{"id":7012,"date":"2022-05-11T10:47:01","date_gmt":"2022-05-11T08:47:01","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7012"},"modified":"2022-05-11T10:47:01","modified_gmt":"2022-05-11T08:47:01","slug":"reconocimiento-de-potencias-de-2","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/reconocimiento-de-potencias-de-2\/","title":{"rendered":"Reconocimiento de potencias de 2"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   esPotenciaDeDos :: Integer -> Bool\n<\/pre>\n<p>tal que <code>(esPotenciaDeDos n)<\/code> se verifica si <code>n<\/code> es una potencia de dos (suponiendo que <code>n<\/code> es mayor que 0). Por ejemplo.<\/p>\n<pre lang=\"text\">\n   esPotenciaDeDos    1        == True\n   esPotenciaDeDos    2        == True\n   esPotenciaDeDos    6        == False\n   esPotenciaDeDos    8        == True\n   esPotenciaDeDos 1024        == True\n   esPotenciaDeDos 1026        == False\n   esPotenciaDeDos (2^(10^8))  == True\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Bits ((.&.))\nimport Data.Numbers.Primes (primeFactors)\nimport Test.QuickCheck (Positive (Positive), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nesPotenciaDeDos1 :: Integer -> Bool\nesPotenciaDeDos1 1 = True\nesPotenciaDeDos1 n\n  | even n    = esPotenciaDeDos1 (n `div` 2)\n  | otherwise = False\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nesPotenciaDeDos2 :: Integer -> Bool\nesPotenciaDeDos2 n = n ==\n  head (dropWhile (<n) potenciasDeDos)\n\n-- potenciasDeDos es la lista de las potencias de dos. Por ejemplo,\n--    take 10 potenciasDeDos  == [1,2,4,8,16,32,64,128,256,512]\npotenciasDeDos :: [Integer]\npotenciasDeDos = iterate (*2) 1\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nesPotenciaDeDos3 :: Integer -> Bool\nesPotenciaDeDos3 x = all (==2) (primeFactors x)\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\n-- Usando la funci\u00f3n (.&.) de la librer\u00eda Data.Bits. Dicha funci\u00f3n\n-- calcula el n\u00famero correspondiente a la conjunci\u00f3n de las\n-- representaciones binarias de sus argumentos. Por ejemplo,\n--    6 .&. 3 == 2\n-- ya que\n--    la representaci\u00f3n binaria de 6 es     [1,1,0]\n--    la representaci\u00f3n binaria de 3 es       [1,1]\n--    la conjunci\u00f3n es                        [1,0]\n--    la representaci\u00f3n decimal de [1,0] es   2\n--\n-- Otros ejemplos:\n--    4 .&. 3 ==   [1,0,0] .&.   [1,1] == 0\n--    8 .&. 7 == [1,0,0,0] .&. [1,1,1] = 0\n\nesPotenciaDeDos4 :: Integer -> Bool\nesPotenciaDeDos4 n = n .&. (n-1) == 0\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_esPotenciaDeDos :: Positive Integer -> Bool\nprop_esPotenciaDeDos (Positive n) =\n  all (== esPotenciaDeDos1 n)\n      [ esPotenciaDeDos2 n\n      , esPotenciaDeDos3 n\n      , esPotenciaDeDos4 n\n      ]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_esPotenciaDeDos\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> esPotenciaDeDos1 (2^(3*10^5))\n--    True\n--    (3.51 secs, 5,730,072,544 bytes)\n--    \u03bb> esPotenciaDeDos2 (2^(3*10^5))\n--    True\n--    (3.12 secs, 5,755,639,952 bytes)\n--    \u03bb> esPotenciaDeDos3 (2^(3*10^5))\n--    True\n--    (2.92 secs, 5,758,872,040 bytes)\n--    \u03bb> esPotenciaDeDos4 (2^(3*10^5))\n--    True\n--    (0.03 secs, 715,152 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Reconocimiento_de_grandes_potencias_de_2.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n esPotenciaDeDos :: Integer -> Bool tal que (esPotenciaDeDos n) se verifica si n es una potencia de dos (suponiendo que n es mayor que 0). Por ejemplo. esPotenciaDeDos 1 == True esPotenciaDeDos 2 == True esPotenciaDeDos 6 == False esPotenciaDeDos 8 == True esPotenciaDeDos 1024 == True esPotenciaDeDos 1026 == False esPotenciaDeDos&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[2],"tags":[41,571,501,59,91,71,50,11,247,6,521,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7012"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=7012"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7012\/revisions"}],"predecessor-version":[{"id":7013,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7012\/revisions\/7013"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7012"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7012"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7012"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}