build: merge the great pkg/ rename into dev.power64

This also removes pkg/runtime/traceback_lr.c, which was ported
to Go in an earlier commit and then moved to
runtime/traceback.go.

Reviewer: rsc@golang.org
          rsc: LGTM

18 files changed, 8903 insertions, 0 deletions

diff --git a/src/math/big/arith.go b/src/math/big/arith.go
new file mode 100644
index 000000000..3d5a8682d
--- /dev/null
+++ b/src/math/big/arith.go
@@ -0,0 +1,240 @@
+// 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.
+
+// This file provides Go implementations of elementary multi-precision
+// arithmetic operations on word vectors. Needed for platforms without
+// assembly implementations of these routines.
+
+package big
+
+// A Word represents a single digit of a multi-precision unsigned integer.
+type Word uintptr
+
+const (
+	// Compute the size _S of a Word in bytes.
+	_m    = ^Word(0)
+	_logS = _m>>8&1 + _m>>16&1 + _m>>32&1
+	_S    = 1 << _logS
+
+	_W = _S << 3 // word size in bits
+	_B = 1 << _W // digit base
+	_M = _B - 1  // digit mask
+
+	_W2 = _W / 2   // half word size in bits
+	_B2 = 1 << _W2 // half digit base
+	_M2 = _B2 - 1  // half digit mask
+)
+
+// ----------------------------------------------------------------------------
+// Elementary operations on words
+//
+// These operations are used by the vector operations below.
+
+// z1<<_W + z0 = x+y+c, with c == 0 or 1
+func addWW_g(x, y, c Word) (z1, z0 Word) {
+	yc := y + c
+	z0 = x + yc
+	if z0 < x || yc < y {
+		z1 = 1
+	}
+	return
+}
+
+// z1<<_W + z0 = x-y-c, with c == 0 or 1
+func subWW_g(x, y, c Word) (z1, z0 Word) {
+	yc := y + c
+	z0 = x - yc
+	if z0 > x || yc < y {
+		z1 = 1
+	}
+	return
+}
+
+// z1<<_W + z0 = x*y
+// Adapted from Warren, Hacker's Delight, p. 132.
+func mulWW_g(x, y Word) (z1, z0 Word) {
+	x0 := x & _M2
+	x1 := x >> _W2
+	y0 := y & _M2
+	y1 := y >> _W2
+	w0 := x0 * y0
+	t := x1*y0 + w0>>_W2
+	w1 := t & _M2
+	w2 := t >> _W2
+	w1 += x0 * y1
+	z1 = x1*y1 + w2 + w1>>_W2
+	z0 = x * y
+	return
+}
+
+// z1<<_W + z0 = x*y + c
+func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
+	z1, zz0 := mulWW(x, y)
+	if z0 = zz0 + c; z0 < zz0 {
+		z1++
+	}
+	return
+}
+
+// Length of x in bits.
+func bitLen_g(x Word) (n int) {
+	for ; x >= 0x8000; x >>= 16 {
+		n += 16
+	}
+	if x >= 0x80 {
+		x >>= 8
+		n += 8
+	}
+	if x >= 0x8 {
+		x >>= 4
+		n += 4
+	}
+	if x >= 0x2 {
+		x >>= 2
+		n += 2
+	}
+	if x >= 0x1 {
+		n++
+	}
+	return
+}
+
+// log2 computes the integer binary logarithm of x.
+// The result is the integer n for which 2^n <= x < 2^(n+1).
+// If x == 0, the result is -1.
+func log2(x Word) int {
+	return bitLen(x) - 1
+}
+
+// Number of leading zeros in x.
+func leadingZeros(x Word) uint {
+	return uint(_W - bitLen(x))
+}
+
+// q = (u1<<_W + u0 - r)/y
+// Adapted from Warren, Hacker's Delight, p. 152.
+func divWW_g(u1, u0, v Word) (q, r Word) {
+	if u1 >= v {
+		return 1<<_W - 1, 1<<_W - 1
+	}
+
+	s := leadingZeros(v)
+	v <<= s
+
+	vn1 := v >> _W2
+	vn0 := v & _M2
+	un32 := u1<<s | u0>>(_W-s)
+	un10 := u0 << s
+	un1 := un10 >> _W2
+	un0 := un10 & _M2
+	q1 := un32 / vn1
+	rhat := un32 - q1*vn1
+
+	for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
+		q1--
+		rhat += vn1
+		if rhat >= _B2 {
+			break
+		}
+	}
+
+	un21 := un32*_B2 + un1 - q1*v
+	q0 := un21 / vn1
+	rhat = un21 - q0*vn1
+
+	for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
+		q0--
+		rhat += vn1
+		if rhat >= _B2 {
+			break
+		}
+	}
+
+	return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
+}
+
+func addVV_g(z, x, y []Word) (c Word) {
+	for i := range z {
+		c, z[i] = addWW_g(x[i], y[i], c)
+	}
+	return
+}
+
+func subVV_g(z, x, y []Word) (c Word) {
+	for i := range z {
+		c, z[i] = subWW_g(x[i], y[i], c)
+	}
+	return
+}
+
+func addVW_g(z, x []Word, y Word) (c Word) {
+	c = y
+	for i := range z {
+		c, z[i] = addWW_g(x[i], c, 0)
+	}
+	return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+	c = y
+	for i := range z {
+		c, z[i] = subWW_g(x[i], c, 0)
+	}
+	return
+}
+
+func shlVU_g(z, x []Word, s uint) (c Word) {
+	if n := len(z); n > 0 {
+		ŝ := _W - s
+		w1 := x[n-1]
+		c = w1 >> ŝ
+		for i := n - 1; i > 0; i-- {
+			w := w1
+			w1 = x[i-1]
+			z[i] = w<<s | w1>>ŝ
+		}
+		z[0] = w1 << s
+	}
+	return
+}
+
+func shrVU_g(z, x []Word, s uint) (c Word) {
+	if n := len(z); n > 0 {
+		ŝ := _W - s
+		w1 := x[0]
+		c = w1 << ŝ
+		for i := 0; i < n-1; i++ {
+			w := w1
+			w1 = x[i+1]
+			z[i] = w>>s | w1<<ŝ
+		}
+		z[n-1] = w1 >> s
+	}
+	return
+}
+
+func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
+	c = r
+	for i := range z {
+		c, z[i] = mulAddWWW_g(x[i], y, c)
+	}
+	return
+}
+
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+	for i := range z {
+		z1, z0 := mulAddWWW_g(x[i], y, z[i])
+		c, z[i] = addWW_g(z0, c, 0)
+		c += z1
+	}
+	return
+}
+
+func divWVW_g(z []Word, xn Word, x []Word, y Word) (r Word) {
+	r = xn
+	for i := len(z) - 1; i >= 0; i-- {
+		z[i], r = divWW_g(r, x[i], y)
+	}
+	return
+}
diff --git a/src/math/big/arith_386.s b/src/math/big/arith_386.s
new file mode 100644
index 000000000..1b47c898f
--- /dev/null
+++ b/src/math/big/arith_386.s
@@ -0,0 +1,278 @@
+// 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.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+	MOVL x+0(FP), AX
+	MULL y+4(FP)
+	MOVL DX, z1+8(FP)
+	MOVL AX, z0+12(FP)
+	RET
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+	MOVL x1+0(FP), DX
+	MOVL x0+4(FP), AX
+	DIVL y+8(FP)
+	MOVL AX, q+12(FP)
+	MOVL DX, r+16(FP)
+	RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), CX
+	MOVL z_len+4(FP), BP
+	MOVL $0, BX		// i = 0
+	MOVL $0, DX		// c = 0
+	JMP E1
+
+L1:	MOVL (SI)(BX*4), AX
+	RCRL $1, DX
+	ADCL (CX)(BX*4), AX
+	RCLL $1, DX
+	MOVL AX, (DI)(BX*4)
+	ADDL $1, BX		// i++
+
+E1:	CMPL BX, BP		// i < n
+	JL L1
+
+	MOVL DX, c+36(FP)
+	RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBL instead of ADCL and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), CX
+	MOVL z_len+4(FP), BP
+	MOVL $0, BX		// i = 0
+	MOVL $0, DX		// c = 0
+	JMP E2
+
+L2:	MOVL (SI)(BX*4), AX
+	RCRL $1, DX
+	SBBL (CX)(BX*4), AX
+	RCLL $1, DX
+	MOVL AX, (DI)(BX*4)
+	ADDL $1, BX		// i++
+
+E2:	CMPL BX, BP		// i < n
+	JL L2
+
+	MOVL DX, c+36(FP)
+	RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), AX	// c = y
+	MOVL z_len+4(FP), BP
+	MOVL $0, BX		// i = 0
+	JMP E3
+
+L3:	ADDL (SI)(BX*4), AX
+	MOVL AX, (DI)(BX*4)
+	RCLL $1, AX
+	ANDL $1, AX
+	ADDL $1, BX		// i++
+
+E3:	CMPL BX, BP		// i < n
+	JL L3
+
+	MOVL AX, c+28(FP)
+	RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), AX	// c = y
+	MOVL z_len+4(FP), BP
+	MOVL $0, BX		// i = 0
+	JMP E4
+
+L4:	MOVL (SI)(BX*4), DX	// TODO(gri) is there a reverse SUBL?
+	SUBL AX, DX
+	MOVL DX, (DI)(BX*4)
+	RCLL $1, AX
+	ANDL $1, AX
+	ADDL $1, BX		// i++
+
+E4:	CMPL BX, BP		// i < n
+	JL L4
+
+	MOVL AX, c+28(FP)
+	RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+	MOVL z_len+4(FP), BX	// i = z
+	SUBL $1, BX		// i--
+	JL X8b			// i < 0	(n <= 0)
+
+	// n > 0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL s+24(FP), CX
+	MOVL (SI)(BX*4), AX	// w1 = x[n-1]
+	MOVL $0, DX
+	SHLL CX, DX:AX		// w1>>ŝ
+	MOVL DX, c+28(FP)
+
+	CMPL BX, $0
+	JLE X8a			// i <= 0
+
+	// i > 0
+L8:	MOVL AX, DX		// w = w1
+	MOVL -4(SI)(BX*4), AX	// w1 = x[i-1]
+	SHLL CX, DX:AX		// w<<s | w1>>ŝ
+	MOVL DX, (DI)(BX*4)	// z[i] = w<<s | w1>>ŝ
+	SUBL $1, BX		// i--
+	JG L8			// i > 0
+
+	// i <= 0
+X8a:	SHLL CX, AX		// w1<<s
+	MOVL AX, (DI)		// z[0] = w1<<s
+	RET
+
+X8b:	MOVL $0, c+28(FP)
+	RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+	MOVL z_len+4(FP), BP
+	SUBL $1, BP		// n--
+	JL X9b			// n < 0	(n <= 0)
+
+	// n > 0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL s+24(FP), CX
+	MOVL (SI), AX		// w1 = x[0]
+	MOVL $0, DX
+	SHRL CX, DX:AX		// w1<<ŝ
+	MOVL DX, c+28(FP)
+
+	MOVL $0, BX		// i = 0
+	JMP E9
+
+	// i < n-1
+L9:	MOVL AX, DX		// w = w1
+	MOVL 4(SI)(BX*4), AX	// w1 = x[i+1]
+	SHRL CX, DX:AX		// w>>s | w1<<ŝ
+	MOVL DX, (DI)(BX*4)	// z[i] = w>>s | w1<<ŝ
+	ADDL $1, BX		// i++
+	
+E9:	CMPL BX, BP
+	JL L9			// i < n-1
+
+	// i >= n-1
+X9a:	SHRL CX, AX		// w1>>s
+	MOVL AX, (DI)(BP*4)	// z[n-1] = w1>>s
+	RET
+
+X9b:	MOVL $0, c+28(FP)
+	RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), BP
+	MOVL r+28(FP), CX	// c = r
+	MOVL z_len+4(FP), BX
+	LEAL (DI)(BX*4), DI
+	LEAL (SI)(BX*4), SI
+	NEGL BX			// i = -n
+	JMP E5
+
+L5:	MOVL (SI)(BX*4), AX
+	MULL BP
+	ADDL CX, AX
+	ADCL $0, DX
+	MOVL AX, (DI)(BX*4)
+	MOVL DX, CX
+	ADDL $1, BX		// i++
+
+E5:	CMPL BX, $0		// i < 0
+	JL L5
+
+	MOVL CX, c+32(FP)
+	RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL x+12(FP), SI
+	MOVL y+24(FP), BP
+	MOVL z_len+4(FP), BX
+	LEAL (DI)(BX*4), DI
+	LEAL (SI)(BX*4), SI
+	NEGL BX			// i = -n
+	MOVL $0, CX		// c = 0
+	JMP E6
+
+L6:	MOVL (SI)(BX*4), AX
+	MULL BP
+	ADDL CX, AX
+	ADCL $0, DX
+	ADDL AX, (DI)(BX*4)
+	ADCL $0, DX
+	MOVL DX, CX
+	ADDL $1, BX		// i++
+
+E6:	CMPL BX, $0		// i < 0
+	JL L6
+
+	MOVL CX, c+28(FP)
+	RET
+
+
+// func divWVW(z* Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+	MOVL z+0(FP), DI
+	MOVL xn+12(FP), DX	// r = xn
+	MOVL x+16(FP), SI
+	MOVL y+28(FP), CX
+	MOVL z_len+4(FP), BX	// i = z
+	JMP E7
+
+L7:	MOVL (SI)(BX*4), AX
+	DIVL CX
+	MOVL AX, (DI)(BX*4)
+
+E7:	SUBL $1, BX		// i--
+	JGE L7			// i >= 0
+
+	MOVL DX, r+32(FP)
+	RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+	BSRL x+0(FP), AX
+	JZ Z1
+	INCL AX
+	MOVL AX, n+4(FP)
+	RET
+
+Z1:	MOVL $0, n+4(FP)
+	RET
diff --git a/src/math/big/arith_amd64.s b/src/math/big/arith_amd64.s
new file mode 100644
index 000000000..56c4cb050
--- /dev/null
+++ b/src/math/big/arith_amd64.s
@@ -0,0 +1,401 @@
+// 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.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// Literal instruction for MOVQ $0, CX.
+// (MOVQ $0, reg is translated to XORQ reg, reg and clears CF.)
+#define ZERO_CX BYTE $0x48; \
+		BYTE $0xc7; \
+		BYTE $0xc1; \
+		BYTE $0x00; \
+		BYTE $0x00; \
+		BYTE $0x00; \
+		BYTE $0x00
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+	MOVQ x+0(FP), AX
+	MULQ y+8(FP)
+	MOVQ DX, z1+16(FP)
+	MOVQ AX, z0+24(FP)
+	RET
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+	MOVQ x1+0(FP), DX
+	MOVQ x0+8(FP), AX
+	DIVQ y+16(FP)
+	MOVQ AX, q+24(FP)
+	MOVQ DX, r+32(FP)
+	RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), DI
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), R9
+	MOVQ z+0(FP), R10
+
+	MOVQ $0, CX		// c = 0
+	MOVQ $0, SI		// i = 0
+
+	// s/JL/JMP/ below to disable the unrolled loop
+	SUBQ $4, DI		// n -= 4
+	JL V1			// if n < 0 goto V1
+
+U1:	// n >= 0
+	// regular loop body unrolled 4x
+	RCRQ $1, CX		// CF = c
+	MOVQ 0(R8)(SI*8), R11
+	MOVQ 8(R8)(SI*8), R12
+	MOVQ 16(R8)(SI*8), R13
+	MOVQ 24(R8)(SI*8), R14
+	ADCQ 0(R9)(SI*8), R11
+	ADCQ 8(R9)(SI*8), R12
+	ADCQ 16(R9)(SI*8), R13
+	ADCQ 24(R9)(SI*8), R14
+	MOVQ R11, 0(R10)(SI*8)
+	MOVQ R12, 8(R10)(SI*8)
+	MOVQ R13, 16(R10)(SI*8)
+	MOVQ R14, 24(R10)(SI*8)
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $4, SI		// i += 4
+	SUBQ $4, DI		// n -= 4
+	JGE U1			// if n >= 0 goto U1
+
+V1:	ADDQ $4, DI		// n += 4
+	JLE E1			// if n <= 0 goto E1
+
+L1:	// n > 0
+	RCRQ $1, CX		// CF = c
+	MOVQ 0(R8)(SI*8), R11
+	ADCQ 0(R9)(SI*8), R11
+	MOVQ R11, 0(R10)(SI*8)
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $1, SI		// i++
+	SUBQ $1, DI		// n--
+	JG L1			// if n > 0 goto L1
+
+E1:	MOVQ CX, c+72(FP)	// return c
+	RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBQ instead of ADCQ and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), DI
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), R9
+	MOVQ z+0(FP), R10
+
+	MOVQ $0, CX		// c = 0
+	MOVQ $0, SI		// i = 0
+
+	// s/JL/JMP/ below to disable the unrolled loop
+	SUBQ $4, DI		// n -= 4
+	JL V2			// if n < 0 goto V2
+
+U2:	// n >= 0
+	// regular loop body unrolled 4x
+	RCRQ $1, CX		// CF = c
+	MOVQ 0(R8)(SI*8), R11
+	MOVQ 8(R8)(SI*8), R12
+	MOVQ 16(R8)(SI*8), R13
+	MOVQ 24(R8)(SI*8), R14
+	SBBQ 0(R9)(SI*8), R11
+	SBBQ 8(R9)(SI*8), R12
+	SBBQ 16(R9)(SI*8), R13
+	SBBQ 24(R9)(SI*8), R14
+	MOVQ R11, 0(R10)(SI*8)
+	MOVQ R12, 8(R10)(SI*8)
+	MOVQ R13, 16(R10)(SI*8)
+	MOVQ R14, 24(R10)(SI*8)
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $4, SI		// i += 4
+	SUBQ $4, DI		// n -= 4
+	JGE U2			// if n >= 0 goto U2
+
+V2:	ADDQ $4, DI		// n += 4
+	JLE E2			// if n <= 0 goto E2
+
+L2:	// n > 0
+	RCRQ $1, CX		// CF = c
+	MOVQ 0(R8)(SI*8), R11
+	SBBQ 0(R9)(SI*8), R11
+	MOVQ R11, 0(R10)(SI*8)
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $1, SI		// i++
+	SUBQ $1, DI		// n--
+	JG L2			// if n > 0 goto L2
+
+E2:	MOVQ CX, c+72(FP)	// return c
+	RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), DI
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), CX	// c = y
+	MOVQ z+0(FP), R10
+
+	MOVQ $0, SI		// i = 0
+
+	// s/JL/JMP/ below to disable the unrolled loop
+	SUBQ $4, DI		// n -= 4
+	JL V3			// if n < 4 goto V3
+
+U3:	// n >= 0
+	// regular loop body unrolled 4x
+	MOVQ 0(R8)(SI*8), R11
+	MOVQ 8(R8)(SI*8), R12
+	MOVQ 16(R8)(SI*8), R13
+	MOVQ 24(R8)(SI*8), R14
+	ADDQ CX, R11
+	ZERO_CX
+	ADCQ $0, R12
+	ADCQ $0, R13
+	ADCQ $0, R14
+	SETCS CX		// c = CF
+	MOVQ R11, 0(R10)(SI*8)
+	MOVQ R12, 8(R10)(SI*8)
+	MOVQ R13, 16(R10)(SI*8)
+	MOVQ R14, 24(R10)(SI*8)
+
+	ADDQ $4, SI		// i += 4
+	SUBQ $4, DI		// n -= 4
+	JGE U3			// if n >= 0 goto U3
+
+V3:	ADDQ $4, DI		// n += 4
+	JLE E3			// if n <= 0 goto E3
+
+L3:	// n > 0
+	ADDQ 0(R8)(SI*8), CX
+	MOVQ CX, 0(R10)(SI*8)
+	ZERO_CX
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $1, SI		// i++
+	SUBQ $1, DI		// n--
+	JG L3			// if n > 0 goto L3
+
+E3:	MOVQ CX, c+56(FP)	// return c
+	RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
+TEXT ·subVW(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), DI
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), CX	// c = y
+	MOVQ z+0(FP), R10
+	
+	MOVQ $0, SI		// i = 0
+
+	// s/JL/JMP/ below to disable the unrolled loop
+	SUBQ $4, DI		// n -= 4
+	JL V4			// if n < 4 goto V4
+
+U4:	// n >= 0
+	// regular loop body unrolled 4x
+	MOVQ 0(R8)(SI*8), R11
+	MOVQ 8(R8)(SI*8), R12
+	MOVQ 16(R8)(SI*8), R13
+	MOVQ 24(R8)(SI*8), R14
+	SUBQ CX, R11
+	ZERO_CX
+	SBBQ $0, R12
+	SBBQ $0, R13
+	SBBQ $0, R14
+	SETCS CX		// c = CF
+	MOVQ R11, 0(R10)(SI*8)
+	MOVQ R12, 8(R10)(SI*8)
+	MOVQ R13, 16(R10)(SI*8)
+	MOVQ R14, 24(R10)(SI*8)
+
+	ADDQ $4, SI		// i += 4
+	SUBQ $4, DI		// n -= 4
+	JGE U4			// if n >= 0 goto U4
+
+V4:	ADDQ $4, DI		// n += 4
+	JLE E4			// if n <= 0 goto E4
+
+L4:	// n > 0
+	MOVQ 0(R8)(SI*8), R11
+	SUBQ CX, R11
+	MOVQ R11, 0(R10)(SI*8)
+	ZERO_CX
+	RCLQ $1, CX		// c = CF
+
+	ADDQ $1, SI		// i++
+	SUBQ $1, DI		// n--
+	JG L4			// if n > 0 goto L4
+
+E4:	MOVQ CX, c+56(FP)	// return c
+	RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), BX	// i = z
+	SUBQ $1, BX		// i--
+	JL X8b			// i < 0	(n <= 0)
+
+	// n > 0
+	MOVQ z+0(FP), R10
+	MOVQ x+24(FP), R8
+	MOVQ s+48(FP), CX
+	MOVQ (R8)(BX*8), AX	// w1 = x[n-1]
+	MOVQ $0, DX
+	SHLQ CX, DX:AX		// w1>>ŝ
+	MOVQ DX, c+56(FP)
+
+	CMPQ BX, $0
+	JLE X8a			// i <= 0
+
+	// i > 0
+L8:	MOVQ AX, DX		// w = w1
+	MOVQ -8(R8)(BX*8), AX	// w1 = x[i-1]
+	SHLQ CX, DX:AX		// w<<s | w1>>ŝ
+	MOVQ DX, (R10)(BX*8)	// z[i] = w<<s | w1>>ŝ
+	SUBQ $1, BX		// i--
+	JG L8			// i > 0
+
+	// i <= 0
+X8a:	SHLQ CX, AX		// w1<<s
+	MOVQ AX, (R10)		// z[0] = w1<<s
+	RET
+
+X8b:	MOVQ $0, c+56(FP)
+	RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+	MOVQ z_len+8(FP), R11
+	SUBQ $1, R11		// n--
+	JL X9b			// n < 0	(n <= 0)
+
+	// n > 0
+	MOVQ z+0(FP), R10
+	MOVQ x+24(FP), R8
+	MOVQ s+48(FP), CX
+	MOVQ (R8), AX		// w1 = x[0]
+	MOVQ $0, DX
+	SHRQ CX, DX:AX		// w1<<ŝ
+	MOVQ DX, c+56(FP)
+
+	MOVQ $0, BX		// i = 0
+	JMP E9
+
+	// i < n-1
+L9:	MOVQ AX, DX		// w = w1
+	MOVQ 8(R8)(BX*8), AX	// w1 = x[i+1]
+	SHRQ CX, DX:AX		// w>>s | w1<<ŝ
+	MOVQ DX, (R10)(BX*8)	// z[i] = w>>s | w1<<ŝ
+	ADDQ $1, BX		// i++
+	
+E9:	CMPQ BX, R11
+	JL L9			// i < n-1
+
+	// i >= n-1
+X9a:	SHRQ CX, AX		// w1>>s
+	MOVQ AX, (R10)(R11*8)	// z[n-1] = w1>>s
+	RET
+
+X9b:	MOVQ $0, c+56(FP)
+	RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+	MOVQ z+0(FP), R10
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), R9
+	MOVQ r+56(FP), CX	// c = r
+	MOVQ z_len+8(FP), R11
+	MOVQ $0, BX		// i = 0
+	JMP E5
+
+L5:	MOVQ (R8)(BX*8), AX
+	MULQ R9
+	ADDQ CX, AX
+	ADCQ $0, DX
+	MOVQ AX, (R10)(BX*8)
+	MOVQ DX, CX
+	ADDQ $1, BX		// i++
+
+E5:	CMPQ BX, R11		// i < n
+	JL L5
+
+	MOVQ CX, c+64(FP)
+	RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+	MOVQ z+0(FP), R10
+	MOVQ x+24(FP), R8
+	MOVQ y+48(FP), R9
+	MOVQ z_len+8(FP), R11
+	MOVQ $0, BX		// i = 0
+	MOVQ $0, CX		// c = 0
+	JMP E6
+
+L6:	MOVQ (R8)(BX*8), AX
+	MULQ R9
+	ADDQ CX, AX
+	ADCQ $0, DX
+	ADDQ AX, (R10)(BX*8)
+	ADCQ $0, DX
+	MOVQ DX, CX
+	ADDQ $1, BX		// i++
+
+E6:	CMPQ BX, R11		// i < n
+	JL L6
+
+	MOVQ CX, c+56(FP)
+	RET
+
+
+// func divWVW(z []Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+	MOVQ z+0(FP), R10
+	MOVQ xn+24(FP), DX	// r = xn
+	MOVQ x+32(FP), R8
+	MOVQ y+56(FP), R9
+	MOVQ z_len+8(FP), BX	// i = z
+	JMP E7
+
+L7:	MOVQ (R8)(BX*8), AX
+	DIVQ R9
+	MOVQ AX, (R10)(BX*8)
+
+E7:	SUBQ $1, BX		// i--
+	JGE L7			// i >= 0
+
+	MOVQ DX, r+64(FP)
+	RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+	BSRQ x+0(FP), AX
+	JZ Z1
+	ADDQ $1, AX
+	MOVQ AX, n+8(FP)
+	RET
+
+Z1:	MOVQ $0, n+8(FP)
+	RET
diff --git a/src/math/big/arith_amd64p32.s b/src/math/big/arith_amd64p32.s
new file mode 100644
index 000000000..908dbbdc5
--- /dev/null
+++ b/src/math/big/arith_amd64p32.s
@@ -0,0 +1,41 @@
+// Copyright 2013 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.
+
+#include "textflag.h"
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+	JMP ·mulWW_g(SB)
+
+TEXT ·divWW(SB),NOSPLIT,$0
+	JMP ·divWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+	JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+	JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+	JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+	JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+	JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+	JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+	JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+	JMP ·addMulVVW_g(SB)
+
+TEXT ·divWVW(SB),NOSPLIT,$0
+	JMP ·divWVW_g(SB)
+
+TEXT ·bitLen(SB),NOSPLIT,$0
+	JMP ·bitLen_g(SB)
diff --git a/src/math/big/arith_arm.s b/src/math/big/arith_arm.s
new file mode 100644
index 000000000..a4c51c212
--- /dev/null
+++ b/src/math/big/arith_arm.s
@@ -0,0 +1,300 @@
+// 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.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+	ADD.S	$0, R0		// clear carry flag
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R4
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	ADD	R4<<2, R1, R4
+	B E1
+L1:
+	MOVW.P	4(R2), R5
+	MOVW.P	4(R3), R6
+	ADC.S	R6, R5
+	MOVW.P	R5, 4(R1)
+E1:
+	TEQ	R1, R4
+	BNE L1
+
+	MOVW	$0, R0
+	MOVW.CS	$1, R0
+	MOVW	R0, c+36(FP)
+	RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBC instead of ADC and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+	SUB.S	$0, R0		// clear borrow flag
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R4
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	ADD	R4<<2, R1, R4
+	B E2
+L2:
+	MOVW.P	4(R2), R5
+	MOVW.P	4(R3), R6
+	SBC.S	R6, R5
+	MOVW.P	R5, 4(R1)
+E2:
+	TEQ	R1, R4
+	BNE L2
+
+	MOVW	$0, R0
+	MOVW.CC	$1, R0
+	MOVW	R0, c+36(FP)
+	RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R4
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	ADD	R4<<2, R1, R4
+	TEQ	R1, R4
+	BNE L3a
+	MOVW	R3, c+28(FP)
+	RET
+L3a:
+	MOVW.P	4(R2), R5
+	ADD.S	R3, R5
+	MOVW.P	R5, 4(R1)
+	B	E3
+L3:
+	MOVW.P	4(R2), R5
+	ADC.S	$0, R5
+	MOVW.P	R5, 4(R1)
+E3:
+	TEQ	R1, R4
+	BNE	L3
+
+	MOVW	$0, R0
+	MOVW.CS	$1, R0
+	MOVW	R0, c+28(FP)
+	RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R4
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	ADD	R4<<2, R1, R4
+	TEQ	R1, R4
+	BNE L4a
+	MOVW	R3, c+28(FP)
+	RET
+L4a:
+	MOVW.P	4(R2), R5
+	SUB.S	R3, R5
+	MOVW.P	R5, 4(R1)
+	B	E4
+L4:
+	MOVW.P	4(R2), R5
+	SBC.S	$0, R5
+	MOVW.P	R5, 4(R1)
+E4:
+	TEQ	R1, R4
+	BNE	L4
+
+	MOVW	$0, R0
+	MOVW.CC	$1, R0
+	MOVW	R0, c+28(FP)
+	RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+	MOVW	z_len+4(FP), R5
+	TEQ	$0, R5
+	BEQ	X7
+	
+	MOVW	z+0(FP), R1
+	MOVW	x+12(FP), R2
+	ADD	R5<<2, R2, R2
+	ADD	R5<<2, R1, R5
+	MOVW	s+24(FP), R3
+	TEQ	$0, R3	// shift 0 is special
+	BEQ	Y7
+	ADD	$4, R1	// stop one word early
+	MOVW	$32, R4
+	SUB	R3, R4
+	MOVW	$0, R7
+	
+	MOVW.W	-4(R2), R6
+	MOVW	R6<<R3, R7
+	MOVW	R6>>R4, R6
+	MOVW	R6, c+28(FP)
+	B E7
+
+L7:
+	MOVW.W	-4(R2), R6
+	ORR	R6>>R4, R7
+	MOVW.W	R7, -4(R5)
+	MOVW	R6<<R3, R7
+E7:
+	TEQ	R1, R5
+	BNE	L7
+
+	MOVW	R7, -4(R5)
+	RET
+
+Y7:	// copy loop, because shift 0 == shift 32
+	MOVW.W	-4(R2), R6
+	MOVW.W	R6, -4(R5)
+	TEQ	R1, R5
+	BNE Y7
+
+X7:
+	MOVW	$0, R1
+	MOVW	R1, c+28(FP)
+	RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+	MOVW	z_len+4(FP), R5
+	TEQ	$0, R5
+	BEQ	X6
+
+	MOVW	z+0(FP), R1
+	MOVW	x+12(FP), R2
+	ADD	R5<<2, R1, R5
+	MOVW	s+24(FP), R3
+	TEQ	$0, R3	// shift 0 is special
+	BEQ Y6
+	SUB	$4, R5	// stop one word early
+	MOVW	$32, R4
+	SUB	R3, R4
+	MOVW	$0, R7
+
+	// first word
+	MOVW.P	4(R2), R6
+	MOVW	R6>>R3, R7
+	MOVW	R6<<R4, R6
+	MOVW	R6, c+28(FP)
+	B E6
+
+	// word loop
+L6:
+	MOVW.P	4(R2), R6
+	ORR	R6<<R4, R7
+	MOVW.P	R7, 4(R1)
+	MOVW	R6>>R3, R7
+E6:
+	TEQ	R1, R5
+	BNE	L6
+
+	MOVW	R7, 0(R1)
+	RET
+
+Y6:	// copy loop, because shift 0 == shift 32
+	MOVW.P	4(R2), R6
+	MOVW.P	R6, 4(R1)
+	TEQ R1, R5
+	BNE Y6
+
+X6:
+	MOVW	$0, R1
+	MOVW	R1, c+28(FP)
+	RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+	MOVW	$0, R0
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R5
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	MOVW	r+28(FP), R4
+	ADD	R5<<2, R1, R5
+	B E8
+
+	// word loop
+L8:
+	MOVW.P	4(R2), R6
+	MULLU	R6, R3, (R7, R6)
+	ADD.S	R4, R6
+	ADC	R0, R7
+	MOVW.P	R6, 4(R1)
+	MOVW	R7, R4
+E8:
+	TEQ	R1, R5
+	BNE	L8
+
+	MOVW	R4, c+32(FP)
+	RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+	MOVW	$0, R0
+	MOVW	z+0(FP), R1
+	MOVW	z_len+4(FP), R5
+	MOVW	x+12(FP), R2
+	MOVW	y+24(FP), R3
+	ADD	R5<<2, R1, R5
+	MOVW	$0, R4
+	B E9
+
+	// word loop
+L9:
+	MOVW.P	4(R2), R6
+	MULLU	R6, R3, (R7, R6)
+	ADD.S	R4, R6
+	ADC	R0, R7
+	MOVW	0(R1), R4
+	ADD.S	R4, R6
+	ADC	R0, R7
+	MOVW.P	R6, 4(R1)
+	MOVW	R7, R4
+E9:
+	TEQ	R1, R5
+	BNE	L9
+
+	MOVW	R4, c+28(FP)
+	RET
+
+
+// func divWVW(z* Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+	// ARM has no multiword division, so use portable code.
