Source file src/pkg/math/big/rat.go

     1	// Copyright 2010 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	// This file implements multi-precision rational numbers.
     6	
     7	package big
     8	
     9	import (
    10		"encoding/binary"
    11		"errors"
    12		"fmt"
    13		"math"
    14		"strings"
    15	)
    16	
    17	// A Rat represents a quotient a/b of arbitrary precision.
    18	// The zero value for a Rat represents the value 0.
    19	type Rat struct {
    20		// To make zero values for Rat work w/o initialization,
    21		// a zero value of b (len(b) == 0) acts like b == 1.
    22		// a.neg determines the sign of the Rat, b.neg is ignored.
    23		a, b Int
    24	}
    25	
    26	// NewRat creates a new Rat with numerator a and denominator b.
    27	func NewRat(a, b int64) *Rat {
    28		return new(Rat).SetFrac64(a, b)
    29	}
    30	
    31	// SetFloat64 sets z to exactly f and returns z.
    32	// If f is not finite, SetFloat returns nil.
    33	func (z *Rat) SetFloat64(f float64) *Rat {
    34		const expMask = 1<<11 - 1
    35		bits := math.Float64bits(f)
    36		mantissa := bits & (1<<52 - 1)
    37		exp := int((bits >> 52) & expMask)
    38		switch exp {
    39		case expMask: // non-finite
    40			return nil
    41		case 0: // denormal
    42			exp -= 1022
    43		default: // normal
    44			mantissa |= 1 << 52
    45			exp -= 1023
    46		}
    47	
    48		shift := 52 - exp
    49	
    50		// Optimisation (?): partially pre-normalise.
    51		for mantissa&1 == 0 && shift > 0 {
    52			mantissa >>= 1
    53			shift--
    54		}
    55	
    56		z.a.SetUint64(mantissa)
    57		z.a.neg = f < 0
    58		z.b.Set(intOne)
    59		if shift > 0 {
    60			z.b.Lsh(&z.b, uint(shift))
    61		} else {
    62			z.a.Lsh(&z.a, uint(-shift))
    63		}
    64		return z.norm()
    65	}
    66	
    67	// isFinite reports whether f represents a finite rational value.
    68	// It is equivalent to !math.IsNan(f) && !math.IsInf(f, 0).
    69	func isFinite(f float64) bool {
    70		return math.Abs(f) <= math.MaxFloat64
    71	}
    72	
    73	// low64 returns the least significant 64 bits of natural number z.
    74	func low64(z nat) uint64 {
    75		if len(z) == 0 {
    76			return 0
    77		}
    78		if _W == 32 && len(z) > 1 {
    79			return uint64(z[1])<<32 | uint64(z[0])
    80		}
    81		return uint64(z[0])
    82	}
    83	
    84	// quotToFloat returns the non-negative IEEE 754 double-precision
    85	// value nearest to the quotient a/b, using round-to-even in halfway
    86	// cases.  It does not mutate its arguments.
    87	// Preconditions: b is non-zero; a and b have no common factors.
    88	func quotToFloat(a, b nat) (f float64, exact bool) {
    89		// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
    90		alen := a.bitLen()
    91		if alen == 0 {
    92			return 0, true
    93		}
    94		blen := b.bitLen()
    95		if blen == 0 {
    96			panic("division by zero")
    97		}
    98	
    99		// 1. Left-shift A or B such that quotient A/B is in [1<<53, 1<<55).
   100		// (54 bits if A<B when they are left-aligned, 55 bits if A>=B.)
   101		// This is 2 or 3 more than the float64 mantissa field width of 52:
   102		// - the optional extra bit is shifted away in step 3 below.
   103		// - the high-order 1 is omitted in float64 "normal" representation;
   104		// - the low-order 1 will be used during rounding then discarded.
   105		exp := alen - blen
   106		var a2, b2 nat
   107		a2 = a2.set(a)
   108		b2 = b2.set(b)
   109		if shift := 54 - exp; shift > 0 {
   110			a2 = a2.shl(a2, uint(shift))
   111		} else if shift < 0 {
   112			b2 = b2.shl(b2, uint(-shift))
   113		}
   114	
   115		// 2. Compute quotient and remainder (q, r).  NB: due to the
   116		// extra shift, the low-order bit of q is logically the
   117		// high-order bit of r.
