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math/big: implement recursive algorithm for division

The current division algorithm produces one word of result at a time,
using 2-word division to compute the top word and mulAddVWW to compute
the remainder. The top word may need to be adjusted by 1 or 2 units.

The recursive version, based on Burnikel, Ziegler, "Fast Recursive Division",
uses the same principles, but in a multi-word setting, so that
multiplication benefits from the Karatsuba algorithm (and possibly later
improvements).

benchmark                             old ns/op        new ns/op      delta
BenchmarkDiv/20/10-4                  38.2             38.3           +0.26%
BenchmarkDiv/40/20-4                  38.7             38.5           -0.52%
BenchmarkDiv/100/50-4                 62.5             62.6           +0.16%
BenchmarkDiv/200/100-4                238              259            +8.82%
BenchmarkDiv/400/200-4                311              338            +8.68%
BenchmarkDiv/1000/500-4               604              649            +7.45%
BenchmarkDiv/2000/1000-4              1214             1278           +5.27%
BenchmarkDiv/20000/10000-4            38279            36510          -4.62%
BenchmarkDiv/200000/100000-4          3022057          1359615        -55.01%
BenchmarkDiv/2000000/1000000-4        310827664        54012939       -82.62%
BenchmarkDiv/20000000/10000000-4      33272829421      1965401359     -94.09%
BenchmarkString/10/Base10-4           158              156            -1.27%
BenchmarkString/100/Base10-4          797              792            -0.63%
BenchmarkString/1000/Base10-4         3677             3814           +3.73%
BenchmarkString/10000/Base10-4        16633            17116          +2.90%
BenchmarkString/100000/Base10-4       5779029          1793808        -68.96%
BenchmarkString/1000000/Base10-4      889840820        85524031       -90.39%
BenchmarkString/10000000/Base10-4     134338236860     4935657026     -96.33%

Fixes #21960
Updates #30943

Change-Id: I134c6f81a47870c688ca95b6081eb9211def15a2
Reviewed-on: https://go-review.googlesource.com/c/go/+/172018
Reviewed-by: Robert Griesemer <gri@golang.org>
Run-TryBot: Robert Griesemer <gri@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
This commit is contained in:
Rémy Oudompheng 2019-04-14 08:16:13 +02:00 committed by Robert Griesemer
parent b8cb75fb17
commit 194ae3236d
3 changed files with 216 additions and 11 deletions

View File

@ -1829,8 +1829,11 @@ func benchmarkDiv(b *testing.B, aSize, bSize int) {
}
func BenchmarkDiv(b *testing.B) {
min, max, step := 10, 100000, 10
for i := min; i <= max; i *= step {
sizes := []int{
10, 20, 50, 100, 200, 500, 1000,
1e4, 1e5, 1e6, 1e7,
}
for _, i := range sizes {
j := 2 * i
b.Run(fmt.Sprintf("%d/%d", j, i), func(b *testing.B) {
benchmarkDiv(b, j, i)