+	B ·divWVW_g(SB)
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+	// ARM has no multiword division, so use portable code.
+	B ·divWW_g(SB)
+
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+	MOVW	x+0(FP), R1
+	MOVW	y+4(FP), R2
+	MULLU	R1, R2, (R4, R3)
+	MOVW	R4, z1+8(FP)
+	MOVW	R3, z0+12(FP)
+	RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+	MOVW	x+0(FP), R0
+	CLZ 	R0, R0
+	RSB	$32, R0
+	MOVW	R0, n+4(FP)
+	RET
diff --git a/src/math/big/arith_decl.go b/src/math/big/arith_decl.go
new file mode 100644
index 000000000..068cc8d93
--- /dev/null
+++ b/src/math/big/arith_decl.go
@@ -0,0 +1,19 @@
+// Copyright 2010 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+// implemented in arith_$GOARCH.s
+func mulWW(x, y Word) (z1, z0 Word)
+func divWW(x1, x0, y Word) (q, r Word)
+func addVV(z, x, y []Word) (c Word)
+func subVV(z, x, y []Word) (c Word)
+func addVW(z, x []Word, y Word) (c Word)
+func subVW(z, x []Word, y Word) (c Word)
+func shlVU(z, x []Word, s uint) (c Word)
+func shrVU(z, x []Word, s uint) (c Word)
+func mulAddVWW(z, x []Word, y, r Word) (c Word)
+func addMulVVW(z, x []Word, y Word) (c Word)
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word)
+func bitLen(x Word) (n int)
diff --git a/src/math/big/arith_power64x.s b/src/math/big/arith_power64x.s
new file mode 100644
index 000000000..c33a9209f
--- /dev/null
+++ b/src/math/big/arith_power64x.s
@@ -0,0 +1,46 @@
+// Copyright 2013 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.
+
+// +build power64 power64le
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+	BR ·mulWW_g(SB)
+
+TEXT ·divWW(SB),NOSPLIT,$0
+	BR ·divWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+	BR ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+	BR ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+	BR ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+	BR ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+	BR ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+	BR ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+	BR ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+	BR ·addMulVVW_g(SB)
+
+TEXT ·divWVW(SB),NOSPLIT,$0
+	BR ·divWVW_g(SB)
+
+TEXT ·bitLen(SB),NOSPLIT,$0
+	BR ·bitLen_g(SB)
diff --git a/src/math/big/arith_test.go b/src/math/big/arith_test.go
new file mode 100644
index 000000000..3615a659c
--- /dev/null
+++ b/src/math/big/arith_test.go
@@ -0,0 +1,456 @@
+// 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.
+
+package big
+
+import (
+	"math/rand"
+	"testing"
+)
+
+type funWW func(x, y, c Word) (z1, z0 Word)
+type argWW struct {
+	x, y, c, z1, z0 Word
+}
+
+var sumWW = []argWW{
+	{0, 0, 0, 0, 0},
+	{0, 1, 0, 0, 1},
+	{0, 0, 1, 0, 1},
+	{0, 1, 1, 0, 2},
+	{12345, 67890, 0, 0, 80235},
+	{12345, 67890, 1, 0, 80236},
+	{_M, 1, 0, 1, 0},
+	{_M, 0, 1, 1, 0},
+	{_M, 1, 1, 1, 1},
+	{_M, _M, 0, 1, _M - 1},
+	{_M, _M, 1, 1, _M},
+}
+
+func testFunWW(t *testing.T, msg string, f funWW, a argWW) {
+	z1, z0 := f(a.x, a.y, a.c)
+	if z1 != a.z1 || z0 != a.z0 {
+		t.Errorf("%s%+v\n\tgot z1:z0 = %#x:%#x; want %#x:%#x", msg, a, z1, z0, a.z1, a.z0)
+	}
+}
+
+func TestFunWW(t *testing.T) {
+	for _, a := range sumWW {
+		arg := a
+		testFunWW(t, "addWW_g", addWW_g, arg)
+
+		arg = argWW{a.y, a.x, a.c, a.z1, a.z0}
+		testFunWW(t, "addWW_g symmetric", addWW_g, arg)
+
+		arg = argWW{a.z0, a.x, a.c, a.z1, a.y}
+		testFunWW(t, "subWW_g", subWW_g, arg)
+
+		arg = argWW{a.z0, a.y, a.c, a.z1, a.x}
+		testFunWW(t, "subWW_g symmetric", subWW_g, arg)
+	}
+}
+
+type funVV func(z, x, y []Word) (c Word)
+type argVV struct {
+	z, x, y nat
+	c       Word
+}
+
+var sumVV = []argVV{
+	{},
+	{nat{0}, nat{0}, nat{0}, 0},
+	{nat{1}, nat{1}, nat{0}, 0},
+	{nat{0}, nat{_M}, nat{1}, 1},
+	{nat{80235}, nat{12345}, nat{67890}, 0},
+	{nat{_M - 1}, nat{_M}, nat{_M}, 1},
+	{nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, nat{1, 0, 0, 0}, 1},
+	{nat{0, 0, 0, _M}, nat{_M, _M, _M, _M - 1}, nat{1, 0, 0, 0}, 0},
+	{nat{0, 0, 0, 0}, nat{_M, 0, _M, 0}, nat{1, _M, 0, _M}, 1},
+}
+
+func testFunVV(t *testing.T, msg string, f funVV, a argVV) {
+	z := make(nat, len(a.z))
+	c := f(z, a.x, a.y)
+	for i, zi := range z {
+		if zi != a.z[i] {
+			t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+			break
+		}
+	}
+	if c != a.c {
+		t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+	}
+}
+
+func TestFunVV(t *testing.T) {
+	for _, a := range sumVV {
+		arg := a
+		testFunVV(t, "addVV_g", addVV_g, arg)
+		testFunVV(t, "addVV", addVV, arg)
+
+		arg = argVV{a.z, a.y, a.x, a.c}
+		testFunVV(t, "addVV_g symmetric", addVV_g, arg)
+		testFunVV(t, "addVV symmetric", addVV, arg)
+
+		arg = argVV{a.x, a.z, a.y, a.c}
+		testFunVV(t, "subVV_g", subVV_g, arg)
+		testFunVV(t, "subVV", subVV, arg)
+
+		arg = argVV{a.y, a.z, a.x, a.c}
+		testFunVV(t, "subVV_g symmetric", subVV_g, arg)
+		testFunVV(t, "subVV symmetric", subVV, arg)
+	}
+}
+
+// Always the same seed for reproducible results.
+var rnd = rand.New(rand.NewSource(0))
+
+func rndW() Word {
+	return Word(rnd.Int63()<<1 | rnd.Int63n(2))
+}
+
+func rndV(n int) []Word {
+	v := make([]Word, n)
+	for i := range v {
+		v[i] = rndW()
+	}
+	return v
+}
+
+func benchmarkFunVV(b *testing.B, f funVV, n int) {
+	x := rndV(n)
+	y := rndV(n)
+	z := make([]Word, n)
+	b.SetBytes(int64(n * _W))
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		f(z, x, y)
+	}
+}
+
+func BenchmarkAddVV_1(b *testing.B)   { benchmarkFunVV(b, addVV, 1) }
+func BenchmarkAddVV_2(b *testing.B)   { benchmarkFunVV(b, addVV, 2) }
+func BenchmarkAddVV_3(b *testing.B)   { benchmarkFunVV(b, addVV, 3) }
+func BenchmarkAddVV_4(b *testing.B)   { benchmarkFunVV(b, addVV, 4) }
+func BenchmarkAddVV_5(b *testing.B)   { benchmarkFunVV(b, addVV, 5) }
+func BenchmarkAddVV_1e1(b *testing.B) { benchmarkFunVV(b, addVV, 1e1) }
+func BenchmarkAddVV_1e2(b *testing.B) { benchmarkFunVV(b, addVV, 1e2) }
+func BenchmarkAddVV_1e3(b *testing.B) { benchmarkFunVV(b, addVV, 1e3) }
+func BenchmarkAddVV_1e4(b *testing.B) { benchmarkFunVV(b, addVV, 1e4) }
+func BenchmarkAddVV_1e5(b *testing.B) { benchmarkFunVV(b, addVV, 1e5) }
+
+type funVW func(z, x []Word, y Word) (c Word)
+type argVW struct {
+	z, x nat
+	y    Word
+	c    Word
+}
+
+var sumVW = []argVW{
+	{},
+	{nil, nil, 2, 2},
+	{nat{0}, nat{0}, 0, 0},
+	{nat{1}, nat{0}, 1, 0},
+	{nat{1}, nat{1}, 0, 0},
+	{nat{0}, nat{_M}, 1, 1},
+	{nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, 1, 1},
+}
+
+var prodVW = []argVW{
+	{},
+	{nat{0}, nat{0}, 0, 0},
+	{nat{0}, nat{_M}, 0, 0},
+	{nat{0}, nat{0}, _M, 0},
+	{nat{1}, nat{1}, 1, 0},
+	{nat{22793}, nat{991}, 23, 0},
+	{nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0},
+	{nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0},
+	{nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0},
+	{nat{_M << 1 & _M}, nat{_M}, 1 << 1, _M >> (_W - 1)},
+	{nat{_M << 7 & _M}, nat{_M}, 1 << 7, _M >> (_W - 7)},
+	{nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, _M >> (_W - 7)},
+}
+
+var lshVW = []argVW{
+	{},
+	{nat{0}, nat{0}, 0, 0},
+	{nat{0}, nat{0}, 1, 0},
+	{nat{0}, nat{0}, 20, 0},
+
+	{nat{_M}, nat{_M}, 0, 0},
+	{nat{_M << 1 & _M}, nat{_M}, 1, 1},
+	{nat{_M << 20 & _M}, nat{_M}, 20, _M >> (_W - 20)},
+
+	{nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+	{nat{_M << 1 & _M, _M, _M}, nat{_M, _M, _M}, 1, 1},
+	{nat{_M << 20 & _M, _M, _M}, nat{_M, _M, _M}, 20, _M >> (_W - 20)},
+}
+
+var rshVW = []argVW{
+	{},
+	{nat{0}, nat{0}, 0, 0},
+	{nat{0}, nat{0}, 1, 0},
+	{nat{0}, nat{0}, 20, 0},
+
+	{nat{_M}, nat{_M}, 0, 0},
+	{nat{_M >> 1}, nat{_M}, 1, _M << (_W - 1) & _M},
+	{nat{_M >> 20}, nat{_M}, 20, _M << (_W - 20) & _M},
+
+	{nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+	{nat{_M, _M, _M >> 1}, nat{_M, _M, _M}, 1, _M << (_W - 1) & _M},
+	{nat{_M, _M, _M >> 20}, nat{_M, _M, _M}, 20, _M << (_W - 20) & _M},
+}
+
+func testFunVW(t *testing.T, msg string, f funVW, a argVW) {
+	z := make(nat, len(a.z))
+	c := f(z, a.x, a.y)
+	for i, zi := range z {
+		if zi != a.z[i] {
+			t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+			break
+		}
+	}
+	if c != a.c {
+		t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+	}
+}
+
+func makeFunVW(f func(z, x []Word, s uint) (c Word)) funVW {
+	return func(z, x []Word, s Word) (c Word) {
+		return f(z, x, uint(s))
+	}
+}
+
+func TestFunVW(t *testing.T) {
+	for _, a := range sumVW {
+		arg := a
+		testFunVW(t, "addVW_g", addVW_g, arg)
+		testFunVW(t, "addVW", addVW, arg)
+
+		arg = argVW{a.x, a.z, a.y, a.c}
+		testFunVW(t, "subVW_g", subVW_g, arg)
+		testFunVW(t, "subVW", subVW, arg)
+	}
+
+	shlVW_g := makeFunVW(shlVU_g)
+	shlVW := makeFunVW(shlVU)
+	for _, a := range lshVW {
+		arg := a
+		testFunVW(t, "shlVU_g", shlVW_g, arg)
+		testFunVW(t, "shlVU", shlVW, arg)
+	}
+
+	shrVW_g := makeFunVW(shrVU_g)
+	shrVW := makeFunVW(shrVU)
+	for _, a := range rshVW {
+		arg := a
+		testFunVW(t, "shrVU_g", shrVW_g, arg)
+		testFunVW(t, "shrVU", shrVW, arg)
+	}
+}
+
+func benchmarkFunVW(b *testing.B, f funVW, n int) {
+	x := rndV(n)
+	y := rndW()
+	z := make([]Word, n)
+	b.SetBytes(int64(n * _W))
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		f(z, x, y)
+	}
+}
+
+func BenchmarkAddVW_1(b *testing.B)   { benchmarkFunVW(b, addVW, 1) }
+func BenchmarkAddVW_2(b *testing.B)   { benchmarkFunVW(b, addVW, 2) }
+func BenchmarkAddVW_3(b *testing.B)   { benchmarkFunVW(b, addVW, 3) }
+func BenchmarkAddVW_4(b *testing.B)   { benchmarkFunVW(b, addVW, 4) }
+func BenchmarkAddVW_5(b *testing.B)   { benchmarkFunVW(b, addVW, 5) }
+func BenchmarkAddVW_1e1(b *testing.B) { benchmarkFunVW(b, addVW, 1e1) }
+func BenchmarkAddVW_1e2(b *testing.B) { benchmarkFunVW(b, addVW, 1e2) }
+func BenchmarkAddVW_1e3(b *testing.B) { benchmarkFunVW(b, addVW, 1e3) }
+func BenchmarkAddVW_1e4(b *testing.B) { benchmarkFunVW(b, addVW, 1e4) }
+func BenchmarkAddVW_1e5(b *testing.B) { benchmarkFunVW(b, addVW, 1e5) }
+
+type funVWW func(z, x []Word, y, r Word) (c Word)
+type argVWW struct {
+	z, x nat
+	y, r Word
+	c    Word
+}
+
+var prodVWW = []argVWW{
+	{},
+	{nat{0}, nat{0}, 0, 0, 0},
+	{nat{991}, nat{0}, 0, 991, 0},
+	{nat{0}, nat{_M}, 0, 0, 0},
+	{nat{991}, nat{_M}, 0, 991, 0},
+	{nat{0}, nat{0}, _M, 0, 0},
+	{nat{991}, nat{0}, _M, 991, 0},
+	{nat{1}, nat{1}, 1, 0, 0},
+	{nat{992}, nat{1}, 1, 991, 0},
+	{nat{22793}, nat{991}, 23, 0, 0},
+	{nat{22800}, nat{991}, 23, 7, 0},
+	{nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0, 0},
+	{nat{7, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 7, 0},
+	{nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0, 0},
+	{nat{991, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 991, 0},
+	{nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0, 0},
+	{nat{991, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 991, 0},
+	{nat{_M << 1 & _M}, nat{_M}, 1 << 1, 0, _M >> (_W - 1)},
+	{nat{_M<<1&_M + 1}, nat{_M}, 1 << 1, 1, _M >> (_W - 1)},
+	{nat{_M << 7 & _M}, nat{_M}, 1 << 7, 0, _M >> (_W - 7)},
+	{nat{_M<<7&_M + 1<<6}, nat{_M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+	{nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 0, _M >> (_W - 7)},
+	{nat{_M<<7&_M + 1<<6, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+}
+
+func testFunVWW(t *testing.T, msg string, f funVWW, a argVWW) {
+	z := make(nat, len(a.z))
+	c := f(z, a.x, a.y, a.r)
+	for i, zi := range z {
+		if zi != a.z[i] {
+			t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+			break
+		}
+	}
+	if c != a.c {
+		t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+	}
+}
+
+// TODO(gri) mulAddVWW and divWVW are symmetric operations but
+//           their signature is not symmetric. Try to unify.
+
+type funWVW func(z []Word, xn Word, x []Word, y Word) (r Word)
+type argWVW struct {
+	z  nat
+	xn Word
+	x  nat
+	y  Word
+	r  Word
+}
+
+func testFunWVW(t *testing.T, msg string, f funWVW, a argWVW) {
+	z := make(nat, len(a.z))
+	r := f(z, a.xn, a.x, a.y)
+	for i, zi := range z {
+		if zi != a.z[i] {
+			t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+			break
+		}
+	}
+	if r != a.r {
+		t.Errorf("%s%+v\n\tgot r = %#x; want %#x", msg, a, r, a.r)
+	}
+}
+
+func TestFunVWW(t *testing.T) {
+	for _, a := range prodVWW {
+		arg := a
+		testFunVWW(t, "mulAddVWW_g", mulAddVWW_g, arg)
+		testFunVWW(t, "mulAddVWW", mulAddVWW, arg)
+
+		if a.y != 0 && a.r < a.y {
+			arg := argWVW{a.x, a.c, a.z, a.y, a.r}
+			testFunWVW(t, "divWVW_g", divWVW_g, arg)
+			testFunWVW(t, "divWVW", divWVW, arg)
+		}
+	}
+}
+
+var mulWWTests = []struct {
+	x, y Word
+	q, r Word
+}{
+	{_M, _M, _M - 1, 1},
+	// 32 bit only: {0xc47dfa8c, 50911, 0x98a4, 0x998587f4},
+}
+
+func TestMulWW(t *testing.T) {
+	for i, test := range mulWWTests {
+		q, r := mulWW_g(test.x, test.y)
+		if q != test.q || r != test.r {
+			t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+		}
+	}
+}
+
+var mulAddWWWTests = []struct {
+	x, y, c Word
+	q, r    Word
+}{
+	// TODO(agl): These will only work on 64-bit platforms.
+	// {15064310297182388543, 0xe7df04d2d35d5d80, 13537600649892366549, 13644450054494335067, 10832252001440893781},
+	// {15064310297182388543, 0xdab2f18048baa68d, 13644450054494335067, 12869334219691522700, 14233854684711418382},
+	{_M, _M, 0, _M - 1, 1},
+	{_M, _M, _M, _M, 0},
+}
+
+func TestMulAddWWW(t *testing.T) {
+	for i, test := range mulAddWWWTests {
+		q, r := mulAddWWW_g(test.x, test.y, test.c)
+		if q != test.q || r != test.r {
+			t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+		}
+	}
+}
+
+func benchmarkAddMulVVW(b *testing.B, n int) {
+	x := rndV(n)
+	y := rndW()
+	z := make([]Word, n)
+	b.SetBytes(int64(n * _W))
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		addMulVVW(z, x, y)
+	}
+}
+
+func BenchmarkAddMulVVW_1(b *testing.B)   { benchmarkAddMulVVW(b, 1) }
+func BenchmarkAddMulVVW_2(b *testing.B)   { benchmarkAddMulVVW(b, 2) }
+func BenchmarkAddMulVVW_3(b *testing.B)   { benchmarkAddMulVVW(b, 3) }
+func BenchmarkAddMulVVW_4(b *testing.B)   { benchmarkAddMulVVW(b, 4) }
+func BenchmarkAddMulVVW_5(b *testing.B)   { benchmarkAddMulVVW(b, 5) }
+func BenchmarkAddMulVVW_1e1(b *testing.B) { benchmarkAddMulVVW(b, 1e1) }
+func BenchmarkAddMulVVW_1e2(b *testing.B) { benchmarkAddMulVVW(b, 1e2) }
+func BenchmarkAddMulVVW_1e3(b *testing.B) { benchmarkAddMulVVW(b, 1e3) }
+func BenchmarkAddMulVVW_1e4(b *testing.B) { benchmarkAddMulVVW(b, 1e4) }
+func BenchmarkAddMulVVW_1e5(b *testing.B) { benchmarkAddMulVVW(b, 1e5) }
+
+func testWordBitLen(t *testing.T, fname string, f func(Word) int) {
+	for i := 0; i <= _W; i++ {
+		x := Word(1) << uint(i-1) // i == 0 => x == 0
+		n := f(x)
+		if n != i {
+			t.Errorf("got %d; want %d for %s(%#x)", n, i, fname, x)
+		}
+	}
+}
+
+func TestWordBitLen(t *testing.T) {
+	testWordBitLen(t, "bitLen", bitLen)
+	testWordBitLen(t, "bitLen_g", bitLen_g)
+}
+
+// runs b.N iterations of bitLen called on a Word containing (1 << nbits)-1.
+func benchmarkBitLenN(b *testing.B, nbits uint) {
+	testword := Word((uint64(1) << nbits) - 1)
+	for i := 0; i < b.N; i++ {
+		bitLen(testword)
+	}
+}
+
+// Individual bitLen tests.  Numbers chosen to examine both sides
+// of powers-of-two boundaries.
+func BenchmarkBitLen0(b *testing.B)  { benchmarkBitLenN(b, 0) }
+func BenchmarkBitLen1(b *testing.B)  { benchmarkBitLenN(b, 1) }
+func BenchmarkBitLen2(b *testing.B)  { benchmarkBitLenN(b, 2) }
+func BenchmarkBitLen3(b *testing.B)  { benchmarkBitLenN(b, 3) }
+func BenchmarkBitLen4(b *testing.B)  { benchmarkBitLenN(b, 4) }
+func BenchmarkBitLen5(b *testing.B)  { benchmarkBitLenN(b, 5) }
+func BenchmarkBitLen8(b *testing.B)  { benchmarkBitLenN(b, 8) }
+func BenchmarkBitLen9(b *testing.B)  { benchmarkBitLenN(b, 9) }
+func BenchmarkBitLen16(b *testing.B) { benchmarkBitLenN(b, 16) }
+func BenchmarkBitLen17(b *testing.B) { benchmarkBitLenN(b, 17) }
+func BenchmarkBitLen31(b *testing.B) { benchmarkBitLenN(b, 31) }
diff --git a/src/math/big/calibrate_test.go b/src/math/big/calibrate_test.go
new file mode 100644
index 000000000..f69ffbf5c
--- /dev/null
+++ b/src/math/big/calibrate_test.go
@@ -0,0 +1,88 @@
+// 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.
+
+// This file prints execution times for the Mul benchmark
+// given different Karatsuba thresholds. The result may be
+// used to manually fine-tune the threshold constant. The
+// results are somewhat fragile; use repeated runs to get
+// a clear picture.
+
+// Usage: go test -run=TestCalibrate -calibrate
+
+package big
+
+import (
+	"flag"
+	"fmt"
+	"testing"
+	"time"
+)
+
+var calibrate = flag.Bool("calibrate", false, "run calibration test")
+
+func karatsubaLoad(b *testing.B) {
+	BenchmarkMul(b)
+}
+
+// measureKaratsuba returns the time to run a Karatsuba-relevant benchmark
+// given Karatsuba threshold th.
+func measureKaratsuba(th int) time.Duration {
+	th, karatsubaThreshold = karatsubaThreshold, th
+	res := testing.Benchmark(karatsubaLoad)
+	karatsubaThreshold = th
+	return time.Duration(res.NsPerOp())
+}
+
+func computeThresholds() {
+	fmt.Printf("Multiplication times for varying Karatsuba thresholds\n")
+	fmt.Printf("(run repeatedly for good results)\n")
+
+	// determine Tk, the work load execution time using basic multiplication
+	Tb := measureKaratsuba(1e9) // th == 1e9 => Karatsuba multiplication disabled
+	fmt.Printf("Tb = %10s\n", Tb)
+
+	// thresholds
+	th := 4
+	th1 := -1
+	th2 := -1
+
+	var deltaOld time.Duration
+	for count := -1; count != 0 && th < 128; count-- {
+		// determine Tk, the work load execution time using Karatsuba multiplication
+		Tk := measureKaratsuba(th)
+
+		// improvement over Tb
+		delta := (Tb - Tk) * 100 / Tb
+
+		fmt.Printf("th = %3d  Tk = %10s  %4d%%", th, Tk, delta)
+
+		// determine break-even point
+		if Tk < Tb && th1 < 0 {
+			th1 = th
+			fmt.Print("  break-even point")
+		}
+
+		// determine diminishing return
+		if 0 < delta && delta < deltaOld && th2 < 0 {
+			th2 = th
+			fmt.Print("  diminishing return")
+		}
+		deltaOld = delta
+
+		fmt.Println()
+
+		// trigger counter
+		if th1 >= 0 && th2 >= 0 && count < 0 {
+			count = 10 // this many extra measurements after we got both thresholds
+		}
+
+		th++
+	}
+}
+
+func TestCalibrate(t *testing.T) {
+	if *calibrate {
+		computeThresholds()
+	}
+}
diff --git a/src/math/big/example_test.go b/src/math/big/example_test.go
new file mode 100644
index 000000000..078be47f9
--- /dev/null
+++ b/src/math/big/example_test.go
@@ -0,0 +1,51 @@
+// Copyright 2012 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 big_test
+
+import (
+	"fmt"
+	"log"
+	"math/big"
+)
+
+func ExampleRat_SetString() {
+	r := new(big.Rat)
+	r.SetString("355/113")
+	fmt.Println(r.FloatString(3))
+	// Output: 3.142
+}
+
+func ExampleInt_SetString() {
+	i := new(big.Int)
+	i.SetString("644", 8) // octal
+	fmt.Println(i)
+	// Output: 420
+}
+
+func ExampleRat_Scan() {
+	// The Scan function is rarely used directly;
+	// the fmt package recognizes it as an implementation of fmt.Scanner.
+	r := new(big.Rat)
+	_, err := fmt.Sscan("1.5000", r)
+	if err != nil {
+		log.Println("error scanning value:", err)
+	} else {
+		fmt.Println(r)
+	}
+	// Output: 3/2
+}
+
+func ExampleInt_Scan() {
+	// The Scan function is rarely used directly;
+	// the fmt package recognizes it as an implementation of fmt.Scanner.
+	i := new(big.Int)
+	_, err := fmt.Sscan("18446744073709551617", i)
+	if err != nil {
+		log.Println("error scanning value:", err)
+	} else {
+		fmt.Println(i)
+	}
+	// Output: 18446744073709551617
+}
diff --git a/src/math/big/gcd_test.go b/src/math/big/gcd_test.go
new file mode 100644
index 000000000..c0b9f5830
--- /dev/null
+++ b/src/math/big/gcd_test.go
@@ -0,0 +1,47 @@
+// Copyright 2012 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.
+
+// This file implements a GCD benchmark.
+// Usage: go test math/big -test.bench GCD
+
+package big
+
+import (
+	"math/rand"
+	"testing"
+)
+
+// randInt returns a pseudo-random Int in the range [1<<(size-1), (1<<size) - 1]
+func randInt(r *rand.Rand, size uint) *Int {
+	n := new(Int).Lsh(intOne, size-1)
+	x := new(Int).Rand(r, n)
+	return x.Add(x, n) // make sure result > 1<<(size-1)
+}
+
+func runGCD(b *testing.B, aSize, bSize uint) {
+	b.StopTimer()
+	var r = rand.New(rand.NewSource(1234))
+	aa := randInt(r, aSize)
+	bb := randInt(r, bSize)
+	b.StartTimer()
+	for i := 0; i < b.N; i++ {
+		new(Int).GCD(nil, nil, aa, bb)
+	}
+}
+
+func BenchmarkGCD10x10(b *testing.B)         { runGCD(b, 10, 10) }
+func BenchmarkGCD10x100(b *testing.B)        { runGCD(b, 10, 100) }
+func BenchmarkGCD10x1000(b *testing.B)       { runGCD(b, 10, 1000) }
+func BenchmarkGCD10x10000(b *testing.B)      { runGCD(b, 10, 10000) }
+func BenchmarkGCD10x100000(b *testing.B)     { runGCD(b, 10, 100000) }
+func BenchmarkGCD100x100(b *testing.B)       { runGCD(b, 100, 100) }
+func BenchmarkGCD100x1000(b *testing.B)      { runGCD(b, 100, 1000) }
+func BenchmarkGCD100x10000(b *testing.B)     { runGCD(b, 100, 10000) }
+func BenchmarkGCD100x100000(b *testing.B)    { runGCD(b, 100, 100000) }
+func BenchmarkGCD1000x1000(b *testing.B)     { runGCD(b, 1000, 1000) }
+func BenchmarkGCD1000x10000(b *testing.B)    { runGCD(b, 1000, 10000) }
+func BenchmarkGCD1000x100000(b *testing.B)   { runGCD(b, 1000, 100000) }
+func BenchmarkGCD10000x10000(b *testing.B)   { runGCD(b, 10000, 10000) }
+func BenchmarkGCD10000x100000(b *testing.B)  { runGCD(b, 10000, 100000) }
+func BenchmarkGCD100000x100000(b *testing.B) { runGCD(b, 100000, 100000) }
diff --git a/src/math/big/hilbert_test.go b/src/math/big/hilbert_test.go
new file mode 100644
index 000000000..1a84341b3
--- /dev/null
+++ b/src/math/big/hilbert_test.go
@@ -0,0 +1,160 @@
+// 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.
+
+// A little test program and benchmark for rational arithmetics.
+// Computes a Hilbert matrix, its inverse, multiplies them
+// and verifies that the product is the identity matrix.
+
+package big
+
+import (
+	"fmt"
+	"testing"
+)
+
+type matrix struct {
+	n, m int
+	a    []*Rat
+}
+
+func (a *matrix) at(i, j int) *Rat {
+	if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+		panic("index out of range")
+	}
+	return a.a[i*a.m+j]
+}
+
+func (a *matrix) set(i, j int, x *Rat) {
+	if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+		panic("index out of range")
+	}
+	a.a[i*a.m+j] = x
+}
+
+func newMatrix(n, m int) *matrix {
+	if !(0 <= n && 0 <= m) {
+		panic("illegal matrix")
+	}
+	a := new(matrix)
+	a.n = n
+	a.m = m
+	a.a = make([]*Rat, n*m)
+	return a
+}
+
+func newUnit(n int) *matrix {
+	a := newMatrix(n, n)
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			x := NewRat(0, 1)
+			if i == j {
+				x.SetInt64(1)
+			}
+			a.set(i, j, x)
+		}
+	}
+	return a
+}
+
+func newHilbert(n int) *matrix {
+	a := newMatrix(n, n)
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			a.set(i, j, NewRat(1, int64(i+j+1)))
+		}
+	}
+	return a
+}
+
+func newInverseHilbert(n int) *matrix {
+	a := newMatrix(n, n)
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			x1 := new(Rat).SetInt64(int64(i + j + 1))
+			x2 := new(Rat).SetInt(new(Int).Binomial(int64(n+i), int64(n-j-1)))
+			x3 := new(Rat).SetInt(new(Int).Binomial(int64(n+j), int64(n-i-1)))
+			x4 := new(Rat).SetInt(new(Int).Binomial(int64(i+j), int64(i)))
+
+			x1.Mul(x1, x2)
+			x1.Mul(x1, x3)
+			x1.Mul(x1, x4)
+			x1.Mul(x1, x4)
+
+			if (i+j)&1 != 0 {
+				x1.Neg(x1)
+			}
+
+			a.set(i, j, x1)
+		}
+	}
+	return a
+}
+
+func (a *matrix) mul(b *matrix) *matrix {
+	if a.m != b.n {
+		panic("illegal matrix multiply")
+	}
+	c := newMatrix(a.n, b.m)
+	for i := 0; i < c.n; i++ {
+		for j := 0; j < c.m; j++ {
+			x := NewRat(0, 1)
+			for k := 0; k < a.m; k++ {
+				x.Add(x, new(Rat).Mul(a.at(i, k), b.at(k, j)))
+			}
+			c.set(i, j, x)
+		}
+	}
+	return c
+}
+
+func (a *matrix) eql(b *matrix) bool {
+	if a.n != b.n || a.m != b.m {
+		return false
+	}
+	for i := 0; i < a.n; i++ {
+		for j := 0; j < a.m; j++ {
+			if a.at(i, j).Cmp(b.at(i, j)) != 0 {
+				return false
+			}
+		}
+	}
+	return true
+}
+
+func (a *matrix) String() string {
+	s := ""
+	for i := 0; i < a.n; i++ {
+		for j := 0; j < a.m; j++ {
+			s += fmt.Sprintf("\t%s", a.at(i, j))
+		}
+		s += "\n"
+	}
+	return s
+}
+
+func doHilbert(t *testing.T, n int) {
+	a := newHilbert(n)
+	b := newInverseHilbert(n)
+	I := newUnit(n)
+	ab := a.mul(b)
+	if !ab.eql(I) {
+		if t == nil {
+			panic("Hilbert failed")
+		}
+		t.Errorf("a   = %s\n", a)
+		t.Errorf("b   = %s\n", b)
+		t.Errorf("a*b = %s\n", ab)
+		t.Errorf("I   = %s\n", I)
+	}
+}
+
+func TestHilbert(t *testing.T) {
+	doHilbert(t, 10)
+}
+
+func BenchmarkHilbert(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		doHilbert(nil, 10)
+	}
+}
diff --git a/src/math/big/int.go b/src/math/big/int.go
new file mode 100644
index 000000000..e70d0489b
--- /dev/null
+++ b/src/math/big/int.go
@@ -0,0 +1,1024 @@
+// 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.
