rat.go

Documentation: math/big

  // 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.
  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(&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()
  }
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