View File

@ -693,7 +693,7 @@ func putNat(x *nat) {
var natPool sync.Pool
// q = (uIn-r)/vIn, with 0 <= r < y
// q = (uIn-r)/vIn, with 0 <= r < vIn
// Uses z as storage for q, and u as storage for r if possible.
// See Knuth, Volume 2, section 4.3.1, Algorithm D.
// Preconditions:
@ -721,6 +721,30 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
}
q = z.make(m + 1)
if n < divRecursiveThreshold {
q.divBasic(u, v)
} else {
q.divRecursive(u, v)
}
putNat(vp)
q = q.norm()
shrVU(u, u, shift)
r = u.norm()
return q, r
}
// divBasic performs word-by-word division of u by v.
// The quotient is written in pre-allocated q.
// The remainder overwrites input u.
//
// Precondition:
// - len(q) >= len(u)-len(v)
func (q nat) divBasic(u, v nat) {
n := len(v)
m := len(u) - n
qhatvp := getNat(n + 1)
qhatv := *qhatvp
@ -729,7 +753,11 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
for j := m; j >= 0; j-- {
// D3.
qhat := Word(_M)
if ujn := u[j+n]; ujn != vn1 {
var ujn Word
if j+n < len(u) {
ujn = u[j+n]
}
if ujn != vn1 {
var rhat Word
qhat, rhat = divWW(ujn, u[j+n-1], vn1)
@ -752,25 +780,175 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
// D4.
qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
qhl := len(qhatv)
if j+qhl > len(u) && qhatv[n] == 0 {
qhl--
}
c := subVV(u[j:j+qhl], u[j:], qhatv)
if c != 0 {
c := addVV(u[j:j+n], u[j:], v)
u[j+n] += c
qhat--
}
if j == m && m == len(q) && qhat == 0 {
continue
}
q[j] = qhat
}
putNat(vp)
putNat(qhatvp)
}
q = q.norm()
shrVU(u, u, shift)
r = u.norm()
const divRecursiveThreshold = 100
return q, r
// divRecursive performs word-by-word division of u by v.
// The quotient is written in pre-allocated z.
// The remainder overwrites input u.
//
// Precondition:
// - len(z) >= len(u)-len(v)
//
// See Burnikel, Ziegler, "Fast Recursive Division", Algorithm 1 and 2.
func (z nat) divRecursive(u, v nat) {
// Recursion depth is less than 2 log2(len(v))
// Allocate a slice of temporaries to be reused across recursion.
recDepth := 2 * bits.Len(uint(len(v)))
// large enough to perform Karatsuba on operands as large as v
tmp := getNat(3 * len(v))
temps := make([]*nat, recDepth)
z.clear()
z.divRecursiveStep(u, v, 0, tmp, temps)
for _, n := range temps {
if n != nil {
putNat(n)
}
}
putNat(tmp)
}
func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
u = u.norm()
v = v.norm()
if len(u) == 0 {
z.clear()
return
}
n := len(v)
if n < divRecursiveThreshold {
z.divBasic(u, v)
return
}
m := len(u) - n
if m < 0 {
return
}
// Produce the quotient by blocks of B words.
// Division by v (length n) is done using a length n/2 division
// and a length n/2 multiplication for each block. The final
// complexity is driven by multiplication complexity.
B := n / 2
// Allocate a nat for qhat below.
if temps[depth] == nil {
temps[depth] = getNat(n)
} else {
*temps[depth] = temps[depth].make(B + 1)
}
j := m
for j > B {
// Divide u[j-B:j+n] by vIn. Keep remainder in u
// for next block.
//
// The following property will be used (Lemma 2):
// if u = u1 << s + u0
// v = v1 << s + v0
// then floor(u1/v1) >= floor(u/v)
//
// Moreover, the difference is at most 2 if len(v1) >= len(u/v)
// We choose s = B-1 since len(v)-B >= B+1 >= len(u/v)
s := (B - 1)
// Except for the first step, the top bits are always
// a division remainder, so the quotient length is <= n.
uu := u[j-B:]
qhat := *temps[depth]
qhat.clear()
qhat.divRecursiveStep(uu[s:B+n], v[s:], depth+1, tmp, temps)
qhat = qhat.norm()
// Adjust the quotient:
// u = u_h << s + u_l
// v = v_h << s + v_l
// u_h = q̂ v_h + rh
// u = q̂ (v - v_l) + rh << s + u_l
// After the above step, u contains a remainder:
// u = rh << s + u_l
// and we need to substract q̂ v_l
//
// But it may be a bit too large, in which case q̂ needs to be smaller.
qhatv := tmp.make(3 * n)
qhatv.clear()
qhatv = qhatv.mul(qhat, v[:s])
for i := 0; i < 2; i++ {
e := qhatv.cmp(uu.norm())
if e <= 0 {
break
}
subVW(qhat, qhat, 1)
c := subVV(qhatv[:s], qhatv[:s], v[:s])
if len(qhatv) > s {
subVW(qhatv[s:], qhatv[s:], c)
}
addAt(uu[s:], v[s:], 0)
}
if qhatv.cmp(uu.norm()) > 0 {
panic("impossible")
}
c := subVV(uu[:len(qhatv)], uu[:len(qhatv)], qhatv)
if c > 0 {
subVW(uu[len(qhatv):], uu[len(qhatv):], c)
}
addAt(z, qhat, j-B)
j -= B
}
// Now u < (v<<B), compute lower bits in the same way.
// Choose shift = B-1 again.
s := B
qhat := *temps[depth]
qhat.clear()
qhat.divRecursiveStep(u[s:].norm(), v[s:], depth+1, tmp, temps)
qhat = qhat.norm()
qhatv := tmp.make(3 * n)
qhatv.clear()
qhatv = qhatv.mul(qhat, v[:s])
// Set the correct remainder as before.
for i := 0; i < 2; i++ {
if e := qhatv.cmp(u.norm()); e > 0 {
subVW(qhat, qhat, 1)
c := subVV(qhatv[:s], qhatv[:s], v[:s])
if len(qhatv) > s {
subVW(qhatv[s:], qhatv[s:], c)
}
addAt(u[s:], v[s:], 0)
}
}
if qhatv.cmp(u.norm()) > 0 {
panic("impossible")
}
c := subVV(u[0:len(qhatv)], u[0:len(qhatv)], qhatv)
if c > 0 {
c = subVW(u[len(qhatv):B+n], u[len(qhatv):B+n], c)
}
if c > 0 {
panic("impossible")
}
// Done!
addAt(z, qhat.norm(), 0)
}
// Length of x in bits. x must be normalized.

View File

@ -739,3 +739,27 @@ func BenchmarkNatSetBytes(b *testing.B) {
})
}
}
func TestNatDiv(t *testing.T) {
sizes := []int{
1, 2, 5, 8, 15, 25, 40, 65, 100,
200, 500, 800, 1500, 2500, 4000, 6500, 10000,
}
for _, i := range sizes {
for _, j := range sizes {
a := rndNat(i)
b := rndNat(j)
x := nat(nil).mul(a, b)
addVW(x, x, 1)
var q, r nat
q, r = q.div(r, x, b)
if q.cmp(a) != 0 {
t.Fatal("wrong quotient", i, j)
}
if len(r) != 1 || r[0] != 1 {
t.Fatal("wrong remainder")
}
}
}
}