+
+// This file implements signed multi-precision integers.
+
+package big
+
+import (
+	"errors"
+	"fmt"
+	"io"
+	"math/rand"
+	"strings"
+)
+
+// An Int represents a signed multi-precision integer.
+// The zero value for an Int represents the value 0.
+type Int struct {
+	neg bool // sign
+	abs nat  // absolute value of the integer
+}
+
+var intOne = &Int{false, natOne}
+
+// Sign returns:
+//
+//	-1 if x <  0
+//	 0 if x == 0
+//	+1 if x >  0
+//
+func (x *Int) Sign() int {
+	if len(x.abs) == 0 {
+		return 0
+	}
+	if x.neg {
+		return -1
+	}
+	return 1
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Int) SetInt64(x int64) *Int {
+	neg := false
+	if x < 0 {
+		neg = true
+		x = -x
+	}
+	z.abs = z.abs.setUint64(uint64(x))
+	z.neg = neg
+	return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Int) SetUint64(x uint64) *Int {
+	z.abs = z.abs.setUint64(x)
+	z.neg = false
+	return z
+}
+
+// NewInt allocates and returns a new Int set to x.
+func NewInt(x int64) *Int {
+	return new(Int).SetInt64(x)
+}
+
+// Set sets z to x and returns z.
+func (z *Int) Set(x *Int) *Int {
+	if z != x {
+		z.abs = z.abs.set(x.abs)
+		z.neg = x.neg
+	}
+	return z
+}
+
+// Bits provides raw (unchecked but fast) access to x by returning its
+// absolute value as a little-endian Word slice. The result and x share
+// the same underlying array.
+// Bits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (x *Int) Bits() []Word {
+	return x.abs
+}
+
+// SetBits provides raw (unchecked but fast) access to z by setting its
+// value to abs, interpreted as a little-endian Word slice, and returning
+// z. The result and abs share the same underlying array.
+// SetBits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (z *Int) SetBits(abs []Word) *Int {
+	z.abs = nat(abs).norm()
+	z.neg = false
+	return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Int) Abs(x *Int) *Int {
+	z.Set(x)
+	z.neg = false
+	return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Int) Neg(x *Int) *Int {
+	z.Set(x)
+	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
+	return z
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Int) Add(x, y *Int) *Int {
+	neg := x.neg
+	if x.neg == y.neg {
+		// x + y == x + y
+		// (-x) + (-y) == -(x + y)
+		z.abs = z.abs.add(x.abs, y.abs)
+	} else {
+		// x + (-y) == x - y == -(y - x)
+		// (-x) + y == y - x == -(x - y)
+		if x.abs.cmp(y.abs) >= 0 {
+			z.abs = z.abs.sub(x.abs, y.abs)
+		} else {
+			neg = !neg
+			z.abs = z.abs.sub(y.abs, x.abs)
+		}
+	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
+	return z
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Int) Sub(x, y *Int) *Int {
+	neg := x.neg
+	if x.neg != y.neg {
+		// x - (-y) == x + y
+		// (-x) - y == -(x + y)
+		z.abs = z.abs.add(x.abs, y.abs)
+	} else {
+		// x - y == x - y == -(y - x)
+		// (-x) - (-y) == y - x == -(x - y)
+		if x.abs.cmp(y.abs) >= 0 {
+			z.abs = z.abs.sub(x.abs, y.abs)
+		} else {
+			neg = !neg
+			z.abs = z.abs.sub(y.abs, x.abs)
+		}
+	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
+	return z
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Int) Mul(x, y *Int) *Int {
+	// x * y == x * y
+	// x * (-y) == -(x * y)
+	// (-x) * y == -(x * y)
+	// (-x) * (-y) == x * y
+	z.abs = z.abs.mul(x.abs, y.abs)
+	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+	return z
+}
+
+// MulRange sets z to the product of all integers
+// in the range [a, b] inclusively and returns z.
+// If a > b (empty range), the result is 1.
+func (z *Int) MulRange(a, b int64) *Int {
+	switch {
+	case a > b:
+		return z.SetInt64(1) // empty range
+	case a <= 0 && b >= 0:
+		return z.SetInt64(0) // range includes 0
+	}
+	// a <= b && (b < 0 || a > 0)
+
+	neg := false
+	if a < 0 {
+		neg = (b-a)&1 == 0
+		a, b = -b, -a
+	}
+
+	z.abs = z.abs.mulRange(uint64(a), uint64(b))
+	z.neg = neg
+	return z
+}
+
+// Binomial sets z to the binomial coefficient of (n, k) and returns z.
+func (z *Int) Binomial(n, k int64) *Int {
+	var a, b Int
+	a.MulRange(n-k+1, n)
+	b.MulRange(1, k)
+	return z.Quo(&a, &b)
+}
+
+// Quo sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Quo implements truncated division (like Go); see QuoRem for more details.
+func (z *Int) Quo(x, y *Int) *Int {
+	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
+	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+	return z
+}
+
+// Rem sets z to the remainder x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Rem implements truncated modulus (like Go); see QuoRem for more details.
+func (z *Int) Rem(x, y *Int) *Int {
+	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
+	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
+	return z
+}
+
+// QuoRem sets z to the quotient x/y and r to the remainder x%y
+// and returns the pair (z, r) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// QuoRem implements T-division and modulus (like Go):
+//
+//	q = x/y      with the result truncated to zero
+//	r = x - y*q
+//
+// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
+// See DivMod for Euclidean division and modulus (unlike Go).
+//
+func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
+	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
+	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
+	return z, r
+}
+
+// Div sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Div implements Euclidean division (unlike Go); see DivMod for more details.
+func (z *Int) Div(x, y *Int) *Int {
+	y_neg := y.neg // z may be an alias for y
+	var r Int
+	z.QuoRem(x, y, &r)
+	if r.neg {
+		if y_neg {
+			z.Add(z, intOne)
+		} else {
+			z.Sub(z, intOne)
+		}
+	}
+	return z
+}
+
+// Mod sets z to the modulus x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
+func (z *Int) Mod(x, y *Int) *Int {
+	y0 := y // save y
+	if z == y || alias(z.abs, y.abs) {
+		y0 = new(Int).Set(y)
+	}
+	var q Int
+	q.QuoRem(x, y, z)
+	if z.neg {
+		if y0.neg {
+			z.Sub(z, y0)
+		} else {
+			z.Add(z, y0)
+		}
+	}
+	return z
+}
+
+// DivMod sets z to the quotient x div y and m to the modulus x mod y
+// and returns the pair (z, m) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// DivMod implements Euclidean division and modulus (unlike Go):
+//
+//	q = x div y  such that
+//	m = x - y*q  with 0 <= m < |q|
+//
+// (See Raymond T. Boute, ``The Euclidean definition of the functions
+// div and mod''. ACM Transactions on Programming Languages and
+// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
+// ACM press.)
+// See QuoRem for T-division and modulus (like Go).
+//
+func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
+	y0 := y // save y
+	if z == y || alias(z.abs, y.abs) {
+		y0 = new(Int).Set(y)
+	}
+	z.QuoRem(x, y, m)
+	if m.neg {
+		if y0.neg {
+			z.Add(z, intOne)
+			m.Sub(m, y0)
+		} else {
+			z.Sub(z, intOne)
+			m.Add(m, y0)
+		}
+	}
+	return z, m
+}
+
+// Cmp compares x and y and returns:
+//
+//   -1 if x <  y
+//    0 if x == y
+//   +1 if x >  y
+//
+func (x *Int) Cmp(y *Int) (r int) {
+	// x cmp y == x cmp y
+	// x cmp (-y) == x
+	// (-x) cmp y == y
+	// (-x) cmp (-y) == -(x cmp y)
+	switch {
+	case x.neg == y.neg:
+		r = x.abs.cmp(y.abs)
+		if x.neg {
+			r = -r
+		}
+	case x.neg:
+		r = -1
+	default:
+		r = 1
+	}
+	return
+}
+
+func (x *Int) String() string {
+	switch {
+	case x == nil:
+		return "<nil>"
+	case x.neg:
+		return "-" + x.abs.decimalString()
+	}
+	return x.abs.decimalString()
+}
+
+func charset(ch rune) string {
+	switch ch {
+	case 'b':
+		return lowercaseDigits[0:2]
+	case 'o':
+		return lowercaseDigits[0:8]
+	case 'd', 's', 'v':
+		return lowercaseDigits[0:10]
+	case 'x':
+		return lowercaseDigits[0:16]
+	case 'X':
+		return uppercaseDigits[0:16]
+	}
+	return "" // unknown format
+}
+
+// write count copies of text to s
+func writeMultiple(s fmt.State, text string, count int) {
+	if len(text) > 0 {
+		b := []byte(text)
+		for ; count > 0; count-- {
+			s.Write(b)
+		}
+	}
+}
+
+// Format is a support routine for fmt.Formatter. It accepts
+// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
+// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+// Also supported are the full suite of package fmt's format
+// verbs for integral types, including '+', '-', and ' '
+// for sign control, '#' for leading zero in octal and for
+// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
+// respectively, specification of minimum digits precision,
+// output field width, space or zero padding, and left or
+// right justification.
+//
+func (x *Int) Format(s fmt.State, ch rune) {
+	cs := charset(ch)
+
+	// special cases
+	switch {
+	case cs == "":
+		// unknown format
+		fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
+		return
+	case x == nil:
+		fmt.Fprint(s, "<nil>")
+		return
+	}
+
+	// determine sign character
+	sign := ""
+	switch {
+	case x.neg:
+		sign = "-"
+	case s.Flag('+'): // supersedes ' ' when both specified
+		sign = "+"
+	case s.Flag(' '):
+		sign = " "
+	}
+
+	// determine prefix characters for indicating output base
+	prefix := ""
+	if s.Flag('#') {
+		switch ch {
+		case 'o': // octal
+			prefix = "0"
+		case 'x': // hexadecimal
+			prefix = "0x"
+		case 'X':
+			prefix = "0X"
+		}
+	}
+
+	// determine digits with base set by len(cs) and digit characters from cs
+	digits := x.abs.string(cs)
+
+	// number of characters for the three classes of number padding
+	var left int   // space characters to left of digits for right justification ("%8d")
+	var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
+	var right int  // space characters to right of digits for left justification ("%-8d")
+
+	// determine number padding from precision: the least number of digits to output
+	precision, precisionSet := s.Precision()
+	if precisionSet {
+		switch {
+		case len(digits) < precision:
+			zeroes = precision - len(digits) // count of zero padding
+		case digits == "0" && precision == 0:
+			return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
+		}
+	}
+
+	// determine field pad from width: the least number of characters to output
+	length := len(sign) + len(prefix) + zeroes + len(digits)
+	if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
+		switch d := width - length; {
+		case s.Flag('-'):
+			// pad on the right with spaces; supersedes '0' when both specified
+			right = d
+		case s.Flag('0') && !precisionSet:
+			// pad with zeroes unless precision also specified
+			zeroes = d
+		default:
+			// pad on the left with spaces
+			left = d
+		}
+	}
+
+	// print number as [left pad][sign][prefix][zero pad][digits][right pad]
+	writeMultiple(s, " ", left)
+	writeMultiple(s, sign, 1)
+	writeMultiple(s, prefix, 1)
+	writeMultiple(s, "0", zeroes)
+	writeMultiple(s, digits, 1)
+	writeMultiple(s, " ", right)
+}
+
+// scan sets z to the integer value corresponding to the longest possible prefix
+// read from r representing a signed integer number in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined but the returned value is nil. The
+// syntax follows the syntax of integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
+	// determine sign
+	ch, _, err := r.ReadRune()
+	if err != nil {
+		return nil, 0, err
+	}
+	neg := false
+	switch ch {
+	case '-':
+		neg = true
+	case '+': // nothing to do
+	default:
+		r.UnreadRune()
+	}
+
+	// determine mantissa
+	z.abs, base, err = z.abs.scan(r, base)
+	if err != nil {
+		return nil, base, err
+	}
+	z.neg = len(z.abs) > 0 && neg // 0 has no sign
+
+	return z, base, nil
+}
+
+// Scan is a support routine for fmt.Scanner; it sets z to the value of
+// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
+// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+func (z *Int) Scan(s fmt.ScanState, ch rune) error {
+	s.SkipSpace() // skip leading space characters
+	base := 0
+	switch ch {
+	case 'b':
+		base = 2
+	case 'o':
+		base = 8
+	case 'd':
+		base = 10
+	case 'x', 'X':
+		base = 16
+	case 's', 'v':
+		// let scan determine the base
+	default:
+		return errors.New("Int.Scan: invalid verb")
+	}
+	_, _, err := z.scan(s, base)
+	return err
+}
+
+// low32 returns the least significant 32 bits of z.
+func low32(z nat) uint32 {
+	if len(z) == 0 {
+		return 0
+	}
+	return uint32(z[0])
+}
+
+// low64 returns the least significant 64 bits of z.
+func low64(z nat) uint64 {
+	if len(z) == 0 {
+		return 0
+	}
+	v := uint64(z[0])
+	if _W == 32 && len(z) > 1 {
+		v |= uint64(z[1]) << 32
+	}
+	return v
+}
+
+// Int64 returns the int64 representation of x.
+// If x cannot be represented in an int64, the result is undefined.
+func (x *Int) Int64() int64 {
+	v := int64(low64(x.abs))
+	if x.neg {
+		v = -v
+	}
+	return v
+}
+
+// Uint64 returns the uint64 representation of x.
+// If x cannot be represented in a uint64, the result is undefined.
+func (x *Int) Uint64() uint64 {
+	return low64(x.abs)
+}
+
+// SetString sets z to the value of s, interpreted in the given base,
+// and returns z and a boolean indicating success. If SetString fails,
+// the value of z is undefined but the returned value is nil.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) SetString(s string, base int) (*Int, bool) {
+	r := strings.NewReader(s)
+	_, _, err := z.scan(r, base)
+	if err != nil {
+		return nil, false
+	}
+	_, _, err = r.ReadRune()
+	if err != io.EOF {
+		return nil, false
+	}
+	return z, true // err == io.EOF => scan consumed all of s
+}
+
+// SetBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z *Int) SetBytes(buf []byte) *Int {
+	z.abs = z.abs.setBytes(buf)
+	z.neg = false
+	return z
+}
+
+// Bytes returns the absolute value of x as a big-endian byte slice.
+func (x *Int) Bytes() []byte {
+	buf := make([]byte, len(x.abs)*_S)
+	return buf[x.abs.bytes(buf):]
+}
+
+// BitLen returns the length of the absolute value of x in bits.
+// The bit length of 0 is 0.
+func (x *Int) BitLen() int {
+	return x.abs.bitLen()
+}
+
+// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
+// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
+// See Knuth, volume 2, section 4.6.3.
+func (z *Int) Exp(x, y, m *Int) *Int {
+	var yWords nat
+	if !y.neg {
+		yWords = y.abs
+	}
+	// y >= 0
+
+	var mWords nat
+	if m != nil {
+		mWords = m.abs // m.abs may be nil for m == 0
+	}
+
+	z.abs = z.abs.expNN(x.abs, yWords, mWords)
+	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
+	return z
+}
+
+// GCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
+// If either a or b is <= 0, GCD sets z = x = y = 0.
+func (z *Int) GCD(x, y, a, b *Int) *Int {
+	if a.Sign() <= 0 || b.Sign() <= 0 {
+		z.SetInt64(0)
+		if x != nil {
+			x.SetInt64(0)
+		}
+		if y != nil {
+			y.SetInt64(0)
+		}
+		return z
+	}
+	if x == nil && y == nil {
+		return z.binaryGCD(a, b)
+	}
+
+	A := new(Int).Set(a)
+	B := new(Int).Set(b)
+
+	X := new(Int)
+	Y := new(Int).SetInt64(1)
+
+	lastX := new(Int).SetInt64(1)
+	lastY := new(Int)
+
+	q := new(Int)
+	temp := new(Int)
+
+	for len(B.abs) > 0 {
+		r := new(Int)
+		q, r = q.QuoRem(A, B, r)
+
+		A, B = B, r
+
+		temp.Set(X)
+		X.Mul(X, q)
+		X.neg = !X.neg
+		X.Add(X, lastX)
+		lastX.Set(temp)
+
+		temp.Set(Y)
+		Y.Mul(Y, q)
+		Y.neg = !Y.neg
+		Y.Add(Y, lastY)
+		lastY.Set(temp)
+	}
+
+	if x != nil {
+		*x = *lastX
+	}
+
+	if y != nil {
+		*y = *lastY
+	}
+
+	*z = *A
+	return z
+}
+
+// binaryGCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
+func (z *Int) binaryGCD(a, b *Int) *Int {
+	u := z
+	v := new(Int)
+
+	// use one Euclidean iteration to ensure that u and v are approx. the same size
+	switch {
+	case len(a.abs) > len(b.abs):
+		u.Set(b)
+		v.Rem(a, b)
+	case len(a.abs) < len(b.abs):
+		u.Set(a)
+		v.Rem(b, a)
+	default:
+		u.Set(a)
+		v.Set(b)
+	}
+
+	// v might be 0 now
+	if len(v.abs) == 0 {
+		return u
+	}
+	// u > 0 && v > 0
+
+	// determine largest k such that u = u' << k, v = v' << k
+	k := u.abs.trailingZeroBits()
+	if vk := v.abs.trailingZeroBits(); vk < k {
+		k = vk
+	}
+	u.Rsh(u, k)
+	v.Rsh(v, k)
+
+	// determine t (we know that u > 0)
+	t := new(Int)
+	if u.abs[0]&1 != 0 {
+		// u is odd
+		t.Neg(v)
+	} else {
+		t.Set(u)
+	}
+
+	for len(t.abs) > 0 {
+		// reduce t
+		t.Rsh(t, t.abs.trailingZeroBits())
+		if t.neg {
+			v, t = t, v
+			v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
+		} else {
+			u, t = t, u
+		}
+		t.Sub(u, v)
+	}
+
+	return z.Lsh(u, k)
+}
+
+// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
+// If it returns true, x is prime with probability 1 - 1/4^n.
+// If it returns false, x is not prime.
+func (x *Int) ProbablyPrime(n int) bool {
+	return !x.neg && x.abs.probablyPrime(n)
+}
+
+// Rand sets z to a pseudo-random number in [0, n) and returns z.
+func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
+	z.neg = false
+	if n.neg == true || len(n.abs) == 0 {
+		z.abs = nil
+		return z
+	}
+	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
+	return z
+}
+
+// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
+// p is a prime) and returns z.
+func (z *Int) ModInverse(g, p *Int) *Int {
+	var d Int
+	d.GCD(z, nil, g, p)
+	// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
+	// that modulo p results in g*x = 1, therefore x is the inverse element.
+	if z.neg {
+		z.Add(z, p)
+	}
+	return z
+}
+
+// Lsh sets z = x << n and returns z.
+func (z *Int) Lsh(x *Int, n uint) *Int {
+	z.abs = z.abs.shl(x.abs, n)
+	z.neg = x.neg
+	return z
+}
+
+// Rsh sets z = x >> n and returns z.
+func (z *Int) Rsh(x *Int, n uint) *Int {
+	if x.neg {
+		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
+		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
+		t = t.shr(t, n)
+		z.abs = t.add(t, natOne)
+		z.neg = true // z cannot be zero if x is negative
+		return z
+	}
+
+	z.abs = z.abs.shr(x.abs, n)
+	z.neg = false
+	return z
+}
+
+// Bit returns the value of the i'th bit of x. That is, it
+// returns (x>>i)&1. The bit index i must be >= 0.
+func (x *Int) Bit(i int) uint {
+	if i == 0 {
+		// optimization for common case: odd/even test of x
+		if len(x.abs) > 0 {
+			return uint(x.abs[0] & 1) // bit 0 is same for -x
+		}
+		return 0
+	}
+	if i < 0 {
+		panic("negative bit index")
+	}
+	if x.neg {
+		t := nat(nil).sub(x.abs, natOne)
+		return t.bit(uint(i)) ^ 1
+	}
+
+	return x.abs.bit(uint(i))
+}
+
+// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
+// That is, if b is 1 SetBit sets z = x | (1 << i);
+// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
+// SetBit will panic.
+func (z *Int) SetBit(x *Int, i int, b uint) *Int {
+	if i < 0 {
+		panic("negative bit index")
+	}
+	if x.neg {
+		t := z.abs.sub(x.abs, natOne)
+		t = t.setBit(t, uint(i), b^1)
+		z.abs = t.add(t, natOne)
+		z.neg = len(z.abs) > 0
+		return z
+	}
+	z.abs = z.abs.setBit(x.abs, uint(i), b)
+	z.neg = false
+	return z
+}
+
+// And sets z = x & y and returns z.
+func (z *Int) And(x, y *Int) *Int {
+	if x.neg == y.neg {
+		if x.neg {
+			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
+			x1 := nat(nil).sub(x.abs, natOne)
+			y1 := nat(nil).sub(y.abs, natOne)
+			z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
+			z.neg = true // z cannot be zero if x and y are negative
+			return z
+		}
+
+		// x & y == x & y
+		z.abs = z.abs.and(x.abs, y.abs)
+		z.neg = false
+		return z
+	}
+
+	// x.neg != y.neg
+	if x.neg {
+		x, y = y, x // & is symmetric
+	}
+
+	// x & (-y) == x & ^(y-1) == x &^ (y-1)
+	y1 := nat(nil).sub(y.abs, natOne)
+	z.abs = z.abs.andNot(x.abs, y1)
+	z.neg = false
+	return z
+}
+
+// AndNot sets z = x &^ y and returns z.
+func (z *Int) AndNot(x, y *Int) *Int {
+	if x.neg == y.neg {
+		if x.neg {
+			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
+			x1 := nat(nil).sub(x.abs, natOne)
+			y1 := nat(nil).sub(y.abs, natOne)
+			z.abs = z.abs.andNot(y1, x1)
+			z.neg = false
+			return z
+		}
+
+		// x &^ y == x &^ y
+		z.abs = z.abs.andNot(x.abs, y.abs)
+		z.neg = false
+		return z
+	}
+
+	if x.neg {
+		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
+		x1 := nat(nil).sub(x.abs, natOne)
+		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
+		z.neg = true // z cannot be zero if x is negative and y is positive
+		return z
+	}
+
+	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
+	y1 := nat(nil).add(y.abs, natOne)
+	z.abs = z.abs.and(x.abs, y1)
+	z.neg = false
+	return z
+}
+
+// Or sets z = x | y and returns z.
+func (z *Int) Or(x, y *Int) *Int {
+	if x.neg == y.neg {
+		if x.neg {
+			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
+			x1 := nat(nil).sub(x.abs, natOne)
+			y1 := nat(nil).sub(y.abs, natOne)
+			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
+			z.neg = true // z cannot be zero if x and y are negative
+			return z
+		}
+
+		// x | y == x | y
+		z.abs = z.abs.or(x.abs, y.abs)
+		z.neg = false
+		return z
+	}
+
+	// x.neg != y.neg
+	if x.neg {
+		x, y = y, x // | is symmetric
+	}
+
+	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
+	y1 := nat(nil).sub(y.abs, natOne)
+	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
+	z.neg = true // z cannot be zero if one of x or y is negative
+	return z
+}
+
+// Xor sets z = x ^ y and returns z.
+func (z *Int) Xor(x, y *Int) *Int {
+	if x.neg == y.neg {
+		if x.neg {
+			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
+			x1 := nat(nil).sub(x.abs, natOne)
+			y1 := nat(nil).sub(y.abs, natOne)
+			z.abs = z.abs.xor(x1, y1)
+			z.neg = false
+			return z
+		}
+
+		// x ^ y == x ^ y
+		z.abs = z.abs.xor(x.abs, y.abs)
+		z.neg = false
+		return z
+	}
+
+	// x.neg != y.neg
+	if x.neg {
+		x, y = y, x // ^ is symmetric
+	}
+
+	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
+	y1 := nat(nil).sub(y.abs, natOne)
+	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
+	z.neg = true // z cannot be zero if only one of x or y is negative
+	return z
+}
+
+// Not sets z = ^x and returns z.
+func (z *Int) Not(x *Int) *Int {
+	if x.neg {
+		// ^(-x) == ^(^(x-1)) == x-1
+		z.abs = z.abs.sub(x.abs, natOne)
+		z.neg = false
+		return z
+	}
+
+	// ^x == -x-1 == -(x+1)
+	z.abs = z.abs.add(x.abs, natOne)
+	z.neg = true // z cannot be zero if x is positive
+	return z
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const intGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Int) GobEncode() ([]byte, error) {
+	if x == nil {
+		return nil, nil
+	}
+	buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
+	i := x.abs.bytes(buf) - 1            // i >= 0
+	b := intGobVersion << 1              // make space for sign bit
+	if x.neg {
+		b |= 1
+	}
+	buf[i] = b
+	return buf[i:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Int) GobDecode(buf []byte) error {
+	if len(buf) == 0 {
+		// Other side sent a nil or default value.
+		*z = Int{}
+		return nil
+	}
+	b := buf[0]
+	if b>>1 != intGobVersion {
+		return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
+	}
+	z.neg = b&1 != 0
+	z.abs = z.abs.setBytes(buf[1:])
+	return nil
+}
+
+// MarshalJSON implements the json.Marshaler interface.
+func (z *Int) MarshalJSON() ([]byte, error) {
+	// TODO(gri): get rid of the []byte/string conversions
+	return []byte(z.String()), nil
+}
+
+// UnmarshalJSON implements the json.Unmarshaler interface.
+func (z *Int) UnmarshalJSON(text []byte) error {
+	// TODO(gri): get rid of the []byte/string conversions
+	if _, ok := z.SetString(string(text), 0); !ok {
+		return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+	}
+	return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface
+func (z *Int) MarshalText() (text []byte, err error) {
+	return []byte(z.String()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface
+func (z *Int) UnmarshalText(text []byte) error {
+	if _, ok := z.SetString(string(text), 0); !ok {
+		return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+	}
+	return nil
+}
diff --git a/src/math/big/int_test.go b/src/math/big/int_test.go
new file mode 100644
index 000000000..299dc72fb
--- /dev/null
+++ b/src/math/big/int_test.go
@@ -0,0 +1,1601 @@
+// 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.
+
+package big
+
+import (
+	"bytes"
+	"encoding/gob"
+	"encoding/hex"
+	"encoding/json"
+	"encoding/xml"
+	"fmt"
+	"math/rand"
+	"testing"
+	"testing/quick"
+)
+
+func isNormalized(x *Int) bool {
+	if len(x.abs) == 0 {
+		return !x.neg
+	}
+	// len(x.abs) > 0
+	return x.abs[len(x.abs)-1] != 0
+}
+
+type funZZ func(z, x, y *Int) *Int
+type argZZ struct {
+	z, x, y *Int
+}
+
+var sumZZ = []argZZ{
+	{NewInt(0), NewInt(0), NewInt(0)},
+	{NewInt(1), NewInt(1), NewInt(0)},
+	{NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+	{NewInt(-1), NewInt(-1), NewInt(0)},
+	{NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+	{NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
+}
+
+var prodZZ = []argZZ{
+	{NewInt(0), NewInt(0), NewInt(0)},
+	{NewInt(0), NewInt(1), NewInt(0)},
+	{NewInt(1), NewInt(1), NewInt(1)},
+	{NewInt(-991 * 991), NewInt(991), NewInt(-991)},
+	// TODO(gri) add larger products
+}
+
+func TestSignZ(t *testing.T) {
+	var zero Int
+	for _, a := range sumZZ {
+		s := a.z.Sign()
+		e := a.z.Cmp(&zero)
+		if s != e {
+			t.Errorf("got %d; want %d for z = %v", s, e, a.z)
+		}
+	}
+}
+
+func TestSetZ(t *testing.T) {
+	for _, a := range sumZZ {
+		var z Int
+		z.Set(a.z)
+		if !isNormalized(&z) {
+			t.Errorf("%v is not normalized", z)
+		}
+		if (&z).Cmp(a.z) != 0 {
+			t.Errorf("got z = %v; want %v", z, a.z)
+		}
+	}
+}
+
+func TestAbsZ(t *testing.T) {
+	var zero Int
+	for _, a := range sumZZ {
+		var z Int
+		z.Abs(a.z)
+		var e Int
+		e.Set(a.z)
+		if e.Cmp(&zero) < 0 {
+			e.Sub(&zero, &e)
+		}
+		if z.Cmp(&e) != 0 {
+			t.Errorf("got z = %v; want %v", z, e)
+		}
+	}
+}
+
+func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
+	var z Int
+	f(&z, a.x, a.y)
+	if !isNormalized(&z) {
+		t.Errorf("%s%v is not normalized", msg, z)
+	}
+	if (&z).Cmp(a.z) != 0 {
+		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
+	}
+}
+
+func TestSumZZ(t *testing.T) {
+	AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
+	SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
+	for _, a := range sumZZ {
+		arg := a
+		testFunZZ(t, "AddZZ", AddZZ, arg)
+
+		arg = argZZ{a.z, a.y, a.x}
+		testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
+
+		arg = argZZ{a.x, a.z, a.y}
+		testFunZZ(t, "SubZZ", SubZZ, arg)
+
+		arg = argZZ{a.y, a.z, a.x}
+		testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
+	}
+}
+
+func TestProdZZ(t *testing.T) {
+	MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
+	for _, a := range prodZZ {
+		arg := a
+		testFunZZ(t, "MulZZ", MulZZ, arg)
+
+		arg = argZZ{a.z, a.y, a.x}
+		testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
+	}
+}
+
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+	z := make([]byte, len(x)+len(y))
+
+	// multiply
+	k0 := len(z) - 1
+	for j := len(y) - 1; j >= 0; j-- {
+		d := int(y[j])
+		if d != 0 {
+			k := k0
+			carry := 0
+			for i := len(x) - 1; i >= 0; i-- {
+				t := int(z[k]) + int(x[i])*d + carry
+				z[k], carry = byte(t), t>>8
+				k--
+			}
+			z[k] = byte(carry)
+		}
+		k0--
+	}
+
+	// normalize (remove leading 0's)
+	i := 0
+	for i < len(z) && z[i] == 0 {
+		i++
+	}
+
+	return z[i:]
+}
+
+func checkMul(a, b []byte) bool {
+	var x, y, z1 Int
+	x.SetBytes(a)
+	y.SetBytes(b)
+	z1.Mul(&x, &y)
+
+	var z2 Int
+	z2.SetBytes(mulBytes(a, b))
+
+	return z1.Cmp(&z2) == 0
+}
+
+func TestMul(t *testing.T) {
+	if err := quick.Check(checkMul, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+var mulRangesZ = []struct {
+	a, b int64
+	prod string
+}{
+	// entirely positive ranges are covered by mulRangesN
+	{-1, 1, "0"},
+	{-2, -1, "2"},
+	{-3, -2, "6"},
+	{-3, -1, "-6"},
+	{1, 3, "6"},
+	{-10, -10, "-10"},
+	{0, -1, "1"},                      // empty range
+	{-1, -100, "1"},                   // empty range
+	{-1, 1, "0"},                      // range includes 0
+	{-1e9, 0, "0"},                    // range includes 0
+	{-1e9, 1e9, "0"},                  // range includes 0
+	{-10, -1, "3628800"},              // 10!
+	{-20, -2, "-2432902008176640000"}, // -20!
+	{-99, -1,
+		"-933262154439441526816992388562667004907159682643816214685929" +
+			"638952175999932299156089414639761565182862536979208272237582" +
+			"511852109168640000000000000000000000", // -99!
