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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/7857
Reviewed-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>
This commit is contained in:
Robert Griesemer 2015-03-20 13:35:02 -07:00
parent a1bb3030c8
commit e5a0d6399e
31 changed files with 13606 additions and 0 deletions
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16
src/cmd/internal/gc/big/accuracy_string.go Normal file
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// 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]]
}

291
src/cmd/internal/gc/big/arith.go Normal file
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// 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
}

55
src/cmd/internal/gc/big/arith_decl.go Normal file
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// 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)
}

456
src/cmd/internal/gc/big/arith_test.go Normal file
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// 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 * _S))
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) }

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// 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)
}
}
}

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// 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()
}
}

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// 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]
}

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// 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)
}
}
}

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// 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
}

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// 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 float-to-string conversion functions.
package big
import (
"fmt"
"io"
"strconv"
"strings"
)
// SetString sets z to the value of s and returns z and a boolean indicating
// success. s must be a floating-point number of the same format as accepted
// by Scan, with number prefixes permitted.
func (z *Float) SetString(s string) (*Float, bool) {
r := strings.NewReader(s)
f, _, err := z.Scan(r, 0)
if err != nil {
return nil, false
}
// there should be no unread characters left
if _, err = r.ReadByte(); err != io.EOF {
return nil, false
}
return f, true
}
// Scan scans the number corresponding to the longest possible prefix
// of r representing a floating-point number with a mantissa in the
// given conversion base (the exponent is always a decimal number).
// It sets z to the (possibly rounded) value of the corresponding
// floating-point number, and returns z, the actual base b, and an
// error err, if any. If z's precision is 0, it is changed to 64
// before rounding takes effect. The number must be of the form:
//
// number = [ sign ] [ prefix ] mantissa [ exponent ] .
// sign = "+" | "-" .
// prefix = "0" ( "x" | "X" | "b" | "B" ) .
// mantissa = digits | digits "." [ digits ] | "." digits .
// exponent = ( "E" | "e" | "p" ) [ sign ] digits .
// digits = digit { digit } .
// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
//
// The base argument must be 0, 2, 10, or 16. Providing an invalid base
// argument will lead to a run-time panic.
//
// For base 0, the number prefix determines the actual base: A prefix of
// "0x" or "0X" selects base 16, and a "0b" or "0B" prefix selects
// base 2; otherwise, the actual base is 10 and no prefix is accepted.
// The octal prefix "0" is not supported (a leading "0" is simply
// considered a "0").
//
// A "p" exponent indicates a binary (rather then decimal) exponent;
// for instance "0x1.fffffffffffffp1023" (using base 0) represents the
// maximum float64 value. For hexadecimal mantissae, the exponent must
// be binary, if present (an "e" or "E" exponent indicator cannot be
// distinguished from a mantissa digit).
//
// The returned *Float f is nil and the value of z is valid but not
// defined if an error is reported.
//
// BUG(gri) The Float.Scan signature conflicts with Scan(s fmt.ScanState, ch rune) error.
func (z *Float) Scan(r io.ByteScanner, base int) (f *Float, b int, err error) {
prec := z.prec
if prec == 0 {
prec = 64
}
// NaNs ignore sign, mantissa, and exponent so we can set
// them below while having a valid value for z in case of
// errors.
z.SetNaN()
// sign
z.neg, err = scanSign(r)
if err != nil {
return
}
// mantissa
var fcount int // fractional digit count; valid if <= 0
z.mant, b, fcount, err = z.mant.scan(r, base, true)
if err != nil {
return
}
// exponent
var exp int64
var ebase int
exp, ebase, err = scanExponent(r, true)
if err != nil {
return
}
// special-case 0
if len(z.mant) == 0 {
z.prec = prec
z.acc = Exact
z.form = zero
f = z
return
}
// len(z.mant) > 0
// The mantissa may have a decimal point (fcount <= 0) and there
// may be a nonzero exponent exp. The decimal point amounts to a
// division by b**(-fcount). An exponent means multiplication by
// ebase**exp. Finally, mantissa normalization (shift left) requires
// a correcting multiplication by 2**(-shiftcount). Multiplications
// are commutative, so we can apply them in any order as long as there
// is no loss of precision. We only have powers of 2 and 10; keep
// track via separate exponents exp2 and exp10.
// normalize mantissa and get initial binary exponent
var exp2 = int64(len(z.mant))*_W - fnorm(z.mant)
// determine binary or decimal exponent contribution of decimal point
var exp10 int64
if fcount < 0 {
// The mantissa has a "decimal" point ddd.dddd; and
// -fcount is the number of digits to the right of '.'.
// Adjust relevant exponent accodingly.
switch b {
case 16:
fcount *= 4 // hexadecimal digits are 4 bits each
fallthrough
case 2:
exp2 += int64(fcount)
default: // b == 10
exp10 = int64(fcount)
}
// we don't need fcount anymore
}
// take actual exponent into account
if ebase == 2 {
exp2 += exp
} else { // ebase == 10
exp10 += exp
}
// we don't need exp anymore
// apply 2**exp2
if MinExp <= exp2 && exp2 <= MaxExp {
z.prec = prec
z.form = finite
z.exp = int32(exp2)
f = z
} else {
err = fmt.Errorf("exponent overflow")
return
}
if exp10 == 0 {
// no decimal exponent to consider
z.round(0)
return
}
// exp10 != 0
// compute decimal exponent power
expabs := exp10
if expabs < 0 {
expabs = -expabs
}
powTen := nat(nil).expNN(natTen, nat(nil).setUint64(uint64(expabs)), nil)
fpowTen := new(Float).SetInt(new(Int).SetBits(powTen))
// apply 10**exp10
if exp10 < 0 {
z.uquo(z, fpowTen)
} else {
z.umul(z, fpowTen)
}
return
}
// Parse is like z.Scan(r, base), but instead of reading from an
// io.ByteScanner, it parses the string s. An error is also returned
// if the string contains invalid or trailing bytes not belonging to
// the number.
func (z *Float) Parse(s string, base int) (f *Float, b int, err error) {
r := strings.NewReader(s)
if f, b, err = z.Scan(r, base); err != nil {
return
}
// entire string must have been consumed
if ch, err2 := r.ReadByte(); err2 == nil {
err = fmt.Errorf("expected end of string, found %q", ch)
} else if err2 != io.EOF {
err = err2
}
return
}
// ScanFloat is like f.Scan(r, base) with f set to the given precision
// and rounding mode.
func ScanFloat(r io.ByteScanner, base int, prec uint, mode RoundingMode) (f *Float, b int, err error) {
return new(Float).SetPrec(prec).SetMode(mode).Scan(r, base)
}
// ParseFloat is like f.Parse(s, base) with f set to the given precision
// and rounding mode.
func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error) {
return new(Float).SetPrec(prec).SetMode(mode).Parse(s, base)
}
// Format converts the floating-point number x to a string according
// to the given format and precision prec. The format is one of:
//
// 'e' -d.dddde±dd, decimal exponent, at least two (possibly 0) exponent digits
// 'E' -d.ddddE±dd, decimal exponent, at least two (possibly 0) exponent digits
// 'f' -ddddd.dddd, no exponent
// 'g' like 'e' for large exponents, like 'f' otherwise
// 'G' like 'E' for large exponents, like 'f' otherwise
// 'b' -ddddddp±dd, binary exponent
// 'p' -0x.dddp±dd, binary exponent, hexadecimal mantissa
//
// For the binary exponent formats, the mantissa is printed in normalized form:
//
// 'b' decimal integer mantissa using x.Prec() bits, or -0
// 'p' hexadecimal fraction with 0.5 <= 0.mantissa < 1.0, or -0
//
// The precision prec controls the number of digits (excluding the exponent)
// printed by the 'e', 'E', 'f', 'g', and 'G' formats. For 'e', 'E', and 'f'
// it is the number of digits after the decimal point. For 'g' and 'G' it is
// the total number of digits. A negative precision selects the smallest
// number of digits necessary such that ParseFloat will return f exactly.
// The prec value is ignored for the 'b' or 'p' format.
//
// BUG(gri) Float.Format does not accept negative precisions.
func (x *Float) Format(format byte, prec int) string {
const extra = 10 // TODO(gri) determine a good/better value here
return string(x.Append(make([]byte, 0, prec+extra), format, prec))
}
// Append appends the string form of the floating-point number x,
// as generated by x.Format, to buf and returns the extended buffer.
func (x *Float) Append(buf []byte, format byte, prec int) []byte {
// TODO(gri) factor out handling of sign?
// Inf
if x.IsInf() {
var ch byte = '+'
if x.neg {
ch = '-'
}
buf = append(buf, ch)
return append(buf, "Inf"...)
}
// NaN
if x.IsNaN() {
return append(buf, "NaN"...)
}
// easy formats
switch format {
case 'b':
return x.bstring(buf)
case 'p':
return x.pstring(buf)
}
return x.bigFtoa(buf, format, prec)
}
// BUG(gri): Float.String uses x.Format('g', 10) rather than x.Format('g', -1).
func (x *Float) String() string {
return x.Format('g', 10)
}
// bstring appends the string of x in the format ["-"] mantissa "p" exponent
// with a decimal mantissa and a binary exponent, or ["-"] "0" if x is zero,
// and returns the extended buffer.
// The mantissa is normalized such that is uses x.Prec() bits in binary
// representation.
func (x *Float) bstring(buf []byte) []byte {
if x.neg {
buf = append(buf, '-')
}
if x.form == zero {
return append(buf, '0')
}
if debugFloat && x.form != finite {
panic("non-finite float")
}
// x != 0
// adjust mantissa to use exactly x.prec bits
m := x.mant
switch w := uint32(len(x.mant)) * _W; {
case w < x.prec:
m = nat(nil).shl(m, uint(x.prec-w))
case w > x.prec:
m = nat(nil).shr(m, uint(w-x.prec))
}
buf = append(buf, m.decimalString()...)
buf = append(buf, 'p')
e := int64(x.exp) - int64(x.prec)
if e >= 0 {
buf = append(buf, '+')
}
return strconv.AppendInt(buf, e, 10)
}
// pstring appends the string of x in the format ["-"] "0x." mantissa "p" exponent
// with a hexadecimal mantissa and a binary exponent, or ["-"] "0" if x is zero,
// ad returns the extended buffer.
// The mantissa is normalized such that 0.5 <= 0.mantissa < 1.0.
func (x *Float) pstring(buf []byte) []byte {
if x.neg {
buf = append(buf, '-')
}
if x.form == zero {
return append(buf, '0')
}
if debugFloat && x.form != finite {
panic("non-finite float")
}
// x != 0
// remove trailing 0 words early
// (no need to convert to hex 0's and trim later)
m := x.mant
i := 0
for i < len(m) && m[i] == 0 {
i++
}
m = m[i:]
buf = append(buf, "0x."...)
buf = append(buf, strings.TrimRight(x.mant.hexString(), "0")...)
buf = append(buf, 'p')
return strconv.AppendInt(buf, int64(x.exp), 10)
}

