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math/big: rewrite pure Go implementations to use math/bits
While we're here, delete addWW_g and subWW_g, per the TODO. They are now obsolete. Benchmarks on amd64 with -tags=math_big_pure_go. name old time/op new time/op delta AddVV/1-8 5.24ns ± 2% 5.12ns ± 1% -2.11% (p=0.000 n=82+87) AddVV/2-8 6.44ns ± 1% 6.33ns ± 2% -1.82% (p=0.000 n=77+82) AddVV/3-8 7.89ns ± 8% 6.97ns ± 4% -11.71% (p=0.000 n=100+96) AddVV/4-8 8.60ns ± 0% 7.72ns ± 4% -10.24% (p=0.000 n=90+96) AddVV/5-8 10.3ns ± 4% 8.5ns ± 1% -17.02% (p=0.000 n=96+91) AddVV/10-8 16.2ns ± 5% 12.8ns ± 1% -21.11% (p=0.000 n=97+86) AddVV/100-8 148ns ± 1% 117ns ± 5% -21.07% (p=0.000 n=66+98) AddVV/1000-8 1.41µs ± 4% 1.13µs ± 3% -19.90% (p=0.000 n=97+97) AddVV/10000-8 14.2µs ± 5% 11.2µs ± 1% -20.82% (p=0.000 n=99+84) AddVV/100000-8 142µs ± 4% 113µs ± 4% -20.40% (p=0.000 n=91+92) SubVV/1-8 5.29ns ± 1% 5.11ns ± 0% -3.30% (p=0.000 n=87+88) SubVV/2-8 6.36ns ± 4% 6.33ns ± 2% -0.56% (p=0.002 n=98+73) SubVV/3-8 7.58ns ± 5% 6.98ns ± 4% -8.01% (p=0.000 n=97+91) SubVV/4-8 8.61ns ± 3% 7.98ns ± 2% -7.31% (p=0.000 n=95+83) SubVV/5-8 10.6ns ± 2% 8.5ns ± 1% -19.56% (p=0.000 n=79+89) SubVV/10-8 16.3ns ± 4% 12.7ns ± 1% -21.97% (p=0.000 n=98+82) SubVV/100-8 124ns ± 1% 118ns ± 1% -4.83% (p=0.000 n=85+81) SubVV/1000-8 1.14µs ± 5% 1.12µs ± 2% -1.17% (p=0.000 n=97+81) SubVV/10000-8 11.6µs ±10% 11.2µs ± 1% -3.39% (p=0.000 n=100+84) SubVV/100000-8 114µs ± 6% 114µs ± 5% ~ (p=0.396 n=83+94) AddVW/1-8 4.04ns ± 4% 4.34ns ± 4% +7.57% (p=0.000 n=96+98) AddVW/2-8 4.34ns ± 5% 4.40ns ± 5% +1.40% (p=0.000 n=99+98) AddVW/3-8 5.43ns ± 0% 5.54ns ± 2% +1.97% (p=0.000 n=85+94) AddVW/4-8 6.23ns ± 1% 6.18ns ± 2% -0.66% (p=0.000 n=77+78) AddVW/5-8 6.78ns ± 2% 6.90ns ± 4% +1.77% (p=0.000 n=80+99) AddVW/10-8 10.5ns ± 4% 9.9ns ± 1% -5.77% (p=0.000 n=97+69) AddVW/100-8 114ns ± 3% 91ns ± 0% -20.38% (p=0.000 n=98+77) AddVW/1000-8 1.12µs ± 1% 0.87µs ± 1% -22.80% (p=0.000 n=82+68) AddVW/10000-8 11.2µs ± 2% 8.5µs ± 5% -23.85% (p=0.000 n=85+100) AddVW/100000-8 112µs ± 2% 85µs ± 5% -24.22% (p=0.000 n=71+96) SubVW/1-8 4.09ns ± 2% 4.18ns ± 4% +2.32% (p=0.000 n=78+96) SubVW/2-8 4.59ns ± 5% 4.52ns ± 7% -1.54% (p=0.000 n=98+94) SubVW/3-8 5.41ns ±10% 5.55ns ± 1% +2.48% (p=0.000 n=100+89) SubVW/4-8 6.51ns ± 2% 6.19ns ± 0% -4.85% (p=0.000 n=97+81) SubVW/5-8 7.25ns ± 3% 6.90ns ± 4% -4.93% (p=0.000 n=97+96) SubVW/10-8 10.6ns ± 4% 9.8ns ± 2% -7.32% (p=0.000 n=95+96) SubVW/100-8 90.4ns ± 0% 90.8ns ± 0% +0.43% (p=0.000 n=83+78) SubVW/1000-8 853ns ± 4% 857ns ± 2% +0.42% (p=0.000 n=100+98) SubVW/10000-8 8.52µs ± 4% 8.53µs ± 2% ~ (p=0.061 n=99+97) SubVW/100000-8 84.8µs ± 5% 84.2µs ± 2% -0.78% (p=0.000 n=99+93) AddMulVVW/1-8 8.73ns ± 0% 5.33ns ± 3% -38.91% (p=0.000 n=91+96) AddMulVVW/2-8 14.8ns ± 3% 6.5ns ± 2% -56.33% (p=0.000 n=100+79) AddMulVVW/3-8 18.6ns ± 2% 7.8ns ± 5% -57.84% (p=0.000 n=89+96) AddMulVVW/4-8 24.0ns ± 2% 9.8ns ± 0% -59.09% (p=0.000 n=95+67) AddMulVVW/5-8 29.0ns ± 2% 11.5ns ± 5% -60.44% (p=0.000 n=90+97) AddMulVVW/10-8 54.1ns ± 0% 18.8ns ± 1% -65.37% (p=0.000 n=82+84) AddMulVVW/100-8 508ns ± 2% 165ns ± 4% -67.62% (p=0.000 n=72+98) AddMulVVW/1000-8 4.96µs ± 3% 1.55µs ± 1% -68.86% (p=0.000 n=99+91) AddMulVVW/10000-8 50.0µs ± 4% 15.5µs ± 4% -68.95% (p=0.000 n=97+97) AddMulVVW/100000-8 491µs ± 1% 156µs ± 8% -68.22% (p=0.000 n=79+95) Change-Id: I4c6ae0b4065f371aea8103f6a85d9e9274bf01d0 Reviewed-on: https://go-review.googlesource.com/c/go/+/164965 Run-TryBot: Josh Bleecher Snyder <josharian@gmail.com> TryBot-Result: Gobot Gobot <gobot@golang.org> Reviewed-by: Robert Griesemer <gri@golang.org>
This commit is contained in:
Josh Bleecher Snyder
parent
c74659290a
commit
d5edbcac98
@ -19,10 +19,6 @@ const (
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_W = bits.UintSize // word size in bits
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_B = 1 << _W // digit base
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_M = _B - 1 // digit mask
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_W2 = _W / 2 // half word size in bits
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_B2 = 1 << _W2 // half digit base
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_M2 = _B2 - 1 // half digit mask
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)
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// ----------------------------------------------------------------------------
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@ -30,50 +26,18 @@ const (
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//
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// These operations are used by the vector operations below.
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// z1<<_W + z0 = x+y+c, with c == 0 or 1
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func addWW_g(x, y, c Word) (z1, z0 Word) {
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yc := y + c
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z0 = x + yc
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if z0 < x || yc < y {
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z1 = 1
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}
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return
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}
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// z1<<_W + z0 = x-y-c, with c == 0 or 1
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func subWW_g(x, y, c Word) (z1, z0 Word) {
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yc := y + c
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z0 = x - yc
