Raíz cúbica entera

Un número x es un cubo si existe un y tal que x = y^3. Por ejemplo, 8 es un cubo porque 8 = 2^3.

Definir la función

   raizCubicaEntera :: Integer -> Maybe Integer.

1	raizCubicaEntera :: Integer -> Maybe Integer.

tal que (raizCubicaEntera x n) es justo la raíz cúbica del número natural x, si x es un cubo y Nothing en caso contrario. Por ejemplo,

   raizCubicaEntera 8             ==  Just 2
   raizCubicaEntera 9             ==  Nothing
   raizCubicaEntera 27            ==  Just 3
   raizCubicaEntera 64            ==  Just 4
   raizCubicaEntera (2^30)        ==  Just 1024
   raizCubicaEntera (10^9000)     ==  Just (10^3000)
   raizCubicaEntera (5 + 10^9000) ==  Nothing

raizCubicaEntera 8 == Just 2

raizCubicaEntera 9 == Nothing

raizCubicaEntera 27 == Just 3

raizCubicaEntera 64 == Just 4

raizCubicaEntera (2^30) == Just 1024

raizCubicaEntera (10^9000) == Just (10^3000)

raizCubicaEntera (5 + 10^9000) == Nothing

Soluciones

import Test.QuickCheck

-- 1ª definición
-- =============

raizCubicaEntera :: Integer -> Maybe Integer
raizCubicaEntera x = aux 0
  where aux y | y^3 > x   = Nothing
              | y^3 == x  = Just y
              | otherwise = aux (y+1)

-- 2ª definición
-- =============

raizCubicaEntera2 :: Integer -> Maybe Integer
raizCubicaEntera2 x 
  | y^3 == x  = Just y
  | otherwise = Nothing
  where (y:_) = dropWhile (\y -> y^3 < x) [0..]

-- 3ª definición
-- =============

raizCubicaEntera3 :: Integer -> Maybe Integer 
raizCubicaEntera3 1 = Just 1
raizCubicaEntera3 x = aux (0,x)
    where aux (a,b) | d == x    = Just c
                    | c == a    = Nothing
                    | d < x     = aux (c,b)
                    | otherwise = aux (a,c) 
              where c = (a+b) `div` 2
                    d = c^3

-- 4ª definición
-- =============

raizCubicaEntera4 :: Integer -> Maybe Integer
raizCubicaEntera4 x 
  | y^3 == x  = Just y
  | otherwise = Nothing
  where y = floor ((fromIntegral x)**(1 / 3))

-- Nota. La definición anterior falla para números grandes. Por ejemplo,
--    λ> raizCubicaEntera4 (2^30)
--    Nothing
--    λ> raizCubicaEntera (2^30)
--    Just 1024

-- Equivalencia
-- ============

-- La propiedad es                        
prop_raizCubicaEntera :: Integer -> Property
prop_raizCubicaEntera x =
  x >= 0 ==>
  and [raizCubicaEntera x == f x | f <- [ raizCubicaEntera2
                                        , raizCubicaEntera3]]

-- La comprobación es              
--    λ> quickCheck prop_raizCubicaEntera
--    +++ OK, passed 100 tests.

-- Comparación de eficiencia
-- =========================
  
--    λ> raizCubicaEntera (10^18)
--    Just 1000000
--    (1.80 secs, 1,496,137,192 bytes)
--    λ> raizCubicaEntera2 (10^18)
--    Just 1000000
--    (0.71 secs, 712,134,128 bytes)
--    λ> raizCubicaEntera3 (10^18)
--    Just 1000000
--    (0.01 secs, 196,424 bytes)
--
--    λ> raizCubicaEntera2 (5^27)
--    Just 1953125
--    (1.42 secs, 1,390,760,920 bytes)
--    λ> raizCubicaEntera3 (5^27)
--    Just 1953125
--    (0.00 secs, 195,888 bytes)
--
--    λ> raizCubicaEntera3 (10^9000) == Just (10^3000)
--    True
--    (2.05 secs, 420,941,368 bytes)
--    λ> raizCubicaEntera3 (5 + 10^9000) == Nothing
--    True
--    (2.08 secs, 420,957,576 bytes)

import Test.QuickCheck

-- 1ª definición

-- =============

raizCubicaEntera :: Integer -> Maybe Integer

raizCubicaEntera x = aux 0

where aux y | y^3 > x = Nothing

| y^3 == x = Just y

| otherwise = aux (y+1)