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// 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 (
"math"
"strconv"
"testing"
)
func TestFloatSetFloat64String(t *testing.T) {
for _, test := range []struct {
s string
x float64
}{
// basics
{"0", 0},
{"-0", -0},
{"+0", 0},
{"1", 1},
{"-1", -1},
{"+1", 1},
{"1.234", 1.234},
{"-1.234", -1.234},
{"+1.234", 1.234},
{".1", 0.1},
{"1.", 1},
{"+1.", 1},
// various zeros
{"0e100", 0},
{"-0e+100", 0},
{"+0e-100", 0},
{"0E100", 0},
{"-0E+100", 0},
{"+0E-100", 0},
// various decimal exponent formats
{"1.e10", 1e10},
{"1e+10", 1e10},
{"+1e-10", 1e-10},
{"1E10", 1e10},
{"1.E+10", 1e10},
{"+1E-10", 1e-10},
// misc decimal values
{"3.14159265", 3.14159265},
{"-687436.79457e-245", -687436.79457e-245},
{"-687436.79457E245", -687436.79457e245},
{".0000000000000000000000000000000000000001", 1e-40},
{"+10000000000000000000000000000000000000000e-0", 1e40},
// decimal mantissa, binary exponent
{"0p0", 0},
{"-0p0", -0},
{"1p10", 1 << 10},
{"1p+10", 1 << 10},
{"+1p-10", 1.0 / (1 << 10)},
{"1024p-12", 0.25},
{"-1p10", -1024},
{"1.5p1", 3},
// binary mantissa, decimal exponent
{"0b0", 0},
{"-0b0", -0},
{"0b0e+10", 0},
{"-0b0e-10", -0},
{"0b1010", 10},
{"0B1010E2", 1000},
{"0b.1", 0.5},
{"0b.001", 0.125},
{"0b.001e3", 125},
// binary mantissa, binary exponent
{"0b0p+10", 0},
{"-0b0p-10", -0},
{"0b.1010p4", 10},
{"0b1p-1", 0.5},
{"0b001p-3", 0.125},
{"0b.001p3", 1},
{"0b0.01p2", 1},
// hexadecimal mantissa and exponent
{"0x0", 0},
{"-0x0", -0},
{"0x0p+10", 0},
{"-0x0p-10", -0},
{"0xff", 255},
{"0X.8p1", 1},
{"-0X0.00008p16", -0.5},
{"0x0.0000000000001p-1022", math.SmallestNonzeroFloat64},
{"0x1.fffffffffffffp1023", math.MaxFloat64},
} {
var x Float
x.SetPrec(53)
_, ok := x.SetString(test.s)
if !ok {
t.Errorf("%s: parse error", test.s)
continue
}
f, _ := x.Float64()
want := new(Float).SetFloat64(test.x)
if x.Cmp(want).Neq() {
t.Errorf("%s: got %s (%v); want %v", test.s, &x, f, test.x)
}
}
}
const (
below1e23 = 99999999999999974834176
above1e23 = 100000000000000008388608
)
func TestFloat64Format(t *testing.T) {
for _, test := range []struct {
x float64
format byte
prec int
want string
}{
{0, 'f', 0, "0"},
{math.Copysign(0, -1), 'f', 0, "-0"},
{1, 'f', 0, "1"},
{-1, 'f', 0, "-1"},
{1.459, 'e', 0, "1e+00"},
{2.459, 'e', 1, "2.5e+00"},
{3.459, 'e', 2, "3.46e+00"},
{4.459, 'e', 3, "4.459e+00"},
{5.459, 'e', 4, "5.4590e+00"},
{1.459, 'f', 0, "1"},
{2.459, 'f', 1, "2.5"},
{3.459, 'f', 2, "3.46"},
{4.459, 'f', 3, "4.459"},
{5.459, 'f', 4, "5.4590"},
{0, 'b', 0, "0"},
{math.Copysign(0, -1), 'b', 0, "-0"},
{1.0, 'b', 0, "4503599627370496p-52"},
{-1.0, 'b', 0, "-4503599627370496p-52"},
{4503599627370496, 'b', 0, "4503599627370496p+0"},
{0, 'p', 0, "0"},
{math.Copysign(0, -1), 'p', 0, "-0"},
{1024.0, 'p', 0, "0x.8p11"},
{-1024.0, 'p', 0, "-0x.8p11"},
// all test cases below from strconv/ftoa_test.go
{1, 'e', 5, "1.00000e+00"},
{1, 'f', 5, "1.00000"},
{1, 'g', 5, "1"},
// {1, 'g', -1, "1"},
// {20, 'g', -1, "20"},
// {1234567.8, 'g', -1, "1.2345678e+06"},
// {200000, 'g', -1, "200000"},
// {2000000, 'g', -1, "2e+06"},
// g conversion and zero suppression
{400, 'g', 2, "4e+02"},
{40, 'g', 2, "40"},
{4, 'g', 2, "4"},
{.4, 'g', 2, "0.4"},
{.04, 'g', 2, "0.04"},
{.004, 'g', 2, "0.004"},
{.0004, 'g', 2, "0.0004"},
{.00004, 'g', 2, "4e-05"},
{.000004, 'g', 2, "4e-06"},
{0, 'e', 5, "0.00000e+00"},
{0, 'f', 5, "0.00000"},
{0, 'g', 5, "0"},
// {0, 'g', -1, "0"},
{-1, 'e', 5, "-1.00000e+00"},
{-1, 'f', 5, "-1.00000"},
{-1, 'g', 5, "-1"},
// {-1, 'g', -1, "-1"},
{12, 'e', 5, "1.20000e+01"},
{12, 'f', 5, "12.00000"},
{12, 'g', 5, "12"},
// {12, 'g', -1, "12"},
{123456700, 'e', 5, "1.23457e+08"},
{123456700, 'f', 5, "123456700.00000"},
{123456700, 'g', 5, "1.2346e+08"},
// {123456700, 'g', -1, "1.234567e+08"},
{1.2345e6, 'e', 5, "1.23450e+06"},
{1.2345e6, 'f', 5, "1234500.00000"},
{1.2345e6, 'g', 5, "1.2345e+06"},
{1e23, 'e', 17, "9.99999999999999916e+22"},
{1e23, 'f', 17, "99999999999999991611392.00000000000000000"},
{1e23, 'g', 17, "9.9999999999999992e+22"},
// {1e23, 'e', -1, "1e+23"},
// {1e23, 'f', -1, "100000000000000000000000"},
// {1e23, 'g', -1, "1e+23"},
{below1e23, 'e', 17, "9.99999999999999748e+22"},
{below1e23, 'f', 17, "99999999999999974834176.00000000000000000"},
{below1e23, 'g', 17, "9.9999999999999975e+22"},
// {below1e23, 'e', -1, "9.999999999999997e+22"},
// {below1e23, 'f', -1, "99999999999999970000000"},
// {below1e23, 'g', -1, "9.999999999999997e+22"},
{above1e23, 'e', 17, "1.00000000000000008e+23"},
{above1e23, 'f', 17, "100000000000000008388608.00000000000000000"},
// {above1e23, 'g', 17, "1.0000000000000001e+23"},
// {above1e23, 'e', -1, "1.0000000000000001e+23"},
// {above1e23, 'f', -1, "100000000000000010000000"},
// {above1e23, 'g', -1, "1.0000000000000001e+23"},
