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  	"fmt"
    11  	"math"
    12  )
    13  
    14  // A Rat represents a quotient a/b of arbitrary precision.
    15  // The zero value for a Rat represents the value 0.
    16  //
    17  // Operations always take pointer arguments (*Rat) rather
    18  // than Rat values, and each unique Rat value requires
    19  // its own unique *Rat pointer. To "copy" a Rat value,
    20  // an existing (or newly allocated) Rat must be set to
    21  // a new value using the [Rat.Set] method; shallow copies
    22  // of Rats are not supported and may lead to errors.
    23  type Rat struct {
    24  	// To make zero values for Rat work w/o initialization,
    25  	// a zero value of b (len(b) == 0) acts like b == 1. At
    26  	// the earliest opportunity (when an assignment to the Rat
    27  	// is made), such uninitialized denominators are set to 1.
    28  	// a.neg determines the sign of the Rat, b.neg is ignored.
    29  	a, b Int
    30  }
    31  
    32  // NewRat creates a new [Rat] with numerator a and denominator b.
    33  func NewRat(a, b int64) *Rat {
    34  	return new(Rat).SetFrac64(a, b)
    35  }
    36  
    37  // SetFloat64 sets z to exactly f and returns z.
    38  // If f is not finite, SetFloat returns nil.
    39  func (z *Rat) SetFloat64(f float64) *Rat {
    40  	const expMask = 1<<11 - 1
    41  	bits := math.Float64bits(f)
    42  	mantissa := bits & (1<<52 - 1)
    43  	exp := int((bits >> 52) & expMask)
    44  	switch exp {
    45  	case expMask: // non-finite
    46  		return nil
    47  	case 0: // denormal
    48  		exp -= 1022
    49  	default: // normal
    50  		mantissa |= 1 << 52
    51  		exp -= 1023
    52  	}
    53  
    54  	shift := 52 - exp
    55  
    56  	// Optimization (?): partially pre-normalise.
    57  	for mantissa&1 == 0 && shift > 0 {
    58  		mantissa >>= 1
    59  		shift--
    60  	}
    61  
    62  	z.a.SetUint64(mantissa)
    63  	z.a.neg = f < 0
    64  	z.b.Set(intOne)
    65  	if shift > 0 {
    66  		z.b.Lsh(&z.b, uint(shift))
    67  	} else {
    68  		z.a.Lsh(&z.a, uint(-shift))
    69  	}
    70  	return z.norm()
    71  }
    72  
    73  // quotToFloat32 returns the non-negative float32 value
    74  // nearest to the quotient a/b, using round-to-even in
    75  // halfway cases. It does not mutate its arguments.
    76  // Preconditions: b is non-zero; a and b have no common factors.
    77  func quotToFloat32(a, b nat) (f float32, exact bool) {
    78  	const (
    79  		// float size in bits
    80  		Fsize = 32
    81  
    82  		// mantissa
    83  		Msize  = 23
    84  		Msize1 = Msize + 1 // incl. implicit 1
    85  		Msize2 = Msize1 + 1
    86  
    87  		// exponent
    88  		Esize = Fsize - Msize1
    89  		Ebias = 1<<(Esize-1) - 1
    90  		Emin  = 1 - Ebias
    91  		Emax  = Ebias
    92  	)
    93  
    94  	// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
    95  	alen := a.bitLen()
    96  	if alen == 0 {
    97  		return 0, true
    98  	}
    99  	blen := b.bitLen()
   100  	if blen == 0 {
   101  		panic("division by zero")
   102  	}
   103  
   104  	// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
   105  	// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
   106  	// This is 2 or 3 more than the float32 mantissa field width of Msize:
   107  	// - the optional extra bit is shifted away in step 3 below.
   108  	// - the high-order 1 is omitted in "normal" representation;
   109  	// - the low-order 1 will be used during rounding then discarded.
