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// Copyright 2009 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.
// Rational numbers
package bignum
import "fmt"
// Rational represents a quotient a/b of arbitrary precision.
//
type Rational struct {
a *Integer // numerator
b Natural // denominator
}
// MakeRat makes a rational number given a numerator a and a denominator b.
//
func MakeRat(a *Integer, b Natural) *Rational {
f := a.mant.Gcd(b) // f > 0
if f.Cmp(Nat(1)) != 0 {
a = MakeInt(a.sign, a.mant.Div(f))
b = b.Div(f)
}
return &Rational{a, b}
}
// Rat creates a small rational number with value a0/b0.
//
func Rat(a0 int64, b0 int64) *Rational {
a, b := Int(a0), Int(b0)
if b.sign {
a = a.Neg()
}
return MakeRat(a, b.mant)
}
// Value returns the numerator and denominator of x.
//
func (x *Rational) Value() (numerator *Integer, denominator Natural) {
return x.a, x.b
}
// Predicates
// IsZero returns true iff x == 0.
//
func (x *Rational) IsZero() bool { return x.a.IsZero() }
// IsNeg returns true iff x < 0.
//
func (x *Rational) IsNeg() bool { return x.a.IsNeg() }
// IsPos returns true iff x > 0.
//
func (x *Rational) IsPos() bool { return x.a.IsPos() }
// IsInt returns true iff x can be written with a denominator 1
// in the form x == x'/1; i.e., if x is an integer value.
//
func (x *Rational) IsInt() bool { return x.b.Cmp(Nat(1)) == 0 }
// Operations
// Neg returns the negated value of x.
//
func (x *Rational) Neg() *Rational { return MakeRat(x.a.Neg(), x.b) }
// Add returns the sum x + y.
//
func (x *Rational) Add(y *Rational) *Rational {
return MakeRat((x.a.MulNat(y.b)).Add(y.a.MulNat(x.b)), x.b.Mul(y.b))
}
// Sub returns the difference x - y.
//
func (x *Rational) Sub(y *Rational) *Rational {
return MakeRat((x.a.MulNat(y.b)).Sub(y.a.MulNat(x.b)), x.b.Mul(y.b))
}
// Mul returns the product x * y.
//
func (x *Rational) Mul(y *Rational) *Rational { return MakeRat(x.a.Mul(y.a), x.b.Mul(y.b)) }
// Quo returns the quotient x / y for y != 0.
// If y == 0, a division-by-zero run-time error occurs.
//
func (x *Rational) Quo(y *Rational) *Rational {
a := x.a.MulNat(y.b)
b := y.a.MulNat(x.b)
if b.IsNeg() {
a = a.Neg()
}
return MakeRat(a, b.mant)
}
// Cmp compares x and y. The result is an int value
//
// < 0 if x < y
// == 0 if x == y
// > 0 if x > y
//
func (x *Rational) Cmp(y *Rational) int { return (x.a.MulNat(y.b)).Cmp(y.a.MulNat(x.b)) }
// ToString converts x to a string for a given base, with 2 <= base <= 16.
// The string representation is of the form "n" if x is an integer; otherwise
// it is of form "n/d".
//
func (x *Rational) ToString(base uint) string {
s := x.a.ToString(base)
if !x.IsInt() {
s += "/" + x.b.ToString(base)
}
return s
}
// String converts x to its decimal string representation.
// x.String() is the same as x.ToString(10).
//
func (x *Rational) String() string { return x.ToString(10) }
// Format is a support routine for fmt.Formatter. It accepts
// the formats 'b' (binary), 'o' (octal), and 'x' (hexadecimal).
//
func (x *Rational) Format(h fmt.State, c int) { fmt.Fprintf(h, "%s", x.ToString(fmtbase(c))) }
// RatFromString returns the rational number corresponding to the
// longest possible prefix of s representing a rational number in a
// given conversion base, the actual conversion base used, and the
// prefix length. The syntax of a rational number is:
//
// rational = mantissa [exponent] .
// mantissa = integer ('/' natural | '.' natural) .
// exponent = ('e'|'E') integer .
//
// If the base argument is 0, the string prefix determines the actual
// conversion base for the mantissa. A prefix of ``0x'' or ``0X'' selects
// base 16; the ``0'' prefix selects base 8. Otherwise the selected base is 10.
// If the mantissa is represented via a division, both the numerator and
// denominator may have different base prefixes; in that case the base of
// of the numerator is returned. If the mantissa contains a decimal point,
// the base for the fractional part is the same as for the part before the
// decimal point and the fractional part does not accept a base prefix.
// The base for the exponent is always 10.
//
func RatFromString(s string, base uint) (*Rational, uint, int) {
// read numerator
a, abase, alen := IntFromString(s, base)
b := Nat(1)
// read denominator or fraction, if any
var blen int
if alen < len(s) {
ch := s[alen]
if ch == '/' {
alen++
b, base, blen = NatFromString(s[alen:], base)
} else if ch == '.' {
alen++
b, base, blen = NatFromString(s[alen:], abase)
assert(base == abase)
f := Nat(uint64(base)).Pow(uint(blen))
a = MakeInt(a.sign, a.mant.Mul(f).Add(b))
b = f
}
}
// read exponent, if any
rlen := alen + blen
if rlen < len(s) {
ch := s[rlen]
if ch == 'e' || ch == 'E' {
rlen++
e, _, elen := IntFromString(s[rlen:], 10)
rlen += elen
m := Nat(10).Pow(uint(e.mant.Value()))
if e.sign {
b = b.Mul(m)
} else {
a = a.MulNat(m)
}
}
}
return MakeRat(a, b), base, rlen
}
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