cmd/internal/gc/big: vendored math/big for use by gc

This is vendored copy of the pure-Go version of math/big.
To update, run vendor.bash in place.

This will permit the use of the new big.Float functionality in
gc (which is not available in 1.4, the version used for bootstrapping).

Change-Id: I4dcdea875d54710005ca3fdea2e0e30422b1b46d
Reviewed-on: https://go-review.googlesource.com/7857Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>
Run-TryBot: Robert Griesemer <gri@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Russ Cox <rsc@golang.org>

src/cmd/internal/gc/big/accuracy_string.go

+// generated by stringer -type=Accuracy; DO NOT EDIT
+
+package big
+
+import "fmt"
+
+const _Accuracy_name = "ExactBelowAboveUndef"
+
+var _Accuracy_index = [...]uint8{0, 5, 10, 15, 20}
+
+func (i Accuracy) String() string {
+	if i < 0 || i+1 >= Accuracy(len(_Accuracy_index)) {
+		return fmt.Sprintf("Accuracy(%d)", i)
+	}
+	return _Accuracy_name[_Accuracy_index[i]:_Accuracy_index[i+1]]
+}

src/cmd/internal/gc/big/arith.go

+// 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_g(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
+}
+
+// Keep for performance debugging.
+// Using addWW_g is likely slower.
+const use_addWW_g = false
+
+// The resulting carry c is either 0 or 1.
+func addVV_g(z, x, y []Word) (c Word) {
+	if use_addWW_g {
+		for i := range z {
+			c, z[i] = addWW_g(x[i], y[i], c)
+		}
+		return
+	}
+
+	for i, xi := range x[:len(z)] {
+		yi := y[i]
+		zi := xi + yi + c
+		z[i] = zi
+		// see "Hacker's Delight", section 2-12 (overflow detection)
+		c = (xi&yi | (xi|yi)&^zi) >> (_W - 1)
+	}
+	return
+}
+
+// The resulting carry c is either 0 or 1.
+func subVV_g(z, x, y []Word) (c Word) {
+	if use_addWW_g {
+		for i := range z {
+			c, z[i] = subWW_g(x[i], y[i], c)
+		}
+		return
+	}
+
+	for i, xi := range x[:len(z)] {
+		yi := y[i]
+		zi := xi - yi - c
+		z[i] = zi
+		// see "Hacker's Delight", section 2-12 (overflow detection)
+		c = (yi&^xi | (yi|^xi)&zi) >> (_W - 1)
+	}
+	return
+}
+
+// Argument y must be either 0 or 1.
+// The resulting carry c is either 0 or 1.
+func addVW_g(z, x []Word, y Word) (c Word) {
+	if use_addWW_g {
+		c = y
+		for i := range z {
+			c, z[i] = addWW_g(x[i], c, 0)
+		}
+		return
+	}
+
+	c = y
+	for i, xi := range x[:len(z)] {
+		zi := xi + c
+		z[i] = zi
+		c = xi &^ zi >> (_W - 1)
+	}
+	return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+	if use_addWW_g {
+		c = y
+		for i := range z {
+			c, z[i] = subWW_g(x[i], c, 0)
+		}
+		return
+	}
+
+	c = y
+	for i, xi := range x[:len(z)] {
+		zi := xi - c
+		z[i] = zi
+		c = (zi &^ xi) >> (_W - 1)
+	}
+	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
+}
+
+// TODO(gri) Remove use of addWW_g here and then we can remove addWW_g and subWW_g.
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+	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
+}