   110  	exp := alen - blen
   111  	var a2, b2 nat
   112  	a2 = a2.set(a)
   113  	b2 = b2.set(b)
   114  	if shift := Msize2 - exp; shift > 0 {
   115  		a2 = a2.shl(a2, uint(shift))
   116  	} else if shift < 0 {
   117  		b2 = b2.shl(b2, uint(-shift))
   118  	}
   119  
   120  	// 2. Compute quotient and remainder (q, r).  NB: due to the
   121  	// extra shift, the low-order bit of q is logically the
   122  	// high-order bit of r.
   123  	var q nat
   124  	q, r := q.div(a2, a2, b2) // (recycle a2)
   125  	mantissa := low32(q)
   126  	haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
   127  
   128  	// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
   129  	// (in effect---we accomplish this incrementally).
   130  	if mantissa>>Msize2 == 1 {
   131  		if mantissa&1 == 1 {
   132  			haveRem = true
   133  		}
   134  		mantissa >>= 1
   135  		exp++
   136  	}
   137  	if mantissa>>Msize1 != 1 {
   138  		panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
   139  	}
   140  
   141  	// 4. Rounding.
   142  	if Emin-Msize <= exp && exp <= Emin {
   143  		// Denormal case; lose 'shift' bits of precision.
   144  		shift := uint(Emin - (exp - 1)) // [1..Esize1)
   145  		lostbits := mantissa & (1<<shift - 1)
   146  		haveRem = haveRem || lostbits != 0
   147  		mantissa >>= shift
   148  		exp = 2 - Ebias // == exp + shift
   149  	}
   150  	// Round q using round-half-to-even.
   151  	exact = !haveRem
   152  	if mantissa&1 != 0 {
   153  		exact = false
   154  		if haveRem || mantissa&2 != 0 {
   155  			if mantissa++; mantissa >= 1<<Msize2 {
   156  				// Complete rollover 11...1 => 100...0, so shift is safe
   157  				mantissa >>= 1
   158  				exp++
   159  			}
   160  		}
   161  	}
   162  	mantissa >>= 1 // discard rounding bit.  Mantissa now scaled by 1<<Msize1.
   163  
   164  	f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
   165  	if math.IsInf(float64(f), 0) {
   166  		exact = false
   167  	}
   168  	return
   169  }
   170  
   171  // quotToFloat64 returns the non-negative float64 value
   172  // nearest to the quotient a/b, using round-to-even in
   173  // halfway cases. It does not mutate its arguments.
   174  // Preconditions: b is non-zero; a and b have no common factors.
   175  func quotToFloat64(a, b nat) (f float64, exact bool) {
   176  	const (
   177  		// float size in bits
   178  		Fsize = 64
   179  
   180  		// mantissa
   181  		Msize  = 52
   182  		Msize1 = Msize + 1 // incl. implicit 1
   183  		Msize2 = Msize1 + 1
   184  
   185  		// exponent
   186  		Esize = Fsize - Msize1
   187  		Ebias = 1<<(Esize-1) - 1
   188  		Emin  = 1 - Ebias
   189  		Emax  = Ebias
   190  	)
   191  
   192  	// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
   193  	alen := a.bitLen()
   194  	if alen == 0 {
   195  		return 0, true
   196  	}
   197  	blen := b.bitLen()
   198  	if blen == 0 {
   199  		panic("division by zero")
   200  	}
   201  
   202  	// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
   203  	// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
   204  	// This is 2 or 3 more than the float64 mantissa field width of Msize:
   205  	// - the optional extra bit is shifted away in step 3 below.
   206  	// - the high-order 1 is omitted in "normal" representation;
   207  	// - the low-order 1 will be used during rounding then discarded.
   208  	exp := alen - blen
   209  	var a2, b2 nat
   210  	a2 = a2.set(a)
   211  	b2 = b2.set(b)
   212  	if shift := Msize2 - exp; shift > 0 {
   213  		a2 = a2.shl(a2, uint(shift))
   214  	} else if shift < 0 {
   215  		b2 = b2.shl(b2, uint(-shift))
   216  	}
   217  
   218  	// 2. Compute quotient and remainder (q, r).  NB: due to the
   219  	// extra shift, the low-order bit of q is logically the
   220  	// high-order bit of r.
