src/math/big/rat.go - go - Git at Google

 // 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 {
 	// Note that x.b.neg is guaranteed false.
 	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()
 }
	// 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 {
	// Note that x.b.neg is guaranteed false.
	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()
	}