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math/big: rewrite pure Go implementations to use math/bits

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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 2019-03-02 14:42:51 -08:00
parent c74659290a
commit d5edbcac98
2 changed files with 26 additions and 178 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

162
src/math/big/arith.go
View File

@ -19,10 +19,6 @@ const (
_W = bits.UintSize // word size in bits _W = bits.UintSize // word size in bits
_B = 1 << _W // digit base _B = 1 << _W // digit base
_M = _B - 1 // digit mask _M = _B - 1 // digit mask
_W2 = _W / 2 // half word size in bits
_B2 = 1 << _W2 // half digit base
_M2 = _B2 - 1 // half digit mask
) )
// ---------------------------------------------------------------------------- // ----------------------------------------------------------------------------
@ -30,50 +26,18 @@ const (
// //
// These operations are used by the vector operations below. // These operations are used by the vector operations below.
// z1<<_W + z0 = x+y+c, with c == 0 or 1
func addWW_g(x, y, c Word) (z1, z0 Word) {
yc := y + c
z0 = x + yc
if z0 < x || yc < y {
z1 = 1
}
return
}
// z1<<_W + z0 = x-y-c, with c == 0 or 1
func subWW_g(x, y, c Word) (z1, z0 Word) {
yc := y + c
z0 = x - yc
if z0 > x || yc < y {
z1 = 1
}
return
}
// z1<<_W + z0 = x*y // z1<<_W + z0 = x*y
// Adapted from Warren, Hacker's Delight, p. 132.
func mulWW_g(x, y Word) (z1, z0 Word) { func mulWW_g(x, y Word) (z1, z0 Word) {
x0 := x & _M2 hi, lo := bits.Mul(uint(x), uint(y))
x1 := x >> _W2 return Word(hi), Word(lo)
y0 := y & _M2
y1 := y >> _W2
w0 := x0 * y0
t := x1*y0 + w0>>_W2
w1 := t & _M2
w2 := t >> _W2
w1 += x0 * y1
z1 = x1*y1 + w2 + w1>>_W2
z0 = x * y
return
} }
// z1<<_W + z0 = x*y + c // z1<<_W + z0 = x*y + c
func mulAddWWW_g(x, y, c Word) (z1, z0 Word) { func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
z1, zz0 := mulWW_g(x, y) hi, lo := bits.Mul(uint(x), uint(y))
if z0 = zz0 + c; z0 < zz0 { var cc uint
z1++ lo, cc = bits.Add(lo, uint(c), 0)
} return Word(hi + cc), Word(lo)
return
} }
// nlz returns the number of leading zeros in x. // nlz returns the number of leading zeros in x.
@ -83,122 +47,48 @@ func nlz(x Word) uint {
} }
// q = (u1<<_W + u0 - r)/v // q = (u1<<_W + u0 - r)/v
// Adapted from Warren, Hacker's Delight, p. 152.
func divWW_g(u1, u0, v Word) (q, r Word) { func divWW_g(u1, u0, v Word) (q, r Word) {
if u1 >= v { qq, rr := bits.Div(uint(u1), uint(u0), uint(v))
return 1<<_W - 1, 1<<_W - 1 return Word(qq), Word(rr)
}
s := nlz(v)
v <<= s
vn1 := v >> _W2
vn0 := v & _M2
un32 := u1<<s | u0>>(_W-s)
un10 := u0 << s
un1 := un10 >> _W2
un0 := un10 & _M2
q1 := un32 / vn1
rhat := un32 - q1*vn1
for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
q1--
rhat += vn1
if rhat >= _B2 {
break
}
}
un21 := un32*_B2 + un1 - q1*v
q0 := un21 / vn1
rhat = un21 - q0*vn1
for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
q0--
rhat += vn1
if rhat >= _B2 {
break
}
}
return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
} }
// Keep for performance debugging.
// Using addWW_g is likely slower.
const use_addWW_g = false
// The resulting carry c is either 0 or 1. // The resulting carry c is either 0 or 1.
func addVV_g(z, x, y []Word) (c Word) { func addVV_g(z, x, y []Word) (c Word) {
if use_addWW_g { for i := range x[:len(z)] {
for i := range z { zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
c, z[i] = addWW_g(x[i], y[i], c) z[i] = Word(zi)
} c = Word(cc)
return
}
for i, xi := range x[:len(z)] {
yi := y[i]
zi := xi + yi + c
z[i] = zi
// see "Hacker's Delight", section 2-12 (overflow detection)
c = (xi&yi | (xi|yi)&^zi) >> (_W - 1)
} }
return return
} }
// The resulting carry c is either 0 or 1. // The resulting carry c is either 0 or 1.
func subVV_g(z, x, y []Word) (c Word) { func subVV_g(z, x, y []Word) (c Word) {
if use_addWW_g { for i := range x[:len(z)] {
for i := range z { zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
c, z[i] = subWW_g(x[i], y[i], c) z[i] = Word(zi)
} c = Word(cc)
return
}
for i, xi := range x[:len(z)] {
yi := y[i]
zi := xi - yi - c
z[i] = zi
// see "Hacker's Delight", section 2-12 (overflow detection)
c = (yi&^xi | (yi|^xi)&zi) >> (_W - 1)
} }
return return
} }
// The resulting carry c is either 0 or 1. // The resulting carry c is either 0 or 1.
func addVW_g(z, x []Word, y Word) (c Word) { func addVW_g(z, x []Word, y Word) (c Word) {
if use_addWW_g {
c = y
for i := range z {
c, z[i] = addWW_g(x[i], c, 0)
}
return
}
c = y c = y
for i, xi := range x[:len(z)] { for i := range x[:len(z)] {
zi := xi + c zi, cc := bits.Add(uint(x[i]), uint(c), 0)
z[i] = zi z[i] = Word(zi)
c = xi &^ zi >> (_W - 1) c = Word(cc)
} }
return return
} }
func subVW_g(z, x []Word, y Word) (c Word) { func subVW_g(z, x []Word, y Word) (c Word) {
if use_addWW_g {
c = y
for i := range z {
c, z[i] = subWW_g(x[i], c, 0)
}
return
}
c = y c = y
for i, xi := range x[:len(z)] { for i := range x[:len(z)] {
zi := xi - c zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
z[i] = zi z[i] = Word(zi)
c = (zi &^ xi) >> (_W - 1) c = Word(cc)
} }
return return
} }
@ -255,11 +145,11 @@ func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
return return
} }
// TODO(gri) Remove use of addWW_g here and then we can remove addWW_g and subWW_g.
func addMulVVW_g(z, x []Word, y Word) (c Word) { func addMulVVW_g(z, x []Word, y Word) (c Word) {
for i := range z { for i := range z {
z1, z0 := mulAddWWW_g(x[i], y, z[i]) z1, z0 := mulAddWWW_g(x[i], y, z[i])
c, z[i] = addWW_g(z0, c, 0) lo, cc := bits.Add(uint(z0), uint(c), 0)
c, z[i] = Word(cc), Word(lo)
c += z1 c += z1
} }
return return

