{"id":6559,"date":"2022-01-26T17:54:12","date_gmt":"2022-01-26T15:54:12","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6559"},"modified":"2022-03-16T09:21:06","modified_gmt":"2022-03-16T07:21:06","slug":"cuadrado-mas-cercano","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/cuadrado-mas-cercano\/","title":{"rendered":"Cuadrado m\u00e1s cercano"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   cuadradoCercano :: Integer -> Integer\n<\/pre>\n<p>tal que (cuadradoCercano n) es el n\u00famero cuadrado m\u00e1s cercano a n, donde n es un entero positivo. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   cuadradoCercano 2       == 1\n   cuadradoCercano 6       == 4\n   cuadradoCercano 8       == 9\n   cuadradoCercano (10^46) == 10000000000000000000000000000000000000000000000\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ncuadradoCercano1 :: Integer -> Integer\ncuadradoCercano1 n\n  | n - b < c - n = b\n  | otherwise     = c\n  where a = raizEntera1 n\n        b = a^2\n        c = (a+1)^2\n\n-- (raizEntera x) es el mayor entero cuyo cuadrado no es mayor que\n-- x. Por ejemplo,\n--    raizEntera 8   ==  2\n--    raizEntera 9   ==  3\n--    raizEntera 10  ==  3\nraizEntera1 :: Integer -> Integer\nraizEntera1 x =\n  last (takeWhile (\\n -> n^2 <= x) [1..])\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ncuadradoCercano2 :: Integer -> Integer\ncuadradoCercano2 n\n  | n - b < c - n = b\n  | otherwise     = c\n  where a = raizEntera2 n\n        b = a^2\n        c = (a+1)^2\n\nraizEntera2 :: Integer -> Integer\nraizEntera2 x = aux (1,x)\n    where aux (a,b) | d == x    = c\n                    | c == a    = c\n                    | x <= d    = aux (a,c)\n                    | otherwise = aux (c,b)\n            where c = (a+b) `div` 2\n                  d = c^2\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ncuadradoCercano3 :: Integer -> Integer\ncuadradoCercano3 n\n  | n - b < c - n = b\n  | otherwise     = c\n  where a = raizEntera3 n\n        b = a^2\n        c = (a+1)^2\n\nraizEntera3 :: Integer -> Integer\nraizEntera3 0 = 0\nraizEntera3 1 = 1\nraizEntera3 n = until aceptable mejora n\n  where mejora x    = (x + n `div` x) `div` 2\n        aceptable x = x^2 <= n\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\ncuadradoCercano4 :: Integer -> Integer\ncuadradoCercano4 = (^ 2) . round . sqrt . fromIntegral\n\n-- Equivalencia de las definiciones\n-- ================================\n\n-- La propiedad es\nprop_cuadradoCercano :: Positive Integer -> Bool\nprop_cuadradoCercano (Positive x) =\n  all (== cuadradoCercano1 x)\n      [cuadradoCercano2 x,\n       cuadradoCercano3 x,\n       cuadradoCercano4 x]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_cuadradoCercano\n--    +++ OK, passed 100 tests.\n\n-- Aunque ha pasado los 100 tests, las definiciones no son\n-- equivalentes. Por ejemplo,\n--    \u03bb> cuadradoCercano3 (10^46) == cuadradoCercano (10^46)\n--    False\n--    \u03bb> cuadradoCercano3 (10^46)\n--    9999999999999998322278400000000070368744177664\n--    \u03bb> cuadradoCercano (10^46)\n--    10000000000000000000000000000000000000000000000\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> cuadradoCercano1 (10^14)\n--    100000000000000\n--    (4.59 secs, 5,920,475,784 bytes)\n--    \u03bb> cuadradoCercano2 (10^14)\n--    100000000000000\n--    (0.01 secs, 512,472 bytes)\n--    \u03bb> cuadradoCercano3 (10^14)\n--    100000000000000\n--    (0.01 secs, 494,248 bytes)\n--    \u03bb> cuadradoCercano4 (10^14)\n--    100000000000000\n--    (0.01 secs, 475,152 bytes)\n--\n--    \u03bb> length (show (cuadradoCercano2 (10^20000)))\n--    20001\n--    (3.94 secs, 1,446,675,504 bytes)\n--    \u03bb> length (show (cuadradoCercano3 (10^20000)))\n--    20001\n--    (4.50 secs, 926,647,904 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Cuadrado_mas_cercano.hs\">GitHub<\/a><\/p>\n<p>La elaboraci\u00f3n de las soluciones se muestra en el siguiente v\u00eddeo:<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/W6Slw8tcoLM\" title=\"Elaboraci\u00f3n de las soluciones\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n cuadradoCercano :: Integer -> Integer tal que (cuadradoCercano n) es el n\u00famero cuadrado m\u00e1s cercano a n, donde n es un entero positivo. Por ejemplo, cuadradoCercano 2 == 1 cuadradoCercano 6 == 4 cuadradoCercano 8 == 9 cuadradoCercano (10^46) == 10000000000000000000000000000000000000000000000 Soluciones import Test.QuickCheck &#8212; 1\u00aa soluci\u00f3n &#8212; =========== cuadradoCercano1 :: Integer&#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":[11,6],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6559"}],"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=6559"}],"version-history":[{"count":10,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6559\/revisions"}],"predecessor-version":[{"id":6782,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6559\/revisions\/6782"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6559"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}