that its author copyist does not
understand the original code and fails to read or ignores the
comment, it exhibits poor performance, and it has a serious bug!
// libdivide.h
// Copyright 2010  2018 ridiculous_fish
…
// Code taken from Hacker's Delight:
// http://www.hackersdelight.org/HDcode/divlu.c.
// License permits inclusion here per:
// http://www.hackersdelight.org/permissions.htm
static uint64_t libdivide_128_div_64_to_64(uint64_t u1, uint64_t u0, uint64_t v, uint64_t *r) {
1 const uint64_t b = (1ULL << 32); // Number base (16 bits)
2 uint64_t un1, un0; // Norm. dividend LSD's
2 uint64_t vn1, vn0; // Norm. divisor digits
2 uint64_t q1, q0; // Quotient digits
3 uint64_t un64, un21, un10; // Dividend digit pairs
uint64_t rhat; // A remainder
#ifdef OPTIMIZE
uint64_t qhat; // A quotient
uint64_t p; // A product
#endif
int32_t s; // Shift amount for norm
// If overflow, set rem. to an impossible value,
// and return the largest possible quotient
if (u1 >= v) {
if (r != NULL)
*r = (uint64_t) 1;
return (uint64_t) 1;
}
// count leading zeros
s = libdivide__count_leading_zeros64(v);
if (s > 0) {
// Normalize divisor
v = v << s;
5 un64 = (u1 << s)  ((u0 >> (64  s)) & (s >> 31));
4 un10 = u0 << s; // Shift dividend left
} else {
// Avoid undefined behavior
6 un64 = u1  u0;
4 un10 = u0;
}
// Break divisor up into two 32bit digits
7 vn1 = v >> 32;
7 vn0 = v & 0xFFFFFFFF;
// Break right half of dividend into two digits
7 un1 = un10 >> 32;
7 un0 = un10 & 0xFFFFFFFF;
// Compute the first quotient digit, q1
#ifdef OPTIMIZE
qhat = un64 / vn1;
rhat = un64 % vn1;
p = qhat * vn0;
while (qhat >= b  p > b * rhat + un1) {
qhat = qhat  1;
rhat = rhat + vn1;
if (rhat >= b)
break;
p = p  vn0;
}
q1 = qhat & 0xFFFFFFFF;
#else // ORIGINAL
q1 = un64 / vn1;
8 rhat = un64  q1 * vn1;
9 while (q1 >= b  q1 * vn0 > b * rhat + un1) {
q1 = q1  1;
rhat = rhat + vn1;
if (rhat >= b)
break;
}
#endif
// Multiply and subtract
un21 = un64 * b + un1  q1 * v;
// Compute the second quotient digit
#ifdef OPTIMIZE
qhat = un21 / vn1;
rhat = un21 % vn1;
p = qhat * vn0;
while (qhat >= b  p > b * rhat + un0) {
qhat = qhat  1;
rhat = rhat + vn1;
if (rhat >= b)
break;
p = p  vn0;
}
q0 = qhat & 0xFFFFFFFF;
#else // ORIGINAL
q0 = un21 / vn1;
8 rhat = un21  q0 * vn1;
9 while (q0 >= b  q0 * vn0 > b * rhat + un0) {
q0 = q0  1;
rhat = rhat + vn1;
if (rhat >= b)
break;
}
#endif
// If remainder is wanted, return it
if (r != NULL)
*r = (un21 * b + un0  q0 * v) >> s;
return q1 * b + q0;
}
…
This implementation has an obvious bug and multiple deficiencies (in
order of appearance):
normalised(least significant) digits of dividend and divisor as well as the final digits of the quotient fit in 32 bits, i.e. the variables
un1
, un0
,
vn1
, vn0
, q1
and
q0
can and should of course be declared as
uint32_t
, not as uint64_t
.
un32
, not un64
.
un6432
and
un10
are superfluous, the dividend u1
and
u0
can and should of course be normalised in place,
like the divisor v
.
