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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
Robert Griesemer
parent
b8cb75fb17
commit
194ae3236d
@ -1829,8 +1829,11 @@ func benchmarkDiv(b *testing.B, aSize, bSize int) {
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}
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func BenchmarkDiv(b *testing.B) {
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min, max, step := 10, 100000, 10
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for i := min; i <= max; i *= step {
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sizes := []int{
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10, 20, 50, 100, 200, 500, 1000,
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1e4, 1e5, 1e6, 1e7,
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}
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for _, i := range sizes {
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j := 2 * i
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b.Run(fmt.Sprintf("%d/%d", j, i), func(b *testing.B) {
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benchmarkDiv(b, j, i)
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@ -693,7 +693,7 @@ func putNat(x *nat) {
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var natPool sync.Pool
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// q = (uIn-r)/vIn, with 0 <= r < y
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// q = (uIn-r)/vIn, with 0 <= r < vIn
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// Uses z as storage for q, and u as storage for r if possible.
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// See Knuth, Volume 2, section 4.3.1, Algorithm D.
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// Preconditions:
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@ -721,6 +721,30 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
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}
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q = z.make(m + 1)
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if n < divRecursiveThreshold {
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q.divBasic(u, v)
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} else {
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q.divRecursive(u, v)
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}
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putNat(vp)
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q = q.norm()
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shrVU(u, u, shift)
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r = u.norm()
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return q, r
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}
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// divBasic performs word-by-word division of u by v.
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// The quotient is written in pre-allocated q.
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// The remainder overwrites input u.
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//
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// Precondition:
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// - len(q) >= len(u)-len(v)
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func (q nat) divBasic(u, v nat) {
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n := len(v)
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m := len(u) - n
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qhatvp := getNat(n + 1)
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qhatv := *qhatvp
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@ -729,7 +753,11 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
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for j := m; j >= 0; j-- {
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// D3.
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qhat := Word(_M)
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if ujn := u[j+n]; ujn != vn1 {
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var ujn Word
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if j+n < len(u) {
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ujn = u[j+n]
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}
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if ujn != vn1 {
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var rhat Word
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qhat, rhat = divWW(ujn, u[j+n-1], vn1)
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@ -752,25 +780,175 @@ func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
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// D4.
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qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
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c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
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qhl := len(qhatv)
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if j+qhl > len(u) && qhatv[n] == 0 {
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qhl--
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}
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c := subVV(u[j:j+qhl], u[j:], qhatv)
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if c != 0 {
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c := addVV(u[j:j+n], u[j:], v)
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u[j+n] += c
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qhat--
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}
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if j == m && m == len(q) && qhat == 0 {
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continue
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}
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q[j] = qhat
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}
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putNat(vp)
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putNat(qhatvp)
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}
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q = q.norm()
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shrVU(u, u, shift)
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r = u.norm()
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const divRecursiveThreshold = 100
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return q, r
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// divRecursive performs word-by-word division of u by v.
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// The quotient is written in pre-allocated z.
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// The remainder overwrites input u.
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//
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// Precondition:
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// - len(z) >= len(u)-len(v)
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//
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// See Burnikel, Ziegler, "Fast Recursive Division", Algorithm 1 and 2.
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func (z nat) divRecursive(u, v nat) {
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// Recursion depth is less than 2 log2(len(v))
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// Allocate a slice of temporaries to be reused across recursion.
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recDepth := 2 * bits.Len(uint(len(v)))
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// large enough to perform Karatsuba on operands as large as v
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tmp := getNat(3 * len(v))
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temps := make([]*nat, recDepth)
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z.clear()
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z.divRecursiveStep(u, v, 0, tmp, temps)
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for _, n := range temps {
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if n != nil {
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putNat(n)
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}
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}
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putNat(tmp)
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}
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func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
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u = u.norm()
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v = v.norm()
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if len(u) == 0 {
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z.clear()
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return
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}
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n := len(v)
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if n < divRecursiveThreshold {
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z.divBasic(u, v)
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return
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}
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m := len(u) - n
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if m < 0 {
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return
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}
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// Produce the quotient by blocks of B words.
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// Division by v (length n) is done using a length n/2 division
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// and a length n/2 multiplication for each block. The final
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// complexity is driven by multiplication complexity.
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B := n / 2
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// Allocate a nat for qhat below.
