511 lines
12 KiB
Go
511 lines
12 KiB
Go
// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// This file implements multi-precision rational numbers.
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package big
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import (
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"fmt"
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"math"
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)
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// A Rat represents a quotient a/b of arbitrary precision.
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// The zero value for a Rat represents the value 0.
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type Rat struct {
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// To make zero values for Rat work w/o initialization,
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// a zero value of b (len(b) == 0) acts like b == 1.
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// a.neg determines the sign of the Rat, b.neg is ignored.
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a, b Int
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}
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// NewRat creates a new Rat with numerator a and denominator b.
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func NewRat(a, b int64) *Rat {
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return new(Rat).SetFrac64(a, b)
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}
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// SetFloat64 sets z to exactly f and returns z.
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// If f is not finite, SetFloat returns nil.
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func (z *Rat) SetFloat64(f float64) *Rat {
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const expMask = 1<<11 - 1
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bits := math.Float64bits(f)
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mantissa := bits & (1<<52 - 1)
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exp := int((bits >> 52) & expMask)
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switch exp {
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case expMask: // non-finite
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return nil
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case 0: // denormal
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exp -= 1022
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default: // normal
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mantissa |= 1 << 52
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exp -= 1023
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}
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shift := 52 - exp
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// Optimization (?): partially pre-normalise.
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for mantissa&1 == 0 && shift > 0 {
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mantissa >>= 1
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shift--
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}
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z.a.SetUint64(mantissa)
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z.a.neg = f < 0
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z.b.Set(intOne)
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if shift > 0 {
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z.b.Lsh(&z.b, uint(shift))
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} else {
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z.a.Lsh(&z.a, uint(-shift))
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}
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return z.norm()
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}
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// quotToFloat32 returns the non-negative float32 value
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// nearest to the quotient a/b, using round-to-even in
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// halfway cases. It does not mutate its arguments.
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// Preconditions: b is non-zero; a and b have no common factors.
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func quotToFloat32(a, b nat) (f float32, exact bool) {
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const (
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// float size in bits
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Fsize = 32
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// mantissa
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Msize = 23
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Msize1 = Msize + 1 // incl. implicit 1
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Msize2 = Msize1 + 1
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// exponent
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Esize = Fsize - Msize1
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Ebias = 1<<(Esize-1) - 1
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Emin = 1 - Ebias
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Emax = Ebias
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)
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// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
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alen := a.bitLen()
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if alen == 0 {
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return 0, true
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}
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blen := b.bitLen()
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if blen == 0 {
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panic("division by zero")
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}
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// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
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// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
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// This is 2 or 3 more than the float32 mantissa field width of Msize:
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// - the optional extra bit is shifted away in step 3 below.
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// - the high-order 1 is omitted in "normal" representation;
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// - the low-order 1 will be used during rounding then discarded.
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exp := alen - blen
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var a2, b2 nat
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a2 = a2.set(a)
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b2 = b2.set(b)
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if shift := Msize2 - exp; shift > 0 {
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a2 = a2.shl(a2, uint(shift))
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} else if shift < 0 {
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b2 = b2.shl(b2, uint(-shift))
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}
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// 2. Compute quotient and remainder (q, r). NB: due to the
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// extra shift, the low-order bit of q is logically the
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// high-order bit of r.
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var q nat
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q, r := q.div(a2, a2, b2) // (recycle a2)
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mantissa := low32(q)
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haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
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// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
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// (in effect---we accomplish this incrementally).
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if mantissa>>Msize2 == 1 {
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if mantissa&1 == 1 {
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haveRem = true
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}
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mantissa >>= 1
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exp++
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}
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if mantissa>>Msize1 != 1 {
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panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
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}
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// 4. Rounding.
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if Emin-Msize <= exp && exp <= Emin {
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// Denormal case; lose 'shift' bits of precision.
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shift := uint(Emin - (exp - 1)) // [1..Esize1)
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lostbits := mantissa & (1<<shift - 1)
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haveRem = haveRem || lostbits != 0
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mantissa >>= shift
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exp = 2 - Ebias // == exp + shift
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}
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// Round q using round-half-to-even.
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exact = !haveRem
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if mantissa&1 != 0 {
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exact = false
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if haveRem || mantissa&2 != 0 {
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if mantissa++; mantissa >= 1<<Msize2 {
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// Complete rollover 11...1 => 100...0, so shift is safe
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mantissa >>= 1
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exp++
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}
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}
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}
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mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
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f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
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if math.IsInf(float64(f), 0) {
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exact = false
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}
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return
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}
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// quotToFloat64 returns the non-negative float64 value
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// nearest to the quotient a/b, using round-to-even in
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// halfway cases. It does not mutate its arguments.
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// Preconditions: b is non-zero; a and b have no common factors.