   221  	var q nat
   222  	q, r := q.div(a2, a2, b2) // (recycle a2)
   223  	mantissa := low64(q)
   224  	haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
   225  
   226  	// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
   227  	// (in effect---we accomplish this incrementally).
   228  	if mantissa>>Msize2 == 1 {
   229  		if mantissa&1 == 1 {
   230  			haveRem = true
   231  		}
   232  		mantissa >>= 1
   233  		exp++
   234  	}
   235  	if mantissa>>Msize1 != 1 {
   236  		panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
   237  	}
   238  
   239  	// 4. Rounding.
   240  	if Emin-Msize <= exp && exp <= Emin {
   241  		// Denormal case; lose 'shift' bits of precision.
   242  		shift := uint(Emin - (exp - 1)) // [1..Esize1)
   243  		lostbits := mantissa & (1<<shift - 1)
   244  		haveRem = haveRem || lostbits != 0
   245  		mantissa >>= shift
   246  		exp = 2 - Ebias // == exp + shift
   247  	}
   248  	// Round q using round-half-to-even.
   249  	exact = !haveRem
   250  	if mantissa&1 != 0 {
   251  		exact = false
   252  		if haveRem || mantissa&2 != 0 {
   253  			if mantissa++; mantissa >= 1<<Msize2 {
   254  				// Complete rollover 11...1 => 100...0, so shift is safe
   255  				mantissa >>= 1
   256  				exp++
   257  			}
   258  		}
   259  	}
   260  	mantissa >>= 1 // discard rounding bit.  Mantissa now scaled by 1<<Msize1.
   261  
   262  	f = math.Ldexp(float64(mantissa), exp-Msize1)
   263  	if math.IsInf(f, 0) {
   264  		exact = false
   265  	}
   266  	return
   267  }
   268  
   269  // Float32 returns the nearest float32 value for x and a bool indicating
   270  // whether f represents x exactly. If the magnitude of x is too large to
   271  // be represented by a float32, f is an infinity and exact is false.
   272  // The sign of f always matches the sign of x, even if f == 0.
   273  func (x *Rat) Float32() (f float32, exact bool) {
   274  	b := x.b.abs
   275  	if len(b) == 0 {
   276  		b = natOne
   277  	}
   278  	f, exact = quotToFloat32(x.a.abs, b)
   279  	if x.a.neg {
   280  		f = -f
   281  	}
   282  	return
   283  }
   284  
   285  // Float64 returns the nearest float64 value for x and a bool indicating
   286  // whether f represents x exactly. If the magnitude of x is too large to
   287  // be represented by a float64, f is an infinity and exact is false.
   288  // The sign of f always matches the sign of x, even if f == 0.
   289  func (x *Rat) Float64() (f float64, exact bool) {
   290  	b := x.b.abs
   291  	if len(b) == 0 {
   292  		b = natOne
   293  	}
   294  	f, exact = quotToFloat64(x.a.abs, b)
   295  	if x.a.neg {
   296  		f = -f
   297  	}
   298  	return
   299  }
   300  
   301  // SetFrac sets z to a/b and returns z.
   302  // If b == 0, SetFrac panics.
   303  func (z *Rat) SetFrac(a, b *Int) *Rat {
   304  	z.a.neg = a.neg != b.neg
   305  	babs := b.abs
   306  	if len(babs) == 0 {
   307  		panic("division by zero")
   308  	}
   309  	if &z.a == b || alias(z.a.abs, babs) {
   310  		babs = nat(nil).set(babs) // make a copy
   311  	}
   312  	z.a.abs = z.a.abs.set(a.abs)
   313  	z.b.abs = z.b.abs.set(babs)
   314  	return z.norm()
   315  }
   316  
   317  // SetFrac64 sets z to a/b and returns z.
   318  // If b == 0, SetFrac64 panics.
   319  func (z *Rat) SetFrac64(a, b int64) *Rat {
   320  	if b == 0 {
   321  		panic("division by zero")
   322  	}
   323  	z.a.SetInt64(a)
   324  	if b < 0 {
   325  		b = -b
   326  		z.a.neg = !z.a.neg
   327  	}
   328  	z.b.abs = z.b.abs.setUint64(uint64(b))
   329  	return z.norm()
   330  }
   331  
   332  // SetInt sets z to x (by making a copy of x) and returns z.
