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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 (
"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.
// 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).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)
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 {
if x == y {
// a squared Rat is positive and can't be reduced
z.a.neg = false
z.a.abs = z.a.abs.sqr(x.a.abs)
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, 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()
}