libgo: update to Go1.17rc2

Reviewed-on: https://go-review.googlesource.com/c/gofrontend/+/341629

51 files changed, 1571 insertions, 504 deletions

diff --git a/libgo/go/math/all_test.go b/libgo/go/math/all_test.go
index 3aae037..55c805e 100644
--- a/libgo/go/math/all_test.go
+++ b/libgo/go/math/all_test.go
@@ -2067,6 +2067,21 @@ var fmaC = []struct{ x, y, z, want float64 }{
 	{-7.751454006381804e-05, 5.588653777189071e-308, -2.2207280111272877e-308, -2.2211612130544025e-308},
 }
 
+var sqrt32 = []float32{
+	0,
+	float32(Copysign(0, -1)),
+	float32(NaN()),
+	float32(Inf(1)),
+	float32(Inf(-1)),
+	1,
+	2,
+	-2,
+	4.9790119248836735e+00,
+	7.7388724745781045e+00,
+	-2.7688005719200159e-01,
+	-5.0106036182710749e+00,
+}
+
 func tolerance(a, b, e float64) bool {
 	// Multiplying by e here can underflow denormal values to zero.
 	// Check a==b so that at least if a and b are small and identical
@@ -3181,6 +3196,34 @@ func TestFloatMinMax(t *testing.T) {
 	}
 }
 
+func TestFloatMinima(t *testing.T) {
+	if q := float32(SmallestNonzeroFloat32 / 2); q != 0 {
+		t.Errorf("float32(SmallestNonzeroFloat32 / 2) = %g, want 0", q)
+	}
+	if q := float64(SmallestNonzeroFloat64 / 2); q != 0 {
+		t.Errorf("float64(SmallestNonzeroFloat64 / 2) = %g, want 0", q)
+	}
+}
+
+var indirectSqrt = Sqrt
+
+// TestFloat32Sqrt checks the correctness of the float32 square root optimization result.
+func TestFloat32Sqrt(t *testing.T) {
+	for _, v := range sqrt32 {
+		want := float32(indirectSqrt(float64(v)))
+		got := float32(Sqrt(float64(v)))
+		if IsNaN(float64(want)) {
+			if !IsNaN(float64(got)) {
+				t.Errorf("got=%#v want=NaN, v=%#v", got, v)
+			}
+			continue
+		}
+		if got != want {
+			t.Errorf("got=%#v want=%#v, v=%#v", got, want, v)
+		}
+	}
+}
+
 // Benchmarks
 
 // Global exported variables are used to store the
diff --git a/libgo/go/math/arith_s390x.go b/libgo/go/math/arith_s390x.go
index c28651e..6352734 100644
--- a/libgo/go/math/arith_s390x.go
+++ b/libgo/go/math/arith_s390x.go
@@ -2,78 +2,172 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
 
 package math
 
 import "internal/cpu"
 
+func expTrampolineSetup(x float64) float64
+func expAsm(x float64) float64
+
+func logTrampolineSetup(x float64) float64
+func logAsm(x float64) float64
+
+// Below here all functions are grouped in stubs.go for other
+// architectures.
+
+const haveArchLog10 = true
+
+func archLog10(x float64) float64
 func log10TrampolineSetup(x float64) float64
 func log10Asm(x float64) float64
 
+const haveArchCos = true
+
+func archCos(x float64) float64
 func cosTrampolineSetup(x float64) float64
 func cosAsm(x float64) float64
 
+const haveArchCosh = true
+
+func archCosh(x float64) float64
 func coshTrampolineSetup(x float64) float64
 func coshAsm(x float64) float64
 
+const haveArchSin = true
+
+func archSin(x float64) float64
 func sinTrampolineSetup(x float64) float64
 func sinAsm(x float64) float64
 
+const haveArchSinh = true
+
+func archSinh(x float64) float64
 func sinhTrampolineSetup(x float64) float64
 func sinhAsm(x float64) float64
 
+const haveArchTanh = true
+
+func archTanh(x float64) float64
 func tanhTrampolineSetup(x float64) float64
 func tanhAsm(x float64) float64
 
+const haveArchLog1p = true
+
+func archLog1p(x float64) float64
 func log1pTrampolineSetup(x float64) float64
 func log1pAsm(x float64) float64
 
+const haveArchAtanh = true
+
+func archAtanh(x float64) float64
 func atanhTrampolineSetup(x float64) float64
 func atanhAsm(x float64) float64
 
+const haveArchAcos = true
+
+func archAcos(x float64) float64
 func acosTrampolineSetup(x float64) float64
 func acosAsm(x float64) float64
 
+const haveArchAcosh = true
+
+func archAcosh(x float64) float64
 func acoshTrampolineSetup(x float64) float64
 func acoshAsm(x float64) float64
 
+const haveArchAsin = true
+
+func archAsin(x float64) float64
 func asinTrampolineSetup(x float64) float64
 func asinAsm(x float64) float64
 
+const haveArchAsinh = true
+
+func archAsinh(x float64) float64
 func asinhTrampolineSetup(x float64) float64
 func asinhAsm(x float64) float64
 
+const haveArchErf = true
+
+func archErf(x float64) float64
 func erfTrampolineSetup(x float64) float64
 func erfAsm(x float64) float64
 
+const haveArchErfc = true
+
+func archErfc(x float64) float64
 func erfcTrampolineSetup(x float64) float64
 func erfcAsm(x float64) float64
 
+const haveArchAtan = true
+
+func archAtan(x float64) float64
 func atanTrampolineSetup(x float64) float64
 func atanAsm(x float64) float64
 
+const haveArchAtan2 = true
+
+func archAtan2(y, x float64) float64
 func atan2TrampolineSetup(x, y float64) float64
 func atan2Asm(x, y float64) float64
 
+const haveArchCbrt = true
+
+func archCbrt(x float64) float64
 func cbrtTrampolineSetup(x float64) float64
 func cbrtAsm(x float64) float64
 
-func logTrampolineSetup(x float64) float64
-func logAsm(x float64) float64
+const haveArchTan = true
 
+func archTan(x float64) float64
 func tanTrampolineSetup(x float64) float64
 func tanAsm(x float64) float64
 
-func expTrampolineSetup(x float64) float64
-func expAsm(x float64) float64
+const haveArchExpm1 = true
 
+func archExpm1(x float64) float64
 func expm1TrampolineSetup(x float64) float64
 func expm1Asm(x float64) float64
 
+const haveArchPow = true
+
+func archPow(x, y float64) float64
 func powTrampolineSetup(x, y float64) float64
 func powAsm(x, y float64) float64
 
+const haveArchFrexp = false
+
+func archFrexp(x float64) (float64, int) {
+	panic("not implemented")
+}
+
+const haveArchLdexp = false
+
+func archLdexp(frac float64, exp int) float64 {
+	panic("not implemented")
+}
+
+const haveArchLog2 = false
+
+func archLog2(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchMod = false
+
+func archMod(x, y float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchRemainder = false
+
+func archRemainder(x, y float64) float64 {
+	panic("not implemented")
+}
+
 // hasVX reports whether the machine has the z/Architecture
 // vector facility installed and enabled.
 var hasVX = cpu.S390X.HasVX
diff --git a/libgo/go/math/asin.go b/libgo/go/math/asin.go
index 46a5fe9..ac70bcb 100644
--- a/libgo/go/math/asin.go
+++ b/libgo/go/math/asin.go
@@ -16,14 +16,13 @@ package math
 // Special cases are:
 //	Asin(±0) = ±0
 //	Asin(x) = NaN if x < -1 or x > 1
-
-//extern asin
-func libc_asin(float64) float64
-
 func Asin(x float64) float64 {
 	return libc_asin(x)
 }
 
+//extern asin
+func libc_asin(float64) float64
+
 func asin(x float64) float64 {
 	if x == 0 {
 		return x // special case
@@ -54,14 +53,13 @@ func asin(x float64) float64 {
 //
 // Special case is:
 //	Acos(x) = NaN if x < -1 or x > 1
-
-//extern acos
-func libc_acos(float64) float64
-
 func Acos(x float64) float64 {
 	return libc_acos(x)
 }
 
+//extern acos
+func libc_acos(float64) float64
+
 func acos(x float64) float64 {
 	return Pi/2 - Asin(x)
 }
diff --git a/libgo/go/math/atan.go b/libgo/go/math/atan.go
index 4c9eda4..e4c997e 100644
--- a/libgo/go/math/atan.go
+++ b/libgo/go/math/atan.go
@@ -90,12 +90,8 @@ func satan(x float64) float64 {
 // Atan returns the arctangent, in radians, of x.
 //
 // Special cases are:
-//	Atan(±0) = ±0
-//	Atan(±Inf) = ±Pi/2
-
-//extern atan
-func libc_atan(float64) float64
-
+//      Atan(±0) = ±0
+//      Atan(±Inf) = ±Pi/2
 func Atan(x float64) float64 {
 	if x == 0 {
 		return x
@@ -103,6 +99,9 @@ func Atan(x float64) float64 {
 	return libc_atan(x)
 }
 
+//extern atan
+func libc_atan(float64) float64
+
 func atan(x float64) float64 {
 	if x == 0 {
 		return x
diff --git a/libgo/go/math/big/arith.go b/libgo/go/math/big/arith.go
index 750ce8a..8f55c19 100644
--- a/libgo/go/math/big/arith.go
+++ b/libgo/go/math/big/arith.go
@@ -170,12 +170,16 @@ func shrVU_g(z, x []Word, s uint) (c Word) {
 	if len(z) == 0 {
 		return
 	}
+	if len(x) != len(z) {
+		// This is an invariant guaranteed by the caller.
+		panic("len(x) != len(z)")
+	}
 	s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
 	ŝ := _W - s
 	ŝ &= _W - 1 // ditto
 	c = x[0] << ŝ
-	for i := 0; i < len(z)-1; i++ {
-		z[i] = x[i]>>s | x[i+1]<<ŝ
+	for i := 1; i < len(z); i++ {
+		z[i-1] = x[i-1]>>s | x[i]<<ŝ
 	}
 	z[len(z)-1] = x[len(z)-1] >> s
 	return
@@ -263,20 +267,6 @@ func divWW(x1, x0, y, m Word) (q, r Word) {
 	return Word(qq), Word(r0 >> s)
 }
 
-func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
-	r = xn
-	if len(x) == 1 {
-		qq, rr := bits.Div(uint(r), uint(x[0]), uint(y))
-		z[0] = Word(qq)
-		return Word(rr)
-	}
-	rec := reciprocalWord(y)
-	for i := len(z) - 1; i >= 0; i-- {
-		z[i], r = divWW(r, x[i], y, rec)
-	}
-	return r
-}
-
 // reciprocalWord return the reciprocal of the divisor. rec = floor(( _B^2 - 1 ) / u - _B). u = d1 << nlz(d1).
 func reciprocalWord(d1 Word) Word {
 	u := uint(d1 << nlz(d1))
diff --git a/libgo/go/math/big/arith_amd64.go b/libgo/go/math/big/arith_amd64.go
index 35164f3..8e4e1a9 100644
--- a/libgo/go/math/big/arith_amd64.go
+++ b/libgo/go/math/big/arith_amd64.go
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-// +build ignore_for_gccgo
-// +build !math_big_pure_go
+//go:build ignore
+// +build ignore
 
 package big
 
diff --git a/libgo/go/math/big/arith_decl.go b/libgo/go/math/big/arith_decl.go
index a14fda3..d80b92b 100644
--- a/libgo/go/math/big/arith_decl.go
+++ b/libgo/go/math/big/arith_decl.go
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
-// +build !math_big_pure_go
 
 package big
 
diff --git a/libgo/go/math/big/arith_decl_pure.go b/libgo/go/math/big/arith_decl_pure.go
index ac78985..cbfd1dd 100644
--- a/libgo/go/math/big/arith_decl_pure.go
+++ b/libgo/go/math/big/arith_decl_pure.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//-go:build math_big_pure_go
 // -build math_big_pure_go
 
 package big
diff --git a/libgo/go/math/big/arith_decl_s390x.go b/libgo/go/math/big/arith_decl_s390x.go
index 712f115..96445b0 100644
--- a/libgo/go/math/big/arith_decl_s390x.go
+++ b/libgo/go/math/big/arith_decl_s390x.go
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
-// +build !math_big_pure_go
 
 package big
 
diff --git a/libgo/go/math/big/arith_s390x_test.go b/libgo/go/math/big/arith_s390x_test.go
index be1ca7d..bda4811 100644
--- a/libgo/go/math/big/arith_s390x_test.go
+++ b/libgo/go/math/big/arith_s390x_test.go
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
-// +build s390x,!math_big_pure_go
 
 package big
 
diff --git a/libgo/go/math/big/arith_test.go b/libgo/go/math/big/arith_test.go
index 2aca0ef..7b3427f 100644
--- a/libgo/go/math/big/arith_test.go
+++ b/libgo/go/math/big/arith_test.go
@@ -671,3 +671,27 @@ func BenchmarkDivWVW(b *testing.B) {
 		})
 	}
 }
+
+func BenchmarkNonZeroShifts(b *testing.B) {
+	for _, n := range benchSizes {
+		if isRaceBuilder && n > 1e3 {
+			continue
+		}
+		x := rndV(n)
+		s := uint(rand.Int63n(_W-2)) + 1 // avoid 0 and over-large shifts
+		z := make([]Word, n)
+		b.Run(fmt.Sprint(n), func(b *testing.B) {
+			b.SetBytes(int64(n * _W))
+			b.Run("shrVU", func(b *testing.B) {
+				for i := 0; i < b.N; i++ {
+					_ = shrVU(z, x, s)
+				}
+			})
+			b.Run("shlVU", func(b *testing.B) {
+				for i := 0; i < b.N; i++ {
+					_ = shlVU(z, x, s)
+				}
+			})
+		})
+	}
+}
diff --git a/libgo/go/math/big/int.go b/libgo/go/math/big/int.go
index 65f3248..7647346 100644
--- a/libgo/go/math/big/int.go
+++ b/libgo/go/math/big/int.go
@@ -425,7 +425,7 @@ func (z *Int) SetString(s string, base int) (*Int, bool) {
 	return z.setFromScanner(strings.NewReader(s), base)
 }
 