src/cmd/internal/gc/big/arith_decl.go

+// Copyright 2015 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
+
+func mulWW(x, y Word) (z1, z0 Word) {
+	return mulWW_g(x, y)
+}
+
+func divWW(x1, x0, y Word) (q, r Word) {
+	return divWW_g(x1, x0, y)
+}
+
+func addVV(z, x, y []Word) (c Word) {
+	return addVV_g(z, x, y)
+}
+
+func subVV(z, x, y []Word) (c Word) {
+	return subVV_g(z, x, y)
+}
+
+func addVW(z, x []Word, y Word) (c Word) {
+	return addVW_g(z, x, y)
+}
+
+func subVW(z, x []Word, y Word) (c Word) {
+	return subVW_g(z, x, y)
+}
+
+func shlVU(z, x []Word, s uint) (c Word) {
+	return shlVU_g(z, x, s)
+}
+
+func shrVU(z, x []Word, s uint) (c Word) {
+	return shrVU_g(z, x, s)
+}
+
+func mulAddVWW(z, x []Word, y, r Word) (c Word) {
+	return mulAddVWW_g(z, x, y, r)
+}
+
+func addMulVVW(z, x []Word, y Word) (c Word) {
+	return addMulVVW_g(z, x, y)
+}
+
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
+	return divWVW_g(z, xn, x, y)
+}
+
+func bitLen(x Word) (n int) {
+	return bitLen_g(x)
+}