& (s >> 31)
is superfluous: in the
original code it is used to avoid the conditional expression
if (s > 0) … else …
and evaluates to
either & 0
or & 1
(alias
& 0xFFFFFFFF
or & ~0
) there; here
it always evaluates to & 1
(alias
& 0xFFFFFFFFFFFFFFFF
or & ~0LL
).
 u0
produces wrong results and
must be removed.
un1
, un0
, vn1
and vn0
are superfluous, they can and of course should
be replaced by the expressions
(uint32_t) (un10 >> 32)
,
(uint32_t) (un10 & 0xFFFFFFFF)
,
(uint32_t) (v >> 32)
and
(uint32_t) (v & 0xFFFFFFFF)
respectively.
rhat
can and should of course be computed
as un6432 % vn1
and
un21 % vn1
respectively, not as
un6432  q1 * vn1
and
un21  q0 * vn1
using expensive 64×64bit
multiplications: both hardware division instructions and software
division routines typically return quotient and remainder together.
q1 * vn0
and q0 * vn0
inside the
while …
loops should be moved outside the loops.
RunTime Library
RunTime Library…
…
(*
 u128_div_u64_to_u64 [local]

 Divides unsigned 128bit integer by unsigned 64bit integer.
 Returns 64bit quotient and reminder.

 This routine is used here only for splitting specially prepared unsigned
 128bit integer into two 64bit ones before converting it to ASCII.

**)
function u128_div_u64_to_u64( const xh, xl: qword; const y: qword; out quotient, reminder: qword ): boolean;
var
b, // Number base
1 v, // Norm. divisor
2 un1, un0, // Norm. dividend LSD's
2 vn1, vn0, // Norm. divisor digits
q1, q0, // Quotient digits
3 un64, un21, un10, // Dividend digit pairs
rhat: qword; // A remainder
s: integer; // Shift amount for norm
begin
// Overflow check
if ( xh >= y ) then
begin
u128_div_u64_to_u64 := false;
exit;
end;
// Count leading zeros
s := 63  BSRqword( y ); // 0 <= s <= 63
// Normalize divisor
1 v := y shl s;
// Break divisor up into two 32bit digits
4 vn1 := hi(v);
4 vn0 := lo(v);
// Shift dividend left
1 un64 := xh shl s;
if ( s > 0 ) then
1 un64 := un64 or ( xl shr ( 64  s ) );
1 un10 := xl shl s;
// Break right half of dividend into two digits
4 un1 := hi(un10);
4 un0 := lo(un10);
// Compute the first quotient digit, q1
3 q1 := un64 div vn1;
5 rhat := un64  q1 * vn1;
b := qword(1) shl 32; // Number base
6 while ( q1 >= b )
7 or ( q1 * vn0 > b * rhat + un1 ) do
begin
dec( q1 );
inc( rhat, vn1 );
6 if rhat >= b then
break;
end;
// Multiply and subtract
3 un21 := un64 * b + un1  q1 * v;
// Compute the second quotient digit, q0
q0 := un21 div vn1;
5 rhat := un21  q0 * vn1;
6 while ( q0 >= b )
7 or ( q0 * vn0 > b * rhat + un0 ) do
begin
dec( q0 );
inc( rhat, vn1 );
6 if ( rhat >= b ) then
break;
end;
// Result
8 reminder := ( un21 * b + un0  q0 * v ) shr s;
8 quotient := q1 * b + q0;
u128_div_u64_to_u64 := true;
end;
…
This implementation has multiple deficiencies, especially for
32bit machines; 64bit machines typically support
128÷64bit division in hardware and don’t need this
function:
v
,
un6432
and
un10
are superfluous, the divisor y
as
well as the dividend xh
and xl
can and
should of course be normalised in place.
normalised(least significant) digits of dividend and divisor fit in 32 bits, i.e. the variables
un1
,
un0
, vn1
and vn0
can and
should of course be declared as dword
, not as
qword
.
un32
, not un64
.
un1
, un0
, vn1
and vn0
are superfluous, they can and of course should
be replaced by the expressions hi(un10)
,
lo(un10)
, hi(v)
and low(v)
respectively.
rhat
can and should of course be computed
as un6432 mod vn1
and
un21 mod vn1
respectively, not as
un6432  q1 * vn1
and
un21  q0 * vn1
using expensive 64×64bit
multiplications: both hardware division instructions and software
division routines typically return quotient and remainder together.
q1 >= b
, q0 >= b
and rhat >= b
can and should of course be replaced
by hi(q1) <> 0
, hi(q0) <> 0
and hi(rhat) <> 0
respectively.
q1 * vn0
and q0 * vn0
inside the
while …
loops should be moved outside the loops.
q1
and q0
can and should of
course be replaced wherever possible by lo(q1)
and
lo(q0)
respectively to avoid expensive 64×64bit
multiplications.