-// setFromScanner implements SetString given an io.BytesScanner.
+// setFromScanner implements SetString given an io.ByteScanner.
 // For documentation see comments of SetString.
 func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
 	if _, _, err := z.scan(r, base); err != nil {
diff --git a/libgo/go/math/big/link_test.go b/libgo/go/math/big/link_test.go
index 42f9cef..6e33aa5 100644
--- a/libgo/go/math/big/link_test.go
+++ b/libgo/go/math/big/link_test.go
@@ -42,7 +42,7 @@ func main() {}
 	if err != nil {
 		t.Fatalf("nm: %v, %s", err, nm)
 	}
-	const want = "runtime.(*Frames).Next"
+	const want = "runtime.main"
 	if !bytes.Contains(nm, []byte(want)) {
 		// Test the test.
 		t.Errorf("expected symbol %q not found", want)
diff --git a/libgo/go/math/big/nat.go b/libgo/go/math/big/nat.go
index bbd6c88..140c619 100644
--- a/libgo/go/math/big/nat.go
+++ b/libgo/go/math/big/nat.go
@@ -631,48 +631,6 @@ func (z nat) mulRange(a, b uint64) nat {
 	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
 }
 
-// q = (x-r)/y, with 0 <= r < y
-func (z nat) divW(x nat, y Word) (q nat, r Word) {
-	m := len(x)
-	switch {
-	case y == 0:
-		panic("division by zero")
-	case y == 1:
-		q = z.set(x) // result is x
-		return
-	case m == 0:
-		q = z[:0] // result is 0
-		return
-	}
-	// m > 0
-	z = z.make(m)
-	r = divWVW(z, 0, x, y)
-	q = z.norm()
-	return
-}
-
-func (z nat) div(z2, u, v nat) (q, r nat) {
-	if len(v) == 0 {
-		panic("division by zero")
-	}
-
-	if u.cmp(v) < 0 {
-		q = z[:0]
-		r = z2.set(u)
-		return
-	}
-
-	if len(v) == 1 {
-		var r2 Word
-		q, r2 = z.divW(u, v[0])
-		r = z2.setWord(r2)
-		return
-	}
-
-	q, r = z.divLarge(z2, u, v)
-	return
-}
-
 // getNat returns a *nat of len n. The contents may not be zero.
 // The pool holds *nat to avoid allocation when converting to interface{}.
 func getNat(n int) *nat {
@@ -693,276 +651,6 @@ func putNat(x *nat) {
 
 var natPool sync.Pool
 
-// 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:
-//    len(vIn) >= 2
-//    len(uIn) >= len(vIn)
-//    u must not alias z
-func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
-	n := len(vIn)
-	m := len(uIn) - n
-
-	// D1.
-	shift := nlz(vIn[n-1])
-	// do not modify vIn, it may be used by another goroutine simultaneously
-	vp := getNat(n)
-	v := *vp
-	shlVU(v, vIn, shift)
-
-	// u may safely alias uIn or vIn, the value of uIn is used to set u and vIn was already used
-	u = u.make(len(uIn) + 1)
-	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
-
-	// z may safely alias uIn or vIn, both values were used already
-	if alias(z, u) {
-		z = nil // z is an alias for u - cannot reuse
-	}
-	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:
-// - q is large enough to hold the quotient u / v
-//   which has a maximum length of len(u)-len(v)+1.
-func (q nat) divBasic(u, v nat) {
-	n := len(v)
-	m := len(u) - n
-
-	qhatvp := getNat(n + 1)
-	qhatv := *qhatvp
-
-	// D2.
-	vn1 := v[n-1]
-	rec := reciprocalWord(vn1)
-	for j := m; j >= 0; j-- {
-		// D3.
-		qhat := Word(_M)
-		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, rec)
-
-			// x1 | x2 = q̂v_{n-2}
-			vn2 := v[n-2]
-			x1, x2 := mulWW(qhat, vn2)
-			// test if q̂v_{n-2} > br̂ + u_{j+n-2}
-			ujn2 := u[j+n-2]
-			for greaterThan(x1, x2, rhat, ujn2) {
-				qhat--
-				prevRhat := rhat
-				rhat += vn1
-				// v[n-1] >= 0, so this tests for overflow.
-				if rhat < prevRhat {
-					break
-				}
-				x1, x2 = mulWW(qhat, vn2)
-			}
-		}
-
-		// D4.
-		// Compute the remainder u - (q̂*v) << (_W*j).
-		// The subtraction may overflow if q̂ estimate was off by one.
-		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
-		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)
-			// If n == qhl, the carry from subVV and the carry from addVV
-			// cancel out and don't affect u[j+n].
-			if n < qhl {
-				u[j+n] += c
-			}
-			qhat--
-		}
-
-		if j == m && m == len(q) && qhat == 0 {
-			continue
-		}
-		q[j] = qhat
-	}
-
-	putNat(qhatvp)
-}
-
-const divRecursiveThreshold = 100
-
-// 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)
-}
-
-// divRecursiveStep computes the division of u by v.
-// - z must be large enough to hold the quotient
-// - the quotient will overwrite z
-// - the remainder will overwrite u
-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)-s >= 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 subtract 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 - 1
-	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):], u[len(qhatv):], c)
-	}
-	if c > 0 {
-		panic("impossible")
-	}
-
-	// Done!
-	addAt(z, qhat.norm(), 0)
-}
-
 // Length of x in bits. x must be normalized.
 func (x nat) bitLen() int {
 	if i := len(x) - 1; i >= 0 {
@@ -1170,19 +858,6 @@ func (z nat) xor(x, y nat) nat {
 	return z.norm()
 }
 
-// greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
-func greaterThan(x1, x2, y1, y2 Word) bool {
-	return x1 > y1 || x1 == y1 && x2 > y2
-}
-
-// modW returns x % d.
-func (x nat) modW(d Word) (r Word) {
-	// TODO(agl): we don't actually need to store the q value.
-	var q nat
-	q = q.make(len(x))
-	return divWVW(q, 0, x, d)
-}
-
 // random creates a random integer in [0..limit), using the space in z if
 // possible. n is the bit length of limit.
 func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
diff --git a/libgo/go/math/big/natdiv.go b/libgo/go/math/big/natdiv.go
new file mode 100644
index 0000000..882bb6d
--- /dev/null
+++ b/libgo/go/math/big/natdiv.go
@@ -0,0 +1,884 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+/*
+
+Multi-precision division. Here be dragons.
+
+Given u and v, where u is n+m digits, and v is n digits (with no leading zeros),
+the goal is to return quo, rem such that u = quo*v + rem, where 0 ≤ rem < v.
+That is, quo = ⌊u/v⌋ where ⌊x⌋ denotes the floor (truncation to integer) of x,
+and rem = u - quo·v.
+
+
+Long Division
+
+Division in a computer proceeds the same as long division in elementary school,
+but computers are not as good as schoolchildren at following vague directions,
+so we have to be much more precise about the actual steps and what can happen.
+
+We work from most to least significant digit of the quotient, doing:
+
+ • Guess a digit q, the number of v to subtract from the current
+   section of u to zero out the topmost digit.
+ • Check the guess by multiplying q·v and comparing it against
+   the current section of u, adjusting the guess as needed.
+ • Subtract q·v from the current section of u.
+ • Add q to the corresponding section of the result quo.
+
+When all digits have been processed, the final remainder is left in u
+and returned as rem.
+
+For example, here is a sketch of dividing 5 digits by 3 digits (n=3, m=2).
+
+	                 q₂ q₁ q₀
+	         _________________
+	v₂ v₁ v₀ ) u₄ u₃ u₂ u₁ u₀
+	           ↓  ↓  ↓  |  |
+	          [u₄ u₃ u₂]|  |
+	        - [  q₂·v  ]|  |
+	        ----------- ↓  |
+	          [  rem  | u₁]|
+	        - [    q₁·v   ]|
+	           ----------- ↓
+	             [  rem  | u₀]
+	           - [    q₀·v   ]
+	              ------------
+	                [  rem   ]
+
+Instead of creating new storage for the remainders and copying digits from u
+as indicated by the arrows, we use u's storage directly as both the source
+and destination of the subtractions, so that the remainders overwrite
+successive overlapping sections of u as the division proceeds, using a slice
+of u to identify the current section. This avoids all the copying as well as
+shifting of remainders.
+
+Division of u with n+m digits by v with n digits (in base B) can in general
+produce at most m+1 digits, because:
+
+  • u < B^(n+m)               [B^(n+m) has n+m+1 digits]
+  • v ≥ B^(n-1)               [B^(n-1) is the smallest n-digit number]
+  • u/v < B^(n+m) / B^(n-1)   [divide bounds for u, v]
+  • u/v < B^(m+1)             [simplify]
+
+The first step is special: it takes the top n digits of u and divides them by
+the n digits of v, producing the first quotient digit and an n-digit remainder.
+In the example, q₂ = ⌊u₄u₃u₂ / v⌋.
+
+The first step divides n digits by n digits to ensure that it produces only a
+single digit.
+
+Each subsequent step appends the next digit from u to the remainder and divides
+those n+1 digits by the n digits of v, producing another quotient digit and a
+new n-digit remainder.
+
+Subsequent steps divide n+1 digits by n digits, an operation that in general
+might produce two digits. However, as used in the algorithm, that division is
+guaranteed to produce only a single digit. The dividend is of the form
+rem·B + d, where rem is a remainder from the previous step and d is a single
+digit, so:
+
+ • rem ≤ v - 1                 [rem is a remainder from dividing by v]
+ • rem·B ≤ v·B - B             [multiply by B]
+ • d ≤ B - 1                   [d is a single digit]
+ • rem·B + d ≤ v·B - 1         [add]
+ • rem·B + d < v·B             [change ≤ to <]
+ • (rem·B + d)/v < B           [divide by v]
+
+
+Guess and Check
+
+At each step we need to divide n+1 digits by n digits, but this is for the
+implementation of division by n digits, so we can't just invoke a division
+routine: we _are_ the division routine. Instead, we guess at the answer and
+then check it using multiplication. If the guess is wrong, we correct it.
+
+How can this guessing possibly be efficient? It turns out that the following
+statement (let's call it the Good Guess Guarantee) is true.
+
+If
+
+ • q = ⌊u/v⌋ where u is n+1 digits and v is n digits,
+ • q < B, and
+ • the topmost digit of v = vₙ₋₁ ≥ B/2,
+
+then q̂ = ⌊uₙuₙ₋₁ / vₙ₋₁⌋ satisfies q ≤ q̂ ≤ q+2. (Proof below.)
+
+That is, if we know the answer has only a single digit and we guess an answer
+by ignoring the bottom n-1 digits of u and v, using a 2-by-1-digit division,
+then that guess is at least as large as the correct answer. It is also not
+too much larger: it is off by at most two from the correct answer.
+
+Note that in the first step of the overall division, which is an n-by-n-digit
+division, the 2-by-1 guess uses an implicit uₙ = 0.
+
+Note that using a 2-by-1-digit division here does not mean calling ourselves
+recursively. Instead, we use an efficient direct hardware implementation of
+that operation.
+
+Note that because q is u/v rounded down, q·v must not exceed u: u ≥ q·v.
+If a guess q̂ is too big, it will not satisfy this test. Viewed a different way,
+the remainder r̂ for a given q̂ is u - q̂·v, which must be positive. If it is
+negative, then the guess q̂ is too big.
+
+This gives us a way to compute q. First compute q̂ with 2-by-1-digit division.
+Then, while u < q̂·v, decrement q̂; this loop executes at most twice, because
+q̂ ≤ q+2.
+
+
+Scaling Inputs
+
+The Good Guess Guarantee requires that the top digit of v (vₙ₋₁) be at least B/2.
+For example in base 10, ⌊172/19⌋ = 9, but ⌊18/1⌋ = 18: the guess is wildly off
+because the first digit 1 is smaller than B/2 = 5.
+
+We can ensure that v has a large top digit by multiplying both u and v by the
+right amount. Continuing the example, if we multiply both 172 and 19 by 3, we
+now have ⌊516/57⌋, the leading digit of v is now ≥ 5, and sure enough
+⌊51/5⌋ = 10 is much closer to the correct answer 9. It would be easier here
+to multiply by 4, because that can be done with a shift. Specifically, we can
+always count the number of leading zeros i in the first digit of v and then
+shift both u and v left by i bits.
+
+Having scaled u and v, the value ⌊u/v⌋ is unchanged, but the remainder will
+be scaled: 172 mod 19 is 1, but 516 mod 57 is 3. We have to divide the remainder
+by the scaling factor (shifting right i bits) when we finish.
+
+Note that these shifts happen before and after the entire division algorithm,
+not at each step in the per-digit iteration.
+
+Note the effect of scaling inputs on the size of the possible quotient.
+In the scaled u/v, u can gain a digit from scaling; v never does, because we
+pick the scaling factor to make v's top digit larger but without overflowing.
+If u and v have n+m and n digits after scaling, then:
+
+  • u < B^(n+m)               [B^(n+m) has n+m+1 digits]
+  • v ≥ B^n / 2               [vₙ₋₁ ≥ B/2, so vₙ₋₁·B^(n-1) ≥ B^n/2]
+  • u/v < B^(n+m) / (B^n / 2) [divide bounds for u, v]
+  • u/v < 2 B^m               [simplify]
+
+The quotient can still have m+1 significant digits, but if so the top digit
+must be a 1. This provides a different way to handle the first digit of the
+result: compare the top n digits of u against v and fill in either a 0 or a 1.
+
+
+Refining Guesses
+
+Before we check whether u < q̂·v, we can adjust our guess to change it from
+q̂ = ⌊uₙuₙ₋₁ / vₙ₋₁⌋ into the refined guess ⌊uₙuₙ₋₁uₙ₋₂ / vₙ₋₁vₙ₋₂⌋.
+Although not mentioned above, the Good Guess Guarantee also promises that this
+3-by-2-digit division guess is more precise and at most one away from the real
+answer q. The improvement from the 2-by-1 to the 3-by-2 guess can also be done
+without n-digit math.
+
+If we have a guess q̂ = ⌊uₙuₙ₋₁ / vₙ₋₁⌋ and we want to see if it also equal to
+⌊uₙuₙ₋₁uₙ₋₂ / vₙ₋₁vₙ₋₂⌋, we can use the same check we would for the full division:
+if uₙuₙ₋₁uₙ₋₂ < q̂·vₙ₋₁vₙ₋₂, then the guess is too large and should be reduced.
+
+Checking uₙuₙ₋₁uₙ₋₂ < q̂·vₙ₋₁vₙ₋₂ is the same as uₙuₙ₋₁uₙ₋₂ - q̂·vₙ₋₁vₙ₋₂ < 0,
+and
+
+	uₙuₙ₋₁uₙ₋₂ - q̂·vₙ₋₁vₙ₋₂ = (uₙuₙ₋₁·B + uₙ₋₂) - q̂·(vₙ₋₁·B + vₙ₋₂)
+	                          [splitting off the bottom digit]
+	                      = (uₙuₙ₋₁ - q̂·vₙ₋₁)·B + uₙ₋₂ - q̂·vₙ₋₂
+	                          [regrouping]
+
+The expression (uₙuₙ₋₁ - q̂·vₙ₋₁) is the remainder of uₙuₙ₋₁ / vₙ₋₁.
+If the initial guess returns both q̂ and its remainder r̂, then checking
+whether uₙuₙ₋₁uₙ₋₂ < q̂·vₙ₋₁vₙ₋₂ is the same as checking r̂·B + uₙ₋₂ < q̂·vₙ₋₂.
+
+If we find that r̂·B + uₙ₋₂ < q̂·vₙ₋₂, then we can adjust the guess by
+decrementing q̂ and adding vₙ₋₁ to r̂. We repeat until r̂·B + uₙ₋₂ ≥ q̂·vₙ₋₂.
+(As before, this fixup is only needed at most twice.)
+
+Now that q̂ = ⌊uₙuₙ₋₁uₙ₋₂ / vₙ₋₁vₙ₋₂⌋, as mentioned above it is at most one
+away from the correct q, and we've avoided doing any n-digit math.
+(If we need the new remainder, it can be computed as r̂·B + uₙ₋₂ - q̂·vₙ₋₂.)
+
+The final check u < q̂·v and the possible fixup must be done at full precision.
+For random inputs, a fixup at this step is exceedingly rare: the 3-by-2 guess
+is not often wrong at all. But still we must do the check. Note that since the
+3-by-2 guess is off by at most 1, it can be convenient to perform the final
+u < q̂·v as part of the computation of the remainder r = u - q̂·v. If the
+subtraction underflows, decremeting q̂ and adding one v back to r is enough to
+arrive at the final q, r.
+
+That's the entirety of long division: scale the inputs, and then loop over
+each output position, guessing, checking, and correcting the next output digit.
+
+For a 2n-digit number divided by an n-digit number (the worst size-n case for
+division complexity), this algorithm uses n+1 iterations, each of which must do
+at least the 1-by-n-digit multiplication q̂·v. That's O(n) iterations of
+O(n) time each, so O(n²) time overall.
+
+
+Recursive Division
+
+For very large inputs, it is possible to improve on the O(n²) algorithm.
+Let's call a group of n/2 real digits a (very) “wide digit”. We can run the
+standard long division algorithm explained above over the wide digits instead of
+the actual digits. This will result in many fewer steps, but the math involved in
+each step is more work.
+
+Where basic long division uses a 2-by-1-digit division to guess the initial q̂,
+the new algorithm must use a 2-by-1-wide-digit division, which is of course
+really an n-by-n/2-digit division. That's OK: if we implement n-digit division
+in terms of n/2-digit division, the recursion will terminate when the divisor
+becomes small enough to handle with standard long division or even with the
+2-by-1 hardware instruction.
+
+For example, here is a sketch of dividing 10 digits by 4, proceeding with
+wide digits corresponding to two regular digits. The first step, still special,
+must leave off a (regular) digit, dividing 5 by 4 and producing a 4-digit
+remainder less than v. The middle steps divide 6 digits by 4, guaranteed to
+produce two output digits each (one wide digit) with 4-digit remainders.
+The final step must use what it has: the 4-digit remainder plus one more,
+5 digits to divide by 4.
+
+	                       q₆ q₅ q₄ q₃ q₂ q₁ q₀
+	            _______________________________
+	v₃ v₂ v₁ v₀ ) u₉ u₈ u₇ u₆ u₅ u₄ u₃ u₂ u₁ u₀
+	              ↓  ↓  ↓  ↓  ↓  |  |  |  |  |
+	             [u₉ u₈ u₇ u₆ u₅]|  |  |  |  |
+	           - [    q₆q₅·v    ]|  |  |  |  |
+	           ----------------- ↓  ↓  |  |  |
+	                [    rem    |u₄ u₃]|  |  |
+	              - [     q₄q₃·v      ]|  |  |
+	              -------------------- ↓  ↓  |
+	                      [    rem    |u₂ u₁]|
+	                    - [     q₂q₁·v      ]|
+	                    -------------------- ↓
+	                            [    rem    |u₀]
+	                          - [     q₀·v     ]
+	                          ------------------
+	                               [    rem    ]
+
+An alternative would be to look ahead to how well n/2 divides into n+m and
+adjust the first step to use fewer digits as needed, making the first step
+more special to make the last step not special at all. For example, using the
+same input, we could choose to use only 4 digits in the first step, leaving
+a full wide digit for the last step:
+
+	                       q₆ q₅ q₄ q₃ q₂ q₁ q₀
+	            _______________________________
+	v₃ v₂ v₁ v₀ ) u₉ u₈ u₇ u₆ u₅ u₄ u₃ u₂ u₁ u₀
+	              ↓  ↓  ↓  ↓  |  |  |  |  |  |
+	             [u₉ u₈ u₇ u₆]|  |  |  |  |  |
+	           - [    q₆·v   ]|  |  |  |  |  |
+	           -------------- ↓  ↓  |  |  |  |
+	             [    rem    |u₅ u₄]|  |  |  |
+	           - [     q₅q₄·v      ]|  |  |  |
+	           -------------------- ↓  ↓  |  |
+	                   [    rem    |u₃ u₂]|  |
+	                 - [     q₃q₂·v      ]|  |
+	                 -------------------- ↓  ↓
+	                         [    rem    |u₁ u₀]
+	                       - [     q₁q₀·v      ]
+	                       ---------------------
+	                               [    rem    ]
+
+Today, the code in divRecursiveStep works like the first example. Perhaps in
+the future we will make it work like the alternative, to avoid a special case
+in the final iteration.
+
+Either way, each step is a 3-by-2-wide-digit division approximated first by
+a 2-by-1-wide-digit division, just as we did for regular digits in long division.
+Because the actual answer we want is a 3-by-2-wide-digit division, instead of
+multiplying q̂·v directly during the fixup, we can use the quick refinement
+from long division (an n/2-by-n/2 multiply) to correct q to its actual value
+and also compute the remainder (as mentioned above), and then stop after that,
+never doing a full n-by-n multiply.
+
+Instead of using an n-by-n/2-digit division to produce n/2 digits, we can add
+(not discard) one more real digit, doing an (n+1)-by-(n/2+1)-digit division that
+produces n/2+1 digits. That single extra digit tightens the Good Guess Guarantee
+to q ≤ q̂ ≤ q+1 and lets us drop long division's special treatment of the first
+digit. These benefits are discussed more after the Good Guess Guarantee proof
+below.
+
+
+How Fast is Recursive Division?
+
+For a 2n-by-n-digit division, this algorithm runs a 4-by-2 long division over
+wide digits, producing two wide digits plus a possible leading regular digit 1,
+which can be handled without a recursive call. That is, the algorithm uses two
+full iterations, each using an n-by-n/2-digit division and an n/2-by-n/2-digit
+multiplication, along with a few n-digit additions and subtractions. The standard
+n-by-n-digit multiplication algorithm requires O(n²) time, making the overall
+algorithm require time T(n) where
+
+	T(n) = 2T(n/2) + O(n) + O(n²)
+
+which, by the Bentley-Haken-Saxe theorem, ends up reducing to T(n) = O(n²).
+This is not an improvement over regular long division.
+
+When the number of digits n becomes large enough, Karatsuba's algorithm for
+multiplication can be used instead, which takes O(n^log₂3) = O(n^1.6) time.
+(Karatsuba multiplication is implemented in func karatsuba in nat.go.)
+That makes the overall recursive division algorithm take O(n^1.6) time as well,
+which is an improvement, but again only for large enough numbers.
+
+It is not critical to make sure that every recursion does only two recursive
+calls. While in general the number of recursive calls can change the time
+analysis, in this case doing three calls does not change the analysis:
+
+	T(n) = 3T(n/2) + O(n) + O(n^log₂3)
+
+ends up being T(n) = O(n^log₂3). Because the Karatsuba multiplication taking
+time O(n^log₂3) is itself doing 3 half-sized recursions, doing three for the
+division does not hurt the asymptotic performance. Of course, it is likely
+still faster in practice to do two.
+
+
+Proof of the Good Guess Guarantee
+
+Given numbers x, y, let us break them into the quotients and remainders when
+divided by some scaling factor S, with the added constraints that the quotient
+x/y and the high part of y are both less than some limit T, and that the high
+part of y is at least half as big as T.
+
+	x₁ = ⌊x/S⌋        y₁ = ⌊y/S⌋
+	x₀ = x mod S      y₀ = y mod S
+
+	x  = x₁·S + x₀    0 ≤ x₀ < S    x/y < T
+	y  = y₁·S + y₀    0 ≤ y₀ < S    T/2 ≤ y₁ < T
+
+And consider the two truncated quotients:
+
+	q = ⌊x/y⌋
+	q̂ = ⌊x₁/y₁⌋
+
+We will prove that q ≤ q̂ ≤ q+2.
+
+The guarantee makes no real demands on the scaling factor S: it is simply the
+magnitude of the digits cut from both x and y to produce x₁ and y₁.
+The guarantee makes only limited demands on T: it must be large enough to hold