// {fdiv(5e-304, 1e20), 'g', -1, "5e-324"},
// {fdiv(-5e-304, 1e20), 'g', -1, "-5e-324"},
// {32, 'g', -1, "32"},
// {32, 'g', 0, "3e+01"},
// {100, 'x', -1, "%x"},
// {math.NaN(), 'g', -1, "NaN"},
// {-math.NaN(), 'g', -1, "NaN"},
{math.Inf(0), 'g', -1, "+Inf"},
{math.Inf(-1), 'g', -1, "-Inf"},
{-math.Inf(0), 'g', -1, "-Inf"},
{-1, 'b', -1, "-4503599627370496p-52"},
// fixed bugs
{0.9, 'f', 1, "0.9"},
{0.09, 'f', 1, "0.1"},
{0.0999, 'f', 1, "0.1"},
{0.05, 'f', 1, "0.1"},
{0.05, 'f', 0, "0"},
{0.5, 'f', 1, "0.5"},
{0.5, 'f', 0, "0"},
{1.5, 'f', 0, "2"},
// http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
// {2.2250738585072012e-308, 'g', -1, "2.2250738585072014e-308"},
// http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
// {2.2250738585072011e-308, 'g', -1, "2.225073858507201e-308"},
// Issue 2625.
{383260575764816448, 'f', 0, "383260575764816448"},
// {383260575764816448, 'g', -1, "3.8326057576481645e+17"},
} {
f := new(Float).SetFloat64(test.x)
got := f.Format(test.format, test.prec)
if got != test.want {
t.Errorf("%v: got %s; want %s", test, got, test.want)
}
if test.format == 'b' && test.x == 0 {
continue // 'b' format in strconv.Float requires knowledge of bias for 0.0
}
if test.format == 'p' {
continue // 'p' format not supported in strconv.Format
}
// verify that Float format matches strconv format
want := strconv.FormatFloat(test.x, test.format, test.prec, 64)
if got != want {
t.Errorf("%v: got %s; want %s (strconv)", test, got, want)
}
}
}
func TestFloatFormat(t *testing.T) {
for _, test := range []struct {
x string
prec uint
format byte
digits int
want string
}{
{"0", 10, 'f', 0, "0"},
{"-0", 10, 'f', 0, "-0"},
{"1", 10, 'f', 0, "1"},
{"-1", 10, 'f', 0, "-1"},
{"1.459", 100, 'e', 0, "1e+00"},
{"2.459", 100, 'e', 1, "2.5e+00"},
{"3.459", 100, 'e', 2, "3.46e+00"},
{"4.459", 100, 'e', 3, "4.459e+00"},
{"5.459", 100, 'e', 4, "5.4590e+00"},
{"1.459", 100, 'E', 0, "1E+00"},
{"2.459", 100, 'E', 1, "2.5E+00"},
{"3.459", 100, 'E', 2, "3.46E+00"},
{"4.459", 100, 'E', 3, "4.459E+00"},
{"5.459", 100, 'E', 4, "5.4590E+00"},
{"1.459", 100, 'f', 0, "1"},
{"2.459", 100, 'f', 1, "2.5"},
{"3.459", 100, 'f', 2, "3.46"},
{"4.459", 100, 'f', 3, "4.459"},
{"5.459", 100, 'f', 4, "5.4590"},
{"1.459", 100, 'g', 0, "1"},
{"2.459", 100, 'g', 1, "2"},
{"3.459", 100, 'g', 2, "3.5"},
{"4.459", 100, 'g', 3, "4.46"},
{"5.459", 100, 'g', 4, "5.459"},
{"1459", 53, 'g', 0, "1e+03"},
{"2459", 53, 'g', 1, "2e+03"},
{"3459", 53, 'g', 2, "3.5e+03"},
{"4459", 53, 'g', 3, "4.46e+03"},
{"5459", 53, 'g', 4, "5459"},
{"1459", 53, 'G', 0, "1E+03"},
{"2459", 53, 'G', 1, "2E+03"},
{"3459", 53, 'G', 2, "3.5E+03"},
{"4459", 53, 'G', 3, "4.46E+03"},
{"5459", 53, 'G', 4, "5459"},
{"3", 10, 'e', 40, "3.0000000000000000000000000000000000000000e+00"},
{"3", 10, 'f', 40, "3.0000000000000000000000000000000000000000"},
{"3", 10, 'g', 40, "3"},
{"3e40", 100, 'e', 40, "3.0000000000000000000000000000000000000000e+40"},
{"3e40", 100, 'f', 4, "30000000000000000000000000000000000000000.0000"},
{"3e40", 100, 'g', 40, "3e+40"},
// TODO(gri) need tests for actual large Floats
{"0", 53, 'b', 0, "0"},
{"-0", 53, 'b', 0, "-0"},
{"1.0", 53, 'b', 0, "4503599627370496p-52"},
{"-1.0", 53, 'b', 0, "-4503599627370496p-52"},
{"4503599627370496", 53, 'b', 0, "4503599627370496p+0"},
// issue 9939
{"3", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"03", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"3.", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"3.0", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"3.00", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"3.000", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
{"3", 350, 'p', 0, "0x.cp2"},
{"03", 350, 'p', 0, "0x.cp2"},
{"3.", 350, 'p', 0, "0x.cp2"},
{"3.0", 350, 'p', 0, "0x.cp2"},
{"3.00", 350, 'p', 0, "0x.cp2"},
{"3.000", 350, 'p', 0, "0x.cp2"},
{"0", 64, 'p', 0, "0"},
{"-0", 64, 'p', 0, "-0"},
{"1024.0", 64, 'p', 0, "0x.8p11"},
{"-1024.0", 64, 'p', 0, "-0x.8p11"},
// unsupported format
{"3.14", 64, 'x', 0, "%x"},
} {
f, _, err := ParseFloat(test.x, 0, test.prec, ToNearestEven)
if err != nil {
t.Errorf("%v: %s", test, err)
continue
}
got := f.Format(test.format, test.digits)
if got != test.want {
t.Errorf("%v: got %s; want %s", test, got, test.want)
}
// compare with strconv.FormatFloat output if possible
// ('p' format is not supported by strconv.FormatFloat,
// and its output for 0.0 prints a biased exponent value
// as in 0p-1074 which makes no sense to emulate here)
if test.prec == 53 && test.format != 'p' && f.Sign() != 0 {
f64, acc := f.Float64()
if acc != Exact {
t.Errorf("%v: expected exact conversion to float64", test)
continue
}
got := strconv.FormatFloat(f64, test.format, test.digits, 64)
if got != test.want {
t.Errorf("%v: got %s; want %s", test, got, test.want)
}
}
}
}

