(window.webpackJsonp=window.webpackJsonp||[]).push([[651],{1059:function(t,s,a){"use strict";a.r(s);var n=a(31),e=Object(n.a)({},(function(){var t=this,s=t.$createElement,a=t._self._c||s;return a("ContentSlotsDistributor",{attrs:{"slot-key":t.$parent.slotKey}},[a("h1",{attrs:{id:"expression-templates"}},[a("a",{staticClass:"header-anchor",attrs:{href:"#expression-templates"}},[t._v("#")]),t._v(" Expression templates")]),t._v(" "),a("h2",{attrs:{id:"a-basic-example-illustrating-expression-templates"}},[a("a",{staticClass:"header-anchor",attrs:{href:"#a-basic-example-illustrating-expression-templates"}},[t._v("#")]),t._v(" A basic example illustrating expression templates")]),t._v(" "),a("p",[t._v("An expression template is a compile-time optimization technique used mostly in scientific computing. It's main purpose is to avoid unnecessary temporaries and optimize loop calculations using a single pass (typically when performing operations on numerical aggregates). 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The most important point to keep in mind is that the expression has not been evaluated yet. It merely holds references to its operands.')]),t._v(" "),a("p",[t._v("In fact, nothing stops you from instantiating the "),a("code",[t._v("MatrixSum<>")]),t._v(" expression template as follows:")]),t._v(" "),a("div",{staticClass:"language-cpp extra-class"},[a("pre",{pre:!0,attrs:{class:"language-cpp"}},[a("code",[t._v("MatrixSum"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("<")]),t._v("Matrix"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("<")]),a("span",{pre:!0,attrs:{class:"token keyword"}},[t._v("double")]),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v(">")]),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(",")]),t._v(" Matrix"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("<")]),a("span",{pre:!0,attrs:{class:"token keyword"}},[t._v("double")]),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v(">")]),t._v(" "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v(">")]),t._v(" "),a("span",{pre:!0,attrs:{class:"token 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sometimes quite expensive expressions. It is widely used in different domains, for example in implementation of linear algebra libraries.")]),t._v(" "),a("p",[t._v("For this example, consider the context of linear algebraic computations. More specifically, computations involving only "),a("strong",[t._v("element-wise operations")]),t._v(". This kind of computations are the most basic applications of "),a("strong",[t._v("ETs")]),t._v(", and they serve as a good introduction to how "),a("strong",[t._v("ETs")]),t._v(" work internally.")]),t._v(" "),a("p",[t._v("Let's look at a motivating example. Consider the computation of the expression:")]),t._v(" "),a("div",{staticClass:"language-cpp extra-class"},[a("pre",{pre:!0,attrs:{class:"language-cpp"}},[a("code",[t._v("Vector vec_1"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(",")]),t._v(" vec_2"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(",")]),t._v(" vec_3"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(";")]),t._v("\n\n"),a("span",{pre:!0,attrs:{class:"token comment"}},[t._v("// Initializing vec_1, vec_2 and vec_3.")]),t._v("\n\nVector result "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("=")]),t._v(" vec_1 "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("+")]),t._v(" vec_2"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("*")]),t._v("vec_3"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(";")]),t._v("\n\n")])])]),a("p",[t._v("Here for the sake of simplicity, I'll assume that the class "),a("code",[t._v("Vector")]),t._v(" and operation + (vector plus: element-wise plus operation) and operation * (here means vector inner product: also element-wise operation) are both correctly implemented, as how they should be, mathematically.")]),t._v(" "),a("p",[t._v("In a conventional implementation without using "),a("strong",[t._v("ETs")]),t._v(" (or other similar techniques), "),a("strong",[t._v("at least five")]),t._v(" constructions of "),a("code",[t._v("Vector")]),t._v(" instances take place in order to obtain the final "),a("code",[t._v("result")]),t._v(":")]),t._v(" "),a("ol",[a("li",[t._v("Three instances corresponding to "),a("code",[t._v("vec_1")]),t._v(", "),a("code",[t._v("vec_2")]),t._v(" and "),a("code",[t._v("vec_3")]),t._v(".")]),t._v(" "),a("li",[t._v("A temporary "),a("code",[t._v("Vector")]),t._v(" instance "),a("code",[t._v("_tmp")]),t._v(", representing the result of "),a("code",[t._v("_tmp = vec_2*vec_3;")]),t._v(".")]),t._v(" "),a("li",[t._v("Finally with proper use of "),a("strong",[t._v("return value optimization")]),t._v(", the construction of final "),a("code",[t._v("result")]),t._v(" in "),a("code",[t._v("result = vec_1 + _tmp;")]),t._v(".")])]),t._v(" "),a("p",[t._v("Implementation using "),a("strong",[t._v("ETs")]),t._v(" can "),a("strong",[t._v("eliminate the creation of temporary")]),t._v(" "),a("code",[t._v("Vector _tmp")]),t._v(" in 2, thus leaving only "),a("strong",[t._v("four")]),t._v(" constructions of "),a("code",[t._v("Vector")]),t._v(" instances. More interestingly, consider the following expression which is more complex:")]),t._v(" "),a("div",{staticClass:"language-cpp extra-class"},[a("pre",{pre:!0,attrs:{class:"language-cpp"}},[a("code",[t._v("Vector result "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("=")]),t._v(" vec_1 "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("+")]),t._v(" "),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v("(")]),t._v("vec_2"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("*")]),t._v("vec3 "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("+")]),t._v(" vec_1"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(")")]),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("*")]),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v("(")]),t._v("vec_2 "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("+")]),t._v(" vec_3"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("*")]),t._v("vec_1"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(")")]),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(";")]),t._v("\n\n")])])]),a("p",[t._v("There will also be four constructions of "),a("code",[t._v("Vector")]),t._v(" instances in total: "),a("code",[t._v("vec_1, vec_2, vec_3")]),t._v(" and "),a("code",[t._v("result")]),t._v(". In other words, in this example, "),a("strong",[t._v("where only element-wise operations are involved")]),t._v(", it is guaranteed that "),a("strong",[t._v("no temporary objects will be created from intermediate calculations")]),t._v(".")]),t._v(" "),a("p",[a("strong",[t._v("How do ETs work")])]),t._v(" "),a("p",[t._v("Basically speaking, "),a("strong",[t._v("ETs for any algebraic computations")]),t._v(" consist of two building blocks:")]),t._v(" "),a("ol",[a("li",[a("strong",[t._v("Pure algebraic expressions")]),t._v(" ("),a("strong",[t._v("PAE")]),t._v("): they are proxies / abstractions of algebraic expressions. A pure algebraic does not do actual computations, they are merely abstractions / modeling of the computation work-flow. A PAE can be a model of "),a("strong",[t._v("either the input or the output of any algebraic expressions")]),t._v(". Instances of "),a("strong",[t._v("PAE")]),t._v("s are often considered cheap to copy.")]),t._v(" "),a("li",[a("strong",[t._v("Lazy evaluations")]),t._v(": which are implementation of real computations. In the following example, we will see that for expressions involving only element-wise operations, lazy evaluations can implement actual computations inside the indexed-access operation on the final result, thus creating a scheme of evaluation on-demand: a computation is not performed only if the final result is accessed / asked for.")])]),t._v(" "),a("p",[t._v("So, specifically how do we implement "),a("strong",[t._v("ETs")]),t._v(" in this example? Let's walk through it now.")]),t._v(" "),a("p",[t._v("Consider always the following code snippet:")]),t._v(" "),a("div",{staticClass:"language-cpp extra-class"},[a("pre",{pre:!0,attrs:{class:"language-cpp"}},[a("code",[t._v("Vector vec_1"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(",")]),t._v(" vec_2"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(",")]),t._v(" vec_3"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(";")]),t._v("\n\n"),a("span",{pre:!0,attrs:{class:"token comment"}},[t._v("// Initializing vec_1, vec_2 and vec_3.")]),t._v("\n\nVector result "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("=")]),t._v(" vec_1 "),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("+")]),t._v(" vec_2"),a("span",{pre:!0,attrs:{class:"token operator"}},[t._v("*")]),t._v("vec_3"),a("span",{pre:!0,attrs:{class:"token punctuation"}},[t._v(";")]),t._v("\n\n")])])]),a("p",[t._v("The expression to compute result can be decomposed further into two sub-expressions:")]),t._v(" "),a("ol",[a("li",[t._v("A vector plus expression (denoted as "),a("strong",[t._v("plus_expr")]),t._v(")")]),t._v(" "),a("li",[t._v("A vector inner product expression (denoted as "),a("strong",[t._v("innerprod_expr")]),t._v(").")])]),t._v(" "),a("p",[t._v("What "),a("strong",[t._v("ETs")]),t._v(" do is the following:")]),t._v(" "),a("li",[t._v("\nInstead of compute right away each sub-expression, **ETs** first model the whole expression using a graphical structure. Each node in the graph represents a **PAE**. The edge connection of the nodes represent the actual computation flow. 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