Note: the highlighted parts show the
optimisations and simplifications missed by the author(s) of the
Div64
function!
// Copyright 2017 The Go Authors. All rights reserved.
…
// Div64 returns the quotient and remainder of (hi, lo) divided by y:
// quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
// half in parameter hi and the lower half in parameter lo.
// Div64 panics for y == 0 (division by zero) or y <= hi (quotient overflow).
func Div64(hi, lo, y uint64) (quo, rem uint64) {
const (
two32 = 1 << 32
mask32 = two32  1
)
if y == 0 {
panic(divideError)
}
if y <= hi {
panic(overflowError)
}
s := uint(LeadingZeros64(y))
y <<= s
1 yn1 := uint32(y >> 32)
1 yn0 := uint32(y & mask32)
un32 := hi<<s  lo>>(64s)
un10 := lo << s
1 un1 := uint32(un10 >> 32)
1 un0 := uint32(un10 & mask32)
q1 := un32 / yn1
2 rhat := un32  q1*yn1
3 for q1 >= two32  uint32(q1)*yn0 > two32*rhat+un1 {
q1
rhat += yn1
if rhat >= two32 {
break
}
}
3 un21 := un32*two32 + un1  uint32(q1)*y
q0 := un21 / yn1
2 rhat = un21  q0*yn1
3 for q0 >= two32  uint32(q0)*yn0 > two32*rhat+un0 {
q0
rhat += yn1
if rhat >= two32 {
break
}
}
3 return q1*two32 + q0, (un21*two32 + un0  uint32(q0)*y) >> s
}
…
This implementation has multiple deficiencies, especially for
32bit machines; 64bit machines typically support
128÷64bit division in hardware and don’t need this
function:
normalised(least significant) digits of dividend and divisor fit in 32 bits, i.e. the variables
yn1
,
yn0
, un1
and un0
can and
should of course be declared as uint32
, not as
uint64
.
rhat
can and should of course be computed
as un32 % yn1
and un21 % yn1
respectively,
not as un32  q1*yn1
and
un21  q0*yn1
using expensive 64×64bit
multiplications: both hardware division instructions and software
division routines typically return quotient and remainder together.
q1
and q0
can and
should of course be cast to uint32
wherever necessary
to avoid expensive 64×64bit multiplications.
Note: the type digit_t
is the same as
uintptr_t
.
// Copyright 2017 the V8 project authors. All rights reserved.
…
// Returns the quotient.
// quotient = (high << kDigitBits + low  remainder) / divisor
BigInt::digit_t MutableBigInt::digit_div(digit_t high, digit_t low,
digit_t divisor, digit_t* remainder) {
DCHECK(high < divisor);
#if V8_TARGET_ARCH_X64 && (__GNUC__  __clang__)
…
#elif V8_TARGET_ARCH_IA32 && (__GNUC__  __clang__)
…
#else
static const digit_t kHalfDigitBase = 1ull << kHalfDigitBits;
// Adapted from Warren, Hacker's Delight, p. 152.
int s = base::bits::CountLeadingZeros(divisor);
DCHECK_NE(s, kDigitBits); // {divisor} is not 0.
divisor <<= s;
digit_t vn1 = divisor >> kHalfDigitBits;
digit_t vn0 = divisor & kHalfDigitMask;
// {s} can be 0. {low >> kDigitBits} would be undefined behavior, so
// we mask the shift amount with {kShiftMask}, and the result with
// {s_zero_mask} which is 0 if s == 0 and all 1bits otherwise.