42
src/math/big/arith_test.go
View File

@ -14,48 +14,6 @@ import (
var isRaceBuilder = strings.HasSuffix(testenv.Builder(), "-race") var isRaceBuilder = strings.HasSuffix(testenv.Builder(), "-race")
type funWW func(x, y, c Word) (z1, z0 Word)
type argWW struct {
x, y, c, z1, z0 Word
}
var sumWW = []argWW{
{0, 0, 0, 0, 0},
{0, 1, 0, 0, 1},
{0, 0, 1, 0, 1},
{0, 1, 1, 0, 2},
{12345, 67890, 0, 0, 80235},
{12345, 67890, 1, 0, 80236},
{_M, 1, 0, 1, 0},
{_M, 0, 1, 1, 0},
{_M, 1, 1, 1, 1},
{_M, _M, 0, 1, _M - 1},
{_M, _M, 1, 1, _M},
}
func testFunWW(t *testing.T, msg string, f funWW, a argWW) {
z1, z0 := f(a.x, a.y, a.c)
if z1 != a.z1 || z0 != a.z0 {
t.Errorf("%s%+v\n\tgot z1:z0 = %#x:%#x; want %#x:%#x", msg, a, z1, z0, a.z1, a.z0)
}
}
func TestFunWW(t *testing.T) {
for _, a := range sumWW {
arg := a
testFunWW(t, "addWW_g", addWW_g, arg)
arg = argWW{a.y, a.x, a.c, a.z1, a.z0}
testFunWW(t, "addWW_g symmetric", addWW_g, arg)
arg = argWW{a.z0, a.x, a.c, a.z1, a.y}
testFunWW(t, "subWW_g", subWW_g, arg)
arg = argWW{a.z0, a.y, a.c, a.z1, a.x}
testFunWW(t, "subWW_g symmetric", subWW_g, arg)
}
}
type funVV func(z, x, y []Word) (c Word) type funVV func(z, x, y []Word) (c Word)
type argVV struct { type argVV struct {
z, x, y nat z, x, y nat