+	},
+}
+
+func TestMulRangeZ(t *testing.T) {
+	var tmp Int
+	// test entirely positive ranges
+	for i, r := range mulRangesN {
+		prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
+		if prod != r.prod {
+			t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
+		}
+	}
+	// test other ranges
+	for i, r := range mulRangesZ {
+		prod := tmp.MulRange(r.a, r.b).String()
+		if prod != r.prod {
+			t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
+		}
+	}
+}
+
+var stringTests = []struct {
+	in   string
+	out  string
+	base int
+	val  int64
+	ok   bool
+}{
+	{in: "", ok: false},
+	{in: "a", ok: false},
+	{in: "z", ok: false},
+	{in: "+", ok: false},
+	{in: "-", ok: false},
+	{in: "0b", ok: false},
+	{in: "0x", ok: false},
+	{in: "2", base: 2, ok: false},
+	{in: "0b2", base: 0, ok: false},
+	{in: "08", ok: false},
+	{in: "8", base: 8, ok: false},
+	{in: "0xg", base: 0, ok: false},
+	{in: "g", base: 16, ok: false},
+	{"0", "0", 0, 0, true},
+	{"0", "0", 10, 0, true},
+	{"0", "0", 16, 0, true},
+	{"+0", "0", 0, 0, true},
+	{"-0", "0", 0, 0, true},
+	{"10", "10", 0, 10, true},
+	{"10", "10", 10, 10, true},
+	{"10", "10", 16, 16, true},
+	{"-10", "-10", 16, -16, true},
+	{"+10", "10", 16, 16, true},
+	{"0x10", "16", 0, 16, true},
+	{in: "0x10", base: 16, ok: false},
+	{"-0x10", "-16", 0, -16, true},
+	{"+0x10", "16", 0, 16, true},
+	{"00", "0", 0, 0, true},
+	{"0", "0", 8, 0, true},
+	{"07", "7", 0, 7, true},
+	{"7", "7", 8, 7, true},
+	{"023", "19", 0, 19, true},
+	{"23", "23", 8, 19, true},
+	{"cafebabe", "cafebabe", 16, 0xcafebabe, true},
+	{"0b0", "0", 0, 0, true},
+	{"-111", "-111", 2, -7, true},
+	{"-0b111", "-7", 0, -7, true},
+	{"0b1001010111", "599", 0, 0x257, true},
+	{"1001010111", "1001010111", 2, 0x257, true},
+}
+
+func format(base int) string {
+	switch base {
+	case 2:
+		return "%b"
+	case 8:
+		return "%o"
+	case 16:
+		return "%x"
+	}
+	return "%d"
+}
+
+func TestGetString(t *testing.T) {
+	z := new(Int)
+	for i, test := range stringTests {
+		if !test.ok {
+			continue
+		}
+		z.SetInt64(test.val)
+
+		if test.base == 10 {
+			s := z.String()
+			if s != test.out {
+				t.Errorf("#%da got %s; want %s", i, s, test.out)
+			}
+		}
+
+		s := fmt.Sprintf(format(test.base), z)
+		if s != test.out {
+			t.Errorf("#%db got %s; want %s", i, s, test.out)
+		}
+	}
+}
+
+func TestSetString(t *testing.T) {
+	tmp := new(Int)
+	for i, test := range stringTests {
+		// initialize to a non-zero value so that issues with parsing
+		// 0 are detected
+		tmp.SetInt64(1234567890)
+		n1, ok1 := new(Int).SetString(test.in, test.base)
+		n2, ok2 := tmp.SetString(test.in, test.base)
+		expected := NewInt(test.val)
+		if ok1 != test.ok || ok2 != test.ok {
+			t.Errorf("#%d (input '%s') ok incorrect (should be %t)", i, test.in, test.ok)
+			continue
+		}
+		if !ok1 {
+			if n1 != nil {
+				t.Errorf("#%d (input '%s') n1 != nil", i, test.in)
+			}
+			continue
+		}
+		if !ok2 {
+			if n2 != nil {
+				t.Errorf("#%d (input '%s') n2 != nil", i, test.in)
+			}
+			continue
+		}
+
+		if ok1 && !isNormalized(n1) {
+			t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n1)
+		}
+		if ok2 && !isNormalized(n2) {
+			t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n2)
+		}
+
+		if n1.Cmp(expected) != 0 {
+			t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n1, test.val)
+		}
+		if n2.Cmp(expected) != 0 {
+			t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n2, test.val)
+		}
+	}
+}
+
+var formatTests = []struct {
+	input  string
+	format string
+	output string
+}{
+	{"<nil>", "%x", "<nil>"},
+	{"<nil>", "%#x", "<nil>"},
+	{"<nil>", "%#y", "%!y(big.Int=<nil>)"},
+
+	{"10", "%b", "1010"},
+	{"10", "%o", "12"},
+	{"10", "%d", "10"},
+	{"10", "%v", "10"},
+	{"10", "%x", "a"},
+	{"10", "%X", "A"},
+	{"-10", "%X", "-A"},
+	{"10", "%y", "%!y(big.Int=10)"},
+	{"-10", "%y", "%!y(big.Int=-10)"},
+
+	{"10", "%#b", "1010"},
+	{"10", "%#o", "012"},
+	{"10", "%#d", "10"},
+	{"10", "%#v", "10"},
+	{"10", "%#x", "0xa"},
+	{"10", "%#X", "0XA"},
+	{"-10", "%#X", "-0XA"},
+	{"10", "%#y", "%!y(big.Int=10)"},
+	{"-10", "%#y", "%!y(big.Int=-10)"},
+
+	{"1234", "%d", "1234"},
+	{"1234", "%3d", "1234"},
+	{"1234", "%4d", "1234"},
+	{"-1234", "%d", "-1234"},
+	{"1234", "% 5d", " 1234"},
+	{"1234", "%+5d", "+1234"},
+	{"1234", "%-5d", "1234 "},
+	{"1234", "%x", "4d2"},
+	{"1234", "%X", "4D2"},
+	{"-1234", "%3x", "-4d2"},
+	{"-1234", "%4x", "-4d2"},
+	{"-1234", "%5x", " -4d2"},
+	{"-1234", "%-5x", "-4d2 "},
+	{"1234", "%03d", "1234"},
+	{"1234", "%04d", "1234"},
+	{"1234", "%05d", "01234"},
+	{"1234", "%06d", "001234"},
+	{"-1234", "%06d", "-01234"},
+	{"1234", "%+06d", "+01234"},
+	{"1234", "% 06d", " 01234"},
+	{"1234", "%-6d", "1234  "},
+	{"1234", "%-06d", "1234  "},
+	{"-1234", "%-06d", "-1234 "},
+
+	{"1234", "%.3d", "1234"},
+	{"1234", "%.4d", "1234"},
+	{"1234", "%.5d", "01234"},
+	{"1234", "%.6d", "001234"},
+	{"-1234", "%.3d", "-1234"},
+	{"-1234", "%.4d", "-1234"},
+	{"-1234", "%.5d", "-01234"},
+	{"-1234", "%.6d", "-001234"},
+
+	{"1234", "%8.3d", "    1234"},
+	{"1234", "%8.4d", "    1234"},
+	{"1234", "%8.5d", "   01234"},
+	{"1234", "%8.6d", "  001234"},
+	{"-1234", "%8.3d", "   -1234"},
+	{"-1234", "%8.4d", "   -1234"},
+	{"-1234", "%8.5d", "  -01234"},
+	{"-1234", "%8.6d", " -001234"},
+
+	{"1234", "%+8.3d", "   +1234"},
+	{"1234", "%+8.4d", "   +1234"},
+	{"1234", "%+8.5d", "  +01234"},
+	{"1234", "%+8.6d", " +001234"},
+	{"-1234", "%+8.3d", "   -1234"},
+	{"-1234", "%+8.4d", "   -1234"},
+	{"-1234", "%+8.5d", "  -01234"},
+	{"-1234", "%+8.6d", " -001234"},
+
+	{"1234", "% 8.3d", "    1234"},
+	{"1234", "% 8.4d", "    1234"},
+	{"1234", "% 8.5d", "   01234"},
+	{"1234", "% 8.6d", "  001234"},
+	{"-1234", "% 8.3d", "   -1234"},
+	{"-1234", "% 8.4d", "   -1234"},
+	{"-1234", "% 8.5d", "  -01234"},
+	{"-1234", "% 8.6d", " -001234"},
+
+	{"1234", "%.3x", "4d2"},
+	{"1234", "%.4x", "04d2"},
+	{"1234", "%.5x", "004d2"},
+	{"1234", "%.6x", "0004d2"},
+	{"-1234", "%.3x", "-4d2"},
+	{"-1234", "%.4x", "-04d2"},
+	{"-1234", "%.5x", "-004d2"},
+	{"-1234", "%.6x", "-0004d2"},
+
+	{"1234", "%8.3x", "     4d2"},
+	{"1234", "%8.4x", "    04d2"},
+	{"1234", "%8.5x", "   004d2"},
+	{"1234", "%8.6x", "  0004d2"},
+	{"-1234", "%8.3x", "    -4d2"},
+	{"-1234", "%8.4x", "   -04d2"},
+	{"-1234", "%8.5x", "  -004d2"},
+	{"-1234", "%8.6x", " -0004d2"},
+
+	{"1234", "%+8.3x", "    +4d2"},
+	{"1234", "%+8.4x", "   +04d2"},
+	{"1234", "%+8.5x", "  +004d2"},
+	{"1234", "%+8.6x", " +0004d2"},
+	{"-1234", "%+8.3x", "    -4d2"},
+	{"-1234", "%+8.4x", "   -04d2"},
+	{"-1234", "%+8.5x", "  -004d2"},
+	{"-1234", "%+8.6x", " -0004d2"},
+
+	{"1234", "% 8.3x", "     4d2"},
+	{"1234", "% 8.4x", "    04d2"},
+	{"1234", "% 8.5x", "   004d2"},
+	{"1234", "% 8.6x", "  0004d2"},
+	{"1234", "% 8.7x", " 00004d2"},
+	{"1234", "% 8.8x", " 000004d2"},
+	{"-1234", "% 8.3x", "    -4d2"},
+	{"-1234", "% 8.4x", "   -04d2"},
+	{"-1234", "% 8.5x", "  -004d2"},
+	{"-1234", "% 8.6x", " -0004d2"},
+	{"-1234", "% 8.7x", "-00004d2"},
+	{"-1234", "% 8.8x", "-000004d2"},
+
+	{"1234", "%-8.3d", "1234    "},
+	{"1234", "%-8.4d", "1234    "},
+	{"1234", "%-8.5d", "01234   "},
+	{"1234", "%-8.6d", "001234  "},
+	{"1234", "%-8.7d", "0001234 "},
+	{"1234", "%-8.8d", "00001234"},
+	{"-1234", "%-8.3d", "-1234   "},
+	{"-1234", "%-8.4d", "-1234   "},
+	{"-1234", "%-8.5d", "-01234  "},
+	{"-1234", "%-8.6d", "-001234 "},
+	{"-1234", "%-8.7d", "-0001234"},
+	{"-1234", "%-8.8d", "-00001234"},
+
+	{"16777215", "%b", "111111111111111111111111"}, // 2**24 - 1
+
+	{"0", "%.d", ""},
+	{"0", "%.0d", ""},
+	{"0", "%3.d", ""},
+}
+
+func TestFormat(t *testing.T) {
+	for i, test := range formatTests {
+		var x *Int
+		if test.input != "<nil>" {
+			var ok bool
+			x, ok = new(Int).SetString(test.input, 0)
+			if !ok {
+				t.Errorf("#%d failed reading input %s", i, test.input)
+			}
+		}
+		output := fmt.Sprintf(test.format, x)
+		if output != test.output {
+			t.Errorf("#%d got %q; want %q, {%q, %q, %q}", i, output, test.output, test.input, test.format, test.output)
+		}
+	}
+}
+
+var scanTests = []struct {
+	input     string
+	format    string
+	output    string
+	remaining int
+}{
+	{"1010", "%b", "10", 0},
+	{"0b1010", "%v", "10", 0},
+	{"12", "%o", "10", 0},
+	{"012", "%v", "10", 0},
+	{"10", "%d", "10", 0},
+	{"10", "%v", "10", 0},
+	{"a", "%x", "10", 0},
+	{"0xa", "%v", "10", 0},
+	{"A", "%X", "10", 0},
+	{"-A", "%X", "-10", 0},
+	{"+0b1011001", "%v", "89", 0},
+	{"0xA", "%v", "10", 0},
+	{"0 ", "%v", "0", 1},
+	{"2+3", "%v", "2", 2},
+	{"0XABC 12", "%v", "2748", 3},
+}
+
+func TestScan(t *testing.T) {
+	var buf bytes.Buffer
+	for i, test := range scanTests {
+		x := new(Int)
+		buf.Reset()
+		buf.WriteString(test.input)
+		if _, err := fmt.Fscanf(&buf, test.format, x); err != nil {
+			t.Errorf("#%d error: %s", i, err)
+		}
+		if x.String() != test.output {
+			t.Errorf("#%d got %s; want %s", i, x.String(), test.output)
+		}
+		if buf.Len() != test.remaining {
+			t.Errorf("#%d got %d bytes remaining; want %d", i, buf.Len(), test.remaining)
+		}
+	}
+}
+
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []struct {
+	x, y int64
+	q, r int64 // T-division
+	d, m int64 // Euclidian division
+}{
+	{5, 3, 1, 2, 1, 2},
+	{-5, 3, -1, -2, -2, 1},
+	{5, -3, -1, 2, -1, 2},
+	{-5, -3, 1, -2, 2, 1},
+	{1, 2, 0, 1, 0, 1},
+	{8, 4, 2, 0, 2, 0},
+}
+
+func TestDivisionSigns(t *testing.T) {
+	for i, test := range divisionSignsTests {
+		x := NewInt(test.x)
+		y := NewInt(test.y)
+		q := NewInt(test.q)
+		r := NewInt(test.r)
+		d := NewInt(test.d)
+		m := NewInt(test.m)
+
+		q1 := new(Int).Quo(x, y)
+		r1 := new(Int).Rem(x, y)
+		if !isNormalized(q1) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+		}
+		if !isNormalized(r1) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+		}
+		if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+		}
+
+		q2, r2 := new(Int).QuoRem(x, y, new(Int))
+		if !isNormalized(q2) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+		}
+		if !isNormalized(r2) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+		}
+		if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+		}
+
+		d1 := new(Int).Div(x, y)
+		m1 := new(Int).Mod(x, y)
+		if !isNormalized(d1) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d1)
+		}
+		if !isNormalized(m1) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+		}
+		if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+		}
+
+		d2, m2 := new(Int).DivMod(x, y, new(Int))
+		if !isNormalized(d2) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d2)
+		}
+		if !isNormalized(m2) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+		}
+		if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
+		}
+	}
+}
+
+func checkSetBytes(b []byte) bool {
+	hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
+	hex2 := hex.EncodeToString(b)
+
+	for len(hex1) < len(hex2) {
+		hex1 = "0" + hex1
+	}
+
+	for len(hex1) > len(hex2) {
+		hex2 = "0" + hex2
+	}
+
+	return hex1 == hex2
+}
+
+func TestSetBytes(t *testing.T) {
+	if err := quick.Check(checkSetBytes, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+func checkBytes(b []byte) bool {
+	b2 := new(Int).SetBytes(b).Bytes()
+	return bytes.Equal(b, b2)
+}
+
+func TestBytes(t *testing.T) {
+	if err := quick.Check(checkSetBytes, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+func checkQuo(x, y []byte) bool {
+	u := new(Int).SetBytes(x)
+	v := new(Int).SetBytes(y)
+
+	if len(v.abs) == 0 {
+		return true
+	}
+
+	r := new(Int)
+	q, r := new(Int).QuoRem(u, v, r)
+
+	if r.Cmp(v) >= 0 {
+		return false
+	}
+
+	uprime := new(Int).Set(q)
+	uprime.Mul(uprime, v)
+	uprime.Add(uprime, r)
+
+	return uprime.Cmp(u) == 0
+}
+
+var quoTests = []struct {
+	x, y string
+	q, r string
+}{
+	{
+		"476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
+		"9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
+		"50911",
+		"1",
+	},
+	{
+		"11510768301994997771168",
+		"1328165573307167369775",
+		"8",
+		"885443715537658812968",
+	},
+}
+
+func TestQuo(t *testing.T) {
+	if err := quick.Check(checkQuo, nil); err != nil {
+		t.Error(err)
+	}
+
+	for i, test := range quoTests {
+		x, _ := new(Int).SetString(test.x, 10)
+		y, _ := new(Int).SetString(test.y, 10)
+		expectedQ, _ := new(Int).SetString(test.q, 10)
+		expectedR, _ := new(Int).SetString(test.r, 10)
+
+		r := new(Int)
+		q, r := new(Int).QuoRem(x, y, r)
+
+		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
+			t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+		}
+	}
+}
+
+func TestQuoStepD6(t *testing.T) {
+	// See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
+	// a code path which only triggers 1 in 10^{-19} cases.
+
+	u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
+	v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
+
+	r := new(Int)
+	q, r := new(Int).QuoRem(u, v, r)
+	const expectedQ64 = "18446744073709551613"
+	const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
+	const expectedQ32 = "4294967293"
+	const expectedR32 = "39614081266355540837921718287"
+	if q.String() != expectedQ64 && q.String() != expectedQ32 ||
+		r.String() != expectedR64 && r.String() != expectedR32 {
+		t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
+	}
+}
+
+var bitLenTests = []struct {
+	in  string
+	out int
+}{
+	{"-1", 1},
+	{"0", 0},
+	{"1", 1},
+	{"2", 2},
+	{"4", 3},
+	{"0xabc", 12},
+	{"0x8000", 16},
+	{"0x80000000", 32},
+	{"0x800000000000", 48},
+	{"0x8000000000000000", 64},
+	{"0x80000000000000000000", 80},
+	{"-0x4000000000000000000000", 87},
+}
+
+func TestBitLen(t *testing.T) {
+	for i, test := range bitLenTests {
+		x, ok := new(Int).SetString(test.in, 0)
+		if !ok {
+			t.Errorf("#%d test input invalid: %s", i, test.in)
+			continue
+		}
+
+		if n := x.BitLen(); n != test.out {
+			t.Errorf("#%d got %d want %d", i, n, test.out)
+		}
+	}
+}
+
+var expTests = []struct {
+	x, y, m string
+	out     string
+}{
+	// y <= 0
+	{"0", "0", "", "1"},
+	{"1", "0", "", "1"},
+	{"-10", "0", "", "1"},
+	{"1234", "-1", "", "1"},
+
+	// m == 1
+	{"0", "0", "1", "0"},
+	{"1", "0", "1", "0"},
+	{"-10", "0", "1", "0"},
+	{"1234", "-1", "1", "0"},
+
+	// misc
+	{"5", "-7", "", "1"},
+	{"-5", "-7", "", "1"},
+	{"5", "0", "", "1"},
+	{"-5", "0", "", "1"},
+	{"5", "1", "", "5"},
+	{"-5", "1", "", "-5"},
+	{"-2", "3", "2", "0"},
+	{"5", "2", "", "25"},
+	{"1", "65537", "2", "1"},
+	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+	{"0x8000000000000000", "2", "6719", "4944"},
+	{"0x8000000000000000", "3", "6719", "5447"},
+	{"0x8000000000000000", "1000", "6719", "1603"},
+	{"0x8000000000000000", "1000000", "6719", "3199"},
+	{"0x8000000000000000", "-1000000", "6719", "1"},
+	{
+		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+	},
+}
+
+func TestExp(t *testing.T) {
+	for i, test := range expTests {
+		x, ok1 := new(Int).SetString(test.x, 0)
+		y, ok2 := new(Int).SetString(test.y, 0)
+		out, ok3 := new(Int).SetString(test.out, 0)
+
+		var ok4 bool
+		var m *Int
+
+		if len(test.m) == 0 {
+			m, ok4 = nil, true
+		} else {
+			m, ok4 = new(Int).SetString(test.m, 0)
+		}
+
+		if !ok1 || !ok2 || !ok3 || !ok4 {
+			t.Errorf("#%d: error in input", i)
+			continue
+		}
+
+		z1 := new(Int).Exp(x, y, m)
+		if !isNormalized(z1) {
+			t.Errorf("#%d: %v is not normalized", i, *z1)
+		}
+		if z1.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z1, out)
+		}
+
+		if m == nil {
+			// the result should be the same as for m == 0;
+			// specifically, there should be no div-zero panic
+			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+			z2 := new(Int).Exp(x, y, m)
+			if z2.Cmp(z1) != 0 {
+				t.Errorf("#%d: got %s want %s", i, z1, z2)
+			}
+		}
+	}
+}
+
+func checkGcd(aBytes, bBytes []byte) bool {
+	x := new(Int)
+	y := new(Int)
+	a := new(Int).SetBytes(aBytes)
+	b := new(Int).SetBytes(bBytes)
+
+	d := new(Int).GCD(x, y, a, b)
+	x.Mul(x, a)
+	y.Mul(y, b)
+	x.Add(x, y)
+
+	return x.Cmp(d) == 0
+}
+
+var gcdTests = []struct {
+	d, x, y, a, b string
+}{
+	// a <= 0 || b <= 0
+	{"0", "0", "0", "0", "0"},
+	{"0", "0", "0", "0", "7"},
+	{"0", "0", "0", "11", "0"},
+	{"0", "0", "0", "-77", "35"},
+	{"0", "0", "0", "64515", "-24310"},
+	{"0", "0", "0", "-64515", "-24310"},
+
+	{"1", "-9", "47", "120", "23"},
+	{"7", "1", "-2", "77", "35"},
+	{"935", "-3", "8", "64515", "24310"},
+	{"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
+	{"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
+
+	// test early exit (after one Euclidean iteration) in binaryGCD
+	{"1", "", "", "1", "98920366548084643601728869055592650835572950932266967461790948584315647051443"},
+}
+
+func testGcd(t *testing.T, d, x, y, a, b *Int) {
+	var X *Int
+	if x != nil {
+		X = new(Int)
+	}
+	var Y *Int
+	if y != nil {
+		Y = new(Int)
+	}
+
+	D := new(Int).GCD(X, Y, a, b)
+	if D.Cmp(d) != 0 {
+		t.Errorf("GCD(%s, %s): got d = %s, want %s", a, b, D, d)
+	}
+	if x != nil && X.Cmp(x) != 0 {
+		t.Errorf("GCD(%s, %s): got x = %s, want %s", a, b, X, x)
+	}
+	if y != nil && Y.Cmp(y) != 0 {
+		t.Errorf("GCD(%s, %s): got y = %s, want %s", a, b, Y, y)
+	}
+
+	// binaryGCD requires a > 0 && b > 0
+	if a.Sign() <= 0 || b.Sign() <= 0 {
+		return
+	}
+
+	D.binaryGCD(a, b)
+	if D.Cmp(d) != 0 {
+		t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, D, d)
+	}
+}
+
+func TestGcd(t *testing.T) {
+	for _, test := range gcdTests {
+		d, _ := new(Int).SetString(test.d, 0)
+		x, _ := new(Int).SetString(test.x, 0)
+		y, _ := new(Int).SetString(test.y, 0)
+		a, _ := new(Int).SetString(test.a, 0)
+		b, _ := new(Int).SetString(test.b, 0)
+
+		testGcd(t, d, nil, nil, a, b)
+		testGcd(t, d, x, nil, a, b)
+		testGcd(t, d, nil, y, a, b)
+		testGcd(t, d, x, y, a, b)
+	}
+
+	quick.Check(checkGcd, nil)
+}
+
+var primes = []string{
+	"2",
+	"3",
+	"5",
+	"7",
+	"11",
+
+	"13756265695458089029",
+	"13496181268022124907",
+	"10953742525620032441",
+	"17908251027575790097",
+
+	// http://code.google.com/p/go/issues/detail?id=638
+	"18699199384836356663",
+
+	"98920366548084643601728869055592650835572950932266967461790948584315647051443",
+	"94560208308847015747498523884063394671606671904944666360068158221458669711639",
+
+	// http://primes.utm.edu/lists/small/small3.html
+	"449417999055441493994709297093108513015373787049558499205492347871729927573118262811508386655998299074566974373711472560655026288668094291699357843464363003144674940345912431129144354948751003607115263071543163",
+	"230975859993204150666423538988557839555560243929065415434980904258310530753006723857139742334640122533598517597674807096648905501653461687601339782814316124971547968912893214002992086353183070342498989426570593",
+	"5521712099665906221540423207019333379125265462121169655563495403888449493493629943498064604536961775110765377745550377067893607246020694972959780839151452457728855382113555867743022746090187341871655890805971735385789993",
+	"203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123",
+}
+
+var composites = []string{
+	"21284175091214687912771199898307297748211672914763848041968395774954376176754",
+	"6084766654921918907427900243509372380954290099172559290432744450051395395951",
+	"84594350493221918389213352992032324280367711247940675652888030554255915464401",
+	"82793403787388584738507275144194252681",
+}
+
+func TestProbablyPrime(t *testing.T) {
+	nreps := 20
+	if testing.Short() {
+		nreps = 1
+	}
+	for i, s := range primes {
+		p, _ := new(Int).SetString(s, 10)
+		if !p.ProbablyPrime(nreps) {
+			t.Errorf("#%d prime found to be non-prime (%s)", i, s)
+		}
+	}
+
+	for i, s := range composites {
+		c, _ := new(Int).SetString(s, 10)
+		if c.ProbablyPrime(nreps) {
+			t.Errorf("#%d composite found to be prime (%s)", i, s)
+		}
+		if testing.Short() {
+			break
+		}
+	}
+}
+
+type intShiftTest struct {
+	in    string
+	shift uint
+	out   string
+}
+
+var rshTests = []intShiftTest{
+	{"0", 0, "0"},
+	{"-0", 0, "0"},
+	{"0", 1, "0"},
+	{"0", 2, "0"},
+	{"1", 0, "1"},
+	{"1", 1, "0"},
+	{"1", 2, "0"},
+	{"2", 0, "2"},
+	{"2", 1, "1"},
+	{"-1", 0, "-1"},
+	{"-1", 1, "-1"},
+	{"-1", 10, "-1"},
+	{"-100", 2, "-25"},
+	{"-100", 3, "-13"},
+	{"-100", 100, "-1"},
+	{"4294967296", 0, "4294967296"},
+	{"4294967296", 1, "2147483648"},
+	{"4294967296", 2, "1073741824"},
+	{"18446744073709551616", 0, "18446744073709551616"},
+	{"18446744073709551616", 1, "9223372036854775808"},
+	{"18446744073709551616", 2, "4611686018427387904"},
+	{"18446744073709551616", 64, "1"},
+	{"340282366920938463463374607431768211456", 64, "18446744073709551616"},
+	{"340282366920938463463374607431768211456", 128, "1"},
+}
+
+func TestRsh(t *testing.T) {
+	for i, test := range rshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		out := new(Int).Rsh(in, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+	}
+}
+
+func TestRshSelf(t *testing.T) {
+	for i, test := range rshTests {
+		z, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		z.Rsh(z, test.shift)
+
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
+		if z.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z, expected)
+		}
+	}
+}
+
+var lshTests = []intShiftTest{
+	{"0", 0, "0"},
+	{"0", 1, "0"},
+	{"0", 2, "0"},
+	{"1", 0, "1"},
+	{"1", 1, "2"},
+	{"1", 2, "4"},
+	{"2", 0, "2"},
+	{"2", 1, "4"},
+	{"2", 2, "8"},
+	{"-87", 1, "-174"},
+	{"4294967296", 0, "4294967296"},
+	{"4294967296", 1, "8589934592"},
+	{"4294967296", 2, "17179869184"},
+	{"18446744073709551616", 0, "18446744073709551616"},
+	{"9223372036854775808", 1, "18446744073709551616"},
+	{"4611686018427387904", 2, "18446744073709551616"},
+	{"1", 64, "18446744073709551616"},
+	{"18446744073709551616", 64, "340282366920938463463374607431768211456"},
+	{"1", 128, "340282366920938463463374607431768211456"},
+}
+
+func TestLsh(t *testing.T) {
+	for i, test := range lshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		out := new(Int).Lsh(in, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+	}
+}
+
+func TestLshSelf(t *testing.T) {
+	for i, test := range lshTests {
+		z, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		z.Lsh(z, test.shift)
+
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
+		if z.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z, expected)
+		}
+	}
+}
+
+func TestLshRsh(t *testing.T) {
+	for i, test := range rshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		out := new(Int).Lsh(in, test.shift)
+		out = out.Rsh(out, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if in.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+	for i, test := range lshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		out := new(Int).Lsh(in, test.shift)
+		out.Rsh(out, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if in.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+}
+
+var int64Tests = []int64{
+	0,
+	1,
+	-1,
+	4294967295,
+	-4294967295,
+	4294967296,
+	-4294967296,
+	9223372036854775807,
+	-9223372036854775807,
+	-9223372036854775808,
+}
+
+func TestInt64(t *testing.T) {
+	for i, testVal := range int64Tests {
+		in := NewInt(testVal)
+		out := in.Int64()
+
+		if out != testVal {
+			t.Errorf("#%d got %d want %d", i, out, testVal)
+		}
+	}
+}
+
+var uint64Tests = []uint64{
+	0,
+	1,
+	4294967295,
+	4294967296,
+	8589934591,
+	8589934592,
+	9223372036854775807,
+	9223372036854775808,
+	18446744073709551615, // 1<<64 - 1
+}
+
+func TestUint64(t *testing.T) {
+	in := new(Int)
+	for i, testVal := range uint64Tests {
+		in.SetUint64(testVal)
+		out := in.Uint64()
+
+		if out != testVal {
+			t.Errorf("#%d got %d want %d", i, out, testVal)
+		}
+
+		str := fmt.Sprint(testVal)
+		strOut := in.String()
+		if strOut != str {
+			t.Errorf("#%d.String got %s want %s", i, strOut, str)
+		}
+	}
+}
+
+var bitwiseTests = []struct {
+	x, y                 string
+	and, or, xor, andNot string
+}{
+	{"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
+	{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
+	{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
+	{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+	{"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
+	{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
+	{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
+	{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
+	{"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
+	{"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
+	{"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
+	{"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
+	{"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
+	{
+		"0x1000009dc6e3d9822cba04129bcbe3401",
+		"0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"0x1000001186210100001000009048c2001",
+		"0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+		"0x8c40c2d8822caa04120b8321400",
+	},
+	{
+		"0x1000009dc6e3d9822cba04129bcbe3401",
+		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"0x8c40c2d8822caa04120b8321401",
+		"-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
+		"-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
+		"0x1000001186210100001000009048c2000",
+	},
+	{
+		"-0x1000009dc6e3d9822cba04129bcbe3401",
+		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+		"-0x1000001186210100001000009048c2001",
+		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+		"0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
+	},
+}
+
+type bitFun func(z, x, y *Int) *Int
+
+func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+	expected := new(Int)
+	expected.SetString(exp, 0)
+
+	out := f(new(Int), x, y)
+	if out.Cmp(expected) != 0 {
+		t.Errorf("%s: got %s want %s", msg, out, expected)
+	}
+}
+
+func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+	self := new(Int)
+	self.Set(x)
+	expected := new(Int)
+	expected.SetString(exp, 0)
+
+	self = f(self, self, y)
+	if self.Cmp(expected) != 0 {
+		t.Errorf("%s: got %s want %s", msg, self, expected)
+	}
+}
+
+func altBit(x *Int, i int) uint {
+	z := new(Int).Rsh(x, uint(i))
+	z = z.And(z, NewInt(1))
+	if z.Cmp(new(Int)) != 0 {
+		return 1
+	}
+	return 0
+}
+
+func altSetBit(z *Int, x *Int, i int, b uint) *Int {
+	one := NewInt(1)
+	m := one.Lsh(one, uint(i))
+	switch b {
+	case 1:
+		return z.Or(x, m)
+	case 0:
+		return z.AndNot(x, m)
+	}
+	panic("set bit is not 0 or 1")
+}
+
+func testBitset(t *testing.T, x *Int) {
+	n := x.BitLen()
+	z := new(Int).Set(x)
+	z1 := new(Int).Set(x)
+	for i := 0; i < n+10; i++ {
+		old := z.Bit(i)
+		old1 := altBit(z1, i)
+		if old != old1 {
+			t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
+		}
+		z := new(Int).SetBit(z, i, 1)
+		z1 := altSetBit(new(Int), z1, i, 1)
+		if z.Bit(i) == 0 {
+			t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
+		}
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
+		}
+		z.SetBit(z, i, 0)
+		altSetBit(z1, z1, i, 0)
+		if z.Bit(i) != 0 {
+			t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
+		}
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
+		}
+		altSetBit(z1, z1, i, old)
+		z.SetBit(z, i, old)
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
+		}
+	}
+	if z.Cmp(x) != 0 {
+		t.Errorf("bitset: got %s want %s", z, x)
+	}
+}
+
+var bitsetTests = []struct {
+	x string
+	i int
+	b uint
+}{
+	{"0", 0, 0},
+	{"0", 200, 0},
+	{"1", 0, 1},
+	{"1", 1, 0},
+	{"-1", 0, 1},
+	{"-1", 200, 1},
+	{"0x2000000000000000000000000000", 108, 0},
+	{"0x2000000000000000000000000000", 109, 1},
+	{"0x2000000000000000000000000000", 110, 0},
+	{"-0x2000000000000000000000000001", 108, 1},
+	{"-0x2000000000000000000000000001", 109, 0},
+	{"-0x2000000000000000000000000001", 110, 1},
+}
+
+func TestBitSet(t *testing.T) {
+	for _, test := range bitwiseTests {
+		x := new(Int)
+		x.SetString(test.x, 0)
+		testBitset(t, x)
+		x = new(Int)
+		x.SetString(test.y, 0)
+		testBitset(t, x)
+	}
+	for i, test := range bitsetTests {
+		x := new(Int)
+		x.SetString(test.x, 0)
+		b := x.Bit(test.i)
+		if b != test.b {
+			t.Errorf("#%d got %v want %v", i, b, test.b)
+		}
+	}
+	z := NewInt(1)
+	z.SetBit(NewInt(0), 2, 1)
+	if z.Cmp(NewInt(4)) != 0 {
+		t.Errorf("destination leaked into result; got %s want 4", z)
+	}
+}
+
+func BenchmarkBitset(b *testing.B) {
+	z := new(Int)
+	z.SetBit(z, 512, 1)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		z.SetBit(z, i&512, 1)
+	}
+}
+
+func BenchmarkBitsetNeg(b *testing.B) {
+	z := NewInt(-1)
+	z.SetBit(z, 512, 0)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		z.SetBit(z, i&512, 0)
+	}
+}
+
+func BenchmarkBitsetOrig(b *testing.B) {
+	z := new(Int)
+	altSetBit(z, z, 512, 1)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		altSetBit(z, z, i&512, 1)
+	}
+}
+
+func BenchmarkBitsetNegOrig(b *testing.B) {
+	z := NewInt(-1)
+	altSetBit(z, z, 512, 0)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		altSetBit(z, z, i&512, 0)
+	}
+}
+
+func TestBitwise(t *testing.T) {
+	x := new(Int)
+	y := new(Int)
+	for _, test := range bitwiseTests {
+		x.SetString(test.x, 0)
+		y.SetString(test.y, 0)
+
+		testBitFun(t, "and", (*Int).And, x, y, test.and)
+		testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
+		testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+		testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+		testBitFun(t, "or", (*Int).Or, x, y, test.or)
+		testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
+		testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
+		testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
+	}
+}
+
+var notTests = []struct {
+	in  string
+	out string
+}{
+	{"0", "-1"},
+	{"1", "-2"},
+	{"7", "-8"},
+	{"0", "-1"},
+	{"-81910", "81909"},
+	{
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"-298472983472983471903246121093472394872319615612417471234712062",
+	},
+}
+
+func TestNot(t *testing.T) {
+	in := new(Int)
+	out := new(Int)
+	expected := new(Int)
+	for i, test := range notTests {
+		in.SetString(test.in, 10)
+		expected.SetString(test.out, 10)
+		out = out.Not(in)
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+		out = out.Not(out)
+		if out.Cmp(in) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+}
+
+var modInverseTests = []struct {
+	element string
+	prime   string
+}{
+	{"1", "7"},
+	{"1", "13"},
+	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
+}
+
+func TestModInverse(t *testing.T) {
+	var element, prime Int
+	one := NewInt(1)
+	for i, test := range modInverseTests {
+		(&element).SetString(test.element, 10)
+		(&prime).SetString(test.prime, 10)
+		inverse := new(Int).ModInverse(&element, &prime)
+		inverse.Mul(inverse, &element)
+		inverse.Mod(inverse, &prime)
+		if inverse.Cmp(one) != 0 {
+			t.Errorf("#%d: failed (e·e^(-1)=%s)", i, inverse)
+		}
+	}
+}
+
+var encodingTests = []string{
+	"-539345864568634858364538753846587364875430589374589",
+	"-678645873",
+	"-100",
+	"-2",
+	"-1",
+	"0",
+	"1",
+	"2",
+	"10",
+	"42",
+	"1234567890",
+	"298472983472983471903246121093472394872319615612417471234712061",
+}
+
+func TestIntGobEncoding(t *testing.T) {
+	var medium bytes.Buffer
+	enc := gob.NewEncoder(&medium)
+	dec := gob.NewDecoder(&medium)
+	for _, test := range encodingTests {
+		medium.Reset() // empty buffer for each test case (in case of failures)
+		var tx Int
+		tx.SetString(test, 10)
+		if err := enc.Encode(&tx); err != nil {
+			t.Errorf("encoding of %s failed: %s", &tx, err)
+		}
+		var rx Int
+		if err := dec.Decode(&rx); err != nil {
+			t.Errorf("decoding of %s failed: %s", &tx, err)
+		}
+		if rx.Cmp(&tx) != 0 {
+			t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+		}
+	}
+}
+
+// Sending a nil Int pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilIntInSlice(t *testing.T) {
+	buf := new(bytes.Buffer)
+	enc := gob.NewEncoder(buf)
+	dec := gob.NewDecoder(buf)
+
+	var in = make([]*Int, 1)
+	err := enc.Encode(&in)
+	if err != nil {
+		t.Errorf("gob encode failed: %q", err)
+	}
+	var out []*Int
+	err = dec.Decode(&out)
+	if err != nil {
+		t.Fatalf("gob decode failed: %q", err)
+	}
+	if len(out) != 1 {
+		t.Fatalf("wrong len; want 1 got %d", len(out))
+	}
+	var zero Int
+	if out[0].Cmp(&zero) != 0 {
+		t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+	}
+}
+
+func TestIntJSONEncoding(t *testing.T) {
+	for _, test := range encodingTests {
+		var tx Int
+		tx.SetString(test, 10)
+		b, err := json.Marshal(&tx)
+		if err != nil {
+			t.Errorf("marshaling of %s failed: %s", &tx, err)
+		}
+		var rx Int
+		if err := json.Unmarshal(b, &rx); err != nil {
+			t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+		}
+		if rx.Cmp(&tx) != 0 {
+			t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+		}
+	}
+}
+
+var intVals = []string{
+	"-141592653589793238462643383279502884197169399375105820974944592307816406286",
+	"-1415926535897932384626433832795028841971",
+	"-141592653589793",
+	"-1",
+	"0",
+	"1",
+	"141592653589793",
+	"1415926535897932384626433832795028841971",
+	"141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+func TestIntJSONEncodingTextMarshaller(t *testing.T) {
+	for _, num := range intVals {
+		var tx Int
+		tx.SetString(num, 0)
+		b, err := json.Marshal(&tx)
+		if err != nil {
+			t.Errorf("marshaling of %s failed: %s", &tx, err)
+			continue
+		}
+		var rx Int
+		if err := json.Unmarshal(b, &rx); err != nil {
+			t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+			continue
+		}
+		if rx.Cmp(&tx) != 0 {
+			t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+		}
+	}
+}
+
+func TestIntXMLEncodingTextMarshaller(t *testing.T) {
+	for _, num := range intVals {
+		var tx Int
+		tx.SetString(num, 0)
+		b, err := xml.Marshal(&tx)
+		if err != nil {
+			t.Errorf("marshaling of %s failed: %s", &tx, err)
+			continue
+		}
+		var rx Int
+		if err := xml.Unmarshal(b, &rx); err != nil {
+			t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+			continue
+		}
+		if rx.Cmp(&tx) != 0 {
+			t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+		}
+	}
+}
+
+func TestIssue2607(t *testing.T) {
+	// This code sequence used to hang.
