std/rationals

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This module implements rational numbers, consisting of a numerator and a denominator. The denominator can not be 0.

Example:

import std/rationals
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

Types

Rational[T] = object
  num*, den*: T
A rational number, consisting of a numerator num and a denominator den.   Source   Edit

Procs

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.

Example:

var r = Rational[int](num: 2, den: 4) # 1/2
reduce(r)
doAssert r.num == 1
doAssert r.den == 2
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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.

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func `//`[T](num, den: T): Rational[T]
A friendlier version of initRational.

Example:

let x = 1 // 3 + 1 // 5
doAssert x == 8 // 15
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func `$`[T](x: Rational[T]): string
Turns a rational number into a string.

Example:

doAssert $(1 // 2) == "1/2"
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func toRational[T: SomeInteger](x: T): Rational[T]
Converts some integer x to a rational number.

Example:

doAssert toRational(42) == 42 // 1
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func toRational(x: float; n: int = high(int) shr 32): Rational[int] {.
    ...raises: [], tags: [].}

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.

Example:

let x = 1.2
doAssert x.toRational.toFloat == x
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func toFloat[T](x: Rational[T]): float
Converts a rational number x to a float.   Source   Edit
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.   Source   Edit
func `+`[T](x, y: Rational[T]): Rational[T]
Adds two rational numbers.   Source   Edit
func `+`[T](x: Rational[T]; y: T): Rational[T]
Adds the rational x to the int y.   Source   Edit
func `+`[T](x: T; y: Rational[T]): Rational[T]
Adds the int x to the rational y.   Source   Edit
func `+=`[T](x: var Rational[T]; y: Rational[T])
Adds the rational y to the rational x in-place.   Source   Edit
func `+=`[T](x: var Rational[T]; y: T)
Adds the int y to the rational x in-place.   Source   Edit
func `-`[T](x: Rational[T]): Rational[T]
Unary minus for rational numbers.   Source   Edit
func `-`[T](x, y: Rational[T]): Rational[T]
Subtracts two rational numbers.   Source   Edit
func `-`[T](x: Rational[T]; y: T): Rational[T]
Subtracts the int y from the rational x.   Source   Edit
func `-`[T](x: T; y: Rational[T]): Rational[T]
Subtracts the rational y from the int x.   Source   Edit
func `-=`[T](x: var Rational[T]; y: Rational[T])
Subtracts the rational y from the rational x in-place.   Source   Edit
func `-=`[T](x: var Rational[T]; y: T)
Subtracts the int y from the rational x in-place.   Source   Edit
func `*`[T](x, y: Rational[T]): Rational[T]
Multiplies two rational numbers.   Source   Edit
func `*`[T](x: Rational[T]; y: T): Rational[T]
Multiplies the rational x with the int y.   Source   Edit
func `*`[T](x: T; y: Rational[T]): Rational[T]
Multiplies the int x with the rational y.   Source   Edit
func `*=`[T](x: var Rational[T]; y: Rational[T])
Multiplies the rational x by y in-place.   Source   Edit
func `*=`[T](x: var Rational[T]; y: T)
Multiplies the rational x by the int y in-place.   Source   Edit
func reciprocal[T](x: Rational[T]): Rational[T]
Calculates the reciprocal of x (1/x). If x is 0, raises DivByZeroDefect.   Source   Edit
func `/`[T](x, y: Rational[T]): Rational[T]
Divides the rational x by the rational y.   Source   Edit
func `/`[T](x: Rational[T]; y: T): Rational[T]
Divides the rational x by the int y.   Source   Edit
func `/`[T](x: T; y: Rational[T]): Rational[T]
Divides the int x by the rational y.   Source   Edit
func `/=`[T](x: var Rational[T]; y: Rational[T])
Divides the rational x by the rational y in-place.   Source   Edit
func `/=`[T](x: var Rational[T]; y: T)
Divides the rational x by the int y in-place.   Source   Edit
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
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func `<`(x, y: Rational): bool
Returns true if x is less than y.   Source   Edit
func `<=`(x, y: Rational): bool
Returns tue if x is less than or equal to y.   Source   Edit
func `==`(x, y: Rational): bool
Compares two rationals for equality.   Source   Edit
func abs[T](x: Rational[T]): Rational[T]
Returns the absolute value of x.

Example:

doAssert abs(1 // 2) == 1 // 2
doAssert abs(-1 // 2) == 1 // 2
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func `div`[T: SomeInteger](x, y: Rational[T]): T
Computes the rational truncated division.   Source   Edit
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.   Source   Edit
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.

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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.

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func hash[T](x: Rational[T]): Hash
Computes the hash for the rational x.   Source   Edit