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if temps[depth] == nil {
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temps[depth] = getNat(n)
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} else {
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*temps[depth] = temps[depth].make(B + 1)
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}
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j := m
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for j > B {
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// Divide u[j-B:j+n] by vIn. Keep remainder in u
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// for next block.
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//
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// The following property will be used (Lemma 2):
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// if u = u1 << s + u0
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// v = v1 << s + v0
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// then floor(u1/v1) >= floor(u/v)
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//
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// Moreover, the difference is at most 2 if len(v1) >= len(u/v)
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// We choose s = B-1 since len(v)-B >= B+1 >= len(u/v)
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s := (B - 1)
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// Except for the first step, the top bits are always
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// a division remainder, so the quotient length is <= n.
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uu := u[j-B:]
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qhat := *temps[depth]
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qhat.clear()
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qhat.divRecursiveStep(uu[s:B+n], v[s:], depth+1, tmp, temps)
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qhat = qhat.norm()
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// Adjust the quotient:
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// u = u_h << s + u_l
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// v = v_h << s + v_l
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// u_h = q̂ v_h + rh
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// u = q̂ (v - v_l) + rh << s + u_l
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// After the above step, u contains a remainder:
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// u = rh << s + u_l
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// and we need to substract q̂ v_l
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//
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// But it may be a bit too large, in which case q̂ needs to be smaller.
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qhatv := tmp.make(3 * n)
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qhatv.clear()
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qhatv = qhatv.mul(qhat, v[:s])
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for i := 0; i < 2; i++ {
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e := qhatv.cmp(uu.norm())
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if e <= 0 {
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break
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}
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subVW(qhat, qhat, 1)
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c := subVV(qhatv[:s], qhatv[:s], v[:s])
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if len(qhatv) > s {
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subVW(qhatv[s:], qhatv[s:], c)
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}
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addAt(uu[s:], v[s:], 0)
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}
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if qhatv.cmp(uu.norm()) > 0 {
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panic("impossible")
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}
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c := subVV(uu[:len(qhatv)], uu[:len(qhatv)], qhatv)
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if c > 0 {
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subVW(uu[len(qhatv):], uu[len(qhatv):], c)
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}
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addAt(z, qhat, j-B)
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j -= B
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}
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// Now u < (v<<B), compute lower bits in the same way.
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// Choose shift = B-1 again.
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s := B
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qhat := *temps[depth]
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qhat.clear()
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qhat.divRecursiveStep(u[s:].norm(), v[s:], depth+1, tmp, temps)
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qhat = qhat.norm()
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qhatv := tmp.make(3 * n)
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qhatv.clear()
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qhatv = qhatv.mul(qhat, v[:s])
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// Set the correct remainder as before.
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for i := 0; i < 2; i++ {
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if e := qhatv.cmp(u.norm()); e > 0 {
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subVW(qhat, qhat, 1)
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c := subVV(qhatv[:s], qhatv[:s], v[:s])
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if len(qhatv) > s {
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subVW(qhatv[s:], qhatv[s:], c)
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}
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addAt(u[s:], v[s:], 0)
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}
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}
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if qhatv.cmp(u.norm()) > 0 {
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panic("impossible")
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}
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c := subVV(u[0:len(qhatv)], u[0:len(qhatv)], qhatv)
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if c > 0 {
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c = subVW(u[len(qhatv):B+n], u[len(qhatv):B+n], c)
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}
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if c > 0 {
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panic("impossible")
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}
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// Done!
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addAt(z, qhat.norm(), 0)
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}
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// Length of x in bits. x must be normalized.
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@ -739,3 +739,27 @@ func BenchmarkNatSetBytes(b *testing.B) {
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})
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}
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}
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func TestNatDiv(t *testing.T) {
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sizes := []int{
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1, 2, 5, 8, 15, 25, 40, 65, 100,
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200, 500, 800, 1500, 2500, 4000, 6500, 10000,
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}
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for _, i := range sizes {
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for _, j := range sizes {
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a := rndNat(i)
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b := rndNat(j)
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x := nat(nil).mul(a, b)
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addVW(x, x, 1)
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var q, r nat
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q, r = q.div(r, x, b)
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if q.cmp(a) != 0 {
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t.Fatal("wrong quotient", i, j)
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}
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if len(r) != 1 || r[0] != 1 {
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t.Fatal("wrong remainder")
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}
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}
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}
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}
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