+	n := NewInt(10)
+	n.Rand(rand.New(rand.NewSource(9)), n)
+}
diff --git a/src/math/big/nat.go b/src/math/big/nat.go
new file mode 100644
index 000000000..16a87f5c5
--- /dev/null
+++ b/src/math/big/nat.go
@@ -0,0 +1,1508 @@
+// 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.
+
+// Package big implements multi-precision arithmetic (big numbers).
+// The following numeric types are supported:
+//
+//	- Int	signed integers
+//	- Rat	rational numbers
+//
+// Methods are typically of the form:
+//
+//	func (z *Int) Op(x, y *Int) *Int	(similar for *Rat)
+//
+// and implement operations z = x Op y with the result as receiver; if it
+// is one of the operands it may be overwritten (and its memory reused).
+// To enable chaining of operations, the result is also returned. Methods
+// returning a result other than *Int or *Rat take one of the operands as
+// the receiver.
+//
+package big
+
+// This file contains operations on unsigned multi-precision integers.
+// These are the building blocks for the operations on signed integers
+// and rationals.
+
+import (
+	"errors"
+	"io"
+	"math"
+	"math/rand"
+	"sync"
+)
+
+// An unsigned integer x of the form
+//
+//   x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
+//
+// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
+// with the digits x[i] as the slice elements.
+//
+// A number is normalized if the slice contains no leading 0 digits.
+// During arithmetic operations, denormalized values may occur but are
+// always normalized before returning the final result. The normalized
+// representation of 0 is the empty or nil slice (length = 0).
+//
+type nat []Word
+
+var (
+	natOne = nat{1}
+	natTwo = nat{2}
+	natTen = nat{10}
+)
+
+func (z nat) clear() {
+	for i := range z {
+		z[i] = 0
+	}
+}
+
+func (z nat) norm() nat {
+	i := len(z)
+	for i > 0 && z[i-1] == 0 {
+		i--
+	}
+	return z[0:i]
+}
+
+func (z nat) make(n int) nat {
+	if n <= cap(z) {
+		return z[0:n] // reuse z
+	}
+	// Choosing a good value for e has significant performance impact
+	// because it increases the chance that a value can be reused.
+	const e = 4 // extra capacity
+	return make(nat, n, n+e)
+}
+
+func (z nat) setWord(x Word) nat {
+	if x == 0 {
+		return z.make(0)
+	}
+	z = z.make(1)
+	z[0] = x
+	return z
+}
+
+func (z nat) setUint64(x uint64) nat {
+	// single-digit values
+	if w := Word(x); uint64(w) == x {
+		return z.setWord(w)
+	}
+
+	// compute number of words n required to represent x
+	n := 0
+	for t := x; t > 0; t >>= _W {
+		n++
+	}
+
+	// split x into n words
+	z = z.make(n)
+	for i := range z {
+		z[i] = Word(x & _M)
+		x >>= _W
+	}
+
+	return z
+}
+
+func (z nat) set(x nat) nat {
+	z = z.make(len(x))
+	copy(z, x)
+	return z
+}
+
+func (z nat) add(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+
+	switch {
+	case m < n:
+		return z.add(y, x)
+	case m == 0:
+		// n == 0 because m >= n; result is 0
+		return z.make(0)
+	case n == 0:
+		// result is x
+		return z.set(x)
+	}
+	// m > 0
+
+	z = z.make(m + 1)
+	c := addVV(z[0:n], x, y)
+	if m > n {
+		c = addVW(z[n:m], x[n:], c)
+	}
+	z[m] = c
+
+	return z.norm()
+}
+
+func (z nat) sub(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+
+	switch {
+	case m < n:
+		panic("underflow")
+	case m == 0:
+		// n == 0 because m >= n; result is 0
+		return z.make(0)
+	case n == 0:
+		// result is x
+		return z.set(x)
+	}
+	// m > 0
+
+	z = z.make(m)
+	c := subVV(z[0:n], x, y)
+	if m > n {
+		c = subVW(z[n:], x[n:], c)
+	}
+	if c != 0 {
+		panic("underflow")
+	}
+
+	return z.norm()
+}
+
+func (x nat) cmp(y nat) (r int) {
+	m := len(x)
+	n := len(y)
+	if m != n || m == 0 {
+		switch {
+		case m < n:
+			r = -1
+		case m > n:
+			r = 1
+		}
+		return
+	}
+
+	i := m - 1
+	for i > 0 && x[i] == y[i] {
+		i--
+	}
+
+	switch {
+	case x[i] < y[i]:
+		r = -1
+	case x[i] > y[i]:
+		r = 1
+	}
+	return
+}
+
+func (z nat) mulAddWW(x nat, y, r Word) nat {
+	m := len(x)
+	if m == 0 || y == 0 {
+		return z.setWord(r) // result is r
+	}
+	// m > 0
+
+	z = z.make(m + 1)
+	z[m] = mulAddVWW(z[0:m], x, y, r)
+
+	return z.norm()
+}
+
+// basicMul multiplies x and y and leaves the result in z.
+// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
+func basicMul(z, x, y nat) {
+	z[0 : len(x)+len(y)].clear() // initialize z
+	for i, d := range y {
+		if d != 0 {
+			z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
+		}
+	}
+}
+
+// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
+// Factored out for readability - do not use outside karatsuba.
+func karatsubaAdd(z, x nat, n int) {
+	if c := addVV(z[0:n], z, x); c != 0 {
+		addVW(z[n:n+n>>1], z[n:], c)
+	}
+}
+
+// Like karatsubaAdd, but does subtract.
+func karatsubaSub(z, x nat, n int) {
+	if c := subVV(z[0:n], z, x); c != 0 {
+		subVW(z[n:n+n>>1], z[n:], c)
+	}
+}
+
+// Operands that are shorter than karatsubaThreshold are multiplied using
+// "grade school" multiplication; for longer operands the Karatsuba algorithm
+// is used.
+var karatsubaThreshold int = 40 // computed by calibrate.go
+
+// karatsuba multiplies x and y and leaves the result in z.
+// Both x and y must have the same length n and n must be a
+// power of 2. The result vector z must have len(z) >= 6*n.
+// The (non-normalized) result is placed in z[0 : 2*n].
+func karatsuba(z, x, y nat) {
+	n := len(y)
+
+	// Switch to basic multiplication if numbers are odd or small.
+	// (n is always even if karatsubaThreshold is even, but be
+	// conservative)
+	if n&1 != 0 || n < karatsubaThreshold || n < 2 {
+		basicMul(z, x, y)
+		return
+	}
+	// n&1 == 0 && n >= karatsubaThreshold && n >= 2
+
+	// Karatsuba multiplication is based on the observation that
+	// for two numbers x and y with:
+	//
+	//   x = x1*b + x0
+	//   y = y1*b + y0
+	//
+	// the product x*y can be obtained with 3 products z2, z1, z0
+	// instead of 4:
+	//
+	//   x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
+	//       =    z2*b*b +              z1*b +    z0
+	//
+	// with:
+	//
+	//   xd = x1 - x0
+	//   yd = y0 - y1
+	//
+	//   z1 =      xd*yd                    + z2 + z0
+	//      = (x1-x0)*(y0 - y1)             + z2 + z0
+	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
+	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
+	//      = x1*y0                 + x0*y1
+
+	// split x, y into "digits"
+	n2 := n >> 1              // n2 >= 1
+	x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
+	y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
+
+	// z is used for the result and temporary storage:
+	//
+	//   6*n     5*n     4*n     3*n     2*n     1*n     0*n
+	// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
+	//
+	// For each recursive call of karatsuba, an unused slice of
+	// z is passed in that has (at least) half the length of the
+	// caller's z.
+
+	// compute z0 and z2 with the result "in place" in z
+	karatsuba(z, x0, y0)     // z0 = x0*y0
+	karatsuba(z[n:], x1, y1) // z2 = x1*y1
+
+	// compute xd (or the negative value if underflow occurs)
+	s := 1 // sign of product xd*yd
+	xd := z[2*n : 2*n+n2]
+	if subVV(xd, x1, x0) != 0 { // x1-x0
+		s = -s
+		subVV(xd, x0, x1) // x0-x1
+	}
+
+	// compute yd (or the negative value if underflow occurs)
+	yd := z[2*n+n2 : 3*n]
+	if subVV(yd, y0, y1) != 0 { // y0-y1
+		s = -s
+		subVV(yd, y1, y0) // y1-y0
+	}
+
+	// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
+	// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
+	p := z[n*3:]
+	karatsuba(p, xd, yd)
+
+	// save original z2:z0
+	// (ok to use upper half of z since we're done recursing)
+	r := z[n*4:]
+	copy(r, z[:n*2])
+
+	// add up all partial products
+	//
+	//   2*n     n     0
+	// z = [ z2  | z0  ]
+	//   +    [ z0  ]
+	//   +    [ z2  ]
+	//   +    [  p  ]
+	//
+	karatsubaAdd(z[n2:], r, n)
+	karatsubaAdd(z[n2:], r[n:], n)
+	if s > 0 {
+		karatsubaAdd(z[n2:], p, n)
+	} else {
+		karatsubaSub(z[n2:], p, n)
+	}
+}
+
+// alias returns true if x and y share the same base array.
+func alias(x, y nat) bool {
+	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+}
+
+// addAt implements z += x<<(_W*i); z must be long enough.
+// (we don't use nat.add because we need z to stay the same
+// slice, and we don't need to normalize z after each addition)
+func addAt(z, x nat, i int) {
+	if n := len(x); n > 0 {
+		if c := addVV(z[i:i+n], z[i:], x); c != 0 {
+			j := i + n
+			if j < len(z) {
+				addVW(z[j:], z[j:], c)
+			}
+		}
+	}
+}
+
+func max(x, y int) int {
+	if x > y {
+		return x
+	}
+	return y
+}
+
+// karatsubaLen computes an approximation to the maximum k <= n such that
+// k = p<<i for a number p <= karatsubaThreshold and an i >= 0. Thus, the
+// result is the largest number that can be divided repeatedly by 2 before
+// becoming about the value of karatsubaThreshold.
+func karatsubaLen(n int) int {
+	i := uint(0)
+	for n > karatsubaThreshold {
+		n >>= 1
+		i++
+	}
+	return n << i
+}
+
+func (z nat) mul(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+
+	switch {
+	case m < n:
+		return z.mul(y, x)
+	case m == 0 || n == 0:
+		return z.make(0)
+	case n == 1:
+		return z.mulAddWW(x, y[0], 0)
+	}
+	// m >= n > 1
+
+	// determine if z can be reused
+	if alias(z, x) || alias(z, y) {
+		z = nil // z is an alias for x or y - cannot reuse
+	}
+
+	// use basic multiplication if the numbers are small
+	if n < karatsubaThreshold {
+		z = z.make(m + n)
+		basicMul(z, x, y)
+		return z.norm()
+	}
+	// m >= n && n >= karatsubaThreshold && n >= 2
+
+	// determine Karatsuba length k such that
+	//
+	//   x = xh*b + x0  (0 <= x0 < b)
+	//   y = yh*b + y0  (0 <= y0 < b)
+	//   b = 1<<(_W*k)  ("base" of digits xi, yi)
+	//
+	k := karatsubaLen(n)
+	// k <= n
+
+	// multiply x0 and y0 via Karatsuba
+	x0 := x[0:k]              // x0 is not normalized
+	y0 := y[0:k]              // y0 is not normalized
+	z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
+	karatsuba(z, x0, y0)
+	z = z[0 : m+n]  // z has final length but may be incomplete
+	z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
+
+	// If xh != 0 or yh != 0, add the missing terms to z. For
+	//
+	//   xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
+	//   yh =                         y1*b (0 <= y1 < b)
+	//
+	// the missing terms are
+	//
+	//   x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
+	//
+	// since all the yi for i > 1 are 0 by choice of k: If any of them
+	// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
+	// be a larger valid threshold contradicting the assumption about k.
+	//
+	if k < n || m != n {
+		var t nat
+
+		// add x0*y1*b
+		x0 := x0.norm()
+		y1 := y[k:]       // y1 is normalized because y is
+		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
+		addAt(z, t, k)
+
+		// add xi*y0<<i, xi*y1*b<<(i+k)
+		y0 := y0.norm()
+		for i := k; i < len(x); i += k {
+			xi := x[i:]
+			if len(xi) > k {
+				xi = xi[:k]
+			}
+			xi = xi.norm()
+			t = t.mul(xi, y0)
+			addAt(z, t, i)
+			t = t.mul(xi, y1)
+			addAt(z, t, i+k)
+		}
+	}
+
+	return z.norm()
+}
+
+// mulRange computes the product of all the unsigned integers in the
+// range [a, b] inclusively. If a > b (empty range), the result is 1.
+func (z nat) mulRange(a, b uint64) nat {
+	switch {
+	case a == 0:
+		// cut long ranges short (optimization)
+		return z.setUint64(0)
+	case a > b:
+		return z.setUint64(1)
+	case a == b:
+		return z.setUint64(a)
+	case a+1 == b:
+		return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
+	}
+	m := (a + b) / 2
+	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.make(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.make(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
+}
+
+// q = (uIn-r)/v, with 0 <= r < y
+// 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(v) >= 2
+//    len(uIn) >= len(v)
+func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
+	n := len(v)
+	m := len(uIn) - n
+
+	// determine if z can be reused
+	// TODO(gri) should find a better solution - this if statement
+	//           is very costly (see e.g. time pidigits -s -n 10000)
+	if alias(z, uIn) || alias(z, v) {
+		z = nil // z is an alias for uIn or v - cannot reuse
+	}
+	q = z.make(m + 1)
+
+	qhatv := make(nat, n+1)
+	if alias(u, uIn) || alias(u, v) {
+		u = nil // u is an alias for uIn or v - cannot reuse
+	}
+	u = u.make(len(uIn) + 1)
+	u.clear()
+
+	// D1.
+	shift := leadingZeros(v[n-1])
+	if shift > 0 {
+		// do not modify v, it may be used by another goroutine simultaneously
+		v1 := make(nat, n)
+		shlVU(v1, v, shift)
+		v = v1
+	}
+	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+	// D2.
+	for j := m; j >= 0; j-- {
+		// D3.
+		qhat := Word(_M)
+		if u[j+n] != v[n-1] {
+			var rhat Word
+			qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1])
+
+			// x1 | x2 = q̂v_{n-2}
+			x1, x2 := mulWW(qhat, v[n-2])
+			// test if q̂v_{n-2} > br̂ + u_{j+n-2}
+			for greaterThan(x1, x2, rhat, u[j+n-2]) {
+				qhat--
+				prevRhat := rhat
+				rhat += v[n-1]
+				// v[n-1] >= 0, so this tests for overflow.
+				if rhat < prevRhat {
+					break
+				}
+				x1, x2 = mulWW(qhat, v[n-2])
+			}
+		}
+
+		// D4.
+		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
+
+		c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
+		if c != 0 {
+			c := addVV(u[j:j+n], u[j:], v)
+			u[j+n] += c
+			qhat--
+		}
+
+		q[j] = qhat
+	}
+
+	q = q.norm()
+	shrVU(u, u, shift)
+	r = u.norm()
+
+	return q, r
+}
+
+// Length of x in bits. x must be normalized.
+func (x nat) bitLen() int {
+	if i := len(x) - 1; i >= 0 {
+		return i*_W + bitLen(x[i])
+	}
+	return 0
+}
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 'z' - 'a' + 10 + 1 // = hexValue('z') + 1
+
+func hexValue(ch rune) Word {
+	d := int(MaxBase + 1) // illegal base
+	switch {
+	case '0' <= ch && ch <= '9':
+		d = int(ch - '0')
+	case 'a' <= ch && ch <= 'z':
+		d = int(ch - 'a' + 10)
+	case 'A' <= ch && ch <= 'Z':
+		d = int(ch - 'A' + 10)
+	}
+	return Word(d)
+}
+
+// scan sets z to the natural number corresponding to the longest possible prefix
+// read from r representing an unsigned integer in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined. The syntax follows the syntax of
+// unsigned integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z nat) scan(r io.RuneScanner, base int) (nat, int, error) {
+	// reject illegal bases
+	if base < 0 || base == 1 || MaxBase < base {
+		return z, 0, errors.New("illegal number base")
+	}
+
+	// one char look-ahead
+	ch, _, err := r.ReadRune()
+	if err != nil {
+		return z, 0, err
+	}
+
+	// determine base if necessary
+	b := Word(base)
+	if base == 0 {
+		b = 10
+		if ch == '0' {
+			switch ch, _, err = r.ReadRune(); err {
+			case nil:
+				b = 8
+				switch ch {
+				case 'x', 'X':
+					b = 16
+				case 'b', 'B':
+					b = 2
+				}
+				if b == 2 || b == 16 {
+					if ch, _, err = r.ReadRune(); err != nil {
+						return z, 0, err
+					}
+				}
+			case io.EOF:
+				return z.make(0), 10, nil
+			default:
+				return z, 10, err
+			}
+		}
+	}
+
+	// convert string
+	// - group as many digits d as possible together into a "super-digit" dd with "super-base" bb
+	// - only when bb does not fit into a word anymore, do a full number mulAddWW using bb and dd
+	z = z.make(0)
+	bb := Word(1)
+	dd := Word(0)
+	for max := _M / b; ; {
+		d := hexValue(ch)
+		if d >= b {
+			r.UnreadRune() // ch does not belong to number anymore
+			break
+		}
+
+		if bb <= max {
+			bb *= b
+			dd = dd*b + d
+		} else {
+			// bb * b would overflow
+			z = z.mulAddWW(z, bb, dd)
+			bb = b
+			dd = d
+		}
+
+		if ch, _, err = r.ReadRune(); err != nil {
+			if err != io.EOF {
+				return z, int(b), err
+			}
+			break
+		}
+	}
+
+	switch {
+	case bb > 1:
+		// there was at least one mantissa digit
+		z = z.mulAddWW(z, bb, dd)
+	case base == 0 && b == 8:
+		// there was only the octal prefix 0 (possibly followed by digits > 7);
+		// return base 10, not 8
+		return z, 10, nil
+	case base != 0 || b != 8:
+		// there was neither a mantissa digit nor the octal prefix 0
+		return z, int(b), errors.New("syntax error scanning number")
+	}
+
+	return z.norm(), int(b), nil
+}
+
+// Character sets for string conversion.
+const (
+	lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz"
+	uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+)
+
+// decimalString returns a decimal representation of x.
+// It calls x.string with the charset "0123456789".
+func (x nat) decimalString() string {
+	return x.string(lowercaseDigits[0:10])
+}
+
+// string converts x to a string using digits from a charset; a digit with
+// value d is represented by charset[d]. The conversion base is determined
+// by len(charset), which must be >= 2 and <= 256.
+func (x nat) string(charset string) string {
+	b := Word(len(charset))
+
+	// special cases
+	switch {
+	case b < 2 || MaxBase > 256:
+		panic("illegal base")
+	case len(x) == 0:
+		return string(charset[0])
+	}
+
+	// allocate buffer for conversion
+	i := int(float64(x.bitLen())/math.Log2(float64(b))) + 1 // off by one at most
+	s := make([]byte, i)
+
+	// convert power of two and non power of two bases separately
+	if b == b&-b {
+		// shift is base-b digit size in bits
+		shift := trailingZeroBits(b) // shift > 0 because b >= 2
+		mask := Word(1)<<shift - 1
+		w := x[0]
+		nbits := uint(_W) // number of unprocessed bits in w
+
+		// convert less-significant words
+		for k := 1; k < len(x); k++ {
+			// convert full digits
+			for nbits >= shift {
+				i--
+				s[i] = charset[w&mask]
+				w >>= shift
+				nbits -= shift
+			}
+
+			// convert any partial leading digit and advance to next word
+			if nbits == 0 {
+				// no partial digit remaining, just advance
+				w = x[k]
+				nbits = _W
+			} else {
+				// partial digit in current (k-1) and next (k) word
+				w |= x[k] << nbits
+				i--
+				s[i] = charset[w&mask]
+
+				// advance
+				w = x[k] >> (shift - nbits)
+				nbits = _W - (shift - nbits)
+			}
+		}
+
+		// convert digits of most-significant word (omit leading zeros)
+		for nbits >= 0 && w != 0 {
+			i--
+			s[i] = charset[w&mask]
+			w >>= shift
+			nbits -= shift
+		}
+
+	} else {
+		// determine "big base"; i.e., the largest possible value bb
+		// that is a power of base b and still fits into a Word
+		// (as in 10^19 for 19 decimal digits in a 64bit Word)
+		bb := b      // big base is b**ndigits
+		ndigits := 1 // number of base b digits
+		for max := Word(_M / b); bb <= max; bb *= b {
+			ndigits++ // maximize ndigits where bb = b**ndigits, bb <= _M
+		}
+
+		// construct table of successive squares of bb*leafSize to use in subdivisions
+		// result (table != nil) <=> (len(x) > leafSize > 0)
+		table := divisors(len(x), b, ndigits, bb)
+
+		// preserve x, create local copy for use by convertWords
+		q := nat(nil).set(x)
+
+		// convert q to string s in base b
+		q.convertWords(s, charset, b, ndigits, bb, table)
+
+		// strip leading zeros
+		// (x != 0; thus s must contain at least one non-zero digit
+		// and the loop will terminate)
+		i = 0
+		for zero := charset[0]; s[i] == zero; {
+			i++
+		}
+	}
+
+	return string(s[i:])
+}
+
+// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
+// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
+// repeated nat/Word division.
+//
+// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
+// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
+// Recursive conversion divides q by its approximate square root, yielding two parts, each half
+// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
+// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
+// is made better by splitting the subblocks recursively. Best is to split blocks until one more
+// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
+// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
+// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
+// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
+// specific hardware.
+//
+func (q nat) convertWords(s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) {
+	// split larger blocks recursively
+	if table != nil {
+		// len(q) > leafSize > 0
+		var r nat
+		index := len(table) - 1
+		for len(q) > leafSize {
+			// find divisor close to sqrt(q) if possible, but in any case < q
+			maxLength := q.bitLen()     // ~= log2 q, or at of least largest possible q of this bit length
+			minLength := maxLength >> 1 // ~= log2 sqrt(q)
+			for index > 0 && table[index-1].nbits > minLength {
+				index-- // desired
+			}
+			if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
+				index--
+				if index < 0 {
+					panic("internal inconsistency")
+				}
+			}
+
+			// split q into the two digit number (q'*bbb + r) to form independent subblocks
+			q, r = q.div(r, q, table[index].bbb)
+
+			// convert subblocks and collect results in s[:h] and s[h:]
+			h := len(s) - table[index].ndigits
+			r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
+			s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
+		}
+	}
+
+	// having split any large blocks now process the remaining (small) block iteratively
+	i := len(s)
+	var r Word
+	if b == 10 {
+		// hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
+		for len(q) > 0 {
+			// extract least significant, base bb "digit"
+			q, r = q.divW(q, bb)
+			for j := 0; j < ndigits && i > 0; j++ {
+				i--
+				// avoid % computation since r%10 == r - int(r/10)*10;
+				// this appears to be faster for BenchmarkString10000Base10
+				// and smaller strings (but a bit slower for larger ones)
+				t := r / 10
+				s[i] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
+				r = t
+			}
+		}
+	} else {
+		for len(q) > 0 {
+			// extract least significant, base bb "digit"
+			q, r = q.divW(q, bb)
+			for j := 0; j < ndigits && i > 0; j++ {
+				i--
+				s[i] = charset[r%b]
+				r /= b
+			}
+		}
+	}
+
+	// prepend high-order zeroes
+	zero := charset[0]
+	for i > 0 { // while need more leading zeroes
+		i--
+		s[i] = zero
+	}
+}
+
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
+// Benchmark and configure leafSize using: go test -bench="Leaf"
+//   8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
+//   8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
+var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
+
+type divisor struct {
+	bbb     nat // divisor
+	nbits   int // bit length of divisor (discounting leading zeroes) ~= log2(bbb)
+	ndigits int // digit length of divisor in terms of output base digits
+}
+
+var cacheBase10 struct {
+	sync.Mutex
+	table [64]divisor // cached divisors for base 10
+}
+
+// expWW computes x**y
+func (z nat) expWW(x, y Word) nat {
+	return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
+}
+
+// construct table of powers of bb*leafSize to use in subdivisions
+func divisors(m int, b Word, ndigits int, bb Word) []divisor {
+	// only compute table when recursive conversion is enabled and x is large
+	if leafSize == 0 || m <= leafSize {
+		return nil
+	}
+
+	// determine k where (bb**leafSize)**(2**k) >= sqrt(x)
+	k := 1
+	for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
+		k++
+	}
+
+	// reuse and extend existing table of divisors or create new table as appropriate
+	var table []divisor // for b == 10, table overlaps with cacheBase10.table
+	if b == 10 {
+		cacheBase10.Lock()
+		table = cacheBase10.table[0:k] // reuse old table for this conversion
+	} else {
+		table = make([]divisor, k) // create new table for this conversion
+	}
+
+	// extend table
+	if table[k-1].ndigits == 0 {
+		// add new entries as needed
+		var larger nat
+		for i := 0; i < k; i++ {
+			if table[i].ndigits == 0 {
+				if i == 0 {
+					table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
+					table[0].ndigits = ndigits * leafSize
+				} else {
+					table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
+					table[i].ndigits = 2 * table[i-1].ndigits
+				}
+
+				// optimization: exploit aggregated extra bits in macro blocks
+				larger = nat(nil).set(table[i].bbb)
+				for mulAddVWW(larger, larger, b, 0) == 0 {
+					table[i].bbb = table[i].bbb.set(larger)
+					table[i].ndigits++
+				}
+
+				table[i].nbits = table[i].bbb.bitLen()
+			}
+		}
+	}
+
+	if b == 10 {
+		cacheBase10.Unlock()
+	}
+
+	return table
+}
+
+const deBruijn32 = 0x077CB531
+
+var deBruijn32Lookup = []byte{
+	0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
+	31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
+}
+
+const deBruijn64 = 0x03f79d71b4ca8b09
+
+var deBruijn64Lookup = []byte{
+	0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
+	62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
+	63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
+	54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func trailingZeroBits(x Word) uint {
+	// x & -x leaves only the right-most bit set in the word. Let k be the
+	// index of that bit. Since only a single bit is set, the value is two
+	// to the power of k. Multiplying by a power of two is equivalent to
+	// left shifting, in this case by k bits.  The de Bruijn constant is
+	// such that all six bit, consecutive substrings are distinct.
+	// Therefore, if we have a left shifted version of this constant we can
+	// find by how many bits it was shifted by looking at which six bit
+	// substring ended up at the top of the word.
