Automated reasoning in F#, Scala, Haskell, C++, and Julia

<div dir="ltr" style="text-align: left;" trbidi="on"> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">We need to simplify the following expression:</span></div> <div style="color: #333333; text-align: justify;"> <span class="mi" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;"><br /></span></div> <div style="color: #333333; text-align: center;"> <span class="mi" id="MathJax-Span-3" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">e</span><span class="mo" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.289em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">=</span><span class="mo" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.289em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">(</span><span class="mn" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">1</span><span class="mo" id="MathJax-Span-7" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">+</span><span class="mn" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">0</span><span class="mo" id="MathJax-Span-9" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">×</span><span class="mi" id="MathJax-Span-10" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">x</span><span class="mo" id="MathJax-Span-11" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">)</span><span class="mo" id="MathJax-Span-12" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">×</span><span class="mn" id="MathJax-Span-13" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">3</span><span class="mo" id="MathJax-Span-14" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">+</span><span class="mn" id="MathJax-Span-15" style="border: 0px; display: inline; font-family: Verdana, sans-serif; margin: 0px; padding: 0px 0px 0px 0.207em; position: static; transition: none; vertical-align: 0px; white-space: nowrap; word-spacing: normal;">12.</span></div> <div style="color: #333333; text-align: center;"> <br /></div> <div style="color: #333333; text-align: left;"> <span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">Luckily for us, we won't have to remember any elementary school arithmetic, because Harris's </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">excellent </span><a href="http://www.cambridge.org/ca/academic/subjects/computer-science/programming-languages-and-applied-logic/handbook-practical-logic-and-automated-reasoning" style="color: #00905d; font-family: Verdana, sans-serif; text-align: justify; text-decoration: none; white-space: nowrap; word-spacing: normal;"><i>Handbook of </i></a><br /> <a href="http://www.cambridge.org/ca/academic/subjects/computer-science/programming-languages-and-applied-logic/handbook-practical-logic-and-automated-reasoning" style="color: #00905d; font-family: Verdana, sans-serif; text-align: justify; text-decoration: none; white-space: nowrap; word-spacing: normal;"><i>Practical Logic and Automated Reasoning</i></a><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;"> </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">begins with a simple algorithm to </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">do exactly that. </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">It's not complicated, but </span><br /> <span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">it's a pretty good barometer of how painful a programming </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">language will be for </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">the kind of hybrid (probabilistic logic, </span><br /> <span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">or statistical relational) approaches I work </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">with. Here, I compare </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">the implementations of Harris' simple algorithm in </span><br /> <span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">F#, Scala, Haskell, C++,  </span><span style="font-family: Verdana, sans-serif; text-align: justify; white-space: nowrap; word-spacing: normal;">and Julia. </span><span style="font-family: Verdana, sans-serif; text-align: justify;">No programming languages were hurt while writing this post. It's not a competition, and I avoided talking about languages I dislike. Sum types are discussed in length because they are awesome and useful for this problem (and many, many others).</span></div> <div style="color: #333333; text-align: left;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;">The ML family</span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOfS32MVSkDFqy4xlNZh9BNzapSWdBkFGfyNc1ohmxIa634l5h_0KMubb-UmsPPqAazBLgD6nt01PFRuq910YMc5hjBb60M2hMtZk6zx0fhWhhLDR1Vt_MSBji7KOMlPMKNEprBFJ6D6c/s1600/the-next-great-functional-programming-language-2-638.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="354" loading="lazy" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOfS32MVSkDFqy4xlNZh9BNzapSWdBkFGfyNc1ohmxIa634l5h_0KMubb-UmsPPqAazBLgD6nt01PFRuq910YMc5hjBb60M2hMtZk6zx0fhWhhLDR1Vt_MSBji7KOMlPMKNEprBFJ6D6c/s640-rw/the-next-great-functional-programming-language-2-638.jpg" width="640" /></a></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Harris' book uses OCaml, a popular language for solvers. F#, Haskell, and Scala all share roots with OCaml, with F# being the closest thing to an OCaml dialect. I'll start with F#:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="fsharp hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span style="color: #738191; font-family: Verdana, sans-serif;">/// A sum type for the expression.</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span style="color: #738191;">/// An expression is either a var (which is a string), a constant</span></div> <span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// (which is an integer), an addition (made of two expressions)</div> </span><span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// or a multiplication (also made of two expressions).</div> </span><span class="hljs-class"><div style="text-align: justify;"> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">type</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Expr</span> </span>=</div> </span><div style="text-align: justify;"> | Var <span class="hljs-keyword" style="font-weight: bold;">of</span> string</div> <div style="text-align: justify;"> | Const <span class="hljs-keyword" style="font-weight: bold;">of</span> int</div> <div style="text-align: justify;"> | Add <span class="hljs-keyword" style="font-weight: bold;">of</span> Expr * Expr</div> <div style="text-align: justify;"> | Mul <span class="hljs-keyword" style="font-weight: bold;">of</span> Expr * Expr</div> <div style="text-align: justify;"> </div> <span class="hljs-comment" style="color: #738191;">/// Simplify a single component of the expression. This function</span> <div style="text-align: justify;"> <span style="color: #738191;">/// takes an expression and use pattern matching to select the</span></div> <span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// right approach based on type and value. For example, if we</div> </span><span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// add a constant 0 to some x (which can be expression), then</div> </span><span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// we return x.</div> </span><span class="hljs-keyword" style="font-weight: bold;"><div style="text-align: justify;"> <span class="hljs-keyword">let</span> simplify1 e =</div> </span><div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">match</span> e <span class="hljs-keyword" style="font-weight: bold;">with</span></div> <div style="text-align: justify;"> | Add (Const <span class="hljs-number">0</span>, x)</div> <div style="text-align: justify;"> | Add (x, Const <span class="hljs-number">0</span>)</div> <div style="text-align: justify;"> | Mul (x, Const <span class="hljs-number">1</span>)</div> <div style="text-align: justify;"> | Mul (Const <span class="hljs-number">1</span>, x) -> x</div> <div style="text-align: justify;"> | Mul (x, Const <span class="hljs-number">0</span>)</div> <div style="text-align: justify;"> | Mul (Const <span class="hljs-number">0</span>, x) -> Const <span class="hljs-number">0</span></div> <div style="text-align: justify;"> | Add (Const a, Const b) -> Const (a + b)</div> | Mul (Const a, Const b) -> Const (a * b) <div style="text-align: justify;"> <span style="color: #738191;">/// Recursive function to simplify an entire expression.</span></div> | _ -> e <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">let</span> <span class="hljs-keyword" style="font-weight: bold;">rec</span> simplify e = <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">match</span> e <span class="hljs-keyword" style="font-weight: bold;">with</span></div> <div style="text-align: justify;"> | Add (x, y) -> Add (simplify x, simplify y)</div> | Mul (x, y) -> Mul (simplify x, simplify y) <div style="text-align: justify;"> <span style="color: #738191;">/// Return the value string if the expression can be reduced to a constant.</span></div> | _ -> e |> simplify1 <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">let</span> exprStr e = <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">match</span> e <span class="hljs-keyword" style="font-weight: bold;">with</span></div> <div style="text-align: justify;"> | Const x -> string x</div> <div style="text-align: justify;"> | _ -> <span class="hljs-string" style="color: #0048ab;">"The expression could not be simplified to a constant."