+the quotient x/y, and y₁ must have roughly the same size.
+
+To apply to the earlier discussion of 2-by-1 guesses in long division,
+we would choose:
+
+	S  = Bⁿ⁻¹
+	T  = B
+	x  = u
+	x₁ = uₙuₙ₋₁
+	x₀ = uₙ₋₂...u₀
+	y  = v
+	y₁ = vₙ₋₁
+	y₀ = vₙ₋₂...u₀
+
+These simpler variables avoid repeating those longer expressions in the proof.
+
+Note also that, by definition, truncating division ⌊x/y⌋ satisfies
+
+	x/y - 1 < ⌊x/y⌋ ≤ x/y.
+
+This fact will be used a few times in the proofs.
+
+Proof that q ≤ q̂:
+
+	q̂·y₁ = ⌊x₁/y₁⌋·y₁                      [by definition, q̂ = ⌊x₁/y₁⌋]
+	     > (x₁/y₁ - 1)·y₁                  [x₁/y₁ - 1 < ⌊x₁/y₁⌋]
+	     = x₁ - y₁                         [distribute y₁]
+
+	So q̂·y₁ > x₁ - y₁.
+	Since q̂·y₁ is an integer, q̂·y₁ ≥ x₁ - y₁ + 1.
+
+	q̂ - q = q̂ - ⌊x/y⌋                      [by definition, q = ⌊x/y⌋]
+	      ≥ q̂ - x/y                        [⌊x/y⌋ < x/y]
+	      = (1/y)·(q̂·y - x)                [factor out 1/y]
+	      ≥ (1/y)·(q̂·y₁·S - x)             [y = y₁·S + y₀ ≥ y₁·S]
+	      ≥ (1/y)·((x₁ - y₁ + 1)·S - x)    [above: q̂·y₁ ≥ x₁ - y₁ + 1]
+	      = (1/y)·(x₁·S - y₁·S + S - x)    [distribute S]
+	      = (1/y)·(S - x₀ - y₁·S)          [-x = -x₁·S - x₀]
+	      > -y₁·S / y                      [x₀ < S, so S - x₀ < 0; drop it]
+	      ≥ -1                             [y₁·S ≤ y]
+
+	So q̂ - q > -1.
+	Since q̂ - q is an integer, q̂ - q ≥ 0, or equivalently q ≤ q̂.
+
+Proof that q̂ ≤ q+2:
+
+	x₁/y₁ - x/y = x₁·S/y₁·S - x/y          [multiply left term by S/S]
+	            ≤ x/y₁·S - x/y             [x₁S ≤ x]
+	            = (x/y)·(y/y₁·S - 1)       [factor out x/y]
+	            = (x/y)·((y - y₁·S)/y₁·S)  [move -1 into y/y₁·S fraction]
+	            = (x/y)·(y₀/y₁·S)          [y - y₁·S = y₀]
+	            = (x/y)·(1/y₁)·(y₀/S)      [factor out 1/y₁]
+	            < (x/y)·(1/y₁)             [y₀ < S, so y₀/S < 1]
+	            ≤ (x/y)·(2/T)              [y₁ ≥ T/2, so 1/y₁ ≤ 2/T]
+	            < T·(2/T)                  [x/y < T]
+	            = 2                        [T·(2/T) = 2]
+
+	So x₁/y₁ - x/y < 2.
+
+	q̂ - q = ⌊x₁/y₁⌋ - q                    [by definition, q̂ = ⌊x₁/y₁⌋]
+	      = ⌊x₁/y₁⌋ - ⌊x/y⌋                [by definition, q = ⌊x/y⌋]
+	      ≤ x₁/y₁ - ⌊x/y⌋                  [⌊x₁/y₁⌋ ≤ x₁/y₁]
+	      < x₁/y₁ - (x/y - 1)              [⌊x/y⌋ > x/y - 1]
+	      = (x₁/y₁ - x/y) + 1              [regrouping]
+	      < 2 + 1                          [above: x₁/y₁ - x/y < 2]
+	      = 3
+
+	So q̂ - q < 3.
+	Since q̂ - q is an integer, q̂ - q ≤ 2.
+
+Note that when x/y < T/2, the bounds tighten to x₁/y₁ - x/y < 1 and therefore
+q̂ - q ≤ 1.
+
+Note also that in the general case 2n-by-n division where we don't know that
+x/y < T, we do know that x/y < 2T, yielding the bound q̂ - q ≤ 4. So we could
+remove the special case first step of long division as long as we allow the
+first fixup loop to run up to four times. (Using a simple comparison to decide
+whether the first digit is 0 or 1 is still more efficient, though.)
+
+Finally, note that when dividing three leading base-B digits by two (scaled),
+we have T = B² and x/y < B = T/B, a much tighter bound than x/y < T.
+This in turn yields the much tighter bound x₁/y₁ - x/y < 2/B. This means that
+⌊x₁/y₁⌋ and ⌊x/y⌋ can only differ when x/y is less than 2/B greater than an
+integer. For random x and y, the chance of this is 2/B, or, for large B,
+approximately zero. This means that after we produce the 3-by-2 guess in the
+long division algorithm, the fixup loop essentially never runs.
+
+In the recursive algorithm, the extra digit in (2·⌊n/2⌋+1)-by-(⌊n/2⌋+1)-digit
+division has exactly the same effect: the probability of needing a fixup is the
+same 2/B. Even better, we can allow the general case x/y < 2T and the fixup
+probability only grows to 4/B, still essentially zero.
+
+
+References
+
+There are no great references for implementing long division; thus this comment.
+Here are some notes about what to expect from the obvious references.
+
+Knuth Volume 2 (Seminumerical Algorithms) section 4.3.1 is the usual canonical
+reference for long division, but that entire series is highly compressed, never
+repeating a necessary fact and leaving important insights to the exercises.
+For example, no rationale whatsoever is given for the calculation that extends
+q̂ from a 2-by-1 to a 3-by-2 guess, nor why it reduces the error bound.
+The proof that the calculation even has the desired effect is left to exercises.
+The solutions to those exercises provided at the back of the book are entirely
+calculations, still with no explanation as to what is going on or how you would
+arrive at the idea of doing those exact calculations. Nowhere is it mentioned
+that this test extends the 2-by-1 guess into a 3-by-2 guess. The proof of the
+Good Guess Guarantee is only for the 2-by-1 guess and argues by contradiction,
+making it difficult to understand how modifications like adding another digit
+or adjusting the quotient range affects the overall bound.
+
+All that said, Knuth remains the canonical reference. It is dense but packed
+full of information and references, and the proofs are simpler than many other
+presentations. The proofs above are reworkings of Knuth's to remove the
+arguments by contradiction and add explanations or steps that Knuth omitted.
+But beware of errors in older printings. Take the published errata with you.
+
+Brinch Hansen's “Multiple-length Division Revisited: a Tour of the Minefield”
+starts with a blunt critique of Knuth's presentation (among others) and then
+presents a more detailed and easier to follow treatment of long division,
+including an implementation in Pascal. But the algorithm and implementation
+work entirely in terms of 3-by-2 division, which is much less useful on modern
+hardware than an algorithm using 2-by-1 division. The proofs are a bit too
+focused on digit counting and seem needlessly complex, especially compared to
+the ones given above.
+
+Burnikel and Ziegler's “Fast Recursive Division” introduced the key insight of
+implementing division by an n-digit divisor using recursive calls to division
+by an n/2-digit divisor, relying on Karatsuba multiplication to yield a
+sub-quadratic run time. However, the presentation decisions are made almost
+entirely for the purpose of simplifying the run-time analysis, rather than
+simplifying the presentation. Instead of a single algorithm that loops over
+quotient digits, the paper presents two mutually-recursive algorithms, for
+2n-by-n and 3n-by-2n. The paper also does not present any general (n+m)-by-n
+algorithm.
+
+The proofs in the paper are remarkably complex, especially considering that
+the algorithm is at its core just long division on wide digits, so that the
+usual long division proofs apply essentially unaltered.
+*/
+
+package big
+
+import "math/bits"
+
+// div returns q, r such that q = ⌊u/v⌋ and r = u%v = u - q·v.
+// It uses z and z2 as the storage for q and r.
+func (z nat) div(z2, u, v nat) (q, r nat) {
+	if len(v) == 0 {
+		panic("division by zero")
+	}
+
+	if u.cmp(v) < 0 {
+		q = z[:0]
+		r = z2.set(u)
+		return
+	}
+
+	if len(v) == 1 {
+		// Short division: long optimized for a single-word divisor.
+		// In that case, the 2-by-1 guess is all we need at each step.
+		var r2 Word
+		q, r2 = z.divW(u, v[0])
+		r = z2.setWord(r2)
+		return
+	}
+
+	q, r = z.divLarge(z2, u, v)
+	return
+}
+
+// divW returns q, r such that q = ⌊x/y⌋ and r = x%y = x - q·y.
+// It uses z as the storage for q.
+// Note that y is a single digit (Word), not a big number.
+func (z nat) divW(x nat, y Word) (q nat, r Word) {
+	m := len(x)
+	switch {
+	case y == 0:
+		panic("division by zero")
+	case y == 1:
+		q = z.set(x) // result is x
+		return
+	case m == 0:
+		q = z[:0] // result is 0
+		return
+	}
+	// m > 0
+	z = z.make(m)
+	r = divWVW(z, 0, x, y)
+	q = z.norm()
+	return
+}
+
+// modW returns x % d.
+func (x nat) modW(d Word) (r Word) {
+	// TODO(agl): we don't actually need to store the q value.
+	var q nat
+	q = q.make(len(x))
+	return divWVW(q, 0, x, d)
+}
+
+// divWVW overwrites z with ⌊x/y⌋, returning the remainder r.
+// The caller must ensure that len(z) = len(x).
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
+	r = xn
+	if len(x) == 1 {
+		qq, rr := bits.Div(uint(r), uint(x[0]), uint(y))
+		z[0] = Word(qq)
+		return Word(rr)
+	}
+	rec := reciprocalWord(y)
+	for i := len(z) - 1; i >= 0; i-- {
+		z[i], r = divWW(r, x[i], y, rec)
+	}
+	return r
+}
+
+// div returns q, r such that q = ⌊uIn/vIn⌋ and r = uIn%vIn = uIn - q·vIn.
+// It uses z and u as the storage for q and r.
+// The caller must ensure that len(vIn) ≥ 2 (use divW otherwise)
+// and that len(uIn) ≥ len(vIn) (the answer is 0, uIn otherwise).
+func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
+	n := len(vIn)
+	m := len(uIn) - n
+
+	// Scale the inputs so vIn's top bit is 1 (see “Scaling Inputs” above).
+	// vIn is treated as a read-only input (it may be in use by another
+	// goroutine), so we must make a copy.
+	// uIn is copied to u.
+	shift := nlz(vIn[n-1])
+	vp := getNat(n)
+	v := *vp
+	shlVU(v, vIn, shift)
+	u = u.make(len(uIn) + 1)
+	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+	// The caller should not pass aliased z and u, since those are
+	// the two different outputs, but correct just in case.
+	if alias(z, u) {
+		z = nil
+	}
+	q = z.make(m + 1)
+
+	// Use basic or recursive long division depending on size.
+	if n < divRecursiveThreshold {
+		q.divBasic(u, v)
+	} else {
+		q.divRecursive(u, v)
+	}
+	putNat(vp)
+
+	q = q.norm()
+
+	// Undo scaling of remainder.
+	shrVU(u, u, shift)
+	r = u.norm()
+
+	return q, r
+}
+
+// divBasic implements long division as described above.
+// It overwrites q with ⌊u/v⌋ and overwrites u with the remainder r.
+// q must be large enough to hold ⌊u/v⌋.
+func (q nat) divBasic(u, v nat) {
+	n := len(v)
+	m := len(u) - n
+
+	qhatvp := getNat(n + 1)
+	qhatv := *qhatvp
+
+	// Set up for divWW below, precomputing reciprocal argument.
+	vn1 := v[n-1]
+	rec := reciprocalWord(vn1)
+
+	// Compute each digit of quotient.
+	for j := m; j >= 0; j-- {
+		// Compute the 2-by-1 guess q̂.
+		// The first iteration must invent a leading 0 for u.
+		qhat := Word(_M)
+		var ujn Word
+		if j+n < len(u) {
+			ujn = u[j+n]
+		}
+
+		// ujn ≤ vn1, or else q̂ would be more than one digit.
+		// For ujn == vn1, we set q̂ to the max digit M above.
+		// Otherwise, we compute the 2-by-1 guess.
+		if ujn != vn1 {
+			var rhat Word
+			qhat, rhat = divWW(ujn, u[j+n-1], vn1, rec)
+
+			// Refine q̂ to a 3-by-2 guess. See “Refining Guesses” above.
+			vn2 := v[n-2]
+			x1, x2 := mulWW(qhat, vn2)
+			ujn2 := u[j+n-2]
+			for greaterThan(x1, x2, rhat, ujn2) { // x1x2 > r̂ u[j+n-2]
+				qhat--
+				prevRhat := rhat
+				rhat += vn1
+				// If r̂  overflows, then
+				// r̂ u[j+n-2]v[n-1] is now definitely > x1 x2.
+				if rhat < prevRhat {
+					break
+				}
+				// TODO(rsc): No need for a full mulWW.
+				// x2 += vn2; if x2 overflows, x1++
+				x1, x2 = mulWW(qhat, vn2)
+			}
+		}
+
+		// Compute q̂·v.
+		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
+		qhl := len(qhatv)
+		if j+qhl > len(u) && qhatv[n] == 0 {
+			qhl--
+		}
+
+		// Subtract q̂·v from the current section of u.
+		// If it underflows, q̂·v > u, which we fix up
+		// by decrementing q̂ and adding v back.
+		c := subVV(u[j:j+qhl], u[j:], qhatv)
+		if c != 0 {
+			c := addVV(u[j:j+n], u[j:], v)
+			// If n == qhl, the carry from subVV and the carry from addVV
+			// cancel out and don't affect u[j+n].
+			if n < qhl {
+				u[j+n] += c
+			}
+			qhat--
+		}
+
+		// Save quotient digit.
+		// Caller may know the top digit is zero and not leave room for it.
+		if j == m && m == len(q) && qhat == 0 {
+			continue
+		}
+		q[j] = qhat
+	}
+
+	putNat(qhatvp)
+}
+
+// greaterThan reports whether the two digit numbers x1 x2 > y1 y2.
+// TODO(rsc): In contradiction to most of this file, x1 is the high
+// digit and x2 is the low digit. This should be fixed.
+func greaterThan(x1, x2, y1, y2 Word) bool {
+	return x1 > y1 || x1 == y1 && x2 > y2
+}
+
+// divRecursiveThreshold is the number of divisor digits
+// at which point divRecursive is faster than divBasic.
+const divRecursiveThreshold = 100
+
+// divRecursive implements recursive division as described above.
+// It overwrites z with ⌊u/v⌋ and overwrites u with the remainder r.
+// z must be large enough to hold ⌊u/v⌋.
+// This function is just for allocating and freeing temporaries
+// around divRecursiveStep, the real implementation.
+func (z nat) divRecursive(u, v nat) {
+	// Recursion depth is (much) less than 2 log₂(len(v)).
+	// Allocate a slice of temporaries to be reused across recursion,
+	// plus one extra temporary not live across the recursion.
+	recDepth := 2 * bits.Len(uint(len(v)))
+	tmp := getNat(3 * len(v))
+	temps := make([]*nat, recDepth)
+
+	z.clear()
+	z.divRecursiveStep(u, v, 0, tmp, temps)
+
+	// Free temporaries.
+	for _, n := range temps {
+		if n != nil {
+			putNat(n)
+		}
+	}
+	putNat(tmp)
+}
+
+// divRecursiveStep is the actual implementation of recursive division.
+// It adds ⌊u/v⌋ to z and overwrites u with the remainder r.
+// z must be large enough to hold ⌊u/v⌋.
+// It uses temps[depth] (allocating if needed) as a temporary live across
+// the recursive call. It also uses tmp, but not live across the recursion.
+func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
+	// u is a subsection of the original and may have leading zeros.
+	// TODO(rsc): The v = v.norm() is useless and should be removed.
+	// We know (and require) that v's top digit is ≥ B/2.
+	u = u.norm()
+	v = v.norm()
+	if len(u) == 0 {
+		z.clear()
+		return
+	}
+
+	// Fall back to basic division if the problem is now small enough.
+	n := len(v)
+	if n < divRecursiveThreshold {
+		z.divBasic(u, v)
+		return
+	}
+
+	// Nothing to do if u is shorter than v (implies u < v).
+	m := len(u) - n
+	if m < 0 {
+		return
+	}
+
+	// We consider B digits in a row as a single wide digit.
+	// (See “Recursive Division” above.)
+	//
+	// TODO(rsc): rename B to Wide, to avoid confusion with _B,
+	// which is something entirely different.
+	// TODO(rsc): Look into whether using ⌈n/2⌉ is better than ⌊n/2⌋.
+	B := n / 2
+
+	// Allocate a nat for qhat below.
+	if temps[depth] == nil {
+		temps[depth] = getNat(n) // TODO(rsc): Can be just B+1.
+	} else {
+		*temps[depth] = temps[depth].make(B + 1)
+	}
+
+	// Compute each wide digit of the quotient.
+	//
+	// TODO(rsc): Change the loop to be
+	//	for j := (m+B-1)/B*B; j > 0; j -= B {
+	// which will make the final step a regular step, letting us
+	// delete what amounts to an extra copy of the loop body below.
+	j := m
+	for j > B {
+		// Divide u[j-B:j+n] (3 wide digits) by v (2 wide digits).
+		// First make the 2-by-1-wide-digit guess using a recursive call.
+		// Then extend the guess to the full 3-by-2 (see “Refining Guesses”).
+		//
+		// For the 2-by-1-wide-digit guess, instead of doing 2B-by-B-digit,
+		// we use a (2B+1)-by-(B+1) digit, which handles the possibility that
+		// the result has an extra leading 1 digit as well as guaranteeing
+		// that the computed q̂ will be off by at most 1 instead of 2.
+
+		// s is the number of digits to drop from the 3B- and 2B-digit chunks.
+		// We drop B-1 to be left with 2B+1 and B+1.
+		s := (B - 1)
+
+		// uu is the up-to-3B-digit section of u we are working on.
+		uu := u[j-B:]
+
+		// Compute the 2-by-1 guess q̂, leaving r̂ in uu[s:B+n].
+		qhat := *temps[depth]
+		qhat.clear()
+		qhat.divRecursiveStep(uu[s:B+n], v[s:], depth+1, tmp, temps)
+		qhat = qhat.norm()
+
+		// Extend to a 3-by-2 quotient and remainder.
+		// Because divRecursiveStep overwrote the top part of uu with
+		// the remainder r̂, the full uu already contains the equivalent
+		// of r̂·B + uₙ₋₂ from the “Refining Guesses” discussion.
+		// Subtracting q̂·vₙ₋₂ from it will compute the full-length remainder.
+		// If that subtraction underflows, q̂·v > u, which we fix up
+		// by decrementing q̂ and adding v back, same as in long division.
+
+		// TODO(rsc): Instead of subtract and fix-up, this code is computing
+		// q̂·vₙ₋₂ and decrementing q̂ until that product is ≤ u.
+		// But we can do the subtraction directly, as in the comment above
+		// and in long division, because we know that q̂ is wrong by at most one.
+		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
+	}
+
+	// TODO(rsc): Rewrite loop as described above and delete all this code.
+
+	// Now u < (v<<B), compute lower bits in the same way.
+	// Choose shift = B-1 again.
+	s := B - 1
+	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):], u[len(qhatv):], c)
+	}
+	if c > 0 {
+		panic("impossible")
+	}
+
+	// Done!
+	addAt(z, qhat.norm(), 0)
+}
diff --git a/libgo/go/math/bits/bits.go b/libgo/go/math/bits/bits.go
index 0bfe90c..d8fd6ae 100644
--- a/libgo/go/math/bits/bits.go
+++ b/libgo/go/math/bits/bits.go
@@ -8,7 +8,7 @@
 // functions for the predeclared unsigned integer types.
 package bits
 