+	// (Knuth, volume 4, section 7.3.1)
+	switch _W {
+	case 32:
+		return uint(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
+	case 64:
+		return uint(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
+	default:
+		panic("unknown word size")
+	}
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func (x nat) trailingZeroBits() uint {
+	if len(x) == 0 {
+		return 0
+	}
+	var i uint
+	for x[i] == 0 {
+		i++
+	}
+	// x[i] != 0
+	return i*_W + trailingZeroBits(x[i])
+}
+
+// z = x << s
+func (z nat) shl(x nat, s uint) nat {
+	m := len(x)
+	if m == 0 {
+		return z.make(0)
+	}
+	// m > 0
+
+	n := m + int(s/_W)
+	z = z.make(n + 1)
+	z[n] = shlVU(z[n-m:n], x, s%_W)
+	z[0 : n-m].clear()
+
+	return z.norm()
+}
+
+// z = x >> s
+func (z nat) shr(x nat, s uint) nat {
+	m := len(x)
+	n := m - int(s/_W)
+	if n <= 0 {
+		return z.make(0)
+	}
+	// n > 0
+
+	z = z.make(n)
+	shrVU(z, x[m-n:], s%_W)
+
+	return z.norm()
+}
+
+func (z nat) setBit(x nat, i uint, b uint) nat {
+	j := int(i / _W)
+	m := Word(1) << (i % _W)
+	n := len(x)
+	switch b {
+	case 0:
+		z = z.make(n)
+		copy(z, x)
+		if j >= n {
+			// no need to grow
+			return z
+		}
+		z[j] &^= m
+		return z.norm()
+	case 1:
+		if j >= n {
+			z = z.make(j + 1)
+			z[n:].clear()
+		} else {
+			z = z.make(n)
+		}
+		copy(z, x)
+		z[j] |= m
+		// no need to normalize
+		return z
+	}
+	panic("set bit is not 0 or 1")
+}
+
+func (z nat) bit(i uint) uint {
+	j := int(i / _W)
+	if j >= len(z) {
+		return 0
+	}
+	return uint(z[j] >> (i % _W) & 1)
+}
+
+func (z nat) and(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+	if m > n {
+		m = n
+	}
+	// m <= n
+
+	z = z.make(m)
+	for i := 0; i < m; i++ {
+		z[i] = x[i] & y[i]
+	}
+
+	return z.norm()
+}
+
+func (z nat) andNot(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+	if n > m {
+		n = m
+	}
+	// m >= n
+
+	z = z.make(m)
+	for i := 0; i < n; i++ {
+		z[i] = x[i] &^ y[i]
+	}
+	copy(z[n:m], x[n:m])
+
+	return z.norm()
+}
+
+func (z nat) or(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+	s := x
+	if m < n {
+		n, m = m, n
+		s = y
+	}
+	// m >= n
+
+	z = z.make(m)
+	for i := 0; i < n; i++ {
+		z[i] = x[i] | y[i]
+	}
+	copy(z[n:m], s[n:m])
+
+	return z.norm()
+}
+
+func (z nat) xor(x, y nat) nat {
+	m := len(x)
+	n := len(y)
+	s := x
+	if m < n {
+		n, m = m, n
+		s = y
+	}
+	// m >= n
+
+	z = z.make(m)
+	for i := 0; i < n; i++ {
+		z[i] = x[i] ^ y[i]
+	}
+	copy(z[n:m], s[n:m])
+
+	return z.norm()
+}
+
+// greaterThan returns true iff (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 {
+	if alias(z, limit) {
+		z = nil // z is an alias for limit - cannot reuse
+	}
+	z = z.make(len(limit))
+
+	bitLengthOfMSW := uint(n % _W)
+	if bitLengthOfMSW == 0 {
+		bitLengthOfMSW = _W
+	}
+	mask := Word((1 << bitLengthOfMSW) - 1)
+
+	for {
+		switch _W {
+		case 32:
+			for i := range z {
+				z[i] = Word(rand.Uint32())
+			}
+		case 64:
+			for i := range z {
+				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
+			}
+		default:
+			panic("unknown word size")
+		}
+		z[len(limit)-1] &= mask
+		if z.cmp(limit) < 0 {
+			break
+		}
+	}
+
+	return z.norm()
+}
+
+// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
+// otherwise it sets z to x**y. The result is the value of z.
+func (z nat) expNN(x, y, m nat) nat {
+	if alias(z, x) || alias(z, y) {
+		// We cannot allow in-place modification of x or y.
+		z = nil
+	}
+
+	// x**y mod 1 == 0
+	if len(m) == 1 && m[0] == 1 {
+		return z.setWord(0)
+	}
+	// m == 0 || m > 1
+
+	// x**0 == 1
+	if len(y) == 0 {
+		return z.setWord(1)
+	}
+	// y > 0
+
+	if len(m) != 0 {
+		// We likely end up being as long as the modulus.
+		z = z.make(len(m))
+	}
+	z = z.set(x)
+
+	// If the base is non-trivial and the exponent is large, we use
+	// 4-bit, windowed exponentiation. This involves precomputing 14 values
+	// (x^2...x^15) but then reduces the number of multiply-reduces by a
+	// third. Even for a 32-bit exponent, this reduces the number of
+	// operations.
+	if len(x) > 1 && len(y) > 1 && len(m) > 0 {
+		return z.expNNWindowed(x, y, m)
+	}
+
+	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
+	shift := leadingZeros(v) + 1
+	v <<= shift
+	var q nat
+
+	const mask = 1 << (_W - 1)
+
+	// We walk through the bits of the exponent one by one. Each time we
+	// see a bit, we square, thus doubling the power. If the bit is a one,
+	// we also multiply by x, thus adding one to the power.
+
+	w := _W - int(shift)
+	// zz and r are used to avoid allocating in mul and div as
+	// otherwise the arguments would alias.
+	var zz, r nat
+	for j := 0; j < w; j++ {
+		zz = zz.mul(z, z)
+		zz, z = z, zz
+
+		if v&mask != 0 {
+			zz = zz.mul(z, x)
+			zz, z = z, zz
+		}
+
+		if len(m) != 0 {
+			zz, r = zz.div(r, z, m)
+			zz, r, q, z = q, z, zz, r
+		}
+
+		v <<= 1
+	}
+
+	for i := len(y) - 2; i >= 0; i-- {
+		v = y[i]
+
+		for j := 0; j < _W; j++ {
+			zz = zz.mul(z, z)
+			zz, z = z, zz
+
+			if v&mask != 0 {
+				zz = zz.mul(z, x)
+				zz, z = z, zz
+			}
+
+			if len(m) != 0 {
+				zz, r = zz.div(r, z, m)
+				zz, r, q, z = q, z, zz, r
+			}
+
+			v <<= 1
+		}
+	}
+
+	return z.norm()
+}
+
+// expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
+func (z nat) expNNWindowed(x, y, m nat) nat {
+	// zz and r are used to avoid allocating in mul and div as otherwise
+	// the arguments would alias.
+	var zz, r nat
+
+	const n = 4
+	// powers[i] contains x^i.
+	var powers [1 << n]nat
+	powers[0] = natOne
+	powers[1] = x
+	for i := 2; i < 1<<n; i += 2 {
+		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
+		*p = p.mul(*p2, *p2)
+		zz, r = zz.div(r, *p, m)
+		*p, r = r, *p
+		*p1 = p1.mul(*p, x)
+		zz, r = zz.div(r, *p1, m)
+		*p1, r = r, *p1
+	}
+
+	z = z.setWord(1)
+
+	for i := len(y) - 1; i >= 0; i-- {
+		yi := y[i]
+		for j := 0; j < _W; j += n {
+			if i != len(y)-1 || j != 0 {
+				// Unrolled loop for significant performance
+				// gain.  Use go test -bench=".*" in crypto/rsa
+				// to check performance before making changes.
+				zz = zz.mul(z, z)
+				zz, z = z, zz
+				zz, r = zz.div(r, z, m)
+				z, r = r, z
+
+				zz = zz.mul(z, z)
+				zz, z = z, zz
+				zz, r = zz.div(r, z, m)
+				z, r = r, z
+
+				zz = zz.mul(z, z)
+				zz, z = z, zz
+				zz, r = zz.div(r, z, m)
+				z, r = r, z
+
+				zz = zz.mul(z, z)
+				zz, z = z, zz
+				zz, r = zz.div(r, z, m)
+				z, r = r, z
+			}
+
+			zz = zz.mul(z, powers[yi>>(_W-n)])
+			zz, z = z, zz
+			zz, r = zz.div(r, z, m)
+			z, r = r, z
+
+			yi <<= n
+		}
+	}
+
+	return z.norm()
+}
+
+// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
+// If it returns true, n is prime with probability 1 - 1/4^reps.
+// If it returns false, n is not prime.
+func (n nat) probablyPrime(reps int) bool {
+	if len(n) == 0 {
+		return false
+	}
+
+	if len(n) == 1 {
+		if n[0] < 2 {
+			return false
+		}
+
+		if n[0]%2 == 0 {
+			return n[0] == 2
+		}
+
+		// We have to exclude these cases because we reject all
+		// multiples of these numbers below.
+		switch n[0] {
+		case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53:
+			return true
+		}
+	}
+
+	const primesProduct32 = 0xC0CFD797         // Π {p ∈ primes, 2 < p <= 29}
+	const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
+
+	var r Word
+	switch _W {
+	case 32:
+		r = n.modW(primesProduct32)
+	case 64:
+		r = n.modW(primesProduct64 & _M)
+	default:
+		panic("Unknown word size")
+	}
+
+	if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 ||
+		r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 {
+		return false
+	}
+
+	if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 ||
+		r%43 == 0 || r%47 == 0 || r%53 == 0) {
+		return false
+	}
+
+	nm1 := nat(nil).sub(n, natOne)
+	// determine q, k such that nm1 = q << k
+	k := nm1.trailingZeroBits()
+	q := nat(nil).shr(nm1, k)
+
+	nm3 := nat(nil).sub(nm1, natTwo)
+	rand := rand.New(rand.NewSource(int64(n[0])))
+
+	var x, y, quotient nat
+	nm3Len := nm3.bitLen()
+
+NextRandom:
+	for i := 0; i < reps; i++ {
+		x = x.random(rand, nm3, nm3Len)
+		x = x.add(x, natTwo)
+		y = y.expNN(x, q, n)
+		if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
+			continue
+		}
+		for j := uint(1); j < k; j++ {
+			y = y.mul(y, y)
+			quotient, y = quotient.div(y, y, n)
+			if y.cmp(nm1) == 0 {
+				continue NextRandom
+			}
+			if y.cmp(natOne) == 0 {
+				return false
+			}
+		}
+		return false
+	}
+
+	return true
+}
+
+// bytes writes the value of z into buf using big-endian encoding.
+// len(buf) must be >= len(z)*_S. The value of z is encoded in the
+// slice buf[i:]. The number i of unused bytes at the beginning of
+// buf is returned as result.
+func (z nat) bytes(buf []byte) (i int) {
+	i = len(buf)
+	for _, d := range z {
+		for j := 0; j < _S; j++ {
+			i--
+			buf[i] = byte(d)
+			d >>= 8
+		}
+	}
+
+	for i < len(buf) && buf[i] == 0 {
+		i++
+	}
+
+	return
+}
+
+// setBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z nat) setBytes(buf []byte) nat {
+	z = z.make((len(buf) + _S - 1) / _S)
+
+	k := 0
+	s := uint(0)
+	var d Word
+	for i := len(buf); i > 0; i-- {
+		d |= Word(buf[i-1]) << s
+		if s += 8; s == _S*8 {
+			z[k] = d
+			k++
+			s = 0
+			d = 0
+		}
+	}
+	if k < len(z) {
+		z[k] = d
+	}
+
+	return z.norm()
+}
diff --git a/src/math/big/nat_test.go b/src/math/big/nat_test.go
new file mode 100644
index 000000000..a2ae53385
--- /dev/null
+++ b/src/math/big/nat_test.go
@@ -0,0 +1,771 @@
+// 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.
+
+package big
+
+import (
+	"io"
+	"runtime"
+	"strings"
+	"testing"
+)
+
+var cmpTests = []struct {
+	x, y nat
+	r    int
+}{
+	{nil, nil, 0},
+	{nil, nat(nil), 0},
+	{nat(nil), nil, 0},
+	{nat(nil), nat(nil), 0},
+	{nat{0}, nat{0}, 0},
+	{nat{0}, nat{1}, -1},
+	{nat{1}, nat{0}, 1},
+	{nat{1}, nat{1}, 0},
+	{nat{0, _M}, nat{1}, 1},
+	{nat{1}, nat{0, _M}, -1},
+	{nat{1, _M}, nat{0, _M}, 1},
+	{nat{0, _M}, nat{1, _M}, -1},
+	{nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
+	{nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
+}
+
+func TestCmp(t *testing.T) {
+	for i, a := range cmpTests {
+		r := a.x.cmp(a.y)
+		if r != a.r {
+			t.Errorf("#%d got r = %v; want %v", i, r, a.r)
+		}
+	}
+}
+
+type funNN func(z, x, y nat) nat
+type argNN struct {
+	z, x, y nat
+}
+
+var sumNN = []argNN{
+	{},
+	{nat{1}, nil, nat{1}},
+	{nat{1111111110}, nat{123456789}, nat{987654321}},
+	{nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
+	{nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
+	{nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
+}
+
+var prodNN = []argNN{
+	{},
+	{nil, nil, nil},
+	{nil, nat{991}, nil},
+	{nat{991}, nat{991}, nat{1}},
+	{nat{991 * 991}, nat{991}, nat{991}},
+	{nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
+	{nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
+	{nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
+	// 3^100 * 3^28 = 3^128
+	{
+		natFromString("11790184577738583171520872861412518665678211592275841109096961"),
+		natFromString("515377520732011331036461129765621272702107522001"),
+		natFromString("22876792454961"),
+	},
+	// z = 111....1 (70000 digits)
+	// x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
+	// y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
+	{
+		natFromString(strings.Repeat("1", 70000)),
+		natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
+		natFromString(strings.Repeat("1", 700)),
+	},
+	// z = 111....1 (20000 digits)
+	// x = 10^10000 + 1
+	// y = 111....1 (10000 digits)
+	{
+		natFromString(strings.Repeat("1", 20000)),
+		natFromString("1" + strings.Repeat("0", 9999) + "1"),
+		natFromString(strings.Repeat("1", 10000)),
+	},
+}
+
+func natFromString(s string) nat {
+	x, _, err := nat(nil).scan(strings.NewReader(s), 0)
+	if err != nil {
+		panic(err)
+	}
+	return x
+}
+
+func TestSet(t *testing.T) {
+	for _, a := range sumNN {
+		z := nat(nil).set(a.z)
+		if z.cmp(a.z) != 0 {
+			t.Errorf("got z = %v; want %v", z, a.z)
+		}
+	}
+}
+
+func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
+	z := f(nil, a.x, a.y)
+	if z.cmp(a.z) != 0 {
+		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
+	}
+}
+
+func TestFunNN(t *testing.T) {
+	for _, a := range sumNN {
+		arg := a
+		testFunNN(t, "add", nat.add, arg)
+
+		arg = argNN{a.z, a.y, a.x}
+		testFunNN(t, "add symmetric", nat.add, arg)
+
+		arg = argNN{a.x, a.z, a.y}
+		testFunNN(t, "sub", nat.sub, arg)
+
+		arg = argNN{a.y, a.z, a.x}
+		testFunNN(t, "sub symmetric", nat.sub, arg)
+	}
+
+	for _, a := range prodNN {
+		arg := a
+		testFunNN(t, "mul", nat.mul, arg)
+
+		arg = argNN{a.z, a.y, a.x}
+		testFunNN(t, "mul symmetric", nat.mul, arg)
+	}
+}
+
+var mulRangesN = []struct {
+	a, b uint64
+	prod string
+}{
+	{0, 0, "0"},
+	{1, 1, "1"},
+	{1, 2, "2"},
+	{1, 3, "6"},
+	{10, 10, "10"},
+	{0, 100, "0"},
+	{0, 1e9, "0"},
+	{1, 0, "1"},                    // empty range
+	{100, 1, "1"},                  // empty range
+	{1, 10, "3628800"},             // 10!
+	{1, 20, "2432902008176640000"}, // 20!
+	{1, 100,
+		"933262154439441526816992388562667004907159682643816214685929" +
+			"638952175999932299156089414639761565182862536979208272237582" +
+			"51185210916864000000000000000000000000", // 100!
+	},
+}
+
+func TestMulRangeN(t *testing.T) {
+	for i, r := range mulRangesN {
+		prod := nat(nil).mulRange(r.a, r.b).decimalString()
+		if prod != r.prod {
+			t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
+		}
+	}
+}
+
+// allocBytes returns the number of bytes allocated by invoking f.
+func allocBytes(f func()) uint64 {
+	var stats runtime.MemStats
+	runtime.ReadMemStats(&stats)
+	t := stats.TotalAlloc
+	f()
+	runtime.ReadMemStats(&stats)
+	return stats.TotalAlloc - t
+}
+
+// TestMulUnbalanced tests that multiplying numbers of different lengths
+// does not cause deep recursion and in turn allocate too much memory.
+// Test case for issue 3807.
+func TestMulUnbalanced(t *testing.T) {
+	defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
+	x := rndNat(50000)
+	y := rndNat(40)
+	allocSize := allocBytes(func() {
+		nat(nil).mul(x, y)
+	})
+	inputSize := uint64(len(x)+len(y)) * _S
+	if ratio := allocSize / uint64(inputSize); ratio > 10 {
+		t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
+	}
+}
+
+func rndNat(n int) nat {
+	return nat(rndV(n)).norm()
+}
+
+func BenchmarkMul(b *testing.B) {
+	mulx := rndNat(1e4)
+	muly := rndNat(1e4)
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		var z nat
+		z.mul(mulx, muly)
+	}
+}
+
+func toString(x nat, charset string) string {
+	base := len(charset)
+
+	// special cases
+	switch {
+	case base < 2:
+		panic("illegal base")
+	case len(x) == 0:
+		return string(charset[0])
+	}
+
+	// allocate buffer for conversion
+	i := x.bitLen()/log2(Word(base)) + 1 // +1: round up
+	s := make([]byte, i)
+
+	// don't destroy x
+	q := nat(nil).set(x)
+
+	// convert
+	for len(q) > 0 {
+		i--
+		var r Word
+		q, r = q.divW(q, Word(base))
+		s[i] = charset[r]
+	}
+
+	return string(s[i:])
+}
+
+var strTests = []struct {
+	x nat    // nat value to be converted
+	c string // conversion charset
+	s string // expected result
+}{
+	{nil, "01", "0"},
+	{nat{1}, "01", "1"},
+	{nat{0xc5}, "01", "11000101"},
+	{nat{03271}, lowercaseDigits[0:8], "3271"},
+	{nat{10}, lowercaseDigits[0:10], "10"},
+	{nat{1234567890}, uppercaseDigits[0:10], "1234567890"},
+	{nat{0xdeadbeef}, lowercaseDigits[0:16], "deadbeef"},
+	{nat{0xdeadbeef}, uppercaseDigits[0:16], "DEADBEEF"},
+	{nat{0x229be7}, lowercaseDigits[0:17], "1a2b3c"},
+	{nat{0x309663e6}, uppercaseDigits[0:32], "O9COV6"},
+}
+
+func TestString(t *testing.T) {
+	for _, a := range strTests {
+		s := a.x.string(a.c)
+		if s != a.s {
+			t.Errorf("string%+v\n\tgot s = %s; want %s", a, s, a.s)
+		}
+
+		x, b, err := nat(nil).scan(strings.NewReader(a.s), len(a.c))
+		if x.cmp(a.x) != 0 {
+			t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+		}
+		if b != len(a.c) {
+			t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, len(a.c))
+		}
+		if err != nil {
+			t.Errorf("scan%+v\n\tgot error = %s", a, err)
+		}
+	}
+}
+
+var natScanTests = []struct {
+	s    string // string to be scanned
+	base int    // input base
+	x    nat    // expected nat
+	b    int    // expected base
+	ok   bool   // expected success
+	next rune   // next character (or 0, if at EOF)
+}{
+	// error: illegal base
+	{base: -1},
+	{base: 1},
+	{base: 37},
+
+	// error: no mantissa
+	{},
+	{s: "?"},
+	{base: 10},
+	{base: 36},
+	{s: "?", base: 10},
+	{s: "0x"},
+	{s: "345", base: 2},
+
+	// no errors
+	{"0", 0, nil, 10, true, 0},
+	{"0", 10, nil, 10, true, 0},
+	{"0", 36, nil, 36, true, 0},
+	{"1", 0, nat{1}, 10, true, 0},
+	{"1", 10, nat{1}, 10, true, 0},
+	{"0 ", 0, nil, 10, true, ' '},
+	{"08", 0, nil, 10, true, '8'},
+	{"018", 0, nat{1}, 8, true, '8'},
+	{"0b1", 0, nat{1}, 2, true, 0},
+	{"0b11000101", 0, nat{0xc5}, 2, true, 0},
+	{"03271", 0, nat{03271}, 8, true, 0},
+	{"10ab", 0, nat{10}, 10, true, 'a'},
+	{"1234567890", 0, nat{1234567890}, 10, true, 0},
+	{"xyz", 36, nat{(33*36+34)*36 + 35}, 36, true, 0},
+	{"xyz?", 36, nat{(33*36+34)*36 + 35}, 36, true, '?'},
+	{"0x", 16, nil, 16, true, 'x'},
+	{"0xdeadbeef", 0, nat{0xdeadbeef}, 16, true, 0},
+	{"0XDEADBEEF", 0, nat{0xdeadbeef}, 16, true, 0},
+}
+
+func TestScanBase(t *testing.T) {
+	for _, a := range natScanTests {
+		r := strings.NewReader(a.s)
+		x, b, err := nat(nil).scan(r, a.base)
+		if err == nil && !a.ok {
+			t.Errorf("scan%+v\n\texpected error", a)
+		}
+		if err != nil {
+			if a.ok {
+				t.Errorf("scan%+v\n\tgot error = %s", a, err)
+			}
+			continue
+		}
+		if x.cmp(a.x) != 0 {
+			t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+		}
+		if b != a.b {
+			t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.base)
+		}
+		next, _, err := r.ReadRune()
+		if err == io.EOF {
+			next = 0
+			err = nil
+		}
+		if err == nil && next != a.next {
+			t.Errorf("scan%+v\n\tgot next = %q; want %q", a, next, a.next)
+		}
+	}
+}
+
+var pi = "3" +
+	"14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651" +
+	"32823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461" +
+	"28475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920" +
+	"96282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179" +
+	"31051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798" +
+	"60943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901" +
+	"22495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837" +
+	"29780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083" +
+	"81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909" +
+	"21642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151" +
+	"55748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035" +
+	"63707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104" +
+	"75216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992" +
+	"45863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818" +
+	"34797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548" +
+	"16136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179" +
+	"04946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886" +
+	"26945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645" +
+	"99581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745" +
+	"53050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382" +
+	"68683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244" +
+	"13654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767" +
+	"88952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288" +
+	"79710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821" +
+	"68299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610" +
+	"21359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435" +
+	"06430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675" +
+	"14269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207734672" +
+	"21825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539" +
+	"05796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007" +
+	"23055876317635942187312514712053292819182618612586732157919841484882916447060957527069572209175671167229109816" +
+	"90915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398" +
+	"31501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064" +
+	"20467525907091548141654985946163718027098199430992448895757128289059232332609729971208443357326548938239119325" +
+	"97463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100" +
+	"44929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915" +
+	"44110104468232527162010526522721116603966655730925471105578537634668206531098965269186205647693125705863566201" +
+	"85581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318" +
+	"58676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797" +
+	"27082668306343285878569830523580893306575740679545716377525420211495576158140025012622859413021647155097925923" +
+	"09907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111" +
+	"79042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043" +
+	"06332175182979866223717215916077166925474873898665494945011465406284336639379003976926567214638530673609657120" +
+	"91807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862" +
+	"94726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939" +
+	"78054193414473774418426312986080998886874132604721569516239658645730216315981931951673538129741677294786724229" +
+	"24654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001" +
+	"59377647165122893578601588161755782973523344604281512627203734314653197777416031990665541876397929334419521541" +
+	"34189948544473456738316249934191318148092777710386387734317720754565453220777092120190516609628049092636019759" +
+	"88281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267" +
+	"94561275318134078330336254232783944975382437205835311477119926063813346776879695970309833913077109870408591337"
+
+// Test case for BenchmarkScanPi.
+func TestScanPi(t *testing.T) {
+	var x nat
+	z, _, err := x.scan(strings.NewReader(pi), 10)
+	if err != nil {
+		t.Errorf("scanning pi: %s", err)
+	}
+	if s := z.decimalString(); s != pi {
+		t.Errorf("scanning pi: got %s", s)
+	}
+}
+
+func TestScanPiParallel(t *testing.T) {
+	const n = 2
+	c := make(chan int)
+	for i := 0; i < n; i++ {
+		go func() {
+			TestScanPi(t)
+			c <- 0
+		}()
+	}
+	for i := 0; i < n; i++ {
+		<-c
+	}
+}
+
+func BenchmarkScanPi(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		var x nat
+		x.scan(strings.NewReader(pi), 10)
+	}
+}
+
+func BenchmarkStringPiParallel(b *testing.B) {
+	var x nat
+	x, _, _ = x.scan(strings.NewReader(pi), 0)
+	if x.decimalString() != pi {
+		panic("benchmark incorrect: conversion failed")
+	}
+	b.RunParallel(func(pb *testing.PB) {
+		for pb.Next() {
+			x.decimalString()
+		}
+	})
+}
+
+func BenchmarkScan10Base2(b *testing.B)     { ScanHelper(b, 2, 10, 10) }
+func BenchmarkScan100Base2(b *testing.B)    { ScanHelper(b, 2, 10, 100) }
+func BenchmarkScan1000Base2(b *testing.B)   { ScanHelper(b, 2, 10, 1000) }
+func BenchmarkScan10000Base2(b *testing.B)  { ScanHelper(b, 2, 10, 10000) }
+func BenchmarkScan100000Base2(b *testing.B) { ScanHelper(b, 2, 10, 100000) }
+
+func BenchmarkScan10Base8(b *testing.B)     { ScanHelper(b, 8, 10, 10) }
+func BenchmarkScan100Base8(b *testing.B)    { ScanHelper(b, 8, 10, 100) }
+func BenchmarkScan1000Base8(b *testing.B)   { ScanHelper(b, 8, 10, 1000) }
+func BenchmarkScan10000Base8(b *testing.B)  { ScanHelper(b, 8, 10, 10000) }
+func BenchmarkScan100000Base8(b *testing.B) { ScanHelper(b, 8, 10, 100000) }
+
+func BenchmarkScan10Base10(b *testing.B)     { ScanHelper(b, 10, 10, 10) }
+func BenchmarkScan100Base10(b *testing.B)    { ScanHelper(b, 10, 10, 100) }
+func BenchmarkScan1000Base10(b *testing.B)   { ScanHelper(b, 10, 10, 1000) }
+func BenchmarkScan10000Base10(b *testing.B)  { ScanHelper(b, 10, 10, 10000) }
+func BenchmarkScan100000Base10(b *testing.B) { ScanHelper(b, 10, 10, 100000) }
+
+func BenchmarkScan10Base16(b *testing.B)     { ScanHelper(b, 16, 10, 10) }
+func BenchmarkScan100Base16(b *testing.B)    { ScanHelper(b, 16, 10, 100) }
+func BenchmarkScan1000Base16(b *testing.B)   { ScanHelper(b, 16, 10, 1000) }
+func BenchmarkScan10000Base16(b *testing.B)  { ScanHelper(b, 16, 10, 10000) }
+func BenchmarkScan100000Base16(b *testing.B) { ScanHelper(b, 16, 10, 100000) }
+
+func ScanHelper(b *testing.B, base int, x, y Word) {
+	b.StopTimer()
+	var z nat
+	z = z.expWW(x, y)
+
+	var s string
+	s = z.string(lowercaseDigits[0:base])
+	if t := toString(z, lowercaseDigits[0:base]); t != s {
+		b.Fatalf("scanning: got %s; want %s", s, t)
+	}
+	b.StartTimer()
+
+	for i := 0; i < b.N; i++ {
+		z.scan(strings.NewReader(s), base)
+	}
+}
+
+func BenchmarkString10Base2(b *testing.B)     { StringHelper(b, 2, 10, 10) }
+func BenchmarkString100Base2(b *testing.B)    { StringHelper(b, 2, 10, 100) }
+func BenchmarkString1000Base2(b *testing.B)   { StringHelper(b, 2, 10, 1000) }
+func BenchmarkString10000Base2(b *testing.B)  { StringHelper(b, 2, 10, 10000) }
+func BenchmarkString100000Base2(b *testing.B) { StringHelper(b, 2, 10, 100000) }
+
+func BenchmarkString10Base8(b *testing.B)     { StringHelper(b, 8, 10, 10) }
+func BenchmarkString100Base8(b *testing.B)    { StringHelper(b, 8, 10, 100) }
+func BenchmarkString1000Base8(b *testing.B)   { StringHelper(b, 8, 10, 1000) }
+func BenchmarkString10000Base8(b *testing.B)  { StringHelper(b, 8, 10, 10000) }
+func BenchmarkString100000Base8(b *testing.B) { StringHelper(b, 8, 10, 100000) }
+
+func BenchmarkString10Base10(b *testing.B)     { StringHelper(b, 10, 10, 10) }
+func BenchmarkString100Base10(b *testing.B)    { StringHelper(b, 10, 10, 100) }
+func BenchmarkString1000Base10(b *testing.B)   { StringHelper(b, 10, 10, 1000) }
+func BenchmarkString10000Base10(b *testing.B)  { StringHelper(b, 10, 10, 10000) }
+func BenchmarkString100000Base10(b *testing.B) { StringHelper(b, 10, 10, 100000) }
+
+func BenchmarkString10Base16(b *testing.B)     { StringHelper(b, 16, 10, 10) }
+func BenchmarkString100Base16(b *testing.B)    { StringHelper(b, 16, 10, 100) }
+func BenchmarkString1000Base16(b *testing.B)   { StringHelper(b, 16, 10, 1000) }
+func BenchmarkString10000Base16(b *testing.B)  { StringHelper(b, 16, 10, 10000) }
+func BenchmarkString100000Base16(b *testing.B) { StringHelper(b, 16, 10, 100000) }
+
+func StringHelper(b *testing.B, base int, x, y Word) {
+	b.StopTimer()
+	var z nat
+	z = z.expWW(x, y)
+	z.string(lowercaseDigits[0:base]) // warm divisor cache
+	b.StartTimer()
+
+	for i := 0; i < b.N; i++ {
+		_ = z.string(lowercaseDigits[0:base])
+	}
+}
+
+func BenchmarkLeafSize0(b *testing.B)  { LeafSizeHelper(b, 10, 0) } // test without splitting
+func BenchmarkLeafSize1(b *testing.B)  { LeafSizeHelper(b, 10, 1) }
+func BenchmarkLeafSize2(b *testing.B)  { LeafSizeHelper(b, 10, 2) }
+func BenchmarkLeafSize3(b *testing.B)  { LeafSizeHelper(b, 10, 3) }
+func BenchmarkLeafSize4(b *testing.B)  { LeafSizeHelper(b, 10, 4) }
+func BenchmarkLeafSize5(b *testing.B)  { LeafSizeHelper(b, 10, 5) }
+func BenchmarkLeafSize6(b *testing.B)  { LeafSizeHelper(b, 10, 6) }
+func BenchmarkLeafSize7(b *testing.B)  { LeafSizeHelper(b, 10, 7) }
+func BenchmarkLeafSize8(b *testing.B)  { LeafSizeHelper(b, 10, 8) }
+func BenchmarkLeafSize9(b *testing.B)  { LeafSizeHelper(b, 10, 9) }
+func BenchmarkLeafSize10(b *testing.B) { LeafSizeHelper(b, 10, 10) }
+func BenchmarkLeafSize11(b *testing.B) { LeafSizeHelper(b, 10, 11) }
+func BenchmarkLeafSize12(b *testing.B) { LeafSizeHelper(b, 10, 12) }
+func BenchmarkLeafSize13(b *testing.B) { LeafSizeHelper(b, 10, 13) }
+func BenchmarkLeafSize14(b *testing.B) { LeafSizeHelper(b, 10, 14) }
+func BenchmarkLeafSize15(b *testing.B) { LeafSizeHelper(b, 10, 15) }
+func BenchmarkLeafSize16(b *testing.B) { LeafSizeHelper(b, 10, 16) }
+func BenchmarkLeafSize32(b *testing.B) { LeafSizeHelper(b, 10, 32) } // try some large lengths
+func BenchmarkLeafSize64(b *testing.B) { LeafSizeHelper(b, 10, 64) }
+
+func LeafSizeHelper(b *testing.B, base Word, size int) {
+	b.StopTimer()
+	originalLeafSize := leafSize
+	resetTable(cacheBase10.table[:])
+	leafSize = size
+	b.StartTimer()
+
+	for d := 1; d <= 10000; d *= 10 {
+		b.StopTimer()
+		var z nat
+		z = z.expWW(base, Word(d))            // build target number
+		_ = z.string(lowercaseDigits[0:base]) // warm divisor cache
+		b.StartTimer()
+
+		for i := 0; i < b.N; i++ {
+			_ = z.string(lowercaseDigits[0:base])
+		}
+	}
+
+	b.StopTimer()
+	resetTable(cacheBase10.table[:])
+	leafSize = originalLeafSize
+	b.StartTimer()
+}
+
+func resetTable(table []divisor) {
+	if table != nil && table[0].bbb != nil {
+		for i := 0; i < len(table); i++ {
+			table[i].bbb = nil
+			table[i].nbits = 0
+			table[i].ndigits = 0
+		}
+	}
+}
+
+func TestStringPowers(t *testing.T) {
+	var b, p Word
+	for b = 2; b <= 16; b++ {
+		for p = 0; p <= 512; p++ {
+			x := nat(nil).expWW(b, p)
+			xs := x.string(lowercaseDigits[0:b])
+			xs2 := toString(x, lowercaseDigits[0:b])
+			if xs != xs2 {
+				t.Errorf("failed at %d ** %d in base %d: %s != %s", b, p, b, xs, xs2)
+			}
+		}
+		if b >= 3 && testing.Short() {
+			break
+		}
+	}
+}
+
+func TestLeadingZeros(t *testing.T) {
+	var x Word = _B >> 1
+	for i := 0; i <= _W; i++ {
+		if int(leadingZeros(x)) != i {
+			t.Errorf("failed at %x: got %d want %d", x, leadingZeros(x), i)
+		}
+		x >>= 1
+	}
+}
+
+type shiftTest struct {
+	in    nat
+	shift uint
+	out   nat
+}
+
+var leftShiftTests = []shiftTest{
+	{nil, 0, nil},
+	{nil, 1, nil},
+	{natOne, 0, natOne},
+	{natOne, 1, natTwo},
+	{nat{1 << (_W - 1)}, 1, nat{0}},
+	{nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
+}
+
+func TestShiftLeft(t *testing.T) {
+	for i, test := range leftShiftTests {
+		var z nat
+		z = z.shl(test.in, test.shift)
+		for j, d := range test.out {
+			if j >= len(z) || z[j] != d {
+				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+				break
+			}
+		}
+	}
+}
+
+var rightShiftTests = []shiftTest{
+	{nil, 0, nil},
+	{nil, 1, nil},
+	{natOne, 0, natOne},
+	{natOne, 1, nil},
+	{natTwo, 1, natOne},
+	{nat{0, 1}, 1, nat{1 << (_W - 1)}},
+	{nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
+}
+
+func TestShiftRight(t *testing.T) {
+	for i, test := range rightShiftTests {
+		var z nat
+		z = z.shr(test.in, test.shift)
+		for j, d := range test.out {
+			if j >= len(z) || z[j] != d {
+				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+				break
+			}
+		}
+	}
+}
+
+type modWTest struct {
+	in       string
+	dividend string
+	out      string
+}
+
+var modWTests32 = []modWTest{
+	{"23492635982634928349238759823742", "252341", "220170"},
+}
+
+var modWTests64 = []modWTest{
+	{"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
+}
+
+func runModWTests(t *testing.T, tests []modWTest) {
+	for i, test := range tests {
+		in, _ := new(Int).SetString(test.in, 10)
+		d, _ := new(Int).SetString(test.dividend, 10)
+		out, _ := new(Int).SetString(test.out, 10)
+
+		r := in.abs.modW(d.abs[0])
+		if r != out.abs[0] {
+			t.Errorf("#%d failed: got %d want %s", i, r, out)
+		}
+	}
+}
+
+func TestModW(t *testing.T) {
+	if _W >= 32 {
+		runModWTests(t, modWTests32)
+	}
+	if _W >= 64 {
+		runModWTests(t, modWTests64)
+	}
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+	x := Word(1)
+	for i := uint(0); i <= _W; i++ {
+		n := trailingZeroBits(x)
+		if n != i%_W {
+			t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
+		}
+		x <<= 1
+	}
+
+	y := nat(nil).set(natOne)
+	for i := uint(0); i <= 3*_W; i++ {
+		n := y.trailingZeroBits()
+		if n != i {
+			t.Errorf("got 0x%s.trailingZeroBits() = %d; want %d", y.string(lowercaseDigits[0:16]), n, i)
+		}
+		y = y.shl(y, 1)
+	}
+}
+
+var expNNTests = []struct {
+	x, y, m string
+	out     string
+}{
+	{"0", "0", "0", "1"},
+	{"0", "0", "1", "0"},
+	{"1", "1", "1", "0"},
+	{"2", "1", "1", "0"},
+	{"2", "2", "1", "0"},
+	{"10", "100000000000", "1", "0"},
+	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+	{"0x8000000000000000", "2", "6719", "4944"},
+	{"0x8000000000000000", "3", "6719", "5447"},
+	{"0x8000000000000000", "1000", "6719", "1603"},
+	{"0x8000000000000000", "1000000", "6719", "3199"},
+	{
+		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+	},
+}
+
+func TestExpNN(t *testing.T) {
+	for i, test := range expNNTests {
+		x, _, _ := nat(nil).scan(strings.NewReader(test.x), 0)
+		y, _, _ := nat(nil).scan(strings.NewReader(test.y), 0)
+		out, _, _ := nat(nil).scan(strings.NewReader(test.out), 0)
+
+		var m nat
+
+		if len(test.m) > 0 {
+			m, _, _ = nat(nil).scan(strings.NewReader(test.m), 0)
+		}
+
+		z := nat(nil).expNN(x, y, m)
+		if z.cmp(out) != 0 {
+			t.Errorf("#%d got %s want %s", i, z.decimalString(), out.decimalString())
+		}
+	}
+}
+
+func ExpHelper(b *testing.B, x, y Word) {
+	var z nat
+	for i := 0; i < b.N; i++ {
+		z.expWW(x, y)
+	}
+}
+
+func BenchmarkExp3Power0x10(b *testing.B)     { ExpHelper(b, 3, 0x10) }
+func BenchmarkExp3Power0x40(b *testing.B)     { ExpHelper(b, 3, 0x40) }
+func BenchmarkExp3Power0x100(b *testing.B)    { ExpHelper(b, 3, 0x100) }
+func BenchmarkExp3Power0x400(b *testing.B)    { ExpHelper(b, 3, 0x400) }
+func BenchmarkExp3Power0x1000(b *testing.B)   { ExpHelper(b, 3, 0x1000) }
+func BenchmarkExp3Power0x4000(b *testing.B)   { ExpHelper(b, 3, 0x4000) }
+func BenchmarkExp3Power0x10000(b *testing.B)  { ExpHelper(b, 3, 0x10000) }
+func BenchmarkExp3Power0x40000(b *testing.B)  { ExpHelper(b, 3, 0x40000) }
+func BenchmarkExp3Power0x100000(b *testing.B) { ExpHelper(b, 3, 0x100000) }
+func BenchmarkExp3Power0x400000(b *testing.B) { ExpHelper(b, 3, 0x400000) }
diff --git a/src/math/big/rat.go b/src/math/big/rat.go
new file mode 100644
index 000000000..e6ab0bb48
--- /dev/null
+++ b/src/math/big/rat.go
@@ -0,0 +1,713 @@
+// Copyright 2010 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.