STATIC_ASSERT(sizeof(intptr_t) == sizeof(digit_t));
const int kShiftMask = kDigitBits  1;
digit_t s_zero_mask =
static_cast<digit_t>(static_cast<intptr_t>(s) >> (kDigitBits  1));
digit_t un32 =
(high << s)  ((low >> ((kDigitBits  s) & kShiftMask)) & s_zero_mask);
digit_t un10 = low << s;
digit_t un1 = un10 >> kHalfDigitBits;
digit_t un0 = un10 & kHalfDigitMask;
digit_t q1 = un32 / vn1;
digit_t rhat = un32  q1 * vn1;
while (q1 >= kHalfDigitBase  q1 * vn0 > rhat * kHalfDigitBase + un1) {
q1;
rhat += vn1;
if (rhat >= kHalfDigitBase) break;
}
digit_t un21 = un32 * kHalfDigitBase + un1  q1 * divisor;
digit_t q0 = un21 / vn1;
rhat = un21  q0 * vn1;
while (q0 >= kHalfDigitBase  q0 * vn0 > rhat * kHalfDigitBase + un0) {
q0;
rhat += vn1;
if (rhat >= kHalfDigitBase) break;
}
*remainder = (un21 * kHalfDigitBase + un0  q0 * divisor) >> s;
return q1 * kHalfDigitBase + q0;
#endif
}
…
Algorithm Dfor unsigned 128÷64bit division on 32bit machines is based on a 64÷32bit division returning a 64bit quotient and a 32bit remainder, trivially implemented per
long(alias
schoolbook) division using a
narrowing64÷32bit division returning a 32bit quotient and a 32bit remainder.
// Copyleft © 20112023, Stefan Kanthak <stefan.kanthak@nexgo.de>
// Divide a 128bit integer dividend, supplied as a pair of 64bit
// integers in u0 and u1, by a 64bit integer divisor, supplied in v;
// return the 64bit quotient and optionally the 64bit remainder in *r.
unsigned long long divllu(unsigned long long u0,
unsigned long long u1,
unsigned long long v,
unsigned long long *r) {
unsigned long long qhat; // A quotient.
unsigned long long rhat; // A remainder.
unsigned long long uhat; // A dividend digit pair.
unsigned q0, q1; // Quotient digits.
unsigned s; // Shift amount for norm.
if (u1 >= v) { // If overflow, set rem.
if (r != NULL) // to an impossible value,
*r = ~0ULL; // and return the largest
return ~0ULL; // possible quotient.
}
s = __builtin_clzll(v); // 0 <= s <= 63.
if (s != 0U) {
v <<= s; // Normalize divisor.
u1 <<= s; // Shift dividend left.
u1 = u0 >> (64U  s);
u0 <<= s;
}
// Compute high quotient digit.
qhat = u1 / (unsigned) (v >> 32);
rhat = u1 % (unsigned) (v >> 32);
while ((unsigned) (qhat >> 32) != 0U 
// Both qhat and rhat are less 2**32 here!
(unsigned long long) (unsigned) (qhat & ~0U) * (unsigned) (v & ~0U) >
((rhat << 32)  (unsigned) (u0 >> 32))) {
qhat = 1U;
rhat += (unsigned) (v >> 32);
if ((unsigned) (rhat >> 32) != 0U) break;
}
q1 = (unsigned) (qhat & ~0U);
// Multiply and subtract.
uhat = ((u1 << 32)  (unsigned) (u0 >> 32))  q1 * v;
// Compute low quotient digit.
qhat = uhat / (unsigned) (v >> 32);
rhat = uhat % (unsigned) (v >> 32);
while ((unsigned) (qhat >> 32) != 0U 
// Both qhat and rhat are less 2**32 here!
(unsigned long long) (unsigned) (qhat & ~0U) * (unsigned) (v & ~0U) >
((rhat << 32)  (unsigned) (u0 & ~0U))) {
qhat = 1U;
rhat += (unsigned) (v >> 32);
if ((unsigned) (rhat >> 32) != 0U) break;
}
q0 = (unsigned) (qhat & ~0U);
if (r != NULL) // If remainder is wanted, return it.
*r = (((uhat << 32)  (unsigned) (u0 & ~0U))  q0 * v) >> s;
return ((unsigned long long) q1 << 32)  q0;
}
long(alias
schoolbook) division for divisors less than 2^{32} and dividends less than 2^{96}.