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func quotToFloat64(a, b nat) (f float64, exact bool) {
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const (
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// float size in bits
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Fsize = 64
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// mantissa
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Msize = 52
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Msize1 = Msize + 1 // incl. implicit 1
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Msize2 = Msize1 + 1
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// exponent
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Esize = Fsize - Msize1
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Ebias = 1<<(Esize-1) - 1
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Emin = 1 - Ebias
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Emax = Ebias
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)
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// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
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alen := a.bitLen()
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if alen == 0 {
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return 0, true
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}
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blen := b.bitLen()
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if blen == 0 {
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panic("division by zero")
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}
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// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
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// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
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// This is 2 or 3 more than the float64 mantissa field width of Msize:
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// - the optional extra bit is shifted away in step 3 below.
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// - the high-order 1 is omitted in "normal" representation;
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// - the low-order 1 will be used during rounding then discarded.
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exp := alen - blen
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var a2, b2 nat
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a2 = a2.set(a)
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b2 = b2.set(b)
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if shift := Msize2 - exp; shift > 0 {
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a2 = a2.shl(a2, uint(shift))
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} else if shift < 0 {
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b2 = b2.shl(b2, uint(-shift))
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}
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// 2. Compute quotient and remainder (q, r). NB: due to the
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// extra shift, the low-order bit of q is logically the
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// high-order bit of r.
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var q nat
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q, r := q.div(a2, a2, b2) // (recycle a2)
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mantissa := low64(q)
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haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
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// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
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// (in effect---we accomplish this incrementally).
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if mantissa>>Msize2 == 1 {
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if mantissa&1 == 1 {
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haveRem = true
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}
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mantissa >>= 1
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exp++
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}
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if mantissa>>Msize1 != 1 {
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panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
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}
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// 4. Rounding.
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if Emin-Msize <= exp && exp <= Emin {
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// Denormal case; lose 'shift' bits of precision.
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shift := uint(Emin - (exp - 1)) // [1..Esize1)
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lostbits := mantissa & (1<<shift - 1)
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haveRem = haveRem || lostbits != 0
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mantissa >>= shift
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exp = 2 - Ebias // == exp + shift
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}
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// Round q using round-half-to-even.
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exact = !haveRem
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if mantissa&1 != 0 {
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exact = false
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if haveRem || mantissa&2 != 0 {
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if mantissa++; mantissa >= 1<<Msize2 {
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// Complete rollover 11...1 => 100...0, so shift is safe
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mantissa >>= 1
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exp++
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}
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}
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}
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mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
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f = math.Ldexp(float64(mantissa), exp-Msize1)
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if math.IsInf(f, 0) {
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exact = false
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}
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return
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}
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// Float32 returns the nearest float32 value for x and a bool indicating
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// whether f represents x exactly. If the magnitude of x is too large to
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// be represented by a float32, f is an infinity and exact is false.
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// The sign of f always matches the sign of x, even if f == 0.
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func (x *Rat) Float32() (f float32, exact bool) {
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b := x.b.abs
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if len(b) == 0 {
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b = b.set(natOne) // materialize denominator
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}
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f, exact = quotToFloat32(x.a.abs, b)
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if x.a.neg {
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f = -f
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}
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return
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}
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// Float64 returns the nearest float64 value for x and a bool indicating
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// whether f represents x exactly. If the magnitude of x is too large to
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// be represented by a float64, f is an infinity and exact is false.
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// The sign of f always matches the sign of x, even if f == 0.
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func (x *Rat) Float64() (f float64, exact bool) {
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b := x.b.abs
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if len(b) == 0 {
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b = b.set(natOne) // materialize denominator
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}
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f, exact = quotToFloat64(x.a.abs, b)
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if x.a.neg {
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f = -f
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}
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return
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}
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// SetFrac sets z to a/b and returns z.
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func (z *Rat) SetFrac(a, b *Int) *Rat {
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z.a.neg = a.neg != b.neg
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babs := b.abs
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if len(babs) == 0 {
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panic("division by zero")
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}
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if &z.a == b || alias(z.a.abs, babs) {
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babs = nat(nil).set(babs) // make a copy
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}
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z.a.abs = z.a.abs.set(a.abs)
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z.b.abs = z.b.abs.set(babs)
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return z.norm()
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}
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// SetFrac64 sets z to a/b and returns z.
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func (z *Rat) SetFrac64(a, b int64) *Rat {
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z.a.SetInt64(a)
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if b == 0 {
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panic("division by zero")
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}
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if b < 0 {
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b = -b
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z.a.neg = !z.a.neg
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}
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z.b.abs = z.b.abs.setUint64(uint64(b))
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return z.norm()
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}
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// SetInt sets z to x (by making a copy of x) and returns z.
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func (z *Rat) SetInt(x *Int) *Rat {
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z.a.Set(x)
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z.b.abs = z.b.abs[:0]
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return z
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}
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// SetInt64 sets z to x and returns z.
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func (z *Rat) SetInt64(x int64) *Rat {
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z.a.SetInt64(x)
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z.b.abs = z.b.abs[:0]
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return z
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}
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// Set sets z to x (by making a copy of x) and returns z.