   118		var q nat
   119		q, r := q.div(a2, a2, b2) // (recycle a2)
   120		mantissa := low64(q)
   121		haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
   122	
   123		// 3. If quotient didn't fit in 54 bits, re-do division by b2<<1
   124		// (in effect---we accomplish this incrementally).
   125		if mantissa>>54 == 1 {
   126			if mantissa&1 == 1 {
   127				haveRem = true
   128			}
   129			mantissa >>= 1
   130			exp++
   131		}
   132		if mantissa>>53 != 1 {
   133			panic("expected exactly 54 bits of result")
   134		}
   135	
   136		// 4. Rounding.
   137		if -1022-52 <= exp && exp <= -1022 {
   138			// Denormal case; lose 'shift' bits of precision.
   139			shift := uint64(-1022 - (exp - 1)) // [1..53)
   140			lostbits := mantissa & (1<<shift - 1)
   141			haveRem = haveRem || lostbits != 0
   142			mantissa >>= shift
   143			exp = -1023 + 2
   144		}
   145		// Round q using round-half-to-even.
   146		exact = !haveRem
   147		if mantissa&1 != 0 {
   148			exact = false
   149			if haveRem || mantissa&2 != 0 {
   150				if mantissa++; mantissa >= 1<<54 {
   151					// Complete rollover 11...1 => 100...0, so shift is safe
   152					mantissa >>= 1
   153					exp++
   154				}
   155			}
   156		}
   157		mantissa >>= 1 // discard rounding bit.  Mantissa now scaled by 2^53.
   158	
   159		f = math.Ldexp(float64(mantissa), exp-53)
   160		if math.IsInf(f, 0) {
   161			exact = false
   162		}
   163		return
   164	}
   165	
   166	// Float64 returns the nearest float64 value for x and a bool indicating
   167	// whether f represents x exactly. The sign of f always matches the sign
   168	// of x, even if f == 0.
   169	func (x *Rat) Float64() (f float64, exact bool) {
   170		b := x.b.abs
   171		if len(b) == 0 {
   172			b = b.set(natOne) // materialize denominator
   173		}
   174		f, exact = quotToFloat(x.a.abs, b)
   175		if x.a.neg {
   176			f = -f
   177		}
   178		return
   179	}
   180	
   181	// SetFrac sets z to a/b and returns z.
   182	func (z *Rat) SetFrac(a, b *Int) *Rat {
   183		z.a.neg = a.neg != b.neg
   184		babs := b.abs
   185		if len(babs) == 0 {
   186			panic("division by zero")
   187		}
   188		if &z.a == b || alias(z.a.abs, babs) {
   189			babs = nat(nil).set(babs) // make a copy
   190		}
   191		z.a.abs = z.a.abs.set(a.abs)
   192		z.b.abs = z.b.abs.set(babs)
   193		return z.norm()
   194	}
   195	
   196	// SetFrac64 sets z to a/b and returns z.
   197	func (z *Rat) SetFrac64(a, b int64) *Rat {
   198		z.a.SetInt64(a)
   199		if b == 0 {
   200			panic("division by zero")
   201		}
   202		if b < 0 {
   203			b = -b
   204			z.a.neg = !z.a.neg
   205		}
   206		z.b.abs = z.b.abs.setUint64(uint64(b))
   207		return z.norm()
   208	}
   209	
   210	// SetInt sets z to x (by making a copy of x) and returns z.
   211	func (z *Rat) SetInt(x *Int) *Rat {
   212		z.a.Set(x)
   213		z.b.abs = z.b.abs.make(0)
   214		return z
   215	}
   216	
   217	// SetInt64 sets z to x and returns z.
   218	func (z *Rat) SetInt64(x int64) *Rat {
   219		z.a.SetInt64(x)
   220		z.b.abs = z.b.abs.make(0)
   221		return z
   222	}
   223	
   224	// Set sets z to x (by making a copy of x) and returns z.
   225	func (z *Rat) Set(x *Rat) *Rat {
   226		if z != x {
   227			z.a.Set(&x.a)
   228			z.b.Set(&x.b)
   229		}
   230		return z
   231	}
   232	
   233	// Abs sets z to |x| (the absolute value of x) and returns z.