-const uintSize = 32 << (^uint(0) >> 32 & 1) // 32 or 64
+const uintSize = 32 << (^uint(0) >> 63) // 32 or 64
 
 // UintSize is the size of a uint in bits.
 const UintSize = uintSize
diff --git a/libgo/go/math/bits/bits_errors.go b/libgo/go/math/bits/bits_errors.go
index f174bd1..53dc113 100644
--- a/libgo/go/math/bits/bits_errors.go
+++ b/libgo/go/math/bits/bits_errors.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build !compiler_bootstrap
 // +build !compiler_bootstrap
 
 package bits
diff --git a/libgo/go/math/bits/bits_errors_bootstrap.go b/libgo/go/math/bits/bits_errors_bootstrap.go
index 5df5738..4d610d3 100644
--- a/libgo/go/math/bits/bits_errors_bootstrap.go
+++ b/libgo/go/math/bits/bits_errors_bootstrap.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build compiler_bootstrap
 // +build compiler_bootstrap
 
 // This version used only for bootstrap (on this path we want
diff --git a/libgo/go/math/bits/bits_tables.go b/libgo/go/math/bits/bits_tables.go
index f1e15a0..f869b8d 100644
--- a/libgo/go/math/bits/bits_tables.go
+++ b/libgo/go/math/bits/bits_tables.go
@@ -6,78 +6,74 @@
 
 package bits
 
-var ntz8tab = [256]uint8{
-	0x08, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x06, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x07, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x06, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x05, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-	0x04, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00, 0x03, 0x00, 0x01, 0x00, 0x02, 0x00, 0x01, 0x00,
-}
+const ntz8tab = "" +
+	"\x08\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x05\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x06\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x05\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x07\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x05\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x06\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x05\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00" +
+	"\x04\x00\x01\x00\x02\x00\x01\x00\x03\x00\x01\x00\x02\x00\x01\x00"
 
-var pop8tab = [256]uint8{
-	0x00, 0x01, 0x01, 0x02, 0x01, 0x02, 0x02, 0x03, 0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04,
-	0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
-	0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
-	0x01, 0x02, 0x02, 0x03, 0x02, 0x03, 0x03, 0x04, 0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
-	0x02, 0x03, 0x03, 0x04, 0x03, 0x04, 0x04, 0x05, 0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06,
-	0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
-	0x03, 0x04, 0x04, 0x05, 0x04, 0x05, 0x05, 0x06, 0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07,
-	0x04, 0x05, 0x05, 0x06, 0x05, 0x06, 0x06, 0x07, 0x05, 0x06, 0x06, 0x07, 0x06, 0x07, 0x07, 0x08,
-}
+const pop8tab = "" +
+	"\x00\x01\x01\x02\x01\x02\x02\x03\x01\x02\x02\x03\x02\x03\x03\x04" +
+	"\x01\x02\x02\x03\x02\x03\x03\x04\x02\x03\x03\x04\x03\x04\x04\x05" +
+	"\x01\x02\x02\x03\x02\x03\x03\x04\x02\x03\x03\x04\x03\x04\x04\x05" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x01\x02\x02\x03\x02\x03\x03\x04\x02\x03\x03\x04\x03\x04\x04\x05" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x03\x04\x04\x05\x04\x05\x05\x06\x04\x05\x05\x06\x05\x06\x06\x07" +
+	"\x01\x02\x02\x03\x02\x03\x03\x04\x02\x03\x03\x04\x03\x04\x04\x05" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x03\x04\x04\x05\x04\x05\x05\x06\x04\x05\x05\x06\x05\x06\x06\x07" +
+	"\x02\x03\x03\x04\x03\x04\x04\x05\x03\x04\x04\x05\x04\x05\x05\x06" +
+	"\x03\x04\x04\x05\x04\x05\x05\x06\x04\x05\x05\x06\x05\x06\x06\x07" +
+	"\x03\x04\x04\x05\x04\x05\x05\x06\x04\x05\x05\x06\x05\x06\x06\x07" +
+	"\x04\x05\x05\x06\x05\x06\x06\x07\x05\x06\x06\x07\x06\x07\x07\x08"
 
