Nim/lib/pure/rationals.nim

#
#
#            Nim's Runtime Library
#        (c) Copyright 2015 Dennis Felsing
#
#    See the file "copying.txt", included in this
#    distribution, for details about the copyright.
#


## This module implements rational numbers, consisting of a numerator `num` and
## a denominator `den`, both of type int. The denominator can not be 0.

import math
import hashes

type Rational*[T] = object
  ## a rational number, consisting of a numerator and denominator
  num*, den*: T

proc initRational*[T:SomeInteger](num, den: T): Rational[T] =
  ## Create a new rational number.
  assert(den != 0, "a denominator of zero value is invalid")
  result.num = num
  result.den = den

proc `//`*[T](num, den: T): Rational[T] = initRational[T](num, den)
  ## A friendlier version of `initRational`. Example usage:
  ##
  ## .. code-block:: nim
  ##   var x = 1//3 + 1//5

proc `$`*[T](x: Rational[T]): string =
  ## Turn a rational number into a string.
  result = $x.num & "/" & $x.den

proc toRational*[T:SomeInteger](x: T): Rational[T] =
  ## Convert some integer `x` to a rational number.
  result.num = x
  result.den = 1

proc toRationalSub(x: float, n: int): Rational[int] =
  var
    a = 0
    b, c, d = 1
  result = 0 // 1   # rational 0
  while b <= n and d <= n:
    let ac = (a+c)
    let bd = (b+d)
    # scale by 1000 so not overflow for high precision
    let mediant = (ac/1000) / (bd/1000)
    if x == mediant:
      if bd <= n:
        result.num = ac
        result.den = bd
        return result
      elif d > b:
        result.num = c
        result.den = d
        return result
      else:
        result.num = a
        result.den = b
        return result
    elif x > mediant:
      a = ac
      b = bd
    else:
      c = ac
      d = bd
  if (b > n):
    return initRational(c, d)
  return initRational(a, b)

proc toRational*(x: float, n: int = high(int)): Rational[int] =
  ## Calculate the best rational numerator and denominator
  ## that approximates to `x`, where the denominator is
  ## smaller than `n` (default is the largest possible
  ## int to give maximum resolution)
  ##
  ## The algorithm is based on the Farey sequence named
  ## after John Farey
  ##
  ## .. code-block:: Nim
  ##  import math, rationals
  ##  for i in 1..10:
  ##    let t = (10 ^ (i+3)).int
  ##    let x = toRational(PI, t)
  ##    let newPI = x.num / x.den
  ##    echo x, " ", newPI, " error: ", PI - newPI, "  ", t
  if x > 1:
    result = toRationalSub(1.0/x, n)
    swap(result.num, result.den)
  elif x == 1.0:
    result = 1 // 1
  else:
    result = toRationalSub(x, n)

proc toFloat*[T](x: Rational[T]): float =
  ## Convert a rational number `x` to a float.
  x.num / x.den

proc toInt*[T](x: Rational[T]): int =
  ## Convert a rational number `x` to an int. Conversion rounds towards 0 if
  ## `x` does not contain an integer value.
  x.num div x.den

proc reduce*[T:SomeInteger](x: var Rational[T]) =
  ## Reduce rational `x`.
  let common = gcd(x.num, x.den)
  if x.den > 0:
    x.num = x.num div common
    x.den = x.den div common
  elif x.den < 0:
    x.num = -x.num div common
    x.den = -x.den div common
  else:
    raise newException(DivByZeroError, "division by zero")

proc `+` *[T](x, y: Rational[T]): Rational[T] =
  ## Add two rational numbers.
  let common = lcm(x.den, y.den)
  result.num = common div x.den * x.num + common div y.den * y.num
  result.den = common
  reduce(result)

proc `+` *[T](x: Rational[T], y: T): Rational[T] =
  ## Add rational `x` to int `y`.
  result.num = x.num + y * x.den
  result.den = x.den

proc `+` *[T](x: T, y: Rational[T]): Rational[T] =
  ## Add int `x` to rational `y`.
  result.num = x * y.den + y.num
  result.den = y.den

proc `+=` *[T](x: var Rational[T], y: Rational[T]) =
  ## Add rational `y` to rational `x`.
  let common = lcm(x.den, y.den)
  x.num = common div x.den * x.num + common div y.den * y.num
  x.den = common
  reduce(x)

proc `+=` *[T](x: var Rational[T], y: T) =
  ## Add int `y` to rational `x`.
  x.num += y * x.den

proc `-` *[T](x: Rational[T]): Rational[T] =
  ## Unary minus for rational numbers.
  result.num = -x.num
  result.den = x.den

proc `-` *[T](x, y: Rational[T]): Rational[T] =
  ## Subtract two rational numbers.
  let common = lcm(x.den, y.den)
  result.num = common div x.den * x.num - common div y.den * y.num
  result.den = common
  reduce(result)

proc `-` *[T](x: Rational[T], y: T): Rational[T] =
  ## Subtract int `y` from rational `x`.
  result.num = x.num - y * x.den
  result.den = x.den

proc `-` *[T](x: T, y: Rational[T]): Rational[T] =
  ## Subtract rational `y` from int `x`.
  result.num = - x * y.den + y.num
  result.den = y.den

proc `-=` *[T](x: var Rational[T], y: Rational[T]) =
  ## Subtract rational `y` from rational `x`.
  let common = lcm(x.den, y.den)
  x.num = common div x.den * x.num - common div y.den * y.num
  x.den = common
  reduce(x)