   234	func (z *Rat) Abs(x *Rat) *Rat {
   235		z.Set(x)
   236		z.a.neg = false
   237		return z
   238	}
   239	
   240	// Neg sets z to -x and returns z.
   241	func (z *Rat) Neg(x *Rat) *Rat {
   242		z.Set(x)
   243		z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
   244		return z
   245	}
   246	
   247	// Inv sets z to 1/x and returns z.
   248	func (z *Rat) Inv(x *Rat) *Rat {
   249		if len(x.a.abs) == 0 {
   250			panic("division by zero")
   251		}
   252		z.Set(x)
   253		a := z.b.abs
   254		if len(a) == 0 {
   255			a = a.set(natOne) // materialize numerator
   256		}
   257		b := z.a.abs
   258		if b.cmp(natOne) == 0 {
   259			b = b.make(0) // normalize denominator
   260		}
   261		z.a.abs, z.b.abs = a, b // sign doesn't change
   262		return z
   263	}
   264	
   265	// Sign returns:
   266	//
   267	//	-1 if x <  0
   268	//	 0 if x == 0
   269	//	+1 if x >  0
   270	//
   271	func (x *Rat) Sign() int {
   272		return x.a.Sign()
   273	}
   274	
   275	// IsInt returns true if the denominator of x is 1.
   276	func (x *Rat) IsInt() bool {
   277		return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
   278	}
   279	
   280	// Num returns the numerator of x; it may be <= 0.
   281	// The result is a reference to x's numerator; it
   282	// may change if a new value is assigned to x, and vice versa.
   283	// The sign of the numerator corresponds to the sign of x.
   284	func (x *Rat) Num() *Int {
   285		return &x.a
   286	}
   287	
   288	// Denom returns the denominator of x; it is always > 0.
   289	// The result is a reference to x's denominator; it
   290	// may change if a new value is assigned to x, and vice versa.
   291	func (x *Rat) Denom() *Int {
   292		x.b.neg = false // the result is always >= 0
   293		if len(x.b.abs) == 0 {
   294			x.b.abs = x.b.abs.set(natOne) // materialize denominator
   295		}
   296		return &x.b
   297	}
   298	
   299	func (z *Rat) norm() *Rat {
   300		switch {
   301		case len(z.a.abs) == 0:
   302			// z == 0 - normalize sign and denominator
   303			z.a.neg = false
   304			z.b.abs = z.b.abs.make(0)
   305		case len(z.b.abs) == 0:
   306			// z is normalized int - nothing to do
   307		case z.b.abs.cmp(natOne) == 0:
   308			// z is int - normalize denominator
   309			z.b.abs = z.b.abs.make(0)
   310		default:
   311			neg := z.a.neg
   312			z.a.neg = false
   313			z.b.neg = false
   314			if f := NewInt(0).binaryGCD(&z.a, &z.b); f.Cmp(intOne) != 0 {
   315				z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
   316				z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
   317				if z.b.abs.cmp(natOne) == 0 {
   318					// z is int - normalize denominator
   319					z.b.abs = z.b.abs.make(0)
   320				}
   321			}
   322			z.a.neg = neg
   323		}
   324		return z
   325	}
   326	
   327	// mulDenom sets z to the denominator product x*y (by taking into
   328	// account that 0 values for x or y must be interpreted as 1) and
   329	// returns z.
   330	func mulDenom(z, x, y nat) nat {
   331		switch {
   332		case len(x) == 0:
   333			return z.set(y)
   334		case len(y) == 0:
   335			return z.set(x)
   336		}
   337		return z.mul(x, y)
   338	}
   339	
   340	// scaleDenom computes x*f.
   341	// If f == 0 (zero value of denominator), the result is (a copy of) x.