-var rev8tab = [256]uint8{
-	0x00, 0x80, 0x40, 0xc0, 0x20, 0xa0, 0x60, 0xe0, 0x10, 0x90, 0x50, 0xd0, 0x30, 0xb0, 0x70, 0xf0,
-	0x08, 0x88, 0x48, 0xc8, 0x28, 0xa8, 0x68, 0xe8, 0x18, 0x98, 0x58, 0xd8, 0x38, 0xb8, 0x78, 0xf8,
-	0x04, 0x84, 0x44, 0xc4, 0x24, 0xa4, 0x64, 0xe4, 0x14, 0x94, 0x54, 0xd4, 0x34, 0xb4, 0x74, 0xf4,
-	0x0c, 0x8c, 0x4c, 0xcc, 0x2c, 0xac, 0x6c, 0xec, 0x1c, 0x9c, 0x5c, 0xdc, 0x3c, 0xbc, 0x7c, 0xfc,
-	0x02, 0x82, 0x42, 0xc2, 0x22, 0xa2, 0x62, 0xe2, 0x12, 0x92, 0x52, 0xd2, 0x32, 0xb2, 0x72, 0xf2,
-	0x0a, 0x8a, 0x4a, 0xca, 0x2a, 0xaa, 0x6a, 0xea, 0x1a, 0x9a, 0x5a, 0xda, 0x3a, 0xba, 0x7a, 0xfa,
-	0x06, 0x86, 0x46, 0xc6, 0x26, 0xa6, 0x66, 0xe6, 0x16, 0x96, 0x56, 0xd6, 0x36, 0xb6, 0x76, 0xf6,
-	0x0e, 0x8e, 0x4e, 0xce, 0x2e, 0xae, 0x6e, 0xee, 0x1e, 0x9e, 0x5e, 0xde, 0x3e, 0xbe, 0x7e, 0xfe,
-	0x01, 0x81, 0x41, 0xc1, 0x21, 0xa1, 0x61, 0xe1, 0x11, 0x91, 0x51, 0xd1, 0x31, 0xb1, 0x71, 0xf1,
-	0x09, 0x89, 0x49, 0xc9, 0x29, 0xa9, 0x69, 0xe9, 0x19, 0x99, 0x59, 0xd9, 0x39, 0xb9, 0x79, 0xf9,
-	0x05, 0x85, 0x45, 0xc5, 0x25, 0xa5, 0x65, 0xe5, 0x15, 0x95, 0x55, 0xd5, 0x35, 0xb5, 0x75, 0xf5,
-	0x0d, 0x8d, 0x4d, 0xcd, 0x2d, 0xad, 0x6d, 0xed, 0x1d, 0x9d, 0x5d, 0xdd, 0x3d, 0xbd, 0x7d, 0xfd,
-	0x03, 0x83, 0x43, 0xc3, 0x23, 0xa3, 0x63, 0xe3, 0x13, 0x93, 0x53, 0xd3, 0x33, 0xb3, 0x73, 0xf3,
-	0x0b, 0x8b, 0x4b, 0xcb, 0x2b, 0xab, 0x6b, 0xeb, 0x1b, 0x9b, 0x5b, 0xdb, 0x3b, 0xbb, 0x7b, 0xfb,
-	0x07, 0x87, 0x47, 0xc7, 0x27, 0xa7, 0x67, 0xe7, 0x17, 0x97, 0x57, 0xd7, 0x37, 0xb7, 0x77, 0xf7,
-	0x0f, 0x8f, 0x4f, 0xcf, 0x2f, 0xaf, 0x6f, 0xef, 0x1f, 0x9f, 0x5f, 0xdf, 0x3f, 0xbf, 0x7f, 0xff,
-}
+const rev8tab = "" +
+	"\x00\x80\x40\xc0\x20\xa0\x60\xe0\x10\x90\x50\xd0\x30\xb0\x70\xf0" +
+	"\x08\x88\x48\xc8\x28\xa8\x68\xe8\x18\x98\x58\xd8\x38\xb8\x78\xf8" +
+	"\x04\x84\x44\xc4\x24\xa4\x64\xe4\x14\x94\x54\xd4\x34\xb4\x74\xf4" +
+	"\x0c\x8c\x4c\xcc\x2c\xac\x6c\xec\x1c\x9c\x5c\xdc\x3c\xbc\x7c\xfc" +
+	"\x02\x82\x42\xc2\x22\xa2\x62\xe2\x12\x92\x52\xd2\x32\xb2\x72\xf2" +
+	"\x0a\x8a\x4a\xca\x2a\xaa\x6a\xea\x1a\x9a\x5a\xda\x3a\xba\x7a\xfa" +
+	"\x06\x86\x46\xc6\x26\xa6\x66\xe6\x16\x96\x56\xd6\x36\xb6\x76\xf6" +
+	"\x0e\x8e\x4e\xce\x2e\xae\x6e\xee\x1e\x9e\x5e\xde\x3e\xbe\x7e\xfe" +
+	"\x01\x81\x41\xc1\x21\xa1\x61\xe1\x11\x91\x51\xd1\x31\xb1\x71\xf1" +
+	"\x09\x89\x49\xc9\x29\xa9\x69\xe9\x19\x99\x59\xd9\x39\xb9\x79\xf9" +
+	"\x05\x85\x45\xc5\x25\xa5\x65\xe5\x15\x95\x55\xd5\x35\xb5\x75\xf5" +
+	"\x0d\x8d\x4d\xcd\x2d\xad\x6d\xed\x1d\x9d\x5d\xdd\x3d\xbd\x7d\xfd" +
+	"\x03\x83\x43\xc3\x23\xa3\x63\xe3\x13\x93\x53\xd3\x33\xb3\x73\xf3" +
+	"\x0b\x8b\x4b\xcb\x2b\xab\x6b\xeb\x1b\x9b\x5b\xdb\x3b\xbb\x7b\xfb" +
+	"\x07\x87\x47\xc7\x27\xa7\x67\xe7\x17\x97\x57\xd7\x37\xb7\x77\xf7" +
+	"\x0f\x8f\x4f\xcf\x2f\xaf\x6f\xef\x1f\x9f\x5f\xdf\x3f\xbf\x7f\xff"
 
-var len8tab = [256]uint8{
-	0x00, 0x01, 0x02, 0x02, 0x03, 0x03, 0x03, 0x03, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04, 0x04,
-	0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05, 0x05,
-	0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06,
-	0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06, 0x06,
-	0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
-	0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
-	0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
-	0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07, 0x07,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-	0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08, 0x08,
-}
+const len8tab = "" +
+	"\x00\x01\x02\x02\x03\x03\x03\x03\x04\x04\x04\x04\x04\x04\x04\x04" +
+	"\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05\x05" +
+	"\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06" +
+	"\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06\x06" +
+	"\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07" +
+	"\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07" +
+	"\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07" +
+	"\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07\x07" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08" +
+	"\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08\x08"
diff --git a/libgo/go/math/bits/make_examples.go b/libgo/go/math/bits/make_examples.go
index 1d3ad53..ac4004d 100644
--- a/libgo/go/math/bits/make_examples.go
+++ b/libgo/go/math/bits/make_examples.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
 
 // This program generates example_test.go.
diff --git a/libgo/go/math/bits/make_tables.go b/libgo/go/math/bits/make_tables.go
index b068d5e..867025e 100644
--- a/libgo/go/math/bits/make_tables.go
+++ b/libgo/go/math/bits/make_tables.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
 
 // This program generates bits_tables.go.
@@ -47,16 +48,15 @@ func main() {
 }
 
 func gen(w io.Writer, name string, f func(uint8) uint8) {
-	fmt.Fprintf(w, "var %s = [256]uint8{", name)
+	// Use a const string to allow the compiler to constant-evaluate lookups at constant index.
+	fmt.Fprintf(w, "const %s = \"\"+\n\"", name)
 	for i := 0; i < 256; i++ {
-		if i%16 == 0 {
-			fmt.Fprint(w, "\n\t")
-		} else {
-			fmt.Fprint(w, " ")
+		fmt.Fprintf(w, "\\x%02x", f(uint8(i)))
+		if i%16 == 15 && i != 255 {
+			fmt.Fprint(w, "\"+\n\"")
 		}
-		fmt.Fprintf(w, "%#02x,", f(uint8(i)))
 	}
-	fmt.Fprint(w, "\n}\n\n")
+	fmt.Fprint(w, "\"\n\n")
 }
 
 func ntz8(x uint8) (n uint8) {
diff --git a/libgo/go/math/cmplx/huge_test.go b/libgo/go/math/cmplx/huge_test.go
index f8e60c2..78b4231 100644
--- a/libgo/go/math/cmplx/huge_test.go
+++ b/libgo/go/math/cmplx/huge_test.go
@@ -5,6 +5,7 @@
 // Disabled for s390x because it uses assembly routines that are not
 // accurate for huge arguments.
 
+//go:build !s390x
 // +build !s390x
 
 package cmplx
diff --git a/libgo/go/math/const.go b/libgo/go/math/const.go
index 0fc8715..5ea935f 100644
--- a/libgo/go/math/const.go
+++ b/libgo/go/math/const.go
@@ -28,15 +28,19 @@ const (
 // Max is the largest finite value representable by the type.
 // SmallestNonzero is the smallest positive, non-zero value representable by the type.
 const (
-	MaxFloat32             = 3.40282346638528859811704183484516925440e+38  // 2**127 * (2**24 - 1) / 2**23
-	SmallestNonzeroFloat32 = 1.401298464324817070923729583289916131280e-45 // 1 / 2**(127 - 1 + 23)
+	MaxFloat32             = 0x1p127 * (1 + (1 - 0x1p-23)) // 3.40282346638528859811704183484516925440e+38
+	SmallestNonzeroFloat32 = 0x1p-126 * 0x1p-23            // 1.401298464324817070923729583289916131280e-45
 
-	MaxFloat64             = 1.797693134862315708145274237317043567981e+308 // 2**1023 * (2**53 - 1) / 2**52
-	SmallestNonzeroFloat64 = 4.940656458412465441765687928682213723651e-324 // 1 / 2**(1023 - 1 + 52)
+	MaxFloat64             = 0x1p1023 * (1 + (1 - 0x1p-52)) // 1.79769313486231570814527423731704356798070e+308
+	SmallestNonzeroFloat64 = 0x1p-1022 * 0x1p-52            // 4.9406564584124654417656879286822137236505980e-324
 )
 