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// 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_test
import (
"fmt"
"math"
"math/big"
)
func ExampleFloat_Add() {
// Operating on numbers of different precision.
var x, y, z big.Float
x.SetInt64(1000) // x is automatically set to 64bit precision
y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
z.SetPrec(32)
z.Add(&x, &y)
fmt.Printf("x = %s (%s, prec = %d, acc = %s)\n", &x, x.Format('p', 0), x.Prec(), x.Acc())
fmt.Printf("y = %s (%s, prec = %d, acc = %s)\n", &y, y.Format('p', 0), y.Prec(), y.Acc())
fmt.Printf("z = %s (%s, prec = %d, acc = %s)\n", &z, z.Format('p', 0), z.Prec(), z.Acc())
// Output:
// x = 1000 (0x.fap10, prec = 64, acc = Exact)
// y = 2.718281828 (0x.adf85458248cd8p2, prec = 53, acc = Exact)
// z = 1002.718282 (0x.faadf854p10, prec = 32, acc = Below)
}
func Example_Shift() {
// Implementing Float "shift" by modifying the (binary) exponents directly.
for s := -5; s <= 5; s++ {
x := big.NewFloat(0.5)
x.SetMantExp(x, x.MantExp(nil)+s) // shift x by s
fmt.Println(x)
}
// Output:
// 0.015625
// 0.03125
// 0.0625
// 0.125
// 0.25
// 0.5
// 1
// 2
// 4
// 8
// 16
}
func ExampleFloat_Cmp() {
inf := math.Inf(1)
zero := 0.0
nan := math.NaN()
operands := []float64{-inf, -1.2, -zero, 0, +1.2, +inf, nan}
fmt.Println(" x y cmp eql neq lss leq gtr geq")
fmt.Println("-----------------------------------------------")
for _, x64 := range operands {
x := big.NewFloat(x64)
for _, y64 := range operands {
y := big.NewFloat(y64)
t := x.Cmp(y)
fmt.Printf(
"%4s %4s %5s %c %c %c %c %c %c\n",
x, y, t.Acc(),
mark(t.Eql()), mark(t.Neq()), mark(t.Lss()), mark(t.Leq()), mark(t.Gtr()), mark(t.Geq()))
}
fmt.Println()
}
// Output:
// x y cmp eql neq lss leq gtr geq
// -----------------------------------------------
// -Inf -Inf Exact ● ○ ○ ● ○ ●
// -Inf -1.2 Below ○ ● ● ● ○ ○
// -Inf -0 Below ○ ● ● ● ○ ○
// -Inf 0 Below ○ ● ● ● ○ ○
// -Inf 1.2 Below ○ ● ● ● ○ ○
// -Inf +Inf Below ○ ● ● ● ○ ○
// -Inf NaN Undef ○ ● ○ ○ ○ ○
//
// -1.2 -Inf Above ○ ● ○ ○ ● ●
// -1.2 -1.2 Exact ● ○ ○ ● ○ ●
// -1.2 -0 Below ○ ● ● ● ○ ○
// -1.2 0 Below ○ ● ● ● ○ ○
// -1.2 1.2 Below ○ ● ● ● ○ ○
// -1.2 +Inf Below ○ ● ● ● ○ ○
// -1.2 NaN Undef ○ ● ○ ○ ○ ○
//
// -0 -Inf Above ○ ● ○ ○ ● ●
// -0 -1.2 Above ○ ● ○ ○ ● ●
// -0 -0 Exact ● ○ ○ ● ○ ●
// -0 0 Exact ● ○ ○ ● ○ ●
// -0 1.2 Below ○ ● ● ● ○ ○
// -0 +Inf Below ○ ● ● ● ○ ○
// -0 NaN Undef ○ ● ○ ○ ○ ○
//
// 0 -Inf Above ○ ● ○ ○ ● ●
// 0 -1.2 Above ○ ● ○ ○ ● ●
// 0 -0 Exact ● ○ ○ ● ○ ●
// 0 0 Exact ● ○ ○ ● ○ ●
// 0 1.2 Below ○ ● ● ● ○ ○
// 0 +Inf Below ○ ● ● ● ○ ○
// 0 NaN Undef ○ ● ○ ○ ○ ○
//
// 1.2 -Inf Above ○ ● ○ ○ ● ●
// 1.2 -1.2 Above ○ ● ○ ○ ● ●
// 1.2 -0 Above ○ ● ○ ○ ● ●
// 1.2 0 Above ○ ● ○ ○ ● ●
// 1.2 1.2 Exact ● ○ ○ ● ○ ●
// 1.2 +Inf Below ○ ● ● ● ○ ○
// 1.2 NaN Undef ○ ● ○ ○ ○ ○
//
// +Inf -Inf Above ○ ● ○ ○ ● ●
// +Inf -1.2 Above ○ ● ○ ○ ● ●
// +Inf -0 Above ○ ● ○ ○ ● ●
// +Inf 0 Above ○ ● ○ ○ ● ●
// +Inf 1.2 Above ○ ● ○ ○ ● ●
// +Inf +Inf Exact ● ○ ○ ● ○ ●
// +Inf NaN Undef ○ ● ○ ○ ○ ○
//
// NaN -Inf Undef ○ ● ○ ○ ○ ○
// NaN -1.2 Undef ○ ● ○ ○ ○ ○
// NaN -0 Undef ○ ● ○ ○ ○ ○
// NaN 0 Undef ○ ● ○ ○ ○ ○
// NaN 1.2 Undef ○ ● ○ ○ ○ ○
// NaN +Inf Undef ○ ● ○ ○ ○ ○
// NaN NaN Undef ○ ● ○ ○ ○ ○
}
func mark(p bool) rune {
if p {
return '●'
}
return '○'
}

