big: Add Rat type

Implementations are pretty rough and simple at this point, but it's a start.

R=gri
CC=golang-dev
https://golang.org/cl/1250043

4 files changed, 459 insertions, 1 deletions

diff --git a/src/pkg/big/Makefile b/src/pkg/big/Makefile
index 520c2b852a..d858e5a687 100644
--- a/src/pkg/big/Makefile
+++ b/src/pkg/big/Makefile
@@ -7,8 +7,9 @@ include ../../Make.$(GOARCH)
 TARG=big
 GOFILES=\
 	arith.go\
-	nat.go\
 	int.go\
+	nat.go\
+	rat.go\
 
 OFILES=\
 	arith_$(GOARCH).$O\
diff --git a/src/pkg/big/nat.go b/src/pkg/big/nat.go
index aa021e8794..19c3d88f73 100755
--- a/src/pkg/big/nat.go
+++ b/src/pkg/big/nat.go
@@ -10,6 +10,7 @@
 // The following numeric types are supported:
 //
 //	- Int	signed integers
+//	- Rat	rational numbers
 //
 // All methods on Int take the result as the receiver; if it is one
 // of the operands it may be overwritten (and its memory reused).
@@ -39,6 +40,7 @@ type nat []Word
 var (
 	natOne = nat{1}
 	natTwo = nat{2}
+	natTen = nat{10}
 )
 