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func (z *Rat) Set(x *Rat) *Rat {
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if z != x {
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z.a.Set(&x.a)
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z.b.Set(&x.b)
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}
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return z
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}
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// Abs sets z to |x| (the absolute value of x) and returns z.
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func (z *Rat) Abs(x *Rat) *Rat {
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z.Set(x)
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z.a.neg = false
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return z
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}
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// Neg sets z to -x and returns z.
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func (z *Rat) Neg(x *Rat) *Rat {
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z.Set(x)
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z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
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return z
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}
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// Inv sets z to 1/x and returns z.
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func (z *Rat) Inv(x *Rat) *Rat {
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if len(x.a.abs) == 0 {
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panic("division by zero")
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}
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z.Set(x)
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a := z.b.abs
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if len(a) == 0 {
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a = a.set(natOne) // materialize numerator
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}
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b := z.a.abs
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if b.cmp(natOne) == 0 {
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b = b[:0] // normalize denominator
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}
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z.a.abs, z.b.abs = a, b // sign doesn't change
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return z
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}
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// Sign returns:
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//
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// -1 if x < 0
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// 0 if x == 0
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// +1 if x > 0
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//
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func (x *Rat) Sign() int {
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return x.a.Sign()
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}
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// IsInt reports whether the denominator of x is 1.
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func (x *Rat) IsInt() bool {
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return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
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}
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// Num returns the numerator of x; it may be <= 0.
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// The result is a reference to x's numerator; it
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// may change if a new value is assigned to x, and vice versa.
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// The sign of the numerator corresponds to the sign of x.
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func (x *Rat) Num() *Int {
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return &x.a
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}
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// Denom returns the denominator of x; it is always > 0.
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// The result is a reference to x's denominator; it
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// may change if a new value is assigned to x, and vice versa.
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func (x *Rat) Denom() *Int {
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x.b.neg = false // the result is always >= 0
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if len(x.b.abs) == 0 {
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x.b.abs = x.b.abs.set(natOne) // materialize denominator
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}
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return &x.b
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}
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func (z *Rat) norm() *Rat {
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switch {
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case len(z.a.abs) == 0:
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// z == 0 - normalize sign and denominator
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z.a.neg = false
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z.b.abs = z.b.abs[:0]
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case len(z.b.abs) == 0:
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// z is normalized int - nothing to do
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case z.b.abs.cmp(natOne) == 0:
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// z is int - normalize denominator
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z.b.abs = z.b.abs[:0]
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default:
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neg := z.a.neg
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z.a.neg = false
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z.b.neg = false
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if f := NewInt(0).binaryGCD(&z.a, &z.b); f.Cmp(intOne) != 0 {
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z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
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z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
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if z.b.abs.cmp(natOne) == 0 {
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// z is int - normalize denominator
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z.b.abs = z.b.abs[:0]
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}
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}
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z.a.neg = neg
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}
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return z
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}
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// mulDenom sets z to the denominator product x*y (by taking into
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// account that 0 values for x or y must be interpreted as 1) and
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// returns z.
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func mulDenom(z, x, y nat) nat {
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switch {
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case len(x) == 0:
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return z.set(y)
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case len(y) == 0:
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return z.set(x)
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}
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return z.mul(x, y)
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}
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// scaleDenom computes x*f.
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// If f == 0 (zero value of denominator), the result is (a copy of) x.
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func scaleDenom(x *Int, f nat) *Int {
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var z Int
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if len(f) == 0 {
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return z.Set(x)
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}
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z.abs = z.abs.mul(x.abs, f)
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z.neg = x.neg
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return &z
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}
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// Cmp compares x and y and returns:
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//
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// -1 if x < y
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// 0 if x == y
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// +1 if x > y
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//
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func (x *Rat) Cmp(y *Rat) int {
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return scaleDenom(&x.a, y.b.abs).Cmp(scaleDenom(&y.a, x.b.abs))
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}
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// Add sets z to the sum x+y and returns z.
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func (z *Rat) Add(x, y *Rat) *Rat {
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a1 := scaleDenom(&x.a, y.b.abs)
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a2 := scaleDenom(&y.a, x.b.abs)
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z.a.Add(a1, a2)
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z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
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return z.norm()
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}
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// Sub sets z to the difference x-y and returns z.
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func (z *Rat) Sub(x, y *Rat) *Rat {
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a1 := scaleDenom(&x.a, y.b.abs)
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a2 := scaleDenom(&y.a, x.b.abs)
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z.a.Sub(a1, a2)
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z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
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return z.norm()
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}
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// Mul sets z to the product x*y and returns z.
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func (z *Rat) Mul(x, y *Rat) *Rat {
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z.a.Mul(&x.a, &y.a)
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z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
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return z.norm()
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}
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// Quo sets z to the quotient x/y and returns z.
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// If y == 0, a division-by-zero run-time panic occurs.
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func (z *Rat) Quo(x, y *Rat) *Rat {
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if len(y.a.abs) == 0 {
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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()
|
|
}
|