190
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// 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 'e', 'f', 'g' floating-point formats.
// It is closely following the corresponding implementation in
// strconv/ftoa.go, but modified and simplified for big.Float.
// Algorithm:
// 1) convert Float to multiprecision decimal
// 2) round to desired precision
// 3) read digits out and format
package big
import "strconv"
// TODO(gri) Consider moving sign into decimal - could make the signatures below cleaner.
// bigFtoa formats a float for the %e, %E, %f, %g, and %G formats.
func (f *Float) bigFtoa(buf []byte, fmt byte, prec int) []byte {
if debugFloat && !f.IsFinite() {
panic("non-finite float")
}
// 1) convert Float to multiprecision decimal
var mant nat
if f.form == finite {
mant = f.mant
}
var d decimal
d.init(mant, int(f.exp)-f.mant.bitLen())
// 2) round to desired precision
shortest := false
if prec < 0 {
shortest = true
panic("unimplemented")
// TODO(gri) complete this
// roundShortest(&d, f.mant, int(f.exp))
// Precision for shortest representation mode.
switch fmt {
case 'e', 'E':
prec = len(d.mant) - 1
case 'f':
prec = max(len(d.mant)-d.exp, 0)
case 'g', 'G':
prec = len(d.mant)
}
} else {
// round appropriately
switch fmt {
case 'e', 'E':
// one digit before and number of digits after decimal point
d.round(1 + prec)
case 'f':
// number of digits before and after decimal point
d.round(d.exp + prec)
case 'g', 'G':
if prec == 0 {
prec = 1
}
d.round(prec)
}
}
// 3) read digits out and format
switch fmt {
case 'e', 'E':
return fmtE(buf, fmt, prec, f.neg, d)
case 'f':
return fmtF(buf, prec, f.neg, d)
case 'g', 'G':
// trim trailing fractional zeros in %e format
eprec := prec
if eprec > len(d.mant) && len(d.mant) >= d.exp {
eprec = len(d.mant)
}
// %e is used if the exponent from the conversion
// is less than -4 or greater than or equal to the precision.
// If precision was the shortest possible, use eprec = 6 for
// this decision.
if shortest {
eprec = 6
}
exp := d.exp - 1
if exp < -4 || exp >= eprec {
if prec > len(d.mant) {
prec = len(d.mant)
}
return fmtE(buf, fmt+'e'-'g', prec-1, f.neg, d)
}
if prec > d.exp {
prec = len(d.mant)
}
return fmtF(buf, max(prec-d.exp, 0), f.neg, d)
}
// unknown format
return append(buf, '%', fmt)
}
// %e: -d.ddddde±dd
func fmtE(buf []byte, fmt byte, prec int, neg bool, d decimal) []byte {
// sign
if neg {
buf = append(buf, '-')
}
// first digit
ch := byte('0')
if len(d.mant) > 0 {
ch = d.mant[0]
}
buf = append(buf, ch)
// .moredigits
if prec > 0 {
buf = append(buf, '.')
i := 1
m := min(len(d.mant), prec+1)
if i < m {
buf = append(buf, d.mant[i:m]...)
i = m
}
for ; i <= prec; i++ {
buf = append(buf, '0')
}
}
// e±
buf = append(buf, fmt)
var exp int64
if len(d.mant) > 0 {
exp = int64(d.exp) - 1 // -1 because first digit was printed before '.'
}
if exp < 0 {
ch = '-'
exp = -exp
} else {
ch = '+'
}
buf = append(buf, ch)
// dd...d
if exp < 10 {
buf = append(buf, '0') // at least 2 exponent digits
}
return strconv.AppendInt(buf, exp, 10)
}
// %f: -ddddddd.ddddd
func fmtF(buf []byte, prec int, neg bool, d decimal) []byte {
// sign
if neg {
buf = append(buf, '-')
}
// integer, padded with zeros as needed
if d.exp > 0 {
m := min(len(d.mant), d.exp)
buf = append(buf, d.mant[:m]...)
for ; m < d.exp; m++ {
buf = append(buf, '0')
}
} else {
buf = append(buf, '0')
}
// fraction
if prec > 0 {
buf = append(buf, '.')
for i := 0; i < prec; i++ {
ch := byte('0')
if j := d.exp + i; 0 <= j && j < len(d.mant) {
ch = d.mant[j]
}
buf = append(buf, ch)
}
}
return buf
}
func min(x, y int) int {
if x < y {
return x
}
return y
}