</span></div> <div style="text-align: justify;"> </div> <span class="hljs-comment" style="color: #738191;">/// The |> operator sends the result on its left to its right, for example</span> <div style="text-align: justify;"> <span style="color: #738191;">/// "5.0 |> log |> sqrt" computes log(5.0) and then the square root of the</span></div> <span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// result. This is nice because it allows a more natural left-to-right</div> </span><span class="hljs-comment" style="color: #738191;"><div style="text-align: justify;"> /// flow for functional programming.</div> </span><span class="hljs-annotation"><div style="text-align: justify;"> [<EntryPoint>]</div> </span><span class="hljs-keyword" style="font-weight: bold;"><div style="text-align: justify;"> <span class="hljs-keyword">let</span> main argv =</div> </span><div style="text-align: justify;"> Add (Mul (Add (Const <span class="hljs-number">1</span>, Mul (Const <span class="hljs-number">0</span>, Var <span class="hljs-string" style="color: #0048ab;">"x"</span>)), Const <span class="hljs-number">3</span>), Const <span class="hljs-number">12</span>)</div> <div style="text-align: justify;"> |> simplify</div> |> exprStr <div style="text-align: justify;"> |> printf <span class="hljs-string" style="color: #0048ab;">"%s"</span></div> <div style="text-align: justify;"> <span class="hljs-number">0</span> <span class="hljs-comment" style="color: #738191;">/// F#'s main returns 0 for success à la C</span></div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">It's almost the same as the OCaml version in Harris' book. The key trick is to define an expression (Expr) as a variable (string) <b>or</b> a constant (integer) <b>or</b> an addition <b>or</b> a multiplication (both made of two expressions). The <b>or</b> is important, object-oriented programming languages focuses on hierarchies of objects, while sum types define a type as a series of alternatives. Sum types are important for another reason: they provide an easy way to express things like "this function <i>might</i> return an integer", for example in Haskell if we want a data structure that maps keys to values:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="haskell hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-family: Verdana, sans-serif; font-weight: bold;">import</span><span style="font-family: Verdana, sans-serif;"> Data.Map </span><span class="hljs-container" style="font-family: Verdana, sans-serif;">(<span class="hljs-type">Map</span>)</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">import</span> <span class="hljs-keyword" style="font-weight: bold;">qualified</span> Data.Map <span class="hljs-keyword" style="font-weight: bold;">as</span> Map</div> <div style="text-align: justify;"> </div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">capitals</span> = <span class="hljs-type">Map</span>.fromList [(<span class="hljs-string" style="color: #0048ab;">"Finland"</span>, <span class="hljs-string" style="color: #0048ab;">"Helsinki"</span>), (<span class="hljs-string" style="color: #0048ab;">"France"</span>, <span class="hljs-string" style="color: #0048ab;">"Paris"</span>), <div style="text-align: justify;"> (<span class="hljs-string" style="color: #0048ab;">"Japan"</span>, <span class="hljs-string" style="color: #0048ab;">"Tokyo"</span>), (<span class="hljs-string" style="color: #0048ab;">"South Korea"</span>, <span class="hljs-string" style="color: #0048ab;">"Seoul"</span>), (<span class="hljs-string" style="color: #0048ab;">"Arrakis"</span>, <span class="hljs-string" style="color: #0048ab;">"Arrakeen"</span>)]</div> <div style="text-align: justify;"> </div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">lookupCapitals</span> country = <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Map</span>.lookup country capitals <span class="hljs-keyword" style="font-weight: bold;">of</span> <div style="text-align: justify;"> <span class="hljs-type">Just</span> capital -> <span class="hljs-string" style="color: #0048ab;">"The capital of "</span> ++ country ++ <span class="hljs-string" style="color: #0048ab;">" is "</span> ++ capital ++ <span class="hljs-string" style="color: #0048ab;">"."</span></div> <div style="text-align: justify;"> <span class="hljs-type">Nothing</span> -> <span class="hljs-string" style="color: #0048ab;">"Is "</span> ++ country ++ <span class="hljs-string" style="color: #0048ab;">" even a country?"</span></div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">The point is that a key-value store will only return a value if the key is present. In this example, the map takes a country (string) and returns its capital (string). However, when we try to take a value from the map with the lookup function, Haskell returns a <b>Maybe</b> type with either <b>Just String</b>, if the string provided is found in the map, or <b>Nothing</b> if the key is absent. We then use pattern matching to deal with these possibilities in the lookupCapitals function. One of the most common mistake in programming is to return a null and not deal with it properly. The solution with sum types is to return a wrapped value and handling possibilities explicitly with pattern matching. It solves with types what many languages would solve with exceptions and try-catch apparatuses.</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="haskell hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-family: Verdana, sans-serif; font-weight: bold;">ghci</span><span style="font-family: Verdana, sans-serif;">> lookupCapitals </span><span class="hljs-string" style="color: #0048ab; font-family: Verdana, sans-serif;">"Arrakis"</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span style="color: #0048ab;">"The capital of Arrakis is Arrakeen."</span></div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;"><div style="text-align: justify;"> <span class="hljs-title">ghci</span><span style="color: #00193a;">> lookupCapitals </span><span class="hljs-string">"Canada"</span></div> </span><span class="hljs-string" style="color: #0048ab;"><div style="text-align: justify;"> "Is Canada even a country?"</div> </span></span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Speaking of Haskell, the code for the algorithm is:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="haskell hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-family: Verdana, sans-serif; font-weight: bold;">data</span><span style="font-family: Verdana, sans-serif;"> </span><span class="hljs-type" style="font-family: Verdana, sans-serif;">Expr</span><span style="font-family: Verdana, sans-serif;"> =</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span class="hljs-type">Var</span> <span class="hljs-type">String</span></div> <div style="text-align: justify;"> | <span class="hljs-type">Const</span> <span class="hljs-type">Int</span></div> <div style="text-align: justify;"> | <span class="hljs-type">Add</span> <span class="hljs-type">Expr</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> | <span class="hljs-type">Mult</span> <span class="hljs-type">Expr</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> </div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">simplify1</span> :: <span class="hljs-type">Expr</span> -> <span class="hljs-type">Expr</span> <div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">simplify1</span> e = <span class="hljs-keyword" style="font-weight: bold;">case</span> e <span class="hljs-keyword" style="font-weight: bold;">of</span></div> <div style="text-align: justify;"> <span class="hljs-type">Add</span> (<span class="hljs-type">Const</span> <span class="hljs-number">0</span>) x -> x</div> <div style="text-align: justify;"> <span class="hljs-type">Add</span> x (<span class="hljs-type">Const</span> <span class="hljs-number">0</span>) -> x</div> <div style="text-align: justify;"> <span class="hljs-type">Add</span> (<span class="hljs-type">Const</span> a) (<span class="hljs-type">Const</span> b) -> <span class="hljs-type">Const</span> $ a + b</div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> x (<span class="hljs-type">Const</span> <span class="hljs-number">0</span>) -> <span class="hljs-type">Const</span> <span class="hljs-number">0</span></div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> (<span class="hljs-type">Const</span> <span class="hljs-number">0</span>) x -> <span class="hljs-type">Const</span> <span class="hljs-number">0</span></div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> x (<span class="hljs-type">Const</span> <span class="hljs-number">1</span>) -> x</div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> (<span class="hljs-type">Const</span> <span class="hljs-number">1</span>) x -> x</div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> (<span class="hljs-type">Const</span> a) (<span class="hljs-type">Const</span> b) -> <span class="hljs-type">Const</span> $ a * b</div> <div style="text-align: justify;"> _ -> e</div> <div style="text-align: justify;"> </div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">simplify</span> :: <span class="hljs-type">Expr</span> -> <span class="hljs-type">Expr</span> <div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">simplify</span> e = <span class="hljs-keyword" style="font-weight: bold;">case</span> e <span class="hljs-keyword" style="font-weight: bold;">of</span></div> <div style="text-align: justify;"> <span class="hljs-type">Add</span> x y -> simplify1 $ <span class="hljs-type">Add</span> (simplify x) (simplify y)</div> <div style="text-align: justify;"> <span class="hljs-type">Mult</span> x y -> simplify1 $ <span class="hljs-type">Mult</span> (simplify x) (simplify y)</div> _ -> simplify1 e <div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">e</span> = <span class="hljs-type">Add</span> (<span class="hljs-type">Mult</span> (<span class="hljs-type">Add</span> (<span class="hljs-type">Const</span> <span class="hljs-number">1</span>) (<span class="hljs-type">Mult</span> (<span class="hljs-type">Const</span> <span class="hljs-number">0</span>) (<span class="hljs-type">Var</span> <span class="hljs-string" style="color: #0048ab;">"x"</span>))) (<span class="hljs-type">Const</span> <span class="hljs-number">3</span>)) (<span class="hljs-type">Const</span> <span class="hljs-number">12</span>)</div> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">s</span> = simplify e <div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">main</span> = putStrLn $ <span class="hljs-keyword" style="font-weight: bold;">case</span> s <span class="hljs-keyword" style="font-weight: bold;">of</span></div> <span class="hljs-type">Const</span> x -> show x <div style="text-align: justify;"> _ -> <span class="hljs-string" style="color: #0048ab;">"Could not simplify the expression to a constant."