+
+// This file implements multi-precision rational numbers.
+
+package big
+
+import (
+	"encoding/binary"
+	"errors"
+	"fmt"
+	"math"
+	"strings"
+)
+
+// A Rat represents a quotient a/b of arbitrary precision.
+// The zero value for a Rat represents the value 0.
+type Rat struct {
+	// To make zero values for Rat work w/o initialization,
+	// a zero value of b (len(b) == 0) acts like b == 1.
+	// a.neg determines the sign of the Rat, b.neg is ignored.
+	a, b Int
+}
+
+// NewRat creates a new Rat with numerator a and denominator b.
+func NewRat(a, b int64) *Rat {
+	return new(Rat).SetFrac64(a, b)
+}
+
+// SetFloat64 sets z to exactly f and returns z.
+// If f is not finite, SetFloat returns nil.
+func (z *Rat) SetFloat64(f float64) *Rat {
+	const expMask = 1<<11 - 1
+	bits := math.Float64bits(f)
+	mantissa := bits & (1<<52 - 1)
+	exp := int((bits >> 52) & expMask)
+	switch exp {
+	case expMask: // non-finite
+		return nil
+	case 0: // denormal
+		exp -= 1022
+	default: // normal
+		mantissa |= 1 << 52
+		exp -= 1023
+	}
+
+	shift := 52 - exp
+
+	// Optimization (?): partially pre-normalise.
+	for mantissa&1 == 0 && shift > 0 {
+		mantissa >>= 1
+		shift--
+	}
+
+	z.a.SetUint64(mantissa)
+	z.a.neg = f < 0
+	z.b.Set(intOne)
+	if shift > 0 {
+		z.b.Lsh(&z.b, uint(shift))
+	} else {
+		z.a.Lsh(&z.a, uint(-shift))
+	}
+	return z.norm()
+}
+
+// quotToFloat32 returns the non-negative float32 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases.  It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat32(a, b nat) (f float32, exact bool) {
+	const (
+		// float size in bits
+		Fsize = 32
+
+		// mantissa
+		Msize  = 23
+		Msize1 = Msize + 1 // incl. implicit 1
+		Msize2 = Msize1 + 1
+
+		// exponent
+		Esize = Fsize - Msize1
+		Ebias = 1<<(Esize-1) - 1
+		Emin  = 1 - Ebias
+		Emax  = Ebias
+	)
+
+	// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+	alen := a.bitLen()
+	if alen == 0 {
+		return 0, true
+	}
+	blen := b.bitLen()
+	if blen == 0 {
+		panic("division by zero")
+	}
+
+	// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+	// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+	// This is 2 or 3 more than the float32 mantissa field width of Msize:
+	// - the optional extra bit is shifted away in step 3 below.
+	// - the high-order 1 is omitted in "normal" representation;
+	// - the low-order 1 will be used during rounding then discarded.
+	exp := alen - blen
+	var a2, b2 nat
+	a2 = a2.set(a)
+	b2 = b2.set(b)
+	if shift := Msize2 - exp; shift > 0 {
+		a2 = a2.shl(a2, uint(shift))
+	} else if shift < 0 {
+		b2 = b2.shl(b2, uint(-shift))
+	}
+
+	// 2. Compute quotient and remainder (q, r).  NB: due to the
+	// extra shift, the low-order bit of q is logically the
+	// high-order bit of r.
+	var q nat
+	q, r := q.div(a2, a2, b2) // (recycle a2)
+	mantissa := low32(q)
+	haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+	// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+	// (in effect---we accomplish this incrementally).
+	if mantissa>>Msize2 == 1 {
+		if mantissa&1 == 1 {
+			haveRem = true
+		}
+		mantissa >>= 1
+		exp++
+	}
+	if mantissa>>Msize1 != 1 {
+		panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+	}
+
+	// 4. Rounding.
+	if Emin-Msize <= exp && exp <= Emin {
+		// Denormal case; lose 'shift' bits of precision.
+		shift := uint(Emin - (exp - 1)) // [1..Esize1)
+		lostbits := mantissa & (1<<shift - 1)
+		haveRem = haveRem || lostbits != 0
+		mantissa >>= shift
+		exp = 2 - Ebias // == exp + shift
+	}
+	// Round q using round-half-to-even.
+	exact = !haveRem
+	if mantissa&1 != 0 {
+		exact = false
+		if haveRem || mantissa&2 != 0 {
+			if mantissa++; mantissa >= 1<<Msize2 {
+				// Complete rollover 11...1 => 100...0, so shift is safe
+				mantissa >>= 1
+				exp++
+			}
+		}
+	}
+	mantissa >>= 1 // discard rounding bit.  Mantissa now scaled by 1<<Msize1.
+
+	f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
+	if math.IsInf(float64(f), 0) {
+		exact = false
+	}
+	return
+}
+
+// quotToFloat64 returns the non-negative float64 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases.  It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat64(a, b nat) (f float64, exact bool) {
+	const (
+		// float size in bits
+		Fsize = 64
+
+		// mantissa
+		Msize  = 52
+		Msize1 = Msize + 1 // incl. implicit 1
+		Msize2 = Msize1 + 1
+
+		// exponent
+		Esize = Fsize - Msize1
+		Ebias = 1<<(Esize-1) - 1
+		Emin  = 1 - Ebias
+		Emax  = Ebias
+	)
+
+	// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+	alen := a.bitLen()
+	if alen == 0 {
+		return 0, true
+	}
+	blen := b.bitLen()
+	if blen == 0 {
+		panic("division by zero")
+	}
+
+	// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+	// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+	// This is 2 or 3 more than the float64 mantissa field width of Msize:
+	// - the optional extra bit is shifted away in step 3 below.
+	// - the high-order 1 is omitted in "normal" representation;
+	// - the low-order 1 will be used during rounding then discarded.
+	exp := alen - blen
+	var a2, b2 nat
+	a2 = a2.set(a)
+	b2 = b2.set(b)
+	if shift := Msize2 - exp; shift > 0 {
+		a2 = a2.shl(a2, uint(shift))
+	} else if shift < 0 {
+		b2 = b2.shl(b2, uint(-shift))
+	}
+
+	// 2. Compute quotient and remainder (q, r).  NB: due to the
+	// extra shift, the low-order bit of q is logically the
+	// high-order bit of r.
+	var q nat
+	q, r := q.div(a2, a2, b2) // (recycle a2)
+	mantissa := low64(q)
+	haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+	// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+	// (in effect---we accomplish this incrementally).
+	if mantissa>>Msize2 == 1 {
+		if mantissa&1 == 1 {
+			haveRem = true
+		}
+		mantissa >>= 1
+		exp++
+	}
+	if mantissa>>Msize1 != 1 {
+		panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+	}
+
+	// 4. Rounding.
+	if Emin-Msize <= exp && exp <= Emin {
+		// Denormal case; lose 'shift' bits of precision.
+		shift := uint(Emin - (exp - 1)) // [1..Esize1)
+		lostbits := mantissa & (1<<shift - 1)
+		haveRem = haveRem || lostbits != 0
+		mantissa >>= shift
+		exp = 2 - Ebias // == exp + shift
+	}
+	// Round q using round-half-to-even.
+	exact = !haveRem
+	if mantissa&1 != 0 {
+		exact = false
+		if haveRem || mantissa&2 != 0 {
+			if mantissa++; mantissa >= 1<<Msize2 {
+				// Complete rollover 11...1 => 100...0, so shift is safe
+				mantissa >>= 1
+				exp++
+			}
+		}
+	}
+	mantissa >>= 1 // discard rounding bit.  Mantissa now scaled by 1<<Msize1.
+
+	f = math.Ldexp(float64(mantissa), exp-Msize1)
+	if math.IsInf(f, 0) {
+		exact = false
+	}
+	return
+}
+
+// Float32 returns the nearest float32 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float32, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float32() (f float32, exact bool) {
+	b := x.b.abs
+	if len(b) == 0 {
+		b = b.set(natOne) // materialize denominator
+	}
+	f, exact = quotToFloat32(x.a.abs, b)
+	if x.a.neg {
+		f = -f
+	}
+	return
+}
+
+// Float64 returns the nearest float64 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float64, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float64() (f float64, exact bool) {
+	b := x.b.abs
+	if len(b) == 0 {
+		b = b.set(natOne) // materialize denominator
+	}
+	f, exact = quotToFloat64(x.a.abs, b)
+	if x.a.neg {
+		f = -f
+	}
+	return
+}
+
+// SetFrac sets z to a/b and returns z.
+func (z *Rat) SetFrac(a, b *Int) *Rat {
+	z.a.neg = a.neg != b.neg
+	babs := b.abs
+	if len(babs) == 0 {
+		panic("division by zero")
+	}
+	if &z.a == b || alias(z.a.abs, babs) {
+		babs = nat(nil).set(babs) // make a copy
+	}
+	z.a.abs = z.a.abs.set(a.abs)
+	z.b.abs = z.b.abs.set(babs)
+	return z.norm()
+}
+
+// SetFrac64 sets z to a/b and returns z.
+func (z *Rat) SetFrac64(a, b int64) *Rat {
+	z.a.SetInt64(a)
+	if b == 0 {
+		panic("division by zero")
+	}
+	if b < 0 {
+		b = -b
+		z.a.neg = !z.a.neg
+	}
+	z.b.abs = z.b.abs.setUint64(uint64(b))
+	return z.norm()
+}
+
+// SetInt sets z to x (by making a copy of x) and returns z.
+func (z *Rat) SetInt(x *Int) *Rat {
+	z.a.Set(x)
+	z.b.abs = z.b.abs.make(0)
+	return z
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Rat) SetInt64(x int64) *Rat {
+	z.a.SetInt64(x)
+	z.b.abs = z.b.abs.make(0)
+	return z
+}
+
+// Set sets z to x (by making a copy of x) and returns z.
+func (z *Rat) Set(x *Rat) *Rat {
+	if z != x {
+		z.a.Set(&x.a)
+		z.b.Set(&x.b)
+	}
+	return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Rat) Abs(x *Rat) *Rat {
+	z.Set(x)
+	z.a.neg = false
+	return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Rat) Neg(x *Rat) *Rat {
+	z.Set(x)
+	z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
+	return z
+}
+
+// Inv sets z to 1/x and returns z.
+func (z *Rat) Inv(x *Rat) *Rat {
+	if len(x.a.abs) == 0 {
+		panic("division by zero")
+	}
+	z.Set(x)
+	a := z.b.abs
+	if len(a) == 0 {
+		a = a.set(natOne) // materialize numerator
+	}
+	b := z.a.abs
+	if b.cmp(natOne) == 0 {
+		b = b.make(0) // normalize denominator
+	}
+	z.a.abs, z.b.abs = a, b // sign doesn't change
+	return z
+}
+
+// Sign returns:
+//
+//	-1 if x <  0
+//	 0 if x == 0
+//	+1 if x >  0
+//
+func (x *Rat) Sign() int {
+	return x.a.Sign()
+}
+
+// IsInt returns true if the denominator of x is 1.
+func (x *Rat) IsInt() bool {
+	return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
+}
+
+// Num returns the numerator of x; it may be <= 0.
+// The result is a reference to x's numerator; it
+// may change if a new value is assigned to x, and vice versa.
+// The sign of the numerator corresponds to the sign of x.
+func (x *Rat) Num() *Int {
+	return &x.a
+}
+
+// Denom returns the denominator of x; it is always > 0.
+// The result is a reference to x's denominator; it
+// may change if a new value is assigned to x, and vice versa.
+func (x *Rat) Denom() *Int {
+	x.b.neg = false // the result is always >= 0
+	if len(x.b.abs) == 0 {
+		x.b.abs = x.b.abs.set(natOne) // materialize denominator
+	}
+	return &x.b
+}
+
+func (z *Rat) norm() *Rat {
+	switch {
+	case len(z.a.abs) == 0:
+		// z == 0 - normalize sign and denominator
+		z.a.neg = false
+		z.b.abs = z.b.abs.make(0)
+	case len(z.b.abs) == 0:
+		// z is normalized int - nothing to do
+	case z.b.abs.cmp(natOne) == 0:
+		// z is int - normalize denominator
+		z.b.abs = z.b.abs.make(0)
+	default:
+		neg := z.a.neg
+		z.a.neg = false
+		z.b.neg = false
+		if f := NewInt(0).binaryGCD(&z.a, &z.b); f.Cmp(intOne) != 0 {
+			z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
+			z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
+			if z.b.abs.cmp(natOne) == 0 {
+				// z is int - normalize denominator
+				z.b.abs = z.b.abs.make(0)
+			}
+		}
+		z.a.neg = neg
+	}
+	return z
+}
+
+// mulDenom sets z to the denominator product x*y (by taking into
+// account that 0 values for x or y must be interpreted as 1) and
+// returns z.
+func mulDenom(z, x, y nat) nat {
+	switch {
+	case len(x) == 0:
+		return z.set(y)
+	case len(y) == 0:
+		return z.set(x)
+	}
+	return z.mul(x, y)
+}
+
+// scaleDenom computes x*f.
+// If f == 0 (zero value of denominator), the result is (a copy of) x.
+func scaleDenom(x *Int, f nat) *Int {
+	var z Int
+	if len(f) == 0 {
+		return z.Set(x)
+	}
+	z.abs = z.abs.mul(x.abs, f)
+	z.neg = x.neg
+	return &z
+}
+
+// Cmp compares x and y and returns:
+//
+//   -1 if x <  y
+//    0 if x == y
+//   +1 if x >  y
+//
+func (x *Rat) Cmp(y *Rat) int {
+	return scaleDenom(&x.a, y.b.abs).Cmp(scaleDenom(&y.a, x.b.abs))
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+	a1 := scaleDenom(&x.a, y.b.abs)
+	a2 := scaleDenom(&y.a, x.b.abs)
+	z.a.Add(a1, a2)
+	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+	return z.norm()
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Rat) Sub(x, y *Rat) *Rat {
+	a1 := scaleDenom(&x.a, y.b.abs)
+	a2 := scaleDenom(&y.a, x.b.abs)
+	z.a.Sub(a1, a2)
+	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+	return z.norm()
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Rat) Mul(x, y *Rat) *Rat {
+	z.a.Mul(&x.a, &y.a)
+	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+	return z.norm()
+}
+
+// Quo sets z to the quotient x/y and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+func (z *Rat) Quo(x, y *Rat) *Rat {
+	if len(y.a.abs) == 0 {
+		panic("division by zero")
+	}
+	a := scaleDenom(&x.a, y.b.abs)
+	b := scaleDenom(&y.a, x.b.abs)
+	z.a.abs = a.abs
+	z.b.abs = b.abs
+	z.a.neg = a.neg != b.neg
+	return z.norm()
+}
+
+func ratTok(ch rune) bool {
+	return strings.IndexRune("+-/0123456789.eE", ch) >= 0
+}
+
+// Scan is a support routine for fmt.Scanner. It accepts the formats
+// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
+func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
+	tok, err := s.Token(true, ratTok)
+	if err != nil {
+		return err
+	}
+	if strings.IndexRune("efgEFGv", ch) < 0 {
+		return errors.New("Rat.Scan: invalid verb")
+	}
+	if _, ok := z.SetString(string(tok)); !ok {
+		return errors.New("Rat.Scan: invalid syntax")
+	}
+	return nil
+}
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s can be given as a fraction "a/b" or as a floating-point number
+// optionally followed by an exponent. If the operation failed, the value of
+// z is undefined but the returned value is nil.
+func (z *Rat) SetString(s string) (*Rat, bool) {
+	if len(s) == 0 {
+		return nil, false
+	}
+
+	// check for a quotient
+	sep := strings.Index(s, "/")
+	if sep >= 0 {
+		if _, ok := z.a.SetString(s[0:sep], 10); !ok {
+			return nil, false
+		}
+		s = s[sep+1:]
+		var err error
+		if z.b.abs, _, err = z.b.abs.scan(strings.NewReader(s), 10); err != nil {
+			return nil, false
+		}
+		return z.norm(), true
+	}
+
+	// check for a decimal point
+	sep = strings.Index(s, ".")
+	// check for an exponent
+	e := strings.IndexAny(s, "eE")
+	var exp Int
+	if e >= 0 {
+		if e < sep {
+			// The E must come after the decimal point.
+			return nil, false
+		}
+		if _, ok := exp.SetString(s[e+1:], 10); !ok {
+			return nil, false
+		}
+		s = s[0:e]
+	}
+	if sep >= 0 {
+		s = s[0:sep] + s[sep+1:]
+		exp.Sub(&exp, NewInt(int64(len(s)-sep)))
+	}
+
+	if _, ok := z.a.SetString(s, 10); !ok {
+		return nil, false
+	}
+	powTen := nat(nil).expNN(natTen, exp.abs, nil)
+	if exp.neg {
+		z.b.abs = powTen
+		z.norm()
+	} else {
+		z.a.abs = z.a.abs.mul(z.a.abs, powTen)
+		z.b.abs = z.b.abs.make(0)
+	}
+
+	return z, true
+}
+
+// String returns a string representation of x in the form "a/b" (even if b == 1).
+func (x *Rat) String() string {
+	s := "/1"
+	if len(x.b.abs) != 0 {
+		s = "/" + x.b.abs.decimalString()
+	}
+	return x.a.String() + s
+}
+
+// RatString returns a string representation of x in the form "a/b" if b != 1,
+// and in the form "a" if b == 1.
+func (x *Rat) RatString() string {
+	if x.IsInt() {
+		return x.a.String()
+	}
+	return x.String()
+}
+
+// FloatString returns a string representation of x in decimal form with prec
+// digits of precision after the decimal point and the last digit rounded.
+func (x *Rat) FloatString(prec int) string {
+	if x.IsInt() {
+		s := x.a.String()
+		if prec > 0 {
+			s += "." + strings.Repeat("0", prec)
+		}
+		return s
+	}
+	// x.b.abs != 0
+
+	q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)
+
+	p := natOne
+	if prec > 0 {
+		p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil)
+	}
+
+	r = r.mul(r, p)
+	r, r2 := r.div(nat(nil), r, x.b.abs)
+
+	// see if we need to round up
+	r2 = r2.add(r2, r2)
+	if x.b.abs.cmp(r2) <= 0 {
+		r = r.add(r, natOne)
+		if r.cmp(p) >= 0 {
+			q = nat(nil).add(q, natOne)
+			r = nat(nil).sub(r, p)
+		}
+	}
+
+	s := q.decimalString()
+	if x.a.neg {
+		s = "-" + s
+	}
+
+	if prec > 0 {
+		rs := r.decimalString()
+		leadingZeros := prec - len(rs)
+		s += "." + strings.Repeat("0", leadingZeros) + rs
+	}
+
+	return s
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const ratGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Rat) GobEncode() ([]byte, error) {
+	if x == nil {
+		return nil, nil
+	}
+	buf := make([]byte, 1+4+(len(x.a.abs)+len(x.b.abs))*_S) // extra bytes for version and sign bit (1), and numerator length (4)
+	i := x.b.abs.bytes(buf)
+	j := x.a.abs.bytes(buf[0:i])
+	n := i - j
+	if int(uint32(n)) != n {
+		// this should never happen
+		return nil, errors.New("Rat.GobEncode: numerator too large")
+	}
+	binary.BigEndian.PutUint32(buf[j-4:j], uint32(n))
+	j -= 1 + 4
+	b := ratGobVersion << 1 // make space for sign bit
+	if x.a.neg {
+		b |= 1
+	}
+	buf[j] = b
+	return buf[j:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Rat) GobDecode(buf []byte) error {
+	if len(buf) == 0 {
+		// Other side sent a nil or default value.