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// 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) }

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// 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)
}
}

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src/cmd/internal/gc/big/int.go Normal file
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@ -0,0 +1,845 @@
// 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
}
// 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 between 2 and 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.ReadByte()
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
if z.neg && len(mWords) > 0 {
// make modulus result positive
z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
z.neg = false
}
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. n must be > 0.
func (x *Int) ProbablyPrime(n int) bool {
if n <= 0 {
panic("non-positive n for ProbablyPrime")
}
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 ring ℤ/nℤ
// and returns z. If g and n are not relatively prime, the result is undefined.
func (z *Int) ModInverse(g, n *Int) *Int {
var d Int
d.GCD(z, nil, g, n)
// x and y are such that g*x + n*y = d. Since g and n are
// relatively prime, d = 1. Taking that modulo n results in
// g*x = 1, therefore x is the inverse element.
if z.neg {
z.Add(z, n)
}
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).sub(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
}

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// 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 int-to-string conversion functions.
package big
import (
"errors"
"fmt"
"io"
)
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.ByteScanner, base int) (*Int, int, error) {
// determine sign
neg, err := scanSign(r)
if err != nil {
return nil, 0, err
}
// determine mantissa
z.abs, base, _, err = z.abs.scan(r, base, false)
if err != nil {
return nil, base, err
}
z.neg = len(z.abs) > 0 && neg // 0 has no sign
return z, base, nil
}
func scanSign(r io.ByteScanner) (neg bool, err error) {
var ch byte
if ch, err = r.ReadByte(); err != nil {
return false, err
}
switch ch {
case '-':
neg = true
case '+':
// nothing to do
default:
r.UnreadByte()
}
return
}
// byteReader is a local wrapper around fmt.ScanState;
// it implements the ByteReader interface.
type byteReader struct {
fmt.ScanState
}
func (r byteReader) ReadByte() (byte, error) {
ch, size, err := r.ReadRune()
if size != 1 && err == nil {
err = fmt.Errorf("invalid rune %#U", ch)
}
return byte(ch), err
}
func (r byteReader) UnreadByte() error {
return r.UnreadRune()
}
// 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(byteReader{s}, base)
return err
}

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// 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 (
"bytes"
"fmt"
"testing"
)
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)
}
}
}

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// 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 (
"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, false)
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 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) {
// test 0 case explicitly
if n := trailingZeroBits(0); n != 0 {
t.Errorf("got trailingZeroBits(0) = %d; want 0", n)
}
x := Word(1)
for i := uint(0); i < _W; i++ {
n := trailingZeroBits(x)
if n != i {
t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
}
x <<= 1
}
// test 0 case explicitly
if n := nat(nil).trailingZeroBits(); n != 0 {
t.Errorf("got nat(nil).trailingZeroBits() = %d; want 0", n)
}
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.hexString(), 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 := natFromString(test.x)
y := natFromString(test.y)
out := natFromString(test.out)
var m nat
if len(test.m) > 0 {
m = natFromString(test.m)
}
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) }
func fibo(n int) nat {
switch n {
case 0:
return nil
case 1:
return nat{1}
}
f0 := fibo(0)
f1 := fibo(1)
var f2 nat
for i := 1; i < n; i++ {
f2 = f2.add(f0, f1)
f0, f1, f2 = f1, f2, f0
}
return f1
}
var fiboNums = []string{
"0",
"55",
"6765",
"832040",
"102334155",
"12586269025",
"1548008755920",
"190392490709135",
"23416728348467685",
"2880067194370816120",
"354224848179261915075",
}
func TestFibo(t *testing.T) {
for i, want := range fiboNums {
n := i * 10
got := fibo(n).decimalString()
if got != want {
t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
}
}
}
func BenchmarkFibo(b *testing.B) {
for i := 0; i < b.N; i++ {
fibo(1e0)
fibo(1e1)
fibo(1e2)
fibo(1e3)
fibo(1e4)
fibo(1e5)
}
}
var bitTests = []struct {
x string
i uint
want uint
}{
{"0", 0, 0},
{"0", 1, 0},
{"0", 1000, 0},
{"0x1", 0, 1},
{"0x10", 0, 0},
{"0x10", 3, 0},
{"0x10", 4, 1},
{"0x10", 5, 0},
{"0x8000000000000000", 62, 0},
{"0x8000000000000000", 63, 1},
{"0x8000000000000000", 64, 0},
{"0x3" + strings.Repeat("0", 32), 127, 0},
{"0x3" + strings.Repeat("0", 32), 128, 1},
{"0x3" + strings.Repeat("0", 32), 129, 1},
{"0x3" + strings.Repeat("0", 32), 130, 0},
}
func TestBit(t *testing.T) {
for i, test := range bitTests {
x := natFromString(test.x)
if got := x.bit(test.i); got != test.want {
t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
}
}
}
var stickyTests = []struct {
x string
i uint
want uint
}{
{"0", 0, 0},
{"0", 1, 0},
{"0", 1000, 0},
{"0x1", 0, 0},
{"0x1", 1, 1},
{"0x1350", 0, 0},
{"0x1350", 4, 0},
{"0x1350", 5, 1},
{"0x8000000000000000", 63, 0},
{"0x8000000000000000", 64, 1},
{"0x1" + strings.Repeat("0", 100), 400, 0},
{"0x1" + strings.Repeat("0", 100), 401, 1},
}
func TestSticky(t *testing.T) {
for i, test := range stickyTests {
x := natFromString(test.x)
if got := x.sticky(test.i); got != test.want {
t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
}
if test.want == 1 {
// all subsequent i's should also return 1
for d := uint(1); d <= 3; d++ {
if got := x.sticky(test.i + d); got != 1 {
t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
}
}
}
}
}