</span></div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">It's quite similar to F#. I decided to add types explicitly to <i>simplify1</i> and <i>simplify</i>, but Haskell is smart enough to deduce the type without this. Arguably the only thing worth explaining is the $ operator. The operator forces Haskell to evaluate the expression to the right of the operator in priority, and if it reminds you of parentheses, you are absolutely right. <i>x</i> and <i>y</i> have the same value here:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="haskell hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-family: Verdana, sans-serif; font-weight: bold;">x</span><span style="font-family: Verdana, sans-serif;"> = log (sqrt (exp </span><span class="hljs-number" style="font-family: Verdana, sans-serif;">5.0</span><span style="font-family: Verdana, sans-serif;">))</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">y</span> = log $ sqrt $ exp <span class="hljs-number">5.0</span></div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">The operator is there to reduce visual clutter. In my opinion, F# is easier to read because the |> operator enforces left-to-right reading, which is more natural than reading code inside-out:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="fsharp hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em; text-align: justify;"><span style="font-family: Verdana, sans-serif;"><span class="hljs-keyword" style="font-weight: bold;">let</span> z = exp <span class="hljs-number">5.0</span> |> sqrt |> log</span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Although it's trivial to simulate this operator in Haskell:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="haskell hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span style="font-family: Verdana, sans-serif;">(|>) :: t0 -> (t0 -> t1) -> t1</span></div> <span style="font-family: Verdana, sans-serif;">(|>) x f = f x <div style="text-align: justify;"> </div> <span class="hljs-comment" style="color: #738191;">-- Now valid Haskell:</span> <div style="text-align: justify;"> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">z</span> = exp <span class="hljs-number">5.0</span> |> sqrt |> log</div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">And now for something a bit different: Scala. It's also a static functional programming language with sum types, but its greater integration with the object-oriented paradigm is evident:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="scala hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-family: Verdana, sans-serif; font-weight: bold;">object</span><span style="font-family: Verdana, sans-serif;"> </span><span class="hljs-title" style="color: #0048ab; font-family: Verdana, sans-serif; font-weight: bold;">Simplify</span><span style="font-family: Verdana, sans-serif;"> {</span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">sealed</span> <span class="hljs-keyword" style="font-weight: bold;">abstract</span> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">class</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Expr</span> {</span> <span class="hljs-keyword" style="font-weight: bold;">override</span> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">def</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">toString</span> =</span> show(<span class="hljs-keyword" style="font-weight: bold;">this</span>) }</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">class</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Variable</span>(</span>name: <span class="hljs-type">String</span>) <span class="hljs-keyword" style="font-weight: bold;">extends</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">class</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Const</span>(</span>value: <span class="hljs-type">Int</span>) <span class="hljs-keyword" style="font-weight: bold;">extends</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">class</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Add</span>(</span>left: <span class="hljs-type">Expr</span>, right: <span class="hljs-type">Expr</span>) <span class="hljs-keyword" style="font-weight: bold;">extends</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-class"><span class="hljs-keyword" style="font-weight: bold;">class</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">Mult</span>(</span>left: <span class="hljs-type">Expr</span>, right: <span class="hljs-type">Expr</span>) <span class="hljs-keyword" style="font-weight: bold;">extends</span> <span class="hljs-type">Expr</span></div> <div style="text-align: justify;"> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">def</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">evalOne</span>(</span>e: <span class="hljs-type">Expr</span>): <span class="hljs-type">Expr</span> = e <span class="hljs-keyword" style="font-weight: bold;">match</span> {</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Add</span>(<span class="hljs-type">Const</span>(<span class="hljs-number">0</span>), r) => r</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Add</span>(l, <span class="hljs-type">Const</span>(<span class="hljs-number">0</span>)) => l</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Add</span>(<span class="hljs-type">Const</span>(a), <span class="hljs-type">Const</span>(b)) => <span class="hljs-type">Const</span>(a + b)</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(<span class="hljs-type">Const</span>(<span class="hljs-number">0</span>), r) => <span class="hljs-type">Const</span>(<span class="hljs-number">0</span>)</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(l, <span class="hljs-type">Const</span>(<span class="hljs-number">0</span>)) => <span class="hljs-type">Const</span>(<span class="hljs-number">0</span>)</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(<span class="hljs-type">Const</span>(<span class="hljs-number">1</span>), r) => r</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(l, <span class="hljs-type">Const</span>(<span class="hljs-number">1</span>)) => l</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(<span class="hljs-type">Const</span>(a), <span class="hljs-type">Const</span>(b)) => <span class="hljs-type">Const</span>(a * b)</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> _ => e</div> } <div style="text-align: justify;"> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">def</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">eval</span>(</span>e: <span class="hljs-type">Expr</span>): <span class="hljs-type">Expr</span> = e <span class="hljs-keyword" style="font-weight: bold;">match</span> {</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Add</span>(l, r) => evalOne(<span class="hljs-type">Add</span>(eval(l), eval(r)))</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Mult</span>(l, r) => evalOne(<span class="hljs-type">Mult</span>(eval(l), eval(r)))</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> _ => e</div> } <div style="text-align: justify;"> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">def</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">show</span>(</span>e: <span class="hljs-type">Expr</span>) = e <span class="hljs-keyword" style="font-weight: bold;">match</span> {</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> <span class="hljs-type">Const</span>(x) => print(x)</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">case</span> _ =></div> <div style="text-align: justify;"> print(<span class="hljs-string" style="color: #0048ab;">"The expression could not be simplified to a constant."</span>)</div> <div style="text-align: justify;"> }</div> <div style="text-align: justify;"> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">def</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">main</span>(</span>args: <span class="hljs-type">Array</span>[<span class="hljs-type">String</span>]) {</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">var</span> e = <span class="hljs-type">Add</span>(<span class="hljs-type">Mult</span>(<span class="hljs-type">Add</span>(<span class="hljs-type">Const</span>(<span class="hljs-number">1</span>), <span class="hljs-type">Mult</span>(<span class="hljs-type">Const</span>(<span class="hljs-number">0</span>), <span class="hljs-type">Variable</span>(<span class="hljs-string" style="color: #0048ab;">"x"</span>))),</div> <div style="text-align: justify;"> <span class="hljs-type">Const</span>(<span class="hljs-number">3</span>)), <span class="hljs-type">Const</span>(<span class="hljs-number">12</span>))</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">var</span> s = eval(e)</div> print(s) <div style="text-align: justify;"> }</div> <div style="text-align: justify;"> }</div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Everything is an object in Scala. Thus, we have to define the functions to simplify as methods inside a singleton object. I named the functions <i>evalOne</i> and <i>eval</i> since it has a bit odd to have a function named <i>simplify</i> inside a Simplify object.