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if z0 > x || yc < y {
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z1 = 1
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}
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return
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}
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// z1<<_W + z0 = x*y
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// Adapted from Warren, Hacker's Delight, p. 132.
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func mulWW_g(x, y Word) (z1, z0 Word) {
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x0 := x & _M2
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x1 := x >> _W2
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y0 := y & _M2
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y1 := y >> _W2
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w0 := x0 * y0
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t := x1*y0 + w0>>_W2
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w1 := t & _M2
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w2 := t >> _W2
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w1 += x0 * y1
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z1 = x1*y1 + w2 + w1>>_W2
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z0 = x * y
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return
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hi, lo := bits.Mul(uint(x), uint(y))
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return Word(hi), Word(lo)
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}
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// z1<<_W + z0 = x*y + c
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func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
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z1, zz0 := mulWW_g(x, y)
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if z0 = zz0 + c; z0 < zz0 {
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z1++
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}
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return
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hi, lo := bits.Mul(uint(x), uint(y))
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var cc uint
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lo, cc = bits.Add(lo, uint(c), 0)
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return Word(hi + cc), Word(lo)
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}
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// nlz returns the number of leading zeros in x.
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@ -83,122 +47,48 @@ func nlz(x Word) uint {
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}
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// q = (u1<<_W + u0 - r)/v
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// Adapted from Warren, Hacker's Delight, p. 152.
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func divWW_g(u1, u0, v Word) (q, r Word) {
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if u1 >= v {
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return 1<<_W - 1, 1<<_W - 1
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}
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s := nlz(v)
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v <<= s
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vn1 := v >> _W2
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vn0 := v & _M2
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un32 := u1<<s | u0>>(_W-s)
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un10 := u0 << s
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un1 := un10 >> _W2
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un0 := un10 & _M2
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q1 := un32 / vn1
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rhat := un32 - q1*vn1
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for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
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q1--
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rhat += vn1
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if rhat >= _B2 {
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break
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}
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}
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un21 := un32*_B2 + un1 - q1*v
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q0 := un21 / vn1
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rhat = un21 - q0*vn1
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for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
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q0--
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rhat += vn1
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if rhat >= _B2 {
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break
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}
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}
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return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
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qq, rr := bits.Div(uint(u1), uint(u0), uint(v))
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return Word(qq), Word(rr)
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}
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// Keep for performance debugging.