; Copyright © 20112023, Stefan Kanthak <stefan.kanthak@nexgo.de>
.386
.model flat, C
.code
divllu proc public
push ebx
push ebp
push edi
push esi
…
mov esi, [esp+40]
mov edi, [esp+36] ; esi:edi = divisor
bsr ecx, esi ; ecx = index of most significant '1' bit in high dword of divisor
jnz NORMALIZE ; high dword of divisor (and high dword of dividend) = 0?
; "long" division (dividend < 2**96, divisor < 2**32)
DIVISION:
mov edx, [esp+28]
mov eax, [esp+24] ; edx:eax = "inner" qword of dividend
div edi ; eax = high dword of quotient,
; edx = high dword of dividend'
mov ebx, eax ; ebx = high dword of quotient
mov eax, [esp+20] ; edx:eax = low qword of dividend'
div edi ; eax = low dword of quotient,
; edx = (low dword) of remainder
or esi, [esp+44] ; esi = address of remainder
jz @F ; address of remainder = 0?
xor ecx, ecx ; ecx:edx = remainder
mov [esi], edx
mov [esi+4], ecx ; store remainder
@@:
mov edx, ebx ; edx:eax = quotient
pop esi
pop edi
pop ebp
pop ebx
ret
NORMALIZE:
mov edx, [esp+32]
mov eax, [esp+28] ; edx:eax = high qword of dividend
xor ecx, 31 ; ecx = number of leading '0' bits in (high dword of) divisor
jz @F ; number of leading '0' bits = 0?
shld esi, edi, cl
shl edi, cl ; esi:edi = divisor'
mov [esp+40], esi
mov [esp+36], edi ; save normalised divisor'
mov ebx, [esp+24]
mov ebp, [esp+20] ; ebx:ebp = low qword of dividend
shld edx, eax, cl
shld eax, ebx, cl
shld ebx, ebp, cl
shl ebp, cl ; edx:eax:ebx:ebp = dividend'
mov [esp+32], edx
mov [esp+28], eax
mov [esp+24], ebx
mov [esp+20], ebp ; save normalised dividend'
@@:
push ecx ; save number of leading '0' bits in divisor
DIVISION_1:
cmp edx, esi
jnb OVERFLOW_1 ; quotient overflow with normal division?
; "normal" division
NORMAL_1:
div esi ; eax = (low dword of) quotient',
; edx = (low dword of) remainder'
mov ebp, edx ; ebp = (low dword of) remainder'
mov ecx, eax
xor ebx, ebx ; ebx:ecx = quotient'
jmp CHECK_1
if 0 ; "long" division
OVERFLOW_1:
mov ecx, eax
mov eax, edx
xor edx, edx
div esi ; eax = high dword of quotient',
; edx = high dword of remainder'
mov ebx, eax ; ebx = high dword of quotient'
mov eax, ecx
div esi ; eax = low dword of quotient',
; edx = (low dword of) remainder'
mov ebp, edx ; ebp = (low dword of) remainder'
mov ecx, eax ; ebx:ecx = quotient'
else
OVERFLOW_1:
sub edx, esi ; edx:eax = high qword of dividend'
;  high dword of divisor'
div esi ; eax = (low dword of) quotient',
; edx = (low dword of) remainder'
mov ebp, edx ; ebp = (low dword of) remainder'
mov ecx, eax
mov ebx, 1 ; ebx:ecx = quotient'
endif
ADJUST_1:
add ecx, 1
adc ebx, 1 ; ebx:ecx = quotient'  1
add ebp, esi ; ebp = (low dword of) remainder'
; + high dword of divisor'
jc BREAK_1 ; remainder' >= 2**32?
AGAIN_1:
test ebx, ebx
jnz ADJUST_1 ; quotient' >= 2**32?