</span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-align: left; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-align: left; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-align: left; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-align: left; text-indent: -1em;">C++</span></div> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4KwrdZs3hvnRXlugNh54DDIPmjCGh0Fx7L5ivSiA7MNw8BHec3oONh7lDovI-T449i4rTLxByQoxN5qbqq9mhd-_okQ069Z60yvocYn8ehtJNQ9KVdXXLm0sVKNO0jP9aCnUBfY5nC1Y/s1600/C%252B%252B-unofficial.sh-600x600.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" loading="lazy" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4KwrdZs3hvnRXlugNh54DDIPmjCGh0Fx7L5ivSiA7MNw8BHec3oONh7lDovI-T449i4rTLxByQoxN5qbqq9mhd-_okQ069Z60yvocYn8ehtJNQ9KVdXXLm0sVKNO0jP9aCnUBfY5nC1Y/s320-rw/C%252B%252B-unofficial.sh-600x600.png" width="320" /></a></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Few understand every corner of C++'s monstrous standard. It's huge. Surely, with so many features, there must be something a quick way to solve this problem in C++. Well... no. It's a well-known lacuna with C++, see <a href="https://parasol.tamu.edu/~yuriys/pm/" style="color: #00905d; text-decoration: none;"><i>Open and Efficient Type Switch for C++</i></a> for a library built to implement pattern matching (the effort is directed by the creator of the C++ language). That said, here I'll use the boost library (A) because solutions based only on the standard library are contrived and (B) because boost is almost standard, and I don't want to rely on third-party libraries.</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="cpp hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span style="color: #4c81c9; font-family: Verdana, sans-serif;">#</span><span class="hljs-keyword" style="color: #4c81c9; font-family: Verdana, sans-serif; font-weight: bold;">include</span><span style="color: #4c81c9; font-family: Verdana, sans-serif;"> <iostream></span></div> <span style="font-family: Verdana, sans-serif;"><div style="text-align: justify;"> <span style="color: #4c81c9;">#</span><span class="hljs-keyword" style="color: #4c81c9; font-weight: bold;">include</span><span style="color: #4c81c9;"> <string></span></div> <span class="hljs-preprocessor" style="color: #4c81c9;"><div style="text-align: justify;"> #<span class="hljs-keyword" style="font-weight: bold;">include</span> <boost/variant></div> </span> <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> Add; <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">struct</span> Mult;</div> <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">using</span> Expr = boost::variant< <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">int</span>,</div> <div style="text-align: justify;"> <span class="hljs-built_in" style="color: #0048ab;">std</span>::<span class="hljs-built_in" style="color: #0048ab;">string</span>,</div> <div style="text-align: justify;"> boost::recursive_wrapper<Add>,</div> <div style="text-align: justify;"> boost::recursive_wrapper<Mult>>;</div> <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> Add { <div style="text-align: justify;"> Expr left, right;</div> <div style="text-align: justify;"> Add(<span class="hljs-keyword" style="font-weight: bold;">const</span> Expr &left_, <span class="hljs-keyword" style="font-weight: bold;">const</span> Expr &right_) : left(left_), right(right_) {</div> } }; <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> Mult { <div style="text-align: justify;"> Expr left, right;</div> <div style="text-align: justify;"> Mult(<span class="hljs-keyword" style="font-weight: bold;">const</span> Expr &left_, <span class="hljs-keyword" style="font-weight: bold;">const</span> Expr &right_) : left(left_), right(right_) {</div> } }; <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> add_visit : <span class="hljs-keyword" style="font-weight: bold;">public</span> boost::static_visitor<Expr> { <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">int</span> l, <span class="hljs-keyword" style="font-weight: bold;">int</span> r)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> Expr(l + r);</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">int</span> l, <span class="hljs-keyword" style="font-weight: bold;">const</span> X &x)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> l == <span class="hljs-number">0</span>? Expr(x) : Add(Expr(l), Expr(x));</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x, <span class="hljs-keyword" style="font-weight: bold;">int</span> r)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> r == <span class="hljs-number">0</span>? Expr(x) : Add(Expr(r), Expr(x));</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X, <span class="hljs-keyword" style="font-weight: bold;">class</span> Y></div> <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x, <span class="hljs-keyword" style="font-weight: bold;">const</span> Y &y)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> Add(Expr(x), Expr(y));</div> } }; <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> mul_visit : <span class="hljs-keyword" style="font-weight: bold;">public</span> boost::static_visitor<Expr> { <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">int</span> l, <span class="hljs-keyword" style="font-weight: bold;">int</span> r)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> Expr(l * r);</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">int</span> l, <span class="hljs-keyword" style="font-weight: bold;">const</span> X &x)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> l == <span class="hljs-number">0</span>? Expr(<span class="hljs-number">0</span>) : (l == <span class="hljs-number">1</span>? Expr(x) : Mult(Expr(l), Expr(x)));</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x, <span class="hljs-keyword" style="font-weight: bold;">int</span> r)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> r == <span class="hljs-number">0</span>? Expr(<span class="hljs-number">0</span>) : (r == <span class="hljs-number">1</span>? Expr(x) : Mult(Expr(r), Expr(x)));</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X, <span class="hljs-keyword" style="font-weight: bold;">class</span> Y></div> <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x, <span class="hljs-keyword" style="font-weight: bold;">const</span> Y &y)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> Mult(Expr(x), Expr(y));</div> } }; <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> simplify1 : <span class="hljs-keyword" style="font-weight: bold;">public</span> boost::static_visitor<Expr> { <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> Add &a)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> boost::apply_visitor(add_visit(), a.left, a.right);</div> } <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> Mult &m)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> boost::apply_visitor(mul_visit(), m.left, m.right);</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> Expr(x);</div> } }; <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">struct</span> simplify : <span class="hljs-keyword" style="font-weight: bold;">public</span> boost::static_visitor<Expr> { <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> Add &a)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> left = boost::apply_visitor(simplify(), a.left);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> right = boost::apply_visitor(simplify(), a.right);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> add_lr = boost::apply_visitor(add_visit(), left, right);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> boost::apply_visitor(simplify1(), add_lr);</div> } <div style="text-align: justify;"> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> Mult &m)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> left = boost::apply_visitor(simplify(), m.left);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> right = boost::apply_visitor(simplify(), m.right);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> mul_lr = boost::apply_visitor(mul_visit(), left, right);</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> boost::apply_visitor(simplify1(), mul_lr);</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">template</span><<span class="hljs-keyword" style="font-weight: bold;">class</span> X> <span class="hljs-function">Expr <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> X &x)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> x;</div> } <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">struct</span> print_expr : <span class="hljs-keyword" style="font-weight: bold;">public</span> boost::static_visitor<<span class="hljs-built_in" style="color: #0048ab;">std</span>::<span class="hljs-built_in" style="color: #0048ab;">string</span>> {</div> }; <div style="text-align: justify;"> </div> <span class="hljs-built_in" style="color: #0048ab;">std</span>::<span class="hljs-function"><span class="hljs-built_in" style="color: #0048ab;">string</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">int</span> x)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{ <span class="hljs-keyword" style="font-weight: bold;">return</span> <span class="hljs-built_in" style="color: #0048ab;">std</span>::to_string(x); }; <div style="text-align: justify;"> <span class="hljs-built_in" style="color: #0048ab;">std</span>::<span class="hljs-function"><span class="hljs-built_in" style="color: #0048ab;">string</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">operator</span><span class="hljs-params">()</span><span class="hljs-params">(<span class="hljs-keyword" style="font-weight: bold;">const</span> Expr &e)</span> <span class="hljs-keyword" style="font-weight: bold;">const</span> </span>{</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> <span class="hljs-string" style="color: #0048ab;">"The expression could not be simplified to a constant."