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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 and
## a denominator. The denominator can not be 0.
runnableExamples:
let
r1 = 1 // 2
r2 = -3 // 4
doAssert r1 + r2 == -1 // 4
doAssert r1 - r2 == 5 // 4
doAssert r1 * r2 == -3 // 8
doAssert r1 / r2 == -2 // 3
import std/[math, hashes]
when defined(nimPreviewSlimSystem):
import std/assertions
type Rational*[T] = object
## A rational number, consisting of a numerator `num` and a denominator `den`.
num*, den*: T
func reduce*[T: SomeInteger](x: var Rational[T]) =
## Reduces the rational number `x`, so that the numerator and denominator
## have no common divisors other than 1 (and -1).
## If `x` is 0, raises `DivByZeroDefect`.
##
## **Note:** This is called automatically by the various operations on rationals.
runnableExamples:
var r = Rational[int](num: 2, den: 4) # 1/2
reduce(r)
doAssert r.num == 1
doAssert r.den == 2
if x.den == 0:
raise newException(DivByZeroDefect, "division by zero")
let common = gcd(x.num, x.den)
if x.den > 0:
x.num = x.num div common
x.den = x.den div common
when T isnot SomeUnsignedInt:
if x.den < 0:
x.num = -x.num div common
x.den = -x.den div common
func initRational*[T: SomeInteger](num, den: T): Rational[T] =
## Creates a new rational number with numerator `num` and denominator `den`.
## `den` must not be 0.
##
## **Note:** `den != 0` is not checked when assertions are turned off.
assert(den != 0, "a denominator of zero is invalid")
result.num = num
result.den = den
reduce(result)
func `//`*[T](num, den: T): Rational[T] =
## A friendlier version of `initRational <#initRational,T,T>`_.
runnableExamples:
let x = 1 // 3 + 1 // 5
doAssert x == 8 // 15
initRational[T](num, den)
func `$`*[T](x: Rational[T]): string =
## Turns a rational number into a string.
runnableExamples:
doAssert $(1 // 2) == "1/2"
result = $x.num & "/" & $x.den
func toRational*[T: SomeInteger](x: T): Rational[T] =
## Converts some integer `x` to a rational number.
runnableExamples:
doAssert toRational(42) == 42 // 1
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 approximation of `x`,
## where the denominator is smaller than `n`
## (default is the largest possible `int` for maximal resolution).
##
## The algorithm is based on the theory of continued fractions.
# David Eppstein / UC Irvine / 8 Aug 1993
# With corrections from Arno Formella, May 2008
runnableExamples:
let x = 1.2
doAssert x.toRational.toFloat == x
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 =
## Converts a rational number `x` to a `float`.
x.num / x.den
func toInt*[T](x: Rational[T]): int =
## Converts 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 `+`*[T](x, y: Rational[T]): Rational[T] =
## Adds 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] =
## Adds the rational `x` to the int `y`.
result.num = x.num + y * x.den
result.den = x.den
func `+`*[T](x: T, y: Rational[T]): Rational[T] =
## Adds the int `x` to the rational `y`.
result.num = x * y.den + y.num
result.den = y.den
func `+=`*[T](x: var Rational[T], y: Rational[T]) =
## Adds the rational `y` to the rational `x` in-place.
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) =
## Adds the int `y` to the rational `x` in-place.
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] =
## Subtracts 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] =
## Subtracts the int `y` from the rational `x`.
result.num = x.num - y * x.den
result.den = x.den
func `-`*[T](x: T, y: Rational[T]): Rational[T] =
## Subtracts the rational `y` from the int `x`.
result.num = x * y.den - y.num
result.den = y.den
func `-=`*[T](x: var Rational[T], y: Rational[T]) =
## Subtracts the rational `y` from the rational `x` in-place.
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) =
## Subtracts the int `y` from the rational `x` in-place.
x.num -= y * x.den
func `*`*[T](x, y: Rational[T]): Rational[T] =
## Multiplies 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] =
## Multiplies the rational `x` with the int `y`.
result.num = x.num * y
result.den = x.den
reduce(result)
func `*`*[T](x: T, y: Rational[T]): Rational[T] =
## Multiplies the int `x` with the rational `y`.
result.num = x * y.num
result.den = y.den
reduce(result)
func `*=`*[T](x: var Rational[T], y: Rational[T]) =
## Multiplies the rational `x` by `y` in-place.
x.num *= y.num
x.den *= y.den
reduce(x)
func `*=`*[T](x: var Rational[T], y: T) =
## Multiplies the rational `x` by the int `y` in-place.
x.num *= y
reduce(x)
func reciprocal*[T](x: Rational[T]): Rational[T] =
## Calculates the reciprocal of `x` (`1/x`).
## If `x` is 0, raises `DivByZeroDefect`.
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] =
## Divides the rational `x` by the rational `y`.
result.num = x.num * y.den
result.den = x.den * y.num
reduce(result)
func `/`*[T](x: Rational[T], y: T): Rational[T] =
## Divides the rational `x` by the int `y`.
result.num = x.num
result.den = x.den * y
reduce(result)
func `/`*[T](x: T, y: Rational[T]): Rational[T] =
## Divides the int `x` by the rational `y`.
result.num = x * y.den
result.den = y.num
reduce(result)
func `/=`*[T](x: var Rational[T], y: Rational[T]) =
## Divides the rational `x` by the rational `y` in-place.
x.num *= y.den
x.den *= y.num
reduce(x)
func `/=`*[T](x: var Rational[T], y: T) =
## Divides the rational `x` by the int `y` in-place.
x.den *= y
reduce(x)
func cmp*(x, y: Rational): int =
## Compares two rationals. Returns
## * a value less than zero, if `x < y`
## * a value greater than zero, if `x > y`
## * zero, if `x == y`
(x - y).num
func `<`*(x, y: Rational): bool =
## Returns true if `x` is less than `y`.
(x - y).num < 0
func `<=`*(x, y: Rational): bool =
## Returns tue if `x` is less than or equal to `y`.
(x - y).num <= 0
func `==`*(x, y: Rational): bool =
## Compares two rationals for equality.
(x - y).num == 0
func abs*[T](x: Rational[T]): Rational[T] =
## Returns the absolute value of `x`.
runnableExamples:
doAssert abs(1 // 2) == 1 // 2
doAssert abs(-1 // 2) == 1 // 2
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 the hash for the 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
func `^`*[T: SomeInteger](x: Rational[T], y: T): Rational[T] =
## Computes `x` to the power of `y`.
##
## The exponent `y` must be an integer. Negative exponents are supported
## but floating point exponents are not.
runnableExamples:
doAssert (-3 // 5) ^ 0 == (1 // 1)
doAssert (-3 // 5) ^ 1 == (-3 // 5)
doAssert (-3 // 5) ^ 2 == (9 // 25)
doAssert (-3 // 5) ^ -2 == (25 // 9)
if y >= 0:
result.num = x.num ^ y
result.den = x.den ^ y
else:
result.num = x.den ^ -y
result.den = x.num ^ -y
# Note that all powers of reduced rationals are already reduced,
# so we don't need to call reduce() here
|