src/cmd/internal/gc/big/bits_test.go

+// Copyright 2015 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 the Bits type used for testing Float operations
+// via an independent (albeit slower) representations for floating-point
+// numbers.
+
+package big
+
+import (
+	"fmt"
+	"sort"
+	"testing"
+)
+
+// A Bits value b represents a finite floating-point number x of the form
+//
+//	x = 2**b[0] + 2**b[1] + ... 2**b[len(b)-1]
+//
+// The order of slice elements is not significant. Negative elements may be
+// used to form fractions. A Bits value is normalized if each b[i] occurs at
+// most once. For instance Bits{0, 0, 1} is not normalized but represents the
+// same floating-point number as Bits{2}, which is normalized. The zero (nil)
+// value of Bits is a ready to use Bits value and represents the value 0.
+type Bits []int
+
+func (x Bits) add(y Bits) Bits {
+	return append(x, y...)
+}
+
+func (x Bits) mul(y Bits) Bits {
+	var p Bits
+	for _, x := range x {
+		for _, y := range y {
+			p = append(p, x+y)
+		}
+	}
+	return p
+}
+
+func TestMulBits(t *testing.T) {
+	for _, test := range []struct {
+		x, y, want Bits
+	}{
+		{nil, nil, nil},
+		{Bits{}, Bits{}, nil},
+		{Bits{0}, Bits{0}, Bits{0}},
+		{Bits{0}, Bits{1}, Bits{1}},
+		{Bits{1}, Bits{1, 2, 3}, Bits{2, 3, 4}},
+		{Bits{-1}, Bits{1}, Bits{0}},
+		{Bits{-10, -1, 0, 1, 10}, Bits{1, 2, 3}, Bits{-9, -8, -7, 0, 1, 2, 1, 2, 3, 2, 3, 4, 11, 12, 13}},
+	} {
+		got := fmt.Sprintf("%v", test.x.mul(test.y))
+		want := fmt.Sprintf("%v", test.want)
+		if got != want {
+			t.Errorf("%v * %v = %s; want %s", test.x, test.y, got, want)
+		}
+
+	}
+}
+
+// norm returns the normalized bits for x: It removes multiple equal entries
+// by treating them as an addition (e.g., Bits{5, 5} => Bits{6}), and it sorts
+// the result list for reproducible results.
+func (x Bits) norm() Bits {
+	m := make(map[int]bool)
+	for _, b := range x {
+		for m[b] {
+			m[b] = false
+			b++
+		}
+		m[b] = true
+	}
+	var z Bits
+	for b, set := range m {
+		if set {
+			z = append(z, b)
+		}
+	}
+	sort.Ints([]int(z))
+	return z
+}
+
+func TestNormBits(t *testing.T) {
+	for _, test := range []struct {
+		x, want Bits
+	}{
+		{nil, nil},
+		{Bits{}, Bits{}},
+		{Bits{0}, Bits{0}},
+		{Bits{0, 0}, Bits{1}},
+		{Bits{3, 1, 1}, Bits{2, 3}},
+		{Bits{10, 9, 8, 7, 6, 6}, Bits{11}},
+	} {
+		got := fmt.Sprintf("%v", test.x.norm())
+		want := fmt.Sprintf("%v", test.want)
+		if got != want {
+			t.Errorf("normBits(%v) = %s; want %s", test.x, got, want)
+		}
+
+	}
+}
+
+// round returns the Float value corresponding to x after rounding x
+// to prec bits according to mode.
+func (x Bits) round(prec uint, mode RoundingMode) *Float {
+	x = x.norm()
+
+	// determine range
+	var min, max int
+	for i, b := range x {
+		if i == 0 || b < min {
+			min = b
+		}
+		if i == 0 || b > max {
+			max = b
+		}
+	}
+	prec0 := uint(max + 1 - min)
+	if prec >= prec0 {
+		return x.Float()
+	}
+	// prec < prec0
+
+	// determine bit 0, rounding, and sticky bit, and result bits z
+	var bit0, rbit, sbit uint
+	var z Bits
+	r := max - int(prec)
+	for _, b := range x {
+		switch {
+		case b == r:
+			rbit = 1
+		case b < r:
+			sbit = 1
+		default:
+			// b > r
+			if b == r+1 {
+				bit0 = 1
+			}
+			z = append(z, b)
+		}
+	}
+
+	// round
+	f := z.Float() // rounded to zero
+	if mode == ToNearestAway {
+		panic("not yet implemented")
+	}
+	if mode == ToNearestEven && rbit == 1 && (sbit == 1 || sbit == 0 && bit0 != 0) || mode == AwayFromZero {
+		// round away from zero
+		f.SetMode(ToZero).SetPrec(prec)
+		f.Add(f, Bits{int(r) + 1}.Float())
+	}
+	return f
+}
+
+// Float returns the *Float z of the smallest possible precision such that
+// z = sum(2**bits[i]), with i = range bits. If multiple bits[i] are equal,
+// they are added: Bits{0, 1, 0}.Float() == 2**0 + 2**1 + 2**0 = 4.
+func (bits Bits) Float() *Float {
+	// handle 0
+	if len(bits) == 0 {
+		return new(Float)
+	}
+	// len(bits) > 0
+
+	// determine lsb exponent
+	var min int
+	for i, b := range bits {
+		if i == 0 || b < min {
+			min = b
+		}
+	}
+
+	// create bit pattern
+	x := NewInt(0)
+	for _, b := range bits {
+		badj := b - min
+		// propagate carry if necessary
+		for x.Bit(badj) != 0 {
+			x.SetBit(x, badj, 0)
+			badj++
+		}
+		x.SetBit(x, badj, 1)
+	}
+
+	// create corresponding float
+	z := new(Float).SetInt(x) // normalized
+	if e := int64(z.exp) + int64(min); MinExp <= e && e <= MaxExp {
+		z.exp = int32(e)
+	} else {
+		// this should never happen for our test cases
+		panic("exponent out of range")
+	}
+	return z
+}
+
+func TestFromBits(t *testing.T) {
+	for _, test := range []struct {
+		bits Bits
+		want string
+	}{
+		// all different bit numbers
+		{nil, "0"},
+		{Bits{0}, "0x.8p1"},
+		{Bits{1}, "0x.8p2"},
+		{Bits{-1}, "0x.8p0"},
+		{Bits{63}, "0x.8p64"},
+		{Bits{33, -30}, "0x.8000000000000001p34"},
+		{Bits{255, 0}, "0x.8000000000000000000000000000000000000000000000000000000000000001p256"},
+
+		// multiple equal bit numbers
+		{Bits{0, 0}, "0x.8p2"},
+		{Bits{0, 0, 0, 0}, "0x.8p3"},
+		{Bits{0, 1, 0}, "0x.8p3"},
+		{append(Bits{2, 1, 0} /* 7 */, Bits{3, 1} /* 10 */ ...), "0x.88p5" /* 17 */},
+	} {
+		f := test.bits.Float()
+		if got := f.Format('p', 0); got != test.want {
+			t.Errorf("setBits(%v) = %s; want %s", test.bits, got, test.want)
+		}
+	}
+}