   333  func (z *Rat) SetInt(x *Int) *Rat {
   334  	z.a.Set(x)
   335  	z.b.abs = z.b.abs.setWord(1)
   336  	return z
   337  }
   338  
   339  // SetInt64 sets z to x and returns z.
   340  func (z *Rat) SetInt64(x int64) *Rat {
   341  	z.a.SetInt64(x)
   342  	z.b.abs = z.b.abs.setWord(1)
   343  	return z
   344  }
   345  
   346  // SetUint64 sets z to x and returns z.
   347  func (z *Rat) SetUint64(x uint64) *Rat {
   348  	z.a.SetUint64(x)
   349  	z.b.abs = z.b.abs.setWord(1)
   350  	return z
   351  }
   352  
   353  // Set sets z to x (by making a copy of x) and returns z.
   354  func (z *Rat) Set(x *Rat) *Rat {
   355  	if z != x {
   356  		z.a.Set(&x.a)
   357  		z.b.Set(&x.b)
   358  	}
   359  	if len(z.b.abs) == 0 {
   360  		z.b.abs = z.b.abs.setWord(1)
   361  	}
   362  	return z
   363  }
   364  
   365  // Abs sets z to |x| (the absolute value of x) and returns z.
   366  func (z *Rat) Abs(x *Rat) *Rat {
   367  	z.Set(x)
   368  	z.a.neg = false
   369  	return z
   370  }
   371  
   372  // Neg sets z to -x and returns z.
   373  func (z *Rat) Neg(x *Rat) *Rat {
   374  	z.Set(x)
   375  	z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
   376  	return z
   377  }
   378  
   379  // Inv sets z to 1/x and returns z.
   380  // If x == 0, Inv panics.
   381  func (z *Rat) Inv(x *Rat) *Rat {
   382  	if len(x.a.abs) == 0 {
   383  		panic("division by zero")
   384  	}
   385  	z.Set(x)
   386  	z.a.abs, z.b.abs = z.b.abs, z.a.abs
   387  	return z
   388  }
   389  
   390  // Sign returns:
   391  //   - -1 if x < 0;
   392  //   - 0 if x == 0;
   393  //   - +1 if x > 0.
   394  func (x *Rat) Sign() int {
   395  	return x.a.Sign()
   396  }
   397  
   398  // IsInt reports whether the denominator of x is 1.
   399  func (x *Rat) IsInt() bool {
   400  	return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
   401  }
   402  
   403  // Num returns the numerator of x; it may be <= 0.
   404  // The result is a reference to x's numerator; it
   405  // may change if a new value is assigned to x, and vice versa.
   406  // The sign of the numerator corresponds to the sign of x.
   407  func (x *Rat) Num() *Int {
   408  	return &x.a
   409  }
   410  
   411  // Denom returns the denominator of x; it is always > 0.
   412  // The result is a reference to x's denominator, unless
   413  // x is an uninitialized (zero value) [Rat], in which case
   414  // the result is a new [Int] of value 1. (To initialize x,
   415  // any operation that sets x will do, including x.Set(x).)
   416  // If the result is a reference to x's denominator it
   417  // may change if a new value is assigned to x, and vice versa.
   418  func (x *Rat) Denom() *Int {
   419  	// Note that x.b.neg is guaranteed false.
   420  	if len(x.b.abs) == 0 {
   421  		// Note: If this proves problematic, we could
   422  		//       panic instead and require the Rat to
   423  		//       be explicitly initialized.
