1 files changed, 544 insertions, 0 deletions
diff --git a/src/math/big/rat.go b/src/math/big/rat.go
new file mode 100644
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--- /dev/null
+++ b/src/math/big/rat.go
@@ -0,0 +1,544 @@
+// 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 (
+	"fmt"
+	"math"
+)
+
+// A Rat represents a quotient a/b of arbitrary precision.
+// The zero value for a Rat represents the value 0.
+//
+// Operations always take pointer arguments (*Rat) rather
+// than Rat values, and each unique Rat value requires
+// its own unique *Rat pointer. To "copy" a Rat value,
+// an existing (or newly allocated) Rat must be set to
+// a new value using the Rat.Set method; shallow copies
+// of Rats are not supported and may lead to errors.
+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. At
+	// the earliest opportunity (when an assignment to the Rat
+	// is made), such uninitialized denominators are set to 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 = natOne
+	}
+	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 = natOne
+	}
+	f, exact = quotToFloat64(x.a.abs, b)
+	if x.a.neg {
+		f = -f
+	}
+	return
+}
+
+// SetFrac sets z to a/b and returns z.
+// If b == 0, SetFrac panics.
+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.
+// If b == 0, SetFrac64 panics.
+func (z *Rat) SetFrac64(a, b int64) *Rat {
+	if b == 0 {
+		panic("division by zero")
+	}
+	z.a.SetInt64(a)
+	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.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.abs = z.b.abs.setWord(1)
+	return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Rat) SetUint64(x uint64) *Rat {
+	z.a.SetUint64(x)
+	z.b.abs = z.b.abs.setWord(1)
+	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)
+	}
+	if len(z.b.abs) == 0 {
+		z.b.abs = z.b.abs.setWord(1)
+	}
+	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.
+// If x == 0, Inv panics.
+func (z *Rat) Inv(x *Rat) *Rat {
+	if len(x.a.abs) == 0 {
+		panic("division by zero")
+	}
+	z.Set(x)
+	z.a.abs, z.b.abs = z.b.abs, z.a.abs
+	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, unless
+// x is an uninitialized (zero value) Rat, in which case
+// the result is a new Int of value 1. (To initialize x,
+// any operation that sets x will do, including x.Set(x).)
+// If 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 {
+		// Note: If this proves problematic, we could
+		//       panic instead and require the Rat to
+		//       be explicitly initialized.
+		return &Int{abs: nat{1}}
+	}
+	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
+		fallthrough
+	case len(z.b.abs) == 0:
+		// z is integer; normalize denominator
+		z.b.abs = z.b.abs.setWord(1)
+	default:
+		// z is fraction; normalize numerator and denominator
+		neg := z.a.neg
+		z.a.neg = false
+		z.b.neg = false
+		if f := NewInt(0).lehmerGCD(nil, nil, &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)
+		}
+		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 && len(y) == 0:
+		return z.setWord(1)
+	case len(x) == 0:
+		return z.set(y)
+	case len(y) == 0:
+		return z.set(x)
+	}
+	return z.mul(x, y)
+}
+
+// scaleDenom sets z to the product x*f.
+// If f == 0 (zero value of denominator), z is set to (a copy of) x.
+func (z *Int) scaleDenom(x *Int, f nat) {
+	if len(f) == 0 {
+		z.Set(x)
+		return
+	}
+	z.abs = z.abs.mul(x.abs, f)
+	z.neg = x.neg
+}
+
+// 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 {
+	var a, b Int
+	a.scaleDenom(&x.a, y.b.abs)
+	b.scaleDenom(&y.a, x.b.abs)
+	return a.Cmp(&b)
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+	var a1, a2 Int
+	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 {
+	var a1, a2 Int
+	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 {
+	if x == y {
+		// a squared Rat is positive and can't be reduced (no need to call norm())
+		z.a.neg = false
+		z.a.abs = z.a.abs.sqr(x.a.abs)
+		if len(x.b.abs) == 0 {
+			z.b.abs = z.b.abs.setWord(1)
+		} else {
+			z.b.abs = z.b.abs.sqr(x.b.abs)
+		}
+		return z
+	}
+	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, Quo panics.
+func (z *Rat) Quo(x, y *Rat) *Rat {
+	if len(y.a.abs) == 0 {
+		panic("division by zero")
+	}
+	var a, b Int
+	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()
+}