 
diff --git a/src/pkg/big/rat.go b/src/pkg/big/rat.go
new file mode 100644
index 0000000000..f35df4b461
--- /dev/null
+++ b/src/pkg/big/rat.go
@@ -0,0 +1,270 @@
+// 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 "strings"
+
+// A Rat represents a quotient a/b of arbitrary precision. The zero value for
+// a Rat, 0/0, is not a legal Rat.
+type Rat struct {
+	a Int
+	b nat
+}
+
+
+// NewRat creates a new Rat with numerator a and denominator b.
+func NewRat(a, b int64) *Rat {
+	return new(Rat).SetFrac64(a, b)
+}
+
+
+// SetFrac sets z to a/b and returns z.
+func (z *Rat) SetFrac(a, b *Int) *Rat {
+	z.a.Set(a)
+	z.a.neg = a.neg != b.neg
+	z.b = z.b.set(b.abs)
+	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 {
+		z.b.setUint64(uint64(-b))
+		z.a.neg = !z.a.neg
+		return z.norm()
+	}
+	z.b = z.b.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 = z.b.setWord(1)
+	return z
+}
+
+
+// SetInt64 sets z to x and returns z.
+func (z *Rat) SetInt64(x int64) *Rat {
+	z.a.SetInt64(x)
+	z.b = z.b.setWord(1)
+	return z
+}
+
+
+// Num returns the numerator of z; it may be <= 0.
+// The result is a reference to z's numerator; it
+// may change if a new value is assigned to z.
+func (z *Rat) Num() *Int {
+	return &z.a
+}
+
+
+// Demom returns the denominator of z; it is always > 0.
+// The result is a reference to z's denominator; it
+// may change if a new value is assigned to z.
+func (z *Rat) Denom() *Int {
+	return &Int{false, z.b}
+}
+
+
+func gcd(x, y nat) nat {
+	// Euclidean algorithm.
+	var a, b nat
+	a = a.set(x)
+	b = b.set(y)
+	for len(b) != 0 {
+		var q, r nat
+		_, r = q.div(r, a, b)
+		a = b
+		b = r
+	}
+	return a
+}
+
+
+func (z *Rat) norm() *Rat {
+	f := gcd(z.a.abs, z.b)
+	if len(z.a.abs) == 0 {
+		// z == 0
+		z.a.neg = false // normalize sign
+		z.b = z.b.setWord(1)
+		return z
+	}
+	if f.cmp(natOne) != 0 {
+		z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f)
+		z.b, _ = z.b.div(nil, z.b, f)
+	}
+	return z
+}
+
+
+func mulNat(x *Int, y nat) *Int {
+	var z Int
+	z.abs = z.abs.mul(x.abs, y)
+	z.neg = len(z.abs) > 0 && 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) (r int) {
+	return mulNat(&x.a, y.b).Cmp(mulNat(&y.a, x.b))
+}
+
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+	a1 := mulNat(&x.a, y.b)
+	a2 := mulNat(&y.a, x.b)
+	z.a.Add(a1, a2)
+	z.b = z.b.mul(x.b, y.b)
+	return z.norm()
+}
+
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Rat) Sub(x, y *Rat) *Rat {
+	a1 := mulNat(&x.a, y.b)
+	a2 := mulNat(&y.a, x.b)
+	z.a.Sub(a1, a2)
+	z.b = z.b.mul(x.b, y.b)
+	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 = z.b.mul(x.b, y.b)
+	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")
+	}
+	z.a.abs = z.a.abs.mul(x.a.abs, y.b)
+	z.b = z.b.mul(x.b, y.a.abs)
+	z.a.neg = x.a.neg != y.a.neg
+	return z.norm()
+}
+
+
+// Neg sets z to -x (by making a copy of x if necessary) and returns z.
+func (z *Rat) Neg(x *Rat) *Rat {
+	z.a.Neg(&x.a)
+	z.b = z.b.set(x.b)
+	return z
+}
+
+
+// Set sets z to x (by making a copy of x if necessary) and returns z.
+func (z *Rat) Set(x *Rat) *Rat {
+	z.a.Set(&x.a)
+	z.b = z.b.set(x.b)
+	return z
+}
+
+
+// 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 decimal number "a.b".
+// If the operation failed, the value of z is undefined.
+func (z *Rat) SetString(s string) (*Rat, bool) {
+	if len(s) == 0 {
+		return z, false
+	}
+
+	// Check for a decimal point
+	sep := strings.Index(s, ".")
+	if sep < 0 {
+		// Check for a quotient
+		sep = strings.Index(s, "/")
+		if sep < 0 {
+			// Just read in the string as an integer
+			if _, ok := z.a.SetString(s, 10); !ok {
+				return z, false
+			}
+			z.b = z.b.setWord(1)
+			return z, true
+		}
+		if _, ok := z.a.SetString(s[0:sep], 10); !ok {
+			return z, false
+		}
+		s = s[sep+1:]
+		var n int
+		if z.b, _, n = z.b.scan(s, 10); n != len(s) {
+			return z, false
+		}
+
+		return z.norm(), true
+	}
+
+	s = s[0:sep] + s[sep+1:]
+	if _, ok := z.a.SetString(s, 10); !ok {
+		return z, false
+	}
+	z.b = z.b.expNN(natTen, nat{Word(len(s) - sep)}, nil)
+
+	return z.norm(), true
+}
+
+
+// String returns a string representation of z in the form "a/b".
+func (z *Rat) String() string {
+	s := z.a.String()
+	if len(z.b) == 1 && z.b[0] == 1 {
+		return s
+	}
+	return s + "/" + z.b.string(10)
+}
+
+
+// FloatString returns a string representation of z in decimal form with prec
+// digits of precision after the decimal point and the last digit rounded.
+func (z *Rat) FloatString(prec int) string {
+	q, r := nat{}.div(nat{}, z.a.abs, z.b)
+
+	s := ""
+	if z.a.neg {
+		s = "-"
+	}
+	s += q.string(10)
+
+	if len(z.b) == 1 && z.b[0] == 1 {
+		return s
+	}
+
+	p := nat{}.expNN(natTen, nat{Word(prec)}, nil)
+	r = r.mul(r, p)
+	r, r2 := r.div(nat{}, r, z.b)
+
+	// See if we need to round up
+	r2 = r2.mul(r2, natTwo)
+	if z.b.cmp(r2) <= 0 {
+		r = r.add(r, natOne)
+	}
+
+	rs := r.string(10)
+	leadingZeros := prec - len(rs)
+	s += "." + strings.Repeat("0", leadingZeros) + rs
+	s = strings.TrimRight(s, "0")
+
+	return s
+}
diff --git a/src/pkg/big/rat_test.go b/src/pkg/big/rat_test.go
