math/big: remove bounds checks in pure Go implementations

These routines are quite sensitive to BCE.

This change eliminates bounds checks from loops.
It does so at the cost of a bit of safety:
malformed input will now return incorrect answers
instead of panicking.

This isn't as bad as it sounds: math/big has very good
test coverage, and the alternative implementations are in
assembly, which could do much worse things with malformed input.

If the compiler's BCE improves, so could these routines.

Notable BCE improvements for these routines would be:

* Allowing and propagating more cross-slice length hints.
  Then hints like _ = y[:len(z)] would eliminate bounds checks for y[i].

* Propagating enough information so that we could do
  n := len(x)
  if len(z) < n {
    n = len(z)
  }
  and then have i < n eliminate the same bounds checks as
  i < len(x) && i < len(z) currently does.

* Providing some way to do BCE for unrolled loops.
  Now that we have math/bits implementations,
  it is possible to write things like ADC chains in
  pure Go, if you can reasonably unroll loops.

Benchmarks below are for amd64, using -tags=math_big_pure_go.

name            old time/op    new time/op    delta
AddVV/1-8         5.15ns ± 3%    4.65ns ± 4%   -9.81%  (p=0.000 n=93+86)
AddVV/2-8         6.40ns ± 2%    5.58ns ± 4%  -12.78%  (p=0.000 n=90+95)
AddVV/3-8         7.07ns ± 2%    6.66ns ± 2%   -5.88%  (p=0.000 n=87+83)
AddVV/4-8         7.94ns ± 5%    7.41ns ± 4%   -6.65%  (p=0.000 n=94+98)
AddVV/5-8         8.55ns ± 1%    8.80ns ± 0%   +2.92%  (p=0.000 n=87+92)
AddVV/10-8        12.7ns ± 1%    12.3ns ± 1%   -3.12%  (p=0.000 n=83+71)
AddVV/100-8        119ns ± 5%     117ns ± 4%   -1.60%  (p=0.000 n=93+90)
AddVV/1000-8      1.14µs ± 4%    1.14µs ± 5%     ~     (p=0.812 n=95+91)
AddVV/10000-8     11.4µs ± 5%    11.3µs ± 5%     ~     (p=0.503 n=97+96)
AddVV/100000-8     114µs ± 4%     113µs ± 5%   -0.98%  (p=0.002 n=97+90)

name            old time/op    new time/op    delta
SubVV/1-8         5.23ns ± 5%    4.65ns ± 3%  -11.18%  (p=0.000 n=89+91)
SubVV/2-8         6.49ns ± 5%    5.58ns ± 3%  -14.04%  (p=0.000 n=92+94)
SubVV/3-8         7.10ns ± 3%    6.65ns ± 2%   -6.28%  (p=0.000 n=87+80)
SubVV/4-8         8.04ns ± 1%    7.44ns ± 5%   -7.49%  (p=0.000 n=83+98)
SubVV/5-8         8.55ns ± 2%    8.32ns ± 1%   -2.75%  (p=0.000 n=84+92)
SubVV/10-8        12.7ns ± 1%    12.3ns ± 1%   -3.09%  (p=0.000 n=80+75)
SubVV/100-8        119ns ± 0%     116ns ± 3%   -1.83%  (p=0.000 n=87+98)
SubVV/1000-8      1.13µs ± 5%    1.13µs ± 3%     ~     (p=0.082 n=96+98)
SubVV/10000-8     11.2µs ± 1%    11.3µs ± 3%   +0.76%  (p=0.000 n=87+97)
SubVV/100000-8     112µs ± 2%     113µs ± 3%   +0.55%  (p=0.000 n=76+88)

name            old time/op    new time/op    delta
AddVW/1-8         4.30ns ± 4%    3.96ns ± 6%  -8.02%  (p=0.000 n=89+97)
AddVW/2-8         5.15ns ± 2%    4.91ns ± 1%  -4.56%  (p=0.000 n=87+80)
AddVW/3-8         5.59ns ± 3%    5.75ns ± 2%  +2.91%  (p=0.000 n=91+88)
AddVW/4-8         6.20ns ± 1%    6.03ns ± 1%  -2.71%  (p=0.000 n=75+90)
AddVW/5-8         6.93ns ± 3%    6.49ns ± 2%  -6.35%  (p=0.000 n=100+82)
AddVW/10-8        10.0ns ± 7%     9.6ns ± 0%  -4.02%  (p=0.000 n=98+74)
AddVW/100-8       91.1ns ± 1%    90.6ns ± 1%  -0.55%  (p=0.000 n=84+80)
AddVW/1000-8       866ns ± 1%     856ns ± 4%  -1.06%  (p=0.000 n=69+96)
AddVW/10000-8     8.64µs ± 1%    8.53µs ± 4%  -1.25%  (p=0.000 n=67+99)
AddVW/100000-8    84.3µs ± 2%    85.4µs ± 4%  +1.22%  (p=0.000 n=89+99)

