std/rationals
Source Edit
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
Imports
math, hashes
Types
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Rational[T] = object
num*, den*: T
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A rational number, consisting of a numerator
num and a denominator den. Source Edit
Procs
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func `$`[T](x: Rational[T]): string
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Turns a rational number into a string.
Example:
doAssert $(1 // 2) == "1/2"
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func `*`[T](x, y: Rational[T]): Rational[T]
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Multiplies two rational numbers. Source Edit
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func `*`[T](x: Rational[T]; y: T): Rational[T]
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Multiplies the rational
x with the int y. Source Edit
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func `*`[T](x: T; y: Rational[T]): Rational[T]
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Multiplies the int
x with the rational y. Source Edit
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func `*=`[T](x: var Rational[T]; y: Rational[T])
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Multiplies the rational
x by y in-place. Source Edit
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func `*=`[T](x: var Rational[T]; y: T)
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Multiplies the rational
x by the int y in-place. Source Edit
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func `+`[T](x, y: Rational[T]): Rational[T]
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Adds two rational numbers. Source Edit
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func `+`[T](x: Rational[T]; y: T): Rational[T]
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Adds the rational
x to the int y. Source Edit
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func `+`[T](x: T; y: Rational[T]): Rational[T]
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Adds the int
x to the rational y. Source Edit
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func `+=`[T](x: var Rational[T]; y: Rational[T])
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Adds the rational
y to the rational x in-place. Source Edit
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func `+=`[T](x: var Rational[T]; y: T)
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Adds the int
y to the rational x in-place. Source Edit
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func `-`[T](x, y: Rational[T]): Rational[T]
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Subtracts two rational numbers. Source Edit
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func `-`[T](x: Rational[T]): Rational[T]
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Unary minus for rational numbers. Source Edit
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func `-`[T](x: Rational[T]; y: T): Rational[T]
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Subtracts the int
y from the rational x. Source Edit
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func `-`[T](x: T; y: Rational[T]): Rational[T]
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Subtracts the rational
y from the int x. Source Edit
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func `-=`[T](x: var Rational[T]; y: Rational[T])
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Subtracts the rational
y from the rational x in-place. Source Edit
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func `-=`[T](x: var Rational[T]; y: T)
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Subtracts the int
y from the rational x in-place. Source Edit
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func `/`[T](x, y: Rational[T]): Rational[T]
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Divides the rational
x by the rational y. Source Edit
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func `/`[T](x: Rational[T]; y: T): Rational[T]
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Divides the rational
x by the int y. Source Edit
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func `/`[T](x: T; y: Rational[T]): Rational[T]
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Divides the int
x by the rational y. Source Edit
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func `//`[T](num, den: T): Rational[T]
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A friendlier version of initRational.
Example:
let x = 1 // 3 + 1 // 5
doAssert x == 8 // 15
Source Edit
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func `/=`[T](x: var Rational[T]; y: Rational[T])
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Divides the rational
x by the rational y in-place. Source Edit
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func `/=`[T](x: var Rational[T]; y: T)
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Divides the rational
x by the int y in-place. Source Edit
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func `<`(x, y: Rational): bool
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Returns true if
x is less than y. Source Edit
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func `<=`(x, y: Rational): bool
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Returns tue if
x is less than or equal to y. Source Edit
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func `==`(x, y: Rational): bool
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Compares two rationals for equality. Source Edit
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func abs[T](x: Rational[T]): Rational[T]
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Returns the absolute value of
x.
Example:
doAssert abs(1 // 2) == 1 // 2
doAssert abs(-1 // 2) == 1 // 2
Source Edit
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func cmp(x, y: Rational): int
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Compares two rationals. Returns
- a value less than zero, if
x < y
- a value greater than zero, if
x > y
- zero, if
x == y
Source Edit
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func `div`[T: SomeInteger](x, y: Rational[T]): T
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Computes the rational truncated division. Source Edit
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func floorDiv[T: SomeInteger](x, y: Rational[T]): T
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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.
Source Edit
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func floorMod[T: SomeInteger](x, y: Rational[T]): Rational[T]
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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.
Source Edit
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func hash[T](x: Rational[T]): Hash
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Computes the hash for the rational
x. Source Edit
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func initRational[T: SomeInteger](num, den: T): Rational[T]
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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.
Source Edit
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func `mod`[T: SomeInteger](x, y: Rational[T]): Rational[T]
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Computes the rational modulo by truncated division (remainder). This is same as
x - (x div y) * y. Source Edit
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func reciprocal[T](x: Rational[T]): Rational[T]
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Calculates the reciprocal of
x (1/x). If x is 0, raises DivByZeroDefect. Source Edit
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func reduce[T: SomeInteger](x: var Rational[T])
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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
Source Edit
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func toFloat[T](x: Rational[T]): float
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Converts a rational number
x to a float. Source Edit
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func toInt[T](x: Rational[T]): int
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Converts a rational number
x to an int. Conversion rounds towards 0 if x does not contain an integer value. Source Edit
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func toRational(x: float; n: int = high(int) shr 32): Rational[int] {.
...raises: [], tags: [], forbids: [].}
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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 toRational[T: SomeInteger](x: T): Rational[T]
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Converts some integer
x to a rational number.
Example:
doAssert toRational(42) == 42 // 1
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