</span>;</div> <div style="text-align: justify;"> };</div> }; <div style="text-align: justify;"> </div> <span class="hljs-function"><span class="hljs-keyword" style="font-weight: bold;">int</span> <span class="hljs-title" style="color: #0048ab; font-weight: bold;">main</span><span class="hljs-params">()</span> </span>{ <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> e = Expr(Add(</div> <div style="text-align: justify;"> Expr(Mult(Expr(Add(Expr(<span class="hljs-number">1</span>), Expr(Mult(Expr(<span class="hljs-number">0</span>), Expr(<span class="hljs-string" style="color: #0048ab;">"x"</span>))))),</div> <div style="text-align: justify;"> Expr(<span class="hljs-number">3</span>))), Expr(<span class="hljs-number">12</span>)));</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">auto</span> s = boost::apply_visitor(simplify(), e);</div> <div style="text-align: justify;"> <span class="hljs-built_in" style="color: #0048ab;">std</span>::<span class="hljs-built_in" style="color: #0048ab;">cout</span> << boost::apply_visitor(print_expr(), s) << <span class="hljs-built_in" style="color: #0048ab;">std</span>::endl;</div> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">return</span> <span class="hljs-number">0</span>;</div> <div style="text-align: justify;"> }</div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">This is boost::variant in action. My biggest qualm with this type of clever header-heavy code is that you get to see a big chunk of the developers' lifework unroll before your eyes every time a small mistake is made. If you want to know how this code works, you need to read a bit on the visitor pattern.</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">C++ template metaprogramming is both awesome and terrifying. Mostly terrifying. Someone might find a simple way to use C++11/C++14 features to build a small and nice matching library.<a href="https://github.com/jbandela/simple_match" style="color: #00905d; text-decoration: none;">This library</a> is halfway there (with boost::variant, it might already allow matching on type but I have yet to try). C++ is great for scientific computing, with tons of awesome libraries, so I'd love to see a nice way to simulate pattern matching and sum types.</span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;">Julia</span></div> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDwoAMyRTzY_iYyvbSuP_wIV5EM9kL_Im5uxd5ANO51zi6qADa4O0NGsYn4tcEya6BL4QkSAglHmdK2x4yf-XuuhYZsdDORJlQhF-Ml6qOxE7Xhq0joyW6waOsrLsihOdUySzXaj5mSlE/s1600/learning-and-modern-programming-languages-37-638.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" loading="lazy" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDwoAMyRTzY_iYyvbSuP_wIV5EM9kL_Im5uxd5ANO51zi6qADa4O0NGsYn4tcEya6BL4QkSAglHmdK2x4yf-XuuhYZsdDORJlQhF-Ml6qOxE7Xhq0joyW6waOsrLsihOdUySzXaj5mSlE/s1600-rw/learning-and-modern-programming-languages-37-638.jpg" /></a></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Julia is an attempt to build a fast and flexible replacement for R/Python/Matlab. An issue with most dynamic languages is that there is no elegant way to switch on type. To be fair, you cannot really do it with most static languages either, see previous section... However, Julia supports multiple-dispatch based on type annotation. To be clear, it's quite different from the F#/Scala/Haskell approach. In these languages, it is possible to define sum types and do pattern matching on their constructors. With Julia, we define a function with type annotation and let the interpreter dispatch based on runtime type information. Multiple dispatch is supported in Julia for performance: it allows the interpreter to compile optimized functions and use the best one, adding predictability while keeping the language dynamic (for some reason...). Here's the algorithm in Julia:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="julia hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span class="hljs-keyword" style="font-family: Verdana, sans-serif; font-weight: bold;">abstract</span><span style="font-family: Verdana, sans-serif;"> </span><span class="hljs-built_in" style="color: #0048ab; font-family: Verdana, sans-serif;">Expr</span></div> <span style="font-family: Verdana, sans-serif;"> <div style="text-align: justify;"> </div> <span class="hljs-keyword" style="font-weight: bold;">type</span> Const <span class="hljs-subtype"><:</span> <span class="hljs-built_in" style="color: #0048ab;">Expr</span>; val<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Int</span> <span class="hljs-keyword" style="font-weight: bold;">end</span> <div style="text-align: justify;"> <span class="hljs-keyword" style="font-weight: bold;">type</span> Var <span class="hljs-subtype"><:</span> <span class="hljs-built_in" style="color: #0048ab;">Expr</span>; name<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">String</span> <span class="hljs-keyword" style="font-weight: bold;">end</span></div> <span class="hljs-keyword" style="font-weight: bold;"><div style="text-align: justify;"> <span class="hljs-keyword">type</span> Add <span class="hljs-subtype"><:</span> <span class="hljs-built_in" style="color: #0048ab;">Expr</span>; left<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>; right<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span> <span class="hljs-keyword">end</span></div> </span><span class="hljs-keyword" style="font-weight: bold;"><div style="text-align: justify;"> <span class="hljs-keyword">type</span> Mult <span class="hljs-subtype"><:</span> <span class="hljs-built_in" style="color: #0048ab;">Expr</span>; left<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>; right<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span> <span class="hljs-keyword">end</span></div> </span> <div style="text-align: justify;"> add(x<span class="hljs-type-annotation">::</span>Const, y<span class="hljs-type-annotation">::</span>Const) = Const(x.val + y.val)</div> <div style="text-align: justify;"> add(x<span class="hljs-type-annotation">::</span>Const, y<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = x.val == <span class="hljs-number">0</span>? y : Add(x, y)</div> <div style="text-align: justify;"> add(x<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>, y<span class="hljs-type-annotation">::</span>Const) = add(y, x)</div> <div style="text-align: justify;"> add(x<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>, y<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = Add(x, y)</div> <div style="text-align: justify;"> mult(x<span class="hljs-type-annotation">::</span>Const, y<span class="hljs-type-annotation">::</span>Const) = Const(x.val * y.val)</div> <div style="text-align: justify;"> mult(x<span class="hljs-type-annotation">::</span>Const, y<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = x.val == <span class="hljs-number">1</span>? y : (x.val == <span class="hljs-number">0</span>? Const(<span class="hljs-number">0</span>) : Mult(x, y))</div> <div style="text-align: justify;"> mult(x<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>, y<span class="hljs-type-annotation">::</span>Const) = mult(y, x)</div> <div style="text-align: justify;"> mult(x<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>, y<span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = Mult(x, y)</div> <div style="text-align: justify;"> simplify1(a<span class="hljs-type-annotation">::</span>Add) = add(a.left, a.right)</div> <div style="text-align: justify;"> simplify1(m<span class="hljs-type-annotation">::</span>Mult) = mult(m.left, m.right)</div> <div style="text-align: justify;"> simplify1(<span class="hljs-literal" style="color: #0048ab;">e</span><span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = <span class="hljs-literal" style="color: #0048ab;">e</span></div> <div style="text-align: justify;"> simplify(a<span class="hljs-type-annotation">::</span>Add) = simplify1(Add(simplify(a.left), simplify(a.right)))</div> <div style="text-align: justify;"> simplify(m<span class="hljs-type-annotation">::</span>Mult) = simplify1(Mult(simplify(m.left), simplify(m.right)))</div> <div style="text-align: justify;"> simplify(<span class="hljs-literal" style="color: #0048ab;">e</span><span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) = <span class="hljs-literal" style="color: #0048ab;">e</span></div> <div style="text-align: justify;"> printExpr(c<span class="hljs-type-annotation">::</span>Const) = print(c.val)</div> <div style="text-align: justify;"> printExpr(<span class="hljs-literal" style="color: #0048ab;">e</span><span class="hljs-type-annotation">::</span><span class="hljs-built_in" style="color: #0048ab;">Expr</span>) =</div> <div style="text-align: justify;"> print(<span class="hljs-string" style="color: #0048ab;">"The expression could not be simplified to a constant."