   342	func scaleDenom(x *Int, f nat) *Int {
   343		var z Int
   344		if len(f) == 0 {
   345			return z.Set(x)
   346		}
   347		z.abs = z.abs.mul(x.abs, f)
   348		z.neg = x.neg
   349		return &z
   350	}
   351	
   352	// Cmp compares x and y and returns:
   353	//
   354	//   -1 if x <  y
   355	//    0 if x == y
   356	//   +1 if x >  y
   357	//
   358	func (x *Rat) Cmp(y *Rat) int {
   359		return scaleDenom(&x.a, y.b.abs).Cmp(scaleDenom(&y.a, x.b.abs))
   360	}
   361	
   362	// Add sets z to the sum x+y and returns z.
   363	func (z *Rat) Add(x, y *Rat) *Rat {
   364		a1 := scaleDenom(&x.a, y.b.abs)
   365		a2 := scaleDenom(&y.a, x.b.abs)
   366		z.a.Add(a1, a2)
   367		z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   368		return z.norm()
   369	}
   370	
   371	// Sub sets z to the difference x-y and returns z.
   372	func (z *Rat) Sub(x, y *Rat) *Rat {
   373		a1 := scaleDenom(&x.a, y.b.abs)
   374		a2 := scaleDenom(&y.a, x.b.abs)
   375		z.a.Sub(a1, a2)
   376		z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   377		return z.norm()
   378	}
   379	
   380	// Mul sets z to the product x*y and returns z.
   381	func (z *Rat) Mul(x, y *Rat) *Rat {
   382		z.a.Mul(&x.a, &y.a)
   383		z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   384		return z.norm()
   385	}
   386	
   387	// Quo sets z to the quotient x/y and returns z.
   388	// If y == 0, a division-by-zero run-time panic occurs.
   389	func (z *Rat) Quo(x, y *Rat) *Rat {
   390		if len(y.a.abs) == 0 {
   391			panic("division by zero")
   392		}
   393		a := scaleDenom(&x.a, y.b.abs)
   394		b := scaleDenom(&y.a, x.b.abs)
   395		z.a.abs = a.abs
   396		z.b.abs = b.abs
   397		z.a.neg = a.neg != b.neg
   398		return z.norm()
   399	}
   400	
   401	func ratTok(ch rune) bool {
   402		return strings.IndexRune("+-/0123456789.eE", ch) >= 0
   403	}
   404	
   405	// Scan is a support routine for fmt.Scanner. It accepts the formats
   406	// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
   407	func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
   408		tok, err := s.Token(true, ratTok)
   409		if err != nil {
   410			return err
   411		}
   412		if strings.IndexRune("efgEFGv", ch) < 0 {
   413			return errors.New("Rat.Scan: invalid verb")
   414		}
   415		if _, ok := z.SetString(string(tok)); !ok {
   416			return errors.New("Rat.Scan: invalid syntax")
   417		}
   418		return nil
   419	}
   420	
   421	// SetString sets z to the value of s and returns z and a boolean indicating
   422	// success. s can be given as a fraction "a/b" or as a floating-point number
   423	// optionally followed by an exponent. If the operation failed, the value of
   424	// z is undefined but the returned value is nil.
   425	func (z *Rat) SetString(s string) (*Rat, bool) {
   426		if len(s) == 0 {
   427			return nil, false
   428		}
   429	
   430		// check for a quotient
   431		sep := strings.Index(s, "/")
   432		if sep >= 0 {
   433			if _, ok := z.a.SetString(s[0:sep], 10); !ok {
   434				return nil, false
   435			}
   436			s = s[sep+1:]
   437			var err error
   438			if z.b.abs, _, err = z.b.abs.scan(strings.NewReader(s), 10); err != nil {
   439				return nil, false
   440			}
   441			return z.norm(), true
   442		}
   443	
   444		// check for a decimal point
   445		sep = strings.Index(s, ".")
   446		// check for an exponent
   447		e := strings.IndexAny(s, "eE")
   448		var exp Int
   449		if e >= 0 {
   450			if e < sep {
   451				// The E must come after the decimal point.