 // Integer limit values.
 const (
+	intSize = 32 << (^uint(0) >> 63) // 32 or 64
+
+	MaxInt    = 1<<(intSize-1) - 1
+	MinInt    = -1 << (intSize - 1)
 	MaxInt8   = 1<<7 - 1
 	MinInt8   = -1 << 7
 	MaxInt16  = 1<<15 - 1
@@ -45,6 +49,7 @@ const (
 	MinInt32  = -1 << 31
 	MaxInt64  = 1<<63 - 1
 	MinInt64  = -1 << 63
+	MaxUint   = 1<<intSize - 1
 	MaxUint8  = 1<<8 - 1
 	MaxUint16 = 1<<16 - 1
 	MaxUint32 = 1<<32 - 1
diff --git a/libgo/go/math/const_test.go b/libgo/go/math/const_test.go
new file mode 100644
index 0000000..170ba6a
--- /dev/null
+++ b/libgo/go/math/const_test.go
@@ -0,0 +1,47 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math_test
+
+import (
+	"testing"
+
+	. "math"
+)
+
+func TestMaxUint(t *testing.T) {
+	if v := uint(MaxUint); v+1 != 0 {
+		t.Errorf("MaxUint should wrap around to zero: %d", v+1)
+	}
+	if v := uint8(MaxUint8); v+1 != 0 {
+		t.Errorf("MaxUint8 should wrap around to zero: %d", v+1)
+	}
+	if v := uint16(MaxUint16); v+1 != 0 {
+		t.Errorf("MaxUint16 should wrap around to zero: %d", v+1)
+	}
+	if v := uint32(MaxUint32); v+1 != 0 {
+		t.Errorf("MaxUint32 should wrap around to zero: %d", v+1)
+	}
+	if v := uint64(MaxUint64); v+1 != 0 {
+		t.Errorf("MaxUint64 should wrap around to zero: %d", v+1)
+	}
+}
+
+func TestMaxInt(t *testing.T) {
+	if v := int(MaxInt); v+1 != MinInt {
+		t.Errorf("MaxInt should wrap around to MinInt: %d", v+1)
+	}
+	if v := int8(MaxInt8); v+1 != MinInt8 {
+		t.Errorf("MaxInt8 should wrap around to MinInt8: %d", v+1)
+	}
+	if v := int16(MaxInt16); v+1 != MinInt16 {
+		t.Errorf("MaxInt16 should wrap around to MinInt16: %d", v+1)
+	}
+	if v := int32(MaxInt32); v+1 != MinInt32 {
+		t.Errorf("MaxInt32 should wrap around to MinInt32: %d", v+1)
+	}
+	if v := int64(MaxInt64); v+1 != MinInt64 {
+		t.Errorf("MaxInt64 should wrap around to MinInt64: %d", v+1)
+	}
+}
diff --git a/libgo/go/math/dim.go b/libgo/go/math/dim.go
index cf06f30..6a857bb 100644
--- a/libgo/go/math/dim.go
+++ b/libgo/go/math/dim.go
@@ -33,6 +33,9 @@ func Dim(x, y float64) float64 {
 //	Max(+0, ±0) = Max(±0, +0) = +0
 //	Max(-0, -0) = -0
 func Max(x, y float64) float64 {
+	if haveArchMax {
+		return archMax(x, y)
+	}
 	return max(x, y)
 }
 
@@ -62,6 +65,9 @@ func max(x, y float64) float64 {
 //	Min(x, NaN) = Min(NaN, x) = NaN
 //	Min(-0, ±0) = Min(±0, -0) = -0
 func Min(x, y float64) float64 {
+	if haveArchMin {
+		return archMin(x, y)
+	}
 	return min(x, y)
 }
 
diff --git a/libgo/go/math/dim_noasm.go b/libgo/go/math/dim_noasm.go
new file mode 100644
index 0000000..9a052c0
--- /dev/null
+++ b/libgo/go/math/dim_noasm.go
@@ -0,0 +1,20 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !amd64 && !arm64 && !riscv64 && !s390x
+// -build !amd64,!arm64,!riscv64,!s390x
+
+package math
+
+const haveArchMax = false
+
+func archMax(x, y float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchMin = false
+
+func archMin(x, y float64) float64 {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/exp.go b/libgo/go/math/exp.go
index f3d3cb6..7ab80b4 100644
--- a/libgo/go/math/exp.go
+++ b/libgo/go/math/exp.go
@@ -11,14 +11,13 @@ package math
 //	Exp(NaN) = NaN
 // Very large values overflow to 0 or +Inf.
 // Very small values underflow to 1.
-
-//extern exp
-func libc_exp(float64) float64
-
 func Exp(x float64) float64 {
 	return libc_exp(x)
 }
 
+//extern exp
+func libc_exp(float64) float64
+
 // The original C code, the long comment, and the constants
 // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
 // and came with this notice. The go code is a simplified
@@ -139,6 +138,9 @@ func exp(x float64) float64 {
 //
 // Special cases are the same as Exp.
 func Exp2(x float64) float64 {
+	if haveArchExp2 {
+		return archExp2(x)
+	}
 	return exp2(x)
 }
 
diff --git a/libgo/go/math/exp2_noasm.go b/libgo/go/math/exp2_noasm.go
new file mode 100644
index 0000000..9b81544
--- /dev/null
+++ b/libgo/go/math/exp2_noasm.go
@@ -0,0 +1,14 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !arm64
+// -build !arm64
+
+package math
+
+const haveArchExp2 = false
+
+func archExp2(x float64) float64 {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/exp_asm.go b/libgo/go/math/exp_amd64.go
index 11e0a61..654ccce 100644
--- a/libgo/go/math/exp_asm.go
+++ b/libgo/go/math/exp_amd64.go
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build amd64
 // +build amd64
-// +build ignore
 
 package math
 
diff --git a/libgo/go/math/exp_noasm.go b/libgo/go/math/exp_noasm.go
new file mode 100644
index 0000000..4f90604
--- /dev/null
+++ b/libgo/go/math/exp_noasm.go
@@ -0,0 +1,14 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !amd64 && !arm64 && !s390x
+// -build !amd64,!arm64,!s390x
+
+package math
+
+const haveArchExp = false
+
+func archExp(x float64) float64 {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/expm1.go b/libgo/go/math/expm1.go
index 9c6e080..bf685ae 100644
--- a/libgo/go/math/expm1.go
+++ b/libgo/go/math/expm1.go
@@ -121,10 +121,6 @@ package math
 //	Expm1(-Inf) = -1
 //	Expm1(NaN) = NaN
 // Very large values overflow to -1 or +Inf.
-
-//extern expm1
-func libc_expm1(float64) float64
-
 func Expm1(x float64) float64 {
 	if x == 0 {
 		return x
@@ -132,6 +128,9 @@ func Expm1(x float64) float64 {
 	return libc_expm1(x)
 }
 
+//extern expm1
+func libc_expm1(float64) float64
+
 func expm1(x float64) float64 {
 	const (
 		Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
diff --git a/libgo/go/math/floor.go b/libgo/go/math/floor.go
index 9115968..963c858 100644
--- a/libgo/go/math/floor.go
+++ b/libgo/go/math/floor.go
@@ -57,8 +57,10 @@ func ceil(x float64) float64 {
 //	Trunc(±0) = ±0
 //	Trunc(±Inf) = ±Inf
 //	Trunc(NaN) = NaN
-
 func Trunc(x float64) float64 {
+	if haveArchTrunc {
+		return archTrunc(x)
+	}
 	return trunc(x)
 }
 
diff --git a/libgo/go/math/floor_noasm.go b/libgo/go/math/floor_noasm.go
new file mode 100644
index 0000000..bcdc5b0
--- /dev/null
+++ b/libgo/go/math/floor_noasm.go
@@ -0,0 +1,26 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !386 && !amd64 && !arm64 && !ppc64 && !ppc64le && !s390x && !wasm
+// -build !386,!amd64,!arm64,!ppc64,!ppc64le,!s390x,!wasm
+
+package math
+
+const haveArchFloor = false
+
+func archFloor(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchCeil = false
+
+func archCeil(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchTrunc = false
+
+func archTrunc(x float64) float64 {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/fma.go b/libgo/go/math/fma.go
index db78dfa..ca0bf99 100644
--- a/libgo/go/math/fma.go
+++ b/libgo/go/math/fma.go
@@ -128,7 +128,7 @@ func FMA(x, y, z float64) float64 {
 	pe -= int32(is62zero)
 
 	// Swap addition operands so |p| >= |z|
-	if pe < ze || (pe == ze && (pm1 < zm1 || (pm1 == zm1 && pm2 < zm2))) {
+	if pe < ze || pe == ze && pm1 < zm1 {
 		ps, pe, pm1, pm2, zs, ze, zm1, zm2 = zs, ze, zm1, zm2, ps, pe, pm1, pm2
 	}
 
diff --git a/libgo/go/math/frexp.go b/libgo/go/math/frexp.go
index 4ad0aee..3c8a909 100644
--- a/libgo/go/math/frexp.go
+++ b/libgo/go/math/frexp.go
@@ -14,6 +14,9 @@ package math
 //	Frexp(±Inf) = ±Inf, 0
 //	Frexp(NaN) = NaN, 0
 func Frexp(f float64) (frac float64, exp int) {
+	if haveArchFrexp {
+		return archFrexp(f)
+	}
 	return frexp(f)
 }
 
diff --git a/libgo/go/math/huge_test.go b/libgo/go/math/huge_test.go
index 9448edc..ec81a4a 100644
--- a/libgo/go/math/huge_test.go
+++ b/libgo/go/math/huge_test.go
@@ -5,6 +5,7 @@
 // Disabled for s390x because it uses assembly routines that are not
 // accurate for huge arguments.
 
+//go:build !s390x
 // +build !s390x
 
 package math_test
diff --git a/libgo/go/math/log.go b/libgo/go/math/log.go
index cf242e8..f1c609d 100644
--- a/libgo/go/math/log.go
+++ b/libgo/go/math/log.go
@@ -77,14 +77,13 @@ package math
 //	Log(0) = -Inf
 //	Log(x < 0) = NaN
 //	Log(NaN) = NaN
-
-//extern log
-func libc_log(float64) float64
-
 func Log(x float64) float64 {
 	return libc_log(x)
 }
 
+//extern log
+func libc_log(float64) float64
+
 func log(x float64) float64 {
 	const (
 		Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
diff --git a/libgo/go/math/mod.go b/libgo/go/math/mod.go
index 436788f..e223c7a 100644
--- a/libgo/go/math/mod.go
+++ b/libgo/go/math/mod.go
@@ -18,14 +18,13 @@ package math
 //	Mod(x, 0) = NaN
 //	Mod(x, ±Inf) = x
 //	Mod(x, NaN) = NaN
-
-//extern fmod
-func libc_fmod(float64, float64) float64
-
 func Mod(x, y float64) float64 {
 	return libc_fmod(x, y)
 }
 
+//extern fmod
+func libc_fmod(float64, float64) float64
+
 func mod(x, y float64) float64 {
 	if y == 0 || IsInf(x, 0) || IsNaN(x) || IsNaN(y) {
 		return NaN()
diff --git a/libgo/go/math/modf.go b/libgo/go/math/modf.go
index b59d4b7..bf08dc6 100644
--- a/libgo/go/math/modf.go
+++ b/libgo/go/math/modf.go
@@ -11,6 +11,9 @@ package math
 //	Modf(±Inf) = ±Inf, NaN
 //	Modf(NaN) = NaN, NaN
 func Modf(f float64) (int float64, frac float64) {
+	if haveArchModf {
+		return archModf(f)
+	}
 	return modf(f)
 }
 
diff --git a/libgo/go/math/modf_noasm.go b/libgo/go/math/modf_noasm.go
new file mode 100644
index 0000000..42d4734
--- /dev/null
+++ b/libgo/go/math/modf_noasm.go
@@ -0,0 +1,14 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !arm64 && !ppc64 && !ppc64le
+// -build !arm64,!ppc64,!ppc64le
+
+package math
+
+const haveArchModf = false
+
+func archModf(f float64) (int float64, frac float64) {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/rand/export_test.go b/libgo/go/math/rand/export_test.go
new file mode 100644
index 0000000..560010b
--- /dev/null
+++ b/libgo/go/math/rand/export_test.go
@@ -0,0 +1,17 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+func Int31nForTest(r *Rand, n int32) int32 {
+	return r.int31n(n)
+}
+
+func GetNormalDistributionParameters() (float64, [128]uint32, [128]float32, [128]float32) {
+	return rn, kn, wn, fn
+}
+
+func GetExponentialDistributionParameters() (float64, [256]uint32, [256]float32, [256]float32) {
+	return re, ke, we, fe
+}
diff --git a/libgo/go/math/rand/gen_cooked.go b/libgo/go/math/rand/gen_cooked.go
index 0afc10d..7950e09 100644
--- a/libgo/go/math/rand/gen_cooked.go
+++ b/libgo/go/math/rand/gen_cooked.go
@@ -2,6 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
+//go:build ignore
 // +build ignore
 