</span>)</div> <div style="text-align: justify;"> </div> <span class="hljs-literal" style="color: #0048ab;">e</span> = Add(Mult(Add(Const(<span class="hljs-number">1</span>), Mult(Const(<span class="hljs-number">0</span>), Var(<span class="hljs-string" style="color: #0048ab;">"x"</span>))), Const(<span class="hljs-number">3</span>)), Const(<span class="hljs-number">12</span>)) <div style="text-align: justify;"> s = simplify(<span class="hljs-literal" style="color: #0048ab;">e</span>)</div> <div style="text-align: justify;"> printExpr(s)</div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Unlike pattern matching, we can only dispatch on type, so we need an if expression (the ? operator in Julia, just like C), or I could've used the match macro, but it's overkill here. It's not too inelegant, and at first I thought it was a good enough way to simulate sum types and pattern matching. It matters to the Julia ecosystem because these features are very useful to build solvers, logic and theorem proving systems, etc etc. Pretty nice for a technical computing platform. Unfortunately, while Julia does well with this simple example, I think an oddity with the language would soon bite us: return type declarations are not allowed, and yes, it <i>is</i> a big deal.</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">First, it's a question of correctness: you can return a float thinking you're returning an integer. That's awful. Also, since annotation is not allowed for the return value, it's also impossible to add annotation for a higher-order function (a function taking functions as input). As a concrete example, first-order logic has <i>predicates</i> mapping objects to a boolean, and <i>functions</i> mapping objects to objects. We'd like to do:</span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <pre style="color: #333333;"><code class="julia hljs" style="background: rgb(234, 238, 243); color: #00193a; display: block; overflow-x: auto; padding: 0.5em;"><div style="text-align: justify;"> <span style="font-family: Verdana, sans-serif;">solve(pre</span><span class="hljs-type-annotation" style="font-family: Verdana, sans-serif;">::</span><span style="font-family: Verdana, sans-serif;">(Object -> </span><span class="hljs-built_in" style="color: #0048ab; font-family: Verdana, sans-serif;">Bool</span><span style="font-family: Verdana, sans-serif;">), ...)</span></div> <span style="font-family: Verdana, sans-serif;"> <div style="text-align: justify;"> solve(fun<span class="hljs-type-annotation">::</span>(Object -> Object), ...)</div> </span></code></pre> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">But instead, we'd have to test the type of the return value inside the function. That said, Julia is young and <a href="https://github.com/JuliaLang/julia/issues/1090" style="color: #00905d; text-decoration: none;">it might get return type declarations at some point.</a></span></div> <div style="color: #333333; text-align: justify;"> <br /></div> <div style="color: #333333; text-align: justify;"> <span style="color: #00905d; font-family: Verdana, sans-serif; text-indent: -1em;">Conclusion</span></div> <div> <span style="font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="font-family: Verdana, sans-serif;">Sum types and pattern matching are awesome.</span></div> <div style="color: #333333; text-align: justify;"> <span style="background-color: orange; font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="background-color: orange; font-family: Verdana, sans-serif;"><br /></span></div> <div style="color: #333333; text-align: justify;"> <span style="background-color: orange; color: white; font-family: 'Ubuntu Mono', 'Source Han Sans', 'AR PL UKai CN'; text-align: center;">let world = "世界" in print $ "Hello " ++ world ++ "!"</span></div> <span style="font-family: Verdana, sans-serif;"><span class="MathJax_Preview"></span></span><br /> <span style="font-family: 'Ubuntu Mono', 'Source Han Sans', 'AR PL UKai CN'; text-align: center;">let world = "世界" in print $ "Hello " ++ world ++ "!"</span><br /> <div id="hidden" style="text-align: justify;"> <span style="font-family: Verdana, sans-serif;">let world = "世界"</span></div> </div>

Automated reasoning in F#, Scala, Haskell, C++, and Julia

We need to simplify the following expression: e = ( 1 + 0 × x ) × 3 + 12. Luckily for us, we won't have to remember any elemen...

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How to Sync Your Android Phone and iPhone With Windows 10

<div dir="ltr" style="text-align: left;" trbidi="on"> <div style="background-color: white; border: 0px; font-family: Ubuntu, Arial, sans-serif; font-size: 18px; font-stretch: inherit; line-height: 28.7999992370605px; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: purple;"><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></em></span></div> <div style="text-align: justify;"> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6yOvDnwGKniu-DTu9LS1k4XfjhK7BSF2cI7YxlArXDEyrIaqbuQAyBgPx5IIG74yfE7ljHSKbcCFrMEMBoVNhYTd_Fw4vfVQT0mfrZrS4rR_upaV2LOK9vXda4oyKVSJTxmO4e_MiRvIX/s1600/ubuntuone.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" loading="lazy" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6yOvDnwGKniu-DTu9LS1k4XfjhK7BSF2cI7YxlArXDEyrIaqbuQAyBgPx5IIG74yfE7ljHSKbcCFrMEMBoVNhYTd_Fw4vfVQT0mfrZrS4rR_upaV2LOK9vXda4oyKVSJTxmO4e_MiRvIX/s1600-rw/ubuntuone.png" /></a></div> <span style="color: purple;"><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 600; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="http://allbabouttechbyspr.blogspot.in/" target="_blank">SPR</a>:</span> With Windows 10, there comes a pre-installed app called Windows 10 Phone Companion app. This app helps you to sync your PC and phone data seamlessly. </em></span></div> <div style="text-align: justify;"> <span style="color: purple;"><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><br /></em></span></div> <br /> <div class="su-row" style="background-color: white; border: 0px; clear: both; font-family: Ubuntu, Arial, sans-serif; font-size: 18px; font-stretch: inherit; line-height: 28.7999992370605px; margin: 0px 0px 1.5em; padding: 0px; vertical-align: baseline; zoom: 1;"> <div class="su-column su-column-size-1-5" style="border: 0px; box-sizing: border-box; float: left; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 134.390625px;"> <div class="su-column-inner su-clearfix" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"> </div> </div> <div class="su-column su-column-size-4-5" style="border: 0px; box-sizing: border-box; float: right; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px 0px 0px 32px; padding: 0px; vertical-align: baseline; width: 633.59375px;"> <div class="su-column-inner su-clearfix" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"> <span class="su-dropcap su-dropcap-style-flat" style="background: rgb(68, 68, 68); border: 0px; display: inline-block; float: left; font-family: inherit; font-size: 1.5em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2em; line-height: 2em; margin: 0px 0.5em 0.2em 0px; overflow: hidden; padding: 0px; text-align: center; text-transform: uppercase; vertical-align: baseline; width: 2em;"><span style="color: white;">E</span></span><span style="color: purple;"></span><br /> <div style="text-align: center;"> <span style="color: purple;"><span style="font-family: inherit; font-size: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit;">very Windows release brings lots of excitement and</span><span style="font-family: inherit; font-size: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit;"> </span><span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="http://allbabouttechbyspr.blogspot.com/2015/06/windows-10-most-important-features-you.html" target="_blank">new features</a></span><span style="font-family: inherit; font-size: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit;">. And, with Windows 10, Microsoft has brought something for your Android devices and iPhones too. With your Windows 10 upgrade, you get the pre-installed Windows 10 Phone Companion app. This app lets you sync your Android phone, iPhone and Windows phone with your Windows 10 PC.</span></span></div> <span style="color: purple;"> </span><br /> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">This Windows 10 Companion App is basically a tool to configure and install Microsoft’s apps and services on all your devices, and then integrate everything. With its help, now you can use OneDrive’s automatic photo backup, OneNote Mobile, Skype, Office Mobile, Outlook, Cortana, and listen to your songs on OneDrive from any device. The two functionalities, <a href="http://allbabouttechbyspr.blogspot.com/2015/06/windows-10-pcs-are-getting-separate.html" target="_blank">Cortana </a>and songs on OneDrive, are currently unavailable on Android and iPhone currently, and they are labeled as <em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coming Soon</em>.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;"><br /></span></div> <h2 style="border: 0px; font-family: inherit; font-size: 34px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 400; letter-spacing: -0.7px; line-height: 1.3em; margin: 0px 0px 14px; padding: 0px; text-align: justify; vertical-align: baseline; word-wrap: break-word;"> <span style="color: purple;">How to Sync Your Android Phone and iPhone With Windows 10 Phone Companion app?</span></h2> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">To sync the data from your Android Phone, iPhone or Windows Phone with Windows 10, you need to open Windows 10 Phone Companion app on your PC. To use this app, make sure that your are logged in using your Microsoft email account.