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// Using addWW_g is likely slower.
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const use_addWW_g = false
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// The resulting carry c is either 0 or 1.
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func addVV_g(z, x, y []Word) (c Word) {
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if use_addWW_g {
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for i := range z {
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c, z[i] = addWW_g(x[i], y[i], c)
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}
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return
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}
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for i, xi := range x[:len(z)] {
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yi := y[i]
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zi := xi + yi + c
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z[i] = zi
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// see "Hacker's Delight", section 2-12 (overflow detection)
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c = (xi&yi | (xi|yi)&^zi) >> (_W - 1)
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for i := range x[:len(z)] {
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zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
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z[i] = Word(zi)
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c = Word(cc)
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}
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return
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}
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// The resulting carry c is either 0 or 1.
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func subVV_g(z, x, y []Word) (c Word) {
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if use_addWW_g {
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for i := range z {
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c, z[i] = subWW_g(x[i], y[i], c)
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}
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return
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}
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for i, xi := range x[:len(z)] {
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yi := y[i]
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zi := xi - yi - c
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z[i] = zi
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// see "Hacker's Delight", section 2-12 (overflow detection)
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c = (yi&^xi | (yi|^xi)&zi) >> (_W - 1)
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for i := range x[:len(z)] {
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zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
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z[i] = Word(zi)
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c = Word(cc)
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}
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return
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}
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// The resulting carry c is either 0 or 1.
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func addVW_g(z, x []Word, y Word) (c Word) {
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if use_addWW_g {
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c = y
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for i := range z {
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c, z[i] = addWW_g(x[i], c, 0)
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}
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return
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}
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c = y
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for i, xi := range x[:len(z)] {
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zi := xi + c
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z[i] = zi
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c = xi &^ zi >> (_W - 1)
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for i := range x[:len(z)] {
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zi, cc := bits.Add(uint(x[i]), uint(c), 0)
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z[i] = Word(zi)
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c = Word(cc)
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}
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return
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}
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func subVW_g(z, x []Word, y Word) (c Word) {
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if use_addWW_g {
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c = y
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for i := range z {
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c, z[i] = subWW_g(x[i], c, 0)
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}
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return
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}
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c = y
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for i, xi := range x[:len(z)] {
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zi := xi - c
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z[i] = zi
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c = (zi &^ xi) >> (_W - 1)
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for i := range x[:len(z)] {
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zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
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z[i] = Word(zi)
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c = Word(cc)
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}
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return
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}
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@ -255,11 +145,11 @@ func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
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return
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}
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// TODO(gri) Remove use of addWW_g here and then we can remove addWW_g and subWW_g.
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func addMulVVW_g(z, x []Word, y Word) (c Word) {
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for i := range z {
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z1, z0 := mulAddWWW_g(x[i], y, z[i])
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c, z[i] = addWW_g(z0, c, 0)
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lo, cc := bits.Add(uint(z0), uint(c), 0)
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c, z[i] = Word(cc), Word(lo)
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c += z1
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}
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return
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@ -14,48 +14,6 @@ import (
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var isRaceBuilder = strings.HasSuffix(testenv.Builder(), "-race")
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type funWW func(x, y, c Word) (z1, z0 Word)
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type argWW struct {
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x, y, c, z1, z0 Word
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}
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var sumWW = []argWW{
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{0, 0, 0, 0, 0},
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{0, 1, 0, 0, 1},
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{0, 0, 1, 0, 1},
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{0, 1, 1, 0, 2},
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{12345, 67890, 0, 0, 80235},
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{12345, 67890, 1, 0, 80236},
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{_M, 1, 0, 1, 0},
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{_M, 0, 1, 1, 0},
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{_M, 1, 1, 1, 1},
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{_M, _M, 0, 1, _M - 1},
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{_M, _M, 1, 1, _M},
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}
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func testFunWW(t *testing.T, msg string, f funWW, a argWW) {
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z1, z0 := f(a.x, a.y, a.c)
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if z1 != a.z1 || z0 != a.z0 {
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t.Errorf("%s%+v\n\tgot z1:z0 = %#x:%#x; want %#x:%#x", msg, a, z1, z0, a.z1, a.z0)
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}
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}
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func TestFunWW(t *testing.T) {
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for _, a := range sumWW {
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arg := a
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testFunWW(t, "addWW_g", addWW_g, arg)
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arg = argWW{a.y, a.x, a.c, a.z1, a.z0}
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testFunWW(t, "addWW_g symmetric", addWW_g, arg)
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arg = argWW{a.z0, a.x, a.c, a.z1, a.y}
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testFunWW(t, "subWW_g", subWW_g, arg)
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arg = argWW{a.z0, a.y, a.c, a.z1, a.x}
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testFunWW(t, "subWW_g symmetric", subWW_g, arg)
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}
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}
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type funVV func(z, x, y []Word) (c Word)
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type argVV struct {
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z, x, y nat
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