 // This program computes the value of rngCooked in rng.go,
diff --git a/libgo/go/math/rand/race_test.go b/libgo/go/math/rand/race_test.go
index 186c716..e7d1036 100644
--- a/libgo/go/math/rand/race_test.go
+++ b/libgo/go/math/rand/race_test.go
@@ -2,9 +2,10 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package rand
+package rand_test
 
 import (
+	. "math/rand"
 	"sync"
 	"testing"
 )
diff --git a/libgo/go/math/rand/rand.go b/libgo/go/math/rand/rand.go
index d6422c9..13f20ca 100644
--- a/libgo/go/math/rand/rand.go
+++ b/libgo/go/math/rand/rand.go
@@ -2,7 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-// Package rand implements pseudo-random number generators.
+// Package rand implements pseudo-random number generators unsuitable for
+// security-sensitive work.
 //
 // Random numbers are generated by a Source. Top-level functions, such as
 // Float64 and Int, use a default shared Source that produces a deterministic
@@ -11,11 +12,9 @@
 // The default Source is safe for concurrent use by multiple goroutines, but
 // Sources created by NewSource are not.
 //
-// Mathematical interval notation such as [0, n) is used throughout the
-// documentation for this package.
-//
-// For random numbers suitable for security-sensitive work, see the crypto/rand
-// package.
+// This package's outputs might be easily predictable regardless of how it's
+// seeded. For random numbers suitable for security-sensitive work, see the
+// crypto/rand package.
 package rand
 
 import "sync"
@@ -104,7 +103,7 @@ func (r *Rand) Int() int {
 	return int(u << 1 >> 1) // clear sign bit if int == int32
 }
 
-// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n).
+// Int63n returns, as an int64, a non-negative pseudo-random number in the half-open interval [0,n).
 // It panics if n <= 0.
 func (r *Rand) Int63n(n int64) int64 {
 	if n <= 0 {
@@ -121,7 +120,7 @@ func (r *Rand) Int63n(n int64) int64 {
 	return v % n
 }
 
-// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n).
+// Int31n returns, as an int32, a non-negative pseudo-random number in the half-open interval [0,n).
 // It panics if n <= 0.
 func (r *Rand) Int31n(n int32) int32 {
 	if n <= 0 {
@@ -138,7 +137,7 @@ func (r *Rand) Int31n(n int32) int32 {
 	return v % n
 }
 
-// int31n returns, as an int32, a non-negative pseudo-random number in [0,n).
+// int31n returns, as an int32, a non-negative pseudo-random number in the half-open interval [0,n).
 // n must be > 0, but int31n does not check this; the caller must ensure it.
 // int31n exists because Int31n is inefficient, but Go 1 compatibility
 // requires that the stream of values produced by math/rand remain unchanged.
@@ -162,7 +161,7 @@ func (r *Rand) int31n(n int32) int32 {
 	return int32(prod >> 32)
 }
 
-// Intn returns, as an int, a non-negative pseudo-random number in [0,n).
+// Intn returns, as an int, a non-negative pseudo-random number in the half-open interval [0,n).
 // It panics if n <= 0.
 func (r *Rand) Intn(n int) int {
 	if n <= 0 {
@@ -174,7 +173,7 @@ func (r *Rand) Intn(n int) int {
 	return int(r.Int63n(int64(n)))
 }
 
-// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0).
+// Float64 returns, as a float64, a pseudo-random number in the half-open interval [0.0,1.0).
 func (r *Rand) Float64() float64 {
 	// A clearer, simpler implementation would be:
 	//	return float64(r.Int63n(1<<53)) / (1<<53)
@@ -200,7 +199,7 @@ again:
 	return f
 }
 
-// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0).
+// Float32 returns, as a float32, a pseudo-random number in the half-open interval [0.0,1.0).
 func (r *Rand) Float32() float32 {
 	// Same rationale as in Float64: we want to preserve the Go 1 value
 	// stream except we want to fix it not to return 1.0
@@ -213,7 +212,8 @@ again:
 	return f
 }
 
-// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n).
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers
+// in the half-open interval [0,n).
 func (r *Rand) Perm(n int) []int {
 	m := make([]int, n)
 	// In the following loop, the iteration when i=0 always swaps m[0] with m[0].
@@ -321,31 +321,31 @@ func Int31() int32 { return globalRand.Int31() }
 // Int returns a non-negative pseudo-random int from the default Source.
 func Int() int { return globalRand.Int() }
 
-// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n)
+// Int63n returns, as an int64, a non-negative pseudo-random number in the half-open interval [0,n)
 // from the default Source.
 // It panics if n <= 0.
 func Int63n(n int64) int64 { return globalRand.Int63n(n) }
 
-// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n)
+// Int31n returns, as an int32, a non-negative pseudo-random number in the half-open interval [0,n)
 // from the default Source.
 // It panics if n <= 0.
 func Int31n(n int32) int32 { return globalRand.Int31n(n) }
 
-// Intn returns, as an int, a non-negative pseudo-random number in [0,n)
+// Intn returns, as an int, a non-negative pseudo-random number in the half-open interval [0,n)
 // from the default Source.
 // It panics if n <= 0.
 func Intn(n int) int { return globalRand.Intn(n) }
 
-// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0)
+// Float64 returns, as a float64, a pseudo-random number in the half-open interval [0.0,1.0)
 // from the default Source.
 func Float64() float64 { return globalRand.Float64() }
 
-// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0)
+// Float32 returns, as a float32, a pseudo-random number in the half-open interval [0.0,1.0)
 // from the default Source.
 func Float32() float32 { return globalRand.Float32() }
 
-// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n)
-// from the default Source.
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers
+// in the half-open interval [0,n) from the default Source.
 func Perm(n int) []int { return globalRand.Perm(n) }
 
 // Shuffle pseudo-randomizes the order of elements using the default Source.
diff --git a/libgo/go/math/rand/rand_test.go b/libgo/go/math/rand/rand_test.go
index e037aae..462de8b 100644
--- a/libgo/go/math/rand/rand_test.go
+++ b/libgo/go/math/rand/rand_test.go
@@ -2,7 +2,7 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package rand
+package rand_test
 
 import (
 	"bytes"
@@ -11,6 +11,7 @@ import (
 	"internal/testenv"
 	"io"
 	"math"
+	. "math/rand"
 	"os"
 	"runtime"
 	"testing"
@@ -21,6 +22,9 @@ const (
 	numTestSamples = 10000
 )
 
+var rn, kn, wn, fn = GetNormalDistributionParameters()
+var re, ke, we, fe = GetExponentialDistributionParameters()
+
 type statsResults struct {
 	mean        float64
 	stddev      float64
@@ -503,7 +507,7 @@ func TestUniformFactorial(t *testing.T) {
 				fn   func() int
 			}{
 				{name: "Int31n", fn: func() int { return int(r.Int31n(int32(nfact))) }},
-				{name: "int31n", fn: func() int { return int(r.int31n(int32(nfact))) }},
+				{name: "int31n", fn: func() int { return int(Int31nForTest(r, int32(nfact))) }},
 				{name: "Perm", fn: func() int { return encodePerm(r.Perm(n)) }},
 				{name: "Shuffle", fn: func() int {
 					// Generate permutation using Shuffle.
diff --git a/libgo/go/math/remainder.go b/libgo/go/math/remainder.go
index 0e2903e..bf8bfd5 100644
--- a/libgo/go/math/remainder.go
+++ b/libgo/go/math/remainder.go
@@ -35,6 +35,9 @@ package math
 //	Remainder(x, ±Inf) = x
 //	Remainder(x, NaN) = NaN
 func Remainder(x, y float64) float64 {
+	if haveArchRemainder {
+		return archRemainder(x, y)
+	}
 	return remainder(x, y)
 }
 
diff --git a/libgo/go/math/sin.go b/libgo/go/math/sin.go
index 03fd14c..bd8b9a7 100644
--- a/libgo/go/math/sin.go
+++ b/libgo/go/math/sin.go
@@ -114,14 +114,13 @@ var _cos = [...]float64{
 // Special cases are:
 //	Cos(±Inf) = NaN
 //	Cos(NaN) = NaN
-
-//extern cos
-func libc_cos(float64) float64
-
 func Cos(x float64) float64 {
 	return libc_cos(x)
 }
 
+//extern cos
+func libc_cos(float64) float64
+
 func cos(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
@@ -181,14 +180,13 @@ func cos(x float64) float64 {
 //	Sin(±0) = ±0
 //	Sin(±Inf) = NaN
 //	Sin(NaN) = NaN
-
-//extern sin
-func libc_sin(float64) float64
-
 func Sin(x float64) float64 {
 	return libc_sin(x)
 }
 
+//extern sin
+func libc_sin(float64) float64
+
 func sin(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
diff --git a/libgo/go/math/sinh.go b/libgo/go/math/sinh.go
index 993424d..9fe9b4e 100644
--- a/libgo/go/math/sinh.go
+++ b/libgo/go/math/sinh.go
@@ -23,6 +23,13 @@ package math
 //	Sinh(±Inf) = ±Inf
 //	Sinh(NaN) = NaN
 func Sinh(x float64) float64 {
+	if haveArchSinh {
+		return archSinh(x)
+	}
+	return sinh(x)
+}
+
+func sinh(x float64) float64 {
 	// The coefficients are #2029 from Hart & Cheney. (20.36D)
 	const (
 		P0 = -0.6307673640497716991184787251e+6
@@ -68,6 +75,13 @@ func Sinh(x float64) float64 {
 //	Cosh(±Inf) = +Inf
 //	Cosh(NaN) = NaN
 func Cosh(x float64) float64 {
+	if haveArchCosh {
+		return archCosh(x)
+	}
+	return cosh(x)
+}
+
+func cosh(x float64) float64 {
 	x = Abs(x)
 	if x > 21 {
 		return Exp(x) * 0.5
diff --git a/libgo/go/math/stubs.go b/libgo/go/math/stubs.go
new file mode 100644
index 0000000..fe56800
--- /dev/null
+++ b/libgo/go/math/stubs.go
@@ -0,0 +1,161 @@
+// Copyright 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//-go:build !s390x
+// -build !s390x
+
+// This is a large group of functions that most architectures don't
+// implement in assembly.
+
+package math
+
+const haveArchAcos = false
+
+func archAcos(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAcosh = false
+
+func archAcosh(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAsin = false
+
+func archAsin(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAsinh = false
+
+func archAsinh(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAtan = false
+
+func archAtan(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAtan2 = false
+
+func archAtan2(y, x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchAtanh = false
+
+func archAtanh(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchCbrt = false
+
+func archCbrt(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchCos = false
+
+func archCos(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchCosh = false
+
+func archCosh(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchErf = false
+
+func archErf(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchErfc = false
+
+func archErfc(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchExpm1 = false
+
+func archExpm1(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchFrexp = false
+
+func archFrexp(x float64) (float64, int) {
+	panic("not implemented")
+}
+
+const haveArchLdexp = false
+
+func archLdexp(frac float64, exp int) float64 {
+	panic("not implemented")
+}
+
+const haveArchLog10 = false
+
+func archLog10(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchLog2 = false
+
+func archLog2(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchLog1p = false
+
+func archLog1p(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchMod = false
+
+func archMod(x, y float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchPow = false
+
+func archPow(x, y float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchRemainder = false
+
+func archRemainder(x, y float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchSin = false
+
+func archSin(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchSinh = false
+
+func archSinh(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchTan = false
+
+func archTan(x float64) float64 {
+	panic("not implemented")
+}
+
+const haveArchTanh = false
+
+func archTanh(x float64) float64 {
+	panic("not implemented")
+}
diff --git a/libgo/go/math/tan.go b/libgo/go/math/tan.go
index ca91d71..eea29d3 100644
--- a/libgo/go/math/tan.go
+++ b/libgo/go/math/tan.go
@@ -79,14 +79,13 @@ var _tanQ = [...]float64{
 //	Tan(±0) = ±0
 //	Tan(±Inf) = NaN
 //	Tan(NaN) = NaN
-
-//extern tan
-func libc_tan(float64) float64
-
 func Tan(x float64) float64 {
 	return libc_tan(x)
 }
 
+//extern tan
+func libc_tan(float64) float64
+
 func tan(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
diff --git a/libgo/go/math/tanh.go b/libgo/go/math/tanh.go
index 009a206..a825678 100644
--- a/libgo/go/math/tanh.go
+++ b/libgo/go/math/tanh.go
@@ -72,6 +72,13 @@ var tanhQ = [...]float64{
 //	Tanh(±Inf) = ±1
 //	Tanh(NaN) = NaN
 func Tanh(x float64) float64 {
+	if haveArchTanh {
+		return archTanh(x)
+	}
+	return tanh(x)
+}
+
+func tanh(x float64) float64 {
 	const MAXLOG = 8.8029691931113054295988e+01 // log(2**127)
 	z := Abs(x)
 	switch {