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: purple; font-family: inherit; font-size: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; text-align: justify;">As you’ll open Windows 10 Phone Companion app, you can see three options to connect a Windows Phone, Android, and iPhone/iPad. If you are using a Windows Phone, Windows 10 Phone companion app would be already busy syncing your stuff using the same Microsoft email account.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">To use your Android and Apple devices, you need to just click on some button and it’s done. At the bottom of the welcome screen, you can see the Windows 10 Phone Companion app prompting you to connect a device manually. This is pretty useful as you can move a couple of files back and forth, or just charge your phone’s battery.</span></div> <div class="wp-caption aligncenter" id="attachment_16107" style="background: rgb(241, 241, 241); border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px auto 14px; max-width: 100%; padding: 5px 0px 0px; text-align: center; vertical-align: baseline; width: 1050px;"> <div style="text-align: justify;"> <a data-rel="lightbox-2" href="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215201/windows-10-phone-companion-app-1.jpg" style="border-bottom-style: none; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-top-width: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;"><span style="color: purple;"></span></a></div> <div class="separator" style="clear: both;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1K7jjXPBWZ4CWrj-xVLg-28WN3ElZDrd68Pfgi1ASvdjHqHkLuZr6oG939U0jl1AMHQgMKhtbFQBSvmhU-IzHaEnJn1g8q8sk80ENjO6lq1F8aaAm_jQDOd7Oe42TKlYpF7bRWLsROAGu/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="505" loading="lazy" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1K7jjXPBWZ4CWrj-xVLg-28WN3ElZDrd68Pfgi1ASvdjHqHkLuZr6oG939U0jl1AMHQgMKhtbFQBSvmhU-IzHaEnJn1g8q8sk80ENjO6lq1F8aaAm_jQDOd7Oe42TKlYpF7bRWLsROAGu/s640-rw/Capture.PNG" width="640" /></a></div> <div class="wp-caption-text" style="border: 0px; font-family: inherit; font-size: 13px; font-stretch: inherit; font-style: italic; font-variant: inherit; font-weight: inherit; line-height: inherit; padding: 0px 0px 8px; vertical-align: baseline;"> <span style="color: purple; font-family: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit;">Windows 10 Phone Companion app</span></div> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">As you’ll connect your device, the Windows 10 Phone Companion app will show information like charging and storage status. Right from this screen, you can import photos and videos to the Windows 10 Photos app. There is also an option to transfer other files using your PC’s File Explorer in Windows 10 Phone Companion app.</span></div> <div class="wp-caption aligncenter" id="attachment_16105" style="background: rgb(241, 241, 241); border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px auto 14px; max-width: 100%; padding: 5px 0px 0px; text-align: center; vertical-align: baseline; width: 1048px;"> <div style="text-align: justify;"> <a data-rel="lightbox-3" href="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215146/windows-10-phone-companion-app.jpg" style="border-bottom-style: none; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-top-width: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;"><span style="color: purple;"><img alt="windows-10-phone-companion-app" class="lazy wp-image-16105 size-full data-lazy-ready" data-lazy-src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215146/windows-10-phone-companion-app.jpg" data-lazy-type="image" height="722" loading="lazy" src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215146/windows-10-phone-companion-app.jpg" style="border-radius: 2px; border: 0px; display: inline; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: auto; line-height: inherit; margin: 0px; max-width: 100%; padding: 0px; transition: opacity 0.2s ease; vertical-align: baseline;" width="1038" /></span></a></div> <div class="wp-caption-text" style="border: 0px; font-family: inherit; font-size: 13px; font-stretch: inherit; font-style: italic; font-variant: inherit; font-weight: inherit; line-height: inherit; padding: 0px 0px 8px; vertical-align: baseline;"> <span style="color: purple;">Windows 10 Phone Companion app</span></div> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">To get started with syncing, click on Android or iPhone icon to reveal the available options. Here, you can see different services and apps from Microsoft. To sync the apps and services between your device and Windows 10 PC, click on any of these and proceed in Windows 10 Phone Companion app.</span></div> <div class="wp-caption aligncenter" id="attachment_16106" style="background: rgb(241, 241, 241); border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px auto 14px; max-width: 100%; padding: 5px 0px 0px; text-align: center; vertical-align: baseline; width: 1048px;"> <div style="text-align: justify;"> <a data-rel="lightbox-4" href="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215152/windows-10-phone-companion-app-.jpg" style="border-bottom-style: none; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-top-width: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;"><span style="color: purple;"><img alt="windows-10-phone-companion-app" class="lazy wp-image-16106 size-full data-lazy-ready" data-lazy-src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215152/windows-10-phone-companion-app-.jpg" data-lazy-type="image" height="724" loading="lazy" src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215152/windows-10-phone-companion-app-.jpg" style="border-radius: 2px; border: 0px; display: inline; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: auto; line-height: inherit; margin: 0px; max-width: 100%; padding: 0px; transition: opacity 0.2s ease; vertical-align: baseline;" width="1038" /></span></a></div> <div class="wp-caption-text" style="border: 0px; font-family: inherit; font-size: 13px; font-stretch: inherit; font-style: italic; font-variant: inherit; font-weight: inherit; line-height: inherit; padding: 0px 0px 8px; vertical-align: baseline;"> <span style="color: purple;">Windows 10 Phone Companion app</span></div> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">Windows 10 Phone Companion app will take you to a new window, where you’ll be asked to enter an email address. This will be used to send a link to your Android phone or iPhone. Enter an email address that can easily checked on your phone or tablet. Alternatively, you can download the apps on your phone via your phone’s app store.</span></div> <div class="wp-caption aligncenter" id="attachment_16109" style="background: rgb(241, 241, 241); border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px auto 14px; max-width: 100%; padding: 5px 0px 0px; text-align: center; vertical-align: baseline; width: 1046px;"> <div style="text-align: justify;"> <a data-rel="lightbox-5" href="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215213/windows-10-phone-companion-app-2.jpg" style="border-bottom-style: none; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-top-width: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;"><span style="color: purple;"><img alt="windows-10-phone-companion-app" class="lazy wp-image-16109 size-full data-lazy-ready" data-lazy-src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215213/windows-10-phone-companion-app-2.jpg" data-lazy-type="image" height="598" loading="lazy" src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/05215213/windows-10-phone-companion-app-2.jpg" style="border-radius: 2px; border: 0px; display: inline; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: auto; line-height: inherit; margin: 0px; max-width: 100%; padding: 0px; transition: opacity 0.2s ease; vertical-align: baseline;" width="1036" /></span></a></div> <div class="wp-caption-text" style="border: 0px; font-family: inherit; font-size: 13px; font-stretch: inherit; font-style: italic; font-variant: inherit; font-weight: inherit; line-height: inherit; padding: 0px 0px 8px; vertical-align: baseline;"> <span style="color: purple;">Windows 10 Phone Companion app</span></div> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">Now, you can download the app on your phone and start syncing the files and photos. If you are using Microsoft apps and services on multiple devices, all your files will be stored at the same place. Thus, you can access it from any place and device.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">While you still have the option of using Google or <a href="http://allbabouttechbyspr.blogspot.com/2015/06/google-microsoft-apple-and-mozilla.html" target="_blank">Apple’s services </a>on your PC and phone for syncing purposes, it’s great to have an option from Microsoft to integrate all your devices.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">Did you find this information helpful? Tell us in the comments below.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; text-align: justify; vertical-align: baseline;"> <span style="color: purple;">For more updates, subscribe to our <a href="http://allbabouttechbyspr.blogspot.com/" target="_blank">newsletter</a>.</span></div> </div> </div> </div> </div>

How to Sync Your Android Phone and iPhone With Windows 10

SPR :  With Windows 10, there comes a pre-installed app called Windows 10 Phone Companion app. This app helps you to sync your PC and ...