src/cmd/internal/gc/big/calibrate_test.go

+// 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()
+	}
+}

src/cmd/internal/gc/big/decimal.go

+// Copyright 2015 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 decimal numbers.
+// The implementation is for float to decimal conversion only;
+// not general purpose use.
+// The only operations are precise conversion from binary to
+// decimal and rounding.
+//
+// The key observation and some code (shr) is borrowed from
+// strconv/decimal.go: conversion of binary fractional values can be done
+// precisely in multi-precision decimal because 2 divides 10 (required for
+// >> of mantissa); but conversion of decimal floating-point values cannot
+// be done precisely in binary representation.
+//
+// In contrast to strconv/decimal.go, only right shift is implemented in
+// decimal format - left shift can be done precisely in binary format.
+
+package big
+
+// A decimal represents a floating-point number in decimal representation.
+// The value of a decimal x is x.mant * 10 ** x.exp with 0.5 <= x.mant < 1,
+// with the most-significant mantissa digit at index 0.
+type decimal struct {
+	mant []byte // mantissa ASCII digits, big-endian
+	exp  int    // exponent, valid if len(mant) > 0
+}
+
+// Maximum shift amount that can be done in one pass without overflow.
+// A Word has _W bits and (1<<maxShift - 1)*10 + 9 must fit into Word.
+const maxShift = _W - 4
+
+// TODO(gri) Since we know the desired decimal precision when converting
+// a floating-point number, we may be able to limit the number of decimal
+// digits that need to be computed by init by providing an additional
+// precision argument and keeping track of when a number was truncated early
+// (equivalent of "sticky bit" in binary rounding).
+
+// TODO(gri) Along the same lines, enforce some limit to shift magnitudes
+// to avoid "infinitely" long running conversions (until we run out of space).
+
+// Init initializes x to the decimal representation of m << shift (for
+// shift >= 0), or m >> -shift (for shift < 0).
+func (x *decimal) init(m nat, shift int) {
+	// special case 0
+	if len(m) == 0 {
+		x.mant = x.mant[:0]
+		return
+	}
+
+	// Optimization: If we need to shift right, first remove any trailing
+	// zero bits from m to reduce shift amount that needs to be done in
+	// decimal format (since that is likely slower).
+	if shift < 0 {
+		ntz := m.trailingZeroBits()
+		s := uint(-shift)
+		if s >= ntz {
+			s = ntz // shift at most ntz bits
+		}
+		m = nat(nil).shr(m, s)
+		shift += int(s)
+	}
+
+	// Do any shift left in binary representation.
+	if shift > 0 {
+		m = nat(nil).shl(m, uint(shift))
+		shift = 0
+	}
+
+	// Convert mantissa into decimal representation.
+	s := m.decimalString() // TODO(gri) avoid string conversion here
+	n := len(s)
+	x.exp = n
+	// Trim trailing zeros; instead the exponent is tracking
+	// the decimal point independent of the number of digits.
+	for n > 0 && s[n-1] == '0' {
+		n--
+	}
+	x.mant = append(x.mant[:0], s[:n]...)
+
+	// Do any (remaining) shift right in decimal representation.
+	if shift < 0 {
+		for shift < -maxShift {
+			shr(x, maxShift)
+			shift += maxShift
+		}
+		shr(x, uint(-shift))
+	}
+}
+
+// Possibly optimization: The current implementation of nat.string takes
+// a charset argument. When a right shift is needed, we could provide
+// "\x00\x01...\x09" instead of "012..9" (as in nat.decimalString) and