-- 2ª definición

-- =============

raizCubicaEntera2 :: Integer -> Maybe Integer

raizCubicaEntera2 x

| y^3 == x = Just y

| otherwise = Nothing

where (y:_) = dropWhile (\y -> y^3 < x) [0..]

-- 3ª definición

-- =============

raizCubicaEntera3 :: Integer -> Maybe Integer

raizCubicaEntera3 1 = Just 1

raizCubicaEntera3 x = aux (0,x)

where aux (a,b) | d == x = Just c

| c == a = Nothing

| d < x = aux (c,b)

| otherwise = aux (a,c)

where c = (a+b) `div` 2

d = c^3

-- 4ª definición

-- =============

raizCubicaEntera4 :: Integer -> Maybe Integer

raizCubicaEntera4 x

| y^3 == x = Just y

| otherwise = Nothing

where y = floor ((fromIntegral x)**(1 / 3))

-- Nota. La definición anterior falla para números grandes. Por ejemplo,

-- λ> raizCubicaEntera4 (2^30)

-- Nothing

-- λ> raizCubicaEntera (2^30)

-- Just 1024

-- Equivalencia

-- ============

-- La propiedad es

prop_raizCubicaEntera :: Integer -> Property

prop_raizCubicaEntera x =

x >= 0 ==>

and [raizCubicaEntera x == f x | f <- [ raizCubicaEntera2

, raizCubicaEntera3]]

-- La comprobación es

-- λ> quickCheck prop_raizCubicaEntera

-- +++ OK, passed 100 tests.

-- Comparación de eficiencia

-- =========================

-- λ> raizCubicaEntera (10^18)

-- Just 1000000

-- (1.80 secs, 1,496,137,192 bytes)

-- λ> raizCubicaEntera2 (10^18)

-- Just 1000000

-- (0.71 secs, 712,134,128 bytes)

-- λ> raizCubicaEntera3 (10^18)

-- Just 1000000

-- (0.01 secs, 196,424 bytes)

-- λ> raizCubicaEntera2 (5^27)

-- Just 1953125

-- (1.42 secs, 1,390,760,920 bytes)

-- λ> raizCubicaEntera3 (5^27)

-- Just 1953125

-- (0.00 secs, 195,888 bytes)

-- λ> raizCubicaEntera3 (10^9000) == Just (10^3000)

-- True

-- (2.05 secs, 420,941,368 bytes)

-- λ> raizCubicaEntera3 (5 + 10^9000) == Nothing

-- True

-- (2.08 secs, 420,957,576 bytes)

Pensamiento

Tras el vivir y el soñar,
está lo que más importa:
despertar.

Antonio Machado

3 Comentarios

import Math.NumberTheory.Powers.Cubes (exactCubeRoot)
 
raizCubicaEntera1 :: Integer -> Maybe Integer
raizCubicaEntera1 x = exactCubeRoot x
 
raizCubicaEntera2 :: Integer -> Maybe Integer
raizCubicaEntera2 x
  | null zs   = Nothing
  | otherwise = Just (head zs)
  where factores x = [z | z <- [1..x], mod x z == 0]
        zs         = [y | y <- factores x, y^3 == x]

import Math.NumberTheory.Powers.Cubes (exactCubeRoot)

raizCubicaEntera1 :: Integer -> Maybe Integer

raizCubicaEntera1 x = exactCubeRoot x

raizCubicaEntera2 :: Integer -> Maybe Integer

raizCubicaEntera2 x

| null zs = Nothing

| otherwise = Just (head zs)

where factores x = [z | z <- [1..x], mod x z == 0]

zs = [y | y <- factores x, y^3 == x]

