Module rationals

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

Imports

math, hashes

Types

Rational[T] = object
  num*, den*: T

a rational number, consisting of a numerator and denominator

Procs

proc initRational[T: SomeInteger](num, den: T): Rational[T]

Create a new rational number.

proc `//`[T](num, den: T): Rational[T]

A friendlier version of initRational. Example usage:

var x = 1//3 + 1//5

proc `$`[T](x: Rational[T]): string

Turn a rational number into a string.

proc toRational[T: SomeInteger](x: T): Rational[T]

Convert some integer x to a rational number.

proc toRational(x: float; n: int = high(int) shr 32): Rational[int] {...}{.raises: [], tags: [].}

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.

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

proc toFloat[T](x: Rational[T]): float

Convert a rational number x to a float.

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.

proc reduce[T: SomeInteger](x: var Rational[T])

Reduce rational x.

proc `+`[T](x, y: Rational[T]): Rational[T]

Add two rational numbers.

proc `+`[T](x: Rational[T]; y: T): Rational[T]

Add rational x to int y.

proc `+`[T](x: T; y: Rational[T]): Rational[T]

Add int x to rational y.

proc `+=`[T](x: var Rational[T]; y: Rational[T])

Add rational y to rational x.

proc `+=`[T](x: var Rational[T]; y: T)

Add int y to rational x.

proc `-`[T](x: Rational[T]): Rational[T]

Unary minus for rational numbers.

proc `-`[T](x, y: Rational[T]): Rational[T]

Subtract two rational numbers.

proc `-`[T](x: Rational[T]; y: T): Rational[T]

Subtract int y from rational x.

proc `-`[T](x: T; y: Rational[T]): Rational[T]

Subtract rational y from int x.

proc `-=`[T](x: var Rational[T]; y: Rational[T])

Subtract rational y from rational x.

proc `-=`[T](x: var Rational[T]; y: T)

Subtract int y from rational x.

proc `*`[T](x, y: Rational[T]): Rational[T]

Multiply two rational numbers.

proc `*`[T](x: Rational[T]; y: T): Rational[T]

Multiply rational x with int y.

proc `*`[T](x: T; y: Rational[T]): Rational[T]

Multiply int x with rational y.

proc `*=`[T](x: var Rational[T]; y: Rational[T])

Multiply rationals y to x.

proc `*=`[T](x: var Rational[T]; y: T)

Multiply int y to rational x.

proc reciprocal[T](x: Rational[T]): Rational[T]

Calculate the reciprocal of x. (1/x)

proc `/`[T](x, y: Rational[T]): Rational[T]

Divide rationals x by y.

proc `/`[T](x: Rational[T]; y: T): Rational[T]

Divide rational x by int y.

proc `/`[T](x: T; y: Rational[T]): Rational[T]

Divide int x by Rational y.

proc `/=`[T](x: var Rational[T]; y: Rational[T])

Divide rationals x by y in place.

proc `/=`[T](x: var Rational[T]; y: T)

Divide rational x by int y in place.

proc cmp(x, y: Rational): int {...}{.procvar.}

Compares two rationals.

proc `<`(x, y: Rational): bool

proc `<=`(x, y: Rational): bool

proc `==`(x, y: Rational): bool

proc abs[T](x: Rational[T]): Rational[T]

proc `div`[T: SomeInteger](x, y: Rational[T]): T

Computes the rational truncated division.

proc `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.

proc 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.

proc 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 proc behaves the same as the % operator in python.

proc hash[T](x: Rational[T]): Hash

Computes hash for rational x