+		*z = Rat{}
+		return nil
+	}
+	b := buf[0]
+	if b>>1 != ratGobVersion {
+		return errors.New(fmt.Sprintf("Rat.GobDecode: encoding version %d not supported", b>>1))
+	}
+	const j = 1 + 4
+	i := j + binary.BigEndian.Uint32(buf[j-4:j])
+	z.a.neg = b&1 != 0
+	z.a.abs = z.a.abs.setBytes(buf[j:i])
+	z.b.abs = z.b.abs.setBytes(buf[i:])
+	return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface
+func (r *Rat) MarshalText() (text []byte, err error) {
+	return []byte(r.RatString()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface
+func (r *Rat) UnmarshalText(text []byte) error {
+	if _, ok := r.SetString(string(text)); !ok {
+		return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Rat", text)
+	}
+	return nil
+}
diff --git a/src/math/big/rat_test.go b/src/math/big/rat_test.go
new file mode 100644
index 000000000..598eac8cc
--- /dev/null
+++ b/src/math/big/rat_test.go
@@ -0,0 +1,1159 @@
+// Copyright 2010 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 big
+
+import (
+	"bytes"
+	"encoding/gob"
+	"encoding/json"
+	"encoding/xml"
+	"fmt"
+	"math"
+	"strconv"
+	"strings"
+	"testing"
+)
+
+func TestZeroRat(t *testing.T) {
+	var x, y, z Rat
+	y.SetFrac64(0, 42)
+
+	if x.Cmp(&y) != 0 {
+		t.Errorf("x and y should be both equal and zero")
+	}
+
+	if s := x.String(); s != "0/1" {
+		t.Errorf("got x = %s, want 0/1", s)
+	}
+
+	if s := x.RatString(); s != "0" {
+		t.Errorf("got x = %s, want 0", s)
+	}
+
+	z.Add(&x, &y)
+	if s := z.RatString(); s != "0" {
+		t.Errorf("got x+y = %s, want 0", s)
+	}
+
+	z.Sub(&x, &y)
+	if s := z.RatString(); s != "0" {
+		t.Errorf("got x-y = %s, want 0", s)
+	}
+
+	z.Mul(&x, &y)
+	if s := z.RatString(); s != "0" {
+		t.Errorf("got x*y = %s, want 0", s)
+	}
+
+	// check for division by zero
+	defer func() {
+		if s := recover(); s == nil || s.(string) != "division by zero" {
+			panic(s)
+		}
+	}()
+	z.Quo(&x, &y)
+}
+
+var setStringTests = []struct {
+	in, out string
+	ok      bool
+}{
+	{"0", "0", true},
+	{"-0", "0", true},
+	{"1", "1", true},
+	{"-1", "-1", true},
+	{"1.", "1", true},
+	{"1e0", "1", true},
+	{"1.e1", "10", true},
+	{in: "1e", ok: false},
+	{in: "1.e", ok: false},
+	{in: "1e+14e-5", ok: false},
+	{in: "1e4.5", ok: false},
+	{in: "r", ok: false},
+	{in: "a/b", ok: false},
+	{in: "a.b", ok: false},
+	{"-0.1", "-1/10", true},
+	{"-.1", "-1/10", true},
+	{"2/4", "1/2", true},
+	{".25", "1/4", true},
+	{"-1/5", "-1/5", true},
+	{"8129567.7690E14", "812956776900000000000", true},
+	{"78189e+4", "781890000", true},
+	{"553019.8935e+8", "55301989350000", true},
+	{"98765432109876543210987654321e-10", "98765432109876543210987654321/10000000000", true},
+	{"9877861857500000E-7", "3951144743/4", true},
+	{"2169378.417e-3", "2169378417/1000000", true},
+	{"884243222337379604041632732738665534", "884243222337379604041632732738665534", true},
+	{"53/70893980658822810696", "53/70893980658822810696", true},
+	{"106/141787961317645621392", "53/70893980658822810696", true},
+	{"204211327800791583.81095", "4084226556015831676219/20000", true},
+}
+
+func TestRatSetString(t *testing.T) {
+	for i, test := range setStringTests {
+		x, ok := new(Rat).SetString(test.in)
+
+		if ok {
+			if !test.ok {
+				t.Errorf("#%d SetString(%q) expected failure", i, test.in)
+			} else if x.RatString() != test.out {
+				t.Errorf("#%d SetString(%q) got %s want %s", i, test.in, x.RatString(), test.out)
+			}
+		} else if x != nil {
+			t.Errorf("#%d SetString(%q) got %p want nil", i, test.in, x)
+		}
+	}
+}
+
+func TestRatScan(t *testing.T) {
+	var buf bytes.Buffer
+	for i, test := range setStringTests {
+		x := new(Rat)
+		buf.Reset()
+		buf.WriteString(test.in)
+
+		_, err := fmt.Fscanf(&buf, "%v", x)
+		if err == nil != test.ok {
+			if test.ok {
+				t.Errorf("#%d error: %s", i, err)
+			} else {
+				t.Errorf("#%d expected error", i)
+			}
+			continue
+		}
+		if err == nil && x.RatString() != test.out {
+			t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+		}
+	}
+}
+
+var floatStringTests = []struct {
+	in   string
+	prec int
+	out  string
+}{
+	{"0", 0, "0"},
+	{"0", 4, "0.0000"},
+	{"1", 0, "1"},
+	{"1", 2, "1.00"},
+	{"-1", 0, "-1"},
+	{".25", 2, "0.25"},
+	{".25", 1, "0.3"},
+	{".25", 3, "0.250"},
+	{"-1/3", 3, "-0.333"},
+	{"-2/3", 4, "-0.6667"},
+	{"0.96", 1, "1.0"},
+	{"0.999", 2, "1.00"},
+	{"0.9", 0, "1"},
+	{".25", -1, "0"},
+	{".55", -1, "1"},
+}
+
+func TestFloatString(t *testing.T) {
+	for i, test := range floatStringTests {
+		x, _ := new(Rat).SetString(test.in)
+
+		if x.FloatString(test.prec) != test.out {
+			t.Errorf("#%d got %s want %s", i, x.FloatString(test.prec), test.out)
+		}
+	}
+}
+
+func TestRatSign(t *testing.T) {
+	zero := NewRat(0, 1)
+	for _, a := range setStringTests {
+		x, ok := new(Rat).SetString(a.in)
+		if !ok {
+			continue
+		}
+		s := x.Sign()
+		e := x.Cmp(zero)
+		if s != e {
+			t.Errorf("got %d; want %d for z = %v", s, e, &x)
+		}
+	}
+}
+
+var ratCmpTests = []struct {
+	rat1, rat2 string
+	out        int
+}{
+	{"0", "0/1", 0},
+	{"1/1", "1", 0},
+	{"-1", "-2/2", 0},
+	{"1", "0", 1},
+	{"0/1", "1/1", -1},
+	{"-5/1434770811533343057144", "-5/1434770811533343057145", -1},
+	{"49832350382626108453/8964749413", "49832350382626108454/8964749413", -1},
+	{"-37414950961700930/7204075375675961", "37414950961700930/7204075375675961", -1},
+	{"37414950961700930/7204075375675961", "74829901923401860/14408150751351922", 0},
+}
+
+func TestRatCmp(t *testing.T) {
+	for i, test := range ratCmpTests {
+		x, _ := new(Rat).SetString(test.rat1)
+		y, _ := new(Rat).SetString(test.rat2)
+
+		out := x.Cmp(y)
+		if out != test.out {
+			t.Errorf("#%d got out = %v; want %v", i, out, test.out)
+		}
+	}
+}
+
+func TestIsInt(t *testing.T) {
+	one := NewInt(1)
+	for _, a := range setStringTests {
+		x, ok := new(Rat).SetString(a.in)
+		if !ok {
+			continue
+		}
+		i := x.IsInt()
+		e := x.Denom().Cmp(one) == 0
+		if i != e {
+			t.Errorf("got IsInt(%v) == %v; want %v", x, i, e)
+		}
+	}
+}
+
+func TestRatAbs(t *testing.T) {
+	zero := new(Rat)
+	for _, a := range setStringTests {
+		x, ok := new(Rat).SetString(a.in)
+		if !ok {
+			continue
+		}
+		e := new(Rat).Set(x)
+		if e.Cmp(zero) < 0 {
+			e.Sub(zero, e)
+		}
+		z := new(Rat).Abs(x)
+		if z.Cmp(e) != 0 {
+			t.Errorf("got Abs(%v) = %v; want %v", x, z, e)
+		}
+	}
+}
+
+func TestRatNeg(t *testing.T) {
+	zero := new(Rat)
+	for _, a := range setStringTests {
+		x, ok := new(Rat).SetString(a.in)
+		if !ok {
+			continue
+		}
+		e := new(Rat).Sub(zero, x)
+		z := new(Rat).Neg(x)
+		if z.Cmp(e) != 0 {
+			t.Errorf("got Neg(%v) = %v; want %v", x, z, e)
+		}
+	}
+}
+
+func TestRatInv(t *testing.T) {
+	zero := new(Rat)
+	for _, a := range setStringTests {
+		x, ok := new(Rat).SetString(a.in)
+		if !ok {
+			continue
+		}
+		if x.Cmp(zero) == 0 {
+			continue // avoid division by zero
+		}
+		e := new(Rat).SetFrac(x.Denom(), x.Num())
+		z := new(Rat).Inv(x)
+		if z.Cmp(e) != 0 {
+			t.Errorf("got Inv(%v) = %v; want %v", x, z, e)
+		}
+	}
+}
+
+type ratBinFun func(z, x, y *Rat) *Rat
+type ratBinArg struct {
+	x, y, z string
+}
+
+func testRatBin(t *testing.T, i int, name string, f ratBinFun, a ratBinArg) {
+	x, _ := new(Rat).SetString(a.x)
+	y, _ := new(Rat).SetString(a.y)
+	z, _ := new(Rat).SetString(a.z)
+	out := f(new(Rat), x, y)
+
+	if out.Cmp(z) != 0 {
+		t.Errorf("%s #%d got %s want %s", name, i, out, z)
+	}
+}
+
+var ratBinTests = []struct {
+	x, y      string
+	sum, prod string
+}{
+	{"0", "0", "0", "0"},
+	{"0", "1", "1", "0"},
+	{"-1", "0", "-1", "0"},
+	{"-1", "1", "0", "-1"},
+	{"1", "1", "2", "1"},
+	{"1/2", "1/2", "1", "1/4"},
+	{"1/4", "1/3", "7/12", "1/12"},
+	{"2/5", "-14/3", "-64/15", "-28/15"},
+	{"4707/49292519774798173060", "-3367/70976135186689855734", "84058377121001851123459/1749296273614329067191168098769082663020", "-1760941/388732505247628681598037355282018369560"},
+	{"-61204110018146728334/3", "-31052192278051565633/2", "-215564796870448153567/6", "950260896245257153059642991192710872711/3"},
+	{"-854857841473707320655/4237645934602118692642972629634714039", "-18/31750379913563777419", "-27/133467566250814981", "15387441146526731771790/134546868362786310073779084329032722548987800600710485341"},
+	{"618575745270541348005638912139/19198433543745179392300736", "-19948846211000086/637313996471", "27674141753240653/30123979153216", "-6169936206128396568797607742807090270137721977/6117715203873571641674006593837351328"},
+	{"-3/26206484091896184128", "5/2848423294177090248", "15310893822118706237/9330894968229805033368778458685147968", "-5/24882386581946146755650075889827061248"},
+	{"26946729/330400702820", "41563965/225583428284", "1238218672302860271/4658307703098666660055", "224002580204097/14906584649915733312176"},
+	{"-8259900599013409474/7", "-84829337473700364773/56707961321161574960", "-468402123685491748914621885145127724451/396955729248131024720", "350340947706464153265156004876107029701/198477864624065512360"},
+	{"575775209696864/1320203974639986246357", "29/712593081308", "410331716733912717985762465/940768218243776489278275419794956", "808/45524274987585732633"},
+	{"1786597389946320496771/2066653520653241", "6269770/1992362624741777", "3559549865190272133656109052308126637/4117523232840525481453983149257", "8967230/3296219033"},
+	{"-36459180403360509753/32150500941194292113930", "9381566963714/9633539", "301622077145533298008420642898530153/309723104686531919656937098270", "-3784609207827/3426986245"},
+}
+
+func TestRatBin(t *testing.T) {
+	for i, test := range ratBinTests {
+		arg := ratBinArg{test.x, test.y, test.sum}
+		testRatBin(t, i, "Add", (*Rat).Add, arg)
+
+		arg = ratBinArg{test.y, test.x, test.sum}
+		testRatBin(t, i, "Add symmetric", (*Rat).Add, arg)
+
+		arg = ratBinArg{test.sum, test.x, test.y}
+		testRatBin(t, i, "Sub", (*Rat).Sub, arg)
+
+		arg = ratBinArg{test.sum, test.y, test.x}
+		testRatBin(t, i, "Sub symmetric", (*Rat).Sub, arg)
+
+		arg = ratBinArg{test.x, test.y, test.prod}
+		testRatBin(t, i, "Mul", (*Rat).Mul, arg)
+
+		arg = ratBinArg{test.y, test.x, test.prod}
+		testRatBin(t, i, "Mul symmetric", (*Rat).Mul, arg)
+
+		if test.x != "0" {
+			arg = ratBinArg{test.prod, test.x, test.y}
+			testRatBin(t, i, "Quo", (*Rat).Quo, arg)
+		}
+
+		if test.y != "0" {
+			arg = ratBinArg{test.prod, test.y, test.x}
+			testRatBin(t, i, "Quo symmetric", (*Rat).Quo, arg)
+		}
+	}
+}
+
+func TestIssue820(t *testing.T) {
+	x := NewRat(3, 1)
+	y := NewRat(2, 1)
+	z := y.Quo(x, y)
+	q := NewRat(3, 2)
+	if z.Cmp(q) != 0 {
+		t.Errorf("got %s want %s", z, q)
+	}
+
+	y = NewRat(3, 1)
+	x = NewRat(2, 1)
+	z = y.Quo(x, y)
+	q = NewRat(2, 3)
+	if z.Cmp(q) != 0 {
+		t.Errorf("got %s want %s", z, q)
+	}
+
+	x = NewRat(3, 1)
+	z = x.Quo(x, x)
+	q = NewRat(3, 3)
+	if z.Cmp(q) != 0 {
+		t.Errorf("got %s want %s", z, q)
+	}
+}
+
+var setFrac64Tests = []struct {
+	a, b int64
+	out  string
+}{
+	{0, 1, "0"},
+	{0, -1, "0"},
+	{1, 1, "1"},
+	{-1, 1, "-1"},
+	{1, -1, "-1"},
+	{-1, -1, "1"},
+	{-9223372036854775808, -9223372036854775808, "1"},
+}
+
+func TestRatSetFrac64Rat(t *testing.T) {
+	for i, test := range setFrac64Tests {
+		x := new(Rat).SetFrac64(test.a, test.b)
+		if x.RatString() != test.out {
+			t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+		}
+	}
+}
+
+func TestRatGobEncoding(t *testing.T) {
+	var medium bytes.Buffer
+	enc := gob.NewEncoder(&medium)
+	dec := gob.NewDecoder(&medium)
+	for _, test := range encodingTests {
+		medium.Reset() // empty buffer for each test case (in case of failures)
+		var tx Rat
+		tx.SetString(test + ".14159265")
+		if err := enc.Encode(&tx); err != nil {
+			t.Errorf("encoding of %s failed: %s", &tx, err)
+		}
+		var rx Rat
+		if err := dec.Decode(&rx); err != nil {
+			t.Errorf("decoding of %s failed: %s", &tx, err)
+		}
+		if rx.Cmp(&tx) != 0 {
+			t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+		}
+	}
+}
+
+// Sending a nil Rat pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilRatInSlice(t *testing.T) {
+	buf := new(bytes.Buffer)
+	enc := gob.NewEncoder(buf)
+	dec := gob.NewDecoder(buf)
+
+	var in = make([]*Rat, 1)
+	err := enc.Encode(&in)
+	if err != nil {
+		t.Errorf("gob encode failed: %q", err)
+	}
+	var out []*Rat
+	err = dec.Decode(&out)
+	if err != nil {
+		t.Fatalf("gob decode failed: %q", err)
+	}
+	if len(out) != 1 {
+		t.Fatalf("wrong len; want 1 got %d", len(out))
+	}
+	var zero Rat
+	if out[0].Cmp(&zero) != 0 {
+		t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+	}
+}
+
+var ratNums = []string{
+	"-141592653589793238462643383279502884197169399375105820974944592307816406286",
+	"-1415926535897932384626433832795028841971",
+	"-141592653589793",
+	"-1",
+	"0",
+	"1",
+	"141592653589793",
+	"1415926535897932384626433832795028841971",
+	"141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+var ratDenoms = []string{
+	"1",
+	"718281828459045",
+	"7182818284590452353602874713526624977572",
+	"718281828459045235360287471352662497757247093699959574966967627724076630353",
+}
+
+func TestRatJSONEncoding(t *testing.T) {
+	for _, num := range ratNums {
+		for _, denom := range ratDenoms {
+			var tx Rat
+			tx.SetString(num + "/" + denom)
+			b, err := json.Marshal(&tx)
+			if err != nil {
+				t.Errorf("marshaling of %s failed: %s", &tx, err)
+				continue
+			}
+			var rx Rat
+			if err := json.Unmarshal(b, &rx); err != nil {
+				t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+				continue
+			}
+			if rx.Cmp(&tx) != 0 {
+				t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+			}
+		}
+	}
+}
+
+func TestRatXMLEncoding(t *testing.T) {
+	for _, num := range ratNums {
+		for _, denom := range ratDenoms {
+			var tx Rat
+			tx.SetString(num + "/" + denom)
+			b, err := xml.Marshal(&tx)
+			if err != nil {
+				t.Errorf("marshaling of %s failed: %s", &tx, err)
+				continue
+			}
+			var rx Rat
+			if err := xml.Unmarshal(b, &rx); err != nil {
+				t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+				continue
+			}
+			if rx.Cmp(&tx) != 0 {
+				t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+			}
+		}
+	}
+}
+
+func TestIssue2379(t *testing.T) {
+	// 1) no aliasing
+	q := NewRat(3, 2)
+	x := new(Rat)
+	x.SetFrac(NewInt(3), NewInt(2))
+	if x.Cmp(q) != 0 {
+		t.Errorf("1) got %s want %s", x, q)
+	}
+
+	// 2) aliasing of numerator
+	x = NewRat(2, 3)
+	x.SetFrac(NewInt(3), x.Num())
+	if x.Cmp(q) != 0 {
+		t.Errorf("2) got %s want %s", x, q)
+	}
+
+	// 3) aliasing of denominator
+	x = NewRat(2, 3)
+	x.SetFrac(x.Denom(), NewInt(2))
+	if x.Cmp(q) != 0 {
+		t.Errorf("3) got %s want %s", x, q)
+	}
+
+	// 4) aliasing of numerator and denominator
+	x = NewRat(2, 3)
+	x.SetFrac(x.Denom(), x.Num())
+	if x.Cmp(q) != 0 {
+		t.Errorf("4) got %s want %s", x, q)
+	}
+
+	// 5) numerator and denominator are the same
+	q = NewRat(1, 1)
+	x = new(Rat)
+	n := NewInt(7)
+	x.SetFrac(n, n)
+	if x.Cmp(q) != 0 {
+		t.Errorf("5) got %s want %s", x, q)
+	}
+}
+
+func TestIssue3521(t *testing.T) {
+	a := new(Int)
+	b := new(Int)
+	a.SetString("64375784358435883458348587", 0)
+	b.SetString("4789759874531", 0)
+
+	// 0) a raw zero value has 1 as denominator
+	zero := new(Rat)
+	one := NewInt(1)
+	if zero.Denom().Cmp(one) != 0 {
+		t.Errorf("0) got %s want %s", zero.Denom(), one)
+	}
+
+	// 1a) a zero value remains zero independent of denominator
+	x := new(Rat)
+	x.Denom().Set(new(Int).Neg(b))
+	if x.Cmp(zero) != 0 {
+		t.Errorf("1a) got %s want %s", x, zero)
+	}
+
+	// 1b) a zero value may have a denominator != 0 and != 1
+	x.Num().Set(a)
+	qab := new(Rat).SetFrac(a, b)
+	if x.Cmp(qab) != 0 {
+		t.Errorf("1b) got %s want %s", x, qab)
+	}
+
+	// 2a) an integral value becomes a fraction depending on denominator
+	x.SetFrac64(10, 2)
+	x.Denom().SetInt64(3)
+	q53 := NewRat(5, 3)
+	if x.Cmp(q53) != 0 {
+		t.Errorf("2a) got %s want %s", x, q53)
+	}
+
+	// 2b) an integral value becomes a fraction depending on denominator
+	x = NewRat(10, 2)
+	x.Denom().SetInt64(3)
+	if x.Cmp(q53) != 0 {
+		t.Errorf("2b) got %s want %s", x, q53)
+	}
+
+	// 3) changing the numerator/denominator of a Rat changes the Rat
+	x.SetFrac(a, b)
+	a = x.Num()
+	b = x.Denom()
+	a.SetInt64(5)
+	b.SetInt64(3)
+	if x.Cmp(q53) != 0 {
+		t.Errorf("3) got %s want %s", x, q53)
+	}
+}
+
+// Test inputs to Rat.SetString.  The prefix "long:" causes the test
+// to be skipped in --test.short mode.  (The threshold is about 500us.)
+var float64inputs = []string{
+	// Constants plundered from strconv/testfp.txt.
+
+	// Table 1: Stress Inputs for Conversion to 53-bit Binary, < 1/2 ULP
+	"5e+125",
+	"69e+267",
+	"999e-026",
+	"7861e-034",
+	"75569e-254",
+	"928609e-261",
+	"9210917e+080",
+	"84863171e+114",
+	"653777767e+273",
+	"5232604057e-298",
+	"27235667517e-109",
+	"653532977297e-123",
+	"3142213164987e-294",
+	"46202199371337e-072",
+	"231010996856685e-073",
+	"9324754620109615e+212",
+	"78459735791271921e+049",
+	"272104041512242479e+200",
+	"6802601037806061975e+198",
+	"20505426358836677347e-221",
+	"836168422905420598437e-234",
+	"4891559871276714924261e+222",
+
+	// Table 2: Stress Inputs for Conversion to 53-bit Binary, > 1/2 ULP
+	"9e-265",
+	"85e-037",
+	"623e+100",
+	"3571e+263",
+	"81661e+153",
+	"920657e-023",
+	"4603285e-024",
+	"87575437e-309",
+	"245540327e+122",
+	"6138508175e+120",
+	"83356057653e+193",
+	"619534293513e+124",
+	"2335141086879e+218",
+	"36167929443327e-159",
+	"609610927149051e-255",
+	"3743626360493413e-165",
+	"94080055902682397e-242",
+	"899810892172646163e+283",
+	"7120190517612959703e+120",
+	"25188282901709339043e-252",
+	"308984926168550152811e-052",
+	"6372891218502368041059e+064",
+
+	// Table 14: Stress Inputs for Conversion to 24-bit Binary, <1/2 ULP
+	"5e-20",
+	"67e+14",
+	"985e+15",
+	"7693e-42",
+	"55895e-16",
+	"996622e-44",
+	"7038531e-32",
+	"60419369e-46",
+	"702990899e-20",
+	"6930161142e-48",
+	"25933168707e+13",
+	"596428896559e+20",
+
+	// Table 15: Stress Inputs for Conversion to 24-bit Binary, >1/2 ULP
+	"3e-23",
+	"57e+18",
+	"789e-35",
+	"2539e-18",
+	"76173e+28",
+	"887745e-11",
+	"5382571e-37",
+	"82381273e-35",
+	"750486563e-38",
+	"3752432815e-39",
+	"75224575729e-45",
+	"459926601011e+15",
+
+	// Constants plundered from strconv/atof_test.go.
+
+	"0",
+	"1",
+	"+1",
+	"1e23",
+	"1E23",
+	"100000000000000000000000",
+	"1e-100",
+	"123456700",
+	"99999999999999974834176",
+	"100000000000000000000001",
+	"100000000000000008388608",
+	"100000000000000016777215",
+	"100000000000000016777216",
+	"-1",
+	"-0.1",
+	"-0", // NB: exception made for this input
+	"1e-20",
+	"625e-3",
+
+	// largest float64
+	"1.7976931348623157e308",
+	"-1.7976931348623157e308",
+	// next float64 - too large
+	"1.7976931348623159e308",
+	"-1.7976931348623159e308",
+	// the border is ...158079
+	// borderline - okay
+	"1.7976931348623158e308",
+	"-1.7976931348623158e308",
+	// borderline - too large
+	"1.797693134862315808e308",
+	"-1.797693134862315808e308",
+
+	// a little too large
+	"1e308",
+	"2e308",
+	"1e309",
+
+	// way too large
+	"1e310",
+	"-1e310",
+	"1e400",
+	"-1e400",
+	"long:1e400000",
+	"long:-1e400000",
+
+	// denormalized
+	"1e-305",
+	"1e-306",
+	"1e-307",
+	"1e-308",
+	"1e-309",
+	"1e-310",
+	"1e-322",
+	// smallest denormal
+	"5e-324",
+	"4e-324",
+	"3e-324",
+	// too small
+	"2e-324",
+	// way too small
+	"1e-350",
+	"long:1e-400000",
+	// way too small, negative
+	"-1e-350",
+	"long:-1e-400000",
+
+	// try to overflow exponent
+	// [Disabled: too slow and memory-hungry with rationals.]
+	// "1e-4294967296",
+	// "1e+4294967296",
+	// "1e-18446744073709551616",
+	// "1e+18446744073709551616",
+
+	// http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+	"2.2250738585072012e-308",
+	// http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+	"2.2250738585072011e-308",
+
+	// A very large number (initially wrongly parsed by the fast algorithm).
+	"4.630813248087435e+307",
+
+	// A different kind of very large number.
+	"22.222222222222222",
+	"long:2." + strings.Repeat("2", 4000) + "e+1",
+
+	// Exactly halfway between 1 and math.Nextafter(1, 2).
+	// Round to even (down).
+	"1.00000000000000011102230246251565404236316680908203125",
+	// Slightly lower; still round down.
+	"1.00000000000000011102230246251565404236316680908203124",
+	// Slightly higher; round up.
+	"1.00000000000000011102230246251565404236316680908203126",
+	// Slightly higher, but you have to read all the way to the end.
+	"long:1.00000000000000011102230246251565404236316680908203125" + strings.Repeat("0", 10000) + "1",
+
+	// Smallest denormal, 2^(-1022-52)
+	"4.940656458412465441765687928682213723651e-324",
+	// Half of smallest denormal, 2^(-1022-53)
+	"2.470328229206232720882843964341106861825e-324",
+	// A little more than the exact half of smallest denormal
+	// 2^-1075 + 2^-1100.  (Rounds to 1p-1074.)
+	"2.470328302827751011111470718709768633275e-324",
+	// The exact halfway between smallest normal and largest denormal:
+	// 2^-1022 - 2^-1075.  (Rounds to 2^-1022.)
+	"2.225073858507201136057409796709131975935e-308",
+
+	"1152921504606846975",  //   1<<60 - 1
+	"-1152921504606846975", // -(1<<60 - 1)
+	"1152921504606846977",  //   1<<60 + 1
+	"-1152921504606846977", // -(1<<60 + 1)
+
+	"1/3",
+}
+
+// isFinite reports whether f represents a finite rational value.
+// It is equivalent to !math.IsNan(f) && !math.IsInf(f, 0).
+func isFinite(f float64) bool {
+	return math.Abs(f) <= math.MaxFloat64
+}
+
+func TestFloat32SpecialCases(t *testing.T) {
+	for _, input := range float64inputs {
+		if strings.HasPrefix(input, "long:") {
+			if testing.Short() {
+				continue
+			}
+			input = input[len("long:"):]
+		}
+
+		r, ok := new(Rat).SetString(input)
+		if !ok {
+			t.Errorf("Rat.SetString(%q) failed", input)
+			continue
+		}
+		f, exact := r.Float32()
+
+		// 1. Check string -> Rat -> float32 conversions are
+		// consistent with strconv.ParseFloat.
+		// Skip this check if the input uses "a/b" rational syntax.
+		if !strings.Contains(input, "/") {
+			e64, _ := strconv.ParseFloat(input, 32)
+			e := float32(e64)
+
+			// Careful: negative Rats too small for
+			// float64 become -0, but Rat obviously cannot
+			// preserve the sign from SetString("-0").
+			switch {
+			case math.Float32bits(e) == math.Float32bits(f):
+				// Ok: bitwise equal.
+			case f == 0 && r.Num().BitLen() == 0:
+				// Ok: Rat(0) is equivalent to both +/- float64(0).
+			default:
+				t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+			}
+		}
+
+		if !isFinite(float64(f)) {
+			continue
+		}
+
+		// 2. Check f is best approximation to r.
+		if !checkIsBestApprox32(t, f, r) {
+			// Append context information.
+			t.Errorf("(input was %q)", input)
+		}
+
+		// 3. Check f->R->f roundtrip is non-lossy.
+		checkNonLossyRoundtrip32(t, f)
+
+		// 4. Check exactness using slow algorithm.
+		if wasExact := new(Rat).SetFloat64(float64(f)).Cmp(r) == 0; wasExact != exact {
+			t.Errorf("Rat.SetString(%q).Float32().exact = %t, want %t", input, exact, wasExact)
+		}
+	}
+}
+
+func TestFloat64SpecialCases(t *testing.T) {
+	for _, input := range float64inputs {
+		if strings.HasPrefix(input, "long:") {
+			if testing.Short() {
+				continue
+			}
+			input = input[len("long:"):]
+		}
+
+		r, ok := new(Rat).SetString(input)
+		if !ok {
+			t.Errorf("Rat.SetString(%q) failed", input)
+			continue
+		}
+		f, exact := r.Float64()
+
+		// 1. Check string -> Rat -> float64 conversions are
+		// consistent with strconv.ParseFloat.
+		// Skip this check if the input uses "a/b" rational syntax.
+		if !strings.Contains(input, "/") {
+			e, _ := strconv.ParseFloat(input, 64)
+
+			// Careful: negative Rats too small for
+			// float64 become -0, but Rat obviously cannot
+			// preserve the sign from SetString("-0").
+			switch {
+			case math.Float64bits(e) == math.Float64bits(f):
+				// Ok: bitwise equal.
+			case f == 0 && r.Num().BitLen() == 0:
+				// Ok: Rat(0) is equivalent to both +/- float64(0).
+			default:
+				t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+			}
+		}
+
+		if !isFinite(f) {
+			continue
+		}
+
+		// 2. Check f is best approximation to r.
+		if !checkIsBestApprox64(t, f, r) {
+			// Append context information.
+			t.Errorf("(input was %q)", input)
+		}
+
+		// 3. Check f->R->f roundtrip is non-lossy.
+		checkNonLossyRoundtrip64(t, f)
+
+		// 4. Check exactness using slow algorithm.
+		if wasExact := new(Rat).SetFloat64(f).Cmp(r) == 0; wasExact != exact {
+			t.Errorf("Rat.SetString(%q).Float64().exact = %t, want %t", input, exact, wasExact)
+		}
+	}
+}
+
+func TestFloat32Distribution(t *testing.T) {
+	// Generate a distribution of (sign, mantissa, exp) values
+	// broader than the float32 range, and check Rat.Float32()
+	// always picks the closest float32 approximation.
+	var add = []int64{
+		0,
+		1,
+		3,
+		5,
+		7,
+		9,
+		11,
+	}
+	var winc, einc = uint64(1), 1 // soak test (~1.5s on x86-64)
+	if testing.Short() {
+		winc, einc = 5, 15 // quick test (~60ms on x86-64)
+	}
+
+	for _, sign := range "+-" {
+		for _, a := range add {
+			for wid := uint64(0); wid < 30; wid += winc {
+				b := 1<<wid + a
+				if sign == '-' {
+					b = -b
+				}
+				for exp := -150; exp < 150; exp += einc {
+					num, den := NewInt(b), NewInt(1)
+					if exp > 0 {
+						num.Lsh(num, uint(exp))
+					} else {
+						den.Lsh(den, uint(-exp))
+					}
+					r := new(Rat).SetFrac(num, den)
+					f, _ := r.Float32()
+
+					if !checkIsBestApprox32(t, f, r) {
+						// Append context information.
+						t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+							b, exp, f, f, math.Ldexp(float64(b), exp), r)
+					}
+
+					checkNonLossyRoundtrip32(t, f)
+				}
+			}
+		}
+	}
+}
+
+func TestFloat64Distribution(t *testing.T) {
+	// Generate a distribution of (sign, mantissa, exp) values
+	// broader than the float64 range, and check Rat.Float64()
+	// always picks the closest float64 approximation.
+	var add = []int64{
+		0,
+		1,
+		3,
+		5,
+		7,
+		9,
+		11,
+	}
+	var winc, einc = uint64(1), 1 // soak test (~75s on x86-64)
+	if testing.Short() {
+		winc, einc = 10, 500 // quick test (~12ms on x86-64)
+	}
+
+	for _, sign := range "+-" {
+		for _, a := range add {
+			for wid := uint64(0); wid < 60; wid += winc {
+				b := 1<<wid + a
+				if sign == '-' {
+					b = -b
+				}
+				for exp := -1100; exp < 1100; exp += einc {
+					num, den := NewInt(b), NewInt(1)
+					if exp > 0 {
+						num.Lsh(num, uint(exp))
+					} else {
+						den.Lsh(den, uint(-exp))
+					}
+					r := new(Rat).SetFrac(num, den)
+					f, _ := r.Float64()
+
+					if !checkIsBestApprox64(t, f, r) {
+						// Append context information.
+						t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+							b, exp, f, f, math.Ldexp(float64(b), exp), r)
+					}
+
+					checkNonLossyRoundtrip64(t, f)
+				}
+			}
+		}
+	}
+}
+
+// TestSetFloat64NonFinite checks that SetFloat64 of a non-finite value
+// returns nil.
+func TestSetFloat64NonFinite(t *testing.T) {
+	for _, f := range []float64{math.NaN(), math.Inf(+1), math.Inf(-1)} {
+		var r Rat
+		if r2 := r.SetFloat64(f); r2 != nil {
+			t.Errorf("SetFloat64(%g) was %v, want nil", f, r2)
+		}
+	}
+}
+
+// checkNonLossyRoundtrip32 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip32(t *testing.T, f float32) {
+	if !isFinite(float64(f)) {
+		return
+	}
+	r := new(Rat).SetFloat64(float64(f))
+	if r == nil {
+		t.Errorf("Rat.SetFloat64(float64(%g) (%b)) == nil", f, f)
+		return
+	}
+	f2, exact := r.Float32()
+	if f != f2 || !exact {
+		t.Errorf("Rat.SetFloat64(float64(%g)).Float32() = %g (%b), %v, want %g (%b), %v; delta = %b",
+			f, f2, f2, exact, f, f, true, f2-f)
+	}
+}
+
+// checkNonLossyRoundtrip64 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip64(t *testing.T, f float64) {
+	if !isFinite(f) {
+		return
+	}
+	r := new(Rat).SetFloat64(f)
+	if r == nil {
+		t.Errorf("Rat.SetFloat64(%g (%b)) == nil", f, f)
+		return
+	}
+	f2, exact := r.Float64()
+	if f != f2 || !exact {
+		t.Errorf("Rat.SetFloat64(%g).Float64() = %g (%b), %v, want %g (%b), %v; delta = %b",
+			f, f2, f2, exact, f, f, true, f2-f)
+	}
+}
+
+// delta returns the absolute difference between r and f.
+func delta(r *Rat, f float64) *Rat {
+	d := new(Rat).Sub(r, new(Rat).SetFloat64(f))
+	return d.Abs(d)
+}
+
+// checkIsBestApprox32 checks that f is the best possible float32
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox32(t *testing.T, f float32, r *Rat) bool {
+	if math.Abs(float64(f)) >= math.MaxFloat32 {
+		// Cannot check +Inf, -Inf, nor the float next to them (MaxFloat32).
+		// But we have tests for these special cases.
+		return true
+	}
+
+	// r must be strictly between f0 and f1, the floats bracketing f.
+	f0 := math.Nextafter32(f, float32(math.Inf(-1)))
+	f1 := math.Nextafter32(f, float32(math.Inf(+1)))
+
+	// For f to be correct, r must be closer to f than to f0 or f1.
+	df := delta(r, float64(f))
+	df0 := delta(r, float64(f0))
+	df1 := delta(r, float64(f1))
+	if df.Cmp(df0) > 0 {
+		t.Errorf("Rat(%v).Float32() = %g (%b), but previous float32 %g (%b) is closer", r, f, f, f0, f0)
+		return false
+	}
+	if df.Cmp(df1) > 0 {
+		t.Errorf("Rat(%v).Float32() = %g (%b), but next float32 %g (%b) is closer", r, f, f, f1, f1)
+		return false
+	}
+	if df.Cmp(df0) == 0 && !isEven32(f) {
+		t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+		return false
+	}
+	if df.Cmp(df1) == 0 && !isEven32(f) {
+		t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+		return false
+	}
+	return true
+}
+
+// checkIsBestApprox64 checks that f is the best possible float64
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox64(t *testing.T, f float64, r *Rat) bool {
+	if math.Abs(f) >= math.MaxFloat64 {
+		// Cannot check +Inf, -Inf, nor the float next to them (MaxFloat64).
+		// But we have tests for these special cases.
+		return true
+	}
+
+	// r must be strictly between f0 and f1, the floats bracketing f.
+	f0 := math.Nextafter(f, math.Inf(-1))
+	f1 := math.Nextafter(f, math.Inf(+1))
+
+	// For f to be correct, r must be closer to f than to f0 or f1.
+	df := delta(r, f)
+	df0 := delta(r, f0)
+	df1 := delta(r, f1)
+	if df.Cmp(df0) > 0 {
+		t.Errorf("Rat(%v).Float64() = %g (%b), but previous float64 %g (%b) is closer", r, f, f, f0, f0)
+		return false
+	}
+	if df.Cmp(df1) > 0 {
+		t.Errorf("Rat(%v).Float64() = %g (%b), but next float64 %g (%b) is closer", r, f, f, f1, f1)
+		return false
+	}
+	if df.Cmp(df0) == 0 && !isEven64(f) {
+		t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+		return false
+	}
+	if df.Cmp(df1) == 0 && !isEven64(f) {
+		t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+		return false
+	}
+	return true
+}
+
+func isEven32(f float32) bool { return math.Float32bits(f)&1 == 0 }
+func isEven64(f float64) bool { return math.Float64bits(f)&1 == 0 }
+
+func TestIsFinite(t *testing.T) {
+	finites := []float64{
+		1.0 / 3,
+		4891559871276714924261e+222,
+		math.MaxFloat64,
+		math.SmallestNonzeroFloat64,
+		-math.MaxFloat64,
+		-math.SmallestNonzeroFloat64,
+	}
+	for _, f := range finites {
+		if !isFinite(f) {
+			t.Errorf("!IsFinite(%g (%b))", f, f)
+		}
+	}
+	nonfinites := []float64{
+		math.NaN(),
+		math.Inf(-1),
+		math.Inf(+1),
+	}
+	for _, f := range nonfinites {
+		if isFinite(f) {
+			t.Errorf("IsFinite(%g, (%b))", f, f)
+		}
+	}
+}