new file mode 100644
index 0000000000..cd47cbd553
--- /dev/null
+++ b/src/pkg/big/rat_test.go
@@ -0,0 +1,185 @@
+// 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 "testing"
+
+
+type setStringTest struct {
+	in, out string
+}
+
+var setStringTests = []setStringTest{
+	setStringTest{"0", "0"},
+	setStringTest{"1", "1"},
+	setStringTest{"-1", "-1"},
+	setStringTest{"2/4", "1/2"},
+	setStringTest{".25", "1/4"},
+	setStringTest{"-1/5", "-1/5"},
+}
+
+func TestRatSetString(t *testing.T) {
+	for i, test := range setStringTests {
+		x, _ := new(Rat).SetString(test.in)
+
+		if x.String() != test.out {
+			t.Errorf("#%d got %s want %s", i, x.String(), test.out)
+		}
+	}
+}
+
+
+type floatStringTest struct {
+	in   string
+	prec int
+	out  string
+}
+
+var floatStringTests = []floatStringTest{
+	floatStringTest{"0", 0, "0"},
+	floatStringTest{"0", 4, "0"},
+	floatStringTest{"1", 0, "1"},
+	floatStringTest{"1", 2, "1"},
+	floatStringTest{"-1", 0, "-1"},
+	floatStringTest{".25", 2, "0.25"},
+	floatStringTest{".25", 1, "0.3"},
+	floatStringTest{"-1/3", 3, "-0.333"},
+	floatStringTest{"-2/3", 4, "-0.6667"},
+}
+
+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)
+		}
+	}
+}
+
+
+type ratCmpTest struct {
+	rat1, rat2 string
+	out        int
+}
+
+var ratCmpTests = []ratCmpTest{
+	ratCmpTest{"0", "0/1", 0},
+	ratCmpTest{"1/1", "1", 0},
+	ratCmpTest{"-1", "-2/2", 0},
+	ratCmpTest{"1", "0", 1},
+	ratCmpTest{"0/1", "1/1", -1},
+	ratCmpTest{"-5/1434770811533343057144", "-5/1434770811533343057145", -1},
+	ratCmpTest{"49832350382626108453/8964749413", "49832350382626108454/8964749413", -1},
+	ratCmpTest{"-37414950961700930/7204075375675961", "37414950961700930/7204075375675961", -1},
+	ratCmpTest{"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)
+		}
+	}
+}
+
+
+type ratBinFun func(z, x, y *Rat) *Rat
+type ratBinArg struct {
+	x   string
+	y   string
+	out string
+}
+
+func testRatBin(t *testing.T, f ratBinFun, a []ratBinArg) {
+	for i, test := range a {
+		x, _ := NewRat(0, 1).SetString(test.x)
+		y, _ := NewRat(0, 1).SetString(test.y)
+		expected, _ := NewRat(0, 1).SetString(test.out)
+		out := f(NewRat(0, 1), x, y)
+
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d got %s want %s", i, out, expected)
+		}
+	}
+}
+
+
+var ratAddTests = []ratBinArg{
+	ratBinArg{"0", "0", "0"},
+	ratBinArg{"0", "1", "1"},
+	ratBinArg{"-1", "0", "-1"},
+	ratBinArg{"-1", "1", "0"},
+	ratBinArg{"1", "1", "2"},
+	ratBinArg{"1/2", "1/2", "1"},
+	ratBinArg{"1/4", "1/3", "7/12"},
+	ratBinArg{"2/5", "-14/3", "-64/15"},
+	ratBinArg{"4707/49292519774798173060", "-3367/70976135186689855734", "84058377121001851123459/1749296273614329067191168098769082663020"},
+	ratBinArg{"-61204110018146728334/3", "-31052192278051565633/2", "-215564796870448153567/6"},
+}
+
+func TestRatAdd(t *testing.T) {
+	testRatBin(t, (*Rat).Add, ratAddTests)
+}
+
+
+var ratSubTests = []ratBinArg{
+	ratBinArg{"0", "0", "0"},
+	ratBinArg{"0", "1", "-1"},
+	ratBinArg{"-1", "0", "-1"},
+	ratBinArg{"-1", "1", "-2"},
+	ratBinArg{"1", "1", "0"},
+	ratBinArg{"1/2", "1/3", "1/6"},
+	ratBinArg{"1/4", "1/3", "-1/12"},
+	ratBinArg{"2/5", "-14/3", "76/15"},
+	ratBinArg{"4707/49292519774798173060", "-3367/70976135186689855734", "250026291202747299816479/1749296273614329067191168098769082663020"},
+	ratBinArg{"-27/133467566250814981", "-18/31750379913563777419", "-854857841473707320655/4237645934602118692642972629634714039"},
+	ratBinArg{"27674141753240653/30123979153216", "-19948846211000086/637313996471", "618575745270541348005638912139/19198433543745179392300736"},
+}
+
+func TestRatSub(t *testing.T) {
+	testRatBin(t, (*Rat).Sub, ratSubTests)
+}
+
+
+var ratMulTests = []ratBinArg{
+	ratBinArg{"0", "0", "0"},
+	ratBinArg{"0", "1", "0"},
+	ratBinArg{"-1", "0", "0"},
+	ratBinArg{"-1", "1", "-1"},
+	ratBinArg{"1", "1", "1"},
+	ratBinArg{"1/2", "1/2", "1/4"},
+	ratBinArg{"1/4", "1/3", "1/12"},
+	ratBinArg{"2/5", "-14/3", "-28/15"},
+	ratBinArg{"-3/26206484091896184128", "5/2848423294177090248", "-5/24882386581946146755650075889827061248"},
+	ratBinArg{"26946729/330400702820", "41563965/225583428284", "224002580204097/14906584649915733312176"},
+	ratBinArg{"-8259900599013409474/7", "-84829337473700364773/56707961321161574960", "350340947706464153265156004876107029701/198477864624065512360"},
+}
+
+func TestRatMul(t *testing.T) {
+	testRatBin(t, (*Rat).Mul, ratMulTests)
+}
+
+
+var ratQuoTests = []ratBinArg{
+	ratBinArg{"0", "1", "0"},
+	ratBinArg{"0", "-1", "0"},
+	ratBinArg{"-1", "1", "-1"},
+	ratBinArg{"1", "1", "1"},
+	ratBinArg{"1/2", "1/2", "1"},
+	ratBinArg{"1/4", "1/3", "3/4"},
+	ratBinArg{"2/5", "-14/3", "-3/35"},
+	ratBinArg{"808/45524274987585732633", "29/712593081308", "575775209696864/1320203974639986246357"},
+	ratBinArg{"8967230/3296219033", "6269770/1992362624741777", "1786597389946320496771/2066653520653241"},
+	ratBinArg{"-3784609207827/3426986245", "9381566963714/9633539", "-36459180403360509753/32150500941194292113930"},
+}
+
+func TestRatQuo(t *testing.T) {
+	testRatBin(t, (*Rat).Quo, ratQuoTests)
+}