name            old time/op    new time/op    delta
SubVW/1-8         4.28ns ± 2%    3.82ns ± 3%  -10.63%  (p=0.000 n=91+89)
SubVW/2-8         4.61ns ± 1%    4.48ns ± 3%   -2.67%  (p=0.000 n=94+96)
SubVW/3-8         5.54ns ± 1%    5.81ns ± 4%   +4.87%  (p=0.000 n=92+97)
SubVW/4-8         6.20ns ± 1%    6.08ns ± 2%   -1.99%  (p=0.000 n=71+88)
SubVW/5-8         6.91ns ± 3%    6.64ns ± 1%   -3.90%  (p=0.000 n=97+70)
SubVW/10-8        9.85ns ± 2%    9.62ns ± 0%   -2.31%  (p=0.000 n=82+62)
SubVW/100-8       91.1ns ± 1%    90.9ns ± 3%   -0.14%  (p=0.010 n=71+93)
SubVW/1000-8       859ns ± 3%     867ns ± 1%   +0.98%  (p=0.000 n=99+78)
SubVW/10000-8     8.54µs ± 5%    8.57µs ± 2%   +0.38%  (p=0.007 n=98+92)
SubVW/100000-8    84.5µs ± 3%    84.6µs ± 3%     ~     (p=0.334 n=95+94)

name                old time/op    new time/op    delta
AddMulVVW/1-8         5.43ns ± 3%    4.36ns ± 2%  -19.67%  (p=0.000 n=95+94)
AddMulVVW/2-8         6.56ns ± 4%    6.11ns ± 1%   -6.90%  (p=0.000 n=91+91)
AddMulVVW/3-8         8.00ns ± 1%    7.80ns ± 4%   -2.52%  (p=0.000 n=83+95)
AddMulVVW/4-8         9.81ns ± 2%    9.53ns ± 1%   -2.86%  (p=0.000 n=77+64)
AddMulVVW/5-8         11.4ns ± 3%    11.3ns ± 5%   -0.89%  (p=0.000 n=95+97)
AddMulVVW/10-8        18.9ns ± 5%    19.1ns ± 5%   +0.89%  (p=0.000 n=91+94)
AddMulVVW/100-8        165ns ± 5%     165ns ± 4%     ~     (p=0.427 n=97+98)
AddMulVVW/1000-8      1.56µs ± 3%    1.56µs ± 4%     ~     (p=0.167 n=98+96)
AddMulVVW/10000-8     15.7µs ± 5%    15.6µs ± 5%   -0.31%  (p=0.044 n=95+97)
AddMulVVW/100000-8     156µs ± 3%     157µs ± 8%     ~     (p=0.373 n=72+99)

Change-Id: Ibc720785d5b95f6a797103b1363843205f4d56bf
Reviewed-on: https://go-review.googlesource.com/c/go/+/164966
Run-TryBot: Josh Bleecher Snyder <josharian@gmail.com>
Reviewed-by: Robert Griesemer <gri@golang.org>

src/math/big/arith.go

@@ -21,6 +21,18 @@ const (
 	_M = _B - 1        // digit mask
 )
 
+// Many of the loops in this file are of the form
+//   for i := 0; i < len(z) && i < len(x) && i < len(y); i++
+// i < len(z) is the real condition.
+// However, checking i < len(x) && i < len(y) as well is faster than
+// having the compiler do a bounds check in the body of the loop;
+// remarkably it is even faster than hoisting the bounds check
+// out of the loop, by doing something like
+//   _, _ = x[len(z)-1], y[len(z)-1]
+// There are other ways to hoist the bounds check out of the loop,
+// but the compiler's BCE isn't powerful enough for them (yet?).
+// See the discussion in CL 164966.
+
 // ----------------------------------------------------------------------------
 // Elementary operations on words
 //
@@ -54,7 +66,8 @@ func divWW_g(u1, u0, v Word) (q, r Word) {
 
 // The resulting carry c is either 0 or 1.
 func addVV_g(z, x, y []Word) (c Word) {
-	for i := range x[:len(z)] {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
 		zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
 		z[i] = Word(zi)
 		c = Word(cc)
@@ -64,7 +77,8 @@ func addVV_g(z, x, y []Word) (c Word) {
 
 // The resulting carry c is either 0 or 1.
 func subVV_g(z, x, y []Word) (c Word) {
-	for i := range x[:len(z)] {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
 		zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
 		z[i] = Word(zi)
 		c = Word(cc)
@@ -75,7 +89,8 @@ func subVV_g(z, x, y []Word) (c Word) {
 // The resulting carry c is either 0 or 1.
 func addVW_g(z, x []Word, y Word) (c Word) {
 	c = y
-	for i := range x[:len(z)] {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x); i++ {
 		zi, cc := bits.Add(uint(x[i]), uint(c), 0)
 		z[i] = Word(zi)
 		c = Word(cc)
@@ -85,7 +100,8 @@ func addVW_g(z, x []Word, y Word) (c Word) {
 
 func subVW_g(z, x []Word, y Word) (c Word) {
 	c = y
-	for i := range x[:len(z)] {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x); i++ {
 		zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
 		z[i] = Word(zi)
 		c = Word(cc)
@@ -139,14 +155,16 @@ func shrVU_g(z, x []Word, s uint) (c Word) {
 
 func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
 	c = r
-	for i := range z {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x); i++ {
 		c, z[i] = mulAddWWW_g(x[i], y, c)
 	}
 	return
 }
 
 func addMulVVW_g(z, x []Word, y Word) (c Word) {
-	for i := range z {
+	// The comment near the top of this file discusses this for loop condition.
+	for i := 0; i < len(z) && i < len(x); i++ {
 		z1, z0 := mulAddWWW_g(x[i], y, z[i])
 		lo, cc := bits.Add(uint(z0), uint(c), 0)
 		c, z[i] = Word(cc), Word(lo)