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How To Make 3D Hologram At Home By the Easiest Method

<div dir="ltr" style="text-align: left;" trbidi="on"> <h1 class="post-title" style="background-color: white; border: 0px; font-family: Ubuntu, Arial, sans-serif; font-size: 38px; font-stretch: inherit; font-weight: 400; letter-spacing: -1px; line-height: 1.3em; margin: 0px 0px 10px; padding: 0px; vertical-align: baseline; word-wrap: break-word;"> <span style="color: #0b5394;">How To Make 3D Hologram At Home By the Easiest Method</span></h1> <div> <span style="color: #0b5394;"><br /></span></div> <div> <div style="background-color: white; border: 0px; font-family: Ubuntu, Arial, sans-serif; font-size: 18px; font-stretch: inherit; line-height: 28.7999992370605px; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><span style="color: #0b5394;"><span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 600; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="http://fossbytes.com/how-to-make-3d-hologram-at-home/smartphone_3d_hologram/" rel="attachment wp-att-15982" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;"><img alt="Smartphone_3D_hologram" class="lazy aligncenter wp-image-15982 data-lazy-ready" data-lazy-src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/04180943/Smartphone_3D_hologram.jpg" data-lazy-type="image" height="455" loading="lazy" src="http://d2brer6wwumtdu.cloudfront.net/wp-content/uploads/2015/08/04180943/Smartphone_3D_hologram.jpg" style="border-radius: 2px; border: 0px; clear: both; display: block; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: auto; line-height: inherit; margin: 18px auto; max-width: 100%; padding: 0px; transition: opacity 0.2s ease; vertical-align: baseline;" width="805" /></a>SPR :</span> Using some stationary items and glass/plastic sheet, you can make 3D hologram at home. Place a pyramidal frustum over hologram specific video and let physics do the rest.</span></em></div> <div class="su-row" style="background-color: white; border: 0px; clear: both; font-family: Ubuntu, Arial, sans-serif; font-size: 18px; font-stretch: inherit; line-height: 28.7999992370605px; margin: 0px 0px 1.5em; padding: 0px; vertical-align: baseline; zoom: 1;"> <div class="su-column su-column-size-1-5" style="border: 0px; box-sizing: border-box; float: left; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline; width: 134.390625px;"> <div class="su-column-inner su-clearfix" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"> </div> </div> <div class="su-column su-column-size-4-5" style="border: 0px; box-sizing: border-box; float: right; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px 0px 0px 32px; padding: 0px; vertical-align: baseline; width: 633.59375px;"> <div class="su-column-inner su-clearfix" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"> <span class="su-dropcap su-dropcap-style-default" style="background: linear-gradient(rgb(69, 72, 77) 0%, rgb(0, 0, 0) 100%); border-radius: 50%; border: 0px; display: inline-block; float: left; font-family: inherit; font-size: 1.5em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2em; line-height: 2em; margin: 0px 0.5em 0.2em 0px; overflow: hidden; padding: 0px; text-align: center; text-shadow: rgb(0, 0, 0) -1px -1px 0px; text-transform: uppercase; vertical-align: baseline; width: 2em;"><span style="color: #0b5394;">Y</span></span><span style="color: #0b5394;">ou won’t call any concept a science fiction unless its characters are flying through space-time, or have phenomenal powers. Your brain also plays with the idea of some futuristic technology and importantly long distance call that include holograms.</span><div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> </div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">While other stuff would be difficult to own, but you can definitely make 3D hologram out of two-dimensional videos now. Holograms are the part of the new technological era and occupy technical enthusiast’s mind. On one hand, we have the <span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="http://allbabouttechbyspr.blogspot.com/2015/07/microsoft-shows-magical-future-of.html" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;" target="_blank">Microsoft HoloLens</a>,</span> and on other we have <span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="http://allbabouttechbyspr.blogspot.com/2015/07/ghost-this-technology-literally-comes.html" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; transition-duration: 0.2s; transition-timing-function: ease; vertical-align: baseline;" target="_blank">GHOST</a></span> that literally sucks out data from your screen and converts into a 3D hologram.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="font-family: inherit; font-size: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit;"><span style="color: #0b5394;">But, now to make a 3D hologram, you don’t need to buy any expensive gadget. Just a few stationary items and a clean transparent glass or plastic sheet will do. You will require:</span></span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px 0px 0px 30px; vertical-align: baseline;"> <span style="color: #0b5394;"><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Graph Paper: To get precise measurements</em><br /><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Plastic/Glass Sheet (Transparent): Main Apparatus</em><br /><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tape</em><br /><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Pen</em><br /><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Knife/Glass Cutter</em><br /><em style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Smartphone/Tablet</em></span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">Once you have collected all these things, you are just 15 minutes away to make a 3D hologram at your home.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">Cut the sheet precisely into 4 congruent trapeziums and join them with a transparent tape to create a simple pyramidal frustum. Your set-up is ready. Now, all you need are some hologram specific videos that you will easily get from the YouTube.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">So, just place your pyramidal frustum set-up on your smartphone screen and let the physics do the rest as it makes 3-D hologram for you.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">You can check out this YouTube video for reference:</span></div> <div class="video-container" style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: inherit; margin: 0px; overflow: hidden; padding: 0px 0px 356.390625px; position: relative; vertical-align: baseline;"> <iframe allowfullscreen="" frameborder="0" height="405" src="https://www.youtube.com/embed/7YWTtCsvgvg?feature=oembed&wmode=opaque" style="border-style: initial; border-width: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 356.390625px; left: 0px; line-height: inherit; margin: 0px; max-width: 100%; padding: 0px; position: absolute; top: 0px; vertical-align: baseline; width: 633.59375px;" width="720"></iframe></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">Did you find this DIY helpful? Tell us in the comments below.</span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">PS: We tried it at our workplace and the apparatus worked <img alt="😀" class="emoji" draggable="false" loading="lazy" src="http://s.w.org/images/core/emoji/72x72/1f600.png" style="background: none !important; border-radius: 2px; border: none !important; box-shadow: none !important; display: inline !important; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1em !important; line-height: inherit; margin: 0px 0.07em !important; max-width: 100%; padding: 0px !important; vertical-align: -0.1em !important; width: 1em !important;" /></span></div> <div style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin-bottom: 1em; padding: 0px; vertical-align: baseline;"> <span style="color: #0b5394;">For more updates and interesting stories, subscribe to SPRTEch newsletter.</span></div> </div> </div> </div> </div> </div>

How To Make 3D Hologram At Home By the Easiest Method

How To Make 3D Hologram At Home By the Easiest Method SPR :  Using some stationary items and glass/plastic sheet, you can make 3D hol...

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