CHECK_1:
mov eax, edi ; eax = low dword of divisor'
mul ecx ; edx:eax = low dword of divisor'
; * low dword of quotient'
ifndef JccLess
cmp edx, ebp
jb BREAK_1
ja ADJUST_1
cmp eax, [esp+28]
ja ADJUST_1
else
cmp [esp+28], eax
mov eax, ebp
sbb eax, edx
jb ADJUST_1
endif
BREAK_1:
push ecx ; save (low dword of) quotient'
mov eax, edi ; eax = low dword of divisor'
mul ecx ; edx:eax = low dword of divisor'
; * (low dword of) quotient'
imul ecx, esi ; ecx = (low dword of) quotient'
; * high dword of divisor'
add ecx, edx ; ecx:eax = divisor'
; * (low dword of) quotient'
mov ebx, eax ; ecx:ebx = divisor'
; * (low dword of) quotient'
mov eax, [esp+32]
mov edx, [esp+36] ; edx:eax = "inner" qword of dividend'
sub eax, ebx
sbb edx, ecx ; edx:eax = "inner" qword of dividend"
; = intermediate (normalised) remainder
push eax ; save low dword of "inner" qword of dividend"
DIVISION_0:
cmp edx, esi
jnb OVERFLOW_0 ; quotient overflow with normal division?
; "normal" division
NORMAL_0:
div esi ; eax = (low dword of) quotient",
; edx = (low dword of) remainder"
mov ebp, edx ; ebp = (low dword of) remainder"
mov ecx, eax
xor ebx, ebx ; ebx:ecx = quotient"
jmp CHECK_0
if 0 ; "long" division
OVERFLOW_0:
mov ecx, eax
mov eax, edx
xor edx, edx
div esi ; eax = high dword of quotient",
; edx = high dword of remainder"
mov ebx, eax ; ebx = high dword of quotient"
mov eax, ecx
div esi ; eax = low dword of quotient",
; edx = (low dword of) remainder"
mov ebp, edx ; ebp = (low dword of) remainder"
mov ecx, eax ; ebx:ecx = quotient"
else
OVERFLOW_0:
sub edx, esi ; edx:eax = "inner" qword of dividend"
;  high dword of divisor'
div esi ; eax = (low dword of) quotient",
; edx = (low dword of) remainder"
mov ebp, edx ; ebp = (low dword of) remainder"
mov ecx, eax
mov ebx, 1 ; ebx:ecx = quotient"
endif
ADJUST_0:
add ecx, 1
adc ebx, 1 ; ebx:ecx = quotient"  1
add ebp, esi ; ebp = (low dword of) remainder"
; + high dword of divisor'
jc BREAK_0 ; remainder" >= 2**32?
AGAIN_0:
test ebx, ebx
jnz ADJUST_0 ; quotient" >= 2**32?
CHECK_0:
mov eax, edi ; eax = low dword of divisor'
mul ecx ; edx:eax = low dword of divisor'
; * low dword of quotient"
ifndef JccLess
cmp edx, ebp
jb BREAK_0
ja ADJUST_0
cmp eax, [esp+32]
ja ADJUST_0
else
cmp [esp+32], eax
mov eax, ebp
sbb eax, edx
jb ADJUST_0
endif
BREAK_0:
pop ebx ; ebx = low dword of "inner" qword of dividend"
push ecx ; save (low dword of) quotient"
mov ebp, [esp+56] ; ebp = address of remainder
test ebp, ebp
jz QUOTIENT ; address of remainder = 0?
REMAINDER:
mov eax, edi ; eax = low dword of divisor'
mul ecx ; edx:eax = low dword of divisor'
; * (low dword of) quotient"
imul ecx, esi ; ecx = (low dword of) quotient"
; * high dword of divisor'
add edx, ecx ; edx:eax = (low dword of) quotient"
; * divisor'
mov edi, [esp+32]
mov esi, ebx
sub edi, eax
sbb esi, edx ; esi:edi = (normalised) remainder
mov ecx, [esp+8] ; ecx = number of leading '0' bits in divisor
; = shift count
if 0
jecxz @F ; shift count = 0?
else
test ecx, ecx
jz @F ; shift count = 0?
endif
shrd edi, esi, cl
shr esi, cl ; esi:edi = remainder
@@:
mov [ebp], edi
mov [ebp+4], esi ; store remainder
QUOTIENT:
pop eax
pop edx ; edx:eax = quotient
pop ecx
pop esi
pop edi
pop ebp
pop ebx
ret
divllu endp
end
Note: this code runs on 35 year old
Intel^{®} 80386 microprocessors, and of
course on current compatible processors too; it is more than twice
as fast as the
ANSI C
version, and less than 1.5× slower than the native
128÷64bit division on 64bit processors (about 100 to 120
versus 80 to 95 processor clock cycles).
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