495
src/cmd/internal/gc/big/natconv.go Normal file
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@ -0,0 +1,495 @@
// 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 nat-to-string conversion functions.
package big
import (
"errors"
"fmt"
"io"
"math"
"sync"
)
// MaxBase is the largest number base accepted for string conversions.
const MaxBase = 'z' - 'a' + 10 + 1
// maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M.
// For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word.
// In other words, at most n digits in base b fit into a Word.
// TODO(gri) replace this with a table, generated at build time.
func maxPow(b Word) (p Word, n int) {
p, n = b, 1 // assuming b <= _M
for max := _M / b; p <= max; {
// p == b**n && p <= max
p *= b
n++
}
// p == b**n && p <= _M
return
}
// pow returns x**n for n > 0, and 1 otherwise.
func pow(x Word, n int) (p Word) {
// n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1
// thus x**n == product of x**(2**i) for all i where bi == 1
// (Russian Peasant Method for exponentiation)
p = 1
for n > 0 {
if n&1 != 0 {
p *= x
}
x *= x
n >>= 1
}
return
}
// scan scans the number corresponding to the longest possible prefix
// from r representing an unsigned number in a given conversion base.
// It returns the corresponding natural number res, the actual base b,
// a digit count, and a read or syntax error err, if any.
//
// number = [ prefix ] mantissa .
// prefix = "0" [ "x" | "X" | "b" | "B" ] .
// mantissa = digits | digits "." [ digits ] | "." digits .
// digits = digit { digit } .
// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
//
// Unless fracOk is set, the base argument must be 0 or a value between
// 2 and MaxBase. If fracOk is set, the base argument must be one of
// 0, 2, 10, or 16. Providing an invalid base argument leads to a run-
// time panic.
//
// For base 0, the number prefix determines the actual base: A prefix of
// ``0x'' or ``0X'' selects base 16; if fracOk is not set, the ``0'' prefix
// selects base 8, and a ``0b'' or ``0B'' prefix selects base 2. Otherwise
// the selected base is 10 and no prefix is accepted.
//
// If fracOk is set, an octal prefix is ignored (a leading ``0'' simply
// stands for a zero digit), and a period followed by a fractional part
// is permitted. The result value is computed as if there were no period
// present; and the count value is used to determine the fractional part.
//
// A result digit count > 0 corresponds to the number of (non-prefix) digits
// parsed. A digit count <= 0 indicates the presence of a period (if fracOk
// is set, only), and -count is the number of fractional digits found.
// In this case, the actual value of the scanned number is res * b**count.
//
func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) {
// reject illegal bases
baseOk := base == 0 ||
!fracOk && 2 <= base && base <= MaxBase ||
fracOk && (base == 2 || base == 10 || base == 16)
if !baseOk {
panic(fmt.Sprintf("illegal number base %d", base))
}
// one char look-ahead
ch, err := r.ReadByte()
if err != nil {
return
}
// determine actual base
b = base
if base == 0 {
// actual base is 10 unless there's a base prefix
b = 10
if ch == '0' {
count = 1
switch ch, err = r.ReadByte(); err {
case nil:
// possibly one of 0x, 0X, 0b, 0B
if !fracOk {
b = 8
}
switch ch {
case 'x', 'X':
b = 16
case 'b', 'B':
b = 2
}
switch b {
case 16, 2:
count = 0 // prefix is not counted
if ch, err = r.ReadByte(); err != nil {
// io.EOF is also an error in this case
return
}
case 8:
count = 0 // prefix is not counted
}
case io.EOF:
// input is "0"
res = z[:0]
err = nil
return
default:
// read error
return
}
}
}
// convert string
// Algorithm: Collect digits in groups of at most n digits in di
// and then use mulAddWW for every such group to add them to the
// result.
z = z[:0]
b1 := Word(b)
bn, n := maxPow(b1) // at most n digits in base b1 fit into Word
di := Word(0) // 0 <= di < b1**i < bn
i := 0 // 0 <= i < n
dp := -1 // position of decimal point
for {
if fracOk && ch == '.' {
fracOk = false
dp = count
// advance
if ch, err = r.ReadByte(); err != nil {
if err == io.EOF {
err = nil
break
}
return
}
}
// convert rune into digit value d1
var d1 Word
switch {
case '0' <= ch && ch <= '9':
d1 = Word(ch - '0')
case 'a' <= ch && ch <= 'z':
d1 = Word(ch - 'a' + 10)
case 'A' <= ch && ch <= 'Z':
d1 = Word(ch - 'A' + 10)
default:
d1 = MaxBase + 1
}
if d1 >= b1 {
r.UnreadByte() // ch does not belong to number anymore
break
}
count++
// collect d1 in di
di = di*b1 + d1
i++
// if di is "full", add it to the result
if i == n {
z = z.mulAddWW(z, bn, di)
di = 0
i = 0
}
// advance
if ch, err = r.ReadByte(); err != nil {
if err == io.EOF {
err = nil
break
}
return
}
}
if count == 0 {
// no digits found
switch {
case base == 0 && b == 8:
// there was only the octal prefix 0 (possibly followed by digits > 7);
// count as one digit and return base 10, not 8
count = 1
b = 10
case base != 0 || b != 8:
// there was neither a mantissa digit nor the octal prefix 0
err = errors.New("syntax error scanning number")
}
return
}
// count > 0
// add remaining digits to result
if i > 0 {
z = z.mulAddWW(z, pow(b1, i), di)
}
res = z.norm()
// adjust for fraction, if any
if dp >= 0 {
// 0 <= dp <= count > 0
count = dp - count
}
return
}
// 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[:10])
}
// hexString returns a hexadecimal representation of x.
// It calls x.string with the charset "0123456789abcdef".
func (x nat) hexString() string {
return x.string(lowercaseDigits[:16])
}
// 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 || b > 256:
panic("invalid character set length")
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 {
bb, ndigits := maxPow(Word(b))
// 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
}

425
src/cmd/internal/gc/big/natconv_test.go Normal file
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@ -0,0 +1,425 @@
// 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 (
"io"
"strings"
"testing"
)
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[:8], "3271"},
{nat{10}, lowercaseDigits[:10], "10"},
{nat{1234567890}, uppercaseDigits[:10], "1234567890"},
{nat{0xdeadbeef}, lowercaseDigits[:16], "deadbeef"},
{nat{0xdeadbeef}, uppercaseDigits[:16], "DEADBEEF"},
{nat{0x229be7}, lowercaseDigits[:17], "1a2b3c"},
{nat{0x309663e6}, uppercaseDigits[:32], "O9COV6"},
}
func TestString(t *testing.T) {
// test invalid character set explicitly
var panicStr string
func() {
defer func() {
panicStr = recover().(string)
}()
natOne.string("0")
}()
if panicStr != "invalid character set length" {
t.Errorf("expected panic for invalid character set")
}
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), false)
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
frac bool // fraction ok
x nat // expected nat
b int // expected base
count int // expected digit count
ok bool // expected success
next rune // next character (or 0, if at EOF)
}{
// error: no mantissa
{},
{s: "?"},
{base: 10},
{base: 36},
{s: "?", base: 10},
{s: "0x"},
{s: "345", base: 2},
// error: incorrect use of decimal point
{s: ".0"},
{s: ".0", base: 10},
{s: ".", base: 0},
{s: "0x.0"},
// no errors
{"0", 0, false, nil, 10, 1, true, 0},
{"0", 10, false, nil, 10, 1, true, 0},
{"0", 36, false, nil, 36, 1, true, 0},
{"1", 0, false, nat{1}, 10, 1, true, 0},
{"1", 10, false, nat{1}, 10, 1, true, 0},
{"0 ", 0, false, nil, 10, 1, true, ' '},
{"08", 0, false, nil, 10, 1, true, '8'},
{"08", 10, false, nat{8}, 10, 2, true, 0},
{"018", 0, false, nat{1}, 8, 1, true, '8'},
{"0b1", 0, false, nat{1}, 2, 1, true, 0},
{"0b11000101", 0, false, nat{0xc5}, 2, 8, true, 0},
{"03271", 0, false, nat{03271}, 8, 4, true, 0},
{"10ab", 0, false, nat{10}, 10, 2, true, 'a'},
{"1234567890", 0, false, nat{1234567890}, 10, 10, true, 0},
{"xyz", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, true, 0},
{"xyz?", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, true, '?'},
{"0x", 16, false, nil, 16, 1, true, 'x'},
{"0xdeadbeef", 0, false, nat{0xdeadbeef}, 16, 8, true, 0},
{"0XDEADBEEF", 0, false, nat{0xdeadbeef}, 16, 8, true, 0},
// no errors, decimal point
{"0.", 0, false, nil, 10, 1, true, '.'},
{"0.", 10, true, nil, 10, 0, true, 0},
{"0.1.2", 10, true, nat{1}, 10, -1, true, '.'},
{".000", 10, true, nil, 10, -3, true, 0},
{"12.3", 10, true, nat{123}, 10, -1, true, 0},
{"012.345", 10, true, nat{12345}, 10, -3, true, 0},
}
func TestScanBase(t *testing.T) {
for _, a := range natScanTests {
r := strings.NewReader(a.s)
x, b, count, err := nat(nil).scan(r, a.base, a.frac)
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)
}
if count != a.count {
t.Errorf("scan%+v\n\tgot count = %d; want %d", a, count, a.count)
}
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, false)
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, false)
}
}
func BenchmarkStringPiParallel(b *testing.B) {
var x nat
x, _, _, _ = x.scan(strings.NewReader(pi), 0, false)
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[:base])
if t := toString(z, lowercaseDigits[: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, false)
}
}
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[:base]) // warm divisor cache
b.StartTimer()
for i := 0; i < b.N; i++ {
_ = z.string(lowercaseDigits[: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[:base]) // warm divisor cache
b.StartTimer()
for i := 0; i < b.N; i++ {
_ = z.string(lowercaseDigits[: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[:b])
xs2 := toString(x, lowercaseDigits[: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
}
}
}