+// avoid the repeated +'0' and -'0' operations in decimal.shr (and do a
+// single +'0' pass at the end).
+
+// shr implements x >> s, for s <= maxShift.
+func shr(x *decimal, s uint) {
+	// Division by 1<<s using shift-and-subtract algorithm.
+
+	// pick up enough leading digits to cover first shift
+	r := 0 // read index
+	var n Word
+	for n>>s == 0 && r < len(x.mant) {
+		ch := Word(x.mant[r])
+		r++
+		n = n*10 + ch - '0'
+	}
+	if n == 0 {
+		// x == 0; shouldn't get here, but handle anyway
+		x.mant = x.mant[:0]
+		return
+	}
+	for n>>s == 0 {
+		r++
+		n *= 10
+	}
+	x.exp += 1 - r
+
+	// read a digit, write a digit
+	w := 0 // write index
+	for r < len(x.mant) {
+		ch := Word(x.mant[r])
+		r++
+		d := n >> s
+		n -= d << s
+		x.mant[w] = byte(d + '0')
+		w++
+		n = n*10 + ch - '0'
+	}
+
+	// write extra digits that still fit
+	for n > 0 && w < len(x.mant) {
+		d := n >> s
+		n -= d << s
+		x.mant[w] = byte(d + '0')
+		w++
+		n = n * 10
+	}
+	x.mant = x.mant[:w] // the number may be shorter (e.g. 1024 >> 10)
+
+	// append additional digits that didn't fit
+	for n > 0 {
+		d := n >> s
+		n -= d << s
+		x.mant = append(x.mant, byte(d+'0'))
+		n = n * 10
+	}
+
+	trim(x)
+}
+
+func (x *decimal) String() string {
+	if len(x.mant) == 0 {
+		return "0"
+	}
+
+	var buf []byte
+	switch {
+	case x.exp <= 0:
+		// 0.00ddd
+		buf = append(buf, "0."...)
+		buf = appendZeros(buf, -x.exp)
+		buf = append(buf, x.mant...)
+
+	case /* 0 < */ x.exp < len(x.mant):
+		// dd.ddd
+		buf = append(buf, x.mant[:x.exp]...)
+		buf = append(buf, '.')
+		buf = append(buf, x.mant[x.exp:]...)
+
+	default: // len(x.mant) <= x.exp
+		// ddd00
+		buf = append(buf, x.mant...)
+		buf = appendZeros(buf, x.exp-len(x.mant))
+	}
+
+	return string(buf)
+}
+
+// appendZeros appends n 0 digits to buf and returns buf.
+func appendZeros(buf []byte, n int) []byte {
+	for ; n > 0; n-- {
+		buf = append(buf, '0')
+	}
+	return buf
+}
+
+// shouldRoundUp reports if x should be rounded up
+// if shortened to n digits. n must be a valid index
+// for x.mant.
+func shouldRoundUp(x *decimal, n int) bool {
+	if x.mant[n] == '5' && n+1 == len(x.mant) {
+		// exactly halfway - round to even
+		return n > 0 && (x.mant[n-1]-'0')&1 != 0
+	}
+	// not halfway - digit tells all (x.mant has no trailing zeros)
+	return x.mant[n] >= '5'
+}
+
+// round sets x to (at most) n mantissa digits by rounding it
+// to the nearest even value with n (or fever) mantissa digits.
+// If n < 0, x remains unchanged.
+func (x *decimal) round(n int) {
+	if n < 0 || n >= len(x.mant) {
+		return // nothing to do
+	}
+
+	if shouldRoundUp(x, n) {
+		x.roundUp(n)
+	} else {
+		x.roundDown(n)
+	}
+}
+
+func (x *decimal) roundUp(n int) {
+	if n < 0 || n >= len(x.mant) {
+		return // nothing to do
+	}
+	// 0 <= n < len(x.mant)
+
+	// find first digit < '9'
+	for n > 0 && x.mant[n-1] >= '9' {
+		n--
+	}
+
+	if n == 0 {
+		// all digits are '9's => round up to '1' and update exponent
+		x.mant[0] = '1' // ok since len(x.mant) > n
+		x.mant = x.mant[:1]
+		x.exp++
+		return
+	}
+
+	// n > 0 && x.mant[n-1] < '9'
+	x.mant[n-1]++
+	x.mant = x.mant[:n]
+	// x already trimmed
+}
+
+func (x *decimal) roundDown(n int) {
+	if n < 0 || n >= len(x.mant) {
+		return // nothing to do
+	}
+	x.mant = x.mant[:n]
+	trim(x)
+}
+
+// trim cuts off any trailing zeros from x's mantissa;
+// they are meaningless for the value of x.
+func trim(x *decimal) {
+	i := len(x.mant)
+	for i > 0 && x.mant[i-1] == '0' {
+		i--
+	}
+	x.mant = x.mant[:i]
+}