Responder

import Data.Numbers.Primes (primeFactors)
import Data.List (nub, group, genericLength)

raizCubicaEntera :: Integer -> Maybe Integer
raizCubicaEntera n | c n == False = Nothing
                   | otherwise = Just (e n)

a :: Integer -> [(Integer, Integer)]
a n = zip (map genericLength (group (primeFactors n))) (nub (primeFactors n))

b :: [(Integer, Integer)] -> Bool
b xs = and [mod a 3 == 0 | (a,b) <- xs]

c :: Integer -> Bool
c n = b (a n)

d :: Integer -> [(Integer, Integer)]
d n = zip [div a 3 | (a,b) <- a n] (nub (primeFactors n))

e ::  Integer -> Integer
e n = product [b^a | (a,b) <- d n]

import Data.Numbers.Primes (primeFactors)

import Data.List (nub, group, genericLength)

raizCubicaEntera :: Integer -> Maybe Integer

raizCubicaEntera n | c n == False = Nothing

| otherwise = Just (e n)

a :: Integer -> [(Integer, Integer)]

a n = zip (map genericLength (group (primeFactors n))) (nub (primeFactors n))

b :: [(Integer, Integer)] -> Bool

b xs = and [mod a 3 == 0 | (a,b) <- xs]

c :: Integer -> Bool

c n = b (a n)

d :: Integer -> [(Integer, Integer)]

d n = zip [div a 3 | (a,b) <- a n] (nub (primeFactors n))

e :: Integer -> Integer

e n = product [b^a | (a,b) <- d n]

Responder

raizCubicaEntera :: Integer -> Maybe Integer
raizCubicaEntera x 
                   | f x == False = Nothing
                   | abc x  == False = Nothing
                   | otherwise = Just (product (d x))


c :: Eq a => a -> [a] -> Integer
c _ [] = 0
c x (y:ys) | x == y = 1 + c x ys
           | otherwise = c x ys

                   

nc :: Integer -> [Integer]
nc n = [ c x (primeFactors n) | x <- (nub (primeFactors n))]

abc :: Integer -> Bool
abc n = and [ multiplo3 x == True | x <- nc n]

multiplo3 :: Integer -> Bool
multiplo3 n | rem n 3 == 0 = True
            | otherwise = False

d :: Integer -> [Integer]
d n  = [ a^(div b 3) | (a,b) <- zip (nub (primeFactors n)) (nc n)]

contar :: Integer -> Integer
contar n | n < 10 = 1
                  | otherwise = 1 + contar (n `div` 10)

e :: Integer -> Bool
e x | multiplo3 (contar x -1) == True = True 
    | otherwise = False
      
f :: Integer -> Bool
f x | e x == True && x < ((10^(div (contar x -1) 3)+1)^3) && x /= (10^(contar x -1)) = False
    | otherwise = True

raizCubicaEntera :: Integer -> Maybe Integer

raizCubicaEntera x

| f x == False = Nothing

| abc x == False = Nothing

| otherwise = Just (product (d x))

c :: Eq a => a -> [a] -> Integer

c _ [] = 0

c x (y:ys) | x == y = 1 + c x ys

| otherwise = c x ys

nc :: Integer -> [Integer]

nc n = [ c x (primeFactors n) | x <- (nub (primeFactors n))]

abc :: Integer -> Bool

abc n = and [ multiplo3 x == True | x <- nc n]

multiplo3 :: Integer -> Bool

multiplo3 n | rem n 3 == 0 = True

| otherwise = False

d :: Integer -> [Integer]

d n = [ a^(div b 3) | (a,b) <- zip (nub (primeFactors n)) (nc n)]

contar :: Integer -> Integer

contar n | n < 10 = 1

| otherwise = 1 + contar (n `div` 10)

e :: Integer -> Bool

e x | multiplo3 (contar x -1) == True = True

| otherwise = False

f :: Integer -> Bool

f x | e x == True && x < ((10^(div (contar x -1) 3)+1)^3) && x /= (10^(contar x -1)) = False

| otherwise = True

Responder

Raíz cúbica entera

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