   424  		return &Int{abs: nat{1}}
   425  	}
   426  	return &x.b
   427  }
   428  
   429  func (z *Rat) norm() *Rat {
   430  	switch {
   431  	case len(z.a.abs) == 0:
   432  		// z == 0; normalize sign and denominator
   433  		z.a.neg = false
   434  		fallthrough
   435  	case len(z.b.abs) == 0:
   436  		// z is integer; normalize denominator
   437  		z.b.abs = z.b.abs.setWord(1)
   438  	default:
   439  		// z is fraction; normalize numerator and denominator
   440  		neg := z.a.neg
   441  		z.a.neg = false
   442  		z.b.neg = false
   443  		if f := NewInt(0).lehmerGCD(nil, nil, &z.a, &z.b); f.Cmp(intOne) != 0 {
   444  			z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
   445  			z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
   446  		}
   447  		z.a.neg = neg
   448  	}
   449  	return z
   450  }
   451  
   452  // mulDenom sets z to the denominator product x*y (by taking into
   453  // account that 0 values for x or y must be interpreted as 1) and
   454  // returns z.
   455  func mulDenom(z, x, y nat) nat {
   456  	switch {
   457  	case len(x) == 0 && len(y) == 0:
   458  		return z.setWord(1)
   459  	case len(x) == 0:
   460  		return z.set(y)
   461  	case len(y) == 0:
   462  		return z.set(x)
   463  	}
   464  	return z.mul(x, y)
   465  }
   466  
   467  // scaleDenom sets z to the product x*f.
   468  // If f == 0 (zero value of denominator), z is set to (a copy of) x.
   469  func (z *Int) scaleDenom(x *Int, f nat) {
   470  	if len(f) == 0 {
   471  		z.Set(x)
   472  		return
   473  	}
   474  	z.abs = z.abs.mul(x.abs, f)
   475  	z.neg = x.neg
   476  }
   477  
   478  // Cmp compares x and y and returns:
   479  //   - -1 if x < y;
   480  //   - 0 if x == y;
   481  //   - +1 if x > y.
   482  func (x *Rat) Cmp(y *Rat) int {
   483  	var a, b Int
   484  	a.scaleDenom(&x.a, y.b.abs)
   485  	b.scaleDenom(&y.a, x.b.abs)
   486  	return a.Cmp(&b)
   487  }
   488  
   489  // Add sets z to the sum x+y and returns z.
   490  func (z *Rat) Add(x, y *Rat) *Rat {
   491  	var a1, a2 Int
   492  	a1.scaleDenom(&x.a, y.b.abs)
   493  	a2.scaleDenom(&y.a, x.b.abs)
   494  	z.a.Add(&a1, &a2)
   495  	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   496  	return z.norm()
   497  }
   498  
   499  // Sub sets z to the difference x-y and returns z.
   500  func (z *Rat) Sub(x, y *Rat) *Rat {
   501  	var a1, a2 Int
   502  	a1.scaleDenom(&x.a, y.b.abs)
   503  	a2.scaleDenom(&y.a, x.b.abs)
   504  	z.a.Sub(&a1, &a2)
   505  	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   506  	return z.norm()
   507  }
   508  
   509  // Mul sets z to the product x*y and returns z.
   510  func (z *Rat) Mul(x, y *Rat) *Rat {
   511  	if x == y {
   512  		// a squared Rat is positive and can't be reduced (no need to call norm())
   513  		z.a.neg = false
   514  		z.a.abs = z.a.abs.sqr(x.a.abs)
   515  		if len(x.b.abs) == 0 {
   516  			z.b.abs = z.b.abs.setWord(1)
   517  		} else {
   518  			z.b.abs = z.b.abs.sqr(x.b.abs)
   519  		}
   520  		return z
   521  	}
   522  	z.a.Mul(&x.a, &y.a)
   523  	z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
   524  	return z.norm()
   525  }
   526  
   527  // Quo sets z to the quotient x/y and returns z.
   528  // If y == 0, Quo panics.
   529  func (z *Rat) Quo(x, y *Rat) *Rat {
   530  	if len(y.a.abs) == 0 {
   531  		panic("division by zero")
   532  	}
   533  	var a, b Int
   534  	a.scaleDenom(&x.a, y.b.abs)
   535  	b.scaleDenom(&y.a, x.b.abs)
   536  	z.a.abs = a.abs
   537  	z.b.abs = b.abs
   538  	z.a.neg = a.neg != b.neg
   539  	return z.norm()
   540  }
   541
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