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#
#
# 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
func 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
func `//`*[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
func `$`*[T](x: Rational[T]): string =
## Turn a rational number into a string.
result = $x.num & "/" & $x.den
func toRational*[T: SomeInteger](x: T): Rational[T] =
## Convert some integer `x` to a rational number.
result.num = x
result.den = 1
func toRational*(x: float,
n: int = high(int) shr (sizeof(int) div 2 * 8)): Rational[int] =
## Calculates 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 theory of continued fractions.
##
## .. 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
# David Eppstein / UC Irvine / 8 Aug 1993
# With corrections from Arno Formella, May 2008
var
m11, m22 = 1
m12, m21 = 0
ai = int(x)
x = x
while m21 * ai + m22 <= n:
swap m12, m11
swap m22, m21
m11 = m12 * ai + m11
m21 = m22 * ai + m21
if x == float(ai): break # division by zero
x = 1/(x - float(ai))
if x > float(high(int32)): break # representation failure
ai = int(x)
result = m11 // m21
func toFloat*[T](x: Rational[T]): float =
## Convert a rational number `x` to a float.
x.num / x.den
func 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
func 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(DivByZeroDefect, "division by zero")
func `+` *[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)
func `+` *[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
func `+` *[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
func `+=` *[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)
func `+=` *[T](x: var Rational[T], y: T) =
## Add int `y` to rational `x`.
x.num += y * x.den
func `-` *[T](x: Rational[T]): Rational[T] =
## Unary minus for rational numbers.
result.num = -x.num
result.den = x.den
func `-` *[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)
func `-` *[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
func `-` *[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
func `-=` *[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)
func `-=` *[T](x: var Rational[T], y: T) =
## Subtract int `y` from rational `x`.
x.num -= y * x.den
func `*` *[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)
func `*` *[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)
func `*` *[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)
func `*=` *[T](x: var Rational[T], y: Rational[T]) =
## Multiply rationals `y` to `x`.
x.num *= y.num
x.den *= y.den
reduce(x)
func `*=` *[T](x: var Rational[T], y: T) =
## Multiply int `y` to rational `x`.
x.num *= y
reduce(x)
func 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(DivByZeroDefect, "division by zero")
func `/`*[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)
func `/`*[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)
func `/`*[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)
func `/=`*[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)
func `/=`*[T](x: var Rational[T], y: T) =
## Divide rational `x` by int `y` in place.
x.den *= y
reduce(x)
func cmp*(x, y: Rational): int =
## Compares two rationals.
(x - y).num
func `<` *(x, y: Rational): bool =
(x - y).num < 0
func `<=` *(x, y: Rational): bool =
(x - y).num <= 0
func `==` *(x, y: Rational): bool =
(x - y).num == 0
func abs*[T](x: Rational[T]): Rational[T] =
result.num = abs x.num
result.den = abs x.den
func `div`*[T: SomeInteger](x, y: Rational[T]): T =
## Computes the rational truncated division.
(x.num * y.den) div (y.num * x.den)
func `mod`*[T: SomeInteger](x, y: Rational[T]): Rational[T] =
## Computes the rational modulo by truncated division (remainder).
## This is same as ``x - (x div y) * y``.
result = ((x.num * y.den) mod (y.num * x.den)) // (x.den * y.den)
reduce(result)
func floorDiv*[T: SomeInteger](x, y: Rational[T]): T =
## Computes the rational floor division.
##
## Floor division is conceptually defined as ``floor(x / y)``.
## This is different from the ``div`` operator, which is defined
## as ``trunc(x / y)``. That is, ``div`` rounds towards ``0`` and ``floorDiv``
## rounds down.
floorDiv(x.num * y.den, y.num * x.den)
func floorMod*[T: SomeInteger](x, y: Rational[T]): Rational[T] =
## Computes the rational modulo by floor division (modulo).
##
## This is same as ``x - floorDiv(x, y) * y``.
## This func behaves the same as the ``%`` operator in python.
result = floorMod(x.num * y.den, y.num * x.den) // (x.den * y.den)
reduce(result)
func 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
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