src/cmd/internal/gc/big/decimal_test.go

+// Copyright 2015 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 "testing"
+
+func TestDecimalString(t *testing.T) {
+	for _, test := range []struct {
+		x    decimal
+		want string
+	}{
+		{want: "0"},
+		{decimal{nil, 1000}, "0"}, // exponent of 0 is ignored
+		{decimal{[]byte("12345"), 0}, "0.12345"},
+		{decimal{[]byte("12345"), -3}, "0.00012345"},
+		{decimal{[]byte("12345"), +3}, "123.45"},
+		{decimal{[]byte("12345"), +10}, "1234500000"},
+	} {
+		if got := test.x.String(); got != test.want {
+			t.Errorf("%v == %s; want %s", test.x, got, test.want)
+		}
+	}
+}
+
+func TestDecimalInit(t *testing.T) {
+	for _, test := range []struct {
+		x     Word
+		shift int
+		want  string
+	}{
+		{0, 0, "0"},
+		{0, -100, "0"},
+		{0, 100, "0"},
+		{1, 0, "1"},
+		{1, 10, "1024"},
+		{1, 100, "1267650600228229401496703205376"},
+		{1, -100, "0.0000000000000000000000000000007888609052210118054117285652827862296732064351090230047702789306640625"},
+		{12345678, 8, "3160493568"},
+		{12345678, -8, "48225.3046875"},
+		{195312, 9, "99999744"},
+		{1953125, 9, "1000000000"},
+	} {
+		var d decimal
+		d.init(nat{test.x}.norm(), test.shift)
+		if got := d.String(); got != test.want {
+			t.Errorf("%d << %d == %s; want %s", test.x, test.shift, got, test.want)
+		}
+	}
+}
+
+func TestDecimalRounding(t *testing.T) {
+	for _, test := range []struct {
+		x              uint64
+		n              int
+		down, even, up string
+	}{
+		{0, 0, "0", "0", "0"},
+		{0, 1, "0", "0", "0"},
+
+		{1, 0, "0", "0", "10"},
+		{5, 0, "0", "0", "10"},
+		{9, 0, "0", "10", "10"},
+
+		{15, 1, "10", "20", "20"},
+		{45, 1, "40", "40", "50"},
+		{95, 1, "90", "100", "100"},
+
+		{12344999, 4, "12340000", "12340000", "12350000"},
+		{12345000, 4, "12340000", "12340000", "12350000"},
+		{12345001, 4, "12340000", "12350000", "12350000"},
+		{23454999, 4, "23450000", "23450000", "23460000"},
+		{23455000, 4, "23450000", "23460000", "23460000"},
+		{23455001, 4, "23450000", "23460000", "23460000"},
+
+		{99994999, 4, "99990000", "99990000", "100000000"},
+		{99995000, 4, "99990000", "100000000", "100000000"},
+		{99999999, 4, "99990000", "100000000", "100000000"},
+
+		{12994999, 4, "12990000", "12990000", "13000000"},
+		{12995000, 4, "12990000", "13000000", "13000000"},
+		{12999999, 4, "12990000", "13000000", "13000000"},
+	} {
+		x := nat(nil).setUint64(test.x)
+
+		var d decimal
+		d.init(x, 0)
+		d.roundDown(test.n)
+		if got := d.String(); got != test.down {
+			t.Errorf("roundDown(%d, %d) = %s; want %s", test.x, test.n, got, test.down)
+		}
+
+		d.init(x, 0)
+		d.round(test.n)
+		if got := d.String(); got != test.even {
+			t.Errorf("round(%d, %d) = %s; want %s", test.x, test.n, got, test.even)
+		}
+
+		d.init(x, 0)
+		d.roundUp(test.n)
+		if got := d.String(); got != test.up {
+			t.Errorf("roundUp(%d, %d) = %s; want %s", test.x, test.n, got, test.up)
+		}
+	}
+}

src/cmd/internal/gc/big/example_test.go

+// 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
+}

src/cmd/internal/gc/big/gcd_test.go

+// 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) }

src/cmd/internal/gc/big/roundingmode_string.go

+// generated by stringer -type=RoundingMode; DO NOT EDIT
+
+package big
+
+import "fmt"
+
+const _RoundingMode_name = "ToNearestEvenToNearestAwayToZeroAwayFromZeroToNegativeInfToPositiveInf"
+
+var _RoundingMode_index = [...]uint8{0, 13, 26, 32, 44, 57, 70}
+
+func (i RoundingMode) String() string {
+	if i+1 >= RoundingMode(len(_RoundingMode_index)) {
+		return fmt.Sprintf("RoundingMode(%d)", i)
+	}
+	return _RoundingMode_name[_RoundingMode_index[i]:_RoundingMode_index[i+1]]
+}