   452				return nil, false
   453			}
   454			if _, ok := exp.SetString(s[e+1:], 10); !ok {
   455				return nil, false
   456			}
   457			s = s[0:e]
   458		}
   459		if sep >= 0 {
   460			s = s[0:sep] + s[sep+1:]
   461			exp.Sub(&exp, NewInt(int64(len(s)-sep)))
   462		}
   463	
   464		if _, ok := z.a.SetString(s, 10); !ok {
   465			return nil, false
   466		}
   467		powTen := nat(nil).expNN(natTen, exp.abs, nil)
   468		if exp.neg {
   469			z.b.abs = powTen
   470			z.norm()
   471		} else {
   472			z.a.abs = z.a.abs.mul(z.a.abs, powTen)
   473			z.b.abs = z.b.abs.make(0)
   474		}
   475	
   476		return z, true
   477	}
   478	
   479	// String returns a string representation of z in the form "a/b" (even if b == 1).
   480	func (x *Rat) String() string {
   481		s := "/1"
   482		if len(x.b.abs) != 0 {
   483			s = "/" + x.b.abs.decimalString()
   484		}
   485		return x.a.String() + s
   486	}
   487	
   488	// RatString returns a string representation of z in the form "a/b" if b != 1,
   489	// and in the form "a" if b == 1.
   490	func (x *Rat) RatString() string {
   491		if x.IsInt() {
   492			return x.a.String()
   493		}
   494		return x.String()
   495	}
   496	
   497	// FloatString returns a string representation of z in decimal form with prec
   498	// digits of precision after the decimal point and the last digit rounded.
   499	func (x *Rat) FloatString(prec int) string {
   500		if x.IsInt() {
   501			s := x.a.String()
   502			if prec > 0 {
   503				s += "." + strings.Repeat("0", prec)
   504			}
   505			return s
   506		}
   507		// x.b.abs != 0
   508	
   509		q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)
   510	
   511		p := natOne
   512		if prec > 0 {
   513			p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil)
   514		}
   515	
   516		r = r.mul(r, p)
   517		r, r2 := r.div(nat(nil), r, x.b.abs)
   518	
   519		// see if we need to round up
   520		r2 = r2.add(r2, r2)
   521		if x.b.abs.cmp(r2) <= 0 {
   522			r = r.add(r, natOne)
   523			if r.cmp(p) >= 0 {
   524				q = nat(nil).add(q, natOne)
   525				r = nat(nil).sub(r, p)
   526			}
   527		}
   528	
   529		s := q.decimalString()
   530		if x.a.neg {
   531			s = "-" + s
   532		}
   533	
   534		if prec > 0 {
   535			rs := r.decimalString()
   536			leadingZeros := prec - len(rs)
   537			s += "." + strings.Repeat("0", leadingZeros) + rs
   538		}
   539	
   540		return s
   541	}
   542	
   543	// Gob codec version. Permits backward-compatible changes to the encoding.
   544	const ratGobVersion byte = 1
   545	
   546	// GobEncode implements the gob.GobEncoder interface.
   547	func (x *Rat) GobEncode() ([]byte, error) {
   548		buf := make([]byte, 1+4+(len(x.a.abs)+len(x.b.abs))*_S) // extra bytes for version and sign bit (1), and numerator length (4)
   549		i := x.b.abs.bytes(buf)
   550		j := x.a.abs.bytes(buf[0:i])
   551		n := i - j
   552		if int(uint32(n)) != n {
   553			// this should never happen
   554			return nil, errors.New("Rat.GobEncode: numerator too large")
   555		}
   556		binary.BigEndian.PutUint32(buf[j-4:j], uint32(n))
   557		j -= 1 + 4
   558		b := ratGobVersion << 1 // make space for sign bit
   559		if x.a.neg {
   560			b |= 1
   561		}
   562		buf[j] = b
   563		return buf[j:], nil
   564	}
   565	
   566	// GobDecode implements the gob.GobDecoder interface.
   567	func (z *Rat) GobDecode(buf []byte) error {
   568		if len(buf) == 0 {
   569			return errors.New("Rat.GobDecode: no data")
   570		}
   571		b := buf[0]
   572		if b>>1 != ratGobVersion {
   573			return errors.New(fmt.Sprintf("Rat.GobDecode: encoding version %d not supported", b>>1))
   574		}
   575		const j = 1 + 4
   576		i := j + binary.BigEndian.Uint32(buf[j-4:j])
   577		z.a.neg = b&1 != 0
   578		z.a.abs = z.a.abs.setBytes(buf[j:i])
   579		z.b.abs = z.b.abs.setBytes(buf[i:])
   580		return nil
   581	}

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