proc `-=` *[T](x: var Rational[T], y: T) =
  ## Subtract int `y` from rational `x`.
  x.num -= y * x.den

proc `*` *[T](x, y: Rational[T]): Rational[T] =
  ## Multiply two rational numbers.
  result.num = x.num * y.num
  result.den = x.den * y.den
  reduce(result)

proc `*` *[T](x: Rational[T], y: T): Rational[T] =
  ## Multiply rational `x` with int `y`.
  result.num = x.num * y
  result.den = x.den
  reduce(result)

proc `*` *[T](x: T, y: Rational[T]): Rational[T] =
  ## Multiply int `x` with rational `y`.
  result.num = x * y.num
  result.den = y.den
  reduce(result)

proc `*=` *[T](x: var Rational[T], y: Rational[T]) =
  ## Multiply rationals `y` to `x`.
  x.num *= y.num
  x.den *= y.den
  reduce(x)

proc `*=` *[T](x: var Rational[T], y: T) =
  ## Multiply int `y` to rational `x`.
  x.num *= y
  reduce(x)

proc reciprocal*[T](x: Rational[T]): Rational[T] =
  ## Calculate the reciprocal of `x`. (1/x)
  if x.num > 0:
    result.num = x.den
    result.den = x.num
  elif x.num < 0:
    result.num = -x.den
    result.den = -x.num
  else:
    raise newException(DivByZeroError, "division by zero")

proc `/`*[T](x, y: Rational[T]): Rational[T] =
  ## Divide rationals `x` by `y`.
  result.num = x.num * y.den
  result.den = x.den * y.num
  reduce(result)

proc `/`*[T](x: Rational[T], y: T): Rational[T] =
  ## Divide rational `x` by int `y`.
  result.num = x.num
  result.den = x.den * y
  reduce(result)

proc `/`*[T](x: T, y: Rational[T]): Rational[T] =
  ## Divide int `x` by Rational `y`.
  result.num = x * y.den
  result.den = y.num
  reduce(result)

proc `/=`*[T](x: var Rational[T], y: Rational[T]) =
  ## Divide rationals `x` by `y` in place.
  x.num *= y.den
  x.den *= y.num
  reduce(x)

proc `/=`*[T](x: var Rational[T], y: T) =
  ## Divide rational `x` by int `y` in place.
  x.den *= y
  reduce(x)

proc cmp*(x, y: Rational): int {.procvar.} =
  ## Compares two rationals.
  (x - y).num

proc `<` *(x, y: Rational): bool =
  (x - y).num < 0

proc `<=` *(x, y: Rational): bool =
  (x - y).num <= 0

proc `==` *(x, y: Rational): bool =
  (x - y).num == 0

proc abs*[T](x: Rational[T]): Rational[T] =
  result.num = abs x.num
  result.den = abs x.den

proc hash*[T](x: Rational[T]): Hash =
  ## Computes hash for rational `x`
  # reduce first so that hash(x) == hash(y) for x == y
  var copy = x
  reduce(copy)

  var h: Hash = 0
  h = h !& hash(copy.num)
  h = h !& hash(copy.den)
  result = !$h

when isMainModule:
  var
    z = Rational[int](num: 0, den: 1)
    o = initRational(num=1, den=1)
    a = initRational(1, 2)
    b = -1 // -2
    m1 = -1 // 1
    tt = 10 // 2

  assert( a     == a )
  assert( (a-a) == z )
  assert( (a+b) == o )
  assert( (a/b) == o )
  assert( (a*b) == 1 // 4 )
  assert( (3/a) == 6 // 1 )
  assert( (a/3) == 1 // 6 )
  assert( a*b   == 1 // 4 )
  assert( tt*z  == z )
  assert( 10*a  == tt )
  assert( a*10  == tt )
  assert( tt/10 == a  )
  assert( a-m1  == 3 // 2 )
  assert( a+m1  == -1 // 2 )
  assert( m1+tt == 16 // 4 )
  assert( m1-tt == 6 // -1 )

  assert( z < o )
  assert( z <= o )
  assert( z == z )
  assert( cmp(z, o) < 0 )
  assert( cmp(o, z) > 0 )

  assert( o == o )
  assert( o >= o )
  assert( not(o > o) )
  assert( cmp(o, o) == 0 )
  assert( cmp(z, z) == 0 )
  assert( hash(o) == hash(o) )

  assert( a == b )
  assert( a >= b )
  assert( not(b > a) )
  assert( cmp(a, b) == 0 )
  assert( hash(a) == hash(b) )

  var x = 1//3

  x *= 5//1
  assert( x == 5//3 )
  x += 2 // 9
  assert( x == 17//9 )
  x -= 9//18
  assert( x == 25//18 )
  x /= 1//2
  assert( x == 50//18 )

  var y = 1//3

  y *= 4
  assert( y == 4//3 )
  y += 5
  assert( y == 19//3 )
  y -= 2
  assert( y == 13//3 )
  y /= 9
  assert( y == 13//27 )

  assert toRational(5) == 5//1
  assert abs(toFloat(y) - 0.4814814814814815) < 1.0e-7
  assert toInt(z) == 0

  assert toRational(0.98765432) == 12345679 // 12500000
  assert toRational(0.1, 1000000) == 1 // 10
  assert toRational(0.9, 1000000) == 9 // 10
  assert toRational(PI) == 80143857 // 25510582