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// 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"
)
// 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[: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[: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[: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 reports whether 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[: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[: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[: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()
}
// 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[: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
}

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// 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"
"math"
"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)
}
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)
}
}
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)
}
}
}

251
src/cmd/internal/gc/big/ratconv.go Normal file
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@ -0,0 +1,251 @@
// 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 rat-to-string conversion functions.
package big
import (
"errors"
"fmt"
"io"
"strconv"
"strings"
)
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
}
// len(s) > 0
// parse fraction a/b, if any
if sep := strings.Index(s, "/"); sep >= 0 {
if _, ok := z.a.SetString(s[:sep], 0); !ok {
return nil, false
}
s = s[sep+1:]
var err error
if z.b.abs, _, _, err = z.b.abs.scan(strings.NewReader(s), 0, false); err != nil {
return nil, false
}
if len(z.b.abs) == 0 {
return nil, false
}
return z.norm(), true
}
// parse floating-point number
r := strings.NewReader(s)
// sign
neg, err := scanSign(r)
if err != nil {
return nil, false
}
// mantissa
var ecorr int
z.a.abs, _, ecorr, err = z.a.abs.scan(r, 10, true)
if err != nil {
return nil, false
}
// exponent
var exp int64
exp, _, err = scanExponent(r, false)
if err != nil {
return nil, false
}
// there should be no unread characters left
if _, err = r.ReadByte(); err != io.EOF {
return nil, false
}
// correct exponent
if ecorr < 0 {
exp += int64(ecorr)
}
// compute exponent power
expabs := exp
if expabs < 0 {
expabs = -expabs
}
powTen := nat(nil).expNN(natTen, nat(nil).setWord(Word(expabs)), nil)
// complete fraction
if exp < 0 {
z.b.abs = powTen
z.norm()
} else {
z.a.abs = z.a.abs.mul(z.a.abs, powTen)
z.b.abs = z.b.abs[:0]
}
z.a.neg = neg && len(z.a.abs) > 0 // 0 has no sign
return z, true
}
// scanExponent scans the longest possible prefix of r representing a decimal
// ('e', 'E') or binary ('p') exponent, if any. It returns the exponent, the
// exponent base (10 or 2), or a read or syntax error, if any.
//
// exponent = ( "E" | "e" | "p" ) [ sign ] digits .
// sign = "+" | "-" .
// digits = digit { digit } .
// digit = "0" ... "9" .
//
// A binary exponent is only permitted if binExpOk is set.
func scanExponent(r io.ByteScanner, binExpOk bool) (exp int64, base int, err error) {
base = 10
var ch byte
if ch, err = r.ReadByte(); err != nil {
if err == io.EOF {
err = nil // no exponent; same as e0
}
return
}
switch ch {
case 'e', 'E':
// ok
case 'p':
if binExpOk {
base = 2
break // ok
}
fallthrough // binary exponent not permitted
default:
r.UnreadByte()
return // no exponent; same as e0
}
var neg bool
if neg, err = scanSign(r); err != nil {
return
}
var digits []byte
if neg {
digits = append(digits, '-')
}
// no need to use nat.scan for exponent digits
// since we only care about int64 values - the
// from-scratch scan is easy enough and faster
for i := 0; ; i++ {
if ch, err = r.ReadByte(); err != nil {
if err != io.EOF || i == 0 {
return
}
err = nil
break // i > 0
}
if ch < '0' || '9' < ch {
if i == 0 {
r.UnreadByte()
err = fmt.Errorf("invalid exponent (missing digits)")
return
}
break // i > 0
}
digits = append(digits, byte(ch))
}
// i > 0 => we have at least one digit
exp, err = strconv.ParseInt(string(digits), 10, 64)
return
}
// 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
}

451
src/cmd/internal/gc/big/ratconv_test.go Normal file
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// 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 (
"bytes"
"fmt"
"math"
"strconv"
"strings"
"testing"
)
type StringTest struct {
in, out string
ok bool
}
var setStringTests = []StringTest{
{"0", "0", true},
{"-0", "0", true},
{"1", "1", true},
{"-1", "-1", true},
{"1.", "1", true},
{"1e0", "1", true},
{"1.e1", "10", true},
{in: "1e"},
{in: "1.e"},
{in: "1e+14e-5"},
{in: "1e4.5"},
{in: "r"},
{in: "a/b"},
{in: "a.b"},
{"-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},
{in: "1/0"},
}
// These are not supported by fmt.Fscanf.
var setStringTests2 = []StringTest{
{"0x10", "16", true},
{"-010/1", "-8", true}, // TODO(gri) should we even permit octal here?
{"-010.", "-10", true},
{"0x10/0x20", "1/2", true},
{"0b1000/3", "8/3", true},
// TODO(gri) add more tests
}
func TestRatSetString(t *testing.T) {
var tests []StringTest
tests = append(tests, setStringTests...)
tests = append(tests, setStringTests2...)
for i, test := range tests {
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 (%s) error: %s", i, test.in, err)
} else {
t.Errorf("#%d (%s) expected error", i, test.in)
}
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)
}
}
}
// 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)
}
}
}

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src/cmd/internal/gc/big/roundingmode_string.go Normal file
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// 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]]
}

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src/cmd/internal/gc/big/vendor.bash Executable file
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#!/usr/bin/env bash
# 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.
# Run this script to obtain an up-to-date vendored version of math/big.
BIGDIR=../../../../math/big
# Start from scratch.
rm *.go
# We don't want any assembly files.
cp $BIGDIR/*.go .
# Use pure Go arith ops w/o build tag.
sed 's/^\/\/ \+build math_big_pure_go$//' arith_decl_pure.go > arith_decl.go
rm arith_decl_pure.go
# Test that it works
go test -short